Answers to Problems

CHAPTER 22.1 (b) SijSij ¼ 28, (c) SjiSji ¼ SijSij ¼ 28, (d) SjkSkj ¼ 23, (g) Snmaman ¼ Smnaman ¼ 59.

2.3 (a) b1 ¼ 2; b2 ¼ 2; b3 ¼ 2. (b) s ¼ 6.

2.4 (c) Eij ¼ BmiCmkFkj.

2.7 i ¼ 1 ! a1 ¼ @v1@t

þ v1@v1@x1

þ v2@v1@x2

þ v3@v1@x3

; etc:

2.10 d1 ¼ 6; d2 ¼ �3; d3 ¼ 2.

2.12 (2) For i ¼ k; LS ¼ RS ¼0 if j 6¼ l0 if j ¼ l ¼ i1 if j ¼ l 6¼ i

8<:

2.20 (b) T½ � ¼0 0 1

0 0 �1

�1 1 0

24

35.

2.21 (c) T aþ bð Þ ¼ 10e1.

2.22 T½ � ¼2 0 �1

0 1 3

1 3 0

24

35.

2.23 T½ � ¼�1=2 0 1=2�1=2 0 1=20 0 0

24

35.

2.24 (a) T½ � ¼1 0 0

0 �1 0

0 0 1

24

35, (b) T½ � ¼

1 0 0

0 1 0

0 0 �1

24

35.

2.25 (a) R½ � ¼1 0 0

0 cos y �sin y0 sin y cos y

24

35, (b) R½ � ¼

cos y 0 sin y0 1 0

�sin y 0 cos y

24

35.

2.26 (b) T½ � ¼ 1

3

1 �2 �2

�2 1 �2

�2 �2 1

24

35, (c) Ta ¼ � 3e1 þ 2e2 þ e3ð Þ.

2.27 ½T� ¼ 1

3

1 �2 �2

�2 1 �2

�2 �2 1

24

35.

Copyright © 2010, Elsevier Ltd. All rights reserved.

2.28 (b) n ¼ e1 þ e2 þ e3ð Þ=ffiffiffi3

p.

2.29 T½ � ¼ 1

3

1þ 2 cos y 1� cos yð Þ � ffiffiffi3

psin y 1� cos yð Þ þ ffiffiffi

3p

sin y1� cos yð Þ þ ffiffiffi

3p

sin y 1þ 2 cos yð Þ 1� cos yð Þ � ffiffiffi3

psin y

1� cos yð Þ � ffiffiffiffi3

psin y 1� cos yð Þ þ ffiffiffi

3p

sin y 1þ 2 cos yð Þ

24

35.

2.30 (b) RA ¼ sin yE.

2.31 (a) S½ � ¼1 0 0

0 �1=ffiffiffi2

p �1=ffiffiffi2

p0 �1=

ffiffiffi2

p1=

ffiffiffi2

p

24

35; (b) T½ � ¼

1 0 0

0 �1=ffiffiffi2

p1=

ffiffiffi2

p0 1=

ffiffiffi2

p1=

ffiffiffi2

p

24

35, (d) c½ � ¼

1

1=ffiffiffi2

p5=

ffiffiffi2

p

24

35.

2.37 a ¼ 2e 01.

2.38 (b) a ¼ e 01 þ

ffiffiffi3

pe 02.

2.39 T 011 ¼ 4=5; T 0

12 ¼ �15=ffiffiffi5

p; T 0

31 ¼ 2=5.

2.40 (a) T 0ij

h i¼ T½ � 0 ¼

0 �5 0

�5 1 5

0 5 1

24

35.

2.42 (b) TijTij ¼ 45, (c) T½ � 0 ¼2 5 1

2 3 1

0 0 1

24

35.

2.48 (a) TS� � ¼ 1 3 5

3 5 7

5 7 9

24

35, TA

� � ¼ 0 �1 �2

1 0 �1

2 1 0

24

35, (b) tA ¼ e1 � 2e2 þ e3.

