Question 1. The asymmetric I-beam shown is made of A992 Gr. 50 steel (f_y = 50 ksi).

a. Assuming Elastic-Perfectly Plastic behavior and the same properties in tension and compression, determine the fully plastic moment.

b. If the beam is curved and subjected to the uniform moment, M, shown below. Determine the circumferential stresses (σ_θθ) in the beam at points A and B.

Question 2. The non-symmetric aluminum (E = 70 GPa) channel section shown is used as a simply-supported beam spanning 3 m that must support a vertical load of 800 N (Φ = 90°) at the center of the beam. The maximum moment for this span configuration is M = 600 N-m and is directed as shown on the cross-section. The beam section properties are given below the figure.

a. Determine the principal moments of inertia, I_max and I_min, and the orientation of the principal axes, θ_p.
b. Determine the orientation of the neutral axis, α.
c. Determine the maximum tensile and the maximum compressive stresses in the beam.

Question 3. A fabrication error resulted in an I-beam with an off-center web causing the shear center to no longer be located at the centroid of the web. The cross-section of the I-beam is shown. Because the I-beam still has one axis of symmetry, the x- and y-axes are principal axes. Therefore, when a vertical shear force, V, is applied, the x-axis is the neutral axis.

a. For a vertical shear force of V = 1 kN, determine the shear flow distribution in each segment of the beam. Also, determine the total shear force in each segment of the beam clearly showing the direction of the resultant shear force in each section.

b. Determine the location of the shear center with respect to the centroid of the web, e, for the beam. You should use your results from part a to locate the shear center and verify this result by calculating the location of the shear center using the correct equation from Table 8.1 in the textbook (slides 12 and 13 in lecture 8b).

Question 4. A large crane is supported on steel (E = 29,000 ksi) rails on an elastic foundation. Each wheel carries a load of 100 kip. The steel rails are 6 in. wide at the base, have a moment of inertia of I = 1,200 in⁴ and a section modulus of S = 130 in³. The elastic foundation consists of concrete sleepers over a thick gravel base over soil which results in an effective spring constant of k₀ = 300 psi/in.

a. Determine the maximum displacement of the rail, maximum moment in the rail, and maximum normal stress in the rail due to a single wheel load if the rail and elastic foundation can be considered to be infinite.

b. Determine the maximum displacement of the rail, maximum moment in the rail, and maximum normal stress in the rail, when a single wheel load is placed at the end of a semi-infinite rail.

c. If the foundation is to be modeled with discrete springs, determine the minimum spacing of the discrete springs necessary for "reasonable" results. Also, determine the spring constant, K, that should be used for each discrete spring.