Appendices

Appendices

Appendices 0 ~  are: 1. The elements used to construct the matrix ½A a11 5 2Asl kz2 2 n2 kn2 ðA 2 A Þ 2 ð1 2 ηÞ 1 mt ω2 sl slv 2R2 2R2 21 ðAsl 1 A...

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Appendices 0

~  are: 1. The elements used to construct the matrix ½A a11 5 2Asl kz2 2

n2 kn2 ðA 2 A Þ 2 ð1 2 ηÞ 1 mt ω2 sl slv 2R2 2R2

21 ðAsl 1 Aslv Þikz n 2R 1 k a13 5 2Aslv ikz 1 kRikz3 1 ð1 2 ηÞikz n2 R 2R 1 a21 5 ðAsl 1 Aslv Þikz n 2R a12 5

1 n2 3k a22 5 2 ðAs1 2 Aslv Þkz2 2 Asl 2 1 ð1 2 ηÞkz2 1 mt ω2 2 2 R As1 n 1 2 knð3 2 ηÞkz2 R2 2 1 k a31 5 As1ν ikz 1 kRikz3 1 ð1 2 ηÞikz n2 R 2R 1 k a32 5 2As1 2 n 2 ð3 2 ηÞkz n2 R 2 a23 5 2

a33 5 2

As1 Bs1 2 2 Bs1 4 k kn2 4 2 B k 2 2 k n 2 n 2 1 2 1 mt ω 2 s1 z z R2 R2 R2 R4 R2 0

~  is given by: For a symmetric loading, the matrix ½A 2 3 a11 a12 a13 0 ~  5 4 a21 a22 a23 5 ½A a31 a32 a33

237

238 Appendices 0

~  is given by: For an antisymmetric loading, the matrix ½A 2 3 a11 2a12 a13 0 ~  5 4 2a21 a22 2a23 5 ½A a31 2a32 a33 2. The elements used to construct the matrix ½S are: s11 5 cos nθ; s12 5 0; s13 5 0 s21 5 0; s22 5 sin nθ; s23 5 0 s31 5 0; s32 5 0; s33 5 cos nθ For a symmetric loading, the matrix ½S is given by: 2 3 cos nθ 0 0 ½S 5 4 0 sin nθ 0 5 0 0 cos nθ For an antisymmetric loading, the matrix ½S is given by: 2 3 sin nθ 0 0 ½S 5 4 0 cos nθ 0 5 0 0 sin nθ 3. The elements used to construct matrices ½U and ½UN  are: n n u12 5 Kn ðα1 rÞ 2 α1 Kn11 ðα1 rÞ ; u14 5 Kn ðα2 rÞ 2 α2 Kn11 ðα2 rÞ r r n u16 5 2ikz Kn11 ðβrÞ ; u18 5 Kn ðβrÞ r n n u22 5 2 Kn ðα1 rÞ; u24 5 2 Kn ðα2 rÞ r r n u26 5 2ikz Kn11 ðβrÞ ; u28 5 2 Kn ðβrÞ 1 βKn11 ðβrÞ r u32 5 2ikz Kn ðα1 rÞ ; u34 5 2ikz Kn ðα2 rÞ u36 5 βKn ðβrÞ ; u38 5 0 For a symmetric loading, matrices ½U and ½UN  are given by: 2 3 2 u32 u34 u36 u12 u14 u16 u18 ½U 5 4 u22 u24 u26 u28 5; ½UN  5 4 u22 u24 u26 u32 u34 u36 u38 u12 u14 u16

