ce 332 final may 21 spring 2014

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CE 332 Mechanics of Deformable Bodies Final Exam - Spring 2014

Wednesday May 21, 2014 10:30 AM – 12:45 PM

Open Text Book – No Sharing of Anything – No use of Cell Phones – No use of WiFi

Question 1 - Provide short answers to the following questions (20 points total) 1. What are the three sets of equations that must all be satisfied for any problem in mechanics of materials. (1) 2. Why are members with circular shaped cross-sections preferred for carrying torsional loads (i.e., torques). (1) 3. A laboratory test of concrete cylinder is conducted and it is found that E = 3.5 106 psi and = 0.2. Assuming that the concrete is isotropic, what is the value of its Shear Modulus? (3) 4. A circular cylindrical steel pipe of 36 inch diameter is used for natural gas transmission from Texas to New York. The internal pressure of the gas in the pipe is 1,000 psi. If the allowable hoop stress in the pipe is 16,000 psi determine the required thickness of the pipe. (3) 5. State the three fundamental assumptions of the Euler-Bernoulli Beam Theory. (3) 6. Determine the deflection under the load (in inches) of simply-supported, 30 ft long S 18 70 I-beam (American Standard Shape) made of ASTM A-36 steel subjected to a concentrated load of 10 kips at its center when it is bent about its major axis. (3) 7. Determine the slope (in radians) at the end of a 5.5 by 5.5 inch square (actual size) Douglas Fir timber cantilever beam having a length of 10 ft and subjected to a uniformly distributed load of 300 lbs/ft. (3) 8. Determine the ultimate axial load carrying capacity, Pult, of a 0.25 inch diameter solid steel rod that is 10 inches long, is fixed at both ends, and loaded in axial compression. Given E = 30 106 psi, Y = 75 ksi. Do not use any safety factors in your calculations. (3)

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Question 2 (30 points) A statically indeterminate continuous beam with constant EI is loaded with a uniformly distributed load on one of its equal-length spans as shown in the figure below. You are required to determine the reactions, sketch the elastic curve, and draw the shear force and bending moment diagrams for the continuous beam. Provide your answers in the following steps: 1. Draw a free body diagram of the beam and write the equilibrium equations. (2) 2. Identify the unknown reactions and for each unknown reaction provide its displacement compatibility equation. (3) 3. Solve for the reactions using either the force method (i.e., the redundant force/superposition method) or the displacement method (compatibility and the differential equation of the elastic curve). For the force method, clearly indicate your choice of redundant and the compatibility equation you are using. For the displacement method, clearly indicate the unknowns and the boundary conditions you are using for the differential equation. (20) 4. Sketch the elastic curve of the beam. (3) 5. Using the reactions you have found in Step 3 re-draw a free body diagram of the continuous beam. (2) NOTE: To aid in your solution you are given additional beam tables on the pages that follow Question 4. You will need to use these together with the tables in your text book in Appendix D. In these tables symbols x or max are the vertical deflections yx and ymax at a distance x from the left end of the beam or at the point of maximum deflection. These are POSITIVE DOWNWARDS and the length of the beam is ℓ. In Appendix D in the text book the deflections are given as NEGATIVE DOWNWARDS and the length of the beam is L. EXTRA CREDIT (ONLY DO THIS AFTER FINISHING ALL OTHER QUESTIONS) 6. Draw the Shear Force diagram, V(x), for the continuous beam. (5) 7. Draw the Bending Moment diagram, M(x), for the continuous beam. (5)

24 ft

C B A

24 ft

2 k/ft

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Question 3 (30 points) A beam is subjected to transverse loading which results in the following maximum internal forces (shear and bending moment) and maximum deflections: Mmax = 150 k-ft Vmax = 65 k

inI

0.1620max

y

What is the lightest section W Shape (in Appendix C in your textbook) that can be used using the Allowable Stress Design (ASD) method if the following yield stresses and safety factors (SF), and deflection limit are given: yield = 50 ksi SFbending stress = 2.5 yield = 30 ksi SFshear stress = 3.0 ymax = 0.8 inches SFdeflection = 1.0 NOTE: Neglect the self-weight of the beam.

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Question 4 (20 points) A parallel chord roof truss-joist shown in Figures 1 to 3 below is constructed using L 4 4 ¼ back-to-back equal leg steel angles (double angles) made of ASTM A36 steel (See Appendix C of the text book) for the compression (top) chord.

The back to back angles have a ½ inch gap (g = 0.5 inch) between them to accommodate the vertical and diagonal members. The distance between the panel points of the truss is 7 ft. For your calculations assume that the top chord is pinned at every second panel point for out-of-plane (i.e. sideways, horizontal) displacement of the top chord and pinned at every panel point for in-plane (i.e., up-and-down, vertical) displacement of the top chord Determine the critical buckling load of the compression (top) chord of the truss. 1. Determine IX and IY of the back to back L 4 4 ¼ steel angles. (8) 2. Determine the critical buckling load for the top chord. (10) 3. Why do you think the truss does not have vertical members intersecting with every panel point at the bottom chord the truss? (2)

Truss “panel point” 7 ft 7 ft

Fig. 2 Back-to-back angles

Back-to-back (double angles) angles

Fig. 2 Parallel chord truss-joist

Fig. 3 Details of the truss members

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ce 332 final may 21 spring 2014

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