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M1.3 Use shear stress concepts for four introductory problems. M1.5 For the pin at C, determine the resultant force, the shear
stress, or the minimum required pin diameter for six configuration
variations.

FIGURE M1.5

M1.6 A torque T is transmitted between two flanged shafts by

means of six bolts. If the shear stress in the bolts must be limited to
FIGURE M1.3
a specified value, determine the minimum bolt diameter required
for the connection.
M1.4 Given the areas and allowable normal stresses for mem-
bers (1) and (2), determine the maximum load P that may be sup-
ported by the structure without exceeding either allowable stress.

FIGURE M1.6

FIGURE M1.4

PROBLEMS
P1.1 A stainless steel tube with an outside diameter of 60 mm P1.3 Two solid cylindrical rods (1) and (2) are joined together at
and a wall thickness of 5 mm is used as a compression member. If flange B and loaded as shown in Figure P1.3/4. If the normal stress
the axial normal stress in the member must be limited to 200 MPa, in each rod must be limited to 40 ksi, determine the minimum di-
determine the maximum load P that the member can support. ameter required for each rod.
P1.2 A 2024-T4 aluminum tube with an outside diameter of P1.4 Two solid cylindrical rods (1) and (2) are joined together
2.50 in. will be used to support a 27-kip load. If the axial normal at flange B and loaded, as shown in Figure P1.3/4. The diameter of
stress in the member must be limited to 18 ksi, determine the wall rod (1) is 1.75 in. and the diameter of rod (2) is 2.50 in. Determine
thickness required for the tube. the normal stresses in rods (1) and (2).

16
 15 kips P1.8 Two solid cylindrical rods support a load of P ⫽ 27 kN, as
shown in Figure P1.7/8. Rod (1) has a diameter of 16 mm, and the diam-
A eter of rod (2) is 12 mm. Determine the axial normal stress in each rod.
d1 2.5 m 3.2 m
(1)
30 kips 30 kips A

(1)
B C
4.0 m (2)
d2
2.3 m
(2)

C
B
FIGURE P1.3/4

P
P1.5 Axial loads are applied with rigid bearing plates to the FIGURE P1.7/8
solid cylindrical rods shown in Figure P1.5/6. The diameter of alu-
minum rod (1) is 2.00 in., the diameter of brass rod (2) is 1.50 in., P1.9 A simple pin-connected truss is loaded and supported as
and the diameter of steel rod (3) is 3.00 in. Determine the axial shown in Figure P1.9. All members of the truss are aluminum pipes
normal stress in each of the three rods. that have an outside diameter of 4.00 in. and a wall thickness of
0.226 in. Determine the normal stress in each truss member.

8 kips C 2 kips

A
5 kips
4 kips (1) 4 kips
(3)
7 ft
B (2)
15 kips 15 kips
(2)
20 kips 20 kips A (1) B

C
6 ft 8 ft

(3) FIGURE P1.9

D
P1.10 A simple pin-connected truss is loaded and supported as
shown in Figure P1.10. All members of the truss are aluminum
FIGURE P1.5/6 pipes that have an outside diameter of 60 mm and a wall thickness
of 4 mm. Determine the normal stress in each truss member.

P1.6 Axial loads are applied with rigid bearing plates to the 1.0 m 3.3 m
solid cylindrical rods shown in Figure P1.5/6. The normal stress in
aluminum rod (1) must be limited to 18 ksi, the normal stress in A (3) C
brass rod (2) must be limited to 25 ksi, and the normal stress in steel
rod (3) must be limited to 15 ksi. Determine the minimum diameter 12 kN
required for each of the three rods. 1.5 m (1)
(2) 15 kN
P1.7 Two solid cylindrical rods support a load of P ⫽ 50 kN, as
shown in Figure P1.7/8. If the normal stress in each rod must be
limited to 130 MPa, determine the minimum diameter required for B
each rod. FIGURE P1.10

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P1.11 A simple pin-connected truss is loaded and supported as P1.14 The rectangular bar shown in Figure P1.14 is subjected to
shown in Figure P1.11. All members of the truss are aluminum a uniformly distributed axial loading of w ⫽ 13 kN/m and a con-
pipes that have an outside diameter of 42 mm and a wall thickness centrated force of P ⫽ 9 kN at B. Determine the magnitude of the
of 3.5 mm. Determine the normal stress in each truss member. maximum normal stress in the bar and its location x. Assume a ⫽
0.5 m, b ⫽ 0.7 m, c ⫽ 15 mm, and d ⫽ 40 mm.
A
(3)
1.6 m w
15 kN C
P d

30 kN (1) A B w C c
x
(2) 4.0 m a b

FIGURE P1.14

B P1.15 The solid 1.25-in.-diameter rod shown in Figure P1.15 is

4.5 m
subjected to a uniform axial distributed loading along its length of
w ⫽ 750 lb/ft. Two concentrated loads also act on the rod: P ⫽
FIGURE P1.11 2,000 lb and Q ⫽ 1,000 lb. Assume a ⫽ 16 in. and b ⫽ 32 in. Deter-
mine the normal stress in the rod at the following locations:
P1.12 The rigid beam BC shown in Figure P1.12 is supported
by rods (1) and (2) that have cross-sectional areas of 175 mm2 (a) x ⫽ 10 in.
and 300 mm2, respectively. For a uniformly distributed load of (b) x ⫽ 30 in.
w ⫽ 15 kN/m, determine the normal stress in each rod. Assume
L ⫽ 3 m and a ⫽ 1.8 m.
w
P Q
A D
A B C
x
(1) (2) a b
w
FIGURE P1.15

B C P1.16 Two 6-in.-wide wooden boards are to be joined by

a splice plates that will be fully glued onto the contact surfaces, as
L shown in Figure P1.16. The glue to be used can safely provide a
shear strength of 120 psi. Determine the smallest allowable length
FIGURE P1.12 L that can be used for the splice plates for an applied load of P ⫽
P1.13 Bar (1) in Figure P1.13 has a cross-sectional area of 10,000 lb. Note that a gap of 0.5 in. is required between boards (1)
0.75 in.2. If the stress in bar (1) must be limited to 30 ksi, determine and (2).
the maximum load P that may be supported by the structure.
P P
D L 6 in.

(1) P (1) (2)

0.5 in.
FIGURE P1.16

A B C P1.17 For the clevis connection shown in Figure P1.17, deter-

6 ft 4 ft
mine the maximum applied load P that can be supported by the
10-mm-diameter pin if the average shear stress in the pin must not
FIGURE P1.13 exceed 95 MPa.

18
 the average shear stress in the adhesive must be limited to 400 psi,
P
Clevis determine the minimum lengths L1 and L2 required for the joint if
the applied load P is 5,000 lb.
t
P1.21 A hydraulic punch press is used to punch a slot in a
0.50-in.-thick plate, as illustrated in Figure P1.21. If the plate
Pin P shears at a stress of 30 ksi, determine the minimum force P required
Bar to punch the slot.
FIGURE P1.17

P1.18 For the connection shown in Figure P1.18, determine the

average shear stress produced in the 3/8-in. diameter bolts if the
0.75 in.
applied load is P ⫽ 2,500 lb. P

3.00 in.

Plan view of slug

Punch

Plate

Slug
P
FIGURE P1.18 FIGURE P1.21

P1.19 The five-bolt connection shown in Figure P1.19 must

support an applied load of P ⫽ 265 kN. If the average shear stress in
the bolts must be limited to 120 MPa, determine the minimum bolt
P1.22 The handle shown in Figure P1.22 is attached to a
40-mm-diameter shaft with a square shear key. The forces applied
diameter that may be used for this connection.
to the lever are P ⫽ 1,300 N. If the average shear stress in the key
must not exceed 150 MPa, determine the minimum dimension a
that must be used if the key is 25 mm long. The overall length of
the handle is L ⫽ 0.70 m.

P
P a Shear key
a
FIGURE P1.19

P1.20 A coupling is used to connect a 2-in.-diameter plastic Handle d

pipe (1) to a 1.5-in.-diameter pipe (2), as shown in Figure P1.20. If
Shaft

P Coupling P
(2) L L
2 2
P
(1) FIGURE P1.22

P P
(1) (2) P1.23 An axial load P is supported by the short steel column
shown in Figure P1.23. The column has a cross-sectional area of
L1 L2 14,500 mm2. If the average normal stress in the steel column must
not exceed 75 MPa, determine the minimum required dimension a
Cutaway section of coupling so that the bearing stress between the base plate and the concrete
FIGURE P1.20 slab does not exceed 8 MPa. Assume b ⫽ 420 mm.

19
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P thickness of t ⫽ 10 mm. If the normal stress produced in the rod

by load P is 225 MPa, determine
(a) the bearing stress acting between the support plate and the
rod head.
(b) the average shear stress produced in the rod head.
(c) the punching shear stress produced in the support plate by the
rod head.
a

b
Support
plate
FIGURE P1.23 Hole diameter D

P
P1.24 The two wooden boards shown in Figure P1.24 are
connected by a 0.5-in.-diameter bolt. Washers are installed under
Rod
the head of the bolt and under the nut. The washer dimensions are Head
D ⫽ 2 in. and d ⫽ 5/8 in. The nut is tightened to cause a tensile
stress of 9,000 psi in the bolt. Determine the bearing stress be-
tween the washer and the wood.