3 u38 u28 5 u18

Appendices 239 For an antisymmetric loading, matrix ½UN  is given by: 2 3 2 u32 u12 u14 2 u16 2 u18 5 4 ½  ½U 5 4 2 u22 2 u24 u26 u28 ; UN 5 2 u22 u32 u34 2 u36 2 u38 u12

u34 2 u24 u14

2 u36 u26 2 u16

4. The elements used to construct matrices ½T and ½TN  are: " ! # n2 2 n α1 2 2 1 α1 2 αf kf In ðα1 rÞ 2 2μ In11 ðα1 rÞ t11 5 2μ r2 r " ! # n2 2 n α1 1 α1 2 2 αf kf2 Kn ðα1 rÞ 1 2μ Kn11 ðα1 rÞ t12 5 2μ 2 r r " ! # n2 2 n α2 t13 5 2μ 1 α2 2 2 αs ks2 In ðα2 rÞ 2 2μ In11 ðα2 rÞ 2 r r " ! # n2 2 n α2 1 α2 2 2 αs ks2 Kn ðα2 rÞ 1 2μ Kn11 ðα2 rÞ t14 5 2μ r2 r t15 5 22μikz βIn ðβrÞ 1 2μikz t16 5 2μikz βKn ðβrÞ 1 2μikz

n11 In11 ðβrÞ r

n11 Kn11 ðβrÞ r

t17 5 2μ

n2 2 n n In ðβrÞ 1 2μ βIn11 ðβrÞ r2 r

t18 5 2μ

n2 2 n n Kn ðβrÞ 2 2μ βKn11 ðβrÞ r2 r

t21 5 22μ

n2 2 n n In ðα1 rÞ 2 2μ α1 In11 ðα1 rÞ 2 r r

t22 5 22μ

n2 2 n n Kn ðα1 rÞ 1 2μ α1 Kn11 ðα1 rÞ 2 r r

t23 5 22μ

n2 2 n n In ðα2 rÞ 2 2μ α2 In11 ðα2 rÞ 2 r r

t24 5 22μ

n2 2 n n Kn ðα2 rÞ 1 2μ α2 Kn11 ðα2 rÞ 2 r r

3 2 u38 u28 5 2 u18

240 Appendices t25 5 2μikz βIn ðβrÞ 1 2μikz

n11 In11 ðβrÞ r

n11 Kn11 ðβrÞ r ! n2 2 n β t27 5 22μ 2 2 μβ 2 In ðβrÞ 1 2μ In11 ðβrÞ r r ! n2 2 n β t28 5 22μ 2 2 μβ 2 Kn ðβrÞ 2 2μ Kn11 ðβrÞ r r n t31 5 22μikz In ðα1 rÞ 2 2μikz α1 In11 ðα1 rÞ r n t32 5 22μikz Kn ðα1 rÞ 1 2μikz α1 Kn11 ðα1 rÞ r n t33 5 22μikz In ðα2 rÞ 2 2μikz α2 In11 ðα2 rÞ r n t34 5 22μikz Kn ðα2 rÞ 1 2μikz α2 Kn11 ðα2 rÞ r n t35 5 2μ βIn ðβrÞ 2 μðkz2 1 β 2 ÞIn11 ðβrÞ r n t36 5 μ βKn ðβrÞ 2 μðkz2 1 β 2 ÞKn11 ðβrÞ r n n t37 5 2μikz In ðβrÞ; t38 5 2 μikz Kn ðβrÞ r r " # n2 2 n α1 t41 5 22μ 2 2 αf kf2 In ðα1 rÞ 1 2μ In11 ðα1 rÞ r r " # n2 2 n α1 t42 5 22μ 2 2 αf kf2 Kn ðα1 rÞ 2 2μ Kn11 ðα1 rÞ r r t26 5 μikz βKn ðβrÞ 1 2μikz

Appendices 241 "

# n2 2 n α2 t43 5 22μ 2 2 αs ks2 In ðα2 rÞ 1 2μ In11 ðα2 rÞ r r " # n2 2 n α2 t44 5 22μ 2 2 αs ks2 Kn ðα2 rÞ 2 2μ Kn11 ðα2 rÞ r r n11 n11 In11 ðβrÞ; t46 5 22μikz Kn11 ðβrÞ t45 5 22μikz r r t47 5 22μ

