The evaluation of am proceeds similarly by

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Unformatted text preview: y be integrated exactly (cf. Problem 1 at the end of this section) to yield AM (V U ) = dj 1 dj ]Mj cj 1cj j ; ; Mj = qhj 2 1 6 12 (1.3.13) where Mj is called the element mass matrix because, as noted, it often arises from inertial loading. 1.3. A Simple Finite Element Problem 13 The nal integral (1.3.9e) cannot be evaluated exactly for arbitrary functions f (x). Without examining this matter carefully, let us approximate it by its linear interpolant f (x) fj 1 ; j ; 1 (x) + fj j (x) x 2 xj 1 xj ] (1.3.14) ; where fj := f (xj ). Substituting (1.3.14) and (1.3.10b) into (1.3.9e) and evaluating the integral yields (V f )j Zx j dj 1 dj ] ; xj;1 j ; j 1 j fj 1 dx = d d ]l j1 jj fj j] 1 ; ; ; (1.3.15a) where lj = hj 2jfj 1 1++2fjj : f 6f (1.3.15b) ; ; The vector lj is called the element load vector and is due to the applied loading f (x). The next step in the process is the substitution of (1.3.12), (1.3.13), and (1.3.15) into (1.3.9a) and the summation over the elements. Since this our rst example, we'll simplify matter...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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