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Unformatted text preview: CEE 379 Assignment 2 Autumn 2008 Due October 8, 2008 This assignment begins with a review problem involving straightforward integration of the fundamental differential equation for beams: EIv = q ( x ). The machinery associated with setting up and solving this kind of problem will be very important to us in deriving stiffness properties for beam elements. The remaining problems are exercises involving the analysis of 1-D spring structures. You will solve the same problems in several different ways showing the consistency of the approaches we have learned so far. 1. Determine an expression for the displaced shape of the beam shown, v ( x ), and determine the end moments and end shears. The supports are fixed at both ends. EI, L w Solution: Solutions to this kind of problem can be found in a typical Mechanics of Materials textbook. For purposes of review, heres how the solutions go: EIv =- w v ( x ) =- w x 4 24 EI + c 1 x 3 + c 2 x 2 + c 3 x + c 4 The boundary conditions are v (0) = v (0) = v ( L ) = v ( L ) = 0, from which we get the following set of equations for the integration constants: v (0) = c 4 = 0 v (0) = c 3 = 0 v ( L ) =- w L 4 24 EI + c 1 L 3 + c 2 L 2 = 0 v ( L ) =- w L 3 6 EI + 3 c 1 L 2 + 2 c 2 L = 0 in which we have used the results of the first two equations in simplifying the final two. Doing the algebra and cleaning up leads us to the following result: v ( x ) = w x 2 24 EI ( L- x ) 2 To determine the end reactions, we can use the following expressions: V (0) =- EIv (0) = w L 2 M (0) =- EIv (0) = w L 2 12 V ( L ) =- EIv ( L ) =- w L 2 M ( L ) =- EIv ( L ) = w L 2 12 2. Consider the simple spring structure shown below: 1 CEE 379 2 1 2 3 k a k b k c...
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- Fall '08