SAMPLE QUESTIONS AND ANSWERS
Lecture 2: Tension in Structural Members
Axial Load
1. A steel rod is 2.2m long and must not
stretch more than 1.2mm when a 8.5-kN
is applied to it. Knowing that
E=200GPa.
a. Determine the smallest diameter that
should be used. 9.96mm
b. Compute the corresponding normal
stress caused by the load. 109.1MPa
5. A 9 m length of 6mm diameter length
steel wire is to be used in a hanger. It is
noted that the wire stretches 18mm when
tensile force speed is applied knowing
that E=200GPa.
a. Determine the magnitude of the
force P
11.31kN
b. Compute for the corresponding
normal stress in the wire.
400MPa
2. A 1.5 m long steel wire of 6mm
diameter steel wire is subjected to a
3.5kN tensile load. Knowing that E =
200GPa.
a. Determine the elongation of the steel
wire.
0.928mm
b. Compute for the corresponding
normal stress in the wire. 123.8MPa
6. A 1.4m aluminum pipe should not
stretch more than 1.3mm when it is
subjected to a tensile load. Knowing that
E=70GPa and that the allowable tensile
load strength is 96.5MPa.
a. Determine the maximum allowable
length of the pipe,
943mm
b. Compute for the required area of the
pipe if the tensile load is 580kN.
6010mm2
3. Two gage marks are placed exactly
254mm apart on a 12mm diameter
aluminum rod with E=70GPa and an
ultimate strength of 110MPa. Knowing
that the distance between the gage marks
is 254.23mm after a load is applied.
a. Determine the stress in the rod.
64.4MPa
b. Compute for the factor of safety.
1.71
7. A nylon tread is subjected to a 8.5-N
tension force. Knowing that E=3.3GPa
and that the length of the thread
increases by 1.1%.
a. Determine the diameter of the thread.
0.55mm
b. Compute for the stress in the tread.
36.3MPa
4. The control rod made of yellow brass
must not stretch more than 3mm when
the extension in the wire is 4kn.
Knowing that E=105GPa and the
maximum allowable normal stress is
180MPa.
a. Determine the smallest diameter that
can be selected for the rod.
5.32mm
b. Compute for the corresponding
maximum length of the rod.
1.75mm
8. A cast iron is used to support a
compressive load. Knowing that
E=70GPa and that the maximum
allowable change in length is 0.025%.
a. Determine the maximum normal
stress in the tube,
17.5MPa
b. Compute for the minimum wall
thickness for a load of 7.3kN if the
outside diameter tube is 50mm.
44.37mm
1
9. A block of 250mm Length and
45mmX40mm cross-section is to support
centric compressive load P. The material
to be used in a bronze for which E =
70GPa. Determine the largest load that
can be applied, knowing that the normal
stress must not exceed 124MPa and that
the decrease in length of the block
should be at most 0.12% of it original
length.
48.4kN
10. A 9-kN tensile load will be applied to a
50-m length of steel wire with
E=200GPa. Determine the smallest
diameter wire that can be used knowing
that the normal stress must not exceed
150 MPa and that the increase in the
length of the wire should be at most
25mm.
10.70mm
Figure P13
a. Determine the value of Q so that the
deflection at a is zero.
32.8mm
b. Compute for the corresponding
deflection at B.
0.073mm
c. Knowing that P = 6kN, determine
the deflection at point A 0.018mm
d. If P change is to 6kN compute the
new deflection at point B. -0.09mm
11. A single axial load of magnitude
P=58kN is applied at end C of brass rod
ABC.
Knowing
that
E=105GPa
determine the diameter d of portion BC
for which the deflection of point C will
be 3mm.16.52mm
12. Both portions of the rod ABC are made
of an aluminum for E=73Gpa.knopwing
that the diameter of portion BC is
d=20mm, determine the largest force P
that can be applied if σall = 160MPa and
the corresponding deflection at point C
is not to exceed 4mm.
50.30kN
14. A 1.2-m section of aluminum pipe of
cross-sectional area of 1100 mm2 rests
on a fixed support at A. The 15mm
diameter steel rod BC hangs from a rigid
bar that rests on the top of the pipe at B.
Knowing that the modulus of elasticity is
200GPa for steel and 72GPa for
aluminum, determine the deflection of
point C when a 60kN force is applied at
C.
4.47mm
Figure P12
13. Both positions on the rod ABC are
made of an aluminum for which
E=10GPa. The magnitude of P is
4kN.
2
Figure P27
a. Determine the
members AB.
b. Determine the
members AD.
Figure P14
15. The steel frame (E=200 GPa) shown has
a diagonal brace BD with an area of
190mm2.
Determine
the
largest
allowable load P if the change in length
of number PD is not to exceed 1.6mm.
50.4kN
deformations of
2.11mm
deformations of
2.03mm
17. A homogeneous cable L and uniform
cross-section is suspended from one end.
Denote ρ, as the density (mass per unit
volume) of the cable and E, its modulus
of elasticity.
a. Determine the elongation of the
cable due to its own weight.
gL2
2E
b. Assuming now the cable to be
horizontal, determine the force that
should be applied to each end of the
cable to obtain the same elongation
1
as in (a).
2w
18. Denoting by e as the engineering strain
in a tensile specimen, derive the formula
for the true strain in terms of the
engineering stain.
 t ln1  e
Figure P15
16. For the steel truss (E=200GPa) and
loading shown as shown in Figure P27,
knowing that their cross-sectional area
are
2400mm2
and
1800mm2,
respectively.
19. An axial centric force of magnitude P =
450kN is applied to the composite block
shown by means of a rigid end plate, and
that h=10mm.
3
Figure P20
a. Determine the change in length of
rod EF.
0.076mm
b. Compute for the stress in the rod.
AB.
30.50MPa
c. Compute for the stress in the rod.
CD.
30.50MPa
d. Compute for the stress in the rod.
EF.
38.10MPa
Figure P19
a. Determine the normal stress in the
brass core.
140.60MPa
b. Determine the normal in the
aluminum plates.
93.75MPa
c. Determine the value of h if the
portion of the load carried by the
aluminum plate is half the portion of
the load carried by the brass core.
15mm
d. Compute for the total load of the
stress in the brass is 80MPa.
288kN
21. A steel tube (E=200GPa) with a 32-mm
outer diameter and a 4-mm thickness is
praised in a vise that is adjusted so that
its jaws just touch the ends of the tube
without exerting any presser on them.
The two forces shown are then applied
to the tube.
I.
After these forces are applied, the
vise is adjusted to decrease the
distance between its jaws by
0.2mm.
20. Three steel rod (E =200GPa) support
36kN Load P. Each of the rods AB
and CD has a 200mm2 crosssectional area and rod EF has
625mm2 cross-sectional area. Use
the
Figure P21
a. Determine the forces exerted by the
vise on the tube at A.
-76.6kN
b. Determine the forces exerted by the
vise on the tube at D,
-64.6kN
c. Compute for the change in length
after of the portion BC of the tube.
-0.039mm
4
II.
Assuming that after the forces have been
applied, the vise is adjusted to decrease
the distance between its jaws by 0.1mm.
a. Determine the forces exerted by the
vise on the tube at A.
47.30kN
b. Determine the forces exerted by the
vise on the tube at D,
-35.30kN
c. Compute for the change in length
after of the portion BC of the tube.
-0.006mm
Determine the tension in each wire A caused
by the load P.
0.200P
Determine the tension in each wire B caused
by the load P.
0.525P
Determine the tension in each wire C caused
by the load P.
0.275P
Determine the tension in each wire D caused
by the load P.
0.275P
Thermal Deformation
24. A
steel
railroad
track
having
6 o
E  200GPa   11.7 X 10 / C was
laid out at a temperature of 6oC.
a. Determine the normal stress in the
rails when the temperature reached
48oC, assuming that the rails are
welded to form a continuous track.
-98.30MPa
b. Determine the normal stress in the
rails when the temperature reached
48oC, assuming that the rails are 10m
long with 3mm gap between them. 38.30MPa
22. The rigid rod ABC is suspended from
three wires of the same material. The
cross-sectional area of the wire at B is
equal to half of the cross-sectional area
of the wire at A and C.


