Problem 1: The grooved circular shaft shown in Figure 1 consists of two segments of diameter D = 440 mm joined by a groove of diameter d = 400 mm with groove radius r = 20 mm. The shaft is loaded simultaneously by an axial force P = 50 kN along the x-axis, a bending moment M= 10 kN-m about the z-axis, and a torque T= 20 kN-m about the x-axis.

a) Calculate the maximum and minimum principal stresses, and describe where they occur.

b) Calculate the maximum shear stress, and describe where it occurs.

c) Calculate the maximum octahedral shear stress, and describe where it occurs.

d) Calculate the normal stress on the plane with the maximum octahedral shear stress.

Problem 2: An electrical contact contains a flat spring in the form of a cantilever beam of length L = 50 mm, shown in Figure 2. The cross section of the beam is rectangular with thickness t = 1 mm (shown) and width b (not shown). The end load P is equal to the sum W_o + F, where W_o = 5 N and F varies continuously in time from 0 to F_max. The yield strength of the material is σ_y = 800 MPa, the ultimate strength is σ_u =1200 MPa, and the endurance limit is σ_e = 400 MPa. Assume that the material is ductile and will fatigue due to yielding.

a) Use the Soderberg criterion to find the minimum value of the width b required to prevent fatigue failure. Take F_max = 10 N.

b) Use the Modified Goodman criterion (for yield) to find the minimum value of the width b required to prevent fatigue failure. Take F_max = 10 N.

c) If the value of F_max is increased to 15 N, and the value of b is the one you obtained in part a), then extend the Soderberg criterion appropriately to predict the fatigue life of the beam.

Problem 3: A machine component of channel cross-sectional area is loaded by equal and opposite forces P of magnitude 3 kN, as shown in Figure 3. All dimensions of length in the figure are in millimeters. Point O is located at the center of curvature of the unstressed semi-circular segment of the component. Use the following notation: r is the radial distance from point O to any point on the cross-section in the semi-circular segment of the component, R_c is the radial distance from point 0 to the centroid of any cross-section in the semi-circular segment of the component, and R_n is the radial distance from point O to the neutral fibers (due to the bending moment, alone) on any cross-section in the semi-circular segment of the component.

a) Calculate R_c, R_n and the eccentricity e of the cross section.

b) Calculate the normal force and the bending moment at the cross section containing points A and B shown in the figure. Indicate clearly whether the normal force is tensile or compressive, and whether the bending moment is positive or negative according to the sign convention introduced in class.

c) Calculate the normal stress (σ_θθ)_A at point A, and show that the contribution to (σ_θθ)_A from the normal force at the cross section is small compared to the contribution from the bending moment.

d) Calculate the normal stress (σ_θθ)_B at point B, and show that the contribution to (σ_θθ)_B from the normal force at the cross section is small compared to the contribution from the bending moment.

e) On the section A-B in Figure 3, neglect the contribution to σ_θθ(r) from the normal force, and find the variation of the radial stress σ_rr(r) with radial distance r from the center of curvature of the component. Plot the variation of σ_rr(r)with r. In particular, find the radial stress σ_rr at r = 40mm, r = 45mm, r = 50^- mm, r = 50⁺ mm, r = 65 mm, and r = 80 mm. Find the maximum value of σ_rr and the value of r at which it occurs.