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WORKS OF PROF. C. E. GREENE
PUBLISHED BV
JOHN WILEY & SONS.
Graphics for Engineers, Architects, and Builders.
A Manual for Designers, and a Text-book for
Scientific Schools.
Trusses and Arches. Analyzed and Discussed by
Graphical Methods. In Three Parts.
Part I. Roof Trusses. Diagrams for Steady Load,
Snow, and Wind. 8vo, cloth, $1.25.
Part II. Bridge Trusses. Single, Continuous,
and Draw Spans; Single and Multiple Systems;
Straight and Inclined Chords. 8vo, cloth, $2.50.
Part III. Arches in 'Wood, Iron, and Stone. For
Roofs, Bridges, and Wall Openings ; Arched Ribs
and Braced Arches ; Stresses from Wind and
Change of Temperature. 8vo, cloth, $2.50.
Published by the A uthor, at Ann A rbor., Mich,
Structural Mechanics: The Action of Materials
Under Stress. A work on the Strength and Resist-
ance of Materials and the Elements of Structural
Design. Ann Arbor, Mich., 1897. Printed for the
author. 8vo, 300 pp., 100 illustrations. Price $3.00.
0|rc^^-'*-^
®rapl)ics for (Engineers, 3rcl)itcct0, anb i3uUbcr3:
^ MANUAL FOR DESIGNERS, AND A TEXT-BOUK FOR TECHNICAL SCHOOLS
TRUSSES AND AECHES
ANALYZED AND DISCUSSED BY GEAPBICAL METHODS
CHARLES E. GREENE, A.M., C.E.,
PROFBSSOR OF CIVIL ENGINEERING, UNIVERSITY OF MICHIGAN ; CONSULTING ENQIMEEB.
IN THREE PARTS.
I.
ROOF-TRUSSES : Diagrams for Steady Load, Snow, and Wind.
iL
BRIDGE-TRUSSES : Single, Continuous, and Draw Spans ; Sinoi^k
AND Multiple Systems; Straight and Inclined Chords.
III.
ARCHES, IN "Wood, Iron, and Stone, for Roofs, Bridges, and Wall-
Openings ; Arched Ribs and Braced Arches ; Stresses from Wind
AND Change of Temperature; Stiffened Suspension Bridges.
Pakt I.— roof-trusses.
FOUB FOLDING PLATES.
REVISED EDITION.
FIFTH THOUSAND.
NEW YORK:
JOHN WILEY & SONS.
London: CHAPMAN & HALL, Limited.
1903.
Engineering
Ubvary
COPTRISHT, 1890,
By CHARLES E. GREENE.
En{nneeiii|g
V. I
PREFACE TO PART I.
The use of Graphical Analysis for the solution of problems
in construction has become of late years very wide-spread.
The representation to the eye of the forces which exist in the
several parts of a frame possesses many advantages over their
determination by calculation. The accuracy of the figures is
readily tested by numerous checks. Any designer who fairly
tries the method will be pleased with the simplicity and
directness of the analysis, even for frames of apparently com-
plex forms. Those persons who prefer ariihmetical computa-
tion will find a diagram a useful check on their calculations.
Being founded on principles absolutely correct, these diagrams
give results depending for their accuracy on the exactness
with which the lines have been drawn, and on the scale by
which they are to be measured. With ordinary care the dif-
ferent forces may be obtained much more accurately than the
several parts of the frame can be proportioned.
It is advisable to draw the figure of the frame to quite a
large scale, as the lines of the stress diagram are drawn paral-
lel to the several pieces of the frame. If it is objected by any
that a slight deviation from the exact directions will materially
change the lengths of some of the lines, and therefore give
erroneous results, it may be suggested that just so much
change in the form of the frame will produce this change in
the forces ; one is therefore warned where due allowance for
V
VI PKEFACE TO PART I.
such deformation should be made by the proper distribution
of material. The comparison of different types of truss for
the same locality can be made with ease, and the changes pro-
duced in all of the forces in any frame by a modification of a
few of its pieces can be readily shown. By applying each new
principle to a new form of truss, quite a variety of patterns
have been treated without an undue multiplication of figures.
The method of notation used was introduced by Mr. Bow,
in his "Economics of Construction." The diagrams, as here
developed, are credited in England to Prof. Clerk-Maxwell,
and the method is known by his name. The arrangement of
the subjects, the application of the method, and the minor
details have been carefully studied by the author. A very
limited knowledge of Mechanics will enable the reader to
■snderstand the method of treatment here carried out.
NOTE TO KEVISED EDITION.
The reception of this Part at tlie hands of teachers and
designers, since its first appearance as a reprint of a series of
articles in " Eiigineering News," has been so hearty and sus-
tained, that it has been thought best to put Egof-Tkusses in a
uniform dress and agreement with Bridge-Trusses and Arches.
The opportunity has been seized to arrange the material in a
more systematic order, introduce some additional problems,
and improve, as it is thought, in some matters of detail.
Quite a modification has been made in the way of regarding
trusses which exert horizontal thrust, and Chapter YIIL,
Special Solutions, is new. The solution by reversal of a diag-
onal has been used in the author's class-room for several years.
The concluding example of this chapter will afford a good test
of the reader's mastery of the preceding principles.
C. E. G.
Am? Akbor, Mich.. March 11, 1890.
vu
TABLE OF CONTENTS.
CHAPTER I.
PAGES
General Principles. Triangle of Forces ; Notation ; Illustrations, . 1-6
CHAPTER II.
Trusses '^ith Straight Rafters ; Vertical Forces. Triangular, King-
post, and Fink Trusses, 7-15
CHAPTER III.
Trusses for Flat Roofs. Queen-post, "Warren, and Howe Trusses, . 16-21
CHAPTER IV.
"Wind Pressure on Pitched or Gable Roofs. Formula for Wind-pres-
sure ; Examples with Roller Bearings; Wind on Alt ernite Sides, 23-32
[CHAPTER V.
"Wind Pressure on Curb (or Mansard) and Curved Roofs. Examples
with and without Rollers, 33-43
CHAPTER VI.
Trusses with Horizontal Thrust. Scissor and Hammer-beam Trusses, 44-49
CHAPTER VII.
Forces not Applied at Joints, 50-52
CHAPTER VIII.
Special Solutions. Reversal of Diagonal; Trial and Error; Example, 53-58
viii
TABLE OF CONTENTS. ix
CHAPTER IX.
Bending Moment and Moment of Resistance. Equilibrium Polygon ;
Graphical Solution for Moment of Resistance, .... 59-71
CHAPTER X.
Load and Details. "Weight of Materials ; Allowable Stresses ; Ties,
, Struts, Beams, Details, 72-77
ROOF-TRUSSES.
CHAPTER I.
GENERAL PRINCIPLES.
1. Aim of the Book. — It is proposed, in this volume, to
explain and illustrate a simple method for finding the stresses
in all of the pieces of such roof or other trusses, under the
action of a steady load, as permit of an exact analysis ; to
show how the wind or any oblique force alters the amount of
the stresses arising from the weight ; to add a device for
solving some systems of trussing which otherwise appear
insoluble by the above method ; and to conclude with such
an explanation of bending moments and moments of resist-
ance as will make this part reasonably complete for roof
designing.
2. Triangle of Forces. — Taking it for granted that, if two
forces, acting at a common point, are represented in length
and direction by the two adjacent sides of a parallelogram c a
and c n, Fig, 2, their resultant will be equal to the diagonal c b
of the figure, drawn from the same point, — it follows that a
force equal to this resultant, and acting in the opposite direc-
tion, will balance the first two forces. Hence, considering
one-half of the parallelogram, we have the well-known propo-
sition that, if three forces in equilibrium act at a single point,
and a triangle be drawn with sides parallel to the three forces,
these sides will be proportional in length, by a definite scale,
to these forces. The forces will also be found to act in order
2 EOOF-TKIISSES.
round tlie triangle, and must necessarily lie in one plane. If
the magnitude of one force is known, the other two can be
readily determined.
For example : — Let a known weight be suspended from the
points 1 and 2, Fig. 1, by the cords 1-3, 3-2, and 3-4. Draw
c b vertically to represent the weight by any convenient scale
of pounds to the inch. This line will then be parallel to, and
will equal the tension in 3-4. Draw c a parallel to 1-3, and
6 a parallel to S-2. Then will the sides of the triangle cha
represent the forces which act on the point 3, and they will be
found to follow one another round the triangle, as shown by
the arrows.
3. Notation. — A notation will now be introduced which
will be found very convenient when applied to trusses and
diagrams. In the frame diagram write a capital letter in
every space which is cut oflf from the rest of the figure by
lines, real or imaginary, along which forces act. See Fig. 2
and following figures. Thus D represents the space within
the triangular frame, A the space limited by the external
forces acting at 1 and 2, B the space between the line to 2
and the line which carries the weight. Then let that piece of
the frame or that force which lies between any two letters be
called by those letters ; thus, the upper bar of the triangle is
AD, the right hand bar is B D, the cord to the point 1 is A C,
that to the weight, or the weight itself, is C B, etc. In the
diagrams drawn to determine the magnitude and kind of the
several forces acting upon or in the frames the corresponding
small letters wall be used ; thus c b will be the vertical line
representing the force in C B, & a the tension of the cord B A,
and ac the pull on 1.
4. External Forces. — Eeturning to Fig. 1, let us suppose
that a rigid, triangular frame is made fast to those cords, so
that, as shown by Fig. 2, the cords are attached to the vertices
of the triangle, while their directions are undisturbed. It is
evident that the same stresses still exist in those cords, if
the frame has no weight, and that the portion of the cords
ROOF-TRUSSES. 3
within the triangle may be cut away without destroying the
equilibrium of this combination. Hence we see that the equi-
librium of this frame is assured, if the directions of these
cords, or forces external to the frame, meet, if prolonged, at a
common point.
The external forces C B, B A and A C, taken in the order
C B A, or passing around the exterior of the triangle in a direc-
tion contrary to the movement of the hands of a watch, give
the triangle of forces c ha, in which c b acting in a known direc-
tion, i.e. downwards, determines the direction of & a and a c in
relation to their points of application to the frame, since for
equilibrium, by § 2, they must follow one another in order
round the stress triangle.
5. Stresses in the Frame. — Consider the left-hand ajDex
of the triangle. This point is in equilibrium under the action
of three forces, viz., those in A C, C D, and D A, which we read
around the point in the same order as be/ore ; we found the
direction and magnitude of A C in the previous section, and
the inclinations of the other two are known. The three forces
at this joint must therefore be equal to the three sides of a
stress triangle, as before.
Begin with A C, the fully known force, and pass from a to c,
because that is the direction of the action of the force A C on
the joint under consideration. Next, from c, draw c d parallel
to C D, prolonging it until a line from its extremity d, parallel
to the piece D A, will strike or close on a. The stress c cZ is
found in C D, and the stress d a exists in D A. The direction
in which we passed around acd, that is, from c to d, and then
to a, shows that C D and D A both exert tension on the joint
where they meet.
Next take the lowest joint. Kemembering again to take the
three forces in equilibrium here in the order in which the
external forces were taken, and commencing with the first
known one, we go, in the stress diagram, from d to c; because,
since we have just found that c d represents the pull of C D on
the left-hand apex of the frame, d c must be the equal and op-
4 KOOF-TRUSSES.
posite pull of D C on the lowest joint. Next comes c h, along
wliich we pass doivn, the direction in which the weight acts ;
and finally we draw from b,bd parallel to the piece B D. This
last line will close on the point d, if the construction has been
carefully made, and the direction in which we pass over it,
from b to d, shows that the piece B D exerts tension on the
lowest joint. If the reader will now run over the triangle
dba, which must belong to the right-hand joint, he will see
that the directions just given are again complied with.
The reader can invert Fig. 2 ; then the weight will press
down upon the upper apex of the triangle, and he will find,
upon drawing the stress diagram, that the three external
forces are thrusts, and that compression exists in each piece
of the frame.
6. Second Illustration: External Forces. — In order to
make these first principles more plain let us take another
case. Suppose a triangular frame. Fig. 3, to rest against a
wall by one angle, to have a weight of known amount sus-
pended from the outer corner, and to be sustained by a cord
attached to the third angle and secured to a point 2. Since
this frame is at rest under the action of three external forces
which are not parallel, their lines of action must, by § 2, meet
at one common point ; and since the known directions of two
of these forces, AC and CB, will meet at 4, if prolonged, the
force exerted on the frame by the wall at 1 must have the
direction of the line 1-4. The magnitude and kind of the
two unknown external forces therefore will be found by the
following construction, observing the rules of interpretation
already laid down : —
Draw ac, vertically down, equal to the known weight and
force A C ; next, from c, a line parallel to the cord and force
CB, and prolong it until, from its extremity b, a line may be
drawn parallel to BA, to strike a. As we went from c to b,
and from 6 to a, C B must pull on, and B A must thrust against,
the frame.
7. Stresses in the Frame. — Take whichever joint is most
ROOF-TRUSSES. 5
convenient, for instance the one wliere the weight is attached ;
pass down ac for the external force and then, obser\ing the
order in which the triangle of external forces was drawn, draw
cd parallel to C D and da parallel to DA. Since cd, in the
triangle acd (made up of forces ac,cd, and d a), must represent
a force acting upwards, C D exerts tension on this joint ; and,
similarly, d a (not a d) shows that D A thrusts against the same
joint.
Take next the joint at 1. Here the reaction, as before as-
certained, is h a ; next comes a d, the thrust of the piece A D
against this joint ; and lastly d h, drawn parallel to D B, to
close on h the point of beginning, shows that D B also thrusts
with this amount at 1.
8. Third Illustration. — Once more, suppose that the tri-
angular frame, Fig. 4, has a weight attached to its lowest
angle and that the two other points are supj)orted by inclined
posts. The forces 1-4 and 2-4 must intersect 3-4 at the same
point. Draw a b vertically downwards, and equal to the given
weight ; draw b c jDarallel to 2-4 or B C and c a parallel to 1-4
or C A. Hence be and ca are thrusts. For the lowest joint,
after passing down a b for the weight, draw b d parallel to B D
and d a parallel to D A, thus finding that B D and D A both
pull on the joint A B, and hence are tension members. As in
former cases, find d c, which proves to be compression.
9. General Application. — Since, in Mechanics, the poly-
gon of forces follows naturally from the triangle of forces,
being simply a combination of several triangles, the same
rules will apply when we have to deal with several external
forces or a number of pieces meeting at one joint. 1°. Draw
the polygon of external forces for the whole frame, taking
them in order round the truss, either to the left or right, as
may seem convenient, 2°. Take any joint where not more
than two stresses in the pieces are unknown, and draw the
polygon of forces for it. Treat the pieces and external forces
which meet at the joint in that order, to the left or right, in
which the external forces were taken, and begin, if possible,
6 ROOF-TRUSSES.
witli tlie first kno\vn force, so that tlie two unknown forces will
be the last two sides of that particular pol^^gon. 3°. The di-
rection in which any line is passed over, in going round the
polygon as above directed, shows whether the stress in the
piece to which that line was drawn parallel acts towards or
from the joint to which the polygon belongs, and hence is
compression or tension. The reader must understand this
lDrinci]3le in order to correctly interpret his diagrams.
10. Reciprocal Figures. — Prof. Clerk-Maxwell called the
frame and stress diagrams reciprocal figures ; for, referring to
the figures already drawn, we see that the forces which meet
at one point in the frame diagram give us a triangle or closed
polygon in the stress diagram, and the pieces which make the
triangular frame have their stresses represented by the lines
which meet at one point in the stress diagram. The same
reciprocity will exist in more complex figures, and it is one
of the checks which we have upon the correctness of our
diagrams.
The convenience of the notation explained in § 3 depends
upon the above property.
CHAPTEK II.
TRUSSES WITH STRAIGHT RAFTERS; VERTICAL FORCES.
11. Triangular Truss; Inclined Reactions. — Suppose
that the roof represented in Fig. 5 has a certain load jjer foot
over each rafter, and let the whole weight be denoted by W.
It is evident that one-half of the load on the rafter C F will
be supported by the joint B and one-half by the ujDper joint ;
the same will be true for the rafter D F ; therefore the joint
B will carr}' ^ W, the upper joint ^ W, and the joint at E ^^ W.
The additional stress produced in C F by the bending action
of the load which it carries is not considered at this time, but
must be noticed and allowed for separately. (See Chap. IX.)
Taking the external forces in order from right to left over the
roof, lay off ed, or |-W, vertically, to represent the weight
E D acting downward at the joint E, nest d c equal to |- W,
for the weight D C, and lastly c b for the weight at B. Call
eb the load line.
Let the two reactions or supporting forces for the present
be considered as a little inclined from the vertical, as shown
by the arrows B A and A E. Since the truss is symmetrical
and symmetrically loaded, the resultant of the load must j)ass
through the apex of the roof, and, as the two supporting forces
must meet this resultant at one point, the two reactions must
be equally inclined. Then, to complete the polygon of ex-
ternal forces : — as we have drawn ed, dc, and c 6 in order,
passing over the frame to the left, — draw next b a, ujd from the
extremity b of the load line, and parallel to the upward reac-
tion B A ; and lastly a line a e, parallel to the other reaction
A E, to close on e, the point of beginning.
12. Triangular Truss : Stresses. — While in this truss we
might find the stresses at any joint, let us begin at B. Here
7
8 ROOF-TEUSSES.
we have equilibrium under the action of four forces, of which
the two external ones are known. Taking the latter in the
same order as above, and beginning at c (§ 9, 2°), pass over
ch downwards and ha upwards; then draw af parallel to
AF, in such a direction that/c, drawn from /parallel to F C,
will strike c, the point of beginning. Because we passed from
a to/, AF will pull on the joint B, and as we then passed
from/ to c, F C will exert a thrust on B. (It is usual to draw
a/ from a and /c from c till they meet at/; but to determine
the kind of stress, one must pass over the lines in the direc-
tions noted.)
Passing next to the apex of the roof, and again taking the
forces in the same order, pass down the line dc for the ex-
ternal force, thence up to / for the thrust c/ and finally
draw fd parallel to F D, thus determining the thrust of that
rafter against the top joint. If this line does not close on d,
the drawing has not been made with care. As all the stresses
are now found we need not examine the remaining joint. It
may again be noted that we pass over a stress line in one
direction when we analyze the stresses at the joint at one end
of the piece to which the line is parallel, and in the reverse
direction when we consider the joint at the other end of the
same j^iece.
13. Effect of Inclined Reactions. — If the supporting
forces had been more inclined from the vertical, the point a,
of their meeting in the stress diagram, would have been nearer
/, thus diminishing the tension in A F, but not affecting the
compression in the rafters. The inclination might be so much
increased that o, would fall on / when the piece A F would
have no stress, the thrust of the rafters being balanced with-
out it. If a fell to the right of / af would be a thrust.
14. Triangular Truss : Vertical Reactions.— If the two
reactions are vertical, as will be the case when the roof truss
is simply placed upon the wall, B A and A E, Fig. 6, will each
be -1^ W, and the point a will therefore be found at the middle
of e h. The polygon of external forces has closed up and be-
ROOF-TRUSSES. 9
come a straight line, but in the analysis it must still be used.
Thus we pass down ed-\-dc-^cb for the weights at the joints
and back over ba-\-ae for the reactions. The rest of the
diagram follows from § 12.
The diagrams which the reader draws may be inked in black
and red, one denoting compression, the other tension, or the
two kinds of stress may be indicated by the signs -j- and — .
15. King-post Truss. — In the truss of Fig. 7 the rafters are
supported at points midway between their extremities. Each
point of junction of two or more pieces is considered a joint
around which the pieces would be free to turn were they not
restrained by their connections with other points. Whatever
stiffness the joint may possess from friction between its parts,
or from the continuity of a piece, such as a rafter, through the
joint, is not taken into account, and may add somewhat to the
strength of the truss.
In this example, therefore, half of the uniform load on C L
will be carried at B, and be represented by the arrow B C ; the
other half together with half of the uniform load on D K will
make the force C D, and so on, three of the joints carrying
each one-quarter of the whole load, and the two extreme ones
one-eighth each.
On a vertical line lay off gf=^Vl,fe = ed = dc = ^W
and c & = |- W ; then ba = ag = ^'W, the two supporting forces.
In the order shown by the arrow, for the joint B we have c b
external load, b a reaction ; then draw a I, tension, § 9, 3°, par-
allel to AL and Ic, compression, parallel to LC. At the
joint C D the unknown forces now are those in L K and K D.
Begin with the load dc, following with cl, the stress just
found in C L ; then draw I Jc, compression, parallel to L K,
and k d, compression, parallel to K D, to close on d. Passing
next to the joint D E, ed is the load, d k the thrust of D K on
this joint, k i the tension in K I,* and i e, to close on e, is the
compression in I E. Take next the joint in the middle of the
* It will be seen that K I is a tension member or tie, and not a post as would
be inferred from the name given to this truss by old builders.
10 EOOF-TRTJSSES.
lower tie ; we know i h, k I, and I a ; the next stress lies in
AH; as we have just arrived at a from I, we must pass back
horizontally until a line from h parallel to H I \d\\ close on i, the
point from which we started. The remaining line lif is easily-
determined by taking either the joint E F or the one at G.
It will be noticed that, since the truss is symmetrically
made and loaded, the stress diagram is symmetrical ; k i must
be bisected by « ? ; dk and e i must intersect on a I. Atten-
tion to such points ensures the accuracy of the drawing.
A truss, Fig. 8, is now submitted, which the reader is advised
to analyze for himself, as a test whether the principles thus far
explained are clearly understood.
16. Wooden Truss with Frequent Joints. — The truss
represented by Fig. 9, a simple extension of Fig. 7, is one well
adapted for construction in timber, the verticals alone being
made of iron. It can be used for roofs of large span. In any
actual case, before beginning to draw the diagram, assume an
approximate value for the weight of the truss, add so much of
the weight of the purlins, small rafters, boards and slates, or
other covering, as is supported by one truss, and divide this
total weight by the number of equal parts, such as D I or E L,
in the two rafters. We thus obtain the weight which is sup-
posed to act at each joint where two pieces of the rafter meet.
The weight at each abutment joint will be half as much. If
the rafter is not supported at equidistant points, divide the
total load by the combined length of both rafters, to obtain the
load per foot of rafter, and then multiply the load per foot by
the distance from the middle of one piece of the rafter to the
middle of the next, to obtain the load on the joint which
connects them. Numerical values will be introduced in later
chapters.
Draw the vertical load line equal to the total weight, and
beginning with 6 c as the load on B from one-half of C H,
space off the weights cd, de, etc., in succession, closing at p
with a half load as at b. The point of di^dsion «, at the
middle of p b, marks off the two supporting forces p a and a &,
EOOF-TRUSSES. 11
which close the polygon of external forces. Beginning now
at B, draw, as heretofore directed, § 9, abcha for this joint.
The order of these letters gives the directions of the forces on
the joint B. Then for the joint C D we have h c d i h ; for H K
we have a h i k a ; for D E we have k i d e I k, etc. Observe
that, by taking the joints in this order, first the one on the
rafter, and then the one below it on the tie, we have in each
case only two unknown forces, out of, at some joints, five
forces. We repeat, also, the remark that it is expedient, when
possible, first to pass over all the known forces at any joint,
taking them in the order observed with the external forces
when laying off the load line. The rest of the diagram pre-
sents no difficulty.
After the stress in N O is obtained, the diagram will begin
to repeat itself inversely, the stress in O G being equal to that
in F N. It is therefore unnecessary to draw more than one-
half of this figure, except for a check on the accuracy of the
drawing by the intersections which are seen on inspection of
this diagram. Noting the stresses found in the several poly-
gons, we see that all the inclined pieces are in compression,
while the horizontal and vertical members are in tension.
17. Superfluous Pieces. — Sometimes a vertical rod is in-
troduced in the first and last triangles, where dotted lines are
drawn. It is e%'ident that this rod will be of no service if all
the load is assumed to be concentrated on the joints of the
rafters, and this fact can be determined from the stress dia-
gram as well. Thus, taking the joint below H, Fig. 9, we
have three forces in equilibrium ; begin at a in the stress dia-
gram and pass to h along the line already found for A H ; then
we are required to draw a vertical line from h and, from its
extremity, a horizontal line to close on the point a from which
we started ; the vertical line therefore can have no length.
All that this vertical rod can do is to keep the horizontal tie
from sagging, by sustaining whatever small weight is found
at its foot.
Therefore, whenever there are at a joint but three pieces or
12 ROOF-TRUSSES.
lines along wliich forces can act, and two of these pieces lie
in one straight line, it follows from the above that the third
piece must be without stress, and that the first two pieces or
lines will have the same stress. Thus, L K of Fig. 7 and
H I of Fig. 9 would have no compression if the external load
C D were removed. This fact will often prove of ser^ace in
analysis.
