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For the two-story, two-bay rigid frame shown in Figure P5–40,
determine (1) the nodal displacement components and (2) the shear force and
bending moments in each member. Let E = 200 GPa;I = 2
10
-4
m
4
for each horizontal member and I = 1:5
10
4
m
4
for each vertical member.

For the two-story, three-bay rigid frame shown in Figure
P5–41, determine (1) the nodal displacements and (2) the member end shear
forces and bending moments. (3) Draw the shear force and bending moment
diagrams for each member. Let E = 200 GPa; I = 1:29
10
-4
m
4
for the beams and I = 0:462
10
-4
m
4
for
the columns. The properties for I correspond to a W 610
155
and a W 410
114 wide-flange section,
respectively, in metric units.

For the rigid frame shown in Figure P5–42, determine (1) the
nodal displacements and rotations and (2) the member shear forces and bending
moments. Let E = 200 GPa, I = 0:795
10
-4
m
4
for the horizontal members, and I = 0:316
10
-4
m
4
for
the vertical members. These I values correspond to a W 460
158
and a W 410
85 wide-flange section,
respectively.

For the rigid frame shown in Figure P5–43, determine (1) the
nodal displacements and rotations and (2) the shear force and bending moments
in each member. Let E = 29
106 psi, I = 3100 in:
4
for the horizontal members and I = 1110 in:
4
for the vertical
members. The I values correspond to a W 24
104 and a W 16
77:

A structure is fabricated by welding together three lengths
of I-shaped members as shown in Figure P5–44. The yield strength of the members
is 36 ksi, E = 29
106 psi, and Poisson’s ratio is 0.3.
The members all have cross-section properties corresponding to a W18
76.
That is, A = 22:3 in
2
, depth of section is d = 18:21 in., Ix = 1330
in:
4
S
x
= 146 in: 3 Iy = 152 in:
4
and Sy =
27:6 in:
3
Determine whether a load of Q = 10;000 lb downward is
safe against general yielding of the material. The factor of safety against
general yielding is to be 2.0. Also, determine the maximum vertical and
horizontal deflections of the structure.

For the tapered beam shown in Figure P5–45, determine the
maximum deflection using one, two, four, and eight elements. Calculate the
moment of inertia at the midlength station for each element. Let E = 30
10
6
psi, I0 = 100 in
4
, and L = 100 in. Run cases where n = 1; 3, and 7.
Use a beam element. The analytical solution for deflection and slope at the
free end for n = 7 is given by Reference [7] as shown below:

1. Derive the stiffness matrix for the non prismatic torsion
bar shown in Figure P5–46. The radius of the shaft is given by
2. Derive the total potential energy for the prismatic
circular cross-section torsion bar shown in Figure P5–47. Also determine the
equivalent nodal torques for the bar subjected to uniform torque per unit
length (lb-in./in.). Let G be the shear modulus and J be the polar moment of
inertia of the bar.

1. For the grid shown in Figure P5–48, determine the nodal
displacements and the local element forces. Let E = 30
10
6
psi, G = 12
10
6
psi, I = 200 in.
4
,
and J = 100 in.
4
for both elements.
2.
For the grids shown in Figures P5–50 and P5–51,
determine the nodal displacements and the local element forces. Let E = 210
GPa, G = 84 GPa, I =2
10
-4
m
4
,
J = 1
10
-4
m
4
, and A
= 1
10
2
m
2
.

Solve the grid structures shown in Figures P5–52 through
P5–57 using a computer program. For grids P5–52—P5–54, let E = 30
10
6
psi, G = 12
10
6
psi, I = 200 in.
4
,
and J = 100 in.
4
, except as noted in the figures. In Figure P5–54,
let the cross elements
have I = 50 in.
4
and J = 20 in.
4
, with
dimensions and loads as in Figure P5–53. For grids P5–55 through P5–57, let E =
210 GPa, G = 84 GPa, I = 2
104 m
4
, J =1
10
4
m
4
, and A = 1
10
-2
m
2
.

1. Determine the displacements and reactions for the space
frames shown in Figures P5–58 and P5–59. Let I
x
= 100 in.
4
,
I
y
= 200 in.
4
, I
z
= 1000 in.
4
, E =
30,000 ksi, G = 10,000 ksi, and A = 100 in.
2
for both frames.
2.
Design a jib crane as shown in Figure P5–60 that
will support a downward load of 6000 lb. Choose a common structural steel shape
for all members. Use allowable stresses of 0:66Sy (S
y
is the yield
strength of the material) in bending, and 0:60S
y
in
tension on gross areas. The maximum deflection should not
exceed 1=360 of the length of the horizontal beam. Buckling should be checked
using Euler’s or Johnson’s method as applicable.