Engineering Library This volume is damaged or brittle and CANNOT be repaired Handle with EXTREME CARE iStiiy j illllllilllllli<|i>H 2 ELEMEXTAKY APPLIED MECHANICS .Cs? MACMILLAN AND CO., LiMiTEr. LONDON • BOMBAY • CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK - BOSTON • CHICAGO DALLAS • SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO ELEMENTARY APPLIED MECHANICS T. ALEXANDER, C.E., M.Lvsr.C.E.L M.A.I, [hoii. causa) 4th okdku meiji, j.\pan, 18S7, kogaku hakushi, 1915 PKOFESSOll OF EXGINKEKING, TUINITY COLLEGE, DUBLIN AXD A. W. THOMSON, D.Sc. EMERITUS PKUFESSOK OK ESGI.NEERIXG. COLLEGE OF SCIENCE, POONA TFJTH NUMEROUS DIAGRAMS, JNJJ A SERIES OF GRADUATED EXAMPLES CAREFULLY WORKED OUT MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 1916 COPYRIGHT. FIRST EDITION. Part I. By Thomas Alexander. Crown Svo, 1880. Tart II. By Tlioinas Alexander and Arthur Watson Thomson. Crotvn Sro, 1883. SECOND EDITION. In one vol., Sro, 1902. THIRD EDITION. In one vol., Sro, 1916. f^ -^^ -\,*' William John Macqiorn Raxkine, ll.d. From the Preface to La Statique Graphiqtie. Bv ^I. Maurice Lew. 1st edition. " Au moment meme oil nous ecrivons ces lignes, nous recevons de la famille de M. Alacquorn Rankine une lettre de faire part de la mort de I'eminent professeur de I'Universite de Glasgow. Qu'il nous soit permis de lui ofiFrir ici notre tribut de regrets. Sa perte sera ressentie non-seulement par les hommes de science, mais aussi par les ingenieurs et les constructeurs ; car ses recherches ont prcsque toutes un caractere utilitaire. Son Manuel of applied Mechanics notamment est le digne prolongement des Lecons sur la Mecanique industrielle de Poncelet." (Mai 1875.) TO THE MEMORY OF DE. WILLIAM JOHX MACQUORN EANKINE LATE PROFESSOR OF CIVIL ENGINEERING AND MECHANICS IN THE IMVERSITT OF GLASGOW THOMAS ALEXANDER AND ARTHUR WATSON THOMSON BY THE SAME AUTHORS. A SET OF GRAPHICAL EXERCISES, quarter Double Elephant size, with Skeleton Data for the Student to practise upon. Many are printed in two colours. An Essay forms a running com- iiientary on the Exercises, and deals with Reciprocal Figures. Piice OS. GRAPHIC STATICS. 8vo. 2?. PREFACE. This third edition has the honour to be induded in the Dublin University Press Series. Chapter XI is new work, dealing with a Standard Loco- motive on Short Girders. It was read to the Koyal Irish Academy on June 24th, 1912, and includes a description of our Kinematical Model by J. T. Jackson, m.a.i. In the Analysis of the Warren Girder, Chapter XXI, we have replaced Levy's rolling load by an uniform advancing load. The new Chapter XXIII shows diagrams of Eoof Stresses reduced from the Exercises advertised opposite the title-page, with excerpts from the Essay. The work forms an elementary consecutive treatise on the subject of Internal Stress and Strain, based on the late Professor Eankine's treatment of the subject in his Applied Meclicmics and Civil Engineering. The end kept in view is the Scientific and Practical Design of Earthworks, of Linkwork Structures, and of Block work Structures. The whole is illus- trated by a systematic and graduated set of Examples. At every point graphical methods are combined with the analytical, and a feature of the work is, that the diagrams are to scale, and, besides illustrating the text, each diagram suits some of the numerical examples, having printed on its face both the data and results. For the Student with little time to draw, the full-page diagrams should prove useful models, furnishing con- cisely the data and checking the results of his constructions to a bold scale without delay. X PREFACE. In Chapter II a Moving Model of Eaukine's Ellipse of Stress is shown at p. 49. It was exhibited to the Eoyal Irish Academy. The Scientific Design of Masonry Retaining Walls and of their foundations in Chapter IV is an extension of a paper by the authors in Industries of 14th September, 1888. The Rules given in Chapter X, p. 196, for the Maxima Bending Moments caused by a locomotive crossing a bridge were published in Engineering on January 10th and July 25th, 1879, in response to queries in Du Bois' early treatise on Graphic Statics, and are quoted by him in his more recent work, Strains in Framed Girders, 1883. Mention is made of them also in the preface to Levy's La Gh^aphique Statique, second edition.' Hele Shaw's Report on Graphic Methods to the Edinburgh Meeting of the British Association for the Advancement of Science refers to our use of the parabolic set-square for the rapid construction of bending moment diagrams, as introduced in the first edition. We now reduce the solution of this impor- tant problem to the construction of a Diagram of Square Eoots of Bending Moments, with arcs of circles only (see p. 188, and for a cut of our Moving Model of a locomotive on a girder, see pp. 175, 222. Two of those are figured in the correspondence on Mr. Farr's paper on Moving Loads on Bridges published in the Proc. of the Inst, of C. E., vol. cxli, 1900). The arcs of circles replacing the parabolic arcs are also to be seen at page 146 for the more general cases of fixed loads. At the end of Chapter XV, on Stress at an Internal Point of a Beam, we have printed an extract, by the kind permission of the Council of the Phil. Soc. of Glasgow, from the late Professor Peter Alexander's paper on The Uses of the Polariscope in the Practical Determination of Internal Stress and Strain. The lines of stress for a bent-glass prism as drawn by his mechanical pen in conjunction with the polariscope lantern are shown. 1 " Nous saisissons, avec plaisir, cette occasion tie mentionner les Ouvrages ou M^moires suivants, parvenus a notre connaissance depuis la redaction de la Prefiice qui precede : — " Un tr^s beau travail de M. le professeur Thomas Alexander sur le problenie du passage d'un convoi sur une poutre a deux appuis simples, public dans les numeros des 10 Janvier et 25 juillet 1879 de V Engineering et dans son excellent Ouvrage Elementanj applied Mechanics, publie en conimun avec M. le professeur A. W^atson Thomson." PREFACE. Xi In Chapter XVII. the solutions given for the uniform girder fixed horizontally at the ends, and subjected to the transit of a concentrated rolling load in one case, and of a movinti uniform load in the other case, will be found interestin2 (see pp. 321 and 327). The most recent graphical treatment is followed, and the symmetrical form in which the results are shown is new. The analytical result for the rolling load agrees with the unsymmetrical results given by Levy and Du Bois, but we have not seen anywhere an attempt at the solution of the second and more practically important case of the transit of the uniform load shown at p. 327. In Chapter XIX we have greatly extended the part on Long Steel Struts, bringing it up to the most recent practice. We quote two of Fidler's tables for their design. On p. 357 we give a formula (8) for the immediate design of the economical double-tee section of required strength and required local stiffness. Also on the table, p. 360, at iv., we give a close approximate expression independent, like the others, of the thickness of the metal, for the square of the radius of gyration of the tee-section, or angle-iron constrained to bend like it, as it usually is. This is an important addition to Rankine's list, as these sections are of every-day occurrence. The Steel Arched Girder in Chapter XX is treated by Levy's graphical methods. Two numerical examples, one hinged at the ends and another fixed at the ends, are worked out in detail, and the scaled results written on the diagrams (see pp. 376 and 380). In Chapter XXI we follow, on the other hand, his beautiful analysis of the Triangular Trussing, involving the three variables, the number of subdivisions in the span, the form of the triangles, and the ratio of the depth to span. The Table of Volumes of Trusses we show at one opening of the book, and the minimum volumes are placed among the others with a heavy-faced type in such a way that it can be seen whether they occur at depths that can be adopted consistent with the more important requirement of stiftness. In Chapter XXII we have developed the method of con- jugate load areas, given by Eankine in mathematical form difi&cult of practical application, to the equilibrium of arches. Xll PREFACE. We substitute a semigraphical method of constructing the load areas, reducing the mathematics and extending the whole to the complete design of segmental, semicircular, and semi-elliptic arches, with their abutments, spandrils, and piers. Incidentally, the design of sewers, inverts, shafts, and tunnels illustrates the full scope of the method. In a paper to the Eoyal Irish Academy, referred to at bottom of p. 448, we demonstrated the true shape of the equilibrium curve, dividing the family into two groups, the more important of which we venture to call two-nosed catenaries. We then show that these two varieties ofifer a philosophical explanation of the two distinct ways in which it has been found by experiment that masonry arches break up when the abutments are gradually removed. Some of the leading writers in America upon Engineering expressed approbation of our method, and Professor Howe, in his treatise, referred to at bottom of p. 447, has done us the honour of adopting it, and calls it the best method yet pub- lished, when the arch bears only vertical loads. In our present treatment we have both vertical and horizontal loads duly considered, but have insisted upon a central elastic portion of the arch-ring being wholly free from other than vertical loads, except when the live load covers only half the arch when the horizontal reaction of the light elastic spandrils of the opposite side comes into play. Now, with the catenary tables, the segmental arches were limited to this elastic part only, and so the assumption of vertical loads only is justified, and the more especially as the horizontal load indicated in this case is necessarily outwards, and cannot be introduced in any practical way. CONTENTS. CHAPTEK I. LIMEAL STRESS AND 8TKAIX. paok State of Simple Strain — Exs. 1 to 6— Elasticity, Young's Modulus — Exs. 7 to 12— The production of Strain— Exs. 13 to 17— Resili- ence — Exs. 18 to 31, 1 CHAPTEK II. INTERNAL STRESS AND STRAIN, SIMPLE AND COMPOUND. RANKINE'S METHOD OF THE ELLIPSE OF STRESS. Internal Stress at a point in a solid in a simple state of Strain — Do., for a compound state of Strain — Uniplanar Stress — Direct Problem, the Principal Stresses given — Equal Like and Unlike Principal Stresses — Positions of third Plane, for maximum Normal Stress, for maximum Shear, for maximum Obliquity — Exs. 1 to 9 — Inverse Problem, to find the Principal Stresses — Moving Model illustrating the Method— Exs. 10 to 13, . . 20 CHAPTEK III. APPLICATION OF THE ELLIPSE OF STRESS TO THE STABILITY OF EARTHWORK. Friction among loose Earth Particles — Conditions of Equilibrium and Angle of Repose — Earth spread in horizontal layers behind a wall — Depth of Foundation — Earth spread in sloping layers behind a wall — Construction of the Auxiliary Figure to the Ellipse of Stress — Exs. 1 to 6, . . . . . .53 CHAPTEK IV. THE SCIENTIFIC DESIGN OF .MASONRY RETAINING WALLS. Practical considerations — Rectangular and Trapezoidal Walls — Do. do. surcharged — Battering Wall with Stepped Back — Spread and Depth of the Foundation of a Wall — Table of Thicknesses required for Masonry Retaining Walls of moderate heights — Profiles of Do. — Graphic Statics, composition of Plane Forces — Graphical Methods of Designing Retaining Walls — Exs. 1 to 6, . 67 XIV CONTENTS. CHAPTEK V. TKANSVERSE STRESS. Beams or Girders — Varieties of Loads — Conditions of their Equili- brium — Exs. 1 to 13 — Neutral Plane and Axis — Deflection, Slope and Curvature — Elements of the Stress at an Internal Point of a Beam — Resolution into Shearing Force and Bending Moment — Resistance to Shearing and Bending — The Cantilever, CHAPTEK VI. THE PARABOLA. Straight and Parabolic Slopes — Graphic Construction of Parabolic Segment and its Tangent — Parabolic Template, its use like a set- square — Equations to arcs drawn with the Template in various positions — Compounding Straight and Parabolic Slopes, . 104 CHAPTER YII. BENDING MOMENTS AND SHEARING FORCES FOR FIXED LOADS. Definition of Bending Moment and Shearing Force Diagrams — Beam loaded with Unequal Weights at Irregular Intervals — Analytical and Graphical Solutions— Cantilever with irregular Loads — Beam with Load at centre or uniform Load — Cantilever do. do. — Beam with equal Loads at equal intervals — Exs. 1 to 14, . . 115 CHAPTEK VIII. BENDING MOMENTS AND SHEARING FORCES FOR COMBINED FIXED LOADS. Beam with Uniform Load and Load at Centre — The Continuous Beam — Irregular Fixed Loads combined with an Uniform Load — Diagram of Square-roots of Bending Moments for do. by means of Arcs of Circles only —Partial Uniform Loads— Beam loaded in any manner with Fixed Loads— Exs. 1 to 9, . . . 139 CONTENTS. XV CHAPTER TX. BENDING MOMENTS AND SHEARING FORCES FOR MOVING LOADS. PACB Classes of Moving Loads— Beam subjected to an uniform advancing Load as long as the Span— Do., Load shorter than the Span- Beam subjected to a Rolling Load — Two- wheeled Trolly confined to a Girder like a Travelling Crane, Loads on the wheels equal, Loads unequal — Moving Model illustrating the Bending Moments on a Girder Bridge due to a Locomotive passing over it — Exs. 1 to 12, 150 CHAPTER X. BENDING MOMENTS AND SHEARING FORCES DUE TO A TRAVELLING LOAD SYSTEM. Maximum Bending Moments for a beam under a Travelling Load system of unequal weights fixed at irregular intervals apart, the Load confined to the span — "Fields" commanded by the different Loads — Nature of the Bending Moment Diagram and its Graphical Construction with one Parabolic Template — How the Scale is readily determined — Graphical Construction of a Diagram of Square-roots of Bending Moments by means of Circular Arcs only — Shearing Force Diagram — Particular Sys- tems of Equal Loads on wheels equally spaced — Exs. 1 to 5, . 182 CHAPTER XL ON THE GRAPHICAL CONSTRUCTION OF MAXDIUM BENDING MOMENTS ON SHORT GIRDERS DUE TO A LOCOMOTIVE. Reference to Farr's paper — A typical Locomotive — First Method, with one parabola only — Second Method, with circular arcs only — Third Method, being Cul man's rendered precise — Loco, and Dead Load combined — Kinematical Model — Example of the Uniform Load combined with Travelling Load .system, . . 198 XVI CONTENTS. CHAPTER XII. COMBINED LIVE AND DEAD LOADS WITH APPROXIMATION BY MEANS OF AN EQUIVALENT UNIFORM LIVE LOAD. PAGE Reduction of Live Loads to an equivalent uniform Dead Load — Shearing Force Diagram for a beam with uniform Dead and Live Loads — Its importance in the matter of counterbracing — The Live Load reduced instead to a single equivalent Rolling Load — Mr. Farr's paper on "Moving Loads on Railway under Bridges" — His allowance for the Impulse of the Moving Loads, with Tables— Exs. 1 to 9, 224 CHAPTER XIII. RESISTANCE, IN GENERAL, TO BENDING AND SHEARING AT THE VARIOUS CROSS-SECTIONS OF FRAMED GIIiDERS AND OF SOLID BEAMS. Graphical application of Ritter's method of sections to the construc- tion of the Stresses on the booms and braces of Framed Girders — Elevations of Flanged Girders of Uniform Strength — Solid Beams — Their Moments of Resistance to Bending for the Cross- Section either rectangular or triangular — Rectangular Beams of Uniform Strength and Uniform Depth— Do. do. of Uniform Breadth — Approximate Plans and Elevations — Exs. 1 to 10 — Area, Geometrical Moment, and Moment of Inertia — Theorems regarding, and their representation by, the Volume and Statical Moment of Isosceles Wedges, ....... 234 CHAPTER XIV. CROSS-SECTIONS : THEIR RESISTANCE TO BENDING AND SHEARING- AND DISTRIBUTION OF STRESS THEREON. Rectangular and Hollow Rectangular Sections — Tabular Method for Sections built of Rectangles— Graphical Construction of an approximation to the Moment of Inertia of a Section built of Rectangles — Absolute Correction in same by a further extension — Triangular Cross-Section and Sections partly built of Triangles, Hexagon, Diamond, Trapezoid — Circular and Elliptic Cross- Sections — Resistance of Cross-Sections to Shearing and the Dis- tribution of the Shearing Stress thereon, including the Hollow Rectangle, Doul)le T-Section, Triangle. Circle— Exs. 1 to 20, . 252 CONTENTS. xvii CHAPTER XV. STRESS AT AX INTERNAL POINT OF A BEAM. PAGB Curves of Principiil Stress — Detenninjitioii of Internal Stress by the Polariscope — Professor Peter Alexander's Method of drawing the Lines of Stress with a mechanical pen on a revolving disc upon which are thrown, by means of a lantern, the dark lines due to a strained prism of glass rotating in the polariscope attached to the lantern — Exercise, . .... 286 CHAPTEPt XVI. CUrVATURE, SLOPE AND DEFLECTION. Definition, depending on the skin coming to the proof strain at the Cross-Section where the bending moment is greatest — Beam hinged at ends. Section uniform, Load at the centre and again, Load uniformly spread — Two Loads symmetrical about the centre, as a pair of locomotive wheels on a cross-girder — Canti- lever — Beams and Cantilevers of uniform strength — Proportion of the depth of a beam to its span dominating the stiifness of the beam — Slope and Deflection due to a given Load less than Proof Load — Beam supported on three props — Uniform beam uniformly loaded and fixed at one or both ends — Virtual and actual Hinges — Exs. 1 to 12 293 CHAPTER XVII. FIXED AND MOVING LOADS ON A UNIFORM GIRDER WITH ENDS FIXED HORIZONTALLY. A. Fixed Load placed unsymmetrically on the span — Theorems : (a) The sum of the tips ^qj at the ends, had they been hinged, pro- portional to the Area of the Bending-Moment Diagram and (6) their ratio to each other inversely as the ratio of the segments into which a perpendicular from the centre of gravity of that Area divides the Span — Transit of a Rolling Load — Diagram of the Maximum positive and negative Bending Moments at each point of the Span, due to the transit — The corresponding posi- tions of the Rolling Load — Maximum of maxima at nearer Abutment, when the Load trisects the Span — References to Levy's La Statiqiie Graphiqxie — Shearing Force Diagram as XVlll CONTENTS. PAGE modified due to the ends being fixed — Transit of an Uniform Advancing Load — Diagram of Maximum positive and negative Bending Moments and corresponding positions of the Load — The Modified Shearing-Force Diagram — Exs. 1 to 4, with a full- page diagram to scale, . . . . . . . . 31 6 CHAPTEK XVIII. SKELETON SECTIONS FOE BEAMS AND STRUTS — COMBINED THRUST WITH BENDING AND TWISTING WITH BENDING — LONG STRUTS. Formulae for the Strength of thin hollow Cross-Sections of Beams — Cross-Sections of equal strength, Rankine's approximate formula for their design in cast-iron, the difference of the strengths being great — Exs. 1 to 4 — Allowance for Weight of Beam— Resistance to Twisting and Wrenching — Bending and Torsion combined — Thrust or Tension combined with Torsion — Thrust and induced Bending — Table of Loads on Pillars — Table of Breaking Loads for cast-iron and wrought-iron Struts for ratios of 1 : h from 10 to 50— Exs. 5 to 19, . . 335 CHAPTEK XIX. DESIGN OF LONG STEEL STRUTS. The Economical Double T- Section of Uniform Strength to resist Bending — Formulas for its Direct Design with given fractional thicknesses of Web and Flanges to secure a prescribed degree of Local Stiffness — Rankine-Gordon Formula — Table of Areas, Moments of Inertia, and Radii square of Gyration, giving close approximations for the Practical Application of the Rankine formula to Struts of box fianged and T- Section — Quotation of two of Fidler's tabulated results of the formula — Exs. 1 to 8, including a full-page diagram. 354 CHAPTER XX. THE STEEL ARCHED GIRDER. As adapted to Roofs — Dimensions of, and test Loads prescribed for, the St. Pancras Station Roof — Reference to Central Station Roof, Manchester Machinery Halls, Paris and Columbian Exposi- tions, and other Roofs, illustrating different outlines as circled. CONTENTS. xix PIGB parabolic. &c., and both with and without Hinges— The adapta- tion to Bridges, Niagara Falls Bridge, Oporto Bridge, and others — Disposition of Loads^ Levy's Graphical Stress Diagram — The curve of benders and the curve of ttatteners — Weight of metal relatively to three types— Rib of Uniform Section, hinged at the ends — Rib of Uniform Stiffness fixed at the ends — Two pair of full-page diagrams, to scale, with the Stresses scaled off in detail, 361 CHAPTEK XXI. ANALYSIS OF TRIANGULAR TRUSSING ON GIRDERS WITH HORIZONTAL PARALLEL BOOMS. Analyses following Levy in the main — His notation for the Triangu- lated Truss — The Dead Load — The Live Load — General expres- sions for the volume of steel required to resist these Loads — Particular expressions for the Warren and Rectangular Trusses — Discussion of the economical shapes of Triangle and of the economical depth which is consistent with that demanded for joint general and local stiffness — The Fink and Bollman Trusses — Tables of the theoretical volumes of those Trusses, showing at one opening of the book their relative advantages — Allowance to stiffen the long Struts — Exs. 1 to 6, . . . . . 384^ CHAPTEE XXII. THE SCIENTIFIC DESIGN OF MASONRY ARCHES. The Aich-ring — Elastic portion from crown outwards to two joints, practically the joints of Rupture — The segmental and semi- elliptic (false) as rival designs: their outstanding characters — Line of Stress confined to a " Kernel " of the Arch-ring — Its equilibrium and stability — Strength for Granite, Sandstone, and Brick rings — The balanced linear rib or chain — Conjugate Load- areas — Superposition of Loads — Fluid Load of equal or varying potential — Thrust at the crown of a rib — Buried Arches, Shafts, and Sewers — Exs. 1 to 10— The Stereostatic Rib — Horizontal Load-area for semi-circular Rib, loaded uniformly along the Rib — Do., loaded with the area between itself and a horizontal straight line over it — Allowance for excess weight of uniform Arch-ring — Do., for ring thickening outwards — Rankine's point and joint of rupture — Authors' location and modification of it — p XX CONTENTS. PAGS Semi -circular Masonry Arch — Equilibrium Rib or Transformed Catenary — The two-nosed Catenaries — Table of do., modulus, unity — Tables for the immediate design of Segmental Arches in granite, sandstone, or brick, with a minimum real factor of safety of 10 and with the Line of Stress confined to a " Kernel," a middle third, fifth, or ninth — Exs. 11 to 20 — Linear trans- formation of balanced rib — Geostatic Load uniform or varying potential — Approximate Elliptic Rib — Hydrostatic and Geostatic Ribs — Semi-elliptic Masomy Arch — Exs. 21 to 25 — Abutments and Piers — The Tunnel Shell — Allowance for excess-density of Elliptic Arch-ring — Exs. 26 to 31 — Quotations from Rankine's C-E.y and from Simms' Practical Tunndling, .... 413 CHAPTEE XXIII. THE METHOD OF llECIPKOCAL FIGURES. A rival of Ritter's Method of Sections — Suitable for roof-frames and irregular fixed loads — Already used in Levy's constructions, Ch. XX, figs. 2 and 4 — The simplest pair of reciprocal figures — One treated as a self-strained frame and the other as its stress diagram, the reciprocity being perfect — General application to an iron roof-frame — To be statically determinate it is assumed that the storm end, only, is anchored — Reactions at the supports constructed by a Link Polygon — The Stress Diagram constructed, using Levy's scientific notation — Anchor shifted to the lee end, and a new stress diagram inscribed in the first by inspection — These two diagrams are maximum and minimum strains ; in the * second it is the wind that has shifted and the frame viewed from the reverse side — Such a stress diagram is only a part of a reciprocal figure — is always a part if the frame be just rigid or indeformable — rule for pre-determining this — The frame and link polygon made into one geometrical figure joined by inter- cepts on the lines of action of the force ; they mutually strain each other — This figure^ being freed from all idea of extern forces, has one redundant bar and is self-strained — The one and only reciprocal figure to it is the stress diagram augment by the vectors from the pole — The reciprocity is imperfect in the reverse order, as the self-strained frame is one of its many reciprocal figures— Ex. I, Queen-post roof double braced in the centre and with an extra load between the supports — Ex. II, Great curved iron King-post roof — do., double braced like King's Cross Ry. station roof— Ex. III. Compound roof-frame, pull on main tie implicitly given — Note on Ch. X, . . . 498 ELEMENTARY APPLIED MECHANICS. CHAPTER 1. LINEAL STRESS AND STRAIN. Elasticity is a projierty of matter. When dealing with the equilibrium of a body under the action of external forces, in order to find the relations among those external forces, the matter of the body is considered to be perfectly rigid, or, in other words, to have no such property as elasticity. When external forces, the simplest of which are stresses acting really on a part of the surface of a body, are considered to act at points on the surface, it is taken for granted that the matter of the body is infinitely strong at such points. But after considering the equilibrium of the body as a whole, we may consider the equilibrium of all or any of its parts. If we take a part on which an external stress directly acts, equilibrium is maintained between that external stress acting on the free surface and the components parallel to it, of stresses which the cut surface of the remaining part exerts on its cut surface. Let MONQP be a solid in equilibrium under the action of the three external uniform stresses acting on planes of its surface at 0, P, and Q. Let MN be the trace of the plane at under the uniform stress A. The stresses aaa . . . hbb . . . ccc may be represented in amount and direction by the single forces A, B, and C acting at the points 0, P, and Q, rigidly connected. We know that, by the triangle of forces, A, B, and C are proportional to the sides of a triangle DFE drawn with its ^ides parallel to their directions. Also that they are in one plane and meet at one point. Hence B Fig. 1. APPLIED MECHANICS. [chap I. Fig. 2. we infer that the stresses which they represent are all parallel to the plane of the paper, and that the planes of action of h and c are at right angles to the plane of the paper as well as that of o.. Thus we find the relation among the external forces. Let a plane mn divide the solid into two parts (fig. 2). Consider the equilibrium of the part MNmn. Sx, s^, S3 . . . are the stresses exerted at all points of the cut surface of MNmn by the cut surface of the other part. S is the sum of their components parallel to the direction of A, acting through P, the centre of pressure. Because there is equili- brium, S is equal and opposite to A ; and they act in one straight line. Also the remaining rectangular com- ponents of Si, 52, S3 are themselves in equilibrium. Thus we see there is a stress on the plane mn, and know the amount of it in one direction. Had we been considering the equilibrium of the other part of the solid, the stresses on mn (fig. 3) would have been acting on the other surface as t^, U, U,... in opposite directions to Si, So, S3,... and of equal intensities. Thus on the plane mn there are pairs of actions, acting on all points of it, as 53, ^3, at q. These vary in intensity and obliquity to mn at different points of the plane. If another plane, as gh, dividing the solid, pass through q, there will be, similarly, pairs of actions at all points of it, and a pair of definite intensity and direction at the point q. If we know the stress at the point q in intensity and direction on all planes passing through q, we are said to know the intenud stress at the point q of the solid. Similarly for all points of the solid. The pairs of actions as S3, ^3 act respectively on the cut surfaces of the upper and under parts of the solid ; but mn may be considered to be a thin layer of the solid with S3 and ^3 acting on its under and upper surfaces. This layer of the solid must l)e, however, infinitely thin ; otherwise its two surfaces would be different sections of the solid, and S3 and ^3 not necessarily equal and opposite. If gh be also considered a thin layer, and^ and K be the pair of actions on it at the point q on the two sides of it, CHAP. I.] LINEAL STRESS AND STRAIN. Fig. 4. thou will the point q be a solid, in figure a parallelepiped, with a pair of stresses acting upon its opposite pairs of faces. S3 and t^ being equal, *S is now put for each, and H is put for both H and K ; and since q, instead of being a point in both planes, has small surfaces in both, though so infinitely small that the stresses over them do not vary from the intensities at the point q, yet surfaces, the stresses spread over which it is more convenient to represent by sets of equal arrows SSS . . . , HHH .... There are two convenient ways of representing by a diagram the iTiternal stress at q, a point within a solid. One,- as in figure 5, in which the indefi- nitely smallparallel- epiped q is all of the solid to be imme- diately considered ; and the other, as in figure 6, in which sheaves of equal ar- rows stand on small portions of the planes mn and gh in the neighbourhood of q. State of Simple Stkain. Thus we see that, in a solid acted upon by external forces, every particle exerts stress upon all those surrounding it. Such a body is said to be in a state of strain. In solids the phenomenon is marked by an alteration of shape, but not necessarily of bulk, or of both. Let AB and CD (fig. 7) be acted upon by two equal and opposite forces P and P in the direction of their length acting in AB away from each other, and in CD towards each other. If P be uniformly distributed over the area A, the section of AB perpendicular to its length, the intensity of the stress on it is P = P 2" Let the prism be of unit thickness normal to the paper ; then b2 APPLIED MECHANICS. [chap. I. will the line MN be equal to the area of the section of the prism perpendicular to its axis, and P P A At any internal layer the prism, the intensity is also 7?, for the equi- librium of the parts requires it ; and not only is the stress of this same intensity at all points of one such sec- tion, but also upon all such sections. The solids AB and CB are said to be in a state of simple strain, in the case of AB of extension, and in that of CD of compression. It is usual to consider the first as MN perpendicular I. to the axis of H pv "^ ^^ 3f 11 771 LLU J!r PV PV ■r: PP Ap Ph pP 'P Fig. 8. .Fig- 7. positive and the second as negative. The change of dimensions due to a simple state of strain is an alteration of the length of the solid in the direction of the stress with or without an accompanying alteration of its other dimensions. Thus a piece of cork in a state of simple compres- sion has become shorter in the direction of the thrust, yet with scarcely any, certainly without a corresponding, increase of area, normal to the thrust. Again, a piece of indiarubber grows shorter in the direction of the thrust with an almost exactly proportionate increase of area normal to it. The increase of length in the case of an extension is the aiigmentation, in that of a compression it is a negative augmen- tation, and in either case it is the amount of strain. The measure of the strain is the ratio of the augmentation to the original unstrained length. ^ ^ . ,. , . augmentation of length Definition. — Longitudmal stram = — ^ , -. ^— ° length where both are in the same name, that is, both in inches or feet, &c. The foot being the vmit of length, it is most convenient to take both in feet; then - ., ,. , , . augmentation in feet longitudmal strain = — —. -r—. — -z — ° length in feet CHAP. I.] LINEAL STRESS AND STRAIN. 5 Suppose the denominator on the right-hand side of the equation to be unity, then longitudinal strain = augmentation in feet of 1 foot of the substance = augmentation per foot of length, expressed in feet. Hence the total augmentation or amovnt of strain in feet equals the length in feet multiplied by the strain. If the augmentation equal the length, that is, if the piece be stretched to double its original length or compressed to nothing, then from the definition strain = unity. Examples. In the following questions the weight of the material is neglected : — 1. A tie-rod in a roof, M-hose length is 142 feet, stretches 1 inch when bearing its proper stress. What strain is it subjected to ? augmentation = 1 in. unstrained length = 1704 in. augmentation 1 „„„ strain = — ^^ ; = — — or '0006. length 1704 2. How much will a tie-rod 100 feet long stretcb when subjected to "001 of strain r augmentation — ^ ; = strain ; length .•. augmentation = strain x length = -OOl x 100 ft. = •! ft. 3. A cast-iron pillar 18 feet high shrinks to 17"99 feet when loaded. "What is the strain ? augmentation of length = — "01 ft. augmentation - -01 ft. 1 strain = , - = — = or - -0005. length 18 ft. 1800 4. Two wire cables, whose lengths are 100 and 90 fathoms, respectively, while mooring a ship, are stietched, ttie first 3 inches and the second 2-75 inches. What strains do they sustain ? Which sustains the greater ? Give the ratio of the strains. For the 100-fathom cable augmentation 3 in. strain = , = — = -000417. length 7200 m. APPLIED MECHANICS. [CHAP. I For the 90-fathom cable augmentation 2*75 in. „„^ „ strain = ^. ,, - = ^.^^ ■ = -000424. length 6480 in. The 90-fathom cable is the more strained. Eatio of these strains is 417 to 424. 5. A 30-feet suspension rod stretches -^ inch under its load. Find the strain upon it. strain = -00014. 6. How much does another of them, which is 23 feet long, stretch when equally strained ? augmentation = -039 in. Elasticity. The elasticity of a solid is the tendency it has when strained 10 reg'ain its original size and shape. If two equal and similar prisms of ditterent matter be strained similarly and to an equal degree, that which required the greater stress is the more elastic — e.g. a copper wire 1000 inches long was stretched an inch by a weight of 680 lbs. while an iron wire of the same section and length required 1000 lbs. to stretch it an inch. Hence iron is more elastic than copper. If they be strained by equal stresses, that which is the more strained is the less elastic — e.g. the same copper wire is stretched as before 1 inch by a weight of 680 lbs., while the iron one is only stretched a -g^th part of an inch by 680 lbs. Hence the elasticities of different substances are proportional to the stresses applied, and inversely proportional to the accom- panying strains. If similar rods of steel and indiarubber be subjected to the same stress, the indiarubber experiences an immensely greater strain, so that steel is very much more elastic than indiarubber. If two similar rods of the same matter, or the one rod successively, be strained by different stresses, the corresponding strams are proportional to the stresses. Thus, if a 480 lbs. stress stretch a copper wire one inch, then a 960 lbs. stress will stretch it, or a similar rod, two inches. Hooke's Law is " The strain is proportional to the stress." It amounts to — " the effect is proportional to the cause." It is only true for solids within certain limits — e.g., 2400 lbs. should stretch the copper wire mentioned above five inches by Hooke's law, but it would really tear it to pieces ; and although 1920 lbs. CllAI'. 1.] LINEAL STRESS AND STRAIN. 7 applied very gradually will not tear it, yet it will stretch it more than four inches ; and further, when that stress is removed the wire will not contract to its original length again. Strain and stress are mutually cause and effect. The efi'ect of stress upon a solid is to produce strain ; and, conversely, a body in a state of strain exerts stress. The expressions " Strain due to the stress," &c., and " Stress due to the strain," &c., are both correct. If a solid be strained beyond a certain degree, called the proof strain, it does not regain its original length when released from the strain ; in such a case the permanent alteration of length is called a set. Def. — The Proof Load is the stress of greatest intensity which will just produce-a strain having the same ratio to itself which the strains bear constantly to the stresses producing them for all stresses of less intensity. If a stress be applied of very much greater intensity, the piece will break at once ; if of moderately greater intensity, the piece will take a set ; and although only of a little greater intensity, yet if applied for a long time, the piece will ultimately take a set ; and if it be applied and removed many times in succession, the strain will increase each time and the piece ultimately break. For all stresses of intensities less than the proof load the elasticity is constant for the same substance, and the -r, ., r T 1 n ^ .■ ■. iuteuslty of stress Def. — Modulus of elasticity = r— =^^ ^- strain due to it = stress per unit strain. If the denominator on the right-hand side of the equation be unity, then the numerator is the stress which produces unit strain, and Mod. of elasticity = stress which would produce unit strain on supposition of rod not experiencing a set and Hooke's law holding. For most substances the proof stress is a mere fraction of £, the modulus of elasticity. For steel the proof stress is scarcely loo^oo ^h part of F. Hence in the equation above, the word would is employed, as it would be absurd to say that U equalled the stress that will produce unit strain, that being an impossi- bility with most substances ; and even when possible, as in the case of indiarubber, the strains at such a pitch will have ceased to be proportional to the stresses producing them, and hence U 8 APPLIED MECHANICS. [CHAP. I. will be no longer of a constant value. But the definition is quite accurate and definite for all substances amounting to this, that for any substance ^ = 10 times the stress that will produce a strain of -fV^h, if such a pitch of strain be possible and within the limit of strain, that is, not greater than the proof strain. But if not, then, E = 100 times the stress that will produce a strain of xftt^^- if such a pitch of strain be possible and within the limit of strain, that is, not greater than the proof strain. Thus for steel E equals one million times the stress which will produce a strain of one-millionth part. Pliability is a term applied to the property which indiarubber possesses in a higher degree than steel. Examples. 7. A wrought-iron tie-rod has a stress of 18000 lbs. per square inch, of section which produces a strain of -0006. Find the modulus of elasticity of the iron. „ intensity of stress 18000 „ ^„„ ,, . , E = ■/—. = "— — = 30000000 lbs. per square inch. strain -0006 ' ^ 8. A tie -rod 100 feet long has a sectional area of 2 square inches ; it bears a tension of 32,000 lbs., by which it is stretched fths of an inch. Find the intensity of the stress, the strain, and modulus of elasticity. total stress 32000 lbs. stress = = ; = 16000 lbs. per sq. m. area 2 sq. m. aug. of length -76 in. strain = -^-j —5— = — -— - = -000625. length 1200 in. „ stress 16000 E = -—r~ = ^^_,, = 25600000 lbs. per sq. in. strain -000625 ^ 9. A cast-iron pillar one square foot in sectional area bears a weight of 2000 tons ; what strain will this produce, E for cast iron being 17,000,000 lbs. ? total stress = 2000 tons per square foot = 2000 x 2240 lbs. per sq. foot. 2000 X 2240 stress = — = 31111-1 lbs. per sq. in. 144 „ stress 31111 E = — . , or 17000000 = ; strain strain ••• strain = ^^^^=-0018 ft. per ft. of length. CHAP. I.] LINEAL STRESS AND STRAIN. 9 irr- 10. The modulus of elasticity of steel is 30,000,000. How much will a steel rod 50 feet long and of Jth inch sectional area be stretched by a weight of one ton? total stress = 2240 lbs. ; total stress in lbs. stress = -. : = 2240 i- i = 17920 lbs. per sq. inch. area in sq. in. „ stress stress 17920 E = —: .-. strain =———= — =-000.'512: strain £ 35000000 elongation , , = strain ; length .-. elongation = strain \- length = -000512 x ')0 = •02.')6 feet or ^ of an inch. 11. An iron wire 600 yards long and A'th of sq. inch in section, in moving a signal, sustains a pull = 250 lbs. ; liow much will it stretch, assuming £ = 25000000? stress = 20000 lbs. per sq. inch ; strain = -OOOS ; elongation = 1*44 feet. 12. Modulus of elasticity of copper is 17,000,000 : what weight ought to stretch a copper thread, of 12 inches in length and -004 inches in sectional area, i^T7th part of an inch. If after the removal of the weight the thread remains a little stretched, what do you infer about the weight and about the strain to which the thread was subjected ? strain = TaVoth ; stress = 14167 lbs. per sq, inch ; weight = 56*668 lbs. Since this weight causes a set, it is greater than the proof load. The Production of Strain. We have as yet only considered the statical condition of strain, i. e. of a body kept in a state of strain by external forces, these forces being balanced by the reactions of the solid at their places of application due to the elasticity, and the forces exerted on any portion of the solid being in equilibrium with the re- actions of the contingent parts. Thus when we found that 32,000 lbs. produced a strain of -00063 on a tie-rod 100 feet long and 2 square inches in area, in all stretching it f ths of an inch, we meant that the weight ke2}t it at that strain ; the rod is sup- posed to have arrived at that pitch of strain and to be at rest, to be stretched the fth inch, and so (by its elasticity or tendency to regain its original length) to balance the weight. We have taken no notice of the process by which the rod came to the strain, nor do we say it was the weight that brought it to that state, the weight being only a convenient way of giving the value of the stress on the rod when forcibly Jcejyt strained. In fact, an actual weight of 32,000 lbs. is capable of producing 10 APPLIED MECHANICS. [CHAP. I. greater strains on the rod in question, depending upon how it IS applied to the rod as yet unstrained. The weight might be attached by a chain to the end of a rod and let drop from a height; when the chain checked its fall, it would produce a strain on the rod at the instant greater the greater the height through which it dropped. Still, if that strain were not greater than the proof strain, the weight upon finally coming to rest after oscillating a while could only keep the rod at the strain •X)0063. We come now to consider the kinetic relations between the stress and the strain, that is, while the strain is being produced, the matter of the body being then in motion, we are consequently considering the relations among forces acting upon matter in motion. If a simple stress of a specific amount be applied to a body, it produces a certain strain, and in doing so the stress does work, for it acts through a space in the direction of its action equal to the total strain. Bui if this stress is applied gradually, so as not to produce a shock, its value increases gradually from zero to its full value, and the work it does will be equal to its mean value, multiplied by the space through which it has acted. And since the stress increases in proportion to the elongation, its average value will equal half of its full value. For example, if a stress of 30,000 lbs. be applied to a rod and produce a strain of I inch, it will do -^1^ x f = 11,250 inch-lbs. of work, which will be stored up as a potential energy in the stretched rod. Suppose a scale-pan attached to the top of a strut or bottom of a tie and the other end fixed. Let a weight be put in contact with the pan, but be otherwise supported so as to exert no stress on the piece, and the next instant let it rest all its weight on the piece, then will the weight do work against the resistance offered by the straining of the piece till the weight ceases descending and comes to rest, when the piece will be for an instant at the greatest strain under the circumstances, at a strain greater than the weight can keep it at; the unstraining of the piece will therefore cause the weight to ascend again, doing work against it to the amount that the weight did in descending, and so the weight will return to its first position, then begin to descend again, and so oscillate up and down through an amplitude equal to the augmentation. Owing to other properties of the matter, whereby some of the work is dissipated during each strain and restitution, the amplitude diminishes every oscillation, and the weight will finally settle at the middle of the amplitude. A weight applied in this manner is called a live load. A CHAP. I.] LINEAL STRESS A1«D STKAIN. 11 live load produces, the instant it is applieil, an augmentation of length double of that which it can maintain, and therefore causes an insfanta neons strain doul)le the strain due to a stress of the same amount as the load. Let now a weight W be applied in the following way : — Divide W into n equal parts of weight ic each. If A be the strain due to a stress of amount W, and a the strain due to a stress u\ then W ^ mu, and by Hooke's law. A = na. Let the first weight iv be put into the scale-pan. It will produce a strain 2a at once, but the piece will settle at a strain a. Now put on the second weight u\ It will produce at once an additional strain 2^, but only of a additional after the piece settles ; there being now a total strain 2a. Add the third weight u\ It also will produce at first an additional strain 2a, but only of ct after the piece settles, giving a total now of 3a ; and so, adding them one by one, there will be a strain of (n - l)a when the second last one has been added and the piece has settled. Now, upon adding the nth. weight w, it will at first produce an additional strain 2a, but only of a after the piece settles, giving then a total strain na or A. Thus we have brought the piece to a pitch of strain A by means of the weight W, and only at the instant of adding the last part (iv) of it was the piece strained to (n + l)a, or to a more than A. By making the parts more numerous into which we divide W, and so each part lighter and producing a lesser strain per part, we can make the strain a the extent to which the piece is strained beyond A at the instant of adding the last part, as small as we please. By so applying the load W we can bring the piece to the corresponding strain A without at all straining it beyond that. A weight so applied is called a dead load. A live load therefore produces, at the instant of its applica- tion, a strain equal to that due to a dead load of double the amount. In designing, the greatest strain is that for which provision must be made, so that live loads must be doubled in amount, and the strain then reckoned as due to that amount of dead load. The dead load, together with twice the live load, is called the gross load. The weights of a structure and of its pieces are generally dead loads. Stress produced by a screw, as in tightening a tie-rod, is a dead load. The pressure of earth or water gradually filled in behind a retaining wall, and of steam got up slowly, of water upon a floating body at rest in it, &c., are all dead loads. The weight of a man, a cart, or a train coming suddenly upon 12 APPLIED MECHANICS. [CHAP. I. a structure, is a live load ; so is the pressure of steam coming suddenly into a vessel ; so is a portion of the pressure of water upon a floating body which is rolling or plunging. The pressure upon a plunger used to pump water is a live load, but that on a piston when compressing gas is a dead load, the gas being so elastic itself. A load on a chain ascending or descending a pit is a dead load when moving at a constant speed or at rest, but a live load at the starting, and while the speed is increasing, partly a live and partly a dead load. The stress upon the coupling between two railway carriages is a dead load while the speed is uniform, and if the buffers keep the coupling chain tight, the stress is a live load while starting; but if the buffers do not keep it tight, but allow it to hang in a curve when at rest, then the stress upon it at starting will be greater than a live load. JIXAMPLES. 13. An iron rod in a suspension -bridge supports of the roadway 2000 lbs., and wben a load of 3 tons passes over it, bears one-fourth part thereof. Find the gross load. If the rod be!20 feet long, and f of a square inch in section, find the elongation, E being 29,000,000. dead load = 2000; live load = 1680 lbs., equivalent to a dead load of 33G0 lbs. ; .-. gross load = 5360 lbs. ; gross load 5360 _,,_,, stress = ° . - = — - = 7147 lbs. per so. in. section '75 _ _ stress strain ' stress 7147 „,^^ „ elongation .-. strain = = = -000246 = , ° , ; E 29000000 length .-. elongation = length x strain = 20 x -000246 = -00492 ft. = -06 in. 14. A vertical wrought-iron rod 200 feet long has to lift a weight of 2 tons. Find the area of section, first neglecting its own weight ; if the greatest strain to which it is advisable to subject wrought-iron be -0005 and E = 30,000,000. Let A be the sectional area in sq. in. live load = 4480 lbs. is equivalent to a dead load of 8960 lbs. 8960 ^ stress „„ „„„ „„„ 8960 .-. stress = — — ; E = — — , or 30,000,000 = — — . A strain A x -0005 8960 •597 sq. in. -0005 X 30000000 15. Find now the necessary section at top of rod, taking the weight into account, calculated from the section found in last. 200 ft. X -597 sq. in. gives 1433 cubic in. ; reckoned at 480 lbs. per cubic foot gives 398 lbs. CHAl'. 1.] LINEAL STHESS AND STRAIN. ] Hence live load = 4480 lbs. ; dead load = 398 lbs. 9358 „ stress 9358 .-. Rross load = 9358 ; stress = ; E= -, or 30000000 = areii strain area x •000.' 9358 - — = -62 sq. in. •0005 X 30000000 The weight of the rod being greater when calculated at this section, a third approximation to the sectional area might be made. 16. Taking now the sectional area at "G'i sq. in., find average strain and elongation. At lowest point 8960 . 8960 gross load = 8960 lbs. ; stress = -— ; strain = — = -00048 ; while strain at highest point is -0005 ; .-. average strain ^ -00049 ; elongation = -00049 x 200 = -098 ft. = M76 in. 17. A short hollow cast-iron pillar has a sectional area of 12 sq. in. It is advisable only to strain cast-iron to the pitch -0015. If the pillar supports a dead load of 50 tons, being weight of floor of a railway platform, and loaded waggons pass over it, what amount should such load not exceed ? E = 20000000. greatest stress = 30000 lbs. per sq. in. ; gross load = 360000 lbs. deduct dead load = 112000 lbs. ; gives a dead load = 248000 lbs. The live load must not exceed one-half of this. Note. — Other considerations limit the strength of the pillar if it be long. Eesilience. Def. — The Resilience of a body is the amount of work required to produce the proof strain. A weight one-half the proof stress applied as a live load would produce the proof strain ; therefore the work done is this weight multiplied by the elongation at proof strain, the distance which the weight has worked through ; or the resilience of a body = ^ amount of proof stress x elongation at proof strain. For comparison among different substances the resilience is measured by the resilience of one foot of the substance by one square inch in sectional area. .'. ifc = ^ proof stress x proof strain, R being in foot-lbs. when the stress is in lbs. per square inch and the strain in feet. 14 APPLIED MECHANICS. [CHAP. I. And now comparing the amount of resilience of different masses of the same substance : if two be of equal sectional area, that which is twice the length of the other has twice the amount of resilience (the elongation being double) ; also if two be of equal length and one have twice the sectional area of the other, then the amount of its resilience is double (the amount of stress upon it being twice that upon the other). That is, the amounts of the resilience of masses of the same substance are proportional to their volumes. This is true not only for pieces in a state of simple strain with which we are in the meantime occupied, but can be proved to be universally true for those in any state of strain, however complex. For any substance R being the amount of resilience of a prism of that substance one foot long by one square incli in sectional area, it follows from the alcove that the amount of resilience of a cubic inch of the substance will be -j-V-K or that of any volume will he ^R x volume in cubic inches. The resilience of a piece, as defined, is the greatest amount of work which can be done against the elasticity of the piece, without injuring its material. We can find the amounts of work done upon a piece in bringing it to pitches of strain lower than the proof strain. For brevity we will call this also resilience. Thus, for a piece 1 foot long by 1 square inch in section amount of resilience = ^ stress x strain is pro. to (stress)^ the strain being proportional to the stress ; hence amount of resilience for any stress _ (stress)^ the resilience (proof stress)*' .'. amount of resilience = R x i 1 • Vp. stress/ For a piece of volume V cubic inches, at any stress we have either — V amount of resilience = i stress x strain x — , or = I amt. of stress x amt. of strain. The amount of resilience of a piece, at the instant a live load is applied, will be the product of that load and the instantaneous elongation. Let W be a load the elongation due to which is A. CHAP. I.J LINEAL STRESS AND STHAIN. 15 If ]r be applied as a live load, the iii-staiitaneous elungatiou is 2A, and the amount of resilience due to a live load W = W x 2 A. If W be applied as a dead load, the amount of resilience is steadily that of the piece elongated to an amount A, is the same as what it would be for an instant upon the application of W a live load — , or W amount of resilience due to a dead load W = -^ y. A. Therefore, a live load produces for an instant an amount of resilience four times that produced by an equal dead load. Examples. 18. A rod of steel 10 feet long and "5 of a square inch in section is kept at the proof strain by a tension of 25,000 lbs., the modulus of elasticity for steel being 35,000,000. Find the resilience of steel, also the amount of resilience of the rod. 25000 „ . , proof stress = — ^ — = 50000 lbs. per sq. inch. *o „ proof stress E = proof strain = proof strain * proof stress 50000 1 E 35000000 700 = -00143 elongation in ft. per ft. of length. resilience, R = h proof stress x proof strain = \ x 50000 lbs. x -00143 ft. = 35-75 ft. -lbs. of work per vol. of 1 ft. in length by 1 sq. in. in sectional area. amt. of res. of rod = R x (vol. expressed in number of such prisms) = — , i? X vol. in cub. in. = — x 35-75 x 120 in. x -5 sq. in. = 178-75 foot-lbs. of work. Otherwise, to find amount of resilience directly, proof strain = -— - ; total elongation = — ft.; amount of stress = -25000 lbs. 700 " 70 amount of resilience = t amount of stress x elongation, 25000 1 = — -— X — , = 178-6 ft. -lbs. of work. 2 lO 16 APPLIED MECHANICS. [CHAP. I. 19. A series of experiments were made on bars of wrought iron, and it was found that they took a set when strained to a degree greater than that produced by a stress 20,000 lbs. per square inch, but not when strained to a less degree. At that pitch the strain was -0006. Find the resilience of this quality of iron. proof stress = 20000 lbs. per sq. in. proof strain = -0006 ft. per ft. of length. Jt = ^ X 20000 X -0006 = 6 ft. -lbs. 20. Find how m\ich work it would take to bring a rod, of the above iron, 20 feet long and 2 square inches in sectional area, to the proof strain. volume = 480 cub. in. E ft. -lbs. of work brings to proof strain a rod 1 foot long by 1 square inch in area; that is, of volume 12 cubic inches, and amounts of resilience being proportional to the volumes. work required = ^R . vol. in cub. in. = -^^ x 6 x 480 = 240 ft.-Ibs. amount of res. 480 cub. in. or = ; — : — = 40. " It 12 cub. m. .-. amount of res. = iJ x 40 = 6 x 40 = 240 ft. -lbs. 21. A wooden strut 18 square inches in section, and 12 feet long, sustains a stress of 1000 lbs. per square inch. Find the amount of resilience of'the strut, ^heing 1,200,000 lbs. half of total stress = 9000 lbs. ; elongation = -01 ft. ; amount of resilience = 90 ft. -lbs. 22. Steam at a tension of 600 lbs. on the square inch is admitted suddenly upon a piston 18 inches in diameter. If the piston rod be 2 inches in diameter and 7 feet long, what is the amount of its resilience at the instant? £ = 30000000. for live load stress = 97200 lbs. per sq. in. gives instant strain = -00324 ft. per ft. of length. elongation = -02268 ft. resilience of rod = live load x elongation = 486007r lbs. x -02268 ft. = 11025r ft. -lbs. 23. The chain of a crane is 30 feet long and has a sectional area equivalent to A of a square inch : what is the amount of its resilience when a stone of 1 ton weight resting on a wooden frame is lifted by the action of the crane ? £ = 30000000. stress = 4480 lbs. per square inch, strain = -000149. amount of stress = 2240 lbs., elongation = -00447 ft. resilience of chain = ^ amount of stress x elongation = 5 ft.-lhs. 24. If the chain be just tight, but supporting none of the weight of stone, and if now the wooden frame suddenly gives way, what is the amount of resilience of the chain at the instant? Being now a live load, there is an instantaneous strain of double the former amount. instantaneous strain = -000298. instantaneous elongation = -00894 ft. resilience of chain = live load x elongation = 2240 x •00894 = 20 ft. -lbs. CHAF. I.] LINEAL STRESS AND STIiAIN. 17 25. The wire for moving a signal 600 yards distant bus, when the signal is down, u tension upon it of 250 lbs., which is maintained by the back weight of the hand lever; under the circumstances the wire is stretched an amount 1-44 feet, and so the buck weight of the signul, which is 280 lbs., rests portion of its weight upon its bed. The hand lever is suddenly pulled back and locks : the wire being more intensely strained, the signal is raised by the elasticity of the wire partially unstraining. 1 1 tlie point where the wire is attached to the signal moves through ■2 feet, find the range of the point where the wire is attached to the hand lever, also the force which must be exerted there. When the signal settles up, the amount of stress on the wire is 280 lbs. elongation for 280 lbs. _ 280 elongation for 250 lbs. 250 ' 280 elongation for 280 lbs. = -— - x 1'44 = 1-613 ; additional elongation = -173 ft., 2?0 .•. range of point at lever = range of point at signal plus this additional elongation = -2+ 173 = -373 feet. Thus, when tiie lever is put back there is upon the wire for an instant before the signal rises an additional elongation of •373 feet. Hence the tension on the wire the instant the lever is put back will be that due to an elongation of (1 44 + -373) ft. = ^"^'^ + '^'^ 250 lbs. = 314-8 lbs. 1-44 This is the force which must be exerted at the point where the wire is attached to the hand lever. That is, the instantaneous value of the force used to raise the signal is 34-8 lbs. greater than its weight. 26. On a chain 30 feet long, f of a square inch in sectional area and having a modulus of elasticity of 25,000,000 lbs. there is a dead load of 3900 lbs. and a live load of 900 lbs. Find the amount of resilience of chain when dead load only is on, also at instant live load comes on. stress 5200 as strain = -— — = — = -0002, £ 25000000 ' elongation = -006 ft., amount of resilience for dead load = J amount of stress x elongation = J X 3900 X -006 =11-7 ft.-lbs. Live load gives an additional elongation equal to that for a dead load of 1800 lbs. elongation (due to 1800 lbs.) _ 1800 -006 ft. ~ 3900 ' inst. elongation for live load = ~— — x -006 = -00277 ft. jyuu Now both the 3900 lbs. and the 900 lbs. worked through this -00277 ft. .-. additional resilience = 4800 lbs. x '00277 ft.' = 13-3 ft.-lbs. amount of resilience at instant live load comes on = 25 ft.-lbs. 27. A rod 20 feet long and ^ inch in sectional area bears a dead load of 5000 lbs. Find the live load which would produce an instantaneous elongation of another -i'oth inch. E = 30000000. Aus. 3125 lbs. C 18 APPLIED MECHANICS. [r'HAP. I. 28. A rod of iron 1 square inch section and 24 feet long checks a weight of 36 lbs. which drops through 10 feet before beginning to strain it. If ^ = 25000000, find greatest strain. Let p = the stress at instant of greatest strain ; then P 1 • 24» strain = — , elongation = -— , amount of resilience = \ amount of stress x elongation = ?>^'^ = 12$ft.-lbs. 2 E E Work done by weight in falling = 36 lbs. x ^10 + ~^\ ft. = 360 + -^ ft. -lbs. Equating, -^ ^ ~E p"- - lip = 30£; p^ - 72jo + (36)2 = 750001296, ;> - 36 - 27386; ;^ = 27422 lbs. per square inch, strain = -001097 ft. per ft. of length. 29. If the weight in Ex. 28 had fallen through the 10 feet by the time it came first to rest and E = 30000000 lbs., what is the greatest strain :- 12;>* amount of resilience = 360 ft. -lbs., or -— - = 360, E .-. p = 30000 lbs. per square inch, strain = "001 ft. per ft. length. 30. If the proof strain of iron be -001, what is the shortest length of the rod of one square inch in sectional area which will not take a set when subjected to the shock caused by checking a weight of 36 lbs. dropped through 10 feet? IE = 30000000 lbs. per square inch.] Let X = length in feet. By hypothesis it comes to the proof strain ; hence elongation = -OOl x a; ft. proof stress = E x proof strain = 30000 lbs. per sq. inch, amount of stress = 30000 lbs. (inst.) amount of res. = ^ amount of stress x elongation = 15000 x -^ = 152: ft.-lbs. 1000 Equating to work done by weight, 15t = 360 : x = 24 ft. Note. — This is the shortest rod of iron one square inch in sectional area which will bear the shock. The volume of this rod is 288 cubic inches, and a rod of iron which has 288 cubic inches of volume will just bear the shock ; as 48 feet long by I square inch in area or 12 feet long by 2 square inches sectional area. The 10 feet fallen through by the weight includes the elongation of rod. When the question is to find the shortest rod to sustain the shock in the case where the weight falls through 10 feet before it hegim to strain the rod, the volumes of the rods would not he exactly equal for difi"erent sectional areas ; for a long thin rod will sustain a greater elongation than a short thick one, and as the falling weight works through this elongation over and above the 10 feet, the first rod wiU require a greater cubical volume than the second. CHAr. I.] LINEAL STKESS AND STRAIN. 19 3L Find the shortest length of a rod of steel which will just bear without injury the shock caused by checkinj; ii weight of 60 lbs. which falls through 12 feet before beginning to strain the rod. First for a rod of sectional area 2 square inches, and then for a rod of ^ square inch sectional area. Given that for steel E = 30000000 lbs. and E = 16 ft. -lbs. Let A = sectional area in square inches and x = length in feet. proof stress y proof strain = 2R def. proof stress ^ , . ; r- = £ def. proof strain Dividing (proof strain)- = — • I2S I , . X proof strain = /-- = r— rr , elongation = ft. ^ V £■ 1000' ^ 1000 "Work done by the weight in falling = 60 lbs. X (l2 + ^)ft. = 720+l.ft.-lbs. Amount of resilience of rod at proof strain = iZ X I — vol. in cub. in. j = i? X (length in ft. x sec. area in sq. ia.). = I5xa;x2 = 30a; ft. -Ibs.^/s^. 15 and lh>.xy.\ = —-x ft. -lbs, second. 4 Equating for first case, 3 30x = 720 + —X, 1497a; = 36000, oO X = 24-05 ft. length of rod. Equating for second case, — a; = 720 + ^x, 738a: = 144000, 4 oO X = 195-12 ft. length of rod. For the first case length is 288-6 inches, and sectional area 2 square inches gives volume = 577*2 cub. inches. While for second case length is 2341*44 inches, and sectional area \ square inch, giving volume = 585-3 cub. inches, which is a little greater. See Note to Ex. 30. c2 ( 20 ) CHAPTER 11. INTERNAL STRESS AND STRAIN, SIMPLE AND COMPOUND. -<5i. In this Chapter, except where specially stated, we premise that — (a) All forces and stresses are parallel to one plane. (b) That plane is the plane of the paper in all diagrams. Hence planes subjected to the stresses we are considering are shown in diagrams by strong lines, their traces. (c) The diagrams represent slices of solid, of unit thickness normal to the paper ; hence, the lengths of the strong lines are the areas of the planes. (d) The stresses which are normal to the paper are supposed constant both in direction and intensity, or every point on a diagram is in the same circumstances with respect to stress normal to the paper. i>r,r,l.,r,^ "~vJr (e) The relative position of two planes is measured by the angle between their normals. (/) The obliquity of the stress to the plane upon which it acts is the angle its direction makes with the normal to the plane. Internal stress at a point in a solid zr^ in a simple state of strain. Let the axis OX (fig. 1) be drawn in the direction of the stress P. Let A A be any section normal to this axis. Since the stress is uniformly distributed over AA, the intensity of the stress at all points of the plane AA is the same ppp "A Fig. 1. Consider the point 0, the intensity of the stress at on the plane normal to OX is P = total stress area of plane P AA Through draw any oblique plane, BB, whose normal, ON, CHAP. II.] INTERNAL STRESS AND STRAIN. 21 makes the angle 9 with OX. The stress on this plane is in the direction OX, and the amount of stress upon it is P (for the equilibrium of the parts). But the inten- j sityof the stress on 5-6 is less than;;, since f -^ P is spread over a larger area than A A. Since AOB = e, OB.'''' and OA OB - cos AOB, or BB cos 6' Intensity of stress on total stress P area of plane BB AA < cos 6 BB = P AA cos 9 = p AA ^cos 9 cos 9. Hence the internal stress at all points within a solid, in a state of simple strain, is parallel ^ to the direction of that stress — is greatest in intensity on the plane normal to that direction — on any other plane inclined at an angle 9 to last, the intensity is one (cosine 9)th part of that intensity, and zero on any plane parallel to the direction of the stress. The stress ^; cos 6 on BB being oblique to BB, it is con- venient to resolve it into com- ponents normal and tangential to BB respectively. The arrow p cos (fig. 2) represents the stress at the point on the plane BB; from its extremity perpendiculars are dropped on ON and BB, which, by parallelogram of forces, give p„ and pt, the intensities of the stresses upon B B, normal and tangential respectively. 22 APPLIED MECHANICS. [chap. II. Now Vn 2) cos d cos clef. :Pn = p cos' e ; (fig. 3) also Pt p cos = sin 0. Pt=p sin 6/ cos 0. V From the superposition of forces these two sets of forces may be considered independently of each other. For some cases in designing it might only be necessary to consider one set, if it were manifest that in pro- viding for it there would be more than sufficient provision made for the other. It is apparent from symmetry that for the plane CC inclined at the angle on the other side of the axis the stress is the same in all particulars as that on £B. On a pair of planes whose obliquities are together equal to a right angle, the intensities of the tangential stresses are equal, and the sum of the intensities of the normal stresses equals the intensity of the initial stress. Let BB (fig. 4) be inclined at the angle 6, and DD at the angle ^, where e + <i> 2 On BB pn = p cos' 0, pt = p sin cos 0. On DJD p'n = p cos^ (j), p't = p sin ^ cos (j>. But sin = cos <p, and cos ^ = sin ; therefore pt = p't, or the tangential component stresses have the same intensity on both planes. Also Pn + p'n = P (cos- d + COS' <p) = p (cos- 6 + sin^ ^) = P'i or the sum of the intensities of the normal component stresses equals the intensity of the primary stress. CHAP. 11.] INTEUNAL STRESS AND STRAIN. 23 There are therefore at one point (fig. 5) four planes BB, DD, CC, and EE, two inclined on each side of OX, upon which the tangential stress ^ has the same intensity. ,---'■ " - -., Grouping together the ,^ .-^p'^'p'^^ r pair of planes BB and EE, / JS^.. \\'\ ^ ."> the one inclined at on the - /' one side of OX, and the^^ --, other at upon the oppo- ~~ -, site side, and therefore at ^ -I- 0, or 90° to each other, we find that at any point two planes being chosen at right angles to each other, the tangential or shearing stresses are of equal intensity, and the sum of the in- tensities of the normal stresses is equal to the intensity of the primary stress. For all planes such as BB, DD, &c., that which is inclined at 45° sustains the tangential stress of great- • est intensity, for TP jO/ = j? sin cos = ^ sin 20. Therefore pt is greatest when sin '2B is greatest, when sin 20 = 1, when 2B = 90°, or = 45*. The tangential stress on a plane such as BB is called a sJiearing stress. Many substances fracture under a shearing stress very readily. Notably cast iron under a strain of compression fractures by shear- ing along an oblique plane, the one portion sliding upon the other (fig. 6). The resistance then which cast iron ofiers to shearing is that which must be considered in designing short pillars to bear great loads. The planes upon which the intensity of the shearing stress is greatest, that is, planes inclined at 45° to the direct thrust, are those upon which it will shear. As the texture of the material is never homogeneous, it may shear along planes more or less w inclined than 45°, also the toughness of the skin will j cause great irregularity. Brick stalks give way by the mortar shearing, and the upper portion sliding down an oblique section like 24 applied mechanics. [chap. ii. Internal Stress at a Point in a Solid in a Compound State of Strain. A solid is in a compowiid state of strain when subjected to two or more simple stresses in different directions simultaneously. We proceed to consider a solid in such a state of strain without inquiring how it was brought into that state; all its parts being supposed to be at rest, and all the parts into which it may be divided in equilibrium under the stresses exerted among each other, due to their elasticity, and those exerted at the external surface : but at the outset we do not regard those external stresses. Upon any plane passing through a point within the solid, the stress at that point is definite in intensity and direction ; for if along that plane the solid were divided into two parts, the mutual pressures between the cut surfaces at that point (no matter how complicated) can be compounded into one force, definite in amount and direction. Along this plane the intensity and direction of the stress vary, and at the point will only be constant over a very small part of the surface round it. If the stress be stated in lbs. per square inch, the total stress on this small surface which we are considering would be a mere fraction of the intensity. It will be convenient to consider the intensities of these stresses to be expressed, say, in lbs. per millionth part of a square in., so that in the diagrams tivo or three arrows {each representing the intensity) may be drawn to represent the total amount of stress upon such small planes, without leading us to the supposition thnt they are of a few square inches in extent. And yet whatever results we arrive at are equally true for intensities expressed in the usual units, for the intensity at a point on a plane, upon which the intensity varies, can be expressed to any degree of accuracy in lbs. per square inch. Thus, at the point, the intensity of the stress in lbs. per square inch equals roughly, nearly, more nearly, &c. Amount of stress on the square inch surrounding point, roughly. 10 times amount of stress on the iVth of a square inch surrounding point. 100 times amount of stress on the yoT^^ ^^ ^ square inch surrounding point. 1,000,000 times amount of stress on the 10000 77^^ of a square inch surrounding point, &c., &c. CHAP. II.] INTERNAL STRESS AND STRAIN. 25 Let OAO'B (fig. 7) be a small rectangular parallelepiped at the point in a solid in a state of strain. Let q = intensity of X*^ , stress on the faces OA and CB at an obliquity a. Let p = intensity of stress on faces OB and O'A at an obliquity j3. f The normal components ^ are — •/ J)„ = p cos (3, q„ = q cos a. The two sets of forces })„ directly balance each other, and may be removed, and also the two sets q„, leaving the parallelepiped in equilibrium under the action of the tangential components. Pt = p sin j3, qt = q sin a. inUruvUej and. ohhquitLes iitO ff stresj&t upon OA ami OB ^r '•' I I '• pCosfi o' _ f^ 1 </;t * q=qCoo- qe. - — B ifn. NorrruxL componerus A PtOS qtOA 9tOA. The amount of tangen- tial stress on each of the faces OA and O'B is qt. OA. Also the amount on each of the faces OB and O'A is Pt. OB. The two forces qt . OA form a couple with a lever- age OB tending to turn the parallelepiped in the direc- tion in which the hands of a watch turn, while the two forces pt • OB form a couple with a leverage OA tending to turn it in the opposite direction. Since the parallelepiped is in equi- librium under these two actions alone, the moments of these two couples must be equal. o — Tfmr~B Arrwujvcs of tanaaiUaJL. streM A O' Pt Pt Pt Pt^B*^V^'''a /ruerhfU-ces of UmgcnuaL ccmnonenCa Fig. 7. Force. Leverage. Force. Leverage. qt • OA X OB = p, . OB ^ OA. 26 APPLIED MECHANICS. [chap II. Now the area OA multiplied by the length OB gives the volume of the parallelepiped, and the area OB multiplied by the length OA also gives the volume. it Ut=pu Hence, at a point within a solid in a state of strain, the tangential components of the stresses upon any two planes through it at right angles to each other are of an equal intensity. Cor. — If the stress upon any plane through a point be wholly normal, then will the stress be wholly normal upon another plane at right angles to that plane. Such a pair of stresses are called principal stresses, and the planes upon which they act are planes of principal stress. When these two stresses are given, the uniplanar stress at a point is completely given. There is also a third principal stress acting on the plane at right angles to the other two. DiKECT Pkoblem, the Pkikcipal Stresses given. Eqiud-like principal stresses (fig. 8). — If the pair of principal stresses at a point be like (both thrusts or both tensions), and be of equal intensity, the stress on any third plane through the points is of that same intensity, and is normal to the plane. Let AA' and BB' be the planes of principal stress at the point 0, and let the intensities of the principal stresses, p and q, be equal and alike (both thrusts). CC is any third plane through 0, inclined at B to AA\ GAB is a small triangular prism at 0, having its faces in those planes. This prism is in equilibrium under the three forces — the total thrusts upon OA, OB, and AB. The three total pressures F, Q, R on the faces, if represented by straight lines, intersect at the middle point of AB. The parallelogram of forces is, however, drawn from the point 0. Lay off OD = total stress parallel to OX = p . OA, and OH = total stress parallel to OY = q . OB. Complete the parallelogram. CHAP. II.] RANKINE S ELLIPSE OF STRESS. 27 Then RO represents the total stress on AB in direction and amount. „^^ OE q.OB OB tan ROD=-^= j-^ = ^ , since p = rj. .: BOB = OAB = 6; .: OR is upon ON. Hence BO is normal to AB. And BO' = OB- + OB" = f . OA' + q' . OB" = p' {OA' + OB'-) = p- . AB', as p = q. .: .RO = p.AB. amount of stress on AB RO Now area oi AB AB = p or q. '^^JH 6'or.— Every plane through is a plane of principal stress. Each point in a fluid is in this state of strain. Equal-unlike principal stresses (fig. 9). — If the pair of principal stresses at a point be unlike (one a thrust and the other a tension) and be of equal intensity, the stress on any thirdplane through the point is of .< ^ thatsameintensity,and "'-JT is inclined at an angle to the normal to the plane of principal stress, equal to that which the nor- mal to this third plane makes therewith, but upon the opposite side. Let -4-4' and .5^ be Fig. 9. the planes of principal stress at the point ; p and q the intensities of the principal stresses of equal value, p being a thrust and q a tension ; and CC any third plane through inclined at 6 to AA'. OAB, a small triangular prism at bounded by these three planes, is in equilibrium under the three forces, viz., the amount of stress on its faces OA, OB, and AB. Lay off OD = total stress parallel to OX = p . OA, and OF = total stress parallel to F in the direction of q -q.OB. 28 APPLIED MECHANICS. [chap* II. Complete the parallelogram ODRE. Then RO represents the total stress on AB in direction and amount. tan ROD = OE OB OA OB = Z—n-A =02^ ^^^^ ^^^' ^ 9_ OD p ROD = OAB = e. That is, BO is inclined at the same angle to the axis OX as OiV is, but on the opposite side. Hence the inclination of RO to the normal ON is 20. Again RO' = OD^ + OE' =^ f . OA" + q^ . OB' =p'(OA' + OB')^f.AK-; amount of stress on AB RO BO = p . AB, area of AB AB = p or q. PTPV Y Consider the triangle of forces OER, we have OE drawn from in the direction of q, then ER drawn from E in the direction of /; ; hence RO, taken in the same order, is the direction of r. If 6 be greater than 45°, r is like q. If 6 equal 45°, r is entirely tangential to AB. If B be less than 45°, r is like p. Hence, if the principal stresses at a point be equal and unlike, the stress on a third plane is of that same in- tensity, is like the stress on the plane it is least inclined to, and its direction is inclined to the axis at the same angle as the normal is, but upon the opposite side. If the new plane be inclined at 45°, the stress is entirely tangential. Unequal principal stresses (fig. 10). — The principal stresses at a point within a solid in a state of strain being given, to find the intensity and obliquity of the stress at that point on a third plane through it. A A' and BB' are the planes of principal stress at ; p and q are the principal stresses. Let p) be the greater, and let them be both positive, say both tensions. It is pppp Fig. 10. required to find r, the intensity of the stress upon CO', CHAP. II.] H^V:s'KINES ELLIPSE OF STRESS. 29 and y, the angle it makes with 01^, the normal to C'C\ B is the inclination of C'C" to AA', the plane of greatest principal stress. Of two unequal quantities the greater is equal to the sum of their half mm and their half difference, while the lesser equals their dift'erence. Therefore p + q p - q .. ... p = -^ + -^ - , an identity, and ? = p+q p-q , an identity. We may look upon the plane A A' (fig. 11) as bearing two p + q 1 T ~ 9. ■ separate tensions of intensities — x- and — ~ in lieu of the tension of intensity p ; and on the plane BB' as bearing a a. *- ea4i ^ inch Pt.Q Z '■A Fis. 11. <uuch. \l C$\ 'It*; ^1^ rum '"lyterLfum. tension of intensity -^-^ and a thrust of intensity ^ — ^ in heu of the tension of intensity q. We may now group these together in pairs, thus : the tension on AA' of intensity 'P + Q ~Y- alo°R with the tension on BB of intensity ^^-^, and the tension on AA' of intensity -^-^ along with the thrust 30 APPLIED MECHANICS. [chap. II. on BB' of intensity jp-q Then find separately for each pair the stress upon CC\ and finally compound these two stresses on CC by means of the triangle of forces. The first pair is a pair of equal-like principal stresses ( both tensions of intensity — So the consequent stress on CC will be a tension of intensity ^^, and normal to CC (fig. 12). The second pair is a pair of equal- unlike principal stresses of intensity ^ (a tension on AA' and a thrust on BB')^ so the consequent stress on CC will be of intensity A 2)-q Fig. 12. and inclined at an angle vipon the side of OX opposite from that upon which ON lies. The diagram shows these partial resultant stresses on AB, a very small part of CC , at 0. To find the total resultant stress upon CC\ it remains to compound these by the triangle of forces. From (fig. 13) lay off OM = ^^-^ = the intensity of X ^-^ ^^ Fig. 13. the first partial stress and in the direction thereof, i.e., along OA^. From M draw MB = p-q = the intensity of the second partial stress and in the direction thereof, i.e., parallel to OS, which CHAP. II.] KANKINES ELLIPSE OF STRESS. 31 direction is most conveuieutly found by describing from M as centre with radius MO a semicircle QOP, and joining QMP. Then will OR, the third side of the triangle OMR, taken in the opposite order (see arrows) be the direction and intensity of the resultant stress r on CC. The preceding construction, as shown on last figure, is geometrically all that is required, p and q being given to find r\ the text and figures given before being the development and proof. From the construction note that MP = MQ= OmJ^; also and QR.MQ.MR = ^^^^^ PR = MP - MR = p+q p-q = ?; 2 2 RMN =26; R03f =y, the obliquity of r. Normal and tangential components of r, the stress on the third plane CC. Drop RT perpendicular to ON. The tangential component of r is rt = TR = MR sin RMT p-q sin 2B = {p - q) sin 6 cos 6. since sin 20 = 2 sin d cos 0. Fig. 14. Cor. — If DJ)' be the plane at right angles to CC, its inclination to the axis OX being 0' = (0 + 90°), the sine of which equals cos 0, and the cosine of which equals - sin ; the value of Vt for DI/ will be the same as above, that is, the tangential components of the stresses on any pair of rectangular planes is the same. 32 APPLIED MECHANICS. [CHAP. II. The normal component of r is r„= OT=OM-MT = OM - MR cos EMT= 0M+ MB cos EMN. (These two angles have the same cosine, but of opposite sign.) = OM ^ ME cos 26 = ^4^ (cos2 e + sin^ 6) + ^-^ (cos^- Q - sin^ B) = cos^ ( — ^ + ^—^ ) + sm^ ^(^} ~ 9"^ ) = p . COS" d + q . sin^ 6. Cor. — If 5„ be the normal component of stress on DI/, the plane at right angles to CC\ whose inclination to OJC is 0' = (0 + 90°), then Sn = p cos- 6' + q sin* 6'. But cos 0' = - sin ©, and sin 0' = cos 6* ; .•. Sn = p sin- + g' cos'' d. Now, r„ = JO cos^ + q sin^ 0, and adding, we get Sn + I'n = p (sin- B + cos- B) + q (sin- + cos^ B) = /j + ry. That is, the sum of the normal components of the stresses on any pair of rectangular planes is equal to the sum of the principal stresses. As CC moves through all positions, M moves in a circle round 0, and B moves in a circle round M, OM and MB keeping equally inclined to the vertical on opposite sides of it. The diagram shows their positions for eight positions of CHAP. II.] KANKINES ELLIPSE OF STKES8. 33 the plane CC. The locus of B is an ellipse, the major semi- axis being ORi = OM, + MiR, P + 9 . P-fL 0, 9. p; and the minor semi- ,' axis is ) OR^ = OM, - M,R, ^p+q P-1^ 2 2^' The moving model, fig. 31 following, shows these positions nicely. This is called the ellipse of stress for the point within a solid in a state of strain. Its principal axes are the normals to the planes of principal stress, the principal semi-axes be- ing equal to the inten- sities of the principal stresses. The radius- vectors OR., OR,, &c., are the stresses in direc- tion and intensity upon the planes at to which OM:^, OM3, &c., are re- spectively the normals. The ordinary tram- mel (fig. 16) for con- structing ellipses con- sists of a piece like FRQ, whose extremities F and Q slide in two grooves, JCOJC' and YOY', at right angles to each other, while the point R traces an ellipse whose semi-axes are PR and QR. 34 APPLIED MECHANICS [CHAP. II. When Q arrives at 0, J? is at ^ and OA = QR = p ; when P arrives at 0, ^ is at P and OB = PR = q. Taking as origin, the coordinates of R are A. = Ovi ; y -= On; :. x = nR = QR .cos d = p. cos 0, X V and y = mR = PR sinB = q sin 0; .'. - = cos 0, and - = sin 6/ ; p q .-. — + - = cos^fl + sin'0 = 1, P ? the ordinary equation to an ellipse in terms of the semi-axes p and q. If p and q are both thrusts, it is convenient to consider a thrust positive, and the proof is exactly the same, all the sides of OMR representing the opposite kind of stress to what they did in the last case. When p and q are unlike, the kind of stress of which the greater p consists is to be considered positive. Thus, it p > q, and p a tension while $ is a thrust, the preceding proof will hold if q be considered to include its negative sign ; but in this case if (- q) be substituted for q, we have arithmetically OM = P^. and Mll=l^ SO that now OM < MR (see fig. 18). It is important to notice that although OM is always positive, that is like p, the greater principal stress, yet MR is now positive or negative according as d is less or greater than 45°, and r = 031 is positive or negative according as b is of a lesser or greater value than that shown on figure 18. Hence, the proposition is proved generally. An advantage of this geometrical method, the ellipse of stress, is that we are now in a position to examine the value and sign of r, the stress upon a third plane CC, and of its normal and tangential components for special positions of that plane. OM is always normal to CC, while MR generally is resolvable into two components, one tangential to CC and the other normal, which last has to be either added to, or subtracted from, OM to give the total normal component according as OMR is an obtuse or an acute angle. CHAP. II.] RANKINE S ELLIPSE OF STRESS. 35 (a) Portions of GC for which r, the stress upon it, will have the greatest or least value. Since OM and MB are constant, OR increases as the angle 0MB increases, is greatest when 0MB = 180°, and OM and MB are in one straight line, and a continuation one of the other, when OB=OM+ MB; ,. P + 1 ^P -Q P -^=4.5° and OB is least when z 0MB, is zero, and OM and MB are again in one straight line, but MB lapping back on OM, when OB=OM-MB; r = ^ili - ^ = j. Hence the planes of principal stress are themselves the planes of greatest and least stress. (b) Position of CC for which the intensity of the shearing stress has the greatest value. As OM is always normal to CC, it does not give any tangential component, whereas MB assumes all positions as CC changes, and will give a component tangential to CC, which will be the greatest possible when MB is altogether tangential to CC. Hence the position of CC, which makes MR parallel to CC, is that for which the shearing stress has the greatest possible in- tensity (fig. 17). Fig. 17. Hence intensity of shearing stress = MB = - — - • And since MB is parallel to CC and OM normal to it, .-. 0MB = 90°, and the triangle MOP being isosceles, we have B = inclination of CC = MOP = 45°. D 2 36 APPLIED MECHANICS. [chap. II. And we know that the tangential stress is the same on the section perpendicular to CC ; that is, the planes of greatest tangential stress are the two planes inclined at 45° to the axes. (c) Position of CC for which the total stress r wpon it will he entirely tangential. When ci is like 'p, it is impossible for the stress to be entirely tangential to CC', because OM > MB, and, how^ever acute 03IE may be, the normal component of MH, which has to be subtracted from OM to give the total normal stress. tenswn, ■posttbre 'drmst w Fis. 18. cannot be greater than MB itself, and consequently is always less than OM, and so there will always be a remainder ; that is, for all positions of CC there is a normal component stress and the total stress can never be entirely tangential. But when ([ is unlike p, then OM < MB, and for the par- ticular position of CC (fig. 18), when the angle OMB is of such an acuteness that the normal component of MB, which has to be subtracted from OM to give the total normal stress, is exactly of the same length as OM ; then the total normal stress will be zero, and the total stress r entirely tangential. This occurs when B is in one straight line with CC. BOM is then a right angle, making MO, the normal com- ponent of MB, equal and opposite to OM, which it destroys, CIIAl'. II.] RANKINES ELLIPSE OF STRESS. 37 leaving the total stress OE tangential to CO' ; its magnitude is found thus : OE- = 3IE' - OM' or r* = fp_+g \ 2 2 = p -q. i.e., the stress on CC is the geometrical mean of the principal stresses. Also 26 = EMN; .'. cos 26 = cos E3m, MO cos 2e cos EMO = - ;;r = _ {_ ME p + g. which determines 6, the position of CC for which the total stress is tangential. Here we must guard against supposing that the above is the position of CC for which the tangential stress has the greatest intensity ; for case (b) holds for all conditions of p and q ; that is, the tangential stress on CC when inclined at 45°, although only a component of the total stress, will be of greater intensity than the total tangential stress in case (c). (d) Position of CC for vjMch y, the oUiquity of the stress tJiereon, is the greatest possible. When q is unlike f), case (c) is the solution, for in it 7 = EON = 90°, the greatest possible. When q is like jj, OM > ME ; and the obliquity of OE, the stress on f^C, is greatest when y = EOM is the greatest possible of all triangles constructed with OM and ME for two of their sides. This occurs when OEM is a right angle. For suppose the tri- angle OME constructed with OEM not a right angle ; then drop ME' at right angles to OE. It is evident that MR is less than ME. Now sin EOM = ^iv^ is greatest when ME' is greatest ; that 38 APPLIED MECHANICS. [chap. II. is, when MB- = MB ; that is, when 0E3£ is a right angle, and BOM is greatest when its sine is greatest. In this case the intensity of the stress is OB' = OM' - MB' ; fP + 2 p-_q 2 = PI, and \ ^ a geometrical mean between the principal stresses. Also 20 = BMN; cos 20 = cos BMN cos 20 = - cos BMO = -~=- ^-^, OM p + q' which determines 0, the position of CC for which the stress has the greatest obliquity possible. Note that these values of r and cos 20 are the same as those of (c), and that whether jj» and q are like or unlike. But this is not the case with y, the obliquity, which is 90° when p and q are unlike, and has alike. p + q for its sine when p and q are Examples. 1. At a point within a solid in a state of strain the principal 'stresses are tensions of 255 lbs. and 171 lbs. per square inch. Find the stress on a plane inclined at 27° to the plane of greatest principal stress (fig. 20). Data. p = 255, hence p + q g = n\, and fl = 27° ; = 213, and = 42. Construction. — OX and OY, the axes of principal stresses ; draw ON the normal to CC, making XON = 9 = 27°. Lay off along it OM = ?-^ = 213. From M as centre with radius MO, de- scribe semicircle FOQ and join FMQ\ f> -q 2 = 42. This construction makes MR to be inclined to OX at an angle 9 = 27°, l)ut upon the opposite side of it from that of OM. lay off from M towards P, MR = Fig. 20. Looking upon the principal stresses as a pair of like principal stresfes, tensions CHAI'. II.] RANKINES ELLIPSE OF STRESS. .39 of intensities 213, together with a imir of unlike principal stresses, a tension and a thrust of intensities 42. Then OM represents a tension 213 upon plane CC due to first group, and Mli the tension 42 upon CC due to second group ; hence OJt, the third side of the triangle, taken in the opposite direction, represents the total stress upon CC in direction and intensity. OH- = OM- + MR- - 20M . MB cos OMR, cos OMR = - cos RMN = - cos 2fl : .-. 6»i?2 = OM- + MR- + lOM . MR cos 20 ; r- = 45369 + 1764 + 17892 cos 54° = 57649 ; r = 240 lbs. sin 7 sin ROM MR sin~2e but Also sill OMR 42 240 X sin 54°= -14158; and figure shows that »• is upon the same side of the normal as OX. tension, since OR is like OM. 2. In Ex. 1 find thf intensity of the tan- gential stress on that plane thi-oiigh the point upon which the tangential stress is of greatest intensity (fig. 21). The plane is that which is inclined at 45° to the axes of principal stress. Since OMR is 90°, MR is the tangential component of OR, Also r is = MR = 42 lbs. per square inch. 3. In Ex. 1 find the obliquity to the plane of greatest principal stress of that plane, through the point, upon which the stress is more oblique than upon any other ; also find the stress (fig. 22). OJ/and MR being constant, the angle MOR has its greatest value when MRO is a right angle. Construction. — Upon OM describe a semi- circle ; from M as centre, with radius MR, describe an arc cutting the semicircle in li ; join OR. cos 20 = cos RMN = - cos OMR MR __P - q _ 42 ~0M:~~ p + q ~~ 213 = - -19718; .-. 26 = 101" 22' obtuse, cosine being nega- tive ; .-. e = 50° 41', obliquity of CC . r2 = OR""- = OM- - MR- Fig. 21. -H^y-i'^) p • q; = V p . q = v 43605 = 2088 lbs. per square inch of tension like OM. 40 APPLIED MECHANICS. [chap. II. MR And sin y = sm RON = — - = -19718 ; MO 7 = 11° 22', obliquity of r. 4 . The principal stresses at a point being a tension of 300 lbs. and a thrust of 160 lbs. per square inch. Find (a) the intensity, obliquity, and kind of stress on a plane through the point, inclined at 30° to the plane of greatest principal pp pp posituv stress ; (*) find the intensity of tangential stress on the plane upon which that stress is greatest ; and (c) find the "^ inclination to the plane of greatest principal stress of that plane upon which the stress is entirely tangential and the intensity thereof. Data. j? = 300; q = - 160, considering a ten- sion positive ; .'. -7j— = 70 tension like p ; p-q and — = 230 tension like p. (a) Construction. — Draw Fig. 23. ON at 30° to OX (fig. 23). Lay off OM = 70. From ^as centre, with radius MO, describe semicircle TOQ. Lay off MPR = 230. Then OR, the third side of the triangle OMR, taken in the opposite order, is the stress on CO in direction and intensity. OR^ = Oif2 + MR-^ -lOM.MR cos OMR = OM'' + MR"- + 20M.MR cos 29, r2 = 4900 + 52900 + 16100 = 73900, r = 272 lbs. per square inch ; sin 7 sin ROM MR and sin 2fl sin RMO OR 2.30 sin 7 = ;;;;7r sin 60° = -7323. 272 7 = 47° .5', being acute, OR is like OM, a tension. {b) Take 9 = 45° ; vt = MR 230 lbs. (c) On MR (fig. 24) describe a semicircle, and from J/, with raiiin.s MO, describe arc cutting it at 0. CHAP. II.] HANKINES ELLIPSE OF STRESS. 41 IiMX= 26, cos 20 = cos RMN ■■ cos RMO = -^=-~ = - -3044 ; MR 230 28 = 107° 44' ; = 53° 52', obliquity of plane upon which the stress is entirely tangential. r2 = OR- = MR- - OM- = 52900 - 4900 = 48000, r = 219, or r = ^ p . q = v (300 x 160) = 219 lbs. per square inch. Note that, though r is entirely tangential, it is less than r, was in (b). 5. At sam'fe point as in Ex. 4, find intensity, kind, and obliquity of a stress on a plane inclined at 85° to the plane of greatest principal stress (fig. 25). Since > 53° 52', > the obliquity of plane upon which the stress was wholly tangential, OR will make with ON an angle greater than 90°, and OR will be unlike OM, and therefore a thrust. Arts, r = 161*5 lbs. per sq. in. ; y = 165° 41'. 6. The principal stresses on AA' and BB' are thrusts of 60 lbs. per square inc^h. Find direction and intensity of the stress on a third plane CC inclined at 65° to A A'. Ans. A thrust of 60 lbs. per square inch normal to CC. 7. The principal stresses on A A' and BB' are of the equal intensity of 34 lbs. per square inch, being a thrust on AA' and a tension on BB'. Find the direction and intensity of the stress on a third plane CC inclined at 65° to AA'. Ans. A tension of 34 lbs. per square inch, its direction being inclined at 65° upon the other side of OX from that to which ON is inclined. 8. The principal stresses on AA' and BB' at a point are a thrust of 94 lbs. and a thrust of 26 lbs. Find kind, intensity, and obliquity of a stress on a third plane CC inclined at 65° to AA'. Using results of Exs. 6 and 7, r = 46*2 lbs. per square inch thrust, y = 34° 19'. 9. Two unlike principal stresses are: on A A' a thrust of 146 and on BB' a tension of 96 lbs. per square inch. Find the stress on CC, a third plane inclined to ^^'at 50°. p = 146, and g = - 96. Half sum is a thrust like p, 2 ^ p — g half diflf. — r — is a tension like g, since > 45 . Atis. r = 119'22 lbs. per square inch thrust, y = 88° 5'. 42 APPLIED MECHANICS. [chap. II. Inverse Pkoblem, to find the Peincipal Stresses. Given the intensities, obliquities, and kinds of the stresses upon any two planes at a point within a solid, find the principal stresses and their planes. In the general problem we know of the triangle OMR (fig. 13), the parts OR and 7 for two separate positions of the plane CC, and we also know that OM and MR are the same for both. If the two given stresses be alike and unequal (fig. 26), let r and r' be their inten- sities, and 7 and 7' their obliquities upon their re- spective planes CC and BB' . Let r be greater than >■'. Note that it is not necessary to have given the inclina- tion to each other of CC and BB'. Choose any line ON and draw OR = r, and making the angle NOR = 7, also draw OR! = ?■', and making the angle NORil = 7'. Join RR', and from S, the middle point of RR', draw, at right angles to it, SM meeting ON at 31. Then will JIR = NR'. Thus we have found OM comparing we have Fig. 26. and MR to suit both data, and the construction of the direct problem (figs. 3, 4) and therefore 0M.'^\-^, and MB=P-J, p = OM+ MR; q = OM- MR. Consider the triangle OMR' alone, and consider ON' the normal to BB' : then R'JI'N = 29' ; hence OJC, drawn parallel to M'T (the bisector of R'M'N), is the axis of greatest principal CHAP. II.] kankine's ellipse of sthess. 43 stress. Thus we have found the principal stresses p and 7, and the position of their axes OX and OFrelative to DD\ one of the given planes. Since li'JIR = JifMN - JULY = 'IB' - 26 ; .-. liMS = 6' - B, the inclination to each other of CO' and DU' ; hence if tlie other triangle OMU be moved round through this angle, it and consequently CC, to which ON is the normal, will also be in their proper positions with respect to the axes l^X and OY. This triangle might be further turned round till ON is inclined at an angle XON = B on the other side of OX, when CO' would again be in a position for which the stress would be the same as given. This would increase the relative inclination of BD' and CO' by twice JCON or by 2B. Adding this to B' - B gives 6' + B. That is, the inclination of CO' and DD' to each other is (B' - B) = BMS on diagram, or (B' + B) = NlMS on diagram, according as they lie on the same or on opposite sides of OJC, the axis of principal stress. If the two given stresses ^' be unlike and unequal (fig. 27), °' considering r the greater as positive, /■' will be negative. Follow the same construction, only OM' = r' must be laid off from in the opposite direction. Complete the figure as before, and we have from either figure — TrigonometHcally MR- = OM' + OE" - 20M. OR cos MOB = OM^ + r^ -20M . r cos 7. Similarly, MR'^ = 03P + r" + 20M . r cos 7' from figs. 26 and 27 respectively. Subtracting, = r' - r"^ - 2031 (r cos 7 + / cos 7'), and therefore ^^ = OM = — "-^^ ;-, (A) 2 2 (r cos 7 - r cos 7 ) / to include its sign ; 44 APPLIED MECHANICS. [chap. II. also or, P-1 = MR = ^ (03/2 + 7-2 - 20M. r cos 7) = ^(071/2 + /2 _ 2OM. r cos y) (B) a known quantity when the value of OM is substituted from equation (Aj. ;p and q are now obtained by adding and subtracting equations (A) and (B). From R drop RL perpendicular to ON, then ML = OL- OM; MR . cos NMR = OR . cos ROM - OM, ^^cos20 = rcosy-^:-l; cos 26/ = 1r cos y - p - q (C) This gives twice the obliquity of the axis of greatest principal stress to the given plane CC\ and similarly for DD' cos 2H' = 2r' cos y' - p - q p -q These three equations (A), (B), and (C) are the general solution of the inverse problem of the ellipse of stress. fA) and (B) give the intensities of the principal stresses, which will come out with signs showing whether they are like or unlike r, the greater of the given stresses. In some particular cases the construction gives a much simpler figure from which the equations (A), (B), and (C) in their modified form are readily calculated. Particular case [a) (figure 28;. Given the intensities and common obliquity of a pair of conjugate stresses at a point ; find the principal stresses and posi- tion of the axes of principal stress. (Note — There are more than sufficient data.) In this case 7 = 7', and R, S, and R' are in one straight line with 0. Draw any line ON; draw OR, making NOR = 7 = 7'; and Fig. 28. CHAP. II.] It^VNKINE'S ELLIPSE OF STRESS. 45 lay off OR = r and OE' = r in the same or opposite directions according as it is like or unlike r ; and from S, the middle point of BR, draw SM at right angles to it, meeting OiVat M. Join M to R and K. Then -— = cos MOS ; ai/ = ,,7^ = rpTH^ , X)M COSMOS COSMOS or i-^ = (A) 2 2 cos 7 Again MB'- = MS' + BS" = {OM'^ - OS') + BS" = OM^ - {OS' - BS') = OM^ - {|!l±l'J-(^!:^'Jj + /V- (B) = OM' - rr = I 1 \ 2 cos 77 9>. 1 o^ Ir COS J -jp-q and cos 20 = — ~ (C) p-q as in general case, ^OjV' =0' +9 = mis = MSO + MOS = ^ + 7. Hence, the angle between the two normals to the sections CC" and DD' (or the ohtuse angle between CC and 1)1/) exceeds the obliquity by a right angle. This we know ought to be the case from the definition of conjugate stresses. It may be easier to calculate cos 26 = - cos B3I0 in terms of the sides of the triangle 03IB when these have been already found. From M as centre, with radius MB, describe the semicircle HBfBN; then (fig. 28) OH = OM - MB =q; ON ^ OM + MB = p. But ON X OH = OB X OR (Euc. iii. 36), and pq = rr' (B,) may be used instead of (B). 46 APPLIED MECHANICS. [chap. II. Particular case (b) (fig. 29). Given the intensities and obliquities of the stresses on a pair of rectangular planes, find the principal stresses and the position of the axes of principal stress. (Note — There are more than suffi- cient data.) If r and r be like stresses, draw any line ON'. Draw OR = r, making NOR = y, also OR' = r making NOR' = y. Complete the figure as before. The given planes being at right angles are necessarily inclined upon opposite sides of the axis of principal stress; hence NMS = inclination of given planes = 90°, and RSR' is parallel to ON. Therefore MS = RL = r sin y = R'K = r' sin y, or r sin y = r' sin y'. That, is the tangential components of r and r' are equal, p + q OM =^{OL + OK); 2 = ^{r cos 7 + / cos 7'). (A) That is, the sum of the principal stresses is equal to the sum of the normal components of r and r'. (Compare page 32.) MR' = RS' + MS' = (^^Ll^Jy + MS^ (r cos y - r' cos 7')^^ + r' sin'' 7 ; p-q tan 20 (r cos y - r cos y) 4 RL + r- sm-7 I • r sin 7 (B) RL _ ML ~ ^{OL - OK) \ (r cos 7 - r' cos 7') 2r sin 7 r cos 7 - ?•' cos 7' (C) CHA.P. II.] RANKINE'S ellipse OF STRESS. 47 Putting Tt = MS = r sin y = r sin 7' = the common value of the tangential components of r and r ; also r„ = OL = r cos y = norm. comp. of r, Tn = 0K=^ r cos 7' = norm. comp. of r , the equations become and 2 P-1 2 tan 20 = >/ 2 ' + n^ 2r, ?•„ - ? When T and ?'' are unlike stresses, con- sider r, the greater, as positive ; then must OR be laid oft' in the opposite direction from (fig. 30). ^ovfR'K=RL = rt, 4,,^ the common tangen- '. tial component of r and r ; hence LiR' and KL bisect each -fi other at *S' or M, which coincide. Fig. 30. 0M=^ {OL - OK) MR p + q r cos 7 - r cos 7 J/Z^ + RL- = ( ^^ + RL' or ^-2^=J{ (r cos 7 + r' cos 7')- + r' sin ^^ i^Z RL tan 20 = --- = i^Z J/Z UK liOL+OK) 2?' sin 7 r cos 7 + r' cos 7' (Ax) (Bx) (Cx) (A.) (B,) (C,) These three equations (A2), (B2), and (Co) are identical with (A), (B), and (C) with (- /) substituted for /. 48 APPLIED MECHAl^ICS. [CHAP. II. The diagram (fig. 31) represents the kinematical model in the Engineering Laboratory of Trinity College, Dublin, devised by Professors Alexander and Thomson for illustrating Kankine's Method of the Ellipse of Stress for Uniplanar Stress. On the left hand of the diagram, ON is a T square, pivoted to the blackboard at 0, and MR is a pointer pivoted to the blade of the T square at M. On the pivot M a wheel is fixed at the back of the blade, and round the wheel an endless chain is wrapped, which also wraps round a wheel fixed to the board at 0. The wheel at 6* is of a diameter double that of the wheel at M. Hence, when the blade of the J square is turned to the right, the pointer MR automatically turns to the left, so that the angle RMN is always equal to 20 when PON equals Q ; or, in other words, the bisector of RMN is always parallel to OP. A cord is fastened to the pointer at ^ ; it passes through a swivel ring at 0, and is kept tight by a plummet. It is easily seen that the point R describes an ellipse. CC, the head-stock of the T square, represents the trace on the blackboard of any plane through normal to the board, and the vector of the ellipse, consisting of the segment RO of the cord, represents the stress on the plane CC in intensity and direction. The angle PON = 6 is the position of the plane CC relative to the plane of greater principal stress AA, while RON = 7 is the obliquity of the stress r upon CC. The model* shows clearly the interesting positions of the plane CC; thus CC may be turned to coincide with A A, when the cord RO will be found to be normal to CC and to be of a maximum length. On the other hand, if CC be turned to coincide with BB, the cord RO is again normal to CC, but of a minimum length. Again, when CC is so placed that the angle RMO is a right angle, the component of RO parallel to CC is a maximum ; and lastly, if it be turned till MRO is a right angle, then ROM, the obliquity of the cord, is a maximum. The auxiliary figure on the right of the diagram is for solving the general problem of uniplanar stress at a point, viz. given the stress in intensity and obliquity for two positions of CC, to * Tlie model was exhibited to the Royal Irish Academy early in 1891, and described for the first time in the Academy's Tratisactxons. Reference may be made to Rankine's Civil Engineeriixj, or Applied Mechanics, and to Williamson's Treatise on Stress. Numerical examples are liere worked by this method, and in Howe's Retaining Walls. The model is made by Messrs Dixon and Hempenstal, SuEFolk Street, Dublin. In Williamson's Treatise on Elasticity, this model is shown in its proper place relative to the complete systematic treatment of elasticity. CHAP. II.] RANKINES ELLIPSE OF STRESS. 49 find the stress for any third required position of CC. On the auxiliary figure, the T squares for all positions of CC are super- imposed upon each other and are represented by one T square fixed to the board, and only the pointer MR turns. In solving questions on the stability of earthworks, some linear dimension upon the auxiliary figure represents the known weight of a column of earth, while another dimension is the required stress on a retaining wall or on a foundation, &c. Generally two angular quantities also are known, such as KON = (p, the maximum value of 7, this being the steepest possible slope of the loose earth, while HON = 7 may be the known slope of surface of earth. On the other side of the board, another T square is similarly pivoted to the same pin 0, the only difference being that OM is shorter than the pointer MR. It represents the case of Unlike Principal Stresses, and is useful in demonstrating the composition 50 APPLIED MECHANICS. [chap. II. of stresses such as that of thrust and bending on a pillar, or of bending and twisting on a crank-pin. By means of the slots on the easel the board can be placed so that any desired vector of the ellipse may be vertical. Examples. 10. If from external conditions it be known that the stresses on two planes at a point in a solid are thrusts of 54 and 30 lbs. per square inch, and inclined at 10° and 26° respectively to the normals to these planes — find the piincipal stresses at that point ; the position of the axis of greater principal stress relative to the first plane ; and the inclination of the two planes to eacli other. Make and Lay off and NOR =7 = 10°, NOR = y = 26°. OR = r = 54, OS = r = 30. Join RR\ bisect it in S, draw SM at right angles to RR' , meeting ON at M: complete figure. Fig. 32. Then p + q = OM, and p-q = MR = MK, also 2d = NUR. 2 -' 2 or p = (OM->r MR), and q = {OM - MR), Trigonometricatly MR' = OM- + Oi?2 -20M. OR cos MOR = OM^ + r^ - 20M . r cos 7. SimHarly MR- = OJP + r'^ - -lOM. r cos 7'. Therefore subtracting = r^ - j-'2 _ 20M{r cos 7 - r' cos 7') ; r^ - r"^ p+q 2016 OM = - , „. 2 (»• cos 7 - r' cos 7') 2 52-43 MR^ = (38-45)2 + (54)2 _ 2 x 38-45 x 54 cos 10° = 1478-4 + 291G - 4088-8 = 305-6 ; ^-^ = \/305^ = 17-48. = 38-45. The principal stresses are P p + q p — q + -;r— = 55"93 lbs. per square inch thrust, like r, 2 2 p-q 9 = —2~ ~ ~~^ - '-0'97 lbs. per square inch, thrust being -|-. (A) (B) CHAP. II. j RANKINKS ELUPSE OF STRESS. 51 Drop RL perpendicular to ON, ML = OL - 0.\f, or MR cos LMR = OR cos LOR - DM; p + q p-q cos '10 = r cos 7 - (C) e = 16° 17^'= XON, ,03 ,e = °^-^^^f'^^ = -8426 ; .-. 2. = 320 30'; 1 1 '48 the inclinakion of OX, the axis of greatest principal stress, to ON, the normal to the plane for which r was given. Similarly, cos 26' = — "6573 ; .-. 26' = 131° 6' (obtuse for - sign) ; d' = 65° 33', inclination XON'. And inclination of the two planes to each other NON = RMS = {ff -e) = 49° 15f , or = NMS = (6' + e) = 81°50r, according as they are on the same or opposite sides of OX. 11. At a point within a solid a pair of conjugate stresses are thrusts of 40 and 30 lbs. per square inch, and their common obliquity is 10°. Find the principal stresses and tiie angle which normal to plane of greater conjusiate stress makes with the axis of greatest principal stress. Draw OR, making NOR = 7 = y = 10° ; lay off OR = >■ = 40, and OR' = r' = 30. Bisect i?A' in S, draw SM perpendicular to £R'; complete figure. Then ^-i-^ = OM, and ^^ = J/J?, and 20^RMN. OS OM OM. cos 7 !(>• + /) ^ p + q cos 7 ' 2 -9848 MH- = MS- + JiS- = OM^ - [OS^ - RS^) — -[(^r-(4^)i='- .-. ^^ = -/(1263-8 - 1200) = 8. Adding and subtracting (A) and (B), I? = 43*5 a thrust, and j = 27 5 a thrust, 78-8 - 71 cos 20 = ~ = -49. 16 (B) (C) E 2 52 APPUED MECHANICS. [chap. II. Therefore 28 = 60° 40', and = 30° 20'. Or geometrically describe semicircle HL'RN (fig. 33), ON = O.U + MR = p, and OH = OM - MR = 9. ON .OK = OR . OR' (Eiic. iii. 36), or p . fj = rr' = 1200. (B') Now p + ? = 711; (A) .-. p'^Jr2pq+ q^ = b0bb-2\ bnt(B), 4j09 = 4800 ; .-.jt?'- 2;?^ + j^ = 255-2; .-. j»-j=16: adding to and subtracting from (A), .-. 2;? = 71-1 + 16, and 2^ = 71-1-16: .-. jo = 43-55, and ? = 27-55. 12. At a point witliin a solid, a pair of conjugate stresses are 182 (tension) and 116 (ihrust). common obliquity 30°. Find the principal stiesses and the pnsilion of axes (fig. 34). r' is negative p + q 2 p - q = C>ilf= 38-14, = MR = 150-3 ; and p= 188-4 (thrust), .-. q = - 112-2 (tension), cos 20 = -7947 ; .-. e=lS°41'. Fig. 34. Fig. 35. 13. The stresses on two planes at right angles to eacli other being thrusts of 240 aTid 193 lbs. per squaie inch and at obliquities, respectively, 8° and 10°, find tlie principal stresses and tlieir axes (fig. 35). r,, = r cos 7, and »•'„ = r' cos 7' ; also r< = r sin 7 = r' sin 7' ; = 237-6 =190 =33-4 iLJ-1 = OM = 213-8, P - <1 2" = MR = l\ ( ''" ^''" \ +r,'j = v/(566+ 1115) = 41 j9 = 254-8, and 9=172-8; tan 26 26 = 54° 32'; .-. d = 27° 16'. -l'V = 1-4034; In — r„ CHAP. III.] STABILITY OF EARTHWORK. 53 CHAPTER III. APPLICATION OF THE ELLIPSE OF STRESS TO THE STABILITY OF EARTHWORK. Loose earth, built up into a mass on a horizontal plane, will only remain in equilibrium with its faces at slopes whose inclinations to the horizontal plane are less than an angle 0. If the earth be heaped up till the slope is greater, it will inin till the slope is at greatest ^. Moist and compressed masses of earth can be massed up into a heap with slopes greater than ip. and will remain in equilibrium for some time, but will ultimately crumble down till the slopes do not exceed <\,. The surface-soil, which is in a compressed state, may be cut away, leaving banks with slopes much greater than ^. These banks will only remain in equilibrium for a time. Slips will occur till ultimately the slopes are not greater than <p. This angle (p, which is the greatest inclination (of the slopes to the horizontal plane) at which a mass of earth will remain in equilibrium, is called the angle of repose. It has different values for different kinds of earth, and also different values for the same earth kept at different degrees of moistness. Average values of (p for different kinds of earth have been ascertained by experiment and observation, and are tabulated. If two particles of earth are pressed together by a pair of equal thrusts p and p normal to their surface of contact, it requires a pair of equal thrusts q and q' tangential to that surface to make them p slide upon each other. For the same material, when q is just sufficient to make them slide, it is a constant frac- . tion of ;;. The fraction which §- requires --^^ -^t ^ ^amtact to be of ;; just to cause slipping is called the coefficient of friction for that mate- rial. Hence the coefficient of friction ? Fig. 1. P Figure 2 is a section of two troughs enclosing earth, and 54 APPLIED MECHANICS. [chap. III. pressed together with a thrust of intensity^ normal to MN, the plane where the troughs are just not in contact, and P is the amount of this thrust. A thrust of intensity q tangential to the plane MN tends to cause the earth to slide in two parts along MN, also Q is the amount of this thrust. If Q be just sufficient to cause slipping along MN, then the coefficient of friction of the earth is u- f^ = Q P If U on AB and CD there be a thrust of intensity p inclined at an angle (j) to the normal, we know that for equilibrium of the prism ABCD there must be a stress q upon the faces AC and BD, whose tangential component equals that oi p; but as far as stability along the plane 3/iVis concerned, we may neglect q, whose normal components destroy each other through the material of the trough, and the tangential ones are at right angles to MN. Considering the components of P, the amount of jo, we have P cos ^ normal to MN. If slipping is just about to take place, then A* = P sin ^ P cos ^ tan (p. It is apparent that ^ is the same angle we were before considering; for, if P be due to the weight of the material, the figure ought to be turned till the direction of P is vertical, when MN, the plane of slipping, will be inclined at ^ to the horizontal. The relation between the coefficient of friction and the angle of repose is f^i = tan <p. Note. — If it were not upon the supposition that the two troughs (being very rigid compared to the earth) transmitted the equal and opposite forces tangential to Jl/^A'" without causing CHAP. III.] STABILITY OF EARTHWORK. 55 lateral compression of the earth, we could not neglect 9. From this result we learn that the tendency to slip along the plane MN, due to p, depends entirely upon the obliquity of p, and not at all upon its intensity. Thus, if p be inclined at an angle less than (p, slipping will not occur though j) be ever so great ; but, if p be inclined at an angle greater than ^.slipping will take place, though -p be ever so small. Consider now the equilibrium of a small prism at a point within a mass of earth in a compound state of strain. The earth will have a tendency to slip along any plane through the point (as there is no artificial envelope), except along the planes of principal stress at the point ; and the tendency to slip will be greater along the plane upon which the resultant stress is more oblique, and greatest along the pair of planes upon which the residtant stress is most oblique, it being of no consequence how intense the stresses upon these various planes may be, but only how oblique. If the stresses upon the pair of planes, for which the resultant stress is more oblique than that upon any other plane through the point, be themselves less oblique than ^, no slipping will occur upon any plane through that point ; but if more oblique than (f>, slipping will take place along one or both of those planes. The condition of equilibrium of a mass of earth in a com- pound state of strain is that, at every point, the obliquity of the stress on the plane upon which, of all others through the point, the resultant stress is most oblique, shall itself not be greater than (f>. Since earth can only sustain thrusts, the principal stresses at a point will be both thrusts and so excludes case (c), and if 7 be the obliquity of the resultant stress upon the plane through the point upon which the stress is most oblique, then by case (d) (fig. 19, Ch. ii), p - Q p 1 + sin 7 sm 7 - ' '- ; .