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1. For the plane truss shown in Figure P15–6, bar element 2
is subjected to a uniform temperature drop of T = 20 C. Let E = 70 GPa, A = 4
10
-2
m
2
, and a = 23
10
-6
(mm/mm)/
C. Determine the stresses in each bar and the displacement of node 1.
2. For the bar structure shown in Figure P15–7, element 1 is
subjected to a uniform temperature rise of T = 30 C. Let E = 210 GPa, A = 3
10
-2
m
2
, and a = 12
10
-6
(mm/mm)/
C. Determine the displacement of node 1 and the stresses in each bar.

A bar assemblage consists of two outer steel bars and an
inner brass bar. The threebar assemblage is then heated to raise the
temperature by an amount T = 40 F. Let all cross-sectional areas be A = 2 in
2
and L = 60 in., Esteel = 30
10
6
psi, Ebrass
= 15
10
6
psi, asteel = 6:5
10‑/
F, and abrass = 10
10
-6
/ F. Determine (a)
the displacement of node 2 and (b) the stress in the steel and brass bars. See
Figure P15–8.

For the plane truss shown in Figure P15–9, bar element 2 is
subjected to a uniform temperature rise of T = 10 C. Let E = 210 GPa, A = 12:5
cm
2
, and a = 12
10
-6
/ C. What
temperature change is needed in bars 1 and 3 to remove the stress due to the
uniform temperature rise in bar 2? Show enough work to prove your answer. Use a
longhand solution.

1. When do stresses occur in a body made of a single
material due to uniform temperature change in the body? Consider Problem 15.1
and also compare the solution to Example 15.1 in this chapter.
2. Consider two thermally incompatible materials, such as
steel and aluminum, attached together as shown in Figure P15–11. Will there be
temperature-induced stress in each material upon uniform heating of both
materials to the same temperature when the boundary conditions are simple
supports (a pin and a roller such that we have a statically determinate
system)? Explain. Let there be a uniform temperature rise of T = 50 F.

A bimetallic thermal control is made of cold-rolled yellow
brass and magnesium alloy bars (Figure P15–12). The bars are arranged with a
gap of 0.005 in. between them at 72F. The brass bar has a length of 1.0 in. and
a cross-sectional area of 0.10 in.2, and the magnesium bar has a length of 1.5
in. and a cross-sectional area of 0.15 in.2. Determine (a) the axial
displacement of the end of the brass bar and (b) the stress in each bar after
it has closed up due to a temperature increase of 100F. Use at least one
element for each bar in your finite element model.

For the plane stress element shown in Figure P15–13
subjected to a uniform temperature rise of T = 50 F, determine the thermal
force matrix f fTg.
Let E = 10
10
6
psi, n =
0:30, and a = 12:5
10
-6
(in./in.)/ F. The
coordinates (in inches) are shown in the figure. The element thickness is t = 1
in.

1. For the plane stress element shown in Figure P15–14
subjected to a uniform temperature rise of T =30 C, determine the thermal force
matrix ( fTg). Let E = 70 GPa, n = 0:3, a = 23
10
-6
(mm/mm)/ C, and t = 5 mm. The coordinates (in millimeters) are shown in the
figure.
2. For the plane stress element shown in Figure P15–15
subjected to a uniform temperature rise of T = 100 F, determine the thermal
force matrix (fT ). Let E = 30
10
6
psi, n =
0:3, a = 7:0
10
-6
(in./in.)/ F, and t
= 1 in. The coordinates (in inches) are shown in the figure.

For the plane stress element shown in Figure P15–16
subjected to a uniform temperature drop of T = 20 C, determine the thermal
force matrix (fT). Let E = 210 GPa, n = 0:25, and a = 12
10
-6
(mm/mm)/ C. The coordinates (in millimeters) are shown in the figure. The
element thickness is 10 mm.

For the plane stress plate fixed along the left and right
sides and subjected to a uniform temperature rise of 50 F as shown in Figure
P15–17, determine the stresses in each element. Let E = 10
10
6
psi, n = 0:30, a = 12:5
10
-6
(in./in.)/
F, and t = 1 4 in. The coordinates (in inches) are shown in the figure. (Hint:
The nodal displacements are all equal to zero. Therefore, the stresses can be
determined from {
} =-{DeT}.)