hw4 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 4 Distributed : Friday, April 2, 2004 Due : Friday, April 9, 2004 Problem 1 (20 points) Note: for reference material, consult the laboratory write-up on elastic-plastic beam bending Consider the square cross-section beam shown, of dimensions h by h , subject to “diamond- orientation” bending in the plane shown (neutral axis: plane y = 0). The beam can be considered to be composed of an elastic/perfectly-plastic material having Young’s modulus E , and tensile yield strength σ y . 1. Using the standard assumptions of engineering beam theory, evaluate the magnitude of applied moment, M y , just suﬃcient to bring the most highly-stressed region to the verge of yielding. Express your answer in terms of h and material properties, as appropriate. (Aside: are you “surprised” by the value you got for I = y 2 dA in this orientation?) 2. If the applied curvature is increased to very large values, the elastic/plastic bound- aries (tension and compression sides) in this geometry, like those in the bending of rectangular cross-sections studied earlier, will move inward, toward the neutral axis. At “infinite” curvature, the boundaries will reach opposite sides of the y = 0 surface, resulting in tensile yielding stress values of magnitude σ y in one “triangle” half of the cross-section, and compressive yielding stress values of magnitude σ y in the other triangular half of the cross-section. At this point, the bending moment carried by the cross-section reaches a limiting value, M L . Evaluate M L for this section. 3. Using your answers to the two previous questions, evaluate the ratio M L /M y for bend- ing of this section. How does this value compare with the ratio for bending of this same cross-section, but on rotated axes, so that the cross-section appears as a square?

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