-. - = :, -. — '- ■ p + g g 1 - sin y By increasing 7, the numerator of the term on right-hand side of equation increases, while the denominator decreases, and so the ratio - increases. But ^ is the greatest value of 7 for which equilibrium is just possible. p 1 + sin q 1 - sin (f) is the greatest ratio of ^ to ^r consistent with equilibrium ; hence 56 APPLIED MECHANICS. [chap. III. radam The condition of equilibrium of a mass of earth is most conveniently stated thus : that at every point the ratio of the greatest to the least prhicipal stress shall not exceed that of (1 + sin <p) to (1 - sin <p). Or geometrically, let OM = ^4"^, niake MOB = <p. Drop 3/R perpendicular to OR. Describe the semicircle HEN. Because MOR = obliquity of thrust on plane which sustains most oblique strain, Fig. 4. and ORM = 90°. and MR=p;'^. Therefore ON = (OM + MR) =^ p, [ OH = {OM - MR) = q ; p ON OM+MR OM+OM sin (f> ' q~ 0H~ OM - MR OM - OM sin ,p 1 + sin ^ 1 - sin ^ For earth whose upper surface is horizontal, the vertical stress due to the weight and the horizontal stress are for all points the principal stresses, and their intensities are the same for all points on the same horizontal plane. Generally the vertical is the greater principal stress in any ratio not exceeding the above ; whenever it exceeds the horizontal thrust by a greater ratio the earth spreads. But the horizontal thrust may be artificially increased till it exceeds the vertical in any ratio not exceeding the above. Whenever it exceeds the vertical by a greater ratio, the earth heaves up. The third axis of principal stress, which we are all along neglecting, is also horizontal. When the earth is in horizontal layers with a horizontal surface, all vertical planes are sym- metrical, and the three planes of principal stress are any two vertical planes at right angles to each other and the hori- zontal plane. The stress on the two vertical planes being equal, the ellipsoid of stress becomes a spheroid. When, howe\er, the horizontal thrust on one vertical plane is artificially increased, CHAP. III.] STABILITY OF EARTHWORK. 57 surftLce. of earth that plane becomes (me of the planes of principal stress, and the stress may be different on all three. See the definition of a geostatic load in a chapter followin<^, the numerical examples there, and especially the quotations from Simras on Ihinnelling. Earth in horizontal layer a loaded ivith its ovm weight, to find the pressure against a retaining vmll with vertical hack. Let w = weight in lbs. of a cub. ft. of earth, ^ = its angle of repose, D = depth of cutting. Consider a layer 1 foot thick normal to paper, and choose a small rect- angular prism at depth x feet. Let If p ^ intensity of vertical pressure at depth x feet in lbs. per square foot = weight of a volume of earth one square foot in section, x feet high = wx lbs. per square foot. q = least horizontal stress which will give equi- librium, we have ? = 1 + sin0 1 - sin 1 - sin ? = 1 - sin P- 1 + sin (p w . X lbs. per square foot 1 + sin , intensity of horizontal pressure on wall at depth x. On the right side of equation all is constant but x ; hence q is proportional to x, is zero at the top, and uniformly increases to ? = 1 - sin 1 + sin </» w . D a.t the bottom, and therefore 1 - sin 1 + sin wD —^ = average intensity of pressure upon wall. 58 APPLIED MECHANICS. [CHAP. III. And the area exposed to this pressure is D square feet. Hence the total pressure on wall is Q = average intensity of pressure x area. 1 - sin d> roD^ ,, = -. — - . -7- lbs. 1 + sm 2 This tends to make the wall slide as a whole along MP : for equiHbriurn the weight of the wall, multiplied by the coefficient of friction at the bed-joint there, must be greater than Q. If PM be laid off to represent the horizontal pressure at P, and M be joined to A, then MA gives the liorizontal thrusts at all points as shown by arrows ; Q, the resultant of all these, is horizontal, and passes through the centre of gravity of the triangle APM : it therefore acts at a point C called the centre of pressure, and •* 3 Q tends to overturn the wall with a moment M = Qy. leverage about P. ^ D 1 - sin A wD'^ P ^ IT. = Q'o = ^TT'^-^ • -T foot-lbs. 6 1 + Sin o Let K be the centre of the combined pressure (due to the weight of the wall and horizontal pressure of earth) on the bed- joint at M\, also let the vertical line, drawn through G the centre of gravity of the wall, cut the joint at S; then for equilibrium the moment, weight of wall x leverage KS, must be greater than M the overturning moment. It is generally sufficient to ascertain if this lowest bed-joint be stable : but for some forms of wall it is necessary to go through all calculations for each bed-joint considered in turn as bottom of wall. In a wall of uniform thickness throughout its height, the weight increases as D; whereas the force Q increases as i)', and the lowest bed-joint is most severely taxed. Similarly, for overturning, KS being constant, the product, KS x weight of wall, increases as D while M increases as D\ K would be the extreme outside of the wall if the material were perfectly strong; for stone retaining walls SK the distance from the middle of base of wall to the centre of pressure at that base is -Iths or -^ths of the thickness. CHAP. HI.] STAIJIUTY OF EARTH WOIIK. 59 Depth to which the foundation of a xvall mtist, at least, be siink in earth laid in horizontal layers consistent with the equi- Uhnum of earth. Consider one lineal foot of wall, normal to paper. V = vol. of wall in cub. ft., W = weight of wall per „ h = height of wall in feet, b = breadth of wall in feet, d = required depth of foundation, IV ^ weight per cubic feet of earth, <f> = its angle of repose. P "^ sartaceoffeirth 9' -^p' — Y — ' Fig. 6. When the wall has just stopped subsiding, the earth on each side is on the point of heaving up, so at the horizontal layer at the depth of d, for points in contact with the bottom of foundation, p exceeds q in the greatest possible limit, that earth being on the point of spreading, or 1 + sin </) 1 - sin <^' while, for points just clear of it, })' exceeds q in that limit ; »' 1 + sin <6 , pp —. = I : — .. and — ; 1 - sin <^' qq 1 + sin ^ 1 - sin ^ Now p' = q, being horizontal thrust on same horizontal layer, cancel these and substitute the values weight of wall WV area exposed to y> b ' q = weight of column of earth = wd ; hence bwd \1 - sin 0/ ' ' ' wb \1 -- sm (f>/ 60 APPLIED MECHANICS. [chap. III. Fig. 7. Earth spread in layers at a uniform slope, and loaded with its own weight, to find the pressure against a retaining wall with a vertical hack. The simplest (com- monest in practice) case is when the vertical face of wall is at right angles to the section showing greatest decHvity of free surface. Let the paper be that section ; then AB is the trace of the upper surface, and 7 is its greatest inclination to the horizon. This inclination must be less than the angle of repose, or the earth would run over the wall. In an extreme case they may be equal. Generally y < (p. Take a slice one foot normal to paper ; suppose the earth to be spread behind the wall in layers sloping at the angle y, consider a small parallelepiped in the layer of depth D having vertical faces. At this depth JD, the intensity in lbs. per square foot of the vertical pressure due to the weight of earth above, on a horizontal surface, would be the weight of a cubic foot of earth multiplied by the depth I) in feet. Hence w . D lbs. per square foot = intensity of vertical pressure on parallelepiped had its surface been horizontal ; but the sloping surface 3fN is greater than the corresponding horizontal surface that supports the same earth ; so the vertical stress thereon will be less than wB, (fig. 2, Ch. II), and will be r = wD cos y lbs. per square foot. This is the intensity of the pressure upon the faces MN and KL, and its direction is vertical and therefore parallel to any pair of vertical faces of the parallepiped ; hence the pressure on any pair of vertical faces is in its turn parallel to the face llIN; that is, every vertical plane is conjugate to the free surface. CIIAI". III.] STABILITY OF EARTHWOIiK. 61 Now, as we have selected the faces of MNLK, the pressure on the faces parallel to the paper when drawn parallel to the free surface will be horizontal, so that the stress normal to the paper is a principal stress, and the plane of the paper is the plane of the other two principal stresses. We can apply there- fore our preceding results. Let r be the stress on the vertical faces ]\IK and NL : it must be parallel to the free surface, and so its direction is that of the sloping layer, so that every point in that layer is in the same state of strain, and r' is transmitted along the layer to act on the wall. To find out the ratio of the pair of conjugate stresses r and r' whose common obliquity is -y. Consider The Auxiliaky Figure to Ellipse of Stress. Let (fig. 8) OM = ^^ ; make MOK = 0, Li the angle of repose of earth. Drop Jf.S' perpendicular on 0K\ then MR = p-q_ (Case (d), Ch. ii.) i-'iK. 8. Draw semicircle OH ^OM- MR = ?, ON = 0M+ MN^p. Draw OR'R, making NOR = 7, the common obliquity of the conjugate thrusts r and r', and OR ORf = r. (Case (a), Ch. il) 62 APPLIED MECHANICS, [chap. III. The relations among those are easily expressed trigono- metrically by supposing OM proportional to unity, when OM prop, to 1 ; radius p prop, to sin ^ ; OS prop, to cos 7 ; MS prop, to sin y. BS = ^{MR' - MS-), or y{p^ - MS') prop, to v/(sin-^ - sin^y), or v^(cos^7 - cos'^). }) or ON = OM + p, prop, to (1 + sin<^), q or OH = OM - p, prop, to (1 - sin <^), r or OR' = OS - BS, prop, to {cosy - v^(cos-y - cos-<^)!, and r or OB = OS + BS, prop, to {cos y + -^/(cos^y - cos"^^)}. r' cos y - \/ (cos"y - cos^^) r cos y + v/(cOS'y - COS'^) ' p 1 + sin (fy r cos y + v^(cos^y - cos*^) ' q 1 - sin <f> r cos y 4- v^(cos^y - cos^</>)' The axis of ^? makes an angle = ^BMN, with OiV the normal to the (sloping layer) plane upon which r acts and on the same side. Also cos 26 = 2r cos y - p = q Ij-q (Case (a), Ch. 11.) Returning to the problem on page 60, and substituting the value of r, we have the least intensity of the conjugate thrust at the depth B, r = wj) cosy cos y - 'v/(cOS^y - C0S-(/)) cos y + v/'(cOS'y - COS^0) ' and its direction is parallel to the upper free surface. On the right-hand side of equation everything is constant except D^ so that r' varies as the depth. CHAP. 111.] STABIUTY OF EARTHWORK. 63 Let D be depth of vertical face of wall. Lay off CT to represent r'. Join AT, and this locus will represent the thrust on the wall. The average intensity . r, dL and the total thrust is 18 R' = average intensity x area exposed r' = - lbs. per sq. ft. x D sq. ft. wD^ cos 7 - v/(cos'7 - cos-</)) ,, = -7r-cos7 '- 77 r- ttIds. 2 ' C0S7+ v/(cos-7-cos-0) and it is parallel to free surface, and passes through the centre of gravity of the triangle A TC so that EC = - • ResoMng B' into horizontal and vertical components, H = R' cos 7, V = R' sin 7. H tends to make the wall slide as a whole alon^ the bed- joint at C; and for equilibrium of the wall, weight of wall x coefficient of friction at bed-joint must be greater than R. H tends to overturn the wall with a moment = h(^\ - M = H V3 wD^ J cos 7 - -v/(cos-7 - cos'^) v/(cos'7 - cos'^) COS' 7 6 cos 7 ft.-lbs. For equilibrium of wall, its weight multiplied by KC feet must exceed M. For position of K see fig. 5 and foot of page 58. Note. — V, the tangential component of the pressure of earth on the back of wall multiplied by KC, tends to resist M and to increase effective weight of wall, but the friction of the earth there is liable to be destroyed by water lodging, and it is not always safe to rely on it. Geometrical Solution. — r = vjD cos 7, being the vertical con- jugate thrust, on a layer at depth I), due to the weight of the earth, to find in terms of r, r', the conjugate thrust parallel to layer. p and q, the principal stresses in the plane of paper. 64 APPLIED MECHANICS. [chap. m. 6 - y, the inclination to the direction of r {i.e., the vertical), of the axis of p. And the tliird principal stress normal to plane of paper. Since the earth is upon the point of spreading, the principal stress normal to the paper will be the least possible, that is, it will be equal to q. r=OR - r -OR Fig. 10. Hence this is the horizontal stress on vertical face of a wall running up the steepest declivity. From a point on layer at depth D draw ON (^g. 10) the normal to layer. Lay oft OM, &c.. complete construction as in last, but now in its proper position, properly oriented and to scale.* Draw OF parallel to 3IT, the bisector of RMN\ this and OQ are the axes of the ellipse of stress parallel to plane of paper. * In Professor Malvarn A. Howe's Treatise on Relainivg Walls, in which he adopts tliis method of Rimkine's as developed by us, lie conii.ares B;iiiscliiiiger's construction with that of tig. 10, supra, and siiows that they me ideniictil. Bau- Bchinger's construction is very arbiti iiry, and fails lo recommend itself as Uankine's does by rational steps readily remembered. CHAP. 111.] STABILITY OF EARTHWORK. 65 Lay otf OP = ON and OQ = OH, and draw ellipse ; since MR is always less than OM for like principal stresses, MliO > y, .-. NMR> 2-y, .•. > y, and OF is always in the acute angle EO W between the vertical and the line of greatest declivity, and making (0 - y) with vertical. Fig. 11. Since the third principal stress normal to paper is also OQ, then if the ellipse revolves about POP it will sweep out a spheroid. Its trace on a plane parallel to the free surface is the ellipse R'QR' (fig. 11), some vector of which is the stress on any vertical plane. Examples. 1 . Th« weight of a certain earth is 120 lbs. per cubic foot, its angle of repose 2.5°. It is spread in horizontal layers. Find the average intensity of the pressure against a retaining wall with vertical back and 4 feet in depth. Also, find total pressure against a slice of wall 1 foot in the direction of the length of the wall and the overturning moment of the earth about the lowest point. p = iw = 480 lbs. per square foot, 1 — sin <f) , p = 194 '8 lbs. per square foot. 1 + 8in (^ average pressure = \q = 97 "4 lbs. per square foot, total pressure Q = 974 lbs. per square foot x 4 square feet = 389'6 lbs. ^ overturning moment M = Q lbs. x f ft. = 519o ft. -lbs. F 66 APPLIED MECHANICS. [CHAP. III. 2. Gravel is heaped against a vertical wall to a height of 3 feet : weight of gravel 94 lbs. per cubic foot : angle of repose 38°. Find horizontal thrust per lineal foot of wall, also overturning moment. Q = 100-5 lbs. ; M = 100-5 ft.-lbs. 3. A ditch 6 feet deep is cut with vertical faces in clay. These are shored up with boards, a strut being put across from board to board 2 feet from bottom at intervals of 5 feet apart. The coefficient of friction of the moist clay is -287, and it weighs 120 lbs. per cubic foot. Find the thrust on a strut ; al.«o find the greatest thrust which might be put upon the struts before the adjoining earth would heave up. Since tan<^ = -287; .-. sin (^ = -276; therefore Q = 1225-5 lbs. per lineal foot. Thrust per strut = 6127-5 lbs., just to prevent earth from falling in. Greatest thrust which might be artificially put upon each strut before earth would heave up = 19029 lbs. 4. A wall 10 ft. high and 2 ft. thick, and weighing 144 lbs. per cubic foot, is founded in earth 112 lbs. per cubic foot, and whose angle of repose is 32°. Find least depth of foundation. p = intensity of vertical pressure below bottom of foundation = 144 X 10 = 1440 lbs. per square foot, q' = intensity of vertical pressure at same depth clear of foundation = 112 . «f, d. 112 p \1 + sin<^/ , _ = -094; .-. d = 1-21 ft. 1440 Note.— The height of wall above ground is 10 - d = 8-79 ft. 5. The slope of a cutting being one in one and a half, weight of earth being 120 lbs. per cubic foot, and its angle of repose 36°, find average intensity, amount of liorizontal component, and overturning moment of the thrust upon a 3 -feet retaining wall at bottom of slope. 1 tan 7 = - — ^=-6666; .-. 7 = 33° 42', <?> = 36°, and m- = 120 lbs., 1 . o i> = 3 feet; .-. r = wL cos 7 = 299 lbs. per square foot. »•' cos 7 — v'(cos^7 — COS-rf)) - = ^7^ — ,r ^ = '62 ; .-. r = 299 x -62 = 185-4, r COS 7 + v{cos''7 — cos^<p) and average intensity of stress = 92*7 lbs. per square foot, r' 2 M = H X - = 231-6 ft.-lbs. o 6. A cutting having 3-foot retaining walls is made on ground sloping at 20° to the liorizon. Weight of earth is 120 lbs. per cubic foot, audits angle of repose 30°. Find the horizontal thrust and the overturning moment — 1st, when cutting runs horizontal ; 2nd, when cutting runs up steepest declivity. Data : D = 5 feet, 7 = 20°, w = 120 lbs., cp = 30°. CHAP. IV.] KKTAlNlNti WALLS. 67 (^Itl) r = wD cos 7 = 338 lbs. per square foot = stress on sloping layer at depth B, being vertical, »■' _ cos 7 - v/(cos-7 - C08^(^) _ 'oTG _ r cos 7 + V(tos''7 — cos-<f)) 1-304 .-. >•' = 338 X '442 = 149'4 lbs. per square foot = conjugate stress on vertical face of wall, being in sloping layer inclined at 7, »•' cos 7 = 140*4 11)S. per square foot = horizontal thrust on wall, ut foot of wall. Average do. = 70-2. Total do. = average intensity x area = 70-2 x 3 = 210*6 lbs. per lineal foot of wall. Moment = 210-6 x - = 210-6 ft.-lbs. 3 I. j\ 9 ^ ~ ^^^ 'P '^ r cos 7 + V(cos-7 - co.s2(f>) 1-304 .-. y = 338 x -383 = 129-3 lbs. per square foot = least principal stress in section on greatest declivity = also third principal stress which is horizontal on face of vertical wall. Average do. = 65 lbs. per square foot. Total do. = 65 x area = 65 x 3 = 195 lbs. per lineal foot of wall. D Moment = 195 lbs. x — = 195 ft.-lbs. CHAPTER IV. THE SCIENTIFIC DESIGN OF MASONRY RETAINING WALLS. In treating this subject analytically, we will consider retaining walls as being built of blocks which touch each other at the joints, and which can exert pressure and friction, but not tension. Some cements are so strong that the whole structure may be considered as one piece, in which case arise questions of strength. In what follows we do not take account of this action of the cement, but consider the joints as being able to resist pressure only. The two conditions which must be fulfilled for a joint of this kind are : (1) the resultant pressure on the joint should fall well within that joint ; and (2) the line of action of this pressure should not be inclined to the normal to the joint at an angle exceeding the angle of repose for masonry. When these two conditions are fulfilled, the f2 68 APPLIED MECHANICS. [CHAP. IV. joint is said to have stability of position and stability of friction. In order to find the direction and amount of earth pressure on the wall, Eankine's method of the ellipse of stress is employed ; and from the results obtained for earths whose natural slopes are ^ = 30° and (/» = 45°, and whose free surfaces are horizontal and inclined at the natural slope (p, the thicknesses of wails of depth 20 feet are calculated. The angle of repose for earth is its natural slope, and is the greatest inclination to the horizon at which its free surface will ])ermanently remain ; and we assume for earth what is true for a granular mass, that " It is necessary for stability that the direction of the pressure between the portions into which a mass of earth may be divided by any plane, should not at any point make with the normal to that plane an angle exceeding the angle of repose." Rectangular Wall. — ACFK, in fig. la, represents the wall in cross-section ; depth, c? = 20 feet ; length, 1 = 1 foot ; it supports a bank of earth whose upper surface is horizontal, and whose natural slope is 30°. Let w = weight of masonry = 140 lbs. per cubic foot, w' = weight of earth = 120 lbs. per cubic foot, (p = angle of repose of earth = 30', t = thickness of wall at base in feet. In this case r and r' are principal stresses, and for distinction may be replaced by p and q ; ^ ^ 1 - sin ^ ^ 1 p 1 + sin 3 ■ ^ ^ That is, the horizontal pressure at any point of AC, the back of the wall, is one-third of the vertical pressure due to that depth of earth. At C, the base of the wall, the vertical and horizontal pressures of earth are therefore 2J = 20 X 120 = 2400 lbs. per square foot, q= \ = 800 lbs. per square foot. o This amount q is represented by OT. By drawing AT,d. triangle is formed ; and the horizontal earth pressure at any point of AG \^ given by the line drawn from the point to meet AT^ and parallel to CT. The area of ACT represents the total overturning force axi AC due to the earth pressure, and it CHAP. IV.J RETAINING WALLS. 69 may be taken as acting through the centre of gravity of ACT; thus Q = «^-i-« -. 8000 Ite. acting at -£',6^ feet above the base C. The ellipse of stress for a point at the average depth of 10 feet is drawn ; OJC, OF(fig. lb) are the principal a.xes of stress ; the semi-diameters represent p and q, now of half the values above ; OC represents a portion of the plane AC on which the intensity of pressure is required. ON is drawn at right angles to CC, and along it, OJf is taken equal in length to ^'-^ = 800 lbs. ; MB = ^^-^ = 400 lbs. is drawn so that BMN = 26 where the angle XON = 6 ; in this case 9 = 90°, and JIR lies in MO ; the point B thus found lies on the ellipse of stress, and the line BO represents the intensity and direction of pressure on CC at the average depth of 10 feet. To find the thickness of wall required for stability of position, let G (fig. la) be the centre of gravity of wall, and from C draw downwards a vertical line ; produce the line of action of Q through IJ, and let R be the point of intersection ; from II draw HV = Q = 8000 lbs., and ED = W = weight of wall (not yet determined) ; complete the parallelogram of forces and draw the diagonal HZ, producing it, if necessary, to cut the base in L; draw LU dX right ■i;^ angles to HV. The point L just found must lie within the base FC\ and in order that the bed-joints near C, the heel of the wall, should not have any tendency to open or to crush at F, the toe, it is necessary to have the point L not further from the centre of FC than about troths of that base ; that is, the distance from centre of base of wall to the centre of pressure v; * should not exceed -'Zt. Taking moments round L, we have * ^ Qx LU= TVx LI, .,fl - sin d)\ \1 + sin (fij t is nearly 8 feet. (2) ^ X -' = c? X ^ X 140 X 'Zt. Fig. 1*. 70 APPLIED MECHANICS. [chap. IV. For stability of friction, considering the masonry built in horizontal courses, the angle LHI should not exceed the angle of repose for masonry — 38°. Now, tan LHI Q ^ 8000 W~ 22400 = -357 < LHI is 20° nearly, a quantity well within the assigned limit for friction. Trapezoidal Wall (fig. 2a). — Front and back of wall inclined at 80° to horizon; upper surface of earth horizontal. The ellipse of stress (fig. 2b) for a point 10 feet deep is drawn in position, and is similar to that shown in fig. \h ; CC is now inclined at 80° to the horizon. When the triangle OM/i is constructed as described for the previous case, OB = r = 445'8 lbs. per square foot, and y = BOM = 17° 52'. The stress on AC (fig. 2a) is represented by the triangle ACT; it is zero at A and increases to CT = 891-2 lbs. per square foot at base of wall ; AC = 20-3 feet, and the total pres- sure on ^6' is i^ = 9050 lbs. Let LU and CJ be perpendiculars on the line of action of B, and LS be perpendicular to CJ; then taking moments round L, as before, we have WxLI, (2) Fi£ •3^, Bx LU observing that LU=^CJ- CS, B X {GE cos 17° 52' - -8^; sin 27° 52') = [t- 3-52) X 20 X 140 X from which t = 82 ft. = thickness at base of wall, t - 3-52 X 2 = 1-2 ft. = thickness at top of wall. For this case the angle LHI = 25°, a quantity less than the angle of repose for masonry. Surcharged Bcctangular Wall. — The earth is surcharged at its natural slope ^ = 30° (fig. 3rfc),and the conjugate pressures are equal {OK, fig. 8, Ch. III). The ellipse of stress (fig. 36) CHAP. IV. ] RETAINING WALLS. 71 is drawn for a point 10 feet deep, and it may be noted^that the major axis of the ellipse is midway between the directions of r and r ; that is, the major diameter is inclined to the vertical at the angle 45^ - I = 30°. The intensity of the earth pressure on a horizontal surface at a depth d, due to the weight of the column above it, is wd ; on a plane inclined to the horizon at the angle ^, the intensity is diminished to u-d cos ^ ; and thus the intensity of the two equal conjugate pressures r and /, for a point 10 feet deep, is 1039-2 lbs. per square foot. In drawing the triangle OMR, proceed as in first case : <RMX= '29 = 120°; <ROM=r^ = 30°; OR = 1039-2 ; OM = ^J- = 1200 ; 2 600 as found by calculation or graphic construction ; from which we have;) = 1800, q = 600 lbs. per square foot as the greatest and least conjugate stresses at the point (), that is, at a point 10 feet deep. The triangle ACT (fig. 3a) represents the pressure on^Cas before; (7r=2078-41bs. per square foot at base, and the total earth pressure on AC is R = 207841bs. Taking moments round X, R X LU= W X LI, (2) and, as before, LU = CJ - CS = CBcos-SO° - -St sin 30^ W^UOtd, Ll^-'M Fig. 3«. Fig. 3*. 20784 (5-77 - -M) = 840^^ ^ = 8 ft. 72 APPLIED MECHANICS. [chap. IV. The angle LHI = 29°, a quantity only a few degrees within the assigned limit for friction. Surcharged Trapezoidal Wall.~ln fig. 4a the front and back of wall are inclined at 80° to the horizon ; the earth is surcharged at its natural slope rp = 30°. The ellipse of stress (fig. 4:h) for a point 10 feet deep i.s drawn in position, and IS similar to that shown in fig. 35 ; CC is now inclined at 80° to the horizon. When the triangle OMR is con- structed as described for fig. Ih, OM = 1200, MR = 600, i?0 = 124.51bs., <XOiV=0=5O°, RMN =29 = 100°, ROM = y = 28° 20'. In fig. 4a, the earth pres- sure on ^Cis represented as before by the triangle ACT; CT = 2490 lbs. per square foot at base of wall, and the total earth pressure R = 25286 lbs. inclined at 38° 20' to the horizon. Taking moments round L, we have RxLU=W xLI, (2) - -^t sin 38° 20', J^=(^-3-52)x 20 xl40, ^ = 8-9ft. = thickness at base, t - 3-52 X 2 = 1-8 ft. = thick- ness at top of wall. The angle LHI = 33^ a quantity exactly equal to the angle of repose for masonry ; the courses of masonry should therefore have their bed-joints dipping from front to back of wall at an angle of say 10° to the horizon. ' Trapezoidal Wall. — In fia. Fis. 4ff. Fig. 4A. 5 the back of the wall IS CHAP. IV.] RETAINING WALLS 73 vertical, and the face batters ; upper surface of earth is hori- zontal. The wall here represented is that shown in fig. la with the wedge, whose cross-section is KK'F (fig. 5), removed. The centre of gravity of the triangle KK'F is vertically above L, and the moment of stability of the wall is not altered if this wedge be removed. To find the position of K in K A, where K is vertically above F, take K K= SFL = -(it. The thickness of the wall will there- fore be / = 8 ft. at base, t - -Qt = 3-2 ft. at top. The angle £HI = 27'', a few degrees within the limit for friction. Battering Wall of Uniform Thick- ness. — In fig. 6, the face and back of wall incline backwards, making an angle a = 10° with the vertical ; earth surface is horizontal. If we suppose AC, the back of the wall, to be made up of a number of rectangular steps, vertical and horizontal, the horizontal earth pressure on a vertical face at a depth d will be, as before, , ^ 1 - sin (h iv . d . : — ^ ; 1 + sin this horizontal pressure will tend to cause the earth to spread upwards, and the vertical earth pressure on a horizontal face looking downwards at depth d. will be «-^-£: Fig. 6. v/ . d . l-sin0 l-sin'/< , , /l-sin0 ^ ^ — 7. q . — =w.d.( r-^ l + sin0 1 + sm^ \l + sm0 The straight line AC may now be considered as the limit of these steps, and the horizontal and vertical pressures on AC will be represented by the triangles ACT and C'CT ; the horizontal pressure is zero at A, and increases to C'T =120 X 20 X ^ = 800 lbs. per square foot at the base ; the vertical pressure is zero at A, and increases to CT' = 120 x 20 x ^ = 267 lbs. per square foot at the base. The total horizontal pressure on AC is Q = 8000 lbs. as for fig. la ; the total vertical pressure on ^ C is Q' = ^CC" xCT' = ^x 3-52 X 267 = 470 lbs., and the points of application are F and F'. 74 APPLIED MECHANICS. [CHAP. IV. Taking moments round L, W X leverage = ^ x leverage + $' x leverage d J J 04. ^ ^ \ , , 1 - sin d> c? w . a .t\'6t -^^ - tan a\=wcl : — - -z \ 2 / 1 + sm ^ 2 , /I - sin aV c? . tan o . _^ ,,^ . + 10 . d.i ; — i (-U + \d tan a), \1 + sm <j)J 2 ^ ^ ^ taking w = 140, and ?// = 120 lbs. per cubic foot, <b = 30°, and <^ = 20 feet, t = 8 v/l + 4 tan^ a - 15-4 tan a. (3) The vertical through G, the centre of gravity of wall, must fall within the base ; and if it be not allowed to deviate further from the centre of the base than f^ths of the breadth of base, we obtain '3t tan a = j-^ = '03^, when d = 20 feet ; putting this value in equation (3), we get ^ = 8 (1 + 4 X -0009^0^ - 15-4 X -03^, t = 5-78 ft. = FC, the tlnckness of wall. The angle a is 10° nearly ; that is to say, the back of the wall should not be inclined to the vertical at an angle greater than 10°. In fig. 6, G' is the centre of vertical forces, viz., the down- ward weight of the wall, and the upward pressure of earth ; HD is the vertical drawn through G', and is equal to W - Q' or 15713 lbs. The angle ZJII = 27°, a few degrees less than the limit for friction. Wall loith Vertical Face and Stepped Back. — In fig. 7, the steps are taken at vertical intervals of 5 feet ; the upper surface of earth is horizontal. The base of the wall FC sup- ports the masonry A'CFK, and the earth A' AC vertically above that base; and the triangle ACT represents the hori- zontal earth pressure on the back of the wall ; G is the centre of gravity, and W is the weight of masonry and earth vertically above FC. CHAP. IV.] RETAINING WALLS. 75 FiK. 7. Taking moments round L as before, Q xLU = W X LI. (2) To find the thickness of upper 5 feet of wall, proceed as for tig. \a ; and obtain 1^ = 2 feet. For /in, thickness at 10 feet deep, we have ^10 = 2000 lbs. acting at -LQ- foot above this assumed level; and taking moments round the point corresponding to L Q,, X Y = 10 X itg X 140 X (-54 - •2^,o) + (t,, - U) (140 + 120) (-3^,0 + -bh) from which ^i,, = 4'1 ft.; similarly ^15 = 6"3 ft.; and tw = 8-4 ft., the thickness at base. The angle LHI = 20"", a quantity well under the limit for friction. Wall ivith Battering Face and Stepped Back. — In fig. 8 the steps are taken 2 feet wide, and at vertical intervals of 5 feet; the batter of face is 1 in 12, and the earth is surcharged at its natural slope 9 = 30°. The ellipse of stress (fig. 3&) applies to this case ; the depth AC \Q inci eased to 23'5 feet by the earth slope ; the pressure at base, CT = 2440 lbs. per square foot ; the total earth pressure on AC\^R= 28600 lbs. acting at E. On account of the battering face, ihe point F projects 1"7 feet beyond the vertical through K; and on account of the steps of the back of wall, the base pro- jects 6 feet beyond the ver- tical through .j' ; the thick- ness of wall at base may be represented thus / = 1-7 + ^' + 6 = ^' + 7-7 ft. In order to find the value of t', take the moments round F, thus : — Weight of masonry = 2800^' + 10780. 76 APPLIED MECHANICS. [CHAP, IV. Moment of masonry round F= UOOd'^ + 18160^' + 36500. Weight of earth vertically over FC = 8400. Moment of earth round F = 8400 + 44800. Moment of earth and masonry round L = 840^' + 13436^' + 51950. Equating this to lixIU^ 28600 S6-77 - "4 {f + 7-7) ) , we get t' = 2 feet, t = 97 feet, thickness of wall at base. The angle LHI = 33°, a quantity exactly equal to the angle of repose for masonry ; the courses of masoniy should have their bed-joints dipping from front to back of wall at an angle of, say, lO*^ to the horizon, or at right angles to FK the battering face. Tabulated Dimensions. Since the stability of a wall is proportional to d . f, where d = depth and t = thickness, and since the overturning pressure of earth is proportional to d?, it follows that the thickness of a wall should be proportional to its depth. Having calculated the thickness for a given depth d = 20, it is easy to fix on the thickness required for any other depth. If the depth of wall be taken as 100, the accompanying table gives the other dimen- sions of walls, as deduced from the results given in this Chapter, when the earth has for its angle of repose ^ = 30*^, or ^ = 45°, or slopes of 1 vertical in 1"73 horizontal, and 1 in 1. For comparison, corresponding values are given for water whose heaviness is 62*5 lbs. per cubic foot. It will be observed that the thickness of a wall for resisting water pressure is much greater than for the varieties of earth considered; this, of course, is caused by ^ becoming zero in the case of water. It is, therefore, of the utmost impor- tance that the earth behind a retaining wall should be kept dry; and for this purpose weeping holes through the walls are formed near the level of the original surface of ground, and a diy stone backing from 12 inches to 18 inches thick is laid beliind the wall for conveying the water easily to the weeping holes. CHAP. IV.] RETAINING WALLS. 77 Spread and Depth of Foundation. . Because of the obliquity of the downward thrust on the foundation, the concrete or masonry forming the substantial part of it must spread out in front of the wall in steps as it goes deeper. The bottom of the trench may have to be con- solidated by driving packing 'piles, or hearing piles may have to be driven to a firm stratum, and surmounted by a staging generally dipping back as the friction is then also precarious. In old consolidated earth two conditions serve to determine the spread and depth of the trench (see first wall, fig. 12). The total vertical tiirust of the bed of the trench on the base of the concrete must equal the weight of the wall and the concrete. Also the centre of this upward thrust must be vertically below (i^'on fig. 12, but L on figs. 8, 7) the assumed centre of stress at the lowest bed joint of the wall. But besides these two conditions which concern the equilibrium of the wall we have the limiting conditions concerning the equilibrium of the earth surrounding the concrete, namely, the upward thrust of the unit- cube under the toe of the concrete is at most nine times a column of earth the depth of the bed of trench below the surface of the earth in front of wall, which is then on the point of heaving-up. And for the unit-cube at heel of trench, it is as a practical lower limit, one-third of the column of earth behind the wall. The two equations obtained can best be solved by trial and error. Taking the height of the wall as sensibly 3^, where t is its thickness, we will try the spread 45 per cent, of the thickness, and the depth 20 per cent, of the depth, or 60 per cent, of the thickness. Weight of wall and concrete is wt^ (3 + 1'45 x -6) = 3"87 ivt^. Upward stress at toe of trench is 9 x -Qtvt = 4-32 wt ; while at the heel it is ^ (3 + -6) w't = -96^^. Multiplying their average value by 1"45^, we get the total upward stress = SSdwt^ which satisfies the first condition. If the centre of this upward stress be distant ya and yj, from the heel and toe of bed of trench respectively, then for the centre of gravity of the trapezium of stress drawn below first wall, fig. 12, we have, ya-.yb--: (4-32 + | x -96) : (i x 4-32 + -96), and ya + yb = 1-45^, so that ya = 'S8t or ^t nearly, which satisfies the second con- dition, as F on fig. 12 is at most ^t from back of wall. H P O O CO l-J <1 . P g S" o in < P^ o Q W P5 t— I P w P5 C5 t— I w w H O CO H w CO W o t-H W H O w pq H «- II « "5 «N •<»• O C^ t-« CO H * c^ ^ S -ja -e- W w <M 00 lO CI r-i CO * o (M O o Tf 1—1 CO CO O Q UJ O 3t3 ?5 e tr. tc O Tl in o c/: < O CO O ^^^O.^ CHAP. IV.] RETAINING WALLS. 81 Graphical Solution. Graphic statics is a study by itself, and in this book we only intend to use the very first elements. In graphic statics a force is represented by a pair of lines ; one a short thick finite line giving the magnitude of the force upon a scale of forces ; the ends of this thick line are marked by the centres of two little rings ; in a hand-made drawing the centres are pricked on the paper. The other is a long thin line giving the actual position of the force on the plane of the paper relative to some structure or solid body upon which the force acts, and relative to other forces also acting on it. It will be seen then that a second scale of feet is necessary to measure the distances along the solid body and among the forces. These force lines are furnished with a barb or arrow-head to indicate the sense of the force in that line ; near the barb stands a numeral which is really the suffix of a letter such as P^ ; the force is called 3 when simply being referred to, but when speaking about it as a quantity it is P,. Often the force line is so long that it reaches to the edge of the paper, and although only a portion may be ultimately inked in, still a little left at each margin may be indicated. The line is fine and long for the practical purpose of setting the rollers or T square accurately parallel to it. A plane set of forces all acting on one body, which may at first be assumed to be the sheet of paper, are numbered in any order ; but for a successful issue they must be numbered in cyclic order. The thick lines drawn parallel to the thin, each to each, form a polygon round which the corresponding numerals, printed heavier or larger, run consecutively ; for some purposes it is convenient to put half barbs on the thick lines. This is the " Polygon of Forces." When the forces are all vertical the sides would lap on each other, but this is avoided by slightly displacing some, and drawing a polygon which often looks like a gridiron pendulum ; the " eyes," however, should all be in one vertical line. In this case, the downward forces being all vertical, the polygon is often called " the load line " ; while the closing upward sides constitute the reactions or supports. Given, graphically, fig. 10, a plane set of forces to construct the balancing force or resultant. On the upper left-hand corner is shown an analytical definition of five plane forces in terms of intercepts and angles and lbs., agreeable to which the force polygon and lines of action are drawn to a pair of scales. 82 APPLIED MECHANICS. [CHAP. IV. Construction. — Draw a line closing the " force polygon," and scale off its length in lbs. on the force scale for the magnitude of Bs. Eeckon its " sense " in the same order round the force polygon as the other forces. Choose any pole (not on the closing side of the force polygon), and, to that pole, construct a " link polygon " A B C D E F amoDg the lines of action. Through F, the closing point of the link polygon, draw the line of action of B^. parallel to the closing side of the force polygon. When scaled off, the values of B(„ x^, and A^ should be nearly the same as the values given on fig. 10 which were calculated by trigonometry. Pkoof. — Consider the set of six forces, 1, 2, 3, 4, 5, 6 ; they are a balanced set of forces. For, add a pair of balanced forces 7 and 7' equal and opposite to each other, having the link A B for their common line of action, and having the vector, which comes to the junction of 1 and 2, for their common magnitude. Add another pair of balanced forces of 8 and 8', with BC for their common line of action, and the vector, coming to the junction of 2 and 3, for their common magnitude. Add in the same way the pairs of forces 9 and 9', 10 and 10', 11 and 11', 12 and 12'. We have now altogether a set of eighteen forces 1,7,12'; 2,8,7'; 3,9,8'; 4,10,9'; 5, 10', 11 ; 6,12,11'; which are exactly an equivalent set to the set of six forces with which we began, since all the forces we added balanced in pairs. Now, of the set of eighteen, the first group of three, 1, 7, 12', act at the same point A and have magnitudes proportional to the homologous sides of a triangle; they are, therefore, a balanced set of three; similarly the second group of three, 2, 8, 7', act at one point B and have magnitudes proportional to the three homologous sides of the triangle. In the same way each group of three is balanced ; hence the set of eighteen is balanced ; hence the original set of six is balanced. Cor. — The graphical conditions of equilibrium of a plane set of forces are two in number. The force polygon must close. A link polygon must close. The proof given here is very important, for, in the first place, conceive the rigid body to be the paper, say, a sheet of brass, and let the six balanced forces be attached to it at the points A, B, C, D, E, and F. Suppose the brass all cut away except the narrow strips under the thick lines A B C D E F, which may further be supposed to be pin-jointed at these points. These strips of brass actually apply to the pins, the pairs of forces given by the vectors, viz., 7, 8, 9, 10, 11, and 12, and CHAP. IV.] RETAINING WALLS. 83 we have designed an articulated structure in equilibrium under the given load system. This is a balanced polygonal lineal frame or rib, and if each strip of brass be sectioned for the load on it, we have then an actual balanced frame. A model of this frame sits on a horizontal table in the Engineering Laboratory of Trinity ' College. It is in unstable equilibrium, for a sharp blow on the table causes it to distort. The pair of diagrams, figs. 12 and 13, show designs for retaining walls. The load on the back of the wall is constructed by the method of the ellipse of stress and its auxiliary figure. This load is reduced so as to be expressed in terms of the \\ eight w of a cubic foot of masonry. Triangular or rectangular blocks of masonry are added one after another, and the partial resultant constructed graphically till the last resultant, that with the greatest number of barbs, at last passes through a centre of stress deemed to be sufficiently far in from the face of the wall. The data and construction are sufficiently given on the face of the diagrams, which, however, are small, having been reduced half the lineal size of a set of graphical exercises, published by Macmillan & Co. for the authors. It wiU be seen that the slopes of the earth, both actual and limiting, are given 3 to 1 and 2 to 1 which are nearly the same as 7 = 20°, and <j) = 30°. The steps of the calculations corresponding to those graphical solutions are as follows. The fourth is only approximate so that one auxiliary figure may serve for two walls. In the process of designing either by the equations (a), (b), etc., and especially by the graphic method shown on figs. 12 and 13, it is sufficient to get the centre of stress to pass through the toe of the wall and then adding a slice to the face of the wall to throw this centre of stress in a suitable distance from the new toe. This addition will not sensibly shift the centre of stress, as may be shown thus : — Minimum distance of the centre of stress from the heel of the wall. For a very thin wall (fig. 11a) the weight to be compounded with the load is so small that the centre of stress lies far out beyond the face and such a wall might be of metal and sunk firmly in the earth. As slice after slice is added to the wall, the centre of stress comes nearer and nearer to the heel till G 2 84 APPLIED MECHANICS. [chap. IV. (fig. lib) it passes through the toe ; but for further slices added begins to move away from the heel, at first very slowly, as is always the case in the neighbourhood of a minimum. t A ■■'"} I \> f / w " / 1 3 / , ^ ■„ ,Jbs. / / rieei. ~aneai ^ Minimum. Increasing. Fig. 11 — rt, b, c. For the same reason wedges can be removed from the face of the rectangular wall to give suitable batters without disturbing the centre of stress. Examples. 1. For the first wall on figure 12, the corresponding analytical solution is tan-ii = 26° 34' ; k= ! ^'" "^ = -382. 1 + sin ip p = \5tv' ; q = kp = b'lZw'. E = Q = y X 30= 171-9!<.''= 137-5t<^^, or 138 times the weight of a cubic foot of masonry. M = \Q H = Wibw; M=ZOtci x^t = U'2bwf. Equating, t- = 122, and ^ = 1 1 feet nearly. 2. Second wall on figure 12. Approximate prnciical solution. In the last suppose an upward force S to act through the centre of gravity of the wedge removed from the hack of the wall equal to the excess weight of masonry over earth. 5 = ^x30x5x I5w, and its lever about the centre of stress is (it — -f) = 8 nearly as t is almost 11. Correcting the equation of moments ll->5icl- = \37bw + low X 8, or <*= 133 and <= 11-5 feet. 3. Second wall on figure 12, solution exa<tly corresponding to the graphic solution, p + g iv' ■ ^ = — (15+ .5-73)= 10-365et7'. 0M = MR = ' 4-635M7'. 2 2 OM"^ = 107-.'). MK^ = 21 -0 ; dropping it'. OR- = OM- + MR^ +20M . MR cos 20. = _(15-.V73) E*RTH. TIm wclihl of on* CuMc Foot Is (aur- nttto that ol inuon- ,«» '^ DESIGNS RETAINING WALLS. -30 J'oof ■ Ca/umn of JCarfh <| AUXILIARY FiaURE tor First ELLIPSE Fig. 12. 86 APPLIED MECHANICS. [CHAP. IV. Now e = P = cot-^l = 80° 33', 80 that 20 = 161° 06', the sup. of which is 18° 54'. OiJ2 = 129 - 96-07 X -946 = 38-12. r= 0R = 6-174U'', and CC the hack of the wall is 30 cosec fi = 30-4 square feet. R = 30-4>- = 187-7^' = 150w. sin 7 MR . 4-635 ^^e = -OR-^ .•.sm7 = ^:^x-324=-245, , =. 14° 11'. To 7 add 9° 27', the complement of 6, and we obtain 23° 38' as the inclination of R to the horizon. Resolving R into horizontal and vertical components H= R . COB 23° 38' = 150m> x -916 = 137-4m-. r= 7? • sin 23° 38' = 150m; x -401 = 60-15mj. The equation of moments about F, the centre of stress, is 1374 = 30 (!! - 5) (f (! - f ) + 75 (-^< - iJt) + 60 (|.< - -|), <2 _ 1-I6i; = I'^O, or <= 11-5 feet. 4. The first m all on figure 13 has the additional datum 7 = tan-' i = 18° 26'. r = low' cos 7 = \bw' X -9487 = 14-23w'. , _ cos7-V(cos^7-cos''(^) _ -9487 - v(-9000 - -8000) _ 9487 - 3162 _ . ~ cos 7 + V(cos^7 - cos*^) ~ -9487 + \J{-^mQ - -8000) ~ 9487 + 3162 ~ /•' = k'r = 7-115if', also 7' = 7 = 18" 26'. R' -- 30/ = 214m;' = 171m; ; E' = R' cosy' = 162m;. r' = iE'sin7' = 54t<;. Also JF = SOwt. Equation of moments about F, the centre of stress, at ^th of t in from the face of the wall. lOJy = 7Fx|< 4 F'x|<, 1620 = 11-25i!2 + 47-1^ )/, and < = 9-3 feet. 5. Approximately for second wall on figure 13, by removing a wedge at back. 1620 = 11-25^- + 47-25<- 15 X 8, and t = 10-5 feet. 6. Detailed calculations for second wall on figure 13 corresponding to the graphical solution shown on it. p 1 + 8in^ 14 -4472 , ,^^ — ~ = 1"144. r cos 7 4 v'Ccos'27 - cos2(^) -9487 4 -3162 p = l-144r = 1-144 X 14-23m;' = 16-27m'. q = kp = -382 X 16-17m;' = 6-21m'. OM = ^(p + q) = 11-24M; ; 3IR = Up - q) = 5-03«;'. ain 20 OR r 1423 sin y ~ MR~ ^ (/? - y) " 5-03 ' . , 1423 sin 2a = -— X -3162 = -8956; 26 = 63° 34' : 6 = 31° 47'. -j^- _ aoFnof C»Tumn of £(vth -tj AUXILIARY FiaURE tor Second ELLIPSE ' ' ' "" Cubiv net of Masohhy. too POO 300 TTTTTt fiv/ I L' :UIUU.i:| /cii ^oo 3c^ 4cn .ill. . -^...Li j-i-j^i-i >-. I -,-:-i=r: 40 feet Fig. 13. 88 APPLIED MECHANICS. [CHAP. V Now the angle between the normals ON' and ON" can be expressed in two ways. e + e" = 7 + /3, or 6" = 18° 26' + 80^ 33'- 31° 47'- 67° 12'. •lef' = 134° 24', the sup. of which is 45° 36'. [OR'Y = OM- 4 iMR^ + 20M. MR cos 6". •24^ + O-03- - 2 X 11-24 x 5-03 x -6997 = 72-53. (^)'-- r" = 8-516«f''; R' = >•" x 304 = 259tt' = 207-2w. sin-y" MR" 5-03 . „ 5030 „, , ,„, , „ „ ,„, ■ ; „ = — -„- = —,—7; .-. sin 7" = -;— X -7145 = -422, and 7" = 24° 58'. sm 26" OR" 8-516 ' 8516 ' Obliquity of OK' to the vertical is ;8 - 7" = 80° 33' - 24° 58' = 55° 35-. R" = K sin 55° 35' = 171m^, and F" = R" cos 55° 35' = l\lw. lOH" = 30,^ {t-o)(ft-A) + nw (it - la) + V" (it - %], or i- + 3-27t = 158-2, and t = ll'OSfeet. CHAPTER V. TKANSVERSE STRESS. In the preceding chapters we have considered the internal stress at any point within a solid, and have shown that it can be expressed by means of three principal stresses. We began with one principal stress, the other two being zero ; this was illustrated by pieces strained under one direct simple stress, such as tie rods and struts ; and at each point in these pieces the strain was similar in every respect. We next considered two principal stresses, the third being zero or identical with one of those two ; this was illustrated by small rectangular prisms of earth under foundations, or loaded with the weight of superincumbent earth, the prism being strained by two (or three) direct simple stresses upon its pairs of opposite faces. 'I'here we saw that the strain at all points, in certain parallel planes, was similar in every respect; varying, however, as we passed from points in one to points in another of those parallel planes. It was pointed out that earth might have the stress in one horizontal direction artificially increased by a direct external stress, in which case there would be three principal stresses at each point, the intensities of which might be difterent at different points. CHAP, v.] TRANSVERSE STRESS. 89 In all such examples, the internal stresses were due to strain produced in the simplest manner possible, viz , by direct external stresses ; and in many the stresses at internal points were given, without specifying what the solid was, or in what manner it was strained. These exercises served to illustrate methods, but it will afterwards appear that the data specifying the stress at such points were obtained by supposing that the body was strained by external stresses, definite though by no means either simple or direct. We now come to consider the stresses at points within solids, due to strains produced in the next simplest manner, viz., by external stresses which are all parallel. Pieces under such stresses are called beams, and the stress is called transverse stress. The case in which both ends of the beam are simply supported will be primarily considered. For simplicity, the external stresses, as shown on the diagrams, are all vertical ; they consist of the two upward thrusts concentrated at the extremi- ties, and the loads concentrated on intermediate portions and acting downwards. These external stresses are uniform in the direction normal to the paper ; and whatever be the breadth of the beam, they may be replaced by forces all in one plane, the plane of the paper. On fig. 1, AA' B' B is the longitudinal section of a beam of length 2c, depth A, and breadth h, and OX is any line chosen as axis. W^ is a force in the plane of the paper, replacing a stress spread uniformly over the breadth of the beam, as shown w, 1" 4-13. w, ._ll -.Q Fitr. 1. on the cross-section below it. Similarly P and Q are forces at the extremities and in the plane of the paper. In order to have these forces specified, it is necessary to know their amounts, and the distances measured from some origin O, say at one end of 90 APPLIED MECHANICS. [CHAP. V. the beam, to the points where their lines of action cross OX. Such distances are called the abscisste of the places of applica- tion of the loads. Thus P acts at 0, W^ at rci, and Q at 2c. C is the centre of span ; its abscissa is c. The varieties of load to be considered are : — V. Loads concentrated at one or more points of the span as Wu 2°. Loads uniformly spread over the whole or parts of the span, as w lbs. per running foot. Such a load is represented on fig. 1 by a set of arrows, each equal to v), and consequently they are one foot apart ; to save trouble, it is more convenient, as on fig. 2, to represent .such a load by means of a parallelo- gram surrounding all the arrows. 3°. Combinations of such loads. The loads concentrated at points might be the ends of cross- beams resting on such points, or the wheels of carriages, &c. The weight of the beam itself is often to be considered as a load spread uniformly over the span. To find the relations among the external forces, we consider the equilibrium of the beam as a whole. The beam is to be consideied as perfectly rigid and indefinitely strong. In order to find the supporting forces P and Q, we require to know the amounts and positions of the loads, and the length of the beam. Since the forces are all parallel and in one plane, there are two conditions of equilibrium : — I. The algebraic sum of the forces is zero. II. The algebraic sum of the moments of the forces about any point is zero. From the first condition we have P+Q= W,+ W, +W, + &c. =. ^(W) (1) where 2 ( W) represents the sum of all the quantities W„ W,, W„ &c. If we take moments about 0, then P has no moment, and Q tends to turn the beam in one direction about 0, while the loads all tend to turn it in the other direction. By the second con- dition the sum of these moments is zero, and we may, if we CHAP, v.] TRANSVEKSE STRESS. 91 choose, put the moment of Q equal to the sum of the moments of the loads, thus Q X leverage = sum of the products got by multiplying each load by its leverage, or ' Q.2c= Jr,.r. + W^x^ + W^^ + &c. = 2 ( Wx) ; hence Q = -^—\ ,2) where 2 ( Wx) represents the sum of all the quantities W]Xu WzXi, &c. P may be found in a similar manner by taking moments about the other end ; or it may be found at once, since we know P + ^ by equation (1). An uniform load, such as tv lbs. per running foot, spread over a portion of span, is to be treated as one force equal to the amount, and concentrated at the middle of that portion. Examples. 1. The span of a beam is 20 feet, iind there is a load of 80 tons at 5 feet from the left end. Find the supporting foi ces. 2c =20; JFi = 80, and a;i = 5 ; Q . 2c = JFiXi. Othei wise, IFiZi 400 Q = -— = -— = 20 tons, /* + e = 80 tons, P= 60 tons. load 80 Q = X segment remote from Q = — x 5 = 20 tons ; span 20 load , ^ 80 and 1' = X segment remote from T— — x 15 = 60 tons. span 20 2. A beam of span 24 feet supports loads of 20, 30, and 40 tons concentrated, in order, at points which divide its length into four equal parts. Find the sup- porting lorces. »^i = 20 ; 7r2 = 30 ; TFs = iO ; zi = 6 ; x^ = 12 ; 2:3 = 18 ; 2c = 24. ^ :S,(TFx) 20 X 6 + 30 X 12 + 40 X 18 Q = -^ — ' = = 50 tons. 2c 24 P = 2 ( ;r) - Q = (20 + 30 + 40) - 50 = 40 tons. 3. A beam 30 feet span supports three wheels of a locomotive v hich transmit each 6, 14, and 8 tons ; the distances measured from the left end of the beam to the wheels are 8, 18, and 24 feet respectively. Find the supporting forces. JFi = 6; Jr2 = U; JF3 = 8; 2 ( ^) = 28 tons ; an = 8 ; 2:2 = 18 ; xs = 24 ; 2c = 30 feet. Atis. P= 11*6 tons; Q = 164 tons. 92 APPLIED MECHANICS. [chap. V. 4. The span of a beam is 60 feet ; an uniform load of 2000 lbs. per running foot is spread over the portion of the span beginning at 40 and ending at 50 feet from the left end. Find the supporting forces. See fig. 1, and suppose the spread load alone on the beam. Replacing ww ... by W2 = w (50 - 40) = 2000 X 10 = 20000 lbs., concentrated at a;2 = i (40 + 50) = 45 feet, F = 5000, and Q = 15000 lbs. 5. A beam 60 feet spnn, and weighing 100 tons, supports an uniform load of 2 tons per running foot, which extends from the left end of the span to a point 20 feet therefrom. Find the supporting forces. M? = 2 ; JFi = w X 20 = 40 tons ; x\ = 10, the middle point of uniform load. "Weight of beam, TF^ = 100 tons ; 3:2 = c = 30. Aus. P= 83^ tons ; Q = 56| tons. 6. A beam 60 feet span, and weighing 100 tons, supports a locomotive as in example 3, and an uniform load of 2 tons per running foot which extends from the middle to the right end of the span. Find the supporting forces. Q X 60 = 6 X 8 + 14 X 18 + 8 X 24 + 60 X 45 + 100 X 30. Aus. 0=103-2, and P = 84-8 tons. 14 8 vy,vy 100 Of— ;^<— --^V-^-^'b X^ I M 60 1* iwibsprft X iQ Fig. 2. 7. A beam 36 feet span weighs one ton per lineal foot. Tlie first half is loaded uniformly with 2 tons, and the second half with 3 tons per running foot. Find Pand Q. Alls. P= 58-5, and Q = 67-5 tons. 8. A beam 42 feet spun supports five wheels of a heavy locomotive. The tore wheel is one foot from the left end, and the distances between the wheels, in order, are 5, 8, 10, and 7 feet, and the loads transmitted, in order, are 5, 5, 11, 12, and 9 tons. Find the supporting forces. See fig. 3. lFi = 5; ir2=o', 7r3=ll; ;r4=12; 7/'5= 9; 2(7r) = 42. .ri = 1 ; a:2 = 6; ars = 14 ; J* = 24; 3:5 = 31; 2c = 42. .-. Q X 42 = 5 X I + 6 X 6 J- 11 X 14 + 12 X 24 + 9 X 31. Ans. P=24, and Q= 18 tons. CHAP, v.] TRANSVERSE STRESS. 93 For a system of loads such as Wi, Wi, &c., there is a point at which, if they were all concentrated, the supports would share the load as they do for the acttnal distrihutiDH at different points. This point is called the centre ot gravitij of the load system ; its position will he marked G, ami its ahscissa 00 will be denoted by X. Hente, snpposing the total force 2 ( ^F) concentrated at O, we have Q .'Ic^liW) .X for the single force 2 ( 7F). Q . 2(; = 2 ( Wx) for the actual distribution ; see equation 2 on page 91. Hence :S.{W) . x = :Z{Wx), or x = -^^ , which gives the position of G. Having calculated .r, we can now find P and Q as for the single load 2 ( JF) at X from the left end ; see example 1, setond method. , . , , totiil load The supporting force at either end = x remote segment, (3) span P = 2c (2c - X) ; 2c 9. Solve exercise 102 by the method just described. We have _ _-Z{Wx) _ 5 X 1 + 5 X 6 + 11 X 14 -f 12 X 24 + 9 X 31 ^^ ■S.[W) ~ 5 + 5 + 11 + 12+9 and the other segment (2c - i) = 24 feet. = 18 feet, and total load span 42 X remote segment = — x 24 = 24 tons, total load 42 „ „ X remote segment = — x 18 = 18 tons. span 42 The position of G relative to the loads W\, W^, . . . can be found, although the span of the beam and the position of tlie loads upon it be unknown, provided the amounts of the loads and their distances apart be given. On fig. 3 let S be 111 42 tons is. 9 tons ' i^iy^ffj;^^f~\^^rr^jj^ o-v s> ■V* G C o -v^ Fig. 3. the point where the first load W\ is situated, and let a\ (= 0), a2,ra3, a, Ui, 05 be- the distances from »S to W\, Wt, JF3 G, JF4, W^, respectively. Taking moments about S, the moment of 2 (fF) ucting at a will equal the sum of the moments of ^1, JFi . . acting at m, rto . . . . 94 APPLIED MECHANICS. [CHAP. V. That is 2(?r)x«- = 2(^«); .-. « = ||,^ gives position of G measured from S the left end of the load. 10. In example 9, find G the position of the centre of gravity of the load measured from tlie fore wheel. Wi = 5, ^2 = 5, TTs = 11, Wi = 12, JTs = 9, 2 (R^) = 42. «i = 0, «2 = 5, rt3 = 13, (74=23, as = 30. ■S,{Wa) + 5x5 + 11x13+12x23 + 9x30 Then a = = 17 feet. 2(Jr) 6 + 5 + 11 + 12 + 9 11. In example 3, find a the distance of G from the fore wheel. TTi = 6, W2=U, TF3 = 8. m = 0, ^2 = 10, as = 16 ; .-. « = 9-57 feet. It is convenient to calculate a, if it be required to find values of P and Q, as in examples 3 and 8, corresponding to the given load system shifted into some new- position upon the beam ; thus 12. In example 8, find F and Q when the locomotive shifts till its fore wheel is 6 feet from the left end. (7=17: hence adding six feet we have x = 23, so that Pand Q will now be the same as for a single load 2(^r) = 42 tons concentrated at G, a point dividing the span into the segments 23 and 19. ^ load 42 „„ .-. Q = X remote segment = — x 23 = 23 tons. span 42 Similarly 42 P= — X 19 = 19 tons. 42 In like manner P and Q may be found with great convenience for other positions of the locomotive, all of whose wheels must, however, be on the beam, because, if one wheel goes off, the beam is under a different load system altogether. 13. Find P and Q in example 3 when the locomotive is shifted so that its fore wheel is 10 feet from the left end of the beam. Here « = 9-57 feet, and 2 ( ^T) = 28. ^«.9. P=9-73, Q = 18-27 tons. Neutral Plane and Neutral Axis. The phenomena which accompany transverse stress are : — Every horizontal straight line parallel to the axis of the beam becomes a curve, one line on the diagram showing the curved condition of all lines lying in the same horizontal layer. All points in the beam, except those over the supports, arrive at a lower level. CHAP, v.] TRANSVEKSE STRESS. 95 The shorter consequence is, that some and others are longer than top bottom horizontal they were layers before are the stress was applied. The | j^q^-j^^jj^ layer is that which is ^ f compressed 1 «^^«t) I extended J' and one nearer the r top 1 \ bottom / IS more than one not so near. Since this condition of ( compressed | { extended ] being extended diminishes gradually as you pass upwards from layer to layer, and passes into a condition of being compressed, there must be one intermediate layer which is neither extended nor compressed. This layer is indefinitely thin, is in fact a plane, and is called the neutral j^^f^ne of the beam, and the line which is its trace upon the diagram is called the neutral axis of the beam. On fig. 4, the straight line OX, the neutral axis, while the beam is unstrained, is chosen as an axis of reference. Let S be any point on the neutral axis after the beam has been strained, 5 its distance from O measured along the curve ; let S' be the point on OX directly above S, and X its distance from v, measured along OX. It is to be observed that the curvature, although ex- aggerated on the diagrams, is really in practice so slight that X and s will be sensibly equal to each other, and x may be put for the amount of either unless where it is absolutely necessary to dis- tinguish between them. S'S is called the deflection of the point S ; the greatest value of this is called the deflection of the beam, and when the beam is symmetrically loaded, it occurs at the centre of span. Let T be another point on the curve. Draw 8H and TR normals to the curve at 8 and T meeting each other at H ; then H will be the centre of a circle which will coincide with the arc ST. Draw also 8K a tangent at 8 (fig. 5), meeting any horizontal line KL at K. Then ds = arc ST is the small difference between the values of s for the two points T and 8. 96 APPLIED MECHANICS. [chap. ds P dx = S' T' is the small difference of the values of x, the abscissae of T and S. = dx so far as value is concerned. = SH is called the radius of curvahire at aS ; -, its reciprocal, is called the curvature at S\ P and H is called the centre of curvature at ;S'. = the angle SKL is called the slope at S\ its greatest value is at one end, and is called the slope of the beam. = the difference of the slopes at T and S di arc ST ds dx ... _--^ = - = — sensibly. oil p p These angles are in circular measure. Let ya be the height of the top layer A A above, yi, the depth of the bottom layer BB below the neutral axis, and let { + ) yhe the distance of any layer CC \ ^qIq^ j the neutral axis. Fig. :,. The portions of these layers intercepted between the two radii ffS and HT before being strained were all equal to ds in length. Let (ds - a), (ds + |3), and (ds ± y) be their lengths respectively when strained ; then by similar triangles arc A A arc ST ds - a ds CHAP, v.] TRANSVERSE STRESS. 1 .-. ds - a = ds - y„ ^^^'"^ BR- = SR ' ds ' p' or ds or a= y„-; P ds + ft ds P + l/b p ' ds p > .-. ds -^ B ^ ds + ijb . — , P or P _/3_ ^ yb ds p , . arcCC arc^T ^^^^^ CR = SR ' or ds ± y ds p±y P ' .•. ds ± y = ds ± y . — , P or ds y = y - \ ••• p JL = 1.. ds p 97 Now the intensity of the longitudinal strain on the layer AA at the point A is augmentation of arc A A -a original length of arc A A ds \ t'- •/ Similarly the intensity of the longitudinal strain on the layer BB at the point ^ is -^ , and that on CC at any point ± y C is —7^ . Hence, from the above equation, we have Ts'-'ds'-l'-'-^"'-^''-'^' ""' ^^ ^°^^^- The intensity of the longitudinal strain on each layer of the point vjhere it crosses a section AB, is proportional to the distance of the layer from the neutral axis. Elements of the Stress at an Internal Point OF A Beam. To specify the stress at a point within a beam, it is necessary and sufficient to find the intensity and obliquity of the stress at that point upon any two planes through, it ; for convenience we take two planes at right angles to the plane of the paper. In fig. 6, (>X and dY are rectangular axes ; OX coincides with the neutral axis of the beam, and the origin is at the middle of its length. Let distances measured along OX to the j^.^^^ , along OY ^""^l^^^] ^ H 98 APPLIED MECHANICS. [chap. V. be \ P°^^ J:^® i Then c and - c will be the abscissae of the two (^negative J ends of the beam ; x^, x^, &c., the abscissae of the weights. Let H be any point in the beam, its coordinates being x and y ; that is, on the diagram, x is the distance of H to the right or left of 0, and y its distance above or below the neutral axis. Of the planes at right angles to the paper and passing through H, choose two, viz., AB and CD, vertical and horizontal. According to custom, CD may be called the ^j^a^i through H, and AB, the cross-section, or shortly the section ; further, it is called the section at x, meaning that the abscissa of every point in the section is x. H may be any of the points on the cross-section at the distance y from the neutral plane ; and as all these points are exactly under the same stress, it is unnecessary to say which of them H is ; or, in other words, it is not necessary to give the W,, Wi W3 A |C-,D I H y V V* *f u ^ '-v*- B ♦y; ^Neutral Axis of Cross Section Q Fig. 6. third or Z coordinate of H required to specify its distance from the plane of the paper. The trace of the neutral plane upon the cross-section is a horizontal straight line, dividing the cross-section into an upper and under portion, and this line is called the neutral axis of the cross-section. In order to specify the stress at H, we find the intensities and obliquities of the stresses at that point upon the two rectangular planes through it. The stresses upon these two planes are due to the strain upon the beam; that is, to the fact of its being bent at the section AB, leaving out of account the particular forces which actually cause the beam to be bent. We may suppose that the beam is bent by these particular forces, and surrounded by an envelope of some rigid material, and then that these particular forces are removed. CHAP, v.] THANSVERSE STRESS. 99 Upon this consideration it is evident that the stress on CD will have no normal comjjonent. Certainly, if one of the weights of the actual load happened to be at -4, then a normal stress on ri) would be directly transmitted to it; such a stress, however, being accidental is left out of the present investigation. Having remarked this about the plane CD, we leave its further con- sideration for some time, and give our attention to the section A£. At the point // on the section AB,yfe see there will be a normal component stress ; and we know, further, that it is a thrust, since the horizontal fibre through // is compressed. If, however, H be on the neutral axis of the cross-section, there is no normal component stress, since a horizontal fibre through such a point is unstrained. On the other hand, if H be below the neutral axis, that is, if its ordinate y be positive, there is a normal component stress ; and we know, further, that it would be a tension, since the horizontal fibre through such a point is stretched. Generally, then, at a point JI on a, section A B, there will be a normal component stress of opposite signs for points situated on opposite sides of the neutral axis of the cross- section ; and generally, also, there will be a tangential com- ponent stress acting in the same direction for all points on the section, as will afterwards be seen. To ascertain the stresses at points on the section AB ; suppose the beam cut into two portions at that section, and consider the equilibrium of one, say the left portion, as a rigid body (fig. 7). The forces acting on it are P, W^, Wz, shown on H 2 100 APPLIED MECHANICS. [CHAP. V. the figure by strong arrows, together with the stress upon the cut surface, shown by fine arrows. Let P be greater than the sum of W\ and Wn \ then the vertical components of the fine arrows must act downwards to conspire with W^ and W^ in balancing P. For some positions of the section, M\ + Wo + &c., the sum of the loads on the portion of the beam to the left of the section AB, may exceed P ; and on such sections all the vertical components of the fine arrows must act upwards ; while for other positions of the section, the sum of W^, W^, &c., may equal P, and on these sections the fine arrows will be normal to AB. Let ',' be the intensity of the stress on the section at the point H, and -y its obliquity. Eesolve the fine arrows into vertical and horizontal components — that is, resolve the stress at each point into a tangential and normal component stress. On figs. 8 and 9, p and q are the components of r shown separately, one set on each diagram ; pa and Pb are the values of ^ at the highest and lowest points of the cross-section, and the value of p at the neutral axis is zero ; qo is the value of q at the neutral axis, and, as will afterwards appear, the value of q at the highest and lowest points is zero. The equilibrium of these forces gives the three following conditions : — I. The algebraic sum of the arrows p is zero. II. The algebraic sum of the external forces (strong arrows) together with the arrows q is zero. III. The algebraic sum of the moments of all the forces about 0' is zero. Condition I. is equivalent to — The sum of the arrows p which are thrusts acting on the portion of the section above the neutral axis, equals the sum of the arrows p) which are tensions acting on the portion of the section below the neutral axis. Hence the resultant of all the arrows^ is a pair of equal and opposite forces not in one straight line ; or, in other words, a couple in the plane of the paper. Since this couple, by con- dition IIL, balances the sum of the moments of the external forces about (>', therefore the moment of the couple is equal to that sum, and acts in the opposite direction. Definitions. — Fx, the algebraic sum of the external forces acting on a portion of a beam included between one end and the cross section at x, and comprising the supporting force (if any) at that end and the loads on that portion, is called the shear in/f force at that cross-section. CHAP, v.] TRANSVERSE STRESS. 101 Ml, the moment of these external forces about any point on the cross- section at .r, is called the bending moment at that cross-section. F„ the amount of the tangential component stress on the cross-section at x, is called the resistance to shearing of that cross-section. Mj, the couple which is the moment of the total stress on the cross-section at x about any point of the cross-section, or, Iw, p'^ r 1^ — ■(C-X) ■? :....{X.-X) , Result ant of tensions JletulUnt of thrusts M the moment of couple is th» Resultant of nil arrows *j^ F Is the ^'Resultant of all arrows g. Fig. 10. what is the same thing, the moment of the normal component stress on the cross-section about any point in the plane of the paper, is called the moment of resistance to hending of that cross- section. On fig. 10 these four quantities are — F, = P - W,- W.. M, = P[c-x)- W,{x, -x)- W,'x, - x). F ^ = Eesultant of arrows q. M,= Resultant of the pair of equal and opposite forces, one of which is the resultant of all the thrusts p, the other of all the tensions p. The conditions of equilibrium for any portion of the beam comprised between one end anrl a cross-section at x are — F. = F.. (1) M, = M,. (2) Since F^ and 3fx are calculated from external forces alone. 102 APPLIED MECHANICS. [CHAP. V, they are independent of the size or form of the cross-section, and depend only upon the amount and distribution of the load ; and, when calculated at a sufficient number of cross-sections, they form the data from which to design the beam. On the other hand, F^ and M^ depend only upon the size and form of the cross- section at x, and upon the material of which the beam is to be made. Having fixed upon a material for which we know the working strengths to resist tension, thrust, and shear- ing, the two equations above enable us to design the different dimensions of the cross-section at x in any required form. The form to be adopted in any particular case depends in some degree upon the shapes in which the material is naturally obtained, or in which it can be manufactured cheaply ; and when it can with equal facility be manufactured in several forms, that one is to be preferred which takes greatest advan- tage of any difference in the resistance of the material to the various kinds of stress ; since, by doing so, we require the least quantity of material. The form of cross-section chosen must be suitable for the particular nature of the load, and locality of the beam. Having thus designed the cross-sections at a suffi- cient number of places, we are said to have designed the beam so as to have sufficient strength to resist the given loads. Besides the above qualities of suitability, cheapness, and economy of material, a beam must also have sufficient stiffness. We will, further on, derive equations to enable us to find the form of economy mentioned above, and also to select the ratio of the dimensions of the cross-section to fulfil the condition of stiffness. The cross-section at x either being given or having been designed, equations d) and (2) enable us to calculate the elements of the stress at any internal point ff. The Cantilever. A beam may be supported by one prop placed exactly below the centre of gravity of the load, as shown on fig. 11. It is evident from the definitions that, in this case, the shearing force and bending moment are maxima at the point of support ; because the external forces upon one of the portions into which the section through this point divides the beam, say upon the left portion, are the loads on that portion, and they all act in one direction. The bending moment at that section is the arithmrtical sum of cacli of these loads into its leverage ; and for any other section to the left, the number of loads to be reckoned in calculating the bending moment may be fewer, and CHAP, v.] TRANSVKKSE STRESS. 103 at the same time all the leverages are shorter. In considering the left portion alone, it is unnecessary to draw the other ; and instead, the left portion may be supposed, as on the figure, to terminate at its right extremity in a vertical plane ; this plane is supposed to give the necessary resistance to balance the shearing force and bending moment at the point of support. T atfliec" end Fig. 11. Such a piece is called a Cantilever; as shown on fig. 11, the right is itsjixed and the left its/ree end ; its length is c. When the cantilever is strained, that is bent, the tangent to the axis at the fixed end remains horizontal ; the upper layer is stretched while the lower is compressed, and the lettering will be accord- ingly. The deflection v of the cantilever is the difference of level of the two ends, and its slope i is the inclination of the tangent to the axis at its free end with the horizontal. We shall now proceed to calculate the bending moments on, and construct bending moment diagrams for, beams and canti- levers under various loadings ; in such a diagram, the horizontal and vertical dimensions are respectively the span and bending moments at the different points of the span drawn to scale. Each of these diagrams requires two scales — a scale, say, of feet for horizontal, and a scale such as foot-tons fcr vertical, measure- ments ; there may be required, also, a scale, say, of tons for loads, if the loads are drawn to scale. Such diagrams must be drawn upon a large scale if intended to be used as graphical solutions, in which case only the construction requires to be known. The diagrams of this treatise are too small for such purposes, but they serve to show clearly the constructions, and are principally useful as maps upon which to note the analytical results. In every case, these diagrams are constructed so that one of their 104 APPLIED MECHANICS. [chap. VI. boundaries is straight — is, in fact, the span. This is a matter of importance, as the diagram so constructed assumes a suitable form for a practical purpose which will be afterwards pointed out. One boundary of such diagrams is generally a parabolic right segment in certain simple positions ; and we will now give a short chapter on the equations to, and the construction of, this curve. CHAPTER VI. THE PARABOLIC SEGMENT. THE parabola. On tig. 1, Xand Tare two rectangidar axes passing through A, any point of reference. Let distances measured from A to the ( left ) J J. ^^ . p ^ (downwards) ^{ positive | ( right j ' and from the axis of X upwards be negative Fig. 1. The points Pi, P., &c., have their ordinates IPj, 2P2, &c., proportional to their abscissae Al, A2, &c. ; that is, if JTand Y be the coordinates of any point P, then Y = mX, where /» is any number, whole or fractional, and the locus of P is a straight line passing through A. In the figure m = \, and for any point P, X, and Y are both positive or both negative. The coordinates of points Pi, P2, &c., are marked ; e.g. for P3, X=3, and Y= f. On fig. 2, Pi, F2, &c., are points coiTesponding to the points maiked similarly on fig. 1, bnt in this case the points P], Pi, &(;., have their ordinales \P\, 'IPz, &c., proportional to the squares of their ahscissse ; that is 'I'his is the principal equal inn to the pMrabola, and the quantity m is the itwdulus. The locus of this equation, P, is the parabola ; and it is altogether on one side of the axis of X, since, although X, the abscissa of any point P, is positive or negative, the quantity X"^ is always positive. CHAP. VI.] THE PARABOLIC SEGMENT, 105 In the figure m = ^, and for any point 2\ while X may he positive or negative, J" is always positive. The coordinates of tlie points Pi, T2, &c., arc marked; e.ff. for F3, X = 3, and Y = 2^. The point A is called the vertex, and the line A Y + X Fig. 2. the axis of the parabola ; the curve is symmetrical about this axis. The points P\, P-2, &c.. thus found are points on the curve ; and if a sufficient number of such points be found and a fair curve drawn through them, the curve will be sensibly a parabola. Fig. 3. Whenever ■\e have an equation of the above form, we conclude that the locus -a parabola vbose vertex is at the origin, and whose axis lies along the axis of Y. 106 applied mechanics. [chap. vi. Parabolic Segment. Any line as BC, fig. 3, meeting the curve in two points, and drawn parallel to the axis of X, may he called a right chord ; and the tigure enclosed between thi& chord and the curve may he called a parabolic right segment. This chord is the base of thf! segment, and AO, the distance of the vertex A from the hase, is the height of the segment. The following is a convenient construction for drawing a parabolic right segment of a given height and on a given base £C, fig. 3. Plot A at the required height above the nii<idle of the base, and complete the rectangle BOAS. Let it he required to construct accurately twelve points at equal horizontal intervals. Divide AH and SJi each into six equal intervals, and number them as on the figure. From each number on AH draw vertical lines, and from the point A draw a. ray to each number on HB. Then Pi, the point of intersection of the vertical through 1 and tlie ray Al, is a point on the parabola , so also is F2, the point of intersection of the vertical through 2 and the ray A2 ; &c. It is evident that, for instance, H/'i, is piopoitional to the square of AD ; for, calling IPi unity, then JjJ = 4, and T)Pi = 4 DJ = 16 = 4-, while the abscissae of the points Pi and P4 are proportional to 1 and 4 ; similarly with the other points. Hence one-fourth is the common ratio of the depth of each point below AH to the square of its distance laterally from A 0. We will return to the subject of drawing a parabolic segment, and will now show how to draw a TANGENT AT ANY I'OINT OF A PARABOLA, say P4 ; on Pi A and P4D construct a parallelogram and draw its diagonal P4£: this line will he the tangent required. P4^ is the same fraction of F^D that P4L is of Pi A, and ZP3 is parallel to P4-fir. If KP3 were parallel to P4i, then would KL be a parallelogram similar to DA, and so P3 would be on the diagonal PiE. But since KP3 is not parallel to P4X, but coTiverges towards it, the point P3 where EP3 meets LG lies to the right of the line PiE. Produce ^P4 through P4. PjiV is tlie same fraction of /'il) that PiM is of P4^ ; if NPt were parallel to PiM, then P5 would he on the diagonal PP4 produced ; but, since NP^ diverges from PiM, itie point Ps where iVPs meets MP5 lies t<> the right of the line PiE produced through P4. In the same manner every point on the curve, either above or below P4, lies to the right of I'iE; that is, PiJE is a tangent at the jioint P4. Now the diagonal PfE bisects the other diagonal AT) in P, and the most convenient way of dra>ving a tangent at anj- point P4 remote from the apex is to project P4 on the hoiizontal tlirough the apex ^ ; bisect AD in F, and draw P4P; this is the tangent required. It is readily shown that PiE is parallel to any chord as PsPa with its ends equidistant horizontally from P4. To plot the locus of an equation of the form F= »«X* ; in other words, to draw the parabola whose modulus is m. For instance, let ni = ^. Draw any lii\e liC parallel to the axis of X, fig. 3, as hase; lay off OA equal to ^05-; then dmw the segment by plotting accurately a number of points Pi P2, &c., by the con- struction already given, and draw a fair curve through them. Usually we fix only a few points on the curve accurately, and from these the rest of the curve is sketched in. By making the number of these points sufficiently great, we can. draw the curve as accurately as we please. A parabolic segment might be constructed on, and cut out of, a piece of cardboard, and used exactly like a set square. The parabola could then be quickly drawn on our diagrams thus : — If the apex A be given, place the parallel rollers to the axis of X, shift the rollers, slide the cardboard segment CHAP. VI.] THE PARABOLIC SEGMENT. 107 along them till the apex is at A, and draw the curve. Again, suppose we are given (fig. 3) the axis OJi and a point B on the curve, and that we wish to draw the curve ; place the rollers at right angles to the given axis OE, slide the cardboard segment with its apex on this axis till the curved edge passes through B, and then draw the curve. Instead of cardboard we may use a parabolic segment cut out of a slip of pear-tree or brass, and with one such segment we can, if we choose, draw all parabolas. Thus, suppose our pear-tree segment to have the modulus |, and that we require to draw a segment whose modulus is 1. We may choose, as suitable for horizontals, a scale of 10 parts to an inch ; lay off the base with this scale and draw the curve with the pear-tree segment ; if we now measure the verticals on the scale, we find for every point Y = ^X"^. If we draw a new scale of 40 parts to an inch for vertical measurements, then for every point on the curve we will have, as required, F= 1 X X^ = X\ In like manner the curve drawn by this pear-tree segment may represent any jiarabola, if the verticals be measured on a suitable scale ; this is similar to the common practice of exaggerating the vertical scale for sections. All the diagrams which immediately follow may be very conveniently drawn with one such seijment, since it is much easier to draw an additional scale than to construct a new parabola; for example, suppose the segment, fig. 90, so drawn, and that we wish it to represent the parabola Y = \X'^. Using a horizontal scale of four parts to an inch, we have by measurement OB = 6, and OA or HB = 9 ; that is, HB = {OB' ; but we wish HB to measure ^OB"-, that is 12. It is only neces- sary then to divide HB into 12 equal parts, and lay off a scale of such parts to be used for verticals ; this scale is evidently that of 3 inches divided into 16 equal parts. If the new scale be only slightly finer than the old, then the old scale plotted sloping to the vertical at an angle can be used by ruling horizontals across to it. Equations to the Parabola. In figs. 4 and 5, lei ihe middle of the span be the origin of rectangular coordinates, the span BC being taken as the axis of X ; let distances measured from to the | ^ ■,. \ , and from the axis of A^ | "P^^*" ^ i be | ' „^„,;,.^ } • I right I ' I downwards ) ' ( negative ) In fig. 4, the axis of the curve passes through 0, whereas in fig. 6 it passes to 108 APPLIED MECHANICS. [chap. VI. one side. Let x, y be the coordinates of any point P\ and for the apex let x = K, and y = H. In fig. i K ^ 0, since Kis on the axis of Y; and in order to find the equation to the curve with the origin at 0, we have r = - mX'- Instead of X and Y, we substitute their values, thus (origin at A). H = — mx^ ; y In what immediately follows, let Co, Ci, and Ca be co«s<flM< quantities; then, if we have an equation of tlie form y = Co + (hx^, consisting of a term not con- taining X. and a term in x"^, we conclu<ie ihat the locus is a i>arabola, with its axis veitical, its apex on the axis of Fand at the distance Co from the origin ; and that the modulus of the parabola is C2, the coefficient of x"-. so that the principal equa- tion is y = C-iX'^. For example, suppose we wish to plot a number of points above the span BC, such that the coordinates of each point may fulfil the equation y = ;i (36 - a:2) = 9 - \x'- ; we conclude that all the points are on a parabola whose axis is the vertical through 0; that the apex A is at the height IT = 9, the term not containing x ; that the' modulus is — ^, the coefficient of x- ; and tlierefore that the principal equation is r=-iX2; .-. S=9 = ^0D\ or OD = 6. Fig. 4. To draw the locus, lay off OA = 9, OD = OE = 6, aud con- struct points on the curve, as already shown on fig. 3 ; or more quickly, by means of the pear-tree segment we may draw a curve through the points D and E just found, aud construct a scale for verticals upon which OA measures 9. Again on fig. o, tlie equation to the curve with the oriain at is derived from the principal equation by ."substituting x - K nr\A y - H for X and Y, thus y ~ H= -m {x- A' )■-, or y = H~ mx- + 2mKx - m A'*, or y ^{H - niK^) + Im Kx - mx^, which is an equation of the form ;/ = Co + Cix + dx-, consisting of a term not CHAP. VI.] THE PARABOLIC SEGMENT. 109 containing x, a term in x, iind a term in x- : and we conclude that the locus ia a parabola, with its axis vertical. In this equation, if C1C2 is { ** *" '^^ i , the •^ ' H » ' i J neg:it:ve ) ' apex is to the | .^^ I of the axis of T; if Co - — r is | \- ^ { , the apex is I above ) . . f ,- { 1 > the axis of A. [ under I To find the value of A' and of H, arrange the equation thus: — y = m (2K- X)x -^ H— mlf^. Now when y = JT, its value is a maximum, and the corresponding value of .»• is K; so that to ascertain A', we huve onl\' to tind that value of x which makes y = m ('2K — x) x + S — mK^ = maximum. This is a maximum when {IK- a;) ^is a maximum, since the rest of the expression is constant for all values of x. Suppose that 2 A is the length of a line, which is divided at a point into two segments ; let x be t)ie length of one of them ; then 'IK - a: is the lengtli of the other, and (2A— x)x is the rectangle contained by the two. "We kno.v by Euclid that this rectangle is greatest when the segments are equal, each being half of the line 2A'. So that, wlien x = K, \^1K — x)x is a maximum, m{2K — x)x + H - iiiK'^ is also a maximum, aiid y = S. Fig. 5. For example, suppose we wish to plot a numher of points whose coordiiuites- X, y fulfil the equation y = i (4 - a:)a: + 3 = 3 + a; - ix2 From the form of this equation, we conclude tliat all the points are on a parabola, whose axis is vertical and to the left of ; that the ap»x A is above BC ; that the modulus is - \, the coefficient of x"^, and therefore that the principal equation is The distance K at which the axis lies to the left of is found thus : — the value of X which makes y = ^ (4 — x)x + 3 = a maximum, is that in which \ — r = x, or x = 2 ; that is, A'= 2. Again, to find the height of the segment ; when x = K, y^H, and IT = i (4 - 2) 2 4- 3 = 4. Substituting in the principal equation, we have i = ir= \SI)- ; • therefore haK-base = 6'i) = + 4. 110 APPLIED MECHANICS. [chap. VI. To draw the locus : — lay off OS = 2, and draw a vertical through S ; from JS lay off SD = SB = 4, and SA = 4 ; and construct points on the curve, as in fig. 3 ; or more quickly by means of the pear-tree segment, we may draw a curve through the points I) and U just found, and construct a scale for verticals upon which SA measures 4. Theorem A. — Fig. 6. Tlie quadrant of a parabola and a riglit-angled triangle stand on a common liorizontal base, witli the right angle of each at one end (the right end in the figure) ; let a be the lieight of the apex of the parabola, and b ihe height of the vertex of the triangle, above that end. If, ai each point cf the base, the ordinate of the parabola be added to that of the hypotenuse of the triangle, and a. new curve be plotted ; it will be the sawj^ parabola with its axis still vertical, and having its apex shifted beyond that end (the right in the figure) of the base .ibove which the apex was, and the curve will still pass through the other end of the base. In like manner, if the ordinate of the parabola be deducted from that of the liypotenuse, a similar result is obtained ; the apex of the new figure, however, is filiifted to the other side. The horizontal distance through which the apex shifts is (oC=£base) of -4 Ord Fis the same fraction of the base that the height of the vertex of the triangle is of twice the lieight of the apex of the parabola, ; or if o be the lateral distance through which the apex shifts, then a = — X base of the quadrant. 2a The addition does not afi'ect the coefficient of x"^, so that the modulus is ■unchanged. In short, the algebraic addition of the ordinates of the straight slope makes the parabolic base segment shift on the rollers from the position on tig. 4 to that on fig. 5. Theorem Ji. — If to the ordinates of a paiahola, axis vertical and ape.x to left or right of origin, the ordinates of another parabola, with axis vertical and passing through the origin, be added ; then the new locus is a parabt)la, its modulus is the sum of their moduli, and its axis is vertical ; its npex lies on the same side of the origin as the apex of the first parabola ; and the abscissa of its apex is the same fraction of the abscissa ol the first parabola's apex that the modulus of the first parabola is of the sum of the moduli. (Fig. 7.) CHAP. VI.] TUE I'AHABOLIC SEGMENT. Ill For the parabolas whose apexes are Ai and Ao, respectively, y'l = yo + yi = -fiTfl - "»»^- + « (iTi - mKi^) + 2inKix - mx', or y 1 = (flo + -fiTi - mi'r') + {mo + ♦«) ( "'-"' ^ - z\ x, (1) which is a parabola of modulus (>w„ + m), its axis is vertical, its apex lies on the same side of origin as the apex of the tirst parabola ; and if K'l is the value of x ... , / -ImKi \ which makes | — x \ x greatest, then \}>io + m / ir\ = i)li) + in A', (2) Q. E. T). Fig. 7. Theorem C. — If to the ordinates of a locus consisting of two parabolas, with axes vertical, wiih a common modulus m, and intersecting at F, there be added the ordinates of a parabola, axis vertical, apex above origin, and modulus nia ; then the new locus consists of a pair of parabolas with axes vertical, having the common modulus (;«o + m), and intersecting at F^ on the same vertical as F. (Fig. 7.) For the parabolas whose apexes are Ai and A2, respectively, yi = {H\ - niKi'^) + 2mKix - mx'^ ; 1/2 = (If^ - mK^'^) + iniEiX - mx'i. If f be the abscissa of the point of intersection, then where yi = yz, x = f ; subtracting, we have = (.S"i - Hz) - m(Ei- - Ki-) + 2m[Ki - K^Y, which gives the value of/. ' (3) 112 APPLIED MECHANICS. [chap. VI. Again, by the previous tlieorem, equation (1), y 1 = [Ho ^ Si - mKr) + 2mK\X - (wq + m)x- ; similarly y'2 = (Ho + -ff2 - mKi^) + 2mE2X - (wo + tn^x^. Let/A be the abscissa of the point of intersection, then where y'^\ = y^i, x =/* ; subtracting. Me have = (£"i - Hi) - m{Ki^ - jfo^) + 2m{Ki - Ki)/^ , (4) hence r =/. Q. E. B. (5) Theorem D. — If two parabolas with axes vertical and a common modulus »« intersect at a point, then the horizontal projection of any double chord drawn through that point is constant and is equal to the double horizontal chord drawn through it. And conversely, if a line whose horizontal projection is equiil to the double horizontal chord through the point of intersection be placed with one extremity on each parabola, either the line or the line produced passes through the point of intersection. (Fig. 8.) vn and vb are constant. Fig. 8. The equation to the parabola (fig. 5) is y = (H- mK^) + 2mEx - mx^ ; choosing the origin at the point of intersection, then Hi = mKi\ and Hi = mK->^ ; hence [Hi - )nKi^) = (Hi - mEz') = ; and considering Ei and E2 positive, we have yi = 2mEiX - mx^, (6) 2/2 = - 2mE2X - mx-. (7) The equation to any straight line thiough may be written .'/ = M-^- ; (8) equating (6) and (8), we find for Fthe point of intersection x = 2E\ ; sinii- m larly for N, x = - 2Ei - ^. Hence vn = 2{Ei >t E->^ = ST. (9) Further, if VN be such that vn = 2 (E\ + Ei), and T moves on parabola No. 1, while iV moves on parabola No. 2, it is evident that J'iV always passes through 0. When V arrives at 0, then VN is a tangent at to parabola No. 1; and if V moves to F", then NV produced passes through 0. Q, E. D. CHAP, VI,] THE PARABOLIC SEGMENT. 113 Theorem E. — While the liouble chord 7'OiV turns, if for each position there be taken on it ti point li whose horizontal projection b divides rTt in a constant ratio ; th»'n this point traces out a parabdla whose axis is vertical, which has the same modulus m as the parabolas Nos. 1 and 2, and which passes through the point of intersection 0. (Fig- 8.) X, tj being the coordinates of F; let X, Y be the coordinates of B \ and vb = C a, constant ; then x = X+ C, and Y = -^ X. Now 1/ = 2mKix - mx^, or — = m {2Ki - x) ; •' X therefore Y=m {2Ki - C - X) X, (10) the locus of B ; and it is a parabola with axis vertical of modulus m, and passing through 0. The abscissa of the apex A3 is equal to JS"] - iC. Q. E. I>. O^pe* A pro «»Vy y 14 30 .^ 12 18 &) A 6 X Fig, 9. Fig. 10, Degradation of a Locus. Definition. — If a locus be drawn whose ordinates are proportional to the square roots of those of a given locus, the new locus is the given locus degraded. 2'heorem F. — Fig. 9. If a locus which is a straight slope be degraded, the new locus is a parabola with its axis horizontal, and its apex at the point where the locus crosses the horizontal base. For the straight slope, y = ax, if we take the origin at the point where it cuts the base ; if A be the corresponding ordinate of the new locus, h^ oc y, or ax ; that is, x oc h^, a parabola with axis parallel to OX, and apex at the origin. Theorem G. — Fis;. 10. If a locus, which is a parabola with axis vertical and apex on the base, be degraded, the new locus is a straight slope cro.'ssing the base at the apex. For a parabola with origin at apex, y = mx- ; but A- oc y; therefore h oc \/ mr, a straight line passing through the oiigin. 114 APPLIED MECHANICS. [chap. VI. Theorem H. — Fig. 11. If a locus which is a parabolic segment be degraded, the new locus is an ellipse with its centre at the middle of the base, the base being a major or minor diameter. O \0 <iO 2,0 ^0 ^0 ^oX 2- OA ' 1 "* Fig. 11. Taking the origin at the centre of base, we have for the equation to a parabola, y = m{c'^ — x^), where 2c is the base. Let h' = fih, where /x is a quantity such that h'^ = y; then since h^ oc y, we have — = c- — x'^ /i'2 + a;2 ^2^2 = 1; the central equation to an ellipse whose semidiameters are c and . The ratio of the semidiameters depends on m. On fig. 11, the ordinates of the degraded figure have been multiplied by 4 ; thus OA is 24, or four times the square root of 36. So that BAF i?, the degraded figure if the vertical scale be four times as coarse as the horizontal. If the multi- plier were 10, then BAF would be a semicircle. Hence a figure consisting entirely of arcs of the same parabolic segment can, when degraded, be represented by a series of arcs of circles, provided a suitable vertical scale be constructed. The false semi-ellipse, fig 11, is readily struck from five centres. The first at a distance rz = 60- -^ 24 = 150 below A. With this radius AD is swept out, and BC with n = 24* H- 60 = 10 nearly. While DC is drawn from the centre E defined as shown on the diagram. CHAP. VII,] BENDING MOMENTS AND SHEARING FORCES. 115 CHAPTER VII. BENDING MOMENTS, AND SHEARING FORCES FOR FIXED LOADS, Definition. — The Beriding Moment at any cross-section of a beam, or, as we may more conveniently say, at any point of the span, is — The sum of the moments about that point of all the external forces, acting upon the portion of the span on either side of the point. For convenience in the case of beams supported at both ends, we calculate this bending moment from the forces acting upon the portion to the left of the point. These forces comprise {vide figs. 8 and 10, Ch. Y) the supporting force P acting upwards at the left end, and the loads acting downwards between that end and the point. Taking the centre of span as origin, the abscissa of P is c ; and, if x be the abscissa of the point about which moments are taken, then (c - x) is the leverage of P, and P tends to break the beam at the point by bending the left portion upwards with a moment P{c - x) ; the abscissa of ir, is x^, its leverage is {x^, - x), and it tends to break the beam at the point by bending it dovmwards with a moment Wx{xi - x) ; all the other loads to the left of the point have an effect on the beam similar to that of W^ ; and since, in this case, the left portion of the beam is bent upwards at the point, the moment P{c - x) exceeds the sum of all these moments W^{xx - x) + W^ixz - x), &c. ; and if Mj: represent the bending moment at any point x, then M^ = P{c - x) - JV,{x, - x) - W,(x2 - x), &c, all the loads on the portion of the beam to the left of the point being taken into account, A Bending Moment Diagram is a figure having a horizontal straight line for its base, equal in length to the span on a scale for horizontals which should accompany the diagram. Above this base is an outline or locus consisting of a curve, a polygon with straight sides, or a polygon with curved sides, and such that the height of any point on the outline gives the bending moment at the point of the span over which it stands, measured on a scale (say) of ft.-lbs. or of ft.-tons for verticals which also should accompany the drawing. It is evident that this outline always meets the horizontal base at both ends, since the bending i2 116 APPLIED MECHANICS. [CHAP. VII. moment at each end is zero. It will be seen that x and Mjc are respectively the abscissa and ordinate of a point on this locus or outline ; an equation between those two quantities, such that, when you substitute into it any value for r, it gives you the corresponding value of Mx, is called the equation to the lending moment. An approximate method of drawing a bending moment diagram is, to calculate the bending moments at a number of points of the span, say at equal short intervals ; plot these to scale, and then join the tops of the verticals with straight lines, or draw through these points a fair curve. Such a diagram will give the bending moments accurately at the points which were plotted, and approximately at intermediate points ; and its principal use is to mark the calculated results thereon. On the other hand, if upon investigation we find the locus to be of a form which we can draw readily, then, drawing the diagram first, we may afterwards, by measurement from it, find the value of the bending moment at any point of tlie span. Such a method of proceeding is called a graphical solution. The MAXIMUM BENDING MOMENT is that value of the bending moment, than which no other value is greater ; if this value be at one particular point of the span, that point is called the jwint of maximum bending moment ; sometimes this value ex- tends iDetween two points of the span, in which case any point intermediate may be so called. The determination of the maximum bending moment, and of the point at which it occurs, is of great importance. A graphical method is pecu- liarly successful in giving these, as we know or readily find the highest point on the diagram ; the height or ordinate of this point is, of course, the maximum bending moment, and its abscissa is the point at which it occurs. In all the cases which follow, that point on the diagram is either the angular point or side of a polygon, or the apex of a parabolic right segment. Definition. — The Shearing Force at any cross-section of a beam, or, as we may more conveniently say, at any point of the span, is — 2'he algebraic sum of the external forces acting upon the portion of the beam to either side of the point. We will denote this shearing force by Fx, and calculate its amount systematically from the forces acting on the portion of the beam to the left of the point; thus for a beam Fx = P- W, - W, - &c., (1) the external forces on the left portion. CHAP. VII.] BENDING ilOMENTS AND SHEARING FORCES. 117 Similarly, for a cantilever t\ = - W, - W, - &e., (2) On t!ie beam then, i^^. is P , • | according as P ' I negative) * IS 1^ , I than the sum of the weights to the left of the section at .r. The beam may be divided at a certain point into two segments, such that for every point in the I • , , o ' J f jnght) segment, the shearing force is j^ ^ • I ; at this point the shearing force changes sign. The shearing force on either segment may be considered as positive ; for convenience we have chosen as just stated. Having fixed the sign thus, on a cantilever as shown on the diagrams, that is with its fixed end to the right, the shearing force is everywhere negative ; and in considerations regarding beams atid cantilevers, it is necessary to take this into account ; when, however, the cantilever alone is considered, it will be better to reckon the shearing forces as positive. It may be observed that the symbolical expression for F^ does not de]>end for its form upon the position of the origin ; in this respect it is unlike J/^ A Shearing Force Diagram is a figure having a horizontal base representing the span on a convenient scale, and an outline or locus. For a cantilever this locus lies wholly under the base ; for a beam, it lies above the base for the segment to the left, under the base for the segment to the right, and crosses the base at an intermediate point. The heiglit of any point on the locus, measured on a scale for forces, say tons or lbs., gives the shearing force at the point of the span over which it stands. Both these scales should accompany the drawing. Mcu:imum Shearing Force. — On a cantilever it is evident that the greatest value is at the fixed end. On a beam there are two values each greater than the others near it, one at the left end and positive, one at the right and negative ; the value of the greater of these is the greatest for the whole span. This locus, in the diagrams for all the cases of fixed loads which we consider, consists of straight lines. For a portion of the span between any two adjacent loads, the straight line is evidently horizontal ; for portions uniformly loaded it slopes at 118 APPLIED MECHANICS. [CHAP. VIL a rate given by the number which indicates the intensity of the load; thus if w lbs. per foot be the intensity of the uniform load, then w vertical to one horizontal is the slope. Where a weight is concentrated at a point, the line is vertical ; that is, at such a point the locus makes a sudden change of level, the change being equal to the weight. On a cantilever, the shearing force at the fixed end is equal to the load, and at the free end it is zero ; we can readily draw the straight lines as above described to suit the nature of the load, and so complete the diagram. On a beam the shearing force at the left end is P, at the right end it is - Q, and at some intermediate point the locus intersects the base and changes sign ; the manner of fixing the position of this point will be explained immediately. The whole locus is then readily completed by drawing the lines as above to suit the nature of the load. Theorem. — The Shearing Force at any point of a beam or cantilever is the rate of variation of the bending moment at that point; and on the beam the shearing force changes sign at the point of maximum bending moment. As before i^. = P-S(IF), where ^{W) means the sum of the loads to the left of x ; and M, = P{c-x)- S(ir..r), where '^{W.x) means the sum of the products got by multiplying each load to the left of x by its leverage about x. Hence if we suppose F^ to be positive, and take the bending moment at any interval d further to the right, the second bending moment will exceed the first by F^ . d, if there be no load on the portion d ; and by Fx . d, minus the load over the portion d into its leverage about x, if there be a load on the portion d. Since the leverage of the load which is on the portion d is less than d, we can by taking d small enough make this product as small as we please ; that is, i^^ • ^ is the change of the bending moment in passing from x through a small interval d. Now the rate of change of M^ means the change in passing from ./■ through an unit interval, say one foot, if the change continued uniform throughout that interval, and at the same rate as at x; in otlier words, the change in Mj^ for d reckoned CHAP. VII.] BENDING MOMENTS AND SHEARING FORCES. 119 equal to unity, without taking into consideration any additional loads which may be over that interval ; hence the rate of change of M^ = Fx ; also, as we pass from the left to the right end of the span, '^ ^^ ^^ {neglti^e}' ^^^ ^« { rc^eashl ) ' ^"^^ ^^ ^^^^ P°^^ where Fx changes sign M^ is a maximum. For fixed loads tlie shearing force diagram greatly assists the construction of the bending moment diagram, besides readily locating the maximum bending moment. It is convenient to consider and draw these two diagrams conjointly. Beam with Unequal Weights at Irregular Intervals. Let a beam 42 feet span (fig. 1) support weights, viz., W, = 5, W., = 5, IF3 =11, W,= 12, and W^ = 9 tons at points whose abscissce, reckoned from the centre, are x-^ = 20, X2 = 15, .rj = 7, Xi = - 3, and x^ = - 10 feet. This is the problem of example 8, fig. 3, Ch. V, and we have P = 24 tons, at c = 21 feet, and Q = 18 tons, at - c = - 21 feet. To calculate the bending moment at any point x, x = - S, for instance, we may take the forces upon the right portion, and i/_3 = ^ X 18 - 1^5 X 7 = 18 X 18 - 9 X 7 = 261 ft.-tons. We will now calculate the bending moments at the points where the weights stand, in each case systematically from the forces upon the left portion. Ft.-tons J/,. - P{c-x,); il/20 = 24 X 1 = 24 M^^ = P(c-X2)- W, (a-i - x^) ; i!/,5 = 24 X 6 - 5 X 5 = . . . . . . .119 M^z = P{c-X3) - }F, (x, - X3) - W2 (X2 - x^) ; iJ/, = 24 X 14 - 5 X 13 - 5 X 8 = 231 Jil.^ = P{c~x,)- W,{x,^x,)- W^(x,-x,)- W,{x,-x,); iT/.3 = 24 X 24 - .5 X 23 - 5 X 18 - 11 X 10 = . . . 261 ^Z5 = P{c - X5) W^ (Xi - X,) - Wi {x2 - x^) - &c. ; M_,, = 24 X 31 - 5 X 30 - 5 X 25 - 11 X 17 - 12 X 7 = 198 120 APPLIED MECHANICS. [chap. VII. In the interval between W3 and W^, consider any two sections A and B, the latter being situated further to the right; let the distance between them be d. The leverage of F is greater at B than at A by the quantity d, and the wpward bending moment of P is greater at B than at A by the quantity F . d ; again, the leverages of W^, W^, and W^, respectively, are greater at B than at A by the quantity d, and the dovmward bending moment due to W^, W^, and IFj "V ;-d-< O <V u-d-J ** W=12 W=9fon» 1 18= Q ^ W'9 «='Q Forces on Right Portion of Span W,= 5 |W.= 6 24= P 24 = P .... I Sect/on af) ■ r I-,. U—.^X-XJ^13-- ^^*T^I '^"'■::ri (C-X) = 24-- 24-Pri( L.2..6'. (X!^X)^30 — tion at) ,= -70 i Fig. I. is greater at B than at A by the quantity (Wi + W^ + W3) d. Hence M^ is to be derived from M^ by adding F . d and sub- tracting ( I'T, + W2 + Ws) d ; in our example F>( W^ + W^ + W^), so that the quantity to be added exceeds the quantity to be subtracted ; that is Ms > M,, , by (F-W,- W, - W,) d, a quantity proportional to d. CHA?. VII. J BENDING MOMENTS AND SlIEAUING FORCES. 121 Again, for any two points C and D in the interval between W^ and Wi, Md is to be derived from M c by adding P . d and subtracting ( W^ + W. + W^ + W^) d ; in this case, however, P < {Wi + W-i + Wi + }Vi\ so that the quantity to be added is smaller than the quantity to be subtracted ; that is Jin < Mc, by ( Wi + IF, + TV, + IV, - P) d, a quantity proportional to d. In both cases, as we pass towards the right from one point to another in the interval between two weights, the change in the bending moment is uniform, and is proportional to the I"' I"' I Fiof. 2. horizontal distance passed over — uniformly increasing or uni- formly decreasing according as the supporting force P is greater or less than the sum of all the weights to the left of the points being considered ; and if for some interval, P is equal to the sum of the weights to the left, the bending moment at points in that interval is constant. Bending Moment Diagram. — With a scale of feet for hori- zontals, lay off the span, fig. 2, and plot the positions of the loads ; at each of these points erect a vertical equal to the bending moment thereat upon a suitable scale of, say, ft.-tons 122 APPLIED MECHANICS. [CHAP. VII. for verticals ; join the tops of these ordinates by straight lines, and join each end of span to the top of the nearest ordinate. The lines just drawn give ordinates which vary uniformly in each interval, so that at any point whatever the ordinate gives the bending moment at that point in ft.-tons when measured on the vertical scale. The bending moment at each load having been calculated analytically, a diagram thus constructed is a graphical solution for every other point. Maximum Bending Jloment. — In our example, since P > Wi, the line AB slopes ujj towards the right, that is, the bending moment increases in this interval ; BC also slopes icp towards the right since P > (W^ + W^), and Cl> slopes up towards the right since F > {Wi + W2 + W3) ; BU slopes doivn towards the right since P < ( ^1 + W2 + JV3 + Wt), and lastly, BF slopes down towards the right since F < {Wi + W2 + W^ -1- W^ + W^). That is, beginning at the left end and passing towards the right, we find that the sides of the polygon slope , ^ towards the right, while P is p . | than the sum of the weights we have passed. For some loads there is one interval where P is equal to the sum of the weights passed, and in such cases the side of the polygon above that interval is horizontal. Evidently the highest ordinate is that of the angle of the polygon made by the last side which slopes up to the right, either with the first side which slopes down, or with the horizontal side if there is one. On fig. 2, that angle is D, and its ordinate is the maxi- m\mi bending moment ; so that, in our example, the maximum bending moment occurs at W^, that is at the point a; = - 3, and its value is 261 ft.-tons. If one side of the polygon be hori- zontal, the ordinate to any point of this line is constant, and is a maximum. The Point of Maximimi Bending Moment is found thus — From P subtract the quantities Wi, JV2, W3, &c., in succession until tlie remainder becomes zero, or first negative ; when the remainder becomes zero, the maximum bending moment occurs at the weight last subtracted, at the weight next in order, and at every point between them : when the remainder becomes negative for tlie first time, the maximum occurs at the weight last subtracted, and at that point only. In our example, from P = 24, subtract Wx = 5, Wn = 5, W^ =11, and the remainder is + 3 ; subtract W^ = 12, and the remainder is negative for the first time ; at this weight, that is, at Xi = - 3, the maximum bending moment 261 ft.-tons occurs. CHAP. VII.] BENDING MOMENTS AND SHEARING FORCES. 123 Graphical Solution for Bending Moment Diagram. — The followiug purely graphical solution for the same problem requires no analysis, but must be drawn with accurate in- struments and upon a large scale. Draw the vertical lines Scale for Bending Momemts (Verticals) 2o°Ft.Tons ' pO 60 I I ' I I I [ I n-r ScALE FOR Forces Tons Scale for Dimensions (Horizontals) ■I'll' FiiT. 3. P, W„ W^, Ws, Wi, TFs, and Q, fig. 3, at the given horizontal distances apart upon a scale for dimensions ; draw ab, be, cd, de, ef equal respectively to the forces Wi, W2, W3, W^, W^, upon a scale for forces, that is equal to 5, 5, 11, 12, and 9 tons 124 APPLIED MECHANICS. [CHAP. VII. in our example. Choose any pole 0' , and join it to a, &, c, d, e, and /. From H' any point on P draw H'A parallel to O'a, and meeting W^ at A' ; draw A'B', H'C, CD', D'E', and E'K parallel respectively to O'h, Oc, O'd, (7e, and O'f. Join H'K' ; and draw O'g parallel to K'K', meeting af in g. Then, upon the scale for forces, /^ = Q, and ga = P. In the example,/^ = 18 tons, and ga = 24 tons ; so far this is a graphical solution for finding the supporting forces. The polygon HA' B' C D' E' K' is a bending moment diagram ; but as it is inconvenient to have ^'iiT' sloping, we draw another polygon thus: — Choose a new pole on the horizontal line passing through g, and such that the distance Og is some convenient integral number upon the scale for dimensions, on the diagram Og = 10 ; draw Oa, Oh, Oc, Od, Oe, Of, and Og. From H any point on P draw HA parallel to Oa, and meeting W-^ at A ; similarly, draw AB, BC, CD, DE, and EK parallel respectively to Oh, Oc, Od, Oe, and Of- We have a new polygon, HABGDEK, which is the bending moment diagram, and HKis horizontal as desired. The vertical ordinates give the bending moments at each point of the span, upon a scale for verticals obtained by subdividing the divisions on the scale for forces by the number which Og measures on the scale for dimensions; thus on the figure this new scale is made by subdividing each division on the scale for forces into 10 equal parts, because we made 0^ = 10 on the scale for dimensions. We can now find the bending moment at z, any point of the span, by measuring the ordinate zs upon the proper scale ; for instance, the ordinate at D measures 261, the bending moment in foot-tons at that point. Produce the two sides HA and ICE to meet at h ; then G the point below h is the centre of gravity of the loads JF,, W2, W3, Wi, and W5 ; on fig. 3, HG = 18 on the scale for dimensions : see example 9, fig. 3, Ch. V, p. 93. Proof — Draw the vertical hn, and produce all the sides of the polygon to meet it ; at z, any point of span, draw the vertical zq, and let the sides of the polygon that lie to the left, or those sides produced, meet it in the points r and s ; then sr is called the intercept on the vertical through z, made by the two sides of the polygon which meet at the angle B. Other intercepts on this vertical are zq and rq, made by the pairs of sides from the angles H and A respectively. Again, on the vertical through h we have the intercepts Gh, kh, mk, and nm made respectively by the pairs of sides from the angles H, A, B, and C ; and also the intercepts Gh, ph, and np made respectively by the pairs of sides from the angles K, E, and D. CHAP. VII.] BENDING MOMENTS AND SHEARING FORCES. 