Solid Mechanics 1/8

Midterm ExamName:

Solutions must be worked on the sheets provided. All problems must begin with a FBD or no credit will be given. Answers must be boxed with units. You may not receive or give aid on this exam. After completing the exam, staple your equation sheet to the back.

Statics

1. (10 points). For the overhanging beam shown at right, determine the internal shear, moment, and axial forces in the section AB before the support at B.

Axial Loads

2. (10 Points) The three part axially loaded member in the figure at right consists of a tubular segment (1) with outer diameter do1=1.25 in. and inner diameter di1 = 0.875 in., a solid circular segment (2) with diameter d2 = 1.25 in., and another solid circular rod segment (3) with diameter d3 = 0.875 in. The line of action of each of the three applied loads is along the centroidal axis of the member.

Determine the axial stress2 in the middle, section (2), of the rod ONLY.

Shear Loads

3. (10 points) The hole in the plate is punched out by a hydraulic punch press similar to that described in a homework problem. In this case, the punch is in the shape of a “rectangular hole”, a 1.5 x 0.25”rectangle with two1/4 in. diameter semicircles attached on each end (see figure at right). The dimensions are given in the figure and the plate has an average shearing resistance of 38 ksi.

What is the required force in the punch?

Axial Strain

4. (10 points) Two cables are moored at points A and B and support a load at point C. Under the load the original geometry of the cables, given in dashed lines (C), deforms to the new geometry given by solid lines (C*). The symmetric geometry is given in the figure at right (not to scale). Assume cables AC and BC have modulus E and area, A.

Find an expression for the vertical load at C, Fcy, that is required to move point C to point C* by straining the two cables. Your answer should depend on the geometry (L’s) and cable properties (E and A)..

Shear Strain

5. (10 points) A rectangular body with the dimensions given is subjected to a shearing strain of  = -0.1 %. Sketch and dimension the new shape of the body in the plane. Overemphasize the deformation so that it is visible.

Generalized Hooke’s Law

6. (10 points) A rectangular body of height and depth h and length L is subjected to a uniform stress, o, on its z-faces. It is also constrained by frictionless, rigid surfaces on its y-faces. Using Generalized Hooke’s Law, find an expression for the three normal and three shear stresses (xx ,yy ,zz ,xy ,xz ,yz) in the body if its temperature is raised by an amount T. Assume G, E, , andareknown.

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Axial Strain with Variable Parameters

7. (20 points) The tapered solid stone pier in the figure at right is 16 ft tall and it has a square cross-sectionwith side dimensions that vary linearly from 24 in. at the top to 36 in. at the bottom. Assume that the stone is linearly elastic with modulus of elasticity E = 4,000 ksi. Determine the shortening of the pier due to the load P.
Statically Indeterminant Axial Loading

8. (20 points) Two linearly elastic members are joined together at B, and the resulting two-segment rod is attached to rigid supports at ends A and C. A single external force, Pb = 16 kips, is applied at joint B. Member (1) is steel with E1 = 210 GPa, A1 = 1000 mm2, and L1 = 2 m; and member (2) is a titanium-alloy rod with E2 = 120 GPa, A2 = 1200 mm2, and L2 = 1.8 m. The axial load at B is Pb = 40 kN.