Bolt t
d
Washer
P
a

b
Washer
Nut
d FIGURE P1.26
D
P1.27 The rectangular bar is connected to the support bracket
FIGURE P1.24
with a circular pin, as shown in Figure P1.27. The bar width is
w ⫽ 1.75 in. and the bar thickness is 0.375 in. For an applied load
P1.25 For the beam shown in Figure P1.25, the allowable bear- of P ⫽ 5,600 lb, determine the average bearing stress produced in
ing stress for the material under the supports at A and B is ␴b ⫽ the bar by the 0.625-in.-diameter pin.
800 psi. Assume w ⫽ 2,100 lb/ft, P ⫽ 4,600 lb, a ⫽ 20 ft, and b ⫽
8 ft. Determine the size of square bearing plates required to sup-
Bracket
port the loading shown. Dimension the plates to the nearest 1/2 in.
Pin
P Bar
w w

A B P
FIGURE P1.27
a b

FIGURE P1.25 P1.28 The steel pipe column shown in Figure P1.28 has an
outside diameter of 8.625 in. and a wall thickness of 0.25 in. The
P1.26 The d ⫽ 15-mm-diameter solid rod shown in Figure timber beam is 10.75 in. wide, and the upper plate has the same
P1.26 passes through a D ⫽ 20-mm-diameter hole in the support width. The load imposed on the column by the timber beam is
plate. When a load P is applied to the rod, the rod head rests on 80 kips. Determine the following:
the support plate. The support plate has a thickness of b ⫽ 12 mm. (a) the average bearing stress at the surfaces between the pipe
The rod head has a diameter of a ⫽ 30 mm, and the head has a column and the upper and lower steel bearing plates

20
(b) the length L of the rectangular upper bearing plate if its width P1.30 Rigid bar ABC shown in Figure P1.30 is supported by
is 10.75 in. and the average bearing stress between the steel a pin at bracket A and by tie rod (1). Tie rod (1) has a diameter
plate and the wood beam is not to exceed 500 psi of 5 mm, and it is supported by double-shear pin connections at
(c) the dimension a of the square lower bearing plate if the B and D. The pin at bracket A is a single-shear connection. All
average bearing stress between the lower bearing plate and pins are 7 mm in diameter. Assume a ⫽ 600 mm, b ⫽ 300 mm,
the concrete slab is not to exceed 900 psi h ⫽ 450 mm, P ⫽ 900 N, and ␪ ⫽ 55°. Determine the following:
(a) the normal stress in rod (1)
(b) the shear stress in pin B
10.75 in. (c) the shear stress in pin A

Timber
beam

L 9

Upper
bearing &
plate ]
Steel
pipe
6 7 8

U
Lower V W
bearing
Concrete plate E
slab
FIGURE P1.30
a a

P1.31 The bell crank shown in Figure P1.31 is in equilibrium

FIGURE P1.28 for the forces acting in rods (1) and (2). The bell crank is supported
by a 10-mm-diameter pin at B that acts in single shear. The thick-
ness of the bell crank is 5 mm. Assume a ⫽ 65 mm, b ⫽ 150 mm,
F1 ⫽ 1,100 N, and ␪ ⫽ 50°. Determine the following:
P1.29 A clevis-type pipe hanger supports an 8-in.-diameter pipe,
as shown in Figure P1.29. The hanger rod has a diameter of 1/2 in. (a) the shear stress in pin B
The bolt connecting the top yoke and the bottom strap has a diameter (b) the bearing stress in the bell crank at B
of 5/8 in. The bottom strap is 3/16 in. thick by 1.75 in. wide by 36 in.
long. The weight of the pipe is 2,000 lb. Determine the following:
(a) the normal stress in the hanger rod ;&
(b) the shear stress in the bolt
(c) the bearing stress in the bottom strap V
&

6 7

7ZaaXgVc` W

Hanger rod ' ;'

Bolt 8
Top yoke
FIGURE P1.31
Bottom strap

P1.32 The beam shown in Figure P1.32 is supported by a pin at

C and by a short link AB. If w ⫽ 30 kN/m, determine the average
shear stress in the pins at A and C. Each pin has a diameter of 25 mm.
FIGURE P1.29 Assume L ⫽ 1.8 m and ␪ ⫽ 35°.

21
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(a) The shear stress in the pin may not exceed 40 MPa.
(b) The bearing stress in the bell crank may not exceed 100 MPa.
6
8 (c) The bearing stress in the support bracket may not exceed
165 MPa.

6 &
l &
8 ;&
U
7ZaaXgVc`
-bb
7 8
E
A W

FIGURE P1.32 U Y
7 7

P1.33 The bell-crank mechanism shown in Figure P1.33 is 6

in equilibrium for an applied load of P ⫽ 7 kN applied at A. V
+bb +bb
Assume a ⫽ 200 mm, b ⫽ 150 mm, and ␪ ⫽ 65°. Determine the
minimum diameter d required for pin B for each of the following HjeedgiWgVX`Zi
conditions: FIGURE P1.33

1.5 Stresses on Inclined Sections

In previous sections, normal, shear, and bearing stresses on planes parallel and perpendicu-
lar to the axes of centrically loaded members were introduced. Stresses on planes inclined
to the axes of axially loaded bars will now be considered.
Consider a prismatic bar subjected to an axial force P applied to the centroid of the bar
MecMovies 1.11 is an (Figure 1.7a). Loading of this type is termed uniaxial since the force applied to the bar acts in
animated presentation of the one direction (i.e., either tension or compression). The cross-sectional area of the bar is A. To
theory of stresses on an inclined investigate the stresses that are acting internally in the material, we will cut through the bar at
plane. section a–a. The free-body diagram (Figure 1.7b) exposes the normal stress ␴ that is distrib-
uted over the cut section of the bar. The normal stress magnitude may be calculated from
␴ ⫽ PⲐA, provided that the stress is uniformly distributed. In this case, the stress will be uni-
form because the bar is prismatic and the force P is applied at the centroid of the cross section.
The resultant of this normal stress distribution is equal in magnitude to the applied load P and
has a line of action that is coincident with the axes of the bar, as shown. Note that there will be
no shear stress ␶ since the cut surface is perpendicular to the direction of the resultant force.
Section a–a is unique, however, because it is the only surface that is perpendicular to
In referencing planes, the
the direction of force P. A more general case would consider a section cut through the bar at
orientation of the plane is
specified by the normal to the an arbitrary angle. Consider a free-body diagram along section b–b (Figure 1.7c). Because
plane. The inclined plane shown the stresses are the same throughout the entire bar, the stresses on the inclined surface must
in Figure 1.7d is termed the n be uniformly distributed. Since the bar is in equilibrium, the resultant of the uniformly
face because the n axis is the distributed stress must equal P even though the stress acts on a surface that is inclined.
normal to this surface. The orientation of the inclined surface can be defined by the angle ␪ between the
x axis and an axis normal to the plane, which is the n axis, as shown in Figure 1.7d. A posi-
tive angle ␪ is defined as a counterclockwise rotation from the x axis to the n axis. The t axis
is tangential to the cut surface, and the n–t axes form a right-handed coordinate system.
To investigate the stresses acting on the inclined plane (Figure 1.7d), the components
of resultant force P acting perpendicular and parallel to the plane must be computed. Using
␪ as defined previously, the perpendicular force component (i.e., normal force) is N ⫽ P cos
␪, and the parallel force component (i.e., shear force) is V ⫽ ⫺P sin ␪. (The negative sign
indicates that the shear force acts in the ⫺t direction, as shown in Figure 1.7d.) The area of

22
 MecMovies Example M1.13
The steel bar shown has a 50-mm by 10-mm rectangular cross section. The allowable
normal and shear stresses on the inclined surface must be limited to 40 MPa and
25 MPa, respectively. Determine the magnitude of the maximum axial force of P that can
be applied to the bar.

MecMovies
M Exercises
M1.12 The bar has a rectangular cross section. For a given load M1.13 The bar has a rectangular cross section. The allowable
P, determine the force components perpendicular and parallel to normal and shear stresses on inclined surface a–a are given. Deter-
section a–a, the inclined surface area, and the normal and shear mine the magnitude of the maximum axial force P that can be ap-
stress magnitudes acting on surface a–a. plied to the bar and determine the actual normal and shear stresses
acting on inclined plane a–a.

FIGURE M1.13

FIGURE M1.12

PROBLEMS
P1.34 A structural steel bar with a 25 mm ⫻ 75 mm rectangular P1.35 A steel rod of circular cross section will be used to carry
cross section is subjected to an axial load of 150 kN. Determine the an axial load of 92 kips. The maximum stresses in the rod must be
maximum normal and shear stresses in the bar. limited to 30 ksi in tension and 12 ksi in shear. Determine the re-
quired diameter for the rod.

27
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P1.36 An axial load P is applied to the rectangular bar shown in P

Figure P1.36. The cross-sectional area of the bar is 400 mm2. De-
termine the normal stress perpendicular to plane AB and the shear
stress parallel to plane AB if the bar is subjected to an axial load of
P ⫽ 70 kN. A

P
A 55° 35°
P

B
B

FIGURE P1.36

P1.37 An axial load P is applied to the 1.75-in.-by-0.75-in. rec-

tangular bar shown in Figure P1.37. Determine the normal stress FIGURE P1.39
perpendicular to plane AB and the shear stress parallel to plane AB
if the bar is subjected to an axial load of P ⫽ 18 kips. P1.40 Specifications for the 6 in. ⫻ 6 in. square post shown in
Figure P1.40 require that the normal and shear stresses on plane
A AB not exceed 800 psi and 400 psi, respectively. Determine
the maximum load P that can be applied without exceeding the
P
60° specifications.
1.75 in. P
P
B
FIGURE P1.37

P1.38 A compression load of P ⫽ 80 kips is applied to a A

40°
4-in.-by-4-in. square post, as shown in Figure P1.38. Determine the
normal stress perpendicular to plane AB and the shear stress paral-
lel to plane AB. B
6 in.