125 The triangle Bsr on the base sr, and of height zv, is similar to the triangle Ocb on the base cb, and of height [/(), because their sides are respectively parallel ; the bases of these triangles measured on one scale will be in the same j)roportion to each other as the heights measured on any other scale ; hence ~ , ~j on scale . . ~~ . /^ on scale of , &/ . CO of forces » ' ZV ' yU dimensions J ~T on scale >^ ZZ. on scale of on scale of forces ^ ^" dimensions or sr of forces — — „n j^ale of y^ dimensions _ Wi tons X ZV feet _ moment of W^ about z ~ 10 ^ 10 But sr measures 10 times as much upon the scale for verticals as it does upon the scale for forces, since we made the scale for verticals by .subdividing each division on the scale for forces into ten parts ; therefore sr on vertical scale = moment of Wt about z. That is, measuring on the vertical scale, the intercept on the vertical through any point z, made by the pairs of sides from any angle of the polygon, equals the moment about z of W, the force at that angle. To show that if P = ga, and Q = fy, they will balance JF, + n\ + W^ ^ W,+ W,. It is evident that P + (> = 2 fr ; and if you produce Hli to meet the vertical through K, the intercept by the pair of sides from H equals the moment of ga, that is of P, about K; the five intercepts made on the vertical through K by the pairs of sides from A, B, C, D, and E, are the moments about K of the weights Wi, W^, W3, Wi, and W^ respectively, and it is evident that the first of these inter- cepts is identically equal to the sum of the other five. Hence the moment of F equals the sum of the moments of JF,, JV^, ^V-h Wi, and Wi about K', that is, ga exactly represents the value of P. To show that G is the centre of gravity of the weights Wx, W2, W3, Wi, and PF5. The sum of the moments about G of all the weights to the left of G, that is of W^, W^, and JFj, is hk + km + mn = hn ; again, the sum of the moments about G of all the weights to the right of G, that is of W^ and W^, is np + ph = hn; hence the sum of the moments about G of all the weights to the left equals the sum of the moments about G of all the weights to the right ; and since these moments tend 126 APPLIED MECHANICS. [chap. VII. to turn the beam in opposite directions, the one sum destroys the other, or the sum of the moments of all the weights W about G is zero ; that is, G is the centre of gravity of the weights. To show that zs, the ordinate of the polygon, measured on the vertical scale equals the bending moment at z, zs = zq- qr - rs, all measured on the scale for verticals = Mom. of P - Mom. of TFi - Mom. of W^, all about z = Bending Moment at z. This is Culman's Theorem. It will be seen that this is a special case of fig. 10, Chap. IV. Also HABCDUK is a balanced frame, the vectors from giving the thrust on the bars, &c. w, iw, w. m w. a'^Q Fisr. 4. In all cases of fixed loads the maximum bending moment occurs where the locus of the shearing force diagram crosses its base. Graphical Solution for Shearing Force Diagram. — Fig 4. P and Q are determined graphically by finding the point g, as on fig. 3, or by moments. Through g, fig. 4, draw a horizontal line and produce the lines of action of P, W\, TFj, . . .Q to cut it; join these lines of action in pairs by horizontals CHAP. VII.] BENDING MOMENTS AND SHEARING FORCES. 127 through a, b, c, d, e,f, respectively. The scale used for forces is also the scale for shearing forces. The construction is cyclical ; through each of the points a, h, e, d, e, f, g, a horizontal line being drawn to join the lines of action of the forces in pairs, viz. : — from a joining P and JF,, from h joining TF, and W^, from c joining W^ and W^, from d joining W^ and W^, ho\\\ e joining W^ and W^, from / joining Wi and Q, add from g joining Q and F. Cantilever with Unequal "Weights at .Intervals. A cantilever (fig. 5) 12 feet long supports three weights, ir, = 8, W. = 6, and W^ = 4 tons at points whose abscissae reckoned from the fixed end are Xi = 12, x^ = 10, and Xz = 4^ feet. This loading is shown also on fig. 11, p. 103, but on that figure the cantilever projects beyond Wi, and so c is greater than a;, ; it is evident, however, that the part which so projects is not strained, so that although a cantilever does so project, yet, for purposes of calculation, its length may be considered as the distance from the fixed end to the most remote load; this is shown on fig. 5, where c = a?,. We will now calculate the bending moments at the fixed end, and at points where the weights stand, systematically from the forces upon the left-hand portion. Ft. -tons Mx,,,,^ Mx^ = JV,(x,-x,); 3f,o = Sx2= . . . . 16 Mxs = W, {x, - x^) ^ W, (x, - ^3) ; i^i = 8 x 8 + 6 x 6 = 100 Mo = JV,x, + W,x, + JV,Xs ; = 8 y 12 + 6 X 10 + 4 X 4 = 172 The Bending Moment at any point is the sum of the products got by multiplying each weight to the left of that point by its distance therefrom; the above is an arithmetical sum, since all the weights tend to bend the left portion downwards. Having found the bending moment at one point, we may derive the bending moment at another point nearer the fixed end and such that no weight intervenes, by adding the product of the sum of all the weights to the left into the distance between the two points, since no neiv weights have to be considered, and all the leverages have increased by the distance betv/een the two points. Hence the bending moments increase uniformly in each interval as you move towards the fixed end. 128 APPLIED MECHA^'ICS. [CHAP. VII. W, Wj jW,|W, I i—(X-X>6- ^...(X-XJ~8 4 tons ii •p'' Bending Moment Diagram. — With a scale of feet for horizontals, lay off the length (fig. 5) and plot the positions of the loads ; at each of these points, and at the fixed end, draw a vertical ordinate doivmvards, equal to the bending moment thereat, upon a suitable scale of foot-tons for verticals ; join the ends of the ordinates by straight lines. The ordinates are drawn doivmvards to signify that the moments on a cantilever are of a different sign, as compared to those on a beam. In either case the moments are all of one sign, which will always be reckoned as positive. Maximum Bending Moment. — It is evident that the maximum bending moment is at the fixed end, and is Jfo = S ( Wr). Fig. 5. W, w. (i; «a Graphical Solution for Shearing Force Diagram. — Pig. 6. From a, b, c, d, joints of load-line, draw horizontals joining the lines of action of the forces in pairs, the vertical through the fixed end being reckoned as a line of action. Thus draw horizontals, from a joining the ver- tical through the fixed end and W^, from h joining W^ and W^, from c joining W^ and W^, and from d joining fFs and the vertical through the fixed end. The scale for forces is also the scale for shearing forces. Graphical Solution for Bending Moment Diagram. — Draw a vertical line through K, the fixed end, and draw the vertical lines W^ W^, W^ at the given horizontal distances apart upon a scale for dimensions (fig. 7). Draw ah, he, cd, equal respectively to the forces Jr,, W^, W^ upon a scale for forces; that is, equal to 8, 6, and 4 tons in our example. Choose a pole on the horizontal line passing through a, and such that the distance Oa is some convenient integral number upon the scale for dimensions ; on the diagram Oa = 10 ; and draw Oh, Oc, and Od. From A any point on IF", draw AB .W, -ih --<.c -id Fig. 6. CHAP. VII.] BENDING MOMENTS AND SHEAUING FORCES. 129 parallel to Oh, and meeting Wz at B ; similarly, draw BC, CD, and A K parallel, respectively, to Oc, Od, and Oa. Then AB('1)K is the bending moment diagram ; its vertical ordinates give tlie bending moment at each point of the span upon a scale for verticals obtained by subdividing the divisions on the scale for forces by the number which Oa measures on the scale for dimensions ; thus, on the figure, this new scale is made by subdividing each division on the scale for forces into 10 equal parts, since Oa = 10 on the scale for dimensions. Proof. — At any point e draw the ordinate eh, and produce those sides of the polygon that lie to the left till they meet it. On the figure, they meet it at e, /, g, h. As in the proof to fig. 3, the intercept on the vertical through e made by the pair of sides from any angle of the polygon will, when measured Fig- 7. upon the scale for verticals, give the moment about e of the force at that angle. Now the ordinate at e is the sum of the intercepts made by the pairs of sides from each angle to the left, and so will give on the vertical scale the sum of the moments of the forces to the left of e; that is, the bending moment at e. On fig. 7, ef, the intercept by the pair of sides from A, gives the moment of Wy about e ; fg, the intercept by the pair of sides from B, gives the moment of W^ about e, and gh gives the moment of JFs about e. Hence eh gives the sum of these three moments, that is the bending moment at e. Shearing Force Diagram due to One Load. For a beam with W at the centre, the shearing force diagram consists of two rectangles of height i W, one standing above the left half and the other below the right half of span. For a beam loaded with ^ at a point dividing the span into any two segments, the shearing force diagram consists of two rectangles, one standing above the left segment, the other 130 APPLIED MECHANICS. [chap. VII. below the right segment. The height of each is inversely pro- portional to the length of the segment on which it stands, and the sum of their heights is IF. In these two cases it is evident, and it can easily be proved for a general case, fig. 4, that the area of the part of the / > W 1 ^W • V / ( "1 Fig. 8. shearing force diagram above the base is equal to the area of the part below the base. For a cantilever with W at its free end, the shearing force diagram is evidently a rectangle of height W, standing below the span. Beam Loaded at the Centre. Fig. 9. Let W be the load at the centre. By symmetry P ^ Q = I W. For a section distant from the centre x towards the left, consider the forces on the left- hand portion. The only force is P, and i^ its leverage is {c - x): hence, pjw /\ wiQ M,^P{c-x) =— (c -X), the equation to the bending moment. The value of Mx is zero at the end, that is where x = c; it increases uni- Fig. 9. formly as x decreases, that is, as you approach the centre, and it is greatest where x = Q, that is at the centre ; by symmetry for the other half of the span, the value will decrease uniformly till it is again zero at the right- hand end ; hence the maximum bending moment 1/ ^ Wl. In Rankine's " Applied Mechanics," the maximum bending moment in each case is given in the above form, viz., maximum CHAP. VII.] BENDING MOMENTS AND SHEAKING FORCES. 131 bending moment = constant x total load x span, or Mg = m. W.l, Throughout a great portion of that work, vi stands for this constant, which he calls the numerical coefficient of the maximum hendiiyj moment exjnrssed in terms of the load and span. The value of m depends upon the manner of loading and of support. In the case we have just solved, we specify the mode of support by calling the piece a beam, and the manner of loading when we say that W is at the centre, and we find m = \. Bendin;] Moment Diagram. — Upon a convenient scale of feet, lay off BC, fig. 9, equal to the span ; construct a triangle with its apex above the centre 0. Draw a scale of ft.-lbs. to measure verticals upon, such that OA shall measure upon it one- fourth of the product of the load in lbs. into the span in feet. Graphical Solution. — The simplest graphical solution is to draw the bending moment diagram as above ; one which is purely graphical may be made as in fig. 3, and it will not be necessary to use the first pole 0', &c., as we know that P = Q = h^^' Draw ah vertical and equal to the load W ; from its middle point draw a horizontal line; choose at a distance from ah equal to some convenient integral number on the scale for dimensions, and draw Oa and Oh. Then fig. 9 is constructed l)y drawing BA parallel to Oa, and AC parallel to OB; and a scale for verticals is obtained by subdividing the scale for forces by the number chosen for the distance of from ah. Cantilever loaded at the end. — Fig. 10. To find the bending moment at a section distant x from the fixed end K, consider the loads to the left of that jw section. The only force is fF, and its leverage r about the section at x is (c - x), and we have ^^^ * — M^ = W{c - x), \\^ I the equation to the bending moment. ^<, The value of Mjc is zero when x equals c, that is at the free end; it uniformly in- fw. lo. creases as x decreases, and is a maximum when .r = 0, that is at the fixed end ; the maximum bending moment is Mo^ Wc= W.l, and the value of the constant is m = 1. Bending Moment D.agram. — Upon a convenient scale of fe^et, lay off KA (Hg 10) equal to the length; draw below KA the right-angled triangle FAK, with the right angle at k2 132 APPLIED MECHANICS, [chap. VII. the fixed end, and construct a scale of ft.-lbs. for verticals, such that KF may measure upon it the product of the load in lbs. into the length in feet. Beam Uniformly Loaded. Fig. 11. Let w be the intensity of the uniform load in lbs. per foot of span. This is represented by a load area, consisting of a rectangle of height w feet standing on the span, and weighing one lb. per square foot. The total load W^is the area of this rectangle, so that W = 2wc, or w = — W 2c LOAD AREA P ' =iyc i' luc =Q toe rr,i lw[c-x) IOC* •— {c-aJ->- To find the bending moment at a section distant x from the centre 0. Consider the load area standing upon the portion of the span to the left of that section ; it consists of a rectangle of length (c - x) feet, its area is w{c - x) square feet, and its weight is w{c - x) lbs. This weight may be considered to be concentrated at the centre of gravity of the area — that is at its middle point ; this gives a bending moment about the section equal to that for the actual distribution. We have two forces to the left of the section, P = wc lbs., half the total load acting upwards with a leverage of (c - x) ft., and w{c - x) lbs. acting downwards with a leverage of 2" (c - x) ft. ; hence Fig. 11. Mjt = P{c - x) -w{c-x)x — — = wc (c - x) - — xY -. ^ (c -x) {2c-c-^x)=^(c-x) {c + x) =|'(f= -.r)i; W = — (c- - x^), the equation to the bending moment. 4c The value of J/^ is zero where x -^ ± c, that is at the two- ends ; it increases as x decreases numfically from c and from - c that is from both ends towards the centre; and it is- CHAP. -.1.] BENDING MOMENTS AND SHEARING FORCP^S. 133 greatest where .r = 0, that is at the centre. By making .v = 0, we have .1/,, = \ Wc ; or hy considering the section at O, the maximum hendinar moment is Mn =^ F . c - wc . - = wc .c - IOC = \Wc, or ^Wl, In this case, the value of the constant is m = \. Bending Moment Diar/ ram. — Examining the equation to the bending moment, we see that the ordinate M^ equals a constant term, minus a term in .r ; hence we know that the locus is a parabola, with its axis vertical and with its apex A (tig 12) exactly above the origin O at the height | Wc = I Wl ; and that . W the modulus of the parabola is — , the coefficient of .r^ so that the principal equation to the parabola, that is, taking A as origin, is W 4c Graphical Solution. — With a scale of feet for horizontals, lay off the span BC (fig. 12j and draw a vertical OA up- wards through () ; apply the parallel rollers to HC ; place any parabolic segment cut upon pear-tree or card-board against the rollers (see fig. 4. Ch. VI), with its apex on the vertical through ; shift the rollers till the curved edge passes through B and C, which it will do simul- taneously, and draw the curve BA C ; construct a scale of f t.- Ibs. for verticals, such that OA = i jr . /. Fig. 12. Shearing Force Diagram. — Fig. 13. Draw verticals, one upwards from the left end, another downwards from the right end of span, each equal to | JV; join their extremities with a straight line, which will cut the base at the centre. 134 APPLIED MECHANICS. [CHAP. VIL The shearing force diagram consists of two right-angled triangles, whose common height is i W; one is above the left half, the other is below the right half, of span, and the right angle of each is at the end of span. Cantilever Uniformly Loaded. Fig. 14. — Consider a section at the distance w from the fixed end. The load area standing on the portion to the left of that section is a rectangle of length (c - x) ft. and height w ft. ; its area represents a weight of tv (c - x) lbs., which may be considered to be concentrated at the centre of gravity of the rectangle ; and so to the left of the section, there is to be ^^.^. taken into account only one force \4j!.^^ k IV [c - x) lbs., having a leverage y | f '■q^q area h about the section of | (c - x) ft. ; \n=\ivc °P hence ^ 1___ .^k M-r. = w{c w W 2- (« - ^)' = -9^ (^ - ^y Fi-. 14. the equation to the bending moment. The value of M^. is zero wherp x = c, that is at the freer end ; it increases a 5 x decreases ; and it is greatest where X = 0, that is at the fixed end. Putting a; = we have the greatest value, or directly from the figure we find the maximum bending moment In this case, the value of the constant is m = 5. Bending Moment Diagram. — For the sake of comparison with the diagrams for beams, we may consider the bending moments on a cantilever to be negative, when the equation becomes W W TFc the locus is a parabola (see fig. 5, Ch. VI) with its axis vertical CHAP. VII.] BENDING MOMENTS AND SHEAKINrx FORCES. 135 and to the left of O, and with its apex on the axis of X. The vahie X = c makes (2c - x)x great(\st, and therefore makes Mc = ii positive maximum ; so that the ap(>x lies to the left of () at a distance c^ — that is, the apex is at the free enil. Since the coefficient of x- is — , the principal equation to the parabola is Y 2c ^ X-.-.,^^ Graphical Solution. — With a con- venient scale of feet, lay off KA, tig. 16, equal to the length ; place the parallel rollers to KA ; set any para- bolic segment against the rollers, with its apex at A ; draw the curve AG, till it meets the vertical through K at G. Construct a scale of f t.-lbs. for verticals, such that KG may measure upon it one-half the pro- duct of the total load in lbs. into the span in feet. Shearing Force Diagram. — Fig. 15. Draw a vertical downwards from the fixed end of the base equal to W, and join its extremity to the free end of the base. The shearing force diagram is a right-angled triangle of height ir, standing below the span, and with the right angle at the fixed end. Fig. 16. Beams Subjected to Equal Loads at Equal Intervals. The actual loading on a girder is often of this kind hecause the dead weight of the platform is transmitted to it by equidistant cross-girders. This again may have to be combined with the uniform weight of the girder itself : see such a com- bination following. Al!^o the wheels of rolling stock riding over a girder or stringer while at rest are sensibly uniform loads of uniform di.~:tances. If the rolling stuck move, the problem is changed and will be treated of in a chapter to follow. On tigs. 17ff, 174. Let JF, the total load, be distributed over the span /, in « parts each equal to w, and at equal intervals a apart ; then ntv = W, (n + \)a = 2c — I, P=knw. Take S, a point directly under one of the weights, and let BS = ra, then »• will be a whole number ; and if j- bi- the distance of S from the centre, then m = (c — x). On tlie portion of the beam to the left of S, there are in all r forces, viz., P= \mv acting upwards with a leverage about 5 of ra, and (r - 1) forces each equal to tv and acting downwards; the nearest to S has a leverage a, 136 APPLIED MECHANICS. [chap. VII. the next a leverage '2a, the next a leverage 3«, &c., and the last a leverage (r - 1) a ; hence Mx = F .ra — w . a — w .'la — w .Za . . , —tv .{r — \)a nw = — . »•« - wa (1 + 2 + 3 ... + >•- 1) nw (r - Y)r w = y . »•« - wa — ^ = 2 • '■" (« + 1 - '■)• But W 2c {n + 1) (c - x) substituting, we have M, iV 2« W 2w {c- x) \n+l - w + 1 (« + 1 ) (c - X) 2c n (.-.)(„+l)''_lf=^^±I (,._,.), 2c 4(7 n which'is the equation to the locus of T. This equation gives the bending moment only at points where weisrhts are, that is for values of a; which are multiples of a, but not at intermediate points ; it n equal loads at equal intervals. W= total load, if W be the same in both so also is M„ the ma> Fig. 17a. Fiff. 176. only'differs from the equation we had for an uniform load, by the constant factor t^-__2J^« locus of T, the tops of the ordinates at the points where the weights are situated, is a parabola with its axis vertical, and its apex above O. Putting i/ instead of Mx for the ordinate to the curve at any point, we have y = (c^ - x^), 4c n 80 that y is a maximum where a: = 0, and W n + I .Vn = — . ic V «+ 1 W . I a maximum, and ;he height of the apex A in every case. On fig. 17 it will he seen that if n be, odd, a weight comes exactly at the CHAP. VII.] BENDING MOMENTS AND SHEARING FORCES. 137 centre ; so that j/o, tlie ordinate of the parahola, is tlie maximum bending moment ; hence the maximum bendinu: moment Mo = * '-^-i^ W. I, (« odd), n and the value of the constant is « 4 1 m = _ 1 ff When n is even, the maximum bending moment \Kq is le»s than yo» and equals the ordinate of the parabola at the weight on either side of the centre ; so that to find the value of .l/o, it is only necessary to substitute- tor x half of an interval, that is ha, or , ; hence the maximum benditiar moment Mo = ij - — \c' - I I =5 W.l, (n Qven), (_c_\ ic n \ \«+l/ ) S»+l ' ^ ^' \n * 1 / and the value of the constant is -i '»±±^ '" ~ ^ m + 1* The cords of the parabola give tlie bending moments at points intermediate between the weights, since the bending moment varies uniformly in these intervals. Examples. 1. A beam 20 feet span supports a load of 10 tons uniformly distributed. Find the shearing forces at intervals of 5 feet. Am. Fio = 5 tons ; Fi = 2-5 tons ; Fo = 0. On right half of span, the values are the same, but negative. 2. A betim 20 feet span supports a load of 10 tons concentrated at the centre. Find the shearing forces. Ans. i^iotoo = 5 tons, i^oto-io = - o tons, and the sign changes at the centre. 3. In the example given on page 119 and with the data shown in fig. 1, find the shearing forces. At2s. Fii to 20 = i* =24 tons. i^2otoi5 =24- jr, = 19 „'{ i^l5t0 7 = 19 - ^^2 ■= 14' ,, i^7to-3 = 14 - JTs = 3 „ -F-aio-io = 3 - F'l =- 9 „ /'-ioto-21 = - 9 - TFs = - 18 tons = ( - Q). See figs. 2 and 4, Ch. VII, and note that the shearing force changes sign, and that the bending moment is a maximum at the point a; = — 3. 4. A lieam 24 feet span is loaded with 20, 30, and 40 tons at points dividing it into equal intervals. Find the maximum bending moment and the point where it occurs. As in example No. 2, p. 91, vee have P = 40 tons ; deduct 20 and it leaves 20 : 138 APPLIED MECHANICS. [CHAP. VII. deduct 30 and the remainder is negative for the first time : hence the maximum occurs under the load 30, that is at 12 feet from the left end, and maximum JJfi2 = P X 12 - TTi X 6 = 40 X 12 - 20 X 6 = 360 ft.-tons. 0. In example No, 4, find the bending moments at the other two weights. j{/6 = P X 6 = 40 X 6 = 240 ft.-tons. Jlfis =Px 18- ^Tix 12- TTz X 6 = 40x 18- 20 X 12 - 30x 6= 300 „ or il/i8 = Q X 6 = 50 X 6 »= 300 „ 6. Find the maximum bending moment in example No. 12, Ch. V. An^. Mi9 = 256 ft.-tons, maximum. Or, taking the centre as origin, Mz = 256 ft.-tons, maximum, and is at the wheel transmitting W3 = 11 tons. 7. A beam 40 feet span supports four weights TFi = 50, TF2 = 10, W3 = 20, and TFi = 30 cwts. at points whose abscissaj, measuring from the centre to left and right, are xi = 10, X2 = 2, .rs = - 12, and Xi — — 16 feet. Find the supporting force at the left end, the maximum bending moment, and the place where that maximum occurs. Ans. P = 50 cwts. : and since P — TFi = 0, the maximum bending moment occurs at zi, at X2, and at every intermediate point, and -^10 to 2 = 500 ft. -cwts., maximum. 8. A beam 50 feet span has ^'eights o, 8, 9, 12, 9, 8, and 5 cwts. placed at equal intervals of 6 feet, in order, upon the span, and with the hiad 12 cwts. at the centre. Find the maximum bending moment. Ans. Mo = 500 ft. -cwts., maximum. 9. A cantilever 12 feet long bears four loads Wi = 8, Wi= 6, Wz = 9, and Wi =12 tons at distances from the fixed end of a;i = 12, X2 = 8, Xi = 6, and Xi = 2 ft. Find the bending moment at each weight, and also the maximum bending moment. A71S. J/,2 = 0, Ms = 32, J/e = 60, M^ = 152, and M^ = 222 ft.-tons. 10. A cantilever is loaded with Aveights of 8, 6, and 4 tons at distances of 12, 10, and 4 feet from the fixed end. Find the bending moment at each weight, and draw a bending moment diagram upon a large scale: see fig. 5. 11. Draw also a bending moment diagram by the graphical construction (fig. 7) to large scales. From either diagram, by measurement, find the bending moments at intervals of two feet. Ans. Mo = 172, M2 = 136, J/4 = 100, Me = 72, Mg = 44, Mio = 16, Mi2= ft.-tons. 12. A cantilever 12 feet long is uniformly loaded with 3 cwts. per foot-run. Find the equation to the bending moment. Ans. .l/x= I (12 -.jpft.-cwts. 13. In the previous example, find the bending moments at intervals of two feet by substituting for x into the equation. Ans. Mn = » (12 - 0)= =216 ft.-cwts. 3^ = # (12 - 2)^ = 150 ft. -cwts. Jfi = 96, Ma = 54, Mg = 24, Mio = 6, M12 = 0. 14. A beam 40 feet span supports seven biads, each two tons, and placed symmetrically on tlie span at interviils of five feet. Calculate the maximum CHAr, VIII.] B. M. AND S. F. FOR FIXED LOADS. 139 bending monient by substituting in the proper equation, and calculate at each load the height of the parabola which gives the bending moments. Here ;r= 14 tons, c=20 feet, I = 40 feet, and w = 7. Tlie maximum bending moment is J/o = yo = i ^^-i- TT . / = J X f y 14 X 40 = 80 ft. -tons. The equation to the parabola is y = ^ . "^-- {c- -x-) = U Xf (400 - ^-) = i (400 - X-'-) ; therefore, at the weights, ^ViOT.o = i (400 - 25) = 75 ft. -tons ; Jfio or - io = 60, Jfisor- 15 = 35ft.-tons. CHAPTER Vni. BENDING MOMENTS AND SHEARING FORCES FOR COMBINED FIXED LOADS. Beam uniformly loaded and with a load oi its centre. — Fig. 1. Let U be the amount of the uniform load, then the bending moment at x due to it alone is -r- (c" - a:") ; let W be the load 4c at the centre, then the bending moment at .r due to it alone IS -^ [c - X) ; summing these, we have •^ = Tc ^'' - ^^) + — (^ - ^) = 4^ (^ - ^) (^^ + ^ + -ir} the equation to the bending monient for positive values of r, that is for the left half of the span. Putting y instead of J/^, we have U. ,/ 2Wc\ a curve, the ordinates of which are the bending moments for the left half of span ; this curve is a parabola with its axis vertical, and its apex above BC the span. To find the position of the apex ^,, it is only necessary to find that value of x which makes y greatest ; now y is greatest when . ./ 2Wc (c - ;r) f c + ;r + — 140 APPLIED MECHANICS. [CHAP. Vni. is a maximum; and since the sum of these two factors is constant, their product is greatest when they are equal ; putting then 2JFc ^ JF c + X + —jr=- = c - a*, we have x = - -jfC as the value of x which makes y greatest ; the negative sign denotes that Ai lies to the right of 0, so that OSi is to be laid W off towards the right and equal to -jy^- Thejheight of A,, is the value of y when we substitute this value for x ; that is, byAi = -7- C + ^^C C - -~rC + 4c V U ) \ U U -] ^{\ui) = U J ^^^^' \ u times the height of A^, the apex of the] parabola for the uniform load alone. A^DG is the same parabola with its apex at the symmetrical point Ai and the portion DC gives the bending moments for the right half of the span. Since the coefficient of o:^ is -r- , ° 4c the principal equation to the parabolas AiDB and A^DC, each referred to its own apex as origin, is Ac this is also the principal equation to the parabola BAJJ for tlie uniform load alone, so that all three parabolas are identical. AVe might suppose the diagram for the uniform load alone to consist of two parabolas lying on the top of each other ; and that upon the addition of the load W at the centre, they both move upwards, while the one moves towards the right and the other towards the left. This follows at once from the Thcorein A, Ch. VI. The Bendiivf Moment Diagram is BDC. It can be shown that the tangent at 1) to the parabola A^DC cuts of!" CE equal to the height of A„. ODBC is an approximate bending moment diagram made with straight lines ; and it is safe, since the ordinate of any point on DE is greater than the ordinate for the corresponding point on DC. CHAP. VIII.] B. .M. AND S. F. FOR FIXED LOADS. 141 Graphical Solution. — With a scale of feet for horizontals, lay off BC equal to the span, fig. 1. From the centre O lay otl" W OSi and OSz to the right and left, each equal to — - c, and draw verticals upwards from Si, 0, and S^. Apply the parallel rollers to the span £C ; place ani/ parabolic segment against the rollers^ as in fig. 5, p. 109, with its apex on the vertical through 6', ; shift the rollers till the curved edge passes through i/, the end of the span on the opposite side of the centre from Si, and draw the curve BDAi. Again, place the segment with its apex on the V^^IOHd at centr Fig. 1. vertical through So ; shift the rollers till the curved edge passes- through the end C, and draw the curve CD A.; then BDC is the bending moment diagram. The scale, of say ft.-lbs., for verticals is to be constructed such that OD measures |(Z7+ 2W)l, where U and W are in lbs. and I is in feet. The same scale may be constructed as follows : — Place the parabolic segment with its apex on the vertical through ; shift the rollers till the curved edge passes through B and C; draw the curve BAoC, and make 142 APPLIED MECHANICS. [chap. VIII. a scale of ft. -lbs. for verticals, such that O.I,, measures upon it I Ul, where U is in lbs. and I is in feet. Cor. — For the same uniform load, although different loads be put at the centre, A^DB is always the same parabola; as the load at the centre increases, the apex A^ moves from the centre, and the are DB is a part of the wing of that parabola further from the apex. Now the wing of a parabola gets \ r „ ^mt.ot un.form.o^d. ~\ n flatter as its distance from the apex increases; hence, if U be constant and W be increased, BD becomes flatter and flatter ; and if W De very great compared to U, DB is sensibly a straight line. Beam uniformly loaded atid supported on three props. — Fig. 2. Let BC be a beam with a load U uniformly distributed on it and supported on three props, one at each end and one at the centre ; let P, JF, aud Q be the forces with which they press upwards, so that /'+ W +Q= (7 always, and P= Q by symmetry. If W=Q, the cen- tral prop bears no share, P= Q = \U, and the bending niiiment diagram is the parabola HAK, as on fig. 2. If the central prop bears a share, then the beam is loaded with a uniform load U, and a negative load W at the centre; and the bending moment diagram is two parabolas, each tlie same as HAK, but with its apex away from the centre, and in a direction opposite to that on fig. 1. The horizontal distance through which each apex moves is given by the equation o — — c. Suppose RAK to be two parabolas lying one on the top of the other, and let the prop press up with a greater and greater force, then the two parabolas shift away from each other. When W=^U, Ai is over S\, and Ai is over iSj, the middle points of OH and OK respectively, and the bending moment at is zero ; this is the best value of JF, if the beam may only be bent so that its convex side shall be down. It is evident that t^vi ./i - iff A : and tliat the beam might be CHAP. VIII.] B. M. AND S. F. FOR FIXED LOADS. 143 sawn through at 0. when it would be two beams of span c, each uniformly loaded and supported at the two ends. If the heara may be bent both upwards and downwards, let the central prop press upwards till W > \U, then 0S\ is greater than \0H, or in symbols o > ^c ; and it can be seen troin the figure that the ordinales from H' to K' are negative, from H" to ^positive, and from K' to jf positive, while at M, R', K, and A" the bending moments are zero. Hence the beam will be bent with the convex side downwards from JTto H', and from K' to K, and upwards from H' to K' . A hinge might I'O put on the beam at H" and E? , since there is no lendencj' to bend at these points, which are called points of contrary flexure. There are three maxima bending uuiments, two equal positive ones at ^i and S^, and a negative one at 0. If the material of the beam may be bent upwards and downwards equally well, then the best value of W is that which makes the positive and negative maxima equal, as their common value in this case is less than the greatest value would be in any other; for, suppose them equal, then if /F be increased the parabolas move outwards, and the ordinate at will increase ; while, if W be made smaller, the parabolas will approach and the ordinates at S\ atid S2 will increase. Let the three maxima be equal to each otiier, then OX = SiAi, or JiSi : AiB : : 1 : 2 ; hence SiE' : liX : : 1 : V2 since the curve is a parabola ; or in symbols 3 : a : : 1 : V2. Practically this could be accomplished by causing the props to press upwards till OSi a V2 7r= — -i- U= U= -4 F= -586 Z7; OH o + y3 V2 + 1 or by fixing hinges at the two points IT and K' at the proper distances from 0, the central prop may be made to bear the above share of U. If the prop at the centre press upwards so that TF = U, then P= Q= 0, and 0S\ = OH; A\ coincides with H, and Ai with K; the bending moment is everywhere negative, and its niaximiun value ON equals OA. OS and OK are cantilevers as on fig. 16, p. 135. The Continuous Beajnl Beam uniformly loaded and supported on many props. — Fig. 3. Let BE be a beam bearing an uniform load and supported on many (5 in the figure) props ; this is only an extension of the previous case. The end C, fig. 2, instead of resting 2^^ .^-J.1^^(°<-/3J r . *^ w w w A H. Fig. 3. on a prop, might be hinged to the end of a cantilever, which in turn might have its other edge hinged to abeam, «fcc. ; see fig. 3. Here a double cantilever, as B'C, over each intermediate prop is hinged on each side to the end of a beam ; 144 APPLIED MECHANICS. [CHAP. VIII. each cantilever bears an uniform load over itself, as well ns half the load on the beam concentrated at its end, and is therefore in the condition of the cantilever shown on fig. 2. E<)ih intermediate beam as C' is uniformly loaded, and is supported by a hinge at each end; tlie two end beams B' li and K' E are supported at one end iiy a iiinge, and at the other by one of the extreme props ; each central span L consists of a beam and tsvo cantilevers; the end span / of a beam and a cantilever. If ihe tiinges be put in the positions indicated by H' and K' on fig. "2, then the negative maxima bending moments over the props at 0, T, &c., are equal to the positive maximii bending moments at .S'l, S-i, &c., the centres of the beams ; the common value of all these maxima will be less than the greatest for the hinges in any other position, and I : L : : a+ fi : 2a : : I + V'2 : 2^2 : : -854 : 1. Cantilever unifoi-mly loaded and with a had at its free end. — Fig. 4. As in the previous case, add the bending moments at the section distant x from the fixed end, due to the loads separately ; thus - M^ = — {c - xf +W{c- x). We consider the bending moments on a cantilever negative as compared with those on a biara, and so they will be represented by ordinates drawn down from the span instead of up as in the case of beams. It we further put y instead of Mx for the ordinate at the point on the curve corresponding to any value of x, then y will be Mr only for values of x from to c ; and U U I 2Wc\ U ^ . / 2Wc\ This curve is a parabola with its axis vertical and its apex above the span. To find the position of the apex A, it is only necessary to find that value of x which makes y greatest ; now y is greatest when {c-x) \x-c —^ is a maximum ; and since the sum of these two factors is constant, their product is greatest when they are equal; putting then 2Wc / , ^ \ is the distance of A to the kft of 0, or — - c is the distance of A to the left of E\ W that is, the apex of the parabola is beyond the free end by the distance -^ c, or the same fraction of the span that W the concentrated load is of U the distributed loiad. CHAP. VIII.] B. M. AND S. F. FOR FIXED LOADS. 145 The height of A above OH is the value of y when we substitute the above value for x; or height of ^ is U / U+W \ (11+ W 2WC\ I^V.yrrn f^V- times the maximum bending moment for the uniform load alone. Since the coeflficient of jr- is — , the principal equation lo the parabola AEF is r = u X-; and it is therefore the snme parabola as that for the ui.iform load alone, which Mouhi follow at once from the Theorem A, Ch. VI. 'I'he BeudiKg Moment Buigram is EFO, formed by the parabola EG, whose apex is at E. and which makes the diagram for the uniform load alone, shifted without turninsj till its apex is at A. Graphical Solution. — With a scale of feet for horizontals, lay off 0£ (fig. 4) equal to the length, and produce it t" <S, W so that ES = -jjrC, or is the same fraction of the lengtli as the load at tlie end is of the uniform loa(i, and draw a vertical upwards through 6' ; apply the parallel rollers to the line 0£ ; place any parabolic segment against the rollers with its apex on the vertical through 6' ; sliift the rollers till the curved edge passes through the free end E, and draw the curve AEF ; OEF is tiie hending moment diagram. The scale of, say, ft. -lbs. for verticals is to be constructed such that OF measures (^-)'• where V and W are in height of A is('^)a= SA Fig. 4. lbs., and I is in feet. The same scale may be constructed as follows : — Place the parabolic segment with its apex on the vertical through the free end E, and move the rollers till the apex comes to E; draw the doited curve EG, and make a scale of ft.-lhs. for verticals such that OG measures upon it | Ul, where U is in lbs., and I is in feet. Beam Loaded both Uniformly and with Unequal Weights FIXED AT IrREGULAE INTERVALS. On fig. 5 the uniform load is U = 36 tons or | a ton per foot on a 72-foot beam, which bears also four concentrated loads dividing the span into five bays. BA^C the parabolic segment giving the bending moments for the uniform load alone may be supposed to be five-fold. Each moves up and away into the position shown by dotted lines when the slopes due to the dead load diagram are added; see Theorem A, Ch. VI. £ £ Jl_£ Fig. 6. CHAP. VIII.] 15. M. AND S- F. FOR FIXED LOADS. 147 Graphical Solution. — The concentrated loads Wi = 8, W^ =10, Wi = 18, and Wi - 13 tons in the example on fig. 5 are drawn downwards in cyclic order to a scale of tons. Then h, the joint between the reactions at the supports, is to be found either by drawing a link polygon or by taking moments about one support. The supporting forces are P = 20 and (^ = 29 tons. The uniform load is f a ton per foot, so that tlie supporting forces due to it are 18 tons at each end. The dotted lines show the shearing force diagram for the loads separately standing on the common base BC. The thick lines show the two compounded, and crossing the base at dz the position of the maximum bending moment. The other sloping lines produced to meet the base at f?i, ^2, di, and d^ determine the points over which must stand the vertices of the parabolic arcs for the other " fields." Now draw BC for the base of the bending moment diagram, produce it each way, and from did,. . . . project down on it the points D^D., . . ., &c. Apply the parallel rollers to BC ; place the parabolic segment against them with its apex on the vertical through D^, move the rollers till the curved edge passes through B, and draw BE meeting the vertical through W^ in E, Shift the segment till the apex is on the vertical through D-i ; move the rollers till the curved edge passes through E, and draw EF. Shift the segment till the apex is on the vertical through A ; move the rollers till the curved edge passes through F, and draw EG meeting the vertical through Wz in G, &c. The accuracy of the drawing is checked by obser\dng whether the last curve passes through C, the other end of the span. Lastly, with the same parabolic segment draw BA^C standing on the span, and construct a scale so that the height of its apex A^, shall measure on it | U7, that is, one-eighth of the product of the uniform load and the span. A Diagram of the Square Roots of Bending Moments is shown at the bottom of fig. 5. The parabolic arcs are replaced by arcs of circles, the centres being the points Z>i, D^, &c., where the slopes of the shearing force diagram when produced meet the base. See the Tlicorem G, fig. 9, p. 113, as to the parabolic segment degraded. Beam Uniformly Loaded on a Portion of the Span. On figs. 6 and 7, let 2c be the length of the span, and its centre; let G be the centre of the load area, and Ik the extent of the load. The intensity of the uniform load is w lbs. l2 148 APPLIED MECHANICS. [chap. VIII. per foot-run ; and, as in the previous case, we take w as the height of the load area in feet, so that every square foot of load area represents one Ih. The total load area is a rectangle of height vj ft., and length 2k ft. ; its area is 2wk square feet, so that the total load on the span is 2v:k lbs., which may be supposed to be concentrated at G, the centre of gravity of the load area ; this gives the supporting forces P and Q as for the actual distribution. Let x be the distance of the centre of gravity of the load from the centre of the span. Then 2wk wk P = -^(c + x), Q = — {c- x), are the supports at the left and right ends of the span. They are also the height and depth of the rectangles on the right and left unload fields of the shearing force diagram, fig. 6, so that KM, the sloping locus for the central loaded field, crosses the base at s. Hence *S^ (fig. 7) divides K'F' inversely as P w \ i G S i \n '■Sw ! i f """^;) Fig. 6. rig. to Q. But G divides the span directly at P to Q. Putting h = GS the distance to the point of maximum bending-moment from the centre of the load, we have Q K'S C[G B'G CHAP. VIII.] B, M. AND S. F. FOR FIXED LOADS. 149 or /r + ^ c, + X » hx k - B c - X c ' and the maximum bending moment is wk f . kx\ wk"- JI,^--(c^x)[c-x + --j-^(c:xy -'^(c^'-x-)(2c-k) (^-^)(2-^ W f^ k The locus of the bending moment diagram (fig. 7) is parabolic on the central loaded segment and tangents to the parabola on the end unloaded segments. The modulus of the parabolic segment is — ; for suppose the whole span loaded and two upward forces at the centres of the end fields to be introduced, then the negative locus for those two forces would only introduce sloping loci to be subtracted mechanically from the right parabolic segment due to the whole span loaded. That segment would thus be shifted into the position HAE without change of modulus. Theorem A, Chapter VI. Cor. If k becomes very small, the load is sensibly at the point X = x, and Zc That is for a load, assumed at a point, but really spread over an inch or two, the scalene triangle is really rounded at the apex. Beam Loaded on one Segment. Note. — This case is of importance, with the half span, only, loaded; itis required in the treatment <>f the Iron- Arched Girder. Also th6 load covering a segment varying in length is required in the treatment of the girder with its ends fixed horizontally. Graphical Construction of the SJiearing Force Diagram and Bending J\Ionient Diagrum (figs. 8, 9). Calculate or construct the supporting forces P and Q. To a suitable scale, prick A' 150 APPLIED MECHANICS. [chap. VIIL at the height P above the base of the shearing force diagram at left end, and J/ at the depth Q below it at the right end of the load. Then KsM is the locus for the loaded segment, and a horizontal through J/ is locus for the unloaded segment. Place a parabolic template, with its axis vertical and it& lower edge against the T-square. Push up the T-square and shift along the parabolic set square till its curved edge passes through B, the vertex being on the vertical through s, then W "A' G S — 4vA 1' s"~~- — ^ > Fiar. 8. J/ draw the arc BAE, and join B to C with a straight line. Construct a scale of foot-tons to suit the calculated height of any point, say that of ^ or of ^. As before, s divides the load inversely as G divides the span. Beam uniformly loaded on its Two Segments with loads of different intensities (fig. 10). — This case is not of much practical importance, but serves to illustrate the perfectly general case of fixed loadings which is to follow. To construct the shearing force diagram, calculate P and Q, the left and right supporting forces, by taking moments about one end. Prick Ji' at Ptons above the If ft end of the base, and /* at Q tons below tlieriglil end. Draw the str;iif:ht locus A'M falling Wi tons vertically per font horizontal so that 31 shall bo lf'\ tons lower than K. Draw Mh falling ta tons per foot. The maximum bending moment must occur at t where this locus crosses the base. To construct the bonding moment diagiam (fig. 1 1). If U'l to the right of Z CHAP. VIII.] B. M. AND S. F. FOR FIXED LOADS. 151 were the only load on the s}«in, tlien the locus of the bending moment diagiam on left segment would be a straight slope Ji'E. Suppose, again, that tlie whole span was loaded with u'\ lbs. pfr foot when the bonding; moment diagram would be a parabolic locus standing on h'C, with its vertex under 0, the middle point of the span, the modulus of the parabola being ^«'i. Now remove completely the load to the right of Z, which is the same as adding an upward load ; from B' to /this would require the ordinates of a straight slope to be W, Wi G, )W, Ti?+Tr Fig. 10. subtracted from those of the parabolic arc, which then, by Theorem A, Ch. VI, would still be an arc B'ae of the same parabola of modulus \ w\, but with its vertex ti shifted to the left under Si. Fig. 11 Lastly, adding the ordinates of the loci B'ae and B'K, we have for the left "field" BF the parabolic locus BAiE of modulus ^tvi with its vertex over Ti defined by i where the slope KM of the shearing force diagram crossed the base. 152 APPLIED MECHANICS. [CHAP. VIII. In the same way the locus of the bending moment diagram for the " field " FC is a parabolic locus AiKC oi moiiwlus \wi, and with its vertex over Tj defined by the point on the shearing force diagram where the slope hM produced meets the base. Mnxxmum Bending Moment. — It is easy on any numerical example to find BT\ so that the load on it shall equal the supporting force Pat B. Then M = T\A\ is a maximum and is to be calculated directly at that section. Graphical Construction of Beudini/ Moment Diagram. — Having first constructed the shearing for(;e diagram as explained above, construct two parabolic segments standing on a common base, but with the height of their vertexes in the ratio of Wi to W2. These are to be out out in ciirdboard. The first is to be placed against the T square and pushed up with its vertex im the vertical through T\ (where the slope of the shearing force diagram crossed the base), and shifted till the curved edge passes through B, and the arc^^i^drawn. This same paraholic template is then to be pushed up with its vertex on the verticiil through till its curved edge passes through B and C simnltaneouslv. and its vertex An pricked. Then a scale is to be constructed so that OAo shall read on it \^w\P where I — span, when the maximum T\A\ can he scaled off. To finish the locus for the right segment the second parabolic template must be used, and pushed up with its vertex on the vertical through T2 (where the corresponding slope hM produced on t'ne shearing force diagram meets the base) when its curved edge will simultaneously pass th tough E and C, and the arc EC is then to be drawn. Beam Loaded in any manner with Fixed Loads. Graphical Construction of the Shearing Force Diagram. — The supporting forces P and Q at the left and right supports are to be calculated by taking moments about one end of the span. Or all the distributed loads may temporarily be concen- trated at their centres of gravity, and Pand Q constructed hy drawing a load line and link polygon as on fig. 3, p. 123. To a suitable scale of feet for horizontals, draw a base line for the shearing force diagram, and w ith a suitable scale of tons lay P upwards from the left end, and Q downwards from the riglit end. Starting from the top of P, draw a locus which will cross the base once and end at llie bottom of Q. This locus will cover "field" after "field" of the span thus. ItwiW cvo&s unloaded Jields horizontally ; at a concentrated loud it will drop the amount of that load vertically, and will cross uniformly loaded Jields sloping downwards with a str«ight slope inclined at the given rate of loading. Graphical Construciion of the Bending Moment Diagram. — If there be "fields" of the span, some loaded at one uniform rate, and some at others, then as many parabolic cardboard templates should be prepared as there are varieties of uniform load. These templates should have a common base, and their heights be in a con- tinued proportion the same as tliat of the difiierent intensities of uniform loads (see fig- 11> . . . . .... The base for the bending moment diagram being drawn, it is to be divided into " fields " ; each concentrated load is the boundary bet ween two " fields " ; so also is a point at wliicli tlie intensity of the uniform load suddenly changes, including the ends of unloaded fields. The locus of the bending moment di;igrain will bcirin at one end of the base and end at the other; being a maximum at the same point ar which the shearing force changes sign. This locus will cross tlie fields thus : it will cross unloaded Jields with a straight locus sloping up at a rate given by the absolute height of the shearing force horizontal locus in that field, or doivn at the depth of it. It will cross uniformly loaded fields in a parabolic arc, the vertical axis of the parabola being the verticnl through the point where the sloping locus of the shearing force diagram for that field crosses the base when produced. Also the modulus of the pambola shall be half the intensity of the uniform load in the field. Lastly, the arc shall at its emls liave a cotnmon tangent with the arc in adjoining field, or, if the locus in the adjoining field he straight, it shall be a tangent. At points where a load is concentrated, the loci meet at an angle whose tangent is the CHAP. VIII.J B. M. AND S. F. KOR FIXED LOADS. 153 sudden fall on the shearing force diagram. The locus is readily drawn over tield after tield, using the template for any loaded field already prepared to correspnrul to it. The snale is to be prepared to give the value at some point as calculated th^re, sav at the point where the shearing force diagram crossed tlie liase ; oi- with one of the prepared parabolic segments, a parabolic segment may be drawn standing on the span as base when a scale is to be prepared upon which the height of its veifex shall measure ^ivl-, where tv is the intensity corresponding to the template used. Examples. 1. A beam is uniformly loaded, and a prop in the centre bears one-third of the load. Find the maximum bending moment (fig. 2). Let U = the amount of the uniform load, IF = the upward thrust of central prop = ^ U. Here a = ^c ; and since a < le, the two parabolas intersect above the span, and there are no negative bending moments ; there is a positive minimum at the centre, and a positive maximum at a on each side of the centre. The base of the segment SJiE- = 2)3 = ic = il. Height SiAi : Height OA : : (§)2 : V : therefore SiAi = iOA, Fi-. 12. Ma = ^ X i^V.l = -x^-U.l, maximum. 2. A cantilever 18 feet long is loaded uniformly for two-thirds of its length from the free end, svitli 10 cwt. per foot-run. Find the bending moments at intervals of two feet. (See Hg. 13.) Look upon the loaded part as a cantilever uniformly luaded, then l=\.wk w lbs per-n^i--(c-ky--^ fe— — ;» , M. = -(c x)^ = 5 (18 -xy-; therefore Mi8 = 0, and Ih = 720 =-. BE. '^k Now consider BK?i cantilever load at B with 120 cwt. ; then k Mr W 80 that ic---x\ = 120(12- a;); ifti = 720 = D^, and J/o = 1440 ft.-cwts. Ans. 0, 20, 80, 180, 320, 500, 720 ; 960, 1200, 1440 ft.-cwts. 13 upon a large scale, and A graphical solution is obtained by drawing fi measuring the ordinates at intervals of two feet. 154 APPLIED MECHANICS. [CHAP. VIII. 3. A beam 72 feet span is loaded with 8 and 10 tons at points 18 and 6 feet to the left of the centre, and with 18 and 13 tons at points 12 and 24 feet to the riiiht of the centre ; there is also an uniform load of half a ton per foot of span. Find the position and value of the maximum bending moment. These data are drawn to scale on tig. 5. Taking moments about the right support / P X 72 = 8 X 54 + 10 X 42 + 18 X 24 + 13 X 12 + 36 x 36. So that P= 38 tons, including the reactions for spread load. At the central point the shearing force 13^0 = ^-8-10-18 = 2 tons ; and to bring this to zero, we must pass to a point 4 feet further to the right : hence F.\ = 0. The maximum bending moment will be at a point 4 feet to the right of the centre, and 3f.i = 38 X 40 - 8 X 22 - 10 X 10 - 20 X 20 = 844 ft.-tons. The shearing force changes sign at the point a; = — 4, and at this point the bending moment is a maximum. 4. A beam 54 feet span is loaded uniformly for two-thirds of its length from the left end with 10 cwt. per foot-run. Find the position and magnitude of the maximum bending moment. (See tig. 9.) In this case, c = 27 ft., and OG = 9 ft., measured to the left of 0. From G lay off GS = 6 ft. = §G0, since the load extends over two-thirds of the span, and the maximum moment occurs at S, that is, at 3 feet to the left of the centre. Suppose the whole load TF= 360 cwt. is concentrated at G ; then 360 P= — X 36 = 240 cwt. 54 This, of course, is equal to load up to S, that is, 24 x 10. Taking a section at S, the portion of the span to the left is 24 ft., so that the load upon it is 240 cwt. acting downwards, and if supposed to be concentrated at its centre, its leverage about the section is 12 ft. ; at the same time F acts upwards with a leverage of 24 ft., and Ms = 240 X 24 - 240 x 12 = 2880 ft.-ewt. max. 5. The left half of a beam 32 feet span is uniformly loaded with 1 ton per foot- run. Find the position and magnitude of the maximum bending moment. ^7is. The maximum occurs at the section 4 feet to the left of the centre, and its value is .Ui = 72 ft.-tons. 6. A beam 36 feet span is loaded uniformly from the middle point towards the left to an extent of 12 feet, with 2 tons per foot-run. Find the position and magnitude of the maximum bending moment. (See tig. 7.) In this case, OG = 6 ft., and GS = -^OG = 2 feet, since the extent of load is one-third of span ; the maximum bending moment is at S, 4 ft. to the left of the centre. Suppose the whole load Jf, 24 tons, concentrated at G, we have 24 P= — X 24= 16 tons. 36 Taking a section at S, the extent of the load to the left is 8 ft., and is equivalent to 16 tons acting downwards with a leverage of 4 feet, while Pacts upwards with a leverage of 14 feet ; hence -JA = 16 X 14 - 16 X 4 = 160 ft.-tons. 7. A beam 72 feet span is loaded in two segments ; the left segment is two- thirds of the S[)an, and is loaded uniformly at the rate of two tons per foot. The right segment is uniformly loaded at tlie rate of one ton per foot. Find the position and amount of the maximum bending moment. CHAP. VIII.] B. M. AND S. F. FOR FIXED LOADS, 155 Fig. 11 is this example drawn to scale. The data in symbols are 1e = 72, 2/t| = 48, 1ki = 24, w\ = 2, and xci = 1 : /» X 72 = 96 X 48 + 24 X 12 and P = 68 tons. In order that the shearing force at T\ maybe zero, it is necessary that the load between the left end and T\ should equal F, that is 68 tons ; hence T\ is 34 feet from the left end or 2 leet left of the centre. And ifs = 68 X 34 - 68 X 17 = 1156 ft. -ton maximum. 8. Draw the shearing force diagrams for the beam and cantilever where*^a uniform load is combined with concentrated loads (figs. 14 and 15). |i>/ 1 in- I Fig. 15. Fig. 14 combines figs. 4 and 12, Ch. VII ; fis. 15 combines figs. 6 and 15, Ch. VII. 9. Find, by analysis, the positions of the apexes in the example solved graphically on fig. 5. On the shearing force diagram the locus begins at height 38 ; it then slopes at 5 to 1, so that Bd\ = 76, or Od\ = 40. The second slope being 8 lower, then d\di is 16 and Od% = 24. See the numbers engraved at the centres of the circles at toot of fig. 5. Just as OJo = 324 = i (36)2 = i (^0)2, so also Di^i = i (Bd,f = i(76)2 = 1444, measured on the scale of ft. -tons. So also the height of Ai above E is ^th the square of their known horizontal distance apart. Adding this height to the calculated height of E itself, we have B-iAi = 1044. 156 APPLIED MECHANICS. [CHAP. IX. / CHAPTER IX. BENDI1<G MOMENTS AND SHEARING FOKCES FOR MOVING LOADS. In the first Chapter the action of a live load wlien applied to a tie or strut is described ; the action is somewhat similar when a live load is applied to a beam. Thus for a beam loaded at the centre, the load W may at one instant be in contact with the central point of the beam, and yet not be resting any of its weight on the beam ; the next instant its whole weight may be resting on the beam. It does not follow directly from Hooke's Law, but is a matter for demonstration, that, for an instant, the strain thus produced is douUe that which the dead load pro- duces, provided the greatest strain does not exceed the proof strain. One way of applying the actual weight W to the centre as a dead load is, as in the case of a tie, to put it on bit by bit ; another way is to put the whole weight W on the end of the beam, when the strain is zero, and then push it very slowly towards the centre, when the strain gradually increases to the full intensity due to JF as a dead load. If W, on the other hand, be pushed from the end to the centre in an indefinitely short time, it will be the same as if it had been applied suddenly at the centre ; in this case, then, W is applied as a live load. Definition. — A load which passes along a beam, and which thus occupies at different instants every possible position upon the span, is called a moving or travelling load. A moving load may be dead or live or of intermediate importance, but not of greater importance than a live load. A travelling crane, which moves very slowly, and so as not to set the suspended weight swinging, is practically a dead moving load. The action of a moving load on a railway bridge is of intermediate importance; when the span of the bridge is short, say less than 20 feet, this importance is about equal to that of a live load ; and when the span is long, say more than 200 feet, it may be considered as about midway between a dead load and a live load. The Commissioners on the Application of Iron to Eailway Structures at p. xviii of their report say : — " That as it has appeared that the effect of velocity communicated to a load is CHAP. IX.] 15. .M. AND S. F, KOIJ MOVING LOADS. 157 to increase the deflection that it would produce if set at rest upon the bridge ; also that the dynamical increase in bridges of less than 40 feet in length is of sufficient importance to demand attention, and may, even for lengths of 20 feet, become more than one-half of the statical deflection at high velocities, but can be diminished by increasing the stiffness of the bridge ; it is advisable that, for short bridges especially, the increased deflection should be calculated from the greatest load and highest velocity to which the bridge may be liable ; and that a weight which would statically produce the same deflection should, in estimating the strength of the structure, be considered as the greatest load to which the bridge is subject." In the same way the shearing strain produced by a moving load is greater than that produced by the same load when fixed. In the cases which follow, it is to be understood that the loads as given are dead loads, or the equivalent reduced dead loads. When the load is partly fixed and partly moving, the equivalent dead load is the sum of the actual dead load and the dead load equivalent to the actual moving load. Definition. — For any point oc, the Mange of Shearing Force due to a moving load is its extent ; and the limits of this extent are the maximum positive and maximum negative values which Fx assumes during the transit of the moving load. Classes of Moving Loads. — An uniform load coming on at one end of the span, covering an increasing segment till it is all on, then moving to a central position on the span, and passing off at the other end, is called an advancing load. A train of trucks, shorter than the span of a bridge, coming on at one end, travelling across and going off at the other end of the bridge, is an approximate example of, and is generally to be reckoned as, an advancing load. The reason that it is called approximate, is that although the weight of the trucks may be uniform per foot of length, yet they are not continuovisly in contact with the bridge, but transmit the load thereto by means of wheels at a number of points. An advancing load may be equal in length to the span ; in which case, in passing across, it covers the whole span for an instant. If the load be longer than the span, it will continue to cover it for a definite time v^hile passing, but as time does not come into our consideration, it will be included in the advancing load equal in length to the span. A load concentrated at a point, and which moves backwards and forwards on the span, is called a rolling load ; a wheel which rolls along a beam is a practical example of this. In. 158 APPLIED MECHANICS. [CHAP. IX. reality the load is distributed over a small area, and if now the load be taken to be uniformly distributed over this small area, it may be considered as an advancing load of small extent ; on the diagrams it is represented by a wheel or circle. A TravelliTig Load System is a load transmitted to the beam in definite amounts at points fixed relatively to each other, the whole load moving into all possible positions on the span; a ^xlocomotive engine is a practical example of such a system, and a rolling load is its simplest form. On the diagrams, the load is represented by a number of circles or wheels with their centres fixed on a frame (see fig. 3, Ch. IV), or for ease in drawing by a number of vertical arrows connected by a thick horizontal line (see fig. 7). It will not be necessary to consider moving loads upon cantilevers, as in practice there is seldom such a thing. It is only necessary to suppose the load fixed in the position most remote from the fixed end ; this, it is evident, gives the greatest bending moment at each point, the maximum being at the fixed end. Bending Moments on a beam under an advancing load equal in length to the span. — Suppose the load to come on from the left end and cover a segment of the span, the bending moment diagram is shown on fig. 8, Ch. VIII ; when the whole span is covered, on fig. 12, Ch. VII ; and when the load is passing off, by fig. 8, Ch. VIII, reversed. Since the parabolas in these two figures are the same, it is evident that the apex A is higher on that where the base of the parabolic segment is the whole span ; the ordinate, not only for A, but also for every point. Hence the maximum bending moment at each point of the span occurs when the whole span is loaded ; of these maxima, the maximum is at the centre, and this case resolves into that of a beam uniformly loaded. Shearing Forces on a beam under an advancing load of uniform intensity (fig. 1).— At any point x, the positive maximum shearing force occurs when the front of the load is at the point. Suppose the load to be in the position shown at the top of the diagram, then the shearing force is positive and equal to P. Jf the load move towards the right, P will decrease, and Fx will equal that decreased value. If the load move towards the left, and if the total load be not yet on the span, then the new load is exactly the same as the first with an additional load to the left of r ; this additional load is shared in some manner between the two supports, so that CHAP. IX.] B. M. AND S. F. FOR MOVING LOADS. 159 the increase of P is only a fraction of that added load ; in reckoning Fx, however, we subtract from this increased value of P the whole of the added load, that is, we subtract more than we add. Again, if the load be shorter than the span and be wholly on the span, then, after the advance, the new load is the same as the tirst with a portion added to the left of x and an equal portion taken off at the tail of the load ; the portion added increases by a fraction of itself the value of P, while the portion taken off decreases by a smaller fraction of itself the value of P ; and in reckoning F^, we subtract the whole • of the added portion from this increased value of P. Hence, whatever be the length of the load, Fx is a positive maximum when the front of the load is at X. Similarly, by calculating the nega- tive shearing forces from the supporting force Q, it can be shown that the shear- ing force is a negative maximum when the tail of the load is at the point. At each point, during the transit of the load the shearing force assumes all values between the two maxima, and passes gradually through the whole range in the same time that the load takes to make a transit. Sliearing Force Diagram. — Length of load equal to, or greater than, span (fig. 1). When the front of the load is at the left end of span, the whole span is covered, and P = ^W, where W is the load which covers span ; hence the positive maximum at the left end is h W. When the front of the load is at the right end of span, P is zero, as no load is on the span ; hence the positive maximum at the right end is zero. When the front of load is at any intermediate point, the right segment is loaded, and Fx = P di positive maximum ; the value of this maximum increases as the point approaches the left end of span, because the length of the loaded segment is increasing, and -iW 160 APPLIED MECHANICS. [chap. IX. because its centre of gravity is nearer the left end; or in symbols, this positive maximum load on span length of loaded segment _ ^' / y span 4c Hence, the locus giving F^ the positive maximum shearing force at each point is a parabola with its axis vertical and its apex at the right end of the span, and whose ordinate at the left end is ^W; the locus giving the negative maximum is a similar parabola below the base ajjd with its apex at the left end of span : the range at each point is given by the double ordinate. Graphical Solution. — Length of load equal to, or greater than, span. With any parabolic segment draw a curve with its apex at the right end meeting the vertical through the left end at L (fig. 1) ; draw another below the base, and construct a scale for verticals upon which where W is the load which covers the span. Shearing Force Diagram. — Length of load less than span (fig. 2). Let 2k be the length of the load. While the load advances from the right end, and so long as it completely covers the right segment, that is up to a distance 2k from G, the diagram is a portion of the parabola (fig. 1); for the further advance of the load, the maximum shearing force is increasing, because the centre of gravity of the load is approaching the left end, and the remainder of the locus is therefore a straight line. The straight portion of this locus and the locus for an equal rolling load (see fig. 6) are parallel, and are separated from each other by a distance k measured horizontally ; this straight portion, when produced, cuts the base at a point F such that GF = k, and it is therefore a tangent to the parabola. Graphical Solution. — Length of load less than span (fig. 2). Draw the parabola GD as in the previous case ; construct a vertical scale such that BD = wc, that is equal to half the load CHAP. IX.] B. M. AND S. F. FOR MOVING LOADS. 161 which would be on the span supposing the whole span covered ; ink in CE the portion of this parabola extending from the apex C through a horizontal distance equal to 2k; and draw for the remainder of the span, a tangent to the parabola at the point E ; this tangent is drawn by laying off CF = k, drawing FE, and producing it to L. Bendiug Moment for a Beam under an advancing load less in length than the Span (fig. 3). — Let 2c = spati ; 'Ik = extent of load ; iv = intensity of load ; C the origin, and centre of span; G the centre of load; and let the load be upon the span in any position. Here k < c, and JF= 'Iwk = total load. To find P, we may suppose the whole load concentrated at 0, and we have where x is the abscissa of any point of the span reckoned positive to the left of C, and y is the distance of the same point reckoned positive to the left of G. Taking a section at the point x, we have two forces acting on the portion of the span to the =1 2 (t'fc - J< (c + x-x W=i 2 wk -(c-x-ty) J< (c + x-yj s) Moving Uniform load of intensity to and extent 21c f^-w-t::::?_:iri.__jc V X -1 >|^ Section at x (Span =2 c) Fig. 3. left of the section, viz. P acting upwards with a leverage (c — x), and a load area equivalent to a force w (k - y) acting downwards with a leverage | {k — y); hence the bending moment at this section is Mx = P{c - x) - IV {k - y) y k - wk {c + x-y){c-x)- -^ {k- yY (wk „ „, ivk'^\ w llkx \ — (--^^)--^) + y(- -y)y. (1) As the load moves about, y varies and the bending moment Mr at the section x depends upon the position of the load, that is, upon the value of y. To find the position of the load which gives the greatest bending moment at the point .( , it is only necessary to find the value of y which makes Mx a maximum ; now Mx is M 162 APPLIED MECHANICS. [chap. IX, I2kx \ greatest when the product ( y\ y is greatest ; and as the sum of the two factors of this product is constant, the product is greatest when the factors are equal to each other, that is when Ikx —-- y = y\ so that w • have kz (2) or y : k : : x : e. (3) This proportion expressed in words gives the following : — KuLE. — The greatest bending moment at any point of the span occurs when there is directly over it, that point in the load which is situated in the extent of the load in a position similar to that in which the point is situated in the e> tent of the span. Substituting in (1) the value of y in (2), we have This is the equation to the maxima bending moments, and may be written thus where Co is a constant quantity ; the locus is therefore a parabola with its apex above the centre of span, and the maximum of these maxima — that is, the maximum for the whole span — is at the centre, or where x = 0\ -*^T(^-:-)^4^-')"" = i'^('-*> (5) =2ivk By the preceding rule the maximum bending moment at the centre of the span occurs when the centre of the load is over the centre of the span. Graphical Solution. — Fig. 4. With a scale of feet lay off the span and draw a vertical upwards through the centre ; apply the rollers to the span; place any parabolic segment against the rollers with its apex on the vertical through 0; shift the rollers till the curved edge passes through the ends of the span, which it will do simultaneously, and draw the curve. Construct a scale of' ft. -lbs. for verticals such that OA = \ W{1- k), where W is in lbs., and I and k are in feet. Note that to give the maximum bending moment at any point, the load assumes a different position tor each point according to the above r\ile, and that it is possible to ful- fil the condition of the rule for every point without any of the load going off the span. Co;-. 1. — Suppose the extent of load equal to the span k = c, (2-')=l, then CHAP. IX.] B. M. AND S. F. FOR MOVING LOADS. 163 and we have max.-3/r = -—(c- - X-), and max.-'l^o = ^^.^ ic the same as for the span uniformly loaded (6g. 12, Ch. VII) and as shown in the preceding case. Note further, that the rule for fixing the position of the load so as to give the maximum bending nionieni at any point is fulfilled simultaneously for a/l points of the span ; as it is evident that when the centre of load is over the centre of span, every point of the load is over the corresponding point of the span. Cor. 2. — Suppose the extent of load to be zero ; then -0, (.4) = ., and the load is a rolling load for which TF max.iWx = -^i''' - ^')> a°<i max. -1^0 = ^ ^■^^ When the rule for finding the position of the load which gives the maximum bending moment at any point is applied to this case, it is found that the maximum occurs at any point oi span when the rolling load is at that point. As this is an important case, and leads to cases still more important, we will give a separate investigation. BendtTig Moments for a beam under a rolling load (fig. 5). — Consider any point of the span at the distance x from the centre, distances to the left being reckoned positive. Let B be the amount of the rolling load, and suppose it over the point in consideration. We may calculate the bending moment Mx from either of the two equations Mx = P {c - x), or Mx = Q{c + x). If the load moves to the right, then the upward supporting force F' is less than P, and M\ =^P' {c-x)< Mx ; if now the load moves to the left, the supporting force Q' is less than Q, and M"x= Q' {c + x) < M^; thus Mx decreases whether B moves to the right or left, that is, Mx, the bending moment at any point x, is greatest when B, the rolling load, is over the point. Let B be over the point x, then 7? P max. ^4 = P{C -X) - —{C+X)[C-X)= ~{c^ - X"'). This is the equation to the maxima bending moments ; the bending moment diagram is a parabola, with its axis vertical m2 164 APPLIED MECHANICS. [chap. IX- , t4j' and its apex above the centre of span ; and the maximum of these maxima^ that is the maxi- mum bending moment, for the whole span occurs at the centre when the load is over the centre; putting X = 0, we have the value of the constant m = \, and the principal equation to the parabola is '2c Bending Moment fcnr a beam, under both a roiling and an uniform dead load, — Let R be the dead load equivalent to the actual rolling load, and U the uniform load. For each load separately the bending moment diagram is a parabola with its apex over the centre. For the combined load, the diagram is a parabola whose apex is also over the centre, and whose modulus is the sum of their moduli ; hence jc = (^ + 27) ^'^ ~ ^^^ " — ir~^ ^^ ~ ^^' ™*™^ and M, = - {U -r IB) = ^ {U ^ 2R) .1, max. of maxima, the bending moments being in terms of a dead load throughout. Shearing Forces in a beam under a rolling load (fig. 6). — At any point of the span, F, the shearing force is positive and equal to P, so long as ^ is to the right of the point : since P increases as R moves towards the left support, F, is evidently a positive maximum when R is indefinitdy close to, and on the right side of, the points ^^ When R passes to the left of the point, F^ = P - R= - Q: since Q in- creases as R comes closer to the right support, it is again evident that F, is a negative maximum when R is indefinitely close to, and on the left side of, the point. When R is Fig. 6. inde6nitelj close to the point, P and Q have sensibly the same CHAP. IX,] B. M. AND S. F. FOR MOVING LOADS. 165 values as for R exactly at the point, and we have the follow- ing :— To find the maximum shearing force at any point x, place R over the point, and calculate P and Q for that position of the load ; these are respectively the maximum positive and maxi- mum negative values of Fj^. During the passage of the load, the shearing force assumes all values lying between these maxima ; and it is important to observe that the shearing force not only changes sign at any point as R passes over the point, but that it changes from its greatest positive to its greatest negative value, or vice versa, according as R is moving to left or right, and does so even although the load be moving slowly. If moving quickly, this sudden change produces what is called a hammer blow. The positive maximum at each point is the value of P as R comes to the point ; and since P is proportional to the remote segment, it follows that the positive maximum at each point is proportional to the distance of the point from the right end of span ; it is zero for the right end, and increases uniformly till it is R for the left end ; for, when R is just to the right of the right end of span, P is zero, no load being on the span ; and again, when R is just to the right of the left end of span, P is sensibly equal to R. For a rolling load the range at each point is constant, and is equal to R. Graphical Solution. — From the left end of the base draw upwards a vertical equal to R, and join its extremity to the right end of the base ; similarly from the right end, draw down- wards a vertical equal to R, and join its extremity to the left end of the base ; at each point the ordinate upwards gives the maximum positive, and the ordinate downwards the maximum negative, shearing force ; while the double ordinate gives the range. Two- Wheeled Trolly confined to a Girder like a Travelling Crane. Bending Moments for a. beam, vender a travelling load system of tv)o equal weifjhts at a fixed interval aj)drt (fig. 7). — Let R be the total load, and W^ = W^ the weights numbered from the loft end ; let 4.s be their distance apart, so that if G be the origin for loads, the abscissie of W^ and W^, are 2s and - 2s, respectively ; the origin for the span is the centre, and x is the distance from to G. 166 APPLIED MECHANICS. [chap. IX. First, let Wi be over the point x, the whole load being on the span ; then F may be calculated as if the whole load E were at G, that is, B 7? P=-(c-:r), and ^M^ = P{c - x) = - (c - x){c - x). (1) If the load travels a little to the right, P diminishes, and therefore the bending moment at x diminishes ; if the load travels until Wj is at a small distance a to the left of the section, then (fig. 8) ^ = 0"^ (c - •* + a), and, if x is positive, M'^ = P'{c-x)- W,a = ^ (c - .r + a){c - x) - ^ a 2c 2 -{c-x){c- x) - ^ ao: = ,M^ Pa "2c t..£+j: -2S — 4 '^ ^-£ , f/? J5£ „....€ -X. 2S K.---2S- ^?-H -** (? . X < ,ilf. -•C-X-t-et X ',1V>#72 -Xi -X-H "h: rin — c-x — -V- X-i Fig- 7. Fis. 8. n7j-^s---|---gs-- |ir. \Fy) — <'''- — ^jf (4S=C) } Hence ^M^ is the maximum bending moment for values of X from to c; that is, for any point in the left half of the span the maximum occurs when W^ is over it, provided the whole load be then on the span. By symmetiy the maximum for any point in the right half of the span occurs when W2, the right weight, is over it. Bending Moment Diagram (fig. 9). — Substituting for x its value (2s - x), we have -p i^/x = 2p (c - 2s + X) {c-x), (2) the equation to the maxima bend- ing moments for first half of span. The locus is a parabola with CHAP. IX.] B. M. AND S. F. FOR MOVING LOADS. 167 its axis vertical, and the principal equation is Y = — X^ ; it is therefore the same parabola as for a rolling load H (see fig. 5, p. 164). To find the horizontal distance to the apex Aj, find that value of x which makes the product (c - 2.s + x) (c - x) greatest ; since the sum of the factors is constant, this occurs when they are equal ; thus c-2s + x = c-x, or X = s. (.3) That is, the apex ^i lies to the left of at a distance s, one quarter of the distance between the two weights. To find the height of Ai, put x = s, and ,if, = -^ (c - sf, (4) the maximum of maxima for first half of span. It is evident that Az will lie to the right at a distance s, and that the two parabolas will intersect at I) on the vertical 'JP7 iR -. ir,|-^s-f-2s-JTT^ =^j; -(4S>C). (4S<C) W, ^L f- S -W— S -); 4S»{2W2).2C "f-.^ .t/„ is max: A,hE fA, OR = 0F={4S-C) Fiff. 10«. Fio-. \0b. through the centre. It is convenient to call the first half of the span Jielcl 1, and to say that this field is governed by W^ ; and we observe that the maximum in field 1 occurs when Wi, being in its own field, lies as far to one side of the centre, as G lies to the other. If it be possible for Wi to occupy every point in its field without W^ going off the span, we say that W^ can overtake its field. In the present problem it is necessary that 4s, the distance between the weights, be not greater than c th^ half- span, in order that each weight may be able to overtake its 168 APPLIED MECHANICS. [chap. IX. field. Two cases are shown on figs. lOo and lOj ; in the second the distance between the wheels being greater than the half span, one wheel can stand at the centre alone on the span giving the height E, a maximum and a rival to the maxima at Ai and A-,. The condition that all three maxima may be equal is engraved on the figure. w, f' A, P=Ii Fig. 11. The complete interpretation of the locus BAiDAtC is shown in fig. 11. Suppose the beam to extend beyond the supports at B and C, and to be fixed at these supports so that P and Q may act upwards or downwards. In the figure the travelling load is standing with G over B, so that /*= li, and Q is zero ; hence at L\, the point under W^, the bending moment is zero, and this is the point at which CA-iD meets BC. Let the load move towards the left until TFz is over any point as JT; Q now acts downwards ; at the point K, the beam is bent upwards and the bending moment is negative ; the value of this negative moment is Q .KC, and it is given by the downward ordinate Xb- "When IFz arrives over B, the bending moment a.t B is negative; and its value, Q.BC'= Wi .As, is given by the ordinate Bd. As the load moves farther to the left, the bending moment at each point, as W^ comes over it, is of the constant value Bd ; BC is now u cantilever under the down- ward load Q, and the bending moment at each jioint of BC is now negative, increasing definitely as the load moves towards the left. For all positions of the load, with no restriction on the value of 4.*, .B^i7)^2C' gives the maximum positive bending moment at each point; and the height of .i4i is the greatest positive bending moment tliat can possibly be produced by the load system. Bending Moments for a beam U7ider a travelling load system of two unequal weights at a fixed interval apart (fig. 12). — Let li be the total load ; PFi and JFj the weights numbered from the lef^ end ; let G their centre of gravity be the origin for CHAP. IX.] B. M. AND S. F. FOR MOVING LOADS. 169 loads, 2Ai and - 2/t2 being the abscissae of Wi and Wj, so that the distance between the weights is 2Ai + 2h>. Let Wi be over any point of the span whose abscissa is x measured (positive to i(<;»a--2/(^N m }--g Zhr-^-2h,^^^ — <;-aj— -i- x-^— X — ^ ii: J2i -c-xid-.; ih; '---y--r ■>; Q' Fis. 12. Fig. 13. left) from the centre as origin, and let it be understood all through that the whole load is on the span, or rides out on a continuation of it as on fig. 11. As in the previous case, i^^x ^^(c- ^)(c-x) (1) is the equation to the bending moment at any point x when Wi is over it. If the load travels a little to the right, P diminishes, and therefore 3f^. diminishes ; if the load travels until Wi is at a small distance a to the left of the section, then (fig. 13) M'x = n {c - X + a){c - x) - Wia That is, the bending moment at x any point of the span, when Wi is over it, is greater than when the load is in any other position, provided that the point itself is situated between B the left end, and F a point whose distance from the centre is or from the left end is BF=^^.2c. Jx 170 APPLIED MECHANICS. [CHAP. IX. Further, 01'=^. 2c, and the bending moment is greatest at any point of CF when li\ is over it. BF and CF are fields 1 and 2, and they are commanded by W'l and W2, respectively. Bending Moment Diagram (fig. 14). — For x substitute 2hi - X in equation (1), and we have ^M, = ^^(c-2h,^z)(c-x), (2) the equation to the maxima bending moments for field 1, The locus is a parabola whose axis is vertical, and principal equation is 2c it is therefore the same parabola as for a rolling load B. The abscissa of the apex A^, that is OSi, is found as before by equating the factors of equation (2), thus c - 2hi + X = c - X, or x = hi. That is, the apex Ai lies to the left of (? at a distance OSi = Jh ; similarly Ai lies to the right of at a distance OS2 = ho, each being half the distance between W and G. Putting x = h^ we have SiAi = ~{c- h,)\ (3) zc Similarly, SU. = § (^ - ^Y' (4) If the point /S, does not lie in field 1, the ordinates which are the bending moments for field 1 continually increase from zero at B the left end to their greatest value at F the other end of the field ; if S^ lies in field 1, then .Mni = SiA, -j^{c- Ihf (5) the maximum of maxima for field 1, CHAP. IX.] B. M. AND S. F. FOR MOVING LOADS. 171 Similarly, if aS^o be situated in field 2, the height of A2 will be the maximum bending moment for field 2. Suppose Wz > Wi ; then, since both parabolas are the same as that for the rolling load JR, and are therefore the same as each other, A. is higher than A^ because the quadrant CAnS-, stands on a longer base than BAiSi ; and (6) the maximum of maxima for field 2, and maximum for whole span Now F is both in field 1 and field 2 ; and when Wi arrives at F, the ordinate of the first parabola gives the maximum bending moment at F; again, when W., arrives at F, the ordinate of the second parabola also gives the maximum bend- ing moment at F\ that is, the ordinates at i^are equal, or the two parabolas intersect at D, a point on the vertical through F. It is well to observe that the maximum in either field occurs when the weight commanding the field, while lying in its own field, is as far from the centre of the span upon one side as G is upon the other. If it be impossible in one of the fields for the weight so to lie, then for that field the bending moment contin- uously increases towards the end of the field not coincid- ing with the end of the span. ! '-'^ i '' / '-'Tk ?>; ~\!eld2 1 "1 j 1 // _.i // •'' '^ i ' 1 N %. i 1/ ' -H i 1 i i \ \ B S,FO S-: Fk. 14. C'a^ie l,-2ki + 2/(3 smaller weight X span, or distance between the weights — shorter field. < total weight In this ease it is evident that each weight can overtake its field. Case 2, 2hi + 2^2 > smaller weight X span, or distance between the weights > total weight horter field (fig. 15). — The two apexes Ai and A2 are further apart than in the previous case ; D occupies a lower position, and is no longer on BEC, of which a portion kEh is above BAiBA-iC ; and the diagram showing the maximum bending moment at each point is now BIcEhAzC. The positions of h and h are indicated on the fig. lo. That E may be the same height as Ai we have, equation (4) S-iAi = — ie - hi)- ; and OE == \W2.l=hW2 -c; 2c equating these gives distance between weights = -— ? 2c. C) 172 APPLIED MECHANICS. [chap. IX. Graphical iSo/M<toM. — Construct G, the centre of gravity as indicated on fig. 15. Mark S\ and ^2 at distances half of those at which the wheels are from G. With any paiaboiic riglit segment, cut ont of cardboard or celluloid and used like a set square, draw the three parabolic arcs with their apexes Ai, A2, and An on the verticals through 'Si, ^2, and 0. The arc of the Icit parabola to pass through Ji, and of the right to pass through C, while the central one passes through both. Then construct a scale fur verticals so that the height of Ag shall read ^ Jil where Ji = IFi + TF2, and I is the span. Note thai the trolly on fig. 15 is standing with fVi in its most commanding position, which is neatly expressed by saying that fo^'' 2hr- — -i--,- -2h, f- ) --' \g 'v ) ,'''' ■f-h, •f- h,-', i / y'^ h ^y 4^ K S',"F OW S., %. Oist. between Wta »J I |( -Oiit. between Wta increased In ratio of Wts' Fig. lr>. the wheel is as far from the one support as the common centre of gravity is from the other. A convenient way of drawing the scale for verticals is to select one on which OAq reads a little less than it should read, and slope the scale over till a horizontal from Aq reads correctly, when the required values of SiA\, &c., can be lead off by horizontal lines ruled over to the sloping scale. If the trolly can ride out on the beam prolonged past the supports, fiA\ DA2C is the locus of maximum bending moments at each point: see fig. 11. If the beam be not prolonged, and if the two wheels he fnr apart, as in fig 15, then the remaining arc kEh is to he drawn by a second iiarabolic segment whose height shall bear to OAo the ratio of JFz to ( fF^ + Wi) or Ji. Note that F below the junction D divides the span into Jields proportional to the loads. On fig. 1, Ch. VIII, we had a parabolic locus which we supposed to be duplicate. A load at the centre made them rise and move towards each other. It can readily be shown that a load at i^on fig. 15 of magnitude will in a like manner make the two side parabolas approach CHAP. IX.] B. M. AND S. F. FOR MOVING LOADS. 173 when At, An, and A-, will all coincide. Such a fixed load at F and the trolly give as maxima BAoC, just as a single rolling load Ii = {W,+ W2) would do. Moving model illustrating the bending moments on a girder bridge dne to a trolly passing over it. — A model appeals to some minds to which analysis and descriptive geometry are tiresome. The model consists of two boards about 18 inches square, with a girder a foot long at the top of each. The rails are directly over the girders, so that the boards are spaced about 2 inches apart, the gauge of the rails (hg. 16). The trolly is supposed to weigh 36 tons, and it has three wheels ; the two wheels on the near side, axles 12 feet apart, ride on the near girder, and transmit to it a total load of 18 tons, 12 by the leading and 6 by the trailing wheel. The axle of the single wheel on the remote side is at the centre of gravity, being 4 feet from the leading and 8 feet from the trailing axle, and rides on the remote girder, transmitting 18 tons directly to it. On the face of the model are three hands which turn upon pivots as the trolly is pushed along the bridge. The side hands are pivoted at the ends of a horizontal line representing the span of the girder, 36 feet to a scale of 3 feet to an inch. This line is the base of the bending moment diagram. The central hand is pivoted directly over the point lying 6 feet to the right of the centre of the span, the point dividing the span in the ratio 2 to 1, just as the two wheels share the load. The height of the pivot is 5-^ inches. As the trolly is shoved along the bridge, the hands turn so that their intersections are always directly under the wheels. At the same time the pole, at the extreme left of the model, moves up and down, keeping the three threads attached to it always parallel to the three hands, one to each. The other ends of the threads pass through three eyelet holes in the face of the model, and little weights are attached to them to take up the slack out of sight. The eyelet holes are ranged on a vertical line, which is the load line to a scale of 3 tons to an inch. The polar distance is 6 feet ; the scale for bending moments is conse- quently 18 foot-tons to an inch. While the trolly stands still, the three hands form a polygon of three sides standing upon the base. This polygon is the instan- taneous bending moment diagram. It also is a balanced frame for the loads fixed in that position, and the threads from the pole form the reciprocal figure giving the thrusts along its sides. Between the boards a pantagraph is pivoted loose on the pivot of the central hand. One point of the pantagraph is 174 APPLIED MECHANICS. [CHAP. IX. driven by the trolly, and therefore the middle point of the pantagraph describes a horizontal path, with half the travel of the trolly. From it a thread passing over a pulley makes the pole move up and down with half the speed of the trolly. From the central point of the pantagraph three ivires radiate out — the middle one slides in a tube fixed at right angles on the pivot of the central hand, and rocks the pivot about ; the other two wires in the same way rock two pivots (not seen on the wood- cut), which are placed horizontally right and left of the central one, just as the end eyelet-holes are placed above and below the central one. These end pivots transmit the rocking motion to the pivots of the end hands by means of cranks and connecting rods. The mechanism will be readily understood by supposing the threads from the poles to be wires transpiercing journals at the eyelet-holes, and causing them to rock as the pole goes up and down ; and then supposing these geared to the pivots of the hands, one to each, by pulleys and endless bands between the boards, thus compelling each hand to be always parallel to the corresponding thread. The intersection of the hands below the leading wheel sweeps out a parabola which is painted on the face of the model. It is at the vertex for the position of the locomotive shown on the woodcut. The leading wheel being 16 feet from the left abutment, the centre of gravity is 16 feet from the right abutment. Now the supporting force at the left end is proportional to the second-mentioned 16, being in fact om-ha/f of it, this being the ratio of total load to span. And the lever of the left supporting force to cause bending on the girder under the leading wheel is the first-mentioned 16 feet. So that the bending moment there, for the position of trolly shown, is M_i = i X 16 X 16 = 128 foot-tons, a max. (Compare Example No. 8, p. 180.) Should the trolly move one foot right or left, one of the factors 16 becomes 15, and the other 17, giving a lesser product. Should it move two feet right or left of the position shown on woodcut, the factors become 14 and 18, with a still smaller product by a proposition of Euclid. Hence the locus of the intersection of left and central iiands is a parabolic right seg- ment, half base 16 feet, and passing through the pivots of those two hands. When the trailing wheel is 14 feet from the right abutment, CHAP. IX.] B. M. AND S, F. FOR M0VIN<1 LOADS. 175 the centre of gravity is 1 4 feet from the left abutment, and the intersection of the central and right hands is at a height, 3fi = I X 14 X 14 = 98 foot-tons, a max., and is at the vertex of its path, which is again a parabolic right segment, half base 14 feet, modulus one-half, and passing through the pivots of those two hands. The position of the pivot of the central hand is then on the intersection of those two parabolas. The height of that pivot is readily found thus : Observe that the perpendicular dropped from the pole upon the load line divides it into two segments which are the supporting forces. If the locomotive be moved till the central hand is horizontal, then the central thread being horizontal is perpen- dicular to the load line, and so we have tlie left supporting force 12 tons, two-thirds of the whole load ; hence the centre of gravity of the trolly must now trisect the span, and be 176 APPLIED MECHANICS. [CHAP. IX. 12 feet from the left abutment. The leading wheel then will be 8 feet from left abutment, and the bending moment under it is 12x8, or 96 foot-tons. But the central hand being hori- zontal, this also is the height of its pivot, namely, 5i inches. On the other face of the model not shown there is no central hand nor thread, and the triangle formed over the base with the two end hands is the instantaneous bending moment diagram for the remote girder, upon which only a single load, 18 tons, rolls. This intersection sweeps out a parabolic right segment, having 18 feet for its half base, with the same modulus as before, namely, one-half ; it passes through the pivots of the end hands, and its vertex is over the centre of span. The fact that the polar distance remains constant shows that the central parabola (giving the bending moments on the remote girder due to the 18 tons rolling on one wheel) and the pair of parabolas (giving the bending moments for the near girder) constitute a pair of diagrams to one common scale. Of course the scale for the single central parabola, which alone appears on the back face of the model, is readily constructed. The scale must be such that the height of the vertex of the central parabola shall measure on it \WL= 162 foot-tons. Otherwise, as already stated, the polar distance being 6 feet makes the scale for bending moment 18 foot-tons to the inch, that is, 6 times finer than the scale for tons. The height to the central parabola at a point 6 feet to one side of the centre is one-half oi the product of the two segments into which that point divides the span, or it is | x 24 x 12 = 144 foot-tons. But the height of the central pivot is 96 foot-tons, so that the depth of the central pivot from the central parabola is 48 foot-tons. Now 48 foot-tons is the statical moment or product of 12 tons, the leading wheel into 4 feet its distance from the common centre of gravity. Another way then of stating the position of the central pivot is to say that the vertical through it must di\TLde the span in the ratio 2 to 1 in which the leading and trailing wheels share the total load, and that it must lie on this vertical at a depth below the central parabola by an amount given by the statical moment of either wheel about their common centre of gravity. The model is made by Messrs. Dixon & Hempenstal, Dublin, and may be seen in Dublin, at the Engineering School, Trinity College, or at the College of Science, Poena. The description of this model in our second edition was original, except that the figure appeared on the correspondence CHAP, IX.] B. M. AND S. ¥. FOR MOVING LOADS. 177 in Mr.Farr's paper on "Moving Loads on KailwayUnder-Bridges" in the Transactions of the Institute of Civil Engineers, 1900. A model of ruder construction, with sliding frames, and called " A Bending Moment Delineator," is described in the Transactions of the Engineers and Ship Builders of Scotland, November. 1889, and was exhibited at the Munich Exhibition in 1893, and described in the /Catalog. We add here a photograph of the " Delineator " from a large black-board model (fig. 17). It will be seen that the three arms are constrained bv slots to move exactlv as the hands on Fig. 17. the model 'fig. 16). The loads on the wheels are equal, being 9 tons each, and are due to a uniform load of 1| tons per foot- spread on the wheel base. The bending moments on the wheel base itself are shown by the parabolic segment painted on a " Distorting Table," which consists of a number of vertical strips of wood slightly spaced, and connected to each other by two horizontal lazytongs across their backs, each strip being pinned to a joint of the upper and under lazytongs. This N 178 APPLIED MECHANICS. [CHAP. IX. distorting table has holes, one at each coiner, which slip over four pegs connected to the cross-arm which bears the two knob- handles at the ends. If the distorting table be pulled forward off the pins, it can then be stretched like an accorciion, but, when on the pins, it can only be subjected to sliding displacement. In the middle position the figure painted on the distorting table is a right parabolic segment ; but when the trolly is shoved to one side as shown on the photograph, it is oblique, but of course the lengths of the white strips are unaltered. The two oblique slotted bars are always tangents at the ends of the parabolic segment. If the uniform load drop directly down upon the girder, and move about on it without the intervention of the trolly, then the instantaneous bending moment, as shown on the photograph, is the parabolic locus for the loaded central segment of the span and the tangents to it for the two side unloaded segments. Compare fig. 7, Ch. VIII, which this model illustrates when the load is at rest. In the photograph the load is so disposed on the girder that the left-hand quartering point of the load is directly over the left-hand quartering point of the span. Consider the bending moment at the left quartering point of the span for this position of the load. It is given by the height above the base of the top of the left quartering white stripe painted on the distorting table. We know that this is the maximum bending moment at this point of the span (see figs. 3 and 4, Ch. IX). Now the model exhibits this beautifully ; for if we move the load a little to the right, the " white strip " is replaced by one shorter than itself, for which reason the top is Imver, but the base of the parabolic segment has risen^ for the cross-bar on which it stands has rotated clock- wise about a pivot under the centre of the girder which exactly neutralises the efi'ect of the shorter " white strip " replacing the original one. That is, for a small motion of the load to the right-hand the bending moment at the quartering point of the span (as given by the new instantaneous diagram) is unaltered. Similarly it is unaltered for a slight motion of the load in the other direction. At that point, then, the variation of the bending moment is zero for a small movement which is the criterion for a maximum. This " distorting table " also serves to illustrate the addition of the parabolic slope to the straight slope (see theorems in Chapter XI). It also serves to show how the shearing force diagram painted on it for any moving load is to be distorted and superimposed upon the shearing force diagram for the dead load. CHAP. IX.] 1$. M. AND S. F. FOR MOVING LOADS. 179 EXAMPLKS. I. An advancing load, as long as or longer than the span, and of intensity I.V tons per foot, conies upon a beam 36 feel long. Find the maximum bending moment for the whole span for all positions of the load mas.^^^o = iJr. I = ^54 x 36 = 243 ft. -tons. Find the maxima at intervals of six feet 4c therefore ^f±^8 = 0, .l/±i2 = 135, .¥±6 = 216, and Mq = 243 ft. -tons, all maxima. 2. For the same beam of 36 feet span, with the same intensity of load Ih tons per foot, the whole length of the load being now only 12 feet, find the maximum for the whole span max..l/o = ^TF{l-k) = i X 18 X (36 - 6) = 135 ft. -tons. Find now the maxima at intervals of 9 feet m^xMx = — (c- - x^-){2 - ^] = -i%(324 - z^) ■ and MiB = 0, Mg = lOli, Mn = 135 fu-tons. all maxima. 3. A beam 42 feet span is subject to an advancing load of 3 tons per foot and 12 feet long. Find the maximum bending moment at 7 feet on either side of the centre. Where is the centre of the load situated when this maximum is produced ? Atis. max..ilf±7 = 288 ft. -tons. Five feet from centre of span. 4. In the previous example, find the maximum bending moment for the whole span, for all positions ot the above load. Alls, njax.-il/o = 324 ft.-tons. 5. A beam 30 feet span is subject to a rolling load of 40 tons : find the maximum bending moment for M'hole span. At what point does it occur, and how is the load then situated ? Aus. maL-iVo = j-ff . I = 300 ft.-tons. Ji is at centre. 6. In the previous example, find the maxima bending moments at intervals of 5 feet. How must the load be situated in each case P Am. max-^/r = ^ (c- - ar^) = 1(225 - x^) ; therefore -" M±is = 0, M±io = I665, J/±5 = 266§, and M^ = 300 ft.-tons, aU maxima. The above load of 40 tons in each case is over the point. 7. If the load of 40 tons in the above examples be spread uniformly over 3 inches, instead of being concentrated at a point, how much are the above results in eiTOr ? Ji , „ „ /, extent of load\ raax.i»/^r= - (c- - X-) 2 ) : 4c \ extent ot span/ N 2 180 APPLIED MECHAiNICS. [CHAP. IX. this differs from the above expression by the factor / 2yt\ , / •25\ 239 I (2 - -) = ^2 - -) = -, hence the results above would be in excess by a ^J-gth part, or by -i^j-ths percent. 8. A beam 36 feet span bears a travelling load of 18 tons concentrated on two wheels 12 feet apart, there being 12 tons on the left and 6 tons on the right wheel. Find the maximum bending moment; the equaiiona to, and amounts at intervals of 6 feel of, the maxima bending moments. Data: jr, = 12, W2 = &, if =18 tons, c = 18, 2/(1 =- 4, 2A2 = 8feet. Dividing 36 directly as 1 and 2, we have JiF and FC, fields 1 and 2, equal respectively to 24 and 12. Since the distance between the wheels is not greater than tlie shorter field, the eXMmjde comes under Case I. (see fig. 14), and the maximum bending moment at each point is the locus liDC, M'hether the load is confined to the span or makes a transit. The maximum bending moment for whole span occurs at x = h\ = — 2 ; its value is to be found by supposing the load standing with the left wheel two feet to the left of the centre as on the model (fig. 16), to which the text now refers, and then calculating the moment at that point, P = 8 tons, and \M.2 =128 ft. -tons max. The equations to the maximum bending moment at each point are. For field 1, iM^ = ^ (14 - x) (18 + x), for values of x from - 18 to 6. For field 2, sJ/^ = ^{IQ + x) (18 - x), for values of x from 6 to 18. And evaluating at intervals of six feet, we have J/.18 = 0, Jf.i2 = 78, i/.6 = 120, J/o = 126, J/e = 96, Mn = 66 ft.-tons, &c. Note. — The height of Az may be calculated from fig. 14 thus : — OSi = 2, the base of quadiant then is 16 ; the moiiulus is the load divided by span, that is A ; hence the height i>f apex, or modulus into base squared, is 128. To inake the graphical solution, lay off 0S\ = 2, 08^=4, and dr&w JJIjC as previou.-ly desciilied ; make a vertical scale upon which OAo = 2 x 18 x 18 = 162 (see model, fig. 16). 9. A beam 36 feet span bears a travelling load of 18 tons concentrated in. equal portions on two wheels 12 feet apart. Find the maximum bending moment. In this case, Wi = Wz, and 4s < c, so that (fig. \0a) UHx = OS2 = s. The maximum occurs at 3 feet on either side of the centre, whether the load is confined to the span like a travelling crane or makes a transit like a truck ; its amount is found by assuming the load to be standing with the left wheel 3 feet to left of centre. Aug. 1' = 7^ tons ; iMa = zM.s = 112j ft.-tons, maximum for whole span. That is, half of the square of fifteen (see the model, fig. 17). 10. If the load of Ex. 9 shift till the left wheel is 6 feet from the left end of the beam, calculate the bending moment 9 feet from the left end. P = 12 tons, J/9 = 12 X 9 - 9 X 3 = 81 ft.-tons. This is the position of the load on the model (fig. 17), and is the bending moment on the girder under a point 3 feet to the right of the left wheel. Also J (6* - S"^) or 20J is the bending moment at the same point on the wheel base or 12 foot beam joining the two wheels, due to the uniform load of 1^ tons spread over it. These two added together give 81 4 20^ = 101; ft.-tons. This CHAP. IX.] B. M. AND S. F. FOR MOVINOx LOADS. 181 is the maxiniinn bending niomoiit on the girder at the point 9 feet from the left end, if the uiiiform load rested directly upon it without the intervention of the two-wheeled trolly. Compare this Example with the preceding, and with Exercises 2 and 3. 11. A beam 56 feet span bears a travelling load of 16 tons concentrated on two wheels 32 feet apart, 7 tons being on the left wheel and 9 tons on the right. Find the maximum bending moment during the transit. From equation 7, p. 171, iz.v^:z::5o, = 32feet, which liappens to be the distance between the weights ; hence there will be two equal maxima, one at 7 feet to the right of the centre when the greater load is over it ; the other, at the centre also when the greater load is over it, the smaller load not being then on the span. Place the load in those positions respectively, and calculate the moments. Ans. 2.U.1 = iM'o ■■= 126 ft.-tons. 12. A travelling load of 5 tons concentrated on two wheels 10 feet apart, 1 ton being on the left wheel and 4 tons on the right, passes over a beam of •10 feet span. Find the maxima bending moments at intervals of 4 feet, and the maximum for the whole span. Distance between weights x ratio of weights = 40 feet = span, so that k coincides with B (fig. 15), ;ind therefore arc BhE lies everywhere above arc BB ; that is, the maximum at each point of span occurs when the greater weight is over it. Further, the height of A2 is greater than that of E, since the distance between the weights does not exceed -^„-lhs of the span. Placing the greater load over points at intervals of 4 feet and calculating the bending moments, or substituting into the equations to the loci Bh and hAoC, we have zM'io = 0, zM'iG = 14-4, 2M'i2 = 25-6; 2M^ = 35, -Mi = 42, zMo = 45, 2^-4 = 44, oiW-g = 39, a.V-x. = 30, 2.I/-I6 = 17, 2iIf-2o = ft.-tons. Since the greater weight lies 2 feet to the right of the centre of gravity of the load, the maximum lies at 1 foot to the right of the centre of span, and 2Jlf-i = 45^ ft.-tons maximum for whole span during transit. The equations to the loci from which these may be calculated are for Bk, ,M'. = -A(c- - x^) = -i, (400 - x% for values of x from 20 to 10 ; and for hAiC .Mr = -/o- (.0 + 2/i2 - x){c + X) = ^ (18 - ^) (20 + x) for values of x from 10 to — 20. 182 APPLIED MECHANICS. [chap. X. CHAPTER X. BENDING MOMENTS AND SHEARING FORCES DUE TO A TRAVELLING LOAD SYSTEM. Bending Moments for a beam under a travelling load system of unequal weighU fixed at irregular intervals the load being confined to the span so that no v-eight passes off (fig. 1). — Let R be the total load ; W^, Wi, ... Wr . . . IVn, the weights numbered in order from the left end ; G the centre of gravity and origin for the weights ; 2Ai, 2^2 . . . 2h,. . . . 2h„, the abscissse. of the V.'> Vil- ^n,'^ >< ' %x<»" ]T7| l4 ^4 - ^trl ; llly.^,.^-^-- \s \'- <S.b ^< w; ' 'fVr'^^ p' C*X*Ot y\ nv '■ ■ Wr- .f Fis. 1. weights, those to the right of G including their negative sign ; the origin for, and 2c the length of, the span, distances towards the left being positive; and let cc be the abscissa^of any section. First. — Let the load be in a position such that the i'"" weight is over the section ; cr^, x^ . . . Xr.\, the abscissae of the (r - 1) weights, measured from ; it the abscissa of G, and c that of the left end of span ; then and R rM^ = -{c^x){c-x)- W,{x, -x)- W,{x, - x) -.. .- Wr.^iXr-, - X) - Wr{Xr-x); (1) the last term is zero, since Xr = x, and it may either be expressed as above or omitted. CHAP. X.] B. M. AND S. F. — LOCOMOTIVE. 183 Second. — Let the load be situated at a short distance a to the light of its former position ; then ^' = 2^ (c + ^ - ") ; and M\ = -- (c + X-- a)(c- X) - W,{x, - a - x) - Wi{x2 -a- x) 2c - ... - Wr.i{Xr.i - a -X) 2c = ,.M, -■^(c-x)a + (W,+ W, + ...+ Wr.,)a. (2) <,.!/„ a -^(c-,r)>^rW^^ or if c - -^^ ^ 2c> a; ; 2c Ji that is, the bending moment for the first position is the greater, if the distance of the section to the left of the centre is less than c '—^ — ^ . 2c ; or, what is the same thing, if the distance of the section from the left end is greater than ^ — f-, . 2c. Third. — Let the load be situated at a short distance « to the left of the first position ; then P' = — (c + X + a), and M"x ^ -^ (c + X + a) (c - x) - Wi{xi + a - x) - Wzix, + a - x) 2c - ... - Wr{Xr + a - X) = ,J/, + ^ (c - ,r)o ~{W, i- W,-^...+ Wr)a. (3) R "^ ^{ W^ < rM.„ if - {c-x)< Sx'X W), or if c- ^^^-j^ .2c<x; that is, the bending moment for the first position is greater than for the third, if the distance of the section to the left of the centre is greater than c „ . 2c ; or, what is the same thing, if the distance of the section from the left end is less than M!Q . 2«. 184 APPLIED MECHANICS. [CHAP. X. Hence there is a portion of the span, lying between the point whose distance from the left end is — ^-^ — . 2c, and the '2i'"^( TV) point whose distance from the left end is p — - . 2c, such that the bending moment at any section in that portion is greater when the r^^ weight is over it than for any other position of the load, if for each position all the weights are on the span as premised. This portion of the span we call the r''^ field, and we say it is commanded by the r"" weight. The extent of this r*'' field is ^^J.2.-^p.2c.I^.2c, (4) .he same fraction of the span as the r"* weight is of the total load ; hence, in order to mark the fields, the span is to be divided into as many portions as there are weights, these portions being proportional to the weights and in the same order. The maximum bending moment at any point occurs when the weight, which commands the field in which the point lies, comes over that point. Since no weight is to go off the span, sometimes there is a part of a field which the commanding weight cannot occupy, and then the weight is said not to be able to overtake that part of its field ; as will be proved when we come to the graphical solution, the maximum bending moment for such points occurs when the commanding weight is as close thereto as it can be brought. Into the expression for ,J/, the maximum at any point of the r'* field, substitute as follows : — x = {x- 2hr), {x, -x) = (2h, - 2hr), [x, - .r) = {2h, - 2h,.), &c., . . . !>,._, -x) = {2h,.., - 2//,,) ; and we have rM,= ^^{c + x- 2K) {c-x)- W, (2A, -2/^) - ]V, (2A, - 2A,) - ... - Wr_, (2A,., - 2/^.) = ^ (c + '• - 2hr) (c - X) + 2hr Sr' {W)- Sr\ W.2h) ; (5) this is the equation to the maxima bending moments for the r'* field. The locus is a paral)ola ; its axis is vertical ; its apex is above the span, and may lie to either side of 0, the centre of CHAP. X.] r.. M. AND S. V. — LOCOMOTIVK. 185 span ; its modulus is ^, a quantity which is the same for all fields, and is the same as the modulus of the parabola due to E as a rolling load. The abscissa of A,-, the apex of the /•''' para- bola, is that value of .r which makes ,J\^x or (c -{ x - 2h,.) (c - x) greatest, that is where a; = //,.; (6) hence the apex of each parabola lies on the same side of, and horizontally half as far from the centre of the span as the commanding weight is from G, the centre of gravity of the load. The apexes for some of the fields may lie in their own fields ; and in such fields the maximum of the maxima is given by the ordinate of the apex ; for other fields, the apexes may lie out- side of their own fields, and in these there is no maximum of maxima, but the maxima increase continuously from one end of the field to the other. Bending Moment Diagram (fig. 2). — The locus is the polygon BDtDn, &c., formed with parabolic arcs £Di, Z>iA, &c. ; each parabola being the same as BA^G that for the rolling load B, but lying with their apexes at the distances OSi = h^, OS-. = hz, &c., where hi, lu, &c., are half the respective distances of W^, Wo, &c., from G the centre of gravity of the load. The para- bolas intersect in pairs on the verticals through the junctions of the fields ; that is, through F^, Fo,, &c., points such that BFi : F,F,, &c. : BC : : IF, : W,, &c. : B; for, if F,. be the junction between the r"' field and the (?• + If' field, then Fr is the last point in the '/'' field ; the maximum at Fr occurs when W,- is over it, and is given by the ordinate at F,- to the r''' parabola. Again, Fr is the first point in the (r + 1)''' field ; the maximum at F,. occurs when W,. + , is over it and is given by the ordinate at F,. to the (r + 1)^^ parabola. But the maximum at Fr is some one definite quantity ; hence the two parabolas have a common ordinate at Fr. Maximum Be7uling Moment for whole spaji. — In fig. 2, A-^ and Ai lie respectively in their own fields ; and the one which has the greater ordinate gives the maximum for whole span. In the general case one or more apexes will lie in their own fields ; one at least, as we cannot conceive of such a series of curves lying so that every apex is inside of another of the curves. The ordinate of the apex that lies in its own field, if only one is so situated, is maximum for the whole span ; the .ordinate of the highest apex, if more than one be so situated. 