Civil Engineering Reference
In-Depth Information
The axisymmetric element in Figure P9.16 is subjected to a uniform, normal
pressure
p
0
acting on the surface defined by nodes 1 and 3. Compute the
equivalent nodal forces.
9.17
The axisymmetric element in Figure P9.16 is part of a body rotating about
the
z
axis at a constant rate of 3600 revolutions per minute. Determine the
corresponding nodal forces.
9.18
Consider the higher-order three-dimensional element shown in Figure P9.19,
which is assumed to be subjected to a general state of stress. The element has
20 nodes, but all nodes are not shown for clarity.
a.
9.19
What is the order of the polynomial used for interpolation functions?
b.
How will the strains (therefore, stresses) vary with position in the element?
c.
What is the size of the stiffness matrix?
d.
What advantages and disadvantages are apparent in using this element in
comparison to an eight-node brick element?
Figure P9.19
9.20
Show that in a uniaxial tension test the distortion energy at yielding is given by
Equation 9.128.
9.21
A finite element analysis of a certain component yields the maximum principal
stresses
1
=
200 MPa,
2
=
0,
3
=−
90 MPa
.
If the tensile strength of the
material is 270 MPa, is yielding indicated according to the distortion energy
theory? If not, what is the “safety factor” (ratio of yield strength to equivalent
stress)?
9.22
Repeat Problem 9.21 if the applicable failure theory is the maximum shear stress
theory.
9.23
The torsion problem as developed in Section 9.9 has a governing equation
analogous to that of two-dimensional heat conduction. The stress function is
analogous to temperature, and the angle of twist per unit length term (
2
G
) is
analogous to internal heat generation.
a.
What heat transfer quantities are analogous to the shear stress components
in the torsion problem?
b.
If one solved a torsion problem using finite element software for two-
dimensional heat transfer, how would the torque be computed?
9.24
The torsion problem as developed in Section 9.9 is two-dimensional when posed
in terms of the Prandtl stress function. Could three-dimensional elastic solid
elements (such as the eight-node brick element) be used to model the torsion
problem? If yes, how would a pure torsional loading be applied?
9.25
Figure P9.25 shows the cross section of a hexagonal shaft used in a quick-change
power transmission coupling. The shear modulus of the material is 12
×
10
6
psi
