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For the space trusses shown in Figures P3–40 and P3–41,
determine the nodal displacements and the stresses in each element. Let E = 210
GPa and A = 10 10
4
m
2
for all elements. Verify force
equilibrium at node 1. The coordinates of each node, in meters, are shown in
the figure. All supports are ball-and-socket joints.

For the space truss subjected to a 1000-lb load in the x
direction, as shown in Figure P3–42, determine the displacement of node 5. Also
determine the stresses in each element. Let A = 25 10
4
m
2
and E = 2 10
11
N/m
2
for all elements. The coordinates of
each node, in inches, are shown in the figure. Nodes 1–4 are supported by
ball-and-socket joints (fixed supports).

For the space truss subjected to the 4000-lb load acting as
shown in Figure P3–43, determine the displacement of node 4. Also determine the
stresses in each element. Let A = 6 in
2
and E = 30 10
6
psi for all elements. The coordinates of each node, in inches, are shown in the
figure. Nodes 1–3 are supported by ball-and-socket joints (fixed supports).

1. Derive Eq. (3.7.21) for stress in space truss elements by
a process similar to that used to derive Eq. (3.5.6) for stress in a plane
truss element.
2. For the truss shown in Figure P3–46, use symmetry to
determine the displacements of the nodes and the stresses in each element. All
elements have E = 30 10
6
psi. Elements 1, 2, 4, and 5 have A = 10 in
2
and element 3 has A = 20 in
2
.

For the truss shown in Figure P3–45, use symmetry to
determine the displacements of the nodes and the stresses in each element. All
elements have E = 200 GPa. Elements 1, 2, 4, and 5 have A = 10 10
4
m
2
and element 3 has A = 20 10
4
m
2
. Let dimension a = 2 m
and P = 40 kN. The supports at nodes 1 and 4 are defined.

All elements of the structure in Figure P3–47 have the same
AE except element 1, which has an axial stiffness of 2AE. Find the
displacements of the nodes and the stresses in elements 2, 3, and 4 by using
symmetry. Check equilibrium at node 4. You might want to use the results
obtained from the stiffness matrix of Problem 3.24.

1. For the roof truss shown in Figure P3–48, use symmetry to
determine the displacements of the nodes and the stresses in each element. All
elements have E = 210 GPa and A = 10 10
4
m
2
.
2. For the plane trusses with inclined supports shown in
Figures P3–49 through P3–51, solve for the nodal displacements and element
stresses in the bars. Let A = 2 in2, E = 30 10
6
psi, and L = 30 in.
for each truss.

Use the principle of minimum potential energy developed in Section 3.10 to solve the bar problems shown in Figure P3–52. That is, plot the total potential energy for variations in the displacement of the free end of the bar to determine the minimum potential energy. Observe that the displacement that yields the minimum potential energy also yields the stable equilibrium position. Use displacement increments of 0.002 in., beginning with x = 0:004. Let E = 30 10
6
psi and A = 2 in
2
for the bars.

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.

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.