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Solutions ManualElasticity: Theory, Applications and NumericsSecond EditionByMartin H. SaddProfessorDepartment of Mechanical Engineering & Applied MechanicsUniversity of Rhode IslandKingston, Rhode IslandForewordExercises found at the end of each chapter are an important ingredient of the text as theyprovide homework for student engagement, problems for examinations, and can be usedin class to illustrate other features of the subject matter. This solutions manual isintended to aid the instructors in their own particular use of the exercises. Review of thesolutions should help determine which problems would best serve the goals ofhomework, exams or be used in class.The author is committed to continual improvement of engineering education andwelcomes feedback from users of the text and solutions manual. Please feel free to sendcomments concerning suggested improvements or corrections to [email protected] Suchfeedback will be shared with the text user community via the publisher’s web site.Martin H. SaddJanuary 2009Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-1.(a) aii a11 a 22 a33 1 4 1 6 (scalar)aij aij a11 a11 a12 a12 a13 a13 a 21 a 21 a 22 a 22 a 23 a 23 a 31 a31 a32 a32 a33 a33 1 1 1 0 16 4 0 1 1 25 (scalar) 1 1 1 1 1 1 1 6 4 aij a jk 0 4 2 0 4 2 0 18 10 (matrix) 0 1 1 0 1 1 0 5 3 3 aij b j ai1b1 ai 2 b2 ai 3b3 4 (vector) 2 aij bi b j a11b1b1 a12 b1b2 a13b1b3 a 21b2 b1 a 22 b2 b2 a 23 b2 b3 a31b3b1 a 32 b3 b2 a33 b3b3 1 0 2 0 0 0 0 0 4 7 (scalar) b1b1bi b j b2 b1 b3 b1b1b3 1 0 2 b2 b3 0 0 0 (matrix)b3 b3 2 0 4 bi bi b1b1 b2 b2 b3b3 1 0 4 5 (scalar)b1b2b2 b2b3b2(b) aii a11 a22 a33 1 2 2 5 (scalar)aij aij a11a11 a12 a12 a13 a13 a21a21 a 22 a22 a23 a23 a31a31 a32 a32 a33 a33 1 4 0 0 4 1 0 16 4 30 (scalar) 1 2 0 1 2 0 1 6aij a jk 0 2 1 0 2 1 0 8 0 4 2 0 4 2 0 16 4 aij b j ai1b1 ai 2 b2 ai 3b3 3 (vector) 6 2 4 (matrix)8 aij bi b j a11b1b1 a12b1b2 a13b1b3 a 21b2b1 a 22b2b2 a 23b2b3 a31b3b1 a32 b3b2 a33b3b3 4 4 0 0 2 1 0 4 2 17 (scalar) b1b1 b1b2bi b j b2b1 b2b2 b3b1 b3b2b1b3 4 2 2 b2 b3 2 1 1 (matrix)b3b3 2 1 1 bi bi b1b1 b2b2 b3b3 4 1 1 6 (scalar)Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

