rec
LIBRARY
UNIVERSITY OF CALIFORNIA.
Class
MATHEMATICAL TEXTS
Edited by
PERCEY F. SMITH, PH.D.
Professor of Mathematics in the Sheffield Scientific School of
Yale University
Elements of Differential and Integral Calculus
By W. A. GRANVILLE, Pn.D.
Elements of Analytic Geometry
By P. F. SMITH and A. S. GALE, PH.D.
Introduction to Analytic Geometry
By P. F. SMITH and A. S. GALE, PH.D.
Theoretical Mechanics In press
By P. F. SMITH and O. C. LESTER, PH.D.
Advanced Algebra
By H. E. HAWKES, Pn.D.
Textbook on the Strength of Materials
By S. E. SLOCUM, PH.D., and E. L. HANCOCK, M.Sc.
Problems in Strength of Materials
By WILLIAM KENT SIIEPARD, Ph.D.
PROBLEMS IN
STRENGTH OF MATERIALS
BY
WILLIAM KENT SHEPARD, PH.D.
INSTRUCTOR OF MECHANICS IN THE SHEFFIELD SCIENTIFIC SCHOOL OF
YALE UNIVERSITY
GINN & COMPANY
BOSTON NEW YORK CHICAGO LONDON
\
COPYRIGHT, 1907
BY WILLIAM KENT SHEPARD
ALL RIGHTS RESERVED
67.9
GINN & COMPANY PRO
PRIETORS BOSTON U.S.A.
PKEFACE
For the average student to obtain a working knowledge of any
scientific subject it is necessary that he solve numerous problems.
This is especially true in the study of mechanics. In teaching
Strength of Materials the author has found that the textbooks do
not give a sufficient number of examples to completely familiarize
the student with the application of the theory.
The aim of this book is to furnish a large variety of problems on
each part of the subject, and thus relieve the instructor of tedious
dictation in the class room.
A discussion of riveted joints is given for use in the computation
and design of such joints as are often found in boiler construction.
No definite notation is adopted in order that the book may be
used in connection with a course of lectures or with any textbook
on the subject.
Tables at the back of the book give all the data necessary
for solving the problems, but answers have been omitted in order
to emphasize that the goal is a proper solution and not a mere
numerical answer.
I wish to thank Professor C. B. Eichards for suggesting numerous
examples and for other valuable assistance in compiling this book.
o PT o c
CONTENTS
PROBLEMS PAGE
I. TENSION, COMPRESSION, AND SHEAR .... 136 13
II. ELASTIC DEFORMATION 3785 48
III. THIN CYLINDERS AND SPHERES 86117 911
IV. RIVETED JOINTS 118166 1222
V. CANTILEVER AND SIMPLE BEAMS :
Shear and Moment Diagrams 167206 2325
Neutral Axis and Moments of Inertia . . . 207240 2628
Investigation 241316 2934
Rupture 317327 35
Moving Loads 328333 36
Deflection 334365 3638
VI. OVERHANGING BEAMS 366380 3940
VII. FIXED BEAMS 381391 4142
VIII. CONTINUOUS BEAMS 392404 43
IX. COLUMNS AND STRUTS 405450 4447
X. TORSION 451500 4851
XI. COMBINED STRESSES 501528 5254
XII. COMPOUND COLUMNS AND BEAMS 529545 5556
XIII. THICK CYLINDERS AND GUNS 546560 5758
XIV. FLAT PLATES 561568 59
TABLES 6170
vii
OF THE
{ UNIVERSITY )
OF
PROBLEMS IN STRENGTH OF
MATERIALS
I. TENSION, COMPKESSION, AND SHEAE
1. Find the breaking unitstress for a round rod 1J inches in
diameter which breaks with a tensile load of 67,500 pounds.
2. If a wroughtiron bar 2 x 1 inches section area breaks under
a tensile load of 125,000 pounds, what load will break a wrought
iron rod 1 inches in diameter ?
3. What should be the diameter of a round castiron bar which is
subjected to a tension of 30,000 pounds, if the unitstress is 2400
pounds per square inch ?
4. Calculate the diameter of a round wroughtiron rod which is
under a tension of 85,000 pounds, if the unitstress is one half the
elastic limit.
5. A piece of timber 2J inches thick is under a tension of 9000
pounds. Find its width if the unitstress is to be 30 per cent of the
elastic limit.
6. A bar of structural steel 1J inches in diameter ruptures under
a tension of 100,000 pounds. Find the ultimate tensile strength of
the bar and the tensile force which will rupture a bar of the same
steel whose section area is 2^ x 3 inches.
7. Find the greatest tensile force a copper wire 0.2 inch in
diameter can stand without breaking.
8. Calculate the size of a square wroughtiron bar to stand a pull
of 3000 pounds without breaking.
9. A steel specimen 0.802 inch in diameter and 8 inches long,
in an experiment with the testing machine, reached the elastic limit
under a tensile force of 24,640 pounds and ruptured under a load of
34,800 pounds. The length of the bar at the elastic limit was 8.0122
inches and at rupture 10.25 inches. Calculate the elastic limit, the
1
2 PEOBLEMS IN STRENGTH OF MATERIALS
ultimate tensile strength, the unitelongation for the elastic limit and
for rupture.
10. A castiron bar has an elliptical crosssection with axes 6 and 4
inches. Find the unitstress under a tensile load of 120,000 pounds,
and the factor of safety.
11. A brick column 2 feet square and 8 feet high sustains a load
of 50 tons. What is the factor of safety ?
12. What load can be borne by a brick pier whose crosssection is
21 x 3 feet, with a factor of safety of 15 ?
13. Find the weight of a wroughtiron bar 1^ X 2 inches section
and 12 feet long, and of a castiron bar of the same size.
14. Find the crosssection area of a wooden beam which weighs
10 pounds per foot. What would be the crosssection area of a steel
beam weighing the same per linear foot ?
15. What must be the height of a brick tower if the'com
pressive unitstress on the lowest brick is one third of its ultimate
strength ?
16. A castiron cylindrical rod 1500 feet long is suspended verti
cally from its upper end. What is the unitstress at this end, and the
factor of safety ?
17. Calculate the length of a castiron bar, supported vertically at
its upper end, that will break under its own weight.
18. The maximum steam pressure in a steam engine is 120 pounds
per square inch and the piston area is 200 square inches. Find the
diameter of the steel pistonrod for a factor of safety of 8, if lateral
bending is prevented.
19. Find the height of a brick wall of uniform thickness for a
factor of safety of 15.
20. What must be the diameter of a hard steel pistonrod, if the
piston is 18 inches in diameter and the maximum steam pressure is
110 pounds per square inch? Consider length of rod less than ten
times its diameter.
21. A short wooden post is 6 inches in diameter. What compres
sive load can it bear with a factor of safety of 8 ?
22. A sandstone column bears a load of 6 tons. Find the area
of its base for a factor of safety of 20, if the ultimate compressive
strength is 3600 pounds per square inch.
TENSION, COMPBESSION, AND SHEAE 3
23. Find the safe steady load for a short, hollow, castiron column,
external diameter 10 inches, internal diameter 8 inches.
24. A wrought iron plate  inch thick requires a force of 60,000
pounds to punch a round hole  inch in diameter through it. Find
the ultimate shearing strength of the plate.
25. Calculate the force required to punch a hole 2 inches square
through a castiron plate  inch thick.
26. What force is necessary to punch a hole 1 inch in diameter
through a wroughtiron plate  inch thick ?
27. What must be the least diameter of a steel bolt which is to
safely resist a simple shearing force of 30,000 pounds ?
28. Determine the unit shearing stress tending to shear off the
head of a 1^inch wroughtiron bolt under a tension of 12,000
pounds, if the head is  inch deep. What is the factor of safety ?
29. A wroughtiron bolt 1 inches in diameter has a head 11 inches
deep and a tension of 30,000 pounds applied longitudinally. Com
pute the factors of safety against tension and shear.
30. Determine the depth of head for a wroughtiron IJinch bolt,
if the tensile strength of the bolt is equal to the strength of the head
against shearing.
31. The diameter of a wroughtiron bolt is  inch. What should
be the depth of the bolt head in order that the bolt be equally strong
in tension and in shear ?
32. A wooden rod 4 inches in diameter and 3 feet long is turned
down to 2 inches diameter in the middle so as to leave the enlarged
ends each 6 inches long. Will a steady tensile force rupture the rod
in the middle or shear off the ends ?
33. The head of an engine cylinder, 12 inches inside diameter, is
fastened on by 10 wroughtiron bolts. What should be the diameter
of the bolts if the steam pressure is 90 pounds per square inch and
the allowable unitstress is 2000 pounds per square inch ?
NOTE. The root area should be used in this problem.
34. A cylinder 9^ inches inside diameter contains steam at 180
pounds per square inch pressure. The cylinderhead is held by 6
wroughtiron bolts placed at equal distances from each other on the
circumference. Find the diameter and depth of head of the bolts
for a factor of safety of 10 against tension and shear.
4 PROBLEMS IN STRENGTH OF MATERIALS
35. A cubical steel block  inch square rests on a wrouglitiron
plate \ inch thick and sustains a load of 10,000 pounds. Deter
mine the factor of safety of the plate against being punched through
by the block, and of the block against be
ing crushed.
36. A cylinder 9 inches inside diame
ter contains steam whose maximum pres
sure is 200 pounds per square inch. The
steel pistonrod which projects through
the piston has a nut on one side of the
piston, and a shoulder on the other side,
FIG. 1
which carries the compressive load on the
rod. Find the diameter of the rod through the piston, and that of
the shoulder for a factor of safety of 8 against tension and compression.
II. ELASTIC DEFOEMATION
37. A bar 1 inch in diameter and 8 feet long elongates 0.05 inch
under a tension of 12,000 pounds. How much will a bar of the same
material and diameter, 12 feet long, elongate with a pull of 30,000
pounds ?
38. Compute the modulus of elasticity for the steel specimen of
Problem 9.
39. A copper wire 0.04 inch in diameter and 10 feet long stretches
0.289 inch under a pull of 50 pounds. Find its modulus of elasticity.
40. A wooden specimen 1 inch in diameter and 9 inches long
elongates 0.004 inch when the tension is increased from 500 to 1000
pounds, and 0.10 inch when the tension is increased from 1500 to
5000 pounds. Find the modulus of elasticity.
41. Determine the elongation of a IJinch round wroughtiron rod
10 feet loiag, under a tensile load of 24,600 pounds.
42. A wroughtiron rod 2 inches square and 10 feet long lengthened
0.03 inch by suspending a load from its lower end. Determine the
load.
43. A wroughtiron bar 10 feet long sustains a load ^ as great as
would be required to pull the bar apart. Determine the elongation
of the bar, also the load if the bar is 1^ inches square.
ELASTIC DEFORMATION
44. A wooden post 4 inches square and 2J feet high sustains a
compressive load of 10 tons. How much will the post be shortened ?
Find the proportion of this shortening to that produced hy loading
the post to its elastic limit.
45. Determine the length of a 1^inch round wroughtiron bar,
which would elongate 0.1 inch under a load of 8000 pounds sus
pended from its lower end, and the factor of safety.
46. By what proportion of its own height will a wroughtiron block
be shortened if loaded to one half its elastic limit, and what will be
the factor of safety ?
47. A bar of structural steel 1 inch in diameter is under successive
tensions of 25,000, 30,000, and 35,000 pounds. Calculate the unit
elongations in each case, and determine by this means which loads
give a stress greater than the elastic limit of the bar.
48. How much will a steel punch 2 inches square and 4 inches
long, of uniform size, be shortened by the force required to punch a
2inch square hole through a wroughtiron plate ^ inch thick ?
49. How many 1inch square rods of wroughtiron would be needed
for the suspension of a platform loaded with 20 tons, if the stretching
of the rods be limited to one half their elongation at the elastic limit?
Each rod bears equal shares of the load.
50. A wroughtiron tie rod is  inch diameter. How long must it
be to lengthen  inch under a steady tension of 5000 pounds ?
51. A wooden rod 3 inches in diameter is elongated 0.05 inch by
a force of 2000 pounds. What was its original
length ?
52. How much will a hundredfoot steel tape,
\ inch wide and ^ inch thick, stretch under a
pull of 50 pounds ?
53. A rectangular timber tie is 12 inches deep
and 40 feet long. Find the proper thickness of
the tie, so that its elongation under a pull of
270,000 pounds shall not exceed 1.2 inches.
54. A flanged cylinder 10 inches inside di
ameter and 10 feet long contains steam at a pressure of 150 pounds
per square inch. The heads of the cylinder are held against he
flanges by a single wroughtiron bolt 1 inch in diameter, extending
6 PROBLEMS IN STRENGTH OF MATERIALS
through the axis of the cylinder. How much must the bolt be
stretched by screwing up the nuts in order that the heads may be
held steamtight against the flanges ?
55. Determine the diameter of a steel pistonrod for a piston 20
inches in diameter and a steam pressure 90 pounds per square inch,
if the maximum unitstress in the rod is to be 5000 pounds per
square inch. Find the lengthening and shortening of the rod per
linear foot under the pull and thrust.
56. A steel pistonrod is 9 feet long and 8 inches in diameter;
the diameter of the cylinder is 88 inches, and the maximum effect
ive pressure 40 pounds per square inch. Find the maximum unit
stress in the rod and the total alteration in length during a
revolution.
57. A pistonrod of structural steel is 4 inches in diameter and 6
feet long. If the piston diameter is 30 inches, what maximum steam
pressure may be used with a factor of safety of 10, and what is the
lengthening and shortening of the rod?
58. A tierod 100 feet long and 2 square inches in sectional area
carries a load of 32,000 pounds, by which it is stretched J inch.
Find the unitstress, unitelongation, and the modulus of elasticity.
59. A beam 12 feet long is suspended horizontally by vertical
wroughtiron rods at each end. The rod at left end is 1 inch square
and 12 feet long, while the rod at opposite end is \ inch square
and 3 feet long. What concentrated load should be applied to the
beam, and how far from the right end must it be placed, in order
that each rod shall be stretched to just one half its elastic limit?
Weight of beam and rods neglected. What is the elongation of
each rod?
60. Calculate the elongation of the rod in Problem 16, due to its
own weight.
61. A vertical wooden bar 100 feet long and 6 inches square car
ries a load of 21,000 pounds at its lower end. Find the unitstress
at the upper end and the elongation of the bar due to combined
weight of bar and load.
62. Find the length of a vertical wooden bar 6x4 inches cross
section, and having a load of 17,000 pounds at lower end, so that the
unitstress at upper end, due to the combined weight, shall be one
ELASTIC DEFOKMATION 7
fourth the elastic limit. What is the elongation of the bar due to the
load and that due to its own weight?
63. A vertical wroughtiron bar 60 feet long and 1 inch in diam
eter is fixed at the upper end and carries a load of 4000 pounds at
the lower end. Find the factors of safety for both ends and the
elongation of the bar.
64. Find the length of a vertical wroughtiron rod fixed at its
upper end, if the maximum unitstress in the rod is 8000 pounds
per square inch. What will be its elongation ?
65. Determine the elongation and factor of safety of a vertical
structural steel rod 1 inch in diameter and 50 feet long, under its
own weight and a weight of 20,000 pounds suspended from its
lower end.
66. A wroughtiron bar 20 feet long and 1 inch square is under a
tension of 20,000 pounds. Find the changes in length, section area,
and volume.
67. A structural steel cylinder 2 feet high and 2 inches in diam
eter bears a compressive load of 90,000 pounds. Find the changes
in length, diameter, section area, and volume.
68. A bar of structural steel 4 inches square and 20 feet long is
under a tension of 2 1 6 tons. Calculate the changes in length, section
area, and volume.
69. A wroughtiron bar is 20 feet long at 32 F. How long will
it be at 90 F.?