FIGURE P1.40

A P1.41 A 90-mm-wide bar will be used to carry an axial ten-

sion load of 280 kN, as shown in Figure P1.41. The normal and
shear stresses on plane AB must be limited to 150 MPa and 100 MPa,
35°
respectively. Determine the minimum thickness t required for
the bar.
B

A
P
FIGURE P1.38
90 mm 50°
P1.39 Specifications for the 50 mm ⫻ 50 mm square bar shown
in Figure P1.39 require that the normal and shear stresses on plane
AB not exceed 120 MPa and 90 MPa, respectively. Determine the
P
maximum load P that can be applied without exceeding the B
specifications. FIGURE P1.41

28
P1.42 A rectangular bar having width w ⫽ 6.00 in. and thick- (a) the magnitude of load P.
ness t ⫽ 1.50 in. is subjected to a tension load P, as shown in (b) the shear stress on plane AB.
Figure P1.42/43. The normal and shear stresses on plane AB must (c) the maximum normal and shear stresses in the block at any
not exceed 16 ksi and 8 ksi, respectively. Determine the maximum possible orientation.
load P that can be applied without exceeding either stress limit.
P1.45 The rectangular bar has a width of w ⫽ 100 mm and a
P1.43 In Figure P1.42/43, a rectangular bar having width thickness of t ⫽ 75 mm. The shear stress on plane AB of the
w ⫽ 1.25 in. and thickness t is subjected to a tension load of rectangular block shown in Figure P1.44/45 is 12 MPa when the
P ⫽ 30 kips. The normal and shear stresses on plane AB must not load P is applied. Determine
exceed 12 ksi and 8 ksi, respectively. Determine the minimum thick- (a) the magnitude of load P.
ness t required for the bar. (b) the normal stress on plane AB.
(c) the maximum normal and shear stresses in the block at any
possible orientation.
A
P
1 w
3 t

B
A 3
FIGURE P1.42/43 P
4
w

P1.44 The rectangular bar has a width of w ⫽ 3.00 in. and a B

thickness of t ⫽ 2.00 in. The normal stress on plane AB of the
rectangular block shown in Figure P1.44/45 is 6 ksi (C) when the
load P is applied. Determine FIGURE P1.44/45

29
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PROBLEMS
P2.1 When an axial load is applied to the ends of the bar shown
in Figure P2.1, the total elongation of the bar between joints A and
C is 0.15 in. In segment (2), the normal strain is measured as (1) (1)
1,300 ␮in./in. Determine (2)
(a) the elongation of segment (2). L1
(b) the normal strain in segment (1) of the bar. L2

A B C
P (1) (2) P Rigid bar

A B C
P
40 in. 90 in.
FIGURE P2.3
FIGURE P2.1

P2.4 A rigid bar ABCD is supported by two bars, as shown in Fig-

P2.2 The two bars shown in Figure P2.2 are used to support a ure P2.4. There is no strain in the vertical bars before load P is applied.
load P. When unloaded, joint B has coordinates (0, 0). After load After load P is applied, the normal strain in rod (1) is ⫺570 ␮m/m.
P is applied, joint B moves to the coordinate position (0.35 in., Determine
⫺0.60 in.). Assume a ⫽ 11 ft, b ⫽ 6 ft, and h ⫽ 8 ft. Determine
the normal strain in each bar. (a) the normal strain in rod (2).
(b) the normal strain in rod (2) if there is a 1-mm gap in the
connection at pin C before the load is applied.
(c) the normal strain in rod (2) if there is a 1-mm gap in the
a b connection at pin B before the load is applied.

A C

y h
(2)

x 1,500 mm
B 240 mm 360 mm 140 mm

P Rigid bar
FIGURE P2.2
A B C D

900 mm
P2.3 A rigid steel bar is supported by three rods, as shown in (1) P
Figure P2.3. There is no strain in the rods before the load P is
applied. After load P is applied, the normal strain in rods
(1) is 860 ␮m/m. Assume initial rod lengths of L1 ⫽ 2,400 mm and FIGURE P2.4
L2 ⫽ 1,800 mm. Determine
(a) the normal strain in rod (2).
(b) the normal strain in rod (2) if there is a 2-mm gap in the
connections between the rigid bar and rods (1) at joints A and P2.5 In Figure P2.5, rigid bar ABC is supported by a pin con-
C before the load is applied. nection at B and two axial members. A slot in member (1) allows
(c) the normal strain in rod (2) if there is a 2-mm gap in the the pin at A to slide 0.25 in. before it contacts the axial member.
connection between the rigid bar and rod (2) at joint B before If the load P produces a compression normal strain in member (1)
the load is applied. of ⫺1,300 ␮in./in., determine the normal strain in member (2).

38
 Sanding sleeve

(2)
D
160 in.
20 in.

B
C
9 in. Mandrel
12 in.
P FIGURE P2.6
(1)
P2.7 The normal strain in a suspended bar of material of vary-
A ing cross section due to its own weight is given by the expression
␥ y3E, where ␥ is the specific weight of the material, y is the dis-
tance from the free (i.e., bottom) end of the bar, and E is a material
32 in. 0.25 in.
constant. Determine, in terms of ␥, L, and E the following:
FIGURE P2.5 (a) the change in length of the bar due to its own weight
(b) the average normal strain over the length L of the bar
(c) the maximum normal strain in the bar
P2.6 The sanding-drum mandrel shown in Figure P2.6 is P2.8 A steel cable is used to support an elevator cage at the
made for use with a hand drill. The mandrel is made from a bottom of a 2,000-ft-deep mineshaft. A uniform normal strain of
rubber-like material that expands when the nut is tightened to 250 ␮in./in. is produced in the cable by the weight of the cage. At
secure the sanding sleeve placed over the outside surface. If the each point, the weight of the cable produces an additional normal
diameter D of the mandrel increases from 2.00 in. to 2.15 in. as strain that is proportional to the length of the cable below the point.
the nut is tightened, determine If the total normal strain in the cable at the cable drum (upper end
(a) the average normal strain along a diameter of the mandrel. of the cable) is 700 ␮in./in., determine
(b) the circumferential strain at the outside surface of (a) the strain in the cable at a depth of 500 ft.
the mandrel. (b) the total elongation of the cable.

2.3 Shear Strain

A deformation involving a change in shape (distortion) can be used to illustrate a shear y
strain. An average shear strain ␥avg associated with two reference lines that are orthogonal ␥xy
in the undeformed state (two edges of the element shown in Figure 2.4) can be obtained by ␦x
dividing the shear deformation ␦ x (displacement of the top edge of the element with respect
to the bottom edge) by the perpendicular distance L between these two edges. If the defor-
mation is small, meaning that sin ␥  tan ␥  ␥ and cos ␥  1, then shear strain can be
L
defined as
␪⬘
␦x x
␥avg ⫽ (2.3) O
L
FIGURE 2.4 Shear strain.
For those cases in which the deformation is nonuniform, the shear strain at a point, ␥xy(O),
associated with two orthogonal reference lines x and y is obtained by measuring the shear
deformation as the size of the element is made smaller and smaller. In the limit,
⌬␦x d␦
␥xy (O) ⫽ lim ⫽ x (2.4)
⌬L 0 ⌬L dL

39
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PROBLEMS
P2.9 The 16-mm by 22-mm by 25-mm rubber blocks shown in P2.11 A thin polymer plate PQR is deformed so that corner Q
Figure P2.9 are used in a double-U shear mount to isolate the vibra- is displaced downward 1.0 mm to new position Q ⬘ as shown in
tion of a machine from its supports. An applied load of P ⫽ 690 N Figure P2.11. Determine the shear strain at Q ⬘ associated with the
causes the upper frame to be deflected downward by 7 mm. Deter- two edges (PQ and QR).
mine the average shear strain and the shear stress in the rubber
blocks.

y
Double U
anti-vibration
shear mount 120 mm 750 mm

x
P R
25

300 mm
P
Q
22
16
1.0 mm
Q⬘
Rubber block FIGURE P2.11
dimensions

Shear deformation
of blocks
FIGURE P2.9
P2.12 A thin square plate is uniformly deformed as
shown in Figure P2.12. Determine the shear strain ␥xy after
deformations
P2.10 A thin polymer plate PQR is deformed such that corner Q
is displaced downward 1/16-in. to new position Q ⬘ as shown in (a) at corner P, and
Figure P2.10. Determine the shear strain at Q⬘ associated with the (b) at corner Q.
two edges (PQ and QR).

y
y
25 in. 4 in. 25 mm
75 mm

R S
x
P R

10 in.
110 mm 100 mm
Q

—1 in. x
16 P Q
Q⬘ 100 mm

FIGURE P2.10 FIGURE P2.12

42
P2.13 A thin square plate is uniformly deformed as shown in P2.14 A thin square plate PQRS is symmetrically deformed into
Figure P2.13. Determine the shear strain ␥xy after deformations the shape shown by the dashed lines in Figure P2.14. For the de-
(a) at corner R, and formed plate, determine
(b) at corner S. (a) the normal strain of diagonal QS.
(a) the shear strain ␥xy at corner P.

y S Undeformed
120 mm
S
Deformed
R 25 mm
P R
249.7 mm x
75 mm
100 mm

Q Q
x
P 250 mm
100 mm
251.2 mm
FIGURE P2.13 FIGURE P2.14

2.4 Thermal Strain

When unrestrained, most engineering materials expand when heated and contract when A material of uniform
cooled. The thermal strain caused by a one-degree (1°) change in temperature is designated composition is called a
by the Greek letter ␣ (alpha) and is known as the coefficient of thermal expansion. The homogeneous material. In
strain due to a temperature change of ⌬T is materials of this type, local
variations in composition
can be considered negligible
␧T ⫽ ␣ ⌬ T (2.6)
for engineering purposes.
Furthermore, homogeneous
The coefficient of thermal expansion is approximately constant for a considerable range of
materials cannot be
temperatures. (In general, the coefficient increases with an increase of temperature.) For a
mechanically separated
uniform material (termed a homogeneous material) that has the same mechanical into different materials
properties in every direction (termed an isotropic material), the coefficient applies to all (e.g., carbon fibers in a
dimensions (i.e., all directions). Values of the coefficient of expansion for common materi- polymer matrix). Common
als are included in Appendix D. homogeneous materials are
metals, alloys, ceramics,
glass, and some types
Total Strains of plastics.
Strains caused by temperature changes and strains caused by applied loads are essentially
independent. The total normal strain in a body acted on by both temperature changes and
applied load is given by
An isotropic material has the
same mechanical properties in
␧total ⫽ ␧␴ ⫹ ␧T (2.7)
all directions.
Since homogeneous, isotropic materials, when unrestrained, expand uniformly in all direc-
tions when heated (and contract uniformly when cooled), neither the shape of the body nor
the shear stresses and shear strains are affected by temperature changes.

