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assing 1M - 2 The moment of inertia of the beam is I all...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.001 Mechanics and Materials I Fall 2006 Problem Set 11 Distributed: Wednesday, November 29, 2006 Due: Wednesday, December 6, 2006 Problem 1: Hibbeler 12-15 Problem 2: Calculate the deflection v(x) of the beam below by direct integration. What is the deflection of the beam at point A? The beam has a uniform Young’s modulus of E, and the moment of inertia of the cross section is I. w B C A x L L Problem 3: Calculate the deflection and slope at the center of the beam below using superposition and the results in Appendix C. L/2 A B P C L/2 P x Problem 4: Using superposition, calculate the deflection in the center (point B) of the beam shown below. The beam has a moment of inertia I and a Young’s modulus E. P A x B C L/2 L/2
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Problem 5: Calculate the deflection at the tip of the cantilever below. The half near the support has a Young’s modulus of E 1 , and the half away from the support has a Young’s modulus of E
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Unformatted text preview: 2 . The moment of inertia of the beam is I all the way along its length. You are welcome to use the table in Appendix C whenever it would be useful. A B C M E 1 E 2 L/2 L/2 Problem 6: Consider the composite beam cross section shown below. The top part of the beam (shaded) has a Young’s modulus of E 1 , and the bottom part of the beam (hatched) has a Young’s modulus of E 2 = 2E 1 . The directions of y and z are marked; it’s up to you to decide where to locate the origin. First locate the neutral axis in the beam. You will now examine the values of stress and strain on the dotted line indicated in the figure. Calculate the strain ε xx and the stress σ xx (as a function of the distance y off the neutral axis) that would result from an applied moment M about the z axis. Then graph both ε xx and σ xx vs. y. b/2 b b/4 b/4 Direction of y a a Direction of z a...
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