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1. For the beam shown in Figure 4–17, determine the displacement and slope at the center and the reactions. The load is symmetrical with respect to the center of the beam. Assume EI constant throughout the beam.
2. For the beam subjected to the linearly varying line load w shown in Figure P4–18, determine the right-end rotation and the reactions. Assume EI constant throughout the beam.

1. For the beams shown in Figures P4–19 through P4–24,
determine the nodal displacements and slopes, the forces in each element, and
the reactions.
2. For the beams shown in Figures P4–25 through P4–30,
determine the maximum deflection and maximum bending stress. Let E = 200 GPa or
30
10
6
psi for all beams as
appropriate for the rest of the units in the problem. Let c be the half-depth
of each beam.
For the beam design problems shown in Figures P4–31 through
P4–36, determine the size of beam to support the loads shown, based on requirements
listed next to each beam.

1. Design a beam of ASTM A36 steel with allowable bending
stress of 160 MPa to support the load shown in Figure P4–31. Assume a standard
wide flange beam from Appendix F, or some other source can be used.
2. Select a standard steel pipe from Appendix F to support
the load shown. The allowable bending stress must not exceed 24 ksi, and the
allowable deflection must not exceed L/360 of any span.

1. Select a rectangular structural tube from Appendix F to
support the loads shown for the beam in Figure P4–33. The allowable bending
stress should not exceed 24 ksi.
2. Select a standard W section from Appendix F or some other
source to support the loads shown for the beam in Figure P4–34. The bending
stress must not exceed 160 MPa.

1. For the beam shown in Figure P4–35, determine a suitable
sized W section from Appendix F or from another suitable source such that the
bending stress does not exceed 150 MPa and the maximum deflection does not
exceed L/360 of any span.
2. For the stepped shaft shown in Figure P4–36, determine a
solid circular cross section for each section shown such that the bending
stress does not exceed 160 MPa and the maximum deflection does not exceed L/360
of the span.

1. For the beam shown in Figure P4–37 subjected to the
concentrated load P and distributed load w, determine the midspan displacement
and the reactions. Let EI be constant throughout the beam.
2. For the beam shown in Figure P4–38 subjected to the two
concentrated loads P, determine the deflection at the midspan. Use the
equivalent load replacement method. Let EI be constant throughout the beam.

1. For the beam shown in Figure P4–39 subjected to the concentrated load P and the linearly varying line load w, determine the free-end deflection and rotation and the reactions. Use the equivalent load replacement method. Let EI be constant throughout the beam.
2. For the beams shown in Figures P4–40 through P4–42, with internal hinge, determine the deflection at the hinge. Let E = 210 GPa and I = 2
10
-4
m
4
.

1. Derive the stiffness matrix for a beam element with a
nodal linkage—that is, the shear is 0 at node i, but the usual shear and moment
resistance are present at node j (see Figure P4–43).
2. Develop the stiffness matrix for a fictitious pure shear
panel element (Figure P4–44) in terms of the shear modulus, G, the shear web
area, AW , and the length, L. Notice that Y and v are the shear force and
transverse displacement at each node, respectively.

1. Explicitly evaluate pp of E
q
. (4.7.15); then
differentiate pp with respect to v
1
;
1
;
v
2
, and v
2
and set each of these equations to zero (that
is, minimize pp) to obtain the four element equations for the beam element.
Then express these equations in matrix form.
2. Determine the free-end deflection for the tapered beam
shown in Figure P4–46. Here I(x) = I
0
(1 + n/x=L) where I0 is the
moment of inertia at x = 0. Compare the exact beam theory solution with a
two-element finite element solution for n = 7[12.

1. Derive the equations for the beam element on an elastic
foundation (Figure P4–47) using the principle of minimum potential energy. Here
kf is the subgrade spring constant per unit length. The potential energy of the
beam is
2. Derive the equations for the beam element on an elastic
foundation (see Figure P4–47) using Galerkin’s method. The basic differential
equation for the beam on an elastic foundation is