18. Problem. — Draw the stress diagram for the truss illus-
trated by Fig. 10, which is supported on a shoulder at the
wall and by an overhead tie running from the right end. It
will be convenient to imagine that tie replaced by the inclined
reaction shown by the arrow at the right, as thus the reaction
is kept on the right of the load at that joint. The reaction at
the wall will cut the tie where the resultant of the load cuts
it ; if the load is uniform over the rafter, that intersection is
at the middle of the tie.
Next, try this problem with the two inclined diagonals
reversed, so as to slant up to the right. Notice the upper
left-hand joint. Compare the two cases, as to difference in
magnitude and kind of stress.
19. Joints where three Forces are Unknown. — It ap-
pears impracticable to determine the stresses at any joint where
more than two forces are unknown. In Fig. 9, we could not
start with the joint C D or at D E ; for we should know only
the external force or load, and have three unknown stresses to
find ; therefore our quadrilateral, of wliich one side is known,
might have the other sides of various lengths, but still parallel
to the original pieces of the frame. When the joints were
taken in the order observed this difficulty was not met with.
When, in some cases, we find three or more apparently un-
known forces at a joint we may have some knowledge of the
proportion which exists between one or more of them and a
known force, and can thus determine the proper length of the
line in the stress diagram. An example of such a case will be
given in Fig. 11. In Chapter YIII. will be found a treatment
KOOF-TRUSSES. 13
that is applicable to certain trusses which otherwise offer diffi-
culties in solution.
20. Polonceau or Fink Truss. — Fig. 11 shows a truss
which is often built in iron. The loads at the several joints
of the rafters are found by the method prescribed in § 16.
It will be unnecessary to dwell ujjon the manner of finding
the stresses at the joints B, C D, and H K, for which the
stresses will be ch, It a, nk, ki, hi and id. But when we
attempt to analyze the joint D E, we find that, with the ex-
ternal load, we have six forces in equilibrium, of which those
along E M, M L, and L K are unknown. If we try the joint
L A we find four forces, three of which are at present unknown.
We are therefore obliged to seek some other way of determin-
ing one of the stresses.
It will be seen, upon inspection, that the joint E F is like
the joint C D ; and it will appear reasonable that X M should
have an equal stress with I H. We may then expect that
there must be as much and the same kind of stress exerted by
M L to keep the foot of the strut N M from moving laterally
as is found necessary in K I to restrain the foot of I H.
Returning then to the joint D E, and beginning with k iy
pass next over i d, then d e, then draw e m, parallel to E M, to
such a point m, that (ha^-ing drawn m I until its extremity I
comes in the middle of what will be the space between e m
and fn, or until m I equals in leng-th i k), the line I k shall close
on k whence we started. The ties and struts can be readily
selected by the direction of movement over these lines in
reference to the joint D E. The remaining joints when taken
in the usual order of succession offer no difiiculty, and the
other half of the diagram need not be added, unless one de-
sires a check on the results.
This truss will be treated again in § 7-4.
The polygon which we have just traced, kidemlk, affords
a good illustration of the rule that the forces which meet at a
joint make a closed polygon in the stress diagram. The sym-
metry of the triangles hik and mnl, and their resemblance to
14 EOOF-TRUSSES.
k 1 0, are wortli noting, and will assist one in drawing diagrams
for trusses of this type.
21. Cambering the Lower Tie. — Sometimes it is tliought
desirable to raise the tie A O, either to give more height be-
low the truss or to improve its appearance. The effect on the
stresses of such an alteration is very readily traced, and one
then can judge how much change it is exj)edieut to make.
Let it be proposed to raise the portion A O of the tie to the
position indicated by the dotted line, and thus to introduce
such changes in the other members that they shall coincide
with the other dotted lines in Fig. 11, while the load remains
unchanged.
The line c li for joint B now becomes cli', being prolonged
until h' a can be drawn parallel to H A in its new position.
Next come h' i' and i' d; then we easily draw i'k', h'V, I'm',
m' n', etc. The struts H I, K L, and M N are the only pieces
in this half of the truss unaffected by the change ; the amount
of increase, and the serious increase, of the other stresses for
any considerable elevation of the lower member can be readily
seen.
22. Load on all Joints. — If one prefers to consider that
a portion of the weight of the truss, or that a floor, ceiling
or other load is supported at the lower joints, the load
may be distributed as in Fig. 12. Here the joints Q R and
R S carry their share of the weight of the pieces which touch
these joints, as well as such other load as may properly be
put there. Each supporting force, if the load is symmetrical,
will still be one-half the total load, but the two will no longer
divide the load line equally, nor can the load line be at once
measured off as equal to the total weight.
Begin, if convenient, with the extremity H of the truss, and
lay off hi, ik, kl, etc., downwards, ending with op. Passing
on, around the truss, lay off next the reaction p q upwards,
equal to one-half the total weight, then q r and r s downwards,
and finally s h upwards, for the other supporting force, to close
on h. The polygon of external forces, therefore, doubles back
EOOF-TRUSSES. 15
on itself as it were, and lip is still the load on the exterior of
the roof. The diagram can now be drawn, by taking three
joints on the rafter in succession before trying the joint Q II ;
when taking that joint remember that there is a load upon it.
The loads on the horizontal tie cause the stresses in its three
parts to be drawn as three separate lines, instead of being
superimposed as in the figures before given.
A diagram may now be drawn for Fig. 13. The upper part
of the roof, dotted in the figure, throws its load, through the
small rafters, on the upper joints of the truss.
23. Stresses by Calculation — It is evident, from insi:)ection of the pre-
ceding diagrams, that the stresses may be calculated by means of the
known inclinations of the parts of the trusses. The degree of accuracy
with which they can be scaled equals, however, if it does not exceed the
approximation which designing and actual construction make to the theo-
retical structure.
24. Distribution of Load on the Joints. — In Unwin's " Iron Bridges and
Roofs" the rafter is treated as a beam continuous over three or more
supports, and the distribution of the load on the several joints is there
determined by that hypothesis. That such an analysis may be true, it is
necessary that all the points at which the rafter is supported shall remain
in definite positions, usually a straight line. As slight deformations
of the truss and unequal loading of the joints will prevent the realiza-
tion of that assumption, a division of the load at any point of a rafter or
other piece so that the joints at its two ends shall be loaded in the inverse
ratio of the two segments into which the point divides the piece will best
represent the case. Uniform loads will be distributed easily by § 16. A
different distribution of the load, however, if one prefers it, will only re-
quire a corresponding division of the load line. (See Part II., Bridge
Trusses, Chaps. VIII. and IX.)
w&fAKTMENT OF CIVIL ENGINEE^"|MU
CHAPTEK III.
TRUSSES FOR FLAT ROOFS.
25. Trapezoidal Truss; Equal Loads. — A consideration
of tlie trapezoidal, or queen-post, truss, rej^resented by Fig. 14,
will bring out two or three points whicb will be of use in the
analysis of other trusses. In this case, let us suppose the
load to be on the lower part, or bottom chord, of the truss.
In order to separate the supporting forces from the small
weights on the ends of the truss, and to permit them to come
consecutively with the other weights in the load line, let us
draw the supporting forces above the tie, instead of below as
before. The rectangle formed by the two vertical and two
horizontal pieces might become distorted ; we will therefore
introduce the brace H I, represented by the full line. The
rectangle is thus divided into two triangles and movement pre-
vented. The dotted line shows a piece which might have been
introduced in place of the other.
If the truss is symmetrically loaded, or C D = D E, we shall
get the first stress diagram. The stress in each vertical is
here seen to be the load at its foot. The stress in the piece
H I proves to be zero. If the load had been on the upper
joints, no stress would have been found in the verticals also.
(See § 17.) It is evident that a trapezoidal truss, when sym-
metrically loaded, requires no interior bracing. This fact
might readily be seen if we considered the form assumed by a
cord, suspended from two points on a level, and carrying two
equal weights symmetrically placed.
26. Trapezoidal Truss; Unequal Loads. — The second
stress diagram will be drawn when the weight C D is less
than D E. Let us suppose that b c and ef are of the same
16
EOOF-TRUSSES. 17
magnitude as in the first diagram, and let the span of the
truss, or distance between supports, which we shall denote by
Z, be di\dded by the joints into three equal parts. The first
step is to find the suj^porting forces. If each external force
be multiplied by the perpendicular distance of its line of ac-
tion from any one assumed point, which distance may be called
its leverage, and all the products added together, those which
tend to produce rotation about this point in one direction
being called plus, and those tending the other way minus, it
is necessary for equilibrium that the sum of these products
shall be zero ; otherwise the rotation can take place. A con-
venient point to which to measure the distances will be one of
the points of support, for instance the right-hand one. Then
we shall have
Ar.Z-rE.Z-ED.tZ-DC.iZ-CB.O + BA.O = 0,
or
AF.Z = rE.Z + ED.tZ + DC.i?;
therefore
AF = FE + tED+iDO.
If E D be taken as 3 D C,
AF = FE + |ED.
It will be seen that the object in taking the point or axis at B
is to eliminate B A, and have only one unknown quantity, A F.
This method of determination is called taking moments, and is at
once the simplest and most generally aj)plicable. Lay off the
above reaction at fa ; a b will be the reaction at the right
support. One cause of a diagram's failure to close, when drawn
by a beginner, is carelessness in placing the reactions on the
load line in the wrong order.
The point a being now located, we can proceed to draw the
second diagram. The construction requires no explanation ;
but we will call attention to the fact that a compressive stress
here exists in H I. If, in place of the diagonal represented
by the full line, the one shown by the dotted line is now sup-
plied, the reader can without difficulty trace out for himself
18 KOOF-TRUSSES.
the change in the diagram, which is denoted by the dotted
lines and the letters marked by accents. The stress in this
diagonal will be seen to be tensile. Changing the diagonal
reverses its stress.
It is also worthy of notice that the only pieces affected by
the substitution of one diagonal for the other are those which
form the quadrilateral enclosing the diagonals. This fact
will be of service later.
27. Use of Two Diagonals.— If, at another time, this ex-
cess of load might fall on C D in place of D E, the stress on
either diagonal would be reversed : that is, if it sloped down
to the right it would be a tie ; if to the left, a strut. As a ten-
sion diagonal is likely to be a slender iron rod, which is of no
practical value to resist a thrust, while the compression mem-
ber, unless made fast at its extremities, will not transmit ten-
sion, a weight or force which may be shifted from one joint to
another may require the designer to introduce two diagonals
in the same rectangle or trapezium, or else to so proportion
and fasten one diagonal as to withstand either kind of stress.
Where both diagonals occur the diagram can still be drawn.
Determine which kind of stress, tension or compression, the
two shall be designed to resist, and then, when drawing the
diagram, upon arriving at a particular panel or quadrilateral,
try to proceed as if only one of the diagonals existed. If a
contrary kind of stress to the one desired is found to be
needed, erase the lines for this panel only, and take the other
diagonal. In the treatment for wind pressure, this method
becomes serviceable, since the wind may blow on either side of
the roof.
This truss can be used for a bridge of short span.
28. Trusses for Halls. — It is sometimes the case that, in
covering a large building, it is desired to have the interior
clear from columns or partitions, while a roof of very slight
pitch is all that is needed. As it is not expedient to have a
truss of much depth, since the space occupied by it is not
generally available for other purposes, one of several types of
ROOF-TRUSSES. 19
parallel-chord bridge trusses may be employed, for instance
the " Warren Girder," of Fig. 15, which is an assemblage of
isosceles triangles. In a j^ublic hall, galleries may be sus-
pended from the roof, and the weight of a heavy panelled or
otherwise ornamented ceiling may be added to what the truss
is ordinarily expected to carry. The depth ma}- be less than
here drawn, but, for clearness of figure, we have not made the
truss shallow.
If the roof pitches both ways from the middle of the span,
the top chord may conform to the slope, making the truss
deeper at the middle than at the ends ; l)ut a light frame may
be placed above, as shown by the dotted lines, and supported
at each joint of the top chord. The straight-chord truss is
more easily framed. If the roof pitches slightly transversely
to the trusses, it will be convenient to make them all of the
same depth and put on some upper works to give the proper
slope. The ends of the truss could readily be adapted to a
mansard roof.
29. Warren Girder. — In Fig. 15, each top joint is sup-
posed to be loaded with the weight of its share of roof, in
which case the joint LM or PQ will have three-quarters of
the weight on N O or O P, if the roof is carried out to the
eaves as marked on ihe left ; or practically the same as N O,
if the roof follows the line I L. The bottom joints are sup-
posed to carry the weight of the ceiling, and in addition the
tension of a suspending rod to a gallery on each side. The
load line will be equal to the weight on the upper part of the
truss, and the polygon of external forces will overlap, as in
Fig. 12, previously exj^lained, § 22. We go from k to r, for
the loads on the exterior in sequence, then up to s for the
left-hand reaction, then down to lo for the loads on the
interior, and finally close on k with the right-hand reaction.
Upon drawing the diagram it -^dll be seen that the stress is
compression in the top chord and tension in the bottom chord ;
that the stresses in the chords increase from the supports to
the middle ; that the stresses in the braces decrease from the
20 KOOF-TKUSSES.
ends of the truss to the middle, and that alternate ones are in
compression and in tension, those which slant up from the
abutment towards the centre being compressed, and those
which incline in the other direction being in tension. The
tie-braces are, therefore, A B, C D, F G, and HI. A decrease
of depth in the truss will increase the stresses iu the chords.
30. Howe Truss; Determination of Diagonals. — A
truss with parallel chords may be employed, in which the
braces are alternately vertical and inclined. The designer
will choose whether the verticals shall be ties and the diag-
onals struts, in which case the type is called the "Howe
Truss," Fig. 16, or the verticals struts and the diagonals ties,
when it is known as the "Pratt Truss." There is an advan-
tage in having the struts as short as possible, but, if one
desires to use but little iron, the Howe is a good form.
To decide which diagonal of the rectangle shall be occupied
by the piece : — Start from the wall as a fixed point ; it is evi-
dent that, to keep the load C D from sinking, C Q must be a
strut. If we wish to put a tie in this panel, it must lie in the
other diagonal, shown by the dotted line. CD now being
held in place, P O as a strut will uphold D E. We thus may
work out from each wall until we have passed as much load
as equals the amount supported, or the reaction, at that wall.
If the last load passed exactly comj)letes the amount required
to equal the reaction, no diagonal will be required in the next
panel. We might draw diagonals, one in each panel, sloping
in either direction as we pleased, and then construct the stress
diagram. If we found a stress in any diagonal opposite to
the stress we desired, § 27, we could then erase that diagonal
and substitute the other, erasing also so much of the diagram
as referred to the pieces in that panel. Were the chords not
parallel, this method might be necessary (see Fig. 20), but in
the present case it is better to draw the load line fi.rst, find the
dividing point ff, Fig. 16, for the two reactions, see what load
it cuts, and then incline the diagonals from each wall either
up or down, as preferred, towards that loaded joint.
ROOF-TRUSSES. 21
31. Howe Truss ; Diagram. — In the present example C D
is supj)osed to be four times D E, etc. A tower on that end
of the truss or some suspended load will account for the dif-
ference. Eecalling the manner in which the supporting forces
were found when the load was unsymmetrical, § 2G, use a
panel as a unit of distance, call a panel length p and the ordi-
nary weight on a joint lo. Then we shall have, taking moments
about H,
w .2)0- +[2 + 3) + 4 ?^ . 4^? + I- li; . 0J5 = R . o^^, or R = 4.9 w,
the reaction at B, or a h. The two supporting forces will then
be A a and a h. Draw the stress diagram as usual ; the di-
agonals will all come in compression as intended, and the
verticals will be ties. There will plainly be no stress in the
dotted vertical O N. The stress in the chords is inversely
proj)ortional to the depth of the truss, and economy of ma-
terial in the chords will be served by making the depth as
much as j)ossible, within reasonable limits. In bridge trusses
this depth is seldom less than from one-sixth to one-eighth of
the span.
32. Moving Load. — If the joint D E also might become
hea^dly loaded, we could draw another diagram for that case,
and, as the joints in succession had their loads increased, we
might make as many diagrams. From a collection of dia-
grams for all positions of a mo^ang load, we could select the
maximum stress for each piece. A truss designed to resist
such stresses would answer for a bridge. We should find that
the greatest stresses in the chords occurred in all panels when
the bridge was hea^dly loaded throughout, and that the great-
est stress in a diagonal was found when the bridge was heaAdly
loaded from this piece to one end only, that end generally
being the more distant one. As we have more expeditious
methods of analyzing a bridge truss, this one is not used.
The graphical treatment of bridge trusses is found in Part XL
of this work.
CHAPTEE ly.
WIND PRESSUEE ON PITCHED ROOFS.
33. Action of Wind. — The forces liitlierto considered have
been vertical ; the wind jDressure on a roof is inclined. It was
once usual to deal with the pressure of the wind as a vertical
load, added to the weight of the roof, snow, etc., and the
stresses were obtained for the aggregate pressure. This treat-
ment manifestly cannot be correct. The wind may be taken
without error as blowing in a horizontal direction ; it exerts its
greatest pressure when blowing in a direction at right angles
to the side of a building ; it consequently acts ujDon but one
side of the roof, loads the truss unsymmetrically, and some-
times causes stresses of an opposite kind, in parts of the
frame, from those due to the steady load. Braces which are
inactive under the latter weight may therefore be necessary
to resist the force of the wind.
It will not be right to design the roof to sustain the whole
force of the wind, considered as horizontal ; nor wdll it be cor-
rect to decompose this horizontal force into two rectangular
components, one perpendicular to the roof, and the other
along its surface, and then take the perpendicular or normal
comj)onent as the one to be considered ; for the pressure of
the wind arises from tlie imj)act of particles of air moving
■with a certain velocity, and these particles are not arrested,
but only de\dated from their former direction upon striking
the roof. Yet the analysis aj^plicable to a jet of water striking
an inclined surface cannot be used here, for water escapes
laterally against the air, a comparatively unresisting medium,
while the wind particles, if we may so term them, deflected by
the roof, are turned off against a stream of similar air, also in
motion, which retards their lateral progress and thus causes
22
EOOF-TRUSSES. 23
them to press more strongly against the roof. We are obliged,
therefore, to have recourse to experiments for our data, and
from them to deduce a formula.
34. Formula for Wind Pressure. — It appears that, for a
given pressure exerted by a horizontal wind current on any
square foot of a vertical plane, the pressure against a plane
inclined to its direction is perj^endicular to the inclined sur-
face, and is greater than the normal component of the given
horizontal pressure. Unwin quotes Hutton's experiments as
showing that, if P equal the horizontal force of the wind on a
square foot of a vertical plane, the perpendicular or norma]
pressure on a square foot of a roof surface inclined at an
angle i to the horizon may be expressed by the empirical
formula
P sin ^■l•**'=°s'■-^
If, then, the maximum force of the wind be taken as 40
pounds on the square foot, representing a velocity of from 80
to 90 miles per hour, the normal pressure per square foot on
surfaces inclined at different angles to the horizon will be :
Angle of
Roof.
Normal
Pressure.
Angle of
Roof.
Normal
Pressure.
5°
5.2 lbs.
35°
30.1 lbs.
10
9.6
40
33.4
15
14.0
45
36.1
20
18.3
50
38.1
25
22.5
55
39.6
30
26.4
■60
40.0
For steeper pitches the pressure may be taken as 40 pounds.
Any component in the plane of the roof, from the friction
of the air as it passes up along the surface, or from pressure
against the biitts of the shingles or slates, is too slight to be
of any consequence.
Duchemin's formula, with the above notation,
P . 2 sin' i ^ (1 + sin^ i),
gives smaller values of normal wind pressure.
35. Example : Steady Load.— The truss of Fig. 17 is
supposed to be under the action of wind pressure from the
24 EOOF-TEUSSES,
left. If the truss is 67 feet span, and the height jj 15 feet, the
angle of inclination will be 2-1° 7', and the normal wind press-
ure, interpolated from the table, will be 21.8 pounds per square
foot. The rafter will be 36.7 feet long. If the trusses are 10
feet apart, the normal wind pressure on one side will be
36.7 X 10 X 21.8 = 8000 lbs.
For steady load of slates, boards, rafters, purlins, and truss,
let us assume 11 pounds per square foot of roof, or
36.7 X 10 X 2 X 11 = 8074 lbs., total vertical load.
The truss is here drawn to a scale of 30 feet to an inch, and
both diagrams are drawn to a scale of 6000 pounds to an inch.
In actual practice these figures should be much larger, the
diagrams showing perhaps 1000 pounds or 800 pounds to an
inch.
We will, in the present case, treat the two kinds of external
force separately. The diagram on the right for steady load
needs no description. Each supporting force will be 4037
pounds, and the weights at the joints of the rafters will be,
673 pounds for the end ones, and 1346 pounds for each of the
others. The above weights are laid off on a vertical load line
and the diagram then drawn. The stresses in the various
pieces for half of the truss are given in the table to follow,
the sign -\- denoting compression, and the sign — , tension.
36. Wind Diagram ; Reactions. — The normal pressure of
8000 pounds distributed uniformly over the whole of the left
side of the roof, and on that alone, will have its resultant, shown
by the dotted arrow, at the middle of that rafter. To find
the supporting force on the right we may take moments about
the left-hand wall, remembering to multiply each force by the
lever arm drawn perpendicular to its direction : or
AP X HT =8000 X HK,
or
AP X 61.15 = 8000 X 18.35;
■whence A P = 2400 pounds, and A H = 5600 pounds.
ROOF-TRUSSES. 25
But since these arms, H T and H K, are proportional to the
span and the left part of the horizontal tie cut off by the re-
sultant, an easier way to get the supporting pressures due to
an inclined force is to prolong this force until it cuts the
horizontal line joining the two abutments, when the two reac-
tions will be inversely proportional to the two segments into
which the horizontal line is thus divided, the larger force
being on the side of the shorter segment, or, for ordinary
pitches, on the side on which the wind blows.
The pressures on the joints will be 2667 pounds each on
IK and KL, and 1333 pounds each on HI and LM, as de-
noted by the arrows. Draw m h by scale, equal to 8000
pounds, so inclined as to be in the direction of the given
forces, that is, perpendicular to the roof ; divide the reactions
of the supports by means of the point a, and lay off the joint
forces in their proper order, m ?, / k, k i and / h. Before going
further be sure that the external forces and the reactions
follow one another in their proper order, down and up the
load line ; for, through heedlessness, the reactions are some-
times interchanged.
37. Wind Diagram; Stresses. — Proceed with the con
struction of the diagram by the usual rules, remembering that
wind alone is being treated. After the joint K L has given
Ikcdel, the joint EA gives eda/e. Taking next the apex
L M, and passing along ml,le and e/, we find that there will
be no line parallel to F G, since g m, parallel to G M, will
exactly close on m, the point of beginning. As no stress passes
through F G, the remainder of the bracing on this side can
experience no stress, and therefore the compression g m affects
the whole of the right-hand rafter while the tension a/is
found in the remainder of the horizontal tie. The stress tri-
angle for the point P will therefore be m g a m. That the
above result is true will be seen if we notice that the piece
Q K, having no wind pressure at its upper end, can, by § 17,
have no stress. Then it follows that ES is now free from
stress, and next SG and lastly GF, all by § 17. Further:
26
ROOF-TRUSSES.
imagine all of the braces in the right half to be removed ; it is
evident that the right rafter is a sufficient support to the joint
L M, conveying to the wall the stress g m which compresses
its upper end, while the tie A F keeps the truss from spread-
ing. If the lower tie or the rafter was not straight, some of
the braces would come into action, as will be seen later.
38. Remarks. — At another time the wind may blow on the
right side. Then the braces on the right will be strained as
those on the left now are, and those on the left will be un-
strained. The wind stresses are j)laced in the third column
of the table. As in this truss they are all of the same kind,
in the respective j^ieces, as those from the steady load, they
are added to give the total or maximum stresses. The force
g m, being smaller than, while it is of the same kind as I e, is of
no consequence ; for, with wind on the right, M G would have
to resist a stress equal to I e.
A combination of the two components of the supporting
forces at each end, as shown in the figure, by either the
parallelogram or triangle of force, will give the direction and
amount of each reaction from the combined load. Wind on
the other side will exactly reverse the amounts and bring
them on the opposite side of the vertical line.
Table of Stresses for Fig. 17.
Piece.
steady Load.
Wind.
Total.
( AB
- 7520 lbs.
10,440 lbs.
17,960 lbs.