186 APPLIED MECHANICS. [CHAP. X. Suppose the first curve continued to the right, past i>, ; the second past D2, &c. ; then each parabola is the locus of the bending moment at each point as the corresponding weight comes over it, the whole load being on the span. If the load stands still in any position, as, for instance, that in which the load is drawn in fig. 2, the bending moment at the point where Wi stands is the ordinate there of the first parabola, at the point where W^ stands the ordinate of the second parabola, &c. if then the ordinates of these points be drawn each to the proper parabola, and the tops of the ordinates be joined, we will have the bending moment diagram for that set of fixed loads ; we will have in fact the diagram shown in fig. 2, Ch. VII, because the load is now the fixed load shown in fig. 3, Ch. V. Further, for the position of the load shown in fig. 2, the straight line joining the tops of the two ordinates, one drawn to parabola 2 from the point where Wn, stands, and the other to parabola 3 from the point where W3 stands, will pass through D^ the intersection of these parabolas, because the horizontal projection of that joining line is constant, being equal to the distance between W^ and W^ ; and we know that one end of this joining line coincides with Do when either of the weights W2 or JF3 is over F^ ; hence by the theorem (D, Ch. VI} it will always pass through D2. Now, for the position of the load shown, the joining line gives the bending moments at all inter- mediate points, so that the ordinate of D^ is the bending moment at F,. for that position of the load ; and similarly for any other position for which F2 lies between the weights W2 and JF^j. In other words, the bending moment at F the junction of two fields is the same, whether the weight com- manding the field on either side is over it, or whether the load stands in any position for which F lies between those weights. When the load stands in a position where those two weights are both to one side of F, then the joining line prochtccd still passes through Z), but the ordinates of the produced line are not bending moments. For instance, the line joining the tops of the ordinates on fig. 2, one drawn to parabola 4 from the point where W^ stands, and the other to parabola 5 from the point where T[\ stands, will, when produced, pass through Dx, the junction of those parabolas ; however, not the ordinate at Fx to that produced line, but the ordinate to the line wliich termin- ates at C, gives the bending moment there. On fig. 2, observe that if the load moves more than one foot to the left, W^ goes oil' the s[)an ; and there is a portion CHAP. X.J B. M. AND S. F. — LOCOMOTIVE. 187 of field 3 at its left end, which W:^ cannot overtake, and for which the corresponding portion of parabola 3 is dotted, being inadmissible. For the position of the load shown, we saw that the bending moment at each point of that portion of field 3 is given by the ordinate to a straight line from Do to the top of the ordinate of parabola 3, at the point where W3 is stand- ing ; that is, by the chord of parabola 3, from D^ to the top of tliat ordinate. Now, the closer JFs comes to Fo the steeper will that chord be, and consequently the greater the bending moments at all these points. Hence the chord of the dotted or inadmissible part of parabola 3 gives the maximum bending moment at each point of the portion of field 3 which W3 cannot overtake. Similarly for any portion of any field which the commanding weight cannot overtake, the chord of the parabola, instead of the arc, gives the maxima bending moments for the whole load on the span. Fig. 15, p. 172, has already been quoted as an example of this, when we assumed the arc Bk, chord kD, chord Dh, and arc hC to be the diagram of maxima bending moments for load confined to the span. Graphical Solution for Bcndinq Moment Diagram (fig. 2). — Lay off the wheel base and find G the centre of gravity of the load by analysis as at fig. 3, p. 93, or graphically as in fig. 3, p. 123, and indicated in fig. 2. Lay off BC equal to the span ; divide it at the points F into fields proportional to the weights, either by arithmetic or as indicated on the diagram, and draw vertical lines through them to separate the fields. Lay o&OSi equal to half the distance of W^ from G, OS2 equal to half the distance of Wo from G, &c. — each point S being on that side of the centre on which the corresponding weight lies with respect to G ; draw verticals through the points ;S' ; apply the parallel rollers to BC; place any parabolic segment against the rollers with its apex on the vertical through S^ ; shift the rollers till the curved edge passes through B, and draw the arc BDi, stopping at the vertical through F^ ; shift the segment till the apex is on the vertical through So, move the rollers till the curved edge passes through A, and draw the arc D-J)>, stopping at the vertical through F2. Similarly draw arc after arc in succession for each field, and if the arc for the last field passes through C the extremity of the span, it checks the accuracy of the drawing. Lastly, shift the segment till the apex is on the vertical through the centre ; move the rollers till the curved edge passes through the two extremities, mark A^, and con- struct a scale for verticals and bending moments such that OAq = \R.l. Find by inspection the portions of the fields I I t '' 'm ■ ; I ■» \' i \t. field l-'jL-field 2-jh field 3 j|l-^ field 4 » field 5 i VERTICALS. HORIZONTALS _2?0FT TONS. I I I I I I I I I I -^ CHAP. X.] H. M. AND S. F. — LOCOMOTIVE. 189 which the commanding weights cannot overtake, and over such portions replace the arcs by chords. Then the locus gives tlie maximum bending moment at each point for all possible posi- tions of the load, the load being confined to the span. Graphical Solution for a Diagram of the Square Hoots of Bending Monieiits. — Divide the span into " fields " proportional to the loads on the wheels. From the middle point of the span prick points at distances one half of those at which the wheels lie from the centre of gravity of the load. From those points as centres draw circular arcs one over each fiekl, and we have a diagram of the square roots of the maximum bending moments. It only remains to construct a scale. On the span draw a semicircle and construct a scale such that ««, the height of the crown of the semicircle, shall measure on it the square root of a fourth of the product of the total load and total span. The height of ffi may be scaled off the drawing, and its value as compared to the known height of a^ found directly by the rule of three without constructing a scale.* Shearing Force Diagram (figs. 3 and 6). — At any point the shearing force increases as each weight in succession approaches from the right; and when a weight passes the point, it suddenly diminishes by an amount equal to that weight. At each point there is a maximum when a weight is just to the right of the point, and a minimum when it is just to the left. In order to find for any point the maximum and minimum corresponding to a particular weight : — Place the load system so that this weight is over the point ; from P subtract the weights to the left of that weight for the maximum, and further subtract that weight for the" minimum. In figs. 3 and 6, a locus giving the maximum at each point due to a particular weight approaching, is shown by a full line ; the locus giving the minimum due to the same weight is evidently parallel to the first, and below it at a constant distance equal to that weight ; this parallel locus is shown by a dotted line. In each of these diagrams there are five loci drawn in full lines, and giving the maximum at each point due to the approach from the right of each of the five weights respectively ; of the five dotted loci giving the minimum at each point due to the * This simple and elegant graphical solution of so complex a problem by means of circular arcs only was published in a note in the Transactions of the Institute of Civil Engineers for 1900. 190 APPLIED MECHANICS. [CHAP. X. receding of each weight respectively, only one is shown, and it is drawn parallel to the locus shown by the lowest full line, and at a constant depth below it equal to the weight at the right end of the load. Having determined the full lines or positive loci, it is easy to draw the others. The first locus Aa is the value of F when JFj is over any point ; so that as the load comes on from the right end, and so long as Wi alone is on the span, Aa is a portion of the diagram due to TFj as a rolling load ; that is, it slopes at an angle whose tangent is W^ -f 2c, or, in other words, at a rate proportional to Wi ; further, Aa extends over a horizontal distance equal to that between IV^ and TF2. When IV. comes over any point, W, and W2 alone being on the span, then F^ is calculated by finding F and subtracting the constant quantity TVt ; F is now increasing as for a rolling load ( Wi + W^) con- centrated at its centre of gravity, so that the second link aa slopes at an angle whose tangent is ( W^ + Wo) - 2c, or, in other words, at a rate proportional to Wi + W^ ; further, aa extends from the point where the preceding link ended, and continues through a horizontal distance equal to that between W2 and W3. Thus, the locus Aaaa . . . begins at the right end of the span, each link sloping more and more at rates in direct proportion to the sum of the weights on the span, and extending respec- tively over horizontal distances equal to those between the weights; the last link extends over a distance which is the excess of the span over the extent of the load. On fig. 6, the first four links are short and equal, the last one is long. The other loci Bbh . . . , Ccc . . . , &c., consist of links sloping more and more as weights come on at the right end, and less and less as they go off at the left ; for instance, Ccc . . . (fig. 6) consists of two equal short links and a long one increasing in slope, and two equal short ones decreasing in slope ; the rale of slope of each link is directly proportional to the sum of the weights on the span at the time corresponding. After having drawn the first locus Aaa . . ., the initial points B, C, D, &c., of the other loci are found as follows : — The ordinate of B represents the value of the shearing force at the right end of span when W^, is just to the right of that point ; if the load be placed in this position, the shearing force at the right end of span, and in the interval between that end and the point where W^ stands, is constant; the value of the shearing force just to the left of Jr, is given by the ordinate of a, the left extremity of the first link Aa, and this quantity diminished by JF, is the value required. Hence CHAP. X.] U. M. AND S. F. — LOCOMOTIVE. 191 B is on the vertical through the right end of span, and at a depth W^ below the level of the first joint a on the locus Aaa . . .; similarly G is on the same vertical and at a depth W-i below the level of the first joint h on the locus Bhb .... Graphical Solution for Shearing Force Dia<jram (fig. 3) — The example shown in the figure is the same as that for which ON THE SPAN Fig. 3. the bending moment diagram is given in fig. 2. Lay the weights up, in order, on the vertical through the left end of span, and then down again in order ; draw a ray from the right end of span to each junction ; rays to joints in ascending order are drawn in full lines, and to joints in descending order are drawn in dotted lines. Draw another line Ae equal to the span, so as not to complicate the figure. Draw the locus Aaa . . . link after link parallel to the slopes in ascending order, each link extending in order for a horizontal distance equal to that between the weights in pairs, the last link completing the locus ; in the figure, the first link Aa extends for a distance equal to that between W-^ and W^ ; the second link aa to that between W^ and Wz, and so on ; the last link extends for a distance equal to the ditt'erence between the span and the length of the load. Plot B on the vertical through the right 192 APPLIED MECHANICS. [CHAP. X. end, at a depth equal to Wi below the level of the first johit a in the locus already drawn; draw the locus ^&& ... parallel to the respective slopes in ascending order ; the first link Bb is parallel to the slope for (ir, + W-,), each link extends till a new weight comes on, the second last link extends till Wi goes off, and the last link is drawn parallel to the slope for (W2 + W3+ W\ + Wi). Plot C at a depth equal to W^ below the level of the first joint b, and draw the locus Ccc . . . ; the first link Cc is parallel to the slope for {Wi + Wz+ W^); the other links are drawn parallel respectively to the slopes in ascending order, and when these are exhausted the remaining links are drawn parallel to the slopes in descending order ; the extent of each link is determined as each weight after JF3. comes on, and then as weight after weight goes off. In the same way, for each weight, a locus is drawn in full lines, consisting of as many links as there are weights ; the highest ordinate at any point gives the maximum positive shearing force thereat, for transit of load. Dotted loci are drawn, one parallel to each locus shown by a full line, and below it at a distance equal to the weight to which it corresponds ; the deepest ordinate at each point gives the negative maximum. On the diagram, ee'...E' is drawn parallel to ee...U; and since, in this case, the locus e'e'...E' gives the maximum shearing force for every point of span, the other four dotted loci are not shown. Bending Moments for a Beam under a travelling load si/stem of equal iceighta fixed at equal xnlervals and confined to the span (figs. 4, 5). — The locus BD\I)iDzO drawn as in the general case, will be symmetrical about the centre. If the number of weights be n, and their distance apart be ith of the span, then it is evident that the span will be divided into « equal fields whose common extent is the same as the distance between two weights, and that each weight will just be able to overtake its field ; it the distance between two weights be less than ith of the span, each weight is still able to overtake its own field. There- fore, for the common distance between the n weights equal to or less than ^^th of the span, the bending moment diagram is the locus BD\D%Di . . . C everywhere following the curves ; the maximum bending moment is at the centre, or at one quarter of the common interval on either side of the centre, according as w is odd or even. On the otlier hand, when the common interval between the n weights is greater than ith of the span, each weight will always be in its own fieM, and will only be able to overtake a portion of its field ; the bending moment diagram is the locus BDyDiBz . . . C following the arcs for portions of fields overtaken by the weights commanding, and the chords for the remainder. If « be odd, the middle weight can always be placed at the centre of the span, and at that point the maximum bending moment will occur; if n be even, and it be possible for a wei^ihl to come as close to the centre as or closer than a quarter of the conimon interval, then the inaximun) bending moment will be on both sides of the centre, and at one quarter of the common interval therefrom ; if a weight cannot come so close, the maximum will still be on both sides of the centre, and nt points as near thereto as the weight on either side of it may approach. CHAP. X.] B. M. AND S. F. — LOCOMOTIVE. 193 For n even, the proof that the maxinmm is at the point, a quarter of the cominon interval <m either side of the centre may be shown thus :— Let 6 be H'/ = \w.= \m = Im k D.. ^ 4 ,_ Span No.of wts. Fig. 4. over the centre : then the weight nearest to the centre will he distant one-half of an interval ; if the end weight be distant from the end at least a quarter of an interval, it will be possible for a weisht to approach the centre, a quarter interval ; hencH, the span must equal the w — 1 equal intervals, and at least two quarter intervals more, that is = _ 2« 2c > (» - 1) s + |s ; 2c > — - - 1 — s. It is readily shown for a system of n equal weights at a common interval not greater than Hh of the span and confined to the span, that the locus] of.- the maximum bending moment at each point will entirely include the loci due to any fTn { ir.. __J'I|- i 1 i iw; = 1 11', = H h>: m a" O Fig. .5. smaller number of the same equal weights at the same common interval. Each passes through the points T)\ of the last, and if the common interval he less each figure is inside the last without touching. For intervals greater than Ath of the span the figures cut each other as shown on fig. -5. O 194 APPLIED MECHANICS. [chap. X. Also fig. 6 shows tl;e shearing force (iiajrram already described conjointlv with fig. 3. -4^ Fig. 6. Examples. 1. A beam of 42 feet span supports the five wheels of the locomotive shown on fig. 2 : find the locus of the maximum bending moment at each point for all positions of the load, it being understood that no wheel moves off the span (fig. 2). Loads 5, ■'), 11, 12, 9 tons = 42 tons. Intervals o, 8, 10, 7 feet = 30 feet. Distances from G, 17, 12, 4, -6, - 13 feet. Dividing the span in the ratio of the weights, we have the extent of the fields as follows : — 1st field from 21 to 16 ; 2nd field from 1(5 to 11 ; 3rd field from II to ; 4th fiehl from to - 12 ; .5th field from - 12 to - 21. Substituting in the general equation ,J/, = £ (c + ,■ - 2f<r) (c-x)-\- •IKS.rK ^n - 21*-' (/'■ ■ 2/0, we have I Jf^ = _ (c + a- - 2Ai) (c - a:) = (4 + x) (21 - .r), 2f for values of x from 21 to 16; R iM^ = _ (c + X - 2/12) {c - x) + ihi . U'l - }}\ . -Ihx 'Ic = (9 + z)(21 -x) - 26, CHAP. X.] B. M, AND S. F. — LOCOMOTIVE. 195 for values of x from 16 to 11 R 3,V, ^ — {c + x- 2/13) (c -x) + 2/(3 ( JFi + TTo) - ( JFi . 2hi + Wn . 2h) 2 c = (17 +x){2l-x)- lor., admissible for values of x from 8 to 0. lif. = |- (c + a; - 2/(4) {c - r) + 2/m ( Wi + W^ + W3) -{JFi . 2hi + TFi . 2hz + /Fs . 2hz) = (27 + a;) (21 _ a-) - 315, admissible for values of x from — 2 to — 12 ; sMr = — {c-\-x- 2/(5) (c - X) + 2/t5 ( Wv + JTo + 7F3 + Wi) -(TFi .2hi+ Wi . 2A2 + W3 . 2/t3 + Wi . 2hv = (34 +a;)(21 - x) - 546. for values of x from — 12 to - 21. To find the ordinates of the apexes ; substitute x = In = S'o in the first equa- tion, a; = A2 = 6 in the second, and so on, and we have 1^8.5=156-25, 2^6 = 200 3iI/2 = 256 4^1/"-:. = 261, 5y-6. 5= 210-25. Only two are expressed by the letter M, as they alone of the five are bending moments ; the others do not lie in their own fields. Hence the maximum bending moment for the whole span is 4^_3 = 261 ft. -tons. 2. Calculate the maxima positive and negative shearing forces at intervals during the transit of the load (fig. 3). The slopes of the links are 5, 10,' 21, 33, 42 ; 37, 32, 21, and 9 tons per 42 feet ; 42 feet heing the span. The coordinates of the joints of the links are : — Links. Ft. Tons. Ft. Tons. Ft. Tons. Ft. Tons. Ft. Tons Ft.Tons Aaa . — 21 — oo-o — 16 06 -8 25 2 7-5 9 130 21 25 Bbb .. —21 — 44 —13— 2-5 -3 2-5 4 8-0 16 20 21 24-4 Ccc . —21 — 7-5 — II — 2-5 -4 30 8 15-0 13 19-4 21 255 DM .. —21 —13-5 — 14 — 80 — 2 4-0 3 8-4 II I4'5 21 19-5 E^e . . — 21 290 — 9—17 -4 — 126 4 -6-5 14—1-5 21 OQ-O For po.sitive maxima, the locus Ccc ... is to be taken between the left end of span and the point a; = — 1, and the locus Aaa of span ; for negative maxima, the locus ^"ee span. o 2 from that !)oint to the right end . is 10 be taken for the whole 196 APPLIED MECHANICS. [CHAP. X. 3. Find tlie equation to the maxima bending moments in the third field (Ex. 1) directly, and without using the general formula. Choose liny point whose absciss-a is x, such that when the load is placed with the 3rd weight over it, the whole loud will be on the span. The distance of G the centre of gravity from the right end will be (c + a; — 2A3), or (17 + a;), and P=ff X (17-t-x) = (17 +a;). Taking a section at x and considering the forces on the left side of it, we have 3^/, = P{2\ -x)- Wi{b + 8) - ^2 X 8 = (17 + a:)(21 -.r) - 105. 4. Find the maximum bending moment at any given points, say z = 8, and X = 10, in Ex. 1 ; and find the maximum for the wliole span without using the general equations (fig. 2).* 1°. To find tlie maximum bending moment jit any given point, divide the span into fields proportional to the weights ; consider which field the point lies in, an<l place the load so that the corresponding weight is over the point. Then if the whole load be on the spar, the bending moment calculated at the point for the load fixed in this position is a maximum. If when the load is so placed, some weights are off the span, move the load the least distance vhich will bring them all on the span, and calculate the bending moment at the point for the load fixed in that position. 2^. To find the miiximuni bending moment for the whole span. — By inspection, plai e the load so that any paiticular weight and the centre of gravity may be upon different sides of the centre of span and at equal distances therefrom ; then if the whole load be on the span and the weight be in its <iwri field, calculate the bending moment at the point where the weight stands tor the load fixed in this position, and it will be the maximum at the point. For each weight which can be placed so as to fulfil tliese conditions, there is a maximum, and having calculated these maxima, the greatest is the maximum for the whole span. In Ex. 1, the point x = 8 lies in the 3rd field ; place the load so that Wz is over it, and it will be tbund that all the wheels are on the span; calculating the bending moment at a; = 8 with the load in this position, we have 3M8 = 220 ft. -tons maximum at point. The same result may he obtained by substituting a: = 8 into zMx. Again, the jjoint 10 lies in field 3, but when Wz is placed over it, W\ is not on the span ; moving tlie load two feet to the right brings W\ just <in the span, while Wz is at 8 ; that is, the load is in the veiy same position as previously. Calculat- ing the bending moment at a; = 10, with the load in this position, we have J/'io = 190 ft. -tons maximum at point. The same result may lie obtained by substituting a; = 8, a!id x = 11, into zMx, and then adding one-third of their difference to the smaller. In Ex. 1, we find that hy placing Wz at the point x — 2, it is in its own field while G is at the point .r = — 2, and the whole load is on the span ; calculating tlie bending moment at 2 for the load in that position we have zM-i = 256. Again, by placirg Jf'i at the point x = — 3, it is in its own field while <? is at a- = 3, and the whole loa<i is on the span ; calculating the bending moment at - 3 for the load in that position, we have (.see example, fig. 1, p. 109) 4JI/.3 = 261. Now since lu) other weifilit can be i)laced according; to the rule, it follows tliat 4J/.3 = 261 ft. -tons is the maximum for span not only for the position on that figure, but for all possible positions of the load on tlie span. • These Kiiles were first piibli.-hed in " En<;iiicering,'' Jan. 10th and July 26th, 1879, previous to the conception of our grapliical constructions. CHAP. X.] B. M. AND S. V. — r,OCOM()TIVE. 197 0. Find the maxima bending moments at tlie junctions of the " fields." That is, tind the heights of tlie points Di, 1)2, &c., on tig. 2. FiBi = i.lfie = 100 ft.-tons ; FiDz = •-.Vu = 175 ft. -tons ; F3D3 = i^fo = 252 ft.-tons ; F^Ih = iM_io = 180 ft.-tons. Where 1.1/^16 means the hending moment at the point 16 feet left of centre when the first wheel is ovei* it ; 2-/'Ai tlie bending moment at 11 feet left of centre when the second wheel is over it, &e. The eciuation to the dotted parabola BAoO is ^^ {c"- - X-) = C2V - x^'), so that the ordinates at Fi, Fz, F3, and -F4 to this dotted parabola are to be found by substituting for x the values 16, 11, 0, and - 12, respeciively. They are there- fore 185, 320,^441, and 297 ft.-tons. Hence the depths of the points 7)|, !)■•, T)^, and D^ (which are the junctions of the " fields '') below the dotted parabola got by subtracting are 85, 145, 189, and 117 ft.-tons. It will now be seen that the depth of T)\ below the dotted parabola is Wi . 2hi = 5x17= 85 ; of Da is JFi . 2/(1 + IF. . 2hi = 5 x 17 + 5 x 12 = 145 ; of 7)3 is /Fi . 2/«i + lF-> . 2/(0 + JFz . 2/(3 = 145 + 11 x 4 = 189 ; and of 7)4 it is Jr, . 2/h . . . TFi . 2/(4 = 189 - 12 x 6 = 117. A general proof can readily be made of this remarkable theorem. That the junctions of the parabolic arcs (fig. 2) lie on vertiial lines dividing the span in segments which are proportional to the individual load on the wheels, and that the depths of any one of these junctions I) below the parabola which is t*ie bending moment diagram for the wliole load rolling on one wheel are given by 'S,[W . 2A)i'", that is by the geometrical moment of the weights about their centre of gravity from the first up to and including all those that command fields left of that junction. A beam 20 feet span is subject to the transit of 5 weights, each 2 tons and fixed at intervals of 3 feet. Find the maximum bending moment at each point during the transit. All the loads may be on the span at once, therefore n — b\ since the common interval is less than a fifth of the span, the locus is that for the whole load on the span (fig. 4). For the left half of span biv iM^= — (c + x - 2s){c -x) = h(i: + a;)(10 - x), 2c for values of x from 10 to 6. 2.1/";, = i (10 + a; - 3)(10 -a-)-2x3 = i(7+ x){lO - x) - &, for values of x from 6 to 2. iMx=h (10 + a; - 0) (10 - a.) - 2 x 6 - 2 X 3 ^\{10 - x) - IS, for values of x from 2 to 0. By symmetry the values for the right half of the span may be obtained, and the maximum for span during transit is zMq — 32 ft.-tons. 198 APPLIED MECHANICS. [CHAP. XI. CHAPTER XI. ON THE GRAPHICAL CONSTRUCTION OF MAXIMUM BENDING MOMENTS ON SHORT GIRDERS DUE TO A LOCOMOTIVE. Much attention has recently been given to the bending effects upon bridges of short span, due to the concentration of the loads on the wheels of locomotives. The subject assumes special importance on account of the ever-increasing weight of the rolling-stock. In a paper by W. B. Farr, read before the Institute of Civil Engineers in 1900, the subject was discussed at great length, as at that time the Board of Trade had required all the railway companies to strengthen their bridges. Farr, in his paper, contends that a period has arrived when the weight concentrated on any wheel cannot further be increased, so that any further increase of weight must be spread over a longer wheel-base and a greater number of wheels. This would, he held, narrow the problem to that of finding the maximum bending moment for a span accommodating a loco- motive and tender, and a like moment for a series of decreasing spans accommodating portions of the locomotive. Then, for each span, an equivalent uniform load was calculated and the rate of this uniform load tabulated for use in designing. Much of the discussion on this papti turned upon the question as to what was the best method of estimating such an equivalent uniform load. Each type of locomotive and tender gave a special table of its own ; and these tables were given in the paper for a large series of locomotives, and were amplified to allow for shock and other important practical considerations, again leading to much important discussion. It was suggested, too, that each railway engineer might use a table derived from a hypothetical locomotive, which was an average of their actual passenger engines. In calculating the bending moments for each span of one series, Farr used Culman's original method of drawing a link- polygon for the locomotive standing still, then moving the span about within the polygon, and when in a promising position projecting its ends up to the polygon, which is then closed by CHAP. XI.] LOCOMOTIVE ON SHORT GIRDERS. 199 an oblique chord, having the span for its horizontal projection. The highest ordinate of this closed polygon is scaled off, and gives a maxinmm bending moment for that particular position of the span placed under the locomotive or under some portion of it. Taking a number of those positions, by a sort of trial and error, an approximate value of the maximum of maxima is obtained. As this method is laborious and not quite certain, Farr thought of trying the authors' method, which required the use of a parabolic set-square, but found a difficulty in adapting it to the continual change of span. In the correspondence on Farr's paper the authors proposed their method of using circular arcs, and received inquiries as to that method from engineers both at home and in the colonies. For the purposes of this chapter the ideal locomotive weigh- ing 42 tons is adopted, having its weight divided among five wheels as shown on the under line, while the spacing of the wheels is shown on the upper line, thus — 5 5 5 10 11 12 = 30 feet. 9 = 42 tons. In the first instance a span of 42 feet is chosen, as it greatly simplifies the description of the graphical constructions to have the span and total load given by a common number. The centre of gravity of the locomotive falls in the 10-foot space, and is 6 feet from the 12-ton wheel and 4 feet from the 11-ton wheel. The locomotive is shown both on fig. 1 and fig. 2 standing in 200 APPLIED MECHANICS. [CHAP. XI. the same particular position. On fig. 1, the positions of the wheels are given left and right of 0, the centre of the span. An instantaneous bending moment diagram ABCDEF is drawn to scale for that particular position of the locomotive. On the other hand, fig. 2 gives the position of the wheels left and right of G, the centre of gravity of the load. By- inspecting these two figures it will be seen that the wheel Wx= 12 tons, which we will call the rulinrj-vjhed, since under it, as will be shown subsequently, the greatest bending moment occurs, lies ^4 = 3 feet to the right of the middle of the span, while G lies .3 feet to its left. On fig. 2 the bending moment under the ruling- wheel is given to scale by Sx At, so that A^ is identical with the apex D on fig. 1. We now contemplate moving the locomotive left and right of the position on the figures, so as to find the locus of the apex D on fig. 1. It is well at this stage to conceive the girder as extending some distance beyond each support as shown on the model, fig. 10, at the end of the chapter. On fig. 2 the locomotive stands with the ruling-wheel at a distance (21 - 3) = 18 feet from the right abutment, while the centre of gravity G^ is 18 feet from the left abutment. Since the load and span have the common value of 42, therefore the push-up at the right abutment is 18 tons, because G is 18 feet from the other abutment. The bending moment under the ruling-vjheel is 18 tons multiplied by 18 feet less 63 foot-tons, the product of the next wheel load 9 tons by its distance 7 feet from the ruling-wheel. The height of D, fig. 1, and of Ai, fig. 2, is tons. ft. 4if_3 = 18 X 18 - 63 = 261 foot-tons. The subscript figures meaning that the fourth wheel stands 3 feet to the right of the middle point of the span. If the locomotive moves one foot either to the left or right, one of the " eighteens " in the above product becomes seventeen and the other nineteen, and tons. ft. tons. ft. M.2 = 19 X 17 - 63 or ,]\L, = 17 x 19 - 63. These are the common height of the locus of D, fig. l,at one foot left and right of its figureil position according as the ruling- wheel Wi arrives at the one or other point. Hence Ai, fig. 2, is the vertex of a parabolic right segment, its half-base being 18, and its height, 18 x 18 ; but placed with its vertical axis 3 feet CHAP. XL] LOCOMOTIVE ON SHOUT GIRDKR.S. 201 to the right of the middle of the span and with its base lowered 63 units below BC, the base of the bending moment diagram. It will be seen that the point 6 feet to the right of the middle and the middle point of the span itself are equidistant about the axis of the parabola, so that the bending moment is the VERTICALS. 1 1C0 HORIZONTALS Fij) same at the two points when the wheel Wi arrives at them respectively. To find the locus of the apex C, fig. 1, the locomotive should be shifted till the wheel W^= 11 tons is two feet to the left of 202 APPLIED MECHANICS. [CHAP. XI. the centre, when C will coincide with A^, fig. 2 ; and the height S^Ai will be calculating from the left end, tons ft. 3il/. = 19 X 19 - (5 X 8 + 5 X 13) = 256 foot-tons. For a movement of one foot left or right the moment has the common value tons. ft. tons. ft. ,J/3 = 20 X 18 - 105 and 3J/, = 18 x 20 - 105. So that ^3 is the vertex of a right parabolic segment ; half base, 19 ; height, 19 x 19 ; placed with its vertical axis 2 feet to the left of the middle of the span, and with its base lowered 105 units below the base of the diagram. The apexes A, B, C, D, and E of the instantaneous bending moment diagram, fig. 1, describe the five parabolic loci whose vertexes are Ai, Ao, A3, A^, and A-^, shown on fig. 2. These have their five vertical axes placed about the middle of the span in the same way that the five wheels are placed about the centre of gra\'ity of the locomotive, but at half the distance in each case thus — 0S\ OSn OS3 OSi OS5 8-5 6 2 - 3 - 6i feet. The dotted parabola BA„C standing on the span is the locus of the apex of the triangle which is the instantaneous bending moment diagram for a single load of 42 tons rolling on the 42-foot span. This triangle is scalene for every position of the single load, except when the single load is at the middle of the span, when the triangle is isosceles, and its height is then OA^ = \ load X .span = 21 x 21 = 441 foot-tons. We now see that all six parabolic loci have this in common that each parabolic right segment has its height numerically equal to the square of its half base. If a teynplate of the parabolic right segment BA^C be made, then the other five segments are drawn from portions of that template. If the template be only a quadrant OAX', it is then called i\ pambolic- set-sqnarc. A parabolic template, such as BAX', whose height OAo is 441, the square of its half base, which is OC = 21, is said to be of morJulvs unity ; and, further, the lieight of the template at any other point is the product of the two segments of the base BC into which it is divided at that point. Such a CHAI'. XI. I LOCOMOTIVE ON SHOKT GIRDERS. 203 parabolic template or set-square would be far too lofty for actual use ; but any template or set-square of convenient proportions can be used by employing two scales — one for horizontal or feet measurements, and another for vertical or foot-ton measure- ments. The modulus of the template for drawing the six parabolic loci in tig. 2 is unity, because the ratio of the load to the span is unity. Had the locomotive only weighed half as much, say 21 tons, then the modulus must have been one-half; still the same template would serve by employing a new vertical scale for foot-tons — in that case twice as coarse. We may now consider the point Di on fig. 2 where the fourth and fifth parabolas intersect. At the point F^ the bending moment will have a common value F^D^ whether the locomotive be moved so as to bring Wi or W-„ to that point. If we put Z for BFx in the first case, the distance from B to G will be (Z - 13) feet so that the push-up of the abutment at C is (Z - 13) tons and the bending moment at Ft is iZ - 13) (42 - Z) foot-tons. In the second case the distance from £ to G is {Z - 6), and the bending moment at F^ is (Z - 6) (42 - Z) - 9 X 7. Equating these, gives ^ = 33 feet, which is numerically equal to the sum of the loads on the first four wheels. In this way (Ex. 1, p. 194) it is shown that the first parabola is above all the others for the first 5 feet of the span, the second parabola for the next 5 feet, the third parabola for the next 11 feet : that is, each wheel commands a portion of the span or a " field," which is the same fraction of the span that the load on that wheel is of the total load. As the load and span have been taken numerically equal, the five fields into which the span is divided are 5, 5, 11, 12, and 9 feet respectively. At any point in any field the maximum bending moment occurs when the commanding weight is over that point. In the two fields, one on each side of the middle of the span, there is a maximum of the maxima SzA^, 2 feet to the left of the middle of the span, and SiA^, 3 feet to its right. Or in symbols ■iM, = 256 foot-tons and iM_^ = 261 foot-tons. As the second of these is the greater, we have called the wheel Wi = 12 tons the ruling-ivheel of the locomotive when riding on a girder or bridge of span 42 feet. To find an uniformly distributed load which will give the same bending moment, 261 foot-tons, at the same point, 3 feet to the right of the middle of the span, we can assume a parabolic locus like BAqC standing on the span, but passing through Ai. 204 APPLIED MECHANICS. [chap. XI. This locus representing the bending moments due to an uniformly- distributed load of intensity v.' tons per foot, gives if we put c = 21 the half span, SU* = ^^xBF, xCF,= '^(c + 3)(c - 3); and equating this to 261, we get vj = 1-209 tons per foot of span. On tig. 3 we now show a method of drawing the five loci without considering the junctions of the fields. It does not give so good a definition, but is instructive. Thus, the locomo- tive is to be fixed with its centre of gravity over the middle point of the span, and the instantaneous bending moment polygon BahcdeC drawn to the vertical scale, upon which the Fig. 3. height of A» reads 441 foot-tons. Now on page 201 it was pointed out that the locus whose vertex is Ai has equal heights at 6 feet to the right of the middle point of the span and at the middle point itself. But on fig. 3 the wheel JFi is 6 feet to the right of the middle of the span. So that the perpendicular from d on fig. 3, dropped on the vertical line through the middle of the span, is the base of a small part of the locus whose vertex is At, which can be drawn with the template. In the same way perpendiculars on the middle vertical line from a, b, c, and e give bases to guide the template to draw parts of CHAl'. XI.] LOCOMOTIVE ON SHORT GIRDERS. 205 the other four loci. Note that to the common height of caiul d it is only necessary to add the square of the half of the small bases, thus : 252 + 4 = 256, and 252 +- 9 = 261 are the heiglits of A3 and Ai respectively. The height at F^, the junction of the fourth and fifth fields, to the dotted parabola BA^fi, fig. 2, is the product BFi X CFi = 33 X 9. In a preceding paragraph we had FiDi = (33 - 13) X 9 ; hence the height from A to the dotted parabola is (9 x 13), that is, the load on the wheel J/'j = 9 tons multiplied by its distance from G, the centre of gravity of the locomotive. In the same way it is found that -Os^o is the sum of the moments of the two weights W^ and Wi about G. Generally the depths of the junctions of the parabolic arcs in pairs below the dotted parabola are given by the moments of the weights of the wheels about their common centre of gravity. The numerical value for i?i is 5 x 17, for Di it is greater by 5 x 12, for A it is still greater by 11 x 4, while for Di it is lesser by 12 x 6, calculating them in order from the left end. Depths from the dotted parabola BAf^C, due to the 42 tons rolling on one wheel, to the junction of the arcs of parabolas giving the locus due to the 42-ton locomotive crossing the 42-foot span, are (Ex., p. 197) : — At Z>, n. D; D, 85 145 189 117 foot-tons. On fig. 4 are shown the two parabolic right segments BAoC standing on the span, and c'Ajj', the arc BjAiB^ of which is the locus of the maximum bending moments for the fourth field commanded by Wi. Let us suppose (see the dis- torting-table, fig. 17, p. 177) c'mnh' to be the end of a pack of cards stacked vertically into a rectangle, and having the right parabolic segment painted upon it, each vertical hatch- ment being on the edge of a card. The pack is then to be distorted and packed into the paiallelogram cmnh, and lifted up till Dz coincides with A^, and D^ coincides with di. We shall then have the arc of the oblique parabolic segment CA^ttidJ) coinciding at every point with the arc of the segment BA^C. Also F will be above at a height DsAq =189 foot- tons, and F will be above F^ at a height Didi =117 foot- tons. Hence FF is an oblique base whose horizontal range is OFi =12 feet, the extent of the fourth field, and the vertical ordinates from that oblique base measured up to the segment BAqC give the maximum bending moments at each 206 APPLIED MECHANICS. [chap. XI. point of the fourth field individually as W^ comes over it. The maximum of these maxima is given by StC/i = SiAt = 261. foot-tons. If each of the five ritrht parabolic segments on fig. 2 be distorted and lifted up in a like manner, we then have, on fig. 5, only one parabolic right segment of the height 441 foot- tons standing on the span as a base, and a polygon ACDEFB standing on the same base, and on the same side of it, having SA^Sa =26] futons. its four apexes on the lines dividing the five fields from each other. The heights of these apexes are : — G 85 D 145 E 189 F 117 foot-tons, being the same as the depths of the junctions of the arcs i),, D^, 1)^, and D^ below the dotted parabola on fig. 2. Fig. 4 furnishes a neat geometrical proof of the heights of the apexes G, D, E, and F. Consider the quadrant Aiiu ; its base is 3 feet, being half the distance of the wheel W^^ 12 tons from the centre of gravity of the locomotive. The height oi the segment is 1» foot-tons, being the square of its lialf base. OHAP. XI.] LOCOMOTIVE ON SHORT GIRDERS. 207 Now, the vertical through A^) meets the tangent m^i at a point twice as high above the base. Hence nui slopes at an angle to the horizontal, whose tangent is twice the square of the half base of the segment A^a^ divided by its half l)ase. So that the tangent of the slope of EF to the horizontal is given numerically by the distance of Wi, the 12-ton wheel from tiie centre of gravity of the locomotive. To get the amount that E is higher than F it is only necessary to multiply OFt by this tangent, when we get the product 12 x 6 = 72 foot- tons. In the same way the height of F above BC is the tangent of the slope of FC, that is, the distance of the wheel TVs = 9 tons from the centre of gravity multiplied by FiC when we have the product ofl3x9 = 117 foot- tons. Then the maximum bending moment at each point of the span is given by the vertical height from the polygon to the parabola. The maximum of maxima in any field — say, the fourth field — is to be found by producing FF to meet the parabola at b and c, then he is to be bisected at the black spot where the height to the parabola gives 261 foot-tons, the maximum of maxima ; provided that the bisecting point falls, as it does, on the side J^JF, and all the wheels are on the span. In the same way for the third field, the height at the centre of the chord of the parabola given by 1)F produced (fig. 5) when measured vertically to the parabola gives 256 foot-tons. For the other three sides of the polygon produced to give chords of the parabola, the bisecting points do not fall on the sides. Observe, too, that the black spot bisecting the chord cFFb falls 3 feet, measured horizontally, to the right of the vertical through the middle of the span, so that the graphical diagram, fig. 5, gives the maximum of maxima for the 42- ton loco., crossing the 42-foot girder, to be 4if_3 =261 foot-tons. that is, when the fourth or 12-ton wheel stands 3 feet to the right of the middle point of the girder. The polygon ACDEFB, which is mechanically subtracted from the parabolic locus on fig. 5, is the bending moment diagram for four fixed forces acting upwards at the junctions of the fields ; 5 tons at C, 8 tons at D, 10 tons at E, and 7 tons at F, as these forces would give the moments 85, 145, 189, and 117 foot-tons of negative bending at those points. Generally, then, the polygon is that due to a set of upward fictitious forces at the junctions of the fields whose magnitudes 208 APPLIED MECHANICS. [CHAP. XI. are numerically the same as the distances between the wheels muUiplied by the ratio of the iveight of the locomotive to the length of the span. First Graphical Method with one Parabola only. For the 42-ton locomotive on fig. 5, after finding the centre of gravity G, choose the smallest span which will accommodate the loco, standing with G and the ruling-wheel W^ equidistant from the two abutments. That span is 40 feet ; but we have taken 42 feet, which is only slightly greater, and makes the steps of the construction more evident, and simplifies any arithmetical checks. Gonstructio7i, fig. 5. — Lay down the span AB = 42 feet on a suitable scale of feet, and divide it into f\\& fields proportional to the weights on the wheels of the locomotive. On the left side draw the load-line equal to 42 tons, the weight of the locomotive to a scale of tons judiciously chosen, so that with a polar distance of 10 feet a well-conditioned isosceles triangle is formed. Join the pole to the ends of the load-line. Construct the apex of an isosceles triangle standing on the base AB, and having its sides parallel to the extreme vectors from the pole. The height of this apex above the middle of the base should measure 441 foot-tons, being a fourth of the product of 42 tons aud 42 feet on a scale ten times finer than the ton scale, the polar distance being ten. Construct the scale of foot-tons by renumbering the ton scale accordingly. Taking the apex of the isosceles triangle as a vertex, construct the parabolic right segment standing on the span AB. Construct the points close together near the vertex, but more sparsely well out from it. A construction is shown, fig. 3, Ch. VI. It will be seen at this stage that the shape of this parabolic segment depends on the choice of scales. To construct the polygon ACDEFB, draw vertically npn-ards the lines of action of iowv fictitious forces through the junctions of the five fields. On the right side draw upwards tlieir load- line 7, 10, 8, and 5 tons, being numerically the distances between the wheels multiplied by unity, the ratio of the weight of the locomotive to the length of the span. To a trial pole draw the dotted link-polygon among the four upward forces and the two holding-down forces at A and B. A vector from the trial pole parallel to the closing sides gives the junction between the magnitudes of the two holding-down forces. From this junction lay oil" horizontally 10 feet for the true pole, and construct the CHAP. XI.] LOCOMOTIVE ON SHOKT GIRDERS. 209 210 APPLIED MECHANICS. [CHAP. XI. link-polygon ACDEFB. Otherwise the height of F may be laid up directly to scale as 9 tons x 13 feet = 117 foot-tons, being the product of the 9-ton wheel-load and its distance from G. Then the height of ^ is 189 foot-tons, being 12 x 6 greater than the last, while the height of D is to be 145 or 11 x 4 less than the last, also the height of C is 85, or 5 x 12 lesser again. In this way by taking the moments of the weights of the wheels about G, their common centre of gravity, the polygon ACDEFB may be plotted, or, being drawn as first described, it can be checked in this way. Still otherwise, if the parabola be engraved on the paper or drawn first of all with a celluloid template. The span AB is, io be placed on it and divided into fields. The fictitious forces on the right are now drawn upwards to a scale for which the height of the vertex is 441. Now from the junction of the two holding-down forces lay up one-half of the span and lay down the other half, and from tlie two ends draw vectors parallel to the two sides of the isosceles triangle on the base AB and with its apex at the vertex. These will meet and determine the polar distance and the pole. To scale off the Maximum of Maxima.. Produce EF, the oblique base of the fourth field, to meet the parabola at c and d, bisect cd at the black spot, and scale off the vertical height to the parabola, which find to be 261 foot-tons; also find the horizontal distance of the black spot from the vertical through the middle of the span to be 3 feet, and we have iM-z = 261 foot-tons. A rival maximum, to be found by producing BE the oblique base of the third field, gives 3M2 = 256 foot-tons. For the other three chords AC, CD, and BF, the bisecting points do not fall in the fields. The fourth weight W^ = 12 is the riding-icheel, and the symbol ^M.3 means the bending moment at 3 feet to the right of the middle of the girder when the loco- motive stands with its fourth wheel at that point, provided all the wheels of the locomotive are actually on the girder. To deal with shorter spans which only accommodate a part of the locomotive. Diop ofl' the 0-ton wheel at the left end of the locomotive,, and drop ott' 5 feet from the left end of the span. Joining B to C CHAP. XI.] LOCOMOTIVE ON SHORT GIRDERS. 211 gives CDEFBC, the polygon for the reduced span of o7 feet. But G is to be projected up to 6" and tlie polygon completed on the oblique base C'B. Next the side corresponding to EF is to be produced both ways to h' and d, then h'd is bisected at the black spot, and the vertical height to the parabola scaled ofi" 214"o foot-tons. Also the horizontal distance of the black spot from the open ring at the middle of the oblique base C'Z? should scale 1"85 feet, being half the distance of the ruling-wheel, 12 tons from the centre of gravity of the group of wheels 5, 11, 12, 9 tons. In like manner another wheel 5 tons is dropped off the loco- motive, and 5 feet is taken off the span. Joining D to B gives DEFBD, the polygon for the reduced span of 32 feet. Project D up to D' on the parabola, and D'B is the oblique base. Com- plete the polygon ; produce the side corresponding to EF both ways to meet the parabola. Bisect this chord at the black spot, and scale off' the height to the parabola 170 foot-tons. Also find the horizontal distance of the black spot from the ring at the middle of the oblique base to be "734 of a foot. Further, removing the trailing wheel 9 tons, we have remaining the two driving-wheels 11 and 12 tons. Removing 9 feet from the right end of the span, then DEED is the polygon for the span, 23 feet. Project both D and F up to the parabola at D' and F', then D'F' is the oblique base. Complete the polygon, and produce the side corresponding to EF to df ; bisect F'd' at the black spot, and scale off the vertical height to the parabola as 83 foot-tons. Scale off' the horizontal distance of the black spot from the ring at the middle of the oblique span F'd', and find it to be 2-4 feet. To find the equivalent uniform rate of loading to give the maximum bending moment on each span at the same point of the span, we must double the moment, and divide by the difference of the squares of the half-span and the displacement from the middle of the span of the point at which the maximum bending occurs. Span. Moment. Displacf.ment. Rate of Load. Feet. Foot-tous. 42 261 37 214-3 32 170 23 83 It will be seen that the rate of loading on the 23- foot span p2 Feet. Tons per foot. 3-00 . 1-209 1-85 1-265 •73 1-331 2-40 1-312 212 APPLIED MECHANICS. [CHAP. XI. is less than that on the 32-foot span. But the rate should constantly increase as the span decreases. By inspection it will be found that the two driving-wheels, 11 and 12 tons, can be accommodated on a span of 15 feet, instead of 23 feet. To interpolate spaiis, slightly smaller or larger than those given hy dropjnng ojf parts of the span proportional to the loads mi the wheels dropped off the locomotive. Thus to determine the max. bending moment for a span of 20 feet loaded in the most trying way by the group of wheels 11, 12, 1> tons. Consider the parabolic segment and polygon standing on the oblique base B'B, the horizontal projection of which is 32 feet. By adopting a coarser scale for feet we can make this horizontal projection measure 20 feet instead of 32. Now, however, the height of the parabolic segment must also be measured on a coarser vertical scale in the like ratio assumed to save the trouble of re-drawing the parabola. But the polygon must be re-drawn. On fig. 5, then, the ruling-side of that polygon is shown by a hatched line with its two ends set up higher from the oblique base D'B' in the ratio of 32 to 20, and produced each way to h and Iz ; then hh is bisected by a black spot the height from which to the parabola measures 119 on the old scale. This is to be decreased in the ratio 20 to 32, when we have 74*4 foot-tons, and the horizontal distance of the black spot from the ring at the middle of the oblique base D'B measures on the old scale 1 -19, but when altered in the ratio of 20 to 32, we get -734 feet. Again, if the original locomotive is to ride on a span of 50 feet, instead of 42 feet, it is only necessary to lower the side of the polygon EF in the ratio of 42 to 50, shown by a hatched line ; produce it each way to m and n ; then bisecting mn in the black spot, and reading the height to the parabola, we have 288-6, which is to be increased in the ratio of 50 to 42, giving 343-56 foot-tons. Also the horizontal distance of the black spot from the middle of the span measures 2-52, which, increased in the ratio 50 to 42, brings it to 3 feet. In this way the shortest span to accommodate any group of wheels may be interpolated. It will require in general two trials, for at the first trial we may find when the deviation of the point of max. bending from the middle point of the span is measured, that placing tlie ruling- wheel, there sets an end CHAl'. XI.] LOCOMOTIVE ON SllOKT GIKDKKS. 213 wheel oH' the span altogether. We have then to slightly increase the span and proceed again. Span. MOMKNT. D SPLACEMENT. Rate of T,oad. Feet. Foot-tons. Feet. Tons per foot. 20 74-4 •734 1-495 50 343-56 3 1115 Second Graphical Method with Circular Arcs only. We have defined a parabolic right segment of modulus unity as having the height at any point nvimerically equal to the product of the two segments into which the point divides the -^c- ii-Tz!!. Fig. 6. base. And we have also pointed out that it represents the maximum bending moments for the transit of a single rolling- load, the height of the segment being made to scale in foot- tons an amount equal to one-fourth the product of the load and the span. Now, by Euclid II, 14, a semicircle will serve as a diagram of the square roots of the maximum bending moments in the same case to a vertical scale upon which the height of the crown of the semicircle shall measure the square root of the above product. Compare fig. 2, Ch. X, and Theorem H, Ch. VI. If we suppose every vertical height on tig. 2 to be replaced by a height equal numerically to its square root, we will have the diagram fig. 6, all the parabolas becoming circular arcs with centres at the points S-^, S^, &c., as shown on fig. 6, each arc beginning on the vertical line through the junction of the fields 214 APPLIED MECHANICS. [CHAP. XI. where the last arc ended. On fig. 6 the vertical scale is the same as the horizontal scale because of the load and span being numerically equal. The height of oti will scale 16"16 parts on the scale upon which the height of the crown of the semicircle scales 21. Then iM.^ = (16-16)- = 261 foot-tons. On fig. 7 is shown the 42-ton locomotive standing on the 42-foot girder. On j&g. 8 the span is divided into five fields proportional to the loads. Then the centres for the arcs are set off about the middle of the span at half the distances at which the wheels stand from the centre of gravity of the locomotive. The centre of gravity of the locomotive is defined by Gn on fig. 7, where the end links of the link-polygon meet, the link- polygon being drawn to a polar distance of 10 feet. The five circular arcs are then drawn on fig. 8, each beginning at the junction of a field where the last arc ended. The vertical scale is made so that the height of the crown of the semicircle shall scale 21. Next the crown of the circle in the fourth field is ruled over to the scale and reads 16*16, and this when squared gives ,M_^ =261 foot-tons. The front wheel, 5 tons, is now dropped off when Gy„ fig. 7, defines the centre of gravity of the remaining four wheels ; from Ga- a perpendicular is dropped on the vertical from the ruling- wheel 12 tons ; this perpendicular is bisected, and the half scales 1*85 feet. On fig. 8 the new span, with the 5 feet at the left end left off, is bisected at the ring, and the centre for the arc corresponding to the wheel, 12 tons, is laid at 1"85 feet to the right of the ring, and the centres for the other three arcs spaced relative to it. The arcs are then drawn for the second, third, fourth, and fifth fields, and the crown of the arc on the fourth field is ruled over to the scale where its height reads 14*64 ; and squaring this, we have M = 214 foot-tons, a maximum, at 1'85 feet to the right of the centre of the 37-foot span when the 12- ton wheel stands over it. Dropping off the second 5-ton wheel we have the centre of gravity of the remaining three wheels defined at Gxi ; and the perpendicular from this point upon the vertical from the ruling-wheel, 12 tons, gives, when bisected, 0*734 feet. This distance is laid off to the right of the ring marking the middle of the 32-foot span and the two remaining centres placed about CHAP. XI.] LOCOMOTIVK ON SHORT GIRDERS. 215 it as before. The three arcs are then drawn, the highest crown ruled over to the scale, where it leads 13"04. Squaring this we get M = 170 foot-tons, a maximum, at a point 734 feet to the right of the centre of the 32-foot span w[ien the 12-ton wheel stands over it. In the same way fig. 8 shows the solution for various spans loaded with various groups of the wheels of the locomotive ; and a table is shown giving the equivalent uniform loading. Third Graphical Method, being Culman's Method rendered precise. Only our original construction, fig. 2, drawn with a para- bolic segment, and the diagram, fig. 8, of circles derived from it show which is the ruling -ivheel. For this purpose it would be well to draw the diagram, fig. 8. in conjunction with Culman's method. Hence we have placed fig. 8 under fig. 7, which is Culman's method rendered precise, so that there is no searching about by trial and error. Lay down on fig. 7 the load-line 5, 5, 11, 12, 9 tons. Choose a polar distance, some round number — say, 10 feet. The two scales, one for tons and the other for feet, should be so related that the two end-vectors may meet at the pole at a well-conditioned angle. Draw the vertical lines of action through the wheels of the lf)Comotive, and among them draw the link-polygon. Produce the two end-links to meet at G^i, which determines the position of the vertical through the centre of gravity of the locomotive. Either the 11- or the 12-ton wheel will be the ruling-wlieel. By drawing the semicircles on fig. 8 and the locus of circular arcs, we at once find the driving-wheel carrying 12 tons to be the ruling-wheel. From Gil drop a perpendicular on the 12- ton load. Bisect the perpendicular, and from its middle point lay off 21 feet horizontally on each side, that is, half the span. Project the two ends down on to the two end-links, and draw the oblique base closing the polygon. This closed polygon is the instan- taneous bending moment diagram when the locomotive is standing in the most trying position, that is, with the 12- ton wheel 3 feet to the right of the middle of the span. Scale off the depth from the apex on the 12-ton load down to the oblique base, and find it to read on the ton scale 261 ; multiply by 10 feet the polar distance, and we have ^M,^ = 261 foot- tons. Drop off the 5- ton wheel from the left end of the locomotive, and leave off 5 feet from the left end of the span. For the o oof oos ooe oo* suox-iJ' a or OS OP o-t- tiuoj, n I I M I I I I n I I I H I I I I I I I I H I I I 1 M I I » I I TTTR '""IV }""3 "I'fU ^S "■'/" ' '"J .£1 QL SL ■S 4 ^ 5 "^ s a. 5S •o S 5 •n fl _g> • II \ II II •J o 1 s s : : ~ •\ •§. ° :S ~ ' - - fci 5 • i fe >.^ ^ > * » i * -0 i « f. « e ^ 1 i5 3 ^ ui S (A < ■yf;MDJ£) yb >Jiu?Q uowiuo} jpyi viojf si; sj^sy^H Smpuods^JMj inoqD p>3Djd }jlv sij y jpjnpjij it/t J'o suiusj »y_^ 218 APPLIED MECHANICS. [CHAP. XI. remainder of the locomotive the two end-links meet at G3,^. From this point drop a perpendicular upon the 12-ton load, and bisect it ; from the bisecting point lay off horizontally 18*5 feet on each side. Project the ends down upon the two end-links, and draw the oblique base, closing a polygon which is the instantaneous bending moment diagram for the remainder of the locomotive standing on the 37-foot span in the most trying position, that is, with the 12-ton wheel at a point 1"85 feet to the right of the middle of the span. Scale ofi' the depth from the apex on the 12-ton load to the oblique base. Measure it 2142 on the ton-scale, and multiply by 10 feet the polar distance, and we get 21-1'2 foot-tons. In the same way fig. 7 shows the maximum of maxima constructed for other spans and loads obtained by dropping off some wheels from one or other end. or from both ends of the locomotive, and taking off corresponding segments from the original span. In some cases observe that the 11-ton driving-wheel becomes the ruling-wlieel. To find the maxmucm of maxima bending movients where there is a given uniform load and tlie locomotive. Eeturning to fig. 5, suppose that on the 42-foot span AB there are 16 tons uniformly spread, which would give a parabolic locus the same as that for 8 tons rolling on one wheel, the height of the locus being ^ x 8 tons x 42 feet or 84 foot-tons. To allow for this it is only necessary to adopt a finer scale so that the height of the parabolic segment in fig. 5 shall measure 441 + 84 foot-tons instead of 441. Adopting this finer scale for verticals, it becomes necessary to lower the chord EF down to the position shown hatched, that is, lower in the ratio of 42 to 50, just as we already noted when we passed from the 42-foot span to a 50-foot span. So that by bisecting mn at the black spot, measuring the vertical height to the parabola on the original vertical scale, we get 288-6, but multiplying by the ratio 50 to 42 it becomes 343-56 foot-tons. Rut in this case the original horizontal scale still obtains, so that the liorizontal distance of the black spot from the middle point of AB is only 2-52 feet. In the former case, where mn was the oblique base for a 50-foot span bearing the locomotive only, this -'-52 had to be multiplied by the ratio 50 to 42, which made a product 3 feet, as it must, for on any span that bears the locomotive only, 3 feet to the right of the middle is the most critical pouit, for then the ruling-wheel, 12 tons, and the centre of gravity of CHAP. XI.] LOCOMOTIVE ON SHOUT GIRDERS. 219 the locomotive are equidistant from the two abutments, or are each 3 feet from the middle of tlie span. But in the case we are now considering, the ruling-wheel, 12 tons, must stand 2"52 feet to the right of the centre ; and it will be found that the centre of gravity, not of the locomotive but of the load made up of the 16 tons spread uniformly, together with that of the locomotive, is 2-52 feet to the left of the middle of the span. Mr. J. T. Jackson has pointed out that the similarity between the above construction and that described on p. 208 at once suggests that it must be possible to devise an ideal locomotive which shall produce the same maximum bending moments at every point of the span as are actually due to the combined effects of the real locomotive and the uniform load. To see how this may be done let us compare the modification of the bending moment diagram in fig. 5 to allow for change of span with that for the addition of an uniform load. (1.) Change of Span. — The ordinates of the ^olygonACDEFB, the unit of length and the unit of bending moment are all altered inversely as the span, while the unit of load is unaltered. (2.) Addition of Uniform, Load. — The ordinates of the poly- gon AODEFB, the unit of load, and the unit of bending moment are all altered inversely as the sum of the rolling load and half the uniform load, while the unit of length is unaltered. The effect on the fortn of the diagram in fig. 5 of an increase of span from, say, 42 to 56 feet, would then be the same as that due to the addition of an uniform load of 2 x (56 - 42) or 28 tons ; i.e. to a load of two-thirds of a ton per foot-run on a girder of 42-feet span. The effect on the scales would, however, be different : in the case of the increase of span the scales both of length and bending moments would be made finer in the ratio of 42 to 56, or 3 to 4, while in the case of the addition of the uniform load the length-scale would be unchanged, while the load and bending moment scales are made finer in the above ratio. It is evident that the diagram of maximum bending moments for the 30-foot 42-ton locomotive of fig. 5 running on a 42-foot span which already carries an uniformly distributed load of 28 tons and drawn to scales of, say, 3 feet, 4 tons, and 48 foot-tons to the inch, is identical in form and size with the diagram for the same locomotive crossing a span of 56 feet, and drawn to scales of 4 feet, 3 tons, and 48 foot- tons to the inch. Now, on changing the scales of the latter figure to 3 feet and 4 tons to 220 APPLIED MECHANICS. [chap. XI. the inch, it is seen to represent equally well the diagram of maximum bending moment for a 22^--foot 56-ton locomotive crossing a 42- foot span. So that the effect on the diagram of maximum bending moments of the addition of an uniform load is A SHORTER, HEAVIER IDEAL LOCOMOTIVE, EQUIVALENT TO REAL LOCOMOTIVE H AND THE DEAD LOAD S 6 Tons. \& 3^' -rector cecc CMS 9~A\ 7- >< . 3oFf. -^'"^ (^^^2 Tons, Decffi Model '^'TON ^^'PER FOOT. Ol4) Fig. 9. the same as that prcjduced by nhorteniifKi the intervals between the wheels in the ratio of the loco, weight to the sum of the loco, weight and half the uniform load, and making the several wheel loads Iteavier in the reciprocal ratio. This is illustrated in fig. 9, where the derivation of the diagram of square roots of CHAP. XI.] LOCOMOTIVE ON SHORT GIRDERS. 221 luaxiiuuui beudiug moments for the loco, of figs. 1-5 is effected when the loco, is supposed to cross a bridge of span 42 feet already carrying an uniform dead load of two-thirds of a ton per foot. The method is useful as illustrating clearly the eftect on the bending moment diagram of the addition of an uniform load ; but it requires to be applied with some degree of caution, as it must be remembered that the change in the character of the diagram which takes place when a wheel goes off the span occurs when a wheel of the real locomotive iroes off, and as the ideal locomotive is shorter than the real, it might readily be forgotten in examining a particular portion of the locomotive that one wheel was off, since the corresponding wheel of the ideal locomotive might be well within the span. Kifiematical Model, demonstrating the variatioTis in bending moment at all points of a girder-bridge as a locomotive comes across the bridge. This model is specially designed to exhibit the manner in which the diagram of maximum bending moments, consisting of arcs of parabolas (as shown in fig. 2), is traced out by the vertices of the link-polygon corresponding to the loads on the wheels of the locomotive, as the locomotive moves across the span. The model (see fig. 10) consists of two parallel plates of mahogany each 18^ in. x 15 in., the front or face-plate being f in. and the back plate \ in. thick ; these plates are set I5 in. apart. On the top of the plates is the model girder constructed to a span of 42 feet on a scale of 4 feet to an inch. Running on top of the girder is the locomotive, with a wheel- base of 1\ in. (to represent 30 feet on the assumed scale). The girder does not terminate at the supports, but overhangs each support by an amount sufficient to prevent any wheel of the locomotive from running off the girder so long as the centre of gravity of the locomotive remains between the supports. The object of this is to avoid the change in the character of the bending- moment dia2;ram which would ensue if an^• wheel were regarded as passing completely off the girder. Passing round the frame and tightly stretched over four small pulleys set on the back plate near the corners is an endless chain. A stud projecting downwards from the loco- motive is attached to the upper horizontal side of this chain, so that the chain carries the locomotive with it as it moves. A detailed description, suitable for the model-maker, is given 0')9 APPLIED MECHANICS. [chap. XI. in a paper to the Royal Irish Academy hy Mr. Jackson, assistant to the professor of engineering in Trinity College, Dublin. He Fig. 10. designed tliis model modified from the Rim]dei' model described at fig. 16, Chap. IX. The load and si)an being numerically CHAP. XI.] LOCOMOTIVE ON SHORT (JIUDKRS. 223 equal, the span divided into fields servos as the load-line, turned through ninety degrees. The rocking pivots are at the junctions of the ^fields. Tlie chain moves the pole at the same rate as the locomotive, but in an opposite direction. The wires (vectors) from the pole rock the pivots which again turn the hands, one directly above each pivot, at the junctions 2),, Z>2, &c., of the parabolic arcs. In this way each hand is always at right angles to the rocking- wire, and so parallel to the thread (vector) in the load diagram on the left. These models were made by Messrs. Dixon and Hempenstall, Dublin, through the liberality of the Board of Trinity College, Dublin. Examples. 1. Suppose with the weights in Example 1, Ch. X, and shown at fig. 9, we combine an uniform load of -|rds of a ton per foot of span, that is, a total load U = 28 tons, and find the equations to the maxima bending moments for the various fields. See Ex. 4, Ch. Xyil. Summing the moduli for R rolling and for ZT" spread uniformly, we have M I7'_jr+2i?_28 + S4_4 2c "*" 4c ~ 4c ~ ~84 ~ 3' the modulus of the parabolas for combined load. The modulus for R rolling, divided by the sum of the moduli, is \2c/ " \2c'^ ic) 2E 84 3 U+2R 28 + 84 4 Substituting into the general equation, lifx = 1(21 +x-ly 17)(21 -x) = f (8-25 + /)(21 - x) for values of x from 21 to 16. oilfx = 1(21 +x-^x r2)(21 - a;) + 12 X 5 - 5 X 17 = |(12 + .r)(21 - x) - 25,, for values of x from 16 to 11. sMx = 1(21 + a: - I X 4)(21 - a:) + 4(5 + 5) - (5 X 17 + 5 X 12) = A(18 -t a:)(21 - x) - 105, for values of x from 8 to 0, the chord being taken for values from 11 to 8. 4ylfx= A(21 +a; + | X 6)(21-a;)-6(5+ 5 + 11) - (5 x 17 4 5 x 12 + 11 x 4) = A(22-5 + a;)(21 - z) - 315, for values of a: from — 2 to — 12, the chord being taken for values from to — 2. sifx - 1(21 + a; + I X 13)(21 - x) - 13(5 + 5 + 11 + 12) - (5 x 17 + 5 X 12+ 11 X 4- 12 X 6) = 1(30-75 + a:)(21 - x) - 546, for values of x from - 12 to - 21. 224 APPLIED MECHANICS. [CHAP. XII. The abscissae of the apexes are a X 8-.) = 6-375 ; ^ x 6 = 4-5 ; | x 2 = 1-5, lies in field 3 ; J x (- 3) = - 2-25, lies in field 4 ; | x (- 6-5) = - 4-875. Here only two apexes lie in the corresponding fields ; substituting for these, we have 3M1.S = 402 maximum in field 3 ; and 4i^/.2.25= 40575 maximum in field 4 ; and maximum for span ; the greatest bending moment occurs at 2^ ft. to the right of the centre, when the fourth weight is over it. The joint centre of gravity is 2^ ft. left of centre. CHAPTER XII. COMBINED LIVE AND DEAD LOADS WITH APPROXIMATION BY MEANS OF AN EQUIVALENT UNIFORM LIVE LOAD. We already found that when the loads were all dead or fixed the joint construction of the shearing force diagram and the bending moment diagram recommended itself, the one aiding in the construction of the other. The shearing force diagram in itself, easy to draw, helped to construct the more difficult bending moment diagram. For moving loads we continued to discuss the two diagrams together, although they no longer had much in common, owing to the fact that the i^osition of the load for a maximum bending moment and for a maximum shearing force at the same point was not the same position. In the combination of live with dead loads to be dealt with in this Chapter matters are reversed, for the bending moment diagram is easily dealt with, while the shearing force diagram demands thorough investigation. For practical purposes the dead load is to be reckoned as of uniform intensity per foot of span. The live load is also reckoned as uniform per foot of length, and as of sufficient length to cover the whole span. The actual live load, no matter how it may be disposed, in positions concentrated on wheels regularly or irregularly spaced is to be replaced by an equivalent uniform load. That is, an uniform advancing load (at least as long as the span) which, in its transit, produces the same maximum bending moment CHAP. XII.] COMBINED LIVE AND DEAD LOADS. 225 at the central point of the span as the actual load produces at or near the centre. It is to be noted that although irregular load systems produce their maximum effect at a section a few feet from the centre, yet this is covered by the fact that in the practical design of girders the cross-sections at the centre and for a few feet on each side of it are identical. The equivalent load also produces the same shearing forces at the abutments as the actual load. Again, it is necessary to augment the actual moving load to allow for the impulsive effect due to high velocities in a large proportion for short spans, and in a lesser proportion for longer spans. This we will afterwards illustrate by tables from a recent paper in the Transactions of the Institute of Civil Engineers. The moving load when thus augmented and equivalated to an uniform intensity may now be called the effective equivalent uniform live load, but for brevity it will now be called simply the live load. Seeing that both the dead and live loads are now uniform, the maximum bending moment will simply be a right parabolic segment standing on the span, for the live load must be put covering the span for maxima. The bending moment diagram is now a parabola, the height of its vertex being one-eighth of the product of the total load and the span ; where the total load means the sum of the dead load and of the equivalent live load covering the span. If u be the dead load per foot of span, and ui the live load per foot, the equation to the bending moment diagram is u + w , M, = —^ (c* - *2), where 2c = ^ is the span, and x the distances of any cross- section from the centre. Now we shall investigate the important practical case of the Shearing Force for a beam under a fixed uniform load together with an advancing load of uniform intensity (fig. 1). — Let u be the intensity of the fixed uniform load, and w the intensity of the advancing load in terms of its equivalent dead load ; as in fig. 13, Ch. VII, draw DE for the uniform dead load, and from DE as a sloping base plot the ordinates of the two parabolas (fig. 1, Ch. IX) up and down respectively ; the two loci FLEqmA DKG give the maximum and minimum, or the positive and negative maximum, shearing force at each point. Q 226 APPLIED MECHANICS. [chap. XII. In fig. 1, Ch. TX, the parabola DC may be supposed to be drawn on the " Distorting Table " described at fig. 17, Ch. IX. The table is then to be distorted till its base is at the slope DE (fig. 1), when we have FLE as the locus for the joint loads ; this is still an arc of the same parabola with its vertex at Az, for which see Theorem A, Ch. VI. The ordinates to DKN, the portion of the curve which extends over the left half of span, are derived by taking the ordinates of the slope 01) due to the dead load alone, and subtracting the ordinates of DN; or, what is the same thing, by adding the ordinates of the slope OD, and of the parabolic segment JDN, and then subtracting the constant quantity ON. When the ordinates of the slope and of the segment are added ii^-^-'^-- UC or i V wc or i W Fig. 1. by the I'heorem A, Ch. YI, the resulting locus is a portion of the same parabola as DN, but with its apex to the left of B at a distance BD \U ^'^=^" = 2^-^^==2Tllr 2c. w (1) When the constant quantity ON is subtracted, the locus unaltered in form moves vertically downwards through that distance ; the apex still remains to the left of B at the distance a ; the curve passes tlirough the point U whose height is BD = ?<c, and through the point G whose depth is GO = (w + w)c. Similarly ELF is symmetrical, left and right, up and down. CHAP. XII.] COMBINED LIVE AND DEAD LOADS. 227 Taking the centre of span as origin, the equation to the parabola AxT>G is wc '2u -{■'WW. , - ,, = --^-^x--x\ (2) The point K is found by making y = 0, and finding the corre- sponding value of X ; thus OK.c-ojf^--'^-)- (3) X^-y w^ w v)j The ordinate to the apex A^ is found by substituting (c + a) for X in equation (2) ; and is ^^1 = [- + 1^ c. (4) \w J Graiihical Solution (fig. 1). — Lay up BD and lay down GE, each equal to half the dead load. Then further lay down EG equal to half the live load. To construct as many points on the parabolic arc DKG as may be required, divide GB in any manner. In fig. 3 it is divided into twelve equal parts corresponding to the l3ays or panels in the girder. More correctly, however, it should have been divided by points over the centres of the hays, that is, it should have been divided into eleven equal central divisions and two half end divisions. Now divide EG similarly to the manner in which BG has been divided ; then vectors drawn from D to the points of division of EG meet verticals drawn through the points of division of BC, and determine the corre- sponding points of the curve DKG. ELF is the same curve transferred by the dividers. To the left of K the shearing force is always positive, to the right of L it is always negative, and between K and L there are both positive and negative maxima. The range at the centre is \ W as for W alone ; if u be great compared to w, the range between K and L is nearly constant and equal to \ W, since the apex is then far out and the portion of the parabola over KL is very flat. The critical points K and L, between which the shearing stress sometimes produces distortion of the girder in one direc- tion and at other times in the opposite direction, according to the position of the live load, are of great importance. Between these points the girder requires counterbracing. When the fig. 1 is drawn graphically, K and L are at once fixed ; while, by Q 2 228 APPLIED MECHANICS. [CHAP. XII. calculation, the exact position of K is given by a modification of equation (3) ; thus BK-2c(E.^--l). (5) This equation is so awkward and the position of K so important that we will determine its position by ascertaining the position of the live load which shall bring the common centre of gravity of the girder and the load on its left segment as near to the left abutment as possible. The centre of gravity of the girder itself is at the middle. As the live load is pushed on from B, the common centre of gravity moves from the middle point nearer and nearer the left abutment B for some time ; then it begins to move back towards the centre 0, where it arrives when the whole girder is covered with the live load. For some segment such as BK loaded, the common centre of gravity of girder and load will be nearer the abutment B than for any other. This is the case when the common centre of gravity is at K the end of the loaded segment. For adding a load to the right of ^will move the centre of gravity to the right of K ; and removing a load to the left of X will have the like effect. To find K then it is only necessary to assume the shorter segment BK to be covered with the live load and at the same time the common centre of gravity to be at K. This is readily remembered, and can be applied at once to any numerical example. That it leads to equation (5) is proved at once, for the sup- porting force at the right end can be calculated in two ways. First, we may suppose the dead load w . 2c concentrated at the centre 0, and the live load vj . BK concentrated at the middle of BK, when the right reaction is Q = uc ■¥ v: . BK^ — = ^ic + — BK- • ^ 2c 4c Again, we may suppose the whole load {u , 2c + w . BK) concentrated at K, when Q={2iui + w. BK) ^ = uBK + '^BK\ ^ 2c 2c Equating these, and solving the quadratic equation, we have the same value for BK as tliat in equation (5). Another way of considering the live load — that which Levy CHAF. XII.] COMBINED LIVP: AND DEAD LOADS. 229 uses in his Graphiquc Statique, and which we follow in a later chapter — is to consider an equivalent rolliiifj load on one wheel which will produce the same bending moment at the centre, and the same shearing force at the abutment as the maxima produced by the actual load. If R be the rolling load and U the dead load, then the bending moment diagram is a parabola, the height of the vertex being {{R + \ U)l. By superposition of the diagrams, figs. 13, Ch. VII, and 6, Ch. IX, first distorting fig. 6, Oh. IX, so that its base will coincide with the slope DE, we have the shearing force diagram, fig. 2. Girder bridges are tested by passing back and forth over them two or three locomotives of the heaviest type and fully loaded, during which the deflections are observed. The eflt'ect is the more marked the shorter the span. With spans above 100 feet the effect is nearly the same as that calculated for a uniform load. That is the fact of the load being concentrated Fig. 2. in parcels at the wheels may be overlooked almost. Compare figs. 4 and 5, Ch. X, and the formulas referring to them. In 1890 the heaviest locomotive and tender used on the Caledonian Railway corresponded very nearly to the following: — Loads on wheels 14, 15, 15, 12, 13, 11 tons = 80 tons. Intervals, 7, 7, 9, 8, 6, 7, 6 feet = 50 feet. Two such locomotives, the leading one running backwards, serve to test all girder bridges from the shortest spans up to a span of 100 feet. That wheel or set of wheels adjacent to each 230 APPLIED MECHANICS. [chap. XII. Other which gives the greatest bending moment is taken for each span, and the greatest bending moment near the centre is calculated in this Chapter by the preceding rules or graphic constructions. Then the equivalent uniform load which would give a like maximum at the centre is calculated. Thus on a span of 5 feet only, one pair of wheels could be on the bridge at one time, that is, the greatest load is the heaviest pair, namely, 15 tons of a rolling load. The equivalent uniform load is 30 tons spread or 6 tons per foot. For a span of 10 feet the 15 tons above the span and at its centre is the same as 30 tons spread or as 3 tons per foot ; but when the 12 tons and 13 tons are both on the span, and placed so that the 13 tons is as far from the right abutment as their common centre of gravity is from the left abutment (see fig. 14, Ch. IX), in which reckon Wi = 12 tons, JFo = 13 tons, and 2h^ + 2^2 = 6 feet ; then the maximum bending moment under Wi is to be calculated, and it will be found to be 4 per cent, greater than with the 15 tons alone at the centre. In each case if vi be the intensity of the equivalent uniform load, we must equate ^wV' to the maximum bending moment for the severest position of the load. The results are given in the following table* : — Span in feet. Maximum in foot-tons. Equivalent uniform load in tons per foot. Span in feet. Maximum in foot-tons Equivalent uniform load in tons per foot. 5 10 20 30 40 50 i8f 3« 104 214 37-' 563 600 3-04 208 191 I 86 I 80 60 70 So 90 100 774 lOIO 1303 1633 2020 172 1-65 163 1-62 162 Within the last few years the weight of rolling stock has greatly increased, and results similar to the above, but on the most elaborate scale, are given in an excellent paper by Mr, W. B. Farr, called " Moving Loads on Eailway Under Bridges," and published in the Transactions of the Institute of • These tabulated results were given by Dr. Thomson in a paper to the students of the Institute of Engineers and Sliiphuilders of Scotland in tlie year 1890. CHAP. XII.] COMBINED LIVE AND DEAD LOADS. 231 Civil Engineers, 1900, No. 3195. Mr. Farr has included the rolling stock of all the British and Belgian Railways. His argument, too, seems excellent, namely, that there is now no further possible increase of the weight on the wheels of rolling stock, so that increased weight of the rolling stock in future must be accomplished by increasing the number of wheels wliich would not affect the equivalent uniform load. Another excellent feature of Mr. Farr's paper is that he gives a suitable augmentation of the equivalent uniform load on a sliding scale to allow for the impulse or impact of the load, so that his tables furnish at once reliable working values for the effective equivalent uniform load which are not likely to alter much in the immediate future. We quote here the summary of his results. Main Gikders. X = maximum uniform load per foot run (40 types of locomotives). y = suggested per cent, increase for impact. Spans X y* x + y 5-0 feet 760 tons 30"oo per cent. 9-88 tons 75 .. 5-55 .. 2750 ,, 7-07 ., 100 ,, 485 .. 25-00 „ 606 „ 150 „ 3*74 >. 22-50 ,, 4-58 „ 20'0 ,, 3 20 ,, 20*00 „ 384 „ 30*0 ,, 2-63 ,, i5"oo ,, 301 „ 400 ,, 240 ,, 14-50 275 .. 6o-o ,, 217 ,, 13-50 2-46 „ 800 ,. 206 ,, I2-0O „ 230 „ 1000 ,, 1-97 .. lO-OO „ 2-i6 „ * "We think Mr. Farr's addition too low, and suggest that 1/ he increased througliout. See Fidler's Bridge Construction, page 241. 232 applied mechanics. Cross-Girders. [chap. XII. X = maximum concentrated loads in tons on each single line. y = suggested per cent, allowance for impact, &c. Distance apart X y x+y 3 feet igoo tons 50 per cent. 28-50 tons 5 .. ig-oo ,, 45 2755 „ 7 .. 1900 „ 40 2660 „ 8 „ 2150 ,, 35 2902 „ 9 „ 23-50 ., 30 3055 ,. lO „ 25-11 25 31-39 « Examples. 1. A bridge 32 feet in span is subject to the transit of a lo-ton road roller . find the equivalent uniform load. Here H = \o tons and I = 3-2 feet, so that Mq = {R( = 120 ft. -tons. Equating this to ^wP, we have Atis. to = if th of a ton per foot run. 2. A beam 36 feet in span is subject to the transit of a load on two wheels, viz., 12 tons on one and 6 tons on the other, spaced 12 feet apart. Find the equi- valent uniform live load (see the model, fig. 16, Oh. IX). 1.V2 = if X 16 X 16 = 128 ft.-tons maximum. Put lu>P = ^ X 36 X 36 = 128. And to is a ^a part of a ton. 3. The beam 42 feet in span, shown on fig. 2, Oh. X, is subject to the transit of the locomotive shown on that figure. Find the equivalent uniform live load. From the figure iM.a = 261 ft.-tons is the maximum bending moment. Equating this to ^w x 42 x 42, we find w equal to 1-184 tons. Compare p. 204. 4. A beam 30 feet span is subject to the transit of five weights, each 3 tons, fixed at intervals of 7 feet. Find the equivalent uniform live load. We have -iMo = 49-5 ft.-tons. Equating ^wF to this, we find tv - -44 ton per foot. 5. An advancing load in length not less than the span, and of uniform intensity 3 tons per foot, passes over a beam 42 feet span. Find the maxima shearing fones positive and negative at the points .r = 21, 14, 7, and 0. Positive tons. Negative tons. Range tons. Ans. F21 .. 630 .. 00 .. 630. Fu .. 43-75 .. 1-75 .. 45-5. F-. .. 28-0 .. 7-0 .. 36-0. /■n ., 15-75 .. 15-76 .. 31-5. 6. In the previous example, find the maximum value of the positive and negative shearing forces at the point z = 7, directly. CHAP. XII.] COMBINED LIVE AND DEAD LOADS. 233 Let the segment to the riglit of the point be loaded ; the amount of load will then be 28 X 3 = 84 tons; suppose this concentrated at the centre of the 1 laded segment and + Ft — P = (84 -^ 4) x 14 = 28 tons ; asiaiii let the segment to the left of the point be loaded, and - Ft = Q= (42 4^ 42) x 7 = 7 tons. 7. Find, for the previous example, the points between which the shearing force is sometimes positive and sometimes negative ; and tind the coordinates of the apex A\. See tig. 1. Ans. 0K= 21 - 42 {^(^ + i) - j} = 7 feet ; 0L = -1 feet. For the apex Ai, OS = z = c -\ '2c = 35 feet on horizontal scale, w SAi = 1/ = ( h " ) e = 28 tons on vertical scale. By using the coordinates of ^i, the graphical construction can be made with greater accuracy, more especially if the poiiiis Ai and D aie situated near each other. 8. A beam 24 feet span bears a load of 6 tons uniformly distributed, and is subject to a rolling load of 6 tons. Find the amounts and the range of the shearing forces at intervals of 2 feet. Since the span is a little greater than 20 feet, the dead rolling load equivalent to the actual roUing load of 6 tons will be, say, 12 tons. For the uniform loiid, F varies uniformly from 3 tons at the left end of span, to — 3 tons at the right ; for the rolling load, i^ varies uniformly from 12 tons to 0, positive from left to right, and negative from right to left; and we have for both loads Fin = 15, - Fi2 = ; Fw = 13-5, - J'lo = ; Fb = 12, - i^s = J?4 = 9, - Ft = 3 Fs = 10-5, --Fe = 1-5 Fo = 7-5, - Fi = 4-5 Fu = 6, - Fo = 6 tons. The results for the right half of the span are similar to the above, the signs alone being changed. The range at intervals of 2 feet for the left half of the span, beginning at the point of support, is 15, 13*5, 12, 12, 12, 12, and 12 tons. 9. A beam 24 feet span bears an uniform dead load of 100 lbs. per foot, and is subject to an advancing load as long as the span, and of intensity 400 lbs. pei foot. Find the shearing forces, and their range, on the left halt of span, at intervals of 2 feet. Find the critical point K, and the coordinates of the apex Ai. See tig. 1, Ch. XIII. Since the span is a little greater than 20 feet, the dead advancing load equiva- lent to the actual advancing load of 400 lbs. per foot will be, say, 800 lbs. per foot. Taking this value, we have u = 100, and w = 800 lbs. : c = 12 feet. Substituting in equation (2), p. 227, Fvi = 10800, - Fi2 = ; Fio = 9067, - Fig = ; Fs = 7467, - Fs = I Fc = COOO, - Fe =0 ; Fi = 4667, - Ft = 667 ; Ft = 3467, - F2 = 1467 ; Fo = 2400, - Fo = 2400 lbs. The range at intervals of 2 feet beginning at left end of span is 10800, 9067, 7467, 6000, 5334, 4934, and 4800 lbs. OK = 6 feet ; OS = 15 feet ; 6'^i = 1350 lbs. 234 APPLIED MECHANICS. [CHAP. XIII. CHAPTER XIII. RESISTANCE, IN GENERAL, TO BENDING AND SHEARING AT THE VARIOUS CROSS-SECTIONS OF FRAMED GIRDERS AND OF SOLID BEAMS. Framed beams are built of pieces either freely jointed, or so slightly connected at their joints that they may be considered as freely jointed. If the load be applied at the joints, each piece is either a strut or a tie ; and if the load be applied at intermediate points on, or distributed over a portion of, a piece, it (the load) is to be replaced by a pair of equivalent forces at the ends, evidently equal and opposite to the supporting forces of that piece looked upon as a short beam. Such a piece has two duties to perform, viz., to resist the bending moments and shearing forces as a small beam, and to act as a strut or tie in the built beam considered as loaded at the joints only; by making such pieces short enough, the duty they have to dis- charge as beams can be made so small compared to what they have to perform as struts or ties, that when designed to fulfil the latter duty they will also be able to fulfil the former ; by making the pieces continuous at their joints, they are greatly strengthened for their duties as short beams. If a cross-section cuts not more than three pieces, the unknown stresses are not more than three in number, and the three conditions of equilibrium (fig. 10, Ch. V) enable us to calculate the stress on each ; or using the bending moment and shearing force diagrams, we can readily apply the conditions to as many sections as necessary, and so design the whole beam. The sections chosen for this purpose should lie just to one side of the joints for bending moment diagram, but for the shearing force diagram they should more correctly be at the centre of the bays, as the stress on a diagonal is constant and equal to the shearing force at centre of the bay multiplied by the secant of its slope. The variations of the shearing force for the half bay on each side of the centre are borne by the boom or stringer bridging across from joint to joint. For any given system of loading a beam is said to be of CHAT. XIII.] RESISTANCE TO BKNDING AND SHEARING. 2'M unifonn .strength when the cross-section of each piece is such that the ratio of the ultimate or proof resistance of the material Advancing Loart equivalent to 800 lbs. p. ft Uniform Load 100 lbs. p. ft. Bending Moment divided by dipth gives amount of stress on Horizontal boom. The Shearing Force is the vertical component of the load on diagonal. ^- Two distinct diagonals may resist the positive '' and negative shearing force. If one resists both, St be able to act as a strut, and the load is to be reckoned as the sum of the two loads. Vf3, hft. COUNTERBRACING KMn 11000 iir.'ijo igsoo sjooo iioooG „tlO Jl'IMO/^i- r,<;(j(i *c. V ' / % / '^\ % w. V. w^ 50.ooo|-ff(.;6s. I '•'""" "■-"" -"""' Warren Girder Load on flange is B. M. divided by depth. fLoad on Web is the L « fiange of S.F. reck- (» oned uniformly ) i> distributed. Fig. 1. and the tension or compression which it has to bear is constant. 236 APPLIED xMECHAKICS. [CHAP. XIII. an allowance being made, if necessary, for pieces which also act as small beams. The term " flanged girder " is employed to denote all girders consisting of a web and one flange, or of two flanges connected together either by a continuous web or by open lattice work ; in bridges, one at least of the flanges is usually straight, and also horizontal. In figs. 8 and 9, Ch. V, the stress at A is horizontal and is denoted by 2^a ', if the flange is thin, as is often the case in iron bridges, this stress is sensibly constant ; and since the intensity of the stress to which a piece is exposed should not exceed the strength of its material, we have Pa ^ ./, (1) where / is the working or proof stress as may be desired ; and if < = amount of stress on the horizontal flange, and S = the cross-sectional area of that flange, then t ^ S.f. (2) When there is only one set of triangles, as is shown in the lowest of the three systems of bracing (figure 1), and shown also in the two upper systems if we neglect the counter-bracing, the amount of stress on the straight boom may be found thus : — Take a cross-section at a point, say F, just on either side of a joint in the boom opposite the straight boom (in the figure both booms are straight) ; take moments round the point F, and since we neglect the counter-brace, the only member not passing through the point F is the upper boom ; the product t . h gives the moment where h is the depth of the beam at the point ; this is equal to the l:)endiiig moment, and we have by substituting the value of t given in equation (2) t.h = S.f.h = M. (3) The quantity M is different at difi'erent points of the beam, and / is a constant quantity ; if the above equation is to be fulfilled for every point, we make aS . h vary as 31 ; in practice one of these two factors is generally kept constant, and the other is made to vary. If we make //. constant, then both booms are liori/.ontal, and S varies as the bending moment ; hence the bending moment diagram gives, upon a suitable scale, the area of the boom at each point. The vertical component of the stress on any CHAP. XIII.] RESISTANCE TO ItKNDING AND SHEAHING. 237 diagonal is the amount of the shearing force at that end of the diagonal where the shearing force is greatest ; hence if t be (jhe P I rectftngu Load on two nvheets. ' The curves are arcs of circles with their centres on the verticals through the apexes of p^trabolas. ELEVATIONS OF THIN FLANGED GIRDERS OF UNIFORM STRENGTH, derived from Bending Moment Diagrams. Fig. the stress on a diagonal, F the shearing force at the end of the- diagonal where it is greatest and as given by the shearing 238 APPLIED MECHANICS. [CHAP. XIII. force diagram, and 6 the angle made by the diagonal with the vertical, then r = i^ sec e. (4) The depth h is chosen from ^th to xr ^^ ^^ ^^^ ^P^'^ ^^ ensure stiffness ; in fig. 1 the depth h is taken at 3 feet, that is ^th of span, and that figure shows how the stresses on the booms and on the diagonals are found. Between the points K and L, counter-bracing is required ; this is accomplished in the two upper girders by introducing ^mrs of diagonals between the.se points, both being ties or both struts as the case may be ; the one diagonal resists the positive, the other resists the negative shearing force. In the Warren girder, the third in the figure, one diagonal resists both the positive and negative shearing force, the one being applied suddenly after the other; each diagonal should be designed to bear the stress due to the sum of these stresses. The lowest girder shown in the figure is one with thin flanges and a thin continuous web ; part of the bend- ing moment is resisted by the web and part of the shearing is resisted by the flanges : these parts, however, are small ; prac- tically the flanges are considered, as in the case of open beams, to resist the whole of the bending moment, and the web is con- sidered to resist the whole of the shearing force : an approximate result is obtained if we consider that the shearing force is uniformly distributed over the cross-sectional area of the web. If we make S constant, then h varies as M \ and (fig. 2) the elevation of the beam will correspond with the bending moment diagram ; h, the depth of the beam at the centre is, as in the previous case, to be taken sufficient to ensure stiffness. The curved boom will bear a share of the shearing force ; this compounded with the stress on the horizontal boom at the same section will give the resultant stress on the curved boom. It is usual in practice to make the area of the curved and of the straight booms uniform, and to make the diagonals sufficientl)'- strong to resist the whole of the shearing force as in the previous case. Where the curved member slopes considerably, as in the bowstring girder, it is made sufficiently strong to bear the whole shearing force, the diagonals being intended for another purpose, viz., to distribute partial loads in a sensibly uniform manner. The theoretical elevations reduce to a height zero at the ends, and so give no material to resist the shearing force at the point where it is greatest ; sufficient material is generally CHAP. XIII.] RESISTANCE TO BENDING AND SHEARING, 239 allowed at the ends either by making the span of the girder exceed the clear span, or by departing from the theoretical form along a tangent near the end. Further, whatever the curves may be — and, as we have seen, they are generally parabolas — they are usually replaced by circles which nearly coincide therewith ; when the iigure passes from one curve to another, the passage is made along a tangent, as will be seen on some of the figures. Approximate forms, consisting entirely of straight lines enveloping the bending moment diagram, are sometimes adopted. Moment of Eesistance to Bending of Eectangular and Triangular Cross-Sections. The moment of resistance to bending we have defined as the moment of the total stress upon the cross-section about any point in it ; and this we have shown (figs. 7, 8, 9, Ch. V) to be equal to the couple which is the moment of the normal stress on the cross-section. The stress (fig. 8, Ch. V) might be artificially produced by building on the portion O'A, columns of a material tending to gravitate towards the left ; and on the portion O'B, columns gravitating towards the right. These columns standing on very small bases, being of uniform density, and of the proper height to produce the intensity at each point, will, if we suppose them to become one solid, form a wedge with a stepped or notched sloping surface ; the more slender the columns are, the more accurately do they give the stress at each point, and the smaller are the notches on the wedge; hence tvjo right wedges exactly represent the normal stress on the cross-section. Such a stress is called an uniformly varying stress. Taking the density of the wedges as unity, the height of one will be expressed hj 2Ja, and the other by jib ', the volumes of the wedges will give the two normal forces ffig. 10, Ch. Y) the resultants of the thrusts and tensions respectively ; these forces are equal ; the volumes of the two wedges must therefore be equal, and the position of the neutral axis of the cross-section is thus determined. Further, each wedge, instead of distributing its weight over its base, may be supposed to stand on the point below its centre of gravity ; this enables us to find the positions of the normal forces (fig, 10, Ch. V), and gives us the arm of the couple ; if we multiply the volume of either wedge by this arm, we have M the moment of resistance to bending. 240 APPLIED MECHANICS. [CHAP. XIII, For a rectangular cross-section, the neutral axis is at the centre since the wedges are equal, and pa equals ])}, ; and if the common volumes of the right wedges be represented by V, then V = ^Pa >i h bh = Ipabh ; where h is the breadth and h is the depth of the beam, as shown in fig, 1, Ch. V. Each wedge stands on a rectangular base, so that the point on the cross-section below its centre of gravity- is distant from 0' by two-thirds of O'A, or ^h ; hence the arm of the couple is §/«, and M = iPabh X 1^ = iPabh\ By increasing the bending moment we can increase M till Pu becomes equal to /, the resistance of the material to direct tension or thrust, but no further ; because if Pa becomes greater than/, the fibres at the skin will be injured ; hence M = iph' (1) is the ultimate, proof, or working resistance to bending, according as / is the ultimate, proof, or working strength of the material. Since /is generally expressed in tons or lbs., per square inch, it is necessary to express b and h also in inches ; in which case M will be in mc7t-tons or lbs. ; on the other hand, M \s> generally expressed in foot-ion^ or lbs. ; and it is well to observe that, if such be the case, M requires to be multiplied by 12 to reduce it to inch-ton^ or lbs. before equating to M. For all cross-sections, if b and h are the dimensions of the circumscribing rectangle, the equation giving the moment of resistance is of the same form, but the numerical coefficient assumes values other than ^-, as will be proved hereafter. Rankine uses n for the value of this constant, which he calls the numerical coefficient of the moment of resistance to bending of any cross-section, and we may put M = nfbh\ We shall now verify this for a triangular cross-section ; an isosceles triangle is taken, but exactly the same result would be obtained ff)r any triangle; the angle A, however, must not be too acute. CHAP. XIIL] resistance TO BENDING AND SHEARING 241 Triaiiffular cross- section (figs. 2 and 3). — Assume that the neutral axis passes through the centre of gravity of the Fi-. 3. triangle, and divide the part below into three areas as shown ; put aS' and V with suffixes for the areas and volumes of the 242 APPLIED MECHANICS. [CHAP. XIII. different parts, put I for the leverages about the neutral axis of their centres of gravity, and, for convenience, suppose h = 36a% and h = 6;/. Vi is a pyramid on S^ as a base, and let/ be its height; F, is a right prism of height |/ ; F3 is a pyramid on S:-, as base and of height 12r. Now the centre of gravity of a pyramid is on the line joining the apex with the centre of base, and at three-quarters of the length of that line from the apex ; and ;S', = i . 24a; . 4y = 48>;?/ ; S^ = 12.r . 4y = 48xy ; Ss ='^ y. 2' V, = \fS, = Wxy- V, = },.^.S, = Ufxy; V,= \.12x.S, = 2fxy= F,; hence F, = F, + F3 + F4, and therefore the assumption as to the position of the neutral axis is correct. Now Zi = 12a:, I2 = Sir, h = li = ^x; and M = V,h + V,h + V,h + F«Z4 = (192 + 96 + 18 + 18)/7/.r = 324/?/a;^ Substituting y = g, and a:' =f — j = ^^^, we have M = i-Jhh\ (2) so that for a triangular section, n = ^. Solid beams are sometimes made of uniform section ; that is, at the point of maximum bending moment the section is made sufficient to resist the bending moment, and this section is adopted along the entire length ; at every other section there- fore, the beain is too strong. This is frequently done with small timber beams cut out of one piece, because the material in excess, even if cut away, would be lost ; the weight of this excess is very little, since timber is light, and probably *he cutting of it away would add to the expense of the beam. When the section of an uniform beam is designed to resist the maximum Ijending moment, it is generally many times more than sufficient to resist the maximum shearing force. Solid rectangular beams are designed of uniform strength to economize material, and to reduce the weiglit when tlie material itself is heavy. CHAP. XIII.] RESISTANCE TO BENDING AND SHEARING. 243 Bectangidar beam of uniform strenrfth and uniforin d^pth. — In this case the breadth is varied so as to make tlie cross- section at every point just sutficient to resist the bending moment. It is evident that the elevation is a rectangle, while the plan will vary according to the load ; the plan, however, is to be symmetrical about a centre line. The theoretical shape of the plan so designed has to be departed from near the ends, so as to make the end sections large enough to resist the shearing force ; and some additional material is required at the €nds of the beam to give it lateral stability ; now nfbh^ = M; therefore h = ^-rr„, nfh- or h is proportional to M, since n, f. and li are constant quantities. Quad, of Par., but circular arc ia close approx. Lo»as/m.ontwoWhe»lt Ip Plans of Rect. Beams of Uniform Strength i Uniform Depth same asB.M Diagrams. Fig. 4. Hence the half plan is the bending moment diagram reduced so that its highest ordinate is equal to ^^o, the half- breadth of the cross-section which can resist the maximum bending moment (see figs. 1 and -i). As has previously been explained, the parabolas being very flat are readily replaced by arcs of circles with their centres on the axes of the parabolas. In designing such beams, then, the uniform depth is fixed as a fraction of the span from an eighth to a fourteenth to ensure the required degree of stiffness ; &,„ the breadth of the cross- section where the bending moment is a maximum, is calculated to make that section sufficiently strong, exactly as in the pre- ceding examples ; then the bending moment diagram is reduced till its highest ordinate is ^^o, and drawn on both sides of a R 2 244 APPLIED MECHANICS. [CHAP. XIII. central line. All the curves may now be replaced by circular arcs, and all sudden changes of breadth bridged over by tan- gents ; the curves are to be departed from, near the ends, to make the sections there strong enough to resist the shearing force. Otherwise, tlie l)readths may be calculated at a number of sections and plotted, and a fair curve drawn through them, &c. Bedcnr/ular beam of uniform strength and nniform hracUh. — It is evident that the plan is a rectangle ; and since ufbh"^ = M, we have h proportional to VtI/. Hence the elevation of the beam is obtained by degrading the bending moment diagram, so that the derived figure has its highest ordinate equal to Ao, the depth required to ensure stifl'ness ; h is then to be made sufficient to resist the maximum bending moment (see Theorems F, G, and IT, Ch. VI). The figures in brackets refer to the corresponding bending moment diagrams. Fig. 5 shows what the bending moment diagrams (figs. 9, 10, lo, and 16, Ch. Vll) become when degraded by the preceding theorems. The elliptical elevation of a beam for an uniform load is readily struck from three centres, as shown in fig. 11, Ch VI ; AD from a centre on AO and with a radius ,.. = ^■' = 60'^-24 = 150; OA BC from a centre K with radius OA- ri = ^ = 24^ -f 60 = 10 nearly ; OB ^ E is found by drawing OE from the first centre, and a circle- about K with a radius 7-, = A.„-/-, = 24-10 = 14; further, if we choose we may retain the single circular arc AD, and depart along the tangent at D. For a beam with the load at the centre, the two parabolas may be replaced by their tangents ; this gives an approximate form, and the beam will now consist of two straight portions tapering so that the depth at each side is \K, K being the depth at the middle (see fig. 3, Ch. VI) ; the area of the theoretical elevation is two-thirds, while that of the approxinuite elevation is three-quariers, of a rectangle of height h„; these areas are as 8 to 9; hence the volume •CHAP. XIII.] ItESISTANCE TO BENDING AND SHEARING, 245 of the approximate form is only one-eiglitli in excess of the theoretical one, and the additional material is well placed to resist shearing ; the approximate form is in some cases preferable, since it has the great advantage of straight boundaries. The same remarks apply to the elevation of a cantilever with the load at the end ; on this principle spokes of wheels, when of uniform thickness, taper to half the depth from boss to tyre. More particularly in timber beams {bcor,unt) and cantilevers are the straight ..-—"X b.m.d boundaries desirable. The material '('-'- is light, and the extra wood at the ends holds the bolts and-^'.-- fastenings securely. ^' ELEVATIONS UNIF. STRENOTH,UNIF. BREADTH B.M. DIAGRAM DEGRADED. Fk Fig. 6. In fig. 6 are shown the effects of degrading such bending moment diagrams as figs. 2, 5, 17, Ch. Vll, diagrams consisting of straight slopes. Producing the slopes to meet the base as at A, B, &c., the elevation for beams of uniform strength and uniform breadth will consist of a series of parabolas with their apexes at A, B, &c., and intersecting on the lines of action of the concentrated weights. An approximate elevation is to be derived thus : — Lay up ho at the point of greatest bending moment, the proper fraction of the span to ensure stiffness, and calculate the breadth of that cross-section as in the preceding examples, so as to give the proper resistance to bending there ; from the top of h^, draw a tangent to the parabola whose apex is A, that is, draw the line which intercepts Via on the vertical through A, and it will cut 246 APPLIED MECHANICS. [CHAl'. Xill, off hi on the line of nearest weight; from the top of A, draw a slope to cut off |/'i on the vertical through ]i, the apex of second parabola and the point where the second slope of the bending moment diagram meets the base; this will be parallel to the tangent to that parabola, &c., &c. It is evident then that these approximate elevations for concentrated loads consist of straight lines, each sloping at half (Jk: rate of the corresponding side of the bending moment diagram ; from this fact they are readily drawn thus : — From the highest point in the bending moment diagram draw a line at half the slope of the adjacent side till it cuts the line of the weic'ht nearest to that point ; from the point thus found draw a line to cut the next w eight at half the slope of the corresponding side of the bending moment diagram, &c. ; lastly, reduce the ordinates so that the highest is Ji„. (bconatant) W^ o When fig. 13, Ch. VIII, is degraded, from K to D it will he a portion of the parabola whose apex is at B the centre of load, and from Z> to 6' a straight slope ; the approximate form is a straight slope to D the end of load, and which, when continued, tapers to Ih^ at B ; then from D a straight slope tapering to zero at the free end. On degrading the two parts of fig 4, Ch. VIII, separately, that for £/" will be a taper from a at the fixed end to zero at the free end ; while the approximate form for W will be a taper from b at the fixed end to |i at the free end ; hence the approximate elevation for the combined load is a taper from (a + J) at the fixed end to ^h at the free end ; sub- stituting for a and b their values, we have H : Ji ■.: 2JV + U : TV. Fig. 7 shows figs. 1, Ch. VIII, and 10, Ch. IX, degraded, the quadrants of parabolas becoming quadrants of ellipses. Ikctawjvlar beam of uniform strength and similar cross- section. — In this case both b and // vary, but they bear to each CHAP. XIII.] RESISTANCE TO BENDING AND SHEAUING. 247 Other a constant ratio ; it is evident that the plan and elevation will ha\e the same form. The plan is always to be symmetrical abont a centre line ; the elevation may either have one straight boundary, or be symmetrical about a centre line. Since now b oc h, then b/r oc b'\ or h^ ; nfbh- = 31, bh' oc M: therefore b and h oc J^l^- That is, both plan and elevation are derived by drawing a locus whose ordinates are proportional to the cube roots of those of the bending: moment diagram. Approx. Uniform Strength Similar Cross Sections. Fig. 8. For bending moment diagrams with straight slopes, that is, for concentrated loads, the degraded figures were parabolas with axes horizontal, &c. ; in the same way, when the new figure is made with its ordinates proportional to the cube roots of the ordinates of the straight slopes, it becomes what is called a cubic parabola ; a property of this curve is that the tangent cuts off tico-thirds of the ordinate upon the vertical through the apex, instead of one-Jialf as in the case of the common parabola. Hence figs. 5 and 6 give approximate elevations and half plans 248 APPLIED MECHANICS. [CHAP. XIII. of beams of uniform strength and similar cross-sections, if we use -f^n instead of hhn in making the construction ; or if we draw the tapers at two-thirds of the slopes of the bending moment diagram instead of one-half. Observe that in this case there is a double taper in planes at right angles to each other. On this principle the crosshead of a piston-rod may have a conical taper, so that the diameter at each end may be two- thirds of the diameter at the centre ; and the spokes of wheels may have a conical taper from the boss to two-thirds (linear dimensions) at the tyre. ExAMPLKS. 1. Find the working moment of resistance to bending of a rectangular section, 10 inches deep and 3 inches broad, the woiking strength of the material being 4 tons per square inch. M = n/b/r- = ^ . 4 . 3 . 100 = 200 inch-tons. Find the same for an isosceles cross-section inscribed in the above rectangle, and with the base horizontal. Ans. M = «/M2 ^ ^\. . 4 . 3 . 100 = .50 inch-tons. 2. Find suitable dimensions for a cast-iron Deam 20 feet span, of uniform and rectangular cross-section, and subject to a load of 10 tons at the centre. Taking h = 20 inches, a twelfth of the span, to ensure stiffness ; / = 2 tons per square inch ; and M = Mo = m Wl, the maximum bending moment. Ans. i . 2 . « . 20^= i . 10 . (12 X 20); .-. i = 4-.5 inches. 3. If the breadth be taken at 6 inches, what depth would give sufficient strength ? Am. i . 2 . 6 . A2 = i . 10 . (12 X 20) ; .-. h = 17-3 inches. 4. If the load is uniformly distributed, and the cross-section is a triangle whose base is horizontal and 8 inches broad, find the height of the triangle. Ans. ^^- . 2 . 8 , //' = 1 . 10 . (12 x 20) ; .-. h = 21-2 inches. 5. Find the greatest cross-section for u wrought-iron beam of rectangular section and 15 feet span, to bear a load of 20 tons uniformly distributed, together with a load of 5 tons at the centre. Take /= 4 tons per square inch, and /» = lo inches to give sufficient stiffness. A7i>. 1.4.*. 1.5- = (i . 20 . 15 + I . o . 15) X 12 ; .-. * = 4-5 inches. 6. Taking the depth one-twelfth of tlie span, and /"- 4 tons per square inch, find the breadth for a wroughi-iron beam of rectangular section, to resist the maximum bending moment in Ex. 9, Ch. IX. Ans. ^ X 4 X 4 X 36» = 112^ x 12 ; .- . b = 1-56 inches. OHAl'. XIII.] RESISTANCE TO BENDING AND SHEAHINO. 249 7. Design a rectangular cantilever 1 feet long of approximately uniform strength and of uniform breadtli, of timber whose working strength is 1 ton per square inch ; the load is 2 tons at the free end, and ho = 15 inches, an eighth of the length. Ans. nfbhu- = m . W . I; i . 1 . i . 15^ = (1 . 2 . 10) 12, gives b = 6*4 inches for uniform breadth ; the depth tapers from 15 inches at tixed end to 7*5 inclies at free end. 8. If an additional uniform load of 4 tons be added, and /(o be still retained 15 inches, find the value now of b, and of h at the free end for Ex. 7. Ans. At the fixed end the bending moment will be double its former amount, so that b will be doubled ; that is, b = 12*8 inches, and 7(0 :/*:: 2 JT + r : Zr :: 15 I/"-:: 2 X 2 + 4 : 2 : 7(10 = 3-7o inches. 9. A wooden cantilever 12 feet long bears 3 tons uniformly distributed on the half next the free end. Design an approximate elevation, supposing the breadth to be uniform, and/= 1 ton per square inch ; take depth at fixed end as 18 inches. Ans. J/o = 27 foot-tons = 324 inch-tons; i . 1 . i . 18^= 324 ; .*. b = 6 inches constant. Between the fixed end and the end of the load next thereto, the elevation will taper at the rate of 1 inch per foot, and thence to zero at the free end : that is, /io = 18, ht = 12, and /ns = inches. 