Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

polymer [E = 370 ksi; α = 39.0 × 10 −6 /°F] cylinder has an outside
diameter of 6.625 in. and a wall thickness of 0.432 in. Assume that
a = 24 in. and b = 28 in. If the temperature of this assembly changes
by DT = 120°F, determine
(a) the normal stress in the polymer cylinder.
(b) the normal strain in the polymer cylinder.
(c) the normal strain in the steel bolts.
p5.51 A load P will be supported by a structure consisting of a
rigid bar ABCD, a steel [E = 29,000 ksi; α = 6.5 × 10 −6 /°F] bar (1),
and an aluminum alloy [E = 10,000 ksi; α = 12.5 × 10 −6 /°F] bar (2),
as shown in Figure P5.51. The bars have cross-sectional areas A 1 =
0.80 in. 2 and A 2 = 1.30 in. 2 , respectively. Dimensions of the structure
are a = 3 ft, b = 3.75 ft, c = 5.0 ft, L 1 = 12 ft, and L 2 = 20 ft. The
bars are unstressed when the structure is assembled. After a concentrated
load P = 35 kips is applied and the temperature has been
increased by 60°F, determine
(a) the normal stresses in bars (1) and (2).
(b) the vertical deflection of joint D.
A
L 1
Rigid bar
(1)
L 2
B C
a b c
FIGURE p5.51
P
(2)
D
Calculate (a) the normal stresses in the aluminum and titanium bars
and (b) the vertical displacement of pin D when the temperature of
the assembly reaches 85°C.
b
(1)
A B C
a
FIGURE p5.52
(2)
p5.53 The pin-connected assembly shown in Figure P5.53 consists
of a rigid bar ABC, a cast iron [E = 24,400 ksi; α = 6.0 ×
10 −6 /°F] bar (1), and an aluminum alloy [E = 10,000 ksi; α = 13.1 ×
10 −6 /°F] bar (2). Bar (1) has cross-sectional area A 1 = 0.50 in. 2 and
length L 1 = 18 in. Bar (2) has cross-sectional area A 2 = 1.60 in. 2 and
length L 2 = 45 in. Dimensions of the assembly are a = 25 in. and
b = 80 in. The bars are unstressed when the structure is assembled
at 75°F. After assembly, the temperature of bar (2) is decreased by
90°F while the temperature of bar (1) remains constant at 75°F.
Determine the normal strains in both bars for this condition.
D
a
(1)
p5.52 The pin-connected assembly shown in Figure P5.52 consists
of two aluminum alloy [E = 69 GPa; α = 23.6 × 10 −6 /°C] bars
(1) and a titanium alloy [E = 114 GPa; α = 9.5 × 10 −6 /°C] bar (2)
that are connected at pin D. Each of the aluminum bars has crosssectional
area A 1 = 300 mm 2 , and the titanium bar has cross-sectional
area A 2 = 500 mm 2 . Dimensions for the assembly are a = 1.80 m
and b = 2.50 m. All bars are unstressed at a temperature of 25°C.
L 2
(1)
L 1
Rigid bar
A
B
a
b
FIGURE p5.53
(2)
C
5.7 Stress concentrations
In the preceding sections, it was assumed that the average normal stress, defined as s = P A,
is the significant or critical stress in an axial member. While this is true for many problems,
the maximum normal stress on a given section may be substantially 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
the structure or machine element a discontinuity that interrupts the stress path (called a
stress trajectory), the stress at the discontinuity may be considerably greater than the
A stress trajectory is a line that
is parallel to the maximum
normal stress everywhere.
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