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8 Pages

### appendix_d

Course: ETD 100698, Fall 2009
School: Virginia Tech
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Word Count: 1494

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D. Appendix Integrals The Psin function are used as trial functions to approximate the displacement field in different models. Let us use u and w as two functions of displacement. 2x 2x 2x 1 u (x ) = 2 cos p L + p cos p L + p = P sin p L b b b (D.1) 2x 2x 2x 1 + q cos q + q = P sinq w(x ) = 2 cos q L L L b b b (D.2) This appendix present some...

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D. Appendix Integrals The Psin function are used as trial functions to approximate the displacement field in different models. Let us use u and w as two functions of displacement. 2x 2x 2x 1 u (x ) = 2 cos p L + p cos p L + p = P sin p L b b b (D.1) 2x 2x 2x 1 + q cos q + q = P sinq w(x ) = 2 cos q L L L b b b (D.2) This appendix present some integrals required when using these functions in a variational model. The first integrals concern one displacement function only. xa + La xa 2 La 2 w(x )dx = Lb G1q (xa , La ) 2 (D.3) xa + La xa 2 La 2 w(x ) 1 dx = G 2q (xa , La ) x 4 (D.4) xa + La 2 xa La 2 2 w(x ) 1 dx = 3 G 3q (xa , La ) 2 x Lb (D.5) These integrals are in term of dimensionless functions G1, G2 and G3, very similar in their form. L 2x 1 sin q a cos q a + q L q Lb b L 2x 1 sin q a cos q a + q q Lb Lb G1q (x a , La ) = 105 106 Appendix D. Integrals Pierre E. Cambou if q = 0 , the first the term is La cos( q ), Lb L if q = 0 , the sec ond term is a cos( q ) Lb (D.6) L 2x G 2q (xa , La ) = sin q a sin q a + q L Lb b L 2x + sin q a sin q a + q L L b b L 2x G 3q (xa , La ) = q sin q a sin q a + q L Lb b L 2x + q sin q a sin q a + q L L b b (D.7) (D.8) The next list of integral permits to determine the coeficient of the mass and stiffness matices: xa + La xa La 2 La 2 u (x )w(x )dx = Lb F 4 pq (xa , La ) 8 (D.9) xa + xa 2 La 2 u (x ) w(x ) 1 dx = F 1pq (xa , La ) 2 Lb x x (D.10) xa + La 2 xa La 2 2u (x ) 2 w(x ) 2 dx = 3 F 2pq (xa , La ) 2 2 Lb x x 1 2 u (x ) w(x )dx = F 5 pq (xa , La ) 2 2 Lb x (D.11) xa + La xa 2 La 2 (D.12) These integrals are in term of four dimentionless functions F4, F1, F2, and F5 very similar in their form: F 4 pq (xa , La ) = SC11pq (xa , La ) + SC12pq (xa , La ) [ ] Pierre E. Cambou Appendix D. Integrals 107 + SC 21pq (xa , La ) + SC 22pq (xa , La ) pq pq a a a a pq pq a a a a [ ] [SC 31 (x , L ) + SC 32 (x , L )] [SC 41 (x , L ) + SC 42 (x , L )] (D.13) F 1pq (xa , La ) = p q SC11pq (xa , La ) + SC12pq (xa , La ) p q pq a a pq a a [ ] [SC 21 (x , L ) + SC 22 (x , L )] + [SC 31 (x , L ) + SC 32 (x , L )] + [SC 41 (x , L ) + SC 42 (x , L )] p q q pq pq a a a a pq pq a a a a p (D.14) 2 F 2pq (xa , La ) = 2 q SC11pq (xa , La ) + SC12pq (xa , La ) p [ ] [SC 31 (x , L ) + SC 32 (x , L )] [SC 41 (x , L ) + SC 42 (x , L )] 2 + 2 q SC 21pq (xa , La ) + SC 22pq (xa , La ) p 2 2 p q 2 q pq pq a a a a pq pq a a a a 2 p [ ] (D.15) F 5pq (xa , La ) = 2 SC11pq (xa , La ) + SC12pq (xa , La ) p 2 p pq a a pq a a 2 p [ ] + [SC 21 (x , L ) + SC 22 (x , L )] [SC 31 (x , L ) + SC 32 (x , L )] [SC 41 (x , L ) + SC 42 (x , L )] pq a a pq a a 2 p pq a a pq a a (D.16) These functions are based on 8 elementary functions SC11, SC12, SC21, SC22, SC31, SC32, SC41, and SC42. They also are very similar and use the Psin coefficients n, n, n, and n. L 2x sin( p + q ) a cos( p + q ) a + ( p + q ) Lb Lb SC11pq (xa ,La ) = p + q for p + q 0 else SC 11pq (xa , La ) = La cos( p + q ) Lb (D.17) 108 Appendix D. Integrals Pierre E. Cambou L 2x sin ( p q ) a cos ( p q ) a + ( p q ) Lb Lb SC12pq (xa , La ) = p q for p q 0 else SC 12pq (xa , La ) = La cos( p q ) Lb (D.18) L 2x sin( p + q ) a cos( p + q ) a + ( p + q ) Lb Lb SC21pq (xa , La ) = p +q for p + q 0 else SC 21pq (xa , La ) = La cos( p + q ) Lb (D.19) L 2x sin ( p q ) a cos ( p q ) a + ( p q ) Lb Lb SC 22pq (xa , La ) = p q for p q 0 SC else 22pq (xa , La ) = La cos( p q ) Lb (D.20) 2x L sin ( p + q ) a cos ( p + q ) a + ( p + q ) Lb Lb SC 31pq (xa , La ) = p +q for p + q 0 else SC 31pq (xa , La ) = La cos( p + q ) Lb (D.21) L 2x sin( p q ) a cos( p q ) a + ( p q ) lb lb SC 32 pq (x a , L a ) = p q for p q 0 else SC 32pq (xa , La ) = La cos( p q ) Lb (D.22) Pierre E. Cambou Appendix D. Integrals 109 2x L sin ( p + q ) a cos ( p + q ) a + ( p + q ) Lb Lb SC 41pq (xa , La ) = p + q for p + q 0 else SC 41pq (xa , La ) = La cos( p + q ) Lb (D.23) SC 42 pq (x a , L a ) = sin ( p q ) La 2xa + ( p q ) cos ( p q ) lb lb p q for p q 0 else SC 42pq (xa , La ) = La cos( p q ) Lb (D.24) Some other integrals are needed especially to compute non-diagonal terms: 1 w(x ) dx = F 6 pq (xa , La ) 4 x xa + La xa La 2 2 La 2 u (x ) (D.25) xa + xa La 2 1 u (x ) 2 w(x ) dx = 2 F 3pq (xa , La ) 2 Lb x x (D.26) These integrals are based on a different type of functions F6 and F3: F 6 pq (xa , La ) = q SS 11pq (xa , La ) + SS 12pq (xa , La ) q pq a a pq a a [ ] [SS 21 (x , L ) + SS 22 (x , L )] + [SS 31 (x , L ) + SS 32 (x , L )] + [SS 41 (x , L ) + SS 42 (x , L )] q pq a a pq a a q pq a a pq a a 2 p q pq a a pq a a (D.27) 2 F 3pq (xa , La ) = p q SS 11pq (xa , La ) + SS 12pq (xa , La ) [ ] + [SS 21 (x , L ) + SS 22 (x , L )] [SS 31 (x , L ) + SS 32 (x , L )] [SS 41 (x , L ) + SS 42 (x , L )] 2 p q 2 q pq pq a a a a pq pq a a a a p (D.28) 110 Appendix D. Integrals Pierre E. Cambou These functions are in term of 8 other elementary functions SS11, SS12, SS21, SS22, SS31, SS32, SS41, and SS42 also based on the Psin coefficients n, n, n, and n. 2x L sin( p + q ) a sin( p + q ) a + ( p + q ) Lb Lb SS11pq (xa ,La ) = p + q for p + q 0 else SS 11pq (xa , La ) = La sin( p + q ) Lb (D.29) 2x L sin ( p q ) a sin( p q ) a + ( p q ) Lb Lb SS 12pq (xa , La ) = p q for p q 0 else SS 12pq (xa , La ) = La sin( p q ) Lb (D.30) L 2x sin( p + q ) a sin( p + q ) a + ( p + q ) Lb Lb SS21pq (xa ,La ) = p +q for p + q 0 else SS 21pq (xa , La ) = La sin( p + q ) Lb (D...

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