2.50 (d) For l ¼ 1; n ¼ a1e1 þ a2e2 � a1 þ a2ð Þe3½ �=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 þ a22 þ a23

q.

2.55 y ¼ 120o.

2.56 (c) For l ¼ 1; n ¼ �e3. (d) For l ¼ �1; n ¼ a1e1 þ a2e2; a21 þ a22 ¼ 1, (e) y ¼ p.

2.59 (a) For l1 ¼ 3; n1 ¼ �e3. For l2 ¼ �3; n2 ¼ � e1 � 2e2ð Þ=ffiffiffi5

p.

2.60 (a) For l1 ¼ 3; n1 ¼ �e1. For l2 ¼ 4; n2 ¼ � e2 þ e3ð Þ=ffiffiffi2

p.

2.61 For l1 ¼ 0; n1 ¼ � e1 � e2ð Þ=ffiffiffi2

p. For l2 ¼ l3 ¼ 2; n ¼ � ae1 þ ae2 þ a3e3ð Þ; 2a2 þ a23 ¼ 1.

2.65 (b) At ð0; 0; 0Þ; ðdf=drÞmax ¼ jrfj ¼ 2 in the direction of n ¼ e3.

At ð1; 0; 1Þ; ðdf=drÞmax ¼ jrfj ¼ 17 in the direction of n ¼ 2e1 þ 3e2 þ 2e3ð Þ=ffiffiffiffiffi17

p.

2.67 (a) q ¼ �3k e1 þ e2ð Þ; (b) q ¼ � 3ke1 þ 6ke2ð Þ.2.69 (a) ½rv�ð1;1;0Þ ¼ 2½I�, (b) rvð Þv ¼ 2e1, (c) div v ¼ 2; curl v ¼ 2e1, (d) dv ¼ 2ds e1 þ e3ð Þ.

2.71 ru½ � ¼�A=r2 �B 0

B A=r2 0

0 0 0

24

35.

2 Answers to Problems

2.72 div u ¼ 3A.

2.73 ru½ � ¼A� 2B=r3 0 0

0 Aþ B=r3 0

0 0 Aþ B=r3

24

35.

2.77 div Tð Þr ¼ div Tð Þy ¼ div Tð Þz ¼ 0.

CHAPTER 3

3.1 (b) v1 ¼ kx11þ kt

; v2 ¼ 0; v3 ¼ 0.

3.2 (a) v1 ¼ a; v2 ¼ v3 ¼ 0; a1 ¼ a2 ¼ a3 ¼ 0;(b) y ¼ A atþ X1ð Þ. Dy=Dt ¼ Aa, (c) y ¼ BX2; Dy=Dt ¼ 0.

3.3 (b) v1 ¼ 0; v2 ¼ 2bX21t; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 2bX2

1; v3 ¼ 0,

(c) v1 ¼ 0; v2 ¼ 2bx21t; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 2bx21; a3 ¼ 0.

3.4 (b) v1 ¼ 2bX22t; v2 ¼ kX2; v3 ¼ 0 and a1 ¼ 2bX2

2; a2 ¼ 0; a3 ¼ 0,

(c) v1 ¼ 2bx22t=ð1þ ktÞ2; v2 ¼ kx2=ð1þ ktÞ; v3 ¼ 0; a1 ¼ 2bx22= 1þ ktð Þ2; a2 ¼ a3 ¼ 0.

3.5 (b) v1 ¼ k sþ X1ð Þ; v2 ¼ 0; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 0; a3 ¼ 0,

(c) v1 ¼ k sþ x1ð Þ= 1þ ktð Þ; v2 ¼ 0; v3 ¼ 0 and a1 ¼ 0; a2 ¼ 0; a3 ¼ 0.