Tie ^AD
— 6020
7,160
13,180
/ AF
- 4520
3,900
8,420
EF
-1830
3,990
5,820
Braces i g^
— 1500
3,280
4,780
+ 1230
2,670
3,900
[de
+ 1840
4,000
5.840
(IB
+ 8240
9.530
17.770
Kafter ^ K C
+ 7690
9,530
17,220
(le
+ 5760
6,550
12,310
If the truss is simply placed upon the wall-plates, and
either of the supporting forces makes a greater angle with the
ROOF-TRUSSES. 27
vertical than the angle of repose between the two surfaces,
the truss should be bolted down to the wall ; otherwise there
will be a teudeucy to slide, diminishing the tension in the tie,
perhaps causing compression in that member, and changing
the action of other parts of the truss. This matter will be
treated of further.
If the weight of snow is also to be provided for, it may
readily be done by taking the proper fraction of the stresses
from the steady load and adding them to the above table.
39. Truss with Roller Bearing ; Dimensions and Load.
— We propose, in the example illustrated by Fig. 18, to con-
sider the truss as supported on a rocker or rollers at the end
T, where the small circle is drawn, to allow for the ex2Dansion
and contraction of an iron frame from changes of temperature.
It is therefore plain that the reaction at T must alwavs be
practically vertical. The truss is supposed to be 79 feet 8
inches in span, and 23 feet in height, which gives an angle of
30° with the horizon, and makes the length of rafter 46 feet.
It would be proper usually to support the rafter at more
numerous points; but our diagram would not then be so
clear, with its small scale, from multiplicity of lines, and one
can readily extend the method to a truss of more pieces.
This frame supports 8 feet of roof, and the steady load per
square foot of roof is taken, including everything, as 14
pounds. The total vertical load will then be
14 X 46 X 2 X 8 = 10,304 lbs.,
or 1717 lbs. on each joint except the extreme ones.
We find, from the table of § 34, that the normal pressure
of the wind, for a horizontal force of 40 j^ounds on the square
foot, may be taken as 26.4 pounds per square foot of a roof
surface inclined at an angle of 30°. The total wind pressure,
normal to the roof, will therefore be
26.4 X 46 X 8 = 9715 lbs.,
or 3238 lbs. and 1619 lbs. on the middle and end joints
38 ROOF-TEUSSES.
respectively' of one rafter. The truss is drawn to a scale of
40 feet to an inch, and the diagrams to that of 8000 j)ounds
to an inch.
40. Diagram for Steady Load. — The diagram for steady
load, ha\dng a vertical load line, is the one above the truss,
and a little more than one-half is shown. The only piece at
all troublesome is G F. On arriving in our analysis at the
apex of the roof, or at the middle joint of the lower member,
we find three pieces whose stresses are undetermined : but as
we have reached the middle of the truss, we know that the
diagram will be symmetrical, and therefore that gf will be
bisected b}' a ?. In the case of an unsymmetrical load we can
recommence at the other point of support and close on the
apex. The stresses caused by this load are given in the first
column of figures in the table in § 44, compression being
marked -j-, and tension — . If it is thought necessary to pro-
vide for snow, in addition to the stresses yet to be found for
wind, make another column in the table, of amounts properly
proportioned to those just found.
41. Wind on the Left; Reactions. — Upon turning our
attention to the other diagrams, we shall find that the rollers
at T cause something more than a reversal of diagram, — often
a considerable variation of stress, when the wind is on differ-
ent sides of the roof. Taking the wind as blowing from the
left, we draw the diagram marked W. L. The line qm, 9715
lbs., § 39, is di^dded and lettered as shown for the four loads
at the joints where arrows are drawn. The resultant of the
wind pressure, at the middle point of the rafter, when pro-
longed by the dotted arrow, will divide the horizontal line or
span in the proportion in which the load line should be
divided to give the two parallel reactions, if there were no
rollers at T. This proportion, for a pitch of 30°, is 2 to 1 ; it
locates the point a', and gives ma' = 64:77 lbs., and a' q =
3238 lbs.
But the reaction at T must be vertical, and consequently
only the vertical component of a' q can be found at T, while
ROOF-TRUSSES. 29
the horizontal component of a' q must come, through the lower
member, from the resistance of the other wall. Therefore
draw a' a horizontally and . we shall get a q as the vertical
reaction at T, while ma, to close this triangle of external
forces, must give the direction and amount of the reaction
atM.
42. Verification. — It may, at first sight, strike the reader
that this analysis will not be correct ; for, if only the vertical
component is resisted at T, and if we decompose the resultant
of the wind pressure at O, where it strikes the roof, into two
components, we get results as follows :
Vert. comp. of 9715 lbs., for angle 30° = 8414 lbs.
Hor. " " " " " =4858 lbs.
The vertical from the middle point of the rafter will divide
the span at \ M T. Therefore, amount of vertical component
carried at T = 2103 lbs., and the remainder is supported at M,
with all of the horizontal component. But take next into
account the moment, or the tendency of the horizontal com-
ponent at O to cause the truss to overturn. It naturally
decreases the pressure at M and increases that at T, or, in
other words, the couple formed by the horizontal component
at O and the equal horizontal reaction at M with an arm of
half the height of the truss must be balanced by an opposite
couple, com23osed of a tension at M and an equal compression
at T, with a leverage of the span. Making the computation
of this tension, or compression T, we have
4858 X 11.5 = T X 79f, or T = 702 lbs.
2103 + 702 = 2805 = i of 8414 lbs.
as obtained by the first process.
Still another way to find the supporting forces is to prolong
the resultant until it intersects the vertical through T, then to
draw a line from M to the point of intersection, and finally to
draw ma and qa parallel to the lines from M and T. This
method depends for its truth on the fact that the three external
30
KOOF-TRUSSES.
forces wliicli keep tlie truss in equilibrium, not being parallel
must meet in one point.
43. Diagram for Wind on Left. — Ha^-ing completed the
triangle of external forces, and laid off tlie pressures on tlie
joints, we can readily draw tlie diagram. It will be found, as
in Fig. 17, § 37, that braces on the right experience no stress,
the lines gf and e q closing the polygon which relates to the
joint P Q. If the lower tie were cambered to the joint D C,
we should find a stress from wind in E F and C D, but not in
B C or C E, as explained in § 37.
Upon combining with the inclined reaction m a the steady
load reaction also marked m a, the direction of the resultant
supporting force at M will be found ; and it may be so much
inclined to the vertical that provision against sliding on the
wall-plate at M should be made. The stresses given by this
diagram for wind on the left are found in the table to follow,
in the column marked W. L, It will be seen that all of them
agree in Jiincl with those for steady load.
44 Diagram for Wind on Right. — This diagram is
marked W. Ft. The supporting force at.T, while still vertical,
Table of Stresses
FOR Fig. 18.
Piece.
steady Load.
W. L.
W. R.
fBS
CR
+ 8570 lbs.
5600 lbs.
8480 lbs.
+ 6850
5600
6540
Rafters J ?J9
+ 5700
5600
5880
J-VCtrJ-L^i. O T T3
+ 5700
5880
5600
KO
+ 6850
6540
5600
,LN
+ 8570
8480
5600
LA
- 7440
11400
Tip J H A
^^^ Ida
- 5450
7050
- 5450
4850
2150
BA
— 7440
4850
6480
EC
+ 1720
3800
CE
+ 1520
3300
EF
— 1000
2150
Braces -
FG
- 2300
2500
2500
GI
- 1000
2150
IK
+ 1520
3300
KL
+ 1720
3800
ROOF-TEUSSES. 31
is greater in amount than before. If diagram W. L. has been
already constructed, the reaction at T can be taken as that
portion of the vertical component of the wind pressure not
included in a g of that figure ; that is, aq-\-ta = vertical
component of qm or pt. If this should be the first diagram
drawn, find the supporting forces in one of the three ways
given above. The reaction at M is rightly denoted by ap, for,
when the wind is on the right, there is no external force to
di^dde the space from M to P.
The point a is moved considerably from its place in diagram
W. L., and this change affects the amounts of stress in the
horizontal member, but not in those pieces which bear similar
relations to the two sides of the truss ; in other words, I P and
E Q interchange stresses, etc. In some forms of truss, how-
ever, we find more material changes. In the present example
it happens that the vertical fg strikes the point a, so that ip,
the stress in the rafter, coincides with ap, the reaction at M ;
the wind on the right consequently causes no stress in L A
and H A. The stresses from this diagram are found in the
last column of the table.
45. Remarks. — There is no need to tabulate the stress in
K H, if that in I G is given, nor gh, ii k i is given. Notice
that the joint K G or C F gives a parallelogram in each
diagram, the stress in K I passing to G H without change, so
that the diagonals which cross may be considered and built
as independent pieces. It will be seen on inspection of the
table that the combination of steady load with wind on the
left gives maximum stresses in I P, K O, L N, L A, HA, T> A,
G I, IK, and K L, while the remainder, with the exception of
F G, have maximum stresses for wind on the right. F G is
strained alike in both cases.
These wind diagrams may be drawn on either side of the
line of wind force, as in the case of steady load, by changing
the order in which the supporting forces are taken, going
round the truss and joints in the opposite direction.
Although there exist two four-sided spaces C and K, the
32 KOOF-TRUSSES.
structure is sufficiently braced against distortion ; for these
spaces are surrounded by triangles on all sides but one.
It may perhaps not be amiss to suggest again how to deter-
mine the kind of stress in any member without retracing the
whole polygon for any joint. Notice, from the load line,
whether the forces were taken in right-hand or left-hand rota-
tion. Read the letters of a piece in that order with reference
to the joint at one end of it ; then read the stress in the
diagram in that same order, and it will show the direction of
the stress in the piece, either to or from that joint. Thus
diagram W. L. is written in left-hand rotation ; K L is then
the reading for that brace at its loioer end, and k I reads down-
ward or is thrust. If we read L K, it must apply to its upper
end, and I k acts upwards or thrusts against the joint near N.
Wind diagrams for the truss of Fig. 21 can now be drawn.
The apex of the roof can be treated first, and the stresses,
obtained in the dotted lines, can then be transferred to the
ends of the upper horizontal member. The truss proper goes
no higher.
CHAPTER V.
WIND PRESSURE ON CURB (OR MANSARD) AND CURVED ROOFS.
46. Truss for Curb Roof; Steady Load Diagram. — To
have a definite problem we will assume that the truss of
Fig. 19, drawn to scale of 20 feet to an inch, is 50 feet in span,
that the height to ridge is 20 feet, to hi23S 14| feet, and that
C D is 14 feet. The sides K B and G E are practically 16f
feet long, at an angle of 60° with the horizon, so that their
horizontal projection is 8^ feet. The upper rafters are 17^
feet long, and therefore make an angle with the horizon of
18° 19'. The trusses are assumed to be 8 feet apart, and are
loaded at the joints only. The rafters in a larger truss would
commonly be supported at intermediate points ; but more
lines would make our diagrams less plain.
The steady load is taken at 12 pounds per square foot of
roof surface, or
(2 X 16t + 3 X 17j)13 X 8 = 6560 lbs., total load.
The joint L will carry one-half the load on KB, or 800
pounds ; the joint I K will carry one-half the load on K B and
one-half of that on I C, or 800 + 840 = 1640 pounds ; IH = 840
-|- 840 = 1680 pounds, etc. These weights are laid off, in the
diagram marked S. L., from I to / by a scale of 4000 pounds
to an inch, and the diagram is drawn. It shows that the
rafters are in compression, marked -\-, and all the braces in
tension, marked — .
47. Snow Diagram. — In treating this truss for snow load,
it is considered that K B and E G are too steep for any weight
of snow to accumulate there, as whatever fell on them would
33
34 KOOF-TRUSSES.
soon slide off. Therefore a weiglit of 12 pounds per horizon-
tal square foot, for the upper rafters only, is taken for the
maximum snow load, and, as the horizontal projection of
I C + D H is 331 feet, that load will be
12 X 33i X 8 = 3200 lbs.,
laid off from k to g, in the diagram marked S. The end por-
tions, Jc i and h g, are each 800 pounds, and i h is 1600 pounds.
The division into two equal reactions at the points of support
gives a. This diagram much resembles the other, but there
is one point worth noticing; the lines of stress, ic and Jid,
cross in the first diagram, but do not in the second ; while the
reverse is the case with ed and be. The result is that the
stress of C D is reversed by the maximum snow load, and, as
this stress is greater in amount than the one for the weight of
roof and truss, C D will be a compression member whenever
such a load of snow falls on the roof ; and will be in tension
when that load is removed. The stresses from these two
diagrams are marked on the truss above each piece on its left
with the usual signs. This strain sheet is more convenient
than the table of § 44.
48. Wind from the Left ; No Roller. — When the rafters
do not slope directly from the ridge to the eaves, but are
broken into tAvo or more planes of descent, we shall have
wind pressures of different directions and intensities on the
two portions, I C and K B. From the table of wind pressures,
§ 34, we see that the intensity of pressure on K B will be 40
pounds, and on I C 16.9 pounds, normally, per square foot of
roof. The total pressure on KB therefore will be 40 X 16f
X 8 = 5333 pounds, of which one-half will be supported at
the joint L, and the other half at the joint J, as indicated by
the two arrows perpendicular to K B. The pressure on I C
will be 16.9 X 17^ X 8 = 2366 pounds, or 1183 pounds on
each joint.
If the truss has no rollers under it, the diagram marked
W. L., I. is obtained. On a scale of 4000 pounds to an inch,
ROOF-TRUSSES. 35
hi = ij = 1183 pounds ; j k = kl = 2667 pounds. For ij and
jk may be substituted ik,ii desired, the resultant of these
two components at J.
To find the supporting forces : — Prolong the resultants of
the wind pressure from the middle point of each rafter to
intersect the span L F. The resultant K will be resisted at
L and F by two reactions parallel to it, and inversely propor-
tional to the two segments into which this resultant divides
L F, as shown in § 36. The same will be true for the result-
ant I. By scale, or from the known angles, it will be found
that resultant K cuts L F at 16f feet, or one-third the span,
from L, and that resultant I cuts it at 22.4: feet from the same
end. Dividing jl at ^ its length, we have la' for one com-
ponent of the reaction at L and a'j for one component of the
22 4
reaction at F. If we divide lij at — -'- of its length, y a" will
oU
be a component of the supporting force at L, and a"li at F.
By drawing the parallelogram a'j a" a we shall bring the com-
ponent reactions for each wall together, and shall have, for
the supporting force at L, or LA, la' and a' a, or their result-
ant I a ; and for that at F, a a" and a"h, which combined give
a Jl, properly called A H in the truss, since the letters from F
to H are not in use at present. Take care to lay off the com-
ponent reactions on the proper ends of the wind-pressure
lines.
The polygon of external forces, when there is no roller
under the truss, is therefore h i, i k, kl,l a, and a h. The com-
pletion of the diagram, by drawing lines parallel to the several
pieces, will be easy without further explanation. That the
point e should apparently fall on i k is accidental. The signs
affixed to the lines will enable one to see readily that the
stresses in B C and E A are now reversed, the pressure I K
obliging us to use a strut to keep that joint in place. The
resultant, however, from the combined stresses in E A is still
tension. The amounts given by diagram W. L., I. have not
been placed on the truss, as we prefer to treat it from another
36 EOOF-TRUSSES.
point of view. Had tliey been used, it would be unnecessary
to draw a diagram for wind on the right, for the different
members of the truss would exchange stresses symmetrically ;
that is, AB would have the stress of EA, and E A that of
A B ; D H of C I, etc., C D remaining the same.
49. Wind from the Left; Roller at Left. — If rollers
are placed at L, to permit of movement resulting from change
of temperature, the supporting forces will be modified, LA
becoming vertical. The diagram marked W. L., II. shows the
effect of this change. So far as drawing the lines of wind
pressure liijkl, the polygon of external forces will be
obtained in the same manner as before. We may then draw
the parallelogram and locate the point here marked a' ; then
draw a' a horizontally, and we shall get I a, the vertical reac-
tion at L, equal to the vertical component of Za of the figure
just j)receding.
In case the former diagram has not been drawn, a readier
way to determine I a will be as follows : — Draw li I, plainly the
resultant of hj and jl; then, having prolonged the dotted
arrows at I and K until they meet, draw a line, parallel to hi,
through their intersection. This line will give the position
of the resultant of the wind pressures, and I h is now to be
divided in the inverse ratio of the two segments into which the
resultant divides the span LF. The point of division will
fall at a", from which draw horizontally a"a, and the reac-
tion I a is thus determined. This method will not answer for
finding the supporting forces if they are both inclined, as it
will make L A and A H parallel to one another. The reac-
tion at L being I a, the one at F is a h, requiring the resistance
at F of the entire horizontal component of the wind pressure.
A comparison of the two W. L. diagrams will show that the
stress in every piece is changed very decidedly in amount,
and that in a number of pieces the stresses are reversed by
rollers at L. These latter stresses are marked on the truss,
at the right of each piece.
ROOF-TEUSSES. 37
50. Wind from the Right. — When the wind blows from
the right, the diagram marked W. K. will be obtained. The
lines ihgf, representing the wind j^ressures, will correspond
in value with hikl of the preceding figure, and, since the other
diagram has been constructed, the vertical reaction at L will
now be obtained by drawing the horizontal line a' a, from either
the angle of the parallelogram or the j^roper point of division
of the resultant if, so as to give a i, the smaller part of the
vertical component of the wind pressure ; that, is I a from W.
L., IL, plus a {from W. K., equals the vertical projection of
the polygon of external forces.
51. Results. — When this diagram is completed by the
customary rules, a comj^arison of it with the one preceding
will make clear the effect of wind on different sides. The.
stress in the rafters is much greater when the wind blows on
the side farther from the rollers, but it is always comj)ressive.
The forces in the braces are all reversed.
The weight of the roof and truss may be the only external
force, or snow may be added ; and, in either case, the wind
may also blow on one side or the other. Selecting then those
stresses which may exist together, we find the maximum tension
and compression marked below each piece. The rafters
are always compressed, and A B is always in tension. The
other pieces must be designed to resist both kinds of stress,
although the compression in D E is quite insignificant.
52. Curved Roof Truss : Example. — If the truss has a
curved exterior outline, the pressure of the wind will make a
different angle wdth the horizon for every point. But there
will be no sensible error if the pressure on each piece is as-
sumed to be normal to the curve at its middle point, or, what
is practically the same thing, perpendicular to the straight
line joining its two extremities. Thus, in the truss of Fig. 20,
the wind pressure on C T is taken as perpendicular to a
straight line from B to the next joint in the rafter.
The span of this truss, drawn on a scale of 30 feet to an
inch, is 60 feet ; height at middle of rafters 15 feet, at middle
38 EOOF-TRUSSES.
of main tie 6 feet. The curves are arcs of circles, the radii of
the uj)per and lower members being respectively 37^ feet and
78 feet. The rafters are spaced off at intervals of 11^ feet
each way from the middle, and the tie is di^ided into 10^ feet
lengths. The end portions will diifer slightly from these
measures. The trusses are to be 10 feet apart. From the
data, radius 37^ feet, and half-chord or sine 5| feet, it is easy
to calculate that the chord of the first piece of rafter from the
middle will make an angle with the horizon of 8° 49^'. The
second piece will be inclined three times as much, or 26° 28',
and the last five times as much, or 44° 6'. The intensity of
normal wind pressure will then be, when interpolated in the
table, § 34, 8.6 pounds per square foot for the upper length,
23.7 pounds for the next length, and 35.6 pounds for the low-
est piece. Multiplying these intensities by 11|- X 10, we get
989 ]30unds, 2725 pounds, and 4094 pounds, respectively, repre-
sented by the small arrows, as if concentrated at the middle
points of E, D, and C. The steady load is taken at a small
figure, 2300 pounds per piece of rafter, to allow the disturbing
effect of the wind to be more marked.
The diagonals in this truss are light iron rods, not adapted
to resist compression, and therefore, if a compressive stress
would occur in a particular diagonal, in case it were alone in a
panel, we substitute the other diagonal, which will then be in
tension. In lettering the figure, that tie which is required for
a particular distribution of load is supposed to be present,
and the other diagonal is not taken account of. Thus, in the
panel through which the dotted arrow is drawn, if the brace
which goes from the top of O P to the bottom of Q R is under
stress, it will be called P Q, while the rafter will be Q E and
the bottom tie PA. If the other diagonal is strained, the
rafter will be called P E and the main tie Q A.
53. Steady-Load Diagram. — The diagram for weight of
roof and truss is drawn on a scale of 8000 pounds to an inch.
The vertical load line is i b, and the polygon for the point of
support ^ is cb ate. On passing to the next joint in the top
ROOF-TRUSSES. 39
or bottom member we find three pieces "wliose stresses are
unknown. Both diagonals R S cannot be in action as ties at
once ; therefore suppress one, for instance that which runs to
the upper end of S T. We then shall have only two unknown
stresses at the upper joint, and can draw t s' and s' d. The
lower joint will then give s't, ta, ar', and r's'. But r's' will
be a compressive stress, as we read from r' to s\ and this
diagonal is not the desired one. Taking the other, and trjdng
the lower joint first, we have t ast, and the uj^per joint then
gives dctsrd, where sr is tension. Notice that change of
diagonal aflfects the stresses in no pieces beyond those which
bound the quadrilateral or panel in which the diagonal is
changed. Analogy will rightly lead us to take the other diag-
onals which slope the same way, that is, down towards the
middle. It is therefore easy, after the first attempt, to decide
which diagonal to reject and which to retain.
54. Remarks. — If d r had been slightly more inclined, so as
to strike s, no diagonal B S would have been reqiiired for this
distribution of load. It will be seen that the stresses, all tensile,
in the bracing are very small as compared with those in the
main members, a fact due to the approximation of the rafter
outline to the equilibrium curve or polygon for a load dis-
tributed as in this case. See § 88. If the outline of a truss
coincides with the equilibrium polygon pertaining to a certain
distribution of load, no interior bracing will theoretically be
needed for such distribution ; but if the distribution or direc-
tion of the external forces is at any time changed, bracing
will be called into action. Further discussion of this subject
comes in Parts II. and III.
The length of hk, etc., as compared with H K, etc., shows
the necessity of drawing the truss skeleton on a large scale,
to secure parallelism of the respective lines in each figure.
As a slight change in the inclinations of the rafter and lower
tie lines will change the magnitude of the stresses in those
pieces quite materially, we are warned by the appearance of
the diagram to provide, by an increase in size of these pieces,
40 ROOF-TKUSSES.
against such a cliange in the truss as would be caused by
slight errors in construction or by deflection under the load.
Stress diagrams are particularly serviceable in this way.
55. Wind and Steady Load. — We might analyze the effect
of the wind separately upon the truss, but, as there is a
likelihood that the wind will reverse the stress in some of the
diagonals which experience tension from the steady load, and
that we shall be obliged, therefore, to substitute the other
diagonals in such panels, it seems better to draw the diagram
for the wind and the weight of the roof in conjunction.
Therefore the two diagrams marked W. R. and W. L. are
drawn for the maximum force of wind on either side, com-
bined with the weight of the roof, etc. The external load line
hi of one case is the exact reverse of ih of the other. An
explanation of the construction of W. R. will suffice for both.
When the wind blows from the right, there is only the
steady load on the left half of the truss. Beginning therefore
with the joint at I, lay off vertically hi = 1150 pounds, or
one-half the load on H K ; next gh = 2300 pounds, load at
G H, and so on to F E, as in the steady-load diagram already
discussed. At FE we find, in addition to 2300 pounds verti-
cal pressure, an inclined force perpendicular to the tangent
at E, or to the chord of the piece, and equal to one-half of 989
pounds, the wind pressure before computed for E. We thus
get the inclined line as far as e in the diagram. The joint
DE gives de, manifestly made up of the other half of 989
pounds, of the vertical 2300 pounds as usual, and finally of
one-half of 2725 pounds from the next length of rafter, and
perpendicular to it. The forces for the remaining joints C D
and B C will be plotted in the same manner, and we therefore
see that, commencing at B, as is proper for this load line, we
lay off the vertical and inclined forces in regular succession
from one side of the truss to the other. If one draws a
straight line from c to d, it will be the resultant of the com-
bined external forces at C D.
EOOF-TRUSSES. 41
56. Reactions and Diagrams. — Connect 6 with i by the
dotted line, which will be the resultant of all these forces.
As the resultant of the dead weight, symmetrically distributed,
acts in the line of the vertical O P, and hence through the
centre of curvature of the rafters, and as the wind pressures
all point to the same centre of the circle, the resultant,
parallel to bi, must pass through the same point. Therefore
draw the dotted arrow through the centre from which the
rafter was struck, and parallel to hi. This arrow cuts the span
B I, by measurement, at 25\ feet from B, or 34f feet from I,
The resultant bi scales 20,620 pounds. If the sup23ortirjg
force at B were parallel to this resultant, it would be found
by taking moments about I, when we should have
Bx60 = 20,620 x34f; or B = 11,943 lbs.