Figure P22
brass
shell
having
6 o

20
.
9
X
10
/
C
is
fully
b
bonded
to
the
steel
core
6 o
 s  11.7 X 10 / C . Determine
the largest allowable increase the
temperature if the stress in the steel
core is not to exceed 55MPa.
75.40oC
a. Determine the tension in each wire A
caused by the load P.
0.200 P
b. Determine the tension in each wire B
caused by the load P.
0.525P
c. Determine the tension in each wire C
caused by the load P.
0.275P
23. The rigid bar ABCD is suspended from
four identical wires. Determine the
tension in each wire caused by the load
P.
Figure P23
5
25. The




27. A rod consisting of two cylindrical
portions AB an BC is restrained t both
ends. Portion AB is made of brass
 Eb  105GPa




6
o
   20.9 X 10 / C  and portion BC


 E a  200GPa

.
is made of brass 
6 o

  a  11.7 X 10 / C 
Assuming that the rod is initially on
stressed and there is a temperature rise
of 42oC.
Figure P25
26. The
concrete
post
having
 Ec  25GPa



   9.9 X 10 6 / o C  is reinforced with
 c

6 steel bars each of 22mm diameter
 E s  200GPa

.
having 
6 o



11
.
7
X
10
/
C
 s

Figure P27
a. Determine the normal stresses
induced in portions AB. -44.4MPa
b. Determine the normal stresses
induced in portions BC. -100MPa
c. Compute for the corresponding
deflection of point B at the same
temperature rise. 0.5mm
28. Using Figure P28, determine
Figure P26
a. Determine the normal stresses
induced in the steel and in the
concrete by a temperature of 35oC.
0.391MPa
b. Determine the normal stresses
induced in the steel and in the
concrete by a temperature of 35oC.
-9.47MPa
Figure P28
a. the compressive force in the bars
shown after temperature rise of 96oC.
217kN
6
b. the corresponding change in length
of the bronze bar.
0.24mm
c. If a 0.5mm gap exist when the
temperature is 20oC.
i. Determine the temperature at
which the normal stress in the
aluminum bar will be 90MPa.
98.6oC
ii. Compute for the corresponding
exact length of the aluminum bar.
450.03mm
Figure P30
a. Determine the change in length of
side AB. 0.075mm
b. Find the change in length of side BC.
0.103mm
c. Compute for the change in length of
diagonal AC. 0.122mm
Poisson Ratio & Young’s Modulus
29. A 2-m length of an aluminum pipe of
240mm outer diameter and 10mm wall
thickness is used as a short column and
carries a centric axial of 640kN.
Knowing that E=73GPa and ν = 0.33.
Use the
31. The brass rod AD is fitted with a
jacket that is used to apply a
hydrostatic pressure of 48MPa to the
250mm-portion BC of the rod.
Knowing that E = 105GPa and ν =
0.33.
Figure P29
a. Determine the change in length of
the pipe.
-2.43mm
b. Find the change in its outer diameter.
0.096mm
c. Compute for the change in its wall
thickness.
0.004mm
Figure P31
a. Determine the change in the total
length AD.
50mm
b. Compute for the change in diameter
of portion BC of the rod. -.0153mm
30. A fabric used in air –inflated structures
is subjected to biaxial loading that
results in normal stresses σx = 120MPa
and σz = 160MPa. Knowing that the
properties of the fabric can be
approximated as E= 87GPa and ν = 0.34.
32. The homogeneous plate ABCD is
subjected to biaxial loading as
shown. It is known that  z   0 and
7
that the change in length of the plate
in x-direction must be zero, that is εx
= 0. If E is the modulus of elasticity
and ν is the Poisson ratio.
36. An elastomeric bearing (G = 0.9 MPa) is
used to support a bridge girder as shown
to provide flexibility during earthquakes.
The beam must not displace more than
10 mm when a 22-kN lateral load is
applied as shown. Assume that the
maximum allowable shearing stress is
420 kPa.
Figure P32
a. Determine the required magnitude of
σx.
 0
b. Compute for the ratio  0  z .
E
1 2
Figure P36
33. Two blocks of rubber are bounded to
rigid support and to the movable plate
AB.
a. Determine the smallest allowable
dimension, b.
262mm
b. Compute for the smallest required
thickness, a.
21.4mm
c. If b = 220mm and a = 30mm,
determine the shear stress 
431kPa
d. Compute for the shearing modulus G
for the maximum lateral load P =
19kN and a maximum displacement
of 12mm.
1.08MPa
37.
I.
Figure 33
34. If a force of magnitude P = 19kN causes
a deflection δ = 3mm when the width w
= 60mm, determine the modulus of
rigidity of the rubber used. 10.26MPa
35. If G=7.5MPa and the width w= 80mm,
determine the effective spring constant,
k  P .. 6.17MN/m
For a rod is made of steel with E
= 200 GPa and v = 0.30 with
200mm length.
Figure P37
8
Determine the dilatation e.
242μ
Compute the change in volume of the rod.
18.40mm3
II.
For a rod is made of aluminum with E =
70 GPa and v = 0.35 with a 200 mm
length.
a. Determine the dilatation e. 519μ
b. Compute the change in volume of
the rod. 39.4mm3
38. From Figure P38;
Figure P39
a. The plate shown has an allowable
stress of 125 MPa, determine the
maximum allowable value of P when
r = 12 mm, 58.3kN
b. If P = 38 kN, determine the
maximum stress when r = 18 mm.
73.9MPa
40. The allowable stress for Figure P89 is
140 MPa. Use Figure P89 to answer
questions 89 to 90.
Figure P38
a. Determine the change in height for
the brass cylinder.
-0.075mm
b. Compute for the change in volume of
the brass cylinder if the loading is
hydrostatic
with
 x   y   z  70MPa -521mm3
Figure P40
41. Determine the maximum allowable
magnitude of the centric load P. 50kN
39. From Figure P39;
42. Compute for the percent change in the
maximum allowable magnitude of P if
the raised portions are removed at the
ends of the specimen. 110%
43. A centric axial force is applied to the
steel bar shown. Knowing that σall is 135
MPa, determine the maximum allowable
load P.
55kN
9
down 9 mm.
concentrations.
Neglecting
stress
Figure P43
44. The 30-mm-square bar AB has a length
L = 2.2 m; it is made of a mild steel that
is assumed to be elastoplastic with E =
200 GPa and σY = 345 MPa. A force P is
applied to the bar until end A has moved
down by an amount 8m.
Figure P45
a. Determine the maximum value of the
force P.
332kN
b. Find the permanent set measured at
points A after the force has been
removed.
6.37mm
c. Find the permanent set measured at
points B after the force has been
removed
0
Figure P44
a. Determine the maximum value of the
force P and the permanent set of the
bar after the force has been removed,
knowing that δm = 4.5 mm
0.705mm
b. Determine the maximum value of the
force P and the permanent set of the
bar after the force has been removed,
knowing that δm = 4.5 mm.
310.5kN
c. Compute for the maximum amount
δm by which the bar should be
stretched if the desired value of δp is
3.5 mm 7.81mm
46. Each of the three 6-mm-diameter steel
cables is made of an elastoplastic
material for which (σY = 345 MPa and E
= 200 GPa. A force P is applied to the
rigid bar ABC until the bar has moved
downward a distance δ = 2 mm.
Knowing that the cables were initially
taut (Hint: In part c, cable BE is not
taut.).
a. Determine the maximum value of P.
23.9kN
b. Find the maximum stress that occurs
in cable AD.
250MPa
c. Compute for the final displacement
of the bar after the load is removed.
0
45. Rod AB and BC are made of mild steel
that is assumed to be elastoplastic with E
= 200 GPa and σY = 345 MPa. The rods
are stretched until the end has moved
10
reaches a maximum value of δm =
0.3 mm and then decreased back to
zero, determine, the maximum value
of P.
990kN
48. Bar AB has a cross-sectional area of
1200 mm2 and is made of a steel that is
assumed to be elastoplastic with E = 200
GPa and (σY = 250 MPa. Knowing that
the force F increases from 0 to 520 kN
and then decreases to zero.
Figure P46
47. Rod AB consists of two cylindrical
portions AC and BC, each with a crosssectional area of 1750 mm2. Portion AC
is made of a mild steel with E = 200 GPa
and (σY = 250 MPa, and portion BC is
made of a high-strength steel with E =
200 GPa and (σY = 345 MPa. A load P is
applied at C as shown. Assume both
steels to be elastoplastic.
Figure P48
a. Determine the permanent deflection
of point C.
0.104mm
b. Find the residual stress in the bar.
-65.2MPa
49. A narrow bar of aluminum is bonded to
the side of a thick steel plate as shown.
Initially, at T1 = 20oC, all stresses are
zero. Assume that the temperature will
be slowly raised to T2 and then reduced
to T1. Also αa = 23.6 X 10-6/oC for the
aluminum and. αs = 11.7 X 10-6/oC for
the steel. Further assume that the
aluminum is elastoplastic, with E = 70
GPa and (σY = 100 MPa. (Hint: Neglect
the small stresses in the plate.)
Figure P47
a. Determine the maximum deflection
of C if P is gradually increased from
zero to 975 kN and then reduced
back to zero.
0.292mm
b. Find the maximum stress in portion
AC of the rod.
250MPa
c. Compute
for
the
permanent
deflection of C.
0.027mm
d. Find the maximum stress in portion
BC of the rod.
-307MPa
e. If P is gradually increased from zero