70. A wroughtiron bar 18 feet long and 1J inches in diameter is
heated to 400 F.; nuts on its ends are then screwed up so as to bear
against the walls of a house which have fallen away from the per
pendicular. Find the pull on the walls when the bar has cooled to
300 F.
71. A wroughtiron bar 2 square inches in crosssection has its
ends fixed immovably between two walls when the temperature is
60 F. What pressure will be exerted on the walls when the temper
ature is 100 F. ?
72. Steel railroad rails, each 30 feet long, are laid at a temperature
of 40 F. What space must be left between them in order that their
ends shall just meet at 90 F.? If the rails had been laid with their
ends in contact, what would be the unitstress in them at 90F. ?
8 PROBLEMS IN STRENGTH OF MATERIALS
73. A wroughtiron tierod 20 feet in length and 2 inches in diam
eter is screwed up to a tension of 10,000 pounds in order to tie to
gether two walls of a building. Find the stress in the rod when the
temperature falls 20 F.
74. A castiron bar is confined between two immovable walls.
Find the unitstress that will be produced by a rise in temperature
of 50F.
75. A structural steel tierod 40 feet in length and 2 inches square
is subjected to a steady stress of 40,000 pounds. Find the elonga
tion and the number of footpounds of work done.
76. How much work is done in subjecting a cube of 125 cubic inches
of wroughtiron to a tensile stress of 10,000 pounds per square inch ?
77. Find the work which is done in stressing bars of castiron,
wroughtiron, and structural steel, each 1 inch in diameter and 2 feet
long, up to their elastic limits.
78. Calculate the work which is required to stress a wroughtiron
bar 2 inches in diameter and 5 feet long from 6000 to 12,000
pounds per square inch.
79. The work done by a gradually applied force in elongating a
1inch square wroughtiron rod 25 feet long is 100 footpounds.
What is the magnitude of the force applied ?
80. A structural steel rod is required to support a suddenly applied
load of 10,000 pounds. What is the minimum diameter of the rod
if a permanent set is avoided ?
81. A vertical rod, 2 square inches sectional area, carries a load of
5000 pounds. If an additional load of 2000 pounds is suddenly
applied, what is the unitstress produced ?
82. Steam at a pressure of 50 pounds per square inch is suddenly
admitted upon a piston 32 inches in diameter. Find the work done
upon the steel pistonrod, which is 4 feet in length and 2 inches in
diameter.
83. A line of steel rails is 10 miles in length when the tempera
ture is 32F. Find the length when the temperature is 102F. and
the work stored up in the rails per square inch of section.
84. A wroughtiron bar 25 feet in length and 1 square inch in
sectional area has its temperature increased 20 F. Determine the
work done.
THIN CYLINDERS AND SPHERES 9
85. Steam at a pressure of 200 pounds per square inch is suddenly
admitted upon a piston 18 inches in diameter. If the steel pistonrod
be 3 inches in diameter and 7 feet long, what is the maximum unit
stress and the work done on the rod at the maximum compression ?
III. THIN CYLINDERS AND SPHERES
86. What internal pressure will burst a wroughtiron cylinder of
20 inches inside diameter and  inch thickness ?
87. Determine the diameter of a wroughtiron cylinder J inch thick
under an internal pressure of 1000 pounds per square inch for a factor
of safety of 5.
88. Find the internal pressure for a castiron water pipe 24 inches
inside diameter and 2 inches thick, for a factor of safety of 10.
89. Determine the thickness of a wroughtiron steam pipe 18 inches
inside diameter to resist a pressure of 200 pounds per square inch
with a factor of safety of 10.
90. Find the factor of safety for an 8inch castiron water main J
inch thick, under a water pressure of 300 pounds per square inch.
91. Determine the thickness of a 6inch castiron water pipe to
carry a steady pressure of 200 pounds per square inch.
92. Find the factor of safety for a castiron water pipe 12 inches
inside diameter and  inch thick, under a head of 400 feet.
93. Calculate the thickness of a 16inch castiron standpipe, which
is subjected to a head of water of 300 feet. Assume that the stress
is steady.
94. Determine the head of water which can be carried by a wrought
iron pipe 20 inches inside diameter and ^ inch thick, with a factor of
safety of 6.
95. A wroughtiron pipe 10 inches inside diameter and ^ inch thick
is subjected to an internal pressure of 500 pounds per square inch.
Find the increase in diameter.
96. Compute the thickness of a castiron water pipe 18 inches in
side diameter, under a head of 200 feet, for a factor of safety of 10.
What is the increase in diameter?
97. What head of water can be carried in a castiron pipe 2 feet
inside diameter and  inch thick, with a factor of safety of 10?
10 PROBLEMS IN STRENGTH OF MATERIALS
98. What internal pressure will burst a castiron sphere 24 inches
inside diameter and 4 inch thick ?
o
99. A castiron sphere 10 inches inside diameter and  inch thick
sustains an internal pressure of 200 pounds per square inch. Find
the factor of safety.
100. What should be the minimum thickness of a castiron sphere
8 inches inside diameter to safely withstand a steady internal pressure
of 200 pounds per square inch?
101. A force of 500 pounds is applied to the pistonhead of a force
pump 1 inch in diameter, which transfers its pressure to a hollow
castiron sphere 10 inches in diameter. What should be the thickness
of the sphere for a factor of safety of 6 ?
102. Determine the pressure for a factor of safety of 5 in a 60inch
wroughtiron boiler shell ^ inch thick, if the efficiency of the joint is
70 per cent.
103. Find the thickness of plates for a boiler shell 8 feet in diam
eter to work at a pressure of 160 pounds per square inch, if efficiency
of joint is 80 per cent, and stress in plates is 5 tons per square inch.
104. A wroughtiron boiler shell 4 feet in diameter sustains a steam
pressure of 120 pounds per square inch. If the efficiency of the riv
eted joint is 60 per cent and the stress steady, what should be the
thickness of the plate?
105. A wroughtiron cylinder 20 inches inside diameter and J inch
thick has hemispherical ends f^ inch thick. Determine the factor
of safety if it is subjected to an internal pressure of 600 pounds per
square inch.
106. A wroughtiron pipe 10 inches inside diameter and ^ inch
thick is 100 feet long when empty. What will be its length when
subjected to an internal pressure of 500 pounds per square inch?
107. A wroughtiron pipe 5 inches inside diameter weighs 12.5
pounds per linear foot. Find its thickness and the internal pressure
it can carry with a factor of safety of 8.
108. Find the elongation and factor of safety for a 6inch wrought
iron pipe lQ inch thick and 50 feet long, under an internal pressure
of 200 pounds per square inch.
NOTE. The following problems are to be solved by Stewart's formulae.*
* Transactions of the American Society of Mechanical Engineers, Vol. XXVII.
THIN CYLINDERS AND SPHERES
J,
P = 1000 (1  ^1  1600 ), (A)
P= 86,670 1386, (B)
a
where P = collapsing pressure in pounds per square inch,
d = outside diameter of tube in inches,
t = thickness of wall in inches.
Formula (^4) should be used for P< 581, or< 0.023.
Formula (B) should be used for values greater than these.
109. What external pressure will collapse a steel tube whose out
side diameter is 6 inches and thickness of wall 0.180 inch?
110. Find the exterior pressure that will collapse a steel tube 8
inches outside diameter and thickness of wall 0.180 inch.
111. Determine the internal and external pressures that will re
spectively rupture and collapse a steel tube 8 inches outside diameter
and 0.20 inch thick.
112. What interior and exterior pressures will respectively rupture
and collapse a steel tube 0.30 inch thick and 10 inches outside
diameter?
113. What thickness of wall should a 4inch boiler tube have in
order to withstand a working pressure of 200 pounds per square inch,
with a factor of. safety of 6 ?
114. In a firetube boiler the tubes are of steel, 2 inches external
diameter and J inch thick. What is the factor of safety for a working
pressure of 200 pounds per square inch?
115. Find the exterior pressure to collapse a wroughtiron tube
4 inches outside diameter and 0.20 inch thick. What should be the
thickness for this tube under a steam pressure of 150 pounds per
square inch with a factor of safety of 6 ?
116. What external pressure can a wroughtiron pipe 3 inches out
side diameter and ^ inch thick safely sustain and be secure against
shocks ?
117. Find the thickness of a boiler tube 3 inches outside diameter
and exposed to an external steam pressure of 150 pounds per square
inch for a factor of safety of 10.
12
PROBLEMS IN STRENGTH OF MATERIALS
IV. RIVETED JOINTS
In structural work, as in girders, trusses, etc., and in many forms
of receptacles, such as tanks, the shells of steam boilers, etc., composed
of plates, the plates are joined together by riveted joints.
When the plates are in tension the rivets transfer the tension
from one plate to another. This brings a stress upon each rivet,
which tends to shear it across in the plane of the surfaces of con
tact of the plates. A compressive stress is also brought upon the
rounded surface of the rivet, where it bears upon the plate, which
tends to crush it against the metal of the plate in front of the rivet.
This is called a bearing stress, and the exact manner in which this
stress acts between the cylindrical surface of the rivet and the hole
in the plate through which the rivet passes is not known. Experi
ment and experience, however, show that for our computations we
may suppose this stress to be uniformly distributed over an area
which is the projection of the curved surface of the rivet hole up
on a plane through its axis. We then compute for this projected
area a working unitstress whose safe value has been determined by
experiment.
The general discussion of riveted joints covers their use in all kinds
of structures, but we shall limit our attention to their use in uniting
plates of pipes and shells which are subjected to internal fluid pres
sure, and have to be designed for tightness as well as strength. The
special case is that of cylindrical boiler shells.
In connecting the plates, the rivets may be arranged in many dif
ferent ways, but in general they are distributed in rows extending
parallel to the edges of the plates that are joined, as is shown in the
diagrams of a few forms of joints (see Figs. 39).
FIG. 3. SingleRiveted Lap Joint
FIG. 4. DoubleChainRiveted Lap Joint
RIVETED JOINTS
13
In each single row the
rivets are spaced uni
formly, although the uni
form spacing in one row
may be different from
that in another row. The
uniform spacing, meas
ured from the center of
one rivet to the center
of the next one in the
same row, parallel to the
edge of the plate, and in
the row in which the
rivets are most widely
spaced, is called the pitch.
By examining the dia
grams it can be seen that
there is in every case a
FIG. 5. Staggered DoubleRiveted Lap Joint
FIG. 6. SingleRiveted TwoStrap Butt Joint
FIG. 7. SingleRiveted SingleStrap Butt Joint FIG. 8. DoubleRiveted TwoStrap Butt Joint
repeating uniformity in the
grouping of the rivets along
the joint, so that the joint may
be divided by lines perpendicu
lar to the edge, into sections
which are in every respect
alike. These are called repeat
ing sections, and in computing
the strength of the joint we
may compute the strength of
one repeating section and
FIG. 9. TripleRiveted TwoStrap Butt Joint
14
PROBLEMS IN STRENGTH OF MATERIALS
assume that the strength of the whole joint is that of the aggregate
of all such sections.
The width of a repeating section will be denoted by p, the
thickness of the plate by t, and the diameter of the rivet holes
by d.
The diameter of the rivet hole is taken instead of the original
diameter of the cold rivet, because the rivet, when properly driven
and headed, completely fills the hole, the size of which therefore
determines the effective di
ameter of the driven rivet.
The cold rivet is usually
about Jg of an inch smaller
than the hole, so that when
heated red hot it may be
easily and quickly inserted.
A riveted joint may fail in
one of several ways.
1. The rivets may be
sheared, as shown in Pig. 10.
2. The plate in front of the rivet may be sheared out, as in a
of Fig. 11.
3. The plate may crush in front of the rivet, as in I or c of Fig. 11.
4 The plate may break
along the rivet holes, as in d,
or along lines from the center
of a rivet in one row to the
center of the next rivet in the
adjacent row, as in e of Fig. 11.
Experiments have shown that
unless the bearing stress be
excessive there is no danger
of the joint failing in the man
ner of 2 or 3, if the " margin," (d) (e)
that 1S, the distance between FIO.H. Tearing or Overstraining the Plate
the edge of the rivet hole and
the edge of the plate, be made sufficiently great. It should be
made at least as great as d.
RIVETED JOINTS. 15.
Let s t = ultimate tensile unitstress of the plate,
let s c = ultimate compressive unitstress of the rivets,
let s s = ultimate shearing unitstress of the rivets.
The efficiency of a joint is the ratio between the strength of the
joint and the strength of the unriveted plate. Consider a single
riveted lap joint. In a repeating section there is here one rivet to be
compressed, one rivet area to be sheared, and the plate is weakened
by one rivet hole ; hence
pts t = strength of unriveted plate,
(P ~~ d)ts t strength of riveted plate, (A)
tds c = compressive strength of rivet,
s s = shearing strength of rivet.
It is evident that the strength of the repeating section will be repre
sented by the least value obtained from these.
We may compute three efficiencies from these expressions, but the
smallest one only will give the true efficiency of the joint.
Or consider the following expressions. If the joint is so propor
tioned that it would fail by the tearing of the plate between the
holes, efficiency would be
  ,
P 
ds
or if by the compression of the rivets, efficiency would be ; (B)
P s t
or if by the shearing of the rivets, efficiency would be
We may therefore compute the efficiency of a joint in two ways :
by dividing the smallest value found from equations (A), by the
strength of the unriveted plate, or by the use of expressions (B), where
the real efficiency of the joint will be the smallest one of the values
thus obtained.
In a repeating section of a staggered doubleriveted lap joint there
are two rivets to transfer the tension ; hence
16 PROBLEMS IN STRENGTH OF MATERIALS
(^9 d) ts t strength of riveted plate,
2 tds c = compressive strength of rivets,
2  s s = shearing strength of rivets.
/Y\  rj
If the joint fail by tearing the plate, efficiency =
2 ds
if by the compression of the rivets, efficiency = ;
P s t
if by shearing the rivets, efficiency = 
pts t
In a butt joint the main plates do not overlap, but cover plates are
used to connect them. When tension is applied to the main plates of
a butt joint having two cover plates, one half of this applied tension
is transferred to each cover plate. Hence, theoretically, the thickness
of each cover plate should be one half that of the main plate ; but
the cover plates, or straps, must be thick enough to remain tight
against leakage arising from their flexure between the rivets, and so
thick that their edges will admit of effective calking. It is customary,
therefore, to make the thickness of each cover plate about five eighths
that of the main plates. In the case of a singlestrap joint, in which
the strap is subject to a bending stress as well as to stress from calk
ing, the strap is made 1J times the thickness of the main plates.
Singlestrap joints ought not to be used for the seams of boiler shells.
In a butt joint with single or double riveting, there are twice as
many rivet sections to be sheared in a repeating section as in the cor
responding case for a lap joint. Hence the strength of the joint against
shearing the rivets is twice as great. The effective rivetbearing sur
faces in a butt joint are those surfaces only which are in front of the
rivets where they pass through the main plate, and their number,
therefore, is equal to the number of rivets in one of the main plates
in the repeating section, one surface for each rivet. It must be recog
nized that in butt joints the number of rivets to be considered in a
repeating section is the number on one side only of the line of
separation of the mam plates. Thus, in Figs. 6 and 7, only one
rivet can be considered, in Fig. 8 two rivets are taken, and in Fig. 9
five rivets.
RIVETED JOINTS 17
For a singleriveted butt joint with two cover plates
(p d)ts t = strength of riveted plate,
tds c = compressive strength of rivets,
7O
2 s s = shearing strength of rivets.
4
For a doubleriveted butt joint with two cover plates
(P ~ ^) ts t = strength of riveted plate,
2 tds c = compressive strength of rivets,
4 s s = shearing strength of rivets.
In any riveted joint let n be the number of rivets and m the num
ber of rivet sections subjected to shearing in a repeating section ; then
(P ~ d) t s t = strength of riveted plate,
ntds c = compressive strength of rivets,
m s s = shearing strength of rivets.
The efficiency of the joint will be the least value obtained from these
expressions, divided by pts t , or the strength of the unriveted plate.