Lay off this force from b to a'. If rollers are placed at B,
that reaction will be vertical, and the horizontal component
of a' b must be resisted at I. Let fall b a vertically, determin-
ing the point a by drawing a' a horizontally, and connect i
with a. The two supporting forces will be ia and ab.
In the W. L, diagram the point a' comes nearer to b than to
i, — that is, the quantity just obtained now applies to the
point of support I, — and a falls very near to, but just outside
of /, in the prolongation of the vertical line.
If there are no rollers under the truss, find the supporting
forces for each oblique pressure separately, as in § 48. The
same course must be pursued when the curve of the rafters
is not circular, as the forces will not then meet at a common
centre. Having thus completed, in either case, the polygon
of external forces, the remainder of the construction will be
made as in any example. After the first trial to ascertain the
proper diagonal, it appears that, in each case, the diagonals
all slant one way ; so that, for wind on one side, one set of
diagonals is in tension, and for wind on the other, all of the
other set are strained.
42 EOOF-TRUSSES.
57. Change of Diagonal. — The effect on the five pieces of
a panel, top, bottom, two sides and the diagonal, of drawing
the diagram so as to give compression in a diagonal, is shown
anew in the W. L. figure for the panel P Q. Instead of op and
qr, we get op' and q'r, considerably increased in amount but
the same in kind ; for ep and aq are substituted eq' and ap'y
unchanged in kind, but having practically what is taken from
one added to the other ; while the diagonal stress is, as we
said, reversed, but very nearly the same in amount.
It might be practicable to deduce some rule for determining
"beforehand the diagonal which would have the desired kind
of stress, but the tentative process seems easy. We find it
convenient to draw the lines j^arallel to the rafter and main
tie first, as ep and ap' , then to sketch roughly two lines for
the suspending piece and diagonal, see whether that diagonal
comes in tension, and finally draw the right ones carefully.
58. Resultant Stresses. — It is not necessary to put the
signs -f- and — on these lines, for it may be seen that all the
rafter is compressed, the whole lower member extended, and
all of the diagonals are in tension, as well as all the suspend-
ing pieces except O P and Q R, which are compressed a trifle
when the maximum wind comes from the right. Such pieces
are easily selected, if one notices that op and g r in the W. R.
diagram are drawn in a direction oi:)posite to the j)revailing
one.
The stresses are given in the following table. The lengths
of rafter are denoted by a single letter. The pieces of the
main tie, having the letter A in common, have also the letters
which stand before the stresses in the proper columns. The
inclination of the diagonal is shown by the sign prefixed to
the stress. The effect of the wind on the roller side is to
materially reduce the stress in a large portion of the main tie.
The light bracing required is a marked feature of this type of
truss, and the predominance of tensile members favors the use
of iron bars. The two compressions, marked -j-, are too in-
significant to require an increase of section.
EOOF-TEUSSES.
Table of Stresses for Fig. 20.
43
!
S. L.
W. R.
W. L.
Max.
+
fc
12.600
18.900
16,200
18,900
D
11,400
17,500
15,600
17,500
Rafters -
E
10,800
15,000
16,200
16,200
F
10,800
13,300
17,900
17,900
G
11,400
12,700
20,100
20.100
.H
12,600
13,100
21,800
21,800
'
K
9,600
K 5,500
K
19,500
19,500
L
9,500
L 5,500
M
18,000
18,000
Mam Tie A -
N
10,400
N 7,200
16,000
16,000
Q
10,400
P 9,000
Q
14,200
14,200
s
9,500
R 10,900
s
12,300
12,300
__
T
9,600
T 12,800
T
12,300
12,800
fLM
\900
\ 1,800
/ 1,800
1 >-Q
Diagonals... -i pQ
"400
/400
"2,100
"2,400
"2,400
"2,200
^RS
"900
"2,200
"2,100
fKL
1,200
700
1,200
1,200
MN
1,000
200
900
1,000
Suspenders. ^ OP
900
+ 100
700
900
1 QR
1,000 ■
+ 50
1,000
1.000
LST
1,200
400
1,600
1,600
If the designer proposes to proportion tlie pieces with re-
gard to minimum as well as maximum stresses, he can readily
select the former from the table.
If a fall of snow is supposed to be uniformly distributed
over the roof, the increased action of the several pieces can
be easily obtained by proportion from column S. L. But, if
it is thought that the inclination of the portions near C and H
is too great to permit of snow accumulating there, a diagram
for snow should be drawn. The horizontal projection of a
piece of the rafter is properly taken when reckoning a snow
load.
We think the reader will have no difficulty in drawing dia-
grams for a truss of similar outline, but with only a system
of simple triangular bracing.
CHAPTEE YI.
TRUSSES WITH HOEIZONTAL THRUST.
59. Scissor Truss.— Wlien it is desired to strengtlien the
rafters in a roof of moderate span by supporting them at their
middle points, a simple means, often employed, is to spike on
a piece from the lower end of one rafter to the middle of the
other, as shown in Fig. 22. The two pieces may or may not
be fastened together where they cross. At the first glance
we should say that, to draw the diagram, we must lay off the
load line ke, di^dde it as usual, and then, beginning at the
joint E, draw a'h' and h'f, parallel to AB and BE. Next, for
the joint F G, we should get the lines h'c' and c'g. For the
apex we should have three lines, viz., h g, g c\ and a line from
c' parallel to C H to strike h. There is evidently something
wrong here. If we start from the other point of support K,
we obtain the remainder of the diagram in dotted lines, and
find that we have two points marked c', some distance apart,
which ought to come together ; we also have two conspiring
forces, gc' and he', whose vertical components ought to bal-
ance hg.
Abandoning this diagram for the present, let us start at the
apex of the roof, where we may feel sure that there are but
two unknown forces. Taking the load h g at that point, draw
the full lines gfc and cli. Next for the joint GF, starting with
c g, pass down (//and draw/6 and h c. The joint H I will simi-
larly give the figure ihcdi. Lastly, the joint AC will add ba
and ad to the stresses d c ancl c b. To close the polygon for
the joint E we must now supply to a bfe the line e o, which
must be the inclined reaction at E, required to keep this truss
44
ROOF-TRUSSES. 45
frtjm sliding outwards ou the wall-plates, on the supposition
that the points of meeting of two or more pieces are true joints
(ones about which the parts are free to turn). As e a may be
decomposed into ea' and a' a, the force a' a is called the hori-
zontal thrust of the truss, which may be resisted by the wall
or by a tie-rod from E to K. The pieces of this truss are all
in compression.
60. Horizontal Thrust or an Additional Member
Necessary. — That the truss is not in equilibrium without
this inclined or horizontal reaction at the walls is seen, if we
suppose that E and K are not prevented from sliding later-
ally ; the joint A C will drop, the joints F G and H I will
approach one another, and the angle at the apex will become
sharper. This change will take place unless the above or
some other restraining force is applied. The trouble arises
from the four-sided space C, which is here free to change its
form. A member added in either diagonal of this space will
cure the evil. One from the apex to the joint C A will plainly
act as a tie, and will be found to supply the missing line c'c'
in the dotted diagram iirst drawn. From this diagram we see
that the stresses in most of the pieces will then be greater
than when the resistance comes from the wall. A strut
between the joints F G and H I will also make the truss
secure ; the reader can try such a diagram, and see what
pieces have their stresses reversed by the change. Either of
the above modifications puts the truss into the class having
vertical reactions.
61. Remarks. — As these trusses are usually made, reliance
against change of form, where little or no horizontal thrust
is supplied by the walls, is placed upon the stiffness of the
rafters, which are of one piece from ridge to eaves, and on
that of the two braces ; but a failure to get a good horizontal
resistance from the walls has sometimes resulted in an
unsightly sagging or springing of rafters and braces. The
bending moments on these pieces are due to the horizontal
46 EOOF-TRUSSES.
tlirust. Bending moments on a rafter or other piece will be
considered later.
It is worthy of notice that c d equals h a, or that the thrust
is constant throughout the brace. Two members crossing as
at A must naturally give a parallelogram in the stress dia-
gram ; the component of the load at H I which starts down
the brace will pass to E without being affected by crossing
the other brace ; yet, to resist the tendency to sag spoken of
above, and for the reason that the braces are better able to
resist thrust by mutually sta^-iug one another, it is advisable
to spike them together at their intersection.
62. Hammer-beam Truss; Curved Members. — Another
example where the horizontal tlirust of the truss against the
walls must be ascertained is shown in Fig. 23. This frame is
called a hammer-beam truss, and is a handsome type often
employed, in this country and abroad, for the support of
church roofs, the bracing being visible from below, and the
spaces containing more or less ornamental work. When the
church has a clear-story, the windows come between the trusses
at B, the truss is supported on columns, and the roof of the side
aisle takes up the horizontal thrust. If there are no side
roofs, the main walls are jDi'operly strengthened by but-
tresses.
It will be well to note in advance that a curved piece in a
truss, so far as the transmission of the force from one joint
to another is concerned, acts as if it lay in the straight line
between the two joints. The curved members in the present
example are the quadrants of a circle. They may have any
other desired curve, depending somewhat upon the pitch of
the roof. If, now, we consider the point of support B P of the
truss, and remember that the curved brace A O transmits the
force between its two extremities as if it were straight, it will
be evident that the thrust of the inclined piece, if any thrust
exists in it, must have a horizontal component which cannot
be neutralized by a vertical supporting force alone. There-
fore, in addition to the reaction of half the weight of the roof
EOOF-TRUSSES. 47
and truss, there must be supplied by the wall, assisted per-
haps by a buttress or a side roof, a certain horizontal thrust.
63. Amount of the Horizontal Thrust. — To determine
the value of this thrust : — Let W equal the weight of truss
and load. We have nine loaded joints, and there is, there-
fore, -JW at each joint except the two extreme ones. The
portion 213 maj be considered a small truss, like Fig. 7,
superimposed on the lower or main truss 4 6 2 3 7 5, and thus
bringing additional loads on the points 2 and 3. If then we
regard the main truss as a trapezoidal truss, and consider
that the pieces L A and Q A are unnecessary because the load
is the same on the two halves of the frame, the trapezoidal
truss will be 4 2 3 5, the brace 4-2 being made up of an assem-
blage of pieces. L A and Q A will be required when wind
acts upon the roof. Considering the trapezoidal truss 42 3 5
alone, the joint 2 will carry a load equal to that on D M, E K,
and F I, or f W, the joint 3 will carry the same amount, while
4 will support i "VV from C N, and 5 the remainder. If then
we lay off on a vertical line f W, for the load on 2, and draw
lines parallel to 2-4 and 2-3 from its extremities, the line
parallel to 2-3 will be the stress in the same, and will also,
since the load is vertical, be the horizontal thrust of the foot
of the compound brace 2-4. This force is marked H in the
dotted triangle drawn below the truss. A reference to § 25,
Fig. 14, may aid one in understanding the above.
64. Stress Diagram. — We now have the data for the
stress diagram, of wliicli one-half is shown. For the point 4,
or B P, we have the upward supporting force bp = ^ W, next
pa = 11, the horizontal thrust just determined of the wall,
etc., against the joint, a o parallel to the line of action of AO,
and finally o h, the pressure of the post O B on 4. The result-
ant oihp and p a, or ha, may of course be used for the reac-
tion of the wall. Taking next the joint 6, we have ch the
load, ho the thrust of B O, and we then draw o n and n c. The
joint C D gives den m d. The joint M A already has the lines
m n, 11 and o a ; since the line which is to close on m must be
48 EOOF-TRUSSES.
parallel to L M, and a is already vertically over m,al can have
no length, and there is no stress in A L, as before assumed.
Upon taking the joint D E we find also that no stress exists
in L K. The reader must not think this fact at variance with
the value H which was said to exist in 2-3 when we consid-
ered the trapezoid alone ; the triangular truss 12 3 will plainly
cause a tension in 2-3, and, with this distribution of load,
such tension will exactly neutralize the compression caused
in the same piece by 4-2. If one will consider the truss as
loaded at 6, 2, 1, 3, and 7 only, thus doing away with N M, K I,
IG, etc., he will find that a diagram will then give some com-
pression in K L.
Another method of treatment will be applied to this truss
later, § 75,
65. Different Horizontal Thrusts Consistent with
Equilibrium. — In studying Fig. 22 we saw that the stresses
in G C and C H were determined by the load G H, and that
the space C would become distorted unless a horizontal
thrust of a definite amount, here a'a, was supplied by the
walls. In Fig. 23 also the same things are true ; the trape-
zoidal truss 4 2 3 5 requires a certain horizontal thrust at the
points 4 and 5 to balance its load ; a greater or less thrust
will cause the truss to rise or fall, so long as L A and Q A are
neglected, for in that case motion can freely occur at joints 2
and 3. If, however, these pieces are under stress, a greater
or less horizontal thrust may be applied, the truss will still
be in equilibrium, and the diagram will close. Indeed a ver-
tical reaction is a supposable one, in which case O A must be
without stress. The same statement applies to Fig. 22, if one
of the diagonals of the space C is put in. As all roof-trusses
of small depth in their middle section, as compared with their
total rise, have a tendency to spread under a load, and hence
to thrust against their supports, their diagrams should be
drawn for a moderate amount of thrust at least, if it is desired
to have them maintain their shape ; and the supports should
be able to offer this resistance, or a tie should be carried across
EOOF-TRUSSES. 49
below. Otherwise, in addition to the sagging, a large increase
of stress is likely to be found in some of the parts as a result
of a vertical reaction. The determination of the horizontal
thrust in a braced frame of this kind is not very simple, but
may be worked out by a method given in Part III, " Arches,"
Chap. XII.
66. Proof. — That such trusses are in equilibrium under a
greater or less amount of horizontal thrust, or even when the
reactions are vertical, provided the pieces are able to with-
stand the resulting stresses, is illustrated by Fig. 24. Here
the load C B is taken as twice D C. The vertical reactions b a
and ad are calculated by the method of § 26. The diagram
with unaccented letters is then drawn and closed as usual.
Next, any horizontal thrust a a' at the points of support is
assumed and the diagram with accented letters is drawn.
This diagram also closes. The reduction of all of the stresses
except that in fg is most marked. We see from these cases
that only when the truss admits of deformation by the distor-
tion of some interior space such as C of Fig. 22, or R of Fig.
27, is the horizontal thrust determinate by the method of
these chapters ; and that moderately inclined reactions or
the tension of a horizontal tie between points of support are
favorable to a reduction of the stresses.
Arched ribs of a nearly constant depth, not infrequently
employed in railroad stations and public halls, will be treated
in Part III.
CHAPTER VII.
FORCES NOT APPLIED AT JOINTS.
67. First Diagram. — In the trusses heretofore treated the
leads have beeu conceutrated at those points only which were
directly supported. It sometimes happens that the cross-
beams or purlins, which connect the trusses and convey the
weight from the secondary rafters to the main rafters, rest
upon the latter at points between the joints. Let us, in Fig.
25, assume that a load rests upon the middle of each of the
upper rafters. If we neglect the bending action of the load
E G u23on the rafter and proceed as usual, we consider that
one-half of the load E G will be supported at each of the
joints C E and G K, and similarly for the load K M. There-
fore, having laid off the weights and the two equal reactions
of the walls on the load-line of the first diagram, we may in-
crease the loads on the joints C E, G K, and M O by the new
points of division, and complete this diagram, taking first B,
then the next joint on the inside, and then the outside one.
It will be noticed that all of the pieces except the rafters are
ties.
68. Supplying Imaginary Forces. — This diagram gives
but one stress along the whole of the upper rafter ; but it is
plain that the vertical force E G must have a component along
the rafter and cause a different stress to exist in E T from
what exists in G T. If, however, we suppose a joint to be at
E G, the transverse component of E G will cause it to yield,
as there is no brace beneath to hold it in place. To secure
equilibrium here we may sujjply an imaginary force EF,
shown by the dotted line, equal and directly opposed to this
50
ROOF-TRUSSES. 61
transverse component. Tliis imaginary force will take the
place of a perpendicular strut, will steady the joint, and will
leave the longitudinal component to affect the rafter. But
the transverse component of FG actually gives a pressure at
the joints C E and G K, while the imaginary force E E, just
added, will lift the ends of this rafter by the same amount ;
therefore we must restore the pressure, and the equilibrium
of the rafter F T as a whole, by adding imaginary forces, each
one-half of E E, at C D and G H. This added sjstem of forces
cannot interfere with the stresses in any other pieces, for they
balance by themselves. Treat the similar load KM in the
same way.
69. Second Diagram. — In the second diagram the two
supporting forces, pa and ah, are each equal to one-half the
total load. Lay off" 6 c as before ; draw the dotted line c d, equal
and parallel to the first imaginary force C D ; then de vertical,
as before ; then ef, equal to, and in the direction of E F ; then
fg, and so on, arriving finally atp, as usual.
The construction of the rest of the diagram presents no
difficult}^ ; the joints are taken in the same order as before,
and, when we have more than one external force on a joint,
we take them in succession, in the order first observed for the
external forces. When we reach the upper rafters, we find
that g falls on the line et; etis, greater and gt in less than the
line for the same piece in the first diagram.
70. Comparison of Results. — Thus it appears that the
first diagram gives the stress which would exist in the whole
length of the rafter E T G, if the load E G were actually at its
extremities ; but, being at its middle point, one-half of the
longitudinal component of EG goes to diminish the compres-
sion otherwise existing in G T, and the other half to increase
the compression in E T. A comparison of the two diagrams
will also show the truth of the former statement, that the
system of imaginary forces does not affect any of the truss
outside of the particular pieces to which it may be applied.
It is still necessary to provide for the bending action of the
52 ROOF-TRUSSES.
transverse portion of F G, or a force equal and opposite to E F
upon the rafter, considered as a beam extending from hip to
apex, a joint of course not being made at E G. This subject
will be treated in Chapter IX.
71. Remarks. — If the action of the wind upon this truss is
considered, it will be seen at once that no special treatment is
needed ; for the wind pressure is normal, and the addition of
the opposite force EE at once balances the force on this
joint, and transfers it to the ends D and H as the first
analysis did. The bending action on the rafter must, how-
ever, be provided for.
The treatment of loads or forces not directly resisted, at
above, is given by Mr. Bow in his " Economics of Construc-
tion," and may be applied to frames where one or more of the
internal spaces are not triangles, but quadrilaterals. If such
spaces are not surrounded by triangular spaces on at least all
sides but one, the truss is liable to distortion, unless the re-
sistance of some of the pieces to bending or the stiffness of
the tJworetical joints is called into play. A use of this treat-
ment at many points in the same diagram will, however, be
apt to make confusion.
Another application of imaginary forces, where a bending
moment exists, will be made at the close of the next chapter.
CHAPTER VIII.
FECIAL SOLUTIONS.
72. Reversal of Diagonal. — Difficulty is sometimes ex-
perienced iu completing the diagram for a truss because,
after passing a certain point, no joint can be found where but
two stresses are unknown ; while yet, judging from the
arrangement of the pieces, the stresses ought apparently, to
be determinate. Such a case was found in Fig. 11, and was
solved in § 20 by what might be called the law of symmetry.
A method of more general application to these cases is what
may be styled Reversal of a Diagonal.
It has been pointed out alreadj^ that, if any quadrangular
figure in a truss is crossed by one diagonal, the other diagonal
of the quadrangle may be substituted for the former without
affecting the stresses in any pieces except those which make
up the quadrangle. See §§ 26 and 53. It will be found that
such a change often reduces the stress in one or more pieces
of the quadrangle to zero, and thus makes the truss solvable
graphically. It will be well, if the reader fails to distinguish
readily the altered truss from the original one, to temporarily
erase from a pencil sketch the pieces thus rendered super-
fluous, or to draw the truss anew with the proper changes as
has been done in Figs. 26 and 27. The modified truss will
then be easily analyzed, and, when the old members are
restored, enough stresses will be known to make the final
solution practicable.
73. Example.— This method will first be applied to the
roof-truss, Fig. 26, of a railroad station at Worcester, Mass.
The span of this roof is 125 feet ; entire height, wall to apex,
53
54 ROOF-TRUSSES.
45 feet ; camber of main tie 8 feet ; rafter divided into six
equal panels ; trusses 50 feet apart.
Under steady load the tie bars S T, T U, U W, WS, whicli
cross the centre line of the truss, will be without stress, as in
Fig. 14, § 25. Indeed, as these two centre ties are indepen-
dent of one another, but one can be in action at a time, as,
for instance, S W and T U when the wind is on the left side.
If we begin our diagram from B with chak c, we meet with
no difficulty until we have passed the joint E F, for which we
drew fen op/. At either of the next joints are three un-
known stresses. As all stresses are determined up to the
piece P Q, change the diagonal Q II in the adjoining quadri-
lateral from the position of the full line to the dotted one.
Then the joint F G, as seen in the sketch below, will give us.
gfpg'g. As the full-lined diagonal has been removed, the
joint R W has disappeared ; for, if three supposed forces are
in equilibrium at one point, § 17, and two of them act in one
line, the third force must be zero, and II S therefore can have
no stress. The stress in SW will also be zero unless it.
resists wind on the left, and the stress in S T is then zero.
In either case we can draw h g g'r'h for the upper joint, and then
find a w and w r', if it exists, at the lower joint. The dotted
peak is not in the main truss, but in the jack-rafters which
transfer their load to G H and H I ; if one prefers, he may
put a load at the peak and draw the triangle of forces for
that point.
After using the above expedient on the other half of the
truss also, if the load is unsymmetrical, we replace the
reversed diagonal and find the true stresses in the pieces
affected by the change,' ^the diagonal and the four sides of the
containing quadrilateral. Hence we may draw ^oa?^^'^ for
the lower joint or hgrsli for the upper joint, and finally
gfV g'^g ^or '^^ left-hand joint of the quadrilateral.
74. Polonceau Truss.— The left half of Fig. 29 is the same
as Fig. 11. It will be remembered that we were stopped at
the piece D E of Fig. 29 by having three unknown stresses at
ROOF-TKUSSES. 55
either end. Change the full line E F to the dotted one. The
stress in F G at once becomes zero, as did K S in Fig. 26.
We may now find the stresses in D E and E L at the joint
K L ; in dotted E F and G M at joint L M, and in A H and
H F at the lower joint. Then the diagonal may be replaced
and the stresses in D E, E F, F G, E H, and F L rectified.
The right half of Fig. 29 may be similarly solved by revers-
ing the diagonal P Q, which change makes the stress in
O P zero.
75. Hammer-Beam Truss, by Reversal of Diagonals.
— The hammer-beam truss of Fig. 27 difters from that of Fig.
23 by the omission of the vertical in the space R. As pointed
out in § 66, this omission renders the horizontal thrust of this
truss definite. In attem23tiug to draw a diagram, however,
we cannot apparently begin at the wall until we know the
horizontal thrust, and, if we begin at F G, we soon meet with
joints where three unknown forces are found. The method
of the preceding sections will first be applied to the right
half. Draw gfr for the upper joint, ligrsTi for joint GH,
and/eg' 7'/ for EF. As joints HI and RA are now insoluble,
draw dotted T W for the full-lined diagonal T W, and do the
same with X Y. The truss will thus be changed to the form
of the sketch below. For, since T A and Y A act in the same
straight line (shown dotted on left half of truss), the stress in
W X is now zero, and T A and Y A have the same stress.
Further, at joint K L there remain K Y, Y L, and the exterior
force or load K L, which latter acts in the vertical line Y L ;
hence the stress in K Y is now zero, and Y L carries K L
only. We can therefore draw ihst'i for joint HI, kit'iv'h
for joint I K, lu't's rq . . . aio' for joint A R, and I k iv'a I for
the abutment. The reaction a I, being thus determined, can
be used to draw the diagram, as in Fig. 23. The diagram for
the left half of the truss is given in full lines, and it may be
seen that A P and A T are now useful.
76. Method of Trial and Error.— Where the unknown
stress in but one piece apjDears to stand in the way of a
56 KOOF-TKTJSSES.
solution, the diagram may sometimes be drawn witli com-
parative ease by trial. Tlius, in the left half of Fig. 27,
we may assume the value of the horizontal thrust or of the
stress in P Q and proceed with the diagram. Upon its failing
to close, we can change the assumed quantity and try again.
Thus, beginning at the apes, draw gfrg, feqrf, and
hgrsh; then assume qp' and its equal st'. The middle
joint will give t'srqp'a't'; the joint DE, p'qedo'p', etc.; and
finally the horizontal line from n' will fail to meet a line
parallel to A M on the load line, to give m h in the post. It
is evident, upon a slight inspection, that qp' is too long.