3.6 (b) For X1;X2;X3ð Þ ¼ 1; 3; 1ð Þ and t ¼ 2; v1 ¼ �4 3ð Þ2 2ð Þ ¼ �72; v2 ¼ �1; v3 ¼ 0:(c) For x1; x2; x3ð Þ ¼ 1; 3; 1ð Þ and t ¼ 2; v1 ¼ �200; v2 ¼ �1; v3 ¼ 0:

3.7 (a) For X1;X2;X3ð Þ ¼ 1; 1; 0ð Þ and t ¼ 2; v1 ¼ 2k; v2 ¼ 2k; v3 ¼ 0:(b) For x1; x2; x3ð Þ ¼ 1; 1; 0ð Þ and t ¼ 2; v1 ¼ 2k= 1þ 4kð Þ; v3 ¼ 0:

3.8 (a) t ¼ 2 ! x1 ¼ 5; x2 ¼ 3; x3 ¼ 0, (b) X1 ¼ �3; X2 ¼ 1; X3 ¼ 2,

(c) a1 ¼ 18; a2 ¼ 0; a3 ¼ 0, (d) a1 ¼ 2; a2 ¼ 0; a3 ¼ 0.

3.9 (b) ai ¼ 0.

3.10 (a) a ¼ �4xex � 4yey, (b) x2 þ y2 ¼ constant ¼ X2 þ Y2.

Or, x ¼ �Y sin 2tþ X cos 2t and y ¼ Y cos 2tþ X sin 2t.

3.11 (a) a ¼ k2 xex þ yey� �

, (b) x ¼ Xekt; y ¼ Ye�kt. Or xy ¼ XY.

3.12 Material description: a ¼ 2k2 x2 þ y2� �

xex þ yey� �

.

3.14 (b) a1 ¼ 0; a2 ¼ �p2 sin ptð Þ sin p X1ð Þ; a3 ¼ 0.

3.15 (b) a ¼ �ða2ffiffiffi2

p=4Þer; DY=Dt ¼ 2ak.

3.16 (b) a ¼ �ða2ffiffiffi2

p=4Þer; DY=Dt ¼ 0.

3.17 (b) ds1=dS1 ¼ ð1=ffiffiffi2

pÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ kð Þ2 þ 1

q¼ ds2=dS2,

cos p=2� gð Þ ¼ sin g ¼ f� 1þ kð Þ2 þ 1g=f 1þ kð Þ2 þ 1g.

Answers to Problems 3

(c) For k ¼ 1; ds1=dS1 ¼ ds2=dS2 ¼ffiffiffiffiffiffiffiffi5=2

p; sin g ¼ �3=5.

For k ¼ 10�2; ds1=dS1 ¼ ds2=dS2 � 1:005; g ¼ �0:0099 rad: (d) 2E 012 ¼ �0:01.

3.19 (a) E½ � ¼0 0 k=20 k 0k=2 0 0

" #, (b) 10�5=2.

3.20 (a) E11 ¼ 5k ¼ 5� 10�4; E22 ¼ 2k ¼ 2� 10�4; 2E12 ¼ k ¼ 10�4rad.

3.21 (a) E 011 ¼ 10�4=3.

3.22 (a) E 011 ¼ ð58=9Þ � 10�4, (b) 2E 0

12 ¼ ð32=ffiffiffiffiffi45

pÞ � 10�4rad:

3.23 (a) E 011 ¼ ð37=25Þ � 10�4, (b) 2E 00

12 ¼ ð72=25Þ � 10�4rad:

3.24 (a) I1 ¼ 11� 10�4; I2 ¼ 31� 10�8; I3 ¼ 17� 10�12.

3.25 I1 ¼ 0; I2 ¼ �t2; I3 ¼ 0.

3.26 At 1; 0; 0ð Þ; lmax ¼ 3k ¼ 3� 10�6.

3.27 (a) D dVð Þ=dV ¼ 0, (b) k1 ¼ 2k2.

3.28 (b) At 1; 2; 1ð Þ; E 011 ¼ k; (c) max elongation ¼ 4k; (d) DV ¼ k.