(c) aii a11 a 22 a33 1 0 4 5 (scalar)aij aij a11a11 a12 a12 a13 a13 a21a21 a 22 a22 a23 a23 a31a31 a32 a32 a33 a33 1 1 1 1 0 4 0 1 16 25 (scalar) 1 1 1 1 1 1 2 2 7 aij a jk 1 0 2 1 0 2 1 3 9 (matrix) 0 1 4 0 1 4 1 4 18 2 aij b j ai1b1 ai 2 b2 ai 3b3 1 (vector) 1 aij bi b j a11b1b1 a12b1b2 a13b1b3 a 21b2b1 a 22b2b2 a 23b2b3 a31b3b1 a32 b3b2 a33b3b3 1 1 0 1 0 0 0 0 0 3 (scalar) b1b1 b1b2bi b j b2b1 b2b2 b3b1 b3b2b1b3 1 1 0 b2 b3 1 1 0 (matrix) b3b3 0 0 0 bi bi b1b1 b2b2 b3b3 1 1 0 2 (scalar)1-2.11(a ) aij (aij a ji ) (aij a ji )22 2 1 1 0 1 1 1 1 1 8 3 1 0 1 22 1 3 2 1 1 0 clearly a( ij ) and a[ ij ] satisfy the appropriate conditions11(b) aij (aij a ji ) (aij a ji )22 2 2 0 0 2 0 1 1 2 4 5 2 0 3 22 0 5 4 0 3 0 clearly a( ij ) and a[ ij ] satisfy the appropriate conditionsCopyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

11(c) aij (aij a ji ) (aij a ji )22 2 2 1 0 0 1 1 1 2 0 3 0 0 1 22 1 3 8 1 1 0 clearly a( ij ) and a[ ij ] satisfy the appropriate conditions1-3.aij bij a ji b ji aij bij 2aij bij 0 aij bij 0From Exercise 1 - 2(a) : a( ij ) a[ ij ] 2 1 1 0 1 1 T 1 tr 1 8 3 1 0 1 04 1 3 2 1 1 0 2 2 0 0 2 0 T 1 From Exercise 1 - 2(b) : a( ij ) a[ ij ] tr 2 4 5 2 0 3 04 0 5 4 0 3 0 T 2 2 1 0 0 1 1 From Exercise 1 - 2(c) : a( ij ) a[ ij ] tr 2 0 3 0 0 1 04 1 3 8 1 1 0 1-4. δ11a1 δ12 a2 δ13 a3 a1 δ ij a j δ i1a1 δ i 2 a2 δ i 3 a3 δ 21a1 δ 22 a2 δ 23 a3 a2 ai δ 31a1 δ 32 a2 δ 33 a3 a3 δ11a11 δ12 a21 δ13 a31 δ11a12 δ12 a22 δ13 a32 δ11a13 δ12 a23 δ13 a33 δ ij a jk a11 a21 a31a12a22a32a13 a23 aija33 Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-5.det(aij ) ε ijk a1i a2 j a3k ε123a11a22 a33 ε 231a12 a23a31 ε312 a13a21a32 ε 321a13a22 a31 ε132 a11a23a32 ε 213a12 a21a33 a11a22 a33 a12 a23a31 a13a21a32 a13a22 a31 a11a23a32 a12 a21a33 a11 (a22 a33 a23a32 ) a12 (a21a33 a23a31 ) a13 (a21a32 a22 a31 )a11 a21a31a12a22a32a13a23a331-6.0 145 o rotation about x 1 - axis Qij 02 /2 0 2 / 20 1From Exercise 1 - 1(a) : bi′ Qij b j 02 /2 0 2 / 2aij′ Qip Q jq a pq0 1 2 /2 0 0 2 / 2aij′ Qip Q jq a pqaij′ Qip Q jq a pq0 2 2 / 2 1 2 / 2 1 T 10 2 0 2 / 2 0 4 1 0 2 1 2 / 2 2 2 0 0 1 2 0 10 2 / 2 0 2 1 02 /22 / 2 0 4 2 0 2 / 20 1From Exercise 1 - 1(c) : bi′ Qij b j 02 /2 0 2 / 20 1 2 /2 0 0 2 / 20 1 1 2 / 2 0 2 2 / 2 2 2 0 1 1 1 10 2 /2 0 4 2 02 /2 2 / 2 0 1 1 0 2 / 20 1 From Exercise 1 - 1(b) : bi′ Qij b j 02 /2 0 2 / 20 1 2 /2 0 0 2 / 20 2 / 2 2 / 2 0 1 2 / 2 1 2 / 2 0 0 1 1 1 10 2 / 2 1 0 2 02 /22 / 2 0 1 4 0 2 / 2T 10 2 2 2 / 2 0 4.5 1.5 0 1.5 0.5 2 / 2 1 2 / 2 2 / 2 T 10 2 0 2 / 2 2 / 2 3.5 2.5 2 / 2 1.5 0.5 2 / 2 Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-7.cos θcos(90 o θ) cos θ sin θ cos( x1′ , x1 ) cos( x1′ , x 2 ) Qij ocos θ sin θ cos θ cos( x 2′ , x1 ) cos( x 2′ , x 2 ) cos(90 θ) cos θ sin θ b1 b1 cos θ b2 sin θ bi′ Qij b j sin θ cos θ b2 b1 sin θ b2 cos θ aij′ Qip Q jq a pq cos θ sin θ a11 sin θ cos θ a 21a12 cos θ sin θ a 22 sin θ cos θ T a cos 2 θ (a12 a 21 ) sin θ cos θ a 22 sin 2 θ a12 cos 2 θ (a11 a 22 ) sin θ cos θ a 21 sin 2 θ 112222 a 21 cos θ (a11 a 22 ) sin θ cos θ a12 sin θ a11 sin θ (a12 a 21 ) sin θ cos θ a 22 cos θ 1-8.a ' δ′ij QipQ jq aδ pq aQip Q jp aδij1-9.α' δ′ij δ′kl β' δ′ik δ′jl γ ' δ′il δ′jk QimQ jnQkpQlq (αδ mn δ pq βδ mp δ nq γδ mq δ np ) αQimQ jmQkpQlp β QimQ jnQkmQln γQimQ jnQknQlm αδij δ kl βδ ik δ jl γδil δ jk1-10.Cijkl αδ ij δ kl βδ ik δ jl γδ il δ jk αδ ij δ kl β(δ ik δ jl δ il δ jk ) αδ kl δ ij β(δ ki δ lj δ kj δ li ) C klij1-11. λ1 0If a 0 λ 2 0 00 0 λ 3 I a aii λ1 λ 2 λ 3II a λ1 0 λ 2 0 λ20λ1 0III a 0 λ 20 00 λ1 0 λ1λ 2 λ 2λ 3 λ1λ 3λ3 0 λ300 λ1λ 2λ 3λ3Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-12. 1 1 0 (a) aij 1 1 0 I a 1, II a 2 , III a 0 0 0 1 Characteristic Eqn is λ3 λ2 2λ 0 λ(λ2 λ 2) 0 λ(λ 2)(λ 1) 0Roots λ1 2 , λ 2 0 , λ 3 1λ1 2 Case : 1 1 0λ2n1(1) n2(1) 01 0 n1(1) n3(1) 01 0 n2(1) 0 n1(1) n2(1) 2 / 2 , n (1) ( 2 / 2)( 1,1, 0 )2220 3 n3(1) n1(1) n2(1) n3(1) 1 0 Case : n1( 2) n2( 2 ) 0 1 1 0 n1 1 1 0 n 0 n1 n2 2 / 2 n ( 2 ) ( 2 / 2)(1,1,0)n3( 2 ) 0 2 222 0 0 1 n3 n1( 2 ) n2( 2 ) n3( 2) 1λ 3 1 Case : 2n1(3) n2(3) 0 2 1 0 n1 1 2 0 n 0 ( 3)( 3)n1( 3) 2n2( 3) 0 n1 n2 0, n3 1 n (0,0,1) 2 222 00 0 n3 n1(3) n2(3) n3( 3) 10 1 1 The rotation matrix is given by Qij 2 / 2 1 10 and 0 0 2 / 2 aij′ Qip Q jp a pq0 1 11 1 102 0 0 2 /0 1 1 0 1 1 1 1 0 1 10 2 0 0 1 0 0 2 /T 2 0 0 0 0 0 0 0 1 2 Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-12. 2 1 0 (b) aij 1 2 0 I a 4 , II a 3 , III a 0 00 0 Characteristic Eqn is λ3 4λ2 3λ 0 λ(λ2 4λ 3) 0 λ (λ 3)(λ 1) 0Roots λ1 3 , λ 2 1, λ 3 0λ1 3 Case : 1 1 0λ2n1(1) n2(1) 01 0 n1(1) n3(1) 01 0 n2(1) 0 n1(1) n2(1) 2 / 2 , n (1) ( 2 / 2)( 1,1, 0 )2220 3 n3(1) n1(1) n2(1) n3(1) 1 1 Case : n1( 2) n2( 2 ) 0 1 1 0 n1 1 1 0 n 0 n1 n2 2 / 2 n ( 2 ) ( 2 / 2)(1,1,0)n3( 2 ) 0 2 222 0 0 1 n3 n1( 2 ) n2( 2 ) n3( 2) 1λ 3 0 Case : 2n1(3) n2(3) 0 2 1 0 n1 1 2 0 n 0 ( 3)( 3)n1( 3) 2n2( 3) 0 n1 n2 0, n3 1 n (0,0,1) 2 222 00 0 n3 n1(3) n2(3) n3( 3) 10 1 1 The rotation matrix is given by Qij 2 / 2 1 10 and 0 0 2 / 2 aij′ Qip Q jp a pq0 1 11 1 102 0 0 2 /0 2 1 0 1 1 1 2 0 1 10 2 00 0 0 0 2 /T 3 0 0 0 1 0 02 0 0 Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-12. 