43
 SOLUTION
The maximum shank outside diameter is 18.000 ⫹ 0.005 mm ⫽ 18.005 mm. The mini-
mum holder inside diameter is 17.950 ⫺ 0.005 mm ⫽ 17.945 mm. Therefore, the inside
diameter of the holder must be increased by 18.005 ⫺ 17.945 mm ⫽ 0.060 mm. To
expand the holder by this amount requires a temperature increase:

0.060 mm
␦T ⫽ ␣ ⌬Td ⫽ 0.060 mm ⬖ ⌬T ⫽ ⫽ 281°C
11.9 ⫻ 10⫺6°C (17.945 mm)

Therefore, the tool holder must attain a minimum temperature of

20°C ⫹ 281°C ⫽ 301°C Ans.

PROBLEMS
P2.15 An airplane has a half-wingspan of 33 m. Determine the 3-mm gap
change in length of the aluminum alloy [␣A ⫽ 22.5 ⫻ 10⫺6°C]
wing spar if the plane leaves the ground at a temperature of 15°C
and climbs to an altitude where the temperature is ⫺55°C.
(1) (2)
P2.16 A square 2014-T4 aluminum alloy plate 400 mm on a side C
B
has a 75-mm-diameter circular hole at its center. The plate is heated A
from 20°C to 45°C. Determine the final diameter of the hole. 540 mm 360 mm

P2.17 A cast iron pipe has an inside diameter of d ⫽ 208 mm FIGURE P2.19
and an outside diameter of D ⫽ 236 mm. The length of the pipe
is L ⫽ 3.0 m. The coefficient of thermal expansion for cast iron
is ␣I ⫽ 12.1 ⫻ 10⫺6°C. Determine the dimension changes P2.20 An aluminum pipe has a length of 60 m at a temperature of
caused by an increase in temperature of 70°C. 10°C. An adjacent steel pipe at the same temperature is 5 mm lon-
ger. At what temperature will the aluminum pipe be 15 mm longer
P2.18 At a temperature of 40°F, a 0.08-in. gap exists between than the steel pipe? Assume that the coefficient of thermal expan-
the ends of the two bars shown in Figure P2.18. Bar (1) is an alumi- sion for the aluminum is 22.5 ⫻ 10⫺6°C and that the coefficient of
num alloy [␣ ⫽ 12.5 ⫻ 10⫺6°F], and bar (2) is stainless steel thermal expansion for the steel is 12.5 ⫻ 10⫺6°C.
[␣ ⫽ 9.6 ⫻ 10⫺6°F]. The supports at A and C are rigid. Determine
the lowest temperature at which the two bars contact each other. P2.21 Determine the movement of the pointer of Figure P2.21
with respect to the scale zero in response to a temperature increase
of 60°F. The coefficients of thermal expansion are 6.6 ⫻ 10⫺6°F
(1)
(2) for the steel and 12.5 ⫻ 10⫺6°F for the aluminum.

B C
A 7.0 in. 1.5 in.

Smooth pins
+
40 in. 55 in.
0
0.08-in. gap
–
FIGURE P2.18
12 in.
P2.19 At a temperature of 5°C, a 3-mm gap exists between two poly-
mer bars and a rigid support, as shown in Figure P2.19. Bars (1) and (2) Steel Aluminum Steel
have coefficients of thermal expansion of ␣1 ⫽ 140 ⫻ 10⫺6°C and
␣2 ⫽ 67 ⫻ 10⫺6°C, respectively. The supports at A and C are rigid.
Determine the lowest temperature at which the 3-mm gap is closed. FIGURE P2.21

45
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P2.22 Determine the horizontal movement of point A of Figure dB ⫽ 299.75 mm and an outside diameter of DB ⫽ 310 mm. The
P2.22 due to a temperature increase of 75°C. Assume that member sleeve is to be placed on a steel [␣S ⫽ 11.9 ⫻ 10⫺6°C] shaft with
AE has a negligible coefficient of thermal expansion. The coeffi- an outside diameter of DS ⫽ 300 mm. If the temperatures of the
cients of thermal expansion are 11.9 ⫻ 10⫺6°C for the steel and sleeve and the shaft remain the same, determine the temperature at
22.5 ⫻ 10⫺6°C for the aluminum alloy. which the sleeve will slip over the shaft with a gap of 0.05 mm.

A P2.24 For the assembly shown in Figure P2.24, bars (1) and
(2) each have cross-sectional areas of A ⫽ 1.6 in.2, elastic moduli
of E ⫽ 15.2 ⫻ 106 psi, and coefficients of thermal expansion of
␣ ⫽ 12.2 ⫻ 10⫺6°F. If the temperature of the assembly is in-
creased by 80°F from its initial temperature, determine the result-
250 mm ing displacement of pin B. Assume h ⫽ 54 in. and ␪ ⫽ 55°.

B Steel C
25 mm A U U C

D Aluminum E (1) (2)

300 mm h

FIGURE P2.22

P2.23 At a temperature of 25°C, a cold-rolled red brass B

[␣B ⫽ 17.6 ⫻ 10⫺6°C] sleeve has an inside diameter of FIGURE P2.24

46
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The total elongation of the bar is the sum of the segment elongations:
u A ⫽ ␦1 ⫹ ␦2 ⫽ 0.0250 in. ⫹ 0.0458 in. ⫽ 0.0708 in. Ans.
Note: If the weight of the bar had not been neglected, the internal force F in both uniform-
width segment (1) and tapered segment (2) would not have been constant, and Equation (5.5)
would be required for both segments. To include the weight of the bar in the analysis, a function
should be derived for each segment, expressing the change in internal force as a function of the
vertical position y. The internal force F at any position y is the sum of a constant force equal to
P and a varying force equal to the self-weight of the axial member below position y. The force
due to self-weight will be a function that expresses the volume of the bar below any position
y, multiplied by the specific weight of the material that the bar is made of. Since the internal
force F varies with y, it must be included inside the integral in Equation (5.5).

MecMovies
M Exercises
M5.1 Use the axial deformation equation for three introductory M5.2 Apply the axial deformation concept to compound axial
problems. members.

FIGURE M5.2

FIGURE M5.1

PROBLEMS
P5.1 A steel [E  200 GPa] rod with a circular cross section is in the rod is 2,200 lb. If the maximum allowable normal stress in
7.5-m long. Determine the minimum diameter required if the rod the rod is 12 ksi, determine
must transmit a tensile force of 50 kN without exceeding an allow- (a) the smallest diameter that can be used for the rod.
able stress of 180 MPa or stretching more than 5 mm. (b) the corresponding maximum length of the rod.
P5.2 An aluminum [E  10,000 ksi] control rod with a circular P5.3 A 12-mm-diameter steel [E  200 GPa] rod (2) is con-
cross section must not stretch more than 0.25 in. when the tension nected to a 30-mm-wide by 8-mm-thick rectangular aluminum

98
[E  70 GPa] bar (1), as shown in Figure P5.3. Determine the force 115 kips
P required to stretch the assembly 10.0 mm.

(1) (2)
(2)
P 14 ft

A B C 155 kips

0.45 m 1.30 m
B
FIGURE P5.3
16 ft (1)
P5.4 A rectangular bar of length L has a slot in the central half
of its length, as shown in Figure P5.4. The bar has width b, thick-
ness t, and elastic modulus E. The slot has width b/3. If L  A
400 mm, b  45 mm, t  8 mm, and E  72 GPa, determine the
overall elongation of the bar for an axial force of P  18 kN.