The reader will find that he can soon bring the diagram to a
closure by diminishing qp' .
By the use of such apjjroximations one of necessity loses
that check on the accuracy of the diagram, of having it close
with reasonable exactness.
Fig. 30, in case one or the other of the dotted diagonals
is used, will serve as an example for the practice of the pre-
ceding suggestions. Which diagonal tie, if either, will be
needed for wind, and which for steady load ?
77. Example. — We will close this branch of the subject
with an example which will introduce one or two new points
in addition to a combination of principles heretofore illus-
trated separately. The example shows the capabilities of
this method in handling complex problems. The structure
drawn in Fig. 28 is to be treated as a whole in its resistance
to wind pressure.
The steady-load diagram would present no difficulty. The
truss is carried upon columns which are hinged at their
lower ends B and P, each being connected by a pin to its
pedestal. The brace at E is therefore necessary to prevent
overturning. The proportions of the frame are as follows :
Distance between columns, 76 ft.; AC = 15 ft.; Q R = 7 ft.;
camber of lower tie, 3 ft.; 1-A = 19 ft.; height of space
1 = 16 ft.; of Y = 7 ft.; extreme height, ground to peak, 48 ft.
KOOF-TEUSSES. 67
Distance between trusses, 12 ft. Scale 40 ft. = 1 in. Scale of
diagram, 8000 lbs. = 1 inch. No wind on C.
Wind pressure on main roof, 12,000 lbs. = hj; therefore /gr,
gh, etc., = 3000 lbs.; wind pressure on KX=3360 Ibs.^J-lO;
on L Y = 3500 lbs. = 10-m. The dotted arrows are resultants
of wind pressure on the sloping surfaces. By moments
about P, or by proportion of segments of span BP, as
in § 48, we find
that 8368 lbs. of bj is carried at B, and 3633 lbs. at P.
that 940 " " 10-m " " " " 2460 " " "
9308 lbs. = 6-9 " " " " 6093 " " "
The horizontal force, y-10, at K, may be supposed to be
resisted equally at each point of support, since the two posts
will be alike. Hence jk = 9-a' = UJ-10) = 1680 lbs. is
carried at B. The moment of this horizontal force K about
B or P, tending to overturn the frame, or the couple formed
by K and the equal reaction in the line P B, will cause an
increased upward vertical force at P and an equal downward
force or diminished pressure at B. Its value, § 42, will be
^ = 1760 pounds = a' a. The reaction at B must
76 ^
balance the components, b-9, 9-«', and a'-a, and hence will
be a b. The reaction at P will then be m (or p) a, which may
be checked in detail, if desired.
The reaction ab, at B, will now be decomposed into its
vertical and horizontal comjjonents ac and cb. The piece AC
can resist a c as a strut or post, but must carry c b 5900 lbs. by
acting like a beam. Were there a real joint at D the struc-
ture would fall. It is therefore necessary to make the post
of one piece, or as one member from B to R. The magni-
tude of the horizontal force at F caused by the 5900 lbs. of
horizontal force at B will be in the ratio of the two segments
of the column (beam) or as 15 to 7, or 12,643 lbs. These two
forces must be balanced at D by a force equal to their sum.
58 ROOF-TRUSSES,
or 18,54i3 lbs. As in § 68, Fig. 25, tliis beam action of tlie post
must be neutralized, before the diagram can be drawn, as
these diagrams take no account of bending moments, for
which see Chap. IX.
We therefore apply at B the imaginary horizontal force
be = 5900 lbs., opposed to the direction of the reaction, and
leaving only a c, the vertical component, which is balanced by
the post; at CD we apply ccZ = 18,543 lbs.; and at EF, we
add ef = 12,643 lbs. The sum of these three imaginary hori-
zontal forces being zero, the stresses in the truss are not dis-
turbed. The same steps must be taken at P, the horizontal
forces mn, no, and op being obtained by the same process
from the horizontal component po ol the reaction p a.
The load line therefore finally becomes bed efg hikl m n op,
the force D E being shifted laterally as shown, and i k being
the resultant of ij andjZ:;. The stress in D Q is readily ob-
tained by drawing deq. Then the point D of the post gives
the figure acdq r a, determining the stresses in the upper
part of the post and the brace R A. The remainder of the
diagram presents no difliculty.
The column must be designed to resist the large bending
moment to which it is liable, as well as the thrust q r. For
bending moments, etc., see the next chapter, and also Part 11.
As this structure is supposed to be open below, the lower
member should be adapted to resist such compression as may
come upon it from the tendency of a gust of wind, entering
beneath, to raise the roof.
CHAPTEK IX.
BENDING MOMENT AND MOMENT OF RESISTANCE,
78. Load between Joints. — Having treated of the action
of external forces upon a great variety of trusses, we propose
now to investigate the graphical determination of the bending
moments which arise from the load on certain pieces, and of
the stresses due to the moments of resistance by which the
bending moments must be met.
To recapitulate some statements of earlier chapters : — In
case the transverse components of the load upon a portion of
a rafter, or other piece of a truss, are not immediately resisted
by the supporting power of some adjacent parts, or, in other
words, unless the load on a structure is actually concentrated
at the several joints, such transverse components will exert a
bending action on the portion in question, and the additional
stress thus caused in the piece may be too great to be safely
neglected. Further, in case the piece makes any other than a
right angle with the line of action of the load, or has an
oblique force acting upor it, the stress along it, given by the
diagram, will be less than the maximum, and will generally be
the mean stress. Lastly, in case a piece is curved, a bending
moment will be exerted upon it by the force acting along the
straight line joining its two ends, this bending moment being
a maximum at the point where the axis or centre line of the
piece is farthest removed from the line drawn between its ends.
79. Example. — To illustrate the former statements by a
simple example : — Suppose the rafters A C and B C, Fig. 31,
to be loaded uniformly over their Avhole extent. Let us
assume, in the first place, that the tie AB is not used, but
59
60 KOOF-TRUSSES.
that the thrust of the rafters is resisted by the walls which
carry the roof. Consider the piece A C. Since the roof is
symmetrically loaded, the thrust at C must be horizontal, and
therefore the reaction which sujjports this end of A C will lie
in the line C E. The centre of gravity of the load on A C is at
D, its middle point, and the resultant of the load will, if pro-
longed upwards, intersect C E at E. Since the rafter is in
equilibrium under the load and the reactions at C and A, the
direction of the reaction of the wall at A must also pass
through E (compare Figs. 3 and 4). Draw A E and prolong
ED to G. Let E G be measured b}^ such a scale as to repre-
sent the load on A C. The three forces meeting in the common
point E will then be equal to the respective sides of the tri-
angle AEG, drawn parallel to them ; and, since A G equals
E C, the reactions at A and C will be A E and C E.
We now decompose AE and CE into components along
and transverse to the rafter, and have AF, direct compression
on the rafter at A, and C F, direct compression at C. The
compression on successive sections of the rafter increases from
C to A by the successive longitudinal components of the load.
The two components A L and C Q, which, combined with A F
and C F, give the original forces A E and C E, are analogous
to the supporting forces of a beam or truss, and through them
we obtain the bending action of the load on this rafter. If,
now, the rafters simply rest on the wall, being secured against
spreading by the tie A B, the reaction A E will be replaced by
the two components, A I, the upward supporting force of the
wall, and A G, the stress exerted by the tie ; these two forces
give the same stress and bending moments on the rafter as
before.
80. Comparison with Diagram.— Consider, next, the
method by diagram. The load is now to be concentrated at
the joints, and in place of E G, we shall have A N and C P,
each one-half of the load on one rafter. Lay oft" 1-2 to repre-
sent the total load on the roof, make 1-3 equal to AN and
1—4 to A I, and draw 3-5 and 4-5 parallel to the rafter and tie.
ROOF-TRUSSES. 61
A G will equal 4-5, and therefore the stress in the tie is given
correctly ; but, since A I— AN = AK = 3-4, 3-5 equals AD,
and this is the stress given bj the diagram as existing from A
to C, a supposition which is true when the load is actually
concentrated at the joints, but is not true for a distributed
load. But A D, or 3-5, is equal to one-half of AF -)- F C, and
is manifestly the value of the direct compression at the middle
j)oiut D of the rafter ; all of the load from A to D was, when
we drew the diagram, considered to be concentrated at the
joint A. To 3-5, or A D, we should add D F, to obtain the
correct compression A F at the lower end ; therefore a piece
which supports a distributed load should have a compression,
equal to the longitudinal com23onent.of so much of the load as
is transferred to its lower end, added to its stress obtained
from the stress diagram. The amount to be added, however,
is generall}^ insignificant as compared with the truss stress.
The load on the principal rafters of a roof-truss is usually
concentrated at series of equidistant points, by means of the
purlins, or short cross-beams which extend from one truss to
another, and which are themselves weighted at a series of
points by the pressure of the second arj^ rafters. These second-
ary rafters, when emplojed, carry the boards, etc., and thus
have a uniformly distributed load. It is only in cases where
purlins rest at other points than the so-called joints that
bending action occurs in the principal rafters, or in very light
trusses where the boards are nailed directly to the main rafters.
"We need to determine the maximum bending moments on
such main rafters, on the purlins and secondary rafters, in
order to intelligently provide sections sufficiently strong to
resist them.
81. Bending Moment,— It will first be well to explain
what bending moment and moment of resistance are. A horizon-
tal beam A B, Fig. 32, supported at its two ends, when loaded
with a series of weights, distributed in any manner, is in
equilibrium under the action of vertical forces, the weights
acting downwards and the two supporting forces acting up-
62 KOOF-TRUSSES.
wards. These supj)ortmg forces are easily calculated by the
principle of the lever, or by taking moments as explained in
§§ 26 and 36. They will be found graphically presently. As
the beam is at rest, there must be no tendency to rotate, and
therefore, if we assume any point for an axis, the sum of the
moments, that is of the products of each force by its distance
from the axis, must equal zero. A moment which tpnds to
produce rotation in one direction being called plus, one which
acts in the other direction is called minus. If then we pass
an imaginary vertical plane of section through any point in
the beam, such as E, the sum of the moments on one side of
the plane of section must balance or equal that on the other.
The sum of these moments on one side or the other is called
the bending moment : the reason for the name will soon be
evident.
82. Moment of Resistance. — These bending moments on
o]3posite sides of the section in question can balance one
another only through the resistance of the material of the
beam at the section where stresses between the particles are
set in action to resist the tendency to bend. The beam
becomes slightly convex, and the particles or fibres on the
convex side are extended, while those on the concave side are
compressed. Experiment shows that, for flexure within such
moderate limits as occur in practice, the horizontal forces
exerted between contiguous particles vary uniformly as we go
from the top of the beam to the bottom, the compressive
stress being most intense on the concave side, diminishing
regularly to zero at some point or horizontal j^lane, called the
neutral axis, then changing to tension and increasing as we
approach the convex side. The two sets of stresses reacting
against each other may be represented to the eye by the
arrows in the vertical section marked E'.
Since all of the external forces are vertical, these internal
stresses, being horizontal, must balance in themselves, or the
total tension must equal the total compression, whence it
follows that the neutral axis must pass through the centre of
EOOF-TRUSSES. 63
gravity of the section. To make this fact clear, let one con-
sider that the distance of the centre of gravity from any as-
sumed axis or the position of the resultant of parallel forces
is found by multiplying each force or weight by its distance
from that axis and dividing by the sum of the forces. Now if
we attempt to lind the centre of gravitj'of a thin cross-section
of this beam, and take our axis through the point where the
centre of gravity happens to lie, the sum of the moments of
the particles on each side will balance or be equal, and we can
see that the distance of each particle from the axis will vary
exactly as these given stresses ; hence the neutral axis must
lie in the centre of gravity of each cross-section.
As these stresses are caused by and resist the external bend-
ing moment on each side of the section, the moment in
the interior of the beam, made up of the sum of the products
of the stress on each particle multiplied by its distance from
the neutral axis, or indeed from any axis, and known as the
Tnoment of resistance, must equal the bending moment at
the given section. As the tensions and compressions on one
side of the plane of section tend to produce rotation about
the neutral axis in the same direction, their moments are
added together.
83. Formula for Bending Moment. — The bending mo-
ment, then, in the beam AB of the figure, at au}- section E,
will be, if Pj is the supporting force on the right, W„ W^,
etc., the weights,
P2 . B E - Wi . C E - W2 . D E ;
or, in general, if L equal the arm of any weight, and 2 be
the sign of summation,
M (the bending moment) = P^ . B E — 2 W . L,
it being remembered always to take only the weights between
one end and the plane of section.
The moment of resistance, being numerically equal to the
bending moment, is therefore equal to the above expression,
and the maximum stress at any section can thence be
64 KOOF-TRUSSES.
determined, or the required cross-section to conform to the
proper working stress for the material. The weights on one
side of the section may all be considered to be concentrated
at their common centre of gravity, or point of application of
their resultant, so far as the bending moment at that section
is concerned ; the load when continuous is always so taken.
If the reader will take a special case, and, having a beam
of known length with weights in given positions, will first
find the supporting forces, and then calculate the bending
moment on either side of a plane of section, he will obtain
the same result with opposite signs, showing that the two
moments balance one another. The numerical result, being
the product of two quantities, is read as so many foot-
pounds or inch-pounds, according to the units employed. As
the stress in any material is usually expressed in pounds on
the square inch, the latter units are the better.
84. Equilibrium Polygon. — Let us suppose that the
weights which, in Fig. 32, rest upon the beam are transferred
to a cord at the several points c, d,f, and g, vertically below
their former positions C, D, F, and G, the cord itself being
attached to two fixed points a and &, at equal distances verti-
cally from A and B. Let us further supj)ose that the amount
of the weight at G alone is at present known. This cord can
be treated as if it were a frame. Taking the joint g into con-
sideration, draw 5-4 vertically, equal to the weight, then 5-0
parallel to ag and 4-0 parallel to gf. The two lines just
drawn must be the tensions in a^ and gf. For the joint /,/gr
is now known ; therefore 4-3 parallel to the weight and 3-0
parallel to fd will determine the other forces at /. The
side 4-3 must equal the weight at F, and must lie in the same
straight line with 5-4 ; for this triangle was constructed on
the side 4-0 previously found. Continuing the construction
for the successive angles of the cord, we find that a vertical
line 5-1 will represent by its several portions the successive
weights, and that the tensions in the diflferent parts of the
cord will be given by the lines parallel to these parts, drawn
EOOF-TRUSSES. 65
from the points of division of the load line, and all converg-
ing to the common point 0. Draw 0-6 horizontally, and
hence parallel to a h ; this line will be the horizontal com-
ponent of the tension at any point of the cord, and is here
denoted by H. The form assumed by the cord for a given
distribution of weights is called the Equilibrium Polygon, as
the system will be in equilibrium or at rest ; and it is also
called in mechanics a funicular polygon. Students of mechan-
ics will recall the fact, so easily shown here, that the hori-
zontal component H is a constant quantity at every point.
85, Reactions. — If now the cord, instead of being fastened
to fixed points at a and b, is attached to the two ends of a
rigid bar a b, and the whole system is then suspended from A
and B by two short cords, its equilibrium will not be dis-
turbed. The pull 5-0 at a will be decomposed into 0-6, com-
pression in ba, and 6-5, tension along a A. Similarly at
h, 0-1 will be decomposed into 1-6 along 6B and 6-0
along a b. 6-0 balances 0-6, while 1-6 and 6-5 must be the
supporting forces at b and a. As the suj)23orting forces do
not dejDeud upon the form of the frame or truss, the reac-
tions which carry the beam at B and A must be these same
quantities.
86. Equilibrium Polygon, General Construction. — We
may make the construction more general by drawing an equi-
librium polygon from any point a', vertically below A, and find-
ing the outline of a cord which will sustain in equilibrium the
given weights at the given horizontal distances from A. Lay
off the weights in succession from 5 to 1 ; assume any point
0' arbitrarily and connect it with all the points of division of
the load line. Begin at a', and draw a'g' parallel to 5-0',
stopping at the vertical dropped from G; then draw g'f
parallel to 4-0', etc., and finally c'b' parallel to 1-0'. That
this will be the figure of a cord suspended from a' and b' fol-
lows from the preceding demonstration. Connect b' with a' ;
a line, parallel to b'a', from 0' must strike the same point 6
which the line from 0, parallel to ha, touched. The sup-
66 KOOF-TKUSSES.
porting forces, if h'a' exists, will be 1-6 and 6-5 as before ;
but 0'-6' will be the horizontal component H' for this cord.
87. The Equilibrium Polygon Gives Bending Mo-
ments. — If we turn again to the first cord, attached at a and
&, the piece a h being dispensed with, the moment of all the
forces on one side of an}^ point, such as e, must be the bend-
ing moment there ; but as the cord is perfectly flexible and at
rest, this bending moment will equal zero. Using, instead of
1-0, its two components 1-6 = P^ and 6-0 = H, multiplying
each force by the perpendicular distance of its line of action
from e, calling the combined moments of the weights on one
side of e ^ W . L as before, and denoting the tendency to pro-
duce rotation in opposite ways by opposite signs, we shall
have, for moments of forces on the right of, and around e,
P2 . 6 A; - :2 W. L — H . e^ = 0,
or
H. eZ; = P2 . 6^-2W. L.
But V,.hh = P, . BE, and P, . BE - JSW.L = M, the bend-
ing moment at the section E of the beam, as shown in § 83 ;
therefore
M = H . eZ;.
By a similar analysis of the lower cord we have
Ps . ?• A' - S W . L = (6-0') . e' Z = M.
From similarity of triangles le'k' and 6'0' 6, we have
e'l : e'7c' = 6'-0' : 6-0',
or
(6-00 .e'Z=(6'-0') .e'k'\
therefore
M=(6'-00 . e'k' = W .e'k',
as in the other case. The solution is therefore general, and
the bending moment at any section of the beam equals the
product of H from the stress diagram 1 5 by the vertical
ordinate, below the section, from the cord to the line connect-
ing its two extremities.
KOOF-TKUSSES, 67
88. Remarks. — The relative situations of a' and h' will de-
pend upon the choice of the position of 0', and this point
may be taken wherever convenient. H' is measured by the
same scale used in plottinji^ 5-1, while e'li! must be measured
by the scale to which AB is laid oj0f. The tAvo scales, one
representing pounds, the other inches, need not be numerically
the same ; their product will be inch-pounds.
A single load on the beam will have for its equilibrium
polygon two straight lines from a' and h' , meeting at a point
vertically under the weight. A uniformly distributed load
will give a parabola with the maximum ordinate at the middle
of the span. This load may be treated as if concentrated at
any convenient number of points along the beam, as we have
done in getting the loads at the several divisions of a rafter,
and the angles of the polygon will lie in the desired parabola.
When the beam is inclined the transverse comj)onents alone of
the load produce any bending, as explained for a uniform
load in § 79. Wind pressure will act as a uniform normal or
transverse load on the piece w^hich directly resists it.
The equilibrium polygon has much more extended applica-
tions in Parts II. and III.
89. Moment of Resistance of Rectangular Cross-Sec-
tion. — Next, to determine the moment of resistance for a par-
ticular form of cross-section : — Consider a beam of rectangular
cross-section, represented by A B C D of Fig. 33. The inten-
sity of stress, as shown at E', Fig. 32, varies uniforml}' each
way from the neutral axis which, lying through the centre of
gravity G of the cross-section, will be at E F, the middle of
the depth. The stress on a square inch will be most intense
on the fibres at the edge A B or C D, and less intense on any
intermediate layer, such as I K, in the proportion of E I to
E A. If then we draw from G the lines G A and G B, and
imagine that the layer I K is replaced by I' K', which has its
breadth diminished in the same proportion, the total stress
on I' K', if of the intensity found at A B, will be equal to the
total stress of less intensity actually existing on I K. The
68 EOOF-TRUSSES.
former stress will also liaise tlie same leverage about E F as
does the actual stress on I K. By the same reasoning for all
layers of the cross-section, we obtain two triangular, shaded
areas, ABG and GDC, which may be termed equivalent areas
of uniform stress of intensity equal to the actual maximum ;
one of them, usually the upper one, when multij^lied by this
maximum intensity of stress, represents the total compression,
and the other the total tension at the section. The moments
of this tension and compression about the neutral axis will be
most readily obtained by considering the stress, which is now
uniformly distributed over the triangle, as concentrated at its
centre of action, the centre of gravity G' of the triangle, dis-
tant two-thirds of its height from the apex G.
Let h represent the breadth and h the height of the cross-
section in inches ; the area of one triangle will be ^h .\h; and
the lever arm about EF will be f . ^/i. Let /represent the
maximum stress on the square inch at AB. Since the tension
and compression tend to produce rotation in the same direc-
tion, we add the moments of the two forces together and have
2 ("2 •/. ^h\ = moment of resistance = ^fhJf.
Putting this value equal to the bending moment M, we obtain
B.'.e'¥=lfbh\
If we select the maximum value of e'k', introduce the safe
working stress/ for the extreme fibres, and assume either & or
h, we can compute the other required dimension, and thus
determine the beam when of uniform section throughout. If
the cross-section is to vary, its moment of resistance at differ-
ent points must at least be equal to the bending moments.
As the stiffness of the beam depends principally upon h, the
depth must not be made too small. If the beam has too little
breadth the compressed edge will yield sideways.
90. Moment of Resistance of T Section. — It is easy to
compute the size of a beam of rectangular cross-section by the
ROOF-TRUSSES. 69
above formula, but for less regular sections the determination
of the moment of resistance by this graphical method may
prove of service. In applying it to a beam of the section
shown in Fig. 34 we must begin by finding the centre of
gravity of the section. By multiplying each rectangular area
by the distance of its centre of gravity from either the top or
the bottom, adding these products, and dividing by the
whole area, we find the distance of the neutral axis from that
edge. If GI = 6, AB = &', GE = A, and C A = h', we have
r^ — ; — jtt; = distance oi neutral axis from G 1.
bh-\- b h
The construction of the shaded area A P B needs no expla-
nation, as it follows the previous example. The stress on the
fibres at the edge G I will not be so great as at the edge A B,
because they are not so far from the neutral axis. If the
fibres at G I were removed to K L, so as to be equally remote
with AB, they would be equally strained. Then to reduce
the layer G I to one which, if it had the same intensity of
stress with A B, would give the same total stress which now
exists on GI, project GI to KL, draw KP and LP,* and GT'
will be the desired reduced length. The remainder of the
shaded area for the lower rectangle follows the usual rule.
In the same way, the fibres at C D will be projected at Q R,
and, by drawing Q P and HP, we determine CD', and thus
complete the shaded portion. These triangles, etc., can be
readily scaled, or computed from the known proportions of
the beam, their centres of gravity found and the moment of
resistance calculated.
91. Moment of Resistance of an Irregular Section. — A
good example of a section whose moment of resistance is not
readily determined by computation alone is afi'orded by a
deck-beam. Fig. 35, often employed in floors and roofs. It is
here drawn to one-quarter scale, showing height of section 6
inches, breadth of flange A B 3| inches, thickness of web |
inch, weight per yard 44 lbs.
* K P and L P should be straight lines, nearly touching C and D.
70 EOOF-TRUSSES.
The readiest way to determine tlie moment of resistance of
such a cross-section is as follows : — Transfer its outlines from
the book of shapes or by such data as you have to a sheet of
heavy paper, and make a tracing for construction purposes.
Cut the section from the heavy paper, balance on a knife-edge
and thus determine the neutral axis C D. Then on the trac-
ing draw K L horizontally at the same distance from C D that
S T is. A B will be projected at K L, and lines from K and
L to P, the middle j)oint of C D, or the centre of gravity of
this section, will cut AB at A' and B', making A'B' the
reduced length of A B, and now considered to have the same
stress per square inch as exists at I G. In the same way the
end M of M N will be projected at O, the point U at Y, and
the lines from O and V to P will cut the horizontal lines
through M and U at new points in the desired curve. Thus
enough points are soon obtained to locate the boundary of
the shaded portion from B' to P. The part of the web with
straight sides gives of course a triangle, found at once by
drawing a line from W to P. The curve A' P corresponds
with B' P. For the lower portion, project E F on T S, draw
lines to P, and get in a similar way enough points for this
curve. Cut out the two shaded figures from the heavy paper,
balance each one over a knife-edge and thus determine their
respective centres of gravity Q and R. Calculate the area of
one ; the area of the other should exactly equal it, for the
total tension equals the total compression. Calling this area
A and the safe working stress on the square inch/, we shall
then have for the moment of resistance
/. A. PQ+/. A. PE=/. A. QR.