10. Design a cantilever for Ex. 7, supposing its section to be a square, taking dimensions to the nearest whole number in inches. Ans. Put ^0 = 7(0 = side of square at fixed end, and equating n/b^ = in . W . I, gives bo = ho = 11-3, say 12 inches. The side of the squai'e at the free end is 612 = hn = 8 inches. Area, Geometrical Moment, and Moment of Inertia. The functions of a plane surface which we require for our investigations regarding the moment of resistance to bending of a cross-section in general, are the area, geometrical moment, and moment of inertia. The area of a rectangle is the product of its two adjacent sides ; the area of any other surface is the sum of all the elementary rectangles into which it may be divided. We take it for granted that the area of the triangle, the circle, the ellipse, and parabolic quadrant are respectively half the product of the height into the l)ase, tt into radius squared, tt into the product of the semi-major and semi-minor diameters, and two- thirds of the product of the circumscribed rectangle. Definition. — The geometrical moment of a surface about any line in its plane as axis, is the sum of the products of each elemental area into its leverage or perpendicular distance from that axis ; the leverages which lie to one side of the axis being reckoned positive, and those to the other side negative. 250 APPLIED MECHANICS. [chap. XI I L It will be convenient for us always to choose a horizontal axis; and if we consider leverages up to be positive, then, when the axis is below the aiea. the geoniftrical moment is positive ; when ahove it, negative; when the axis cuts the area, it will he positive ornetrative accordinji as the axis is near one or other edge; and for one position of the axis, cutting the area, the positive and negative products will destroy each other, and the geometrical moment will he zero. An axis about which the geometrical moment of an area is zero passes through a point called the geo»ietrical centre of the area. From this it appears that the geometrical centre of an area corresponds with the centre of gravity of a thin plate of uniform thickness and of that an a ; and for this reason the geometrical centre of an area is often called its centre of gravity. Thenrem A. — The geometrical moment of a surface about any axis in its plane is equal to the area multiplied by the distance of the geometrical centre from the axis (fig. {')• Suppose G the geometrical centre of the surface to he known ; through G draw 00' parallel to the axis A A ; let s and «' be a pair of elementnl areas, one on each side of 00', such that the sum of their geometrical moments is zero ; that is, i' . a = s .b. It is evident that the whole area can he divided into such pairs from the definition of the geometrical centre. The geometric al moment of *' about AA is s(d + a), that of s is 8[d - b), and their joint moment is («' + s)d + (*'« - sh) = («' + .s)rf, since the second term is zero. In the same way the moment of each pair is their sum multiplied by rf ; \ \ \ hence the geometrical moment of the area about AA j, i i^j— 4 is the sum of all the pairs into d, that is, the area multipiitd by the distance of the geometrical centre Fig. 9. from the axis. Cor. — The geometrical moment of an area which can be divided into simpler figures whose geometrical centres are known, may be found by multiplying the area of each sucli figure by the distance of the axis from its geometrical centre and summing algebraically. T'neorem Ji. — The geometrical moment of an area about an axis in its plane is expressed by the number which denotes the volume of that portion of a right- angled isosceles wedge whose sloping side passes through the axis, and which stands on the area as base. Let AEKFA' (fig. 10) he the wedge, and AF its sloping side passing through the axis A A' ; the angle at A is 4.5° and that at E is 90' ; let BCBE be the area, then the geometrical moment of BCDE rela- tively to the axis of A A' is represented by the volume of l'bckl. If s be an elemental poition of the area BCBE, its geometrical moment about AA' is s multiplied by its distance from AA' \ but the column of the wedge standing on « is sensibly a parallelepiped whose height is the same as the distance of s from AA', and the volume of that column expresses the geometrical moment of s; hence the volume oiUBCKL, a portion of the isosceles wedge, is the geometrical moment of BCDE about A A'. When the axis cuts the area, the plane sloping at 45° will form a wedge above one portion and bclotv the other; by considering these of different signs, their algebraic sum is still the geometiical moment. For example, if we wish to find the geometrii-al moment of the triangle (fig. 3) about the axis JSA ; let /= J//, th.n the plane will slope at 45° ; in the figure, the volume on one side of the axis is equal to the volume on the •O-^ Fijr. 10. CHAr. XIII.] RESISTANCE TO BENDING AND SHEARING, 251 other, tliiit is, the geonietrifiil monieiit is zero; the axis lY^I must pass therefore through the geometrical centre. To find an axis passing through the geometrical centre of a plane area then, it is only necessarj' to draw a plane sloping at 45° which will cut off an equal volume of the wedge above and below ; the intersection of this plane with the area will be the axis re(|uired. By similar triangles a plane through that axis at any slope will cut off equal volumes above and below ; the wedges which represent tlie normal stress (fig. 8, Ch. V) require to be equal ; hence the neiiiral axis of a cross-section passes through the geometrical centre (or centre of gravity) of the cross-section. The distance of the neutral axis from the furtiiest away skin is, in each cross- section, a definite fraction of h the deptli of the circumscribing rectangle ; for instance, for a trianguhir cross-section (figs. 2, 3) OA = %h. Bankine expresses this generally thus OA = m'h. where m' is the fraction rvhich the distance from the neutral axis to the farthest atray skin is of the depth ; for all cross-sections, symmetrical above and below, as a rectangle, ellipse, hollow rectangle, &c., «»' = ^. Definition. — The moment of inertia of a surface, about a line in its plane as axis, is the sum of the products of each elemental area into the square of its distance from the axis. Theorem C. — The moment of inertia of a surface about any axis in its plane equals that about a parallel axis through its geometrical centre, together with the product of the area into the square of the deviation of the axis from the centre. Let s and s' (fig. 9) be a pair of elemental areas, the sum of whose geometrical moments about 00" is zero; and let Xi and X2 be their leverages about A A respectively. The sum of their moments of inertia about A A is s X2- + sxr = s'{d + a)'^ + s{d — b)- = (.«' -f- s)d^ + 2(s'« - sb)d + (.■>■'«- 4 sb-) = (s' + s)cP + (s'a- + sb'), since (s'a — sb) = 0. Summing the leftside for all pairs we have the moment of inertia of the area about AA. The first term on the right side is the area of each pair into the square of the deviation ; and the sum of these for all pairs is the area into the square of the deviation ; the second term on the right side is the moment of inertia of s and 4' about 00", the sum will be the moment of inertia of the whole area about 00'. If I^i and lo represent the moments of inertia i ound the axes A and respectively, then I A = S . cP -f lo, where S is the area of the figure. Cor. — For any set of parallel axes, the moment of inertia about that axis which passes through the geometrical centre is a minimum ; and those axes which give minima values for the moment of inertia intersect at a point. For every cross-section, lo will be of the same form, a constant multiplied by the breadth and multiplied by the cube of the depth of the circumscribing rectangle. Rankine puts generally lo = n'bh'^ where n' is the numerical coefficient of the moment of inertia of the cross-section about its neutral axis, the other factors being the breadth and cube of the depth of the circumscribing rectangle. If through the neutral axis of a cross-section we draw a plane sloping at 45°, it wUl form two isosceles wedges, or two portions, one on each side of the plane ; the sum of the products got by multiplying the volume of each by the distance of the point under its centre of gravity from the neutral axis gives lo for the ciosa- section ; for this purpose we may take each portion as a whole or subdivide it into a number of parts if such is more convenient. 252 APPLIED MECHANICS. [CHAP. XIII. For a rectangular wedge ^EKFBA' (fig. 10) let S = area of hase AEBA', f = height EK, V — volume, xa = distance from A A' to the point which is under the centre of gravity, then V = ^S./; xq = fZt'. For an isosceles-triangular wedge ACCS (6g. 3) let S = area of base ACC, /■= height AS, Xo = distance from CC of the point which is under the centre of gravity, then r= ^S .f; xo = \0A. If any doping plane be drawn through the neutral axis, it will cut off two wedges ; and since the volumes of all such wedges are proportional to their heights, we have F: V::f :f::f -.mh, where V is ilie volume corresponding to/' the new value of EKin fig. 10, and of AS in fig. 3 ; and 7n'h is the height of the isosceles wedge, that is the distance of the skin from the neutral axis. Since the leverages are the same as before, the statical moments of the wedges are also in the above ratio. CHAPTER XIV. c;ross-sectio>'s : their resistance to bending and shearing, AND distribution OF STRESS THEREON. PtESISTANCE TO BENDING OF C ROSS- SECTIONS. We see then that Iq the moment of inertia of the cross-section about its neutral axis is represented by the statical moment of the isosceles wedges made by a plane sloping at 45^ and passing through the neutral axis ; that the highest point of these wedges is ya or 6 = 'ni'h, the distance from the neutral axis to the further skin ; while M , the moment of resistance to bending, is repre- sented by the statical moment of the wedges made by a plane passing through the neutral axis, and sloping so that the height of the highest point of these wedges is^„„r6 =/, the stress on the skin A or B, whichever is further from the neutral axis ; hence 1^ normal stress on skin furthest from neutral axis j distance of skin from neutral axis ° mh ma m k is the proof, or working moment of resistance according as / is the proof, or working strength of tlie material supposed to be the same for both skins, that is. for thrust and tension. Rectangular Section (fig. 1).— Let BCDE be a rectangle. Its moment of inertia about DC one side is found thus : — The CHAP. XIV.] RESISTANCE TO BENDING AND SHEAKING, 253 height of the isosceles wedge is zero at C aiul BC at /> ; the average height is therefore \BC ; its vohmie is ^EB . BC-, and since the point below the centre of gravity of the wedge is two- thirds of BC from DC, we have I DC = h^B . BC^ X ^BC= \EB X BC as the moment of inertia of a rectangle about a side. Now, if BE'' be a rectangular cross-section of which DC is the neutral axis, then its moment of inertia about that axis will be double the above ; and thus the moment of inertia of a rect- angle about an axis through the centre and parallel to a side is In this case m' = M /o = 2 X I . EB . BC = -^b/i\ h, so that f_ X -J^bh^ = i/7>/r. Thus, for a rectangle, m' = -i, n' = yV, and u = -i. Hollow Rectangular Section. — For a hollow rectangular section symmetrical above and below the neutral axis, that is, when the whole rectangle and rectangle removed have their centres on the 12 k h » B |.,, ^^Mkj^^^l T . -+o— M~ -i--o Fijr. 1. Fig. 2 same horizontal axis, the moment of inertia is the difference of the moments of the two rectangles. Let H, h, and B, b be the outer and inner dimension.^ respectively, so that the area is Bff - bh ; then bh' BH fBH\ BR' Hence, for a hollow rectangle, bh' \ BH'j and for the dimensions given in fig. 3, viz., H=30, B= 10, 7t = 24, J = 6 inches, m = ^, n = -jL ( 1 - ^, ), and n = l(l- -^^ 254 APPLIED MECHANICS. [chap. XIV. we obtain lo = 15588, and M = 1039-2/ inch-lbs., if/ be in lbs. Tabular Mdliod. — The following is a convenient form for expressing the area, geometrical moment, and moment of inertia of a rectangle: — Let BCDE (fig. 2) be a rectangle, and A A' an axis parallel to DC, and let Y and y be the distances of its sides EB and DC from AA'. The area is S=h{Y-y\ (1) the breadth into the difference of the ordinates. For EL, part of the isosceles wedge (fig. 10, Ch. XIII) EK = Y, CL = ?/; and the geometrical moment of the rectangle BCDE about ^^' is G , = volume of part of isosceles wedge = h{Y-y) X i(r+ y) = h.h{Y' - f), (2) one-half the breadth into the diflerence of the squares of the ordinates of the sides parallel to the axis. By theorem C, Fig. 3. Ch. XIII, we have the moment of inertia of the rectangle BCDE (fig. 2) about AA' I^ = Io + Sd' = j'jEB . BC + EB . BC (^^)' = ^^h(Y-yY + h{Y-y)^-^^=!j.b.(Y^^y^), (:)) one-third of the breadtli into the diflerence of the cubes of the ordinates of the sitles paraUel to the axis. CHAP. XIV.] RESISTANCE TO BENDING AND SHEAUING. 255 To apply this to the case of a hollow rectangle and a sym- metrical double-T section, which give similar results (tig. ?>). Choose 00 the neutral axis in the centre from symmetry ; and if only the upper half of the section be considered, it consists of two rectangles, the ordinates to the edges being 0, 12, 15 ; for one rectangle Y = 15, ;/ = 12 ; for the other Y = 12, and y = '■ and for each (equation (3), /« = F>(^'^' - y^). h Y j ya ya_v3 WV-y') 10 4 15 3?75 12 1728 1647 1728 5490 2304 - 1 - 1 - *,.=.:« if X:: This tabular method applies to any section made up of rect- angles and which is symmetrical above and below the neutral axis. As we require Gr the geometrical moment of the semi-sec- tion relatively to the neutral axis when we come to resistance to shearing, it is convenient in making the table to find o-*- j=- Gr ; thus, for the symmetrical section, one-half of which is shown in fig. 4, we have the following results : — Fio;. 4. b Y yi Y^-f- \h{Y-'—f) ys y3_^,3 ^A(y3_y3) 10 100 1000 12 8 64 36 216 512 488 1952 9 6 36 28 126 216 296 888 3 36 54 216 216 — - - G, = 396 - — ^^0=3056 Where (r is for semi-section onlv. 256 APPLIED MECHANICS. [chap. XIV. Suppose the working strength / = 4 tons per square inch, then the distance from 00 the neutral axis to the skin is 10 inches ; hence M = -4-, /., = tV X 6112 = 2444-8 inch-tons. m h The use of G" will be shown at the proper place. Symmetrical sections are suitable for materials for which the strengths to resist thrust and tension are equal, as ^^and^* become /'simultaneously (tig. 8, Ch. V). For materials whose strengths are unequal, /„ is put for the greatest value oi ])„ and fb for that of pb ; for wrought iron, for instance, the working resistance to thrust is ./„ = 4 tons per square inch, while to tension it is /j = 5 tons per sq. f^"^ inch. For symmetrical sections in such materials the value of f employed must obviously be the smaller. Because of this property of materials, cross-sections are made unsymmetrical above and below. Unsym met rical Sect ion. — Let the unsymmetrical section (fig. 5) be given, and let/a = 4, and/j = 5 tons per square inch. In order to determine its resistance to bending we find Iq '. we will also ?/^« i^//////m^/AA m. v;r/*--o— ys'^ Jfj- 5 Ions pe r aq. in, Fiir. a m o.'._ Axis Orainatet. A find the area of the section, and G' the geometrical moment of the portion of the section lying to one side of the neutral axis whose position is not yet known. Choose an// line as axis, say, the upper skin ; the diagram and table show the breadths, and ordinates laid down from this line. b Y Y-y Area y^ y2 _ j2 t'^y- y^) ya Y3.y3 i3(y3->3) 12 3 3 36 9 9 54 2 27 108 6 6 3 18 36 27 81 216 189 378 1-5 i6 lO 15 256 220 165 4096 3880 1940 4 i8 2 8 324 68 136 5832 1736 2315 8 2 16 76 304 2l6S 5781 20 400 8000 ~ Ia =10522 S=93 - Ga = 740 CHAP. XIV.] RESISTANCE TO BENDING AND SHEARING. 257 Q Now y„= ^ = W = 8 "iches (by theorem A, Ch. XIII) ; o .-. //6 = 20 - 8 = 12 inches. Agcain, I, = /., - Sija' = 10522 - 93 x (Vj"-)" = 4634. The neutral axis divides the depth almost exactly as 2 to 3 ; and if we had/a r/j : : 2 : 3, then both skins would come to their working stress simultaneously, and we might obtain M by f f multiplying /« either by — or by - ; but since fa'.fb ' '■ -^ ■ o, Va Vb it is evident that both skins cannot come to their working strength at once. In this example, when the skin B comes to fb = o tons, the skin A will be at |/j or 3-^ tons per square inch, and so is not at its full strength 4, yet the stresses are now at their greatest ; on the other hand, the skin A cannot come to its strength /„ = 4, because then the skin B would be at f x 4, or 6 tons, and so be overtaxed ; hence M is to be obtained by f f multiplying /„ by the ratio —, not by — , which would give too yb Va much. Of the two ratios •^ = 4 = -5, and *^ = -^ = -42, select the less, and M Vb X 7o = -42 X 4634 = 1931 inch tons. Taking the neutral axis now as origin, the geometrical moment for the part of the section lying above or below, is (?o = 296 ; a quantity to be used in calculating the resistance to shearing. b Y Y2 Y2-y^ hHY^-y^) i'5 8 §4 64 48 4 10 lOO 36 72 8 44 176 12 144 - — — — Gn 0=296 258 APPLIED MECHANICS. [CHAP. XIV. Graphical Solution (fig. 6 : for the same data as fig. 5), — Eeplace the areas of the rectangles by the forces (1), (2), (3), (4), (5) acting at their centres of gravity, the amount of each force being the same as the number of units in the area corresponding ; draw the first link polygon as in fig. 3, Ch. VII, and Z, the intersection of the end links, gives the centre of the forces and therefore the centre of gravity and neutral axis of the cross-section. The intercepts on 00 are the geometrical moments of the areas respectively, that is, the product of each area into its leverage about 00 \ the scale is found by sub- dividing the scale for areas by 10, since 10 on the scale for dimensions was taken as the polar distance. Considering these intercepts as magnitudes of forces acting in the same lines as before, and drawing a second link polygon, its intercepts on 00 are the products of these forces each into its leverage about 00; or these intercepts are the areas each into the square of its leverage about 00 ; the scale is derived from the previous one by again subdividing by the polar dis- tance, 10 on the scale for dimensions. The proof is given at fig. 3, Ch. VII. The sum of the intercepts made by the second link polygon is nearly /o ; being deficient by one-twelfth of the sum of the products of the breadth of each rectangle into the cube of its depth. Also KL the geometrical moment of areas (1) and (2) is nearly G^n, the deficiency being that of the rectangle lying between the area (2) and the neutral axis ; this rectangle, however, is small and has a short leverage. There is also shown in fig. 7 a link polygon drawn for the three areas (1), (2), (3"), which constitute the portion of the section lying above the neutral axis ; the sum of the intercepts is (?o. ^ Corrections. — The manner of correcting is shown in figs. G and 7. Take the lines of action of the areas first along their (say) n2')'per edges, and construct the first link polygon ; treating the intercepts on 00 as forces acting along the unthr edges of the rectangles, construct a second link polygon ; then to the sum of the intercepts made by the second link polygon in fig. 6, add, as a correction, one-third of its excess above the sum of the intercepts made by the second link polygon in fiu. 7. CHAP. XIV.] RESISTANCE TO BENDING AND SHEARING. 259 The proof of the correction on Z,, which is exact, is shown thus : — For any rectangle Ia = \KY' - f) = hK^- y)ir'+ Yy + 1) = S\d' + ^ {cP - Yy) } = ScP + ;1 (ScP - SYy) ; where S is the area of the rectangle under consideration. The smaller h becomes, the more nearly does the value of Yy approach d-, and the smaller is the correction. The greater the number of rectangles into which the cross-section is divided, the more nearly accurate will be the approximation given by fig. 6 alone. In the example, the correction on /o is about 4 per cent. ; in the rectangle EBFE' (fig, 1), the areas of the two halves each into the square of its leverage about 00 is less than I^ by one-twelfth, or about 8 per cent. Any cross-section can be blocked out into rectangles, and /o and G\ easily calculated for it by the tabular method, if we have a table of squares and cubes. If the cross-section has a very irregular outline, it may require to be blocked into a great many rectangles, and the construction (fig. 6) will probably give a sufficiently close approximation. Definition. — A cross-section for which the neutral axis divides the depth in the same ratio as the strengths of a given material to resist tension and thrust is called a cross-section of uniform strength for that material. The cross-section (fig. 5, for instance) would be of uniform strength for a beam of a material whose resistance to tension and thrust is as 3 to 2, A A being the compressed skin, and BB the stretched skin ; for a cantilever of the same material it would be turned upside down. It is readily seen that the area between the first link polygon and its end vectors produced to meet at L is propor- tional to the moment of inertia. s 2 260 APPLIED MECHANICS. [chap. XIV, Fig. G. CHAP. XIV.] RESISTANCE TO BENDING AND SHEARING. 2G1 Fi-. 7. 262 APPLIED MECHANICS. [chap. XIV, Triangular cross-seclion and sectiotis which can be divided into triangles. — They are of no practical importance, but lead »ip to others which are. Triangular Section. — On making /= f A, the wedges (figs. 2 and 3,;Ch. XIII) become isosceles, and we have by substituting in the expressions given thereat and C = vol. of wedge on triangular portion above NA = -^^hk-. In the following way. /p may be derived from the rectangle (fig. 8)., The moment of inertia of the shaded triangle about the central axis CC is half that of the rectangle ; lor the triangle, then /o = Jc - Sd'- = -i\-bh-- - — X f - j = h ■bh^. M = -^ /o = ^'ffbh\ Hence for a triangular section ».'/ = 3 , «' - "ST) " = "5 4 • Hexagonal Section. — As an example of a built figure, we will t.ike a hexagon about a diameter as axis (fig. 9}. " . / — --\ ' ^ / '• ' / \ / *'■»■* sH ^mm^-^ — r-— V- — '■ — ■ 1' \ » /. ■*!■» / \ ■ * Jr. _ / ^^^^ jt Fig. 8. Fig. 9. Taking a quadrant, we can divide it into three equal triaiigles|;*the breadth of each is \b, and the height \h ; hence the area of ench is "iVMj »"<! tl'e moment of inertia of each about its own neutral axis is aV x \b x \h'^ - -\\ rr-ibh'^ . If 7o be the moment of inertia of the hexagon about 00, we have for a quadrant j/ri = moment of each of the three triangles abotit its own neutral axis, together with the area of each into the square of the distance of its centre from 00, hh^ bh i /h\- /h\' /h\ »i bh\ CHAP. XIV.] EESISTANCE TO BENDING AND SIIEAUING. 263 Dividing the semi-hexagon above the axis into any set of convenient figures, and multiplying the area of each by the deviation of its centre of gravity, we have Also Go = iV*^"' m h ik ■' ® * " Hence for a hexagonal section about a diameter as axis, «,' = ^, n' = A = -05208, H = -A = -10417. Rhomhoidal Section. — As another example, we will take a section in the form of a rhombus, with diagonal as neutral axis (fig. 10). ^•6->k B — 7b m&ka Fig. 10. Fig. 11. bh For upper half section, the area is — , the moment of inertia about its own neutra hence 1 axis is -s^h f - j = ;^„, and the deviation of its centre of gravity is - ; ,^ bh^ /bh\/hy- therefore Jo = -Mh\ and M = £- /o = -rrfbhK Hence for a rhomhoidal section with diagonal as neutral axis, til = ^, n = 4 S". " = "£"-4". Square Section. — A square section lying with its diagonal horizontal is a par- ticular case of this. 264 APPLIED MECHANICS, [CHAP. XIV. In order to compare the strengths of a square section when lying with a side horizontal, and when lying with a diagonal horizontal; let n be the side of the square, then d = aV2 is the diagonal ; for b and /t. substitute a in the one case, and rtV2 in the other ; then M = i >^ M' = ^:V/(« y/2V = Y^/«' ; so that M : M' : : \/2 : 1, being stronger when the side is horizontal. Trapezoidal Section (fig. 11). — Observe that a triangular section is of uniform strength for a material whose strengths to resist tension and thrust are in the ratio 1 : 2, or 2 : 1. It is evident, then, that for a material, the ratio of whose strengths is between 1 and 2, a cross-section of uniform strength can be made by selecting the proper frustum of a triangle ; that is, a trapezoid with the parallel sides horizontal. Suppose the strengths are as 6' : s, not greater than 2:1; then to find the ratio of the parallel sides B and b (fig. 11) ; a well-known construction for finding G is to lay off .B in a continuation of b, and 4 in a continuation of B, one in each direction ; the line joining the extremities cuts the medial line at G ; hence h Tt : S:s; therefore b 2s- S B 2S-S For instance, if the strengths are as 3 : 2, the in the ratio i _ 2 X 2 - 3 , B ~ 2 x~3'-~2 ~ * when the neutral axis will divide the depth as 3 : 2. Though such sections are thus far economical, still they are not the most economical, as too much of tlie material lies near the centre of the section, where it cannot act effectively in resisting bending. Circular Section. — In the circle (fig. 12) inscribe a hexagon with a diameter horizontal ; for the hexagon b = d, h = dj~, and 7, = ^^ {d) ld^\ = -OSSSd*, an approximation on the small side to that for the circle about a diameter. Again, circumscribe a hexagon with a diameter horizontal ; for this hexagon h^d, b = d.~=jJ3, ami I, = ^\(^^^3ydy = -0602d\ an approximation on the large side. CHAP. XIV,] ItESISTANCE TO BENDING AND SHEARING. 265 Taking the average of these, we have for the ciicle where d is the diameter of the circle, ami the breadth and depth •of the circumscribing rectangle. It will be seen tliat n' = '047 is a close approximation, being correct to 2 decimal places. In the same way an approximation to (?'o may l)e found. The exact value of /„ is found thus : — From the centre of the circle (fig. 13) draw three axes OX, OY diameters .at right angles, and OZ normal to the plane of the paper. Let for Circle l-h^ie^ FiK. 12. Fis. 1.3. /, J, K be the moments of inertia of the circle about these axes respectively, and let s be any small elementary area. Now OA, OB, and OS are its leverages about the three axes respectively, and by definition / = (.s X 0A-) summed for each element, J ={sx OW) K=(sx OS^) but, by Euclid i 47, OS- -^ OA' + OB- for each element, hence K = I + J \ and further, since / and J are equal, each being the moment of inertia about a diameter, K = 2/, so that if we find K, the value of / is at once obtained. On the element 5 build a column of material of unit density, and whose height is OS; suppose this column to gravitate tangentially, that is, at right angles to OS, then its statical moment about OZ, that is its volume into OS, gives the moment of inertia of the element about OZ ; all these columns will form 266 APPLIED MECHANICS. [chap. XIV. a solid standing on the circle as base, whose height at the cir- cumference is r, and whose upper surface is a conical surface, apex at 0, and sloping at 45° to the plane of the circle. The volume of this solid is that of a cylinder of height r standing on the circle, minus a cone of equal height and base ; its volume is therefore two-thirds of the cylinder, viz. F = "I X irr- X r = ^rrr = •?■; every particle gravitating tangentially and in the same direction. If we cut this solid (fig. 14) into slices by planes at right angles to the plane of the circle, and passing through consecutive radii, the slice between two adjacent planes will be a pyramid with its apex at 0, and having its base on the cylindrical surface ; one dimension of the base, there- fore, is r, and the other an arc of the circle. Now by taking the slices thin enough, the base / Plane sloping mt 45° Height at K=LK r^Yal.of Cyl.-Vol of Cone Fis. 14. Fis. 15. of each pyramid becomes in the limit a rectangle, and the points in the circle below the centres of gravity of these pyramids will form a circle of radius |r ; the whole weight may be supposed to be applied by a cord on a pulley of radius ^r, and hence K = volume of solid x radius of pulley = f tt?-^ x ir= ^wrK /o = k" or 64 hjr- M yr<^ 32-^ ' or i/». Hence for a circular section. m' = i ,q'-««. « = 32 = -098. To find G\ : — Suppose a wedge standing on tlie quadrant (fig. 15) formed by a 45° sloping plane passing througli OX;. CHAP. XIV.J RESISTANCE TO BENDING AND SlIEAlilNG. 267 the geometrical moment of a semicircle about a diameter will be twice this volume. Cut the wedge into slices by planes at right angles to the plane of the circle, and passing through consecutive radii. Each slice is a pyramid with apex at O, and of height ?•; its base is part of a cylindrical surface, and the upper edge of the base is sloping. Let « be the short arc as shown in the figure, and 8 its projection on OX; then in the limit when the slices are thin, the base becomes a plane rect- angle whose dimensions are a and the height of the sloping plane at K ; but height of ^ = LK = r cos B ; hence volume of pyramid = ^ x r y. a . r cos 6* = ;^r^ . 8, since S = a cos 0. For each pyramid \r- is constant : so that the volume of wedge on the quadrant is ^r"^ multiplied by the sum of the quantities 2 ; this sum is r, whether the slices be thick or indefinitely thin ; hence the volume of wedge = -jr' ; and doubling, we have for semicircle CP, 2 ,^ 3' ' If we divide G\ by the area of the semicircle, / • ; }' we obtain the distance from the centre of ' * the circle to the centre of graA'ity of the semicircle ; k. — r — i < r-b i - =^. ' ' Sir Fi?. 16. Elliptic Section. — The ellipse is immediately derived from the circle thus : — Let the ellipse (fig. 16) have the same minor diameter as the circle above it, so that & = 2r ; all the vertical dimensions of the circle are, however, to be altered in the constant ratio a : r, since the vertical radius of the circle is altered to a. Let the circle be divided into rectangular ele- ments, each with an edge lying on the neutral axis ; when the circle is changed into the ellipse, each element remains of the same breadth h' as before, but its vertical dimension is changed from y to Y, where Y : y : : a : r. 268 APPLIED MECHANICS. [CHAP. XIV. For each element of circle and ellipse respectively / = i7/(?/3 _ o»), and / = ^b\ Y' - 0'). Por each element, and therefore for the whole figure, the moment of inertia has changed in the ratio For circle hence, for ellipse. y3 = ^3 . ^.3^ ^'^Ul-''-')-->'--r^^''' putting ^b for r, and ^h for a, "-i'-- li^'" Hence, for elliptic section, 64' '' " 32' For a serai-ellipse about a diameter as axis, we have 4ff 3^" G^' = f ?'«* ; and // = — . Hollow Circular Section. — For a hollow circle or ellipse the reduction is the same as for the hollow rectangle only — replaces \ ; hence hh' «-i-i^.V-^'- Eesistance of Cross-Sections to Shearing, and Distribution OF Shearing Stress on a Cross-Section. In fig. 17 let AB be the cross-section shown in figs. 6, 7, 8, 9, and 10, Ch. V, and let A'B' be another section lying at a small distance S.v to the left thereof ; in figs. 6 and 7, we CHAP. XIV.J RESISTANCE TO BENDING AND SHEARING. 260 assume AB to be in a position such that F is greater than Wi + IF>, that is, AB lies to the left of the section of maximum bending moment; hence the bending moment on A'B' will be less than that on AB. Consider the horizontal equilibrium of AHH'A' (fig. 17) part of the slice of the beam between these sections, and bounded below by the horizontal face HH' a portion of the plane CD (fig. 6, Ch. V). There is no stress on the free surface. On AH the horizontal stress is indicated by arrows Pay &c. ; and on A'H' by the shorter arrows p'a, &c. (see fig. 8, Ch. V), shorter because the bending moment on AB' is less ; in so far as these affect the horizontal equilibrium of AHH'A', they may be replaced by arrows {pa - p'a), &c., on the face AH alone. For horizontal equilibrium, there must be a tangential •V Pa ^ A' A Pa \ / / \ * < / \ ' ; / \h' X.— \ 8r, / t B' "" B ^ 'A ji]:i'a) A yuJi'^9^ B 0' A_.A B Fiff. 17. stress q acting towards the right on the remaining face H'H ; and since H'H is very small, the stress on it will be of constant intensity q ; therefore q X area of plane H'H = sum of the arrows (pa -p'a), &c., on plane AH] or q . z .})X = volume of frustum of wedge standing on HAA the shaded part of section as base, and of height (Pa - Pa)- Now if G stands for the geometrical moment of the shaded part of the cross-section relatively to the neutral axis, that is, for the- 270 APPLIED MECHANICS. [CHAP. XIV. volume of the frusLum standing on HAA of the isosceles wedge made Ijy a plane sloping at 45° and passing through the neutral axis, then the frustum of the wedge above is the same fraction of G, that iiJa-p'a) its height is of y,, the height of the isosceles wedge, and ^ -, !> , Va-Pa fi (1 . Z . CJO = Cr. Va Xow on the cross-section AB, the bending moment equals the moment of resistance, that is M^Ui-'^L; .: '-'-"- f. (1) For the sections AB and A'B', ?/„ and the moments of inertia are the same, because in the limit when ^x is indefinitely small the sections coincide ; so that if M' be the bending moment at A'B', then Ji = — In, . . - — — —^, anu — — = -— , Va Va A lla ^o ^0 where ^M stands for the small difference of the bending moments on the sections AB and A'B' at the small distance h' apart ; therefore /MA „ ■BM \&v .G q .z.hx = -y- G, or q = — = — - • In the limit when Sa; becomes indefinitely small, so does IM; but their ratio -^ becomes F the shearing force at the section AB, by theorem, page 118, and therefore F G since H' now coincides with H, we are warranted in assuming q constant over IfH. Now q is the intensity of the shearing or tangential stress at the point H in the horizontal plane CD (fig. G, Ch. V) ; but OHAP. XIV.] RESISTANCE TO BENDING AND SHEARING. 271 it is also the intensity of the shearing stress at the point H in the vertical plane AB (see fig 14, Ch. II) ; hence (tig. 9, Ch. Y) F a ^ = 7 • T (2) is tlie intensity of the shearing stress at H, any point of the cross-section AB. "Where F is the shearing force on the cross-section ; /o is the moment of inertia of the cross-section about the neutral axis ; G is the geometrical moment of the portion of the cross-section beyond the point, about the neutral axis ; and z is the breadth of the cross -section at the point. On the cross-section AB, it will be seen that q acts down- wards at any point 1I\ if we choose K oX an indefinitely small distance ahoxe H, the tangential stress on ICK will also be q since it is indefinitely near H'H ; and on ICK t\\Q lower surface of AKK'A', q will act to the right just as on H'H, but on K'K the uirpcr surface of the small prism KHH'K\ q acts to the left. The horizontal stresses q form a couple tending to turn the small prism in the left-handed direction ; hence for equilibrium, the vertical stress q on the faces KH and K'H' tends to turn it in the opposite direction ; so that q on the face KH acts downwards, an assumption made in fig. 9, Ch. V, which is now proved. On the other hand, if P were less than W^ + W^, &c., then the bending moment on A'B' would be the greater ; q on face H'H would act towards the left, the arrows q all round the prism would be reversed, and so q on the face HK would act upwards ; that is, q at any point H of the cross-section would act upwards. Observe that if q be evaluated for two different points of a cross-section of a beam loaded in any manner, F and /o will be the same in both, and the two values of q will therefore be to each other directly as the geometrical moments of the parts of the section beyond the points, and inversely as the breadths at the points respectively. But for any given form of cross-section, as rectangular, circular, elliptical, &c., the ratio of the geometrical moments of the portions beyond two points definitely situated in the section with respect to each other, is the same whether the section be large or small ; whether, for instance, it be a large circle or a small one ; so also is the ratio of the breadths ; and hence the distribution of the shearing stress on a cross- section, or the manner in which jq varies from point to point in the section, depends only upon the form of the cross-section. 272 APPLIED MECHANICS. [chap. XIV. It is instructive to know how the shearing stress is distributed on cross-sections of various forms employed in practice ; and it is of the greatest practical importance to know ichere the intensity is a maximum, the amount of that intensity, and the ratio of its maximum and average values. This ratio is an abstract quantity, and depends only upon the form of the cross-section. In the graphical solutions it is inconvenient to draw the tangential arrows q as in fig. 9, Ch. V, since they interfere with each other ; we will therefore draw them at right angles to AB, when their extremities will give a locus which specifies the distribution. For such a locus the origin will always be at the neutral axis, the abscissae y are measured on the vertical axis, and the ordinates q are measured on the horizontal axis; and \y at any point H whose abscissa // is given, the ordinate q gives the inten- sity of the shearing stress. Rectangular cross-section (fig. 18). — Since z = h = constant, q will vary as G the geometrical moment of the shaded portion of the section, that is as the isosceles ^frustum) wedge standing on that portion ; now this wedge will be greatest when y = 0, for then the shaded portion will be the whole isosceles wedge on the semi-section. If ?/ be Fig. 18. negative, we have more than the semi- section shaded, but the portion lying below the neutral axis gives a negative geometrical moment, and q is again less than ^o; hence g- is a maximum at the neutral axis, it has equal values for equal values of y above and below 0, and is zero at each skin. The maximum value is thus ^^"-M-") \a q^n.»J ^J y^=f^ V 5.=M ^o = /o (r' F u{(|,-o= w^' 3F 2bh' Graphical Solution. — Lay off OA = q„ ; and if we take A as origin, qo - q the ordinate of any point will be proportional to the volume of the isosceles wedge on the semi-section minus tliat on the shaded part, that is to the isosceles wedge on the ])art of the section from to // ; but the breadth being constant, the volume of that wedge is proportional to Oy- ; so that from CHAP. XIV.] RESISTANCE TO BENDING AND SHEARING. 27J A as origin, the ordinate of any point on the curve is propor- tional to .?/-, and the curve is a parabola. The modulus of the parabola is go ^ 4 go and the equation, with as origin, is 4 go fh- h- \ 4 - r Let g'aver. represent the average value of the intensity of the shearing force on the cross-section, then g aver. \ ~ shearinsT force F area bh ' and we have for the ratio of the maximum and average intensity (7 " 2' 1 aver. -■ Hollolo rectaw/idar cross-section, or symmetrical donhle-J sec- tion (fig. 19). — For values of y from ^H to hh, q varies as^ G (, — J5=6...J -,3 20 =H q„nnax. 3 (Bir-bh-XRI{-br > q'^ver. ^(BH--hK'j(B-b) \ ^o max . :A, S:s:KL::B:B-b Figr. 19. exactly as in the previous case, so that the locus from Y to L is a parabola whose apex is ^i ; the modulus of this parabola is greater than that for the solid rectangle since the constant 274 APPLIED MECHANICS. [CHAP. XIV. divisor /„ is now less. For values of // between ^h and 0, consider the eifect upon each of the items that make up q. If the hollow were filled up, SA2 would be part of the para- bola in fig. 18 ; IqIS now less, however, and allowing for that fact alone SA2 would coincide with LA^. The removal of the centre decreases z from B to Zi; this increases each ordinate in the same proportion, so that SA2 is still a parabola. Lastly, for values of y between ^h and 0, the removal of the centre alters the value of G from what it would be for the solid figure, by the geometrical moment of the part of the hollow beyond y ; now the geometrical moment of the part of the hollow is §& I (ihy - i/'\ ', this leaves the equation consisting of a term in y"^ and a constant term, so that SAo is finally a parabola with apex at A2, but with a modulus different from that of ZAi ; and hence q^ = OA2 is the maximum. For semi-section, G\, = l(Bff' - W) ; for total section, Area ,S' = (BH - hh), and z, = {B - h) is the breadth at the neutral axis ; ?'>= T F G\ SF Bff'--hJr 5 lo' z, 2 {BH'-hh'){B-b/ F F aver. ,S (BR - bh) ' . 2oma^ J,{BR'--hh'){BH-hh) • • q\,„, 2{BH' - hh') {B - h) For the dimensions given in fig. 19, %^ = 2-4, and KS : KL : : G : 2. 2 aver. The locus (fig. 10) gives tlie shearing stress on the horizontal layers as well as on the cross-section. Now, as you pass from the horizontal layer K cutting the web to another cutting the liange, there is a sudden change of intensity from KS to CHAP. XIV.] KESISTANCE TO BENDING AND SHEARING. 275 KL ; this change cannot be supposed to take place altogether at K, as the free overlianging surface of the section at that point does not bear any share of the shearing at all, and for a small distance just above K the portion overhanging does not bear its proper share ; in the vicinity of K, therefore, the stress is not constant on the horizontal on the cross-section. The consequence of this is to introduce shearing stress on the vertical plane through K. In order that the intensity of the stress should change from KL to KS absolutely at K, there would require to be an infinite amount of shearing force on the horizontal plane at K ; and since the intensity changes from KL to KS in passing through a small vertical distanc-e at K, there must be a great amount of shearing force on the horizontal plane at K. Hence sections of this shape very readily give way by shearing at K ; cast-iron sections being specially liable to do so under the shearing force developed by irregularities in cooling. Ee-entrant angles, as that at K, are to be rounded off in castings and rolled plates, and filled in with angle irons in built sections ; this allows the breadth to change gradually from B to (B - b), and the intensity of shearing stress to change gradually from KL to KS. It follows as a corollary from tig. 19 that, for symmetrical sections made up of rectangles with breadths diminishing towards the centre, ju is the maximum. Approximate mctJiod. — For a cross-section, such as is shown in fig. 19, the web bears the greater share of the shearing stress; and, moreover, the stress is nearly uniform in its distribution. A close approximation to the resistance to shearing for such a cross-section will therefore be obtained by multiplying the area of the web into qo =f, f being the shearing strength of the material. This is equivalent to considering that the web bears all the shearing stress uniformly distributed over it ; or that the central parabola, in such a diagram as fig. 19, is replaced by a rectangle of height h and breadth q„, and that the upper and lower portions of the diagram are left out of consideration. The area of the web required for a double- T section is readily found by the converse of the above, and is given by the equation F '-?■ where aS* is the area of web in square inches ; F is the amount of the greatest shearing force in lbs. at the section ; and /is the resistance of the material to shearing in lbs. per square inch. T 2 276 APPLIED MECHANICS. [chap. XIV. Circular Cross-section (figs. 20 and 21].— On the shaded sector (fig. 20) suppose an isosceles wedge standing, made by a plane sloping at 45° and intersecting the horizontal radius, and cut up into pyramids as in fig. 15. In this case the sum of the small quantities S is OC = r sin d, and the volume of the wedge is \r x r sin 6/ ; for the sector, the geometrical moment about OX, G = Jr^ sin B ; and deducting for triangle ODB, we have for the geometrical moment of ADB about OX as axis G = \r^ sin B - {I . r sin B . r cos 6) x |/' cos B = ^ sin^ft. Hence for the shaded part of the circle in fig. 21 G = f/-' sin'0 ; : = 2/- sin B ; and — = -^ sm* B, z 6 where B is half the angle subtended at centre. But y - r cos B ; cos = — ; sin^e = r' - W and 6^„ = W-y')\ A D ^^m \d ^ ram 9 ^ ^ \ K "e>^^ .se\ so that Ci OC i(r* - y"'), and the locus is a para- ^ bola with its axis on OQ. Fig. 20. Hence 5,, is the maximum ; and since <t\. = i^ r^ ; Zi = -7 r^ ; aS' = ttf- ; and c, = 2>' ; Vomax. ^0 Lz \F F <2 Elliittical section (fig. 21).— Let the ellipse shown in the figure be derived from the circle by altering in a constant ratio the breadth ~ of the circle at any point, to z' the breadth of the ellipse at the point corresponding, that is, let z = nz . Suppose the shaded part of the circle to be divided by vertical lines into elementary rectangles, and let the ellipse be correspondingly divided ; by considering each of these ele- mentary rectangles, it is easily seen that I, G, S, and z for the CHAP. XIV.] KESISTANCK TO BENDING AND SHEARING. 277 ellipse are derived from the corresponding quantities for the circle by multiplying by n ; for the circle we had F a ^= 77' Fig. 21. so that for the ellipse F iiG F G 4 F 111 nz 1 z 6 rnv 9_ aver. ~ / '■> ~~? ~ q ' 77?'/' q aver. ^ the same as for the circle. St/mmetricnl section of three rectangles. — When the middle rectangle is the narrowest, qo is the maximum, as we saw in the previous case. go„,ax ._ 3 {bB:UB-b)h'\{hII*(B-bjk] q'a.er. 2 { bJli (B-bJ h'l B \ KS -KL ::b : s 3hmax.y f > / Umax.^ 3 (SLh-){bS^(B -h)h\ quaver. 2 bSU(B-b)h' Fig;. 22. 278 APPLIED MECHANICS. [chap. XIV. In fig. 22 the middle rectangle is the broadest ; and in this case, the intensity of the shearing force has two maxima values, one at y = 0, the other at y = -. Their values are shown on the figure. Numerically they are qo = -QUF, and qn = -013^; ^^j = 1-3 ; 4 = 1-2.5. ? aTer. 'i aver. Triangular section (fig. 23). — Let ij, which includes its sign, be the ordinate of tlie base of any portion such as is shaded in the diagram ; then for that portion (?o = ^Y- y).- \!/+l[^-y)} = ^ (2A - 3y)(A + 3y). Now F Go ^ . F . g = -f — , and since — is constant. Go 9 « — , or q x (2A - 3y)(A + 3y). The sum of these factors is constant, and their product is therefore greatest when they are equal ; that is when h {2h - 3y) = (/* + 3y), or y = g ; hence $ is a maximum at middle of depth h, and the locus is a parabola with its axis horizontal. K- I, X Fig. 23. For the portion above this central point ^'=^•2- (6-^*2) ='24-' For the whole section /o = -s^JA* ; heme ■ 24 FiK. 24. Jo • 2 and : 3F bh^ b bh ' ^ ^ 2 CHAP, XIV.] KESISTANXE TO BENDING AND SH BAKING. 279 2^' q aver. iS \bh bk ^'aver. - The equation to the parabola is easily found. Rhomboidal section (fig. 24). — For a portion such as is shaded, in the ui)per half of section, that is for positive values of y, <^.=i(^.)^[..i(^,)j=ijr-.^^,)|. Now G q X — IX h'^ + S>/ (:-')• so that the locus is a parabola with its axis horizontal. The sum of the factors .'/ ( >/) is constant, and the product is greatest when these are equal, so that when y = -, ^ is a maximum. The locus for the upper half is a parabola whose 8 axis is |A above the neutral axis, and for the under half a similar parabola symmetrical below. For that part of the section above y = , o Zh 3b For the cross-section 1 i-section Ko = -aV*y +**2" (^.j =-96= ••• ^» = I8' ?A FG 9F •IF qh 4bh' ^^''^'- ^bk bh ' In . z i bh " ifbh bh. 5aver. Regular hexagonal section, diameter vertical (fig. 25). — If we suppose the sloping Base of each Par.Seg.s ih. Heights as 1'i:9. Fig. 25. sides above and below produced to meet the neutral axis, we have a form like 280 APPLIED MECHANICS. [chap. XIV. fig. 24; and for values of y from \h to \h, everything so far as regards q is the same as if those sides were produced, except the constant /„ which is less ; hence for those values of y, the locus is a parabola with its apex A\ on the horizontal \h above the centre as in the previous case. For values of y from to \h, suppose the vertical sides produced and tlie rectangle completed ; the four triangles which require to be added on to complete that rectangle increase the constant /q, hut so far as that affects q the locus is still a parabola with its apex on the neutral axis ; they also increase G^q for every value of y, augmenting both the constant term and the'term in y^ in the expression for q ; the locus, however, is still a parabola with its axis coinciding with the neuti'al iixis, but the modulus is altered. Now we know that this parabola intersects the other pair at K and i, points beyond their apexes, hence OAi. is greater than ordinate of -4i ; and ?o = 0A% is the maximum. We have readily G\ = -hhli^ ; /o = yfri/t^ ; =1 = * ; and FG\, 28i^ , _ ^ _ 4 -f ?omax: = -T;;^ = X^/' 9 aver: " p " 3 j^ 5 90 max . Examples. 1. Find the working moment of resistance to bending, and the working resistance to shearing of the section, of which the upper half is shown in fig. 26. In wrought-iion built sections, the piercing of holes for rivets reduces the effective area to resist tension but not to resist thrust ; the area as thus reduced ,«j [t— | is to the original area in the ratio of "^ about 4:5, which is the same as the ratio of the strengths ; hence it is usual to make such built sections symmetrical above and below, to consider the woridng resistance to tension and thrust as each eqtial to 4 tons per sq. inch, and then to neglect tlie fact that the holes diminish the effective sectional area; the working resistance to shearing may also be taken at 4 tons per square inch. Fig. 26. 3\ b Old. Dif. Dif X b Ord.- Dif. Dif.xt Otd. JJ,/. Dif.xi 3 12 3 I 10 S 35 2 45 35 24 I3'5 35 100 64 123 36 52 12 216 78 6 1 1000 1 1 512 43 4SS 469 43 '952 469 14 41 X 2=82=S. Gvo=3oo. 2435 X 2=487O=/0. •CHAr. XIV.] KESISTANCE TO BENDING AND SHEAIUNG. 281 M = r,- 1» = -^^r X 4S70 = 1948 inch-tons. F0-, qohzx 4x4870 x 1 ^»=:^= •• ^ = -lF7^ — 300— =*^^^'^"^- 2. If the rivets a, a, fig. 26, be pitched at 4 inches apart, find the diameter -necessary for each rivet. For part of section beyond SS, (^o = ^ x 12 (10= - 9-) = 114 ; ?o = FGq _ 67 X 114 ~hz ~ 4870 X 12 = -13 tons per square inch is the intensity of the sliearing stress on the horizontal plane SS at the cross- section, and it will be sensibly constant on SS for a few inches on either side of cross -section. There is one rivet for each 24 sq. inches of SS; lience a rivet has to bear -13 x 24 = 3*12 tons of shearing force. If the rivet be very tight, this shearing force would be uniformly distributed on its section, and the area required would be found by dinding by /= 4. If the rivet be not perfectly tight, there will be a bending moment on it, and we must consider the shearing stress distributed as in fig. 21 ; in which case </y = 4; .•. (?' = fx4 = 3 tons per square inch average intensity. area = -^ = 1'04, which gives a diameter d = \-2 inches. Taking 1*2 inches as the diameter of the rivets, the six holes reduce the area of the cross-section by 16-8 sq. inches, almost exactly a fifth as we supposed it would do. This is on the supposition that the section bears the full shearing force that it can resist ; at other cross-sections where the shearing force is less, the rivets might either be made smaller in diameter or be more widely pitched. 3. Find the resistance to shearing of a cross-formed section ; width of each pair of wings 5 inches, thickness of metal 1 "5 inches ; /= 4 tons per sq. inch (fig. 27) ; see also fig. 22. For semi-section G'o=lx 1-0 (2-5-- -75-) + ^ X 5(-7o2-0-) = 5-7 G' Q 5"7 and — = — ;r = 1'14 IS a maximum. ^1 i/o = i X 1-5 (2-5» - -753) + i X (-753 - 0^) ; therefore 7u = 16'6. For portion beyond K G^o = ^ X 1-5 (2-5- - -lb-) = 4-27 ; Gq 4-27 and — = = 2"84 is therefore the maximum. : 1-0 F G^, F ^- = ZT" ^=16:6^ ^-^^^ F = 23-4 tons. SI dS T vm ( :)^ /:iy -\S C7 :) ,IZ —.0-75 .-L-n Fie 4. If the section (fig. 27) be built of half-inch plates as shown, and fixed with ^rivets a, a, one-inch diameter, find what should be the pitch near the cross-section 282 APPLIED MECHANICS. [chap. XIV which is under ^ of the full working sheaving force F, if /= 5 tons per square inch. For the part beyond SS, Go = ^ x l-o (2-.5' - -lb-) + J x 5 {-lb-- -252) = o-o ; therefore ?.25 = IF Go _ ^ X 23-4 X 5-5 ; ~ 16-6 X 5 = '5 tons per square inch is the intensity of the shearing stress on the horizontal plane SS. If the working strength of the rivets be 5 tons per square inch, the average resistance to shearing will be | x S = 4 tons per square inch nearly : hence -b -^ 4, that is ^t*" of the area along ST, and normal to the paper, must be pierced with holes. But ST= 2 inches; so that for every four inches measured on iST normal to the paper, there should be one square inch pierced ; that is, rivets about one inch in diameter should be pitched four inches apart. \f.... 4 --*-2..J 9 I U. 9 Semi. Symmetrical Section, Fig. 28. FiK. 29. 5. Find the moment of resistance to bending of the symmetrical section, the upper half of which is shown in fig. 28 ; the material is wrought-iron, for which the weaker working strength is fa = 4 tons per square inch. Consider the two triangles as one of breadth 8 inches ; for a triangle about its own neulral axis, / = -Jg- dh^ = 48 ; hence for semi-section |7o = (48 4 24 X 82) for triangle + -^ x 2(10' x O^) for rectangle ; therefore /„ = •^501 ; and M = -,— Jq = -,*,r x 4501 = 1800 inch-tons. 6. For the cross-section shown in fig. 29, and which is the same as fig. 28, find the working value of M, if the working strengths of the material be f„ = 4, and fb = 5 tons per square inch. Choose ££ as an axis of reference ; then 5 = 24 + 20 = 44 ; Gs = {24 >. S) + ^ \ 2 (10- - 0=) = 292. /^= A/u of Ex. 5 = 2251. Gb 292 yb= -Zr = — = 6-64, and >ia = 10 - 6-64 = 3-36 ; o 44 /« =//?-«. 'Jb- = 315 ; ^" = -^ = 1-19, and -^ ' y„ 3-36 ijb taking the smaller value, we have M = "752 X /„ = 237 inch-tons. (3-G4 = -752: CHAP. XIV.] RESISTANCE TO BENDING AND SHEARING. 283 7. Find M for the section, the upper half of which is shown in fig. 30; the material is cast iron, for which /j = 2 tons per square inch. For the circle about its own neutral axis 1= ^bh^ = -049 X 8< = 201 ; and 6' = 50-26. 64 For the cross-section, Uo = (201 + 50-26 X 8=) + A X 3 (4^ - 0-*) ; .-. Jo = 6964 ; and M =- jy^ /q = -A- x 6964 = 1161 inch-tons. Fi?. 30. Fig. 31. 8. Find M for the section shown in fig. 31, the working strengths /« and/* being 4 and 5 tons per square inch respectively. It is convenient to choose CC the diameter of the ellipse as the axis of reference, and we have Ellipse. 5 = 7rx4x3 + Gc = Middle rect. Lower rect. 3x6 + 10 x2 = 75-7; -t Jx 3(9-- 32) + i X 10 (112-9=) = 308; ic = — X 8 X 63 + 1 x 3 (9» - 33 + J x 10 (IP - 93) = 2793 ; 308 y = = inches sensiblj- ; so that i/a = 7, and tjb-l inches ; S 75-7 J^ = Ic-Sr = 1582. The neutral axis being sensibly in the middle, take ft = 4 the smaller strength, and M = f/o = 904 inch-tons. 9. Find the resistance to bending for the section shown in fig. 32 ; the dimensions are in inches, and the strengths of the material are /« = 4 tons (thrust), and /ft = 5 tons (tension) per square inch. Choose CC the diameter of the ellipse as an axis of reference ; and for semi- ellipse 5 = 23 ; 6 G^c = - 30 ; Ic = 53. Alls. M = -519 Jo = 11G7 inch-tons. 284 APPLIED MECHAKICS. [chap. XIV, 10. Find the resistance to bending of the wrought-iron section, fig. 33 ; the dimensions are in inches, and the metal everywhere is 1 inch thick. 1 ; '_\ ji.. (gi. ) 00-12 y.^as.r. Fig. 33. Choose CCthe diameter of the hollow semicircle as an axis of reference. Ans. M = •512/n = 668 inch-tons. 1 S <= \'^ Hollow semicircle . . Rectangle above CC, First rectangle below CC, Second rectangle below CC, 14 4 10 10 - 41 144 — 8 21 50 333 105 1 1 103 j Whole section .. 38 106 1 1601 11. For a wheel, design an elliptical spoke of approximately uniform strength, of a material whose smaller strength is 2 tons per square inch ; length of spoke 3 feet, load at end due to a force applied to the circumference of wheel o-V*^ ^^ ^ ton, and at each cross-section the depth is to the breadth in the ratio 3:1. M„ Wl = -_,^,- ft. -tons = 1-8 inch-tons ; M = nfbli^ = —x2 X b {3b)- = l-lGb^ inch-tons. Equating these values of M and M, we have 1-7643 = 1-8 ; therefore b^ = 1. Therefore b = I, and A = 3 inches, are the diameters of the elliptic section at boss, while two-thirds of these are the diameters at tyre (see tig. 8, Ch. XIII). CHAP. XIV.] RESISTANCE TO BENDING AND SHEARING. 285 12. Find the thickness of a cast-iron pipe whose external diameter is 2 feet, that it may have a working moment of resistance to bending of 800 inch-tons; the smaller working strength of cast-iron is 2 tons per square inch. Put D = 24 inches, and let d be the inside diameter; then /, = ^(2)*-rf*), and M = ^/,: therefore 800 = -OOSIS [B^ - d*) or {D'- - d*) = 97800. Ans. d = 22 inches ; and for the thickness of the metal, we have t = \ inch. 13. The section of a beam is a rectangle 2 inches broad by 6 inches deep ; the material is wrought-iron whose working resistance to shearing is /= 5 tons per square inch. Find the working resistance to shearing of the cross-section. F will be such that qo the maximum intensity shall attain to /= o tons per square inch ; <?o = 5 ; but <?'„er. = f?o = V ^^^s. F = average intensity x area = q' . bh = ^ x 12 = 40 tons. 14. Find the resistance to shearing of the cross-section, fig. 5, taking/= 4^tons per square inch, ^q is the maximum value. From the tabular form or graphical solution we had h = 4634, (?'o=296, and ri = 1-5 ; F&'o Fx 296 or 4 — F = 94 tons, v-i' 4634 X 1-5' but ^0 15. Find the resistance to shearing of the section shown in fig. 4, taking y the resistance of the material to shearing at 4 tons per square inch. F(?'o , Fx396 _ _. ?o = y^; •^ =fiTT^-^5 ••• F = lSotons. loZ\ 6112 X 3 16. In section (fig. 4), find the ratio of the maximum and average intensity of the shearing stress. FG'o ^ , F 9o = -f — , and q = -x,; -to^i o ., . q^ma^. SG'o 120 X 396 „ ^ therefore = ^^ = FTT, o =2-6. 9 aver. -'o^i 6112 x 3 17. Find F for fig. 19, taking/= 4 tons per square inch ; and find the ratiojof the maximum and average intensity of the shearing force. i7o = i X 6(103-63) +i X 2 (63-0^); .-. /o = 3424 ; (?'o = ^ X 6 (102 - 62) + 1 X 2 (62 - 02) = 228. ^5 = 6(10-6) + 2(6-0); .-. .S=72. Ff" 21 = 2, and we wish to have qo = 4 ; but qo = — — ; .•. F = 120 tons. Jo-i F go max- 4 ? aver. = r-„ = f tons ; and = v = 2-4. 72 q aver. f 286 APPLIED MECHANICS. [chap, XIX. 18. Find the ratio of the maximum and the average intensity of the shearing stress for fig. 31. Take the under half which consists entirely of rectangles, and find G'o = 157'5. Show /o = 1582, jS = 75-7, and ci = 3. ^^^^ <?omax. _ <^.^ Q aver. 19. Compare the resistance to shearing for the cross-section shown in fig. 3 as obtained by the exact and approximate formulae respectively, taking /= 5 ton per square inch. loziqo 15588 X 4 X 5 F = —^ = g^2 = ^^*^ tons (exact). F =/<S' = 5 X 9fi = 480 tons (approx.). 20. Suppose the M-eb of the cross-section shown in fig. 4 to extend to the under side of the upper plate, take /= 4 tons per square inch, and find approxi- mately the resistance to shearing. Compare the approximate result with that obtained for Ex. 15. F = 4 X 48 = 192 tons. CHAPTER XV. STKESS AT AK INTERNAL POINT OF A KEAM. Keturning to fig. 6, Ch. V, we have now found the intensity and obliquity of the stresses at the point H, on AB and CI) the pair of rectangular planes through it, viz. :— On CD the liL = ( R' = q, in figs. 198 to 201 = r,, in fig. 47. N = tangential component stress on the rectangular planes AB, CD. OL = p, in fig. 84 = >„, in fig. 47. = normal component stress on plane AB. Fig. 1. total stress tangential and given by q in such diagrams as fig. 18, Ch. XIV, and on AB the total stress of intensity r at obliquity y (fig. 1) given in terms of its normal component;;, and its tangential component q on such diagrams as fig 18, CHAP. XV.] LINES OF STRESS ON A BEAM. 287 Ch. XIV. The planes of principal stress at H are to be found as on fig. 10, Ch. IX; making that construction, and noting that y is a right angle, we have (fig. 1) OM = \p, and 3IR = jt^ + g^-. These are readily calculated, and become known quantities ; we also have tan 26 = ^^ = -^^-^-Q^^ g (fig- 18, Ch. X IV) ML half-arrow p (fig. 3) giving d the angle which the plane of greater principal stress makes with the plane of cross-section AB^ both planes being normal to the paper. Further OM + MR = intensity of greater principal stress (of same kind as arrowy;, fig. 3) ; OM - MR = intensity of smaller principal stress (of oppo- site kind from arrow p, fig. 3) ; and MR = intensity of greatest tangential stress, being on the planes inclined at 45° to the planes of principal stress ; that is, we have the maximum value of the thrust, tension, and shearing stress at the point H. 288 APPLIED MECHANICS. [CHAP. XV. That the two principal stresses are of opposite kind, follows from the fact that the stress on the plane CD is wholly tan- gential (fig. 14, Ch. II). If the principal planes be drawn through H for a short distance on each side, that upon which the stress is thrust by a full line, and that upon which it is tension by a dotted line, and if this be done for a number of points H on the elevation, then the full lines and the dotted lines will each form a series of polygons with short sides ; and if we take the points U close enough, each polygon becomes a curve. These curves are called lines of principal stress ; and the tangents thereto at any point H, where a dotted line and a full line curve intersect, are the planes of principal stress at H. See Kankine's " Applied Mechanics," sect. 310, and his " Treatise on Shipbuilding." These curves have the following properties : — They cut the neutral axis at ■45'^ ; for, considering a point there, |? = 0, and in figure 1 R will fall on the line R'O produced upsvards, 26* = 90°, and 6 = 0' = 45°. Similarly all lines that meet the end section, that is the section over the point of support, meet it at an inclination of 45° ; for, since the bending moment is zero, we again have p = 0. All lines meet the upper and under skin at right angles, since q = a,{, A and B (Hg. 4). In the elevation of tlie same beam, these curves will be different for different loads, except when the loads are kept in the same positions on the beam and altered in a fixed ratio ; thus, for a rectangular beam — if the load be uniform, then for positions of H ranged on a horizontal line, the arrows p and q, figs. 3 and 4, will both vary, since the bending moment and shearing force alter for each cross-section ; if the load be at the middle of span, p will vary, but q will not, since the shearing force is the same at each cross-section. The planes of principal stress will differ in the elevations of two beams which are loaded alike, and whose cross-sections are different but uniform throughout in each case ; for instance, if the cross-sections be a rectangle and a triangle respectively, we have (/max. f-t the centre of the cross-section in both cases ; the neutral axis, however, is at the centre in one case but not in the other ; so that, though both be loaded alike, the planes of prin- cipal stress will differ in the two elevations. In designing beams, it will be seen that we have followed the usual practice of considering p„ and pb at the section of maximum bending moment to be the greatest value of thrust and tensile stress respectively, and //„ for the section over the greater supporting force to be the greatest intensity of shearing CHAP. XV.] LINES OF STRESS ON A BEAM. 289 stress. Strictly the points at which these maxima occur are to be defined thus : — Let .r, ;// (fig. 6, Ch. V) be the coordinates of any point H referred to the centre of the neutral axis as origin ; then for the cross-section at x M^ = npaW, and -pa = -rf. ; also p,:v^::y:r,a- ,. Vy = ^^^^,- We also have (f = -^, and q^ = — ^^ ; k being the ratio of the maximum and the average intensity of the shearing stress for such cross-section ; q^ is readily derived by considering the manner in which the stress is distributed. Then (fig. 1) ME^ = q\ + ap,y, and by finding the values of x and y which make this a maxi- mum, we get the point at which the intensity of the shearing stress is greatest ; also mi ± MR) = ip, ± Jq\ + [^^py)\ and by finding the values of x and y which make this sum and difference a maximum respectively, we get the maximum value of the intensity of the thrust and of the tension. To find these maxima is difficult even in the easiest cases ; for instance, for a beam of uniform rectangular cross-section, loaded at the centre, we have _ ^ W{c - x) _ 3 W{c - x) _ y_ _ 6 W{c - x)y ^'^ ~ nbh' ~ hh^ ' ^^ " p;^" ~ Ih^ " ' and ME^ = (i^)^/^^ - ^I/J + 16r(c - xf] ; u 290 APPLIED MECHANICS. [CHAP. XV, for every value of y, MBr is greatest when x = 0, and then the greatest vahie of this is obtained by putting h ^ oWc 2/ = ± - , then MR = ^^ at the surface of the beam at centre of span. The quantity Py is greatest when h ^ 6W{c-x)y SWc X = 0, and y = ^l then p, = -j-^ = -^, and since p^y ^^'^^ ^^J^ thus have their greatest values at the same point, viz., at the surface at the middle of the beam, the greatest principal stress is there situated, and its amount is OM+MR = ~j'. oh' Determination of internal stress by the ])olariscope. — On fig. 5 is shown a plate of glass strained in a frame like a beam supported at the ends and loaded in the middle. The plate is then ^^WF placed in a lantern, when a cone , ^ ^,j^ ~:~ , "^ of polarized light throws an ' '"~ image on a cardboard screen. The whole image will appear luminous, except along certain lines, which are the loci of points at which the axis of greatest ^ i^'- •'• or least elasticity is parallel to the vibrations of the rays leaving the polarizer, which lines will appear black. If the plate be E ""i'.iifiiiimM'™'"^"'''^''" — ^'"W- I ^•^ C % * Extract from " How to use the Poliiiiscope in tlie Practical Determination of Internal Stress aiul Strain." By tlie late Prtifes.-or Peter Alexander. .\ paper read liefore the Matheniaiic.il and Pliy&ital Section ot tlie Philo^opllical Society of Glasgow, 3rd February, 18S7, und rejMinti'd in full in our liist edition by the kind ptrmissioa of the Society. CHAP. XV.] LINES OF STRESS ON A BEA.M. 291 now rotated about a line through its centre and perpendicular to its faces, the black lines will assume new positions in tlie plate. Fig. 6 shows the forms and positions of the black line for nine positions of the bar, from the horizontal position to the vertical position and corresponding to the first eight points of the compass. Fig. 6. It will be observed that the central point of the bar is fdivays black ; consequently we conclude that there is 7io stress there. At every other point of the black loci lines of stress cross them horizontally and vertically, that is to say, parallel and perpendicular to the direction of vibration of the polarized ray. u2 292 APPLIED MECHANICS. [chap. XV. A practical way of drawing the lines of stress is to rotate the card in front of the lantern simultaneously with the rotation of the glass bar, so as always to have the image of the glass bar occupying the same position on the card, and let the tracing point, while carried round by the card, at the same time move horizon- tally, so as always to be on the black loais projected on the card. The line drawn in this way v/ill be an accurate delineation of a line of stress. This motion of the tracing point may be accomplished as follows : — Let the tracer be a small toothed wheel, T (fig. 7), capable of rotation about a fixed axis, AA', at the end of a tracing pen, F, and let a heavy load W, be attached rigidly to the pen at the end of a rod, JD, so as to compel the axis, A A', of the tracing wheel to be vertical when in use. The tracing point will thus be incapable of any motion but that communicated to it by the rotation of the card, and the horizontal motion given to it by the rotation of the tracing wheel as the tracing point is made to keep on the black locifs. The network of dotted lines (fig. 8) shows what is the state of internal stress of the glass bar, and consequently that of the loaded beam. The black loci, i, ii, Sec, correspond to the angular positions, 1, II, &c., of the card. The dotted lines are the lines of internal stress. Example. 318. A beam of constant rectangular section 9 ins. broad, 20 ins. deep, and 20 ft. span, bears a load of 96 tons uniformly distributed. At a point in the section half- way between the centre and left end, and half-way between the neutral axis and CHAP. XVI.] CURVATURE, SI.Ol'E, AND UEFLECTION. 293 upper skin, tind the greatest intensities of thrust, of tension, and of sheuriug stress; and find the inclinations of the planes of principal stress at the point. .Vf. = 2160 inch-tons ; irPabh- = M^; ■■ . pa = — — = 3'6, and py = ^pa =1*8 tons per square inch. 2? 24 2 /^ = 24tons; ,' = _ = ^-^ = _ ; ,.= ./= '^ ; and qy = |?o = '15 ton per square inch ; ME- = -lb- + '9^ Therefore MR = "91 ton per square inch max. intensity of shearing stress. OM + MR = "9 + -gi = rSl ton per square inch max. (thrust) ; OM — MR = - -01 ton per square inch max. (tension). Tan 2d = ~= — = 0-16 ; .-. 26 = 9° 28' ; ML -90 6=4° 44', the inclination of the plane of greater principal stress to the cross- section. CHAPTER XVI. CURVATURE, SLOPE, AND DEFLECTION. At any point in a plane curve, the direction of the curve is that of the tangent at the point ; and the curvature is the rate of change of direction at the point. If two points be taken on a circle, the change of direction as you pass along the arc from one to the other is the angle which the tangent at one of the points makes with the tangent at the other ; this angle is equal to the angle at the centre subtended by the arc between the points ; since this change of direction takes place uniformly, the rate of change is found by dividing the total change by the arc ; the total change as just stated is the angle at the centre, and this angle when expressed in circular measure is the ratio of the arc to the radius ; dividing this ratio by the arc, we then have for every point of a circle, 1 curvature = - • (1) r If we take a point in any plane curve, we can find a circle ■which coincides with the curve for a short arc in the vicinity 294 APPLIED MECHANICS. [CHAP. XVI. of the point ; and if p be the radius of that circle, then the curvature of the circle everywhere being 1 ^ |0, it is clear that for that particular point of the plane curve, 1 curvature = -• (-:) P For any point in the neutral axis of a beam (or cantilever), we have (fig. 5, Ch. V), 1 \ds) strain on any horizontal layer curvature = - = = .. ^ „ :— 7 — , p y dist. irom neut, axis to layer \dsj \dsj strain on either skin y„ yj dist. of skin from neut. axis Now the strain on a horizontal layer is equal to the longi- tudinal stress on the layer divided by U the modulus of elasticity of the material (see page 7) ; hence 1 stress on any horizontal layer 1 _p 1 Pa 1 , p dist. of layer from neut. axis £ y' E ya' E' Pa being the normal stress at the skin on the cross-section, and 7/„ the distance of the skin from the neutral axis as shown in fig. 3, Ch. XV, we have Pa ^^ hence at any point of the neutral axis, the curvature due to any load which induces the bending moment M on the cross-section passing through the point, is / being the moment of inertia of the cross-section, and U the modulus of elasticity of the material. Choosing as origin that point where the neutral axis crosses the section of greatest bending moment, we have the curvature at that point for a beam or a cantilever, /o<i -ti io and if we wish this to correspond to the proof or working load. 1 _ 1 ,/■ 1 / . p„ E ' v/o E ' m'h ' 1 E tK°=i_ /. or ^ ~'~- = 1. CHAP. XVI.] CURVATURE, SLOPE, AND DEFLECTION. 295 it is only necessary to make M^ the bending moment due to the one or the other. These vahies of J/„ can be easily obtained from equation 3 by substituting for^;„ the value /corresponding to the proof or working strength of the material. For this cross-section of greatest bending moment, putting t/o instead of Ua, so that Multiplying the value for by this quantity, which being unity will not alter the value, we have (5) The slope of a beam or cantilever at any point whose abscissa is Xi, the origin being at the point where the neutral axis is horizontal, is found as follows : — Let S be any point between the origin and that point ; then at fig. 5, Ch. V, we had for the increment of slope between T and S, supposing these points to be indefinitely close, di = - X clx ; (6) P if we add these increments di from point to point between the origin where the slope is zero and the point Xi, it gives us the slope at that point ; and 1 \E ll\i/, mJ f ^n. p ' Ey,-MJ -I;© dx ; (7) the right-hand side expresses, in the language of the Integral Calculus, the summation of the products equal respectively to these increments. In the case of a beam symmetrically loaded, the origin will be at the centre of the span, and in practice, for any load, the origin will be sensibly in the same position ; in the case of a cantilever, the origin will be at the point of support. 296 APPLIED MECHANICS. [CHAP, XYI. At the point of support of a beam, and at the free end of a cantilever, we thus have the slojx, ^(-)d.v. (7a) V/ that is the integral with respect to ./; of the general expression for the curvature between the limits c and 0. Dejiection of the neutral axis of a heam or cantilever at any point wliose abscissa is Xz. Let Xi be the abscissa of tS (figs. 4 and 5, Ch. V) ; in the case of a beam let 7i be the height of *S^ above the lowest point of the neutral axis, in the case of a cantilever let 2C be the depth below the highest point, and let du be the small difference of level between >S and T, two points indefinitely close ; then -J— = tan ijr = ii (the angle being small), or the difference of height of S and T is du = 2xj . dx^ ; (8) and summing all these increments from point to point, between the point a; = (the lowest point) and the point .Tj, we have the height of Oh above 0, '"{i.)dx,. (9) The height of the end above the centre of a beam, or above the free end of a cantilever, is given by the equation [' {i.) dx, ■ (9«) J this in the case of a beam is the deflection of the centre below the points of support, and in the case of a cantilever it is the deflection of the free end below the fixed end ; so that the dejiection v.) = (t.,) dr.. (10) Again the deflection at any point x. is V^ = Vo - Vr,^ = t^„ - I ' (ir) dx,. (10(/) CHAP, XVI.J CURVATURE, SLOPE, AND DEFLECTION. 297 For ije we may substitute the value in equation 7, and — ) dx . iU\ ; v^ = ?',. - /I ^7 dx . dr,. (11) (11a) Beatn of v.niform section, load at centre. — For all cases of ^uniform section / is constant ; and for load at centre, we have, fig.'9, Ch.YII, M W ^ ( x\_ 1 / / X M. ^'^0 (12) \P) %0 1 - -x\. do: c '2 Ic- Eyo\ c 2 Ut^'^^ In every case the expression for ic the slope can be arranged into these three factors ; a numerical factor depending on the form of the cross-section, in this case \, and for which Eankine / " c puts 7ft" ; a factor — depending on the material, and a factor — depending on the dimensions. The general expression is %c = m"i-^. — : (13a) and 7/t" is called the numerical coefficient for th-e slope. The deflection lA / x/>/ %0 J "' (^1 _ l.r ) dx . dxr \ ^ - \ \x - 1 %~\' dx. = I- [ (x, -^%- O) dx, oJoL c 2J„ %oJoV c 2 ) = Z(^?_l^'_o 'Xx' 1 ^1 T~2't!"3 Jo %oV2 2c 3 (14) In every case the expression for the deflection can be arranged into these three factors : a numerical factor depending on the 298 • PPLIED MECHANICS. [chap. XVI. form of the section, in this case ^, and for which Eankine puts f . c' n" ; a factor V, depending on the material ; and a factor — depending on the dimensions. The general expression is (14a) and w" is called the numerical coefficient for the deflection. In this case, the numerical coefficients are m" = h ; n" = ^s- Observe that / is the working or proof strength of the material, according as you may want the working or proof values of ic and v,- If / be given, as is generally the case, in lbs. per square inch, then £ is to be in lbs. per square inch, and y^ and c in inches ; if we then calculate p or p,,, v or v^^, they will be in inches also. From equation (12), if we put x = 0, we get P."f-"- (15) This will be the expression for p^ in every case, and we shall not repeat it. The inclinations i, when calculated from these formuht, are in circular measure. Beam of uniform section, uniform load. — From fig. 12, Ch. VII^ (16) p £//, ' MJ Uf/o \ <f Ic = Jo \P dx = ■^-— 1 - -, x^ \ dx __f_( _ 1 i' _ a\ ? / ^ ■•"\[r.'e)"-"-=i..[j;^-?")^^-"-^- (17) 'c r 1 x'' f %0 -x' 2 1 x'' ~37'4_ Ui/q\2 3c' 4 J VlE'ij^' (18) CHAP. XVI.] CURVATURE, SLOPE, AND DEFLECTION. 299 The numerical coefficients are 111 = 2. . 3 > TJ* Cantilever of uniform section, load at end. — From tig. 10, Ch. VII, |.,r(.-.).ire=(i-'j) exactly as in the case of a beam loaded at the centre ; and the numerical coefficients are m'' = \ ; n" = i. Cantilever of uniform section, imiform load. — / cancels /^ ; from fig. 16, Ch. VII. M. •J.C (19) i. = -)c^-»=77- [l--Jl> +-X'' \dx pj Mjo J V C C" 2 x^ 1 x^'Y _ / / _ 2 c* 1 6-3 (20) Vn = I I ^ I - ) f^^ • f^-^i = -¥r The numerical coefficients are ^ ( 1 - - X -i- - x'^ \dx , dxi c c- 2 X- I x^ \xi , X ^ % ^ • dxi c 'Z c- 3Jo _ _ 1 ^^ l^x,' 2 c y "*■ 3? T (21) - 1 . ~ 3 5 500 APPLIED MECHANICS. [chap. XVI. Beam of uniform section, bending moment constant. — If a beam be symmetrically placed on, and extend beyond its two points of support ; and if the two projecting parts be loaded symme- trically, while the intermediate portion is unloaded ; the bending moment on the portion of the beam between the points of support is constant, and we have /and /„, J/and .1/,,, cancelling each other. p ^>o ' a constant quantity, so that the neutral axis is circular. (22) Ic = ■y,. = -]d.r \PJ -\ (23) ^1 /1\ f [" [^ -\dx , dxi = -ir— d^: . dxi ^•^^^•"=^1/-^^-'^^' %o 12 Jo (24) The numerical coefficients are m" = 1 ; n" = \. Beam of uniform section, loaded with tujo equal weights at equal distances from the centre of span. — Let Whe the total load, Xr and - Xr the abscissa of the loads. P= ^W; i^/o = i IV (c - Xr), and the bending moment is constant along the central portion of span; for values of x between x,- and c, M^ = ^W{c - x). H T For the central portion of span, -^ = ^\ for the end portion (25) J/^ C - X il/o ~ c - a-,. ' 1 / M P Eyo Mo / For values of x from to Xr, 1 ./■ 1 ■(c -- f) Fl/n C - Xr For values of x from x,. to c. (25a) CHAP. XVI.] CURVATURE, SLOPE, AND DEFLECTION. 1 301 / dx + c - X,. c - X,.\xr -2-|C (c - .r) dx tJy = - .dx . d,i\ = \ d.v .dx^ + \ - dx. dx^ ; J J p J J |0 J -iV J /O Ey,, [Xr XI [" i 1 IT, ) (26) [.*■, - 0] f/.r, + {ex, - I Ci» - ;r,.2) f?.>;i c - X,- J .r^ .[ir.'fH- ^ c - ^,. ^ - l.ri^ - Wx, 9''i 6'M = ^r- + c- .r,. 2 6 2 '^' ' 2 ^'' "^ ' ' "*" ^'^''^ I ' 2(;^ - 3c.r,^ + .g.^ ) / _ jl £: _ i ^ 1 / ^ ''" ^ 6(c-^,.) ]Eyr l*"'3c * cM^"2/o ■i'-*(-'Jli,r (27) Thus, for an unifovm beam, loaded at two points which are equi- distant from the centre, and so that the stress on the outer fibres of the middle cross-section is/, we have m" = i(i + :^); i^'=\-l(l-'' Beam or cantilever of uniform strength and uniform depth. — For the proof or working load, the bending moment at each section equals the proof or working moment of resistance to bending there ; since, however, the depth is constant, the moment- 302 APPLIED MECHANICS. [CHAP. XVI. of resistance to bending at each cross-section is proportional to the breadth, that is the breadth at such sections is proportional to M; and since the depth is constant, / for each section is M . proportional to the breadth ; hence -j is constant at every section. fj.l, and \-I^, (2S) as in the preceding case of a beam of uniform section and bending moment constant ; so that for beams or cantilevers to resist (my load, and made of uniform strength by varying the breadth only, the numerical coefficients for the slope and deflection are 7/i" = 1 ; n" = i. Proportion of the greatest depth of a beam to the span. — Putting the working strength of the material as the value of/, we have, as a general formula, and since 3/0 = m'h^, we have 2c 4r/<,' a h^ Now ^ is the ratio of the deflection to the span, and its recipro- 2c cal — represents the stiffness of the beam ; -^ is the ratio of depth of beam at centre to span, and we have 2c 4m' E h, ^ v,~ n" f2c' or, the stiffness of a beam is proportional to the ratio of the depth at centre to span. For instance, to give a working stiffness 1000 to a wrought- iron beam of uniform symmetrical section uniformly loaded; we have the ratio of depth at centre to span (Eankine, C.F. § 1 70) 2c ^'ni"E'v^ (-iV) 10000 lbs. p er square inch _^^ 1_ " Ti) 30000000 l¥sT per square inch ~ 144 CHAP. XVI.] CURVATURE, SLOPP], AND DEFLECTION. 303 That is to secure the degree of stiffness 1000 required for the wrought-iron beam, the depth must not be less than a fourteenth of the span. In the same way it may be shown that, in general, to give to beams the degree of stiffness that practice shows to be necessary, and that is usually proscribed, the depth at centre must bear to the span a ratio varying from -^th to -,^4-th, accord- ing to the material, form, and manner of loading. Slope and deflection under any load less tlian the proof load. — For the proof load, the stress on the skin furthest from the neutral axis at the cross-section of maximum bending moment is/' and for a smaller load it is pa (fig. 8, Ch. Y) ; now, though the load is less than the proof load, yet being distributed in the same way, the ratio M : liL^ is in no way changed ; / is the same as before, and we have E and y^ constant ; so that But since we have ,,Pa c , „ p^ e E y. E y. mWl . „ no^fi Q mm" Wlc , inn" Wlc^ I . = m'n UbohJ'' " m' n EbJiJ ^O't'o For a beam / = 2c, and for a cantilever I = c\ so that for any load for a beam .. _ 2mm" Wc- , _ 2mn" W(^ ^''~ m'n Ehji/' ^'~^i^EbJJ" 'O'^O and for a cantilever mm" Wc' , mn" Wc^ v„ = m'n EhXy ' m'n EbJiJ> O'-'O The coefficients mm . mn — T- and —r- m n m n assume values which depend on the cross-section and the system of loading. For similar beams similarly loaded we thus have v\, the deflection under any load less than the proof load, propor- tional to W the total load, and to c' or the cube of the leno-th ; 304 APPLIED MECHANICS. [chap. XVI. and inversely as h the breadth at centre of span, and h„'' the cube of depth at centre of span. That v\ is proportional to W does not follow from Hooke's Law, but has been established upon the supposition that the slope is so small that the tangent and circular measure of the slope are sensibly equal. In bending small pieces of wood in a machine which registers the load and the deHection as the coordinates of a line, it is found that the line is straight even when the piece of wood is bent to a considerable amount ; this proves that the formula above is a close approximation, even when the slope is considerable ; it must not be forgotten, how- ever, that the formula is only approximate when the slope is great. It has been stated that where such a machine's register ceases to be straight, the elastic limit has been passed ; no such conclusion can be drawn ; the only just conclusion to draw is that since the slope is visibly great (say 8° or 10°), the formula above has ceased to be a close approximation. Were a second approximation made, it would be found that v\, was not exactly proportional to W; and so long as the register did not depart from the curve which is the locus of that new equation, it would be inaccurate to infer that the elastic limit had been passed. Defiedion of heam suf ported on three props. — Let HK be a beam of uniform cross-section, bearing an uniform load of amount U; and let it be supported on three props, one at each end and one at the centre. Let W be the reaction of the central prop, and P = Q, the reaction of each end prop ; then P+Q+ W=U. Fig. 1. Fig. 2. U. In fig. 1, let W = U, then F = Q = 0, and OFT is a canti- lever of uniform cross-section fixed at and uniformly loaded, for which ?i" = i ; in fig. 2, let W = 0, then P = Q^ ^U, and CHAP. XVI.] CUrxVATURE, SLOPE, AND DEFLECTION. 305 HK is a beam of uniform cross-section uniformly loaded, for which n" = fV ; hence BH{'[ig. l):DO{^g. 2) ::3 : 5. Putting BH = Zz, then DO = 5z. In fig. 1, if we suppose the end props to be pushed up till they support all the load, then 7/0^ will assume the form LOB, where HL = 8z ; and in fig. 2, if we suppose the central prop to be pushed up till it just supports all the load, then HO K will assume the form HTK, where OT = 8z. Hence we have the following theorems. Theorem. — If to an uniform cantilever {OH, fig. 1) loaded uniformly, we apply at the end a load [P = ^U) equal to that uniform load, it will produce an additional deflection in the direction of the applied load {HL = ^BH) equal to eight-thirds of that due to the uniform load alone. In the figure, P = lU \s applied upwards at H, but the theorem holds for P applied downwards, since deflection is sensibly proportional to load ; it being understood that the total deflection in no case exceeds the proof deflection. Cor. — A load P', applied to the end of an uniform cantilever loaded uniformly, will produce an additional deflection in the direction of P' ; and the amount of this additional deflection is proportional to the load P'. Theorem. — If to an uniform beam {HOK, fig. 2) loaded uniformly we apply at the centre a load ( IF = U) equal to the uniform load, it will produce an additional deflection in the direction of the applied load {OT = ^DO) equal to eight-fifths of that due to the uniform load alone. In the figure, W = U\& applied upwards at 0, but the theorem holds for W applied downwards, as for the previous theorem. Cor. — A load IF', applied at the centre of an uniform beam loaded uniformly, will produce an additional deflection in the direction of W ; and the amount of this additional deflection is proportional to the load W. iSee Thomson and Tait's " Natural Philosophy," first edition, §§ 618, 619. Uniform beam uniformly loaded and supported on three props at the same level, one at each end and one at the centre Fig. 3. (fig. 3). — This figure is obtained by pushing up the central prop X 306 APPLIED MECHA^■ICS. [chap. XVI. in fig. 2 till coincides with D. Then since the reaction of the central prop in fifj. 3 has produced the deflection 01), which is five-eighths of OT, it follows that W = ^U, and therefure From fig. 2, Ch. YIII, _ W ^ OS, = jj OH = ^OH; hence OH, = \0l}. At centre 0, Mq = - -^^Ul, max. ; F = - -lVf^ max. At Si and /S'2 the maximum deflection occurs, and, as proved in the next article, n" =-■ \ X ^. = -043. 1024 The central prop increases the strength four times and the stiffness nearly ten times. Uniform beam tmifoi'vily loaded fixed at one end mid sup- ported at the other (fig. 4).— The left half of fig. 3 represents such a beam. Let W = ^U the uniform load, and / = Off the span ; then P = }W ; the shearing force at is the remainder of load, viz. fTF; the shearing force changes sign at S where the bending moment has a maximum value ; and the locus of tlie shearing force diagram is a straight line, since the load is ' uniform. From fig. 2, Ch. VIII, the bending moment diagram is AFBE a parabohx with axis vertical and apex on vertical through S. 'dW S[_ S_W 3^ _ _9_ ^~ ■ ¥ ~ ~8~ • 16 " 128 a positive maximum ; and since AFBE is a parabohi FD : Fa : : DFT- : O'R- : : 25 : 9 ; therefore FD = ^FO', and CE = O'D = FI) - Fa = - ^O'F --^ 111, that is Mo = - ^^'^/. tlie greatest value; this quantity may also be calculated directly by taking moments about 0. 0'F=M,= Fis. 4. Wl. CHAP. XVI.] CUKVATUUE, SLOPE, AND DEFLECTION. 307 This solution is exact, and all the results are readily got by remembering that OH' is a quarter of the span. The approxi- mate solution indicated in Rankine's "Applied Mechanics," sec. 308, assumes H' to be sensibly on the same level as H. To find the deflection at S ; considering the beam HW alone, we have for the deflection at aS^, since SH = f/ = |c; = AZ* ^'= 5_A ^i= ^-A t. ^' 1-2 E ' ?A, 1'^ %> ■ 16 64 E ' y, ' where /, is the intensity of stress at the skin at S. Next consider the cantilever OR', taking into account its uniform load alone. If the cantilever were of length SH, its deflection would be -Iv, ; and since the deflection of a cantilever is proportional to the cube of its length and to the load we have for the deflection of OH for uniform load alone. ^- ~ ^T? • 3-' 1 - U • 8^ • \SH :V, = _ I'l 6 T3 T^l > further, if a load equal to that on OH' be put at the end, it will produce an additional deflection frj ; the load at end of OH, viz., that on SH, will produce a proportionate deflection, and we have for the deflection due to load at end of OH', and the total deflection of OH' = 5vi = Ifvi. The total deflection of the point S is therefore 1 16 _ 35 _ 35 15/, c- _ 175 /,c- ^-^'i-^ 2 27"^ ~ 27''^ ~ 27'64:E'i,'"576Ei: Let/o be the proof strength of the material; when the beam is loaded with the proof load, then /^ = intensity of stress on skin at the fixed end, and /.:/,::i)/,:ilf, ::9:16; hence/, = t%/,; substituting this, we have = ns 9^^ c' _ 175 /, r ^ 576*16 ^'y," 1024*^ >/ 1 175 so that m = -, and n' = = -171. o 1(J24 x2 308 APPLIED MECHANICS. [chap. XVI. Uniform beam uniformly loaded and fixed at both ends (fig. 6). — Suppose the central of the three props that support UK, fig. 5, to push up till is above the level of H and K such a distance that H' and K\ the points of contrary tiexure, shall be on the same level as H and K. Let x = QIC the dis- tance of the point of contrary flexure from 0, and let u be the intensity of the uniform load. Consider OK' alone. It is a cantilever fixed at 0, of length x, loaded uniformly with intensity v, and loaded at X' the free end with 7^ (c - x), half the load spread on K'K. For the uniform load alone, compare OK' with OK (fig, 1), whose deflection is 3z ; their deflections are proportional to the cubes Fig. 5. of their lengths and to the loads ; hence for uniform load alone the deflection of OK (fig. 5) is fOK'^ \0K) 3s = 3.- OK being shown in fig. 5, and OK in fig. 1. By theorem at fig. ], a load at the end of this cantilever, and of amount ux, would produce an additional deflection 8 ( - | z; the load \u{c - x) will produce a proportional deflection ; and we have for the deflection due to the load at end of OK, and for the total deflection of the point K, OD =- (4c - x) z. CHAr. XVI,] CURVATURE, SLOPE, AND DEFLECTION. 309 To pass from fig. 2 to fig. 5, the central prop has been pushed up through OD (fig. 2) together with DO (fig. 5) ; that is through and by the converse's of corollary to theorem at fig. 2, the reaction on the central prop will be 1 / (4c - x)x^\ 8s I c^ ) Now, OS, = \(c + x); and since (fig. 2, Ch. VIII) OS, =--^c, we have i(c + a;)=i(5+^l^i^ Uc-,x)=l[5+^±^l^y; solving this equation we find OK' = x = c(2 - v/3), which determines the position of the points of contrary flexure. Substituting this for x in the expression for W as given above, we have the reaction of central prop If we take OK'K, half of this beam, and suppose the end at fixed, and the end K to be supported by the cantilever O'K similar to OK' in all respects ; we have (fig. 6) a beam fixed at both ends and uniformly loaded ; its semi-span is n^ 3 -v/3 c = OS,= — - — c ; but OK' = {2 -^S)c = (l --yojc'; hence the distance from the centre to a point of contraiy flexure is SK' =^-,0= -289/', where /' is the span. 310 APPLIED MECHANICS. [chap. XVI. Let W be the total load on the beam (fig. 6) ; then the shearing force diagram will vary from ^W to - ^W from left to right end. The bending moment dia- gram, is the same parabola as 'for the beam supported m. ^^ at the ends, except that itn" = j ■passes through the points of contrary flexure ; FO" :FD':: 0"B- : D'L\ FiK. 6. or 0"F is one-third, while 0"D is two-thirds of the maximum bending moment for the beam supported only ; hence at fixed end -'"max. ~ S a" "^ ' TU To find the deflection; consider first the part KK\ for it 5/, SK^ 5 /; c' 5/, c' 12E- y. 12 ■ Ey, 3 36 E ' y„ where /, is the stress on the skin at S. Consider next the cantilever OK', taking into account the uniform load alone ; a cantilever of length SIC would deflect j^v^, and taking account of its length, we have for the deflection of OK' for uniform load alone fOKy 3 A load at end of OK', equal to the uniform load on it, would produce an additional deflection of ^v^ ; but the load at end of OK' is that on K'S, and it will produce a proportionate deflec- tion ; so that we have for the deflection of OK' for load at end fSK'\ 8 fOK'\ 8 '' = \0K')Z'' = \SK')V^' Hence the total deflection of ;S' is L ZfOK'\ SfOK'Vi CHAP. XVI.] CURVATURE, SLOPE, AND DEFLECTION. 311 OK' / 1 \ 1 fOK'y (0K'\^ (^ =(28-16 v/3), and (^^-,j = (6^3 - 10) ; o j 5 4. E y, ^ E y, since J/ at the end and at the centre are in the ratio of 2 and 1 We thus have m = - tV 5 ^^^ ''^" = s- From the above, we see that fixing the ends of an uniform beam which is loaded uniformly increases the strength in the ratio ^ : ■^, or 3:2; stiffness in the ratio j^2" • ¥» or 10:3. As already shown at fig. 2, Ch. VIII, the strength is econo- mised to the greatest extent if by means of hinges, or in some other way, we shift the point of contrary flexure to a distance c -f v^2 from the centre ; we then increase the strength in the ratio 2:1; stiffness in the ratio r-^ : ^^, or 10 : 5*66 (see below). Maximum Stiffness. — For some position of the hinges, the stiffness will be a maximum ; in order to find this point, let /3 be the distance of the hinges from the centre, and/, the stress on the skin at centre ; then as above we have {5 j '\ {i j )''E y. _ 3c^ - 4c''/3 - 6c/3' + 12/3" ^ c^ ^ 12cj3^ E'iJ 312 APPLIED MECHANICS. [CHAP. XVI. where/, is the proof strength so long as 3fniax is at the centre, c that is so long as /3 > — ; putting /3 = c, we have ?/' = -/'j. (Hinges at ends) ; /3 = -rr, we have J2 n" = -^ • (Hinges at "TOTc from centre.) When j3 < -^, then iT/max is at the end, and it is necessary to substitute for /« in terms of /„ the proof stress on skin at end ; thus, if /3 = put f, = \fn, and the above expression gives or ?i _ 1 ~ 8- The stiffness is a maximum when v is a minimum ; this occurs when /3 = ^c, sensibly giving 1 J'^ 1 yo'' //I Ml • • f = ^ ^;— = 4 t;— : or 'Ji = TT sensibly a minimum. Ut/o ' %„ '^ -^ Fixing the ends and placing the two hinges at the quarter points of span increases the stiffness in the ratio ■f'j : ^, or 4 : 1 nearly. Some economy of a material whose strength to resist tension is, say less than that to resist thrust, can be secured by making the constant cross-section of such a form that the upper skin at end section, and under skin at central section, shall simul- taneously come to their working stress. Thus for a material half as strong to resist tension as thrust, and with the hinges fixed so that /3 = — , a section whose neutral axis is half as far from the upper as it is from the under skin, would give considerable economy ; as the upper and under skin at end sections, and under skin at central section would all come to their working strength at one time. CHAP. XVI.] CURVATURE, SLOl'E, AND DEFLECTION. 313 Beam of uniform strength, uniform depth, fixed at the ends. — Suppose that OK'SKO', fig. 6, is the beam. From equation 28, p. 302, the curvature is constant so that OK' is part of a circle whose centre is on tlie vertical through 0, and SK' is part of a circle of the same radius whose centre is on the vertical through S; by symmetry SK' = O/v"', and /iTand K' the points of contrary liexure are midway between the centre and the ends of the span. Since the beam is of uniform strength and depth, the breadth varies as the bending moment ; hence the plan should correspond with the bending-moment diagram. For the case of an uniform load, the plan will correspond with the bending-moment diagram in fig. 6, the curve being drawn on both sides of a centre line, the two parabolas passing through the quarter points of the span ; the breadths are then to be reduced till that at any point (say the end) is just sufficient to give the necessary resistance to bending, and we have the plan of the beam ; see Eankine's "Applied Mechanics," fig. 144. Since 0"B, fig. 6, is half of D'L, and since the load is uniform, 0"F is one-quarter, and 0"D' the maximum bending moment is three- quarters, of FD' the amount of the maximum bending moment for the same beam not fixed at the ends. The plan must allow sufficient breadth at the points of contrary flexure in order to be able to resist the shearing force at these points ; on K', for instance, the shearing force will be the load on SK', and the breadth at K' must be sufficient to resist this amount. EXAMPLKS. 1. A beam 24 feet span, of uniform symmetrical section as shown in fig. 26, Ch. XIV, is made of wrought iron whose working strength /= 4 tons per square inch, and whose modulus of elasticity E = 11600 tons per square inch. Find the radii of curvature at intervals of 4 feet, when loaded uniformly with the working load. Taking equation 16, we have p Eijo \ c' J 11600 X 10 \ 144/ where x and c are to be in one name, 144 p = 29000 — ; , X being in feet. 144 — X' The radii of curvature are po = 29000 ; Pi = 3262.5 ; p^ = .52200 inches : pn = infinity. = 2420 ;• = 2720 ; = 4350 feet. The reason that p is in inches is because vo is in inches, and the proportions •derived at fig. .5, Oh. V, show clearly that p and y are in one name. "We had to put j/oin inches, because for the material/ and E are given in tons on the square inch. 314 APPLIED MECHANICS. [chap. XVI. 2. Calculate the slope and deflection in the previous example 4 144 5 / c- 11600 10 ■5 4 (H4)' •0033; .-. i, = 0° 11' nearly; = 0-30 inch = 0-02.5 foot. 12 2; yo 12 11600 10 Tliis is nearly one-thousandth of tlie span, which is ahout the extreme ratio of working deflection to span allowable in practice (Rankine's "Applied Mechanics," sect. 302) ; and this degree of stiff"ncss is secured by making the depth of this wrought-iron girder a -j-j — j^ part of tlie span. 3. Find the slope and deflection in example 7 by means of a graphical solution. As in fig. 7, which is diawn to a scale of 20 feet to an inch, draw verticals ; one to represent the vertical through the centre of span ; the others to be drawn on each side of the centre, and at 2, 6, 10, and 12 feet therefrom ; the last two will be through the extremities of span. From c\ any point in the central vertical, with radius 24*2 feet, that is a hundredth part of po, describe the arc B'AB between the verticals through B and B' ; produce Bc\ to C2, so that CiB = 27"2, a hundredth part of p4, and about c^ describe the arc BO: similarly (■36'= 43-5; and from D, DE is drawn at right angles to Dc^. On the other side of the point v/ construct AB'E'; draw the horizontal chord EOE', and the tangent E'K. Then EAE' represents the curve assumed Y'l". by the neutral axis of the beam when under the ° proof load, but with its vertical dimensions exaggerated 100 fold. The deflection ^■■'=iTo^^^ 1 100 2'n(by scale) = -025 foot, OK . 4-0 t, = tant<:= --- -f OE' = — - 124 = -003 = 0' 11'. ICO 100 4. If the beam in example 1 be loaded with one ton per foot of span, tiiul the deflfction. Let W" be the working load, then mW'l equals the working value of M,. : that is \W'{1\ y 12) = 1948 (see fig. 26, Ch. XIV) ; therefore W = ')4-l tons. Now the load in this example is 24 tons, ^""^ of W the working load : and since deflection is proportional to load, we have Deflection = v'o = J x -30 = 0-13 inch = 0-011 foot. Oihcrwise, if the working deflection had not been already calculated, we have Wc^ 2inn" Wc^ 2init" Wc^ 'jV '~E~ I2mn"\ Deflection = Vo = I — p- I \ mn I Ehhn^ n'boho^ E 2(i)(A) 24x(144in.)3 /o = 4870 4870 11600 = -13 in. = (HI foot. CHAP. XVI.] CURVATURE, SLOPE, AND DEFLECTION. 315 5. Find the working deflection for a wrouglit-iron beam of uniform sticngth and uniform breadth, and loaded uniformly ; the span is 24 feet, and the upper half of the cross-section at centre of beam is shown in fig. 26, Cb. XIV ; /= 4 tons, and 2.'= 11600 tons per square inch. vo = »" . 4- . - = I „ - 1 1 ^ . - = '5" X :r • = '41 m. = '034 feel. A' I/O \2 / ii Vo 11600 10 6. A M-rought-iron rectangular beam, '20 feet span, 16 inches deep, and 4 inches broad, is loaded at the centre. Calculate the proof deflection if the proof strength of the iron be 7 tons, and its modulus of elasticity 12000 tons per square inch. f c" 7 120- Vo = n" . 4; . — = i . — . = -35 inch = -029 feet. 7. Find the resilience of the above beam (see Chap. I). Let JF be the proof load, .Vq = Mo, or m Wl = nfbh^ ; | TT. 240 = i x 7 x 4 x 16= : therefore 7F = 20 tons nearly. Resilience = \ proof load x proof deflection = J x 20 tons x "029 feet = -29 ft. -tons = 650 ft. -lbs. 8. Find a general expression for the resilience of a rectangular beam loaded at the centre. Let W be the proof load, and Vq the proof of deflection, then _ /2mn"\ \ m'n ) Ebh>' ('})l)i \ c —r- -^rrr:; x W^ ; but m . TF . 2c = nfbh- ; m n I Eoh^ ••• ^= -o ; •■• resilience = -;- . ■— - . -^ = ; .•'— . -Icbh. The first factor depends on the form of cross-section and the distribution of load, and the second upon the material ; the third is the volume of the beam. Hence for a rectangular beam of a given material loaded at the centre with the proof load, the resilience is directly proportional to its volume ; a result correspond- ing with that obtained for direct stress, Chap. I. Suppose for any given material we take a one-inch cube ; then, when loaded at the centre as beam we have resilience per cubic inch = ( , ) "— - ; for a rectangular section n = J, tn' = ^ ; for load at centre 1 ;i " 1 "''" ^ '" = - "^"^ " ='' s;;^' = T8' .1/1 and resilience per cubic inch = — ■ / • "f ~ To ^ P'oof stress x proof strain. IS £ 18 '^ = ^V^,L.^ = il£ ^- . 600 = -156. 316 APPLIED MECHANICS. [CHAP. XVI. In the case of M-rought-iron, for which the proof-stress /= 7, and E= 12000 tons per square inch, and remembering that the deflection is in inches, resilience per cubic inch = 77: x = -000227 inch-tons = -0423 ft. -lbs. For the previous example, the volume of the beam is 15300. cubic inches; multiplying this quantity by '0423, we find the result given there. It. For a rectangular timber beam of uniform section and uniformly loaded, find the ratio of depth to span, so that the working deflection may be a si.x- hundreuth part of the span. The working strength of the MOod is one ton, and its modulus of elasticity is 800 tons per square inch. 2c * in ' £' Vo * i 800 That is, the depth is to be between a sixth and a seventh of the span. 10. What stiffness will be secured for an uniform rectangular beam of the same timber, loaded at centre, by making the depth an eighth of the span ? 15 = 1!!:' L -1 = ^ ^ i- 8 = —. 2c * m' ' E' liQ •* ■ ^ ■ 800 ' 600 That is, the working deflection will be a 600*'' part of the span. 11. Find w" and w", the numerical coefiBcients for slope and deflection, for a beam of uniform section loaded with two equal weights at points which trisect the span. In the expressions following equation (27) substitute |c for Xr ; and 12. A beam of uniform cross-section is loaded with two equal weights symme- trically placed on the span ; the amount of each weight is for each position such that tlie outer fibres of the middle cross-section bear the working stress/. Compare the amount of deflection when the abscissae of the left weight are as follows : — Xr = 0, Xr = he, and Xr = |c. Ahs. For Xr = 0, n" = J- ; for Xr = h, n'' = %% ; for a-, = %c, n" = -?§. CHAPTER XVII. fixed and moving load.s on an uniform girder with ends fixed horizontally. Unsvmmkti;1(Al Fixed Loads. We now consider the detleclioii of an uniform beam and tlie slopes or tip.s-vji at the freely hingetl ends due to loads placed unsymmetrically ui)on it. CHAP. XVII.] GIRDERS FIXED AT THE ENDS. 317 On tij,'. 1 let a load W be concentrated at D a distance z from the centre of the span. D divides the span into two seg- ments III and n. The bending moment diagram is ACB. The height of C is h, and of any other point is y. The origin is at Q the lowest point of the beam which divides the span into segments a and 6. The deflection is u^ = «3, and the tips-up are o and /3. The span is / = 2c. In Chapter XV f. equation (5), / cancels /o, and all the factors are constant except J/ the bending moment, y may be- supposed to include this constant in every case. h 111 ' h b - Xo •ll - ^'' ~ h III The slope at D and its height above Q are ill }o in L 2 Jo 2 Ub 71 = — I (« + '^'i) dxdx = 6 m (m - «)■ (??i + 2a). fi i { ri\ x^ '/ And h = - \ (h - Xj) dxdx =-\ hx - — . dx= - (a - X3) dxdx = oJo Also Ua = 3 w 318 APPLIED MECHANICS. [CHAP. XVII. Equating w„ and v.b we have (fig. 1), Ua = «6 = t'h-n + 'ilOh.n + A', 2ft" = (m - ay {m + 2a) + I6n {m- - a^) + 2nhn, (m + n) (m^ + 2mn - 3f'-) = 0, 3a' = w (/ + n) = {c + z) (3c - =). (1) Substituting we have the deflection of the beam l\l = Ua = Uh = ha^ h-yS . 3m 27 (3c - s) ^y{3e - z)(c + z). (2) (i = 6b = dh-n + - f Q>- ^2) dx=^ {m- - a-) + hx - 2jt.n hf a^ \ h( a? 2 V m / 2 V ?n But by equation (1) (7^ -^ 7n = (/ + ?i) H- 3, so that /3 = ^(2^-70=^^ (3c + .). ha- h ,. . /«- ,., (ft - .r^) rfr» = -- = -(/ + w) = -F^^c - z). 2 ?n 6 And a _ Zc-Z ]3~ Sc + z C — X C + X (3) (•i) (5) where .^ is the distance horizontally of G the centre of gravity of the bending moment diagram from the centre of the span. Also (« + /3) = he = area ACB. (6) The following are perfectly general : — Theorems. — If a girder of uniform section and hinged at the ends be loaded in any manner between them — (ft) The sum of the tij)!i-v2^ at tlie ends is proportional to the area of the bending moment din^ram, and (//) their ratio is the same as that of tiie segments into wliich the perpendicular from the centre of gravity of the bending moment diagram divides the span, but in tlie inverse order. Ecjuations (G) and (5) prove these theorems for one concen- trated load. "CHAW'. XVII.] (HKDKKS FIXKD AT THK ENDS. 319 Let there be two sucli concentrated loads so that a = a, + a, a.n(A /3 = /3i t /Sj ; then, from eqnation (5), or 2t'oi = AiC - AiJ- = AiC - G^ a, + i3, 2c whc'^re Ai is the area of the bending moment diagram for JJ\ aloiu^, while G^ is its geometrical moment about the centre of the sf :>an. For IF-, alone on the girder, we have 2e(r. = A-.c - Gn, and auUling 2ca = Ac- G for the joint bending moment diagram. Similai 'ly, for any number of loads. Pu ttiiig - = 0, in equations (4) and (2), gives us ip = ^hc and u^ = \hr, (7) whicni aie just the same as equations (13) and (14) of Ch. XVII for t he Uniform beam with load at centre if we put /Again put z = c, and from equations (3) and (4), we have /3 = |/ic and a = Ihc. (8) Ttte bending moment diagram is now right-angled at B ; and as ^J^is at B, and over the abutment, there is no strain on the beam, ajad a and /:• are both zero, but still just before W went ofi* they v/ere in that ratio. There is tinother way of loading the beam so that the bend- ing moment diagram shall be right-angled at B, that is, by applying a couple at the end B. Suppose then a right-handed couple Mq applied at B, the hinge A will hold down with a force F, and the hinge B will push up with Q = - P. In this manner the two hinges will constitute a reacting couple, with the span as an arm, and of oppiosite sense, but equal in moment to Mg ; so that The ends of the beam will drooj) by a and /3 at the ends A and B, the droop /3 bt'iug double the droop a by equation (8). Their sum will be the a I'ea of bending moment diagram, viz. the right- angled triangle of height Mg and base 2c. In the same way let a left-handed couple 3[phe applied at the end A. The joint bending moment diagram due to both couples will be of height 3fp and Mq at A and B, with a straight locus MK, fig. 2, between them. If these two end couples be the only loads on 320 APPLIED MECHANICS. [CHAP. XVII. the beam, Mn being bigger than Mp, then the hinge A must hold down and the hinge B hold up with equal forces P= - Q, form ing a reacting couple with the span as arm to balance Mq - Mp, so that P = - Q == (^^<j- ^^p) -^ 2c. The beam will be every whc^re convex upwards, its two ends drooping so that the droops a ai id j3 shall have their sum equal to the area of the bending monif^nt diagram AHKB, and bear to each other the same ratio "mto which the perpendicular from g divides the span only ii'i the inverse order. / Suppose now that the uniform girder hinged at the pnds is loaded between them in any way, as, for instance, merely with the weight W (tig. 2), trisecting the span. The two e^c Is will tip up. By applying suitable end couples Mp and M^, whihh alone would produce droops exactly equal to those tips-up, iwe! may destroy the tips-v/p and make the ends level. \ The conditions are that Mp and Mq are given by A H and BK when HK is drawn across the bending moment diagram ABC, in such a way that the areas AHKB and ACb shall be equal, and have their centres of gravity ij and Q in one vertical line. We have the following graphical solution for a uniform girder with its ends held horizontal and loaded between ihvni in any way. First draw the bending moment diagram for the extern loads as if the girder were hinged at the ends ; next draw a straight line cutting across it so that the four-siderl figure shall include the same area over the span as the bendino- moment diagram, and have its centre of gravity on the st^ue vertical. This line, HK on fig. 2, is the new base cutting the locus in two points S and T between which the bending njoments make the girder convex downwards, beyond which they inake it convex upwards. From S to T ordinates are to be measured iip to SCT for positive bending moments, and from S to J£ and T to K downwards to SA and TB for negative bending i.noments. Beam of Uniform Section fixed Hohizontai.lv at the Ends AND subject to a PiOLLING LOAd. Let W be at any distance z (fig. 2) froi.n the centre 0, then ACB, with C directly under W, is the, bending moment diagram for the ends hinged. Let HK n^ap out the area AHKB = ACB, and let <j and 6', their cent/res of gravity, be in a vertical line. For shortness, put h ■= AH, and /.: = BK\ also put m for the lieight of C from AB, and OA = OB = c, t/he half span. For CENTRE OF HYPS. IS OVER Vat 4 times ITS HEIGHT AND 2 t/JC = {3X-C){C-X) W ■ ® ®.® r^-4-) X2 2C' FEET 10 I I I I I I I I I s POSITION OF THE LOAD Fig. 2. Y 322 APPLIED MECHANICS. [CHAP. XVII. areas equal we have h + Jc = m. For g under G take moments about A, first dividing AH KB into two triangles by the diameter RB ; their areas are he and kc ; their levers about A are ^ of 2c and § of 2c, wliile the area of ACB is 9«c, and its lever c + ^z, so that mc(c + As) = he X fc + ke x *c ; ?;2(3c + z) = 2c(h + 2/i-) ; T c + z ^ ^ c - z therefore k = —rr— m, and h = tn. 2c 2c The equation to HK is 2/, - h h - h X + c 2c Jc - k k + h zm m y'^-2^'''--2r- = 2^^'''-2' " The equation to BC is i/z - m - c - X i/i = m , X - c z - c Let us take the negative bending moment at any point x to the right oi T. It will be the intercept from TK down to TB, and M • " '^' ~ ''-''' " 2& ic - z) r^ ~ *^ ^^' "^ "^^^ " ^^' ^^ ~ "^^i • Now, Oil, the bending moment under W, is the height of C above ^^', or and substituting ancrsiniplifying ^^neg =^(0 + 2)X(2c-2).r-C^). (9) On the girder above T and <S' are shown virtual hinges or points of contrary Hexure. To find |, the abscissa of T, sub- stitute Mneg. = zero, and .r = ? into (9), and change sign of z for r, then CHAP. XVII.] GIRDERS FIXED AT THE ENDS. 323 Now t, is a miiiiuium when z = - c, that is, when W i.« just at left end, and substituting this vahie of z, we get the niininiuni vahie of ^ to be Jc. Hence, for the middle third of the span, the bending moment cannot be negative at any point for any position of the load. For negative values of M, then, we require x >-^. Inequa- tion (9) let X be any constant value between -^c and c, while z varies, Ac' dM W^ ^^"^^^^ ^ (^ - ^) - ^^"")> 4c' dm _ . „ — -7-r = - 6^^ - 2c-. W dz- ±1 irst, z = - c makes —,- = 0, and rrr- —r-ir = - be { .r -~ dz W dz- \ 6 positive quantity, as x is between ^ and c. Hence z = - c makes J/jjeg a minimum. ^ . .,, 2c'- dM _ , 4c^ dW f c Second, with s = c - — -, ^r- = 0, and -jr- -3—7 = - be [x - — S.r d.r ' IF dx' \ 3 a negative quantity, since x > — . Substituting this value of z in equation (9), we have c \» 3fx = JV- ~— , a negative maximum. (11) When the load is at a point. 2c' = = . - 3-. (12) On fig. 2 there are laid doivn to scale the values of Mx at 1, II, III, IV, and v, for the values of x, namely, 2, 3, 4, 5, and 6 twelfths of the span. The corresponding positions of the rolling load are shown with circles round the Eoman numerals agreeable to equation (12). The maximum of negative maxima is at the end when x = c, and z = Jc. At the top of fig. 2, the load W is shown in this position, namely, trisecting the span, and the maximum of negative maxima bending moments at the end is 8 4 3fc= ^r- Wc= ^rr- IVl maxlmum. (13) 27 27 Y 2 324 APPLIED MECHANICS. [CHAP. XVII. By substituting for m in the preceding expression for h and k we have W 3Iq=BK=^Xc^--z')(c^z). (14) M^ = AH= ]^(c^ - z') (c - z). (14a) We come now to consider the positive bending moments. Changing the sign of (9) we have M', = i^(c + zf (r - (2c - -) x) from C to T, or as long as x lies between z and | = ^ _ _ . Now Jf j; is only admissible while ^ :^ ^ ; and as it continually increases with z, M'x is clearly greatest for z = x, that is, when the load is at the section. As the diagram is of the like form right and left of W or C, it follows that the positive maximum bending moment occurs at any point when the load W is over it. Substituting 2 = a; we have W Mj, = —-i{c^ - •«')■> a positive maximum, (15) Cor. — If the girder were only hinged at the ends, the maximum bending moment is 3Io = } Wl ; when fixed horizontally at the ends by equation (13) it is ^S-^^^^ > hence fixing an uniform girder at the ends increases its strength in the ratio 27 to 16 to resist a rolling load. This increase is 69 per cent. With the load confined to the central point the increase would be 100 per cent,, for HK would then be horizontal, and ACB isosceles, and for the two bending moment diagrams of equal area we would then have AH = CD = BK, and all three maxima half of that for the ends hinged. It can readily be proved that the loci of ^S'and T are the pair of hyperbolas shown on fig, 2, The equation to the locus of T is 2i/x = (Sx - c){c - x) W. The principal axes of the pair meet on the vertical OK produced four times that amount. It is interesting to notice from fig. 2, that at a point -^ of the span from the right end, the positive and negative bending moments 25 and 27 are nearly equal. That is, for the end sixth CHAP. XVII.] GIRDERS FIXED AT THE ENDS. 325 of the span the negative niaximuin bending moment is greater than the positive. This point might be found from the equation 4c^ x' 2 It will be found that by trial and error x = -c gives 25 1 -j ^r^ = -7 nearly. 4 X 27 4 "^ Levy gives this point as '171 from the end, a trifle more than I" of span, as the diagram (fig. 2) shov^^s it to be.* Shearing Force. — For the position of the loads shown on figs. 2, 3, the right supporting force Q is composed of two parts, one a major part of W, namely W {c + z) ^ 2c, just as if the girder were simply hinged at the ends. The other part is {Mq - Mp) -^ 2c due to the couple at one end being greater than the other, so that ^= ^-2r^4^3^(^'-^^)- Now Q is the instantaneous shearing force at every point in the shorter segment of the girder from JV to the right abutment B. It is clear, then, for a section x = z, that is, for a section inde- finitely close to W on its right side, that Q is the maximum shearing force. Hence c + X W' . . ,. i\ = yy—^ — + -TV ^{(^ -^') a maximum, 2c 4c^ ^ the load being just to the same side of the section as the more remote abutment IF Fa: = "ttC^ + ^)"(2c - x) maximum. (16) * In Za Statique Graphique, second edition, vol. ii., p. 124, Levy gives equa- tions (20) and (19) derived by constructing graphicully "lines of influence." He finds that (19) supersedes (20) in magnitude for '17 part of the span from each end. In Levy's equations (20) and (19) the origin is at the end of the girder; substituting (c - x) for his x, and 2c for his I, these equations (20) and (19) are identical with our equations (11) and (15) given above in their symmetrical form, the origin at the centre, and the loci of which are plotted on the lower part of our fig-- . . The solution in the same form as Levy's is given in Du Bois on The Stresses in Framed Structures, second edition. 