3.32 E11 ¼ a; E22 ¼ c; E12 ¼ b� aþ cð Þ=2.3.33 (a) E12 ¼ �100� 10�6: (b) For l1 ¼ 261:8� 10�6; y ¼ �31:7o, or

n ¼ 0:851e1 � 0:525e2. For l2 ¼ 38:2� 10�6; y ¼ 58:3o, or n ¼ 0:525e1 þ 0:851e2.

3.34 (a) E12 ¼ 0, (b) Prin. strains are 10�3 in any direction lying on the plane of e1 and e2.

3.35 E11 ¼ a; E22 ¼ 2bþ 2c� að Þ=3; E12 ¼ b� cð Þ=ffiffiffi3

p.

3.36 E11 ¼ 2� 10�6; E22 ¼ 1� 10�6; E12 ¼ ½1=ð2ffiffiffi3

p� � 10�6.

3.37 E11 ¼ 2� 10�3; E22 ¼ 2� 10�3; E12 ¼ 0.

3.38 (a) D½ � ¼0 kx2 0kx2 0 00 0 0

24

35; W½ � ¼

0 kx2 0�kx2 0 00 0 0

24

35, (b) D nð Þ nð Þ ¼ 3k.

3.39 D11 ¼ �a 1þ kð Þ; D 011 ¼ 1þ kð Þ=2.

3.40 (a) D12 ¼ p cos t cos px1ð Þ=2; W12 ¼ �W21 ¼ � p cos t cos px1ð Þ=2.(b) D11 ¼ 0, D22 ¼ 0, D 0

11 ¼ p=2.

3.42 (a) Drr ¼ �1=r2; Dyy ¼ 1=r2; other Dij ¼ 0; W½ � ¼ 0½ �. (b) Drr ¼ �1=r2.

3.43 At r ¼ 2; ar ¼ �18; ay ¼ 0; (b) D½ � ¼ 0 �1

�1 0

� �.

3.44 (a) ar ¼ �ðAr þ B=r2Þ2sin2y=r; ay ¼ �cos y sin yðAr þ B=r2Þ2=r; af ¼ 0.

(b) Drf ¼ �ð3B=2r3Þsiny; Dyf ¼ 0.

4 Answers to Problems

3.45 W½ � ¼ 0:

3.49 k ¼ 1.

3.50 f ¼ g yð Þ=r.3.51 vy ¼ � k=2ð Þsin y=

ffiffir

p.

3.53 v1 ¼ f x2ð Þ; v2 ¼ 0:

3.54 (a) r ¼ ro 1þ ktð Þ�a=k; (b) r ¼ r*xo=x1.

3.55 r ¼ roe�at2 .

3.60 (b) 2kX1X2 ¼ f X2;X3ð Þ þ g X1;X3ð Þ.

3.62@r@t

þ vr@r@r

þ vyr

@r@y

þ vz

@r@z

� �þ r

@vr@r

þ vrrþ 1

r

@vy@y

þ @vz@z

¼ 0.

3.63 (b) U½ � ¼1 0 0

0 2 0

0 0 3

24

35; (c) B½ � ¼

9 0 0

0 1 0

0 0 4

24

35, (d) R½ � ¼

0 0 1

�1 0 0

0 �1 0

24

35,

(e) E*� � ¼ 0 0 0

0 3=2 0

0 0 4

24

35, (f) e*

� � ¼ 4=9 0 0

0 0 0

0 0 3=8

24

35, (g)

DVDVo

¼ 6, (h) dA ¼ �3e3.

3.64 (b) U½ � ¼1 0 0

0 2 0

0 0 3

24

35, (c) B½ � ¼

4 0 0

0 9 0

0 0 1

24

35, (d) R½ � ¼

0 1 0

0 0 1

1 0 0

24

35,

(e) E*� � ¼ 0 0 0

0 3=2 0

0 0 4

24

35, (f) e*

� � ¼ 3=8 0 0

0 4=9 0

0 0 0

24

35, (g)

DVDVo

¼ 6, (h) dA ¼ 3e1.