In this example A = 1.29 sq. inches, P Q = 2.12 inches, and
PR =: 2.66 inches. If therefore for a static load /= 12,000
lbs., the moment of resistance equals
12,000 X 1.29 X 4.78 = 74,000 inch-pounds.
92. Moment of Resistance of I Beam. — In simpler cases
the required size of beam to sustain a given load is more read-
ROOF-TRUSSES. 71
ily found by formula. If I beams are used, the web being
thin, and the top and bottom Hanges alike, an approximate
formula may be used. If F rej)resents the area in square
inches of the cross-section of either flange, W the area of the
web, h the dejjth from centre to centre of flanges or the entire
depth minus thickness of one flange (that is, between centres
of gravity approximately), and/ the safe stress on the square
inch, the moment of resistance is nearly equal to
CHAPTER X.
LOAD AND DETAILS.
93. Lateral Bracing.— The principal trusses, if large,
should be braced together in the planes of the rafters to pre-
vent wind, in a direction perpendicular to the gable ends, from
producing any lateral movement. The roof boards, if laid
close, and well nailed, will stiffen trusses of moderate span.
It is often customary also to fasten the trusses down to the
walls, especially in those buildings where wind may get below
the roof. In such cases it is proper to consider and provide
for the tendency of the wind to reverse the stresses in a roof
which has a light covering.
94. Weight of Materials. — The weight of the roof cover-
ing can be ascertained in advance. The bending moments on
the jack-rafters and the purlins can then be found, their sizes
computed and their weights added in. The weight of the
truss must then be assumed from such data as may be at
hand. After the diagrams have been drawn and the truss has
been roughly designed, its weight should be calculated to see
how well it agrees with the assumed weight. If this agree-
ment is not sufficiently exact, the proper allowance is then to
be made.
Trautwine says that, for spans not exceeding about 75 feet,
and trusses 7 feet apart, of the type shown in Figs. 11 and 29,
the total load per square foot, including the truss itself, pur-
lins, etc., complete, may be taken as follows :
Roof covered with corrugated iron, unboarded, . . 8 lbs.
Same if plastered below the rafters, 18 "
Roof covered with corrugated iron, on boards, . . 11 "
72
EOOF-TEUSSES. 73
Same if plastered below the i*afters, 21 lbs.
Eoof covered with slate, unboarded or on laths, . . 13 "
Same on boards li inches thick, 16 "
Same if plastered below the rafters, 26 "
Eoof covered with shingles on laths, 10 "
For spans from 75 feet to 150 feet it will suffice to add 4 lbs.
to eacli of these totals.
The weight of an ordinary lathed and plastered ceiling is
about 10 lbs. per square foot ; and that of an ordinary floor
of 1-inch boards, together with the usual 2 X 12 inch joists,
12 inches apart from centre to centre, is from 9 to 12 lbs. per
square foot. White pine timber, if dry, may be considered to
weigh about 25 lbs., northern yellow pine 35 lbs., and south-
ern yellow pine 45 lbs. per cubic foot ; if wet, add from 20 to
50 per cent. Oak may be reckoned at from 40 to 50 lbs. per
cubic foot ; cast iron at 450 lbs. per cubic foot ; wrought iron
at 480 lbs. per cubic foot.
The allowance to be made for the weight of snow will
depend upon the latitude ; from 12 to 15 lbs. per square foot
of roof will suffice for most places. In some situations snow
may accumulate in considerable quantities, becoming satu-
rated with water and turning to ice ; but snow saturated with
water will generally slide off from roofs of ordinary pitch.
The weight of a cubic foot varies much ; freshly fallen snow
may weigh from 5 to 12 lbs. ; snow and hail, sleet or ice may
weigh from 30 to 50 lbs. per cubic foot, but the quantity on
a roof will usually be small.
95. Action of Materials under Stress.— After the stresses
in the frame are determined, tlie several parts must be designed
to withstand them. It is not the purpose here to proportion
the members of a truss and work out the details. The action
of materials under applied forces, the method of calculating
beams, ties, and struts, and the proper designing of connec-
tions and details are discussed at length in the author's
" Structural Mechanics."
As materials, if repeatedly strained to an amount at all
approaching the breaking strain, will fail sooner or later, the
74
EOOF-TRUSSES.
severe action weakening them, and as we must provide for
unforeseen and unknown defects of material and workman-
ship, as well as for more or less of shock and \ibration, it is
customary to so proportion the several parts of a structure
that they will be able to resist without failure much larger
forces than those obtained from the stress diagrams. The
smaller the load or stress on a piece the greater number of
applications and removals before the piece is injured or
broken. If the stress is reduced so much by increase of
cross-section of the member that the j)iece will safely sustain
an indefinitely great number of repetitions of it, such cross-
section will be the proper one for a piece in a bridge or
machine.
The stress arising from a stationary load, such as the weight
of the structure, which is constant, is not so trying as repeated
application and release of the same stress. The heavy wind-
stresses determined in the previous chapters are not likely to
occur more than once or twice, if at all, in the life of the
structure. Hence good practice wall authorize the employ-
ment of stresses some fifty per cent, in excess of those consid-
ered allowable in first-class bridge structures and those sub-
jected to frequent change of load, to shock and vibration.
96. Allowable Stresses. — In accordance with this view,
the following values may be used, where the wind-pressure of
Chapter IV. has been allowed for.
Material.
White Oak
Long-leaf Southern Pine. .
Oregon Pine or Fir
White Pine (Eastern)
Spruce
Wrought Iron.
" " best quality.
Soft Steel ,
Medium Steel
Bending
stress.
Tension.
1,600
1,500
1,600
1,400
1,600
1,800
1,400
800
1,200
1,200
10,000
12,000
12,000
15,000
14.000
16,000
16,000
18,000
Compres-
sion with
grain.
1,400
1,400
1,300
1,200
1,200
Compres-
sion across
grain.
400
300
250
200
200
Compression
10,000
12,000
12,500
13,750
Shear with
grain.
180
150
200
100
100
Shear
8,000
10,000
10,000
11,000
ROOF-TRUSSES. 75
The above values must not be applied to parts subjected to
mo^dng loads, such as floor-beams and suspending rods for
same, unless the load is moderate in total amount and very
gradually ajjplied and removed. For bridge work they must
be reduced from '20 to 33 per cent.
97. Tension Members.— Pieces in tension will be liable to
break at the smallest cross-section. It is therefore economi-
cal to enlarge the screw-ends of long iron rods and bolts so
that the cross- section at the bottom of the threads shall be
at least as large as at any other point. It is desirable that
the centre of resistance of the cross- section of struts and ties
shall coincide with the centre of figure, as a deviation from
that j)osition greatly weakens the piece. To calculate the net
or smallest cross-section of a tension member where the pull
is axial or central it is sufficient to divide the force by the safe
working tensile stress. Allowance must be made for diminu-
tion of cross-section by any cutting away, bolt or rivet holes.
98. Compression Members. — For very short pieces or
blocks in compression, whose lengths do not exceed six times
the least dimension, the same process may be followed. But
as the length increases the strut has a tendency to yield
sideways when compressed, and the cross-section must be
increased. Let I be the length of the strut in inches, h ita
least external diameter in inches, and r the least radius of gyra-
tion of its cross-section in inches. Then the safe mean work-
ing compressive stress, to be used as a divisor of the given
force, to find the cross-section of the strut, will be, for piecee
with flat, securely bedded ends, or ends fixed in direction by
bolting or riveting.
Southern Pine 1200 - 12-.
A
White Pine 1000 - loj-.
Soft Steel 12500 - 42-.
r
Medium Steel 13750 - 48-.
r
76 ROOF-TRUSSES.
If the struts are jointed at their ends bj pin connections,
or are so narrow as to readily yield sideways at these points^
double the subtractive term in the preceding formulas.
The hand-books issued by the steel manufacturers give the
sections and weights of the various rolled shapes, the values
of r for different axes, the safe loads for beams of different
spans, details of construction, and miscellaneous useful infor-
mation. The inexperienced designer should exercise great
care in computing compression members, and be sure that the
least radius of gyration is used in the formula.
Pieces subjected alternately to tension and compression
should have a materially larger section than would be required
for either stress alone.
Cast iron is not in favor with the best designers for any
but short compression pieces, packing blocks and pedestals,
although it is still employed for columns. The formula for
I
cast iron may be 15,000 — 50-.
99. Beams. — The values of / to be used in the moment of
resistance, for pieces subjected to bending, are marked bend-
ing stress in the preceding table. In determining the moment
of resistance of a piece exposed to bending, or in calculating-
the cross-section required at the point of maximum bending^
moment, allowance must be made for portions cut away on.
the tension side in attaching fastenings, bolting or riveting
together parts, and also on the compression side unless the
holes, etc., are so tightly filled that the compression can be
fairly considered as resisted by those portions also.
Those pieces which resist both a bending moment and a
direct stress may first be designed to safely carry the bending
moment, and then the dimension transverse to that in which
the piece will bend may be so much increased that the added
slice will resist the direct pull or thrust. If that force is
thrust, it will be well to test the size of the piece by the for-
mula on the preceding page.
100. Pins and Eyes. — A reasonable rule for proportioning;
ROOF-TRUSSES. 77
pins and eyes of tension bars is as follows : — Make the diam-
eter of the pin from three-fourths to four-fifths of the width
of the bar in flats, and one and one-fourth times the diameter
of the bar iu rounds, giving the eye a sectional area of fifty
per cent, in excess of that of the bar. The thickness of flat
bars should be at least one-fourth of the width in order to
secure a good bearing surface on the pin, and the metal at the
eyes should be as thick as the bars. As tlie bending moment
on a pin generally determines its diameter, pieces assembled
on a pin should be packed closely, and thojs:; having ojiposing
stresses should be brought into juxtaposition if possible.
101. Details. — Very close attention must be given to all
minor details ; to so proportion all the parts of a joint that it
will be no more likely to yield in one way than another ; to
■weaken as little as possible the pieces connected at a splice ;
to give suflicient bearing surface so as to bring the intensity
of the comj)ression on the surface within proper limits : to
distribute rivets and bolts so as to give the greatest resist-
ance with the least cutting away of other parts ; to keep the
action line of every piece as near its axis as possible ; and to
examine all sections and parts for tension, compression, and
shear. The failure of a joint or connection is as fatal to a
frame as to have a member too small for the stress upon it.
The following sections are quoted from the author's " Struc-
tural Mechauics " :
102. Framing of Timler: Splices. — Sketches XL to
XVI. in Plate IV. represent diflereut methods of splicing a
timber tie. In each case the smallest cross-section of the
timber determines the amount of tension that can be trans-
mitted. The shoulders are in compression, and the longitu-
dinal planes between the shoulders are in shear. In XI., for
equal strength, the depth of the two opposite shoulders or
indents should be to the remaining depth of the timber as the
safe unit tensile stress is to the safe unit compression along
the grain. The shearing length, on either timber or clamp,
should be to the depth of shoulder as the safe unit compres-
78 ROOr-TKlTSSES.
sion is to the safe unit sliear. In actual practice, unless con-
siderable dependence is placed upon the resistance of the
bolts against shearing through the timber, the splice should
be much longer than shown. If the two clamps are of stronger
wood than the main timber, they need not together have so
much depth as the net depth of the timber. The iron strap
in XIY. illustrates the same principle. The bolts are usually-
small, and serve mainly to balance the moment set up on each
clamp by the pressure on the shoulder and the tension in the
neck. The modification in XII. permits the introduction of the
bolts without reducing the net section of the timber. In XIIL,
each indent is only half the previous depth, with obvious
economy of the main timber, and increase of shearing area of
clamp and timber without lengthening the clamps. It is much
more difiicult to fashion, however, and it is not probable that
both shoulders on one half will bear equally.
XV. and XVI. are scarfed joints. The tension sections,
the compression shoulders and the longitudinal shearing
planes should again be properly proportioned here. In XV.,
but one-third of the timber is available, if unit tension and
compression have the same numerical value, while in XVI.
one-half of the stick is useful; but the latter joint is more
troublesome to fashion. The bolts serve to resist the moment
which tends to open the joint, and, by resisting it, cause a
fairly uniform distribution of stress in the critical section.
The bolt-holes do not weaken the timber. Sometimes the ex-
treme ends of the scarf are undercut to check the tendency to
spring out when the bolts are not used. Keys may be driven
through places cut for them at the shoulders. The joint can
then be readily assembled and forced to place. These sketches
show that timber, although possessing good tensile strength,
is ill-adapted for ties, on account of the great loss of section
in connections and joints.
103. Struts and Ties. — The connection of a strut and
tie in wood is illustrated in II., III., IV. and VII. The shrink-
ago of the pieces of II. in seasoning tends to open botli por-
EOOF-TRUSSES. 79
tions of the joint by changing the angles ; but the bearing of
the strut is still central, if only on a small area. The com-
pression of the tie across the grain may be large in such a
case, and the introduction of a block, as in IV., will remedy
such a difficulty as well as that from shrinkage. The block
below is the wall-plate, for distributing the truss load along
the wall. It is subjected to compression across the grain.
If the shearing area to the left in these four cases is not suf-
ficient, the bolt or strap is a wise provision to take up the
horizontal component. The bolt, if a little oblique to the strut,
as shown, holds at once by tension, to some degree, and not
alone by shear. It also relieves the smallest section of the tie
from a part of the tension. The square shoulders of III. are
good, if the timber is seasoned, as the bearing is then over the
whole end of the strut, and the tie is not weakened any more
than in II., while the joint is more simply laid out. The strap
of VII. gives a satisfactory bearing for the strut, but the fast-
enings of such a strap are often weaker than the strap itself.
The holes in it may well be enlarged hot, without removal of
metal and diminution of cross-section.
In VIII., IX. and X. are shown connections of struts which
may at some time be called on to resist tension, or which may
be relieved of stress and become loose. The tenon in VIII.
must be pinned to carry tension; and the pin will resist but
little before shearing out of the tenon or splitting oflf the side
of the other timber by tension across the grain. The tenon
should be fashioned as indicated, with sufficient area at the
left-hand edge to carry the perpendicular component of the
thrust of the strut as compression across the grain, and suf-
ficient cross-section not to shear off. The size of the strut
must be determined, not only by the column strength, but by
the area necessary to prevent crushing the piece against which
it abuts. This remark applies to IX. and X. also. The abil-
ity of IX. to carry tension depends on the resistance of the
nut, which is slipped into a hole at the side, to shearing out
along the strut, or crushing the fibres on which it bears, the
80 ROOF-TRUSSES.
latter method of failure being the more likely, unless the nut
is quite near the end of the strut. The strap on X. is very
effective, and the arrangement, if inverted, will serve as a sus-
pending piece, although a rod is better. Many of these con-
nections are serviceable in other positions.
To keep a strut from crushing the side of a timber, a con-
nection may be employed, as in the lower part of I. This
device may be economical, if a number of such joints are to
be made, and it is superior to a mortise in work exposed to
the weather, as there is no place for water to lodge. The post
in XVIII. is capped by a similar device for distributing and
thus reducing the unit pressure on the other piece. Lateral
displacement is provided against in both cases by ribs on the
castings.
Strut connections are shown in XIX. and XX., with a tie-
rod in addition. The broad, flat washer reduces the unit com-
pressive stress on the wood under it : the lip keeps water out
of the joint. Shrinkage and a slight deflection of the frame
under a load will cause the mitre joint in XIX to bear at the
top only, throwing the resultant stress out of the axis of the
respective compression members and causing the unit com.
pression at top edge of the joint to be very high. The joint
in XX. gives a better centre pressure, and is easily made ;
the upper piece is simply notched for one-half its depth, and
the upper and lower edges come on the mitre line of XIX.
The connection of XX ., by the insertion of an iron plate or
a block of wood, secures a certain continuity or rigidity in
the joint, to resist a moderate amount of bending moment.
The two pieces might have been halved together. XXYI. is
like VIII., without provision for tension, which is usually un-
necessary. The roof purlin with its block is also shown in
relative position.
104. Beam Connections. — In I. and XVIII. are shown
supports of beams on posts. The double or split cap of I. is
serviceable where several posts are to be connected laterally,
as in a trestle bent, and it is desired to do away with mortises.
ROOF-TRUSSES. 81
Bolts sliould be put transversely through the caps and top
of the post. A comparatively wide bearing for the beam,
without the use of large timber caps, may be here secured.
Lateral bracing, as in XXVIII., will be needed. An indirect
and intermediate support for a beam, by two inclined braces,
is seen in XXV., and the reverse case is represented in XXVII.
A mortise and tenon of usual proportions are shown in XVII.
The ordinary wall bearing for joists may be seen in the lower
left-hand corner. The slanting end is a wise provision to pre-
vent harmful action of the loaded joist on the wall, and it pro-
motes ventilation of the timber.
The usual way of connecting two floor joists or beams,
when their upper surfaces are to be at one level, is drawn in
VI. The nearer the mortises are to the neutral axis, the less
the weakening of the pieces in which they are cut ; on the
other hand, the farther the two tenons are aj^art, the more
firmly is the tenoned joist held against lateral twist. The
shouldered tenon, indicated by the dotted lines at the left, is
designed to attain both objects, to weaken the mortised piece
as little as possible and to have a considerable depth of tenon,
as well as a long tongue projecting entirely through. The
work of framing is considerably more than in the former
case.
105. Wooden Built Beams. — If seasoned material is at
hand, and large timbers are too expensive, a useful beam
may be built up by placing planks, from two to four inches
thick, edge to edge, and then thoroughly nailing or spiking
boards on both sides at an angle of 45° with the length of the
beam, and sloping in opposite directions on the two sides. By
due regard to jointing and nailing a beam of considerable span
may be made at moderate cost. The construction can be
doubled if necessary.
Another compound beam is seen m XXV. The keys and
bolts resist the shear along the neutral axis ; the horizontal
sticks are butted together on the compression side, and are
strapped by the metal clamp indicated to carry tension, if
82 KOOF-TRUSSES.
necessary. The small block behind the clamp keeps it in
place.
106. Curved Beams. — Planks placed side by side, as in
XXII., cut to the form of a curved beam or arched rib, and
bolted together to prevent individual lateral yielding, are quite
effective, if the grain of the wood does not cross the curve too
obliquely. Hence, when the curvature is considerable, it may
be advisable to use short lengths, which must break joint in
the several parallel pieces. It is well to make a deduction of
one piece in computing the strength of the member at any
section. The ratio of strength of this combination, when well
bolted together, to that of a solid stick may be considered to
be as 71 — 1 to n, where n is the number of layers.
If the planks are bent to the curve and laid upon one
another, as in XXIII, this combination is not nearly so effect-
ive as the former, but it can be more cheaply made. The
lack of efficiency arises from the unsatisfactory resistance
offered to shear between the layers by the bolts or spikes.
The strength to resist bending moment will be intermediate
between that of a solid timber and that of the several planks
of which it is composed, with a deduction of one for a prob-
able joint.
If the curved member has a direct force acting upon it
and a moment arising from its curvature, the treatment will
follow the same lines; but the joints, if there are any, will be
more detrimental in case there is tension at any section.
Such curved pieces are sometimes used in open timber trusses
for effect, but their efficiency is low on account of the large
moment due to the curvature. XXIL is the stiffer.
The joints and connecting parts in all timber construction
should be jjroportioned in detail for such tension, compression
and shear as they may have to withstand. Often the three
kinds of stress occur in different parts of one joint or connec-
tion.
107. Iron Roof-truss. — Joints I. to IV., Plate V.,
represent ways of connecting the several pieces of a compara-
ROOF-TUUSSES. 83
tively liglit roof-truss. All the members are made with angles,
and at several points both legs of the tension angles are fast-
ened. Joint I. comes between II. and III., and IV. comes
perpendicularly opposite it. The number of rivets in each of
the ties and centre member of II. depends upon the force in
the particular piece and the rivet shearing value and bearing
value in the thinnest piece. The number of rivets in the raf-
ter likewise depends upon the force it carries, unless the two
rafters are supposed to abut and to transmit so much of the
horizontal component as does not come through the inclined
ties, a treatment not to be commended. The two angle-irons
of the rafter, being in compression, should be connected at
intervals by a rivet and filling piece or thimble. The number
of rivets through the rafter and connection plate at I. need only
be enough to transmit the force from one diagonal to the raf-
ter. Study the necessity for rivets, and do not add all the
rivets in abutting pieces to obtain the number in a main
member.
Similarly, in IV., the first four or possibly five rivets on
the left in the horizontal member balance the rivets in the
inclined tie on the right ; the six remaining rivets seen and
three others unseen, on the left of the splice, balance the same
number in the smaller angle. Note how, by an extension of
the connecting plate and a short plate below, the main tie is
neatly spliced and reduced in section.
The rafter at III. has more rivets than at the upper end
because the thrust is somewhat greater. The rivets in the tie
at that connection will practically equal those at the other end
of the same piece. The black holes at VI. indicate the rivets
to be inserted at the time of erection, and these should, in good
practice, exceed the number called for in joints riveted in the
shop. They must carry the load and resist the moment of the
horizontal component due to the wind pressure, which passes
down the post IX. as shear. The post is subjected to bend-
ing moment as well as compression, and hence has one dimen-
sion much greater than the other. Bracing perpendicular to
84 ROOF-TRUSSES.
the plane of the truss is needed to resist wind pressure on the
end of the structure. Columns and comjDression members, in
structural work of any kind, if joined one to another, must be
thoroughly stayed against lateral movement.
Pin-connected roof-trusses resemble in their details the
joints of the next section.
108. Pin-connected Bridge. — Ordinary details in a pin-
jointed bridge truss of moderate span are shown in VII., VIII.
and XVI. The position of the splice in the top chord is near
the pin. The splice-plate may be extended to reinforce the
pinhole, if required. The ends of the chord pieces are
machined plane and parallel, and only enough rivets are then
used in the splice to insure the alignment. The pin is usually
placed in the centre of gravity of the chord section. The
connection plates are seen below, to keep the sides of the
chord from spreading; the rest of the panel length is usually
laced. Another chord section, employing channels, is drawn
at XI.
XII., XIII. and XIV. show sections for posts. They offer
facilities for the central support of floor-beams. Post flanges
are sometimes turned out, sometimes in. The floor-beam, of
plate-girder type, is riveted at XVI. to the post through the
holes shown. This attachment stiffens the trusses laterally
and is much superior to hangers. Top and bottom lateral
bracing, to convey the wind pressure to the abutments, is
needed in the planes of the chords, and portal bracing at
each end to throw the wind pressure from the top system into
the end posts, which convey it to the abutments as shear,
with the accompanying bending moments in those posts.
The posts go inside of the top chord, as do the main
diagonals or ties, which come next to the posts. The bottom
chord bars are on the outside, one of those running towards
the middle of the span being usually the farthest out.
109. Riveted Bridges. — A riveted Warren girder or
latticed truss is shown below. These details are not for con-
secutive joints. The increase of chord section, when neces-
ROOF-TRUSSES. 86
sary, is indicated at XIX. If the truss is loaded on the top,
interior diagonal bracing, drawn at XXI., must be used.
When the truss is a lattice, the web members are connected
at intersections to stiffen the compression members, as at
XX., or preferably as at V, or at XV., if the web is double.
Horizontal lateral bracing must not be overlooked.
X. is one form of section of a solid bridge floor. Beet-
angular sections are also used.
\
ROOF TRUSSES
ROOF TRUSSES,
ROOF TRUSSES
<^
■■-yf ^
^ --
ROOF TRUSSES.
INDEX.
PAGE
Action of wind 2i
Allowable stresses 74
Analysis, order of 5
Beams, designing 76
Bending moment 61
" " formula 63
" " from equilibri-
um polygon 66
Bending moment on rafter 59
Bracket truss 12
Cambering lower tie, effect of . . . 14
Change of diagonal 18, 42
Compression and tension, to dis-
tinguish between 6, 32
Compression members, designing 75
Curb roof, truss for 33
" " without roller 34
" " with roller 36
Curved members 46
Curved roof-truss 37
Details 77
Diagonal, change of 18, 42
' ' reversal of 53
Diagonals in same quadrilateral,
two 18
Distribution of load 15
Equilibrium polygon 64 to 67
Eyebars 76
Example, general 56
Flat roofs, trusses for 16
Fink truss 13. 54
Forces not applied to joints — 50, 59
Hammer-beam truss 46, 55
" " " amount of
horizontal thrust 47
PAGE
Horizontal thrust .ndeterminate. . 48
" " trusses with 44
Howe truss 20
Imaginary f Drees, use of 50, 58
Inclined forces, trusses under. .22, 33
' ' reactions 7, 8
Irregular section, moment of re-
sistance of 69
I section, moment of resistance, 70
Joints, loads between. 50. 59
" loads on all 14,19
King-post truss 9
Lateral bracing 72
Load and details 72
" between joints 50. 59
" on all joints 14, 19
Lower tie, effect of cambering 14
Materials under stress, action of, 73
" weight of 72
Method of trial and error 55
Moment, bending 61
Moment, formula for bending. ... 63
from equilibrium poly-
gon 66
Moment of resistance 62
" " " irregular sec-
tion 69
Moment of resistance, I section. . . 70
" " " rectangular
section 67
Moment of resistance, T section. .. 68
Moments, reactions found by. .17, 21
Moving load 21
Notation 2
Order of analysis 5
87
88
INDEX.