In our computations we shall use, for iron or soft steel plates and
iron rivets,
s t = 55,000 pounds per square inch,
s c = 80,000 pounds per square inch,
s s = 38,000 pounds per square inch.
If we use a factor of safety of 5, and divide these values just given
by this number, we shall have safe working unitstresses of
11,000 pounds per square inch for tension,
16,000 pounds per square inch for compression,
7,600 pounds per square inch for shearing.
In determining the efficiency of a joint these working unitstresses
may be used in place of the values given for s t , s c , and s s .
Let P t = strength of riveted plate,
P c = compressive strength of rivets,
P s shearing strength of rivets,
P = force transmitted by a repeating section ;
18 PROBLEMS IN STRENGTH OF MATERIALS
then = factor of safety against tearing the plate,
= factor of safety against compressing the rivets,
 = factor of safety against shearing the rivets.
DESIGN OF RIVETED JOINTS
If no other consideration than economy of material in securing
the necessary strength were taken into account in designing a joint,
the relations between t, d, and p ought to be selected so as to make
the values of P t , P g , and P c equal; the efficiency will then be a
maximum.
The process would be to first compute d by setting P s = P c , and
then find p by putting P t = P c . 4 .
Practical considerations, however, such as the stanchness required
in some joints, convenience of construction, economy of labor, etc.,
have led to a diversity of custom in proportioning joints so that they
may be best adapted to the particular conditions of use.
For present purposes it may be assumed that in good American
practice in the design of the joints of steam boiler shells, the diameter
of the rivet hole is arbitrarily selected, and corresponds, practically,
to a value derived from the expression d K^ft, in which K 1.5
for single and double lap joints, and K= 1.3 for doublestrap butt
joints. The dimension p is then computed by using the value of d
thus determined, and putting P t =P s , or P t = P c , selecting the smaller
value of p thus derived as giving the safe dimensions for the pitch.
The efficiency of the joint, if otherwise properly proportioned, will be
pd
P
In staggered doubleriveted joints the distance between the two
rows of holes should be determined by making the diagonal pitch p"
(see Figs. 5 and 8), such that p" d = 0.6 (p d). Experiment has
shown this proportion to be a good one.
For an example we will design a tripleriveted, twostrap butt joint
for inch plates (Fig. 9).
RIVETED JOINTS 19
t =  inch, d = 1.3 V7= .796 ; selecting d to the nearest sixteenth
of an inch, we have d = if inch, n = 5, and m = 9.
p ( = (p  d)ts t = (p if ) 55,000,
P s==m 7 ^ Sg = ^ (.518) 38,000 = 177,666.7,
p c = ntds c = 5  1 if 80,000 = 121,875.
... (p  if ) . 55,000 = 121,875.
Solving p = 6.71, we will then take
p = 6 inches ;
p ^ 05
"T = To8 = ' 88 
The joint then has an efficiency of 88 per cent, and the inner rows
of rivets will have a spacing of 3 inches.
The present discussion of riveted joints is not given with the in
tention of completing the subject in regard to all forms and methods
of construction, but for use in computations and design of such joints
as are often found in boiler construction.
118. Determine the efficiency of a singleriveted lap joint if t == T 3 g
inch, d =  inch, and p = 1 inches.
119. Find the efficiency of a singleriveted lap joint if t = f inch,
d = if inch, and p = 2 T 3 inches.
1 u * 1 D
120. Calculate the efficiency of a singleriveted lap joint if t = 
inch, d = Iy 3 g inches, and p = 2^ inches.
121. Determine the efficiency of a singleriveted lap joint if t = 
inch, d = l^g inches, and p = 2 T 5 g inches.
122. Calculate the efficiency of a singleriveted lap joint if t = j 5 6
inch, d = if inch, and p 2^ inches.
123. A singleriveted lap joint, with t = ^ inch, d = 1 inch, and
p 2J inches, sustains a tension of 5000 pounds on each repeating
section. Compute the efficiency of the joint and the factors of safety.
124. Determine the pitch of a singleriveted lap joint where t = ^
inch, and d = ^ inch, so that the strength of the joint against tearing
the plate between the rivet holes shall equal the shearing strength of
the rivets. Calculate also the efficiency of the joint.
125. Determine the efficiency of a doubleriveted lap joint where
t =  inch, d = l inch, and p = 3 T 7 g inches.
20 PROBLEMS IN STRENGTH OF MATERIALS
126. Calculate the efficiency of a doubleriveted lap joint if t = T T g
inch, d = 1 inch, and p = 3 inches.
127. In a doubleriveted lap joint t = J inch, c = 1J^ inches, and
^> = 3 T 9 g inches. * Find its efficiency.
128. Find the efficiency of a doubleriveted lap joint if t = J inch,
d =  inch, and p = 2 inches.
129. Determine the efficiency of a doubleriveted lap joint if t = f^
inch, e = l inch, and ^? = 2 J inches.
130. Each repeating section of the riveted joint of Problem 129
sustains a tension of 7000 pounds. Find the factors of safety.
131. Find the pitch of a doubleriveted lap joint in which t = J inch
and d =  inch, so that the strength of the joint against tearing the
plates between the rivet holes shall equal the shearing strength of
the rivets. Find also the efficiency of the joint.
132. Determine the efficiency of a singleriveted, twostrap butt
joint if t =  inch, d = 1 inch, and p = 2 inches.
133. Find the efficiency of a singleriveted, twostrap butt joint if
t = J inch, d = l inch, and p = 2 inches.
134. Determine p for a singleriveted, twostrap butt joint in which
t =  inch and d = 1 Jg inches, so that the strength of the joint against
tearing the plates between the rivet holes shall equal the compressive
strength of the rivets. Determine also the efficiency of the joint.
135. Determine the efficiency of a doubleriveted, twostrap butt
joint if t =  inch, d = l inch, and p = 3 J inches.
136. Find the efficiency of a doubleriveted, twostrap butt joint if
t =  inch, d = 1 inch, and p = 3 J inches.
137. Determine the pitch for a doubleriveted, twostrap butt joint
in which t = ^ inch, and ^ = f inch, so that the strength of the
joint against tearing the plates between the, rivet holes shall equal
the compressive strength of the rivets. What is the efficiency of
this joint ?
138. Each repeating section of the riveted joint of Problem 135
sustains a tension of 9000 pounds. Find the factors of safety.
139. Design a singleriveted lap joint for inch plates and find its
efficiency.
140. Design a singleriveted lap joint for inch plates and find its
efficiency.
RIVETED JOINTS 21
141. Design a doubleriveted lap joint for j^inch plates and find
its efficiency.
142. Design a doubleriveted lap joint for inch plates and find its
efficiency.
143. Design a singleriveted, twostrap butt joint for ^ginch plates
and find its efficiency.
144. Design a singleriveted, twostrap butt joint for j^inch plates
and find its efficiency.
145. Design a doubleriveted, twostrap butt joint for ^ginch plates
and find its efficiency*
146. Design a doubleriveted, twostrap butt joint for ^inch plates
and find its efficiency.
147. Determine the efficiency of a tripleriveted, twostrap butt
joint in which t = f^ inch, d =  inch, and p = 6 inches.
148. Find the efficiency of a tripleriveted, twostrap butt joint in
which t = J inch, d 1 inch, and p 7 J inches.
149. Calculate the efficiency of a tripleriveted, twostrap butt
joint in which t = T 9 g inch, d = l^g inches, and p = 7 inches.
150. Design a tripleriveted, twostrap butt joint for j^inch plates
and find its efficiency.
151. Design a tripleriveted, twostrap butt joint for inch plates
and find its efficiency.
152. Show that when s t , s s , and s c have values as given above, if
m = n and K = 1.5 for t < 0.313 inch, then P C >P 8 .
153. Show that if 5 m = 9 n (Fig. 9), and JT= 1.3,
for t > 0.76 inch, then P c < P..
. 154. Show that if m = 2 n and K= 1.3
for t > 0.94 inch, then P c < P s .
155. A boiler shell 4 feet in diameter has longitudinal singleriveted
lap joints for which t = f^ inch, d = ij inch, and p = 2 ^ inches.
Determine the maximum steam pressure which can be used with a
factor of safety of 5.
156. A boiler shell 60 inches in diameter has longitudinal single
riveted lap joints for which t = J inch, d = l inch, and p = 2f^
inches. Calculate the maximum steam pressure which can be used
with a factor of safety of 5.
v
Or THE \
UNIVERSITY )
OF
22 PROBLEMS IN STRENGTH OF MATERIALS
157. A boiler 48 inches in diameter carries a steam pressure of 65
pounds per square inch. It has singleriveted longitudinal lap joints for
which t=\ inch, d=  inch, and p = 2 inches. Find the factor of safety.
158. Determine the steam pressure which will rupture a boiler
shell 5 feet in diameter, with singleriveted longitudinal lap joints
for which t =  inch, d =  inch, and p = 2 inches.
159. A boiler shell 60 inches in diameter, with singleriveted longi
tudinal lap joints, is to carry a steam pressure of 78 pounds per square
inch, with a factor of safety of 5. Determine the thickness of the
shell and fhe pitch of the rivets if the efficiency of the joint is 0.572.
160. Determine the factor of safety when a steam pressure of 80
pounds per square inch is used in a 60inch boiler, with double
riveted, longitudinal lap joints for which t =  inch, d = 1 inch, and
p = 3_7_ inches.
161. What steam pressure will burst a boiler 4 feet in diameter,
with doubleriveted, longitudinal lap joints for which t = ^ inch,
d =  inch, and p = 3^ inches ?
162. A boiler shell 5 feet in diameter has singleriveted, twostrap
butt joints for the longitudinal seams, for which t = J inch, d = i
inch, and p = 2 inches. What steam pressure can it carry with a
factor of safety of 5 ?
163. A boiler 36 inches in diameter has double riveted, twostrap
butt joints for the longitudinal seams, for which t =  inch, d = 1 inch,
and p = 3 1 inches. Find the factor of safety for a steam pressure of
250 pounds per square inch.
164. A boiler 5 feet in diameter, with longitudinal, doubleriveted,
twostrap butt joints/ is to carry a steam pressure of 103 pounds per
square inch, with a factor of safety of 5. Find the thickness of the shell
and the pitch of the rivets if the efficiency of the joints is 75 per cent.
165. A cylindrical standpipe, 100 feet high, inside diameter 18 feet,
has longitudinal, doubleriveted twostrap butt joints at the lowest
part of the pipe, for which t =  inch, d 1 inch, and p 3 J inches.
Compute the factor of safety when the pipe is full of water.
166. A boiler 66 inches in diameter, with longitudinal, triple
riveted, twostrap butt joints, is to carry a steam pressure of 100
pounds per square inch, with a factor of safety of 5. Find the thick
ness of the shell and the pitch of the rivets if the efficiency of the
joints is 80 per cent.
CANTILEVER AND SIMPLE BEAMS 23
V. CANTILEVER AND SIMPLE .BEAMS
SHEAR AND MOMENT DIAGRAMS
'Construct the shear and moment diagrams and find the maximum
shear and moment for the following cases.
167. A cantilever beam of length I, with a uniform load of w pounds
per linear foot.
168. A cantilever beam of length I, with a concentrated load P at
the free end.
169. A cantilever beam of length I, with a uniform load of w pounds
per linear foot, and a concentrated load P at the free end.
170. A simple beam of length /, with a uniform load of w pounds
per linear foot.
171. A simple beam of length /, with a concentrated load P at the
middle.
172. A simple beam of length /, with a uniform load of w pounds
per linear foot, and a concentrated load P at the middle.
173. A cantilever beam of length 12 feet, with a total uniform load
of 240 pounds.
174. A cantilever beam of length 10 feet, with a concentrated load
of 100 pounds at the free end.
175. A wooden cantilever beam, 9 inches deep, 8 inches broad, and
15 feet long, with a concentrated load of 1000 pounds at the free end.
176. A simple beam 20 feet in length, with a uniform load of 30
pounds per linear foot.
177. A simple beam 12 feet in length, with a concentrated load of
1000 pounds at the middle.
178. A simple beam 18 feet in length, with a total uniform load of
180 pounds, and a concentrated load of 800 pounds at the middle.
1 79 . A cantilever beam 1 feet long and weighing 1 2 pounds per linear
foot, with a concentrated load of 80 pounds, 2 feet from the free end.
180. A cantilever beam 12 feet long, weighing 10 pounds per linear
foot, with concentrated loads of 100 and 150 pounds at distances of
4 and 8 feet respectively from the free end.
181. A cantilever beam 10 feet long, with a uniform load of 50
pounds per linear foot, and concentrated loads of 100, 300, and
24 PROBLEMS IN STRENGTH OF MATERIALS
500 pounds at distances of 2, 5, and 8 feet respectively from
the fixed end.
182. Show analytically that the maximum moment occurs in a
cantilever beam and in a simple beam at that section where the shear
passes through zero.
In the following problems find also the position of the danger
section.
183. A simple beam of length I, with two equal concentrated loads
at the quarter points. Neglect weight of beam.
' 184. A simple beam of length /, with two equal concentrated loads
at the quarter points, and a uniform load of w pounds per linear foot.
185. A simple beam of length 10 feet, with 200 pounds 4 feet from
the left end. Weight of beam neglected.
186. A simple beam 12 feet in length, with 300 pounds 4 feet from
the left end, and a uniform load of 20 pounds per linear foot.
187. A simple beam 20 feet long, weighing 12 pounds per linear
foot, with a load of 240 pounds 5 feet from the left end.
188. A simple beam 8 feet in length, with a concentrated load of
1000 pounds 2 feet from the left end, and a uniform load of 500
pounds per linear foot.
189. A simple beam 6 feet long, with concentrated loads of 1000
pounds 2 feet from each end, if weight of beam is neglected.
190. A simple beam 12 feet long, with a uniform load of 40 pounds
per linear foot, and concentrated loads of 2000 pounds at 3 feet from
each end.
191. A simple beam 12 feet in length, with 240 pounds 3 feet from
the left end, and 360 pounds 4 feet from the right end, if weight of
beam is neglected.
192. The beam of Problem 191 has in addition to the concentrated
loads a uniform load of 60 pounds per linear foot.
193. A simple beam 20 feet long, with concentrated loads of 2000
pounds 4 feet from the left end, and 1000 pounds 2 feet from the
right end, and also a uniform load of 100 pounds per linear foot.
194. A simple beam 6 feet in length, there being concentrated loads
of 4000 and 1000 pounds 1 and 2 feet respectively from the left end,
and no uniform load.
CANTILEVER AND SIMPLE BEAMS 25
195. A simple beam 5 feet long, with a uniform load of 50 pounds
per linear foot, and concentrated loads of 50 pounds 2 feet from the
left end, and 75 pounds 1 foot from the right end.
196. A simple beam 16 feet in length, carrying a uniform load of
40 pounds per linear foot, and two concentrated loads, one of 240
pounds 3 feet from the left support, and one of 180 pounds 4 feet from
the right support.
197. A simple beam 12 feet long, there being concentrated loads of
90 and 60 pounds 4 and 7 feet respectively from the left end, and a
uniformly distributed load of 20 pounds per linear foot.
198. A simple beam 30 feet in length, bearing a uniform load of
40 pounds per linear foot, and concentrated loads of 1 and 1.5 tons at
9 and 20 feet respectively from the left end.
199. A simple beam 12 feet long, there being concentrated loads of
240, 90, and 120 pounds at 3, 4, and 8 feet respectively from the left
end, but no uniform load.
200. The beam of Problem 199 has, in addition to the concentrated
loads, a uniform load of 100 pounds per linear foot.
201. A simple beam of 12 feet span, weighing 35 pounds per linear
foot, with concentrated loads of 300, 60, and 150 pounds at 3, 5, and
8 feet respectively from the left support.
202. A simple beam 100 feet between the supports, with three con
centrated loads of 1200 pounds each at distances from the left sup
port of 40, 60, and 80 feet. Neglect weight of beam.
203. A simple beam 12 feet long, bearing concentrated loads of
1, ^, and 3 tons at distances of 3, 6, and 7 feet respectively from the
left support, and a uniform load of ^ ton.