until the deflection of point C
Figure P49
11
a. Determine the highest temperature
T2 that does not result in residual
stresses. 140.04oC
b. Find the temperature T2 that will
result in a residual stress in the
aluminum equal to 100 MPa.
260.1oC
slightly longer than the tube, it is
observed that the cover must be forced
against the rod by rotating it one-quarter
of a turn before it can be tightly closed.
50. The rigid bar ABC is supported by two
links, AD and BE, of uniform 37.5 X 6mm rectangular cross section and made
of a mild steel that is assumed to be
elastoplastic with E = 200 GPa and (σY =
250 MPa. The magnitude of the force Q
applied at B is gradually increased from
zero to 260 kN. Knowing that a = 0.640
m.
Figure P51
a. Determine the average normal stress
in the tube.
67.9MPa
b. Determine the average normal stress
in the rod. -55.6MPa
c. Find the deformations of the tube.
0.2425mm
d. Find the deformations of the rod. 0.1325mm
52. The uniform wire ABC, of unstretched
length 2l, is attached to the supports
shown and a vertical load P is applied at
the midpoint B. Denoting by A the
cross-sectional area of the wire and by E
the modulus of elasticity, show that, for
δ « I, determine the deflection at the
P
  l3
midpoint B.
AE
Figure P50
a. Determine the value of the normal
stress in link AD. 250MPa
b. Find the maximum deflection of
point B. 0.622mm
c. Determine the value of the normal
stress in link AD. 124.3MPa
51. A 250-mm-long aluminum tube (E = 70
GPa) of 36-mm outer diameter and 28mm inner diameter may be closed at
both ends by means of single-threaded
screw-on covers of 1.5-mm pitch. With
one cover screwed on tight, a solid brass
rod (E = 105 GPa) of 25-mm diameter is
placed inside the tube and the second
cover is screwed on. Since the rod is
Figure P52
53. The steel bars BE and AD each have a 6
X 18-mm cross section. Knowing that E
= 200 GPa, determine the deflections of
point B of the rigid bar ABC. 0.296mm
12
Figure P54
a. Determine the diameter d of the
largest bit that can be used if the
allowable load at the hole is not to
exceed that at the fillets. 9mm
b. If the allowable stress in the plate is
145 MPa, what is the corresponding
allowable load P? 62kN
c. For P = 58 kN and d = 12 mm,
determine the maximum stress in the
plate shown. 134.7MPa
d. Solve part P54, assuming that the
hole at A is not drilled. 135.3MPa
Figure P53
54. A hole is to be drilled in the plate at A.
The diameters of the bits available to
drill the hole range from 9 to 27 mm in
6-mm increments.
Lecture 3: Torsion in Shafts
55. Using Figure P55 determine
Figure P55
Figure P56
a. the torque T that causes a maximum
shearing stress of 70 MPa in the steel
cylindrical shaft shown. 641N.m
b. the maximum shearing stress caused
by a torque of magnitude T = 800
N.m. 87.3MPa
57. The torques shown are exerted on
pulleys A and B. Knowing that, each
shaft is solid, determine the
maximum shearing stress in shaft
BC. 36.6MPa
56. . For the hollow shaft and loading
shown, determine the
a. maximum
shearing
stress.
70.5MPa
b. diameter of a solid shaft for which
the maximum shearing stress under
the loading shown is the same as in
part a. 55.8mm
13
the smallest diameter of shaft BC for
which the maximum value of the
shearing stress in the assembly will not
be increased. 42.8mm
61. The allowable stress is 103.5 MPa in the
38mm-diameter steel rod AB and 55
MPa in the 45.7 mm diameter brass rod
BC.
Figure P57
58. In order to reduce the total mass of the
assembly of Prob. 57, a new design is
being considered in which the diameter
of shaft BC will be smaller. Determine
the smallest diameter of shaft BC for
which the maximum value of the
shearing stress in the assembly will not
increase. 39.8mm
Figure P 61
59. Under normal operating conditions, the
electric motor exerts a torque of 2.4 kN.
m on shaft AB. Assume that each shaft
is solid.
a. Neglecting the effect of stress
concentrations,' determine the largest
torque that may be applied at A..
1.04kN.m
b. If the torque has a magnitude of
1.1kN m. is applied at A, determine
the required diameter of rod AB.
38.10 mm
62. The solid rod AB has a diameter dAB =
60 mm and is made of a steel for which
the allowable shearing stress is 75 MPa.
Figure P 59
a. Determine the maximum shearing
stress in shaft AB 77.6MPa
b. Find the maximum shearing stress in
shaft BC. 62.8MPa
c. Compute maximum shearing stress
in shaft CD. 20.9MPa
60. In order to reduce the total mass of the
assembly of Prob. 59, a new design is
being considered in which the diameter
of shaft BC will be smaller. Determine
Figure P62
14
a. The pipe CD, which has an outer
diameter of 90 mm and a wall
thickness of 6 mm, is made of an
steel for which the allowable
shearing stress is 75 MPa. Determine
the largest torque T that may be
applied at A. 3.18kN
b. The pipe CD, which has an outer
diameter of 90 mm and a wall
thickness of 6 mm, is made of an
aluminum for which the allowable
shearing stress is 54 MPa. Determine
the largest torque T that may be
applied at A. 3.37kN.m
Figure P64
a. Determine the maximum shearing
stress in a shaft CD 68.7MPa
b. A torque of magnitude T=1000N.m
is applied at D as shown. Knowing
that the allowable shearing stress is
60MPa in shaft CD, determine the
required diameter of shaft AB,
59.6mm
63. The solid rod BC has a diameter of 30
mm and is made of an aluminum for
which the allowable shearing stress is 25
MPa. Rod AB is hollow and has an outer
diameter of 25 mm; it is made of a brass
for which the allowable shearing stress is
50MPa.
65. The two solid shafts are connected by
gears as shown and are made of steel for
which the allowable shearing stress is
60MPa. Assume that a torque of
magnitude TC = 600N.m is applied at C
and that the assembly is in equilibrium.
Figure P63
a. Determine the largest inner diameter
of rod AB for which the factor of
safety is the same for each rod
15.18mm
b. Compute for the largest torque that
may be applied at A. 132.5N.m
Figure P65
a. Determine the required diameter of
shaft BC. 37.1mm
b. Find the required diameter of shaft
EF. 31.7mm
c. If the allowable shearing stress is
50MPa and the diameters of the two
shafts are, respectively, dBC = 40mm
and dEF = 32mm and that the
assembly
is
in
equilibrium,
64. A torque of magnitude T=1000N.m is
applied at D as shown. Assume that the
diameter of shaft AB is 56mm and the
diameter of shaft CD is 42mm.
15
determine the largest torque TC that
may be applied at C. 515N.m
68. The torques shown are exerted on
pulleys B, C, and D. The entire shaft is
made of aluminum of G = 27 GPa.
66. The shaft shown is made of steel and
has a modulus of rigidity of G = 77
GPa.
Figure P68
a. Determine the angle of twist between
C and B. 8.54o
b. Find the angle of twist between D
and B.
2.11o
Figure P66
a. If the steel shaft is solid, determine
angle of twist at A. 4.21o
b. If the steel shaft is hollow with a 30mm outer diameter and a 20-mm
inner diameter, compute for the
angle of twist at A.
5.25o
69. The solid brass rod AB (G = 39 GPa) is
bonded to the solid aluminum rod BC (G
= 27 GPa).
67. The torques shown are exerted on
pulleys A and B. The shafts are solid and
made of steel of G = 77 GPa.
Figure P69
a. Determine the angle of twist at B.
0.741oC
b. Find the angle of twist at A.
1.573oC
Figure P67
a. Determine the angle of twist between
A and B. 2.53o
b. Find the angle of twist between A
and C.
3.42o
70. Two solid steel shafts (G = 77 GPa) are
connected by the gears shown. Assume
that the radius of gear B is rB = 20 mm.
16
72. The design of the gear-and-shaft system
shown requires that steel shafts of the
same diameter be used for both AB and
CD. It is further required that Tmax < 60
MPa and that the angle CPD through
which end D of shaft CD rotates not
exceed 1.5°. Knowing that G = 77 GPa,
determine the required diameter of the
shafts. 62.9mm
Figure P70
a. Determine the angle of twist at C.
2.06o
b. Find the angle of twist at B. 6.17o
c. Compute for the angle through
which the end A rotates when ' TA =
75 N.m. 7.94o
Figure P72
73. The electric motor exerts a torque of
800 N.m on the steel shaft ABCD
when it is rotating at constant speed.
Design specifications require that the
diameter of the shaft be uniform
from A to D and that the angle of
twist between A and D not exceed
1.5°. Knowing that Tmax < 60 MPa
and G = 77 GPa, determine the
minimum diameter shaft that may be
used. 42.1mm
71. A coder F is used to record in digital
form the rotation of shaft A, is connected
to the shaft by means of the gear train
shown, which consists of four gears and
three solid steel shafts each of diameter
d. Two of the gears have a radius r and
the other two a radius nr. If the rotation
of the coder F is prevented, determine in
terms of T, I, G, J, and n the angle
through
which
end
A
rotate.
TAl  1
1

