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1. Evaluate the nodal forces used to replace the linearly
varying surface traction shown in Figure P9–2. Hint: Use Eq. (9.1.34).
2. For an element of an axisymmetric body rotating with a
constant angular velocity o = 20 rpm as shown in Figure P9–3, evaluate the
body-force matrix. The coordinates of the element are shown in the figure. Let
the weight density p
w
be 0.283 lb/in
3
.

1. For the elements shown in Figure P9–6, evaluate the
stiffness matrices using Eq. (9.2.2). The coordinates (in millimeters) are
shown in the figures. Let E = 210 GPa and n = 0:25 for each element.
2.
For the axisymmetric elements shown in Figure
P9–7, determine the element stresses. Let E = 210 GPa and n = 0:25. The
coordinates (in millimeters) are shown in the figures, and the nodal
displacements for each element are

1. How would you evaluate the circumferential strain, e
y
,
at r = 0? What is this strain in terms of the a’s given in Eq. (9.1.15). Hint:
Elasticity theory tells us that the radial strain must equal the
circumferential strain at r = 0.
2.
The soil mass in Figure P9–13 is loaded by a force
transmitted through a circular footing as shown. Determine the stresses in the
soil. Compare the values of s
r
using an axisymmetric model with the
sy values using a plane stress model. Let E = 3000 psi and n = 0:45 for the
soil mass.

1. Perform a stress analysis of the pressure vessel shown in Figure P9–14. Let E = 5
10
6
psi and n = 0.15 for the concrete, and let E = 29 10
6
psi and n = 0.25 for the steel liner. The steel liner is 2 in. thick. Let the pressure p equal 500 psi. Use a 0.5 in. radius in the re-entrant corners.
2. Perform a stress analysis of the disk shown in Figure P9–16 if it rotates with constant angular velocity of ? = 50 rpm. Let E = 30 10
6
psi, n = 0.25, and the weight density ?
w
= 0.283 lb/in
3
(mass density, ? = ?
g
/ 386 in./s
2
). (Use 8 and then 16 elements symmetrically modeled similar to Example 9.4. Compare the finite element solution to the theoretical circumferential and radial stresses given by

1. For the die casting shown in Figure P9–17, determine the
maximum stresses and their locations. Let E = 30
10
6
psi and n = 0:25. The dimensions are shown in the figure.
2.
For the axisymmetric connecting rod shown in
Figure P9–18, determine the stresses s
z
; s
r
; s
y
,
and t
rz
. Plot stress contours (lines of constant stress) for each of
the normal stresses. Let E = 30
10
6
psi and n =
0:25. The applied loading and boundary conditions are shown in the figure. A
typical discretized rod is shown in the figure for illustrative purposes only.

For the thick-walled open-ended cylindrical pipe subjected
to internal pressure shown in Figure P9–19, use five layers of elements to
obtain the circumferential stress, sy, and the principal stresses and maximum
radial displacement. Compare these results to the exact circumferential stress
and radial displacement equations given by
where
P
i
= inner pressure, P0 = outer pressure (set to
zero in this problem)
a = inner radius of vessel, b = outer radius of vessel, r =
any radial location
Let E = 205 GPa and n = 0.3.

A steel cylindrical pressure vessel with flat plate end caps
is shown in Figure P9–20 with vertical axis of symmetry. Addition of thickened
sections helps to reduce stress concentrations in the corners. Analyze the
design and identify the most critically stressed regions. Note that inside
sharp re-entrant corners produce infinite stress concentration zones, so
refining the mesh in these regions will not help you get a better answer unless
you use an inelastic theory or place small fillet radii there. Recommend any
design changes in your report. Let the pressure inside be 1000 kPa.

For the cylindrical vessel with hemispherical ends (heads)
under uniform internal pressure of intensity p = 500 psi shown in Figure P9–21,
determine the maximum von Mises stress and where it is located. The material is
ASTM—A242 quenched and tempered alloy steel. Use a factor of safety of 3
against yielding. The inner radius is a = 100 inches and the thickness t = 2
in.