204. A simple beam 20 feet in length, there being a uniform load
of 20 pounds per linear foot, and concentrated loads of 200, 100, 400,
and 200 pounds at 4, 6, 8, and 12 feet respectively from the left end.
205. The simple beam of Problem 204, with the same concentrated
loads, but no uniform load.
206. A simple beam 20 feet long, with concentrated loads of 200,
100, 400, and 200 pounds at 4, 6, 8, and 12 feet respectively from
the left end, and a uniform load of 100 pounds per linear foot.
26
PROBLEMS IN STRENGTH OF MATERIALS
FIG. 12
FIG. 13
FIG. 14
FIG. 15
tl i
Rr
FIG. 17
FIG. 18
J
1 J
6 
FIG. 20
FIG. 21
U
FIG. 22
'1
I
I
FIG. 23
Q..J
FIG. 24
CANTILEVER AND SIMPLE BEAMS 27
NEUTRAL Axis AND MOMENTS OF INERTIA
Determine the position of the neutral axis and the moments of
inertia in respect to this axis for the following beamsections.
207. The rectangular section shown in Fig. 12.
208. The circular section shown in Fig. 13.
209. The triangular section shown in Fig. 14.
210. The hollow rectangular section shown in Fig. 15.
211. The square section shown in Fig. 16.
212. The Tsection shown in Fig. 17, if I = 3, ti = 2, t = l, and
d = 8 inches.
213. The Tsection shown in Fig. 17, if d = 12, 6 = 5, t = 1, and
tj = 2 inches.
214. The Tsection shown in Fig. 17, if t l = t = 1, d = 9, and ~b 4
inches.
215. The section of Fig. 15, if d = 6, I = 4, d = 4, and \ = 2 inches.
216. The section of Fig. 19, if d = 6, t x = t 2 = t = 1, and b l = \ = 4
inches.
217. The section of Fig. 19, if d = 8, ^ = * 2 = t = J, and ^ = 6 2 = 4
inches.
218. The section of Fig. 19, if d = 12, ^ = 2 = t = J, and ^ = 6 2 = 5
inches.
219. A section like Fig. 19, if d = 12, \ = 4, 6 2 = 2, * = 1, and
t l= = t 2 = 2 inches.
220. A section like Fig. 19, if d = 10, t = \ t ^ = t 2 = 1, ^ = 4, and
Z> 2 = 2 inches.
221. The section of Fig. 20, if t = J, & = 8, and d = 2 inches.
222. The section of Fig. 20, if t = 1, I = 8, and d = 6 inches.
223. The section of Fig. 20, if t = 1, & = 12, and d = 4 inches.
224. A section like Fig.21, if d = 12, I = 4, = 1, and ^ = 2 inches.
225. A section like Fig. 21, if d = 15, I = 31, and = ^ = J inch.
226. The section of Fig. 22, if I = d = 6, and t = J inch.
227. The section of Fig. 22, if d = 6, I = 4, and t = \ inch.
228. A section like Fig. 23, if d = 6, I = 4, and = 1 inch.
229. A section like Fig. 23, if d = 8, I = 3, and t = J inch.
230. The section of Fig. 24, if d = 6, b = 4, ^ = ^ = 1, and t =
inch.
28
PROBLEMS IN STRENGTH OF MATERIALS
231. A trapezoidal section, if the depth is 8 inches, the longer base
6 inches, and the shorter base 4 inches.
232. A section like Fig. 25, if d = 12 inches, t = \ inch, and each
angle. section 4 x 3 X \ inches, with the longer leg horizontal.
233. A section like Fig. 25, if d = 10 inches, t = ^ inch, and each
angle section 3 X 2^ x \ inches, with the longer leg horizontal.
234. A section like Fig. 25, if d = 10 inches, t =  inch, and each
angle section 5 X 3^ X f inches, with the longer leg horizontal.
9
M*
_i
i rt >
Ui i ifji
FIG. 25
FIG. 26
FIG. 27
FIG. 28
235. The section of Fig. 26, if d 14 inches, t = \ inch, and each
angle section 3J x 3J x \ inches.
236. The section of Fig. 26, if d = 12 inches, t = ^ inch, and each
angle section 4 X 4 x ^ inches.
237. A section like Fig. 27, if 6 = 8 inches, t = J inch, and each
channel is 6 inches deep and weighs 8 pounds per foot. Find also the
moment of inertia in respect to the axis 22.
238. A section like Fig. 27, if b = 10 inches, t = \ inch, and each
channel is 8 inches deep and weighs 11.25 pounds per foot. Find also
the moment of inertia in respect to the axis 22.
239. A section like Fig. 27, if b = 12 inches, t = \ inch, and each
channel is 10 inches deep and weighs 30 pounds per foot. Find also
the moment of inertia in respect to the axis 22.
240. The section of Fig. 28, if I = 12 inches, t = \ inch, and each
channel is 12 inches deep and weighs 20.5 pounds per foot. Find also
the moment of inertia in respect to the axis 22.
CANTILEVER AND SIMPLE BEAMS 29
INVESTIGATION
241. A rectangular, wooden cantilever beam 12 feet long, 4 inches
broad, and 8 inches deep bears a total uniform load of 50 pounds per
linear foot. Find the factor of safety.
242. A wooden cantilever beam 5 feet in length has a rectangular
section 2 inches broad and 3 inches deep. Find the total uniform
load it can carry with a factor of safety of 8.
243. Find the factor of safety for a rectangular, wooden cantilever
beam 12 feet long, 4 inches broad, and 8 inches deep there being a
concentrated load of 300 pounds at the free end. Neglect the weight
of the beam.
244. Determine the factor of safety for the beam in Problem 243,
if the weight of the beam is considered.
245. A castiron bar 1 inch in diameter and 2 feet long is supported
at its middle, and loads of 50 pounds are hung at each end. Find the
factor of safety if the weight of the bar is neglected.
246. A rectangular, wooden cantilever beam 10 feet long and 6 inches
deep is to support a load of 200 pounds at the free end. What should
be its width for steady stress if weight of cantilever is neglected ?
247. A rectangular, wooden cantilever beam 8 feet long, 6 inches
broad, and 8 inches deep carries a load of 300 pounds at the free end
and a total uniform load of 160 pounds. What is the factor of safety?
248. A rectangular, wooden cantilever beam 8 feet in length, 6 inches
broad, and 8 inches deep has a load of 200 pounds at the free end.
What total uniform load can it also carry with a factor of safety of 10 ?
249. A simple wooden beam of rectangular section 8 x 12 inches
and 16 feet long sustains a total uniform load of 500 pounds per linear
foot. Find the factor of safety if the short side is horizontal.
250. Would the beam of Problem 249 be safe if the long side were
horizontal ?
251. A piece of timber 20 feet long, supported at its ends, is to carry
a total uniform load of 4 tons. What should be the size of its square
crosssection for a factor of safety of 10?
252. What should be the depth of a rectangular wooden girder 20
feet long and 4 inches broad to sustain a total uniformly distributed
load of 1600 pounds, with a factor of safety for varying stress ?
30 PEOBLEMS IN STRENGTH OF MATERIALS
253. Find the total uniform load that a wooden floor beam 2 x 10
inches in rectangular section and 1 6 feet long will carry with a factor
of safety of 8.
254. Solve Problem 253 with the 10inch side horizontal.
255. A simple wooden beam 3 inches wide, 4 inches deep, and 16
feet long bears a concentrated load of 140 pounds at the middle.
Determine the factor of safety if the weight of the beam is neglected.
256. Find the factor of safety for the beam of Problem 255, taking
into account the weight of the beam.
257. A piece of scantling 2 inches square and 8 feet long is sup
ported at its ends, and sustains a load of 150 pounds at its middle.
Is it safe ? Neglect weight of beam.
258. A wooden beam 4 inches square, resting on end supports, is
to carry a uniform load of 40 pounds per linear foot, including its
own weight. Find the maximum safe distance between supports.
259. A wooden beam 4 inches broad, 6 inches deep, and 10 feet
long is supported at its ends. Calculate the load it can carry at its
middle point with a factor of safety of 8, if weight of beam is neglected.
260. Solve Problem 259, considering weight of beam.
261. The piston of a steam engine is 14 inches in diameter, and the
steam pressure 80 pounds per square inch. Assuming that the total
pressure on the piston comes on the crank pin at the dead points, and
that the crank pin is a cantilever uniformly loaded, what should be
its diameter if 4 inches long and made of wroughtiron ? Use a factor
of safety of 10.
262. A steel engine shaft resting on bearings 5 feet apart carries a
3ton flywheel midway between the bearings. Find the diameter of
the shaft to carry this load with a factor of safety of 10.
263. A wooden cantilever 8 feet in length and 4 inches broad bears
a total uniform load of 80 pounds per linear foot, and concentrated
loads of 600 and 200 pounds at 3 and 8 feet respectively from the
support. Determine the depth of the beam, using a factor of safety
for steady stress.
264. Find the factor of safety for a structural steel engine shaft
12 inches in diameter, resting in bearings 54 inches apart and carry
ing a flywheel of 40 tons midway between the bearings. Neglect
weight of shaft.
CANTILEVER AND SIMPLE BEAMS 31
265. A hollow, circular, castiron beam, inside diameter 5 inches,
outside diameter 6 inches, rests upon end supports 8 feet apart.
With a factor of safety for varying stress, what is the maximum
safe load that may be concentrated at its center? Neglect weight
of beam.
266. A rectangular wooden beam 14 feet long, 4 inches wide, and
9 inches deep rests on end supports. Find the factor of safety if it
bears a uniform load of 100 pounds per linear foot in addition to its
own weight.
267. Design a rectangular wooden cantilever to project 4 feet from
a wall and bear a load of 500 pounds at its free end, the factor of
safety being 8, and weight of beam neglected.
268. A wooden beam of circular crosssection rests on end sup
ports 10 feet apart. What load may be hung at the middle, if the
radius of beam is 4 inches ? Use factor of safety for steady stress,
and neglect weight of beam.
269. A wooden beam resting on end supports 10 feet apart has a
crosssection which is an isosceles triangle with a 6inch horizontal
base. It carries a uniform load, including its own weight, of 120
pounds per linear foot. What must be the altitude of its crosssection
for a factor of safety of 8 ?
270. Find the uniform load per linear foot which a wooden canti
lever 6 feet in length, rectangular section 2 inches broad and 3 inches
deep, can carry with a maximum fiberstress of 800 pounds per square
inch.
271. Wooden beams 18 feet between supports, 6 inches deep, and
2 niches broad support a floor weighing 100 pounds per square foot.
Neglecting the weight of the beams and using a factor of safety for
varying stress, how far apart should they be spaced?
272. A balcony projecting 6 feet from a wall is supported by
wooden beams 4 inches broad and spaced 3 feet apart. Find the depth
of the beams if the total uniform load is 120 pounds per square foot
and the maximum fiberstress is 800 pounds per square inch.
273. A rectangular, wooden simple beam 9 inches deep and 3 inches
wide supports a load of ^ ton concentrated at the middle of an 8foot
span. Find the maximum fiberstress, considering the weight of the
beam.
32 PROBLEMS IN STRENGTH OF MATERIALS
274. A wroughtiron beam 14 feet long, supported at its ends and
of circular section, is loaded with a total uniform load of 200 pounds
per linear foot. Find its diameter for a factor of safety of 4.
275. A simple wooden beam 20 feet long and 12 inches square
supports a load of 2 tons at the middle. Find the factor of safety,
considering the weight of the beam.
276. A castiron rectangular beam resting upon end supports 12 feet
apart carries a load of 2000 pounds at the center. If the breadth is
one half the depth, find the area of crosssection for a factor of safety
of 4. Neglect weight of beam.
277. Round and square beams of the same material are equal
in length and have the same loading. Find the ratio of the diameter
to the side of the square so that the two beams may be of equal
strength.
278. Compare the relative strengths of a square beam and a circular
beam which is the inscribed cylinder.
279. Compare the strength of a square beam with a side vertical, to
that of the same beam with a diagonal vertical.
280. Compare the relative strengths of a cylindrical beam and the
strongest rectangular beam that can be cut from it.
281. A wroughtiroii beam 8 feet in length and supported at its
ends bears a total uniform load of 2000 pounds per linear foot. Its
section is like Fig. 19, where \ = b% = 4, t 1 = t 2 = t = 1, and d = (>
inches. Find the factor of safety.
282. Find the factor of safety for the beam of Problem 281, if its
section is like Fig. 15, with I = 4, d = 6, \ 2, and d^ = 4 inches.
283. The beam of Problem 281 has a section like Fig. 21, with
I = 4, ^ = t = 1, and d 8 inches. Find the factor of safety.
284. Determine the factor of safety for a wroughtiron beam 8 feet
in length and supported at its ends, with a concentrated load of 8000
pounds 3 feet from the left end. Its section is like Fig. 19, with \ =
& 2 = 4, tfj = Z 2 = t 1, and d = 6 inches. Neglect weight of beam.
285. The beam of Problem 284 has a section like Fig. 15, with
I = 4, d = 6, 6 X = 2, and d^ = 4 inches. Find the factor of safety.
286. Find the factor of safety for a circular wroughtiron simple
beam 3 inches in diameter and 6 feet in length, with concentrated
loads of 1000 pounds 2 feet from each end. Neglect weight of beam.
CANTILEVER AND SIMPLE BEAMS 33
287. A wroughtiron simple beam 6 feet in length, with crosssection
3 inches square, has concentrated loads of 1000 pounds 2 feet from
each end. Neglecting the weight of the beam, find the factor of safety.
288. Solve Problem 287, considering the weight of the beam.
289. Solve Problem 287, if the diagonal is vertical.
290. Determine the factor of safety for a wroughtiron beam 8 feet
in length, supported at its ends, with a total uniform load of 1000
pounds per linear foot and a concentrated load of 1000 pounds at the
middle. Its section is like Fig. 19, with ^ = & 2 = 4, ^ = 2 = t = 1,
and d = 6 inches.
291. A wroughtiron, circular simple beam 6 feet in length has
concentrated loads of 4000 and 1000 pounds at 1 and 2 feet respec
tively from the left end. Is it safe if the diameter is 3 inches ?
292. Find the factor of safety for the beam of Problem 291, if its
section is 3 inches square and a side vertical.
293. A wroughtiron beam 8 feet span and supported at its ends
has an Isection, with &j = & 2 = 4, ^ = 2 = t = 1, and d = 6 inches.
What concentrated load can it carry at its middle point with a factor
of safety of 6 ? Neglect weight of beam.
294. A simple wooden beam 12 feet span, 9 inches deep and 8 inches
wide, carries two equal loads, each 3 feet from the middle, but on
opposite sides. Find these loads for a factor of safety of 10, neglecting
the weight of the beam.
295. Solve Problem 294, considering the weight of the beam.
296. Determine the total uniform load for a castiron beam 12 feet
span and supported at its ends for a factor of safety of 6, if the sec
tion is like Fig. 18, with b = 5, d = 12, t = 1, and ^ = 2 inches.
297. A castiron simple beam 10 feet span has a section like Fig. 18,
with I = 3, d = 8, t = 1, and t 1 = 2 inches. What concentrated load
can it carry at the middle with a factor of safety of 8 ? Neglect weight
of beam.
298. A 12inch steel Ibeam, weighing 35 pounds per linear foot,
of 10 feet span and supported at the ends sustains a total uniform
load of 20 tons. Find the factor of safety.
299. Determine the total uniform load for a 10inch steel Ibeam,
30 pounds per foot, 12 feet span and supported at the ends, for a
factor of safety of 6.
34 PROBLEMS IN STRENGTH OF MATERIALS
300. Find the factor of safety for a 15inch steel Ibeam, 42 pounds
per foot, 16 feet span and supported at the ends, if it bears a concen
trated load of 15,000 pounds at the middle. Neglect weight of beam.
