Category: Writing

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. 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. 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
.

For the linear-strain elements shown in Figure P8–5,
determine the strains e
x
;e
y
, and g
xy
. Evaluate
the stresses s
x
; s
y
, and t
xy
at the centroids.
The coordinates of the nodes are shown in units of inches. Let E = 30 10
6
psi, n = 0:25, and t = 0:25 in. for both elements. Assume plane stress
conditions apply. The nodal displacements are given as

1. For the element of Figure 8–3 (shown again as Figure
P8–3) subjected to the uniform pressure shown acting over the vertical side,
determine the nodal force replacement system using Eq. (6.3.7). Assume an
element thickness of t.
2. For the element of Figure 8–3 (shown as Figure P8–4)
subjected to the linearly varying line load shown acting over the vertical
side, determine the nodal force replacement system using Eq. (6.3.7). Compare
this result to that of Problem 6.11. Are these results expected? Explain.

1. Evaluate the shape functions given by Eq. (8.2.6). Sketch
the variation of each function over the surface of the triangular element shown
in Figure 8–3.
2. Express the strains e
x
;e
y
, and g
xy
for the element of Figure 8–3 by using the results given in Section 8.2.
Evaluate these strains at the centroid of the element; then evaluate the
stresses at the centroid in terms of E and n. Assume plane stress conditions
apply

For the tapered solid circular shaft shown in Figure P9–24,
a semicircular groove has been machined into the side. The shaft is made of a
hot rolled 1040 steel alloy with yield strength of 71,000 psi. The shaft is
subjected to a uniform axial pressure of 4000 psi. Determine the maximum
principal stresses and von Mises stresses at the fillet and at the semicircular
groove. Is the shaft safe from failure based on the maximum distortion energy theory?

For the various parts shown in Figure P7–39 on the next page
determine the best one to relieve stress. Make the original part have a small
radius of 0.1 in. at the inside reentrant corners. Place a uniform pressure
load of 1000 psi on the right end of each part and fix the left end. All units
shown are taken in inches. Let the material be A36 steel.