3.65 (b) U½ � ¼1 0 0

0 2 0

0 0 3

24

35, (c) B½ � ¼

1 0 0

0 9 0

0 0 4

24

35, (d) R½ � ¼

1 0 0

0 0 1

0 �1 0

24

35,

(e) E*� � ¼ 0 0 0

0 3=2 0

0 0 4

24

35, (f) e*

� � ¼ 0 0 0

0 4=9 0

0 0 3=8

24

35, (g)

DVDVo

¼ 6,

(h) dA ¼ �3e3.

3.66 (b) U½ � ¼1 0 0

0 2 0

0 0 3

24

35, (c) B½ � ¼

4 0 0

0 1 0

0 0 9

24

35, (d) R½ � ¼

0 1 0

�1 0 0

0 0 1

24

35,

(e) E*� � ¼ 0 0 0

0 3=2 0

0 0 4

24

35; (f) e*

� � ¼ 3=8 0 0

0 0 0

0 0 4=9

24

35, (g)

DVDVo

¼ 6,

(h) dA ¼ 3e1.

Answers to Problems 5

3.67 (a) C½ � ¼1 3 0

3 10 0

0 0 1

24

35: (b) For l1 ¼ 10:908326; n1 ¼ 0:289785e1 þ 0:957093e2.

For l2 ¼ 0:0916735; n2 ¼ 0:957093e1 � 0:289784e2: For l3 ¼ 1; n3 ¼ e3,

(c) U½ �ni ¼3:30277 0 0

0 0:302774 0

0 0 1

24

35, (d) U½ �ei ¼

0:554704 0:832057 0

0:832057 3:05087 0

0 0 1

24

35,

½U�1�ei ¼3:050852 �0:832052 0

�0:832052 0:554701 0

0 0 1

24

35, (d) R½ �ei ¼

0:55470 0:83205 0

�0:83205 0:55470 0

0 0 1

24

35.

3.70 (a) 3, 2 and 0.6, (b) ds=dSð Þ ¼ffiffiffiffiffiffiffiffiffiffi13=2

p, (c) cos y ¼ 0. No change in angle.

3.71 (a) U½ � ¼ 1ffiffiffi5

p2 1 0

1 3 0

0 0ffiffiffi5

p

24

35, (b)

ffiffiffiffiffiffiffiC22

p¼

ffiffiffi2

p, (c)

ds

dS¼

ffiffiffiffiffiffiffiffi5=2

p, (d) cos y ¼ 1ffiffiffi

2p :

3.72 (a) U½ � ¼ 1ffiffiffi2

p1 1 0

1 3 0

0 0ffiffiffi2

p

24

35, (b)

ffiffiffiffiffiffiffiC22

p¼

ffiffiffi5

p, (c)

ds

dS¼

ffiffiffi5

p, (d) cos y ¼ 2ffiffiffi

5p :

3.77 B�1rr ¼ @ro

@r

2

þ ro@yo@r

2

þ @zo@r

2

; B�1yy ¼ @ro

r@y

2

þ ro@yor@y

2

þ @zor@y

2

.

3.80 C�1royo ¼

ro@yo@r

@ro@r

þ ro@yo

r@y

@ror@y

þ ro@yo

@z

@ro@z

.

3.81 Bry ¼ @r

@X

r@y@X

þ @r

@Y

r@y@Y

þ @r

@Z

r@y@Z

.

3.82 B�1ry ¼ @X

@r

@X

r@y

þ @Y

@r

@Y

r@y

þ @Z

@r

@Z

r@y

.

3.84 (a) B½ � ¼1 0 0

0 1þ rkð Þ2 rk0 rk 1

24

35; (b) C½ � ¼

1 0 0

0 1 rk0 rk 1þ rkð Þ2

24

35.

3.85 (a) B½ � ¼a=rð Þ2 0 0

0 r=að Þ2 0

0 0 1

24

35, (b) det B ¼ 1, no change of volume.