FIGURE P5.6
P P
b b
3 P5.7 Aluminum [E  70 GPa] member ABC supports a load of
28 kN, as shown in Figure P5.7. Determine
(a) the value of load P such that the deflection of joint C is zero.
L L L (b) the corresponding deflection of joint B.
4 2 4
FIGURE P5.4

28 kN
P5.5 An axial member consisting of two polymer bars is sup- (1) A
ported at C, as shown in Figure P5.5. Bar (1) has a cross-sectional C
area of 540 mm2 and an elastic modulus of 28 GPa. Bar (2) has 1.8 m 40 kN
a cross-sectional area of 880 mm2 and an elastic modulus of 32-mm
16.5 GPa. Determine the deflection of point A relative to support C. 1.0 m diameter
B
(2)
(2)
50 kN B 1.2 m

35 kN C
(1) (2) P
1.3 m 50-mm
A diameter 14 kN
B C 1.6 m
50 kN (3)
(1) A
D
0.85 m 1.15 m

FIGURE P5.5 25 kN
FIGURE P5.7 FIGURE P5.8

P5.6 The roof and second floor of a building are supported by P5.8 A solid brass [E  100 GPa] axial member is loaded and
the column shown in Figure P5.6. The column is a structural steel supported as shown in Figure P5.8. Segments (1) and (2) each have
W10  60 wide-flange section [E  29,000 ksi; A  17.6 in.2]. a diameter of 25 mm, and segment (3) has a diameter of 14 mm.
The roof and floor subject the column to the axial forces shown. Determine
Determine (a) the deformation of segment (2).
(a) the amount that the first floor will deflect. (b) the deflection of joint D with respect to the fixed support at A.
(b) the amount that the roof will deflect. (c) the maximum normal stress in the entire axial member.

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P5.9 A hollow steel [E  30,000 ksi] tube (1) with an outside taper of the cone is slight enough for the assumption of a uniform
diameter of 2.75 in. and a wall thickness of 0.25 in. is fastened to a axial stress distribution over a cross section to be valid.
solid aluminum [E  10,000 ksi] rod (2) that has a 2-in.-diameter (a) Determine an expression for the stress distribution on an
and a solid 1.375-in.-diameter aluminum rod (3). The bar is loaded arbitrary cross section at x.
as shown in Figure P5.9. Determine (b) Determine an expression for the elongation of the rod.
(a) the change in length of steel tube (1).
(b) the deflection of joint D with respect to the fixed support at A.
(c) the maximum normal stress in the entire axial assembly.
d0
P

34 kips 18 kips x P 2d0

25 kips
(1) (2) (3) L
C D
A B
34 kips 18 kips
FIGURE P5.12
60 in. 40 in. 30 in.

FIGURE P5.9
P5.13 Determine the extension, due to its own weight, of the
conical bar shown in Figure P5.13. The bar is made of aluminum
alloy [E  10,600 ksi and   0.100 lb/in.3]. The bar has a 2-in.
P5.10 A solid 5/8-in. steel [E  29,000 ksi] rod (1) supports
radius at its upper end and a length of L  20 ft. Assume that the
beam AB as shown in Figure P5.10. If the stress in the rod must
taper of the bar is slight enough for the assumption of a uniform
not exceed 30 ksi and the maximum deformation in the rod must
axial stress distribution over a cross section to be valid.
not exceed 0.25 in., determine the maximum load P that may be
supported.

C
A

y
y
(1)
w
16 ft L
L

P
B B
8 ft
12 ft
x FB
FIGURE P5.10 FIGURE P5.13 FIGURE P5.14

P5.11 A 1-in.-diameter by 16-ft-long cold-rolled bronze P5.14 The wooden pile shown in Figure P5.14 has a diame-
[E  15,000 ksi and   0.320 lb/in.3] bar hangs vertically while ter of 100 mm and is subjected to a load of P  75 kN. Along
suspended from one end. Determine the change in length of the bar the length of the pile and around its perimeter, soil supplies a
due to its own weight. constant frictional resistance of w  3.70 kN/m. The length of
P5.12 A homogeneous rod of length L and elastic modulus E is the pile is L  5.0 m and its elastic modulus is E  8.3 GPa.
a truncated cone with a diameter that varies linearly from d0 at one Calculate
end to 2d0 at the other end. A concentrated axial load P is applied (a) the force FB needed at base of the pile for equilibrium.
to the ends of the rod as shown in Figure P5.12. Assume that the (b) the magnitude of the downward displacement at A relative to B.

100
PROBLEMS
P5.15 Rigid bar ABCD is loaded and supported as shown in a diameter of d2  0.75 in. Aluminum rod (3) has a diameter of
Figure P5.15. Bars (1) and (2) are unstressed before the load P is d3  1.0 in. The yield strength of the bronze is 48 ksi, and the yield
applied. Bar (1) is made of bronze [E  100 GPa] and has a cross- strength of the aluminum is 40 ksi.
sectional area of 520 mm2. Bar (2) is made of aluminum [E  (a) Determine the magnitude of load P that can safely be applied to
70 GPa] and has a cross-sectional area of 960 mm2. After the load the structure if a minimum factor of safety of 1.67 is required.
P is applied, the force in bar (2) is found to be 25 kN (in tension). (b) Determine the deflection of point D for the load determined
Determine in part (a).
(a) the stresses in bars (1) and (2). (c) The pin used at B has an ultimate shear strength of 54 ksi. If a
(b) the vertical deflection of point A. factor of safety of 3.0 is required for this double shear pin
(c) the load P. connection, determine the minimum pin diameter that can be
used at B.

(2)
P
0.8 m

A B C D Aluminum
(2)
Bronze
0.4 m 1.1 m 0.5 m 8 ft
(1)
6 ft
0.6 m
(1)
2.5 ft 1.5 ft

FIGURE P5.15
A B C
P5.16 In Figure P5.16, aluminum [E  70 GPa] links (1) and Aluminum 3 ft
(2) support rigid beam ABC. Link (1) has a cross-sectional area of (3)
300 mm2, and link (2) has a cross-sectional area of 450 mm2. For D
an applied load of P  55 kN, determine the rigid beam deflection
at point B.
P
FIGURE P5.17

P5.18 The truss shown in Figure P5.18 is constructed from three

aluminum alloy members, each having a cross-sectional area of
(2) A  850 mm2 and an elastic modulus of E  70 GPa. Assume that
a  4.0 m, b  10.5 m, and c  6.0 m. Calculate the horizontal dis-
4,000 mm
(1) placement of roller B when the truss supports a load of P  12 kN.
2,500 mm

A C
B C
1,400 mm 800 mm

P
c
FIGURE P5.16

A B
P5.17 Rigid bar ABC is supported by bronze rod (1) and alumi-
y
num rod (2), as shown in Figure P5.17. A concentrated load P is
applied to the free end of aluminum rod (3). Bronze rod (1) has an x
a b
elastic modulus of E1  15,000 ksi and a diameter of d1  0.50 in.
Aluminum rod (2) has an elastic modulus of E2  10,000 ksi and FIGURE P5.18

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P5.19 The rigid beam in Figure P5.19 is supported by links (1) steel with a modulus of elasticity of E  29,000 ksi and a yield
and (2), which are made from a polymer material [E  16 GPa]. strength of Y  36 ksi. For the tie rod, the minimum factor of
Link (1) has a cross-sectional area of 400 mm2, and link (2) has a safety with respect to yield is 1.5 and the maximum allowable
cross-sectional area of 800 mm2. Determine the maximum load P axial elongation is 0.30 in. Assume that a  21 ft, b  9 ft, and
that may by applied if the deflection of the rigid beam is not to c  27 ft.
exceed 20 mm at point C.
(a) Determine the minimum diameter required to satisfy both
constraints for tie rod (1).
(b) Draw a deformation diagram showing the final position of
joint B.

(2) A

1,250 mm

(1) a
A Rigid beam B C

600 mm 300 mm B
1,000 mm (2)
(1) P b
P

C
c
FIGURE P5.19
FIGURE P5.21

P5.20 The pin-connected assembly shown in Figure P5.20 P5.22 Two axial members are used to support a load of
consists of solid aluminum [E  70 GPa] rods (1) and (2) and solid
P  72 kips, as shown in Figure P5.22. Member (1) is 12-ft long, it
steel [E  200 GPa] rod (3). Each rod has a diameter of 16 mm.
has a cross-sectional area of A1  1.75 in.2, and it is made of struc-
Assume that a  2.5 m, b  1.6 m, and c  0.8 m. If the normal
tural steel [E  29,000 ksi]. Member (2) is 16-ft long, it has a
stress in any rod may not exceed 150 MPa, determine
cross-sectional area of A2  4.50 in.2, and it is made of an aluminum
(a) the maximum load P that may be applied at A. alloy [E  10,000 ksi].
(b) the magnitude of the resulting deflection at A.
(a) Compute the normal stress in each axial member.
(b) Compute the deformation of each axial member.
(c) Draw a deformation diagram showing the final position of
a b joint B.
(d) Compute the horizontal and vertical displacements of
C joint B.
(1)
c
P (3)
A B C
c
(2)
D (2)

FIGURE P5.20 55°

(1)

B
P5.21 A tie rod (1) and a pipe strut (2) are used to support A
a load of P  25 kips, as shown in Figure P5.21. Pipe strut (2)
has an outside diameter of 6.625 in. and a wall thickness of P
0.280 in. Both the tie rod and the pipe strut are made of structural FIGURE P5.22

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M5.7 Determine the internal forces and normal stresses in bars M5.8 Determine the internal forces and normal stresses in bars
(1) and (2). Also, determine the deflection of the rigid bar in the x (1) and (2). Also, determine the deflection of the rigid bar in the x
direction at C. direction at C.

FIGURE M5.7 FIGURE M5.8

PROBLEMS
P5.23 The 200  200  1,200-mm oak [E  12 GPa] block (2) P5.24 Two identical steel [E  200 GPa] pipes, each with a
shown in Figure P5.23 is reinforced by bolting two 6  200  cross-sectional area of 1,475 mm2, are attached to unyielding
1,200 mm steel [E  200 GPa] plates (1) to opposite sides of supports at the top and bottom, as shown in Figure P5.24/25. At
the block. A concentrated load of 360 kN is applied to a rigid cap. flange B, a concentrated downward load of 120 kN is applied.
Determine Determine
(a) the normal stresses in the steel plates (1) and the oak block (2). (a) the normal stresses in the upper and lower pipes.
(b) the shortening of the block when the load is applied. (b) the deflection of flange B.