1 1 0 (c) aij 1 1 0 I a 2 , II a 0 , III a 0 0 0 0 Characteristic Eqn is λ3 2λ2 0 or λ2 (λ 2) 0Roots λ1 2 , λ 2 λ 3 0λ1 2 Case : 1 1 0λ2n1(1) n2(1) 01 0 n1(1) n1(1) n2(1) 2 / 2 , n (1) 2 / 2( 1,1, 0 )n3(1) 01 0 n2(1) 0 2220 2 n3(1) n1(1) n2(1) n3(1) 1 λ 3 0 Case : 1 1 0 n1 1 1 0 n 0 n1 n2 0 n n , n 2 1 2n 2 n (k,k, 1-2k 2 )2221231 2 n1 n2 n3 1 0 0 0 n3 for arbitrary k , and thus directions are not uniquely determined. For convenience we may choosek 2 / 2 and 0 to get n ( 2) 2 / 2(1,1, 0 ) and n ( 3) (0 , 0 ,1)0 1 1 The rotation matrix is given by Qij 2 / 2 1 10 0 0 2 /aij′ Qip Q jp a pq0 1 11 1 102 0 0 2 / and 2 0 1 1 0 1 1 1 1 0 1 10 2 0 0 0 0 0 2 /T 2 0 0 0 0 0 0 0 0 2 1-13*.Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-14.(a) u x1e 1 x1 x2 e 2 2 x1 x2 x3 e 3 u u1,1 u 2, 2 u3,3 1 x1 2 x1 x2e1 u / x1x1e2 / x2x1 x2e3 / x3 2 x1 x3 e 1 2 x2 x3 e 2 x2 e 32 x1 x2 x3 2 u 0e 1 0e 2 0e 3 0 1 u x2 2 x2 x30x12 x1 x30 0 , tr ( u) 1 x1 2 x1 x22 x1 x2 (b) u x12 e 1 2 x1 x2 e 2 x33 e 3 u u1,1 u 2, 2 u3,3 2 x1 2 x1 3 x32e1 u / x1x12e2 / x22 x1 x2e3 / x3 0e 1 0e 2 2 x2 e 3x33 2 u 2e 1 0e 2 6 x3 e 3 0 2 x1 u 2 x2 000 2 x1 0 , tr ( u) 4 x1 3 x320 3 x32 (c) u x22 e 1 2 x2 x3 e 2 4 x12 e 3 u u1,1 u 2, 2 u3,3 0 2 x3 0e1 u / x1x22e2 / x22 x 2 x3e3 / x3 2 x2 e 1 8 x1e 2 2 x2 e 34 x12 2 u 2e 1 0e 2 8e 3 0 0 u 0 8 x12 x22 x300 2 x 2 , tr ( u) 3 x30 Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-15.1ai ε ijk a jk2δii11ε imn ai ε ijk ε imn a jk δ ji22δ kiδimδ jmδ kmδin1δ jn a jk (δ jm δ kn δ jn δ km )a jk2δ kn11 (amn anm ) (amn amn ) amn22 a jk ε ijk ai1-16.(a) (φψ) (φψ) ,k φψ ,k φ,k ψ φψ φ ψ 2 (φψ) (φψ) ,kk (φψ ,k φ ,k ψ ) ,k φψ ,kk φ,k ψ ,k φ,k ψ ,k φ ,kk ψ φ,kk ψ φψ ,kk 2φ ,k ψ ,k ( 2 φ)ψ φ( 2 ψ ) 2 φ ψ (φu) (φu k ) ,k φu k ,k φ ,k u k φ u φ( u)(b) (φu) ε ijk ( φu k ) , j ε ijk ( φu k , j φ , j u k ) ε ijk φ , j u k φε ijk u k φ u φ( u) ( u v ) (ε ijk u j vk ) ,i ε ijk (u j vk ,i u j ,i vk ) v k ε ijk u j ,i u j ε ijk vk ,i v ( u) u ( v ) φ ε ijk (φ ,k ) , j ε ijk φ ,kj 0 because of symmetry and antisymmetry in jk φ (φ ,k ) ,k φ ,kk 2 φ(c) ( u) (ε ijk u k , j ) ,i ε ijk u k , ji 0 , because of symmetry and antisymmetry in ij ( u) ε mni (ε ijk u k , j ) ,n ε imn ε ijk u k , jn (δ mj δ nk δ mk δ nj )u k , jn u n ,nm u m ,nn ( u) 2 uu ( u) ε ijk u j (ε kmn u n ,m ) ε kij ε kmn u j u n ,m (δ im δ jn δ in δ jm )u j u n,m u n u n ,i u m ui ,m1 ( u u) u u2Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-17.Cylindrical coordinates : ξ1 r , ξ 2 θ , ξ3 z(ds ) 2 (dr ) 2 ( rdθ) 2 ( dz ) 2 h1 1 , h2 r , h3 1eˆr cos θe 1 sin θe 2 , eˆθ sin