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1. The square plate in Figure P15–22 is subjected to uniform
heating of 80 F. Determine the nodal displacements and element stresses. Let
the element thickness be t = 0:1 in., E = 30
10
6
psi, n = 0:33, and a = 10
10
-6
/ F.
2. The square plate in Figure P15–23 has element 1 made of
steel with E = 30
10
6
psi, n = 0:33, and a
= 10
10
-6
/ F and element 2
made of a material with E = 15
10
6
psi, n =
0:25, and a = 50
10
-6
/ F. Let the plate
thickness be t = 0:1 in. Determine the nodal displacements and element stresses
for element 1 subjected to an 80 F temperature increase and element 2 subjected
to a 50 F temperature increase.

The aluminum tube shown in Figure P15–26 fits snugly into a
hole (with surrounding material aluminum) at room temperature. If the
temperature of the tube is then increased by 40C, determine the deformed
configuration and the stress distribution of the tube. Let E = 70 GPa, n =
0:33, and a = 23
10
-6
/ C for the tube.

For the solid model of a fixture shown in Figure P15–27, the
inside surface of the hole is subjected to a temperature increase of 80 C. The
right end surfaces are fixed. Determine the von Mises stresses throughout the
fixture due to this temperature increase. What is the largest von Mises stress?
Is it a concern against yielding of the material? Assume the material is AISI
1020 cold-rolled steel.

A rectangular slab of concrete in a highway as shown in
Figure P15–29 can be considered to have sides parallel to the road free and
sides perpendicular to the road simply supported. The critical buckling stress
is shown to be
where for medium strength concrete we assume E = 3:5
10
6
psi and v = 0:20. Let the slab dimensions be length a = 120 in., width b = 96
in., and thickness t = 6 in. Determine the uniform temperature increase in the
slab that will buckle it. Use a finite element code to solve this problem.

1. Determine the consistent-mass matrix for the
one-dimensional bar discretized into two elements as shown in Figure P16–1. Let
the bar have modulus of elasticity E, mass density r, and cross-sectional area
A.
2. For the one-dimensional bar discretized into three
elements as shown in Figure P16–2, determine the lumped- and consistent-mass
matrices. Let the bar properties be E; r, and A throughout the bar.

For the one-dimensional bar shown in Figure P16–3, determine
the natural frequencies of vibration, o’s, using two elements of equal length.
Use the consistentmass approach. Let the bar have modulus of elasticity E, mass
density r, and crosssectional area A. Compare your answers to those obtained
using a lumped-mass matrix in Example 16.3.

1. For the one-dimensional bar shown in Figure P16–4, determine
the natural frequencies of longitudinal vibration using first two and then
three elements of equal length. Let the bar have E = 30
10
6
psi, r = 0:00073 lb-s2/in
4
, A=1 in
2
, and L = 60 in.
2. For the spring-mass system shown in Figure P16–5, determine
the mass displacement, velocity, and acceleration for five time steps using the
central difference method. Let k = 2000 lb/ft and m = 2 slugs. Use a time step
of Dt = 0:03 s. You might want to write a computer program to solve this
problem.

1. For the spring-mass system shown in Figure P16–6,
determine the mass displacement, velocity, and acceleration for five time steps
using (a) the central difference method, (b) Newmark’s time integration method,
and (c) Wilson’s method. Let k = 1200 lb/ft and m = 2 slugs.
2. For the bar shown in Figure P16–7, determine the nodal
displacements, velocities, and accelerations for five time steps using two
finite elements. Let E = 30
10
6
psi, r =
0:00073 lb-s2/in
4
, A = 1 in
2
, and L = 100 in.