3.86 C½ � ¼f ðXÞð Þ2 0 0

0 gðYÞð Þ2 0

0 0 hðZÞð Þ2

24

35.

6 Answers to Problems

CHAPTER 44.1 (a) 1 MPa; 4 MPa; 0 MPa: (b) 3:61MPa; 5:39 MPa; 5:83 MPa.

4.2 (a) t ¼ 1=3ð Þ 5e1 þ 6e2 þ 5e3ð Þ: (b) Tn ¼ 3 MPa; Ts ¼ 0:745 MPa:

4.3 (a) t ¼ 3:47e1 � 2:41e2: (b) Tn ¼ 2:21 MPa; Ts ¼ 3:60 MPa:

4.4 t ¼ 25ffiffiffi3

pe1 þ 25e2 � 25

ffiffiffi3

pe3.

4.5 (a) t ¼ e3. (b) n21 � n22 ¼ 0, including n ¼ e3; n ¼ e1 þ e2ð Þ=ffiffiffi2

p; n ¼ e1 � e2ð Þ=

ffiffiffi2

p.

4.6 T 011 ¼ �6:43 MPa; T 0

13 ¼ 18:6 MPa.

4.7 (a) te1 ¼ ax2e1 þ be2. (b) FR ¼ 0e1 þ 4 be2; Mo ¼ � 4a=3ð Þe3.4.8 (a) te1 ¼ ax22e1. (b) FR ¼ 4a=3ð Þe1; Mo ¼ 0.

4.9 (a) te1 ¼ ae1 þ ax3e2. (b) FR ¼ 4ae1; Mo ¼ � 4a=3ð Þe1.4.10 (a) tn1 ¼ 0, tn2 ¼ ax3e2 � ax2e3; tn3 ¼ �ax3e2 þ ax2e3: (b) FR ¼ 0; Mo ¼ 8pae1.

4.11 (b) FR ¼ 0; Mo ¼ �p= 2ffiffiffi2

p �.

4.12 (a) tr S ¼ 0. (b) S½ � ¼0 500 �200

500 �300 400

�200 400 300

24

35kPa:

4.13 (a) 4.

4.14 (b) T12 ¼ T21.

4.17 fmax ¼ 2.

4.21 (a) Ts ¼ 0: (b) For Tmax ¼ 100 MPa; n1 ¼ e1 þ e2ð Þ=ffiffiffi2

p. For Tmin ¼ �100 MPa;

n2 ¼ e1 � e2ð Þ=ffiffiffi2

p: (c) Tsð Þmax ¼ 100 MPa, on the planes e1 and e2.

4.23 (a) Tsð Þmax ¼ 150 MPa; n ¼ ðe1 � e3Þ=ffiffiffi2

p; (b) Tn ¼ 250 MPa:

4.24 T33 ¼ 1 and T11 ¼ 1.

4.25 (a) Tn ¼ 800=9 ¼ 88:89 kPa; Ts ¼ 260 kPa;(b) Tsð Þmax ¼ 300 kPa:

4.26 (a) tn ¼ 5=ffiffiffi2

p �e1 þ e2ð Þ, (b) Tn ¼ 5 MPa;

(d) Tnð Þmax ¼ 5 MPa; n1 ¼ 1=ffiffiffi2

p �e1 þ e2ð Þ;

Tnð Þmin ¼ �3 MPa; n2 ¼ 1=ffiffiffi2

p �e1 � e2ð Þ. Tsð Þmax ¼ 4 MPa, on n ¼ e1 and n ¼ e2.

4.27 (a) For l1 ¼ t; n1 ¼ 1=ffiffiffi2

p �e1 þ e2ð Þ.

For l2 ¼ �t; n2 ¼ 1=ffiffiffi2

p �e1 � e2ð Þ.

For l3 ¼ 0; n3 ¼ e3. (b) Tsð Þmax ¼ t; n ¼ e1 and n ¼ e2.