PAGE
Pins 77
Pulonceau truss 13, 54
Polygou, equilibrium 64, Ho
Pratt truss 20
Pressure, wind- 23
Principle of reciprocity 6
Queen-post truss 16
Kafier, bending moment on 59
Railroad-station roof 53, 56
KeactioQS found by moments,
17, 21, 24
Reactions from wind 24, 28. 35
Reciprocity, principle of 6
Rectangle, moment of resistance, 67
Resistance, moment of 6"i
" " " for various
sections 67 to 70
Reversal of diagonal 53
Roller bearing, effect of 27, 36
Roof, truss to conform to 19
Roof -truss, wooden 10
Scissor truss 44
Snow diagram 33
Snow, weight of 73
Special solutions 53
Stress, action of materials under. . 73
" determining kind of 6, 32
Stresses, allowable 74
" in triangular frame 3, 4
Superfluous pieces 11
Tension and compression, to dis-
tinguish between 6, 32
Tension members, designing 75
Three forces unknown 13, 53
Trapezoidal truss, equal loads. ... 16
" " unequal loads, 16
Trial and error, method of 55
Triangle of forces 1
" " external forces 2, 4
PAGE
Triangular truss. .. . 7
Truss conforming to shape of
equilibrium polygon 16, 39
Truss for curb or mansard roof. .. 33
Truss, Fink 13, 54
Hammer-beam 46, 55
Howe 20
" King-post 9
Pratt 20
" Polonceau 13,54
" Queen-post 16
" Scissor 44
" Trapezoidal 16
' ' Warren 19
Truss to conform to roof 19
" with roller bearing 27, ::6
Trusses for flat roofs 16
" halls 18
" under vertical forces 7, 16
" inclined " ...22,33
" with horizontal thrust 44
T section, moment of resistance, 68
Use of two diagonals in quadri-
lateral 18, 40
Vertical forces, trusses under..?, 16
Warren girder 19
Weight of materials 72
Wind, action of 22
" diagram, reactions, 24, 28. 29, 41
stresses, 25, 30, 34, 43
" on alternate sides, change of
stress 26, 31, 43
Wind-pressure 23
" " on curb or man-
sard roofs 33
Wind-pressure on curved roofs.. 37
" " " pitched or ga-
ble roofs 23
Wooden roof -truss 10
SHORT-TITLE CATALOGUE
OK THE
PUBLICATIONS
OF
JOHN WILEY & SONS.
New York.
Loxdon: chapman ti HALL, Limited.
ARRANGED UNDER SUBJECTS.
Descriptive circulars sent on application. Books marked with an asterisk &t9
sold at net prices only, a double asterisk (**) books sold under the rules of the
American Publishers' Association at net prices subject to an extra charge for
postage. All books are bound in cloth unless otherwise stated.
AGRICULTURE.
Armsbj-'s Manual of Cattle-feeding i2mo, Si 7sr
Principles of Animal Nutrition 8vo, 4 oo
Budd and Hansen's American Horticultural Manual:
Part I. — Propagation, Culture, and Improvement i2mo, i 50
Part II. — Systematic Pomology izmo, i 50
Downing's Fruits and Fruit-trees of America 8vo, 5 oo-
Elliott's Engineering for Land Drainage i2mo, i so-
Practical Farm Drainage lamo, i o
Green's Principles of American Forestry. (Shortly.)
Grotenfelt's Principles of Modern Dairy Practice. (Woll.) i2mo, 2 o
Kemp's Landscape Gardening i2mo, 2 .so
Maynard's Landscape Gardening as Applied to Home Decoration i2mo, i 5
Sanderson's Insects Injurious to Staple Crops i2mo, i 5
Insects Injurious to Garden Crops. (In preparation.)
Insects Injuring Fruits. (In preparation.)
Stockbridge's Rocks and Soils 8vo, 2
Woll's Handbook for Farmers and Dairymen i6mo, i 50
ARCHITECTURE.
Baldwin's Steam Heating for Buildings i2mo, 2 50
Berg's Buildings and Structures of American Railroads 4to, 5 oo-
Birkmire's Planning and Construction of American Theatres 8vo, 3 00
Architectural Iron and Steel 8vo, 3 50
Compound Riveted Girders as Applied in Buildings 8vo, 2 00
Planning and Construction of High Office Buildings 8vo, 3 50
Skeleton Construction in Buildings 8vo, 3 00
Briggs's Modern American School Buildings 8vo, 4 00
Carpenter's Heating and Ventilating of Buildings 8vo, 4 00
Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo, 3 50
Fireproofing of Steel Buildings 8vo, 2 50
French and Ives's Stereotomy 8vo, 2 50
Gerhard's Guide to Sanitary House-inspection i6mo, i 00
Theatre Fires and Panics .... i2mo, i 50-
i
Hatfield's American House Carpenter 8vo, 5 00
Holly's Carpenters' and Joiners' Handbook i8mo, 75
Johnson's Statics by Algebraic and Graphic Methods 8vo, 2 00
Kidder's Architect's and Builder's Pocket-book i6mo, morocco, 4 00
Merrill's Stones for Building and Decoration Svo, 5 00
Monckton's Stair-building 4to, 4 00
Patton's Practical Treatise on Foundations Svo, 5 00
Siebert and Biggin's Modern Stone-cutting and Masonry Svo, i 50
Snow's Principal Species of Wood Svo, 3 50
Sondericker's Graphic Statics with Applications to Trusses, Beams, and Arches.
(Shortly.)
Wait's Engineering 'and Architectural Jurisprudence Svo, 6 00
Sheep, 6 50
Law of Operations PreUminary to Construction in Engineering and Archi-
tecture _ . . Svo, 5 00
Sheep, 5 50
Law of Contracts Svo, 300
Woodbury's Fire Protection of Mills Svo, 2 50
Worcester and Atkinson's Small Hospitals, Establishment and Maintenance,
Suggestions^for Hospital Architecture, with Plans for a Small Hospital.
i2mo, I 25
The World's Columbian Exposition of 1893 Large 4to, i 00
ARMY AND, NAVY.
Bemadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose
Molecule i2mo, 2 50
* Bruff's Text-book Ordnance and Gunnery Svo, 6 00
Chase's Screw Propellers and Marine Propulsion Svo, 3 00
Craig's Azimuth 4to, 3 50
Crehore and Squire's Polarizing Photo-chronograph Svo, 3 00
Cronkhite's Gunnery for Non-commissioned Officers 24mo. morocco, 2 00
* Davis's Elements of Law Svo, 2 50
* Treatise on the Military Law of United States Svo,
* Sheep
De Brack's Cavalry Outpost Duties. (Carr.) 24mo, morocco,
Dietz's Soldier's First Aid Handbook i6mo, morocco,
* Dredge's Modern French Artillery 4to, half morocco,
Durand's Resistance and Propulsion of Ships Svo,
* Dyer's Handbook of Light Artillery. i2mo,
Eissler's Modern High Explosives Svo,
* Fiebeger's Text-book on Field Fortification Small Svo,
Hamilton's The Gunner's Catechism iSmo,
* Hoff's Elementary Naval Tactics Svo,
Ingalls's Handbook of Problems in Direct Fire Svo,
* Ballistic Tables Svo.
* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II . . Svo, each,
* Mahan's Permanent Fortifications. (Mercur.) Svo, half morocco.
Manual for Courts-martial i6mo morocco,
* Mercur's Attack of Fortified Places i2mo,
* Elements of the Art of War Svo,
Metcalf 'sXost of Manufactures — And the Administration of Workshops, Public
and Private Svo,
* Ordnance and Gunnery i2mo,
Murray's Infantry Drill Regulations i8mo, paper,
* Phelps's Practical Marine Surveying Svo,
Powell's Army Officer's Examiner i2mo,
Sharpe's Art of Subsisting Armies in War iSmo, morocco,
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♦ Walke's Lectures on Explosives 8vo, 4 00
* Wheeler's Siege Operations and Military Mining 8vo, 2 00
Winthrop's Abridgment of Military Law i2mo, 2 50
Woodhull's Notes on Military Hygiene i6mo, i 50
Young's Simple Elements of Navigation i6mo. morocco, i 00
Second Edition, Enlarged and Revised i6mo, morocco 2 00
ASSAYING.
Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe.
i2mo, morocco, i 50
Furman's Manual of Practical Assaying 8vo, 3 00
Miller's Manual of Assaying 1 2mo, i 00
O'DriscoU's Notes on the Treatment of Gold Ores 8vo, 2 00
Ricketts and Miller's Notes on Assaying 8vo, 3 00
Ulke's Modern Electrolytic Copper Refining Svo, 3 00
Wilson's Cyanide Processes i2mo, i 50
Chlorination Process 1 2mo . I 50
ASTRONOMY.
Comstock's Field Astronomy for Engineers 8vo, 2 50
raig's Azimuth 4to , 3 50
Doolittle's Treatise on Practical Astronomy 8vo, 4 00
Gore's Elements of Geodesy 8vo, 2 50
Hayford's Text-book of Geodetic Astronomy Svo, 3 00
Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50
* Michie and Harlow's Practical Astronomy Svo, 3 00
* White's Elements of Theoretical and Descriptive Astronomy i2mo, 2 00
BOTANY.
Davenport's Statistical Methods, with Special Reference to Biological Variation.
i6mo, morocco, 1 25
Thome and Bennett's Structural and Physiological Botany i6mo, 2 25
Westermaier's Compendium of General Botany. (Schneider.) Svo, 2 00
CHEMISTRY.
Adriance's Laboratory Calculations and Specific Gravity Tables i2mo, t 25
Allen's Tables for Iron Analysis Svo, 3 00
Arnold's Compendium of Chemistry. (Mandel.) (/n ■preparation.)
Austen's Notes for Chemical Students i2mo, i 50
Bernadou's Smokeless Powder. — Nitro-ceUulose, and Theory of the Cellulose
Molecule i2mo, 2 50
Bolton's Quantitative Analysis Svo, i 50
* Browning's Introduction to the Rarer Elements Svo, i 50
Brush and Penfield's Manual of Determinative Mineralogy Svo, 4 00
Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.) . . . .Svo 3 00
Cohn's Indicators and Test-papers i2mo, 2 00
Tests and Reagents Svo, 3 00
Copeland's Manual of Bacteriology. {In preparation.)
Craft's Short Course in Qualitative Chemical Analysis. (Schaeffer.). . . . i2mo, 2 00
Drechsel's Chemical Reactions. (Merrill.) i2mo, i 25
Duhem's Thermodynamics and Chemistry. (Burgess.) (Shortly.)
Eissler's Modern High Explosives Svo, 4 00
3
Effront's Enzymes and their Applications. (Prescott.) 8vo, 3 00
Erdmann's Introduction to Chemical Preparations. (Dunlap.) i2mo, i 25
Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe.
i2mo, morocco, i 50
Fowler's Sewage Works Analyses 12 mo, 2 00
Fresenius's Manual of Qualitative Chemical Analysis. (Wells.) 8vo, 5 00
Manual of QuaUtative Chemical Analysis. Parti. Descriptive. (Wells.)
» 8vo, 3 00
System of Instruction in Quantitative Chemical Analysis. (Cohn.)
2 vols. (Shortly.)
Fuertes's Water and Public Health i2mo, i 50
Furman's Manual of Practical Assaying 8vo, 3 00
Gill's Gas and Fuel Analysis for Engineers i2mo, i 25
Grotenfelt's Principles of Modern Dairy Practice. ( Woll.) i2mo. 2 00
Hammarsten's Text-book of Physiological Chemistry. (MandeL) 8vo, 4 00
Helm's Principles of Mathematical Chemistry. (Morgan.) i2mo. i 50
Hinds's Inorganic Chemistry 8vo, 3 00
* Laboratory Manual for Students i2mo, 75
Holleman's Text-book of Inorganic Chemistry. (Cooper.) 8vo, 2 50
Text-book of Organic Chemistry. (Walker and Mott.) 8vo, 2 50
Hopkins's Oil-chemists' Handbook 8vo, 3 00
Jackson's Directions for Laboratory Work in Physiological Chemistry. .8vo, r 00
Keep's Cast Iron 8vo, 2 50
Ladd's Manual of Quantitative Chemical Analysis 12 mo i 00
Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 00
Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) i2mo, i 00
Leach's The Inspection and Analysis of Food with Special Reference to State
Control. (In preparation.)
LiJb's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) i2mo, i 00
Mandel's Handbook for Bio-chemical Laboratory i2mo, 1 50
Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.)
3d Edition, Rewritten 8vo, 4 00
Examination of Water. (Chemical and Bacteriological.) i2mo, i 25
Meyer's Determination of Radicles in Carbon Compounds. (Tingle.). . i2mo, 1 00
Miller's Manual of Assaying i2mo, i 00
Mixter's Elementary Text-book of Chemistry 1 2mo , i 50
Morgan's Outline of Theory of Solution and its Results 12 mo, i 00
Elements of Physical Chemistry i2mo. 2 00
Nichols's Water-supply. (Considered mainly from a Chemical and Sanitary
Standpoint, 1883.) 8vo, 2 50
O'Brine's Laboratory Guide in Chemical Analysis 8vo, 2 00
O'Driscoll's Notes on the Treatment of Gold Ores Svo, 2 00
Ost and Kolbeck's Text-book of Chemical Technology. (Lorenz — Bozart.)
(In preparation.)
* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.
8vo, paper, 50
Pictet's The Alkaloids and their Chemical Constitution. (Biddle.) (In
preparation . )
Pinner's Introduction to Organic Chemistry. (Austen.) i2mo, i 50
Poole's Calorific Power of Fuels Svo, 3 00
* Reisig's Guide to Piece-dyeing 8vo, 25 00
Richardsand Woodman's Air ,Water, and Food from a Sanitary Stand point. Svo, 2 00
Richards's Cost of Living as Modified by Sanitary Science i2mo, 1 00
Cost of Food, a Study in Dietaries i2mo, i 00
* Richards and WilUams's The Dietary Computer Svo, i 50
Ricketts and Russell's Skeleton Notes upon Inorganic Chemistry. (Part I. —
Non-metallic Elements.) Svo, morocco, 75
Ricketts and Miller's Notes on Assaying Svo, 3 00
4
d eal's Sewage and the Bacterial Purification of Sewage 8vo, 3 50
Ruddiman's Incompatibilities in Prescriptions 8vo, 2 00
Schimpf's Text-book of Volumetric Analysis izmo, 2 50
Spencer's Handbook for Chemists of Beet-sugar Houses i6pio, morocco, 3 00
Handbook for Sugar Manufacturers and their Chemists. .i6mo, morocco, 2 00
Stockbridge's Rocks and Soils 8vo, 2 50
• Tillman's Elementary Lessons in Heat 8vo, i 50
• Descriptive General Chemistry 8vo 3 00
Treadwell's Qualitative Analysis. (Hall.) 8vo, 3 00
Turneaure and Russell's Public Water-supplies 8vo, 5 00
Van Deventer's Physical Chemistry for Beginners. (Boltwood.) i2mo, i 50
* Walke's Lectures on Explosives 8vo, 4 00
Wells's Laboratory Guide in Qualitative Chemical Analysis 8vo, i 50
Short Course in Inorganic Qualitative Chemical Analysis for Engineering
Students i2mo, i 50
Whipple's Microscopy of Drinking-water 8vo, 3 50
Wiechmann's Sugar Analysis Small 8vo. 2 So
Wilson's Cyanide Processes i2mo, i 50
Chlorination Process 1 2mo i 50
Wulling's Elementary Course in Inorganic Pharmaceutical and Medical Chem-
istry 1 2mo , 2 00
CIVIL ENGINEERING.
BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING.
RAILWAY ENGINEERING.
Baker's Engineers' Surveying Instruments i2mp, 3 00
Bixby's Graphical Computing Table Paper, 19* X 24} inches 25
** Burr's Ancient and Modern Engineering and the Isthmian CanaL (Postage
27 cents additional.) 8vo, n*. 3 50
Comstock's Field Astronomy for Engineers 8vo, 2 50
Davis's Elevation and Stadia Tables 8vo, i 00
EUiott's Engineering for Land Drainage i2mo, i 50
Practical Farm Drainage i2mo, i 00
Folwell's Sewerage. (Designing and Maintenance.) Svo,
Freitag's Architectural Engineering. 2d Edition, Rewritten Svo,
French and Ives's Stereotomy Svo,
Goodhue's Municipal Improvements i2mo,
Goodrich's Economic Disposal of Towns' Refuse Svo,
Gore's Elements of Geodesy Svo,
Hayford's Text-book of Geodetic Astronomy Svo,
Howe's Retaining Walls for Earth i2mo,
Johnson's Theory and Practice of Surveying Small Svo,
Statics by Algebraic and Graphic Methods Svo,
Kiersted's Sewage Disposal i2mo,
Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo,
Mahan's Treatise on Civil Engineering. (1873.) (Wood.) 8vo
* Descriptive Geometry 8vo,
Merriman's Elements of Precise Surveying and Geodesy Svo,
Elements of Sanitary Engineering Svo,
Merriman and Brooks's Handbook for Surveyors i6mo, morocco,
Nugent's Plane Surveying 8 vo ,
Ogden's Sewer Design i2mo,
Patton's Treatise on Civil Engineering Svo, half leather.
Reed's Topographical Drawing and Sketching 4to,
Rideal'slSewage and the Bacterial Purification of Sewage Svo,
Siebert and Biggin's Modern Stone-cutting and Masonry Svo,
Smith's Manual of Topographical Drawing. (McMillan.) Svo,
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Sondericker's Graphic Statics, witn i.pplications to Trusses, Beams, and
Arches. (Shortly.)
* TraiitwLne's Civil Engineer's Pocket-book i6mo, morocco,
Wait's Engineering and Architectural Jixrisprudence 8vo,
Sheep,
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8vo,
Sheep,
Law of Contracts 8vo,
Warren's Stereotomy — Problems in Stone-cutting 8vo,
Webb's Problems in the U?e and Adjustment of Engineering Instruments.
i6mo, morocco,
♦ Wheeler's Elementary Course of Civil Engineering 8vo,
Wilson's Topographic Surveying 8vo,
BRIDGES AND ROOFS.
Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 oo
* Thames River Bridge 4to, paper, s oo
Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and
Suspension Bridges 8vo, 3 50
Du Bois's Mechanics of Engineering. Vol. II Small 4to, 10 00
Foster's Treatise on Wooden Trestle Bridges 4to, 5 00
Fowler's Coffer-dam Process for Piers 8vo, 2 50
Greene's Roof Trusses 8vo, i 25
Bridge Trusses 8vo, 2 50
Arches in Wood, Iron, and Stone 8vo, 2 50
Howe's Treatise on Arches 8vo 4 00
Design of Simple Roof-trusses in Wood and Steel 8vo, 2 00
Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of
Modern Framed Structures Small 4to, 10 00
Merriman and Jacoby's Text-book on Roofs and Bridges:
Part I. — Stresses in Simple Trusses 8vo, 2 50
Part II. — Graphic Statics 8vo, 2 50
Part III. — Bridge Design. 4th Edition, Rewritten 8vo, 2 50
Part IV. — Higher Structures 8vo, 2 50
Morison's Memphis Bridge 4to, 10 00
Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . . i6mo, morocco, 3 00
Specifications for Steel Bridges i2mo, i 25
Wood's Treatise on the Theory of the Construction of Bridges and Roofs.8vo, 2 00
Wright's Designing of Draw-spans:
Part I. — Plate-girder Draws 8vo, 2 50
Part II. — Riveted-truss and Pin-connected Long-span Draws 8vo, 2 50
Two parts in one volume 8vo, 3 50
HYDRAULICS.
Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an
Orifice. (Trautwine. ) 8vo, 2 00
Bovey's Treatise on Hydraulics 8vo, 5 00
Church's Mechanics of Engineering 8vo, 6 00
Diagrams of Mean Velocity of Water in Open Channels paper, i 50
CoflSn's Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50
Flather's Dynamometers, and the Measurement of Power i2mo, 3 00
Folwell's Water-supply Engineering 8vo, 4 00
Frizell's Water-power 8vo, 5 00
Fuertes's Water and Public Health i zmo, i 50
Water-filtration Works i2mo, 2 50
Ganguillet and Kutter's General Formula for the Uniform Flow of Water in
Rivers and Other Channels. (Hering and Trautwine.) 8vo, 4 00
Hazen's Filtration of Public Water-supply 8vo, 3 00
Hazlehurst's Towers and Tanks for Water- works 8vo, 2 50
Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal
Conduits 8vo, 2 00
Mason's Water-supply. (Considered Principally from a Sanitary Stand-
point.) 3d Edition, Rewritten 8vo, 4 00
Merriman's Treatise on Hydraulics, gth Edition, Rewritten 8vo, 5 00
* Michie's Elements of Analytical Mechanics 8vo, 4 00
Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water-
supply Large 8vo, 5 00
** Thomas and Watt's Improvement of Riyers. (Post., 44 c. additional), 4to, 6 00
Turneaure and Russell's Public Water-supplies 8vo. 5 00
Wegmann's Desien and Construction of Dams 4to, 5 00
Water-suoolv of the City of New York from 1658 to 189S 4to, 10 00
Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.) 8vo, 5 00
Wilson's Manual of Irrigation Engineering Small 8vo, 4 00
Wolff's Windmill as a Prime Mover 8vo, ' 3 00
Wood's Turbines 8vo, 2 50
Elements of Analytical Mechanics 8vo, 3 00
MATERIALS OF ENGINEERING.
Baker's Treatise on Masonry Construction 8vo,
Roads and Pavements 8vo,
Black's United States Public Works Oblong 4to,
Bovey's Strength of Materials and Theory of Structures 8vo,
Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edi-
tion, Rewritten 8vo,
Byrne's Highway Construction 8vo,
Inspection of the Materials and Workmanship Employed in Construction.
i6mo.
Church's Mechanics of Engineering 8vo,
Du Bois's Mechanics of Engineering. VoL I Small 4to,
Johnson's Materials of Construction Large 8vo,
Keep's Cast Iron 8vo,
Lanza's Applied Mechanics 8vo,
Martens's Handbook on Testing Materials. (Henning.) 2 vols 8vo,
Merrill's Stones for Building and Decoration 8vo,
Merriman's Text-book on the Mechanics of Materials 8vo, 4 00
Strength of Materials lamo, i 00
Metcalf's Steel. A Manual for Steel-users i2mo, 2 00
Patton's Practical Treatise on Foundations 8vo, 5 00
Rockwell's Roads and Pavements in France i2mo, i 25
Smith's Wire : Its Use and Manufacture Small 4to, 3 00
Materials of Machines 1 2mo, i 00
Snow's Principal Species of Wood 8vo, 3 50
Spalding's Hydraulic Cement i2mo, 2 00
Text-book on Roads and Pavements i2mo, 2 00
Thurston's Materials of Engineering. 3 Parts 8vo, 8 00
Part I. — Non-metaUic Materials of Engineering and Metallurgy 8vo, 2 00
Part II. — Iron and Steel 8vo, 3 50
Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8vo, 2^0
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Thurston's Text-book of the Materials of Construction 8vo, 5 00
TiUson's Street Pavements and Paving Materials 8vo, 4 00
Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.). . i6mo, mor., 3 00
Specifications for Steel Bridges , i2mo, 1 25
"Wood's Treatise on the Resistance of Materials, and an Appendix on the Pres-
ervation of Timber 8vo, 2 00
Elements of Analytical Mechanics 8vo, 3 00
RAILWAY ENGINEERING.