 4  2  1
GJ  n
n

Figure P73
74. The solid cylindrical rod BC is
attached to the rigid lever AB of
length and to the support at C. The
vertical force P applied at A causes a
small displacement at point A.
Determine
the
corresponding
Figure P71
17
maximum shearing stress in the rod.
Gd


2 La
75. The solid cylindrical rod BC of length L
= 610 mm is attached to the rigid lever
AB of length a = 380 mm and to the
support at C. Design specifications
required that the displacement of A not
exceed 25 mm when a 450 N force P is
applied at A. For the material indicated.
Figure P76
77. Assume the bolts in Problem 76 are
slightly undersized and permit a 1.50
rotation of one flange with respect to the
other before the flanges begin to rotate
as a single unit. Determine the maximum
shearing stress in shaft AB when a
torque T of magnitude 500 N.m is
applied to the flange indicated.
a. If the torque T is applied to flange B.
68.8MPa
b. If the torque T is applied to flange C.
10.34MPa
Figure P75
a. Determine the required diameter of
the rod if steel: τall = 103 MPa, G =
69 GPa21.92mm
b. Find the required diameter of the rod
if Aluminum: τall = 69MPa, G = 27
GPa
27.78mm
78. At a time when rotation is prevented at
the lower end of each shaft, a 50N.m
torque is applied to end A of shaft AB.
Assume that G = 77 GPa for both shafts.
76. Two solid steel shafts are fitted with
flanges which are then connected by
fitted bolts so that there is no relative
rotation between the flanges. Assume G
= 77 GPa.
a. Determine the maximum shearing
stress in shaft CD when a torque of
magnitude T = 500 N.m applied to
flange B. 31.7MPa
b. Find the maximum shearing stress in
shaft AB when a torque of
magnitude T = 500 N.m applied to
flange B. 39.6MPa
Figure P78
18
a. Determine the maximum shearing
stress in shaft CD.
47.1MPa
b. Find the angle of rotation at A.
0.779o
c. Assuming that the 50 N.m torque is
applied to end C of shaft CD,
determine the maximum shearing
stress in shaft CD. 70.7MPa
d. Find the corresponding angle of
rotation at A. 1.169o
a. Determine the magnitude and
location of the maximum shearing
stress in the annular plate.
T
 max 
2r12
b. Find the angle through which end B
of the shaft rotates with respect to
end
C
of
the
tube.
T  1
1
 2  2 
 BC 
4Gt  r1 r2 
79. A torque T is applied as shown to a solid
tapered shaft AB. Determine the angle of
7TL
twist at A.
12GC4
81. An annular aluminum plate (G = 27
GPa), of thickness t = 6 mm, is used to
connect the aluminum shaft AB, of
length L1 = 90 mm and radius r1 = 30
mm, to the aluminum tube CD, of length
L2 =150 mm, inner radius r2 = 75 mm
and 4 mm thickness (Fig P80). Knowing
that a torque of magnitude T = 2500 N.m
is applied to end A of shaft AB and that
end D of tube CD is fixed.
a. Determine the maximum shearing
stress in the shaft-plate-tube system.
73.7MPa
b. Find the angle through which end A
rotates.
0.510o
Figure P79
80. An annular plate of thickness t and
modulus of rigidity G is used to connect
shaft AB of radius r1 to tube CD of inner
radius r2. Knowing that a torque T is
applied to end A of shaft AB and that
end D of tube CD is fixed.
82. While a steel shaft of the cross section
shown rotates at 120 rpm, a stroboscopic
measurement indicates that the angle of
twist is 2° in a 4-m length. Using G = 77
GPa, determine the power being
transmitted. 25.6kW
Figure P82
83. A steel pipe of 60-mm outer diameter is
to be used to transmit torque of 350 N.m
without exceeding an allowable shearing
Figure P80
19
stress of 12MPa. A series of 60-mmouter-diameter pipes is available for use.
Knowing that the wall thickness of the
available pipes varies from 4 mm to 10
mm in 2-mm increments, choose the
lightest pipe that can be used. 8mm
85. The diameter of each shaft is as follows:
dAB = 16 mm, dCD = 20 mm, dEF = 28
mm. Knowing that the frequency 0: the
motor is 24 Hz and that the allowable
shearing stress for each shaft is 75MPa,
determine the maximum power that can
be transmitted. 7.11kW
Figure P83
Figure P85
84. Three shafts and four gears are used to
form a gear train that will transmit 7.5
kW from the motor at A to a machine
tool at F. (Bearings for the shafts are
omitted in the sketch.) The frequency of
the motor is 30 Hz and that the allowable
stress for each shaft is 60 MPa.
86. A 1.6-m-long tubular shaft of 42-mm
outer diameter do is to be made of a steel
for which Tall = 75 MPa and G = 77
GPa. Assume that the angle of twist
must not exceed 4° when the shaft is
subjected to a torque of 900 N.m,
determine the largest inner diameter di
which can be specified in the design.
24.9mm
87. A 1.6-m-Iong tubular steel shaft (G = 77
GPa) of 42-mm outer diameter and 30mm inner diameter is to transmit 120
kW between a turbine a generator.
Knowing that the allowable shearing
stress is 65 MPa and that the angle of
twist must not exceed 3o, determine the
minimum frequency at which the shaft
may rotate.
33.54Hz or 2012rpm
Figure P84
88. The stepped shaft shown rotates at
450 rpm Assume an allowable
shearing stress of 45 MPa.
a. Determine the required diameter of
shaft CD. 20.4mm
b. Find the diameter of shaft AB.
15mm
c. Compute for the required diameter of
EF 27.6mm
Figure P88
20
a. Knowing that r = 10mm, determine
the maximum power that can be
transmitted313kW
b. Knowing that r = 4 mm, determine
the maximum power that can be
transmitted. 268kW
a. Determine the torque T when the
angle of twist at A is 25°. 283N.m
b. Find the corresponding diameter of
the elastic core of the shaft
12.95mm
91. A hollow steel shaft is 0.9 m long and
has the cross section shown. The steel is
assumed to be elastoplastic with Ty =
180 MPa and G = 77 GPa. Use Figure
