Author: admin

1. For the bar fixed at both ends and subjected to the
uniformly distributed loading shown in Figure P3–56, determine the displacement
at the middle of the bar and the stress in the bar. Let A = 2 in
2
and E = 30 10
6
psi.
2. For the bar hanging under its own weight shown in Figure
P3–57, determine the nodal displacements using (a) two equal-length elements
and (b) four equal-length elements. Let A = 2 in
2
, E = 30 10
6
psi, and weight density p
w
= 0:283 lb/in
3
. (Hint: The
internal force is a function of x. Use the potential energy approach.)

1. Determine the energy equivalent nodal forces for the
axial distributed loading shown acting on the bar elements in Figure P3–58.
2. Solve Problem 3.55 for the axial displacement in the bar
using collocation, subdomain, least squares, and Galerkin’s methods. Choose a
quadratic polynomial u(x) = c
1
x + c
2
x
2
in each
method. Compare these weighted residual method solutions to the exact solution.

For the tapered bar shown in Figure P3–60 with cross
sectional areas A
1
= 2 in.
2
and A
2
= 1 in.
2
at each end, use the collocation, subdomain, least squares, and Galerkin’s
methods to obtain the displacement in the bar. Compare these weighted residual
solutions to the exact solution. Choose a cubic polynomial u(x) = c
1
x
+ c
2
x
2
+ c
3
x
3
.

For the bar shown in Figure P3–61 subjected to the linear
varying axial load, determine the displacements and stresses using (a) one and
then two finite element models and (b) the collocation, subdomain, least
squares, and Galerkin’s methods assuming a cubic polynomial of the form u(x) =
c
1
x + c
2
x
2
+ c
3
x
3
.

Use a computer program to solve the truss design problems
shown in Figures P3–62 through P3–70. Determine the single most critical
cross-sectional area based on maximum allowable yield strength or buckling
strength (based on either Euler’s or Johnson’s formula as relevant) using a
factor of safety (FS) listed next to each truss. Recommend a common structural
shape and size for each truss. List the largest three nodal displacements and
their locations. Also include a plot of the deflected shape of the truss and a
principal stress plot.

1. Use Eqs. (4.1.7) to plot the shape functions N
1
and N
3
and the derivatives (dN
2
=dx) and (dN
4
=dx),
which represent the shapes (variations) of the slopes
1
and
2
over the length of the
beam element.
2. Derive the element stiffness matrix for the beam element
in Figure 4–1 if the rotational degrees of freedom are assumed positive
clockwise instead of counter clockwise. Compare the two different nodal sign
conventions and discuss. Compare the resulting stiffness matrix to Eq.
(4.1.14).

1. For the beam shown in Figure P4–3, determine the rotation
at pin support A and the rotation and displacement under the load P. Determine
the reactions. Draw the shear force and bending moment diagrams. Let EI be
constant throughout the beam.
2. For the cantilever beam subjected to the free-end load P
shown in Figure P4–4, determine the maximum deflection and the reactions. Let
EI be constant throughout the beam.

For the fixed-fixed beam subjected to the uniform load w
shown in Figure P4–12, determine the midspan deflection and the reactions. Draw
the shear force and bending moment diagrams. The middle section of the beam has
a bending stiffness of 2EI; the other sections have bending stiffnesses of EI.

Determine the midspan deflection and the reactions and draw
the shear force and bending moment diagrams for the fixed-fixed beam subjected
to uniformly distributed load w shown in Figure P4–13. Assume EI constant
throughout the beam. Compare your answers with the classical solution (that is,
with the appropriate equivalent joint forces given in Appendix D).

1. For the beam loaded as shown in Figure P4–15, determine the free-end deflection and the reactions and draw the shear force and bending moment diagrams. Assume EI constant throughout the beam.
2. Using the concept of work equivalence, determine the nodal forces and moments (called equivalent nodal forces) used to replace the linearly varying distributed load shown in Figure P4–16.