P5.25 Solve Problem 5.24 if the lower support in Figure P5.24/25

yields and displaces downward 1.0 mm as the load P is applied.

360 kN

A
B (1)
3.0 m

1.2 m

120 kN
(1) (1) 3.7 m
(2) (2)
A C

FIGURE P5.23 FIGURE P5.24/25

122
P5.26 A composite bar is fabricated by brazing aluminum alloy P5.29 The concrete [E  29 GPa] pier shown in Figure P5.28/29
[E  10,000 ksi] bars (1) to a center brass [E  17,000 ksi] bar (2), is reinforced by four steel [E  200 GPa] reinforcing rods. If the
as shown in Figure P5.26. Assume that w  1.25 in., a  0.25 in., pier is subjected to an axial force of 670 kN, determine the required
and L  40 in. If the total axial force carried by the two aluminum diameter D of each rod so that 20% of the total load is carried by
bars must equal the axial force carried by the brass bar, calculate the steel.
the thickness b required for brass bar (2).

670 kN
250 mm

P L
250 mm
Aluminum (1)
a
Aluminum (1)
a
1.5 m
Brass (2)
b P
w

FIGURE P5.26

P5.27 An aluminum alloy [E  10,000 ksi] pipe with a cross- FIGURE P5.28/29
sectional area of A1  4.50 in.2 is connected at flange B to a steel
[E  30,000 ksi] pipe with a cross-sectional area of A2  3.20 in.2.
The assembly (shown in Figure P5.27) is connected to rigid sup- P5.30 A load of P  100 kN is supported by a structure consist-
ports at A and C. For the loading shown, determine ing of rigid bar ABC, two identical solid bronze [E  100 GPa]
(a) the normal stresses in aluminum pipe (1) and steel pipe (2). rods, and a solid steel [E  200 GPa] rod as shown in Figure P5.30.
(b) the deflection of flange B. Each of the bronze rods (1) has a diameter of 20 mm and is
symmetrically positioned relative to the center rod (2) and the
applied load P. Steel rod (2) has a diameter of 24 mm. All bars are
unstressed before the load P is applied; however, there is a 3-mm
clearance in the bolted connection at B. Determine
45 kips (a) the normal stresses in the bronze and steel rods.
(b) the downward deflection of rigid bar ABC.
(1) (2)
C
A B
45 kips

160 in. 220 in.

(1) (1)
FIGURE P5.27 (2) 3.0 m
1.5 m

P5.28 The concrete [E  29 GPa] pier shown in Figure P5.28/29

A B C
is reinforced by four steel [E  200 GPa] reinforcing rods, each
having a diameter of 19 mm. If the pier is subjected to an axial load
of 670 kN, determine
(a) the normal stress in the concrete and in the steel reinforcing rods. P
(b) the shortening of the pier. FIGURE P5.30

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P5.31 Two steel [E  30,000 ksi] pipes (1) and (2) are connected rod (2) has an outside diameter of 20 mm. The normal stress in the
at flange B, as shown in Figure P5.31. Pipe (1) has an outside dia- aluminum rod must be limited to 160 MPa, and the normal stress in
meter of 6.625 in. and a wall thickness of 0.28 in. Pipe (2) has an the bronze rod must be limited to 110 MPa. Determine
outside diameter of 4.00 in. and a wall thickness of 0.226 in. If the (a) the maximum downward load P that may be applied at
normal stress in each steel pipe must be limited to 18 ksi, determine flange B.
(a) the maximum downward load P that may be applied at flange B. (b) the deflection of flange B at the load that you determined in
(b) the deflection of flange B at the load that you determined in part (a).
part (a).
P5.33 A pin-connected structure is supported as shown in
Figure P5.33/34. Member ABCD is rigid and horizontal before
load P is applied. Bar (1) is made of brass [E  17  106 psi], and
it has a length of L1  3.5 ft. Bar (2) is made of an aluminum alloy
C [E  10  106 psi]. Bars (1) and (2) each have cross-sectional
(2) areas of 0.40 in.2. Assume that a  3.0 ft, b  4.0 ft, c  1.0 ft,
and P  4,000 lb. Determine the maximum length L2 that can be
16 ft used for bar (2) if the normal stress developed in bar (1) must not
exceed ½ of the normal stress in bar (2); that is, 1  0.52.
P5.34 A pin-connected structure is supported as shown in
Figure P5.33/34. Member ABCD is rigid and horizontal before
load P is applied. Bar (1) is made of brass [Y  18,000 psi; E 
B
17  106 psi], and it has a length of L1  8.0 ft. Bar (2) is made
of an aluminum alloy [Y  40,000 psi; E  10 × 106 psi] , and it
has a length of L2  5.5 ft. Bars (1) and (2) each have cross-
10 ft
P sectional areas of 0.75 in.2. Assume that a  4.0 ft, b  6.0 ft, and
c  1.5 ft. If the minimum factor of safety required for bars (1)
(1) and (2) is 2.50, calculate the maximum load P that can be applied
A to the rigid bar at D.

FIGURE P5.31

P5.32 A solid aluminum [E  70 GPa] rod (1) is connected to a

solid bronze [E  100 GPa] rod at flange B as shown in Figure P5.32.
Aluminum rod (1) has an outside diameter of 35 mm, and bronze L1 (1)
L2 (2)

A B C D

C
P
(2) a b c

340 mm FIGURE P5.33/34

B P5.35 The pin-connected structure shown in Figure P5.35/36

consists of a rigid beam ABCD and two supporting bars. Bar (1) is a
bronze alloy [E  105 GPa] with a cross-sectional area of A1 
175 mm P 290 mm2. Bar (2) is an aluminum alloy [E  70 GPa] with a cross-
(1) A sectional area of A2  650 mm2. If a load of P  30 kN is applied at
B, determine
(a) the normal stresses in both bars (1) and (2).
FIGURE P5.32 (b) the downward deflection of point A on the rigid bar.

124
P5.36 The pin-connected structure shown in Figure P5.35/36 P5.39 A load P is supported by a structure consisting of
consists of a rigid beam ABCD and two supporting bars. Bar (1) is rigid bar BDF and three identical 15-mm-diameter steel [E 
a bronze alloy [E  105 GPa] with a cross-sectional area of A1  200 GPa] rods, as shown in Figure P5.39. Use a  2.5 m,
290 mm2. Bar (2) is an aluminum alloy [E  70 GPa] with a cross- b  1.5 m, and L  3 m. For a load of P  75 kN, determine
sectional area of A2  650 mm2. All bars are unstressed before the (a) the tension force produced in each rod.
load P is applied; however, there is a 3-mm clearance in the pin (b) the vertical deflection of the rigid bar at B.
connection at A. If a load of P  85 kN is applied at B, determine
(a) the normal stresses in both bars (1) and (2).
(b) the downward deflection of point A on the rigid bar.

a a

(1)
A C E
2,250 mm
1,150 mm 650 mm
(1) (2) (3)
C D L

A B
480 mm B D F
1,600 mm
P (2) b

FIGURE P5.35/36 P
FIGURE P5.39

P5.37 A pin-connected structure is supported as shown in

Figure P5.37/38. Bar (1) is made of brass [Y  330 MPa; E 
105 GPa]. Bar (2) is made of an aluminum alloy [Y  275 MPa;
E  70 GPa]. Bars (1) and (2) each have cross-sectional areas of P5.40 A uniformly-distributed load w is supported by a struc-
225 mm2. Member ABCD is rigid. If the minimum factor of safety ture consisting of rigid bar BDF and three rods, as shown in
required for bars (1) and (2) is 2.50, calculate the maximum load P Figure P5.40. Rods (1) and (2) are 15-mm diameter stainless steel
that can be applied to the rigid bar at A. rods, each with an elastic modulus of E  193 GPa and a yield
P5.38 A pin-connected structure is supported as shown in strength of Y  250 MPa. Rod (3) is a 20-mm-diameter bronze
Figure P5.37/38. Bar (1) is made of brass [E  105 GPa], and bar (2) rod that has an elastic modulus of E  105 GPa and a yield strength
is made of an aluminum alloy [E  70 GPa]. Bars (1) and (2) each of Y  330 MPa. Use a  1.5 m and L  3 m. If a minimum fac-
have cross-sectional areas of 375 mm2. Rigid bar ABCD is supported tor of safety of 2.5 is specified for the normal stress in each rod,
by a pin in a double-shear connection at B. If the allowable shear calculate the maximum distributed load magnitude w that may be
stress for pin B is 130 MPa, calculate the minimum allowable dia- supported.
meter for the pin at B when P  42 kN.