P188 to answer questions 184 to 193.
89. A torque of magnitude T = 25 N.m is
applied to the stepped shaft shown,
which has a full quarter-circular fillet.
Assume that D = 24 mm.
Figure P91
a. Determine the applied torque at the
onset of yield. 11.714kN.m
b. Find the corresponding angle of twist
at the onset of yield. 3.44o
c. Compute for the applied torque when
the plastic zone is 10 mm deep.
14.12kN.m
d. Evaluate the angle of twist when the
plastic zone is 10 mm deep. 12.82o
e. Determine the angle of twist at
which the section first becomes fully
plastic 14.89kN.m
f. Compute
the
corresponding
magnitude of the applied torque.
8.04o
Figure P89
a. Determine the maximum shearing
stress in the shaft when d = 20 mm.
21.6MPa
b. Find the maximum shearing stress in
the shaft when d = 21.6 mm.
17.9MPa
90. A torque T is applied to the 20-mmdiameter steel rod AB. Assuming the
steel to be elastoplastic with G = 77 GPa
and y = 145 MPa.
92. A 50-mm-diameter cylinder is made of a
brass for which the stress-strain diagram
is as shown. Knowing that the angle of
twist is 5o in 725-mm length, determine
by approximate means the magnitude T
of the torque applied to the shaft.
2.32kN.m
Figure P90
21
b. Find the permanent angle of twist of
the shaft. 2.09o
Figure P94
Figure P92
95. The solid shaft shown is made of a steel
that is assumed to be elastoplastic with
Ty = 145 MPa and G = 77 GPa. The
torque T is increased in magnitude until
the shaft has been twisted through 6°,
and the torque is then removed.
93. The solid circular drill rod AB is made
of a steel that is assumed to be
elastoplastic with Ty = 160 MPa and G
= 77 GPa. Knowing that a torque T = 5
kN.m is applied to the rod and then
removed, determine the maximum
residual shearing stress in the rod.
44.92MPa
Figure P95
a. Determine the magnitude and
location of the maximum residual
shearing stress. 33.5MPa;
24.6MPa
b. Find the permanent angle of twist of
the shaft. 1.019o
Figure P93
94. The hollow shaft AB is made of a steel
that is assumed to be elastoplastic with
Ty = 145 MPa and G = 77 GPa. The
magnitude T of the torque is slowly
increased until the plastic zone first
reaches the inner surface, the torque is
then removed.
a. Determine the maximum residual
shearing, stress. 29.1MPa;
40.5MPa
96. Knowing that T = 800 N.m,
determine for each of the cold-rolled
yellow brass bars shown the
maximum shearing stress and the
angle of twist at end B. Use G =
39GPa.30.8MPa 0.535o 37.9MPa
0.684o
22
99. The torque T causes a rotation of 2°
at end B of the stainless steel bar
shown. Knowing that G = 77 GPa,
determine the maximum shearing
stress in the bar. 60.8MPa
Figure P96
97. Using 'Tall = 50 MPa and G = 39GPa for
each of the cold-rolled yellow brass bars
shown in Fig P96.
a. Determine the largest torque T that
may be applied. 1.3kN.m
1.055kN.m
b. Find the corresponding angle of
twist. 0.869o 0.902o
Figure P99
100. The composite shaft shown is
twisted by applying a torque T at end
A. Knowing that the maximum
shearing stress in the steel is 150
MPa, determine the corresponding
maximum shearing stress in the
aluminum core. Use G = 77 GPa for
steel and G = 27 GPa for aluminum.
39.4MPa
98. A torque of magnitude T = 300 N m is
applied to each of the aluminum bars
shown and that Tall = 60 MPa.
Determine the required dimension b for
each bar. 29.4mm 28.9mm
21.7mm
Figure P100
Figure P203
23
Lecture 4: Bending of Beams
101. Assume that the couple shown acts
in a vertical plane.
couple Mz that can be applied to the bar.
2.38kN.m
Figure P101
Figure P103
a. Determine the stress at point A.
116.4MPa
b. Find the stress at point B. 87.3MPa
104. Two vertical forces are applied to a
beam of the cross section as shown.
102. The wide-flange beam shown is
made of a high-strength, low-alloy steel
for which σY = 345 MPa and σu = 450
MPa.
Figure P104
Figure P102
a. Determine the maximum tensile
stress in portion BC of the beam.
81.8MPa
b. Determine
the
maximum
compressive stress in portion BC of
the beam 67.8MPa
a. Determine the largest couple that can
be applied to the beam when it is
bent about the z axis. Neglect the
effect of fillets. 243kN.m
b. Solve Prob. a, assuming that the
beam is bent about the y axis
56.3kN.m
105. Two equal and opposite couples of
magnitude M = 15 kN.m are applied to
the channel-shaped beam AB. Observing
that the couples cause the beam to bend
in a horizontal plane.
103. A nylon spacing bar has the cross
section shown. Knowing that the
allowable stress for the grade of nylon
used is 24 MPa, determine the larges
24
determine the total force: acting on the
shaded portion of the lower flange.
37.9kN
Use Figure P108 to answer questions 108
and 109.
Figure P105
a. Determine the stress at point C.
83.7MPa
b. Find the stress at point D. 146.4MPa
c. Compute for the stress at point E.
14.67MPa
Figure P108
108. Knowing that for the extruded beam
shown, the allowable stress is 120MPa
in tension 150MPa in compression;
determine the largest couple M that can
be applied. 7.67kN.m
Use Figure P106 to answer questions 106
and 107.
109. Knowing that for the extruded beam
shown, the allowable stress is 120MPa
in tension 150MPa in compression;
determine the largest couple M that can
be applied.
3.79kN.m
Use Figure P110 to answer questions 110
and 111.
Figure P106
106. A beam of the cross section shown is
bent about horizontal axis and that the
bending moment is 8 kN.m, determine
the total force acting on the top flange.
123.8kN
107. 14 A beam of the cross section
shown is bent about vertical axis and
that the bending moment is 4 kN.m,
25
Figure P112
a. Determine
the maximum stress.
6M
 max  3
a
b. Find the curvature of the bar.
1 12M