n2 2 n n In ðβrÞ 2 2μ βIn11 ðβrÞ 2 r r

n2 2 n n Kn ðβrÞ 1 2μ βKn11 ðβrÞ 2 r r n n t51 5 2μikz In ðα1 rÞ; t52 5 2μikz Kn ðα1 rÞ r r n n t53 5 2μikz In ðα2 rÞ; t54 5 2μikz Kn ðα2 rÞ r r n n t55 5 μ βIn ðβrÞ 2 μkz2 In11 ðβrÞ; t56 5 2μ βKn ðβrÞ 2 μkz2 Kn11 ðβrÞ r r n n t57 5 μikz In ðβrÞ 1 μikz βIn11 ðβrÞ; t58 5 μikz Kn ðβrÞ 2 μikz βKn11 ðβrÞ r r t48 5 22μ

t61 5 2ðαs ks2 1 2μkz2 ÞIn ðα1 rÞ;

t62 5 2ðαs ks2 1 2μkz2 ÞKn ðα1 rÞ

t63 5 2ðαs ks2 1 2μkz2 ÞIn ðα2 rÞ;

t64 5 2ðαs ks2 1 2μkz2 ÞKn ðα2 rÞ

t65 5 2μikz βIn ðβrÞ; For a symmetric 2 t11 t12 6 t21 t22 6 6 t31 t32 ½T 5 6 6 t41 t42 6 4 t51 t52 t61 t62

t66 5 22μikz βKn ðβrÞ;

t67 5 0;

loading, matrices ½T and ½TN  are given by: 3 t13 t14 t15 t16 t17 t18 2 t23 t24 t25 t26 t27 t28 7 7 t32 t33 t34 t35 t36 t37 t38 7 7; ½TN  5 4 t22 t43 t44 t45 t46 t47 t48 7 7 t12 t53 t54 t55 t56 t57 t58 5 t63 t64 t65 t66 t67 t68

t68 5 0

t34 t24 t14

t36 t26 t16

3 t38 t28 5 t18

242 Appendices For an antisymmetric loading, matrices ½T and ½TN  are given by: 3 2 t12 t13 t14 2t15 2t16 2t17 2t18 t11 6 2t21 2t22 2t23 2t24 t25 t26 t27 t28 7 7 6 6 t t32 t33 t34 2t35 2t36 2t37 2t38 7 7 6 31 ½T 5 6 7 6 t41 t42 t43 t44 2t45 2t46 2t47 2t48 7 7 6 4 2t51 2t52 2t53 2t54 t55 t56 t57 t58 5 t61 t62 t63 t64 2t65 2 3 t32 t34 2t36 2t38 4 ½TN  5 2t22 2t24 t26 t28 5 t12 t14 2t16 2t18

2t66

2t67

2t68

5. The elements used to construct matrix ½WN  are: n w11 5 μf Kn ðα1 rÞ 2 μf α1 Kn11 ðα1 rÞ; r w13 5 2μt ikz Kn11 ðβrÞ;

n w12 5 μs Kn ðα2 rÞ 2 μs α2 Kn11 ðα2 rÞ; r n w14 5 2μt ikz Kn ðβrÞ; w21 5 2 μf Kn ðα1 rÞ; r

n w22 5 2 μs Kn ðα2 rÞ; w23 5 2μt ikz Kn11 ðβrÞ; r n w24 5 2 μt Kn ðβrÞ 1 μt βKn11 ðβrÞ; w31 5 2μf ikz Kn ðα1 rÞ; r w32 5 2μs ikz Kn ðα2 rÞ; w33 5 μt βKn ðβrÞ; w34 5 0 For a symmetric loading, matrix ½WN  2 w31 ½WN  5 4 w21 w11

is given by: w32 w22 w12

w33 w23 w13

3 w34 w24 5 w14

For an antisymmetric loading, matrix ½WN  is given by: 2 3 w31 w32 2 w33 2 w34 ½WN  5 4 2 w21 2 w22 w23 w24 5 w11 w12 2 w13 2 w14 6. The elements used to construct matrix ½G0  are: g011 5 Mðα 1 μf Þkf2 Kn ðα1 rÞ g012 5 Mðα 1 μs Þks2 Kn ðα2 rÞ g013 5 0; g014 5 0 g015 5 0; g016 5 0 g017 5 0; g018 5 0

