Category: Writing

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.

Determine the size of a solid round shaft such that the
maximum angle of twist between C and B is 0.26 degrees per meter of length and
the deflection of the beam is less than 0.005 inches under the pulley C for the
loads, as shown in Figure P5–66. Assume simple supports at bearings A and B.
Assume the shaft is made from cold-rolled AISI 1020 steel. (Recommended angles
of twist in driven shafts can be found in Machinery’s Handbook, Oberg, E., et.
al., 26th ed., Industrial Press, N.Y., 2000.)

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

A small hydraulic floor crane as shown in Figure P5–65
carries a 5000-lb load. Determine the size of the beam and column needed.
Select either a standard box section or a wide-flange section. Assume a rigid
connection between the beam and column. The column is rigidly connected to the
floor. The allowable bending stress in the beam is 0:60Sy. The allowable
deflection is 1=360 of the beam length. Check the column for buckling.

Determine the nodal displacements and the element stresses,
including principal stresses, due to the loads shown for the thin plates in
Figure P6–14 on the next page. Use E = 210 GPa, n = 0:30, and t = 5 mm. Assume
plane stress conditions apply. The recommended discretized plates are shown in
the figures.

A piston ring (with a split as shown in Figure P5–64) is to
be expanded by a tool to facilitate its installation. The ring is sufficiently
thin (0.2 in. depth) to justify using conventional straight-beam bending
formulas. The ring requires a displacement of 0.1 in. at its separation for
installation. Determine the force required to produce this separation. In
addition, determine the largest stress in the ring. Let E = 18
10
6
psi, G = 7
106 psi, cross-sectional area A =
0:06 in.
2
, and principal moment of inertia I = 4:5
10
-4
in.
4
. The inner radius is 1.85 in., and the outer radius is 2.15 in.
Use models with 4, 6, 8, 10, and 20 elements in a symmetric model until
convergence to the same results occurs. Plot the displacement versus the number
of elements for a constant force F predicted by the conventional beam theory
equation of Reference [8].

Determine the nodal displacements and the element stresses, including principal stresses, for the thin plate of Section 6.5 with a uniform shear load (instead of a tensile load) acting on the right edge, as shown in Figure P6–13. Use E = 30
10
6
psi, n = 0:30, and t = 1 in. (Hint: The =K matrix derived in Section 6.5 and given by Eq. (6.5.22) can be used to solve the problem.)

1. Determine the nodal forces for (1) a linearly varying
pressure px on the edge of the triangular element shown in Figure P6–11(a); and
(2) the quadratic varying pressure shown in Figure P6–11(b) by evaluating the
surface integral given by Eq. (6.3.7). Assume the element thickness is equal to
t.
2. Determine the nodal forces for (1) the quadratic varying
pressure loading shown in Figure P6–12(a) and (2) the sinusoidal varying
pressure loading shown in Figure P6–12(b)
by the work equivalence method (use the surface integral
expression given by Eq. (6.3.7)). Assume the element thickness to be t.

For the plane strain elements shown in Figure P6–10, the
nodal displacements are given as
u
1
= 0:005 mm v
1
= 0:002 mm u
2
= 0:0 mm
v
2
= 0:0 mm u
3
= 0:005 mm v
3
= 0:0 mm
Determine the element stresses s
x
; s
y
;
t
xy
; s
1
, and s
2
and the principal angle y
p
.
Let E = 70 GPa and n = 0:3, and use unit thickness for plane strain. All
coordinates are in millimeters.

For the plane strain elements shown in Figure P6–9, the
nodal displacements are given as
u
1
= 0:001 in: v
1
= 0:005 in: u
2
= 0:001 in:
v
2
= 0:0025 in: u
3
= 0:0 in: v
3
= 0:0 in:
Determine the element stresses s
x
; s
y
;
t
xy
; s
1
, and s
2
and the principal angle y
p
.
Let E = 30
10
6
psi and n = 0:25, and
use unit thickness for plane strain. All coordinates are in inches.