 Ea 4
Figure P110
113. A couple of magnitude M is applied to a
square bar of side a as shown.
Figure P110
Figure P113
110. The beam shown is made of a nylon
for which the allowable stress is 24 MPa
in tension and 30 MPa in compression.
Determine the largest couple M that can
be applied to the beam. 849N.m
111.
a. Determine
the maximum stress.
6 2M
 max 
a3
b. Find the curvature of the bar.
1 12M

 Ea 4
Solve Prob. 110 if d = 80mm.
1.501kN
Use Figure P114 to answer questions 114
and 115.
112. A couple of magnitude M is applied
to a square bar of side a as shown.
26
below, determine the largest permissible
bending moment when the composite
bar is bent about a horizontal axis
1.240kN
117. For the composite bar indicated,
determine the permissible bending
moment when the bar is bent about a
vertical axis. 720N.m
118. A copper strip (Ec = 105 GPa) and an
aluminum strip Ea = 75 GPa) are bonded
together to form the composite bar
shown. Assume that the bar is bent about
a horizontal axis by a couple of moment
35 N.m.
Figure P114
114. 40 Two metal strips are securely
bonded to a metal bar of 30 X 30-mm
square cross section. Using the data
given below, determine the largest
permissible bending moment when the
composite bar is bent about a horizontal
axis. 1.043kN.m
115. For the composite bar indicated,
determine the permissible bending
moment when the bar is bent about a
vertical axis. 855N.m
Figure P118
a. Determine the maximum stress in the
aluminum strip. . -56MPa
b. Find the maximum stress in the
copper strip. 66.4MPa
Use Figure P116 to answer questions 116
and 117.
119. The prismatic rod shown is made of
a steel that is assumed to be elastoplastic
with E = 200 GPa and σY = 280 MPa.
I.
The couples M and M' of moment
525 N.m are applied and maintained
about axes parallel to the x axis.
Figure P117
116. Two metal strips are securely bonded
to a metal bar of 30 X 30-mm square
cross section. Using the data given
Figure P119
27
a. Determine the thickness of the
elastic core. 21.9mm
b. Find the radius of curvature of the
bar. 7.81m
II.
Assuming that the couples M and M' are
applied and maintained about axes
parallel to the y axis.
a. Determine the thickness of the
elastic core. 5.87mm
b. Find the radius of curvature of the
bar.
2.09m
Figure P121
122. Determine the plastic moment Mp of
a steel beam of the cross-section shown,
assuming the steel to be elastoplastic
with a yield strength of 240MPa
2.03kN.m
120. A bar of the cross section shown is
made of a steel that is assumed to be
elastoplastic with E = 200 GPa and τy =
240 MPa. The bending is about the z
axis.
Figure P122
Use Figure P123 to answer questions 123
to 127.
Figure P120
a. Determine the bending moment at
which yield first occurs. 5.65kN.m
b. Find the bending moment at which
the plastic zones at the top and
bottom of the bar are 20 mm thick. '
8.0kN.m
121. Determine the plastic moment Mp of
a steel beam of the cross-section shown,
assuming the steel to be elastoplastic
with a yield strength of 240MPa
19.01kN.m
Figure P123
28
123. Determine the stress at point A for
the loading shown. -8.33MPa
124. Find the stress at point B if the 60kN loads are applied at points 1 and 2
only. -15.97MPa
125. Compute for the stress at point A if
the 60-kN loads are applied at points 1
and 2 only. 4.86MPa
Figure P129
126. Determine the stress at points A if
the 60-kN loads applied at points 2 and 3
are removed. -13.19MPa
a. Determine the stress at point A, for
the loading shown. -37.5MPa
b. Find the stress at point A, if loads are
applied at points 1 and 2 only. 38.4MPa
c. Compute for the stress at point A, if
loads are applied at points 2 and 3
only. -11.62MPa
127. Compute for the stress at point B if
the 60-kN loads are applied at points 2
and 3 only. 7.64MPa
128. Assume that the magnitude of the
horizontal force P is 8 kN.
130. The two forces shown are applied to
a rigid plate supported by a steel pipe of
140-mm outer diameter and 120-mm
inner diameter. Knowing that the
allowable compressive stress is 100
MPa, determine the range of allowable
values of P. 94.8kN < P < 177.3kN
Figure P128
a. Determine the stress at point A. 102.8MPa
b. Find the stress at point B. 80.6MPa
Figure P130
131. Assume that the allowable stress
is 150 MPa in section a-a of the
hanger shown.
129. As many as three axial loads, each of
magnitude P = 50 kN, can be applied to
the end of a W200 X 311 rolled-steel
shape.
29
Figure P132
Figure P131
133. An eccentric axial force P is applied
as shown to a steel bar of 25 X 90-mm
cross section. The strains at A and B
have been measured and found to be are
 A  600 and  B  420 . Assume
that E = 2000Pa.
a. Determine the largest vertical force P
that can be applied at point A.
40.3kN
b. Find the corresponding location of
the neutral axis of section a-a.
6.30mm
132. The C-shaped steel bar is used as a
dynamometer
to
determine
the
magnitude P of the forces shown.
Knowing that the cross section of the bar
is a square of side 40 mm and that strain
on the inner edge was measured and
found to be 450μ, determine the
magnitude P of the forces. Use E = 200
GPa 9.0kN
Figure P133
a. Determine the distance d. 30-mm
b. Find the magnitude of the force P.
94.5kN.m
Use the diagrams below to answer the
following questions:
a. Determine the equations of
the shear and bending
moment curves for the beam
and loading for each diagram.
30
b. Draw the shear and bending-moment
diagrams for the beam and loading
and determine the maximum
absolute value of the shear and
bending moment for each diagram
c. Compute the bending stresses for
each diagram
d. Determine the equations of the slope
and deflection of the beam for each
diagram
137.
138.
V  wL  x  M  
V  wL
134.
M 
w
L  x 2
2
wL2
2
139.
Ans.
135.
V 
Pb
V 
L
2
Pb
L
Pb
x
L
Pab
M 
L
M 
140.
Ans.
V  w L2  x 
w
wL
M  x L  x  V 
2
2
2
wL
M 
8
141.
V  w L2  x  M 
V 
wL
2
w
x L  x 
2
wL2
M 
8
136.
142.
w0 x 2
w
Ans. V  
M   0 x3
2L
6L
2
wL
wL
M  0
V 0
6
2
31
143.
148.
149.
144.
150.
145.
21
146.
23
151.
152.
147.
32
153.
154.
158.
40
159.
155.
160.
156.
161.
157.
162.
33
163.
58
168.
164.
169.
165.
170.
171.
166.
172.
167.
34
173.
179.
174.
95
180.
175.
181.
176.
177.
178.
35
Lecture 5: State of Stress and Strain
Use Figure P182 to answer questions 182
and 183.
Use Figure P186 to answer questions 186
and 187.
Figure P186
Figure P184
186. Determine the principal planes. 37.03o
182. Determine the normal stress exerted
on the oblique face of the shaded
triangular element. -0.521MPa
187. Compute for the maximum principal
stress. -13.6MPa
183. Find the shearing stress exerted on
the oblique face of the shaded triangular
element.
56.4MPa
Use Figure P188 to answer questions 188
and 189.
Use Figure P184 to answer questions 184
and 185.
Figure P188
188.
Figure P184
Determine the principal planes.
-30.96o
189. Compute for the maximum principal
stress.
-84MPa
184. Determine the normal stress exerted
on the oblique face of the shaded
triangular element.
32.9MPa
Use Figure P190 to answer questions 190
to 192.
185. Find the shearing stress exerted on
the oblique face of the shaded triangular
element.
71.0MPa
36
Figure P196
Figure P190
196. Determine the normal and shearing
stress σx’ after the element shown has
been rotated through 25° clockwise. 37.5MPa
190. Determine the orientation of the
planes of maximum in-plane shearing
stress. 7.97o
191. Find the maximum in-plane shearing
stress. 36.4MPa
192. Compute for the