Appendices 243 For both a symmetric loading and an antisymmetric loading, matrix ½G0  is given by:   ½G0  5 g011 g012 g013 g014 g015 g016 g017 g018 7. The elements used to construct matrix ½G are: " # n g11 5 Mðα 1 μf Þkf2 Kn ðα1 rÞ 2 α1 Kn11 ðα1 rÞ r " # n g12 5 Mðα 1 μs Þks2 Kn ðα2 rÞ 2 α2 Kn11 ðα2 rÞ r g13 5 0; g14 5 0; g15 5 0; g16 5 0; g17 5 0;

g18 5 0

For both a symmetric loading and an antisymmetric loading, matrix ½G is given by:   ½G 5 g11 g12 g13 g14 g15 g16 g17 g18 8. Matrices H~ 11 ; H~ 21 ; H~ 22 ; H~ 31 ; H~ 32 ; H~ 41 ; H~ 42 , and H~ 51 , related to the poroelastic soil, take the form: 3 3 2 2 2H~ r H~ r 0 0 0 0 0 0 6 0 6 ~ 0 0 7 0 7 2H~ r 7; H~ 12 5 6 0 H r 0 7 H~ 11 5 6 5 4 0 4 ~ ~ 0 0 2H r 0 0 Hr 0 5 0 0 0 2H~ r 0 0 0 H~ r 2 6 H~ 21 5 4 2 6 H~ 22 5 4

2H~ v cos ϕ 2H~ h sin ϕ

2H~ v 0

H~ r ðR0 2 bb Þsin ϕ H~ v

0

0

0

H~ r at1

H~ r at2

2H~ r at1

H~ v

2

2kr 6 2kr H~ 32 5 6 4 2kr 2kr

H~ v sin ϕ 2H~ h cos ϕ

2H~ r ðR0 2 bb Þsin ϕ 2H~ r ½R0 2 ðR0 2 bb Þcos ϕ 2 3 3 kr 0 0 0 H~ v 60 k 0 07 r 7 6 7 0 5; H~ 31 5 6 7 4 0 0 kr 0 5 2H~ r at2 0 0 0 kr

0

H~ v

2H~ v cos ϕ H~ h sin ϕ

0 0 0 0

3

2kr at1 2kr at2 7 7; kr at1 5 kr at2

2

kn cos ϕ 6 kn 6 6 kn cos ϕ H~ 41 5 6 6 2ks sin ϕ 6 4 0 ks sin ϕ

0 2H~ h

2H~ r bb

2H~ v sin ϕ 2H~ h cos ϕ

2H~ r ½R0 2 ðR0 2 bb Þcos ϕ

3 kn sin ϕ 2kn ðR0 2 bb Þsin ϕ 7 0 0 7 7 2kn sin ϕ kn ðR0 2 bb Þsin ϕ 7; ks cos ϕ ks ½R0 2 ðR0 2 bb Þcos ϕ 7 7 5 ks ks b b ks cos ϕ ks ½R0 2 ðR0 2 bb Þcos ϕ

3 7 5

244 Appendices 2 6 6 6 6 H~ 42 5 6 6 6 6 4

2kn

0

0

0

0

0 0

2kn 0

0 2kn

0 0

0 0

0

0

0

2ks

0

0

0

0

0

2ks

0

0

H~ 627 H~ 727

0 H~ 628 H~ 728

0 H~ 629 H~ 729

H~ 827 H~ 927

H~ 828 H~ 928

H~ 829 H~ 929

H~ 1027 H~ 1127

H~ 1028 H~ 1128

H~ 1029 H~ 1129

0 2 ~ H 626 6 H~ 6 726 6 6 H~ 826 H~ 51 5 6 6 H~ 6 926 6 4 H~ 1026 H~ 1126

0

3

7 7 7 7 7 0 7 7 7 0 5 0 0

2ks H~ 6210 H~ 7210 H~ 8210 H~ 9210 H~ 10210 H~ 11210

9. The elements used to construct matrix Btmn are: 00

Btmn ð1; 1Þ 5 Ztm ðzÞcos nθ 1 Btmn ð2; 2Þ 5 Ztm ðzÞsin0 nθ Rt Btmn ð2; 3Þ 5

1 Ztm ðzÞcos nθ Rt

Btmn ð3; 1Þ 5

1 0 Z ðzÞcos0 nθ Rt tm 0

Btmn ð3; 2Þ 5 Ztm ðzÞsin nθ 00 Btmn ð4; 3Þ 5 2Ztm ðzÞcos nθ 1 Btmn ð5; 2Þ 5 2 Ztm ðzÞsin0 nθ Rt Btmn ð5; 3Þ 5 2