Answers to Problems 7

4.29 T12 � T21 ¼ M*3; T13 � T31 ¼ M*

2 and T23 � T32 ¼ M*1.

4.30 (b) T12 ¼ 2x1 � x2 þ 3.

4.31 T33 ¼ 1þ rg=að Þx3 þ f x1; x2ð Þ.4.32 (a) C ¼ �1, (b) A ¼ 1;B ¼ 2.

4.36Tyrr

þ @Tyr@r

þ Tryr

þ 1

r

@Tyy@y

þ @Tyz@z

þ rBy ¼ ray.

4.39 B ¼ ðpo � piÞr2i r2o=ðr2o � r2i Þ; A ¼ ðpir2i � por2oÞ=ðr2o � r2i Þ.

4.41 A ¼ �ðpor3o � pir3i Þ=ðr3o � r3i Þ; B ¼ �ðpo � piÞr3or3i =½2ðr3o � r3i Þ�.

4.42 (a) To½ � ¼1000=16 0 0

0 0 0

0 0 0

24

35MPa; to ¼ 1000=16ð Þe1.

(b) ~T� � ¼ 1000=256 0 0

0 0 0

0 0 0

24

35MPa; ~t ¼ 1000=256ð Þe1.

4.44 (a) dV ¼ 1=4; dA ¼ 1=16ð Þe1,

(b) To½ � ¼100=16 0 0

0 0 0

0 0 0

24

35MPa; to ¼ 100=16ð Þe1MPa:

(c) ~T� � ¼ 100=64 0 0

0 0 0

0 0 0

24

35MPa; ~t ¼ 100=64ð Þe1MPa; d~f ¼ 100

64

e1.

4.45 (a) dV ¼ dVo ¼ 1; dA ¼ e1 � ke2,

(b) To½ � ¼�100k 100 0

100 0 0

0 0 0

24

35MPa; to ¼ 100 �ke1 þ e2ð Þ MPa; t ¼ 100ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ k2p �ke1 þ e2ð Þ.

(c) ~T� � ¼ �200k 100 0

100 0 0

0 0 0

24

35MPa; ~t ¼ 100 �2ke1 þ e2ð ÞMPa; d~f ¼ 100 �2ke1 þ e2ð Þ.

4.46 (a) dV ¼ 8dVo ¼ 8. dA ¼ 4e1

(b) To½ � ¼400 0 0

0 400 0

0 0 400

24

35MPa; to ¼ 400e1 MPa; t ¼ 100e1 MPa:

(c) ~T� � ¼ 200 0 0

0 200 0

0 0 200

24

35MPa; ~t ¼ 200e1 MPa; d~f ¼ 200e1.

8 Answers to Problems

CHAPTER 55.3 EY=l ! 0; m ! EY=3.

5.4 m ¼ EY

2 1þ nð Þ ; k ¼ 2m 1þ nð Þ3 1� 2nð Þ.

5.9 l ¼ 81:7 GPa 11:8� 106psi� �

; m ¼ 38:4 GPa 5:56� 106psi� �

; k ¼ 107:3 GPa 15:6� 106psi� �

.

5.10 n ¼ 0:27; l ¼ 89:1 GPa 12:9� 106psi� �

; k ¼ 140 GPa 20:3� 106psi� �

.

5.11 T½ � ¼17:7 1:9 4:751:9 18:4 0

4:75 0 16:0

24

35MPa:

5.13 (a) E½ � ¼0:483 0:253 0:3800:253 �1:41 0

0:380 0 1:12

24

35� 10�3,

(b) e ¼ 0:193� 10�3; DV ¼ 24:1� 10�3cm3.

5.14 DV ¼ 2:96� 10�3:

5.17 (a) T11 ¼ T22 ¼ T33 ¼ 0; T12 ¼ T21 ¼ 2mkx3; T13 ¼ T31 ¼ mk 2x1 þ x2ð Þ; T23 ¼ T32 ¼ mk x1 � 2x2ð Þ.

