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The machine shown in Figure P7–33 on page 433 is an overload
protection device that releases the load when the shear pin S fails. Determine
the maximum von Mises stress in the upper part ABE if the pin shears when its
shear stress is 40 MPa. Assume the upper part to have a uniform thickness of 6
mm. Assume plane stress conditions
for the upper part. The part is made of 6061 aluminum alloy.
Is the thickness sufficient to prevent failure based on the maximum distortion
energy theory? If not, suggest a better thickness. (Scale all dimensions as
needed.)

The steel triangular plate
in. thick shown in Figure P7–34 on
page 433 is bolted to a steel column with 3 4-in.-diameter bolts in the pattern
shown. Assuming the column and bolts are very rigid relative to the plate and
neglecting friction forces between the column and plate, determine the highest
load exerted on any bolt. The bolts should not be included in the model, just
fix the nodes around the bolt circles and consider the reactions at these nodes
as the bolt loads. If 3 4-in.-diameter bolts are not sufficient, recommend
another standard diameter. Assume a standard material for the bolts. Compare
the reactions from the finite element results to those found by classical
methods.

A
in. thick machine part supports an
end load of 1000 lb as shown in Figure P7–35. Determine the stress
concentration factors for the two changes in geometry located at the radii
shown on the lower side of the part. Compare the stresses you get to classical
beam theory results with and without the change in geometry, that is, with a
uniform depth of 1 in. instead of the additional material depth of 1.5 in.
Assume standard mild steel is used for the part. Recommend any changes you
might make in the geometry.

A plate with an off-center hole is shown in Figure P7–36.
Determine how close to the top edge the hole can be placed before yielding of
the A36 steel occurs (based on the maximum distortion energy theory). The
applied tensile stress is 10,000 psi, and the plate thickness is
in. Now if the plate is made of
6061-T6 aluminum alloy with a yield strength of 37 ksi, does this change your
answer? If the plate thickness is changed to
in., how does this change the
results? Use same total load as when the plate is
in. thick.

One arm of a crimper tool shown in Figure P7–37 is to be
designed of 1080 as-rolled steel. The loads and boundary conditions are shown
in the figure. Select a thickness
for the arm based on the material not yielding with a factor
of safety of 1.5. Recommend any other changes in the design. (Scale any other
dimensions that you need.) Remember stresses at concentrated loads are false.

Design the bicycle wrench with the approximate dimensions
shown in Figure P7–38. If you need to change dimensions explain why. The wrench
should be made of steel or aluminum alloy. Determine the thickness needed based
on the maximum distortion energy theory. Plot the deformed shape of the wrench
and the principal stress and von Mises stress. The boundary conditions are
shown in the figure, and the loading is shown as a distributed load acting over
the right part of the wrench. Use a factor of safety of 1.5 against yielding.
Round each corner with a 0.1 cm radius.

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.

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

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.

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