Category: Writing

For the symmetric plane truss shown in Figure P3–35,
determine (a) the deflection of node 1 and (b) the stress in element 1. AE=L
for element 3 is twice AE=L for the other elements. Let AE=L = 10
6
lb/in. Then let A = 1 in
2
, L = 10 in., and E = 10 10
6
psi
to obtain numerical results.

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
.

1. For the plane trusses supported by the spring at node 1
in Figure P3–33 (a) and (b), determine the nodal displacements and the stresses
in each element. Let E = 210 GPa and A = 5:0 10
4
m
2
for
both truss elements.
2. For the plane truss shown in Figure P3–34, node 2 settles
an amount d = 0:05 in. Determine the forces and stresses in each element due to
this settlement. Let E = 30 10
6
psi and A = 2 in
2
for
each element.

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.

1. Remove element 1 from Figure P3–30 and solve the problem.
Compare the displacements and stresses to the results for Problem 3.30.
2. For the plane truss shown in Figure P3–32, determine the
nodal displacements, the element forces and stresses, and the support
reactions. All elements have E = 70 GPa and A = 3:0 10
4
m
2
.
Verify force equilibrium at nodes 2 and 4. Use symmetry in your model.

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. Show that for the transformation matrix [T] of Eq.
(3.4.15), [T]
T
= [T]
1
and
hence Eq. (3.4.21) is indeed correct, thus also illustrating that [k]=[ T]
T
k’
[T] is the expression for the global stiffness matrix for an element.
2. For the plane trusses shown in Figures P3–29 and P3–30,
determine the horizontal and vertical displacements of node 1 and the stresses
in each element. All elements have E = 210 GPa and A ¼ 4:0 10
4
m
2
.

For the bar subjected to the uniform line load in the axial
direction shown in Figure P3–55, determine the nodal displacements and axial
stress distribution using (a) two equal-length elements and (b) four
equal-length elements. Compare the finite element results with an exact
solution. Let A = 2 in
2
and E = 30 10
6
psi.

1. Now remove the element connecting nodes 2 and 4 in Figure P3–24. Then determine the nodal displacements and element forces.
2. Determine the displacement components at node 3 and the element forces for the plane truss shown in Figure P3–27. Let A = 3 in
2
and E = 30 106 psi for all elements. Verify force equilibrium at node 3.

1. Derive the stiffness matrix for the non prismatic bar
shown in Figure P3–53 using the principle of minimum potential energy. Let E be
constant.
2. For the bar subjected to the linear varying axial load
shown in Figure P3–54, determine the nodal displacements and axial stress
distribution using (a) two equal-length elements and (b) four equal-length
elements. Let A = 2 in.
2
and E =30 10
6
psi. Compare the
finite element solution with an exact solution.