P
880 mm a 2a

A B A C E
720 mm
260 mm

C (1) (2) (3)

(1) L

400 mm B D F
430 mm

D
(2) w
FIGURE P5.37/38 FIGURE P5.40

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P5.41 The pin-connected structure shown in Figure P5.41 con- with a cross-sectional area of A1  300 mm2 and a length of
sists of two cold-rolled steel [E  30,000 ksi] bars (1) and a bronze L1  720 mm. Link (2) is cold-rolled steel [Y  430 MPa; E 
[E  15,000 ksi] bar (2) that are connected at pin D. All three 210 GPa], with a cross-sectional area of A2  200 mm2 and a length
bars have cross-sectional areas of 0.375 in.2. A load of P  11 kips of L2  940 mm. A factor of safety of 2.5 with respect to yield is
is applied to the structure at pin D. Using a  3 ft and b  5 ft, specified for the normal stresses in links (1) and (2). Furthermore,
calculate the maximum horizontal displacement of the rigid bar at end D may
not exceed 2.0 mm. Calculate the magnitude of the maximum load
(a) the normal stresses in bars (1) and (2).
P that can be applied to the rigid bar at D. Use a  420 mm, b 
(b) the downward displacement of pin D.
420 mm, and c  510 mm.

a a D
P

L1 c
A B C
C
(2)
(1) (1) b (1)
L2 b

B
(2)
D
a

P A
FIGURE P5.41
FIGURE P5.43

P5.42 The pin-connected structure shown in Figure P5.42

consists of a rigid bar ABC, a steel bar (1), and a steel rod (2). The P5.44 A 4.5-m-long aluminum tube (1) is to be connected to a
cross-sectional area of bar (1) is A1  0.5 in.2, and its length is 2.4-m-long bronze pipe (2) at B. When put in place, however, a gap
L1  24 in. The diameter of rod (2) is d2  0.375 in., and its length of 8 mm exists between the two members as shown in Figure P5.44.
is L2  70 in. Assume that E  30,000 ksi for both axial members. Aluminum tube (1) has an elastic modulus of 70 GPa and a cross-
Using a  18 in., b  32 in., c  20 in., and P  7 kips, determine sectional area of 2,000 mm2. Bronze pipe (2) has an elastic modulus
of 100 GPa and a cross-sectional area of 3,600 mm2. If bolts are
(a) the normal stresses in bar (1) and rod (2). inserted in the flanges and tightened so that the gap at B is closed,
(b) the deflection of pin C from its original position. determine
(a) the normal stresses produced in each of the members.
(b) the final position of flange B with respect to support A.
(2)
L2
b
C

B (1)
c C 4.5 m
a P
(1)
A B
8 mm
L1 (2)
2.4 m
FIGURE P5.42
A

P5.43 Links (1) and (2) support rigid bar ABCD shown in
Figure P5.43. Link (1) is bronze [Y  330 MPa; E  105 GPa], FIGURE P5.44

126
P5.45 The assembly shown in Figure P5.45 consists of a steel of 2 mm between the rod flange at B and the tube closure at A. After
[E1  30,000 ksi; A1  1.25 in.2] rod (1), a rigid bearing plate B load P is applied, rod (2) stretches enough so that flange B contacts
that is securely fastened to rod (1), and a bronze [E2  15,000 ksi; the closed end of the tube at A. If the load applied to the lower end
A2  3.75 in.2] post (2). The yield strengths of the steel and bronze of the aluminum rod is P  230 kN, calculate
are 62 ksi and 75 ksi, respectively. A clearance of 0.125 in. exists (a) the normal stress in tube (1).
between the bearing plate B and bronze post (2) before the assem- (b) the elongation of tube (1).
bly is loaded. After a load of P  65 kips is applied to the bearing
plate, determine P5.47 A 0.5-in.-diameter steel [E  30,000 ksi] bolt (1) is
placed in a copper tube (2), as shown in Figure P5.47. The copper
(a) the normal stresses in bars (1) and (2).
[E  16,000 ksi] tube has an outside diameter of 1.00 in., a wall
(b) the factors of safety with respect to yield for each of the
thickness of 0.125 in., and a length of L  8.0 in. Rigid washers,
members.
each with a thickness of t  0.125 in., cap the ends of the copper
(c) the vertical displacement of bearing plate B.
tube. The bolt has 20 threads per inch. This means that each time
the nut is turned one complete revolution, the nut advances 0.05 in.
(i.e., 1/20 in.). The nut is hand-tightened on the bolt until the
bolt, nut, washers, and tube are just snug, meaning that all slack
has been removed from the assembly, but no stress has yet been
C induced. What stresses are produced in the bolt and in the tube if
the nut is tightened an additional quarter turn past the snug-tight
condition?
(1)
14 ft
P
— P
—
2 2
Rigid washer Rigid washer

B 0.125 in. Bolt (1)

3 ft (2)
A Tube (2)

t L t
FIGURE P5.45
FIGURE P5.47

P5.46 In Figure P5.46, the cutaway view shows a solid alumi-

num alloy [L2  600 mm; A2  707 mm2; E2  70 GPa] rod (2) P5.48 A hollow steel [E = 30,000 ksi] tube (1) with an outside
within a closed-end bronze [L1  610 mm; A1  1,206 mm2; E1  diameter of 3.50 in. and a wall thickness of 0.216 in. is fastened to
100 GPa] tube (1). Before the load P is applied, there is a clearance a solid 2-in.-diameter aluminum [E = 10,000 ksi] rod. The assem-
bly is attached to unyielding supports at the left and right ends and
is loaded as shown in Figure P5.48. Determine
Support
(a) the stresses in all parts of the axial structure.
C (b) the deflections of joints B and C.
15 mm 35 mm
610 mm (1) 29 mm

600 mm (2) 17 kips 13 kips

A B
Flange (2) C D
B (1)
(1) (2) (3)
Cross section
A
2 mm 17 kips 13 kips

4 ft 5 ft 5 ft
P
FIGURE P5.46 FIGURE P5.48

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PROBLEMS
P5.49 A 22-mm-diameter steel [E  200 GPa;   11.9  P5.52 A steel [E  29,000 ksi and   6.6  106/°F] rod
106/°C] bolt is used to connect two rigid parts of an assembly, as containing a turnbuckle has its ends attached to rigid walls. During
shown in Figure P5.49. The bolt length is a  150 mm. The nut is the summer when the temperature is 82°F, the turnbuckle is tight-
hand-tightened until it is just snug (meaning that there is no slack ened to produce a stress in the rod of 5 ksi. Determine the stress in
in the assembly, but there is no axial force in the bolt) at a tem- the rod in the winter when the temperature is 10°F.
perature of T  40°C. When the temperature drops to T  10°C,
determine
P5.53 A high-density polyethylene [E  120 ksi and   78 
106/°F] block (1) is positioned in a fixture, as shown in Figure P5.53.
(a) the clamping force that the bolt exerts on the rigid parts. The block is 2-in. by 2-in. square by 32-in.-long. At room temperature,
(b) the normal stress in the bolt. a gap of 0.10 in. exists between the block and the rigid support at B.
(c) the normal strain in the bolt. Determine
(a) the normal stress in the block caused by a temperature
increase of 100°F.
(b) the normal strain in block (1) at the increased temperature.

B
0.10 in.

FIGURE P5.49
(1)
32 in.

P5.50 A 25-mm-diameter by 3.5-m-long steel rod (1) is

stress free after being attached to rigid supports as shown in
Figure P5.50/51. At A, a 16-mm-diameter bolt is used to connect
A
the rod to the support. Determine the normal stress in steel rod (1)
and the shear stress in bolt A after the temperature drops 60°C. Use
E  200 GPa and   11.9  106/°C. FIGURE P5.53

P5.51 A 0.875-in.-diameter by 15-ft-long steel rod (1) is

stress free after being attached to rigid supports. A clevis-and-
bolt connection as shown in Figure P5.50/51 connects the rod P5.54 The assembly shown in Figure P5.54 consists of a brass
with the support at A. The normal stress in the steel rod must be shell (1) fully bonded to a ceramic core (2). The brass shell
limited to 18 ksi, and the shear stress in the bolt must be limited [E  115 GPa;   18.7  106/°C] has an outside diameter of
to 42 ksi. Assume that E  29,000 ksi and   6.6  106/°F, 50 mm and an inside diameter of 35 mm. The ceramic core [E 
and determine 290 GPa;   3.1  106/°C] has a diameter of 35 mm. At a tem-
perature of 15°C, the assembly is unstressed. Determine the largest
(a) the temperature decrease that can be safely accommodated by temperature increase that is acceptable for the assembly if the
rod (1) on the basis of the allowable normal stress. normal stress in the longitudinal direction of the brass shell must
(b) the minimum required diameter for the bolt at A, using the not exceed 80 MPa.
temperature decrease found in part (a).

200 mm

(1) (2) Ceramic core

A B Brass shell (1)

FIGURE P5.50/51 FIGURE P5.54

136
P5.55 At a temperature of 60°F, a 0.04-in. gap exists between P5.57 Rigid bar BCD is supported by a single steel [Y 
the ends of the two bars shown in Figure P5.55. Bar (1) is an alumi- 430 MPa; E  200 GPa;   11.7  106/°C] rod and two identi-
num alloy [E  10,000 ksi;   0.32;   12.5  106/°F] bar with cal aluminum [Y  275 MPa; E  70 GPa;   23.6  106/°C]
a width of 3 in. and a thickness of 0.75 in. Bar (2) is a stainless steel rods, as shown in Figure P5.57. Steel rod (1) has a diameter of
[E  28,000 ksi;   0.12;   9.6  106/°F] bar with a width of 18 mm and a length of a  3.0 m. Each aluminum rod (2) has a
2 in. and a thickness of 0.75 in. The supports at A and C are rigid. diameter of d2  25 mm and a length of b  1.5 m. If a factor of
Determine safety of 2.5 is specified for the normal stress in each rod, deter-
(a) the lowest temperature at which the two bars contact each mine the maximum temperature decrease that is allowable for this
other. assembly.
(b) the normal stress in the two bars at a temperature of
250°F.
(c) the normal strain in the two bars at 250°F. D (2)
(d) the change in width of the aluminum bar at a temperature F
of 250°F. 250 mm
A (1)
C
250 mm
(1) (2) E
(2)
B
a b
3 in. 2 in.
B C
A FIGURE P5.57