326 APPLIED MECHANICS. [chap. XVII. Now Fc = TV at the ends just as if the girder had been hinged at the ends. So also at the centre Fo= h W, as indeed is evident, for with IV at the end there are no end couples ; while with W at the centre the end couples are equal and do not modify the supporting forces. Hence fig. 6, p. 164, is an approximation to the shearing force diagram. On fig. 3 is shown the diagram of maximum shearing forces plotted from equation (16). F(XED Sct F=|j5(c*-x)tk<s.tO 2F \^' X* Fisr. 3. Levy shows this diagram in his fig. 25, p. 125 (see foot-note). From his equation (21 his), identical with our equation (16), he infers that the locus starts horizontally at each end. It is readily shown by differentiating (16) when the differential coefficient of the variable part is 3 (c* - x'^) , then x = ± c makes this zero, so 3 IF that the slope at end is zero, and at centre is tan"'— r— • It is important to observe that at any section the positive maximum shearing force is instantly succeeded by the negative as the load passes it, so that their arithmetical sum or the rcoigc of the shearing force, as we luive called it, is IF, and is in no way altered by the fixing of the ends of the girders. Fig. 4. 328 applied mechanics. [chap. xvii. Beam of Uniform Section fixed hokizontally at the ends AND subject to AN UNIFORM ADVANCING LOAD. For convenience the uniform load may be taken as 1 ton per foot-run. Consider first the negative bending moments. For the middle third of the span they are zero at each point ; for the load, however it be disposed, may be looked upon as a number of concentrated loads, no one of which can produce a negative bending moment on the middle third by the last problem (fig- 2). Let X be the abscissa of any section on the third next the right abutment, measured from the centre as origin, so that x lies between the values \c and c, where c is the semi-span. Let the load advance from the left abutment till \hQ front is at the section z from the centre, where z is measured in the same direction as x, but is less than x. On fig. 4, A VGB is the instantaneous bending moment diagram for the ends hinged ; compare fig. 9, Ch. VIII. A VC is parabolic, but CB is a straight locus. By theorems {a), 'h), HR' is drawn across so that the areas AHH'B and AVCB are equal and have their centres of gravity on one vertical line. As z is less than x, the new base B.H\ for ends fixed, cuts at S on the straight part CB. For ends hinged the height of C above AB is ^ (c + z)\c - z), while the height at x to SB is y = 4^ (<^ + -)" (c - ^)- The bending moment diagram AUCB for the extern load divides into two parts a triangle ACB of area \v:{c->r zf {(' - s), and lever about A of AD = (c + ^z), and a parabolic part AUG of area Yl'U{c + z), and lever about A of AE = \{c + z), where TU = ^w(c + zf, just as for a span (r + z) uniformly loaded. CHAP. XVII.] GIRDEKS FIXED AT THE ENDS. o29 Agaiu, the bending moment diagram AHirB, for the end couples m and n that destroy the tips up, divides into tv^o triangles ; AIIH' of area mc and lever about A of fc, and AH B of urea nc and lever ^e. Equating the areas of these two Ijending moment diagrams and also their moments about A, satisfies the two conditions laid tlown by preceding Theorems [a) and (7;) ; we have the two equations (m + n]c =.-(c + zy{c-z) + — {c + z)\ ;So that m = ^_{c + zf (llr - lOcs + .V-), (17) n = ~{c^zf{^c-^z). (18) The height to HH' at the section x is c — X y = 71 + —^ — (m - n), So that at .r M.,,. = 2/' - 2/ = 4^3 (^ + ^)' (3(3c - z).v - ^0% (19) If iT = ^, when M is zero, we have for S the virtual hinge e _ ^^* ^~ 3(3c-s)' Putting it for the part of equation (19) that varies with z, then ^^ = 12{c + z)- i2cx - d" - o:z), dz ~^=12{c + z) [^cx - 2c* - 2>xz). 330 APPLIED MECHANICS. [CHAP. XVII. z = 2c — , ( 20) X which, observe, is less than x as premised, makes di.t fli( i c \ — = 0, and — = 36c(c + z){-- x], a negative quantity, as we are only dealing with values of x which are greater than \c, in that we have disposed of the middle third of the span already as far as negative bending moments are concerned. Hence this value of z makes il/ucK. ^ maximum for all points in the third of the span towards the right abutment. Substituting and reducing, we have ^^^neg. = -j^ ^ max., (21) when the front of the load is at z, equation 20. We come now to consider the positive bending moments. Let the load cover the span, and simultaneously at all sections the bending moment is given by the dotted parabola on fig. 4, as already explained, measuring up and down from the hori- zontal base across it. The equation to this parabola with the centre of that base as origin is y = ~ (c^ - Sx'). (22) See iig. 4, Ch. VI, for such equations, and fig. 6, Ch. XYI, for the conditions. Now y is positive in the middle third and even further out from the middle, but we shall only consider the middle third. Xow removing any part of the load is the same as if we had left the load all over the span and applied a loail of equal intensity n2waTcl at a part, and this upward load being itself negative cannot cause any positive bending moment in the middle third of the span. Hence the dotted parabola itself gives the maximum bending moment at each section in the middle third, the load being all over the span -'l/pc. = "^ (c^- - 3.r) max. (23) for values of x from to ^c only. For the third of the span next the right abutment, let x be the abscissa of any section as before. Now we have shown that CHAP. XVII.] GIRDERS FIXED AT THE ENDS. 001 •joi equation (20) il/„eg. is greatest for the load extending from the left abutment to z = 2c — . If this load were acting upwards, the bending moment at x would still be the maximum, but of course positive. But before applying this upward load, let the whole span be loaded, and the upward load removes the load from left abutment to z and leaves it from z, to the right abut- ment. Hence the maximum positive bending moment is the F=:^s(cf^yQc% ^cv-jcy SCALE 16 TONS TOAN INCH Fis. 5. same as J/neg. only measured upwards from the dotted parabola (fig. 4), and occurs when the short segment from z is loaded. Hence for values of x from -jc to c ir ^^ / 2 o ox 27WC X - ■ (24) a maximum when the tail of the load is at z, equation 20. The same argument, on the last page, shows the negative or downward intercepts between IIH' and the arc A U, all to be maxima for the whole span loaded. These maxima only reach down to the dotted parabola, and are superseded by the locus of equation 21. 332 APPLIED MECHANICS. [CHAP. XVII. Sliearing Force. — For the ends fixed (fig. 5}, we have ^nd this is the shearing force at x for the position of the load shown. Xow the differential coefficient of Q with respect to z is positive, or Q continually increases with z till ;; = a-. Again z> X means z = .r, and an additional load between x and the right abutment, a share of which is thrown on the left abutment, so that the increase of Q is more than cancelled when from Q we subtract this whole extra load, as we must for the shearing force ; hence the shearing force at any section x has maximum positive and negative values when the segment from x to one or other abutment is loaded, exactly as was the case for the ends hinged (see fig. 1, Ch. IX), only now V) (c + x)\U- + 2cx - x^). (25) Examples. 1. Grapliical construction of the bending moment and shearing force diagrams for a uniform girder of 36 feet span, fixed horizontally at the ends, and loaded with 12 tons at 9 feet from the left end, and 6 tons at 21 feet from the left end (fig- 6)- The load line is laid down to a scale of 10 tons to an inch. From a trial pole a link polygon constructed among the four forces P, W\, JFj, and Q, laid down at 10 feet to an inch, and a vector from the pole parallel to the closing side determines the joint between the reactions, making /"scale \\\ and Q scale 6^ tons, being the values for hinged ends. They may also be found by taking moments about the left hinge. Through this joint the base of the shearing force diagram is drawn horizontally, and the diagram completed for hinged ends. A new pole on the horizontal through tlie joint is chosen at a distance 6 feet, a horizontal base laid off for tlie bending moment diagram (with hinged ends), which is then constructed as a link polygon to the new pole. Thus (ar it will be seen that the load and girder correspond to the moving model (fig. 16, Ch. IX), with the locomotive standing so that its heavy wheel is 9 feet from the left abutment. To return to fig. 6, it will be found that the scale for bending moments is 60 foot-tons to an inch, tliat is, six times as fine as the ton scale. For the ends fixed we must draw a base, cutting across the bending moment diagram, so as to include the same area over the base, and have the centres of gravity in one verti<:al (see Theorems («) and (A)). To do this graphically the bending moment diagram is divided into thru triangles and one rectangle. The first triangle has a base '•• feet and a height 103'o foot-tons, being the bending moment under 7/'i, either as scaled oflF or as calculated. The last triangle has a base Ift feet and a heiglit 97'5 foot-tons. The dimensions of the remaining triangle and the rectangle are readily seen. CHAP. XVII.] GIRDERS FIXED AT THE ENDS. 333 It is convenient to reckon 18 feet the half span as a horizontal unit for these areas, which will then be found to be Ai = 26, A^ = 2, Ai = 65, and Ai = \\. The lines (if action are drawn doivh through the re-pei;tive centres of gravity. Next the lines of action A^ and A,-, are drawn upwards through the trisecting points of the span to represent the areas of the two triangles, into which the bending moment diagram for the pair of end couples can always be divided. A load line of the areas is laid down to the fool-ton scale, a link polygon drawn among the lines of action when a vector from the pole drawn parallel to its closing side divides the areas At and Aa from each other. Since the half-span is unity, it will be seen that Aj and A(, are M^ and Ma on the foot-ton scale. Their values scale 51 and 83 foot-tons, and are pricked up at the ends and the base drawn across. To modify the shearing force diagram M^ - ,1/5 = 31 foot-tons, dividing this by 36 feet, the lever between the supports, we get gths of a ton as the common force by which the right end must hold down and left hold up to resist the left-handed couple J/e - -l^o- Drawing a new base for the shearing force diagram |ths of a ton lower down decreases the one and increases the other support by this amount. 2. To find the righting end couple M^ (fig. 6) by calculation. For equal areas and equal moments about 0, M& 4 3h = 26 + 2 4- 65 + 41 = 134, \1Mz + 24ilf5 = 26 X 6 + 2 X 13 + 65 X 15 + 41 X 26, and Mi = r>\, M(. = 83 ft. -tons. 3. A uniform girder is loaded on the left half only with a uniform load : find the righting end couples to hold the ends horizontal. Substituting for m and n in equations (17) and (18), we get «j = ^^ IV c- and n = -^^ wc'. Now the bending moment at centre for hinged ends is Q a quarter of total load, multiplied by c half span: that is, the height of C (fig. 4) is now \icc- or \%wc-. Hence, if on fig. 4, we suppose the load to be on left half only then m and n are \\ and -^4,- respectively, of the height of Cthe middle point of the bending moment diagram. 4. The case of a uniform girder, uniformly loaded on the left half only, is so important that we shall calculate the righting end couples directly. First since Q is a quarter and F three-quarters of the load, it follows that the shearing force diagram crosses the base at a point |th of the span from A (fig. 4). Also AE is one-half of span so that U and C are the same height and ET= TU are each half the height of C. With c the half span as unity for horizontal areas, the area A VC is %TU or ird height of C. Putting /.• for the height of C, area A UG is \k and its lever about A is +c while area ACB is k, and its lever about A is c. Again for AHH'C divided into two triangles by AH', the area of AER' is m, its lever -3 x 2c; the other AH'B has an area n and lever j x '2c. Equating areas and moments of areas about A k , 2c 4c k c , 11 , , 5 ~ + k; >n-^ + n.~= ^.~ + kc; m = ~~ k and « = -- The graphical construction is shown at top of fig. 3, Ch. XX. Fi- 6. CHAP. XVIII. 1 COMBINED STRESSES. ;J35 CHAPTER XVIII. skeleton sections for beams and struts ; combined thi'.ust with bending and twisting with bending ; long struts. Thin Hollow Cross-sections. Let t be the uniform thickness of a thin hollow cross-section of any form, and let n and n be the numerical coefficients respec- tively of the moment of inertia and the moment of resistance to bending of a solid cross-section of the same form ; let B, H be the breadth and depth of the rectangle circumscribing the section, and h, h the breadth and depth of the rectangle cir- cumscribing the hollow ; then /o = n [BW - hh') = 71 B (IP - h'), since B = h nearly = n' B (H-h) {H- + Hh + h') = n' B.2t. 3H'- since H=h nearly = n' . 6BH-f. M = -4^I. = {-]Qf.B.H.t m H \m J = n . QfBHt (1st approx.) (a) Thin hollow rectangle, M = i . QfBHf =fBHt. Thin hollow circle, M = ^ ^BSt = -Qfr-f, {d = diam.) A closer approximation for any form is obtained thus : — /„ = n' \BH' - hh'\ = n'{ {B - h) m + h {H' - h') | = n {{B-h)H' + h[H - h){W + Hli + h'] | = n' \2t.H' + b. It: . SHh] (approx.) = 2n'H\tH- + '6ht'h\ (approx.) ; .-. M = 2??/ (^^-^ + 3^570 (approx.), - [h) where t is the thickness of each side, and /' the thickness of the top or bottom. To design a thin hollow crosts-section — choose the depth H the proper fraction of the span to give the required degree of 336 APPLIED MECHANICS. [CHAP. XVIII. stiffness ; assume B a suitable fraction of U to give sufficient lateral stiffness, and the above is a simple equation from which to find t; or t may be assumed a multiple of the thickness that the metal plates are usually manufactured, and B found from the formulre. Xovv find the moment of resistance to bending, and the resistance to shearing, by the accurate formulae ; and if this differs from M and F, alter t or B for further approxima- tion. Cross-sections of Equal Strength. When a beam is made of a material whose strengths to resist tension and thrust are different, the area of the upper fiauge is made different from that of the lower (fig. 1), in order that both flanges may be brought to the proof or working stress at the same time. In the case of cast iron, the strengths to resist tension and thrust are as 1 and 6 ; and on this account the area of the upper flange (compressed) is about one-sixth that of the lower flange (extended). This form of cross-section was first proposed by Mr. Hodgkinson. On account of the liability of cast iron to crack if unequally cooled, sudden changes of thick- ness of metal are to be avoided ; on this account the top of the web may be made of the same thickness as the top flange, and the bottom of the web of the same thickness as tlie bottom flange. Double T cross-sectio7i (fig. 1) — The position of the neutral axis, and the moment of inertia about that axis, in terms of the areas and depths of the three rectangles, are expressed as follows, the notation being : .\reas. Depths. Upper flange, . . . A^, hi. Web, A,, 1u_. Lower flange, . . . A^, h^. q^ Totals, . A Fig. 1. Exact Solution. — The height of the neutral axis above the lower side of the cross-section is obtained thus — h {hi + hi) As - [hi + hi) Ax - ijh - hi) A^ ,- . II » =2 2 A " ^^^ CHAP. XVIII.] COMBINED STRESSES. 337 The moment of inertia for each rectangle is shown ofi page 215 ; and letting 2 as before denote the " sum," the moment of inertia for the cross-section is + A,A, {Ju + 2h + IhY + A,Az {K + h)'}. (2) The moment of resistance to bending is as before llh Approximate Solution. — (Eankine's "Civil Engineering," §§ 163, 164). When 1h and A3 are small compared with lu,, we may leave them out of the exact formulae ; and we obtain _k {A.-a;)1u yi>-2 2 A — ' ^^ lo =^' + ^ (^1^2 + 4:A,A, + A,A.). (4) Put h' for hi + ' ^ ^ that is for the distance between the centres of gravity of the flanges ; and let Az be the area of the cross-section of the vertical web measured from centre to centre of the top and bottom flanges ; then, nearly A.Jj-A.vi^A.^. mJ-^A.Z-^A... (5,6) Substituting this value of Az in equation (4), we obtain J/o = h' j/Us + (2/6 -/.) ^ j = K |/„^, + (24 - /,) ^^ j . (7) In designing a beam to resist a given bending moment, the depth li is taken at a fraction, say -^th to -j^-th of the span so as to ensure stiffness ; the thickness of the web is then fixed by considerations of practical convenience, and so as to give suffi- cient resistance to shearing ; and the area of the upper and lower flange can then be found by equations 5 and 6 ; having thus fixed the value of Ax and A^ we can then choose breadths and depths suitable for the flanges. 338 APPLIED MECHANICS. [chap. XVIII. Fof the section as thus fixed calculate M and F by the tabular method shown for figs. 5, 26, Ch. XIV ; if ^i and h^, the depths chosen for A\ and A^, are very small, it will be found that IVI and F are sensibly what is required, and that the neutral axis sensibly divides the depth of section as/o and/^, so that no further calculation is necessary. If, however, one or both of the depths /;,, hi, be not very small, the solution by the tabular method will differ considerably from the data, and further approximation will be necessary. When one of the flanges, as in the case of cast iron, is comparatively deep, the inaccuracy of the results will be considerable, and one or more further ap- proximations may be required. For different examples, the error will be diiferent in amount, and we have no simple means of judging how great this error will be in any particular case. In the equations given above, the results for M and I are close when hi and hs are small ; the result for 7/a will be close although hs is not small, and that for yb will be close although hi is not small; thus, suppose that Aj is small, we have h (hi + h->)Ai 2/4 = 9 hiAi + hiAi 2A (8) putting h' - |Ai = /ij, and h' = h, ,24l + ^2 yb = h' 2A , nearly ; (9) a result similar to that found previously in equation (3), but ya cannot now be found by interchanging Ai and A^. Common forms for cast-iron beams are shown in fig. 2 ; the corresponding equations for these T-shaped sections are derived from the above by putting Ax = 0, or A^ = 0, according as/„ or fh is the greater. Thus for a section of this form, when/,, is greater than fb, the flange will be required on the extended side ; when /i is greater than /„, the flange will be required on the com- pressed side ; and we have A.J-'^-^A,; (10) ovA,J-^A,; (11) [ J as the case may be. Fit,'. '.2. CHAP. XVIII.] COMBINED STRESSES. 339 Similarly for resistance to bending, we have M=h'i2^,-f,)^; (12a) or =h'i2/,-fa)~'; (Ub) as the case may require. In the case of trough -shaped beams the same formulae are applicable, if we consider the web A2 to consist of the two vertical ribs. Examples. 1. Find the moment of resistance to bending of a cast-iron pipe 18" external diameter, metal 1" thick, and/ = 2 tons per square inch, by means of the exact formula ; and compare the result with that obtained by the approximate formulae (a) and (b). M = 430 inch-tons by exact formula = 390 and 429 by approximate formulce. 2. Find the thickness of metal required for a cast-iron pipe 24" external diameter, so that its moment of resistance to bending may be 50 foot-tons, and its resistance to shearing 10 tons ; taking/= 2 tons per square inch, 600 = -6 X 2 X 244 X t; .-. t = 0"-87 nearly. ■On checking the calculation by the accurate formula, and taking t = 0"-87, M = 705 inch-tons ; so that i = 0"-87 is more than sufficient ; and for shearing this thickness will be several times too great. On account of the difficulty of casting so as to have the thickness quite uniform, an allowance has to be made, and t would probably be taken at about 1""25. 3. Find, by the approximate formula, the moment of resistance to bending of the cross-section shown on fig. 3, p. 254, and compare the result with that given in the text at that figure, M = /i(2 . 900 + 3 . 3 . 6 . 24) = 1032/(2 x 7794/^ 15). 4. Design the central cross-section for a wrought iron beam, for which /„ = 4, and/j = 5 tons per square inch, suitable for example 1 and fig. 9, Ch. XI. Since the span is 42 feet, we may take h' = 40 inches ; M= 405-75 ft. -tons = 4863 inch-tons ; the web may be taken as f " thick. From equation 7 we have 4863 = 40 {4^1 + (8 -5) V} ; .-. ^1 = 28-5 sq. in. 4863 = 40 {5^3+ (10-4) ifi^} ; .-. ^3 = 21-3 „ Adopting as a first approximation — breadth of flanges, 21 inches ; thickness of top flange, 1-36 in., of bottom do., 1 in. ; thickness of web, |- in., and depth of girder (outside to outside), 41-2 in. ; we get by applving the exact method shown at p. 256, 4 X 21340 „ . , M = — rj-^T — = 4652 inch-tons, a result differing from what is required by only 4 per cent. z2 340 APPLIED MECHANICS. [CHAP. XVIII, The upper flange may therefore be taken 1| in. thick, the lower flange 1 in. thick, and the breadth of each 22 in. This does not take into account the angle irons, and the loss by rivet-holes. Solve this example by another approximate formula. Jf = A'{/„^i - i(A-/a).:f2}. (7a), 4863 = 40 {4^1 -i (5- 4)15} ; Ai = 31, and similarly A3 = 24 square inches for a first approximation ; we may fix on t„ = 1-25 and th = I inch, and from these obtain suitable breadths, y'„ = |40 =^ 17-78 in., and y'j = 22-22 in., y, = 18-4 in., and yj = 22-72 in. ;. .-. /'„ = 4 (1 - — ^j = 3-86, and /'& = 4-89 tons. We have therefore 4863 = 40 {3-86^1 - ^(4-89 - 3-86) 15} ; ^1 = 30-5 square inches, and ^3 = 25-6 square inches. Adopting the following dimensions, upper flange 24-4 in. x 1-25 in., lower flange 25*6 in. X 1 in., and h = 41-12 in., and solving by the exact method, we obtain 4 X 24250 . , M = — f^.T," = ^^^^ inch-tons, a quantity differing from the required result by less than 4 per cent. Allowance for Weight of Beam. After having designed a beam which is sufficient to bear sn given external load, it is necessary to make an allowance for the weight of the beam itself; especially is this the case for beams of long span, as then the weight of the beam bears a considerable proportion to the amount of the external load. This allowance is readily made by increasing the breadth of the provisional beam sufficient for the external load alone ; since the breadth is a dimension which appears in the first power in the expression for the resistance to bending. Consider the weight of the beam, and the external load reduced to its equivalent dead load, as uniformly distributed, a supposition sufficiently exact for our present purpose ; let b' denote the breadth, and B' the weight of the provisional beam, computed for IV the external load alone ; let b, B, and W CHAP. XVIII.] COMBINED STRESSES. 341 denote the same quantities for the actual beam sufi&cient to bear the external load and its own weight ; then V~ B'~ W' ~ ]r~ IF' W W - B W W ~ W - F W - E Therefore the breadth of beam required, The weight of beam required, The gross load Eesistance to Twisting and Weenching. One end of a cylindrical bar is rigidly fixed, and to the other •end a couple is applied in a plane at right angles to the axis of the bar ; or, what is the same thing, as shown in fig. 3, a pair •of equal and opposite couples are applied to the ends of the bar ; the tendency of these couples is to make the bar rotate about its axis ; and if we suppose the bar to consist of fibres originally straight and parallel to the axis, each of these fibres will now have assumed a spiral form. The moment of each couple is called the twisting moment or moment of torsion applied to the bar, and it is constant for each cross-section ; on account of the bar being uniform, the stress will be similarly distributed on each cross-section ; and since the bar is circular in section, the stress at all points equidistant from the axis will be the same. Suppose two cross-sections to be taken at the distance dx apart ; the twisting moment causes the one section to move relatively to the other through an angle di ; and if we consider two points originally opposite to each other, that is in the same fibre, one in each section and at a distance r from the axis, then these points, relatively to each other," move laterally through a 342 APPLIED MECHANICS. [CHAP. XVIII. distance r .di; and since the two sections are dx apart, the rate of twist is a quantity directly proportional to the distance of the points under consideration from the axis. We have thus at any point in a cross-section, a shearing stress at right angles to the radius drawn to the point, and pro- portional to that radius in intensity ; this may be expressed thus ,-Crl^; (17) where C is the coefficient of transverse elasticity for the material of the cylinder under consideration ; Eankine gives For cast iron, C = 3,000,000 lbs. per square inch (approx.). For wrought iron, C = 9,000.000 The greatest value of q occurs at the surface of the cylinder ; and if / represent the resistance of the material to shearing, and ?'! the radius of the cylinder, then we have 9=/} (18) If we consider s a small portion of the ring of the cross- section, with its middle point at a distance r from the axis, then r . di will be its mean length., and we may denote its Fig. 3. breadth by dr ; its area then is r . di . dr ; the intensity of the shearing stress at s is q; the amount of shearing stress on the small area is therefore / g . r . di , dr = — r"^ . do . dr; r, CHAl'. XVIII.] COMBINED STRESSES. 343 and its moment round the axis, found by multiplying this quantity by r, is therefore ^~ r" ir .di . dr). The quantity within brackets is the small area, and r is its distance from the centre; r^{r . di . dr) is therefore the moment of inertia of the small area about the centre ; summing for every such small area, we have the moment of resistance to torsion for the cylinder M = '--K=^-2I,; (19) where K is the moment of inertia of the surface about the centre, and I^ is the moment of iner about a diameter. We therefore have centre, and I^ is the moment of inertia of the same surface where di is the diameter of the cylinder. For a hollow cylinder, let r^ and Vi, d^ and di, be the internal and external radii or diameters, as the case may be ; let /'o and Iq be the moment of inertia about a diameter of a cylinder equal in radius to Tq and ?'i respectively, then I^ - I\ is the moment of inertia about a diameter of the rinar under consideration ; we have therefore M = ?/ (/„ - /;) = ?^(i7rr,* - iTrn*) ' 1 ^1 = 5/!il-!i' = . 196/1^^'. (21) 2 ri d^ ^ ' Comparing : these equations with those on page 266, we find that for equal values of the limiting stress /, the resistance of a cylinder, solid or hollow, to wrenching is double its resistance to breaking across. The working values of the limiting stress /, suitable for shafts, as given by Eankine, are cast iron, / = 5000 lbs. per square inch ; wrought iron, /= 9000 lbs. per square inch. 344 APPLIED MECHANICS. [CHAP, XVIII. For a cross-section whicli is not circular, the above formulae are inapplicable, since the ratio - is no longer constant. For a square shaft M. de St. Venant gives as the moment of resis- tance to torsion M = 0-28 1/^^ (22) Angle of torsion of an uniform cylindrical shaft. — Let x be the length of the shaft, and i the angle in circular measure through which the one end has turned relatively to the other ; then, since the angle of torsion per unit length is constant, we have, from equation (17), — = i = i-=-/-- A^tl.J^ (C)'X\ dx X Cr Or,' Cr^ Cd\' ^^^ If/ be the working resistance of the material to shearing, we have the same angle, whether the shaft be solid or hollow ; the values of /and of C for cast and wrought iron have already been stated, and for cast iron, ^'=3^0^' ^'^^^ 1 X wrought iron, *=^vm"r' (^^Z*) where i is the angle in circular measure through which the one end of a shaft of length x and diameter d^ has turned relatively to the other end, when the working strain has been produced ; the coefficient for cast iron is somewhat uncertain. When subjected to J/ any twisting moment not greater than the proof moment, we liave for a solid shaft (equation (20)) r, 2 ' r 2 TT/'i and from equation (23) i- ^ *" Cr For a liollow shaft, similarly . q^x 'IMx _ , Mx ,^_. ' ' "-2 CW -,:..')• ^''> CHAP. XVIII.] COMBINED STRESSES. 345 If we make x=l, or, what is the same thins^, if the distance between the two cross-sections which we consider is unity, the stiffness of the shaft will be measured by the I'eciprocal of i, i being the angle in circular measure through which the two cross-sections have turned relatively to each other, when the skin has been brought to the proof strain. For two shafts of the same length and material, but of different diameters, we see from equation (25) that the twisting moment to be applied to each, in order that both may be turned through the same angle of torsion, is proportional to the fourth power of the diameter, the proof stress not being in any case exceeded ; thus »= 10-2^= 10-2^, or 3^, = ^, (27) Bending and Torsion Combined. Let the shaft shown in fig. 4 be acted upon by a bending load and a pair of equal twisting couples ; and at the point H let Ml be the moment of the first, and Mo the moment of the second ; then in order to find the amount and direction of the greatest principal stress, we require to combine the greatest direct stress due to bending with the greatest shearing stress due to twisting ; this is done by the method of the ellipse of stress. At the point H, let ]> be the intensity of thrust (or tension) due to the bending moment il/i, and q the intensity of shearing stress due to the twisting couple M^ ; then we have 4.1/, 2J/. P = — J 1 = — i- (^^) Let pi be the greatest intensity of stress (thrust) at the point ; then fig. 1, Ch XV, shows the construction required to find its amount ; in that figure OL =^:», OR' = q, and we have :P =0M ^ MrJ^^ +k^+,/-- (29) the greatest intensity of shearing stress is represented by MR=J^+q'; (30) 346 APPLIED MECHANICS, [chap. XVIII. and the angle made by the greatest stress pi with the axis of the shaft is given by the equation tan 26 = JRL ML 2q P (31) By substituting for 2^ and q the vahies given in equation 28, we have and for the greatest intensity of shearing stress 2 ME (32) (33) A very important example of this principle is that of a shaft with a crank attached ; in this case we have a force applied to the centre of the crank pin, and resisted by the equal and opposite force at the bearing ^S'. If F represent the force, then the moment of the couple is (fig. 4) M=P .SP; (34) this couple may be resolved into two couples, one a bending couple M, = P .N8=McoQj^ t\\e other a twisting couple, M,^P .NP = M sin /. t if, (35) (36> Cl S) Fi-. 4. The greatest intensity of stress is found by equation 32, 2 Pi =^3 (M cos J + ^3P cos'j + M^ sin^y) 2 2 = — ; {M cos j + M) -- — - M (I + cos ;■) TTVi = -^ M (1 + cos J) ^r. (37> CHAP. XVIII.] COMBINED STRESSES, 847 If instead of px we put / the resistance to torsion or thrust (the smaller), we get 5*1 d^ = — M{1 + cos j) ; (38) which enables us to calculate the diameter required for the shaft. If we put /, for the greatest intensity of the shearing stress, we have d^=^-^M; (39) which also enables us to calculate the diameter required ; and the greater of the two, one got from equation 138, the other from equation 39, is to be adopted. The angle made by the principal stress with the axis of the shaft is given by equation 31, ta„2S-?'' = ^.?^-^ = tani; .-. - i. (40) p il/i COS 7 "^ 2 ^ ' Thrust or Tension combined with Torsion. Let the shaft shown in fig. 5 be acted upon by a thrust (or tension) P and a pair of twisting couples of moment M\ the stress due to P is uniformly distributed, and that due to M is greatest at the skin ; the greatest intensity of stress will therefore be at the skin. If under thrust, the \p length of the shaft is to be so short compared with its fr^ diameter, that the bending action need not be taken '^^^ into account. At the point H we have a thrust (or P 2M tension) p = — ^, and a shearing stress q = • — ^\ pro- ceeding as at fig. 1, Ch. XY, we have X 0^/ = i-2~- («) $^ ^^-J?- = J(a^(iJ- ^-) Fig. The greatest intensity of thrust (or tension) is lh = OM+ MR^ (43) the angle B made by p^. with the axis of the shaft is given by the equation tan20=S=?^. (44) ML p The greatest intensity of shearing stress is MB. 348 AITLIED MECHANICS. [chap. XVIII. Thrust and Induced Bending. When thrust is applied to a pillar or strut whose length is great compared with its diameter, it will collapse not by direct crushing but by bending and breaking across. Let a long thin vertical bar, originally straight, be deflected to an extent not greater than the proof deflection by the appli- cation of a horizontal external force applied, say at its middle, while the ends are guided so that they cannot move laterally, and let it be held in that position ; it will then have a form such as is shown in fig. 6 ; let the load P he now applied, then when the restraint is withdrawn, the bar will tend to assume its original vertical form, it will remain neutral, or it will collapse according to the amount of P ; that is to say, if the moment of P relatively to the centre of the bar, viz. P . v, is less than the moment of resistance of the bar to bend- ing, the bar will tend to right itself. The stress on the cross-section AB consists of one part 2^' due to the load P, and another part p' due to the bending which takes place in the direction in which the pillar is most flexible ; since P is uniformly distributed, we have P P S (45) where S represents the sectional area of the bar ; by equation given on page 240, we have M = ivp'hh', and since M = Pv (46) where h is the smaller, and h is the larger diameter, when these are unequal ; the proof deflection v, Ch. XVI, equation (14rt), is directly proportional to the square of the length and inversely proportional to the depth ; that is to say that is /- Pl^ Pt^ v:r.~, and /' x ^^ ac -^, ; ,, , /* , p p" (^ N' » ccp —■. therefore —i cc (47) that is, the additional stress due to bending is to the stress due CHAP. XVIII.] COMBINED STRESSES. 349 to the direct thrust, as the square of the proportion in which the length of the pillar exceeds the least diameter. The total intensity of stress p + p" must not exceed the strength of the material ; equating that intensity to /, the strength of the material, we have f = p + 1i" = p + o.}/ P F " S 1 + a P = fs . i-'^'T (48) 1' Fig. 7. where a is a constant coefficient to be determined by experi- ment. The above investigation is due to Tredgold, and values of the coefficient a, as determined by Professor Gordon, are given below. For a strut or pillar securely fixed at the ends : — Wrought-iron solid rectangular section, . . . a = yi^Vir' *Angle, channel, cruciform and T-iron (see fig. 8), a = ^-5-5-. Cast-iron, hollow, cylinder, a = tItt* fFor timber struts, oak and pine, « = t-^tt* A pillar rounded at both ends is as flexible as one of double the length fixed at the ends ; so that fS fS 1 + a 21V- h 1 + 4« (49) The strength of a pillar fixed at one end and rounded at the other is a mean between that of a strut fixed at both ends and one rounded at both ends. * As deduced by Unwin. t As deduced by "Weisbach. 350 APPLIED MECHANICS. [chap. xvin. The values of / in lbs. per square inch are given by Kankine as follows : — Breaking load. Proof load. Working load. Wrought-iron solid > rectangular section/ 36,000 18,000 6,000 to 9,000 Wrought-iron cell 27,000 13.500 4.500 ,, 7,000 Cast-iron cylinders 80,000 26,700 13,300 ,, 20,000 British oak, dry 10,000 — 1,000 American oak, dry 6,000 — 600 Red pine and larch, dry 5.400 — 550 For green timber the values of/ should be halved. A pillar or strut securely fixed at both ends corresponds with a beam fixed at the ends (fig. 6, Ch. XVI) ; the points of fracture are the points where in that figure the bending moment is greatest, viz., at the centre and at the ends. A pillar fixed at one end corresponds with the beam shown in fig. 4, Ch. XVI ; the points of fracture being at the fixed end, and at about one- third of the length from the rounded end. A pillar rounded at both ends corresponds with the beam shown in fig. 12, Ch. VI, the point of fracture being at the middle of the length. The following table gives the results of the above formulae for pillars of wrought and cast iron, whose diameter and lengths are in different proportions, and whose ends are securely fixed : — h p Breaking load, lbs. per sq. in.=-=^ h p Breaking load, lbs. per sq.in. = -^ Wrought-iron solid rectangular section. Cast-iron hol- low Cylinder Wrought-iron solid rectangular section. Cast-iron hol- low Cylinder. 10 15 20 26 4 34,840 33.490 31.765 29,230 64,000 51,200 40,000 29,230 30 35 40 50 27,700 25.56c 23,480 19,640 24,620 19,700 16.000 11.030 €HAP. XVIII.] COMBINED STRESSES. 351 From this table it is seen that, so far as the ultimate strength of such pillars is concerned, cast iron is stronger tlian, as strong as, or less strong than, wrought iron when the proporlion , is less than, equal to, or greater than 26-4 ; this result was first pointed out by Professor Gordon. For struts in wrought-iron framework, such cross-sections as are shown in fig. 8 may be chosen on account of their stiffness ; these are called T, angle, channel, and cruciform ra -p^i^ tp^ iron, respectively ; in fix I ing the proportion 7 for Fig- 8. such sections, the value of h is to be taken as the least diameter ; this is marked in the diagram. Since, however, such cross-sections are not made of very large dimensions, struts above a certain size require to be built ; in this case they usually consist of four thin plates forming a square and connected at the corners by angle irons ; of two thin plates held parallel and at a fixed distance apart, by means of a web and angle irons, or by a lattice work of small diagonal bars ; or of two f -irons or channel-irons, with the ribs turned towards each other, and held by a lattice work of small diagonal bars as in the case just stated. Examples. •5. A lattice girder 80 feet span bears a load of 100 tons uniformly distributed; depth from centre to centre of flanges 6 feet, and f — 4 tons per square inch ; the breadth of the flange is 1 ft. 9 in., and is constant ; the thickness, however, varies. Taking the provisional breadth as 1ft. 9 in., find how much this has to be increased so as to allow for the weight of the beam itself. Jfo= 12,000 inch-tons; J/io= 11,250; Jfao = 9000 ; Jfso = 5250 ; Mm = ; since the flange is thin, we have M = fhh't = 4 x 72 x 21 k (!, and we obtain <o=2in., <io = 2 in., U,^ = 1-5 in., <3o = 1 in., and <4o = 0-5 in. say; the average being 1*2 in. nearly. Taking this as the thickness of the flanges, and allowing for bracing, we obtain 10 tons nearlj'^ as the weight of the provisional girder ; so that V = 21 inches, B' = 10 tons, and W = 100 tons ; from which we readily find J = 24 inches, B =\\ tons, and W - \\\ tons for the actual girder. 6. A water wheel of 20 horse power makes 5 devolutions per minute : find the diameter suitable for the malleable iron shaft which transmits this force. For each revolution 132,000 ft. -lbs. of work are performed ; this is equivalent to 21,008 lbs. acting on a wheel of radius one foot, and we have iL = 21,008 ft.-lbs. = 252,096 inch-lbs. 252,096 = -196 x 9000 rfi^ = 1764 «^i=; .-. d = 5-23 inches. 352 APPLIED MECHANICS. [CHAP. XYIII. If this shaft be 12 feet long, what is its angle of torsion when the working moment as above is applied? Take /= 9000, and C = 9,000,000; then 1 144 » = -TTT^ Ti^ = "05507 = 3" 9'. 500 5-23 7. The crank shaft of an engine is 5 inches diameter; the distance from the centre of the bearing to the point opposite the centre of the crank pin, NS in fig. 4, is 12 inches; the half stroke, NP in figure, is 16 inches; and the pressure applied to the crank pin is 5000 lbs. Find the greatest intensity of thrust, tension, and shearing stress ; and 6 the angle made by the line of principal stress with the axis of the shaft, FS = 20 inches ; .•. M= 100000 inch-lbs. ; J/i = 60000, and Mi = 80000. 5-1 pi = -— 100000 (1 + f ) = 6530 lbs. per square inch, the greatest intensity of thrust and of tension, at the bearing, the one being at the one side and the other being at the other side of the shaft. The greatest intensity of shearing stress is ytt 100000 = 4080 lbs. per square inch. The angle e = -^ = 27°. 8. A shaft 8 inches diameter is subjected to a thrust of 100 tons uniformly distributed over its two ends, and a twisting moment of 30 foot-tons. Find the greatest intensity of thrust and shearing stress, and the angle made by the line of principal stress with the axis of the shaft, 100 « 720 „ ,„ » = = 1-99 tons ; q = = 3'o8 tons per square inch ; ^ 50-26 ^ 201-04 ^ ^ /ji = 1-0 + \/-99 + 12*81 = 4-71 tons per square inch ; the greatest intensity of thrust ; the greatest intensity of shearing stress is 3-71 tons per square inch ; and tan 20 = ^ = 3*6 ; 6 = 37°. 9. The diameter of one shaft is double that of another of the same material ; the smaller gave way when subjected to a twisting moment of 2 ft. -tons. What twisting moment will be required to wrench the other? Aus. M = 16 ft. -tons. 10. A shaft 12 feet long and 6 inches diameter is subjected to a twisting moment of 16 ft. -tons, and the two ends are thus twisted through a certain angle; a second shaft of the same material, 16 feet long and 9 inches diameter, is twisted so that its angle of torsion is exactly the same as that of the first : find the twisting moment required to do this, „ 192 X 144 ,^ ^ M X 192 therefore 192j<J44 ^ ^_xJ92 ^jjgj.^.f^j^,j ^ ^ -29 i„..tons = 60-75 ft.-tons. 1296 6561 CHAP. XVIII.] COMBINED STRESSES. 353 11. What thickness of metal is required for a cast-iron hollow sliaft, 10 inches outer diameter, so as to resist a twisting moment of 10 ft. -tons? 10* - ^/„' A>is. M = 120 X 2240 = -196 x 5000 rfu* = 7258 ; therefore d^ = 9-23, and t'ne thickness required is 0'4 inch. 12. A malleable iron shaft 20 feet long and 6 inches diameter is subjected to a moment which twists the ends through an angle of 2' ; taking C the coefficient of transverse ehisticity as 9,000,000, find /, the stress at the skin, 2/* 240 i = -0349 = — — • : therefore / = 3926 lbs. per square inch. 9 X 10« X 6 -^ f H 13. The inner and outer diameters of a hollow steel shaft are 10 and 12 inches, and / = 6 tons per square inch is the working value of the resistance to shearing. What is tiie twisting moment this shaft is capable of transmitting? 12* - 10' M -- -196 X 6 — — ,^ = 10.')2 inch-tons. 14. A cast-iron column, securely fixed at the ends, external diameter 8 inches, length 20 feet, is to bear a steady load of 30 tons. Find the thickness of metal required. Here - = 30, and for that proportion the breaking load is 24,620 lbs. per //. square inch ; taking g for a factor of safety, we get the working stress /= 4100 lbs. = 1*8 tons per square inch; the area of metal required is there- fore 17 inches; this gives 6-7 inches for the internal diameter, or finch nearly for the thickness of metal. Since an allowance has to be made for slight irregularities in casting, the thickness of metal should be 1 inch. 15. Find the working load for a cast-iron pillar 12 inches diameter, 40 feet long, metal 1 inch thick, taking 6 as the factor of safety. Here - = 40 ; 6' = 34'6 square inches ; /" = 1'2 tons per square inch ; h and the steady working load is 42 tons nearly, both ends being securely fixed ; but if the load is such as to cause considerable vibration, from one-half to two-thirds of this amount may be taken. 16. What is the crusliing load for a malleable iron bar 6 in. x Sin. x 10 feet long, securely fixed at one end ? I . . S = 18 square inches ; - = 40, and the ultimate stress corresponding is n 10"5 tons per square inch, wlien fixed at both ends; when rounded at both ends, this reduces to o-l tons; when fixed at one end and rounded at the other, the result is the mean of these quantities, viz. 7*8 tons per square inch. The crushing load is therefore 140 tons steadily applied. 2 A .^54 APPLIED MECHANICS. [CHAP. XIX. 17. Find the working strength of a strut formed of channel irons | inch thick. 6 inches broad, width of each flange (outside) 25 inches, and length 6 feet, fixed securely at both ends. Here 5=5 square inches ; - = 29 nearly, and the working strength is h \\ tons per square inch, or 7 tons nearly. 18. What is the working load for a strut of seasoned American oak firmly fixed at the ends, 20 feet long, and 1 foot square ? Ans. P = 33,000 lbs. nearly, or Hi tons. 19. A strut of red pine whose ends are to be well fixed, is to be 4 inches thick and 6 feet long ; the thrust applied to its ends is calculated to be 4 tons. Find the breadth required. Iiy- 324 \h) ^ "^ 27o S = F = 4 = 37 square inches nearly. and the breadth required is therefore 9^ inches nearly. CHAPTER XIX. DESIGN OF LONG STEEL STRUTS. For long struts in wrought iron and steel, the double T-sec- tion is of great importance, as most struts, where merely booms of girder or themselves girders, can be hlockcd-out into such a form. The extreme shapes where the flanges are equal, and where one flange is zero, that is, double T -sections equal above and below, and the T-iron sections are especially important. The angle-iron too is often constrained to bend like a T-iron by being braced in other directions. We will establish a general expression from which to select an approximate one suitable for designing struts. The sizes are shown clearly on flg. 1. E<j. 9, p. 338. 25 + qc h' 2a + qc h' a + b + qc 2 a + o + qc 2 CHAr. XIX.] LONG STEEL STRUTS. 355 and taking a mean of the term* in ^- repeated twice, once with each of the evident values of 8, for symmetry, we have /„ = afm^ + hM + J^ {aP + hP + qtc^) _thf^ \{2a + qc) (2b + gcf + i(2& + qc){2a + qcf 4 (rt + 6 + qcy - Iqdh'' + -^\qcH + JL (a + &) ^' 4 a + + qc = *' (2« *,c)&.gc) _ _ , ^ ^^ _^ ^ ^ 4 a + h + qc /o = .4 ( //iyi + t;^ ) - \qctli'{li' + 0- (1) /o = ^('m7i + ^^-^-M'c. (2) /„ = ^r/z,?i - ^^'A,'^ nearly. (3) Here we have (3) a remarkable expression for the minimum moment of inertia of the double T-section, the degree of approxi- mation clearly shown by comparison with the exact expression (2). The approximate expression may be given in words thus : — The moment of inertia of a double T-section is sensibly equal to its area multiplied by the product of the two segments into which the centre of gravity divides the depth bctiueen the middle points of its flanges, from which is to be deducted twice the moment of iTiertia of the web considered separately, and as reaching from centre to centre of the flanges. Cor. — For a given area the value of /o and therefore of the stiffness is a maximum when m = 71, that is, with equal Hanges. 2a 2 356 APPLIED MECHANICS. [chap. XIX. By a further reduction we have th'^ fl2ah + 4(« + b)qc + q-r „ .A t^ , , ^ , ,, a + b + qc th'- Vlnh + 4 (« + h) qc + q^c- 12' nearly. (4a) + 2i!' + &c. a + b + qc By an exactly similar reduction t]i-fl2a fi+4(a + ^)qh + qVi' ~ 12 V, a + ^ + qh Or, for the double T-section thickness of Metal constant, , th" 12aZ/ + 4(« + b)h + 1r ^" = 12- • — ^rrm . ' ^^^^ is a sufficient approximate value for designincr struts, even with h for h'. Economical double "X-scction of VMifwm drenffth to resist bending. In fig. 1 suppose the two skins b and a to simultaneously come to their working or proof strengths /j and /„, and putting ft = fb ■ fa their ratio, then p wQl sensibly be equal to the ratio ./"ft :/'„, that is, 7i : on. Just as for f/en em I stiffness h or h' must be a suitable fraction of the span, so also for loccd stiffness must also t\ the thickness of the web, be pi„ i a suitable fraction r of h or h'. The resistance to bending is, by equation (3), t'h''l 6 n < r / > : K-f- ±._.L.;_.Lo' SA n or -7 = Ani - ./ 6 But h' and pA' il/ 1 + p' 1- + p' so that rh'\\ + p) i + jO Op 1 + p 6p Equating zero to the differential coefficient with respect to li\ A _ rh'\\ + p) 1 + p " = ^^ dK /'ft :2/> CHAr. XIX.] LONG STEEL STKUTS. 357 Or the area of the web measured from centre to centre of the flanges, namely, ''"'-iil'.r' <5) makes the strength a maximum. Its vakie is J/ _ 2 Ah' _ (1 + p)rh'' _ (1 -^ p) t'h'^ f\ ~3"(lT7)~ Zp ~ 3p (6) For homogeneous strength p = 1, and the section has then equal flanges. For wrought iron p = 4 : 5 commonly, and is never less than 1 : 2 for steel or steely-iron. For values of p from i to 1 the coetflcient 2p -f (1 + p)* is \ sensibly constant. Hence the condition given by (5) that tlie duuhle "X -section of uniform strength shall he the most economical for a given degree of local stiffness is that the area of the nrb measured from centre to centre of the flanges shall he half of the toted area. Substituting p = 1 in (5) and (6) we get for the most economical section of equal flanges for a given degree of local stiffness the two conditions th' = -A and ~ = — - = -/h-. (7a, 1h) 2 f I o 6 The second may be written M = 4 {^fht'h'-) where the expression in brackets is the share of the resistance to bending exerted by the web measured as before from centre to centre of the flanges. Hence the two conditions (7) for designing the economical cross-section with equal flanges for a given degree of local stiffness can readily be remembered thus — Design the section so that the iveh measuring from centre to centre of the flanges shcdl he half the area and take I'.p a fourth of tJie resistance to hending. For the purpose of direct design, however, (7j is to be modified by putting f'b = —fb ; h' = {h - t); t = sh, and f = rh, when we lm\e, where fb is the lesser strength, 2 2 1/ -rh''=^,r{l-.s)%'=^- (8) bo fb •358 APPLIED MECHANICS. [CHAP. XIX. For comparison put Mi in (6) and M, in (7/^), when Ml - 3f, 1-p 31 1 - M, l-p ,. ^ is the fractional excess of strength of the one section over the other where both have the same area A, the same depth li, or same r/eneral stiffness, and the same vahie of r = t' ~ A', or same luca/ stiffness. In each section the *' web " is half the total area and each is the most economical of its kind. For the ordinary- value /» = 4. the expression (9) is 10 per cent. To design the economical double T-section of uniform strength for a given degree of local stiffness. From the required strength, deduct the percentage indicated by (9), and design the economical section with flanges equal, of the required degree of local stiffness by (8) ; then borrow from the flange a a por- tion, and add it to the flange b, till (a + ^h) : (b + gA) is in the ratio p : 1. Care must be taken that the weaker strength fb in (8; is used always for the compressive strength of wrought iron and steel. Then, too, it must be the weaker or thrust flange that is increased at the expense of the other. Also the beam is to be placed with the proper flange up, and there must be no possibility of the bending reversing. For beams a saving of 10 per cent, may be effected. Such sections are not suited for struts for which the flanges are best to be equal, the bending being equally like to be one way or the other. The T-section and angle-iron constrained to bend like a T-section are often used as struts, as they suit constructional purposes. The angle-iron if not braced so as to bend like a T-iron is very weak as a strut, as it tends to open flat and bend over. These points are best illustrated by the numerical examples to follow. Double T-section vnth equal flanges, fig. 2. — In equation (3) /y = Aran - jit'h'^; we must now put a = b, VI = n = }Ji', t' = qt, and we have A = t{qh' + -lb), (10a) ^0 = -jV ('/''-' + t)6), (10&) CHAr. XIX. J LONG STEEL STRUTS. 359 and the radius sq. of gyration A 12 ^/t^ + 2i ' ^ ^ where ^ is the ratio of the thickness of the web to the common thickness of the flanges. J -section and angle-iron constrained to bend like it (fig. 2, iv). — In equation (4«) put « = 0, and now c = h\ A=t(qc + h). (lla) _ t}i- q-c" + 'khqc _ t'Ti^ qc + 4h ^ ty q c + 4h " " 12 qc + b ~ "12" qc + b ~ 12 qc + b' ^ ■ /o qc^ qc + 45 A 12 qc + b (lie) Angle-iron itnsupported or a crux-section (fig. 2, v.). — Let t' = t, for angle-irons are always of one thickness being rolled from a plate of one thickness. The section is readily identified with a rectangle of breadth B and depth H, where H=hiimd = bh^ yh- + &' /I , 1 and B = t (sec 6 + cosec d) = t ^/h- + ^'^ [J"^l, so that putting Iq = -yj^'^^ 3,nd reducing, we get the values, of little practical use, given in line V. of the accompanying table. In the Table the struts are constrained to bend only about the dot-and-dash lines. Line /. is box or double channel section. Line //. is double T -section of uniform metal. Line ///. has the web half as thick as the flanges ; these lines are got by putting 2, 1, and | for q in equation 10. Line IV. is the uniform metal T or angle-iron constrained to bend like a J-iron. Unity is put for q in equation 11. The approximation is very close, as the terms in t^ and t^ vanish together, in the exact expression expanded in powers of t. It is of great practical use, and is not given in the Table, Eankine's Civil Enyiiutring, p. 523, where there only appears the value in •our Table at Line V. for unconstrained angle-iron a section quite unsuited for struts. 360 APPLIED MECHANICS. [CHAl*. XIX. Table of Akeas, Moments of Inertia, and Uadh Square OF Gyration. SECTIONS. h 7. oL ^ ./;.... jj T ■■■ .. , 'h n V- --- - ^ ■--ir- cz: ...i5.... ! M', ft (■ 2 n K.— Tf—" U--6 ^ _.H.Ht; I ^(^'^*)! ■n; f=<? A <• <(2A' + 26) : t(/t'+2i): <(iA'+2i); ^■1 ', t{h + i) ; 'A-"(2A- + 6J) '£'(/*■ + 6i); j'7(K+6A) M'^ A' + 46 72" ■ ~h'~Vb ' th-h^ h-\-h 12 ' Ti^b- h for A' h- h+U 12 A + A A' A + 66 12 ■ A -r 27/" 12 ' T+ 4'/ ■ A' A + 4* 12 ■ (A + bf A2 _J2_ 12 ' A' + r-" Fio. 2. CHAP. XIX.] LONG STEEL STUUTS. 361 Kankine-CJordon Fokmula. Eankiiie first proved that Gordon's formula, equation 49, Ch. XVIII, was rational (see Eankine, C. E., p. 523). He then substituted v for .-^ , because that is the relation of the radius square of gyration to the depth for the solid rectangular sections upon which the actual experiments were made. He then says the formula is general, only for skeleton sections the proper value of i- must he used. To be of practical use the values of ^- like those in the above table must not contain t, the thickness of the metal, otherwise there would result a quadratic equation to determine its value, since t is generally the unknown quantity in this formula. We prefer to present the formula suitable to a strut with hinged ends, because, as Fidler justly says, in wrought-iron and steel struts, the ends, at best, can only be imperfectly fixed in direction, as the other portions of the steel, in a bridge, for in- stance, to which such struts are fixed, are themselves so elastic, that, in yielding to the loads imposed on them, the end of the struts do deflect. Fidler allows for this imperfect fixing by supposing the virtual length of such a strut to be yV^hs of its actual length. But every strut imperfectly fixed at the ends can be judged on its merits, and a deduction made from the length never more than ^Vths. As the length in the Rankine formula is in inches, it is always wise in making the above deduction to so do it as to leave the virtual length expressed in a round number of inches. The reason that I is in inches is because h and i are always in inches. The Rankine formula then for ends hinged is T „ , 4 tons X A /io \ Iron, F tons = ,, (12a) 1 + -^— - -^ ^ 9 -i Mild Steel, p tons = ^^-^^^^^ ■ (12h) 1 + — '— - ^ ^ 7 ^-2 These, of course, are only good average values. 362 APPLIED MECHANICS. [CIIAP. XIX. For a practical knowledge of this formula for designing long struts Fidler's Treatise on Bridge Design should be studied. The formula for many years was thought to be too general, and, in America, most eminent engineers made formulae of their own, of like form, but varied to suit the different cross-sections, in a way that led to great confusion. Fidler has, in the most skilful way, arranged the Kankine formula, and tabulated results which agree, with great exactness, with the results of the more recent exhaustive and elaborate experiments made on struts with the skeleton sections. His tabulated results agree also with the results of the special formuke for special sections. In Britain again many writers put forward a modification of Euler's formula. In our opinion Euler's formuka is quite unsuited for struts in engineering steel-work, Euler contem- plates a load which will just not bend the strut. Now the whole duty of a steel strut in a structure is to resist bending, and bend it will, no matter how it is designed, for it will take a set to one side or other with its own weight if with nothing else, for the weight of the immense struts in modern structures is very considerable. Also heat will distort a strut ; and in riveted work the straining of the adjacent members sends bending strains along a strut, which must be designed with an ample radius of gyration, and it is undesirable to talk about a load which will just not bend it. Fidler has done great service in vindicating the supreme position of the Eankine-Gordon formula as a master formula for tentative design, though, of course, experiments on the actual strut of any special section will supersede all formula^, provided the experiments are on a proper scale, and in this matter no perfunctory experiment in special struts can compare in im- portance with the life-history of such a strut in some large structure. We quote a part of two of the tables given in Fidler's Practical Treatise on Bridrje Construction. These tables gi\e the coetticient for stiffening long struts, that is, a multiplier for converting the net area of the strut, where quite short, into tlie gross area to resist both thrust and induced bending. In a good design, this multiplier should not exceed 1'5, that is, the extra volume of steel for stiffening the strut sliould not be more tban 50 per cent, of the net volume. Otlierwise the main cross- dimension h should be increased to give a bigger radius of gyration. CHAP. XIX.] LONG STEEL STRUTS. 363 For section (fig. 2 n.) with h = 2I> = IGt and ft, = 4 tons per square inch. Ends fixed, . . 5 feet. 10 feet. 20 feet. 30 feet. 40 feet. Ends hinged. 3 - 6 ., 12 ,, 18 ,. 24 .. /' tons 8 /in sq ill I 04 I-I4 I '45 I -So 2'l8 l6 4 I'02 107 1-25 I 49 174 32 8 I 01 I 04 ri4 128 1-44 48 12 lOI 103 i-io 120 1-32 64 16 I 01 102 i-oS i-i6 I 26 80 20 I 00 I-QI I "06 113 I 21 For section (fig. 2 L) with h = |7; -= 16;! and /j = 4 tons per S(}uare inch. Ends fixed, . . 5 feet. 10 feet. 20 feet. 30 feet. 40 feet. Ends hingeci, 3 -. 6 „ 12 „ 18 „ 24 ., P tons. 16 A" sq.in, 4 I 04 I-I3 1-42 176 2-13 32 8 102 I 07 1-24 1-46 171 48 12 roi 105 117 1-34 1-52 64 16 I'OI 104 I-I3 1-27 1-42 So 20 I 01 103 I'll 1-22 1-35 q6 24 i-oi I "02 i-og i-ig 1-31 Examples. 1 . Compare the strength to resist bendinf,' of four double T cross-sections, all having the common area of 18 square inches, and all having the common local stiffness afforded by the constant thickness of the metal being a tenth part of the depth of the section. The four sections are respectively to have deptlis of 13-4, 12, 10, and 8 inches. The fig. 3 shows the four sections drawn to scale, and the calculations made by the exact tabular method of Chapter XIV, fig. 3. The section iii. has -^ = 49-2 a maximum, and it will be seen that the web is ialf of the area. It is the strongest section with area 18 square inches, and the local stiffness due to ( = -,\,-/'- 364 APPLIED MECHANICS. [chap. XIX. Table of COMPAPw'vTIVE h^TJtENGinS OF FOUR DOUBLE T CKU^S-SECTIONS, WITH Thickness of Metal iVth of the depth. IIai.f-Sections. 5y^ X } X 6 M^f Area. ^ -> 6 7 2. o Area /v=_</j^4: re •^ ^ 1^^ Aroa_ /i=/o .."^o J^ 1 vm L^o [ZIZU^EZISj /8. IK _._ o noo-8 0-0 216-0 110-6 0-0 25-0 64-0 0-0 C4-0 32-8 0-0 300-8 X i X 1-34 = 134-33 i /= 134-33 f I_ 6^7 = 40-1 105-4 X j^ X 2-70 = 94-86 j Jfj / 110-6 X ^ X 1-20 = 44-24 j 7"" "T = 46-4 \1= 139-10 61-0 X ^ X 5-00 - 101-67 64-0 X ^ X 1-00= 21-33 \1= 123-00 31-2 X i X 8-05= 83-80 32-8 X ^ X 0-SO = 8-80 \I= 92-60 Mi I f b = 49 2 ~y "4 = 46-3 Fig. 3. CHAP. XIX.] LONG STEEL STRUTS. 365 The section ii. has the web tliiee-foiirths of the area if it be measured from <entre to centre of the flanges, and is the stijf'est section of area 18 square inches, and local stiffness due to t = y^h. For substituting from (lOrt) into (lOi), we have 7o = "iVA'- (3^ - 2l/i'), which, with A constant and t' a constant r"' fraction of h', give the variable factor /It = SAh'- — 2rh'^. The ditt'erential coefficient with respect to li' equated to zero gives QAh' — 8r/i'^ = 0, or the area of web t'h' = ^^A. See fig. 3, ii. 2. Design the economical double T section with equal flanges in wrought iron lor M'hich fb = 4 tons per square inch. Its strength to be M = 196'8 ft. -tons, and to have the local stift'ness afforded by the thickness of the web, t' being a tenth of its depth h' . The thickness of the flanges to be t = t', that is, " unifoi'm metal.'' We must plan the web measured from centre to centre of the flanges to have its area t'h' a half of the total area J., and its share in the strength ^f'bt'h'- to be a fourth of IVl- This is done bj- putting in (8) M -r fh = 49-2, and s = r = iV when -I . TV(-nrA? = 49-2 ; A» = 1013 ; // = 10. So that t = t' = 1, h' = 9, and c = 8, while t'h' = 9, and so A = 18 square inches. The upper half of this section is shown at fig. 3, iii., and the whole sectkm at fig. 2, ii. 3. Design the economical double T section of uniform strength for wrought iron with y's = 4 and /„ = 5 tons per square inch. For local stiffness the web is to have t' — -xiyh , and the thickness of the flanges t = t' . The required moment of resistance to bending is M = 2I8'6 inch tons. In (9) put p = f , and we tind that the section will be 10 per cent, stronger than if its flanges were equal. From 218*6 deduct 10 per cent., and it leaves 196-8 ft. -tons. In Ex. 2 we have already designed the section with equal flanges and found the values there quoted. It only remains to borrow from one flange and add to the other, so that, with a + l> stUl equal to 10, we get a -{■ he : b + ^c equal to 4 : 5. And « + 4 4 a + b=lO; ——■ = ^ ; + 4 r> or ff = 4 and b = 6 inches. The half-section is seen at fig. 3, iii., if an inch of breadth be taken off the upper flange and added to the lower. The section may now be checked by the tabular method at fig. 5, Ch. XIV, when, with the axis at the lower skin of b, the results are S = 18, Gi, = 81, I„ = 606, and y = 4*5, so that h' is exactly divided as 4 to 5, although h is only nearly divided in that ratio ; further /o = 242, and multiplying this by the smaller of the quantities 4 tons-^ 4i in. or 5 tons -i- 5j in., namely, the first, we get the exact value of M = 215, almost exactly what was prescribed. 4. Design a double T section for a wrought-iron strut 24 feet long and hinged at the ends, the load being 64 tons. The depth should be A = 12 inches for stift'- ness, and with the breadth of the flanges each 6 inches, and flanges and web of one thickness t as in fig. 2, ii., we iiave ., A- h + 6b '--l-2h^2b-^'' I = 288 in. ; /= 4 tons per square inch. A = {2b + h)t = 24< nearly. then being the same degree of approximation as i. 366 APPLIED MECHANICS. [CHAP. XIX. By (12fl) fA 4 X 24« 9000 i* < = 1 inch nearly, but A' = 2it nearly, or A = 22t = 22 square inches. 5. Design the same strut, using Fidler's table, corresponding to fig. 2, ii. ^0 = 64 -f 4 = 16 square inches is the net area. The multiplier is the last in the second last line, that is, 1*26. The gross area A - 1*26 x 16 = 20 square inches. As the relative dimensions on the table are ^ = 2* = 16<, so that A = {h- 2t)t + 2bt = 30(=, and therefore <* = 20 -^ 30 = -67, or (=-816, and A = 16< = 13 inches. This is the nearest that can be done with the table. It is evident that with /i = 12 inches instead of 13 incht-s, then A must be greater, say, 22 square inches. 6. A square box steel strut, virtual length 30 feet, is to take a thrust of 80 tons. Design the section. For stiffness b = h = \& inches might be suitable (fig. 2, i.). „ h- h - 3A 1-=---; r = 44 ; A =2t(b ^- h) = %U ; / = 360 ; 12 A + i f-^ o. 7 X 64i! TTTyr " ^^' 7000 i- As the divisor 1-42 indicates that the excess metal to stiffen the strut is not over 50 per cent., we can feel assured that h = 16 was ample enough. Proceeding t = a- lo = \ inch, and A = \& square inches. 7. Design a T iron or angle iron constrained to bend like one. The strut is !2 feet long, the load 30 tons. Taking A = 8 inches and J = 4 inches, then (fig. 2, IV.) , A3 A + 4A A = t[h^b) = \2t ■ i^ = -- -— — = 7 ; 12 (/♦ + *)- 4 V \2t 48 t " 9000 7 t = -S, and A = 9*6 square inches. 8. If the above strut be an angle iron not constrained in any way, design it. Taking h = b = 10 inches, say, we have by fig. 2, v, ^ = 20^ .-4-2 nearly; .30 . — Lfi^, = i^ ; _1_ 144- 1 00 9000 4-2 and t = •58 = g inches, and A = 201 = 125 square inches. CHAP. XX.] STEEL ARCHED GIRDER. 367 CHAPTER XX. THE STEEL ARCHED GIRDER. For roofs, the steel arched girder is of great importance. It is used as a roof principal to cover the multiple platforms of great railway stations, a gigantic example being the St. Pancras Eail- way Station, London. It is struck in arcs of circles with a slight cusp or peak at the crown, span 240 ft., rise 96 ft. The ribs, wliich are spaced about 30 ft. apart, are each 6 ft. deep, and uni- form except a part of the top where the moment of inertia is of double the value elsewhere. The rib is without hinges either at the crown or springings, and a tie-girder holds the ends together. These tie-girders are underneath the platforms which they support. We shall quote from Walmisley on Iron Eoofs, 2nd edition, p. 83, the test prescribed in the specification of this roof: — "Two main principals of the roof shall be erected at the works upon proper abutments erected for the purpose upon tlie flooring. The principals shall be erected with purlins complete in every respect, and shall be tested in presence of the engineer or assistant appointed by him. First a weight of 200 tons or 100 tons on each principal shall be evenly distributed. Xext 25 tons shall be taken from one side of each principal, leaving 50 tons upon the other side, and making the relative loads 25 tons and 50 tons on the two sides of each principal." We would note, too, that in Mills' Railwai/ Construction, 1898, p. 271, he gives the weight of iron work per square (of 100 square feet) as 2'07 tons for the Central Station, Manchester roof, only 30 feet less in span than the St. Pancras. In Mills' book there are four consecutive pages showing arched roofs, most of them like the St. Pancras without hinges. One, however, of the roof of the Anhalt Railway, Berlin, he shows with a hinge at each end, parabolic in outline, uniform in depth, which is greater in proportion to the span 205 feet than that of the St. Pancras. The Anhalt principals suddenly taper at the ends towards the hinges. Again, arched girder roofs for buildings, often of great span, are generally made in two halves affording three hinges, one at 368 APl'LIED MECHANICS. [CHAP. XX. the crown, as well as the two at springing. The third hinge eliminates to a great extent the temperature effects, and facili- tates the erection. Besides, the two halves can, with economy, be swelled out in the middle, and shaped with a curved lower or inner member, and yet have a polygonal upper member to suit the covering. Two of the finest examples can be seen super- imposed on each other in the World's Columhidn Exjjosition, re- printed from Engineering of April 21st, 1893. They are, one the roof of the Manufacturers' and Liberal Arts Building of the Exposition, and the other the roof of the Machinery Hall, Paris, 1889. The spans have the common value of 368 feet. The Columbian roof has a rise 206 feet, being about 50 per cent, higher than the Paris one, the principals being spaced 50 feet apart. The girders are 8 feet deep at the top, and at least twice as deep as the haunches. Two rival modes were employed in erecting the Columbian roof by two contractors : figures illus- trating them are shown. They are called the Fives Lille mode and the Cail mode. AVe mention those examples, as the drawings can readily be seen by British students. A complete list of roofs and bridges is given by Howe in his Treatise on ArcJies, Wiley and Son, 1897, an excellent and exhaustive treatise on the subject. For bridges, the steel arched girder seems likely to entirely supersede the suspension bridge. A comprehensive view can be had on one page of Engineering, of April 28th, 1899. AVith the exception of the top one, the spans shown are all between 500 and 600 feet, the rises varying from, judging by the eye, one-fifth to two-thirds of the span. Some are circular, some parabolic, some have no hinges, when the girders have a small depth at crown and increase till they are from twice to thrice as deep at the spriugings. Others have a hinge at each springing, and are either uniform with a sudden taper at the hinges or are very deep at the crown and taper gradually to the hinges. Again, some have abutments, others tie-rods suspended from the arch. Also in some of the examples there shown, tlie railway runs over the crown on straight girders supported l»y "Eiffel" towers rising from the arch; in other exampk^s these straight girders are suspended or are partly suspended from crowns and partly supported over the haunches. The figure at the top of the group is the great Niagara Falls bridge, 1898, span 840 feet, rise 150 feet, depth of parallel booms 26 feet. The shape of arched-ril) is parabolic, and it CHAP. XX.] STEEL ARCHED GIRDER. 369 rests on steel pins 12 inches in diameter one at each end. Two arched girders support two railway tracks and side walks. The horizontal girders ride over the arches on which they rest by vertical and horizontal spandril bracing. The live load is pre- scribed in one form as a string of the Ely Tram and Trailer, spaced 200 feet clear apart per track. Such a tram is roughly 40 tons in a 60-foot length. Disposition of loads. — It is evident that the live load all over the span of an arch produces the greatest thrusts on any section of the rib as a whole. But the live load, sa}' half over, produces bending moments which at some sections greatly increase the thrust on one boom, at the same time decreasing it on the other, it may be to such an extent as to change it to tension. The unsymmetrical position of the live load causes the thrust at some sections not to be truly along the rib but oblique. This oblique thrust is then equivalent to a direct component thrust, and a tangential component which distorts the panel or bay of the rib, which must accordingly be braced by one or two diagonal members. These effects would be most pronounced for the middle panel or bay with the live load half over the span. And with this position of the live load these effects are sufficiently pronounced at all bays for practical purposes, the span being very great, although with short spans it might be well to put the load three- fourths over for some bays. From these remarks on bridge arches, and from the specifi- cation of the test on the St. Pancras' roof arch, it will be seen that for large span lofty steel arched girders, either for roof or bridge work, it is practically sufficient to consider only two dispositions of the load — (1) an uniform load all over the span, and (2) an uniform load on each half, but greater on one half of the span than on the other. Levy has given his solution of this problem in the happiest way. He draws the stress diagram for a unit uniform load on left half of the span only, from which by suitable multipliers the stresses on the members cut by any section can be computed for — (1) the live load as well as the dead load all over the span; (2) the live load on the half in which the section lies, but not on the other half ; and (3) the live load on the other half, but not on the half of the span in which the section itself lies. The curve of henders and curve of flatteners. — If an arched rib stand on its ends on a horizontal plane, and if we suppose each 2b 370 APPLIED MECHANICS. [CHAP. XX. end to be a pin furnished with wheels, then if any load be imposed on the back of the arch, it will straddle out and be everywhere flatter than it was before, and the flattening moment at each section can be shown by the diagram which, for a straight girder hinged at the ends bearing the same load, is called the bending moment diagram. We shall now call that figure the curve of flattcners. If we measure the amount by which the arch has straddled or increased the span and remove the load, and if we now put a tie-rod across between the pins and screw it tight, till they are brought nearer to each other by the same amount that they straddled before, the arch will be like a bow a little more bent at each section than it was at first. That is, there is now at each section a bending moment the amount of which is tlie pull on the tie-rod multiplied by the height of the middle point of the section above it as a lever. Hence the curve drawn up the middle of the arched rib is itself the curve of lenders due to the pull on the tie-rod. If the pull on the tie-rod be ascertained in any way, a scale could readily be constructed for this curve of benders. Suppose that the free girder as at first is loaded with the external load which straddles it out, and that then the tie-rod is put across and screwed till the straddle is destroyed, and the hinges brought to the original span. We have now two diagrams, one the curve of flatteners due to the extern load drawn arbi- trarily to any scale, and the other, the middle line up the rib itself, a curve of benders due to the pull on the tie-rod, the scale for which is unknown, unless we should happen to know, say by experiment, the pull on the tie-rod. Suppose for a moment that we did know the pull on the tie-rod and so were enabled to construct the scale for it. We propose to call this scale the normal scale. On looking now at our curve of flatteners, which is hardly likely by accident to be to this very scale, we could reduce it or draw it over again to this normal scale, and in doing so, we could draw it on the chord of the rib as base so as to have both curves, the one of flatteners and the other of benders, on the chord as a common base, and both to the normal scale. We have already shown that a bending moment diagram is itself a balanced linear rib for the load, so that the two curves thus superimposed on the chord as base and to the same scale, may now be distinguislied with propriety as the real rib and the ideal balanced rib. It will be seen that if tlic free unstrained rib be first pro- vided with rigid aliutments that will not yield appreciably, or CHAP. XX.] STEEL AKCHED GIRDER. 371 with a lie-rod which will not stretch an a^jpreciable amount, the load placed on its back produces simultaneously the flatteners (a curve of which we can draw to an arbitrary scale independent of everything but the extern loads) and the benders due to the induced pull on the tie-rod, of which the central line of the rib is itself the curve to the normal scale unknown. This normal unknown scale depends only on the induced pull on the tie-rod, that is, on the straddle of the girder without a tie-rod under the extern load. Now this straddle depends on the variations of the cross-sections of the rib from section to section, so that the normal scale is statically indeterminate as a rule. The exception is the presence of a third hinge, say at the crown. Suppose there had all along been a hinge at the crown, but that it had been securely clamped. The minute it is undamped the normal scale is known, for the ideal rib must pass through this third hinge in order that the flattener and the bender may at that point neutralize each other ; hence the mere reduction of the curve of flatteners so that it may pass through the third hinge changes its arbitrary scale at once into the normal scale which is then known. Eankine advised the adoption of a third hinge at the crown of cast-iron arched girders, and says the assumption of such a hinge is always a good approximation. Levy illustrates his graphical construction for an arch with hinged ends by an application to the Oporto bridge over the Douro. In this bridge the rib is crescent-shaped, and is hinged at the spriugings where it rests on abutments. Span 160, mean rise 42-5, and depth of rib at crown 10 metres. The moments of inertia of the cross-sections from crown out to a hinge vary at intervals as 10, 8, 6, 5, 4. It is described in the Mem. et Compte Rendu dcs Trav. dc la Soc. des Ing. Civils, Sept., 1878, par T. Seyrig. The line up the middle of the rib is an arc of a circle. Levy primarily assumes the rib to be of uniform cross- section, and indicates a correction for the variation. The stresses on the members of this bridge are calculated by Howe in his treatise already referred to, and his results com- pared with those of Seyrig. Howe quotes the moving load covering the entire roadway as 277 tonne or 270 tons, which is roughly half a ton per foot, or half a share of a single row of locomotives. Levy illustrates his graphical construction for an arch with fixed ends by a rib with the same arc of a circle as the other, but with the section increasing from the crown outwards so that 2 b2 372 APPLIED MECHANICS. [CHAP. XX, its moment of inerlia at each section is proportional to the secant of the slope of the rib to the horizon. Such an arched rib is of uniform di/fhcss. This is proved quite simply by Rankine {see " The Sloping Beam," Civil Eng., p. 292). It is evident that this arch should deepen in section as it goes outward, for two reasons. The bending moments at the ends of the straight boom fixed at ends are greatest. Also a broad foot is required to securely fix it to the skewbacks. Lastly, before illustrating Levy's construction, the relative weight of steel on the three types are given by Howe in the following proportion : — (1) Three hinges, ... 1. (2) Two hinges, . . . .1-2. (3) No hinges, .... I'o. The economy of (2) over (3) would, we think, be increased' by the extra material in the skewbacks to supply the reacting couples at the ends. Observe that the type (o) cannot, when circular, subtend much more than 120° at the centre, as the secant of 60° is 2, and beyond 60° it increases enormously. Eoof girders of this type (3) are of ten the complete semicircle with a vertical part in addition at each end. Here only the portion out to 60'^ on each side of the crown should be treated as the actual elastic rib, the lower part on each side being reckoned as part of the vertical post, and is generally along with the post buttressed or braced to a smaller rib rising on the opposite side of the post. Other references are The Engineer, June 30 and July 14, 1899, for a Danish bridge ; April 26, 1907, for a road bridge- over the Usk ; also Engineering, March 30, 1900, for the Ehine bridge at Bonn. See, too, a Mem. cle Viaduct Garaht, par G. Eiflel, 1880. Levy's Graphical Solution of the Arched Bib of Uniform Section Hinged at each end. On fig. 1, 2 is shown to scale a segment of a circle, span 240 feet, rise 80 feet, radius 130 feet. It is the lineal real rib up the centre of the iron arch which consists of two parallel booms 10 feet apart. This, then, is the curve of benders to the unknown normal scale. It is divided into 16 equal parts along the curve, and the ordinates are numbered from to 8 to the right hand, and from to 8' to the left. The fig. 1, 5 shows the load line W = 120 tons drawn to a •CHAP. XX.] STKKL AHCIIKl) GIRUEIt. MTo scale of tons, being one ton per foot on left half of span only. The reactions are Q = ^W and F => f W. A pole Q is chosen arbitrarily, and the curve of f,attciicrs (fig. 1, 1) drawn to it, and consisting of a straight tail over the unloaded half parallel to the vector Qa. A parabolic arc must be drawn over tlie loaded half. The vertex is over the point where the shearing force diagram (dotted) crosses the base, that is, at a point divid- ing the span in the ratio o to 5. The curve of flatteners when tinished is like a great fish or " Jon Dore," and it is best to construct the points on the parabolic arc corresponding to the ordinates of the figure below ruled up. This is done by producing the straight tail till it makes an intercept on the vertical through the left end (see fig. 3, 1), dividing this vertical intercept pro- portional to the way the left half of span is divided when vectors from the top point of the straight tail will cut ofi' the proper heights on the ordinates. Number these ordinates from 1 to 8 right, and from 1 to 8' left (see fig. 3, Ch. VI). The next step is best done with proportional compasses. Double all the ordinates of fig. 1, 2, drawing right and left a series of -fictitious parallel forces 1 to 8 and 1 to 8'. Set the compasses to the convenient ratio 4 to 1, construct the load line (fig. 1, 3) with equal to a quarter of the middle ordinate of the curve of benders, with 1 equal to a quarter of the sum of its ordinates 1 and 1', &c., and to a convenient pole draw the link polygon among the horizontal fictitious forces going left, and produce the two end links to make the first intercept on the chord joining the hinges of the real rib. In the same way lay off the load line (fig. 1, 4) with equal to a quarter of the middle ordinate of the " Jon Dore," with 1 equal to a quarter sum of its ordinates 1 and 1', &c. With the same polar distance as last draw the link polygon among the horizontal fictitious forces going right, and produce the end links to make the second intercept on the chord joining the hinges of the real rib. On fig. 1 this second intercept is about two-thirds the length of the first. Now we know that these two intercepts should be equal, for their algebraic sum equated to zero means that the moment of all the fictitious forces, with respect to the chord joining the hinges, is zero. In this problem three definite integrals are involved, namely, ^Mclx = 0, IxMdx = 0, lyMdx = 0, where x and y are the coordinates of the real rib, and the limits •of the integration being from hinge to hinge. 374 APPLIED MECHANICS. [CHAP. XX. The first two integrals we have fully discussed as regards the straight beam and illustrated by examples in Chapter XVII, at the beginning of which are theorems {a) and (6), corre- sponding to these first two integrals, and for an illustration see fig. 2, Ch. XVII, in which AK is drawn that its area may equal that of ACB, and that its geometrical moment about the vertical through the centre shall equal that of ACB. That is, the joint areas are zero, and the joint geometrical moments are zero. So with the third integral ; 3Idx is the joint algebraical area of the two corresponding strips of the two curves, one of benders and one of flatteners, and by drawing a fictitious force for this partly right and partly left, and giving them the common lever about the chord, namely, the height of the rib itself (only doubled for good definition), we obtain the product Mdz x y for that finite part of the rib by producing the two links that concur on the fictitious force till they make an intercept on the chord. The total intercept must be zero, and it matters not that we have grouped them in two categories according to whether they tend to flatten or further bend the rib. The reason that the two intercepts are not equal is that the arbitrarily chosen pole Q, to which we drew the " Jon Dore," does not make that figure be to the normal scale. It is now necessary to draw the " Jon Dore " or curve of flatteners over again to a vertical scale increased in the ratio in which the first intercept exceeds the second. It is best by trial and error to set the proportional compasses till one end spans one intercept, and the other spans the other intercept. Take the height of the middle ordinate of the " Jon Dore " on the smaller end, and, with the large end, set it up on the middle ordinate of the real rib. On fig. 1, 2 it will be seen to reach a little above the crown of the real rib ; joining this with a straight line to the right hinge we have the tail of the new curve of flatteners, and with the proportional compasses the seven intermediate points of the left half may be set up, or the new parabola may be constructed at those ordinates as at first. Next from L the Ijottom of the load line fig. 1, 5 a line parallel to the tail of the new curve of flatteners svill meet the horizontal from d, the junction of the reactions P and Q, at the new pole. Or dO is shorter than dQ in the ratio of the intercepts. This polar distance, which is 130 feet, determines the normal scale of the joint curves of flatteners and benders, or, as they may now be called, the ideal and real ribs. For it is evident that if the right-hand link polygon were redrawn for CHAP. XX.] STEEL ARCHED GIRDER. 375 the new curve of flatteners, the new intercept 2 would now equal the first intercept. The virtual hinge or point at which there is no bending moment is a little to the right of the crown of the real rib where the tail of the ideal rib crosses it. Had there actually been a hinge at the crown of the rib, the curve of flatteners could have been drawn at once with its tail drawn from the crown to the right end, and the normal scale determined atonce. The elevation, to the double scale, of the steel arched girder is shown on fig. 2, 6 ; it is sketched as if for a roof. It has radial stiffenevH with double diagonal ties in tach square. Counting the crown square as zero, we will now take out the stress at a radial cross-section through the middle of the third square to the right hand. This cross-section is shown at A on fig. 1, 2, and A' is th& corresponding cross-section in the loaded half of the girder. A portion of the arch reaching from A' to A is shown above, furnished with arrows to indicate the stresses on the upper and under booms and the radial or shearing stress. The thrust on the ideal rib at the point below A is given by the vector 10 of fig 1, 5, which is resolved into two components, a tangential thrust 50 tons, and an outward radial force 19 tons. The 50 tons is divided into 25 tons each (of thrust), on the upper and under boom. At ^ the height of the curve of benders or real rib is greater than that of the curve of flatteners or ideal rib. The intercept at A between the ribs is the residual bender ; it measures 5 tons and, when multiplied by the polar distance 130 feet, we have the bender ilf^ equal to 650 ft.-tons. The booms have a lever of 10 feet to resist it, and accordingly there is a pull of 65 tons on the upper and thrust of 65 tons on the lower boom at A due to this bender. This is the most business-like way of taking out the effect of the bending couple at A. But there is another and instructive way of looking at it and of scaling it off. Suppose the thrust hO tons acting on the real rib at the point below A. Then to shift it parallel to itself so as to act at A instead, we must compound with that force 10, a left-handed couple, the common force, of which shall be hO tons and whose arm is the perpendicular distance between the old and new positions of the force before and after the shift. Now this arm is shorter than the vertical intercept at A. It is equal to the vertical intercept multiplied by the cos 0, if d be the slope of the tail of the ideal rib to the horizon. So that if we measure the vertical intercept at A mfeet, we have the arm 376 APPLIED MECHANICS. [chap. XX. Jon OORE. ^Cue tir-Ertern'Xoizd.' draxvTi/ to a.TZ/ arinjrary Scale. J'clc £. tkc"^?! OJia-rTia^ic coiled e^ "the J'u^Mcd, tu/ /i" a^v ujOc. — Fig 3. ^'g^. bearing thosamoTUijnicr: cft}i6Su^7vdfa,pcurcf07-din/J/ea Fii;. 1. CHAP. XX.] STEEL AECHED GIRDER. 377 1 MA _!. ^ Ton PrR F0or 1 1 TiaZf tx^ ttm . h- !-jr 1 TuOy^ ci. ten, 1 Stresses due, te thcXoeiA c cA^* ha^ pf the. Girder: AleidSc Thorrtson. 4f? ^/7 ^a 7& 80 sa Tffc 710 7V/iet^. Fis. 2. 378 APPLIED MECHANICS. [CHAP. XX. of the shifting couple multiplied by sec. 61. But if instead of measuring hO in tons we take instead dO in tons, we have the force of the shifting couple decreased in the exact proportion that the arm was increased. Hence the moment of the shifting couple equals the product of the vertical intercept measured in feet, and of dO measured in tons. But the "product is in no way altered if we interchange the scales upon which we scale off the two factors. Hence what we measured oft before, namely, the intercept at A in tons, and the polar distance dO in feet, give, when multiplied together, the moment of the couple which shifts the thrust on the ideal rib up to act at A on the real rib. "When the thrust arrives at -^, it brings with it this left-handed couple, so that the two booms have not only to oppose the thrust, but have also to exert a right-handed couple to neutralize the shifting couple. This is tlie penalty, as it were, that the real rib pays at A for not coinciding with the ideal one. Again, now that the thrust is at A it is not directly along the rib, and when resolved into a direct thrust along the rib there is also a radial outward or shearing force as v/ell, tending to distort the square at A, and this distorting force is the penalty, as it were, that the real rib pays at A for not being parallel to the ideal one. Note that dO plays two parts ; it is 130 feet as polar distance, but is also 44 tons as the thrust at crown of ideal rib or pull on the tie-rod. At A' the vertical intercept measures 5 tons, and when multiplied by 130 feet we have J/.^i = 650 foot-tons a residual flattener, giving a thrust of 65 tons on upper boom, and a like pull on the lower. A tangent drawn to the parabola at A happens to be horizontal : it resolves into a direct thrust of 43 tons along the real rib at A', or 22 on each boom, together with an inward tangential force of 10 tons. The like quantities are constructed for the pair of points B and B' corresponding to the twelfth square from crown. In drawing the tangent to the parabolic part of the real rib at B' it is best to construct the tangent at B" by ruling it horizontally to meet the vertical tlirough the vertex, setting up a like height above the vertex whicii gives a point to join B" to ; thus Qh' is drawn parallel to this tangent, and, joining h' to 0, we have the tangent to the real rib at B'. When the point B" lies near the vertex, this construction is bad, and a better one in such a case is shown for the point S" on fig. 3, I, namely, to draw a chord from c (the most remote given point on the parabola to the right of *S'") to d, a point lying at the same distance CHAl'. W'.] STEEL AIJCHED (illlDKR. 319 horizontally from S" as c does, but on the left side. The tangent at S" is parallel to dc. The other tangent at K" is drawn from K" to /where Vf = e/'as already described. On fig. 2, 6 three varieties of load are shown. Fig. 2, 7 gives the stress at the third square selected from values at A and A' as follows : the pull on the upper boom at -4 is 65 - 25 = 40, due to a load of 1 ton per foot on left half of span ; but at fig. 2, 7 only half of this or 20 tons is set on the upper boom beside a half barb or arrow-head, because there is only half a ton per foot on the left half of the arch. In the same way we have set 45 as thrust on low boom, and 9 as outward radial stress. These are set beside half barbs to indicate that the stresses are due to the load on left half of arch only. Now if we suppose the figs. 1, 1 and 2 turned over so as to be seen through the paper at the back, A' will have taken the place of A, so that by halving the stresses at A' and writing them on fig. 2, 7 with full barbs, we have the stress due to the load on the right half of the arch. These barbed and half- barbed results, are summed algebraically and written with a black figure a little to the left of the section (fig. 2, 7). On fig. 2, 8 the quantities for half barbs are unaltered, as the load on the left lialf for the second variety is unaltered, but the quantities at the full barbs are halved, as the load on right half is reduced. On the other hand, fig. 2, 9 shows the stresses for the third variety of load. Here the quantities at the full barbs are unaltered and the others halved. On the third square to right hand of the arch itself 33 tons is written on the upper boom, being the greatest of the three varieties. On the lower boom it is 35 tons. On one of the diagonals in the third square is written 10 tons : this is a pull. The other diagonal is dotted to indicate that there is no pull on it for any of the three varieties of load. It will be seen that the gross radial stresses for the three varieties of load are 4, 7, and 0, all outurird. As the cross- section cuts the diagonal at about 45°, and as 7 tons is only one component of the pull on the diagonal, it has to be multiplied by sec. 45° or by 1^ nearly, giving the pull of 10 tons. To make the roof stiff the two diagonals in the third square might both be tightened by a thimble, or one only, till they had an initial pull of 2 tons each, giving 12 and 2 as the crreatest pulls on the two diagonals respectively. "The figs. 1 to 4 are reduced half lineal size from a set of graphical exercises advertised near the title-page. 380 APPLIED MECHANICS. [chap. XX. AtBX.'i y-^^ ■^c? f/vr((^r\'ii Sief\f,i(>e/ injrjaf^s. Fig. CHAP. XX.] STEKL ARGUED GlKDP^Il. 381 • ot^r -znzmrrrtvr. i ' ' ' ' ^' ' ' 1 ' ^ I ' I I M I the Stresses ciae to -^ I mol,on6lo/i the h-ft if Fig. 4. 382 applied mechanics. [chap. xx. Levy's Graphical Solution of the Arched Rib of Uniform Stiffness fixed at the ends and without hinge any- where. The preceding description must serve in detail, and we shall point out the marked differences in this case. We have the same central line for the linear-rib. The booms are now 7 feet apart at crown, and 17 feet apart at springing, and the moment of inertia of the cross-section is everywhere pro- portional to the secant of the slope. The stiffeners are vertical and are prolonged as suspending rods to support the horizontal girders that carry a railway. The first marked difference in the construction is that the real rib is divided into equal parts along the span or horizontal. There are more parts, being 12 in each half. The curve of flatteners is shown at fig. 3, 1, drawn as before to a short load line with a polar distance of 50 feet. Then for theorems (a), (b), Ch. XVII, as applied generally at fig. 4, Ch. XVII (and particularly at Exs. 3, 4, Ch. XVII), we must draw a new base ran across the curve to include the same area an that a Vh contains, and with its centre of gravity on the same vertical line. This is accomplished by niaking am = Woe, and hn = -f-^oc. The graphical construction is shown to left of fig. 3, 1. The half span being reckoned as the horizontal unit, a force is laid down equal to oc to represent the area of the isosceles triangle acb, followed by a force fa^ to represent the area of the para- bolic segment aac. The lines of action of those two areas are drawn downward ; also two lines of action trisecting the span are drawn upwards to represent the two triangles into \vhich the unknown figure an can be divided. A link polygon drawn among these four lines of action is shown, its closing side dotted ; a vector from the pole parallel to this divides the load line into two segments which are the areas of the two triangles amii and (nil). But the half span being unity, those segments are am and hn respectively. MNis drawn across the real rib so that JTmay have the .same area as ACB. To draw MN its height ^ J/ will be a little more than two-thirds of the height of G. For if ACB had been parabolic, it would have been exactly two- thirds. The area ACB can be measured with a planimeter, or calculated from three ordinates in a half of it, using Simpson's multipliers. CHAP. XX.] STEEL ARCHED GIRDEK. 363 In drawing the load line, fig. 3, III., 0, 1, ... 7 go left, as long as the ordinates of the real rib in pairs are above MN, but 8, 9, ... 12 go right, and only onc-lialf of ordinates 12 are to be taken. This load line should be a closed polygon, but it will only nearly close. This must be averaged so that the two end vectors may coincide. When the link polygon to left of fig. 3, II is drawn, its two end links are parallel, and make an intercept on the horizontal, the load line being horizontal ; the two ends links of the right polygon make another intercept on the horizontal. This second intercept, which should be equal to the first, shows that the curve of flatteners is to a scale too small. The propor- tional compasses may be set so that one end shall span the one intercept and the other end the other. The curve of fiatteners is then to be increased and superimposed on the real rib with mn coinciding with MX. To do this from the middle of MN set up oC equal to the increased value of oc. Set downwards NE equal to the increased value of nh, and MF equal to the increased value of nia. Join CE with a straight line and it is the tail of the ideal rib, and the points on the curved half FS'G ■can have their values set up from the chord EC increased from those of aS"c from its chord ac. The load line to a bold scale is drawn on fig, 4, V, the poles Q and determined, and the stresses at four sections ;S', *S", K, K', scaled off as in the last, and then these are manipulated with the proper multipliers as on the hinged rib. The reason that equi-intervals are taken along the span is, that / is not constant, being the integral J (if ^ I) y dx = ; but / cos 6, the projection on the chord, is constant, since I is proportional to the sec 0, so that in taking finite intervals it is necessary to make them have a constant horizontal projection. Numbere engraved along the upper and under booms of both of our girders show how the plates must vary, and it is interest- ing to note the difference in the variation for the two cases. Some booms sustain a pull instead of a thrust for the live load half over, and we note that eight-tenths of this reverse stress is prescribed on the figure of the Niagara bridge as a quantity to be added to the direct stress. 384 APPLIED MECHANICS. [chap, XXI, CHAPTER XXI. ANALYSIS OF TRIANGULAR TRUSSING IN GIRDERS WITH HORIZONTAL PARALLEL BOOMS. The lower member A,A,n (fig. 1) is supported at each end and is divided into 2n equal parts. Upon the first n of these booms as bases similar triangles are erected, the upper booms joining the vertices of the triangles. This constitutes the left half of the girder with which we have principally to deal. The right half of the girder extending from An^ to Ain is the image /^(l_^\ Reaction due to uniform dead load W. xt,(i--^\ Reaction due to rolling load W at A2y. g, A3 Aji-i 2 1+ 1 ^(ztT'^z^"^^ « is No. of parts in half span. r is ratio h to L, depth to span. ;^ c is segment of base of triangle remote from centre of girder. Triangular Trussing.— (Fig. 1.) of the left half, that is, it has the pattern reversed just as an image of the left half would appear in a Aertical mirror forming a right cross-section at the centre. It will be seen that we are dealing with a girder divided into an even number of baysy This is partly because of the comparison to be drawn with the Fink system of trussing, which only admits of even subdivision, and partly because an odd number of bays is not so desirable from an economical point of view. Neither could the notation be conveniently adapted to both the odd and even subdivision of the span. Eeturning to lig. 1, it will be seen that there are 2n - 1 upper booms all equal in/ length with the exception of the central one. At each lowet^ joint or apex the letter A is placed CHAP XXI.] ANALYSIS OF TRUSSES. 385 witli an even suffix, and at the upper apexes with an odd suffix. The depth of the girder is h, and the span is X = A^Ai„, and the common length of all the booms, with the exception of the upper central one, is X -^ 2n. A variable c to regulate the shape or form of the triangle is now introduced in the following manner. Fig. .2 is one of the triangles, singled out from those n triangles forming the left half of the girder. Its base is A.i.-iAii, and from its vertex A.j.i a perpendicular h is dropped dividing the base into two segments ; c is the segment of the base rcrrote from the centre This side of the triangle, and the segment c ot its base, are said to be remote from the centre of girder. Form of the tri- angle varies with c. This side and segment are adjacent to centre of girder. Position of tri- angle varies with The i"' Tkiaxgle from left end of Girder. — (Fig. 2.) of the span (the shorter in the diagram), while {L -^ 2n) - c is the segment adjacent to the centre. The surd ^y {h- + c^) gives the length of the diagonal Aii^iA^i-o, which, in the same way, is said to be remote from the centre of the span, and a similar surd gives the length of the diagonal adjacent to the centre. For all symmetrical loads, such as the dead weight of the girder itself, the diagonals remote from the centre are struts, and those adjacent to it are ties. It will now be seen that the upper cen- tral boom is twice the length of the segment adjacent to the centre. To the variable i are to be assigned the integer values 1, 2, 3, &c., . . . n, when the triangle (fig. 2) will become in succession the triangle at the left support, the second, the third, &c., each and all of the triangles forming the left half of the girder. 2c 386 APPLIED MECHANICS. [CHAP. XXI. Dealing only with the left half of the span, we have Lower apexes, A^A^Ai . . . A^.^A^i . . . Atn- Stress on upper boom opposite, ?'2 7\ . . . r^.i r>i . . . r2„. Upper apexes, AiAi ^21-1 ^2n-i. Stress on lower boom opposite, Vi Ts r;,_i rjn-i- The stresses on diagonals remote from the centre are tAh • • • Ui-i . . . <2n-i, and on the adjacent diagonals they are *2'4f6 • • • ^2i • • • ^2»1- The bending moment at a section through any apex is represented by the letter 3f with the same subscript which the A at the apex bears, and the shearing force, constant for any interval of the span, is represented by the letter F with two subscripts separated by a comma, which subscripts are those borne by the ^'s at the two apexes bounding that interval. If we confine ourselves to the representative triangle (fig. 2), we have generally Bending moment at upper apex, . JAf-i. Stress on a lower boom, . . rji^i = Ma-i -f h. Bending moment at under apex, . Mu. Stress on an upper boom, . . rji = M^i ^ h. Shearing force in the i"' bay, . . i^2.-2, 2.' Stress on a diafjonal remote from , ,,, ,, centre, . . . . . yO'll^ h F,i. ,2)2i Stress on a diagonal adjacent to yM ■In ' centre, > \" ' w . The values are given irrespective of sign ; the first two are obtained by dividing the bending moment at the section through the opposite apex by the depth of the girder. The second pair of quantities are obtained by multiplying the shearing stress in the bay by the secant of the slope of the diagonal brace to the vertical. It is to be remarked here that all loads, even the dead loads, are to be looked upon as something apart altogether from the girder itself, and tliat the loads are all redncal to the joints of the lower boom. For this reason i^is taken as constant from ^2,-.2 U) A-ii. CHAP. XXI.] ANALYSIS OF TRUSSES. .'387 Dexd Load. — Let the dead load be w tons per incli of span uniforndy distributed on the lower boom, then the total load is W = irL tons where all lineal dimensions as L and li are to be taken in inches. This load when reduced to the joints of the lower boom gives a share oi W - 2n concentrated at each of the lower joints except the two end ones, which have only a half share each, and those are directly over the abutments, so that the reaction of the left abutment on the left end of the truss is W uL _W f 1 T -^n ~ 2 \ 2n The shearing forces are W2n-\ W _ W%i-\ W ^ ^ W2n-1 ^. ^^ W W(2n+1 i\ 2 2n 2n 2 \ 2n n) The bending moments are ,, W2n-1 W2n-\(L \ W 2 2% \2n y 2 zn W2n-lf^ Z \ WfL ^ M.= 2. — ^ c] — + 2c 2 2n V 2n 7 27^1 2^^ ,^ W2n-lf^ L \ Wf- L \ M.= ^--^— 4.~ + c--- 1 + 2 + 3— + 4c), 2 In \ In / 2n \ 2n J 2 271 ( ^ ^ 2n ) 2nl 2 2n ^ ' ) 2 c 2 388 APPUED MECHANICS. [CHAP. XXI. The numerical coefficient of the second last term being the sum of the natural numbers from 1 do [i - 2). Then by reduction, WL ,. ,, [In +1 i\ Wc [2n + 1 i Mn-, = -r- (* - 1) — 7r] + '*" \ Zn in) M, 4?t \ '^11 2nJ 2 \ '2n n W 2n -\ Z 2 2n 2n ' '~ 2 2n ■ ^ ■ 2^ ~2n'2n' 2 2n 2n 2n ^ ■ 2n' "■~T ~ll^'^'2ii ~2^x 2 2n' The numerical coefficient of the last term being the sum of the natural numbers from 1 to (t - 1). Then by reduction, M^i = -— 2 1 - — 471 \ In Substituting these values in the preceding expression for the stresses on the booms and diagonals, and arranging in ascending powers of i, we have JVXf 2n_h2 n+ 1 . i- 4:nh \ 2n n 2n Wc (2n + 1 + 2h \ 2n 11 J ' WL I . i' ''''~4:nhV 2n W/2n+l i\ ,,, The strength of the material is/ tons per square inch. It is this unit which demands that the lineal dimensions shall be CHAP. XXI.] ANALYSIS OF TRUSSES. 389 in inches. It is reckoned to be of the same vahie with respect to tension and thrust. It will be shown afterwards how a cor- rection can be made if the material l)e not homogeneous in strength. Dividing r..,_i by / we have, in square inches, the theoretical area of the cross-section of the i'^ lower boom ^.,_2 A^i, reckoning from the left abutment, that is, of the boom which is the base of the /'^ or specimen triangle (fig. 2). Multiplying by L ^ 2//, its length, we have a general expression, containing the integer variable i, for the volume in cubic inches of that boom. To i in that general expression is to be assigned the values 1, 2, . . . n and the whole summed up, giving then tlie volume of the left half of the lower boom. Doubling so as to include the right half we have the theoretical volume of the lower boom in cubic inches to resist the dead load t=»i ~ T ^, . ■ V o^ _ Jill ' under booms — Z^ o J! dead load ,v, L-^" / + 7^-77 2 ^'hi + 1 n 2nfli ^ \ 2n n\ WD (_ 2w + 1 n + 1 n (n + 1) »(» + 1)(2// + 1) 4n»//i ( 2~ ^ ~7i 2 12^1 : WLc\2n + l M(n + l)| (n- l)(-in + 1) + lu summing, terms not containing i are simply to be multi- plied by n ; those containing i involve the summation of the natural numbers from 1 to n, while the term containing t* involves the summation of the squares of the natural numbers from 1 to n. The theoretical volume of the upper booms from the left end to the centre of the span is obtained in the same way, only in summing for values of i from 1 to n we include a portion of the upper central boom extending a distance c past the centre of the girder. Hence a deduction equal to c multiplied by the 390 APPLIED MECHANICS. [chap. XXI. cross-section of the middle upper boom is to be subtracted, before doubling to include the right half of the girder. ' upper booms ~ ■* ^ dead load ,_, _ WU \ n{u + 1) )t{n+ 1)(2» + 1) WD . .... ^. WLc - L _2)t - 2r — n 2 1 ^ in WLc r • i"'! I - 2>i 2nf/i V ~ 2;^\ WLc Infh '^^n-fh 4/A It is remarkable that the sum of the volumes of the booms both upper and under is independent of c, that is, of the shape or form of the triangle. It amounts to WL^ (1) a quantity continually increasing with n, but sensibly constant for values of n above 10. It has its greatest value when n is infinite, that is, for the ideal plate-girder with areas of flanges everywhere proportional to the bending moment. It is verified by considering that particular case in which the load is uniformly distributed, the maximum bending moment one-eighth of the product of the load and the span, and the average bending moment two-thirds of that maximum. Dividing that average bending moment by h and by/, and multiplying by 2Z, the joint WL" length of the flanges, we get • For the diagonals remote from the centre of the girder, '21-1 /rrz r. I remote diagonal dead load. = 21 .=. L ./' w w w (AM r)2 yii' + r j 2n+ 1 ^- -•■)(-.- -^^0 1 ("'-)■ CHAP. XXI.] ANALYSIS OF TRUSSES. And for the adjacent diagonals similarly, 391 adjacent diagonal _ dead load. Wn /r + 2« Adding these four quantities, we find the theoretical volume of the truss in cubic inches, to resist the dead load of W tons uniformly spread on the lower boom, to be, after reduction, dead load Consider now the effect upon this volume of varying only the shape or form of the triangle. The only variable is c, and Fig. 3. it will be seen that the volume is a minimum when the product c r c] is a maximum which occurs when the factors are \2n J equal, that is, when the perpendicular from the vertex upon the base bisects it. Hence the isosceles, or, as it is sometimes called, the Warren truss (fig. 3), is the most economical form of FiK. 4. triangular trussing to resist a dead load. Substitute, then, in (2), c = half a bay = Z ^ 4n, and for convenience put h = rZ, where r is the fractional depth of the span, or ratio of depth to span, and we have, after reduction, V; isosceles dead load WL nr + Sn- + 48^:- 2 1) (3) 392 APPLIED MECHANICS. [chap. XXI. A form of bracing that rivals the isosceles is the rectangular, of which there are two types ; in one the struts are vertical ; in the other, the ties are vertical (figs. 4. 5). Their volumes are obtained from equation (2) by substituting c = 0, and c = L -^ 2n respectively, and, what is remarkable, both have the same value, so that one expression serves for the volume of either. It is y rectangular dead load. WL nr + (» + l)(4»-l) 1 24n* • r (4) To resist a dead load the Warren truss is considerably more economical than the rectangular truss when the number of bays is small. Thus if both have a depth one-tenth of the span and a common span, and if there be only two bays in the span, then Fig. 5. the Warren has an economy of 33 per cent., as may be seen from the tabulated results which follow, or directly by substituting 71= 1 and r = -^-ff"* into equations (3) and (4). When, however, w = 4 or 8, between which it will usually lie in bridge work, the percentage of economy is reduced to 7 and 3. Now the rect- angular joints will probably recover this economy. The economical values of n, and especially of r, the ratio of depth to span, will be discussed after the live loads have been dealt with. Live load. — We shall now consider a live load consisting of an uniform load of u tons per inch per girder which comes on at the right end of tlie deck or platform, passes across the platform, completely covering it, and then passes oft' at the left end. The bending moments everywhere are simultaneously a maximum when the platform is covered. If we put W = v.L, we have the same Vdhime for the upper and under booms conjointly as we liad with the dead load. Vol. of l)ooms =w('-4y- (i*-) CHAP. XXI.] ANALYSIS OF TRUSSES. 393 The maximum sheaving force in any triangle of the left half of tlie girder occurs when all the triangles to its right are loaded and a portion z of its own base. This is illustrated by fig. 6, which sliows the third triauijle BCD resisting a shearing force I\ due to the load standing with its crest at a distance z from D. If we put the base BJD = a, we will see that z = fa makes F3 a maximum, while Fi is a max. for 2 = fa. That is, to get the maxima values of F for each of the eight hays on lig. G, we must lay up from A half the total load W = uL, draw the parabolic concave locus beginning at the right end of the girder, with the span as a horizontal tangent and ending at A with the height huL, which is the max. for the end bay. The span must now be divided into seven equal parts where the heights to the locus give the maxima 2^ositive shear on each bay. Fig. G. On fig. 6, let BCD be the i"' bay from the left end. Let m be the number of equal bays on the span, each bay being a inches in length. Then a {m - i) + z is the length of the loaded segment, while half of this is the distance of the centre of the load from the right end of the girder. And F = P load span X dist. of c of ^ from other end. u {a (m - i) + z) a (m - i) + z ma u 2ma P = ^r^^ {{m - i)a + zf. V I'll '-» ^ ^ ' ' When the loads are apportioned to the joints, we have at each joint to the right of -D a load iv = ua. Let the load uz spread on the segment z divide into x at B and y at D. Then X uz , IIZ'' t— = — and X = -^ Iz a 2a 394 APPLIED MECHANICS. [CHAP. XXI. The shearing force in the i'* bay is Ima 2a when and 2; = — a m - 1 „ uaCm - iy uu (2ni - i)^ ,_. F = — -^ — = — -^ (5> 2 m - 1 2 2n - 1 ^ ^ a maximum, where n is the number of parts in the half span. Now Fi sec a -^ / is the sectional area of BC in square inches, while its length is h sec a, and substituting Ji^ -h [h^ + c-y for sec. a we have Vol. of 5^="^'^^^^-'^^'^^"^' 2/ 2n - 1 h Summing for values of i from 1 to n, and doubling so as to include both half girders, the volume of struts is ^struts = 77 7^ T S (4?i^ + 4?^^ + i^) But S (4w*) = 4?^^ 2 (47it) = 4w . — ^^^ — ^ and substituting and reducing 12/' V ^'' CHAP. XXI.] ANALYSIS OF TRUSSES. 395 For the volume of the ties k is to be substituted for c, and adding, the volume of the diagonals is ^^^^\^'^^'^) <«) Since the sum of c and k is constant, the sum of their squares is a minimum when so that a L The economical shape of triangular trussing is the isosceles triangle. Adding this to the expression (1) for the volume of the l)ooms, we get the volume of the isosceles truss to resist the transite of a unifoi-m load W which can just cover the span Wj'in-l ) WL^ nn- - 1 In- 1 \1 6/ ' ^ '^ "^J^\i^^ Un- Jh T^ WL \7n - 1 16?r + 7n - 5 1j r = r + -— — • — > (7) .isosceles / ( 6 9671" r\ moT. load having substituted h - rL. Again in the expression (6) if we put c = ^ and k = 0, or conversely we obtain the common value of rectangular trussing (figs. 5 and 4) to resist the live load, and V =^fi^z} . 8?r + 77t - 3 1| rectangular / ( 6 4872-' ?' I mov. load In the expressions for the theoretical volumes of trusses given in equations (2), (3), (4), (7j, and (8), if all the quantities be constant except n, the volume will be seen to vary directly with n. Practical considerations fix the values of n between very narrow limits, the chief being the necessity for stiffness, which is of paramount importance, especially with moving loads. The stiffness of a girder is of two kinds — stiffness of the girder as a whole and local stiffness. 