hw2 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF...

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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 2 Distributed : Wednesday, February 18, 2004 Due : Wednesday, February 26, 2004 Problem 1 ( 15 points ) The stiffness of a linear elastic structure subjected to a force of magnitude F , which results in a displacement of the load point in the direction of the force of magnitude , is given by F . k Figure 1: Cantilever beam: a) axial loading; b) lateral loading. Consider the case when the structure is, say, a tip-loaded cantilever beam, as shown in the figure. Depending on the orientation of the applied load with respect to the beam axis, two different stiffness values result. When loading is parallel to the beam axis, the axial stiffness of the structure can be defined as F x k axial , x where it is understo od that x is the axial displacement of the tip under load F x . When the applied load is perpendicular to the axis of the cantilever, a bending stiffness can be defined as F y k bending , y 1 where y is the lateral displacement of the tip under transverse load F y . The beam is of length L , and has cross-sectional area A and area moment of inertia I . Under the usual assumptions of slender beam-bending theory, L is much greater than linear dimensions of the beam cross-section, including both 1 A and 2 I /A ; that is, L 1 and L 2 . Show that the ratio of a slender cantilevers bending stiffness to its axial stiffness is extremely small, and give a precise expression for this ratio in the case of a solid circular cross-section of diameter d L . 2 Problem 2. ( 45 points ) When a cantilever beam is subjected to a temperature change that varies linearly through its thickness (but does not vary along the length of the beam), the beam deects laterally with a constant curvature, (thermal) , with resulting lateral displacement field 1 v ( x ) = (thermal) x 2 ....
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This note was uploaded on 02/23/2012 for the course MECHANICAL 2.002 taught by Professor Davidparks during the Spring '04 term at MIT.

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hw2 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF...

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