θe 1 cos θe 2 , eˆ z e 3 eˆ eˆ eˆ eˆ eˆ eˆr eˆ eˆθ , θ eˆr , r θ z z z 0 θ θ r r r θ z 1 eˆθ eˆ z eˆr r zr θ f f1 f eˆθ eˆ z f eˆr r zr θ1 1 uθ u z u ( rur ) zr rr θ21 f 1 f 2 f 2 f r r r r r 2 θ2 z 2 u 1 u z uθ ˆ u r u z ˆ 1 u e r e θ ( ruθ ) r eˆ zr r r θ z z r θCopyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-18.Spherical coordinates : ξ1 R , ξ 2 φ , ξ3 θx 1 ξ1 sin ξ 2 cos ξ3 , x 2 ξ1 sin ξ 2 sin ξ3 , x 3 ξ1 cos ξ 2Scale factors :( h1 ) 2 x k x k (sin φ cos θ) 2 (sin φ sin θ) 2 cos 2 φ 1 h1 111 ξ ξ( h2 ) 2 x k x k R 2 h2 R22 ξ ξ x k x k R 2 sin 2 φ h3 R sin φ33 ξ ξUnit vectors :eˆ R cos θ sin φe 1 sin θ sin φe 2 cos φe 3eˆφ cos θ cos φe 1 sin θ cos φe 2 sin φe 3( h3 ) 2 eˆθ sin θe 1 cos θe 2 eˆ R eˆ eˆ 0 , R eˆφ , R sin φeˆθ R φ θ eˆφ eˆ eˆ 0 , φ eˆr , φ cos φeˆθ R φ θ eˆθ eˆ eˆ 0 , θ 0 , θ cos φeˆφ R φ θUsing (1.9.12) - (1.9.16) 1 1 eˆφ eˆθ eˆR RR φR sin φ θ f1 f1 f eˆφ eˆz f eˆR RR φR sin φ θ1 1 1 ( R 2 sin φuR ) 2( R sin φuφ ) 2( Ruθ ) u 2R sin φ RR sin φ φR sin φ θ1 1 1 2( R 2uR ) (sin φuφ ) ( uθ )R RR sin φ φR sin φ θ 2 f f 1111 f 2 f (sin φ ) 2() R sin φ 2 φ R sin φ θ sin φ θ R R sin φ φR sin φ R 2 f 2 f1 2 f 11 φ (sin)R φ R 2 sin 2 φ θ2R 2 R R R 2 sin φ φCopyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

1-18. Continued 1 1 [ ( R sin φuθ ) ( Ruφ )] eˆR [ (u R ) ( R sin φuθ )] eˆφ u 2 θ R R sin φ φ R sin φ θ 1 [( Ruφ ) (u R ) eˆθ φ R R u 1 1 u R 1 (sin φuθ ) φ eˆ R ( Ruθ ) eˆφ θ R sin φ θ R R R sin φ φ 1 u ( Ruφ ) R eˆθ φ R RCopyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

2-1.(a) u Axy , v Bxz 2 , w C ( x 2 y 2 )1 2Cx Ay 2 ( Ax Bz )1eij (u i , j u j ,i ) Bxz Cy 0 2 0 1 Cx 0( Ax Bz 2 ) 2 11Bxz Cy ωij (u i , j u j ,i ) ( Ax Bz 2 )02 2Cx Bxz Cy0 (b) u Ax 2 , v Bxy , w Cxyz11 2 Ax 2 By 2 Cyz 11eij (u i , j u j ,i ) BxCxz 22 Cxy 11 By Cyz 022 111ωij (u i , j u j ,i ) By Cxz 022 211 CyzCxz0 22(c) u Ayz 3 , v Bxy 2 , w C ( x 2 z 2 )eij ωij 1(u i , j2 0 u j ,i ) 1(u i , j2 0 1 u j ,i ) ( Az 3 By 2 )) 2 1 (3 Ayz 2 2Cx ) 21( Az 3 By 2 )22 Bxy 1 (3 Ayz 2 2Cx ) 2 0 2Cz 1( Az 3 By 2 )2001 (3 Ayz 2 2Cx ) 2 0 0 Copyright 2009, Elsevier Inc. All rights reserved.Full file at ty-2nd-Edition-Martin-Sa

Elasticity: Theory, Applications and Numerics Second Edition . By . Martin H. Sadd . Professor . Department of Mechanical Engineering & Applied Mechanics . University of Rhode Island . Kingston, Rhode Island . Foreword . Exercises found at the end of each chapter are an important ingredient of the text as they