ES 64 July 25, 2015Third Long Exam

1. A beam is loaded in such a way that a certain point experiences stresses with the following normal and shear stresses, x = 20 MPa tensile, y = 25 MPa compressive, and xy = 12 MPa. Using Mohrs circle, determine:(a) the principal stresses and the orientation in which the stresses exist,(b) the maximum in-plane shear stress and its orientation, and(c) the shear and normal stresses when = 34 measured counterclockwise from the vertical axis.(100 points)

2. A 8 m long steel tube is loaded as shown, having Fx = 10 kN, Fy = 20 kN, and Fz = 17 kN. Inner and outer radii are 100 mm and 120 mm, respectively. Determine (a) the maximum tensile and compressive stresses and the maximum shear stress at the support, and (b) the stress block at points a and b at the support. (E = 200 GPa, = 0.28, y = 300 MPa)(150 points)

3. Based on the stress blocks of points a and b in the previous item, solve for their principal stresses (eigenvalues) and their respective eigenvectors.(150 points)

4. A 10 m steel W-beam is loaded as shown. Determine the maximum deflection, ymax, and its position, x, along the beam. F1 = F2 = 3 kN, F3 = 4kN, w = 2 kN/m, M = 1 kNm, LAC = 3 m, LCD = 3 m, LDE = 2 m.(100 points)

Bonus:If the beam in item 4 is limited to have a maximum deflection of 2 cm, determine w in kN/m. (50 points)