32 in. 44 in.
P5.58 The pin-connected structure shown in Figure P5.58
0.04-in. gap
consists of a rigid bar ABC, a solid bronze [E  100 GPa;  
FIGURE P5.55 16.9  106/°C] rod (1), and a solid aluminum alloy [E  70 GPa;
  22.5  106/°C] rod (2). Bronze rod (1) has a diameter of
24 mm, and aluminum rod (2) has a diameter of 16 mm. The bars
P5.56 An aluminum alloy cylinder (2) is clamped between rigid are unstressed when the structure is assembled at 25°C. After
heads by two steel bolts (1), as shown in Figure P5.56. The steel assembly, the temperature of rod (2) is decreased by 40°C, while the
[E  200 GPa;   11.7  10−6/°C] bolts have a diameter of 16 mm. temperature of rod (1) remains constant at 25°C. Determine the
The aluminum alloy [E  70 GPa;   23.6  10−6/°C] cylinder normal stresses in both rods for this condition.
has an outside diameter of 150 mm and a wall thickness of 5 mm.
Assume that a  600 mm and b  700 mm. If the temperature of this
assembly changes by T  50°C, determine
(a) the normal stress in the aluminum cylinder. 350 mm
(b) the normal strain in the aluminum cylinder. (1)
250 mm
(c) the normal strain in the steel bolts. B C

A
(1) 200 mm (2)
500 mm

(2) c

FIGURE P5.58
a c

(1)
P5.59 Rigid bar ABC is supported by two identical solid bronze
[E  100 GPa;   16.9  106/°C] rods, and a solid steel
b [E  200 GPa;   11.9  106/°C] rod as shown in Figure P5.59.
The bronze rods (1) each have a diameter of 16 mm, and they are
FIGURE P5.56 symmetrically positioned relative to the center rod (2) and the

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applied load P. Steel rod (2) has a diameter of 20 mm. The bars P  26 kips is applied and the temperature is increased to 100°F,
are unstressed when the structure is assembled at 30°C. When the determine
temperature decreases to 20°C, determine (a) the normal stresses in bars (1) and (2).
(a) the normal stresses in the bronze and steel rods. (b) the vertical deflection of joint D.
(b) the normal strains in the bronze and steel rods.

(2)
(1) (2) (1)
0.35 m (1) 8 ft
6 ft
A B C D
A B C

FIGURE P5.59
2.5 ft 3 ft 1.5 ft

P
FIGURE P5.61
P5.60 A steel [E  30,000 ksi;   6.6  106/°F] pipe col-
umn (1) with a cross-sectional area of A1  5.60 in.2 is connected
at flange B to an aluminum alloy [E  10,000 ksi;   12.5 
106/°F] pipe (2) with a cross-sectional area of A2  4.40 in.2.
P5.62 A cylindrical bronze sleeve (2) is held in compression
against a rigid machine wall by a high-strength steel bolt (1), as
The assembly (shown in Figure P5.60) is connected to rigid
shown in Figure P5.62. The steel [E  200 GPa;   11.7 
supports at A and C. It is initially unstressed at a temperature of
106/C] bolt has a diameter of 25 mm. The bronze [E  105 GPa;
90°F.
  22.0  106/C] sleeve has an outside diameter of 75 mm, a
(a) At what temperature will the normal stress in steel pipe (1) be wall thickness of 8 mm, and a length of L  350 mm. The end of
reduced to zero? the sleeve is capped by a rigid washer with a thickness of t  5 mm.
(b) Determine the normal stresses in steel pipe (1) and aluminum At an initial temperature of T1  8°C, the nut is hand-tightened on
pipe (2) when the temperature reaches 10°F. the bolt until the bolt, washers, and sleeve are just snug, meaning
that all slack has been removed from the assembly, but no stress has
yet been induced. If the assembly is heated to T2  80°C, calculate
(a) the normal stress in the bronze sleeve.
(b) the normal strain in the bronze sleeve.
30 kips

(1) (2)
Rigid machine wall
C
A B
30 kips

Sleeve (2)
120 in. 144 in.

FIGURE P5.60 Bolt (1)

t L

P5.61 A load P will be supported by a structure consisting of

a rigid bar ABCD, a polymer [E  2,300 ksi;   2.9  106/°F] FIGURE P5.62
bar (1), and an aluminum alloy [E  10,000 ksi;   12.5 
106/°F] bar (2) as shown in Figure P5.61. Each bar has a cross-
sectional area of 2.00 in.2. The bars are unstressed when the P5.63 The pin-connected structure shown in Figure P5.63
structure is assembled at 30°F. After a concentrated load of consists of a rigid bar ABCD and two axial members. Bar (1) is

138
steel [E  200 GPa;   11.7  10−6/°C], with a cross-sectional P5.65 Rigid bar ABCD is loaded and supported as shown in
area of A1  400 mm2. Bar (2) is an aluminum alloy [E  70 GPa; Figure P5.65. Bar (1) is made of bronze [E  100 GPa;   16.9 
  22.5  10−6/°C], with a cross-sectional area of A2  400 mm2. 10−6/°C] and has a cross-sectional area of 400 mm2. Bar (2) is
The bars are unstressed when the structure is assembled. After a made of aluminum [E  70 GPa;   22.5  10−6/°C] and has a
concentrated load of P  36 kN is applied and the temperature is cross-sectional area of 600 mm2. Bars (1) and (2) are initially un-
increased by 25°C, determine stressed. After the temperature has increased by 40°C, determine
(a) the normal stresses in bars (1) and (2). (a) the stresses in bars (1) and (2).
(b) the deflection of point D on the rigid bar. (b) the vertical deflection of point A.

900 mm

A
(1) (2)
350 mm

B 0.92 m
(2)
36 kN A B C D
600 mm

C D
1m 2m 1m

720 mm 0.84 m
(1)
FIGURE P5.63

FIGURE P5.65
P5.64 The pin-connected structure shown in Figure P5.64 con-
sists of two cold-rolled steel [E  30,000 ksi;   6.5  106/°F]
bars (1) and a bronze [E  15,000 ksi;   12.2  106/°F] bar
(2) that are connected at pin D. All three bars have cross-sectional
areas of 1.250 in.2. Assume an initial geometry of a  10 ft and P5.66 Three rods of different materials are connected and
b  18 ft. A load of P  34 kips is applied to the structure at pin D, placed between rigid supports at A and D, as shown in Figure
and the temperature increases by 60°F. Calculate P5.66/67. Properties for each of the three rods are given in the ac-
companying table. The bars are initially unstressed when the struc-
(a) the normal stresses in bars (1) and (2).
ture is assembled at 70°F. After the temperature has been increased
(b) the downward displacement of pin D.
to 250°F, determine
(a) the normal stresses in the three rods.
a a (b) the force exerted on the rigid supports.
(c) the deflections of joints B and C relative to rigid support A.

A B C
Aluminum (1) Cast Iron (2) Bronze (3)
(2)
b L1  10 in. L2  5 in. L3  7 in.
(1) (1)
A1  0.8 in.2 A2  1.8 in.2 A3  0.6 in.2
E1  10,000 ksi E2  22,500 ksi E3  15,000 ksi
D 1  12.5  10−6/°F 2  7.5  10−6/°F 3  9.4  10−6/°F

P P5.67 Three rods of different materials are connected and placed

FIGURE P5.64 between rigid supports at A and D, as shown in Figure P5.66/67.

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Properties for each of the three rods are given in the accompanying (a) the normal stresses in the three rods.
table. The bars are initially unstressed when the structure is assem- (b) the force exerted on the rigid supports.
bled at 20°C. After the temperature has been increased to 100°C, (c) the deflections of joints B and C relative to rigid support A.
determine

Aluminum (1) Cast Iron (2) Bronze (3)

Cast iron
Aluminum Bronze
(2)
(1) (3) L1  440 mm L2  200 mm L3  320 mm
A1  1,200 mm2 A2  2,800 mm2 A3  800 mm2
A B C D
E1  70 GPa E2  155 GPa E3  100 GPa
L1 L2 L3 1  22.5  10−6/°C 2  13.5  10−6/°C 3  17.0  10−6/°C

FIGURE P5.66/67

5.7 Stress Concentrations

In the preceding sections, it was assumed that the average stress, as determined by the
expression   P兾A, is the significant or critical stress. While this is true for many prob-
lems, the maximum normal stress on a given section may be considerably greater than the
average normal stress, and for certain combinations of loading and material, the maximum
rather than the average normal stress is the more important consideration. If there exists in
A stress trajectory is a line that the structure or machine element a discontinuity that interrupts the stress path (called a
is parallel to the maximum stress trajectory), the stress at the discontinuity may be considerably greater than the aver-
normal stress everywhere. age stress on the section (termed the nominal stress). This is termed a stress concentration
at the discontinuity. The effect of stress concentration is illustrated in Figure 5.12, in which
a type of discontinuity is shown in the upper figure and the approximate distribution of
normal stress on a transverse plane is shown in the accompanying lower figure. The
ratio of the maximum stress to the nominal stress on the section is known as the stress-
concentration factor K. Thus, the expression for the maximum normal stress in an axially
loaded member becomes

␴max ⫽ K␴nom (5.13)

Curves, similar to those shown in Figures 5.13, 5.14, and 5.15,1 can be found in nu-
merous design handbooks. It is important that the user of such curves (or tables of
factors) ascertain whether the factors are based on the gross or net section. In this book,
the stress-concentration factors K are to be used in conjunction with the nominal stresses
produced at the minimum or net cross-sectional area, as shown in Figure 5.12.
The K factors shown in Figures 5.13, 5.14, and 5.15 are based on the stresses at the net
section.

1 Adapted from Walter D. Pilkey, Peterson’s Stress Concentration Factors, 2nd ed. (New York: John Wiley &
Sons, Inc., 1997).

140