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asst3A2008Solution

asst3A2008Solution - CEE 379 Assignment 3 Autumn 2008 Due...

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CEE 379 Assignment 3 Autumn 2008 Due October 15, 2008 This assignment begins with yet another review problem, again involving straightforward integration of the fundamental differential equation for beams: EIv ′′′′ = q ( x ). Note that in this case q ( x ) = 0, and there is an imposed displacement at the left support. The next problems consider member misfit and support settlement. 1. Determine an expression for the displaced shape of the beam shown, v ( x ), and determine the end moments and end shears. There is no rotation at either end of the beam ( v (0) = v ( L ) = 0). EI, L v 0 Solution: EIv ′′′′ = 0 v ( x ) = c 1 x 3 + c 2 x 2 + c 3 x + c 4 In this case the boundary conditions are v (0) = v 0 , v (0) = v ( L ) = v ( L ) = 0. The corresponding expanded expressions become: v (0) = c 4 = v 0 v (0) = c 3 = 0 v ( L ) = c 1 L 3 + c 2 L 2 + v 0 = 0 v ( L ) = 3 c 1 L 2 + 2 c 2 L = 0 Carrying out the indicated algebra leads to the following result: v ( x ) = v 0 bracketleftbigg 1 - 3 parenleftBig x L parenrightBig 2 + 2 parenleftBig x L parenrightBig 3 bracketrightbigg 2. A cast iron rod has restrained ends as indicated. Use the approach in Section 2.5 in the course notes to determine the temperature change necessary to cause the rod to fracture if its tensile strength is F u = 15 ksi. The coefficient of thermal expansion for iron is α = 6 × 10 6 / F , and E = 29 , 000ksi. Note that cast iron is a relatively brittle material, so it will fracture with little plastic deformation.

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asst3A2008Solution - CEE 379 Assignment 3 Autumn 2008 Due...

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