396 APPLIED MECHANICS. [CHAP. XXI. Xow these two affect the question of economy in opposite ways, for increasing the depth of the girder increases the stiff- ness of the girder as a whole, but makes the struts so long that to be effective they must be greatly stiffened individually. Considering these jointly, a depth of girder from one-eighth to one-tenth of the span is most suitable for iron and steel girders. Where the booms ai-e timber which demands a large scantling for strength, and so affording at the same time local stiffness, the depths may be greater in proportion to the span. In the American early practice with timber, depths of even one- fourth the span were used. In the British early practice, in passing from cast iron to solid plate girders, depths as low as one-fourteenth the span were used. The modern practice in both countries is now from one-eighth to one-tenth of the span. Econcyinicul Depth. — Having discussed the economical shape and subdivision, it now remains to consider the variable r in WL the six expressions above. They are all of the form —r- x it where n = ar + Q-, (9) r and, what is more, the theoretical volume of all trasses without redundant bars reduces to this form. There is a value of r which necessarily makes u a minimum, because with very large values of r, u is sensibly «/•, incrcasiivj with r, wliile with small values of v, n is j3 ^ /", increasing as r decreases. Put making u = 2 Jn ./3 a minimum. (11) Tablks uf Theoketical Volumes of Trusses. The quantities n (9) are given in the tables, both for isos- celes or Warren bracing, and for right-angled bracing to resist a rolling load IV, an uniform dead load W, and an uniform moving load. The five rows are for 2, 4, 8, 16, and o2 subdi- visions or bays on the span, and in each pane of the table tliere appear four values of u correspomling to the four depths expressed as fractions of the span, viz., r = -p'^-, yV- -k' o^' T' respectively. In each ' pane ' there appears also a heavy-faced CHAP. XXI.] ANALYSIS OF TRUSSES 397 number, being the niininiuni value of u by equation (11), wliile the corresponding vahie of r by equation (10, appears with an asterisk (*) at it. These heavy-faced numbers are put in their proper sequence among the plain-faced ones, so that one can see at a glance whether this most economical depth is feasible, that is, whether it falls within those demanded by stifihess. This, of course, decides whether the economy of material can be realized or not. For instance, on the firsfe bottom ' pane,' p. 398, *r = -103 while on the third, *r = '096, so that jointly for live and dead load the economical depth lies in the neighbourhood of r = y'o' J^^s^ ^s demanded by stiffness. On the other hand, consider the second last bottom ' pane ' of the second table, p. 399, *r = '258, so that the economical depths for the Bollman trusses are quite out of the question, and for the Fink trusses too they can hardly be realized, if we may use such a word. The tables give the numerical values oiu, see expression (9), WL the other factors being — — - where / being usually expressed in tons per square inch, then W must be in tons and L in inches. Having obtained this theoretical volume of the truss to resist the load, it will be in cubic inches, and we must add 20 to 30 per cent, to stiffen the struts. The tables deal with the isosceles and rectangular trusses resisting an uniform dead load, an uniform moving load as long as the span, and a rolling load. The Fink trusses for a dead and rolling load are also shown for comparison. The values for the rolling load are quoted from the last edition, and are Levy's values. For the isosceles truss with vertical members to transmit part of the load to the upper joints, the formulas are shown and the results tabulated on a separate table along with the Bollman truss. Girder with an Odd Number of Bays. For the span divided into an odd number of parts (2m + 1) we find as in equation (1) that the joint volume of the upper and under booms is 6 A \ (27U + l)'^ And the volumes are as under UNIFORM DEAD LOAD. UNIFOEM MOVING LOAD. Ratio of 1 X ^ 1 ■» ^ 1 V. V) -< 1 i. '/ — 1 ~ 05 depth to c CO ■S C5 •X Si 1 1^ a span, 6 ^ Z do •■^ w 3 •< + w . -< 1 h ■1. u * H a u 7 r- w 1 »■ = V "^ X + U -^ )i< -T U L ^* o u: -h 1 9 1 1 u P< '■r. ^ a :. 1 a: ~ 1- t CO r + vVi r + ii f- 4- -2- J. » + 1 8 r r+ 1 J Is ■h ^ 2-879 ^ 3-817 _ 2-879 o 3-817 Oi -i^.r II ^ 1-975 c 2-600 w 1975 f2 2-600 "S 1 J* 1 -625 ■„ 2-125 1-625 •„ 2-125 g^ i 1 (100 ." 1-2.50 ." 1-000 I 1-250 C4 * 866 '' 1000 * -866 * 1000 09 2r + -A-i •>r J. J"- i Vr -f ./.V ^ Vr + A\ \ ^ T^/f .^ 2-946 _ 3-415 _ 2-996 ., 3 504 P4 ce II ^ 2-075 1-750 ^ 2-388 ■, 2-000 ^ 2-118 • 1-792 Ji 2-456 •„ 2()63 J. ^ ." 1-250 .: 1.375 ." 1-302 I 1-437 Tt< * 1225 '' 1323 * 1 283 * 1393 4r + -.vVJr 4;- ^ -aV i Sr + -;\V i f r + -AV ^ -.\ ^ 2-961 ^ 3-196 3025 „ 3-288 Pi h Tt< t; 2-197 a| 2-353 o 2-266 ;:j 2-442 X II 1-9.37 • 2<;63 2-016 ■|, 2-157 •1 1895 ;3 s i 1696 i 1-768 ." 1808 1 * 1-719 '- 1 781 \ * 1-852 * 1-922 GO 8/- » -.%V i 8r + v'A- i Vr + niii Vr + -3W4- i S -i\- :5141 3258 3-236 3-370 1— 1 W 00 ':; 2-5;j8 2 ^"616 « 2-660 :; 2-750 0« 7i J. II II 'J.. '5 9! II 2-453 II 2-516 11 2-617 f3* :,- 2359 i- 2 411 ^- 2532 i- 2592 CO \ 2 695 2-727 2-991 3-028 16r+-,v;'H- i Ifir + ^AVn-i Vr + itSii 3 '+ 4nt€ r ;2 3-624 3-682 3-800 3-866 P4 l-t CO o r .'5-:304 o r 3-343 g 3552 1 c 3-560 S 3 584 9 3-605 -a II II 3302 II 3341 M II & X s :^'- 3-36.3 i- 3-395 ^- 3-681 »- 3-717 ^ \ 4-682 4-697 1 5-309 5-327 • The value of r (or ininiriniin volume. ILOLLINe LOAD. FINK TEU88. 1 Ratio — ' ^ « 1 i. ■A — 1 ■_ ^ e-i r/;- »0 1 of a J. •:. •/; 1 „ ; y < ^' deptli r^ «;-? ai o 4< 13 C ,7. O ^'i^ oT Se] 1- 'c u O to span, 2 c CJ i^ H 3^ 3 ^ ^ "^ ~ _ A •yj -H O 1 a 1 ' - s = O j '■=1 Si2 8 s: s ^ ^ ■^ £i. CO CO 5 CI 2r + ' Ji- 2r + J i r4 i I 2/- + ^ J; OB ^ 5-758 o 7-633 II o 3-817 _, 7-633 iV 5 3-9A0 g 5.200 ~ 5 2- 600 .9 5-200 -i^r O. ".. 3--'50 ;, 4-250 II - '-^ •■ 4-250 i g ;^ 2-000 ^ 2-5O0 II ^ 1-251) v^ 2-50() X 4 o* * 1732 * 2000 1 ~l * 1000 "^ 2 000 9 or + ^ i 5r + U J I ■2r + -iV i 6r T- ^ ^ OS ^ 5-958 ^ 7-365 II .. 4-821 „ 9-775 -tV »4 - 4-J50 o 5-188 r ?; 3 325 2J 6-850 1 At -. 3 125 o II •,; 2-750 :1 1-750 ■; 5-750 -^ 4-000 i * 2 739 * 3062 c^ * 1581 * 3873 ""^ llr+Ui: llr + -,VK-i CO II 3r+fi I 14r+ K^ i ^ 6-124 ^ 6-944 5-122 11-246 T^> 2 4-H94 2 5-241 ^ ;: 3-582 a5 8-275 iV 04 ; 4-250 i 3977 • 4-688 I 4-268 o II t= 3-000 II f 7-J50 1 6 205 i "3 1* * 4-188 * 4-406 e *^ 2-063 1984 i *"" 6-250 t CO ^3r + ^\2g- J- 23r + H-ii^ 4r + ^",V -> 30r+ 5} J 6-748 7-188 II 5 247 1 12-781 ^ 1 ^ 5-777 ?, 6070 5g 3-720 ! :g 10188 -tV ! Is P4 r 5655 T ^ ?* 1 7" II 5-656 11 5 891 2 11 3-156 ii 9-500 "3 *■ *^ 5 889 II ^T *^ 9287 7-141 7-258 fi 2-:? 28 2 305 10 375 -1 CO 47r + -,W*-^ 47r + /„V),-i II o/-^tV.'5 J ■ 1 8-246 8-458 5-328 15-149 1 6 f S 8005 SS 8180 ^ 30 C3 cd 9 8-108 9 8-250 o ?» 3 830 r 13-544 iV II 11 C>1 CO II " 13495 "3 ,> 8-590 *"- 8-715 , II %- 3-289 1 *-~ 13-625 » 49 13-113 13- 170 5< 2-5S-J 2581 18-438 i eo The hetivy figures give the minimutk volume. 400 APPLIED MECHANICS. [cHAP. XXI. Warren Triiss. — Dead and rolling load, odd number of bays. ^ _WL m(»t + 1) /^ 8m + 7 1\ j^ _ WZ m f 2m-, (13) -. 2(3w + 2) r + ^ o/^ — T^ • - • (14) 1 \ ^ ^ 3 (2m + 1)^ ry ^ ^ Bectangular Trusses. — Dead and rolling loads. „ WL m(m + 1) /_ 4m + 5 1\ .- _. ^d = —r -^ T^ 2r + ,^j^ . - • (15) IFZ m /^,^ ^, Sm^ + 2lm + 10 1 ^r- ,^ A 2(3W + 2) ?• + ^-T^r r— — . - / 2?n + 1\ ^ ^ ^{2m + lf r For (13) and (15) there is no shearing stress in the odd middle bay. In (16) it is assumed that there is only one diagonal in the middle bay. If there are a pair acting only one at a time, say both ties, and incapable of acting as struts, then (8??i2 + 21?^ + 10) is to be replaced by (8m^ + 21?;? + 13). We have not tabulated these quantities, as values sufficiently correct can be found by proportional parts from those which are tabulated for the even division of the span. For the uniform moving load in both trusses the coefficient of r is (14?;^' + 12??^ + 1) -f 6 {2m + 1) ; but for the Warren or isosceles truss the coefficient of - is r 32m.'' + 62m' + 28m + 1 ^ 24(2m. + 1)' ' ^^ For the rectan'nilar truss, assumiuLf two diagonals on the middle rectangle acting fine at a time, the coefficient of - is 16;»=' + 38m' + 23m + 1 .^. Vl{2)v + iy ^ ' CHAP. XXI.] ANALYSIS OF TKUSSES. 401 The Fink Truss. Here the beam A A' (fig. 7) is divided into a number of equal parts by a series of repeated bisections. There are thus 1 primary truss, 2 secondary trusses, 2'^ tertiary trusses, &c., so that the number of trusses of the i*'^ variety is 2""* where n is the number of bisections. Hence summing 2"'' for integer vahies of i from n to 1, we have 1 + 2 + 2- . . . 2"-' = 2" - 1, the total number of apexes of all orders, and therefore also 2" is the number of equal parts into which the boom is divided. The Dead Load. — If W be the dead load uniformly spread on the boom AA', then the load directly over each vertical strut is W ^ L v'+^, and a=^ is the common length of each segment. A A3 / 2 ^ 3 Ai Ao a a' 1 V, ^ ^ ^ J ^^ ^ \ vtx /I S\ ^ 1 \ 7^ 1 \ A \L \ w\ i^ X A B^_, B^ I ''- n is N'o. of bisections. « is No. ot varieties of Trusses. 2'» is No. of parts in span. FixK Tiiuss.— (Fig. 7.) Consider an apex of the order i?„-,-, the stress on the vertical strut is 10 the load on top of it, and this is all for i = 1; but for 1 = 2 there is besides the downward component pull of a pair of tie rods attached to the top of the strut, being between them another v:, or in all 2u\ Now a pair of the steepest ties transmit loads with a vertical component w per pair, the next steepest a vertical load of 2iv per pair, and the third steepest, a vertical component load 2hv per pair, &c. The load on the strut ^„_,-, B„^i (fig. 8) then is 2'-'?r. The pull on each tie from Bn.i is |- x 2'"'w sec 0j. The thrust on all segments of the boom is constant. At the end A a tie rod from one of each variety of apex comes in. The 2d 402 APPLIED MECHANICS. [chap, XXI. horizontal component load on the tie from Bn.i is ^ 2''hi: tan Bu Substituting for tan 0., and summing for all values of i from 1 to 11 so as to include one of each variety, we have thrust on boom = '^ 2" 2^('-i) = ^"^ (1 + 2^ + 2' 111 1 2li 7m 2*" -1 2"u'.2'Va 1-(^)'" WL 1 - (1)^ Dividing this by / gives the constant sectional area of the boom, and further multiplying by its length, we have volume of boom = Wl- ilY' D f 6 h (A) B77, -2 i z w or i w The i'*' Vakiety oi' Thlss. — (Fig. 8.) The volume of the strut A,,., B,,--, is 2'"'?'.7i ^f, and multiply- ing by 2""' gives us 2"'^ ich -f ,/'as the volume of all the struts of the same variety. But the expression being independent of i shows that one strut of the primary truss has the same volume as the Uoo struts of the secondary trusses, as the fi>vr struts of the tertiary trusses, &c. We have to simply multiply the last expression by n, the number of varieties, and volume of struts = (2"K >) nh _ W nh (B) CHAP. XXI.] ANALYSIS OF TRUSSES. 403 Again, tlie volume of the pairs of ties from all apexes />„_,• is stress , , . i2^%sec0i , „ 2 X number x ~—y- x length = 2 x 2"^ x x A sec % = 2" > — SeC-^/ = 2" ' — r r:; / / ^^' Summing for values of i from 1 to n, volume of ties = 2"-^ '^^ + 2"-' ^ t 2M'-i) / ¥ I 1''w nh 2"w a- ,, ^, ^ = + Q + 2- + 2* . -f 2-"--) / 2 f 2h^ ^ _ Wiih PT «^ 2^" - 1 / 2 f 2h 3 / I 2 '^ 6 77 (C) Summing (A), (B), and (C), and putting h = rL, then the volume of the Fink truss to resist a dead load is ^ jvLf 1-ar 1\ Xr'"' — 3 -r)' ^^^^ TA€ Rolling Load.. — When W the rolling load is at the apex An-i, the load on the strut An-i Bn.i is W, which also is the vertical component load on the pair of ties supporting it at Bn-i The stresses in all pieces from the other apexes of the same order, or of higher orders, is i^ero ; while those on struts from apexes of lower order are fractions of W. Multiplying W, the greatest load on a strut, by h its length, and dividing by/, we have its volume, and as they are all alike, we must multiply by 2" - 1 their number. W Volume of struts = (2" - 1) h-- ■ (D) The greatest pull on a tie from J5„.j is hW sec Oi, the area of this tie W sec b ^ 2/" and its length h sec 0„ and there are 2d 2 404 APPUED MECHANICS. [CIIAP. XXI. twice as many as there are apexes of order Bnj, that is, there are 2 x 2""*. The volume then is --r 2"-' sec- Qi = — 2"-' ^ , which has to be summed for vahies of i from 1 to n, and Wh vohime of ties =— r- (1 + 2 + 2= . . . + 2»-^) Wa- fh^- . . . + ^ ; = — |(2"-])/^ + (2"-l)(i)'--^J = _(^(2"-l)/;-f (2"-l)a)"--J. (E) The greatest thrust on the boom occurs when TV is at -4„ the centre of the span. It is ^W ^ \L ^ h, dividing by /and multiplying by L, volume of boom = .- -r -7- • (F) f i h Summing (D), (E), and (F), and putting h = vL, the volume of the Fink truss to resist a rolling load W is ^^.J^(2(,..-l)...iL4ill',i). (20, If the dead load be heloiv the truss, each vertical strut is relieved of a load ic, but is not altogether relieved of load, as Levy assumes. Diminishing then the volume of each strut by w/i ^ f, and multiplying by 2" - 1 the number of struts, then we have the proper deduction to be made from V ^ to be 2" - 1 Wh which caves the volume of the Fink truss to resist a dead load hcloio the truss r. = '*^^{o.-l + (4)")'-^^-.'{ (19') CHAP. XXI.] ANALYSIS OF TRUSSES. 40; And again for the rolling load below the Fink truss, we must not leave out the whole volume of the vertical struts, as Levy inadvertently does. For tlie rolling load JV l)elow the truss, the greatest stress on a strut occurs when W is at the foot of the next adjoining strut. The amount is then 1 Iv ' — I — + - . . —■ — V2 4 8 2*-V = JV I iW) Hence the volume of all such struts is 2.-Mr(i-Qj -HC Summing for all the n varieties, volume of struts = ^<Q = / f 3 1- (D*) Summing (D*), (E), and (F), and putting L = rh, we find the volume of the Fink truss to resist a load PF rolling below it to be WL (/ 2"^^ + (I)"- F,.= - 2 3 - [\Y^ V) 4 "r) (20*) We have not thought it necessary to tabulate the values for (19*) and (20*), as the Fink and Bollman trusses are only given here for comparison, and for an exercise in analysis. The trusses themselves are out of date and far from economical for the II = 2w is No. of parts in span. Bollman Truss. — (Fig. 9.) rolling load. The economy for dead load, too, is only apparent, as it could only be realized with a depth of truss which is im- practicable, as it would involve a stiffening of long struts that would eat up the economy. 406 .APPLIED MECHANICS. [chap. XXI. The Bollman Truss. In this system of trussing (fig. 9) the upper boom AA' is divided into m equal parts ; at each joint is a vertical strut whose foot is tied back to the two abutments. The i"' Truss from Left Abutment. — (Fig. 10.) Dead load. — Consider the i^^ truss from the left abutment (fig. 10). __ , . , ^ length Volume of struts = number x common stress x ° — ,,W h m-1 W , = {m - 1) — - = . -.- .h; m f m f sin ABjA' = sin (6 + cf)) = sin 9 cos </> + cos sin 6, (G) and using the surds in fig. 10, which are derived by the forty- seventh proposition of Euclid I, we have sin(^ + ^) = mha y/h'^ + i-a^ y/li- + (w - ifa^ For equilibrium at Bi we have W , i^i : i!2 : — : : sin AiBiA : sin AiB[A : sin ABiA m : : cos B : cos 9 : sin {Q * (j>). Using the surds in the fig. 10, we have ^ ^ W (m - i) y/rTlv- in mh and multiplying by the length ABt and dividing by/ we have the volume of ABi = '' .(m -z)(/i^ + i'a'), in nj CHAP. XXI.] ANATA'SIS OF TRUSSES. 407 and also the volume of IFi A'Bi = ^77-7 ('''" + (^'^ - *')•«■)• m-lif Their sum is the volume of the pair of ties from the apex Bi and is W —.-, (h^ + i(rii-i)(ir). mlif To sum the first term, which is independent of i, it must be multiplied by {m - 1) the number of apexes ; the other two terms are to be summed for values of i from 1 to (m - 1). Putting L for a m finally jy / »'-i '"-1 vol. of ties = — r-z, (in -l)h- + ma^ V i- a- V v ^V (, ,,-,, o [m - l)m , {m - l)?n{2m - 1) ' (m -l]/r + mar —-^ a- mh/y ' 2 6 Wm-lf^ m + lL'\ / m \ Qm h Part of the thrust on AA' is the horizontal component of ti, which is W m - i . ■ W . . .. ((■ ^1 cos f/» = — . — 7" . ta = — - I [m -%) —• m mil mil L Summing for values of i from 1 to {in - 1) includes the horizontal pull of one rod from each apex, and thrust on boom = — r -7- //^ T ? - 2 i' ) mh X V 1 1 / W a}(m^{m-V) ( m-l)w(2m-l) 6 mil L \ 2 WL m- - 1 6/t nt" Multiplying by L -:-/, we get volume of boom = — ; — -—- — ^ • (I) f D??r li •iOS APPLIED MECHANICS. [CHAP. XXI. Summing 'G), fH), and (I), and putting m = 2n so that n may be the number of parts in half span as in the other trusses, also putting h = rL, we have volume of Bellman truss to resist a dead load W over the truss to be WL 2n-lf 2n+l 1\ ^.^^. / n \ I2n rj Lire load. — Let the live load W^come to the point A,, and it will be seen that the load is now m times what it was for the dead load. Now as W comes to every point in succession, the volume of the struts and of the ties will be obtained by multi- plying the expressions (G) and (H) by m, when we have TF volume of struts = (w - 1) — 7i ; (J) volume of ties = {jii - 1) -y{h+ — — ) • (K) / V 6)/^ h With the boom it is otherwise. When W is at Bi , we get the thrust on the boom by multiplying the former value of the horizontal component of ^i by m, which gives W.. .cr -^.(m-^)-; but this is thrust on the boom only as long as IF remains at Ai, but i(m - i) is greatest when the factors are equal, so that the maximum thrust on the boom occurs when i = m - i or i = hn, that is, when W is at the middle point of the boom. Substituting this we have max. thrust on boom = -; — r v li -i L Dividing this by / to get the sectional area, and multiplying by L, we get , „, JF m'cc' WD volume of boom = — — = — - — • (L) jli 4 f/l 4: Summing (J), (K), and (L), and putting m = 2)i, and 7/ = rL, CHAP. XXI.] ANALYSIS OF Tl'JJSSES, 400 we have the vohune of the Bollman truss to resist a load W rolling over it to be F. = -y- (2 {2n - l)r + ^^^ • " ) • (22) BOLLMAN ISOSCELES TRUSS. TRUSS. Part of Load sent to vertices by vertical rods. i. - ^^T^ : " o r Kstio LOAD 12n s " 1-1 e* , i! 1 9 1 ^ of depth A a + o -S^ a =■ O -e to M ir- oi + 2 -S- K 00 epaD, 1 S 7 . ° ^ o 7 A » i * « i. . b -1 X J_ c« '"Z (S r-t-i i 2r + i i ' r + i i 2r + i i o 3-817 ^ 7-633 3-817 . 7 633 iV o 2-600 o 6-200 o a-600 5 5200 T^r — •„ 2-125 ■ 4-250 t 2-125 - 4-250 i V II : 1-250 ^ 2-500 " 1250 " 2-500 1 * 1-000 • 2000 * 1000 <■ 2000 f (»- + -A- i) er + i ^ ir*^i i ir*H i =» ^'^f ^ 13-525 3-382 ra ''332 -i^r « 3-275 g 9-350 =» 2-338 £i 5-138 -i^. (M •„ 2-688 •- 7>75a " 1-937 " 4-313 i ^ 1, 1-625 * 1369 i 5-00O • 4-683 :: 1-250 * 1-146 ° 3-000 *" 2655 i 5-039 14r + fJ > 24-371 .!r+-,V. i ^»r + -,y.- i ?? 3-456 ;5 17-025 3096 6-844 •^h T 2-844 ^ 14-250 2 2-203 § 5-091 -l^r -f* T^ 1-876 « 4-500 i I] .,>- 1-750 »>- 9-750 9354 11 11 3-967 » 1-516 »i~ 1-406 »> 4031 1 1398 v(»■-^ii y 30/-+ J} i 5-105 45-594 !-^ + -.V. i Vr + iv* i _ 3-508 _^ 32063 3-025 6955 iV CO 5 2-266 g 5-720 ■>';,- 00 • 2-891 -„ 27-000 ?* T 11 i II 11 2-016 t: 5-454 1. ^ » 1-797 •'' 19-125 ,^ 1-808 »'- 5-422 1-578 18675 1-852 6-383 ilir + ii i) 62r+V.'^ J \hr 4. ^aiSjtf 1 ^%--hA% i 5-124 87-805 3-182 7-973 -iV 11^ o CO o 7-500 CO :;; 3-524 o 61-981 :>: 2-597 g 7-509 <^ 11 ." 2-906 " 52-375 " 2-458 11 7-786 *■ *" r-816 • 37-813 * 2,436 • 1-607 37190 2-822 11-280 i If the load be below the truss instead of above it, the volume of the struts (G) and (J) can be left out, as there is no load on them theoretically, so that Vd and V,- are obtained from (21) and (22) by halving the first term. 410 APPLIED MECHANICS. [CHAP. XXI. Examples. 1. A Warren girder has a span of 160 feet and a depth of 20 feet, and is subdivided into 16 eqnal parts of 10 feet each. Find the theoretical volume of wrought iron for which /= 4 tons per square inch, that the girder may resist a rolling load of 80 tons. Here « = 8, r = ^, iJ = 80 tons, /= 4 tons per sq. in. and Z = 1920 inches. From Levy's table, second column, fourth line, we get 5-6o6, and HZ I', = .5-656 — = 217190 cubic inches. 2. Determine a girder of similar type to bear the dead weight of the last, there being 8000 cubic inches of iron to the ton, that is TFi = 217190 -!- 8000 = 27-2 tons, and also ^' = 160 x | = 60 tons, a share of the deck or platform which is three-eighths of a ton per foot. Vi = 2-391 — TT- = 28167 cubic inches, TTj = 3-52 tons, TF'L Yz = 2-391 — r- = 68866 cubic inches, Wi = 8-61 tons. 3. Find now the theoretical weight of the girder to resist the rolling load E, its own weight, and a share of the deck or pl.itform W= ZFi + W2 + TFs = 39-33 ions nearly. But no provision has been made to resist W2 = 3"52 tons. For this V = 2-391 X IFiL 4 /= 4040 cubit; inches, or its weight is half a ton, so that TF= 40 tons. Adding 20 per cent, to stiffen the long struts and allow for rivet heads, &c., the probable weight is 48 tons. 4. Find the weight of the girder if it were braced on the Fink system. The tabular numbers are now 9-500 for the rolling load, and 3-1.56 for the dead load. ^"1 = 45-6, Jr2 = 8-6, 7r3=ll-4, and «• = 0-67, JF= Wx + W% + 7F3 + w = 66-3 tons. Adding 20 per cent, we have SO tons. 5. Find the theoretical volume of the isosceles truss shown on fig. 1 1 . Here n = 3 and r = i. As w = 3 is not given in the table, these values of n and r are to be substituted into the general formula at the top of columns one an<l two, giving 1709 for the dead load, and 3703 for the rolling load, as piinted on tig. 1 1, which see. CHAP. XXI.] ANALYSIS OF TRUSSES. 411 The upper figure shows the "Warren or isosceles girder. The number of bays in the half -span is n (3 on the figure). The span is L inches, so that r = L -^ 4m is half a bay. The total uniform dead load is IF tons, giving a parcel of hv over each abutment, and of w= W~'2n at each intermediate joint between them. The left supporting force is P = \W - ho = tv {n — §). The depth of the girder is A inches, and h = rL, when expressed as a fraction of the span. "We will find ISOSCELKS AND RECTAXOfLAU BkACINO. 4» V = 1.709 — jr~ cub. in. r=// -^ I. = 1. V = 3.-03: i — r~ ■ l> / y / \ \ \ <- — c -> 1 ^ 2n < I 1 f \ \ \/ / 1 c = o r 554 — Y cub. WL cub. in. 11. the bending moment at each of the sections through the upper joints in left half of span, and add them. This sum divided by h gives the sum of the tensions on the n lower booms, and dividing this by/, we have the joint cross-sectional area in square inches of those n booms as if packed in one bundle. Lastly, multiplying by their common length "ic, we have their volume in cubic inches. Using the odd digits for the upper joints, beginning at the left end, the bending moments are — .1/, = F.c. Mz = F.2c-ivic). Ms = P.!ic-u>(c + 3c). Mt = F.lc- w(c+ 3c + 5c). 412 APPLIED MECHANICS. [CHAP. XXI. 5 J/ = Fc {I -r '-i + b to II terms) - my (1 + 4 4- to h - 1 terms) = w {n - ^) or - ur . i- (n - 1) u (in - 1) = l-wcn{in'^ - 1). The sum of the odd digits being the square of the number of terms, while the sum of the squares of the natural numbers up to n is g n (n + 1) (2« + !)• Multi- plying the above sum by 2c -r/%, and doubling to include the other half of the girder, we have for the volume of the lower boom — 2 If - -— »<(4w^ — l)c- cubic inches. 3 jh With zero at the left abutment, and the even digits at the lower joints — Mz = F. 2c. Ml = F.4c- w.'Ic. Ms = F.6i- u-{2c + 4c). Me = F. Sc-w(2e + 4c + 6c). Mi,, = F. 2>ic - 2u-c (1 + 2 to « - 1 terras). This last line is the bending moment at the centre, half of which must be subtracted, so as to exclude half the upper central boom. 2Jf - ^if2„ = 2Fc(l + 2ton terms) - 2t*c (I + 3 + 6 to u - 1 terms) = 2w (« - i) c . in (« 4 I) - 2tvc . ^ {n - 1) n (« + 1) — tv {« — 5)mc 4 u-c , ^ (« - 1) il. = iwc«(4««- 1), which gives the same volume as for the lower boom. The only additional series being 1-I-3 + 6 + 10 + &c., the sum of which is ^» {n + 1) {n + 2), where u is the number of terms. The length of each diagonal brace is ^(c^ 4- h-), and if d be its slope to the vertic il, sec d = V(c* + A*) -f- //. The shearing forces in the bays, beginning at the left abutment, are — K„2 = F\ F2,i = P- w ; Fu a = F- 2k-, 2J'= F. n - :o{l + 2 + 3 to « - 1 terms), This sum, wlien multiplied by sec 0, becomes the joint thrtist on all the diagonal struts in left half of girder; dividing by /gives us their sectional area as if all in one bundle ; multiplying by their common length, and doubling, so as to include both halves ot girder, we have vulume of diagonal struts — u-w* sec X . -yc' + h- = — »j2 (c* + A-). CHAP. XXII.] DESIGN OF MASONRY AKCHES. 413 The same expression is the volume of the diagonal ties. Adding, we get the volume of the Warren or isosceles girder to resist a uniform dead load. ^' = 3^^' ('"'-')'•' + 7^' "'(^' + ''') = — - 1 nh + „- c- W ( , 8w"- + 3« - 2 L- h + 3/t 16/i= WL I 8«- + 3>( - 2 1 + — ' \ nr + / ( iSn- r) 6. A giriier is braced as shown in fig. 1. Span Z = 168 feet, h = 12 feet, and « = 6. The struts are shorter than the ties somewhat in the manner of the Post truss. The shape of each triangle is such that a perpendicular from the vertex divides the base into a short segment c = 6 feet remote from the centre, and a longer one of 9 feet adjacent to it. It will he seen that the struts are 13 feet long, while the ties are 15 feet long. Find the theoretical volume of tlie truss to resist an uniform load of four-sevenths of a ton per toor. That is, 7r= 96 tons or 8 tons concentrated at each lower joint of fig. 1. The jtudent should work this example by taking out the volumes in detail as on last example, and cheek his result with that obtained by substituting JF= 96, / = 4, i = 12 X 168, r = -^^, c = 12 x 5 in equation (2). Atis. 148480 cubic inches. CHAPTER XXII. THE SCIENTIFIC DESIGN OF MASONRY ARCHES, In this chapter we have chiefly in view the design of important Masonry Kailway Bridges and Viaducts, having arches of long span, with as small a rise as is consistent with moderately heavy abutments, and in which the surcharge, over the top of the key- stone, is no more than sufficient for the proper formation of the bed of the railway, which may be taken at about eighteen inches. To successfully resist incessant, swift, heavy traffic, the arch-ring, or at least some central portion of it, must not be restrained by rigid backing. An arch-ring well built of wedges of stone, or voussoirs, as they are called, and more especially one built of brick masonry in concentric rings, as is the most modern practice, has ample elasticity to accommodate itself to all the vagaries of the live load, provided it be not overmuch constrained by a too rigid superstructure. The two joints at equal distances from 414 APPLIED MECHANICS. [chap. XXII. the key-stone, which map out this centra elastic part of the •arch-ring, are practically the joints of rupture ; and a horizon- tal line, through the upper edges of these joints, is the level of heavy backing. The backing above this level should consist of light spandril walls bonded into the arch-ring by through headers, or simply riding on the arch-ring. Their principal function is to give a horizontal passive reaction on the arch- ring when it bulges out due to the live load on the other half. They stand upon heavy spandril walls below them, and stretch horizontally such a distance that the friction on their Ijase shall at least be equal to the excess of the horizontal thrust of the loaded half arch over that of the unloaded half. The light spandrils are best built of rubble masonry, or of brickwork with plenty of mortar in the vertical joints, to allow the arch-ring sufficient play while the passive horizontal re- action is being called out. The spandrils serve also to lighten Fis. I. the superstructure and reduce its average weight per cubic foot, as the spaces can be left void and the walls covered across. The heavy backing consists of thicker walls with vertical- dressed joints, and having the stones or bricks packed close horizontally, so as to yield but little when the centre is struck. The voussoirs, below the point of rupture, have their backs •dressed in a flight of steps, so as to take the horizontal thrust truly. The heavy spandrils must stretch horizontally a sufficient distance to give friction on their base equal to from two-thirds to one-half of the horizontal thrust of the arch at the crown. The earth tilled in behintl tlio spandrils can be relied upon to give promptly the remaining tliird or half, according as it is tilled loose, as in an embankment, or is rammed in layers between the masonry and the face of the excavation, as in some cuttings in old consolidated earth or rock. CHAP. XXII.] DESIGN OF MASONRY AUCIIES. 415 Observe, before the earth is filled, tliat the heavy backing can only take a large fraction of the horizontal load. But the centre of an arch is always struck when the superstructure is only partly built, else the superstructure would crack, due to the after-subsidence, so that only a part of the horizontal load is required. Of course the earth must be filled behind the spandrils before the remainder of the superstructure is built. Two rival designs present themselves for long-span light- surcharged masonry arches — the Segmental Arch and the Semi- elliptic Arch. In the one, the soffit is a segment of a circle, with a rise about a quarter of the span. The joints of rupture should be at the springings, which demands that the total headroom, from springing level to level of rails, should bear to the span the ratio, almost constant, of one to three. In this design, too, it ^z:^^ '2— o,< ' I on ^^rCA''^' \ Fk will be found that the arch-ring must thicken outwards from the crown to the springing, so that a skewback shall be from once and a half to twice as thick as the keystone. This may be disguised on the front of the bridge by an uniform moulding, of the thickness of the keystone, cut upon the face of the pend- ring of stones. In the other design, the soffit is a false semi-ellipse struck from five centres. At the ends of a quadrant the radii are those of the ellipse, being the square of one semi-diameter divided by the first power of the other. Between them a centre is interpolated, as on fig. 2 (see Eank., C.U., p. 420), at the intersection of two arcs, one from each of the main centres, with a radius equal to the difference of that main radius and the rise. It will require three templates to work in these three varieties of groups of voussoirs. In this design the rise of the arch may be about a fifth of the span, and the ring in this case is of uniform thickness. As the thrust at crown will not exceed twice that at springing, the ring can accommodate the line of stress in its middle third. 416 APPLIED MECHANICS. [chap. XXII. It is well to observe that this soffit, although called elliptic by courtesy, approximates in an equal degree to one of the elastic curves, called by Rankine the Hydrostatic Arch, or to that elastic curve moditied and then called by him the Geostatic Arch. (Eank., C.E., pp. 208, 419 ; 212, 420.) We have already given a practical definition of l\\& joint of rupture, and, as Eankine says {ihid., p. 428), the more rigid part of the ring, below it, is really part of the pier or abutment. It will appear as we proceed that this practical definition corre- sponds to Eankine's^wm^ of rupture for a linear rih loaded in any way. By locating these joints as far out from the crown as is consistent with equilibrium, two advantages are gained : one, economy of heavy backing ; the other, a long elastic field. Line of Steess. In calculations connected with the arch, it is convenient to consider one foot of the ring in the direction of its axis, just as if the bridge were only a foot broad, and the same symbol t gives the thickness of the ring in feet and the area in square feet of the joint on which two adjacent voussoirs abut. It is Fig. 3. assumed that their mutual pressure on each otlior is either uniformly distril)uted over that joint, or varies unifi)rmly from a maximum value at the one edge to a minimum at the other, Such a stress is indicated on a drawing by a sheaf of arrows varying in length, and with their feathers ranged in a straight slope, or more simply by a trapezium mapping them out. CHAP. XXII.] DESIGN OF MASONRY ARCHES. 417 Three varieties of this distribution of the stress on the left cheek of the keystone are shown upon fig. o. Each has the same maximum value ./'at the lower edge of the joint, and the minimum values are Ci = 0, c^ = \/, and e-s = |./' at the upper edges respectively, while the average values at the middle point of the joints are respectively 2h, Ih, and ih, equal to 4, 5, and 6 eighths of the maximum. The whole stress may be supposed concentrated at a point called the centre of stress. To find the centre of stress it is only necessary to construct the centre of gravity of the wedge whose face is the trapezium of which c and /'are the parallel sides. A construction shown on fig. 11, Ch. XIY, and fig. 3, Ch. XXII, is— Produce e to the left a distance equal to./', and./' to the right a distance equal to e, and the line joining the extremities cuts the line joining the middle points of e and ./ in the point g. If now S stands for the deviation of the centre of stress from the middle of the joint, we have, by similar triangles, U + S:ht-S::y+/:e+ hf :: p + hf : '2p - If. Since the average value ^j = hf + he, then the ratio — Maximum stress /' 6S Average stress p ' t ' So then, on fig. 3, we have ^1 = g , c\ = — , and ^a = jg , and e^ = 0, £■, = If, and e^ = \f. Let the ring on fig. 3 be that of a small model, the blocks resting on a centre and separate from each other, with square pins to preserve the spacing. The ring being loaded in a suitable manner, the pins are to be moved up and down in the joints till it is found that the centre can be lowered, leaving the ring in equilibrium. Then the curve joining the pins is the line of stress for that manner of loading. The equilibrium is unstable : any slight disturbance would make each pair of blocks to rotate about the pin between them till either their upper or lower edges came in contact. Xote, too, that the thrust between some of the blocks being oblique to the joint, there is a tendency for the pin between them to roll over, if it were too small, or 2e I (1) 418 APPLIED MECHANICS. [CHAP. XXII. had its corners rounded. In the one case the equilibrium is said to be destroyed by heitdinr/, which is only stopped by the edges coming in contact, while in the other the equilibrium is destroyed by slylin;/, which is only stayed by the roughness and squareness of the pins. Equilibkium and Stability. For every alteration or re-arrangement of the load, the pins or fulcrums would require to be shifted into new positions to maintain equilibrium, so that the line of stress changes its form to suit the load. A small extra load laid on the centre of the part of the arch shown on fig. o would slightly flatten the arch, and tend to make the central joints shut at their upper edges, so that the pins there would have to be moved up, and the new line of stress would then be a sharper curve. The same thing liappens in the masonry arch, where the mortar fills the spaces between the joints. Every slight alteration of the load alters the distribution of the stress over the joints in the elastic untrammelled part of the arch. The centres of stress move up and down in the joints, just as we moved the pins in the model, and the line of stress freely changes its shape, and accommodates itself to the change of load. And wherever the ring is it?,Q\i flattened and depressed, there the line of stress becomes sliarper in curvature, and rues into a hif/hcr position in the ring. Also, the friction prevents the stones sliding where the obliquity of the stress is only slicrht, say less than 17°. To render the equilibrium secure it is only necessary to have deep joints, and to limit the displacement of the centre of stress to some central field of the joints. This displacement should not be greater than a sixth of the thickness of the joint, for then the minimum stress at one edge is zero (see fig. 3) ; to force the stress further would mean opening the joints, so that as the live load crossed and re-crossed, the joints would ' work,' the mortar crumble and drop out, and the whole be destroyed. The conditions of statical and dynamical equilibrium of an arch-ring practically reduce to this — That the line of stress shall always be in the middle third of the arch-ring, and the joints be normal to the soffit ; it being then practically im- possible to draw a curve, in the middle third or ' kerne/ ' of the ring, which shall cross any of those joints at an angle greater than 17°. chap. xxii.] design of imasonry arches. 419 Strength. "We will take the ultimate resistance of sandstone to crushing at / = 576,000 lbs. per square foot, and its weight at 140 lbs. per cubic foot. And the least factor of safety shall be ten. The ./ shown in fig. o, indicating the working intensity of the thrust at the keystone, is not to exceed 57,600 lbs. per square foot, if the ring be sandstone. For strong brick we take/= 154,000 lbs. per square foot, and vj = 112 lbs. per cubic foot. Now, as the loads are to be reckoned in cubic feet or columns of the actual substance of the bridges, then, speaking directly as to the strength, but inversely as to the mass, strong brick is one-third as efficient as sandstone, while granite is twice. as efficient. For granite, /= 1,350,000 and vj = 164. For large bridges the question of strength is entirely domi- nated by that of stability ; but for small arches where the s%ir- cliavfje is rclativeli/ larger, the question of strength, on the other hand, dominates that of stability, and it becomes convenient to confine the line of stress to closer limits, and we then take as a ' kernel,' a middle yZ/i^/t of the ring, and as a ' kernel,' a middle ninth (see fig. 3). In this last, the line of stress may be looked upon as practically up the centre of the ring, furnishing no reliable clue to the proper thickness of the ring. But even in this case, the maximum and average stress on the keystone differ by 25 per cent. In designing arches (usually short in span) with heavy surcharges, one is really designing a bent strut — a ^'ery difficult matter in itself. The line of stress is practically up the centre, but with a probable variation of 25 per cent. The difficulty in the one case is, that the factor of safety to be used is left entirely to the judgment, whereas, in the long span lightly surcharged arches, the stability demands so deep an arch-ring, that the factor of safety against direct crushing is beyond what prudence would otherwise dictate. Balanced Linear Rib or Chain. The Linear Eib and Chain are ideal structures, usually curved or polygonal. They are designed to suit or resist a given external set of loads, with the intention of afterwards clothing the sides of arcs in timber, metal, or masonry. They are then designs for actual engineering structures. The pins in the model (fig. 3) map out such a linear rib, and its clothing is masonry. Eib and chain are to distinguish whether the 2e 2 420 APPLIED MECHA^'ICS. [CHAP. XXII. members of the structure are in compression or tension. The rib is of far greater practical importance, but its equilibrium is unstable, until a measure of security is given to it by the material clothing. It is hard for the mind to conceive of the linear rib witliout its clothing ; and still harder to write about its e(iuiliV)rium, clearly and accurately, in words. On the other hand, the linear chain may be regarded as having a slight amount of material, of too little depth to interfere with its flexibility, and of little weight compared to the external loads. It may also be supposed to be furnished with hinges at short intervals, and so be perfectly free to assume the proper shape for equilibriiim, whatever may be the external loads, and to return again to that shape if disturbed. Consider the equili- brium of the pin of one hinge. The two links which it joins must be pulling the pin in opposite directions with equal force. This can only be the case if the mutual interaction of the two links, through the medium of the pin, be along the tangent to the chain at that point. We must assume a hinge at every point, and so — The rjcncral condition of equilibrium is, that the 2ndlor thrust along a linear chain or rib, at each point, is alone/ the tangent. Having designed a linear chain, for given loads, it is only necessary to invert it, or suppose the loads reversed, and we have the corresponding linear rib. The funicular polygon of older writers, and the link polygon (see tig. 10, Ch. IV) of graphic statics, are examples of linear structures. All the ribs we are about to treat of shall be horizontal at the central highest point or croini, and shall spring from two points on the same level. The suspension bridge is the only practical example of a linear chain. In its primitive form, with vertical rods, and free from diagonal bracing or stiffening girders, part of the load was nearly uniform along the chain, and another part uniform along the span or platform. It formed a good example of a balanced chain under vertical loads alone. The two theorems, that the linear chain is in form a catenary when loaded uniformly along the chain, and a 2}<^'>'(t'l'ola when loaded uniforndy along the span, seemed to have an important application to the suspension bridge, and much has been written about them under that misapprehension. It is impossible to liave eitlier of these loads alone, in a useful structure. Still, the two theorems inform us that tlie chain of tlie bridge is, in form, something between the catenary and the parabola. As both of these curves are sharpest at the vertex, they show that this also is the char- acter of the bridge-cliain. But the sharpness at the centre of CHAP. XXII.] DESIGN OF MASONRY AllClIES. 421 the Inidoe-chain is not very marked, because the dip is always a mere fraction of the span, so that the corresponding ares near the vertex of both catenary and parabola are sensibly circular. In the modern suspension bridge, what with oblique rods, stiffening girders, and bracing, the load is no longer wholly vertical, and the chain may have a variety of forms. These two theorems, although of little practical use, are exceedingly convenient as a start-off in the study of the balanced semicircular rib or chain. An elegant proof that the linear chain hangs in the catenary when loaded uniformly along the chain, and that that curve is sharpest at its lowest point, may be found in Williamson's Iniajral Calculus. -^ * ^ ' ' I 'Earih (loose or gunned). Fis. 4. The parabolic chain, balanced under a load uniform along the span, is figured in Kankine's Civil Engineering at page 188. The proof is simple and instructive, and may be given here by supposing fig. 4 modified for that purpose. Let the quadrant ABS be the edge of a thin steel rib or curved ribbon, one foot broad normal to the paper, and supported by a horizontal thrust T^ at the crown hinge A, and a tangential thrust T^ at the springing hinge >S'. The only load on it is to be the load of 2? lbs. per foot, spread uniformly on the horizontal platform ah. This load is to be transmitted down to the rib by vertical struts without weight themselves. The plat- form ah is also one foot broad normal to the paper, and is to be quite flexible, so as to allow the load to be transmitted 422 APPLIED MECHANICS. [CHAP. XXII to the rib without constraint. The parallelogram standing on the platform ah, and having the height p, is called the vertical-load-area. Consider the equilibrium of a portion of the rib from the crown hinge A, out to any other hinge B, and suppose it to be rigid, having all hinges between A. and B clamped. The corresponding part of the vertical-load-area, which is sliaded on fig. 4, may now be supposed to be con- centrated into one force P, acting down through the centre of gravity of that shaded area. AB is balanced by the three forces 1\, T, and P. They must therefore meet at one point. The general condition of equilibrium requires T to act along the tangent at B. If we lower the shaded area till a coincides with the crown A, it becomes apparent that the tangent at B bisects its base ae. The rib is parabolic, for that is the only curve whose tangent at any point behaves in this manner. Now, as B is any point on the rib, it can be in succession each joint from the crown outwards, so every joint can be undamped, still preserving equilibrium. Hence, the particular condition of equilibrium of a linear rib under a vertical-load-area alone is — That the tangent at any point shall meet the crovni tangent on the vertical through the centre of gravity of the 'portion of the load-area from the croicn out to that point. Conjugate-Load- Areas. We have already discussed a linear rib to resist a vertical- load-area alone. The vertical load was spread on a horizontal platform, equal in length to the span of the rib, and one foot broad normal to the paper. The shape of the area was a parallelogram of height |a The rib itself was parabolic, and so peaked at the crown A. On fig. 4 this rib is shown : ABS is a quadrant of it. It is no longer parabolic, but has been deliberately pulled out horizontally at each pair of points, except the springing hinges, until it is a semicircle. It will no longer be in equilibrium ; the vertical load will tend to flatten it at the crown, and spread it out further at each pair of points. To prevent it spreading horizontally, we may suppose a pair of vertical platforms, of which rd is one, each in length equal to a radius, and a foot broad normal to the paper, and furnished with loads gravitating horizontally inwards. One of these loads is indicated on fig. 4 by a parallelogram of height q = /), standing on the platform cd as its base. It is called the conjugate horizuntal-load-area. If we consider the whole circular CHAl'. XXII. J DESIGN OF MASONRY ARCHES. 423 rib, it is almost axiomatic, that the two conjugate-load-areas shouhl be alike, from symmetry. The proof is as follows : — Let the quadrant ABS be rigid, with hinges only at its ends A and S ; then for horizontal component equilibrium, we have the thrust at the crown Tq equal to the whole of the horizontal- load-area on cd. Again, let AB be rigid, with hinges at its ends, and resolve 2\ the thrust along the tangent at B, into its vertical and horizontal components V and H. Then, for horizontal component equilibrium we have T^ equal to H, together with the unshaded part of the horizontal-load- area. It follows then that H must be equal to Q, the shaded part of that area. Also, V is equal to P, the shaded parts of the vertical- load-area. If 6 be the slope of the rib at B to the horizontal, then V must bear to H the ratio of sin Q to cos B, that T may be along the tangent at B, as required by the general condition of equilibriunr. But it is evident that P and Q are also in this ratio, as the bases ac and cf are sin i) and cos B, respectively, when the radius is taken as unity. Hence, with q constant and equal to p, we have the condition of equilibrium satisfied at each point B of the quadrant. For a right circular or other quadrantal linear rib, that is, a quadrant horizontal at the crown and vertical at the springing, which has to resist a pair of conjugate loads, one vertical and the other horizontal, the particular condition of equilibrium is — That the part of the vertical-load-area from the crown out to any point, ami the part of the horizontal-load-area from the springing up to that point, shall hear the same ratio to each other as that of the sine and cosine of the slope of the rib, at that point, to the horizon. On fig. 5 is shown a manner of loading the linear quadrant ABS of a circle with another pair of conjugate loads which balance it. The horizontal conjugate load, being the simpler, may be described first. It is what is called a uniformly varying load. The horizontal-load-area Gj stands on the base Hj and is mapped out by the 45"^, or 1 to 1 sloping line drawn from X, where the bases of the two areas meet. The vertical load con- sists of two parts. The first part is the substance between the horizontal base DCL and ABS the rib itself. The second part is a load distributed itniformly along the rib. The rate of loading along the rib is such that the amount on an arc AB would fill up the area OAB with the same substance as the first part is made of. On the arc AB the first part of the vertical load is a slab ABCD, one foot wide normal to the paper. The second part might be a uniform ring of voussoirs, pinned to the 424 APPLIED MECHANICS. [chap. XXIL arc AB at close intervals by their middle points, as on fig. 3. If the ring of vonssoirs be a foot wide normal to the paper, and be made of the same substance as the slab ABCD (fig. 5), it is clear that the thickness of the roussoirs (t, on fig. 3) must be half the radius, that their volume might complete the slab OBCD. Il will now be seen that P, equal to the shaded area OBCD, is the vertical load from the crown out to B, while the shaded area EFGH equals Q, the horizontal load from the springing up to B. The two shaded areas have their parallel sides equal, each to each, BC = EF, and OD = HG. Their areas are proportional to the distances between the parallel sides, respectively. Hence P : Q : : CD : HE : : sin 6 : cos 0, and the rib is balanced. Sur/a^-e L of C FluU Fig. 5. It is to be observed that only the upper part ABCD of the vertical-load-area truly represents the distribution of the first part of the vertical load. The area GAB merely gives the amount of the second part of the vertical load on AB, but in no ■way represents its distribution ; the arrow P should be further to the left on fig. 5, instead of going through the centre of buoyancy. Just as in Graphic Statics a line is used in the force-polygon to represent a force in the limited degree of magnitude only, so also is the area OBCD used here. Superposition of Loads. It is evident that if a rib be balanced under two systems of loading separately, it will balance under the systems conjointly. CHAR XXII.] DESIGN OF MASONRY ARCHES. 425 The whole load will then be represented by a pair of conjugate- load-areas described by geometrically adding or subtracting the two given pairs of areas. The proof of the balancing of the circular quadrant ABS for the manner of loading shown on fig. 5 is quite independent of the height of the horizontal platform BCL above A, the crown of the rib. Suppose it to be lowered till J) coincides with A, then AJ, the horizontal platform in its new position, cuts off a parallelogram JD from the vertical-load-area. At the same time the new 45° sloping boundary jn cuts off a parallelogram jG, of exactly the same area, from the horizontal conjugate-load-area. This is just the same thing as if we had removed the load shown on fig. 4. So that equilibrium of a circular quadrant ABS for the load on fig. 4 follows as a corollary once the equilibrium for the load fig. 5 has been established. In the same way the load on fig. 4 may be added to that on fig. 5. The rectangle P will sit on the top of CD, while the rectangle Q may either be placed against JIB on its right side, or Q may be distorted till its sides slope at 45° and placed against GF. See Distorting-table <fig. 17, Ch. IX). The practical importance of the load shown on fig. 4 is now evident, and the assumption of weightless struts transmitting the load to the rib is got rid of. The vertical load F is the share of the live load, which falls to a slice of a bridge one foot broad, due to two rows of locomotives covering a thirty-foot broad platform. The height of P is ^j ^ 1"5 feet when reduced to a column of the same material as the superstructure of the bridge. If iv be weight per cubic foot of the superstructure, then 2^ = I'ow. A good average value of ic is about one cwt. The other rectangle Q is the passive additional horizontal reaction with which the backing muse promptly oppose the extra effort of the arch to spread when the live load comes upon it. On the other hand, fig. 5 is a picture of the dead load on the bridge. The level of rails is DCL, while ABS is the line of stress up the middle of the voussoirs. These two lines form the upper and lower boundaries of the great hulk of the dead load. It is convenient to have names for such boundaries. They are called the extrados and the intrados of the load. The other portion of the vertical load, spread uniformly along the line of stress, is the weight of the half of the arch ring itself lying below ABS, and the excess iveight of the half lying above ABS, the material of the ring being always heavier than that of the superstructure by about 50 per cent. The conjugate-area y^'G^ maps out from the cer// croivn down to the springing, the passive 426 APPLIED MECHANICS. [CHAP. XXII. horizontal resistance with which the backing must oppose the tendency of the linear rib to spread due to the dead weight. As far as the masonry arch is concerned, fig. 5 is, as yet, a very imperfect picture of the dead load, the load spread along the rib being out of due proportion. We saw that it would require the thickness of the arch-ring to be half a radius, whereas the actual thickness of the key-stone is only about one-fifteenth of the radius. It is only for the convenience of being able to draw the horizontal-load-area with the 45° set-square, that we assumed this enormous load along the ring, the intention being to afterwards remove the greater part of it, making at the same time the corresponding correction upon the horizontal-load-area. It is better to remove the whole of the load assumed along the ring, as a small proportion can readily be restored again to suit the practical requirement. The next step will be to find the shape of the horizontal- load-area for the circular quadrant ABS bearing an uniform load along the rib of an intensity half its radius. We will calculate its breadths at ten equidistant points of the platform jH, and map out its shape with three straight lines or hatters sufficiently correct for practical purposes. The area is then to be drawn standing on the left side of jH, so as to subtract it from the area already drawn. The treble-batter then furnishes, once and for all, one boundary of the conjugate horizontal-load-area for the linear quadrant ABS bearing the vertical load between itself and the straight line DCL. The other boundary of the horizontal-load-area is the 1 to 1 batter LFG, drawn with the 45° set-square ; this boundary, and this one only, changes its position, as you vary the position of the rails DCL to different heights above the crown, or as you add or take oft' the live load. Fluid Load Uniform or Varying Potential. The pair of conjugate loads shown on fig. 4 can be produced simultaneously by placing the circular ring horizontally in water, the water being excluded from its inside. The platforms now form two pairs of barricades running north to south and east to west. It is evident the water will attack the barricades in the manner shown on the figure. The stresses ^; and q are now both passive. The active stress is the column of water from the surface down to level at which the ring of the cylinder lies. The height of this active column is called the potential, and the load shown on fig. 4 is shortly called a Jiuul load of uMiforni potential. CHAP. XXII.] DESIGN OF MASONRY ARCHES. 427 The stress as depicted on tig. 4 is called imiplauar stress, because the equilibiiuin of bodies subjected to it is in no way aftected by iheiwtnitial strrss vvliich is iionnal to the paper. The whole stress at this depth in the fluid is represented by a sphere. From this it will be seen that the resultant stress on the back of the rib due to the two conjugate equal loads p and q, fig. 4, is a normal stress r =^ p = q as indicated on the S. W. quadrant. For if we remove the four barricades, we know that is how the water will attack the back of the rib. The formal proof is given in Chapter II. In the same way the load on fig. 5 is described as a jiuid load of varjiing jJotcntial together with a uniform load along the rib of an intensity such that the load on any arc equals the weight of the fluid which would fill up the hollow subtended by the arc at the centre. Thrust at the Crown of a Rib. To get the thrust along the rib at any point B of the circular rib, loaded with the normal load of constant intensity r, as shown on fig. 4, we have T- = V^ + H- = ^j^ . ar + q^ . cp = r"W- for !p = q = i\ and the radius R is the hypotenuse of the right-angled triangle with sides equal to ae and cf. The thrust at any point is the product of the radius and the intensity of the external normal stress on the back of the rib at the point. At the crown A the constant value of the thrust is readily found by inspection of the figure, for T(, = q.cd = rR. Now this would be the thrust at the crown, although the rib only extended for a little arc of the circle on each side of it, and would still be the thrust at the crown, if the rib, after that arc, continued on, of any form or shape whatever. At any point of a rib, if we are sure the external load is normal, and if we know the radius of a circle which j^^.'? the rib for a little arc there, we can, by taking their product, get the thrust along the rib. Now the ribs we are considering are all horizontal at their crown, and suffer only a vertical load there, so we have this rule : — The thrust T,, at tJie crown of the rib is the product i^p,^ of the depth of the load at the ci'oimi and the radius of curvatiirc there. If we apply the rule for the load shown on fig. 5, and remember that the part of the vertical-load-area inside the rib is due to roussoirs of thickness h\ we get H^^ = {DA + \r)r, where the weight of a cubic foot of the material is taken as unity. This is exactly the value of the horizontal conjugate- load-area y^6-', which hns to be balanced by Hq, and so the rule is verified. 428 applied mechanics. [chap. xxii. Buried Arches, Shafts, and Sewers. An application to the design of tlie above, chiefly a question of strength, is given in the examples. EXAMPLKS, 1. a biiuk shaft or well, inside diameter 6 feet, is sunk vertically in water which is kept on the outside. h\ a one-foot length, see the east half of fig. 4, at a depth of S2 feet, t = IS" being made up of four rings of common bricks. Find the factor of safety against crushing. JR = 3-75 ft. and q = 32ic, where w is 62 lbs., the weight of a cubic foot of water. To = q . dc = 'S2w . R = 7440 lbs., and as this is uniiormly distributed over To sq. feet of brick, we have then for the intensity of the crushing load 4960 lbs. per sq. foot. The crushing strength of common brick is only half of what we gave for strong brick, that is 77,000 lbs. per sq. foot. The factor of safety against direct crushing is 77,000 4 4960 = 15, which is a prudent value on the lace of so treacherous a load as an actual fluid load. 2. If the same shaft be loaded with earth spread all round it in hoiizontal layers, the earth being filled quite loosely, find the factor of safety against crushing at the same ring. The potential is doubled, as earth is twice as heavy as water. But the passive horizontal stresses p = q are no longer equal to the potential, but are now only one-tliird of it. Hence p = q is now two-thirds of what it was before. Hence the apparent factor of safely, 50 per cent, greater, is 23. Distortion may halve it. 3. If the earth be rammed in 9" layers as it is filled outside the shaft, or punned as it is called, find the factor of safety now. The punning can only make the passive horizontal conjugate stres>es p = q, fig. 4, equal to the potential, so the earth may now be assumed to be pressing like^ a fluid, but like an imaginary fluid twice as heavy as water. Hence the factor of safety is now of half the value in Ex. 1, namely 8. Notwithstanding the fact that tlie factor of safety in this example is only 8, while in Ex. 2 it is 23, yet the absolute element of safety is just as satisfactory, for the punned earth is a steady load. Practically, then, the answers to Exs. 1, 2, and 3, are 15, 11, and 8. 4. If the shaft be bored and lined, discuss the factor of safety. If the shaft be bored in old consolidated earth, and then be lined with brick, the three rings would suflice at the depth of 32 feet. At first, the load on the lining is nominal, but increases gradually as cracks run out into the earth, and suddenly when slips occur. In this way, the potential comes into play, and crowds the earth in behind the shell. At depths beyond 32 feet, the cracks due to this small bore hole would not run out sufficiently far to bring the whole potential into play, so that the three rings of brick might practically suffice down to 64 feet, double the depth. The mere arithmetical calculation would infer that the factor of safety was 4, the half of 8, though, in all probability, it is still 8. Thus it is that with buried arches, the factor of safety may seem to be as low as 4 : see Rankine's Civil Enyiuccring, p. 437. 5. The upper right quadrant of fig. 5 shows, to a scale of 10 feet to one inch, lialf of a semicircular arch ring, 14 feet in span. The ring is 2 feet thick, and made of rou.ssoirs which are 2^ times as heavy as water. Show that the line of stress A.U up the centre of the vnussoirs is balanced under a water load uj) to any level BK, together with the weight of the voussoirs themselves. CHAl'. XXII.] DESIGN OF MASONRY ARCHES. 429 If «e suppose the water to reach from DK down to AM, the excess weight of the upper half of the ring is I J times tliat of water. That is, u share of the weight of that upper half of the ring is already reckoned ; it is, as it were, Inioyed up. The lower half of the ring is, in weight, 2^ times that of water. That is the excess weight of the 2 foot thick ring, as a whole, is twice that of water. That is the same as if the roiissoirs were 4 feet thick, and of the same weight as water. But 4 feet is half the radius of AM, and tliis is exactly the required load along the circular rib which is required in conjunction with llie fluid load of varying potential to give absolute equilibrium. Such an arch-ring requires slight spandrils to balance it where the centre is struck and before the water load is tilled in. 6. A culvert siphon or sewer, of 7 feet inside diameter, with two circled rings of brick, is laid horizontally along the bottom of a sheet of water. Show that the equilibrium is perfect when the inside is empty, if the bricks be '1\ times as dense as water. The section is shown on fig. 5, to a scale of five feet to an incli, >' = 4 and <= 1, both in feet. The problem is that already discussed in Ex. 5, only with the dimensions halved and the circle completed. The equilibrium, as a whole, must be considered. If the cylinder be one foot long, and if iv be the weiglit of a cubic foot of water, the weight of the empty ring is '2irrl x 'I'bw, and its buoyancy is ir(j' + ^ty-ic, which are in the ratio 20 to 20^, so that the empty cylinder would float upwards. Increasing the thickness of the ring by a couple of inches the proportion would now be 23 to 21, and the cylinder would sit on its bed, or if the brickwork were a trifle more than twice and a half the density of water, the same end would be accomplished. 7. If the axis of the cylinder be at a depth of 20 feet, find the thrust at the highest and lowest points A and a, fig. 5, for a ring of the cylinder one foot long. Eo =jG = ^^LH- - ^Lf- = 1 (20= - IS^) iv = 72k- = 72 x 62 lbs. = 4464 lbs., Si = GN= |ZiV^ - ^LH- = h{2i- - 202) j^ = gSjt; = 88 x 62 lbs. = 5456 lbs., and this is spread on one square foot, so that, for ordinary brick of crushing strength 77,000 lbs. per sq. foot, the factor of safety is 14. 8. An empty circular cylinder, 7 feet inside diameter, with a shell 12 to 13 inches thick of brickwork 2i times as dense as water, will, if completely submerged, remain at rest in neutral buoyancy at any depth and in any position. If the axis he vertiial, the surface of stress is the circuLir cylinder up the middle of the brick- work (fig. 4). But what is most remarkable is that this is still the surface of stress if the axis be horizontal (fig. 5). It follows, by induction, that it is the surface of stress for all positions, so that there is no tendency for the tube to collapse under the external water pressure, though it float in any position whatever, completely submerged. 9 If the cylinder be made of a metal very much denser than water, it would be a thin shell, and there would be no necessity to distinguish between the inside and outside diameters. Thus a platinum circular shell need only have a thickness one-twenty -second part of half its radius, so as to displace, when empty, its own weight of water. Such a shell might be S-8 inches in diameter, and 0-1 inches thick. Let it be submerged with its axis horizontal, the water being kept out by face plates at the ends of the same density themselves as water. There is no tendency for the cylinder to collapse even if it were made in staves like a barrel. The staves might be hinjied together, and the whole covered witii a thin coat of water-tiffht material, the hinges to have a little stifiness to give slight stability to the equilibrium. 10. If this hinged cylinder were split into two semicircular troughs, one of them would float with the water up to its lip, and remain semicircular in form in 430 APPLIED MECHANICS. [CHAP. XXII. perfect equilibrium if only the hinges have a little stiffness to give stability. In the same way two vertical plates, submerged in and e.xcluding water from between them, might be arched over and inverted under with the two semicircular troughs.* The practical importance of Exs. 3, 4, and 5, where the shell by its very thickness in proportion to its radius, and by the very density of the brickwork, coi-rccts the tendency of the extern fluid-load to distort its circular form, will appear when the conditions in which culverts, sewers, and inverts are usually built are considered. The circular form recommends itself in situations where great strength is required; the simplicity of this form, too, lends itself to good workmanship. The surrounding load is due to earth, loose or more or less consoli- dated and in a transitional unsettled state. With the gathering of water sapping the earth close to the brickwork, the external load becomes more and more like a fluid load ; and it is well that the very increased thickness of shell required in such exposed situations, to give extra strength, should at the same time render the equilibrium more secure. Stekeostatic Eib. This is the name by which Eankine calls the most general €ase of a rib balanced by two conjugate loads : see his Applied Mechanics, p. 198. Three quantities are involved, the shape of the rib and the shapes of the two component conjugate-load- areas equivalent to the actual distribution of the uniplanar load ; these two may be either oblique or rectangular. The theorem is that it is sufticient and necessary to give, either implicitly or explicitly, two of these shapes to find the third. In most cases the solution ends in integrals which can only be approximated to very roughly and with great labour. We are confiuing ourselves to the particular case where the ribs are complete, that is they are horizontal at the crown, and vertical at the springings. The conjugate loads are vertical and horizontal, and the two quadrants are symmetrical. For one quadrant the load-areas stand on flnite bases, the base or platform * This theorem of the equilibrium of the thin horizontal empty circular cylinder, displacing its own weight of fluid, was given by the authors in a letter to Nature, of 18th February, 1897. From some private correspondence with scientific men interested in it, it appears as if the theorem were new to hydrostatics. FVom criticism of the method of conjugatc-load-areas employed, it seems as if this elegant method, especially lending itself to graphical construction, were little known or understood. The theorem can be proved by the strict but laborious methods of the integral calculus. See letter on I'olygonal Shells, Nature, Ist May, 1902. I CHAP. XXII.J DESIGN OF MASONKY ARCHES. 431 of the vertical-load-area being the half -span of the rib and the rise of the rib being the base or platform of the conjugate horizontal-load-area. We ha\e already solved two important examples (figs. 4 and 5) by the geometry of the areas. One was the direct problem : given the shapes of the rib and the vertical-load-area, to find the shape of the horizontal-load-area. In the other, the shape of the rib is given and the shape of the horizontal-load-area assumed, and a suitable vertical load built up in two portions : tliis is the inverse problem. We established a rule for building up these areas, namely, that the shaded portions of the areas should be in the same projiortion as the sine and cosine of the slope of the rib. Given the rectangular equation to the curve, with the axis of X horizontal, and that of Y vertical. Given also the vertical- load-area so that P is known, and how it changes as we shift from point to point. Let q be the unknown breadth of the horizontal-load-area at the level of B. Let B shift out a little and the decrement of Q is the small parallelogram dQ = q . d//. Also Q: P\ : cos : sin or $ = P cot d. So that horizontal-load-area for semicircular elb, loaded Uniformly along the Eib. A quadrant ACB is shown on fig. 6. It is convenient to take the intensity of the uniform load along the rib as half a radius. We may suppose it to consist of a ring of material of unit density, with a uniform thickness ^r, so that the area of the part of the ring loading any arc AC is, equal to the area OAC, subtended by the arc of the rib at its centre. By the rule for the thrust at crown, we have H^, = ^r . r = \r-. This must be the area of the total horizontal-load-area ahM, standing on the platform ad = r. On fig. 4, if the intensity of the load spread on the horizontal platform were p = ^R, then q = ^R, and the total horizontal-load-area would be ^R', the same in each case. In this new case, however, the boundary hhk no longer maps out a parallelogram. Suppose, at first, the rib is balanced under the uniform load along the rib alone. Its shape, the catenary, would be peaked at the crown A. If it were deliberately pulled into a semicircle, each pair of points on one level, except the springing pair, have been pulled further apart, giving the load an advantage to spread the rib. The horizontal load must, at each level, 2>?'ess 432 APPLIED MECHANICS. [CIIAP. XXII. imvards, to prevent the spreading, most violently at the springing level, and decreasing gradually till it is least at the crown level. To find/y, the breadth of the horizontal-load-area at the level of any point C on the rib, we have for the vertical-load-area from the crown A outwards to 0, a ring of breadth h', and whose length is s = AC the arc. With i for the slope of the curve at C. p r r . r- . r"^ d r- P cot i = -X i cot i ; - (P cot i) = - (cot i - i cosec- i). ^ Ct% -d ,, . di/ . . , di - 1 y = jd = r cos i; -j-.= - r sin i ; and di dij r sin i So that irrespective of sign the value of /r is d ,^ . d ,^ . di q = --(P cot ^) = -- (^ cot ^) — df/ di di/ 7-2 1 = - (cot %-i cosec* %) 2 r sin i sin i cos i-i = r 2 sin^ i This can be modified into a form more convenient for calcu- lation, as it is engraved on fig, 6, . 2i - sin 2i ,.-,, « -> ' 3sin»-Bin3» ''■ ^^^ To find q^ = ha = ^r, the breadth of the area at the level of the crown A, we must substitute i = 0, when the coettieient of r takes the indeterminate form of the ratio of zero to zero. The numerator and denominator are to be difierentiated separately and i = substituted, which has to be repeated three times when the ratio becomes determinate. 1^'or i = 0, we have 21 - sin 2i 2-2 cos 2i 4 sin 2i 3 sin i - sin 3i 3 cos i - S cos '3i - 3 sin i + 9 sin Si _ 8 cos '«^* _ 1 - 3 cos i + 27 cos '3i 3' or tliis may be found by substituting the series for the sines. CHAP. XXII.] DESIGN OF MASONRY ARCHES. 433 y 1 -v i>N > <s • 5^ % 1 >C5» -'.- •iNsr' |~--JS v< ^ - ^ K vi y= p$ Ni ^ '1 F 434 APPLIED MECHAJNICS. [CHAP. XXII. The value of a^ = kd = -r, the breadth of the area at the 4 TT springing level B, is found by substituting i = ^. Since kd acts normal at B, and is the total external load on the back of the rib there, by the rule, the vertical thrust along the rib at B must equal the product of kd and the radius. But the thrust at B is the area of the quadrant of the ring-area whose thickness TT is hr, that is, an area |r x -r, and from this the xaXwe of kd is found by dividing out the radius. The breadths of the horizontal-load-area at ten equidistant points of ad are found by substituting in the expression for/v, ten values of the angle i, whose cosines are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 ninths of r the radius, respectively. They are marked on fig. 6. The exact boundary is a gentle curve, but a second boundary maps out the area with three straight lines hg, gl, and Ik. The first extends downwards for 4 ninths of the radius, giving a breadth of 3 and 4 ninths of the radius at h and g ; the second extends downwards for 4 ninths of the radius, gi^'ing a breadth of 6 ninths at /, and the last extends downwards the remaining ninth of the radius, and gives the final breadth of 7 ninths at k. This approximate boundary begins with the proper breadth one-third of the radius at the top, and encloses the exact area |;'^, which may be found by counting the number of squares on the paper, square-ruled with the sides a ninth of the radius. Thus hg encloses 14, gl encloses 20, and Ik encloses 6h squares, in all 40 1 squares ; but each square is ?•■ ^ 81, so the whole area included is |?■^ It is called shortly the trchle-haffcr-houndarg hglk, and may be seen clearly near the centre of fig. 6, drawn in dotted lines with the hatters written upon them. Beginning at h on the crown level, bg is drawn with a hatter of 1 in If, for the first 4 ninths, then gl with a hatter of 1 in 2 for the next 4 ninths, and lastly, Ik is drawn with a hatter of 1 in 1 for the remaining ninth of the radius. Horizontal-load-area for Semicircular Eib. loaded with THE Area between Itself and a Horizontal Straight Line over it. Let -4 Ci? be the circular quadrant, and D^ the horizontal extrados (fig. 6). Suppose, for a little, that besides the shaded CHAP. XXII.] DESIGN OF MASONRY AKCHES. 435 vertical-load-area bounded thus, there were also, as on fig. 4, a tinifonn load along the rib of intensity half a radius. Then the 45^ slope, or 1 to 1 batter no, gives the horizontal-load-area for the joint-loads. Now remove, on fig. 6, the load along the rib, and at the same time remove the corresponding horizontal- load-area ahfilkd, it is to be drawn on the same base ad, and to be supposed to gravitate horizontally outwards from the rib, while the area already drawn gravitates horizontally imoards towards the rib. Hence, for the given vertical-load-area between the rib and the straight extrados, the horizontal-load-area lies between the 1 to 1 batter uo, and the curve approximated to by the treble- batter hglk. In form it is like a figure 8, as these two boundaries generally cross each other at a pointy. The part below p gravitates horizontally invmrds, and is to be reckoned as positive', the part above p gravitates horizontally oidv:ards, and is negative. They are to have the same density as the given vertical-load-area. Allourincefor the JExccss-iveiyht of a Masonry-riiuj of Uniform thickness. — It is Eankine's practice in the masonry arch to assume as a first approximation, that the line of stress is along the soffit. In the meantime we follow this practice in fig. 6, so that the quadrant of the circle is at once the soffit of the masonry-ring, the intrados of the vertical-load-area, and the linear-rib or line of stress. It will be seen that the weight of the masonry-ring has been included in the vertical-load- area, just as if it were of the same density la as the superstructure. It is usually fifty per cent, denser, and its uniform thickness, see t^ = -ji-r, on the figure, may be taken as a fifteenth of the radius of the soffit, and although its area lies wholly above the .soffit it may be assumed to act along it in the meantime. We have then to consider an additional vertical load uniformly spread along the rib, and mapped out by two concentric circles at a distance apart -^r, and of density hvj. But this is the same as if the two concentric circles were at a distance apart ■one-fifteenth of a half-radius, and the area of the full density w. This is exactly one-fifteenth of the load which for convenience we assumed along the rib, and the removal of which introduced the treble-batter hglk. It is only necessary, then, to restore a fifteenth part of the area ahglM by pricking back bh' = -j^ha and kk' = -yi^kd, and similarly at g and I, and drawing the dotted treble-batter instead. Or, shortly, with the line of stress assumed to be along the soffit, an allowance is to be made for a masonry- ring of an excess density of 50 per cent., and of a uniform thickness a fifteenth of the radius by receding the treble-batter boundarg one-fifteenth. 2f 2 436 APPLIED MECHANICS. [CHAP. XXII. Allowance for the Excess- vciijht of a Masonnj-ring, Thickeniiiff outivards from Keystone to Skevjhcichs. — In the best French and English practice, the skewbacks of the segmental circular arch are from oiice and a half to twice as thick as the keystone. It depends in part on the size of the segment, and it will answer all purposes to suppose the masonry-ring included between two slightly excentric circles, with their centres on one vertical line, with their crowns at a distance apart t^^ and the pair of points 60° out from the crown It^ apart nearly. They are shown on fig. 6, by the softit and the upper dotted circle. The lower dotted circle is concentric with the soffit, and we have already considered the excess-weight of that part. Consider now the two dotted circles. Their crowns toucli, and if completed, their greatest distance apart is at their lowest points, and equals the difference of their diameters. At the points left of the centre, their distance apart is half as much, being nearly the difference of the radii, slightly more or less, according as we measure along a horizontal line through the one or other centre. It will be seen, then, that the distance apart of the two dotted circles on fig. 6, measured along the radius through 0, at any angle i out from the crown is given closely by z = 2tQ (1 - cos i), as at the three cardinal points for ■i = 0, 90°, and 180^ we have 2=0, 2^0' ^"^ '^'^0' t^® proportions indicated, while with i = 60° we have 2: = ^^ as required. An element of the area between these dotted circles \b z .ds or zrdi, and the vertical-load-area from the crown out to i is the definite integral of this between the limits and i. Hence P = r\zdi = 2r/'„J(l - cos i)di = 2rt^ [i - sin t'j = 2^„ [ri - r sin ?'] = 2^, [s - .r]. (4> Interpreting the two terms separately : — The first is a uni- form load along the arc equivalent to the area l)etween two concentric circles 2t„ apart; the second term is a negative load uniform along the span. That is a parallelogram of height 2^0 to be tdhii 0//' the span. But since the excess- weight is only ^10, each of the uniform loads is to be an area of unifttrm breadth t^^, and of the normal density v, the one added along the rib and the other taken oil" the span. The corresponding conjugate-loads we already know. On fig. 6, suppose the load only to reach up to the hori- zontal thrDUgh the crown A, that is, the pair of parallelograms •CHAP. XXII.] DESIGN OK MASONRY AllCIIES. 437 AH and ((0 simultaneously removed. The conju,i!;ate horizontal- load-area is drawn inside the arch in dotted lines. One boundary is the dotted 45° line AgB, and the other the dotted treble-batter boundary h(/lJc. This last is to recede -fV^lis of the area hO, by pricking back hh' = -fjfiA and kk' = -^kO, and similarly at g and I. One-fifteenth part is due to the uniform part of the ring, and the other two-fifteenths to the first term of the spread-out part. At the same time the boundary AB is to recede -p^-th of a radius into the position an, due to the second term. Lastly, a'o is to be added on when we restore AE. The conjugate horizontal-load-area as modified for the thickening-out ring is shown shaded, Q-^ is the portion of it pushing in below ^>", while ^2 is the part pulling out above ;p". Note that at the point r/, where the two dotted boundaries cross, the effect of the weight of the thickening-out part of the arch-ring makes both boundaries move to the right the same distance. For the one moves a -J^th part of r, and the other -fj-ths of the breadth of the figure hO, which at g is almost half of r. Hence j/ and p" are practically at one level, whether the ring be uniform or thicken outwards. Note, however, that the assumption that the area between the dotted circles acts along the soffit makes the approximation much coarser than in the case of the uniform ring. liANKiNE's Point and Joint of Ruptuke. We are now in a position to define Rankine's point of rupture, and to exhibit it to the eye by our graphical con- struction on fig. 6. The point j/ or ;/' projected horizontally on to the circle gives P the point of rupture, that is, the point ■on the rib where the conjugate horizontal-load changes sign. Below P there is required a thrust Qi on the back of the arch, but above P an outward pull Q-,. The angle of rupture is the slope which the tangent to the rib at P makes with the horizon, which in the circle is the same as AOP. For the shaded load, fig. 6, between the circle itself and an extrados -^r over the •crown, together with a slight additional load due to the excess density of the voussoirs, the shaded conjugate horizontal-load- area may be readily drawn on a large scale on square-ruled paper, taking nine sides of the square to represent the radius. P is to be projected from p", and the angle of rupture measured Avith a protractor, when it will be found to be AOP = 4;')" 15'. The joint at P, between two voussoirs there, is the joint of 488 APPLIED MECHANICS. [CHAP. XXII. rupture, and the point where this joint meets the back of the masonry-ring gives Eankine's level of lieavy backing. In the simpler case, with no excess load along the linear rib, p determines the point of rupture : see fig. 6. This is where the 45° line no meets the treble-batter hjlk, or more strictly the gentle curve to which that treble-batter closely approximates. It is instructive to observe how y behaves as the extrados moves lower or higher over the crown. First let the extrados touch the crown A, then the 45° line an meets the treble-batter exactly at the joint //, the positive and negative parts of the conjugate horizontal-load-area are glkn and gob, respectively. They each contain exactly six squares or cjnk=gah = ^i^^=-()Ur Their algebraic sum is zero, and JZ, the thrust at the crown is zero. As the extrados cannot come lower down, g projected over to G gives the lowest possible position of the joint of rupture, and putting % for the corresponding biggest value of the angle of rupture we have the cosine of i, given by the ratio of the height of g above dk to the radius ; but the height of g is five sides of the square, that is, |-r, so that cos % = '5 and % = 56° 15'- As the extrados DiJu is placed higher, |? moves up and P approaches the crown and % is decreasing. It will be seen, by inspection of fig. 6, that the positive part of the conjugate- load- area is increasing and the negative part decreasing. Their algebraic sum expresses H„ the thrust at crown, the unit being the weight of a cubic foot of the superstructure. That sum must be equal to the product of the height of the load over the crown, and the radius there. When the extrados reaches a height one- third of r above the crown A, the conjugate-load-area bglks is wholly positive, the point of rupture has reached the crown, Zo is zero, and counting the squares /rV r- Us ^ „. ^ 27 [^) - .y The area remains wholly positive for all higher positions of the extrados. Since the triangle peg has one side sloping at 1 in 1, and the other at 1 in 4, it follows that the height of the triangle is ^rds of its base eg, that is -irds of AD, tlie height of the load over the crown of the rib. It is convenient to express the height of the load over the crown as a fraction of the radius ; if .s is this important ratio, then <g = AD = sr. CHAP. XXII.] DESIGN OF MASONRY ARCHES. 439 For the height of i) the point of rupture, we have the sum of the heights of fi above OB, and of p above <j. It is r cos t = ^/' + ^$r, and cos -i = -| + -^s (5) is a close approximation to the angle of rupture. For the positive part of the conjugate horizontal-load-area, we have pok = (jnh + (JO + peg = '•= (-tr*¥ + ¥')■ (6) The negative portion equals in area fjah - ea + peg or ?•• (^^ - f « + f «'), (7) giving as their algebraic sum sr, which is the product of the height of the load over the crown and the radius there. Semicircular jVIasonry Arch. In the application to the semicircular arch, the left quadrant of which is shown on fig, 7, the assumed line of stress is the quadrant cm with the centre at d, and having a radius of 48 feet. It is taken up the middle of the masonry-ring, which has a thickness AD = to = S feet at the keystone, and a thickness twice as great, ^o = 6 feet at the joint near C which is 60^ out from the crown a. So far the ring is mapped out by two excentric circles as already explained : one is the soffit AECB drawn from the centre S with radius SA = 43'5 feet, and the other from the centre c with radius eD = 52"5 feet. Beyond the joint C the back of the ring is stepped to receive the heavy backing, and the ring ceases to widen out any further. From a the middle of the keystone ah is laid oft' horizontally and equal to one-third of 48 the mean radius ; then hg is drawn at the batter 1 in 4 till ^r is at a level |-ths of 48 lower than h ; next gl is drawn at the batter 1 in 2 till I is |-ths of 48 lower in level than g, when U: is drawn at the batter 1 in 1, Also agn is drawn through a the crown of the assumed line of stress at the batter 1 in 1. These dotted lines are readily drawn on squared-paper, making da nine sides ; they include the conjugate 440 APPLIED MECHANICS. [CHAP. XXII. horizontal-load-area, neglecting altogether the weight of the half of the ring below the assumed line of stress, and the excess weight of the other half over the average weight of the super- structure, which is v: lbs. per cubic foot. The load between a horizontal through the crown a and the formation is in the meantime also neglected. Now the arch-ring is to be built of masonry half again as heavy as the superstructure, so that the excess density of the lower half is ^v:, and of the upper half it is he, or conjointly the excess weight of the arch-ring is w, and as this is twice as great as it was in fig. 6, the boundaries must now recede twice as much. So that, on fig. 7, the treble- batter-boundary hfjlk recedes -p-ths of the area into the position h'k', while the 45' boundary an recedes y-^-ths of 48 when there is added to it a parallelogram, by drawing uo. equal to the parallelogram between the horizontal through the crown a and the formation through 0, which is at a height aO = 8 feet above it. Xote -^ is still sensibly -JU. The area of ph'o, the positive part, is readily found by measurement or by counting the squares, or it may lie calcu- lated from the manner in which it was constructed. It is almost 400 square feet, so that Qx = 400?r is the inward thrust with which the' solid backing must resist the tendency of the arch to spread at the haunches. For this purpose the square-dressed heavy backing must stretch out at the springing joint B, a distance z = 12 feet. For the weight pressing the backing down on its base is then 12 x 48/r; and taking the coefficient of friction of masonry on masonry at "7, the f rictional stability of the backing at the joint ^ is i^ = y% x 12 x 48?r = 400/f nearly. The level of this heavy backing might be up to the pointy?. Above this level tlie conjugate horizontal-load required is Q^ = 112?'' outward, and if it were practicable to apply this in any way, for instance, by the centering which might still be inside the arch, then the line of stress would be the circle of radius 48 feet as assumed, and the thrust at the crown ffo = Qi - Q-. = 288«\ This is exactly the value of H^, got by the rule for the thrust at tlie crown of the circular linear rib of 48 feet radius with the normal load at a oi AO made up of the two parts IJO = l-5w and AD = 3 x 4"', in all equivalent to Qw, so that ^, = 48 x 6?/' = 288 /r. In practice the negative part Q-i of the horizontal-load-area has to be left out, and for a masonry-ring of uniform thickness the heavy backing may come up to the level of p, as recom- ■ffSi' 442 APPLIED MECHANICS. [CHAP. XXII. mended by Eankine. He gives the joint at 45^, as being always on the safe side. With the arch-ring as in fig. 7, we need only bring the backing up to the level of g, as the very thickening of the ring outwards is sufhcient backing down to that point. It will be seen that we are leaving out a part of the positive area from p down to g, as well as the negative part Qi. This positive part, shown shaded across from p down to g, is not completely left out ; it is only applied lower down, so that the whole inward thrust §, = 400^:' of the square-dressed heavy backing is applied to the back of the arch between the joints B and C. A portion of the negative area from p upwards, 20«- is also shaded across : equal in area to the shaded part below p. In this way we determine fico joints of rupture E and C, one on each side of Eankine's joint. And just as the average position of his joint is 45°, so the average positions of our pair are 30° and 60° out from the crown. The leaving out of the horizontal load above C requires the line of stress from that joint to the same joint on the other side of the crown to be modified. For want of the outward pull Qi, the arch tends to shrink horizontally and the crown to go up. This brings the centre of stress at the crown joint down from the bisecting point to the lower trisecting point. As the thrust ^0 at the crown is now greatly increased, we must divide 400?'/ by AB + BO or %w, when we have the raflius of curvature of the modified line of stress at its crown, yo,, = 67 feet. A circle swept out with this radius, having its centre on the vertical through the crown, and beginning at the lower trisecting point of the crown joint A, crosses the assumed line of stress, and approaches the upper boundary of the middle third of tlie arch- ring. Consider now the thrust at E, the upper end of the portion of the arch-ring CE. At E, its upper end, a thrust a little greater than 400?^ has shifted from the bisecting point to nearly the upper trisecting point of the joint, a distance nearly half a foot. This is the same as applying at E, the upper end of the block CE, a left-handed couple 400?^? x ^ = 2007r, and takes the place of the couple 20u' x 9 feet, constituted by the shaded-across parts of the horizontal-load-area, above and below p, which ought to act on the back of the block CE, but which have been left out as far as that block is concerned. Three points on the left half of the modified line of stress, fig. 7, are the lower trisecting point of the crovn joint A, the upper trisecting point of the first joint of rupture E, and a point slightly above the middle of the second joint of rupture C. This CHAP. XXII.] DESIGN OF MASONKV ARCHES. 443 is to be verified as follows : — Consider the structure rigid from A to C, and examine whether the three forces acting on it meet at a point. Namely, the weight of the structure from C to 680h', acting vertically downwards through the centre of gravity, the horizontal force i/o = 400iy acting horizontally through the centre of the stress at the crown joint assumed to be its lower trisecting point and the oblique thrust at the centre of stress of the joint assumed to be sensibly at its middle point. Taking moments about this last point, we have G80«' x 13 feet and 400?/' X 22 feet sensibly equal. It is only necessary then to show that the whole weight from A down to G is 6807/', and the centre of gravity 15"4 feet measured horizontally from the right vertical boundary. The weights and leverages of four parts into which the area AC can be divided are printed on the right half of fig. 9. Again, consider the structure rigid from A to U when the three forces must again riieet at a point : namely, 287?/' vertical through the centre of gravity, 400-26' horizontal through the lower trisecting point of the joint A, and an oblique force parallel to the tangent of the line of stress acting through the upper trisecting point of F. Taking moments about this last point, we have AOOiv x 7*5 ft. and 287'?/^ x 10-5 ft. sensibly equal. The areas and levers of the various regular figures into which the area under consideration can be divided are printed on the lower part of the right half of fig. 7. In this way we have indirectly designed a Segmental Arch, the half span of soffit being 37*5 feet, the rise of the soffit 21*5 feet, the formation level being 4-5 feet above the crown of soffit, or otherwise the total rise of the formation above the springing level is 26 feet. (See the right half of fig. 7.) We know five points on the line of stress : the lower trisecting point of the crown joint A, the upper trisecting points of the first pair of joints of rupture U and U at about 30° out from the crown one on each side, and a pair of points slightly above the central points of the two springing joints C and C. These five points being just comfortably located in the "Kernel," con- sisting of the middle third of the arch-ring, we will prove that the line of stress lies wholly inside that ■' Kernel." It is not sufficient, as Eankine does in his C. E., p. 421, to ensure the centres of stress to be in the middle third of crown- joint and his (45") joint of rupture, as there is a point of maximum curvature on line of stress near 30°, of which he was unaware. Light elastic rubble spandiils are shown riding out to a 444 APPLIED MECHANICS. [CHAP. XXII. stretch of 5 = 7 feet on the heavy backing or abutments, but they only exert a liorizontal reaction when the live load conies on the opposite half of the arch. They can offer a horizontal resistance 7 x 23?6' x -7 = 113h\ Xow the heicrht of superstruc- ture representing the live load is I'o?/-; this when multiplied by /oo or 67 feet, gives 100 w, the excess of the horizontal thrust •of the loaded half over the unloaded half. Equilibrium Eib or Tkansformed Catenary. It will be seen on fig. 7 that the modified line of stress in the segmental arch-ring CC bears only a vertical load, the bulk of which is given by the area included between itself and the formation level. The remainder of the load is spread in a very similar manner because of the thickening of the arch-ring from the crown outwards. The curve which is a balanced rib for the vertical- load-area alone included between itself and a horizontal straight extrados is known as the Equilibrium Curve. Let AB, fig. 8, be a finite arc of the curve balanced under the vertical-load-area NOAB, so that ^and T meet at the same point .^ on W, drawn vertically through the centre of gravity of that area. H= T cos e, W = T sin ^, W = H tan Q. Expressing tbe area by an integral, and the tangent of the slope at B by a differential coefficient w f ydx = !{-—, ^ dx and differentiating each side wy = H -/^, or // = - -[, d.ar 1IJ d.r TT or with 7>i^ as a short name f(n- — , the differential equation to >r the equilibrium curve is '/ = »' % (8) dx- There are only two functions of x which differ from their original values by a constant m^, when differentiated twice successively, so that their sum is the general solution of this equation. It is r r y = Ai"' + Be''^. (9) CHAP. XXII.] DESIGN OF MASONRY ARCHES. 445 When ^ = 0, y = y,, = OA, giving j/,, = A + B. By differentiating (9), we have n (^y 1 - -- tan e = -- = - {Ai'" - Bi "•). (10) Now when x = 0, tan ^ = 0, giving ^ = A - B, so that A = B = 1^05 ^^^ m - — y = r ^ (e'" + e '"),) where ?/o = OA is put equal to rm. If '/• = 1, the curve is the common catenary. It is shown inverted by a thick line on fig. 8. Any other of the curves can be described by dividing the ordinates of the catenary in a constant ratio r. The catenary might be drawn on a sheet of indiarubber fixed along its edge OX, and previously stretched in the direction OY only. If the sheet were now to contract in that direction only, the catenary would become in succession the Transformed Catenaries AB, ac, hd, &e. In this way a whole family of transformed catenaries is derived from the common catenary by assigning a succession of graduated values to r, the ratio of transformation. There is only one such family, for the catenary itself, like the circle, admits of no variety ; you may make ni larger or smaller, but you only draw the catenary to a coarser or finer scale. It is therefore convenient in tabulating numerical values of the different dimensions of the family of transformed catenaries to assign to m the value unity, and to r a series of fractional values graduated at close intervals. If the curve AB, with its tangents Ag and B(j, be trans- formed towards OX by the shrinking of the stretched sheet of indiarubber, it is e^'ident that g travels on a vertical locus. It follows that the centres of gravity of the areas from ON down to hd, to ac, to AB, &c., all lie on one vertical, and the same is true of the areas included between any pair of the curves. Hence any one of the curves is a balanced rib for the vertical- load-area alone between any other pair. As a matter of fact, hd is the modified line of stress, in the segmental arch, iig. 7. Its soffit is a circle coiiicident 446 APPLIED ■MECHANICS. [chap. XXII. nearly with ac, while ONX is ihe formation level. The portion of extra density lies between another pair, of which ac is one ; the other being a curve a little higher than hd, but not shown on fig. 8, Of course for this comparison between the figs. 7 and 8, the scale of fig. 8 must be such that Oa shall measure absolutely 4*5 feet. We then have Oh = 3-5 feet = ,?/.,, while py is 67 feet, so that for the curve hd we have the ratio 2/o : Po = '05 nearly. ThcWireefrJX . Mo of f/ie ^raUsforfV(d(07i, Ik Now, we only used iir as a convenient abbreviation of H^ -f w, that is the thrust at the crown of the equilibrium curve fexpressed in square units of the vertical-load-area, so that ^0 = ^^^y ^^^t from the general rule for the thrust at the crown of a rib we have H^ = ■2t'//„p,„ so that p^ = ??r -^ 2/o = ^" ^ '". but ?/,, = rm, so that = ?•' = N. (12) That is — The important ratio .9, namely, that of the depth CHAP. XXII.] DESIGN OF MASONRY AKCIIKS. 447 of the load at the crown to the radius of curvature there, is in the transformed catenary, exactly the square of the ratio of transformation. On fig. 8, AB is the transformed catenary for which s = \. All the members of the family of curves below this are like the catenary or parabola in this, that their curvature is sharpest at the vertex. They are of little importance in their application to arches. On the other hand, all the curves for which s is less than one-third, such as ac, bd, differ in form completely from that of the catenary or parabola. They have a flat curvature at the vertex a or b. The curvature becomes sharper and sliarper as you travel out from the crown on both sides, till it is sharpest of all at a pair of points on each curve, one on each side of the vertex, one of the pair being shown with a little ring on fig. 8. From this, as you go outward, the curvature begins to flatten till at d or c it is again exactly the same as it was at the crown. This finite part of the curve, having the same curvature at its middle point and at its two ends, must now engage our attention. Beyond d or c the curves flatten indefinitely outwards. Observe that the value s = J, which divides the transformed catenaries into the two sets, those sharpest at the crown and those flattest at the crown, corresponds exactly to the same value of s in fig. 6. On that figure s ^ J makes the conjugate horizontal-load-area wholly positive. So that in an example like fig. 7, but with the formation level above the crown of the assumed line of stress by a third or more of its radius, the horizontal-load-area would be wholly positive or inwards. If then an upper part were left out, that would decrease the thrust at the crown, so that the modified line of stress would sharpen at the crown instead of flattening. In bridge practice the formation would never be at such great heights. The Two-Nosed Catenapjes. We have ventured to use this name for the important part of the family of transformed catenaries having their ratio of transformation less than the square root of one-third. We shall only give the part of the treatment necessary for the construction of the tables. That their importance is appreciated is shown by their adoption by Professor Howe in the chapters of his elegant treatise which deal with the masonry arch.* We are indebted to the courtesy of the Eoyal Irish Academy for * "A Treatise on Arches," by Malverd A. Howe, C.E., Professor of Civil Engineering, Kose Polytechnic Institute, TeiTe Haute, Indiana. New York: John Wiley & Son. London : Chapman & Hall, Ltd., 1897. 448 APPLIED MECHANICS. [chap. XXII. the use of their blocks, and especially for a photo-block from the design of an arch by this method engraved on a grand scale Ijy the splendid liberality of the Academy.* For simplicity, put m equal to unity, and % = !^(t'-6-0 = tan« = y(y^-r^). dx vw sec^O (if- + 1 - /•-)- or P = (y- + (ty y Fiff. 9. if (1} = 1- r- = 1 - 6\ dy ~ //' d-p ^ v/(?/- + (r) df ~ y --(V--)^-^^- * " On Two-Nosed Catenaries tind their Application to the Design of Segmental Arclies," by T. Alexander, M.A.I., Profissor of Engineering, 'i'rinitj- College, Dublin, ami A. W. Thomson, B.Sc, Lecturer in the Glasgow nnd "West of Scotland Technical College, Transactions of the Jioi/al Jiuh Acadtmij, Vol. xxix.. Part iii., 1888. CHAP. XXII.] DESIGN OF MASONRY ARCHES. 449 Now (2}/- -a-) = makes ^ ^ 0, and as a first step makes (^f' y a positive quantity upon a further substitution from %f - a- = 0. Hence 2y'- - a- = gives the value of y, which makes p a minimum, namely, a pair of points symmetrical about the crown or vertex. They are the noses, and are shown at Bi and B'^ on fig. 9. The abscissa, ordinate, slope, and radius of curvature of the " nose " B^ are designated by ^i, //i, d^, and p,. If p^ be produced to meet the vertical through and Q,, and if now a Directrix Fig. 10. circle be drawn from Q^ with a radius QiBi = ^i, it will touch the curve externally at its two " noses." This is the described circle, and is outside of the finite arc B^, B'^ with which we are concerned. On the other hand, fig. 10 shows a circle drawn from the centre Q. with a radius Q2B2 = B2 passing through the three points B:, A, and B'-. of common curvature. It is the three- point circle. It touches the curve internally at A, and sensibly touches it at B\ and B., as the slope of the curve 62 and that of the circle /3 at B2 never differ by more than two and a half degrees. For all practical purposes this circle is inscribed in the curve. There is evidently a mathematically inscribed circle, but its equations are very troublesome. 2g 450 APPLIED MECHANICS. [CHAP. XXII. For the described-circle, we have 2. 3;, = Pi = '' — - — '- = ^-yAy,- = -^ (1 - s), (14) tan Q, = ^(i/;- -s) = J^-^- (15) So that when s = ^ then ^i = 0, and the crown is the sharpest point. Again, by adding the expressions for // and for its first differential coefficient, we get n' = y + tan 0, . 7/ + tan , v/(l - .s) + v^(l - os). X = log '^- , X, = log -^^ '-^z^^ — — '-' ''' ■/•2s El = Zi cosec ^1, Fq = //, + Bi cos ^i - B^ For the three-point circle, we have sec^ d = py = fj y (r^ + tan^ 6), sec" 9 = p- (r- + tan- 9), bnt p2 == p., = - = - - and ?/o = '" = /^ ', T ^ S .'. r- sec^ 9i = r'^ + tan'- 0,, 9, + 1 = 1 sec^ 02-1 1 , ,. ../>-, Po or - = — —,, -, or - - sec* 9. + sec- 0. + 1 = — • (16) r^ sec- 02-1 s Vn Solving this like a quadratic equation tan* 02 -je-^)-'- (17) So that when s = -L, then 6i = 0, and B. comes to the crown just as Bi did. Also y2 = ys sec^ 02, QO2 — fVi tan 02\ , / , - tan 02\ log - + = log sec* 02 + As 62 = ^ nearly 1 J?. -7 = sec* 03 + sec- 02 + sec 0, = — . ( I8) CHAP. XXII.] By Euc. iii. 35. DESIGN OF MASONRY ARCHES. X\ = (^2 - !/,) \2Ii., - (l/. - %)], 451 ■1/2 -^S ^' - 2(y. -v's) ■ 2 ' 8o = 3/o - Yo, ^2 = Bi - Bn + A cos 02 sensibly, where A = Hi - Bi + ^o- (See fig. 11.) Table A, for each assigned value of s, the following ten quantities are calculated by the above equations in the order y^, pi, 0u Ru Yo, p2, or /3„, 02, Bn, do, ^2. The values assigned to s are at equi- distances apart. Table A. 1 ^ c . = 11 ei e> ^1 i?2 "' 1 11 Yo 60 Si vi •333 a 173 173 173 173 ■577 ■577 •000 000 i •250 19 28 1-97 I 98 I 94 2-00 •500 499 •000 •000 — •180 25 39 224 2-25 213 2-35 ■424 •419 •004 •002 — •170 26 41 229 230 2-15 2-42 •412 ■406 •005 003 — •160 27 42 234 235 2-i8 250 •400 •392 •007 004 — •150 27 44 2 39 2*40 2-20 258 •387 ■378 •009 •005 — •140 28 45 244 2-45 223 2-67 •374 •363 on •006 — •130 28 46 2-50 2-51 2-26 277 •360 •347 •013 •008 — •120 29 48 257 2-58 2-28 2-88 •346 ■330 •016 ■010 i •III 30 49 263 2-63 2-30 3 00 ■333 •313 •019 •012 — •100 30 51 272 272 233 316 ■316 •292 024 •oi5 — •090 31 52 2 So 2 -80 2-36 3'33 •300 •270 029 019 ^ — ■080 31 54 2-89 2-88 239 3 53 ■282 •247 035 •025 — •070 32 55 3'oo 2.99 241 377 •264 •220 •043 •032 J i 063 32 57 3-09 307 2-43 4-00 •250 •199 •051 039 I — •050 33 59 32S 324 246 4-47 •223 •156 •o65 •054 i •040 33 61 3-46 340 2-49 5 '00 •200 •115 •084 073 ^ — •000 35 90 00 00 2 "60 00 •000 — — — 2 G2 452 APPLIED MECHANICS. [chap. XXII. Fig. 11 shows the finite q.xc BzBiAB'iB\ sandwiched between the arcs of these two circles, and So = 2/0 - Yo is their distance apart at the crown, while ^2 is their distance apart at each pesc Qj Pole Fig. 11. end. These distances are so small, that as far as being ^the houndari/ of a load area the arc of the three-point-circle is prac- tically the same as the finite arc of the two-nosed catenary itself. The dotted curve should pass through A. Tables for the Immediate Deskjn of Se(;mental Arches. Table B (p. 457, et seq.) is derived from Table A (p. 451) line by line by dividing the linear quantities in I'able A by the value of ^,. Ifatios r, s, di, and 62 are unaltered. The description of the table is fully given on the face of it. In the Supplementary Table B|, the thickness of the key- stone ^0 is made equal to 3^0- (See DA on fig. 12.) This confines the line of stress to a " kernel," which is the middle third of the arch-ring. The upper limit of this " kernel " is the described-circle of the line of stress. It is still the described- circle of the new line of stress due to a uniform live load all over the span, which is equivalent to raising the directrix. The deptli of the crown of the sojit from the formation-level or directrix is cl = y^ + Sq. (See OA on fi<4S. 7 and 12, and 0(1 on fig. 8.) The soffit is the three-point-circle of a lower- down member of the same family as the line of stress itself. This is accomplished as follows. In any line of Table Bi, take- CHAP. XXII.] DESIGN OF MASONRY ARCHES. 453 the value of d, which divide by the vahie of m in that line produced into Table B. Look for the quotient in Table A, under the heading ?/o ; take the corresponding value of B. and place it in Table Bi, under the heading R, but, of course, first multiplying by m. The rise and span of the soffit are /.' = R vers Q,, ^md 2e = 2R sin B,. The quantities engraved on fig. 12 correspond to those set down on the fourth line from the bottom of Tables Bi and B, only multiplied each by 50-07. This line will serve to explai)i the construction of the Table B,, while inspection of fig. 12 may help to make it clear. The position of the line fourth from the bottom is designated as the line on Table B, with s = -05. In the third column, 104 is the maximum multiplier which may be used on the linear quantities in this line that there may be a factor of safety 10 against crushing a sandstone-key. First, the thrust at the crown wd x p„ = 140 x -089 x 1*3634 = 16-987, and dividing this by ^o = "061, the number of square feet exposed to it, we get the average thrust on the crown-joint to be 278 lbs. per square foot. Multiplying by 2, as the maxi- mum is double the average (see fig. 3), the maximum stress on the cheek of the sandstone-key is 556 lbs. per square foot. Now this increases directly with the multiplier used ; hence dividing 556 into 57,600 lbs. per square foot, the resistance to crushing of sandstone allowing a factor of safety of 10, we get 104. Any smaller multiplier gives a factor of safety proportionately greater. Thus on fig. 12, the multiplier being 50*07, gives the factor of safety 20. In the fourth column 155 feet is the max- mum span between the points of rupture for a sandstone bridge with its proportions as on that line; it is 2c = 1*498 multiplied by 104. The first two columns are the same numbers divided by three for strong brick, while the fifth and sixth columns are the same numbers doubled for granite. Ptefer to p. 419 under " strength." In column seven {cl - Q = -028 is the surcharge ; it is DO on fig. 12, and should be only sufficient to make a soft bed for the railway that the sleepers may not hammer the top of the keystone. It only remains to explain how t. = "123, the thickness of the voussoir at the joint of rupture sloping at B^., that is at the skewback or springer of the segmental arch, is obtained. The radius of the described-circle (fig. 12), as given on the fourth lowest line of Table B is 1. Also QA = R = "869 is the radius of the sojfit. If now a circle were drawn from the centre Q, concentric with the soffit and touching the described circle at the crown, its radius would be {R + 2go) = '869 + 2 x -0204 = -9098. tlO CHAl'. XXII. ] DESIGN OF MASONRY ARCHES, 455 Here, then, we have two circles touching at the crown, the described-circle itself of radius l,and the circle concentric with the soffit of radius "9098, just like the two dotted circles on fig. 6. By the approximation, page 436, the distance apart of these two excentric circles at the springing (fig. 12) is the difi'erence of their radii multiplied by (1 - cos 6^2) where 0, = 59- ol'. Putting Z for this, we have Z=(l- -9098) (1 - -5078) = -044. Add to this the distance between the soffit and the circle we drew concentric to it, namely 28„ = 2 x -0204, we have "084 the distance between the soffit and the described- circle at the springing joint on fig. 14. Increasing this by 50 per cent., we get ti = '126 ; and by a closer approximation we make U = •123. The thickening of the arch -ring from t^, at the crown out- wards to /; at the springing (lower joint of rupture) by this tabular method ensures that the line of stress, beginning at the loicer trisecting-point of the crown-joint A (fig. 12), just reaches the upper trisecting-point of the joint Ji" (higher joint of rupture at 33° 6') ; then it comes back into the white-kernel. This thickening makes the extra load due to the excess density of the ring suited to the line of stress, since the two circles embracing the voussoirs are, approximately, for boundary purposes, two members of the same family of transformed catenaries as the line of stress itself. The Tables Bo and B3 are only wanted for small arches in which the surcharge is relatively larger and the economy of material in the arch-ring has to give place to the consideration of strength. Apparent Factor of Safety. — In looking at fig. 7, it will appear that the design there made by assuming sizes and drawing the conjugate load-area is the same as that made immediately from Table Bj on fig. 12. In fig. 7, the apparent thrust at the crown increased from 288?c- to the actual thrust 400?r from the leaving out of the negative part of the conjugate load, and the radius at crown of line of stress changed from 48 to p^^ = 37 feet. Dividing equation (16) by (18), we have Pq = i^o(sec 02 - 1 + cos 021, (19) which may serve instead of drawing the conjugate load-area or using the tables. Thus taking R^ = 43*5, the radius of soffit roughly, and 9. = 60° ; then p = I0R2 = 65 feet. As 02 is never greater than 60"", it follows that the real factor of safety against crushing the keystone will not be less than tvjo-thirds the apparent factor. SUPPLEMENTAllY TABLE B,. i StrongBrick. Sandstone. Granite. "o U ■! .5 rt =^ ' - 1 "c** "o as' sf a»=112 /= 164000 7c = 140 /= 576000 71. = 164 /= 1350000 o li th of load fi trados (direct crown of sotTi 1° J5| =e = = o f ; o'n"o ^ 1 o ' "o ' 4.1 ^ 4., hfl < E^ E"^ ' ° II ■~ B So*, c • H 5; J " ■" 3 MJ3 ^ "o Z O V P-S ° 1 U O u •ji 2'o ^ t t- 9 rt ^. = "3 c "3 c ■s-. q"*- '^ c u _^_.^ a 1 ^ 1 5 "" a." ■j: E ". E Z 1 >•? E M rt u d-tu- d. /u. h. R. k. 2c. s. hH t^i <r. /^■S. rr. f', ^ ^ w is in lb. I pr cub ft. i 1 /i s in lb. per sq. ft. • Kernel" Middle Third of Arch-ring. 1 Intensity of thr ust on Sandstone The Soffit is the Three-point Circle of a lower £ j Key — • mu St not e ^ 576000 ''"«''r0xT40 1* • 2 ' curve transformed from same Catenary as Line of Stress, the directrix being: surface of Kails. 2 its average \ jeing i its maximum. !! 1 1 1 a i Feet. ' 1 j 1 1 1 1 50 72 •122 •141 •019 •026 •973 •325 1-451 •120 i 55 81 •116 •137 ■021 •029 •970 •333 1-462 •115 61 90 •110 •133 ■023 •033 •966 •341 1-473 •110 Feet. 68 101 •104 •129 •025 •037 •962 •3l9 1^482 ■105 38 56 76 112 •098 •125 •027 •041 •957 •357 1-491 •100 42 63 84 126 •092 •121 •029 •045 ■952 •364 1-497 •095 ; 46 70 92 140 •085 •117 •032 •019 •946 •372 1-504 •090 61 78 102 156 •07'J ■113 •034 •055 •940 •380 1-509 •085 Feet. 19 29 67 86 114 172 •073 •110 •037 •062 •933 •388 1-514 ■080 21 32 63 95 126 190 •006 •106 ■040 •071 •925 •395 1-516 ■075 23 35 69 105 138 210 •059 •103 ■044 •081 •917 •403 1-518 ■070 •25 39 77 116 1.54 232 •051 •099 ■048 ■089 •907 •409 1-516 •065 28 43 85 128 170 256 •043 •095 •052 •097 •896 •416 1-514 •060 31 47 94 141 _ _ ■036 •092 •056 •110 ■8S3 •421 1-506 •055 35 52 104 155 - - •028 •089 ■061 •123 ■869 •427 1-498 •050 38 67 - - - - •018 •085 •067 •141 ■850 ■431 1-480 •045 42 62 - - - - •009 •082 •073 •159 •831 •436 1-462 •040 46 67 — - - •000 ■078 •078 •177 •812 •441 1^444 •036 a.' •1- S Subject to a mult., loss than irivcn max., to give i « •(5 a factor of safety greater than 10 in inverse ratio. "" B O o o • Q TABLE B. \ A Srn'ex o/" " Two-xoskd CATKXAiaics" inscribed n the Circle of Iiadiiis (ii, ) Unihf, and hariur/ Parallel Directrices a t Graduated Distances {h\ + Yu) from its Centre from (1 -t. -187) to (1 + -026). Tl is Table forms a continuation of the Supplementary lables Bi, B,, and B3, and lias then for its purpose the designing of arch-rings, so as to secure the condition of the ine of stress Iving within a A •#>•««/ forming the middle third, fifth, or ninth of tlie an li-rin -, respt ttively. s. Ou ^3. m. lil. It.. pi. pd = P2- yo. To. 5o. 5.2. C "1 ^ « 5 w S ^ V J) 1 u. « -— >. u c •— I. 0, C u 3 „ >y > . 3 3 u u rt 3 V u u a k.T3 c rt "rt s j= " ? «. •3 U M « = .1 •-; > '"' 5 u t = e III c— = = :=£ .0 rt "H 5 t -^ c = •£ ' 5^ ill 3 V " ^:5 B rt V 3 rt J2 i. '^ tn "0 ° in .2 -5 C . c S •=1'= "- X- S « a 1-3 :=.a u i: U ■5 Cm |5 ^ r — -» 7 '^H a. 3 u 'p a c •a. 5 • a C/3 £^- ^^- T° 0- Q" fi- •180 25°37 39°46 •4455 1^ r0024 -9491 1^0501 -1890 •1869 •0021 •ooio •175 25°58 40°2S •4410 r0024 •9452 1 0543 -1845 1 -1821 •0023 -0011 •170 26°20 41°11 •4365 1^0024 -9413 roose •ISOO -1774 -0026 -0013 •165 26°40 4r53 •4319 H-g 1^0024 -9370 1^0635 -1754 •1726 -0028 -001.5 ■160 27°01 42°3G •4273 ^■" 1^0024 -9327 1^0684 •1709 •1678 -0031 -0017 •155 27°20 43°18 •4-226 3 ^ 1^0023 •9279 1^0738 •1664 •16-29 •0034 j -0019 •150 27°40 44°00 •41S0 n ~ 1^0022 •9232 r0793 •1619 ^1581 -0038 : -0021 •145 27°59 44°42 •4132 = 1-0021 •9180 1-0856 •1574 ^1532 -0042 ; -0023 •140 28°1S 45°24 •4085 r ** ^ 1^0020 •9129 1-0919 •1529 ^1484 -0045 -0026 435 28°36 46°06 •4036 .2 y rt 1^0019 •9071 ru990 •1483 •1434 -0049 -0029 •130 28°55 46°49 •3988 1^ rooi7 •9014 M061 •1438 •1384 •0054 ^0032 •125 29°r2 47''31 •o9.SS rooi4 •8952 1-1142 -1392 -1333 •0059 -0035 •rio 29°30 48°14 •3888 1-0011 •SS90 1-1224 -1347 •1'283 -0064 -0039 •115 29°47 4S°o7 •3836 '§^ 1-0007 •8821 M318 -1301 -1231 •0070 -0043 •110 30°(i4 49°41 •3785 1-0004 •8752 1^1412 -1255 •1180 -0076 ; -0048 •105 30°20 50°25 •3731 c ;; tr 0-9998 •8676 M521 •1209 •1127 -0082 0053 •100 30°37 51°09 •3678 1 0-9993 •8600 1^1631 •1163 •1074 -0089 -0058 •095 30°52 51°54 •3622 0-9987 •8517 1^1760 •1116 •1019 •0097 -0064 •090 31°08 o2°40 •3567 0^9981 •8433 1 1889 -1070 •0965 -0105 [ -0071 •085 31°23 53°27 •3508 ^ 0-9972 •8340 1^2043 -1023 •0908 •0114 -0079 •080 31°39 54°14 •3450 0^9963 •8246 1^2197 -0976 •0852 •0124 -0087 •075 31°54 55°03 •33S8 0^9951 •8141 r2384 -0928 •0793 -0134 -0097 •070 32°09 o5°53 •3326 0-9940 •8036 P2o71 -0880 •0735 -0145 1 -0108 •065 3-2°23 56°45 •32o9 0-9925 •7917 1^2803 -0831 -0673 -0158 -0121 •060 32°38 57°38 •3193 0-9910 •7798 1^3036 -0782 •0611 •0172 ^0134 •055 32°52 58°34 •3121 0^9890 •7661 1-3335 -0732 •0544 -0188 -0150 •050 33°06 59^31 •3049 0-9870 •7524 1-3634 •06S2 -0478 -0-204 -0167 •045 33°20 60°34 •2968 0-9842 •7363 1-4037 -0630 -0406 -0223 , -0190 •040 33°33 6r37 •2888 0-9815 •7202 1-4440 -0578 -0334 •0243 ' -0213 •035 33°46 62°49 •2808 . 