Thus write 1212a as the summation of contributions

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Unformatted text preview: = AS (V U ) + AM (V U ) j j AS (V j U) = AM (V j Z U) = (V f )j = xj pV U dx (1.3.9c) qV Udx (1.3.9d) V fdx: (1.3.9e) 0 xj;1 Zx j xj;1 Zx j xj;1 (1.3.9b) 0 It is customary to divide the strain energy into two parts with AS arising from internal j energies and AM arising from inertial e ects or sources of energy. j Matrices are simple data structures to manipulate on a computer, so let us write the restriction of U (x) to xj 1 xj ] according to (1.3.7) as ; U (x) = cj 1 cj ] ; j (x) = j (x) ; 1 j ; 1 (x) j (x)] cj 1 cj ; x 2 xj 1 xj ]: (1.3.10a) ; We can, likewise, use (1.2.3b) to write the restriction of the test function V (x) to xj 1 xj ] in the same form 1( j V (x) = dj 1 dj ] j (xx) = j 1(x) j (x)] dd 1 x 2 xj 1 xj ]: (1.3.10b) ) j j ; ; ; ; ; ; 12 Introduction Our task is to substitute (1.3.10) into (1.3.9c-e) and evaluate the integrals. Let us begin by di erentiating (1.3.10a) while using (1.3.4a) to obtain U (x) = cj 1 cj ] 0 ; ;1=hj = ;1=hj 1=hj ] cjc 1 j x 2 xj 1 xj ]: ; 1=hj...
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