normal stress. -50MPa
197. Find the shearing stress after the
element shown has been rotated through
10° counterclockwise. 50.1MPa
corresponding
198. The grain of a wooden member
forms an angle of 15° with the vertical.
Use Figure P17 to answer questions 17
and 18.
Use Figure P193 to answer questions 193
to 195
Figure P193
193. Determine the orientation of the
planes of maximum in-plane shearing
stress.
14.04o
Figure P198
a. Determine the in-plane shearing
stress parallel to the grain. -0.30MPa
b. Find the normal stress perpendicular
to the grain. -2.92MPa
194. Find the maximum in-plane shearing
stress.
68MPa
195. Compute for the corresponding
normal stress.
-16MPa
199. Two members of uniform cross
section 50mmX50mm are glued together
along the plane a-a, that forms an angle
of 25o with the horizontal. Knowing that
the allowable stresses for the glued joint
are   800kPa and   600kPa ,
Use Figure P196 to answer questions 196
and 197.
37
determine the largest axial load P that
can be applied. 3.9kN
Figure P199
Figure P201
200. Two steel plates of uniform cross
section 10 X 80 mm are welded together
as shown. Knowing that centric 100-kN
forces are applied to the welded plates
and that f3 = 25°. Use Figure P20 to
answer questions 20 and 21.
a. Determine the maximum principal
stress and the maximum shearing
stress at point H. 24.37MPa
11.02MPa 0 0 0 35.39MPa
35.39MPa -35.9MPa 35.9MPa
b. Determine the principal stresses and
the maximum shearing stress at point
K. 24.37MPa 0 36.56MPa 0 36.56MPa 24.37MPa 12.18MPa 48.74MPa 30.46MPa
202. Determine the range of values of σx
for which the maximum in-plane
shearing stress is equal to or less than 50
MPa.
15MPa   x  135MPa
Figure P200
a. Determine the in-plane shearing
stress parallel to the weld.
47.9MPa
b. Find the normal stress perpendicular
to the weld. 102.7MPa
201. The steel pipe AB has a 102-mm
outer diameter and a 6-mm wall
thickness. Knowing that arm CD is
rigidly attached to the pipe.
Figure P202
Use Figure P203 to answer questions 203
and 204.
38
Figure P203
203. Determine the value of τxy for which
the in-plane shearing stress parallel to
the weld is zero. -2.89MPa
Figure P207
207. Determine the maximum shearing
stress when σz = 0. 85MPa
204. Find the corresponding maximum
principal stresses. 12.77MPa 1.226MPa
208. Find the maximum shearing stress
when σz = +45 MPa. 85MPa
Use Figure P205 to answer questions 205
and 206.
209. Compute for the maximum shearing
stress when σz = -45 MPa. 95MPa
Use Figure P210 to answer questions 210
to 212.
Figure P205
205. Determine the maximum shearing
stress when σy = 40 MPa. (Hint:
Consider both in-plane and out-of-plane
shearing stresses.)
94.34MPa
Figure P210
210. Determine the maximum shearing
stress when σz = 0. 97.5MPa
206. Find the maximum shearing stress
when σy = 120 MPa. (Hint: Consider
both in-plane and out-of-plane shearing
stresses.) 105.31MPa
211. Find the maximum shearing stress
when σz = +45 MPa. 85MPa
Use Figure P207 to answer questions 207
to 209.
212. Compute for the maximum shearing
stress when σz = -45 MPa 120MPa
39
213. For the state of stress shown,
determine two values of σy for which
the maximum shearing stress is 75
MPa. 56.88MPa -130MPa
216. For the state of stress shown,
determine the range of values of  xy
for which the maximum shearing
stress is equal to or less than 60
MPa.  40MPa   xy  40MPa
Figure P216
Figure P213
Use Figure P217 to answer questions 217
and 218.
214. For the state of stress shown,
determine the value of '  xy for
which the maximum shearing stress
is 80 MPa. 60MPa
Figure P217
Figure P214
217. Determine the value of σy for which
the maximum shearing stress is as small
as possible. 45.7MPa
215. For the state of stress shown,
determine two values of σy for which
the maximum shearing stress is 80
MPa. -40MPa
130MPa
218. Find the corresponding value of the
shearing stress. 92.9MPa
219. The cylindrical portion of the
compressed air tank shown is fabricated
of 6-mm-thick plate welded along a
helix forming an angle β = 30° with the
horizontal. Knowing that the allowable
stress normal to the weld is 75 MPa,
Figure P215
40
determine the largest gage pressure that
can be used in the tank. 2.95MPa
Figure P221
222. A pressure vessel of 250-mm
inner diameter and 6-mm wall
thickness is fabricated from a 1.2-m
section of spirally welded pipe AB
and is equipped with two rigid end
plates. The gage pressure inside the
vessel is 2 MPa and 45-kN centric
axial forces P and P' are applied to
the end plates.
Figure P219
220. The pipe shown was fabricated by
welding strips of plate along a helix
forming an angle β with a transverse
plane. Determine the largest value of β
that can be used if the normal stress
perpendicular to the weld is not to be
larger than 85 percent of the maximum
stress in the pipe. 56.8o
Figure P222
a. Determine the normal stress
perpendicular
to
the
weld.
21.4MPa
b. Find the shearing stress parallel to
the rod. 14.17MPa
Figure P220
221. A torque of magnitude T = 12 kN.m
is applied to the end of a tank containing
compressed air under a pressure of 8
MPa. Knowing that the tank has a 180mm inner diameter and a l2-mm wall
thickness, determine the maximum
normal stress and the maximum shearing
stress in the tank. 18.277Mpa
60MPa 30MPa 68.64MPa 0
34.32MPa
223. A brass ring of l60-mm outer
diameter fits exactly inside a steel ring
of l60-mm inner diameter when the
temperature of both rings is 5oC.
Assume that the temperature of the rings
is then raised to 55°C.
41
Figure P225
226. The rosette shown has been used to
determine the following strains at a point
on the surface of a crane hook:
 2  450
 1  420 ,
and
 4  165 . Use Figure P51 to answer
questions 51 and 52.
Figure P223
a. Determine the tensile stress in the
steel ring. 28MPa
b. Calculate the corresponding pressure
exerted by the brass ring on the steel
ring. 1.4MPa
224. The strains determined by use of the
rosette shown during the test of a rocker
arm are:  1  600 ;  2  450 and
 3  75 .
Figure P226
a. What should be the reading of gage
3? -300μ
b. Determine the principal strains and
the maximum in-plane shearing
strain. 435μ -315μ 750μ
227. Determine the largest in-plane
normal strain, knowing that the
following strains have been obtained by
use of the rosette shown: ε1 = -50 X 10-6
mm/mm, ε2 = +360 X 10-6 mm/mm and
ε3 =+315 X 10-6 mm/mm. 315
-5 o
410
415
260
-26
Figure P224
a. Determine (a) the in-plane principal
strains.
734μ -84.3μ
b. Calculate the in-plane maximum
shearing strain.
819μ
225. Determine the strain  x , knowing
that the following strains have been
determined by use of the rosette shown:
ε1 = +720 X 10-6 mm/mm, ε2 = -180 X
10-6 mm/mm and ε3 = + 120 X 10-6
mm/mm.. 380X10-6
mm/mm
-6
460X10 mm/mm -1339X10-6
mm/mm
Figure P227
228. Find the sum of the three strain
measurements made with a 60° rosette is
independent of the orientation of the
42
rosette.
 1   2   3  3 ave , where
εave is the abscissa of the center of the
corresponding Mohr's circle for
strain.
Figure P230
231. A single strain gage is cemented to a
solid 96-mm-diameter aluminum shaft at
an angle β = 20° with a line parallel to
the axis of the shaft. Knowing that G =
27 GPa, determine the torque T
corresponding to a gage reading of 400μ.
Figure P228
229. Using a 45° rosette, the strains ε1 ε2
and ε3 have been determined at a given
point. Using Mohr's circle, derive the
equation for the principal strains.
 max,min  12  1   3  
1
2

