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exactly (cf. Problem 1 at the end of this section) to yield AM (V U ) = dj 1 dj ]Mj cj 1cj
; ; 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
(V f )j Zx j dj 1 dj ]
; xj;1 j ; j 1 j fj 1 dx = d d ]l
fj j] 1 ; ; ; (1.3.15a) where lj = hj 2jfj 1 1++2fjj :
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
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14