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

(a) Determine the normal strains e x and e z and the average normal
stress σ y at the ambient temperature.
(b) Determine the normal strains e x and e z and the average normal
stress σ y at a temperature of 150°C.
(c) At what temperature will the average normal stress in the y
direction be reduced to zero?
p13.56 A thin aluminum alloy [E = 69 GPa; ν = 0.33;
α = 23.6 × 10 −6 /°C] plate with dimensions a = 1,700 mm and
b = 1,000 mm is set in a rigid frictionless cavity as shown in Figure
P13.56. At room temperature, there is a gap of c = 2 mm between
the rigid cavity and three sides of the plate as shown in the figure.
The plate is not constrained in the z direction. After the temperature
of the plate has been raised by 140°C, what are the normal stresses
σ x and σ y in the plate?
Rigid
b
Plate
y
a
x
Gap c
(typ.)
σ z
FIGURE p13.60
p13.61 The titanium [E = 16,500 ksi; ν = 0.33] block shown in
Figure P13.61/62/63/64 has dimensions a = 6 in., b = 4 in., and c =
2 in. The block is subjected to triaxial stresses σ x = −45 ksi, σ y =
−25 ksi, and σ z = 15 ksi, acting on the x, y, and z faces, respectively.
Determine (a) the changes ∆a, ∆b, and ∆c in the dimensions of the
block and (b) the change ∆V in the volume of the block.
c
σ y
a
z
y
σ x
x
FIGURE p13.56
Rigid
y
b
p13.57 At a point on the free surface of a ductile cast iron [E =
168 GPa; ν = 0.32; α = 10.8 × 10 −6 /°C] machine part, the measured
strains resulting from both a temperature decrease of 65°C and externally
applied loads are e x = −370 me, e y = −715 me, and γ xy = 0.
Determine the stresses σ x and σ y at the point.
p13.58 A thin plate (σ z = 0) of an aluminum alloy [E = 10,000 ksi;
ν = 0.33; α = 13.1 × 10 −6 /°F] is stretched until the strains in the x
and y directions are 0.0010 in./in. and 0.0015 in./in., respectively.
Then, the plate is rigidly held in the deformed position and heated
until its temperature has increased by 80°F. Determine the final
stresses in the plate.
p13.59 A thin plate (σ z = 0) of an aluminum alloy [E = 10,000 ksi;
ν = 0.33; α = 13.1 × 10 −6 /°F] in the x–y plane is heated from an
ambient temperature of 65°F to a final temperature of 350°F. Then,
the plate is rigidly clamped in position so that it is restrained in
both the x and y directions. Determine the absolute maximum shear
stress in the plate when its temperature cools down to the ambient
temperature.
p13.60 The principal stresses at a point are σ x = 92 MPa,
σ y = 78 MPa, and σ z = 66 MPa, acting as shown in Figure P13.60.
The material is red brass, for which E = 115 GPa and ν = 0.307.
Determine
(a) the principal strains.
(b) the dilatation.
z
x
FIGURE p13.61/62/63/64
p13.62 The malleable cast iron [E = 26,000 ksi; ν = 0.27] block
shown in Figure P13.61/62/63/64 has dimensions a = 9 in., b =
6 in., and c = 3 in. The block is subjected to normal stresses σ y =
−19 ksi and σ z = −12 ksi; however, the block is constrained at its ends
against displacement in the x direction. What normal stress develops
in the x direction, and what are the strains in the y and z directions?
p13.63 The polymer [E = 3.7 GPa; ν = 0.33; α = 85 × 10 −6 /°C]
block shown in Figure P13.61/62/63/64 has dimensions a = 400 mm,
b = 250 mm, and c = 50 mm. The block is subjected to a uniform
temperature increase of 45°C. The block is completely free to expand
in the y and z directions but is constrained at its ends against displacement
in the x direction. Determine (a) the changes ∆b and ∆c in the
dimensions of the block and (b) the normal stress in the x direction.
p13.64 The polymer [E = 3.7 GPa; ν = 0.33; α = 85 × 10 −6 /°C]
block shown in Figure P13.61/62/63/64 has dimensions a = 400 mm,
b = 250 mm, and c = 50 mm. The block is subjected to a uniform
temperature increase of 45°C. The block is completely free to
expand in the z direction but is constrained at its ends against
displacement in the x and y directions. Determine (a) the normal
stresses in the x and y directions and (b) the change in the length of
the block in the z direction.
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