1   2    2   3 
2
2

1
2
Figure P231
232. The strains determine by the use of
rosette attached as shown to the surface
of a structural member are ε1 = 200 X
10-6 mm/mm, ε2 = +425 X 10-6 mm/mm
and ε3 =+480 X 10-6 mm/mm. Determine
(a) the orientation and magnitude of the
principal strains in the plane of the
rosette, (b) the maximum in-plane
shearing strain.
Figure P229
230. The given state of plane stress is
known to exist on the surface of a
machine component. Knowing that E =
200 GPa and G = 77 GPa, determine the
direction and magnitude of the three
principal strains (a) by determining the
corresponding state of strain and then
using Mohr's circle for strain, (b) by
using Mohr's circle for stress to
determine the principal planes and
principal stresses and then determining
the corresponding strains.
-28.15o 820
Figure P232
43
Lecture 6: Failure Criteria
233. The state of plane stress shown
occurs in a machine component made of
a steel with σY = 325 MPa. Use the
maximum-shearing-stress criterion to
answer these questions.
determine the magnitude of the torque T
for which yield occurs when P = 240kN.
717N.m
Figure P234
Figure P233
235. The state of plane stress shown will
occur at a critical point in a cast pipe
made of an aluminum alloy for which
σUT = 75 MPa and σUC = 150 MPa.
Using Mohr's criterion, determine the
shearing stress  o for which failure
should be expected.  49.1MPa
a. Determine whether yield occurs when σo
= 200 MPa. If yield does not occur,
determine the corresponding factor of
safety. 1.083
b. Determine whether yield occurs when σo
= 240 MPa. If yield does not occur,
determine the corresponding factor of
safety. yielding occurs
c. Determine whether yield occurs when σo
= 200 σo = 280 MPa. If yield does not
occur, determine the corresponding
factor of safety. yielding occurs
Figure P235
234. The 38-mm-diameter shaft AB is
made of a grade of steel for which the
yield strength is σY = 250 MPa. Using
the maximum-shearing-stress criterion,
44