301. Solve Problem 300, considering the weight of the beam.
302. Find the concentrated load at the middle of a 9inch steel
Ibeam, 25 pounds per foot, 14 feet span and supported at the ends,
for a factor of safety of 4. Neglect weight of beam.
303. A castiron simple beam of 12 feet span has a section like
Fig. 19 with t 1 = t z = t=l,d=W, and b 1 = \ = 6 inches. Find the
concentrated load it can carry at its middle with a factor of safety
of 6. Neglect weight of beam.
304. Select the proper steel Ibeam of 12 feet span, supported at
its ends, to carry two loads of 5000 pounds each, one at the middle
and the other 2 feet from the left end, with a factor of safety of 4.
Neglect weight of beam.
305. A floor designed to carry a total uniform load of 180 pounds
per square foot is supported by steel Ibeams of 20 feet span and 4
feet apart from center to center. Find the proper beam that should
be used for a factor of safety of 5.
306. A floor which is to carry a total uniform load of 150 pounds
per square foot is supported by 9inch steel Ibeams, 35 pounds per
foot and 15 feet span. Find their distance apart from center to center,
if the factor of safety is 4.
307. A cylindrical wroughtiron simple beam resting on end sup
ports 24 feet apart sustains three concentrated loads of 400 pounds
each at distances of 4, 12, and 16 feet respectively from the left
support. What should be the diameter of the beam for varying stress ?
Neglect weight of beam.
308. A 10inch steel Ibeam, 40 pounds per foot, 15 feet span and
supported at its ends, bears a concentrated load of 5 tons at its center.
Is it safe ?
309. Select a steel Ibeam 1 feet long and supported at its ends to bear
a total uniform load of 1500 pounds per linear foot with varying stress.
310. A simple beam 16 feet span is loaded with 8000 pounds at
the middle and has a section like Fig. 18, with t = t 1 = l, dW,
and I = 6 inches. Neglecting the weight of the beam, determine the
maximum fiberstress, both tensile and compressive.
CANTILEVER AND SIMPLE BEAMS 35
311. A floor is supported by wooden beams, 2 x 10 inches section
and 12 feet span, spaced 16 inches between centers. Find the safe
load per square foot of floor area if the maximum fiberstress is 800
pounds per square inch.
312. A floor is to support a total load of 200 pounds per square foot
of floor area. Determine the proper size for the steel Ibeams, 12 feet
span and spaced 5 feet apart between centers, to support this floor
with a factor of safety of 4.
313. A beam has a section like Fig. 19, with \ = b 2 = 6, t 1 =t 2 = 1,
t=\, and d = 10 inches. Compare its strength to resist bending when
placed like this: I ; and like this: HH .
314. A castiron simple beam of 12 feet span has a section like Fig. 20,
with t = 1, d 6, and 6 = 8 inches. Find the factors of safety against
tension and compression under a total uniform load of 5000 pounds.
315. Select the proper steel Ibeam for Problem 193 for a factor of
safety of 6.
316. Select the proper steel Ibeam for Problem 203 for a factor of
safety of 6.
RUPTURE
317. Find the length of a castiron cantilever beam 2 inches square
that will break under its own weight.
318.. A castiron cantilever beam 2x4 inches section area and
12 feet long, carries a concentrated load at its free end. Find this
load to break the beam, considering the weight of the beam itself.
319. Calculate the length of a wooden cantilever beam 1x2 inches
section area that will break under its own weight.
320. Determine the total uniform load to rupture a castiron canti
lever beam 2 inches square and 10 feet long.
321. Compute the size of a square wooden simple beam 9 feet span
that will break under its own weight.
322. Determine the concentrated load placed at the middle that
will rupture a wooden simple beam 2x4 inches crosssection and
12 feet span. Neglect weight of beam.
323. A castiron simple beam 12 feet long and 3 inches square
carries two equal loads at quarter points. Find the loads that will
rupture the beam, neglecting its own weight.
36 PROBLEMS IN STRENGTH OF MATERIALS
324. Determine the total uniform load that will rupture a wooden
simple beam 8 feet span and 2x6 inches crosssection.
325. Find the load placed at the middle that will break a round
castiron bar 16 feet long and 4 inches in diameter supported at the
ends. Neglect weight of bar.
326. Determine the greatest length of a round castiron bar 1 inch
in diameter that can just carry its own weight when supported at the
ends.
327. A simple beam 6 feet long, 2 inches broad, and 3 inches deep
is broken by a weight of 1200 pounds placed at the center. What
uniformly distributed load will break a simple beam of the same
length and material, if breadth is 3 inches and depth 4 inches ?
MOVING LOADS
328. A load of 500 pounds is rolled over a simple beam 20 feet
long, whose weight is 40 pounds per linear foot. Find the position
of this load for the maximum bending moment and compute its value.
329. Two loads each of 3000 pounds, 5 feet apart, roll over a simple
beam of 15 feet span. Find the position of these loads for the maxi
mum bending moment and find its value.
330. Two wagon wheels, 8 feet apart, roll over a simple beam 24
feet long. If the load on each wheel is 2000 pounds, find their posi
tion for the maximum bending moment and determine its value.
331. Solve Problem 330, if the load on one wheel is 3000 pounds
and on the other 2000 pounds.
332. Three loads of 3000 pounds each, 4 feet apart, roll over a
simple beam of 20 feet span. Find the position of these loads for the
maximum bending moment and compute its value.
333. Three loads 4 feet apart, one being 3000 pounds and the
others 1500 pounds each, roll over a simple beam 19 feet long. Find
their position for the maximum bending moment, and find its value.
DEFLECTION
334. Find the maximum deflection of a wooden cantilever beam
6 inches wide, 8 inches deep, and 10 feet long, due to a total uniform
load of 100 pounds per linear foot.
CANTILEVER AND SIMPLE BEAMS 37
335. A wooden cantilever beam 6 inches wide, 8 inches deep, and
10 feet long supports a weight of 1000 pounds at the free end. Find
the maximum deflection due to this load.
336. Find the maximum deflection for the beam of Problem 334, if
it has a concentrated load of 1000 pounds at the free end in addition
to the uniform load.
337. A castiron bar 2 inches wide, 4 inches deep, and 6 feet long
is balanced upon a support at its middle point, and a weight of 5000
pounds hung at each end. Find the end deflections.
338. Find the maximum deflection of a simple wooden beam 16 feet
long, 2 inches wide, and 4 inches deep, due to a load of 120 pounds
at the middle.
339. Calculate the maximum deflection of a steel bar, supported
at its ends, 1 inch square and 6 feet long, due to a load of 100 pounds
at its center.
340. A steel engine shaft 12 inches in diameter, resting in bearings
4 feet apart, carries a flywheel weighing 30 tons midway between
the bearings. Considering the shaft as a simple beam, find its maxi
mum deflection.
341. A wooden floor beam 2 X 10 inches crosssection and 16 feet
long sustains a total uniform load of 80 pounds per linear foot. Find
the maximum deflection.
342. Determine the maximum deflection of a wooden girder 10 feet
long, 8 inches wide, and 10 inches deep, supported at its ends, if it
carries a total uniform load of 10,000 pounds.
343. A simple wooden beam 8 x 14 inches crosssection and 16 feet
long bears a total uniform load of 100 pounds per linear foot. How
much greater will be the maximum deflection when the short side is
vertical than when the long side is vertical?
344. A castiron rod supported at both ends, 5 feet span, 2 inches
wide and ^ inch deep, has a maximum deflection of ^ inch due to a
weight of 18 pounds at its center. Find its modulus of elasticity.
345. Find the maximum deflection of a simple wooden beam 9 feet
long, due to a concentrated load of 1000 pounds at the middle.
Crosssection^ is an ellipse having axes 6 and 4 inches, with short
axis vertical.
346. Solve Problem 345 if the long axis is vertical.
38 PROBLEMS IN STRENGTH OF MATERIALS
347. A hollow circular castiron beam 8 feet long, inside diameter
5 inches and outside diameter 6 inches, rests upon end supports.
Calculate the maximum deflection due to a load of 4000 pounds at
the middle.
348. A beam is 4 x 12 inches section area and 16 feet long. Another
beam of the same material is 6 x 8 inches section area and 10 feet
long. What is the ratio between the maximum deflections, if the
longer side is vertical in each beam, and the manner of loading and
supporting is the same ?
349. A beam 16 feet long, 2 inches wide, and 6 inches deep has a
maximum deflection of 0.3 inch. Determine the maximum deflec
tion of a beam of the same material 12 feet long, 3 inches wide, and
8 inches deep, with the same loading and manner of support.
350. Find the maximum deflection of a 12inch steel Ibeam, 40
pounds per foot, resting on end supports 20 feet apart, and bearing a
total uniform load of 900 pounds per linear foot.
351. A 9inch steel Ibeam, 21 pounds per foot, supported at its
ends and 10 feet long, bears a concentrated load of 25 tons at its
center. Determine the maximum deflection due to this load.
352. Find the deflection at a point 4 feet from the left end of the
beam in Problem 351.
353. A floor is to support a total uniform load of 100 pounds per
square foot. The 10inch steel Ibeams, 25 pounds per foot, have a
span of 20 feet and are spaced 6.5 feet apart between centers. Does
the maximum deflection of the beams exceed ^^ of the span ?
354. Determine the proper distance from center to center of 12inch
steel Ibeams, 35 pounds per foot, 24 feet span, to support a total
uniform load of 100 pounds per square foot of floor area, with a maxi
mum deflection of gl^ of the span.
355. Solve Problem 354 for 15inch steel Ibeams, 50 pounds per
foot and 26 feet span.
356. Find the distance between supports for 9inch steel Ibeams,
30 pounds per foot, spaced 7.5 feet from center to center, to support
a total uniform load of 100 pounds per square foot of floor area, with
a maximum deflection of gj^ of the span.
357. Solve Problem 356 for 8inch steel Ibeams, 18 pounds per
foot, spaced 9 feet apart from center to center.
UNIVERSITY
OF
OVERHANGING BEAMS 39
358. For the case of 10inch steel Ibeams, 30 pounds per foot,
supported at both ends, and loaded uniformly, determine the span for
which the maximum stress shall be 16,000 pounds per square inch.
and the maximum deflection ^J^ of the span.
359. Solve Problem 358 for 12inch steel Ibeams, 35 pounds per
foot.
360. Find the total uniform load for a 6inch steel Ibeam, 14.75
pounds per foot, resting on end supports 20 feet apart, if the maxi
mum deflection is gl^ of the span and the maximum stress not
greater than 16,000 pounds per square inch.
361. Solve Problem 360 for a 7inch steel Ibeam, 20 pounds per
foot.
362. A wooden cantilever 15 feet long, 3 inches wide, and 4 inches
deep carries a load of 100 pounds 5 feet from the free end. Find the
deflection at the end due to this load.
363. The wooden cantilever of Problem 362 carries a load of 100
pounds 5 feet from the free end, and another load of 100 pounds
10 feet from the free end. Calculate the end deflection.
364. A beam of length I rests on end supports and bears a total
uniform load W. Another support just touches under the middle of
the beam. How much must this middle support be raised in order
that the end supports shall just touch the beam?
365. Solve Problem 364 for a wooden beam 12 feet long, 3 inches
wide, and 4 inches deep, bearing a total uniform load of 400 pounds.
VI. OVERHANGING BEAMS
Draw the shear and moment diagrams, determine the maximum
shear, maximum moment, danger sections, and points of inflection in
the following cases :
t^^^^^^4M^pe^t^^^^^^^\
T
 '_  *J^  10'  >J
367.
^SLOO(ntypei^t^
^0'._
40 PROBLEMS IN STRENGTH OF MATERIALS
368.
1000 lO.per ft. SOOlb.perfe]
U 10'  JC* _y_ J
750 Ib. 125 lb.>
369. I y"" g "1
2000 Z6.
370. pmmmmmmmmammmmmLmmfmmm
L< j 0' >!<=. _ ^. _>.
2000 Ib.
2000 Ib.
*~&~9L
371.
372. A piece of timber is supported at one end and at one other
point. Find the position of this point if the reaction is double that
at the end.
373. r
U^'^U 20'
tlOO Ib. per ft. \
*__ HBBIIMHBHri
_.jL_>l< IQL J^_ _5^_ _J
100 Ib. per ft. \
T T
10' *+ 15 ' > J  e:  5 r ^
Z6.
156 Ib.
376.
2000 Ib.
FIXED BEAMS
2000 Ib.
377.
w
WOO Ib.
I
' *** 5 >
378.
erg^^^l
'___: >U'J
40001500
Ib. Ib. I
16000
Ib.
12000
Ib.
379.
 8' >
4000
500lb.perft. I
U ^dc 10' J< 4  ^>
41
VII. FIXED BEAMS
Draw the shear and moment diagrams, determine the maximum
shear, maximum moment, danger sections, and points of inflection in
the following cases :
381
t^^lOCHt^e^^^^^
12 f
382.
^00 Ib.perft.
3200 Ib.
"4
42 PEOBLEMS IN STEENGTH OF MATEEIALS
4800 Ib.
384
,p_d
U_ &L
540 Ib.
e . 8 '. :
385. P" "' g '" H
76
2000 Z6.
386. 1
k 7'
c~
 JU
387. pi
/00 16. per ft
388.
,200 Ib.
n'
389. A castiron hollow cylindrical beam 30 feet long and fixed at
both ends has an external diameter of 10 inches and an internal
diameter of 10 inches. What load can it support at the middle with a
factor of safety of 6, and what will then be the maximum deflection ?
Neglect weight of beam.
390. What steel Ibeam with fixed ends is required for a span of
20 feet to support a total uniform load of 20,000 pounds, with a
maximum unitstress of 15,000 pounds per square inch? Find also
the maximum deflection.
391. An 8inch steel Ibeam, 18 pounds per foot, with fixed ends
and 8 feet span, is loaded at the middle so that the maximum unit
stress is 16,000 pounds per square inch. Find the maximum deflection.
CONTINUOUS BEAMS
VIII. CONTINUOUS BEAMS
43
Draw the shear and moment diagrams, determine the maximum
shear, maximum moment, danger sections, and points of inflection in
the following cases :
392. 41
w Ib. per ft.
c 1
\
;  J
393.
394.
395,
396.
U 12' >U iz' J
t
120 Ib.perjt.
18'
r f . I
>1< i2 r  J
w Ib. per ft.

z  J
t^^^^^^^^^^^^^M^^^e^/^^^^^^^^^^^^^^l
1 J^g. i J^ 1 _JU f^^J
397. JT
210 Ib.perjt.
398.
J* so' J
U if)' J^ ' Je in' J
399. ^
w^^per^^^^^^^^^^^^^^^j
T" T "t
r   ^^ w' ^U s' ^
8 '   *** 10' * 1 * 70' 3Jc   8'
A 12inch steel Ibeam, 40 pounds per foot, 1 extends over these
supports. Find w for a factor of safety of 4.
400.
1
44 PKOBLEMS IN STRENGTH OF MATERIALS
401. W
vmfr
w Ib.p
1
lhperft
i
i
... ...I. ..... .3
3200 J6.
404
'^
U 10':
J0'
IX. COLUMNS AND STRUTS
405. Find the load to rupture a castiron cylindrical column with
flat ends, 15 feet long and 6 inches in diameter, both by Euler's and
Rankine's formulas.
406. A cylindrical steel column with round ends is 36 feet long
and 6 inches in diameter. Calculate by Euler's formula the axial load
for rupture.
407. Find the buckling load for a steel strut with rounded ends,
3 inches square and 2J feet long, both by Euler's and Rankine's
formulas.
408. Solve Problem 407 for a length of 9 feet.
409. Solve Problem 407 for a length of 15 feet.
410. A square wooden column with fixed ends is 20 feet long and
sustains a load of 10,000 pounds. Find its size by Euler's formula,
with a factor of safety for steady load.
411. Find the breaking load for a solid round wroughtiron pillar
2 inches in diameter and 10 feet in length, with fixed ends.
