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
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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)
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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)
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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...