0-9788 •7040 1-4843 •0526 •0260 -0263 ! -0236 i Independent of /i^. Direct y proportioned to Ji^, and s ubject to any mu Itiplier. SUPPLEMEXTAllY TATTLE B,. — - - il.„ 1 tC :« *— .*? — « , Strons Brick M. = 112 /= 154000 Sandstone. 7,= 140 / = 576000 Granite. 7,. = 164 /= 1350000 O CM a 1 £■" - .S 3 11 1° £ 2 c lii =e = O = iEc o *. so c = 11 3 si I' 5 ^ 3 « 2 a" a-l- ° II o c •- o "a c C3 lilt, to give >f safety lo. ilueofspnn that mult. "3 n in •/3 a M 3 Q s ^X. >.s C X 7< :1 5 1 2-S rs 5 d-l„. d. <0- <2. R. k. 2c. $. s. f. 3^ 3 <=i f-; «5 w is in lb. per cub. ft. /is in lb. per sq. ft. "Kernkl"' Middle P'ifth ok Arch-ring. Intensity of thrust on Sandstone The -Soffit is the Three-point Circle of a loTOer _, curve transformed from same Catenary as Line of Stress, the directrix bein? surf.ice of Rails. p^d , 576000 Key -y- must not exceed jQ-^^-j^ . g its average being % its maximum. 18 20 22 24 27 30 32 35 38 42 46 50 Feet. 17 24 26 29 32 35 39 43 46 50 54 59 64 69 40 45 50 55 61 67 73 81 89 97 106 Feet, 56 63 71 79 87 95 104 116 128 139 161 1 Feet. 52 j 70 •150 59 80 •145 66 90 •139 73 101 •134 80 112 •128 90 126 •122 100 142 •115 110 158 •109 122 174 •103 134 190 •096 146 208 •089 162 232 •082 178 256 •075 - - •068 _ _ •060 _ _ •053 _ _ •044 _ _ •036 •027 •169 •166 •162 •159 ■165 •151 •147 •144 •141 •137 •134 •131 •128 •126 •122 •120 •117 •115 •113 019 021 023 025 027 029 032 035 038 041 045 049 053 057 062 067 073 079 086 026 030 034 038 042 046 050 056 063 070 077 086 095 105 115 128 142 159 176 •972 •968 •964 ■960 •956 •951 •945 •939 •932 •925 •917 •908 •899 •877 •862 •848 •831 •814 273 280 287 294 302 309 316 322 329 335 342 34S 354 359 365 368 372 375 378 351 362 373 383 394 402 410 415 420 424 429 4 '29 429 426 423 414 404 390 376 •150 •145 •140 ■135 • •130 •125 •120 •115 •110 •105 •100 •095 •090 •085 •080 •07'. •070 •065 •060 Subject to a mult., less than eiven max., to Rive factor of safety ureater than 10 in inverse ratio. TAT.LE 1 1. A Series o/ " Two-xosed Catenaries" inscribed ■« the Circle of Radius (i?,^ Utiittj, and hari)ig Parallel Direc trices n t Graduated Distances (ill -^ Yt) ff'om its Centre from (1 ->- •IS 7) to (1 + -026). i Tt is Tahle forms a continuation of the Supplementary Tables Bi. Bj, and Ba, and has then for its purpose th B designing of arcli-ringrs, so as to secure the condition of the ine of stress lying within a kernel formiu!? the middle third, tifth, or ninth of | the s. irch-ring, respectively fli. 02. m. JRi. Hi. pi. P0 = P2- yo. To. 5o. 52. S u a •5 3 a .S~ » » o u O O Ji VI a) w u »2 c rt . o U o 3 e u 01 5 u. 2 2 0— o |5. S.2 o _ ih CO « ill 04 •3 3 O •O w o. = o u •5 ° JO "2 li. = ■5,2 3 -j.t; o o a o 3 Is .= •= li* i'^ ■ 3 ■" 2 U 3 u m o .5 « a u ° o H J3 CO sd • lit o 3 a. 3 Ui "o '5 « c dl •180 25°37 39°46 •4455 u-3 1-0024 •9491 l^OoOl -1890 -1869 -0021 •0010 •175 25°58 40°2S •4410 U jj 1 0024 •9452 1 0543 -1845 -1821 . -0023 -0011 •170 26°20 41°11 •4365 IH-a 1-0024 -9413 r0586 -1800 -1774 -0026 -0013 •165 26°40 41°53 •4319 ^"a 1-0024 •9370 1^0635 •1754 1 •1726 ^0028 ' •OOlo •160 27''01 42°36 •4273 >v- r0024 -9327 1-0684 •1709 •1678 •OOSl ^0017 •155 27°20 43°18 •4226 'S ° 1-0023 •9279 1-0738 •1664 •1629 ^0034 •0019 •150 27''40 44°00 •4180 c iJ O 3 1-0022 •9232 1^0793 •1619 •1581 -0038 i -0021 •145 27°59 44°42 -4132 1-0021 '9180 1-0856 •1574 -1532 ^0042 ' -0023 1 •140 28°1S 45°24 -4085 10020 •9129 1-0919 -1529 -1484 -0045 -0026 1 •135 28°36 46°06 •4036 1-0019 ■9071 1-0990 -1483 -1434 -W49 •00^29 •130 1 ' 2S°55 46°49 -3988 •3 u 1-0017 •9014 1-1001 -1438 -1384 •0054 •0032 •125 29°12 47=31 • -3938 1-0014 •8952 1-1142 •1392 1 -1333 •0059 ' -0035 •120 29°30 48°14 i •3888 S a 1-0011 •S890 1-1224 •1347 ^1283 -0064 i -0039 •115 29°47 48°57 ; •3836 |u{ rooo7 •8821 M318 •1301 -1231 -0070 -0043 •110 30°(i4 49°41 1 -3785 Q u 1^0004 •8752 1-1412 •1255 1-1180 -0076 -(X)48 •105 30°20 50°25 •3731 ■^5 en o 0^9998 •8676 1-1521 •1209 •1127 -0082 0053 •100 30''37 51°09 ^3678 3 0^9993 •8600 1-1631 •1163 -1074 1 -WS9 1 -0058 | •095 30°52 51^54 •3622 0-9987 •8517 1-1760 •1116 -1019 -0097 j -0064 •090 31°08 52°40 •3567 0-9981 -8433 1 1889 •1070 -0965 -0105 ' -0071 •085 31°23 53°27 •3508 j5 0-9972 •8340 1-2043 •1023 -0908 •0114 ^0079 •080 31°39 54°14 •3450 1 0-9963 •8246 r2197 -0976 -0852 -0124 -OCKT •075 31°54 55°03 •3388 1 0-9951 •8141 1-2384 -0928 -0793 -0134 ' -(DO? •070 32°09 55°53 •3326 1 0-9940 -8036 1^2571 -0880 -0735 -0145 ; -0108 •065 32°23 56°45 •3259 I 0-9925 •7917 r2803 -0831 -0673 -0158 i -0121 | •060 32°38 57°38 •3193 0-9910 •7798 1-3036 •0782 -0611 •0172 ^0134 •055 32°52 58°34 •3121 I 0-9890 •7661 1-3335 -0732 -0544 -0188 -0150 •050 33°06 59=31 •3049 1 0-9870 •7524 1-3634 -0682 •0478 -0204 1 -0167 •045 33''20 60°34 •2968 1 0-9842 •7363 1-4037 •0630 -0406 -0223 i -0190 •040 33°33 61°37 •2888 1 0-9815 •7-202 1-4440 •0578 •0334 ■0243 ^0213 •035 33''46 62°49 •2808 0-9788 •7040 1-4843 •0526 •0260 •0263 •0-236 Indep endent of R-^. Directly proportioned to R,, and subject to any multiplier. SUPPLEMEXTAllY TABLE B,. strong Brick. w=lI2 /= 164000 e e M M K S rt s ^ Sandstone. T< = 140 y = 576000 Granite. tc = 164 /= 1350000 S 2 0.3 .5 rt o ■ ^ — ' - ? ° £ c B ■ B-l- => II I 2 - E °3 d-lo. £. ■ur is in lb. per cub ft. /is in lb. per sq. ft. Intensity of thrust on Sandstone p„rf ^ 576000 3* Key -y- must not exceed ^q ^ ^^q . ^• its average being J its maximum. 20 22 25 27 30 33 36 39 42 46 50 55 Feet. 27 30 33 37 40 44 48 52 56 61 66 71 40 44 49 55 61 67 75 Feet, 50 57 64 72 80 89 83 110 91 • 121 99 108 118 132 145 157 58 64 70 80 98 110 122 134 150 166 182 198 Feet. 72 80 88 101 114 129 144 161 178 199 220 242 264 "Kernel" Middle Ninth of Arch-ring. The Soffit is the Three-point Circle of a lower curve transformed from same Catenary as Line of Stress, the directrix being surface of Rails. •178 •197 . •173 •194 •167 •190 •162 •187 •155 •183 •149 •180 •143 •177 •137 •174 •130 •171 •123 •168 •116 •165 •109 •162 •102 •160 •0G5 •158 •088 •156 •080 •154 •072 •152 •064 •1.51 •054 •149 019 1 •028 021 •031 023 •034 025 •037 028 •041 031 •047 034 •054 037 •060 041 •067 045 •073 049 •080 053 •089 058 •098 063 •108 068 •119 074 •131 080 •144 087 •1.58 095 •173 ' 964 •223 963 •230 962 •238 958 •245 953 •252 948 ■258 942 •265 936 •269 929 •274 921 •281 •913 •288 903 •293 893 •298 882 •302 •870 •307 857 •311 844 •315 828 •317 S12 •320 1-234 1-250 1-266 1-278 1-290 1-299 1-309 1-316 1-323 1-327 1-331 1-331 1-332 1-334 1-326 1-320 1-315 1-303 1-291 C4 -180 -175 •170 -165 •160 •165 -150 •145 •140 -136 •130 -125 -120 -115 -110 •105 •100 -09.5 -090 . • • 1- o CO o CO • c Subject to a mule, less tli.m eiven max., to g^ve tactor uf saluly greater than 10 in inverse ratio. TABLE B. A Sen'es o/" " Two-xosfd Catexakies" inscribed in the Cirde of liadiuH {R\) Unity, and having Parallel Directrices at Graduated Distances (lii + To) from its Ce7itre from (1 + -187) ^o (1 + -020). This Table forms a continuation of the Supplementary Tables Bi, Bj, and 0:^. and has then for its purpose the designiiifj; of nrcli-rings, so as to secuie the condition of the line of stress lying within a kernel forming the middle third, fifth, or ninth of the arch-ring, respectively. Independent of A'j. s. 0i. 6i. m. Hx. ■- § c pi. P0 = P2- yo. To. 5o. 8>. i u a rt "5 3 c.ti c-5 <J o O O u ■" ». « o »- . t) ■« « C ,1 3 . U ¥j ill E.2 ^ .O •r o J 0. c ■^ c •£ ° o a 3 > £ Si O "1 3.5 ;5^ t . u ^ ~ 3 «^ Ill i i a .£ H si ^ u S u 'o "o ■51 ?•■" 1-^ JJ, 5 = o o 3 a. 3 'o '5 d c d ~ •180 25°37 39°46 •4455 I'S 1-0024 •9491 1-0501 •1890 •1869 •0021 •0010 •175 25°o8 40°2S •4410 1^0024 •9452 1 0543 •1845 •1821 •0023 •0011 •170 26°20 41°11 •4365 1-0024 •9413 r0586 •1800 •1774 •0026 •0013 •165 26°40 41°53 •4319 ^'b 1-0024 •9370 1^0635 -1754 •1726 •0028 •0015 •160 27°01 42°36 •4273 ^■~ 1^0024 •9327 1-0684 •1709 •1678 •0031 •0017 ■155 27''20 43°18 •4226 r0023 •9279 1^0738 •1664 •1629 •0034 •0019 •150 27°40 44°00 •4 ISO rt ~ c tB o = 1^0022 •9232 1^0793 •1619 •1581 -0038 •0021 •145 27°59 44°42 •4132 1-0021 •9180 1-0856 •1574 •1532 •0042 •0023 •140 28°1S 45°24 •4085 10020 •9129 1^0919 •1529 •1484 •0045 •0026 •135 28°36 46°06 •4036 M O rooi9 •9071 1^0990 •1483 •1434 -0049 •00-29 •130 28°55 46<'49 •3988 1-0017 •9014 M061 •1438 •1384 •0054 •0032 •125 29°12 47°31 •3938 1-0014 •8952 ril42 •1392 •1333 •0059 •0035 •120 29°30 48°14 •3888 ■^ % 1-0011 •f.890 M224 •1347 •1283 •0064 ; •0039 •115 29°47 48°57 •3836 'iM 1-0007 •8821 M318 •1301 •1231 •0070 1 •0043 •110 30°()4 49°41 •3785 1^0004 •8752 1^1412 •1255 •1180 •0076 •0048 •105 30°20 50°25 •3731 M O 0-9998 •8676 M521 •1209 •1127 •0082 0053 •100 30°37 51°09 •3678 3 •g o 0^9993 1 •8600 1-1631 •1163 •1074 •0089 •0058 •095 30°52 51°54 •3622 0^9987 •8517 1-1760 -1116 •1019 •0097 j •0064 ■090 31°08 52°40 •3567 0-9981 •8433 1 1889 •1070 •0965 •0105 ' •0071 ■085 3r23 53°27 •3508 .2 H 0^9972 ; •8340 1-2043 •1023 •0908 •0114 •0079 •080 31°39 54°14 •3450 0-9963 ; •8246 1-2197 -0976 •0852 •0124 •0087 •075 31°54 55°03 •3388 0^995 1 •8141 1-2384 •0928 •0793 •0134 1 •0097 •070 3-2°09 o5°53 •3326 0^9940 •8036 1-2571 -0880 •0735 •0145 ! •0108 •065 32°23 56°45 •32o9 1 0-9925 •7917 1-2803 -0831 •0673 •0158 1 •0121 •060 32°38 57°38 •3193 1 0-9910 •7798 P3036 •0782 •0611 •0172 i 1 •0134 •055 32°.72 58°34 •3121 1 0-9890 •7661 1^3335 •0732 •0544 •0188 • •0150 •050 33''06 59^31 •3049 0-9870 •7524 1-3634 •06«2 •0478 •0204 ; •0167 •045 33°20 60°34 •2968 0-9842 1 •7363 1-4037 -0630 -0406 •0223 1 •0190 •040 33°33 61°37 •2888 0-9815 1 •7202 1-4440 •0578 -0334 •0243 •0213 •035 33°46 62°49 •2808 0-9788 1 •7040 1-4843 1 •0526 •0260 •0263 •0236 Directly proportioned to /?,, and subject to any multiplier. 402 APPLIED MECHANICS. [CHAP. XXII. EXAMPLKS. 11. Design of a sandstone segmeiitul arch with vertical load, directly from the tables, span, 75 feet, and depth of surcharge at crown about 1' 4" : the springing to be the joint of rupture. (See fig. 12, left half.) Here '2c = 75, and d — to = I'S ; their ratio is 56"25. "We find by trials on Table Bi, that 2c -^ {d - to) = 53 occurs on the line where »■ = "05, and the multiplier required on that line to make 2c into 75, is 50'07, about half of the max. mult, given under " sandstone " ; so we shall have a factor of safety of about twice ten, and need not consult the other supplementary Tables, £2, £3. That line gives the following relative and absolute values : — Mult. d-to d ^0 h It k 2c 50-07 j -028 ( 1-41 •089 4-46 •061 3-05 •123 6^15 •869 43-5 •427 21-4 1-498 75 ft Rx Po Yo 5o 62 1 1^3634 •0478 •0204 •0167 50-07 08-265 2-393 1-02 •836 ft. •050 The radius and rise of soffit are 43^5 and 21*4 feet ; the thickness of arch-ring at crown and springing 3' 0" and 6' 2" ; the surcharge being 1'41 feet, or about 1' 4", as required. On the same line, continued on Table B, we have — s Bi Mult. •050 59= 31' 50-07 At tlie crown the thrust on the arch-ring per foot of the breadth is S = wpud = 140 X 68-265 x 4-46 = 42624 lbs. ; the average intensity of the stress is 42624 -^ 3-05 = 13975 ; and double of this 27950 lbs. per sq. ft., is the maximum intensity, giving a factor of safety of 576000 4- 27950 = 20. Otherwise, the multiplier being half the maximum in the Table, the factor of safety is double ten. At the springing, 2'= jff sec ^2 = 8402;j ; the average stress" 84023 -^ 6-15 = 13062 ; and since the deviation of the centre of stress is ^^2 — 5i = 1^025 — -836 = -189 above the centre of the joint ; we can substitute this for 5, the deviation of the centre of stress from the middle of the joint, equation (1), at page 417, Max. stress 65 = 1 + _ = 1-184. Aver, stress ^2 Hence the maximum intensity of the stress on the springing joint is 16180 lbs. per sq. ft. Dividing this out of 576000, the crushing strength of sandstone, we get the factor of safetj* 35. A simpler way to proceed to get the factor of safely against crushing, sufficiently close for all purposes, is to divide the crushing strength 576000 by 13062, the average, and the quotient 42 is the apparent factor of safety against crushing the sandsione-skew-back. For the three deviations of the centres of stress shown on fig. 3, expressed as fractions of t the thickness of the joint, the average bears to the maximum the ratios below. Deviations of c. of stress, iVth, "iVth, ith. llatios of aver, to max., *> «, ^• Now tlie deviation of the centre of stress on the skew-hack (fig. 12) is -180 in 6- 15 ; as this is less than jf , then the factor of safety is at least |ths of 42, that is, there is a real factor of safely greater than 31. lil Pu To «o 52 \ .50-07 1-2384 62-007 •0793 3-970 •0134 •671 •0097 •486 ft, CHAP. XXII.] DESIGN OF MASONRY ARCHES. 463 12. If a live load of 2"20 lbs. per sq. ft. of tlie ])lutfoiiii be all over the span of the bridge (Ex. 11), find the new line of stress in the arch-ring and the intensities of the stresses at crown and springing. (Fig. 12, right half.) The height of superstructure equivalent to this live load is A = 220 -f 140 = 1-571 ft. of sandstone. Here we have to find a new two-nosed catenary still inscribed in the same circle, ii'i = J0'07, forming the upper boundary of "kernel" (middle third) of the arch-ring, as already designed (Ex. 11), but to a directrix, h higher than before. Adding h to the old value of Yq, we get 2-393 + 1'571 or 3-904, which, divided by the multiplier 50-07, gives us -0793 as a nctc relative value of I'o, which is found in the Table B, at the line — .y e-i Mult. •075 55° 3' 50-07 This is a new two-nosed catenary, of a different modulus and of a different family, so that the soffit already designed will not be mathematically the three- point circle of another member of the family of this line of stress, but it "w-iU sensibly be so. The joints of rupture have gone up to 55° 3' ; but this is immaterial, as the line of stress is now closer to the upper "kernel," and will therefore be wholly in the " kernel," down to 59° 31' the springing joint. In designing the abutment, the tangent at the joint 55° 3' should lie in its middle third. At tbe crown, now we have the thrust, 5"' = M;poC«^ 4- A) = 140 x 62-007 (4-46 + 1-57) = 52346 lbs. ; average intensity 52346 -f 3^05 = 17162 ; the devia- tion of centre of stress is \to — 5,. = -509 - -671 = •162 ft. hdow centre of the joint. The apparent factor of safety is 576000 -f 17162 = 33. The fractional deviation of the centre of stress from the middle of joint being -162 in 3"05 or 1 in 18 nearly, the factor of safety is of |ths its apparent value, that is, 27 is the factor of safety at crown when the live load is on the bridge. At the springing joint, T = if ' sec 59° 31' = 103188 lbs., nearly ; the average intensity is 103188 -^ 6^15 = 16778 ; the deviation of centre of stress is -J<2 - 52 = 1"025 — •486 = -539 ft. above the centre of joint. The fractional deviation of centre of stress is -539 in 6-15 or less than 1 iu 10. The apparent factor of safety is 576000 -f 16778 = 34, and the factor of safety not less than fths of this or 21. 13. Let the live load (Ex. 12) cover only one-half of the span: find the horizontal thi-ust to be balanced by the backing of the voussoirs. The horizontal thrust due to dead load is (Ex. 11) 42624 lbs. ; and due to dead and live loads combined it is (Ex. 12) 52346 lbs. ; the difference is 9722 lbs. per ft. of breadth. 14. Suppose the arch-iing, spandiils, &c., of Ex. 11, have, by means of voids in the supersti-ucture, an average density of 100 lbs. per cubic ft. Find results corresponding to those of Ex. 11. For stability, and to give the required surcharge, the dimensions in Ex. 11 are required, just as before, but the stresses will be altered in the ratio 140 : 100. H = 30500 lbs., nearly ; T= 60020 lbs., giving factors of safety, 28 at crown and 49 at springing. These values aie engraved on fig. 12. The voids in the superstructure should be so arranged that their boundary may be roughly a member of the same family as line of stress, by making the ordinates of their boundary a constant fraction of those of the soffit. 464 APPLIED MECHANICS. [CHAP. XXII. 15. Let a live load of 157 lbs. per sq. ft. of platform be over the \rhoIe span of bridge (Ex. 14) : find the line of stress and the intensities of stress at crown and springing. The equivalent height of structure is l'o7 ft., taking the new density into account ; so that the solution is the same as Ex. 12, only we must alter the quantities in the ratio 140 : 100. E' = 37500 lbs., nearly; T' - 73700, nearly, and the factors of safety are increased to 35 at crown and 31 at springing. These values are engraved on fig. 12. 16. Let the live load in Ex. 15 be only over one-half the span: find the amount of horizontal thrust to be balancedby the frictional stability of vault covers butting against the higher voussoirs. Find also the distance back to which the vault covers must extend to balance it. (See tig. 12.) The thrust is, P= E" - H= 37500 - 30500 = 7000 lbs. per foot of breadth. If the underside of the vault covers come up to the level of the crown of the soffit, then the weight per foot of breadth of bridge on the spandrils due to the vault covers, and dead load over them alone, is wdl-=^ 140 x 4-46/ = 624'. Taking the coefficient of friction at "7 ; then -7 x 624/ = 7000 or / = 16 ft. to the neare.'^t foot. The voussoirs near the key-stone should have square-dressed side-joints until the sum of their vertical projection is to, the depth of key-stone, so as to receive the horizontal longitudinal resistant thrust of the vault covers truly; and the covers are to be built with square-dressed side joints set closely so as to yield as little as possible. Light spandril walls may be built up to level of crown of soffit, and extend hack so as to give 16 feet longitudinally of vault covers ; then the spandrils- may step rapidly to lower levels. 17. Design of a semicircular arch-ring of common sandstone, the span to i.e 100 ft., and a surcharge of at least Ijft. being required for the formation of the roadway, laying of gas-pipes, &c. The data are R = 50, and .T( -f (rf — <y) not to be greater than 33. On Table Bi, the lines above that with .< = -OS (in order to make li into 50) require a multiplier greater than the maximum given for sandstone ; these lines are tlierefore excluded on the question of strength, while the lines below that with s = '05 give Ji i- {d — t^) greater than 33, and are excluded by requirements of the roadway. These two limiting lines give — ■08 •05 The upper gives greatest economy of material in arch-riiiff, wliich is only 2 ft. at crown, but kss economy of material in superstructure, as d is larger, and also less economy of solid backing, which has to be built to a joint 5° higher. Hence the line midway between them would be most suitable all round. For a single arch a line a little nearer the upper may be adopted ; and for a series of arches a line nearer the lower, that is, in favour of a heavier arch-ring to withstand the shocks transmitted from arch to arch. The best lines, then, are for a Single Arch, or a Series, respectively — Mult. R d tn 03 Factor of Safety. 53-6 50 5'9 2 54° 14' 57 X 10 57-5 50 0-1 3-5 59° 31' ol'b Mult. R d (n h e-i Factor of Safety. ) X 1 d4"5 5 X 1 55^8" 69 X 10 •07 54526 50 o^6 2-4 44 55° 53' = 12^7 d4^5 •06 55-804 50 6^3 29 5-4 .'57° 38' --^— = l.j 2 CHAP. XXII.] DESIGN OF MASONRY ARCHES. 465 Compare Kunkine's empirical rule, Civ. Eng., Art. 290, giving t^ = v/^12~>r50 and \/T7 x 50 2-45 ,, 2-92 respectively. The solid backinK must be brought up to the point where the joint at Q2 meets the back of the arch-ring, and below that joint the arch-ring may be of the uniform thickness /■.>. The su!)erstriicture may readily be reduced by voids and the employment of material of less density than sandstone, till the average density of the whole is a fifth less than that of sandstone, which would raise the factors of safety at crown to 16 and 19. The factors of safety at joint of rupture are even greater as the centre of stress is nearer the centre of the joint, and is -r <i > sec 6n. By means of the values obtained for w, p^, d, 5j the thrust at crown and joint of rupture, and the centre of stress at joint of rupture are calculated, as in preceding example. A tangent from this last point enables a suitable abutment to be designed. 18. Design of a segmental brick arch, having the joints of rupture at the springing, a span of 30 ft., and a surcharge of 2| ft., slightly more or less. Here the ratio '2c -- (d - i^) is 12. But this ratio (nearly) occurs in the Tables Bi, Bj, and B3, as noted under — d - /,j s Max. Span Brick. Factor of Safety. Table Bi 11-9 •120 12 ft. 10 „ B2 W-h •125 21ft. 10 )> B3 11-5 •130 37 ft. 10 ^2 Mult. d-t^ d (0 h R 46° 49' 22^54 ( -116 t2-61 •165 -05 -08 3-72 1-2 1-8 -913 20-6 po 1-106 24-92 k •29 4^5 2c 1-331 30 ft. Factor of Safety. 37x10 ,„ , = 12-5. 30 It is clear, then, that we cannot adopt the proportions from Table Bi or Bo ; for althougli the surcharge and span would be in the required proportion, the multiplier to make 2c equal to 30 ft. would only give in Table Bi a factor of safety \% x 10 = 4, and in Table B2 a factor |^ x 10 = 7, which are not sufficient. From Table B3, then, ■130 The arch is to spring at about 45°, have a rise of \\ feet, the radius of soffit being 20' 8", while the thickness of the crown is to be 14 inches, and at the springing 50 per cent, greater. These dimensions would suffice for the arch-ring built of the strongest red bricks moulded into the proper wedge-shaped forms. The line of stress may be exactly located, as in previous examples ; but it would be sufficiently near, in this case, to take it as being up the very centre of the arch-ring : and a line drawn normal to the springing-joint from its middle point is sensibly the line of thrust on the abutment. The superstructure could easily be arranged with voids to make the average mass of the whole m; = 90 lbs. per cubic ft., or |ths that of brickwork, which would increase the factor of safety to about 20. The thrust then would be — At crown, B. = xcp^d = 9262 lbs. per ft. of breadth. At springing, jr= ZT sec 62 = 13532 lbs. ,, ,, For ordinary-shaped bricks the arch-iing ought to be built of a uniform thick- ness, the line of stress and thrust remaining sensibly unchanged, as the extra part 2h 466 APPLIED MECHANICS. [CUAP. XXII. of the arch-ring at crown will really form part of the superstructure. Fcir a single arch, an average of to and Cj could he taken ; but for a series of arches the greater had hetter be chosen, giving respectively t= 1-5 and 1-8 feet. Compare Rankine's empirical formula, Civ. Eiig., Art. 290 — t = v/20-6 X ^ and \/20-6 x -17 — 1-57 ,, I '87 respectively. 19. If the bridge, left-half of fig. 12, is to carry a road, approaching with a slope 1 in 100, and passing over with a curved profile sensibly circular; to find position of profile. To make the profile sensibly a member of the same family as line of stress, it is sufficient to derive it from the soffit, which is sensibly a member of the family. Let Mo and ui be depth to profile at ciown and springing. The tangent to profile, being like that to a pai-abola, gives slope of half chord of profile 1 in 200; hence ?<2 - Mo : 37-0 : : 1 : 200, and ii2 : tio : : 4-G + 21-4 : 4-46. Hence >/„ = -04 ft., and «2 = "23 ft. The profile is to be below the directrix about i an inch at crown, and about 3 indies at springing. The load left out between directrix and profile increases the factors of safety as 4*46 ; (4-40 — 0-04). 20. Compare the segmental masonry arch of 75 feet span as already designed in Example 10, for the moderate surcharge of I' 4" with others with the surcharge heavy and light i-espectively. Deriving the dimensions from the three lines of Table Bi where s = 'OS, s = '05, and s = •035, the multipliers are 49-6, 50-07, and 51"94, that in each case the span may be 75 feet. The corresponding maximutn spans for sandstone in the fourth cidumn are 86, 155, and 201 feet; and 10 increased in the ratios which these bear to 75 gives the factors of safety against crushing the keystone it' the ring be sand- stone. The values for strong brick are one-third part, while those for granite are double that for sandstone. The three designs are sketched du fig. 13, and we have the following dimensions and factors of safetj' against the crushing of the key- stone: — Surcharge, . 3' 8" r 4" 0' 0" Span, . 75' 0" 75' 0" 75' 0" Keystone, . r ID" 3' 0" 4' 0" Ske whack, . 3' 1" 6' 2" 9' 0" Rise, . 19' 3" 21' 5" 22' 10" Total rise, . 24' 8" 25' 10" 26' 10" Strong brick, 4 7 9 Sandstone, 12 20 27 Granite, 24 40 54 It is to be observed of Table B, that it gives designs of the utmost economy of the material in the arch-ring itself con- sistent with equilibrium under the vertical-load only. But as the surcharge gets heavier, as, for instance, on the first part of fig. 13, there is not sutiicient strengtli with biick or sandstone, tliough granite would suffice. With still heavier surcharges (see Example 8), we are forced to sacrifice economy of material and use tlie other Tables Bz and B3. The problem is passing CHAP. XXII.] DESIGN OF MASONRY AKCHES. 46^ from one of stabiliti/ to one of mere streivjth. See page 419, under " strength." As the surcharge further increases relative to the span, the line of stress becomes practically up to the very centre of the ring, and we have buried arches as in examples to follow, and in tunnel roofs for the deep lengths. Heavy Surcharge. Directrix Moderate Surcharge. Inspection of the Tables B,, Ba, and B3 shows clearly that for balanced segmental masonry arches strong brick is economi- cal for spans from 20 to 50 feet, sandstone from 50 to 120 feet, and granite from 60 to 250 feet. 2 H 2 468 APPLIED MECHANICS. [CHAP. XXII. Linear Transformation of Balanced Eib. We had an example of this in fig. 8, wliere we supposed one of the curves AB to be painted on a sheet of indiarubber pinned down along its edge OJ^, and already stretched in the vertical direction. The sheet is, of course, made of a purely imaginary kind of indiarubber which can be stretched in one direction without changing in the direction at right angles to it. At the ends of the arc A and B are to be painted the forces which balance it, H and 2' to scale ; while the veitical-load- area, the only load oil the arc AB, is mapped out by the four painted lines, the arc itself and the three straight boundaries OA, NB, and ON. Then, by further stretching the sheet vertically or by allowing it to contract, we obtain a new balanced rib with new forces at its ends and a new load-area. Otherwise the lines may be painted or traced through upon a sheet of glass lying flat on tlie paper and himjed to the paper along its upper edge OX. The glass is then to be tilted up on its edge OX, and the work projected normally on the paper. The sheet of glass, although hinged to the plane of the paper along OX, may extend beyond the hinge. It may also have been originally tilted up when the work was projected normally from the paper on to it, then flattened down before projecting the work back on the paper. This is the simplest case of the orthogonal or linear transformation to obtain, from a rib balanced under a vertical-load-area alone, another rib balanced under the transformed vertical-load-area. In the work which follows, it is more convenient to transform horizontally. In dealing with a rib balanced by a vertical-load- area and horizontal-load-area conjointly, both of these areas themselves may not be painted on the stretching-sheet or tilting-plate. For convenience, the vertical-load-area may be supposed to be painted on the tilting-plate, but after the transformation the density or weight per square foot of area is to be altered. That is, the superstructure, if transformed linearly, must then be reckoned to be of a new material of different density or weight per cubic foot. All forces, thrusts along the rib or their components, cross the hinge at the same points after the transformation as before. Forces or component forces in the direction of the transformation alter in the ratio of the transformation ; tho.se at right angles remain unaltered. Ilesultant stresses or arrows representing the e.xtern-load. CHAP. XXII.] DESIGN OF MASONRY ARCHES. 46!» such as r, r, &c., on the south-west quadrantal-iib, tig. 4, also cross the hinge at the same points after the transfonnatiou and before. Tlie new arrows obtained this way, altliough correct in direction, are not of the proper length. Consider one of the arrows r (fig. 4) ; it represents to scale the extern load normal on a unit tangent-plane to the back of the rib with a certain southern exposure to stress and a certain western exposure. After the transformation the tangent plane is no longer a unit plane, and one exposure to stress has altered, while the other has remained unaltered. But the length of the new arrow must give the extern-load on a unit plane, allowing for every alteration. Let the north-west quadrant, fig. 4, in equilibrium under t\\Q fluid load represented by the two equal rectangles, be pulled out horizontally till it is twice as broad as before. It is now shown on fig. 14. The quadrant of the circle AB'S' has become ABS, the quadrant of an ellipse twice as long east and west as it is north and south. Since Tq has doubled in value, so must also the area of the east load, and as this load stands on the same base or platform as before, so the height of the rectangle must have doubled. On the other hand, T, is unaltered by the transformation, so that the south-load is unaltered in area ; but as it stands now on a platform or base doubled in length, the height of the rectangle is hcdf what it was at first. The ratio oi q to p is now 4, which is the duplicate of the ratio of transformation. The shaded part of the east-load-area before and after the elongation is the horizontal component of T' and of T, respectively. The shaded part has therefore doubled in area. By starting with B at «S' and moving it up to A, the shaded area has added to it strip after strip, each strip being stretched to double what it was at first. AVe see that the new east-load- area is of the same for/n as at first, only with the height at each point of its base increased in the ratio of transformation. In the same way it will be seen that the south-load-area is also of the same original form, but with the height at each corresponding point of its base decreased in the ratio of transformation. The thrusts T and T' along the ribs at B and B' meet at the same point j on the hi7igc Oj. The extern resultant stresses on the backs of the ribs r and r'l also meet at the common point on the hinge. The whole stress at B' for a unit plane, its edge towards you and its face turned in all directions, is given by a circle 470 APPLIED MECHANICS. [chap. XXII. of which r' is a radius normal to the circular rib. The whole stress at B for a unit plane, its edge towards you and its face turned in all directions, is given by an ellipse of which r is tlie radius-vector parallel to BO, and whose major and minor semiaxes are q and p, the heights of the east- and south- load- areas, respectively, for the point B. For clearness, a quadrant of this ellipse is shown with its centre at D, the point on the base of the east-load-area in a line with B. It is called the Ellipse of Stress. It completely determines r in direction as well as in length. Thus through D draw CC, the unit plane, parallel to Bj the tangent to the rib at B; draw JDN normal to GO laying off DM equal to the half sum of g and p, f/onsiant J^otentzaZ, ^JRatto ofCan/uqate 9.P \ /7 / / / / ^ ^Pu C^ Xoad! on ' ^'iffpi^ Fig. 14. make Md isosceles with MB, and lay off MR equal to the half difference of q and p. This gives RD or ?•, which is to be shifted parallel to itself to act at B. It is not necessary to know the point O on the lihifjc through which r must pass ; it is only necessary to know the shape of the rib itself so as to be able to draw a tangent to it at B, and further to know q and p tlie semi-axes of the ellipse of stress at that level. In the example, fig. 14, both of the rectangles, giving the south- and east-loads on tlie elliptic-(]ua(lrant, may be doubled in height, and their density per square foot taken CHAP. XXII. J I»ES1GN UK MASONKY AKCIIE8, 471 as half that of the rectangles giving the loads on the circular (juadrant. The new south-load-area is then obtained simply by projection, while the new east-load-area has been obtained by projection in tlie duplicate ratio. Generally then a balanced quadrantal rib and its vertical- load-area, that is its superstructure, can be projected by a pull- out horizontally, provided that the conjugate horizontal-load- area he pulled out in the duplicate ratio. After which the density of the new load-areas, that is, the weight per cvMc foot of the nev: superstructure, is to he reckoned as less than that of the old super- structure in the ratio of the transformation. (See figs. 14 and 15.) Geostatic Load Unifoem or Vaeying Potential. The pair of conjugate-loads shown on fig. 14 can be produced simultaneously by placing the elliptic ring horizontally in a fric- tional granular mass with a free level surface, the mass being excluded from the inside. The east- and west-load q on the pair of short barricades being uniform, and four times as intense as p> the north- and south-load on the pair of long barricades. Com- pared to the greater, the lesser load is passive. If the shaft or well were a thin elastic cylinder erected first and stayed inside by an elliptic centre, which could be gradually shrunk upon itself, and if now the granular mass were spread loosely around it in horizontal layers, the genesis of the horizontal stresses would be as follows. The potential is the weight of the column of grains from the surface down to ihe ring of the cylinder under consideration ; it is the active force. Every cube of grains around the elliptic ring would try to spread equally in all horizontal directions. The horizontal load on all the four barricades around the elliptic ring would be p, just as on the circular ring, fig. 4. But/? would only be a, fraction of the potential, for the friction among the grains assists in supporting the overhead-load on a cube of grains. Let us suppose that p is about a fourth part of the potential. This would be the proportion if the granular mass were crushed stone or, in a model, small-shot. Any cube of the small-shot would cease spreading from the load overhead, when the horizontal load applied to its four vertical faces was a fourth of the overhead-column. Here we have the elliptic ring, fig. 14, with a fluid-load p one-fourth of the potential outside as in fig. 4, and supported inside by an elliptic centre. If now the centre were gradually shrunk, the elliptic ring would gradually collapse or flatten, 472 APPLIED MECHANICS. [CHAP. XXII. «Uie to the advantage which the tiuid-load has on tlie long l>arricades. In elongating, the elliptic ring would consolidate the granular mass opposite the short V)arricades, till it pressed back with a stress ^ = 4^ = the potential, when the ring would cease elongating and the centre could be removed. The ellipse would now be more than twice as long as it was broad. To remedy this, the granular mass is to be spread around the shaft in thin layers and punned or rammed hard, layer after layer opposite the ends of the ellipse, but filled loose opposite the sides. The granular mass will be earth or shivers in the case of masonry cylinders or wells, and must be filled as described, as the distortion would he fatal to an elliptic brick cylinder long before the earth at its ends would be sufficiently consolidated. The load on the elliptic rib, fig. 14, is called shortly an earth or (jeostatic load of unifoi^m jjotcntial. One ellipse of stress, a quadrant being shown with its centre at D, serves to give completely the stress on the vertical faces of every unit-cube-of- grains all around the rib. If the unit-cube lie north and south, p and (y = 4p) are the loads on its vertical pairs of faces, while the potential, also q, is the load on the horizontal faces. The whole stress at D is given by an oblate sjyJieroid. If the unit-cube be turned till a pair of faces is parallel to CC, then the vector r gives the load on that pair of faces, and is therefore the mutual pressure between the face of the unit-cube and the back of the rib at B which are in contact. In the same way the ribs shown on fig. 15 are subjected to the one a fluid-load, and the other a grostatic-load of uniformly varying 'potential. The vector r' gives the load on the face of a unit-cube parallel to jj at the depth B' below the surface of the earth, that is the stress on the back of the rib at B.' One ellipse of stress serves for all cubes at depth B', :rF being the semi-major axis which is itself the potential. The load on the face of the unit-cube parallel to the paper is equal to the minor semi-diameter, and the whole stress at the depth B' is given by a prolate si^heroid. As the depth of B' increases, this spheroid's axes increase at the same rate ; but tlie ratio of any pair of its axes remains the same. EyuiLiuiauM OK A FuiuTioxAL Ghanilah Mass. If the loads on the elliptic rib, fig. 14, were actually due to a granular mass of crushed stone, filled around the cylinder in the manner described, then, in order that the equilibrium may l)e pernuinent it is necessary that the unit-granular-cube at 7> CHAP. XXII.] DESIGN OF MASONKY ARCHES. 473 should itself be in ec[iulibriuni with the load 7 on its east and west faces 4 times as great as the load j) on its north and south faces. These loads are called the 2^f'^ncipal stresses at D because they are normal to the faces of the cube lying north and south. If we map out a unit-cube at I) with a pair of faces parallel to CO, the load on these faces is r which is no longer normal, but " makes an angle BDM. The manner of constructing r is to lay off Z)il/ equal to the half-sum of the principal stresses along the normal DN, and then to lay off MR equal to their half-difference along Md isosceles to MD. Hence ■?' is most oblique when the unit-cube-of-grains is so mapped out that MBD is a right angle. It is only necessary then to see that grains, in the cube mapped out in this most critical position, shall not slide on each other due to the extreme obliquity RDM of the load on its faces. But when i? is a right angle, then sine RDM equals R.M divided by MD. If we put RDM = ^, the angle of friction between the grains, then for the cquilihrium of the granular mass at D it is necessary that sine (ft shall not he greater than the ratio of the difference of the principal stirsscs to their sum. Or what is the same thing, if the friction, among the particles of the unit-cube-of-grains mapped out by the principal planes (north-and-south and east- and- west on tig. 14) is to be sufficient to enable the minor principal stress to resist the tendency of the major to flatten out the cube, then the ratio of the minor to major stress must not he less than (1 - sin ^) : (1 + sin (p). The value of the angle of friction <f> for a granular mass may be found by shooting the mass out on a level plane, and when the surface of the conical mass ceases to run, its slope to the horizon is to be measured. For crushed stone (p = 37°, and sin = "6, so that (1 - sin 1^) : (1 + sin 0) = 1 : 4 which would exactly suit the conditions required by the rib, fig- 14. For dry granular earth spread in layers horizontally or gently sloping = 30°, so that (1 - sin 0) : (1 + sin 0) = 1 : 3. Compare the preceding chapter on Rankine's Method of the Ellipse of Stress, Chapter II. Approximate Elliptic Pjb. Consider the circular quadrantal rib ACB, fig. 6, loaded with the vertical-load-area between itself and the straight extrados ^r above its crown A. The conjugate horizontal- load-area hglks is wholly positive or inwards, and its area is 474 APPLIED MECHANICS. [chap. XXII. i/-*. On fig. 15 the same quadrant of a circle ARS' is shown, having the same directrix OE at a height AO = \BA = \r as before. The conjugate-load-area is hglke, only it has iDcen turned round so that hglk the treble-batter-boundary is outside, while the other boundary he is now vertical instead of sloping at 45°. The area is still ^r^ and the breadth at each point Hydrostatic ctrtf/ Ceos ta t/c Ribs /o. l/ju forint y- YmyfiKr TofoT(/laL Y CoTfjimai^e S<n/'rrc^"^f Z L\ '^^|.^^^^ ' , | .>^^.a i Fig. 15. exactly the same as before. The slopes of the treble-batter are now, h(j battering at 3 in 4, (jl at 1 in 2, while Ik is vertical. Of course the exact outer boundary is a gentle curve to which that treble-batter is a close approximation. Let now a sinrjle houndarij f'li be drawn so that when produced it will pass through F, and such that the enclosed area hf'h'e shall equal CHAP. XXII.] DESIGN OF MASONRY AUCHES. 475' h(jlh\ that is ^r-. It is only necessary to draw Ff'h' at the l)atter 4 in 10. We have then hf'li'e as a rough approxi- mation to the conjugate horizontal-load-area for the circular ([iiadrantal rib AB'IS' loaded up to a straight extrados at a height OA, one-third of the rise DA above the crown. Suppose now that the quadrant be slightly modified in form so as to exactly balance under these two load-areas. We have A B'S' the quadrant of some curve differing but slightly from a circle, and balancing under the vertical-load- area between itself and a horizontal extrados at a heighi over the crown OA which is one-third of the rise DA, together with the horizontal-load-area standing on the platform he equal to DA the rise, and mapped out by the straight slope Ffli battering^ at 4 to 10. The modified quadrant AB'S' is called a geostatic rib, because both loads can be simultaneously imposed on the rib by immersing it in a frictional granular mass of which OEY is the free horizontal surface, and the friction among the particles such that the unit-cubes mapped out horizontally and vertically are themselves in equilibrium with the horizontal load on each four-tentlu of the vertical. Dry granular earth spread outside the arch in horizontal layers can, as already explained, produce such a load which is shortly called a geostatic load, the potential uniformly increasing with the depth below the free surface. At x, a point at the same depth as B , the ellipse of stress is shown on j&g. 15, FW is an arc, and the semi-diameters are in the ratio of 4 to 10. By transforming horizontally the rib AB'S', in any ratio, we obtain a new rib ABS. If AB'S' be the quadrant of a circle, then ABS is the quadrant of an ellipse, the new hori- zontal load area being bg'l'k'e derived from h[/lke by a horizontal transformation in the diqjlicate ratio. But if AB'S' be a true geostatic quadrantal rib, differing little from a circular quadrant, then the new rib ABS is also the quadrant of a true geostatic rib differing but little from the quadrant of an ellipse, and the horizontal-load-area is bfhe derived from bf'h'e by a horizontal transformation in the duplicate ratio. By choosing a suitable ratio for the horizontal transforma- tion, namely, one for which the duplicate ratio is 10 to 4, we have Efh sloping at the batter 1 to 1, the load is then o, fluid load, and the rib ABS is called a liydrostatic rib. Xow the ratio whose duplicate is 10 to 4 is approximately 3 to 2 ; hence the hydrostatic rib, of which ABS is the left half, has its span three times the rise, and the depth of the load over the crown 476 APPLIED MECHANICS. [CHAP. XXII. one-third ^)a?*^ of the rise. The ellipse of stress has become a circle of stress. In this way we see that a semi-elliptic rib, or one which in courtesy (see fig. 2) is called semi-elliptic, may with like pro- priety be called a complete hydrostatic rib. That if the span be treble the rise the surface of the fluid must be over the crown at a height onc-thircl the rise. It is better now to suppose that, on fig. l.l, we start with this liydrostatic rib ABS loaded outside up to the fluid surface OFF, so that the 45°, or 1 to 1 line Ffh, defines the horizontal- load-area. Tlien to suppose the geostatic rib AB'S' to be derived from ABS, by a crush-in in the ratio 2 to 3, the hori- zontal-load area crushing-in in the ratio 4 to 9, or 4 to 10 nearly, and defined by the 4 to 10 battering line Ffli' : the circle of stress at x becoming the ellipse of stress with diameters in this duplicate ratio. The particular geostatic rib AB'S' derived by this crushing-in of the particular hydrostatic rib ABS is sensibly the quadrant of a circle. One might readily fall into the error of supposing that all hydrostatic ribs might l)e crushed-in so as to be sensibly quadrants of circles by choosing a fractional ratio of transformation to make the semi- span equal to the rise. But by considering the genesis we have employed, it will appear that only those hydrostatic ribs, with the surface of the fluid at a height over the crown at least one-third of the rise, can be crushed-in to give derived geostatic-ribs with half span equal to rise, so that such derived ribs may be, in any other respect, like the quadrant of a circle. Because starting as we do on tig. 15 with a horizontal-load-area hilkc wholly positive or inwards, the rougher approximation hf'h'e is reasonable, but with a lower directrix to start with, the hori- zontal-load-area would be partly positive and partly negative, to which one sloping boundary Ffh' could, in no sense, give an approximation. We have this important practical distinction, that all sensibly semi-elliptic-ribs may be treated as sensibly geostatic whether derived from a hydrostatic rib by a 2^M-out or di push-in, pro- vided the height of the load over the crown be at least one-third of the rise. For loads over the crown less than one-third of the rise tlie hydrostatic rib itself, and geostatic ribs got by a pull- out, are sensibly elliptic, but not those got by a push-in. Hydrostatic and Geostatic Ribs. Let AliS be the quadrant of a hydrostatic rib. fig. 15. The origin is the point over the crown, while the axis OF is the CHAP. XXII.J DESIGN OF MASONRY AKCIIES. 477 surface of the fluid, the vertical is the axis OX. The depths of A and S the crown and springing are a\, and Xi. The rise of the arch is a = (^r, - Xo), and the half-span is //i. The vertical-load-area is OESA, while the horizontal-load-area, mapped out by the 45° or 1 to 1 line is hflie; it is the difference of two half squares, namely h (^i" - x^-), and this multiplied by IV, the weight of a cubic foot of the fluid, gives us the thrust at the crown 7",, equal to }iW{jr^- - x^^). Since the extern load is normal at each point on the back of the rib, the thrust along the rib is constant. So that we have, at *S the springing point, at B any intermediate point, and at the crown A, iij 1\ =T = 1\ = — (a-,' - ;r;-). (20) The rule, p. 427, for the thrust at the crown of a rib applies alike to every point of the hydrostatic rib. To get the radius of curvature at any point of the rib it is only necessary to divide the constant thrust T by the intensity of the extern normal load^ which itself is the same as the mass of the fluid vj multiplied by the depth of the point on the rib below the surface. As w is common to both expressions, the same quotient is obtained by dividing the horizontal-load-area by the depth of the point. The radii of curvature of the points >S', B, and A, are — X\' — .^0 2Ji" Xf, Xi' — pi = ^ » P = ~~ o^ ' l°" " 7,—^ ' (21) The central of these expressions is an equation to the hydrostatic rib, and is to be expressed simply in words thus — at any point of the rib the radius of curvature and the depth of the •point below the surface have a constant product ; — the value of the constant being the horizontal conjugate-load-area for the quadrant. The constant involves two linear quantities called parameters; they are x^ and x-^, or, what is the same, a = {x^ -x^ the rise of the rib, and x^ the depth of the load over the crown. Any two values assigned to those determine a particular hydrostatic rib, when ijx the half-span can be approximated to. In having two parameters this rib is like the ellipse. Further, as the depth to the rib increases continuously from the crown A down to the springing S, it follows that the radius of curvature decreases continuously between those points just as in the quadrant of an ellipse. 478 APl'LIEl* MECHANICS. [CHAP. XXII. Kankine approximates to the half-span ^, as follows : see the lowest part of fig. 17, which shows the quadrant of the hydrostatic rib, and the quadrant of an auxiliary ellipse having tlie same absolute rise a = (.Tj - r„), and having its extreme radii of curvature in the same ratio to each other that they have on the other quadrant. If h is the half-span •of this ellipse, it will be a fair approximation to y^. Now, the extreme radii of curvature of the elliptic quadrant are Jy^ -f a, and «- -^ h, so that their ratio is J^ : a'. For the quadrant of the hydrostatic rib their ratio is /og : p,, or Xx : x^. Equating and arranging we have = a 'j^ . (22) \ «0 That is, an approximation to the half span of the hydrostatic rib is the rise multiplied l^y the cube root of the ratio of the depth of the springing point to that of the crown. That is always about 5 per cent, too great, or, more definitely, from h is to be deducted one-thirtieth of the radius of curvature at the crown of either the auxiliary ellipse or of the hydrostatic rib as it may be convenient. The closer approximation to the half span of the hydrostatic rib is a (23) In this way we shall find the span of the hydrostatic rib having the depth of the springing points four times as great as that of the crown, or, what is the same thing, having the rise of the rib treble the depth of load over crown. Putting unity for the depth of the load at the crown A, fig. 15, then x^ = OA = 1, and the rise is f = DA = 3, while Xi = BS=4:. By equation (22) J = 3 ^J4 = 4-76. Squaring this and dividing by a we get 7*55, the radius of curvature of crown of auxiliary ellii)se. Or for the hydrostatic rib, the horizontal-load-area being ^ (4- - 1-), we divide it by x^, and get p„ = 7*5. Also a thirtieth part of these radii is "252, which is the same as 5 per cent, of b. Keducing h by this amount, we get y^ = 4*51, and doubling, the span of the rib is 0-02. Hence for the hydrostatic rib with the load over tlie crown u third part of the rise the span is three ti/iu\s the rise, CHAP. XXII.] DKSIGN OF MASONRY ARCHES. 479 a result which we arrived at in the last section by a ([uite independent approximation of our own. For this particular hydrostatic rib the approximation given by (Rankine's) equations (22) and (23) are verified graphically on fig. 17, the curve being drawn in small arcs beginning at the crown A. It is now more convenient to have OA = 3 feet, and OD = 12 feet. The scale was double before reduction by photography. Horizontal lines were ruled at different sou)idin;/s below OJkJ the surface of the fluid. The conjugate horizontal-load-area h (144 - 9), or 67'5 square feet, being divided by 3, gives po = 22'5. With this radius the first arc is drawn beginning at A, and ending when it cuts the horizontal at the sounding 3'75. Along the old radius at this point a new shorter radius is laid off, with which a second arc is drawn till it cuts the horizontal through the next sounding 4o. In this way arc after arc is drawn till the last one has a vertical tangent at (script) s, which completes the approximate quadrant. As each arc has been drawn of constant radius, while the radius ought to have decreased, it follows that the semi-diameters of this approximate quadrant are too large. By measurement Ad is found to be 5 per cent, greater than the given rise AD, and we conclude that ds is 5 per cent, too big also ; hence BS is laid off at that reduced value when ABS, the false quadrant, is struck from two main centres, and an intermediate one as already explained at fig. 2, the whole rib being struck from five centres. By measure- ment the half-spans ds and BS verify the values already calculated by the equations (22) and (23). The soundings and radii used in the graphical construction of the curve are given below in two rows. Each radius is found in turn by dividing the sounding out of the area Q7\). These radii begin w4th the largest p,^ = 22*5, and end with jOi = 5-6. 3-00, 3-75, 4-5, 6-0, 7-5, 9-0, 10-5, 11-25, 12. 22-5, 18-0, 15, 11, 9-0, 7*5, 6-5, 6-00, 5-6. Observe that in this particular rib, 5 per cent, of the half- span and -joth of the crown radius are the same, but in other ribs the fraction of the radius is to be preferred. In the same way the spans of two other particular hydro- static ribs may be calculated from the equations (22j and (23), The depth of load over crown (jto) being taken as unity, then for the rise (a = Xi - .r^) three times the load, the span (2_?/i) was found to be 3*01 times the rise. With the rise four 480 APPLIED MECHANICS. [chap. XXII. times the load, the span is 3*22 times the rise, and with the rise five times the load the span is 3"41 times the rise. Exact values, by elliptic functions, are 3"06, 3'28, 3"47.* As already pointed out, on page 476, only the first of these hydrostatic ribs may be transformed by pushing-in. All three may be transformed by pulling them out horizontally, and the resulting geostatic ribs are sensibly semi-elliptic. Let the pull-out, in the first place, be such that the 1 to 1 sloping boundary of the horizontal-load-area becomes a 1^ to 1 sloping boundary. That is, the duplicate of the ratio of transformation is 1*5, so that the ratio of transformation is 1"225, which is the multiplier to give the three new spans. In the second place, the 1 to 1 sloping boundary is to become a 2 to 1 sloping boundary, so that the ratio of transformation is now 1-414, which again is the multiplier to give the three new spans. We have then the following nine geostatic ribs, that is, ribs sensibly semi- elliptic, able to be struck out from five centres as on ficj. 2, and such that, when loaded between the ribs themselves and a straight extrados at the given height above the crown, require for equilibrium a horizontal-load-area mapped out by a sloping line at the batter given in the table. In the following table the depth of the load over the crown is given as a fraction of the rise of the rib, and the span as a multiple of the rise : — Spans in Terms of the Rise of Semi- Elliptic Bibs which are sensibly Geostatic. Boundary of hori- zontal-load-area. Load at Crown |rd of Rise. Load at Crown jth of Rise. Load at Crown |th of Rise. 1 to 1 301 3-22 341 U to 1 3-68 3-94 418 2 to 1 4-25 4-55 4-82 With the semi-elliptic rib, it being desirable to liave the span between four and five times the rise, it appears from the above table that the depth of the load over the crown should be between |th and -J^th of the rise. •Notes on the Hydrostatic Curve, and its Application to the Stability of Elliptic Miisonrv Arches, by J. T. Jackson, m.a.i., Assistant Professor of Civil Engineering, Trinity College, Dublin. Trahs. of the Inst, of Civil Engineers in Ireland, vol. xli, 1915. CHAl'. Wll.] DKSJIGN OF MASONRY AKGIIE8. 481 Semi-Elliptic Masoxrv Auch, 111 this arch tlie soffit is a false semi-ellipse struck from five- centres as on fig. 2. The span being cleterniined upon, the rise of the sotftt may be any fraction of the span from a fourth to a fifth. The radius at the crown is now determined, being the square of the half-span divided by the rise. Then a mean pro- portional between this radius and '12 gives liankine's empirical thickness of the keystone ; to this a suitable sr.rcharge over the keystone being added, we have the depth of the load over the crown, which will be from a third to a fifth of the rise. For example, take the lower part of fig. 16, showing a quadrant of an elliptic masonry arch, half-span of sotiit 38 feet and rise 19 feet, the radius of crown will be 38- ^ 19 = 76 feet. By Eankine's formuke the thickness of the keystone is ^7Q x '12 = 3*02 for a single arch, or for one of a series it is v/76 x -17 = o*6. Now, it is usual to have the ring of stone of uniform thickness, instead of thickening out as in the segmental arch, so that 4 feet had better be allowed for the uniform thickness of the ring. To this a surcharge of not less than a foot and a half is to be added. With this minimum surcharge the depth of load at crown of soffit is sensibly a fourth of the rise. But we have on fig. 16 adopted a surcharge of 3 feet, so that the depth of the load at crown of soffit is 7 feet. This we have chosen, that the total rise may be 26 feet, so that this semi-elliptic arch may have the same span and total rise or head-room as the segmental arch, figs. 12 and 5, the two patterns being rival designs for the same railway viaduct of many arches. The line of stress is to be supposed to be along the soffit in the first place. The soffit is now, instead of elliptical, to be con- sidered as sensibly a geostatic rib. The quadrant is shown on the upper part of fig. 16. The three data are OA = ^Tq = 7, OD = Xi = 26 feet, and the half span c = sy, = 38 feet, where x^ and .Ti are the extreme depths of the hydrostatic rib from which the geostatic rib is derived by a horizontal transformation in the ratio s. The thrust Tq at A the crown of the geostatic rib is the mass of the superstructure to multiplied by the horizontal- load-area hf'h'e, while Ti the thrust at the springing is an sth part of T„, for before the transformation they were equal, and only T^ has increased. We shall return to the strict numerical solution of the arch on fig. 16 among the examples following. 2 I 0, Q, CHAP. XX II. J DESIGN OF MASONRY ARCHES, 483 In the nieautiiue fig. 16 may serve to illustrate tlie quite general treatjiient. Since j-„ = 7 is sensibly ^rtl of a = 19, we may look in the second column of the table, and since the span 76 is exactly 4 expressed in t^rms of the rise a, the slope of the boundary of the horizontal-load-area will lie between 1| to 1 and 2 to 1, just as 4 lies between 3*68 and 4"25. By proportional parts this slope is If to 1. Quite strictly it is 1"853 to 1 : see the numerical examples following. On fig. 16, s- to 1 is taken as sensibly 2 to 1. At A the thrust is T„ = w x area hf'h'c = 627'?t', y.nd dividing by .s-, Ti = 420u' nearly. Consider the quadrant of the masonry ring as rigid, and apply a pair of equal and opposite couples, one at the crown, and the other at the springing. Let the couple at the crown be of a moment such that it shifts T^ just into the middle third of the keystone as shown on the lower part of the figure ; then the other couple will shift Ti well into the middle third. In no ease will 1\ be shifted past the middle third, as 7\ is never twice as great as Tj. The modified line of stress begins and ends in the middle third, and as its curvature is continuously varying, the whole curve is confined to the middle third of the masonry ring, shown white on the figure. For the middle elastic part of the ring, shown dense black, the heavy backing is left out. It stretches between the pair of joints where the tangents to the soffit are inclined at 30° to the horizon. For one thing T,, is now slightly reduced. The soffit for this middle part of the ring is actually all struck from the main centre, and is really a segmental circular masonry ring springing at 30° to the horizon. The modified line of stress of this central part, for the most perfect economy of masonry, would start at the lower limit of the middle third at the keystone and would reach the upper limit of the middle third at the joint of rupture, the springing joint for this black part of the ring, that is at the joint which, when produced, goes through the main centre, and makes 60° with the vertical. This will now determine whether the thickness adopted for the arch-ring is sufficient, for the horizontal through the lower limit of the middle third of the crown joint, and the line at SO'' to the horizon drawn through the upper limit of the joint of rupture must meet on the vertical through the centre of gravity of the corresponding superstructure. The construction will be similar to that shown at the lower right-hand corner of fig. 7. In this case the thickness of the ring is the least for permanent equili- brium, and at both crown and joint of rupture the maximum 2l2 484 APPLIED MECHANICS. [CHAP. XXII. stress is double the average. With a more liberal thickness of ring, different pairs of tangents may be drawn to meet on the proper vertical, and the actual line of stress will be determined by the ceasing of the subsidence of the keystone when the centre is slowly removed. It is always safe to assume that the maximum intensity of the stress at the crown joint is double the average, oc that the real factor of safety is only half the apparent factor against crushing. If a uniform live load be added all over the span equivalent to an additional height of the superstructure, 1| feet on fig. 16, the calculations could be repeated, making 3;,, and Xi greater l)y this amount. The increase of Ty is simply 7v multiplied by the area of the rectangle of base 38 feet, and height 1| feet, or (T/ - Ti) = 48w". And for the increase of To we have (by the rule for the thrust at the crown of a rib, p. 427) merely to- multiply l'2ovj the extra normal load at the crown by 76, the radius of curvature, when we have {TJ - To) = 95?'.'. Light spandrils to resist this load are shown on fig. 16. The depth from the formation to the top of the hea\y spandril walls upon which the light spandrils ride measures 8 feet on the figure. If I be the distance they stretch out longitudinally, then 8ivl is the over-head load pressing the spandrils down on their base. Then with '7 for the coefficient of friction of stone on stone we have the frictional resistance of the light spandrils 5-6 wl equal to 95i/', so that / = 16"9 feet. Examples. 21. In a semi-elliptic masonry arch the span of the soffit is 76 feet and the rise is 19 feet, and the level of rails is 7 feet above the crown of the soffit. The soffit which is to be struck from •') centres is to be taken in tlie fii-si place as the line of stress. Assuming it to be sensibly a gcostatic rib, find the thrust at the crown and springing. The extreme depths of a quadrant of tlie hydrostatic rib from which the gcostatic rib is derived are .To = 7 and x\ = 26. A rough and a close approxima- tion to the half-span of this ky/rostatic rib are by equations (22) and (23), =-;/!= 29-424, b- y, =29-424 -,,Vj,j == 29-424 - 1-Jil8 =27-906. The half-span of the gcostatic rib being given s>j\ = 38, we have then the ratio- of transformation or pull-out by which the half-span 27-906 has become 38 to be 38 , = — -— = 1-3617 27-900 and .-(2 = 1-853. Hence the 1 to 1 boundary of the horizontal-load-arca for the hydrostatic rib- CHAP. XXII.] DESIGN OF MASONRY ARCHES. 485 ha\'ing been puUed-out in the dupUcale ratio l-8o3 to I, this furnishes the slope of the boiiiuJiirv mapping out the horizontal-load-aiea of the geoiflatic rib. Multiplying w the weight ot a cubic foot of superstructure into this area the thrust at the crown is r„= 1-8.53 X ^(26-- 7-) = 5Shr, and Ti c=To^s= ^-— = 427^. l"oo2 Note. — These are the more exact values which should have appeared on fig. 16, only that the slope of the boundary of the geostatic rib was taken 2 to 1 instead of 1-853 to l^for clearness of explanation. 22. If a live-load equivalent to an additional height of superstructure of 1-25 feet be added all over the masonry arch in last example, find now the thrusts at crown and springing. - We have now Xo = 8-25 and .ri = 27'25, 6= 193/^1^=28-297, \ 8-2o yi = 28-297 - 3V ^ = 26-892, og - = 1-414 and *- = 2, 26-892 w To = 2- (27-252 - 8-252) ^ 67o?c, Ti' = To ^s = il8w. Note that now tlie boundary of the conjugate horizontal-load-area slopes at 2 to 1. Also To' - To = Glow - oSliv = Sitv, about the product of \-2r)w and 38- -^ 19 already used for an approximation : see fig. 17. Again Ti' - Ti = blw, a little more than the, simple product of l-2bw and 38 the half-span. 23. The pier between two arches is 12 feet thick at the top, and the faces batter 1 in 20 : find the centre of stress at a joint 45 feet below the springing level. That joint is 16 -o feet thick, the weight of tiie pier (;41(i'. The half -arches rest 427ct' and 478u- on its top ; the whole weight above the joint is 1546«-. The excess of the horizontal load on one side over the other is 95«-. If ; be the devia- tion of the centre of stress from the midiileof the joint 45 feet down, the equation of moments is lo-iGtcz = 45 X 9.5ir. ;: = 2-76 feet. This is one-sixth\oi 16-5, so that the centre of stress is just within the middle third of the joint. 24. The abutment pier is 20 feet thick at top, and its faces batter at 1 in 20. At*a joint 45 below the springing, find z the deviation of the centre of stress from the middle, when the pier bears a half-finished arch on one side only. The joint is 24-5 feet thick ; and if we take the mass ot the pier 2m- and reckon a half-finished arch on one side only, then weight of pier is 2002«-, and the equation of moments is 2002it" -r 427«- (0 -f- 8) = 581m- x 45, so that z = 9-3 or three-eiyhths of 24-5 ; the thickness of the joint and the line of stress does not come closer to the face of pier than an eighth of its thickness. 486 APPLIED MECHANICS. [CHAl'. XXIL 25. If. when the viaduct is finished, an arch fiiUs, it leaves a whole arch on one side only of the abutment pier. The first term in the equation above must be halved, ae we cannot now consider the pier of double density ; at the same time we have now a mass of masonry 20 feet by 26 feet over the pier, so we must add 520M-sand deduct lOOlw; from the first term and 152hf: + 427!f(; + Sj = oSlw x 4o. And now z= 11 -0 less than half of 24-5, so that the centre of stress is still within the masonry. Masonry Abutments. In the case of the segmental arch (fig. 12), the abutment is readily designed by building the masonry in benches stepped at the back, the oblique thrust of the arch, shown by a dot-and-dash line, to lie in the middle third of the abutment if it be no deeper than the rise of the arch. The masonry may be entirely in square -dressed course.«^, but the central part may. with advantage, be in radial courses. The abutment is best honeycombed at the back with thick walls for the thin spandrils to ride out upon. The spaces between the thick walls to be thoroughly drained and packed with heavy ballast and broken shivers of stones. The spaces between the thin spandrils to- be coated with waterproof material, arched or bridged over, and preferably left void, drain-pipes and ventilating holes being provided. In the semi-elliptic arch, lower part of fig. 16, the heavy spandrils, with square- dressed joints, are shown in three benches, from tlie springing level up to the joint of rupture. The thickness z, at springing level, is calculated in the first place, as if the backing required were like that of a fiuid load, or as if the horizontal-load-area were mapped out by a 1 to 1 boundary. In the example -7 x 26//'-' = ^?'- (26- - 8-), or / = 17 feet, taking v/ and ii' as e(iual. A right-angled triangle, with the base z, maps out the backing. The re-entrant angles of the benches are to lie on the hypotenuse of this triangle, shown with a dot-and-dash line ; then the middle and top benches are slid along inwards till they bult against the square-dressed backs of the voussoirs. As v' is likely to be 20 per cent, greater than w, that is, the mass of tlie backing greater than that of the superstructure, we will have the reduced value ~ = 14 feet. We have sliown that the conjugate horizontal-load-area for elliptic arclies has its boundary sloping at 2 to 1 at most, that is double the horizontal action of a Huid load. l»ut if the CUAI'. XXII.] DKSICN OF MASONRY ARCHES. 487 abutment, as on the lower part of fig. IG, is built in an excavation in old consolidated earth, then the earth filled between the abutment and the face of the excavation can be matle to press horizontally, like a Huid, by punning it hard in thin horizontal layers. This reinforcing the frictional stability of the masonry abutment, as already designed, supplies the necessary resistance to tlie spreading of the arch. The centre of the arch is to be struck when about half of the superstructure is built, and before the earth is rammed behind the abutment. When the masonry has settled and consolidated, the earth is to be first rammed behind the abutment, and then the superstructure finished. On the other hand, if the abutment is built and a loose embankment built over it, then even by filling the best stuff behind the abutment, it will only press horizontally with half a tiuid load, so that the masonry abutment must now be equal to a 1| to 1 bounded load area. That is, z must be increased by 50 per cent, from 14 feet to 21 feet. Adding to each of these, 4 feet for the thickness of the springing stone, we have the total thickness of an abutment for the semi-elliptical masonry arch (fig. 16) to be 18 feet and 25 feet in the best and worst circimistances respectively. These are almost exactly ^th and ird of 76, the radius of curvature of the crown of the soffit. And so we have verified Eankine's proportion : see his Civil Eng., p. 514: The thickness of the abutments of a masonry bridge are from a third to a fifth of the radius of curvature at the croivn. Masonry Piers and Abutment Piers. The design for a common pier is shown on the left half of fig. 17. It is for a viaduct of many arches, of which half an arch is shown on fig. 16. Every fourth or fifth pier is to be an abutment pier, the design of which is shown on the right half of fig. 17. Both piers are 45 feet high, and are divided into three blocks of 15 feet each. The common pier is 12 feet thick at the top, and the abutment pier is 20 feet thick. Their faces batter at 1 in 20. The arch to the left of the common pier bears a live load e(|uivalent to 1-^ feet of extra height of its superstructure. Hence the excess horizontal thrust of this arch above the one on the right side of the pier is 1-252/; X 76, or 9oiy. It is shown on fig. 17, acting on the horizontal line, through the centre of gravity of the horizontal- load-area, as drawn on the figure above. Then 468w and 420i^; are the vertical loads on the springing stones at the €HAP. XXII.] DESIGN OF MASONIJY ARCHES. 489 •conieis of tlie pier, due to the halt' arch loaded, and the half Arch unloaded. They are quoted from fig. 16. There are also shown four barbs on the vertical line down the middle of the pier, which are the weights of the column of masonry between the two half arches, and of the three blocks of masonry into which we divided the pier itself. The mass of the masonry f is taken as unit. The vertical scales, both for •dimensions and loads, are three times as fine as the horizontal. By drawing a force and a link polygon, the centres of stress at each of the three joints below the springing are defined by the barbs of oblique arrows. Each centre of stress is well witliin the wliite "kernel," which is the middle tliird of the masonry of the pier. It is absolutely necessary, for the permanent stability of the pier, that the line of stress be confined to the middle third, otherwise the joints would open, as the pier was rocked by the live load shifting from one arch to the other, the mortar drop out, and the whole l)e destroyed. The everyday work of the abutment pier is to reflect vibra- tions passing from arch to arch along the viaduct. Once in its history it has to act as an abutment, the arches on one side only being finished, or partly built, Now the arch resting on the abutment pier will have its centre struck when itself half complrterh On fig. 17 the horizontal and vertical loads, 280^(7 and 210^r, at the left corner of the abutment pier, are due to the half finished arch on that side, and are quoted from fig. 16, being slightly modified because the pier is of dense masonry. The smaller force polygon, and its corresponding link polygon, define the centres of stress to be within the white kernel. That is, the centres of stress do not approach so close to the face as onc-cigldli of the thickness, which is a suitable limit, as the load is steady, and only in one direction, just as in retaining walls. If by an accident, such as the failure of a foundation, an arch of the viaduct should fall, the damage should only reach to the nearest abutment pier. The larger polygon shows the full load of an arch on one side, and the oblique dot-and-dash arrows show the centres of stress to be still within the masonry, so that the abutment pier could be expected to sustain the complete arch for a short time till it could be shored up. A study of the construction shows that the second last full line oblique arrow produced to the lowest joint defines the same centre as the last oblique dot-and-dash line. This saves the trouble of drawing the larger polygon. 4ftO APPLIED MECHANICS. [CHAP. XXII. Tunnel Shell. On fig. IS is shown a tunnel shell ; it has nearly the same profile as the Blechingley tunnel (see Plate 1 of Simms' Practical TnniuUinfj). In other respects the shell is designed to illustrate the mutual stability established between the shell and the surrounding cubes of earth. The left part is a half cross-section of {\\e front louith of the tunnel, where a deep cutting is first made in old consolidated earth, the sides as nearly vertical as may be. The thick side- walls are first built, 4 feet thick, and the roof turned on a centre. It is 2 feet thick at the crown, thickening by a ring of bricks at intervals outwards to the haunches. The notches are covered with a band of asphalt to render the roof waterproof. From A, the crown of the soffit, the invert is swept out truly with a suitable versine, to accommodate a covered drain between the rails. The earth is now filled between the back of the walls and the face of the cutting. Up to the level of h it is punned in thin layers, that the cubes may press horizon- tally like a fluid ; above that it is filled more loosely in thick layers, so as to be pressing horizontally, with little more than a third of column overhead. The lowest level of the filling when completed is to be SP, so that Sa may be a third part of 14' 5, the radius of the circular quadrant ah assumed to be the line of stress. This quadrant, then, is a geostatic rib, whose hori- zontal-load-area is bounded by the 4 to 10 sloping line lying close to the ^rd to 1 slope, indicating the minimum horizontal- load necessary for the equilibrium of the cubes of earth. The depths of a and h, below SP, are 5 and 20 feet, so that the horizontal-load-area measures -^-^ . ^ (20' - 5') = 75. The liorizon- tal thrust at a, the centre of tlie crown joint, is lov:. At h the vertical thrust is Ibv), multiplied by ^, the ratio of transformation by which the quadrant ah is converted into the corresponding hydrostatic quadrant with the same extreme depths (see fig. 15). To this vertical thrust of the rib at h there falls to be added a column of earth, 20 feet by 5 feet, making in all 2V2ii\ acting down through h. The total upward tlirust, acting partly on CD, the base of the tb.ick wall, and partly on DE, the half invert, must also be 212/'-. Now, from the manner of con- struction, most of this load is thrown on VI), the invert only ottering the minimum reaction to the swelling of the earth Itoneath it being built later and l^eing elastic and yielding. TUN'NEL SHELL Cut ajxd Core jr. TLo soffit lias neariy the samo jitoGle as tho Bletchingloy Tnnnel : SP is tho (jnidrant o£ a falso ellipso irith gA = 10 and gF 12 feet. Tho fonnalion is 22 feet belo-^r^, and the invert 13 struck from A 03 centre. In the shaUowJength tho roof ia 2 feet thick at tho crown, the sido iralk 4, and lUo'IuTcrl li feot thick. Tho shcU has a uniform thickness of 3 feot on the deep length. ^OfL eselra/'tilina, ^ CeyeL of-'J^illtna, 2iay 7i Si 00 oJls-p. SaJir, EARTH PRESSINgC LIKE A FLUID. ^ % k dJXXXXXuXVIJ Kclow tlio juiut li the earth is rammed in 9-iucU layers hetween tlio face cf tho ex- cavation and tho -wall, above that it is spread, in 12'iiic!> lajers. lun;. Cot. Ben. ISDJ. 30 ziTGach i J ^cLofCoOanjr lilllUIII souieaeh EARTH ON THE POINT OFHEAVING UP. Zfxapei'. 8 W — J20 lbs. r£ET lO aoorr. ^ ^ * « 3 J20nre^ 2 g/^cois\ Fig. IS. 492 APPLIED MECHANICS. [CHAP. XXII. Suppose all the unit-cubes of earth at the level CE, 30 feet below the free surface PS, to be pressed horizontally with 30w, a load equal to the overhead column. Then the least vertical pressure on the 12 unit-cubes (white) under the invert DE, consistent with the stability of those cubes themselves, is lOw, or one-third of column. From the ranmiing of the earth at C, tlie vertical load, on eacli of the unit-cubes (black) under CD the base of the wall, may be taken as column, or 30?/', more or less. This furnishes an equation to verify if the thickness of the walls' foundation is sufficient. For we must have 30?/- X t + 10?/; X 12 = 212w. This gives t nearly 4 feet, and taking t exactly 4 feet, and modifying the equation, we liave the vertical load on the four (black) unit-cubes below the wall to be IZw, and •2Zw X 4 + lOif X 12 = 212^6'. The radius of the line of stress in the invert is 25 feet, and it has the upward load of lOvj uniform along the span, just as in fig. 14, so that the thrust along it is constant, and equal to IQiv x 25 = 250«-. Next to join together these two lines of stress, in roof and invert, by a line of stress confined to the middle third of the lower part of the thick wall, we must consider the horizontal load applied to the liack of the wall by the punned earth l^ehind it. This load is given by an area mapped out by any boundary lying between the 1 to 1 and -^ to 1 boundaries. On fig. 18 these areas are measured up to the 1 to 1 boundary, and at intervals of 2 feet of depth are o^v\ 46?r, b^w. and 54?'". Beginning with 21 2?/', at h, and compounding those in order by means of the force polygon, pole 0, and the link polygon LM, the centres of stresses are (lefined — c by the arrow with two feathers ; d by the arrow with three feathers ; c by the arrow with four feathers; and lastly, B, by the arrow with five feathers. In another example the points c, d, c, &c., are to be laid down one after another, in suitable position, each in its own joint, and the horizontal areas necessary to make them so lie determined in succession, and provided the boundary of those load areas lies between the two practical limits, the 1 to 1 and -J to 1 boundaries ; then hcdeB is a possible line of stress, consistent with tlie equilibrium of the punned earth itself. CilAT. XXII.] DESIGN OF MASONItV AUCHES. 49.'> Addiug 70 /r for the horizontal load below B, we liave the sum of 30h', 46?^', 50?r, 547r, and ^Qiv, equal to 250?t', the thrust at crown of invert. Both the top and bottom quadrant, then, of the left half of tig. 18 are in horizontal and vertical equilibrium, consistent with the equilibrium of the surrounding cubes of earth, and throwing as light a load on the invert as may be. There still remains the equilibriunt of moments of the shell as a whole. This is satisfied by the fact that the principal axes of stress for the cubes of earth are vertical and horizontal, the free surface being horizontal, and as well the axes of the shell are also vertical and horizontal. With the free surface of the earth, at a heavy side-long slope, it may become necessary to find the- cant which the principal axis of the earth-pressure makes with the vertical, which is easily done by the formulae on the model (fig. 31, Ch. II), and then to prevent the shell of the tunnel from twisting like a stick of barley-sugar under the unbalanced couples, one of two things has to be done. One is to build the shell of the tunnel at tlie same cant, an example of which is the Dove Tunnel, a section of which can be seen on the plate xxii of the fourth edition of Simms' Practical lunnellinfj. But the more modern way is to load the shell by an extra mass of masonry at one foot, so as to cant the virtual axes where it may be required. As the filling, over the tunnel shell, gets deeper and deeper^ the problem becomes less and less one of stability, and more and more one of strength. The right half of fig. 18 shows an extra 30 feet of filling. The total depth is now QB = 60 feet, the usual maximum for tunnels built in cut and cover. Even for a tunnel built in a cutting greater than 60 feet in depth, and covered, it is likely that only to this height will the load overhead affect the shell, as the earth itself forms a relieving arch. And, again, in a bored tunnel, the disturbance is only likely to affect the earth to a limited height above the hole. If on the diameter of the hole as base, an isosceles triangle be constructed, with its sides sloping at 75° to the horizon, the vertex gives the probable extent of the disturbance due to the loosening of consolidated earth due to the bore-hole. For cracks will run out on the plane on which the direction of the thrust is most oblique. Thus, in all cases, the height of load shown on the left half of fig. 18 is the probable maximum (see quotations,- p. 497, from Simms' treatise). Consider the right half of fig. 18 : the additional vertical load is a column, 30 feet by 16 feet, or 480;'-. The upward push 494 APPLIED MECHAKICS. [CHAP. XXII. of the unit-cubes under the right half of invert is doubled, as the depth is doubled; and taking the upward load on the four unit-cubes under the thick wall at li as 1207/-, the total extra upward push on the right half of the shell is 10?'- x 12, together with 90 tr x 4, or, in all. 480i'j'. The horizontal load among the cubes, to the right of JS, must be equal to the column QR, so that no one of these sixteen cubes shall have the rectangular pair of loads on it in a ratio exceeding 3 to 1. This requires that the horizontal thrust of the punned earth, at the depth B, shall equal the column, but at the depth 35 feet below Q, it may be less, say, instead of Sou-, it be only 20w, while at the depth 50 below q, it is 50w. Then the load-area standing on the base, from ?>5 to 50, has a height half the sum of 20^/^ and bOw, that is 35?r, and multiplying by the base 15, then we have 525?r as the probable horizontal ■extern load on the upper rigiit quadrant of the shell. With 15 feet as the radius of the new line of stress at the crown, and the load over it at 35a6', we have, by the rule for the thrust at the crown, 35?y x 15 = 525?'^ This establishes the horizontal •equilil)rium of the upper right quadrant of shell. For the lower right quadrant, the horizontal load due to the punned w «arth is — (58- - 50^) = 432?6- nearly, and adding Q8ir for the load below 58 at B, we have in all oOO?'*, the same as the new thrust at crown of invert. Thus the thrust at both crowns is the same, and the shell is now of a uniform thickness of 3 feet. The thrust at crown is o'2dv: ^ 3 = 21000 lbs. per sq. foot of brick, taking w =120 lbs. per cubic foot. The crushing strength of strong brick is 154000 lbs. per square foot, so that the apparent factor of safety against crushing is 8, and the real factor cannot be less than 4, one-half of it. (See p. 427, -and Kankine's Civil Enfjincerinff, p. 514, on buried arches.) Allowance for Excess-load alon<j the Ellijjtic Masonry Ming. — Rankine's assumption of the line of stress along the sofht of the •elliptic arch makes allowance for the excess load of hw along tlie ring, because the softit, not being so elongated as the line up the middle of the ring, the ratio of transformation is greater. Compare the two quadrants ABS (fig. 15) and ABS (fig. 5). In the first the half -span is 50 per cent, greater than the rise, and it balances under the fluid load alone. In the second the half-span equals the rise, and it balances under the fluid load together with a uniform load along the rib. CHAP. XXII.] DESIGN OF MASOXUY AUCIIKS. 495 equivalent to voussoirs of the same mass as the lluid, and of a depth half (the radius) of the rise. By proportional parts the half-span of a quadrant of given rise, to Ijalance under a fluid load, and an excess uniform load along the rib, can be found by reducing the half-span for fluid load above by a proportionate reduction. Thus, in the numerical example (fig. 10), our second approximation is as follows : — Assume the line of stress to be up the middle of the masonry ring: then Xo = 5, x^ = 26, and c^i = 40. For the corresponding hydro-rib : then h = 36-382, and y^ = 34*28. If the ring be 50 per cent, heavier than the superstructure, the excess-mass is u', for the upper half having been included, its excess is hw, and the under half having been excluded, its excess is I'ovj where vj is the weight of a cubic foot of the superstructure. Now, with no load along the rib, in addition to the fluid load, pi = 34'28, with voussoirs 10^ feet deep, that is half of the rise 21, //i = the rise 21, being a reduction of 13*28 feet. Hence with the actual voussoirs of 4 feet in depth, the propor- tionate reduction is 13*28 x 4 -;- 10| = 5*06. 8o that we have the reduced value, i/i = 29*22, giving s = 1*37, and s' = 1*88. The horizontal-load-area is, therefore, 1*88 x | (26- - 6") = 612. Then Tq = 612m?. This is practically the same as the results of Eankine's approximation (see fig. 16). In an aqueduct, when the masonry ring has a mass greatly denser than water, our second approximation becomes essential to accuracy (see numerical example which follows). ExAMrr.Ks. 26. Design of a semi-elliptic masonry arch of sandstone. Span 80 feet, and with a surcharge of I'o feet. Taking the rise a fifth of the span 16 feet, the radius of crown of soffit is 40- -f 16 = 100 feet. For depth of keystone, <o = v(*17 x 100) = 4 feet, say. First approximation. — Take line of stress along the soffit and Xq = vb, x\ = 21'5, a = x\ — Xf) = 16, and 47/1 = 40. The horizontal-load-area is \ {xi- — aro*) = 216 for the corresponding hydro-arch. By equation (22) and (23), 3/ 21 -5 h = 16 /- — = 25-204. V 5-0 y: = 2.5-204- 3! f^ = 23-88i: 40 s = -—■■ = 1-676, s2 = 2-80S. JiO*00 Hence the thrust at the crown is 2-808 x 216, or To = GOGiv. 49 G APPLIED MECHANICS. [CHAP. XXII, 27. The extreme depths of a quadrant of a hydrostatic rib uie xn = 7 anck = 9.ri fpf>h below tiie siirf.ii;« <it tbe fluid ffiir. 161. At !i noint R' Kvo laar 27. The extreme depths of a quadrant of a hydrostatic nb uie xn x\ = 26 feet below the surface ot the fluid (fig. 16). At a point £', lower than the crown, find the slope of the rib to the horizon. five Ceet and if the 28. In a complete geostatic rib, span 76 feet, rise 19 feet, height of enrth load over crown 7 feet (tig. 16), find the sloite of tlie rib at B, a point five feet below the level of the crown. So that xo = 7, ^i = 26, and st/i = 38 feet. As in last example, we must find the slope 8 of the corresponding hydro- arch, iind also s = 1'362, as in example 21, and cot 6' = stotd determines the- slope at Ji. 29. Design of an aqueduct with a semi-elliptic masonry ring twice as dense as- water. The clear span is to be 40 feet, and the depth of water over the keystone 2*5 feet. We proceed by trial and error, assuming values for the rise, and calculating the span. If there were no excess load along the ring, a first trinl value foi- the rise wouM be a tliird part of the span, but a greater vuiue is indicated, because of the large excess load along tlie rib. Try the rise of the soffit 15 feet. This gives the radius of crown of soffit 20- -^ 15 = 27. As an actual fluid is a troublesome load, the greater of Raukine's values is preferable for the thickness of the keystone, or ^0 = V('17 X 27) = 2^ feet nearly. Adopting the value 2^, and assuming tlie line of stress up the centre of the ring, we have x„ = 3'75 and xi = 20, and for the horizontal -load- area ^(20- - 3'75-) = li)2"97. Also the rise of the line- of stress is a = 16' 25, so that A = 16-25^ (xi -fa:o) = 28-39, and i/i = b - s^jh- ^ a = 2G-7i. Now the excess load of the upper part included is w, and of the lower half excluded is 2w, or, on an average, l-5if. This is the same as if the voK.siuiis were of a thickness 1-5 ^o = 3-7-t feet. "With no load along nb >/i = 26-74, with imaginary voiissoirs half the rise, that is Sy feet, yi = the rise = a = 16-25, or is reduced by lO'o feet. The proportionate reduction for vous.soirs 3"7o feet is 4'84, so that the reduced value of the half sp.m of line of stress is y'l = 26-74 - 4-84 = 21-9. The half span of soffit would then be 21-9 - 1-25 = 20-6. This is near enough to 20 feet, the required half-span of soflEt, so that further trial values of the rise of soffit are not required. For the line of stress, then, the horizontal-load-area is 192 '97 square feet, multiplied by w = 64 lbs., gives the thrust at crown ^'o = 12350 lbs., and the factor of safety can be evaluated. 30. A semicircular rib of radius 27 feet is loaded between itself and a hori- zontal straight line 3 feet over the crown. Find the angle of rupture, and the- inward and outward parts of the horizontal-load-aiea. By equations (5), (6), and (7), p. 440, r, - 5 4 1 C«s, = --f-.._^=-.04. / 2 o 1 2 1 \ ,,.,, ^'= (27-^9-9^38l)(-')^'^-=^'^"-- / 2 4 1 2 1 \ ^ V27 9 9 3 8iy ^ -* Their algebraic sum is 81w, the product of the radius and load over crown. CHAP, XXII.] DESIGN OF MASONRY AKCHES. 497 31. One of tlie shafts in the Hoosac Tunnel, Western Massachusetts, is elliptical in form, the axes being 27 by 15 feet (Simms* Tunnelling, 6th edition, p. 527). Show that 2 feet of brick lining is strong enough for such a shaft, surrounded by earth, even at great depths. The major diameter of the bore hole is -M feet, so that 50 feet is about the height of the isosceles triansile, with base 31 feet, and vertical angle 30°. This is the deepest column of earth likely to be disturbed by the bore hole. Consider a ring of briekwork .'jO feet bolow tlie surface of the earth, the ring itself 2 feet thick and 1 foot deep. Assuming the line of stress, in the first place, to be up the middle of the ring, its diameters are 29 and i? feet, their ratio is 1-7, almost e.\aetly %JZ. Hence the diameters of the ellipse of stress for the extern cubes of eai th may be in the ratio 1 to 3. Now the lesser horizontal stn-ss may be a third of the potential oOw, so that the greater horizontal stress is equal to 50w. The horizontal-load-area for a quadrant, in the direction of the major axis, is 50(f x 8-5, or42oM7. This also is the thrust 7'„ isee fig. 14), at the ei.d of the minor axis of the ring. But the area of the section of the ring is 2 square feet, so that the thrust per square foot of brick is 2r2-ow, or 25500 lbs. For strong brick the strength is 154000 lbs. per square foot. Hence the apparent factor of safety is six. QuOXAXtONS FROM RaNKIKe's CiVIL ENGINEERING. Page 425. — To determine, with precision, the depth required for the keystone of an arch by direct deduction from the principles of stability and strength would be an almost impracticable problem from its complexity. That depth is always many times greater than the depth necessary to resist the direct crushing action "of the thrust. The proportion in which it is so in some of the best existing examples has been calculated and found to range from 3 to 70 ... good medium values are those ranging from 20 to 40.* An empirical rule, founded on dimensions of good existing examples of bridges, is — depth of keystone, a mean proportional between the radius of curvature of the intrados at the crown, and a constant "12 or -17 for a single arch, or one of a series. Page 428. — In some of the best examples of bridges, the thickness of the abutments ranges from one-third to one-fifth of the radius of curvature of the arch at its crown. Page 429. — The thickness adopted for piers in practice ranges from one-tenth to one-fourth of the span of the arches ; the latter thickness, and those approaching to it, being suitable for "abutment-piers." The most common thickness for ordinary piers is from one-sixth to one-seventh of the span of the arches. Page 435. — It appears that, in the brickwork of various existing tunnels, the factor of safety is as low as four. This is sufficient, because of the steadiness of the load; but in buried archways, exposed to shocics, like those of culverts under high embankments, the factor of safety should be greater ; say, from eight to ten. QuoxAiiONS FROM SiMMs' Pracxical Tunnelling (4th edition, pp. 202, 203). It is known, too, that the pressure on tunnels in comparatively shallow ground, say, less than 40 feet below the surface, may be localised and concentrated upon the crown of the arch with peculiar severity. Mr. Simms accounts for the greater * Factors of safety calculated from existing bridges are likely to be what we have called apparent ones, so that the actual factors will range from 10 to 20, the 10 being the lowest assumed in our tables. 2 K 498 APPLIED MECHANICS. [CHAP. XXIII. pressure upon the work in shallow ground by the supposition that the whole superincumbent mass acts vertically downwards, whilst at greater depths it is more or less sustained as an arch over the tunnel, which is proportionally relieved of the pressure. In the building of the Stapleton Tunnel the arc)i was first built with only four rings of brickwork in morlar. "When the autumnal rains set in, the ground began to press heavily ; so much so, that the line of the tunnel could be traced as a hollow on the surfiace of the ground, which was not more than 40 feet above the tunnel. A portion of the tunnel fell in, and was rebuilt with five and six rings of brick. The pressure of clay and shale, when disturbed by excavation, is, in some situations, something almost immeasurable. The phenomena of disturbance supply powerful examples of the flow of solids. There is the familiar phenomenon of the bending and snapping of huge poles, and strongly timbered frames, by the moving pressure of clay. In the construction of the Primrose Hill Tunnel, through the London Clay, the lengths were limited to 9 feet, and strongly timbered, till the arching was completed. In virtue of its mobility, however, the moist clay exerted so great a pressure on the brickwork, as to squeeze the mortar from the joints, to bring the itiner edges of the bricks into contact, to grind them to dust by degrees, and to reduce the dimensions of the tunnel slowly but irresistibly. The evil was counteracted by using very hard bricks, laid in lloman cement, which, setting before the back ))ressure accumulated, hardened and resisted the pressure, and so saved the bricks. The thickness of the brickwork was augmented to 27 inches. A similar accident occurred to a portion of the invert of the Netherton Tunnel, built on a foundation of "blue-bind," or marl. Some weeks after it was built, the invert was forced up in several places by the swelling of the ground, and at one point the bricks were crushed almost to powder. The invert was rebuilt with a greater versed sine. CHAPTER XXIII. THE METHOD OF KECIPROCAL FIGURES. This method is a rival to Putter's method of sections. It is especially applicable to irregularly shaped frames such as roof principals. Fig. 1 shows the simplest pair of reciprocal figures, each consisting of the six lines joining four points. The (three) lines forming a (triangle) closed polygon on any one of the figures meet at a point on the other. Also the lines as numbered are parallel. We have chosen the case where the fourth 'point is not inside the triangle joining the other three, so that a pair of the six lines pass each other at E. If the second figure be drawn, then each side bisected, and the dotted lines drawn at right angles to each side through the bisecting points, we have the dotted figure. If it be turned through a right angle, it is similar to the first of the pair of reciprocal figures. Again, drawing the first figure, and looking upon it as a quadrilateral frame with a pair of diagonals passing each other at E, it is evidently able to be self -strained by tightening the diagonal (5). Now draw 1* and Q on the section figure CHAP. XXIII.] METHOD OF RECIPROCAL FIGURES. 499 equal and opposite to each other, and giving to scale the pull on the diagonal (5). Next, a triangle of forces is drawn for the apexes B, C, and I), as shown in fig. 1. In this way the second figure is a stress diagram for the self-strained frame. The frame with the bar (5) removed is said to be i/ideformaUe. In general for an indeformable frame two conditions are required to balance each pin, giving in all twice as many conditions as there are pins. But three of these conditions are first used to balance tlie external loads acting on the frame as a rigid body. Hence we have a rule that the frame may be indeformable, viz. : l^wice the number of joints less three should equal the number of bars. A Pair Of j^Recjprocal Figures. c ^he Of/terttaned (ipsitle (/otY/7 . Jitress JJlagj^ocyn. FiK. 1. The stress diagram for a roof-frame is usually only a 'part of a reciprocal figure, the frame itself, being only a part of the first figure. We now describe at great length the process of drawing it, using the scientific notation of Levy. On figure 2 is shown a skeleton elevation of an iron roof-frame, span 32 feet, rise 12 feet. The rafter is trisected by two isosceles struts PN and FO, also PQU is 60°. The frames or principals are 20 feet apart, so that a fully loaded joint such as R has apportioned to it all the load on an oblong •of the roof-cover 20 feet x 7 feet nearly, since MQ, is 20 feet, being the hypotenuse of a triangle of which the sides are 16 and 12 feet. Taking the weight of the cover at 4 lbs. 2k2 500 APPLIED MECHANICS. [CHAP. XXIIL per square foot, and of one foot deep of snow at 12 lbs. per square foot, the vertical load (5) concentrated at the joint H is 140 ^ 16 = 2240 lbs. = 20 cwts. The vertical loads, then, are 20 cwts. at the joints iV, 0, Q, R, and S, but only 10 cwts. at J)!f and T. The wind-load is on the left side only, and its direction is at 45° to the rafter MQ. We take the normal component wind-loads as 25 per cent, greater than the vertical load. That is, the two loads at N and normal to the rafter are each 25 cwts., those at Q and M being half as much. The Reactions at the Supports. — The frame is securely anchored to the wall at the storm-end M, while the other end T has a free horizontal motion on expansion- rollers on the top of the wall there. This is only the first case ; later on we will develop the procedure if both ends be anchored. The Force-polygon, or load line, is now drawn to a scale of cwts. Beginning at M, and going round the frame in right- cyclic-order, the loads at the joints have the numbers 1 to 7 allotted to them. Thus at M there is 1„ and 1^ for the normal and vertical component loads there. Because of the trolly, the reaction at T is necessarily vertical, and 8 is its number. The reactions at the anchored end M are 9t, and 9^, being vertical and horizontal component reactions, respectively. The sides of the force-polygon are now drawn in the same cyclic order. It will be found to be complete, with one exception, namely, the joint x between the vertical reactions 8 and 9„ is unknown. Because the half-truss is a 3, 4, 5 right-angled triangle, it will be found that all the joints in the force-polygon are horizontally opposite multiples of 10 cwts. on the scale. The Li7i]c-polygo7i. — By consulting fig. 10, Ch. IV, this part of the construction will be more readily followed. On fig. 2 a pole is chosen just opposite the last bend in the force-polygon, and so as, roughly, to be the vertex of an equilateral triangle standing on the force-polygon. Vectors are to be drawn (in pencil only) from the pole to each joint of the force-polygon except X, which is not yet determined. Since the joint x is at the junction of 9, and 8, the vector Ox must be parallel to the link of the link-polygon joining up those two vertical forces. For this reason we should begin the link-polygon at a point on the vertical line of action 9,,, and end it at a point on the line of action of 8. Choose any point i on the line of action of 9^. On fig. 2 it is chosen on 9,„ produced about an inch above 3f, and is marked with a little square, as a memo- randum that i is the beginning point. The link ii' is drawn /CHAP. XXIII.] METHOD OF KECIPUOCAL FIGURES. 501 between the lines of action of 9„ and 9/„ parallel to the vector from tlie pole to the junction of those two forces. In the same way the other links, i'n, aa, ah, he, c'd, dd', d'r, ef,fq, and gh, are ■drawn consecutively between the lines of action of the forces in pairs, each link parallel to that vector which comes from the pole to the junction of the corresponding pair of forcCvS. The line (fh has no finite length, the lines of action of 7 and 8 •coinciding ; hence g and h coincide too, and have a little circle with a little square round their common position. Joining the two little squares i and li with a dot-and-dash line, we have found the closing side of this link-polygon, while a vector (or search-light) drawn from the pole parallel to it determines x the ■closing point of the force-polygon. The judicious choice of the pole, and of the starting-point i, ensure that the link-polygon will 'rainbow' over the truss, out of the way of the further construction, and also that the links will go on continuously without looping back on each other. The Stress Diagram. — Put half-barbs on the sides of the force-polygon, and full-barbs on the lines of action of the forces external to the frame. It will then be seen that at the apex M six forces concur. Four of them are known, that is, they are already drawn to scale on the force-polygon. In naming these four they are to be taken in right-hand cyclic order round M, thus, 9,„ 9a, 1,„ It,. It is convenient to call the remaining two forces exerted on the joint M by the rafter and tie 10 and 11, keeping to the same cyclic order round M. Now, the four-sided open polygon 9„, 9^, 1„, and 1„ is to be completed as a closed six-sided polygon, by drawing a side 10 parallel to the rafter to meet a side 11 drawn backwards from the initial point x. In this way the magnitudes of 10 and 11 are now determined, and by going round the six-sided closed polygon in the direction indicated by the half-barbs, it will be seen that 10 acts along the rafter toicards M, while 11 acts along the tie away from M. On 10 and 11, in the neighbour- hood of M, place half -barbs pointing accordingly ; but at the other end of 10 put a full- barb in the opposite direction from that of the half-barb, so as to indicate the reaction of the rafter there, and a fidl-ho^vh to indicate that it is already drawn to scale on the force-polygons. In the same way put a/w//-barb reversed at the other end of the tie 11. It will now be seen that of the five forces that concur at N, all are furnished with barbs except 12 and 13, which reminds us that all the sides of the five-sided closed polygon for the apex N are already drawn on the force-polygon, and the lines 502 APPLIED MECHANICS. [CHAP. XXIII. drawn parallel to 12 and 13 close it, when the senses of 12 and 13 are then found to be both towards N, so that they are to be furnished with half-barbs in the neighbourhood of N, and with full-barbs reversed at their other ends. The next apex that can be solved is 0, for here the group have all barbs but two. In this way the apexes are solved in the order Q, R, and S, the known and found parts being duly registered in the table for that purpose on fig. 2, while the letters w, n, &c., give the corresponding closing points. The next apex T is abnormal : there all the forces are known but one, namely, xi, which cannot be compelled to close the corresponding polygon. When XI is drawn from ?;, if it only fails to go through s by a little deviation, that is the allowable error of drawing. In this way we have a check upon the construction. If it utterly fails to go through s, we must look back to the construction of the polygons for some point at which we did not keep to strict cyclic order. If with every care to draw exactly it still quite fails to close, that would mean that the truss is deficient in members — that it is not rigid. Of course this could not happen with the truss on fig. 2, which is necessarily rigid, being a set of triangles. Anchor on Lee-sule. — If we suppose the anchor shifted to ^f, that will make no difference in the magnitudes of the vertical reactions of the walls. For one way of calculating the magni- tude of 8 is to take moments about M. Now, the horizontal reaction of the anchor passes through M, whether the anchor be at M or 2\ and so has no moment, and cannot affect the result. There is no need to draw the link-polygon anew, but only to shift or into the new position ;:, and the force-polygon is now 1„, Ip, 2„, 2e„ 3„, 3^, 4„, 4j„ 5, 6, 7, \iUh, viiir, 9. Nor is there any need to redraw the stress diagram, polygon by polygon, for we may suppose the original stress diagram as having been constructed in two steps, as follows : — In the first step the six legs are drawn of indefinite length, one from each joint of the force-polygon. In the second step we might suppose a billiard- ball to start from x, describing the route 11, 13, 15, 16, rebounding from the legs in order at m, n, and o, and arriving on the horizontal at ^), then completing its route xvi, xv, xiii : rebounding from the legs at q, r, and s, and returning along xi to the pocket at x. The dotted route is mapped out at once in this way, only beginning and ending at z. i<calin(j off the results. — The full lines are scaled off' and the results are written on a small elevation of the roof on tig. 2, having the anchor on the left, or storm side. Another reverse Fig. 2. 504 APPLIED MECHANICS. [CHAPs XXIII. elevation is shown with the anchor shifted to the right, or lee side, and on it are written the loads as sealed off the dotted diagram. Xow this is called a reverse elevation, because aft«r all it is not the oMchor which has really shifted, but the wind has, and we have gone round about to look at the back of the roof-frame. Hence this reverse elevation requires to be read through on the back of the paper when comparing the stresses with those marked on the first direct elevation. That is, the quantities scaled off for the bars bearing arable numbers on the first elevation are to be compared with those scaled off for the bars bearing roman numbers on the reverse elevation. Both ends fixed. — If both ends of a truss be fixed, it is quite indeterminate in what proportions the two anchors will take up the horizontal load due to the wind. Suppose for a moment that both ends of the roof, fig. 2, are fixed, and that we assume that the two anchors share equally the horizontal load. It would be solved on this assumption, by simply placing the closing point of the force-polygon at a point midway between X and z, and of course the results would just be an average of those for the full and dotted stress diagrams shown there. These are marked on a half-elevation at bottom of the figure. In every ease, then, it is best to assume an anchor at the storm side for the reactions ; then shift .r horizontally in accordance with any assumed proportion between the holding power of the anchors. With the anchor at the storm side, the stresses are marirna, so that it is a safe assumption to make in all cases. The dotted stress diagram, fig. 2 gives minima stresses; but in some examples these minima are very important. For a member which was a tie for a maximum may have, in the minima diagram, decreases through zero, and become a strut. Figs. 3 and 4 show the King-post frame and its stress diagram drawn as already described, but the loose ends of the loads at K, L, &c., are removed. Fig. 3 is now to be looked upon as a self-strained frame. It is composed of two frames, the link polygon, an open balanced frame (fig. 10, Ch. lY) and the iiuleformahle King-post, each straining the other through the bars 1, 2, 3, &c., joining them point to point, and all idea of external load is dismissed. Fig. 4 is the one and only one reeiprocal fjijnre drawn to fig. 3. The reciprocity in the other direction is not so complete, as fig. 3 is only one of the many reciprocal figures to fig. 4, for we might have begun the link polygon at a different point. /^ Jb A Anchor £ L EVA T /ON. Jfeal and Jc^eczl Frame^ (coTTTZmecf)' ft 10 o LU-LX ' M I I ^ 40 SO ft. I J Fig. 3. Stress JDia^rcuny. Fig. -i. Fig. 5. CIIAr. XXIII.] METHOD OF KKCU'ltOCAL FICfURES. 507 Exercise I. Here the Queen-post Eoof, having a span 56 feet, is shown on fig. 5 with the central rectangle double-braced with ties which can only act one at a time. There is an additional load 9 at the joint I. The truss is anchored at both walls, the strong wall (square anchor) taking two-thirds of the horizontal displacing ellbrt of the wind-load. In the first place assume that the left (square) anchor takes all the horizontal load, and leave out the load 9 at the joint I. After drawing the link polygon we find the point x separating the two vertical reactions at 8* and 10". Next draw a vertical line through the right trisecting point of vy. Here is the true position of 8* and 10". They are to be slightly displaced left and right of each other, and 8" is to be made longer than z.r by two-thirds of 40 cwts. (the value of the extra force 9) and 10" made longer than .ry by one-third of 40 cwts. The force 9 will now fill the gap between the two augmented vertical supports. After completing the solution of apex after apex as shown on the table, the dotted reactions are placed in the vertical through the otlier trisecting point of vij, and augment with the portions of the 40 cwts. in the reverse order, as 9 will now be at the joint U and be called ix, also the counterbrace has replaced the brace, for we suppose the truss turned end for end and the wind to remain as shown by the storm arrow. The dotted stress diagram is now to be drawn. The half roof is a 3, 4, 5 triangle, as the rise 7 is 21 feet. Assume the roof to be rigid, and the whole vertical load 120 cwts. may be concentrated at C, and the whole normal wind-load 90 cwts. at the centre of the rafter AC, and then further decomposed into a vertical component of 72 cwts. and a horizontal component of 54 cwts. Taking the moment of this 54 cwts. about A and U alternately adds 10^ cwts. to 8' and subtracts it from 10". Hence yv = 54, 2-0: = 60 + 18 + 10|, and xy = 60 + 54 - 10^ cwts. Scale 50 cwts. to an inch. C U R 1/ E D ROOT. ft /O O /o ZO Jo ■*o So OoJ^eef. 1 i 1 1 :) , 1 1 1 1 ! 1 I 1 ! 1 'O o C,t/^ so /oo /SO SOO C'U/CS ._t-j .::.■!■!•:■'■ i; 1 1 ■'■ ! Fig. 6. CHAP. XXIII.] METHOD OF RECIPROCAL FIGURES. 509* Exercise II. This is a great cur\ed roof-frame (fig. 6). In drawing the legs 10, 14, 18, they should be drawn at right angles to Mj, Mk, Ml, and the vectors 11, 13, 17 at right angles to Np, Nq, Nr. The legs may be firm lines drawn indefinitely outwards, and numbered boldly near the force-polygon. The vectors are then light lines numbered on an arc of a circle. The indefinite lines may be temporarily numbered again in pencil at their far ends. It will then be easier to work out the contour 12, 15, 16, being verticals and obliques alternately, and joining a leg and a vector alternately. On the dotted figure for the truss turned round, taking the anchor to the other end, the contour XII, XV, XVI, is put in by inspection. Note that 16 is drawn down instead of upwards on the first contour. Hence the stress on the bar HI has changed from a pull to a thrust. The frame is primarily of the King-post pattern. On the first auxiliary figure the scaled results are marked for wind on each side alternately. The ^•ertical bars are all ties except that HI changes with the wind. On the second half auxiliary figure the vertical bars are compelled to be struts by double bracing as on King's Cross Eailway station roof. A thimble acting on a right and left screw on the counterbrace BE puts upon it an initial pull of 80 cwts., which induces a pull of 61 cwts. on the rod CD and thrusts on all four sides of the trapezium. Choose a scale on which BE measures 80 and the lengths of the other bars on this scale give the induced strains, those on the parallel sides being interchanged, see fig. 1. If both BE and CD be furnished with tightening thimbles, they assist in adjusting the shape of the frame. COMPOUND ROOr TRUSS 'iind on Anchored Sida, Mp-, kno-n fcunrl i R 1? ?„ ;>» r LI n c n F JSI6H.ln M X H r 22 23 n lO fl PI ti Mti Sa. m 1 !SK J X 2i> It 30 A I. .» 31 3i. /. 3i IS t? 7 33 a 3i. 3r. .V J7. 3* .K nCJtrdl n free Motion HomontaJly o 10 ^ ^ncAerrd on Lrr Sidr. fret O T.AIwmT Trin. Coll. Hu^Ha ' ' " ' ^---^ 4» reft Fig. 7. CHAP. XXIIl.] MKTHOL) OF KECIPROCAL FIGURES. 511 Exercise III. Here is a compound roof-frame (fig. 7). A vertical section through the crown cuts only one bar, the main tie. Hence the pull on tlie main tie is virtually given when the loads are determined upon. In this example both x and y have to be found by the link -polygon. It is to be drawn passing through the crown H, and may be drawn Hd'd backwards, and Hfg forwards. A vector from the pole parallel to the closing side of the link-polygon gives x, and a vector from the pole parallel to Hz cuts off xy on the horizontal through x, where z is the point where jk cuts the main tie (produced). The span being 56 feet, and the rise 21 feet, then the half frame is a 3, 4, 5 right-angled triangle, which makes an arithmetical check on the values of the supports and component loads easy. Observe the joints of the force-polygon ticked over on to the vertical scale. As on Ex I, with the frame assumed to be rigid, the loads reduce to a vertical load of 80 cwts. at the vertex H, a vertical load of 48 tons at E the middle of the storm rafter AH and also a horizontal load there of 36 cwts. The calculated values of the supporting forces are (lift) = 36, (10) = 40 + 12 + 6|, and (11„) = 40 + 36 - 6| cwts. In this particular pattern the slightest error in the vertical position of x causes a greatly increasing error as you pass from joint to joint. This is to be eliminated by drawing the four parallels 12, 14, 20, 22, with one setting of the parallel rollers or sliding set-square and of an ample indefinite length, and also the four 25, 27, 33, 35. Then, while solving apex after apex, the fact, due to symmetry, that 15 and 23 are in one line, and as well 24 and 35, enabled any slight error in the position of x and of y to be corrected. 512 APPLIED MECHANICS. Note to Chapter X. In this edition we have omitted the transite of the locomotive over the constant span. It is purely of academic interest, and had to give place to the matter in Ch. XI, which is of practical importance. The transite was illustrated by figs. 158 to 166. The fig. 158 showed a number of parabolic segments on a common base equal to the span of the girder but of varying heights, proportional to — 1st wheel only on span, 1st two wheels only on the span, &c. With templates made from these a set of diagrams gave the maximum bending moment at each point of the span, with only the set of wheels on the span corre- sponding to the particular template. Each of these diagrams consisted of arcs for such points as the wheels could actually reach without any other wheel coming on or off, and tangents to the arcs for the other points. 3AyONARA^ Printed by Ponsonby & GiBBS, University Press, Dublin. /, 8i3i TA 350 A5 1916 Alexander, Thomas iilementaiy applied raechanicE c3d ed.-, Eof irx-rrmt PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY III!' HiHilulKitUiI i:!iiH!i