COLUMNS AND STRUTS 45
412. Calculate the breaking load for a wroughtiron column with
fixed ends, 9x4 inches section area and 20 feet long.
413. Find the loads to rupture a round wroughtiron and a round
castiron column each 9 feet in length and 6 inches in diameter, with
flat ends.
414. Determine the buckling load for a castiron strut with rounded
ends, 2x1 inches section area and 16 inches long.
415. Find the breaking load for a cylindrical strut of wroughtiron,
3 inches in diameter and 10 feet long, with rounded ends.
416. What load can be sustained by a castiron column with flat
ends, 14 feet long and 6 inches in diameter, with a factor of safety
of 8?
417. A wooden stick 3x4 inches crosssection and 10 feet long is
used as a column with flat ends. Find the factor of safety under a
load of 2000 pounds.
418. Determine the load for a fixedend timber column, 3x4 inches
crosssection and 10 feet long, for a factor of safety of 10. What would
it be for a length of less than 2 feet ?
419. Find the breaking load for a hollow castiron pillar with fixed
ends, 9 feet in length and 6 inches square, the metal being 1 inch thick.
420. Find the safe steady load for a hollow castiron column with
fixed ends, outside dimensions 8x6 inches, inside dimensions 6x4
inches, and 10 feet long.
421. Solve Problem 420 for a length of 20 feet.
422. Find the safe steady load for a hollow castiron column with
fixed ends, length 20 feet, outside dimensions 4x5 inches, inside
dimensions 3x4 inches.
423. Solve Problem 422 for a length less than 3 feet.
424. A hollow wooden column of rectangular section, 4 x 5 inches
outside dimensions, 3x4 inches inside dimensions, has fixed ends
and a length of 16 feet. Find the factor of safety for an axial load
of 1200 pounds.
425. A wroughtiron pipe 10 feet long, 4 inches external and 3 inches
internal diameter, sustains a load of 14 tons. What is the factor of
safety ?
426. What safe steady load will a hollow castiron column with flat
ends support if it is 14 feet long, outside diameter 10 inches and inside
46 PKOBLEMS IN STRENGTH OF MATERIALS
diameter 8 inches ? Compare the load it could support for a length
less than 8 feet with the result obtained.
427. Determine the load on a hollow, round, castiron column with
flat ends, external diameter 12 inches, thickness 1 inch, and length
14 feet, for a factor of safety of 8.
428. A cylindrical, wroughtiron column with fixed ends is 10 feet
long, 6.4 inches outside diameter, 6 inches inside diameter, and carries
a load of 50,000 pounds. Find its factor of safety.
429. Determine the size of a square, wooden column 30 feet long
with flat ends to safely sustain a steady load of 20 tons.
430. Solve Problem 429 for a steady load of 7 tons.
431. A square, wooden post 12 feet high is to support a load of 16
tons. What must be the size of the post for a factor of safety of 10?
432. Find the size of a square steel strut 8 feet long with round
ends to safely transmit a steady compressive load of 5 tons.
433. Around, solid, wroughtiron pillar 10 feet in height is to support
a load of 40,000 pounds. Find its diameter for a factor of safety of 5.
434. Determine the diameter of a round, solid, steel pillar 16 feet
high to safely support a steady load of 29,000 pounds.
435. A round, solid, castiron strut 15 feet long with rounded ends
bears a compressive load of 10 tons. Find its diameter for a factor
of safety of 6.
436. A hollow, square, wooden column with flat ends is to safely
support a steady load of 12,000 pounds. If the thickness of each
side is 1J inches and length of column 20 feet, what should be the
outside and inside dimensions ?
437. A cylindrical, structural steel connectingrod 7J feet long is
subjected to a maximum compressive load of 21,000 pounds. Consid
ering it to be a column with both ends hinged, determine its diameter
for a factor of safety of 10.
438. A wroughtiron pistonrod has a diameter of 2 inches and a
length of 4 feet. Considering it to be a column with one end flat
and the other round, what is the allowable diameter of the piston,
if the steam pressure is 60 pounds per square inch, for a factor of
safety of 10?
439. The diameter of a piston is 18 inches and the maximum steam
pressure 130 pounds per square inch. Find the proper diameter
COLUMNS AND STRUTS 47
for the structural steel pistonrod if it is 5 feet long and subject
to shocks.
440. The diameter of a piston is 40 inches and the maximum steam
pressure 110 pounds per square inch. The wroughtiron connecting
rod has a rectangular crosssection and a length of 121 feet. Find
the dimensions for the crosssection of the rod for a factor of safety
of 10.
441. A hollow, circular, steel column 28 feet long with fixed ends
is to support a steady load of 30 tons. If the external diameter is 6
inches, determine the thickness of the metal for a factor of safety of 4.
442. A steel Ibeam 15 inches deep, 50 pounds per foot, is used as
a column 10 feet long with fixed ends. Find the load it can bear
with a factor of safety of 4.
443. A column 20 feet long with fixed ends is formed by joining
the legs of two 10inch steel channels, 30 pounds per foot, by two
plates 10 inches wide and ^ inch thick, section as shown in Fig. 27.
Find the load for a factor of safety of 4.
444. A steel column 20 feet long with fixed ends is used in a bridge
under an axial compression of 240,000 pounds. The section is like
Fig. 28, the 12inch channels weigh 20.5 pounds per foot, I = 16, and
t =  inch. Determine the factor of safety.
445. Two 8inch steel Ibeams, 25.25 pounds per foot, are joined
by lattice work to form a column 20 feet long with fixed ends. How
far apart must the beams be placed, center to center, in order that the
column shall be of equal strength to resist buckling in either axial
plane ? What load can the column then stand with a factor of safety
of 5?
446. Solve Problem 445 with two 9inch steel Ibeams, 21 pounds
per foot.
447. A steel column 14 feet long with square ends is formed by
two 12inch steel channels, 20.5 pounds per foot, placed back to back.
Determine the proper spacing of the channels and the load the column
will then carry with a factor of safety of 4.
448. Find the load for a hollow castiron column with fixed ends,
16 feet long, outside dimensions 4x5 inches, inside dimensions 3x4
inches, if the eccentricity of the load is 1J inches and the factor of
safety is 6.
48 PEOBLEMS IN STRENGTH OF MATERIALS
449. Determine the size of a square wooden column, 10 feet long,
with fixed ends, to carry an eccentric load of 15 tons, eccentricity
2 inches, with a factor of safety of 10.
450. A hollow, cylindrical, castiron column with fixed ends, 10
inches external diameter, 8 inches internal diameter, and 10 feet long,
is loaded with 80,000 pounds 2 inches out of center. Determine the
factor of safety.
X. TORSION
451. Find the horsepower that can be transmitted by a castiron
shaft 3 inches in diameter and making 10 R.P.M., with a factor of
safety of 10.
452. Find the diameter of a solid, wroughtiron, circular shaft to
safely transmit 150 H.P. at a speed of 60 R.P.M.
453. Calculate the diameter of a structural steel shaft to transmit
300 H.P. at a speed of 200 R.P.M., with a factor of safety of 6.
454. Find the H.P. that can be safely transmitted by a castiron,
circular shaft 7 inches in diameter and making 80 R.P.M.
455. What should be the diameter of a structural steel shaft to
safely transmit 500 H.P. at 200 R.P.M.?
456. Calculate the H.P. that a round, wroughtiron shaft 8 inches
in diameter and making 150 R.P.M. will transmit with a factor of
safety of 6.
457. Find the diameter of a wroughtiron shaft to transmit 5000
H.P. at 100 R.P.M., with a factor of safety of 10.
458. Calculate the diameter of a structural steel engineshaft to
transmit 4000 H.P. at 50 R.P.M., with a factor of safety of 10.
459. Find the factor of safety for a wroughtiron shaft 3 inches in
diameter when transmitting 40 H.P. at 100 R.P.M.
460. Determine the factor of safety for a structural steel shaft
2 inches in diameter and transmitting 25 H.P. at 100 R.P.M.
461. What H.P. can be transmitted, with a factor of safety of 6, by
a hollow, wroughtiron shaft, external diameter 12 inches, internal
diameter 10 inches, and making 60 R.P.M.
462. Find the speed for a hollow, castiron shaft, 10 inches outside
diameter, 6 inches inside diameter, to transmit 750 H.P. with a factor
of safety of 10.
TOESION 49
463. Find the H.P. that can be safely transmitted by a hollow,
wroughtiron shaft making 100 E.P.M., if the outside diameter is
8 inches and the inside diameter 5 inches.
464. Find the ratio of the strength of a hollow circular shaft to that
of a solid circular one of the same material and the same section area.
465. A solid shaft has a diameter of 12 inches and a hollow shaft
of the same material has an external diameter of 20 inches. Find
the internal diameter of the hollow shaft for the same section area,
and the ratio of their strengths.
466. A hollow and a solid shaft are of the same material and same
section area. If the outside diameter of the hollow shaft is twice its
inside diameter, find the ratio of the strengths of the shafts.
467. A solid shaft of structural steel is to transmit 300 H.P. at
200 RP.M. If the maximum moment is 30 per cent greater than
the average, find the diameter of the shaft for a factor of safety of 6.
468. A hollow, structural steel shaft, outside diameter 6 inches,
transmits 300 H.P. at 200 E.P.M. Find the inside diameter for a
factor of safety of 6.
469. Find the H.P. that can be transmitted by a hollow, structural
steel shaft, 15 inches external diameter and 11 inches internal diam
eter, at a speed of 50 E.P.M., with a factor of safety of 4.
470. A steel wire 0.18 inch in diameter and 20 inches long is
twisted through an angle of 18.5 by a moment of 20 inchpounds.
Determine the shearing modulus of elasticity of the wire.
471. Find the shearing modulus of elasticity of a castiron bar
10 inches long and 0.82 inch in diameter, if twisted through an
angle of 1.3 by a twisting moment of 50 poundfeet.
472. A shaft 15 feet long and 4.5 inches in diameter is twisted
through an angle of 2 by a moment of 2000 poundfeet. Find the
moment which will twist a shaft of the same material, 20 feet long and
7 inches in diameter, through an angle of 2.5.
473. A round, castiron shaft 15 feet in length is acted upon by a
weight of 2000 pounds applied at the circumference of a wheel on
the shaft, whose diameter is 2 feet. Determine the diameter of the
shaft so that the angle of torsion shall not exceed 2.
474. A steel shaft 20 feet in length and 3 inches in diameter trans
mits 50 H.P. at 200 E.P.M. Through what angle is the shaft twisted ?
50 PROBLEMS IINT STRENGTH OF MATERIALS
475. A wroughtiron shaft 20 feet long and 5 inches in diameter is
twisted through an angle of 2. Find the maximum unitstress in
the metal.
476. Find the diameter and angle of twist of a 12foot wroughtiron
shaft transmitting 20 H.P. at 25 R.P.M., with a factor of safety of 6.
477. A structural steel shaft 120 feet long and 16 inches in diam
eter transmits 8000 H.P. at 20 R.P.M. Find the angle of twist and
the factor of safety.
478. A turbine transmits 92 H.P. at 114 R.P.M. through a wrought
iron shaft 8.5 feet in length. Determine the diameter of the shaft so
that the angle of torsion shall not exceed 1.
479. A structural steel shaft 20 feet in length and 3 inches in
diameter transmits 50 H.P. at 200 R.P.M. Through what angle is
the shaft twisted and what is the factor of safety?
480. A structural steel shaft 2 inches in diameter transmits 25 H.P.
at 100 R.P.M. Find the factor of safety and the angle of twist per
linear foot.
481. A castiron shaft in a spinning mill is 84 feet long and trans
mits 270 H.P. at 50 R.P.M. Find its diameter if the stress in the
metal is not to exceed 5000 pounds per square inch and the angle of
torsion is not to exceed 0.1 per linear foot.
482. Determine the diameter and angle of twist of a solid steel
shaft 20 feet long, to transmit 6000 H.P. at 116 R.P.M., the maxi
mum twisting moment being 30 per cent greater than the mean and
the maximum allowable stress 10,000 pounds per square inch.
483. Find the size of a hollow steel shaft to replace the one of
Problem 482, if the inside diameter is f of the outside diameter.
What is the saving in weight in 50 feet of shafting ?
484. Find the diameter and angle of twist per linear foot of a hollow
steel shaft transmitting 5000 H.P. at 70 R.P.M., if the external diam
eter is twice the internal and the maximum stress is 7500 pounds
per square inch.
485. Find the ratio of the strengths of two solid circular shafts of
the same material, and the ratio of their stiffness for the same length.
486. Solve Problem 485, if one diameter is twice the other.
487. The external diameter of a hollow shaft is n times the internal.
Compare its torsional strength with that of a solid circular shaft of
TORSION 51
the same material and same section area, in terms of n. Find also
their stiffness ratio for the same length.
488. Solve Problem 487, if the external diameter of the hollow
shaft is twice its internal diameter.
489. A solid shaft 10 inches in diameter is of the same material
and section area as a hollow shaft whose internal diameter is 5 inches.
Determine the external diameter of the hollow shaft and compare
their torsional strengths and stiffness for the same length.
490. A hollow steel shaft has an external diameter d and an inter
nal diameter  Compare its torsional strength and stiffness with a
Zi
solid steel shaft of the same length and of diameter d.
491. A solid shaft 6 inches in diameter is coupled by bolts 1 inch in
diameter on a flange coupling. The centers of the bolts are 5 inches
from the axis. Find the number of bolts in order that their torsional
strength shall equal that of the shaft.
492. What H.P. can be transmitted with a factor of safety of 6 by
a wroughtiron shaft 4 inches square and making 110 RP.M. ?
493. Find the H.P. that can be transmitted by a 7inch castiron
square shaft making 80 R.P.M., with a factor of safety of 10.
494. A wooden beam 6 inches square projects 4 feet from a wall,
and is acted upon at the free end by a twisting moment of 20,000
poundfeet. Find the angle of twist.
495. What torsional moment can a wroughtiron shaft 10 feet
long and 5 inches square withstand, with the angle of torsion less
than J?
496. A round wroughtiron shaft 3 inches in diameter and 20 feet
long transmits 20 H.P. at 100 RP.M. Find the size of a square
wroughtiron shaft of equal strength, and the angle of twist for each
shaft.
497. A square wooden shaft 8 feet in length is acted upon by a
force of 200 pounds applied at the circumference of an 8foot wheel
on the shaft. Find the size of the shaft in order that the angle of
torsion shall not exceed 2.
498. Determine the factor of safety and the angle of twist per foot
of length for a wooden shaft 12 inches square when transmitting
24 H.P. at 12 E.P.M.
52 PROBLEMS IN STRENGTH OF MATERIALS
499. Compare the strength and stiffness of a square shaft with that
of a round shaft of the same material when a side of the square shaft
is equal to the diameter of the round shaft.
500. Compare the strength and stiffness of a round shaft with that
of a square one of the same material and having the same area of
crosssection.
XL COMBINED STRESSES
501. A 12inch steel Ibeam, 40 pounds per foot, 6 feet span, carries
in addition to its own weight a uniform load of 1200 pounds, and is
subjected to an axial compression of 60,000 pounds. Find the factor
of safety.
502. Find the size of a square, wooden simple beam of 12 feet span
to carry a load of 400 pounds at the middle, when it is also subject
to an axial compression of 3000 pounds, the maximum allowable
compressive stress being 1000 pounds per square inch. Neglect weight
of beam.
503. Determine the factor of safety for a simple wooden beam
8 feet long, 10 inches wide, and 9 inches deep, under an axial com
pression of 40,000 pounds, and bearing a total uniform load of 4200
pounds.
504. A wooden cantilever beam 3 feet long, 3 inches wide, and
4 inches deep has a load of 300 pounds at the free end, and is under
an axial compression of 4500 pounds. Determine the maximum com
pressive unitstress, neglecting weight of beam.
505. A wooden cantilever beam 8 inches wide and 4 feet long
carries a total uniform load of 400 pounds per linear foot, and is sub
jected to an axial compression of 40,000 pounds. Find the depth of
the beam so that the maximum compressive unitstress shall be 1000
pounds per square inch.
