Homework - Mechanics of Solids

Question 1. An elastomeric bearing under a bridge girder is deformed as shown below.

a. Determine the average shear strain at corners A and C.
b. Determine the average normal strain along diagonals AC and BD.

Question 2. The figure shows a schematic representation of a 5 ton capacity jib crane. A, B, and C are pinned connections similar to the detail shown at B, member BC is a solid circular rod, and the beam (AC) is a W10x33. The hook is free to roll along the beam from x = 0 to x = 10 ft.

a. Determine the required diameter of rod BC if the rod is steel with an allowable stress of σ_allow = 30 ksi and round bars are available in 1/16-in increments for diameters up to 1-in. and 1/8-in increments for diameters over 1-in.
b. Determine the required diameter of the bolts at A, B, and C if the bolts have an allowable shear stress of τ_allow = 21 ksi and are available with diameters in 1/8-in increments. Use a single bolt diameter for all joints.
c. Determine the normal stress at the top and bottom of the beam at the location of the maximum bending moment. Also, sketch the stress distribution through the depth of the beam at this location.
d. Determine the shear stress in the web at the top, bottom, and centroid of the web using τ = VQ/It at the location of the maximum shear force and compare those to the average shear

stress of τ_avg = V/A_web

Question 3. An input torque is applied at C to supply output torques at A, B, and D as shown.

a. Determine the required diameter of the shaft if the shaft is prismatic and solid with an allowable shear stress of τ_allow = 85 MPa.
b. Determine the angle of the twist between output gear A and input gear C (Φ_A/C), if the shaft is aluminum for which G = 26 GPa.
c. Determine the percentage of material that can be saved by making the shaft hollow if the maximum outside diameter is 150 mm.
d. Determine the percent increase in Φ_A/C for the hollow shaft.

Question 4. Rigid beam ACE is supported by three rods as shown. Rods AB and EF are an aluminum alloy and rod CD is steel. All three rods are 15 mm in diameter. The material properties for each rod are given in the table below the figure.

a. Determine the maximum force, w, that can be applied be rigid beam ACE without yielding any rod.
b. Determine the stress in each bar if the temperature is increased 60 °C.