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For the cylindrical vessel with ellipsoidal heads shown in
Figure P9–22a under loading p = 500 psi, determine if the vessel is safe
against yielding. Use the same material and factor of safety as in Problem
9.21. Now let a = 100 in. and b = 50 in. Which vessel has the lowest hoop
stress? Recommend the preferred head shape of the two based on your answers.
For modeling purposes, the equation of an ellipse is given
by b
2
x
2
+ a
2
y
2
= a
2
b
2
,
where a is the major axis and b is the minor axis of the ellipse shown in
Figure P9–22(b).

The syringe with plunger is shown in Figure P9–23. The
material of the syringe is glass with E = 69 GPa, n = 0:15, and tensile
strength of 5 MPa. The bottom hole is assumed to be closed under test
conditions. Determine the deformation and stresses in the glass. Compare the
maximum principal stress in the glass to the ultimate tensile strength. Do you
think the syringe is safe? Why?

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?

A steel hole punch is shown in Figure P9–25a. Investigate
the proper material for a hole punch. Tell me what material you used and why?
Model the punch without the side groove (Figure P9–25b) and with the side
groove (Figure P9–25c). Determine the von Mises stress distribution throughout
the punch for both cases. Are the punches safe under the loading shown?

Using the three-noded bar element shown in Figure P10–5 (a)
and (b), with shape functions given by Eq. (10.5.9), determine (a) the
intrinsic coordinate s at point A and (b) the shape functions, N
1
, N
2
,
and N3 at point A. For the displacements of the nodes shown in Figure P10–6,
determine (c) the displacement at A and (d) the axial strain expression in the
element.

For the bar subjected to the linearly varying axial line
load shown in Figure P10–7, use the linear strain (three-noded element) with
two elements in the model, to determine the nodal displacements and nodal
stresses. Compare your answer with that in Figure 3–31 and Eqs. (3.11.6) and
(3.11.7). Let A = 2 in.
2
and E = 30 10
6
psi. Hint: Use
Eq. (10.5.22) for the element sti¤ness matrix.

1. Use the three-noded bar element and find the axial
displacement at the end of the rod shown in Figure P10–8. Determine the stress
at x = 0, x = L=2 and x = L. Let A = 2 10
-4
m
2
, E = 205
GPa, and L = 4 m. Hint: Use Eq. (10.5.22) for the element sti¤ness matrix.
2.
For the element shown in Figure P10–14, replace
the distributed load with the energy equivalent nodal forces by evaluating a
force matrix similar to Eq. (10.2.29). Let h = 0:1 in thick. The global
coordinates (in inches) are shown in Figure P10–14.