Category: Writing

Consider the rectangular plate shown (Figure P7–11) in plane
stress. Using a computer program, verify that the plane stress element of the
code satisfied the patch test. That is, apply constant displacement of u =
0:005 m to the right-side nodes, 3, 6, and 9, and determine the displacement at
interior node 5. Use E = 200 GPa, v = 0:3, and plate thickness of 0.1 m.
Explain your results.

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.

Consider the bar with two elements shown in Figure P7.10.
Perform a patch test using these two elements. Let E = 200 GPa, and A = 1
10
4
m
2
. Use the standard bar element stiffness matrix (Eq. (3.1.14))
derived using the shape functions N
1
(x) = x/L and N
2
(x) =
1 x/L.
a. For the rigid body motion test, set u
1
= 1 m
and u
3
= 1 m and verify that u
2
= 1 m by using the direct
stiffness method.
b. For the constant strain test, assume linear displacement
function u(x) = x for the nodes at the boundaries, such that u
1
= 0
and u
3
= 2 m, and verify that u
2
= 0:6 m.
Solve the following problems using a computer program. In
some of these problems, we suggest that students be assigned separate parts (or
models) to facilitate parametric studies.

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.

1. Which one of the CST models shown in Figure P7–7 is
expected to give the best results for a cantilever beam subjected to an end
shear load? Why?
2. In considering the patch test, answer the following
questions:
a. Can elements of different mechanical properties be used?
Why?
b. Can the patch be arbitrary in shape? Why?
c. Can we mix triangular and quadrilateral elements in the
patch test?
d. Can we mix bar elements with triangular or quadrilateral
elements? Why?
e. When should the patch test be applied?

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.)

1. For the finite element mesh shown in Figure P7–1, comment
on the appropriateness of the mesh. Indicate the mistakes in the model. Explain
and show how to correct them.
2. Comment on the mesh sizing in Figure P7–2. Is it
reasonable? If not, explain why not
3. When do problems occur using the smoothing (averaging of
stress at the nodes from elements connected to the node) method for obtaining
stress results?

1. Determine the principal stresses in the blade implant and
the bony material shown in Figure P7–29 on page 431. Let E
blade
= 20
GPa, v
blade
= 0:30, E
bone
= 12 GPa, and v
bone
= 0:35. Assume plane stress conditions with t = 5 mm.
2. For the 0.5 in. thick canopy hook shown in Figure P7–31
on page 432, used to hold down an aircraft canopy, determine the maximum von
Mises stress and maximum deflection. The hook is subjected to a concentrated
upward load of 22,400 lb as shown. Assume boundary conditions of fixed supports
over the lower half of the inside hole diameter. The hook is made from AISI
4130 steel, quenched and tempered at 400F. (This problem is compliments of Mr.
Steven Miller.)

For the rectangular element of Figure 6–20 on page 367 the
nodal displacements are given by
u
1
= 0 in: v
1
= 0 in: u
2
=
0:005 in:
v
2
= 0:0025 in: u
3
= 0:0025 in: v
3
= 0:0025 in:
u
4
= 0 in: v
4
= 0 in:
For b = 2 in., h = 1 in., E = 30
10
6
psi, and n = 0:3, determine the element strains and stresses at the centroid of
the element and at the corner nodes.

1. For the concrete dam shown subjected to water pressure in
Figure P7–27, determine the principal stresses. Let E = 3:5
10
6
psi and n = 0:15. Assume plane strain conditions. Perform the analysis for
self-weight and then for hydrostatic (water) pressure against the dam vertical
face as shown.
2. Determine the stresses in the wrench shown in Figure
P7–28 on the next page. Let E = 200 GPa and n = 0:25, and assume uniform
thickness t = 10 mm.