506. Solve Problem 501 for an axial tension of 60,000 pounds
instead of the axial compression.
507. Find the size of a square, wooden simple beam of 12 feet
span to carry a load of 400 pounds at the middle, when it is also
subject to an axial tension of 3000 pounds, the maximum allowable
tensile stress being 1000 pounds per square inch. Neglect weight
of beam.
COMBINED STRESSES 53
508. Determine the factor of safety for a simple wooden beam 8 feet
long, 10 inches wide, and 9 inches deep, under an axial tension of
40,000 pounds, and bearing a total uniform load of 4200 pounds.
509. A wooden cantilever beam 3 feet long, 3 inches wide, and 4
inches deep has a load of 300 pounds at the free end, and is under
an axial tension of 4500 pounds. Compute the maximum tensile and
compressive unitstresses, neglecting the weight of the beam.
510. Find the size of a square, wooden simple beam of 12 feet span,
which bears a total uniform load of 50 pounds per linear foot, and at
the same time is under an axial tension of 2000 pounds, the maxi
mum allowable unitstress being 1000 pounds per square inch.
511. A bolt 1 inch in diameter is subjected to a longitudinal tension
of 5000 pounds, and at the same time to a crossshear of 3000 pounds.
Determine the maximum combined tensile and shearing unitstresses,
and the angles they make with the axis of the bolt.
512. Find the maximum unitstresses in a circular steel shaft
6 inches in diameter, resting on supports 10 feet apart, and trans
mitting 50 H.P. at 225 R.P.M., due to the combined bending and
torsional moments.
513. Determine the diameter of a solid wroughtiron shaft 12 feet
between bearings and transmitting 50 H.P. at 130 R.P.M., if pulleys
are placed so as to produce a maximum bending moment of 600
poundfeet at the middle, and the maximum combined unitstress is
10,000 pounds per square inch.
514. A bar of iron is under a direct tensile stress of 5000 pounds
per square inch and a shearing stress of 3500 pounds per square
inch. Find the maximum tensile and shearing unitstresses.
515. A wroughtiron shaft is subjected simultaneously to a bending
moment of 8000 poundinches and a twisting moment of 15,000
poundinches. Determine the least diameter of the shaft if the maxi
mum tensile strength is not to exceed 10,000 pounds per square inch,
and the shearing stress 8000 pounds per square inch.
516. Find the diameter of a wroughtiron shaft to transmit 90 H.P.
at 130 E.P.M., with a factor of safety of 5, if there is also a bending
moment equal to the twisting moment.
517. A wroughtiron shaft 3 inches in diameter and making 140
RP.M. is supported in bearings 16 feet apart. If a load of 210 pounds
54 PROBLEMS IN STRENGTH OF MATERIALS
is brought by a belt and pulley at the middle, what H.P. can be
transmitted with a maximum shearing stress of 8000 pounds per
square inch?
518. Compute the maximum unitstresses for a steel shaft 3 inches
in diameter, in fixed bearings 12 feet apart, which transmits 40 H.P.
at 120 R.P.M., and upon which a load of 800 pounds is brought by a
belt and pulley at the middle.
519. Find the diameter of a steel shaft, in fixed bearings 8 feet
apart, to transmit 90 H.P. at 250 R.P.M., if there is a load of 480
pounds at the middle and the maximum allowable unitstress is 7000
pounds per square inch.
520. Determine the factor of safety for a wroughtiron shaft 3 inches
in diameter, resting in bearings 12 feet apart, when transmitting 25
H.P. at 100 R.P.M., and bearing a load of 200 pounds at the middle.
521. A hollow structural steel shaft, 17 inches outside diameter
and 11 inches inside diameter, with ends fixed in bearings 18 feet
apart, is to transmit 15,000 H.P. at 50 R.P.M. Find the maximum
unitstresses, considering the weight of the shaft.
522. A steel shaft 4 inches in diameter, with ends fixed in bearings
10 feet apart, carries a pulley 14 inches in diameter at its center. If
the tension in the belt on this pulley is 250 pounds, and the shaft
makes 80 R.P.M., how many H.P. is it transmitting, and what is the
maximum unitstress in the shaft ?
523. Find the factor of safety for a vertical wroughtiron shaft
4 feet long and 2 inches in diameter, if it weighs with its loads 6000
pounds, and is subjected to a twisting moment of 1200 poundfeet.
524. Find the maximum horizontal shearing unitstress in a canti
lever beam 6 inches wide, 8 inches deep, and 10 feet long, if it sup
ports a weight of 1000 pounds at its free end.
525. A simple wooden beam 4 inches wide, 12 inches deep, and
14 feet span bears a load of 12,500 pounds at the middle. Find the
maximum horizontal shearing unitstress.
526. A wooden, builtup simple beam 6 inches wide, 12 inches
deep, and 10 feet long is formed by bolting together three 4x6 inch
beams. When the beam supports a load of 2000 pounds at its middle
point, find the maximum unitshear in the planes of contact and the
total horizontal shear on the bolts.
COMPOUND COLUMNS AND BEAMS 55
527. A 10inch steel Ibeam, 30 pounds per foot, resting on sup
ports 20 feet apart, carries a load of 6000 pounds at its middle point.
Determine the maximum horizontal unitshear if the center of gravity
of each half section is 4.5 inches from the neutral axis.
528. An 8inch steel Ibeam, 18 pounds per foot, resting on sup
ports 15 feet apart, carries a load of 5000 pounds at its middle point
Determine the maximum horizontal unitshear, if the center of gravity
of each half section is 3.6 inches from the neutral axis.
XII. COMPOUND COLUMNS AND BEAMS
529. A vertical bar 10 feet long and 1 inch square is compounded
by fastening together rigidly at the two ends a bar of steel and a bar
of copper of equal size. When a load of 12,000 pounds is applied at
the lower end o'f the compound bar, how much of this load will be
sustained by each of the component bars, and what will be the elonga
tion of the compound bar ?
530. A compound column 4 feet in length is formed by bolting two
* inch steel plates, 8 inches wide, to the 8inch sides of a piece of
timber 6x8 inches in section area. When the column sustains an
axial load of 120,000 pounds what is the compressive unitstress in
the steel and in the timber ?
531. A Hitched timber beam 15 feet long, supported at its ends,
has a timber section 8x12 inches, with two steel plates ^ X 9 inches
bolted to the 12inch sides. When the beam supports a total uniform
load of 16,000 pounds, find the factors of safety for the timber and
the steel.
532. Aflitched beam consists of two tiinbers, each 10 inches wide and
14 inches deep, with a steel plate  inch thick and 7 inches wide
bolted between them on the 14inch sides. Find the unitstress in the
steel when the unitstress in the timber is 900 pounds per square inch.
533. A concrete column 12 feet high and 12x12 inches in section
area has four vertical steel rods, each 1^ inches in diameter, placed
near the corners. Compute the unitstresses in the concrete and steel
due to their own weight and to an axial load of 30,000 pounds.
534. Find the load that a short concrete column 24 inches square,
reinforced with 4 round, vertical, steel rods 2^ inches in diameter,
56 PROBLEMS IN STRENGTH OF MATERIALS
can safely carry if the compression in the concrete is limited to 450
pounds per square inch. What is then the unitstress in the steel rods ?
535. What percentage of reinforcement must be introduced into a
concrete column designed to sustain a load of 650 pounds per square
inch, when the compressive unitstress in the concrete is limited to
500 pounds per square inch?
536. Find the safe bending moment for a reinforced concrete beam
16 inches deep and 4 inches wide, having one steel rod J inch in
diameter, with its center 1^ inches above the bottom of the concrete,
if the tensile resistance of the concrete is neglected and the maximum
compressive stress in the concrete is 600 pounds per square inch.
537. Find the safe bending moment for the beam of Problem 536,
if the concrete is to resist part of the tensile stresses with a maximum
tensile stress of 100 pounds per square inch.
538. Determine the maximum bending moment for a reinforced
concrete beam 8 niches wide and 17 inches deep, with a 1inchsquare
steel rod placed with its center 2 inches above the bottom, if the con
crete offers no tensile resistance and the maximum compressive stress
in the concrete is limited to 600 pounds per square inch.
539. Solve Problem 538, if the maximum compressive stress in the
concrete is limited to 500 pounds per square inch.
540. The beam of Problem 538 is supported at the ends of a 20foot
span and bears a total uniform load of 5200 pounds. Determine the
unitstresses in the concrete and in the steel, assuming that the con
crete sustains none of the tensile load.
541. A reinforced concrete beam 5 inches deep, 48 inches wide,
and 6 feet span has 2 square inches of steel placed 1 inch above the
bottom of the concrete and sustains a total uniform load of 6000
pounds. If the beam is supported at the ends and the concrete offers
no tensile resistance, determine the position of the neutral surface
and the maximum unitstresses in the concrete and in the steel.
542. A concretesteel beam 12 inches wide, 13J inches deep, and
14 feet span, with supported ends, has 1 per cent of steel embedded
11 inches above the bottom of the concrete and bears two loads of
1300 pounds each at the third points of the span. Assuming that
the concrete offers no tensile resistance, find the maximum unit
stresses in the steel and in the concrete. Consider weight of beam.
THICK CYLINDERS AND GUNS 57
543. A concrete beam 14 inches deep, 8 inches wide, and 10 feet
span, with supported ends, has two inch square steel rods placed
1 inch from the bottom. Supposing that the concrete offers no tensile
resistance, find the total uniform load for the beam if the maximum
cornpressive unitstress in the concrete is 500 pounds per square inch.
What is then the tensile unitstress in the steel ?
544. Solve Problem 543 for a concrete beam 8 inches broad, 10
inches deep, and 15 feet span, which is reinforced on the tensile side
by six ^inch steel rounds with their centers 2 inches from the
bottom of the beam.
545. A concrete beam 8 inches broad and 10 inches deep is
reinforced by steel rods placed with their centers 2 inches from the
bottom of the beam. Neglecting the tensile strength of the concrete,
tind the area of the steel reinforcement necessary to make the beam
equally strong in tension and compression. What is then the safe
bending moment for a factor of safety of 6 ?
XIII. THICK CYLINDEES AND GUNS
546. A cylinder 1 foot inside and 2 feet outside diameter is sub
jected to an internal pressure of 600 pounds per square inch and an
external pressure of 15 pounds per square inch. Determine the tan
gential unitstresses at the inside and outside surfaces of the cylinder.
547. The steel cylinder of an hydraulic press has an internal diam
eter of 5 inches and an external diameter of 7 inches. How great an
internal pressure can the cylinder withstand with a factor 'of safety
of 4 ?
548. Find the internal pressure to burst a castiron cylinder 10
inches inside diameter and 5 inches thick. Compare the result with
that obtained when considering it a thin cylinder.
549. Determine the internal pressure for a castiron pipe 10 inches
inside diameter and 2 inches thick, for a factor of safety of 8. Con
sider it first as a thick and then as a thin cylinder.
550. If a gun of 3 inches bore has an internal pressure of 2000
pounds per square inch, what should be its thickness so that the
greatest stress in the material shall not exceed 3000 pounds per
square inch?
58 PROBLEMS IN STRENGTH OF MATERIALS
551. The cylinder of an hydraulic press has an internal diameter
of 6 inches. Find its thickness to resist an interior hydrostatic pres
sure of 1200 pounds per square inch, with a maximum stress in the
material of 2000 pounds per square inch.
552. Determine the thickness for a steel locomotive cylinder 22
inches internal diameter, to withstand a maximum steam pressure of
200 pounds per square inch, with a factor of safety of 10.
553. What inside pressure will produce a maximum stress of
20,000 pounds per square inch in a guntube 6 inches inside diameter
and 3 inches thick?
554. A pipe 6 inches inside diameter is to withstand an internal pres
sure of 1000 pounds per square inch. Find its outside diameter, if the
maximum tensile stress in the metal is 3000 pounds per square inch.
555. A wroughtiron cylinder, inside radius 2 inches and outside
radius 3 inches, has no inside pressure but an external pressure of
4200 pounds per square inch. Find the stresses at the inside and
outside surfaces of the cylinder.
556. A guntube 3 inches inside radius and 5 inches outside radius
is hooped so that the tangential compression at the bore is 14,400
pounds per square inch. The inside pressure caused by an explosion
is 25,000 pounds per square inch. Determine the resultant tangential
tension at the bore during the explosion.
557. A guntube 4 inches inside diameter and 6 inches outside
diameter is hooped so that the tangential compression at the inside
surface is 18,000 pounds per square inch. Find the resultant tan
gential stress at the bore during an explosion which causes a pressure
of 25,000 pounds per square inch.
558. A guntube 4 inches inside diameter and 2 inches thick is
hooped so that the tangential compression on the inside surface is
30,000 pounds per square inch. What powder pressure will produce
a resultant tangential tension on the inside surface of 30,000 pounds
per square inch ?
559. A steel hoop whose thickness is 2 inches is shrunk upon a
steel tube whose inside radius is 3 inches and outside radius 5 inches.
Find the stresses produced at the inside and outside surfaces of the
hoop and tube, if the original difference between the outside radius of
the tube and inside radius of the hoop is 0.004 inch.
FLAT PLATES 59
560. A steel tube, outside radius 4 inches and inside radius 2.9984
inches, is shrunk upon another tube, outside radius 3.00098 inches
and inside radius 2 inches. Find the stresses produced in the tubes
at the outside and inside surfaces.
XIV. FLAT PLATES
561. Find the thickness of a fixed castiron cylinderhead 36 inches
in diameter, to sustain a uniform pressure of 250 pounds per square
inch, with a maximum tensile stress of 4000 pounds per square inch.
562. Determine the thickness of a fixed steel cylinderhead 36
inches in diameter to sustain a uniform pressure of 300 pounds per
square inch, with a factor of safety of 5.
563. The cylinder of a locomotive is 20 inches inside diameter.
Find the thickness of the steel endplate to withstand a pressure of
160 pounds per square inch, with a maximum tensile stress of 10,000
pounds per square inch.
564. A circular castiron valvegate ^ inch thick closes an opening
6 inches in diameter. Find the maximum unitstress in the gate if
the pressure against it is 65 pounds per square inch.
565. Find the maximum unitstress in a circular steel plate 1^
inches thick and 24 inches in diameter, bearing a load of 4000 pounds
at its center, if this load is distributed over a circle 3 inches in
diameter.
566. Determine the uniform pressure, with a factor of safety of 4
for an elliptical castiron manhole cover 3 feet long, 18 inches wide,
and 1 inch thick.
567. Find the proper thickness for an elliptical castiron manhole
cover 24 inches long and 16 inches wide, when used in a standpipe
under a head of water of 60 feet, with a factor of safety of 6.
568. Determine the safe uniform pressure for a castiron elliptical
manhole cover 20 inches long, 13 inches wide, and 1^ inches thick,
if the maximum unitstress is limited to 3000 pounds per square inch.
TABLES
AVERAGE PHYSICAL CONSTANTS
MATERIAL
ULTIMATE
TENSILE
STRENGTH
ULTIMATE
COMPRES
SIVE
STRENGTH
ULTIMATE
SHEARING
STRENGTH
MODULUS OF
ELASTICITY
SHEARING
MODULUS OF
ELASTICITY
Pounds per
Pounds per
Pounds per
Pounds per
Pounds per
Square Inch
Square Inch
Square Inch
Square Inch
Square Inch
Hard steel ....
100 000
120 000
80000
30 000 000
12 000 000
Structural steel .
60000
60000
50000
30 000 000
12 000 000
Wroughtiron
50000
50000
40000
25 000 000
10 000 000
Castiron ....
20000
90000
20000
15 000 000
6 000 000
Copper
30000
15 000 000
6 000 000
Timber, with grain .
10000
8000
600
1 500 000
Timber, across grain
3000
400 000
Concrete ....