1 Ztm ðzÞcos00 nθ R2t

Btmn ð6; 2Þ 5

1 0 Z ðzÞsin nθ Rt tm

Btmn ð6; 3Þ 5

1 0 Z ðzÞsin nθ Rt tm

10. The elements used to construct matrix Nsmn are: N11mn 5

R2 2 r 0 Z ðzÞcos nθ R2 2 R1 sm

N14mn 5

r 2 R1 0 Z ðzÞcos nθ R2 2 R1 sm

N22mn 5

R2 2 r Zsm ðzÞsin nθ R2 2 R1

N25mn 5

r 2 R1 Zsm ðzÞsin nθ R2 2 R1

N33mn 5

R2 2 r Zsm ðzÞcos nθ R2 2 R1

N36mn 5

r 2 R1 Zsm ðzÞcos nθ R2 2 R1

3 H~ 6211 H~ 7211 7 7 7 H~ 8211 7 7 H~ 9211 7 7 7 H~ 10211 5 H~ 11211

Appendices 245 11. The elements used to construct matrix Bsmn are: Bsmn ð1; 1Þ 5

R2 2 r 00 Z ðzÞcos nθ R2 2 R1 sm

Bsmn ð1; 4Þ 5

r 2 R1 00 Z ðzÞcos nθ R2 2 R1 sm

Bsmn ð2; 2Þ 5

R2 2 r Zsm ðzÞsin0 nθ rðR2 2 R1 Þ

Bsmn ð2; 3Þ 5

R2 2 r Zsm ðzÞcos nθ rðR2 2 R1 Þ

Bsmn ð2; 5Þ 5

r 2 R1 Zsm ðzÞsin0 nθ rðR2 2 R1 Þ

Bsmn ð2; 6Þ 5

r 2 R1 Zsm ðzÞcos nθ rðR2 2 R1 Þ

Bsmn ð3; 3Þ 5 2 Bsmn ð3; 6Þ 5

1 Zsm ðzÞcos nθ R2 2 R1

1 Zsm ðzÞcos nθ R2 2 R1

Bsmn ð4; 2Þ 5 2

R2 Zsm ðzÞsin nθ rðR2 2 R1 Þ

Bsmn ð4; 3Þ 5

R2 2 r Zsm ðzÞcos0 nθ rðR2 2 R1 Þ

Bsmn ð4; 5Þ 5

R1 Zsm ðzÞsin nθ rðR2 2 R1 Þ

Bsmn ð4; 6Þ 5 2

R1 2 r Zsm ðzÞcos0 nθ rðR2 2 R1 Þ

1 0 Zsm ðzÞcos nθ R2 2 R1 R2 2 r 0 Z ðzÞcos nθ R2 2 R1 sm 1 0 Zsm ðzÞcos nθ R2 2 R1 r 2 R1 0 Z ðzÞcos nθ R2 2 R1 sm R2 2 r 0 Zsm ðzÞcos0 nθ rðR2 2 R1 Þ R2 2 r 0 Z ðzÞsin nθ R2 2 R1 sm r 2 R1 0 Zsm ðzÞcos0 nθ rðR2 2 R1 Þ r 2 R1 0 Z ðzÞsin nθ R2 2 R1 sm

Bsmn ð5; 1Þ 5 2 Bsmn ð5; 3Þ 5 Bsmn ð5; 4Þ 5 Bsmn ð5; 6Þ 5 Bsmn ð6; 1Þ 5 Bsmn ð6; 2Þ 5 Bsmn ð6; 4Þ 5 Bsmn ð6; 5Þ 5