Andrews's Handbook for Street Railway Engineers. 3X5 inches, morocco, i 25
Berg's Buildings and Structures of American Railroads 4to, 5 00
Brooks's Handbook of Street Railroad Location i6mo, morocco, 1 50
Butts's Civil Engineer's Field-book i6mo, morocco, 2 50
Crandall's Transition Curve i6mo, morocco, 1 50
Railway and Other Earthwork Tables 8vo, i 50
Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 4 00
Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 00
* Drinker's Tunneling, Explosive Compounds, and Rock Drills, 4to, half mor., 25 00
Fisher's Table of Cubic Yards Cardboard, 25
Godwin's Railroad Engineers' Field-book and Explorers' Guide i6mo, mor., 2 50
Howard's Transition Curve Field-book i6mo morocco i so
Hudson's Tables for Calculating the Cubic Contents of Excavations and Em-
bankments 8vo, I 00
Molitor and Beard's Manual for Resident Engineers i6mo, i 00
Nagle's Field Manual for Railroad Engineers i6mo, morocco. .^ 00
Philbrick's Field Manual for Engineers i6mo, morocco, 3 00
Pratt and Alden's Street-railway Road-bed Svo, 2 00
Searles's Field Engineering i6mo, morocco, 3 00
Railroad Spiral i6mo, morocco i 50
Taylor's Prismoidal Formulae and Earthwork Svo, 1 50
* Trautwine's Method of Calculating the Cubic Contents of Excavations and
Embankments by the Aid of Diagrams Svo, 2 00
he Field Practice of .Laying Out Circular Curves for Raibroads.
i2mo, morocco, 2 50
* Cross-section Sheet Paper, 25
Webb's Railroad Construction. 2d Edition, Rewritten i6mn. morocco, 5 00
Wellington's Economic Theory of the Location of Railways Small Svo, 5 00
DRAWING.
Barr's Kinematics of Machinery Svo, 2 50
• Bartlett's Mechanical Drawing Svo, 3 00
Coolidge's Manual of Drawing Svo, paper, i 00
Durley's Kinematics of Machines Svo, 4 00
Hill's Text-book on Shades and Shadows, and Perspective Svo, 2 00
Jones's Machine Design:
Part I. — Kinematics of Machinery Svo,
Part II. — Form, Strength, and Proportions of Parts Svo,
MacCord's Elements of Descriptive Geometry Svo,
Kinematics; or. Practical Mechanism Svo,
Mechanical Drawing 4to,
Velocity Diagrams ." Svo,
♦ Mahan's Descriptive Geometry and Stone-cutting Svo,
Industrial Drawing. (Thompson.) Svo,
Reed's Topographical Drawing and Sketching 4to,
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Reid's Cotirse in Mechanical Drawing 8vo,
Text-book of Mechanical Drawing and Elementary Machine Design. .8vo,
Robinson's Principles of Mechanism 8vo,
Smith's Manual of Topographical Drawing. (McMillan.) Svo,
Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. . i2mo,
Drafting Instruments and Operations i2mo,
Manual of Elementary Projection Drawing i2mo.
Manual of Elementary Broblems in the Linear Perspective of Form and
Shadow i2mo,
Plane Problems in Elementary Geometry i2mo,
Primary Geometry 1 2mo,
Elements of Descriptive Geometry, Shadows, and^Perspective 8vo,
General Problems of Shades and Shadows Svo,
Elements of Machine Construction and Drawing Svo,
Problems. Theorems, and Examples in Descriptive Geometry Svo,
Weisbach's Kinematics and the Power of Transmission. v Hermann an'*
Klein.) Svo, 5 00
Whelpley's Practical Instruction in the Art of Letter Engraving i2mo, 2 00
Wilson's Topographic Surveying Svo, 3 50
Free-hand Perspective Svo, 2 50
Free-hand Lettering. {In preparation.)
Woolf's Elementary Course in Descriptive Geometry Large Svo, 3 00
'ELECTRICITY AND PHYSICS.
Anthony and Brackett's Text-book of Physics. (Magie.). ...... .Small Svo. 3 00
Anthony's Lecture-notes on the Theory of Electrical Measurements i2mo, 1 00
Benjamin'slHistory of Electricity Svo, 3 00
Voltaic CelL 8vo, 3 00
Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .Svo, 3 00
Crehore and Squier's Polarizing Photo-chronograph Svo, 3 00
Dawson's "Encineering" and Electric Traction Pocket-book. . lomo, morocco, 4 00
Klather's Dynamometers, and the Measurement of Power i2mo, 3 00
Gilbert's De Magnete. (Mottelay.) Svo, 2 50
Hohnan's Precision of Measurements Svo, 2 00
Telescopic Mirror-scale Method, Adjustments, and Tests Large »vo 75
Lanaauer's Spectrum Analysis. (Tingle.) Svo, 3 00
Le ChateUer's High-temperature Measurements. (Boudouard — iJurgess. )i2mo, 3 00
Lob's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) i2mo, i 00
* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and 11. Svo, each, 6 00
* Michie. Elements of Wave Motion Relating to Sound'and Light. ... ^. .Svo, 4 00
Niaudet's Elementary Treatise on Electric Batteries. (FishoacK. i i2mo, 2 50
* Parshall and Hobart's Electric Generators Small 4to. half morocco, 10 00
* Rosenberg's Electrical Engineering. (HaldaneGee — Kinzbninner.). . . .Svo, i 50
Ryan, Norris, and Hoxie's Electrical Machinery. (In preparalioi'.'
Thurston's Stationary Steam-engines Svo, 2 50
* Tillman's Elementary Lessons in Heat Svo, 1 50
Tory and Pitcher's Manual of Laboratory Physics Small Svo, 2 00
Ulke's Modern Electrolytic Copper Refining Svo, 3 00
LAW.
*^Dayis's Elements of Law Svo, 2 50
• Treatise on the MiUtary Law of United States Svo, 7 00
♦ Sheep, 7 50
Manual for Courts-martial i6mo, morocco, 1 50
6
00
6
so
5
00
5
50
3
00
2
50
Wait's Engineering and Architectural Jurisprudence 8vo,
Sheep,
Law of Operations Preliminary to Construction in Engineering'and Archi-
tecture 8vo,
Sheep,
Law of Contracts 8vo,
Winthrop's Abridgment of Military Law lamo.
MANUFACTURES.
Bernadou's Smokeless Powder — Nitro-cellulose and Theory of the Cellulose
Molecule i2mo, 2 50
Bolland's Iron Founder i2mo, 2 50
" The Iron Founder," Supplement i2mo, 2 50
Encyclopedia of Founding and Dictionary oflFoundry Terms Used in the
Practice of Moulding 1 2mo, 3 00
Eissler's Modem High Explosives 8vo, 4 00
Efifront's Enzymes and their Applications. (Prescott.) 8vo, 3 00
Fitzgerald's Boston Machinist i8mo, i 00
Ford's Boiler Making for Boiler Makers i8mo, i 00
Hopkins's Oil-chemists' Handbook 8vo, 3 00
Keep's Cast Iron 8vo, 2 50
Leach's The Inspection and Analysis of Food with SpeciaI]Reference to State
Control. (In preparation.)
Metcalf 's Steel. A Manual for Steel-users i2mo, 2 00
Metcalfe's Cost of Manufactures — And the Administration of Workshops,
Public and Private 8vo,
Meyer's Modern Locomotive Construction 4to,
* Reisig's Guide to Piece-dyeing 8vo,
Smith's Press-working of Metals 8vo,
Wire: Its Use and Manufacture Small 4to,
Spalding's Hydraulic Cement i2mo,
Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco,
Handboo'K tor bugar Manufacturers ana their Chemists.. . i6mo, morocco,
Thurston's Manual of Steam-boilers, their Designs, Construction and Opera-
tion 8vo,
* Walke's Lectures on Explosives 8vo,
West's American Foundry Practice i2mo.
Moulder's Text-book < i2mo,
Wiechmann's Sugar Analysis Small 8vo,
Wolff's Windtnill as a Prime Mover 8vo,
Woodbury's Fire Protection of Mills 8vo,
MATHEMATICS.
Baker's Elliptic Functions 8vo, i 50
♦ Bass's Elements of Differential Calculus z2mo, 4 00
Briggs's Elements of Plane Analytic Geometry i2mo, i 00
Chapman's Elementary Course in Theory of Equations i2mo, i 50
Compton's Manual of Logarithmic Computations lamo, i 50
Davis's Introduction to the Logic of Algebra 8vo, i 50
♦ Dickson's College Algebra Large i2mo, i 50
♦ Introduction to the Theory of Algebraic Equations Largeliamo, i 25
Halsted's Elements of Geometry Svo, i 75
Elementary Synthetic Geometry Svo, 1 50
10
5
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10
00
25
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3
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3
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2
00
3
00
2
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5
00
4
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2
50
2
50
2
50
♦Johnson's Three-place Logarithmic Tables: Vest-pocket size paper, is
100 copies for 5 00
• ■' Mounted on heavy cardboard, 8 X lo inches, as
10 copies for 2 00
Elementary Treatise on the Integral Calculus Small 8vo, i 50
Curve Tracing in Cartesian Co-ordinates i2mo, i 00
Treatise on Ordinary and Partial Differential Equations Small 8vo, 3 50
Theory of Errors and the Method of Least Squares i2mo, i 50
• Theoretical Mechanics 1 2mo, 3 00
Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) i2mo, 2 00
• Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other
Tables 8vo, 3 00
Trigonometry and Tables published separately Each, 2 00
Maurer's Technical Mechanics. (In preparation.)
Merriman and Woodward's Higher Mathematics 8vo, 5 00
Merriman's Method of Least Squares 8vo, a 00
Rice and Johnson's Elementary Treatise on the Differential Calculus. Sm., 8vo, 3 00
Differential and Integral Calculus. 2 vols, in one Gmall 8vo, 2 50
Wood's Elements of Co-ordinate Geometry 8vo, 2 00
Trigonometry: Analytical, Plane, and Spherical i2mo, i 00
MECHANICAL ENGINEERING.
MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS.
Baldwin's Steam Heating for Buildings i2mo,
Barr's Kinematics of Machinery 8vo,
• Bartlett's Mechanical Drawing 8vo,
Benjamin's Wrinkles and Recipes izmo.
Carpenter's Experimental Engineering 8vo,
Heating and Ventilating Buildings 8vo,
Clerk's Gas and Oil Engine Small 8vo,
Coolidge's Manual of Drawing 8vo, paper,
Cromwell's Treatise on Toothed Gearing i2mo.
Treatise on Belts and Pulleys i2mo,
Durley's Elinematics of Machines 8vo,
Flather's Dynamometers and the Measurement of Power i2mo.
Rope Driving i2mo.
Gill's Gas and Fuel Analysis for Engineers i2mo,
HaU's Car Lubrication i2mo,
Button's The Gas Engine. (In preparation.)
Jones's Machine Design:
Part I. — Kinematics of Machinery v- -Svo, i 50
Part II. — Form, Strength, and Proportions of Parts. 8vo,
Kent's Mechanical Engineer's Pocket-book i6mo, morocco,
Kerr's Power and Power Transmission 8vo,
MacCord's Kinematics; or, Practical Mechanism 8vo,
Mechanical Drawing 4to,
Velocity Diagrams 8vo,
Mahan's Industrial Drawing. (Thompson.) 8vo,
Poole's Calorific Power of Fuels 8vo,
Reid's Course in Mechanical Drawing 8vo. 2 00
Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 3 00
Richards's Compressed Air i2mo, 1 50
Robinson's Principles of Mechanism 8vo, 3 00
Smith's Press-working of Metals - - 8vo 3 00
Thurston's Treatise on Friction and Lost Work in Machinery and Mil
Work 8vo, 3 00
Animal as a Machine and Prime Motor, and the Laws of Energetics. 1 2mo, i 00
11
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50
2
50
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2
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35
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2
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5
00
4
00
I
50
3
50
3
00
Warren's Elements of Machine Construction and Drawing 8vo, 7 50
Weisbach's Kinematics and the Power of Transmission. Herrmann —
Klein.) 8vo, 5 00
Machinery of Transmission and Governors. (Herrmann — Klein.). .8vo, 5 00
Hydraulics and Hydraulic Motors. (Du Bois.) 8vo, 5 00
Wolff's Windmill as a Prime Mover 8vo, 3 00
Wood's Turbines 8vo, 2 50
MATERIALS OF ENGINEERING.
Bovey's Strength of Materials and Theory of Structures 8vo, 7 50
Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition,
Reset 8vo , 750
Church's Mechanics of Engineering 8vo, 6 00
Johnson's Materials of Construction Large Svo, 6 00
Keep's Cast Iron Svo 2 50
Lanza's Applied Mechanics Svo, 7 50
Martens's Handbook on Testing Materials. (Henning.) Svo, 7 50
Merriman's Text-book on the Mechanics of Materials Svo, 4 00
Strength of Materials i2mo, i 00
Metcalf's Steel. A Manual for Steel-users i2mo 2 00
Smith's Wire: Its Use and Manufacture Small 4to, 3 00
Materials of Machines i2mo, i 00
Thurston's Materials of Engineering 3 vols , Svo, 8 00
Part II. — Iron and Steel Svo, 3 50
Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents Svo, 2 50
Text-book of the Materials of Construction Svo 5 00
Wood's Treatise on the Resistance of Materials and an Appendix on the
Preservation of Timber Svo, 2 00
Elements of Analytical Mechanics Svo, 3 00
STEAM-ENGINES AND BOILERS.
Carnot's Reflections on the Motive Power of Heat. (Thurston.) i2mo, i 50
Dawson's "Engineering" and Electric Traction Pocket-book. . T6mo, mor., 4 00
Ford's Boiler Making for Boiler Makers iSmo, i 00
Goss's Locomotive Sparks Svo, 2 00
Hemenway's Indicator Practice and Steam-engine Economy i2mo, 2 00
Button's Mechanical Engineering of Power Plants Svo, 5 00
Heat and Heat-engines Svo, 5 00
Kent's Steam-bo'ler Economy Svo, 4 00
Kneass's Practice and Theory of the Injector Svo. i 50
MacCord's Slide-valves Svo, 2 00
Meyer's Modern Locomotive Construction 4to, 10 00
Peabody's Manual of the Steam-engine Indicator i2mo, i 50
Tables of the Properties of Saturated Steam and Other Vapors Svo, 1 00
Thermodynamics of the Steam-engine and Other Heat-engines Svo, 5 00
Valve-gears for Steam-engines Svo, 2 50
Peabody and Miller's Steam-boilers Svo, 4 00
Pray's Twenty Years with the Indicator Large Svo, 2 50
Pupln's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.
(Osterberg.) i2mo, i 25
Reagan's Locomotives : Simple, Compound, and Electric i2mo, 2 50
Rontgen's Principles of Thermodynamics. (Du Bois.) Svo, 5 00
Sinclair's Locomotive Engine Running and Management i2mo, 2 00
Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50
Snow's Steam-boiler Practice Svo, 3 00
12
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6
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5
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I
50
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5
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5
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4
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Spangler's Valve-gears 8vo, 2 50
Notes on Thermodynamics i2mo, i 00
Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo,
Thurston's Handy Tables 8vo.
Manual of the Steam-engine 2 vols. 8vo,
Part I. — History, Structuce, and Theory 8vo,
Part n. — Design, Construction, and Operation 8vo,
Handbook of Engine and Boiler Trials, and the Use of the Indicator and
the Prony Brake 8vo,
Stationary Steam-engines 8vo,
Steam-boiler Explosions in Theory and in Practice i2mo,
Manual of Steam-boiler? , Their Designs, Construction, and Operation. 8vo,
Weisbach's Heat, Steam, a J Steam-engines. (Du Bois.) 8vo,
Whitham's Steam-engine 1 esign 8vo,
Wilson's Treatise on Steam-boilers. (Flather.) i6mo.
Wood's Thermodynamics. Heat Motors, and Refrigerating Machines. . . .8vo,
MECHANICS A.ND MACHINERY.
Barr's Kinematics ot machinery 8vo, 2 50
Bovey's Strength of Materials and Theory of Structures 8vo, 7 50
Chase's The Art of Pattern-making i2mo, 2 50
Chordal. — Extracts from Letters i2mo, 2 00
Church's Mechanics of Engineering 8vo 6 00
Notes and Examples in Mechanics 8vo, 2 00
Compton's First Lessons in Metal-working i2mo, i 50
Compton and De Groodt's The Speed Lathe i2mo, i so
Cromwell's Treatise on Toothed Gearing i2mo, i 50
Treatise on Belts and Pulleys i2mo, 1 50
Dana's Text-book of Elementary Mechanics for the Use of Colleges and
Schools i2mo, 1 50
Dingey's Machinery Pattern Making i2mo, 2 00
Dredge's Record of the Transportation Exhibits Building of the World's
Columbian Exposition of i8g3 4to, half morocco, 5 00
Du Bois's Elementary Principles of Mechanics:
Vol. I. — Kinematics 8vo,
Vol. II.— Statics 8vo,
Vol. III.— Kinetics 8vo,
Mechanics of Engineering. Vol. I Small 4to,
Vol. II SmaU 4to,
Durley's Kinematics of Machines 8vo,
Fitzgerald's Boston Machinist i6mo, i 00
Flather's Dynamometers, and the Measurement of Power i2mo, 3 00
Rope Driving i2mo, 2 00
Goss's Locomotive Sparks 8vo, 2 00
Hall's Car Lubrication i2mo, i 00
Holly's Art of Saw Filing i8mo 75
♦ Johnson's Theoretical Mechanics i2mo, . 3 00
Statics by Graphic and Algebraic Methods 8vo, 2 00
Jones's Machine Design:
Part I. — Kinematics of Machinery 8vo, i 50
Part II.^Form, Strength, and Proportions of Parts 8vo, 3 00
Kerr's Power and Power Transmission 8vo, 2 00
Lanza's Applied Mechanics 8vo, 7 50
MacCord's Kinematics; or, Practical Mechanism 8vo, 5 00
Velocity Diagrams 8vo, i 50
Maurer's Technical Mechanics. (In preparation^)
13
3
50
4
00
3
50
7
50
10
00
4
00
Merriman's Text-book on the Mechanics of Materials 8vo, 4 00
•^Michie's Elements of Analytical Mechanics 8vo, 4 00
Reagan's Locomotives: Simple, Compound, and Electric 1 2mo, 2 50
Reid's Course^in Mechanical Drawing 8vo, 2 00
Text -book of^Mechanical Drawing and Elementary Machine Design. .8vo, 3 00
Richards's Compressed Air 1 2mo, i 50
Robinson's Principles of Mechanism 8vo, 3 00
Ryan, Norris, and Hoxie's Electrical Machinery. (In preparation.)
Sinclair's Locomotive-engine Running and]Management izmo, 2 00
Smith's Press-working of Metals 8vo, 3 00
< Materials of Machines i2mo, 1 00
Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 00
Thurston's Treatise on Friction and Lost Work in Machinery and Mill
"Work 8vo, 3 00
Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, i 00
Warren's Elements of Machine Construction and Drawing 8vo, 7 50
Weisbach's Kinematics land the Power of Transmission. (Herrmann —
Klein.) 8vo, 5 00
Machinery of Transmission and Governors. (Herrmann — Klein. ).8vo, 5 00
Wood's Elements of Analytical Mechanics 8vo, 3 00
Principles of Elementary Mechanics i2mo, i 25
Turbines 8vo, 2 50
The World's Columbian Exposition of 1893 4to, i 00
METALLURGY.
Egleston's Metallurgy of Silver, Gold, and Mercury:
VoL I. — Silver 8vo, 7 50
VoL n. — Gold and Mercury 8vo, 7 50
** Iles's Lead-smelting. (Postage g cents additional.) i2mo, 2 50
Keep's Cast Iron 8vo, 2 50
Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50
Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.) . i2mo, 3 00
Metcalf's Steel. A Manual for Steel-users i2mo, 2 00
Smith's Materials of Machines i2mo, i 00
Thurston's Materials of Engineering. In Three Parts 8vo, 8 00
Part II. — Iron and Steel 8vo, 3 50
Part III.— A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8vo, 2 50
Hike's Modern Electrolytic Copper Refining 8vo, 3 00
MINERALOGY.
Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50
Boyd's Resources of Southwest Virginia 8vo, 3 00
Map of Southwest Virginia Pocket-book form, 2 00
Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 4 00
Chester's Catalogue of Minerals 8vo, paper, i 00
Cloth, 1 25
Dictionary of the Names of Minerals 8vo, 3 50
Dana's System of Mineralogy Large 8vo, half leather, 12 50
First Appendix to Dana's New "System of Mineralogy.". . . .Large 8vo, i 00
Text-book of Mineralogy 8vo, 4 00
Minerals and How to Study Them . . . : i2mo, 1 50
Catalogue of American Localities of Minerals Large 8vo, i 00
Manual of Mineralog^y and Petrography i2mo, 2 00
Egleston's Catalogue of Minerals and Synonyms 8vo, 2 50
Hussak's The Determination of Rock-forming Minerals. (Smith.) Small 8vo, 2 00
14
♦ Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.
8vo, paper, o 50
Rosenbusch's Microscopical Physiography of the Rock-making Minerals.
(Iddings.) 8vo, 5 00
• Tillman's Text-book of Important Minerals and Docks 8vo, 2 00
Williams's Manual of Lithology 8vo, 3 00
MINING.
Beard's Ventilation of Mines i2mo, 2 so
Boyd's Resources of Southwest Virginia 8vo, 3 00
Map of Southwest Virginia Pocket-book form, 2 00
♦ Drinker's Tunneling, Explosive Compounds, and Rock Drills.
4to, half morocco, 25 00
Eissler's Modem High Explosives 8vo, 4 00
Fowler's Sewage Works Analyses i2mo, 2 00
Goodyear's Coal-mines of the Western Coast of the United States i2mo, 2 50
Ihlseng's Manual of Mining . 8vo, 4 00
** Iles's Lead-smelting. (Postage gc. additional.) i2mo, 2 50
Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50
O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 00
• Walke's Lectures on Explosives 8vo, 4 00
Wilson's Cyanide Processes i2mo, i 50
Chlorination Process i2mo, i 50
Hydraulic and Placer Mining i2mo, 2 00
Treatise on Practical and Theoretical Mine Ventilation i2mo, i 25
SANITARY SCIENCE.
Copeland's Manual of Bacteriology. {In preparation.)
Folwell's Sewerage. (Designing, Construction and Maintenance.; 8vo, 3 00
Water-supply Engineering 8vo, 4 00
Fuertes's Water and Public Health i2mo, i 50
Water-filtration Works i2mo, 2 50
Gerhard's Guide to Sanitary House-inspection i6mo, 1 00
Goodrich's Economical Disposal of Town's Refuse Demy 8vo, 3 50
Hazen's Filtration of Public Water-supplies 8vo, 3 00
Kiersted's Sewage Disposal i2mo, r 25
Leach's The Inspection and Analysis of Food with Special Reference to State
ControL (In preparation.)
Mason's Water-supply. (Considered Principally from a Sanitary Stand-
point.) 3d Edition, Rewritten 8vo, 4 00
Examination of Water. (Chemical and BacteriologicaL) 12 mo, i 25
Merriman's Elements of Sanitary Engineering 8vo, 2 00
Nichols's Water-supply. (Considered Mainly from a Chemical and Sanitary
Standpoint.) (1883.) 8vo, 2 50
Ogden's Sewer Design i2mo, 2 00
* Price's Handbook on Sanitation i2mo, i 50
Richards's Cost of Food. A Study in Dietaries i2mo, i 00
Cost of Living as Modified by Sanitary^Science i2mo, i 00
xUchards and Woodman's Air, Water, and Food from a Sanitary Stand-
point 8vo, 3 00
* Richards and Williams's The DietarylComputer 8vo, i 50
Rideal's Sewage and Bacterial Purification of Sewage 8vo, 3 50
Turneaure and Russell's Public Water-supplies 8vo, 5 00
Whipple's Microscopy of Drinking-water 8vo, 3 50
Woodhull's Notes^and Military Hygiene i6mo, i 50
15
MISCELLANEOUS.
Barker's Deep-sea Soundings 8vo, 2 00
Emmons's Geological Guide-book of the Rocky Mountain Excursion of the
International Congress of Geologists Large 8vo, i 50
Ferrel's Popular Treatise on the Winds. . 8vo, 4 00
Haines's American Railway Management i2mo, 2 50
Mott's Composition.'Digestibility, and Nutritive Value of Food. Mounted chart, i 25
Fallacy of the Present Theory of Sound i6mo, i 00
Ricketts's History of Rensselaer Polytechnic Institute, 1824-1894. Small 8vo, 3 00
Rotherham's Kmphasized New Testament Large 8vo, 2 00
Steel's Treatise on the Diseases of the Dog 8vo, 3 50
Totten's Important Question in Metrology 8vo, 2 50
The World's Columbian Exposition of 1893 4to, i 00
Worcester and Atkinson. Small Hospitals, Establishment and Maintenance,
and Suggestions for Hospital Architecture, with Plans for a Small
Hospital i2mo, i 25
HEBREW AND CHALDEE TEXT-BOOKS.
Green's Grammar of the Hebrew Language 8vo, 3 00
Elementary Hebrew Grammar i2mo, i 25
Hebrew Chrestomathy 8vo , 2 00
Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures.
(Tregelles.) Small 4to, half morocco, 5 00
Letteris's Hebrew Bible 8vo, 2 25
16
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