300
3000
1000
3 000 000
Stone
6 000
1 500
6 000 000
Brick
3 000
1 000
2 000 000
ELASTIC
ULTIMATE
COEFFICIENT
UNIT ELON
GATION AT
LIMIT
FLEX URAL
WEIGHT
OF LINEAR
ELASTIC
STRENGTH
EXPANSION
MATERIAL
LIMIT
Pounds per
Pounds per
Pounds per
For 1
Square Inch
Square Inch
Cubic Foot
Fahrenheit
Inch
Hard steel ....
60000
110000
490
0.000 0065
0.0012
Structural steel. . .
35000
490
0.000 0065
0.0012
Wroughtiron . . .
25000
480
0.000 0067
0.0010
Castiron (tension) .
6000
35000
450
0.000 0062
0.0004
Castiron (compression)
20000
Timber ....'.
3000
9000
40
0.000 0028
0.0020
Concrete (compression)
1 000
700
150
0.000 0055
Stone (compression) .
2000
2000
160
0.000 0050
Brick (compression) .
1000
800
125
0.000 0050
63
DIMENSIONS OF BOLTS
DIAMETER
OF BOLT
DIAMETER
OF BOLT
THREADS
PER INCH
DIAMETER
AT ROOT
AREA OF
BODY
AREA OF
ROOT
Inches
Inches
Number
Inches
Square Inches
Square Inches
i
.125
40
.0122
A
.1875
30
.0276
\
.25
20
.185
.0491
026
A
.3125
18
.24
.0767
.045
1
.376
16
.294
.1104
.068
i 7 *
.4375
14
.346
.150
.093
\
.50
13
.400
.196
.125
A
.5625
12
.454
.249
.162
I
.625
11
.507
.307
.202
ft
.6875
.372
I
.75
10
.620
.442
.302
it
.8125
.518
1
.876
9
.731
.601
.420
it
.9375
.690
1
1.0
* 8
.837
.785
.550
*A
1.0625
.882
11
1.125
7
.940
.994
.694
*A
1.1875
1.110
11
1.25
7
1.065
1.227
.893
*A
1.3125
1.348
it
1.375
6
1.160
1.485
1.067
H
1.50
6
1.284
1.767
1.295
if
1.625
6
1.389
2.074
1.615
if
1.75
5
1.491
2.405
1.744
i
1.875
5
1.615
2.761
2.048
2
2.0
4
1.712
3.142
2.302
2i
2.25
41
1.962
3.976
3.023
2*
2.50
4
2.175
4.909
3.715
2
2.75
4
2.425
5.940
4.619
3
3.0
3
2.629
7.069
5.428
3 i
3.25
31
2.879
8.296
6.510
8*
3.50
i
3.100
9.621
7.548
3f
3.75
3
3.317
11.045
8.641
4
4.0
3
3.567
12.566
9.993
64
FACTORS OF SAFETY
MATERIAL
FOR STEADY STRESS
(BUILDINGS)
FOR VARYING STRESS
(BRIDGES)
FOR SHOCKS
(MACHINES)
Hard Steel ....
5
8
15
Structural Steel . .
4
6
10
WroughtIron . . .
4
6
10
Castiron ....
6
10
20
Timber
8
10
15
Brick and Stone . .
15
25
30
POISSON'S EATIO
Steel
Iron .
Brass
.295
.277
.357
Copper
Lead
Zinc
.340
.375
.205
EANKINE'S COLUMN FORMULA
VALUES OF CONSTANT
MATERIAL
BOTH ENDS FIXED
FIXED AND ROUND
BOTH ENDS ROUND
Steel
1
i
i
^VYou ^htIron
25000
1
14060
6250
1
CastIron
36000
1
20250
1
9000
1
Timber
5000
1
2810
1
1250
1
3 U
1690
7 50
65
PROPERTIES OF STANDARD IBEAMS
p
1 n
"i
J
J 2
J
DEPTH
OF
BEAM
WEIGHT
PER
FOOT
AREA
OF
SECTION
WIDTH
OF
FLANGE
MOMENT
OF
INERTIA
Axis 11
SECTION
VlODULUS
Axis 11
RADIUS
OF
GYRATION
Axis 11
MOMENT
OF
INERTIA
Axis 22
RADIUS
OF
GrY RATION
AXIS 2 2
Inches
Pounds
Square
Inches
Inches
Inches 4
Inches 3
Inches
Inches 4
Inches
3
5.50
1.63
2.33
2.5
1.7
1.23
.46
.53
3
6.50
1.91
2.42
2.7
1.8
1.19
.53
.52
3
7.50
2.21
2.52
2.9
1.9
1.15
.60
.52
4
7.50
2.21
2.66
6.0
3.0
1.64
.77
.59
4
8.50
2.50
2.73
6.4
3.2
1.59
.85
.58
4
9.50
2.79
2.81
6.7
3.4
1.54
.93
.58
4
10.50
3.09
2.88
7.1
3.6
1.52
1.01
.57
5
9.75
2.87
3.00
12.1
4.8
2.05
1.23
.65
5
12.25
3.60
3.15
13.6
5.4
1.94
1.45
.63
5
14.75
4.34
3.29
15.1
6.1
1.87
1.70
.63
6
12.25
3.61
3.33
21.8
7.3
2.46
1.85
.72
6
14.75
4.34
3.45
24.0
8.0
2.35
2.09
.69
6
17.25
5.07
3.57
26.2
8.7
2.27
2.36
.68
7
15.00
4.42
3.66
36.2
10.4
2.86
2.67
.78
7
17.50
5.15
3.76
39.2
11.2
2.76
2.94
.76
7
20.00
5.88
3.87
42.2
12.1
2.68
3.24
.74
8
18.00
5.33
4.00
56.9
14.2
3.27
3.78
.84
8
20.25
5.96
4.08
60.2
15.0
3.18
4.04
.82
8
22.75
6.69
4.17
64.1
16.0
3.10
4.36
.81
8
25.25
7.43
4.26
68.0
17.0
3.03
4.71
.80
9
21.00
6.31
4.33
84.9
18.9
3.67
5.16
.90
9
25.00
7.35
4.45
91.9
20.4
3.54
5.65
.88
9
30.00
8.82
4.61
101.9
22.6
3.40
6.42
.85
9
35.00
10.29
4.77
111.8
24.8
3.30
7.31
.84
10
25.00
7.37
4.66
122.1
24.4
4.07
6.89
.97
10
30.00
8.82
4.80
134.2
26.8
3.90
7.65
.93
10
35.00
10.29
4.95
146.4
29.3
3.77
8.52
.91
10
40.00
11.76
5.10
158.7
31.7
3.67
9.50
.90
12
31.50
9.26
5.00
215.8
36.0
4.83
9.50
1.01
12
35.00
10.29
5.09
228.3
38.0
4.71
10.07
.99
12
40.00
11.76
5.21
245.9
41.0
4.57
10.95
.96
15
42.00
12.48
5.50
441.8
58.9
5.95
14.62
.08
15
45.00
13.24
5.55
455.8
60.8
5.87
15.09
.07
15
50.00
14.71
5.65
483.4
64.5
5.73
16.04
.04
15
55.00
16.18
5.75
511.0
68.1
5.62
17.06
.03
15
60.00
17.65
5.84
538.6
71.8
5.52
18.17
.01
18
55.00
15.93
6.00
795.6
88.4
7.07
21.19
1.15
18
60.00
17.65
6.10
841.8
93.5
6.91
22.38
1.13
18
70.00
20.59
6.26
921.2
102.4
6.69
24.62
1.09
20
65.00
19.08
6.25
1169.5
117.0
7.83
27.86
1.21
20
75.00
22.06
6.40
1268.8
126.9
7.58
30.25
1.17
24
80.00
23.32
7.00
2087.2
173.9
9.46
42.86
1.36
24
90.00
26.47
7.13
2238.4
186.5
9.20
45.70
1.31
24
100.00
29.41
7.25
2379.6
198.3
8.99
48.55
1.28
PROPERTIES OF STANDARD CHANNELS
DEPTH
OF
CHAN
NEL
WEIGHT
PER
FOOT
AREA
OF
SECTION
WIDTH
OF
FLANGE
MOMENT
OF '
INERTIA
Axis 11
RADIUS
OF
GYRATION
Axis 11
MOMENT
OF
INERTIA
Axis 22
RADIUS
OF
GYRATION
Axis 22
OUTSIDE
OF WEB TO
CENTER OF
GRAVITY
Inches
Pounds
Square
Inches
Inches
Inches
Inches
Inches
Inches
Inches
3
4.00
1.19
1.41
1.6
1.17
0.20
.41
.44
3
6.00
1.76
1.60
2.1
1.08
. 0.31
.42
.46
4
5.25
1.55
1.58
3.8
1.56
0.32
.45
.46
4
7.25
2.13
1.73
4.6
1.46
0.44
.46
.46
5
6.50
1.95
1.75
7.4
1.95
0.48
.50
.49
5
11.50
3.38
2.04
10.4
1.75
0.82
.49
.51
6
8.00
2.38
1.92
13.0
2.34
0.70
.54
.52
6
13.00
3.82
2.16
17.3
2.13
1.07
.53
.52
6
15.50
4.56
2.28
19.5
2.07
1.28
.53
.55
7
9.75
2.85
2.09
21.1
2.72
0.98
.59
.55
7
14.75
4.34
2.30
27.2
2.50
1.40
.57
.53
7
19.75
5.81
2.51
33.2
2.39
1.85
.56
.58
8
11.25
3.35
2.26
32.3
3.10
1.33
.63
.58
8
16.25
4.78
2.44
39.9
2.89
1.78
.61
.56
8
21.25
6.25
2.62
47.8
2.76
2.25
.60
.59
9
13.25
3.89
2.43
47.3
3.49
1.77
.67
.61
9
20.00
5.88
2.65
60.8
3.21
2.45
.65
.58
9
25.00
7.35
2.81
70.7
3.10
2.98
.64
.62
10
15.00
4.46
2.60
66.9
3.87
2.30
.72
.64
10
30.00
8.82
3.04
103.2
3.42
3.99
.67
.65
10
35.00
10.29
3.18
115.5
3.35
4.66
.67
.69
12
20.50
6.03
2.94
128.1
4.61
3.91
.81
.70
12
25.00
7.35
3.05
144.0
4.43
4.53
.78
.68
12
35.00
10.29
3.30
179.3
4.17
5.90
.76
.69
. 12
40.00
11.76
3.42
196.9
4.09
6.63
.75
.72
15
33.00
9.90
3.40
312.6
5.62
8.23
.91
.79
15
40.00
11.76
3.52
347.5
5.44
9.39
.89
.78
15
50.00
14.71
3.72
402.7
5.23
11.22
.87
.80
15
55.00
16.18
3.82
430.2
5.16
12.19
.87
.82
07
PROPERTIES OF STANDARD ANGLES
DISTANCE
DISTANCE
DIMEN
SIONS
WEIGHT
PER
FOOT
AREA OF
SECTION
OF CENTER
OF GRAV
ITY FROM
BACK OF
LONGER
MOMENT
OF
INERTIA
Axis 11
RADIUS
OF
GYRA
TION
Axis 11
OF CENTER
OF GRAV
ITY FROM
BACK OF
SHORTER
MOMENT
OF
INERTIA
Axis22
RADI u s
OF
GYRA
TION
Axis22
LEG
LEG
Inches
Pounds
Square
Inches
Inches
Inches*
Inches
Inches
Inches*
Inches
2ix2 xi
6.8
2.00
.63
.64
.56
.88
1.14
.75
3 x2x
8.5
2.50
.75
1.30
.72
1.00
2.08
.91
3^x2ix
9.4
2.75
.70
1.36
.70
1.20
3.24
1.09
3^x3 x^
10.2
3.00
.88
2.33
.88
1.13
3.45
1.07
4 x3 xi
11.1
3.25
.83
2.42
.86
1.33
5.05
1.25
4 x3 xf
16.0
4.69
.92
3.28
.84
1.42
6.93
1.22
5 x3 xi
12.8
3.75
.75
2.58
.83
1.75
9.45
1.59
5 x3 xf
18.6
5.44
.84
3.51
.80
1.84
13.16
1.55
5 x3xl
13.6
4.00
.91
4.05
1.01
1.66
9.99
1.58
C y Ol y 3
19.8
5.82
1.00
5.55
.98
1.75
13.92
1.55
6 x3ixi
15.3
4.50
.83
4.26
.97
2.08
16.59
1.92
6 x 3* x 3
22.4
6.57
.93
5.84
.94
2.18
23.34
1.89
6 x4 x
16.2
4.75
.99
6.27
1.15
1.99
17.40
1.91
6 x4 xf
23.6
6.94
1.08
8.68
1.12
2.08
24.51
1.88
Equal legs
2 x2 x
6.0
1.75
.68
.59
.58
2x2x^
7.7
2.25
.81
1.23
.74
3 x3 x
9.4
2.75
.93
2.22
.90
3x3xi
11.1
3.25
1.06
3.64
1.04
3^x3xf
16.0
4.69
1.15
4.96
1.02
4 x4 x
12.8
3.75
1.18
5.56
1.20
4 x4 Xf
18.5
5.44
1.27
7.66
1.17
6 x6 xl
19.6
5.75
1.68
19.91
1.84
6 x6 xf
28.7
8.44
1.78
28.15
1.81
8 x8 xi
26.4
7.75
2.19
48.65
2.49
8 x8 xf
38.9
11.44
2.28
69.74
2.45
PKOBLEMS OF VAKIOUS SECTIONS
SECTIONS
AREA OF
SECTION
DISTANCE
FROM EX
TREME FIBER
TO NEUTRAL
Axis
MOMENT OF
INERTIA
SECTION
MODULUS
RADIUS OF
GYRATION
*
&d
d
2
bd s
12
~6~
Vl2
\//y///\ t
j
in
i
d
2
d*
~Q
900,7
12
V12
mm 1
/O^N
^p 1

~V2 = ' 7d
12
eV2
OQO fj
V12
&i
d
d* d*
d**}
i i^r^
/ V 3 
*!2i
i
2
12
6d
"N 12
^i v "*
JR
M _ Ml
d
2
6d 3  bid?
6d 3  brf?
6d 3 bid?
12
6d
Nl2(6d6idi)
IZ2221..J
>Jk 4
2d
6d 3
6d 3
d 236d
;^^ri
3
36
24
Vl8
i
^^^
TTd'2
d
TTd*
7rd8 d 3
d
SB
4
2
"6T~' 4
32
4
CL!
(d.*,)
d
ir(d*d*)
^lfc
v^
^pi
4
2
64
32 d
4
jo
TT&d
d
TT&d 3
, M =
d
j j
L*._^^
4
2
64
32
4
w
TT (6d  Mi)
d
2
Tr&dzbrf*)
^(bdzbid*)
1 /6d 3  M?
4
64
32 d 4 ^ 6d  6id!
BENDING MOMENTS AND DEFLECTIONS
METHOD OF LOADING
AND SUPPORTING
MAXIMUM
BENDING
MOMENT
MAXIMUM
DEFLECTION
REMARKS
f , m
1 P/3
PZ
i tr
7 J/X/
3 ^1
end.
i MS
TF7
1 TrZ 3
Cantilever, uniform
JL,
2
PZ^ >FZ
8 El
I Pl s 1 Wl 3
load W.
Cantilever, load at free
t Y /7
H 2
3 JET 8 "/
end and uniform load W.
ej iJ
PZ
T
1 PI*
48EI
Simple beam, load at
middle.
7
1
Wl
5 W7 3
Simple beam, uniform
1
Q
384 El
load W
r ~ J4JP
3 Pi
1 PJ 3
One end fixed, other
end supported, load at
i
16
108 El
11T/73
middle.
//;
rr 6
rKt
v/
///
iL_i_4_iJl
8
PI
185 j;i
i PI S
end supported.
Both ends fixed, load
'///A*. . I JC^//
8
192 .El
at middle.
y//r l *W/.
W7
1 W73
Both ends fixed, uni
fl i
12
384 ^1
form load W.
( UNIVERSITY
OF