Author: admin

Can you use the plane stress or plane strain element to
model the following:
a. a flat slab floor of a building with vertical loading
perpendicular to the slab
b. a wall subjected to wind loading (the wall acts as a
shear wall with loads in the plane of the wall)
c. a tensile plate with a hole drilled transversely through
it
d. an eyebar with loads in the plane of the eyebar
e. a soil mass subjected to a strip footing loading
f. a wrench subjected to a force in the plane of the wrench
g. a wrench subjected to twisting forces (the twisting
forces act out of the plane of the wrench)
h. a triangular plate connection with loads in the plane of
the triangle
i. a triangular plate connection with out-of-plane loads

1. The plane stress element only allows for in-plane
displacements, while the frame or beam element resists displacements and
rotations. How can we combine the plane stress and beam elements and still
ensure compatibility?
2. Compute the stiffness matrix of element 1 of the two-triangle
element model of the rectangular plate in plane stress shown in Figure P6–23.
Then use it to compute the stiffness matrix of element 2.

For the plane structures modeled by triangular elements
shown in Figure P6–20, show that numbering in the direction that has fewer
nodes, as in Figure P6–20(a) (as opposed to numbering in the direction that has
more nodes), results in a reduced bandwidth. Illustrate this fact by filling
in, with X’s, the occupied elements in ½K for each mesh, as was done in
Appendix B.4. Compare the bandwidths for each case.

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 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. 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?

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.

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.