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For the tripod space frame shown in Figure P5–72, determine
standard steel pipe sections such that the maximum bending stress must not
exceed 20 ksi, the compressive stress to prevent buckling must not exceed that
given by the Euler buckling formula with a factor of safety of 2, and the
maximum deflection will not exceed L/360 in any span, L. Assume the three
bottom supports to be fixed. All coordinates shown in units of inches.

The curved semi-circular frame shown in Figure P5–73 is
supported by a pin on the left end and a roller on the right end and is
subjected to a load P ¼ 1000 lb at its apex. The frame has a radius to
centerline cross section of R ¼ 120 in. Select a structural steel W shape from
Appendix F such that the maximum stress does not exceed 20 ksi. Perform a
finite element analysis using 4, 8, and then 16 elements in your finite element
model. Also, determine the maximum deflection for each model. It is suggested
that the finite element answers for deflection be compared to the solution
obtained by classical methods, such as using Castigliano’s theorem. The expression
for deflection under the load is given by using Castigliano’s theorem as
where A is the cross sectional area of the W shape, Av is
the shear area of the W shape (use depth of web times thickness of web for the
shear area); E = 30
10
6
psi; and G = 11:5
106
psi:
Now change the radius of the frame to 20 in. and repeat the
problem. Run the finite element model with the shear area included in your
computer program input and then without. Comment on the difference in results
and compare to the predicted analytical deflection by using the equation above
for d
y
:

1. Sketch the variations of the shape functions Nj and Nm,
given by Eqs. (6.2.18), over the surface of the triangular element with nodes
i; j, and m. Check that Ni + N
j
N
m
+= 1 anywhere on the
element.
2.
For a simple three-noded triangular element, show
explicitly that differentiation of Eq. (6.2.47) indeed results in Eq. (6.2.48);
that is, substitute the expression for ½B and the plane stress condition for
[D] into Eq. (6.2.47), and then differentiate
p
with respect to each nodal degree of freedom in Eq. (6.2.47) to obtain Eq.
(6.2.48).

1. Evaluate the stiffness matrix for the elements shown in Figure P6–3. The coordinates are in units of inches. Assume plane stress conditions. Let E = 30
10
6
psi, n = 0.25, and thickness t = 1 in.
2.
For the elements given in Problem 6.3, the nodal displacements are given as
Determine the element stresses s
x
; s
y
; t
xy
; s
1
, and s
2
and the principal angle y
p
. Use the values of E; n, and t given in Problem 6.3.

1. Evaluate the stiffness matrix for the elements shown in Figure P6–6. The coordinates are given in units of millimeters. Assume plane stress conditions. Let E = 210 GPa, n = 0.25, and t = 10 mm.
2.
For the elements given in Problem 6.6, the nodal displacements are given as
u
1
= 2.0 mm v
1
= 1.0 mm u
2
= 0.5 mm
v
2
= 0.0 mm u
3
= 3.0 mm v
3
= 1.0 mm
Determine the element stresses s
x
; s
y
; t
xy
; s
1
, and s
2
and the principal angle y
p
. Use the values of E; n, and t given in Problem 6.6.

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.

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.

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.

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.)

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.