Author: admin

Determine the Jacobian matrix [J] and its determinant for
the elements shown in Figure P10–11. Show that the determinant of [J] for
rectangular and parallelogram shaped elements is equal to A/4, where A is the
physical area of the element and 4 actually represents the area of the
rectangle of sides 2 2 when b = 1 and h = 1 in Figure 6–20.

1. Use Gaussian quadature with two and three Gauss points
and Table 10–2 to evaluate the following integrals:
Then use the Newton-Cotes quadrature with two and three
sampling points and Table 10–1 to evaluate the same integrals. Compare your
results.
2.
For the quadrilateral elements shown in Figure
P10–16, write a computer program to evaluate the sti¤ness matrices using
four-point Gaussian quadrature as outlined in Section 10.4. Let E = 30 10
6
psi and n = 0:25. The global coordinates (in inches) are shown in the figures.

1. For the rectangular element of Figure 10–20 with the
nodal displacements given by
determine the fsg matrix at s = 0, t = 0 using the
isoparametric formulation described in Section 10.4. (Also see Example 10.5.)
Let E = 30 10
6
psi and n = 0:3.
2.
For the three-noded bar (Figure P10–1), what
Gaussian quadrature rule (how many Gauss points) would you recommend to
evaluate the sti¤ness matrix? Why?

1. Evaluate the stiffness matrix for the elements shown in
Figure P11–1. Let E = 30
10
6
psi and n =
0:3:
2.
For the elements shown in Figure P11–1, assume the
nodal displacements have been determined to be u
Determine the strains and then the stresses in the elements.
Let E = 30
10
6
psi and n = 0:3.
3. Show that for constant body force Z
b
acting on
an element (X
b
= 0 and Y
b
= 0),
where (f
bi
) represents the body forces at node i
of the element with volume V.

1. For the elements shown in Figure P11–6, assume the nodal
displacements have been determined to be
Determine the strains and then the stresses in the elements.
Let E = 210 GPa and n = 0:3.
2.
Figure P11–9 shows how solid and plane elements
may be connected. What restriction must be placed on the externally applied
loads for this connection to be acceptable?

For the linear strain tetrahedral element shown in Figure
P11–8, (a) express the displacement fields u, v, and w in the x, y and z
directions, respectively. Hint: There are 10 nodes each with three
translational degrees of freedom, u
i
, v
i
, and w
i
.
Also look at the linear strain triangle given by Eq. (8.1.2) or the expansion
of Eqs. (11.2.2).

1. Express the explicit shape functions N2 through N8,
similar to N1 given by Eq. (11.3.4), for the linear hexahedral element shown in
Figure 11–5 on page 546.
2. Determine the deflections at the four corners of the free
end of the structural steel cantilever beam shown in Figure P11–13. Also
determine the maximum principal stress. Compare your answer for deflections to
the classical beam theory equation (d = PL
3
/(3EI).

1. Write a computer program to evaluate [k of Eq. (11.3.9a)
using a 2
2
2 Gaussian quadrature rule.
2.
A portion of a structural steel brake pedal in a
vehicle is modeled as shown in Figure P11–14. Determine the maximum deflection
at the pedal under a uniform pressure acting over the pedal totaling 100 N.

For the compressor flap valve shown in Figure P11–15,
determine the maximum operating pressure such that the material yield stress is
not exceeded with a factor of safety of two. The valve is made of hardened 1020
steel with a modulus of elasticity of 30 million psi and a yield strength of
62,000 psi. The valve thickness is a uniform 0.018 in. The valve clip ears
support the valve at opposite diameters. The pressure load is applied uniformly
around the annular region.