5.19 (a) T½ � ¼ 2klx3 mx3 mx2mx3 lx3 mx1mx2 mx1 lþ 2mð Þx3

24

35.

5.21 For n ¼ 1=3; cL=cT ¼ 2; n ¼ 0:49; cL=cT ¼ 7:14; n ¼ 0:499; cL=cT ¼ 22:4.

5.24 (c) a ¼ 1;(d) b ¼ np= 2ℓð Þ; n ¼ 1; 3; 5 . . .

5.25 (c) a ¼ 1;(d) b ¼ np=ℓ; n ¼ 1; 2; 3 . . .

5.28 (d) b ¼ np=ℓ; n ¼ 1; 2; 3 . . .

5.30 (a) u1 ¼ 3e5sin

2pℓf

� �; u2 ¼ 4e

5sin

2pℓf

� �; f x1; x2tð Þ ¼ 3x1

5þ 4x2

5� cLt� �

.

5.32 (a) a2 ¼ a3 ¼ 0, e2 ¼ e1, and (b) a3 ¼ 31:17o; e2 ¼ 0:742e1; e3 ¼ 0:503e1.

5.35 (b) e3=e1 ¼ �sin 2a1=cos a1 � a3ð Þ; e2=e1 ¼ cos a1 þ a3ð Þ=cos a1 � a3ð Þ.

5.38 (a) u1 ¼ a cosox1cL

þ tanoℓcL

sinox1cL

cos ot;

(b) oℓ=cL ¼ np=2; n ¼ 1; 3; 5 . . .

5.40 (a) u3 ¼ a cos ox1=cTð Þ � cot oℓ=cTð Þsin ox1=cTð Þ½ �cos ot,(b) o ¼ npcT=ℓ; n ¼ 1; 2; 3 . . .

5.42 (a) Tnð Þmax ¼ 71:4� 106N; Tsð Þmax ¼ 23:7� 106N; bð Þdℓ ¼ 1:39� 10�3m.

Answers to Problems 9

5.44 (a) Tn ¼ s cos2 a; Ts ¼ s sin 2a=2,(b) (i) a ¼ p=2; Ts ¼ Tn ¼ 0; and iið Þ a ¼ p=4; Ts ¼ Tn ¼ s=2;(c) s � 2to=sin 2a.

5.46 (a) To11 ¼ 2s=3; To

22 ¼ To33 ¼ �s=3; To

12 ¼ To13 ¼ To

23 ¼ 0;

(b) I1 ¼ 0; I2 ¼ �s2=3; I3 ¼ 2s3=27.

5.49 M1 ¼ Mtℓ2= ℓ1 þ ℓ2ð Þ; M2 ¼ Mtℓ1= ℓ1 þ ℓ2ð Þ.

5.51 Tnð Þmax ¼ ½sþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ 4b2r2

q�=2; Ts ¼ ½

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ 4b2r2

q�=2; s ¼ P=A; b ¼ Mt=Ip.

5.53 (a) Mtð Þell= Mtð Þcir ¼ 2, (b) a 0ell=a

0cir ¼ 5=16.

5.54 (b) C ¼ a0=6a; (c) T12 ¼ 0; T13 ¼ 0 at all three corners; along x3 ¼ 0,

T12 ¼ 0; T13 ¼ ma0=2að Þ 2ax2 þ x22� �

, (d) Ts ¼ 3a=2ð Þma0 at x2; x3ð Þ ¼ a; 0ð Þ.

5.57 Mt ¼ ma0

3

2að Þ3 2bð Þ 1� 192

p5a

b

� X1n¼1;3;5

1

n5tan h

npb2a

" #.

5.60 Neutral axis in the direction of M2e2 þ I22=I33ð ÞM3e3.

5.63 (b) T11 ¼ 2a3; T12 ¼ �a2; T22 ¼ 2a1;(c) x1 ¼ 0; t ¼ �2a3e1 þ a2e2; x1 ¼ b; t ¼ 2a3e1 � a2e2;(d) T33 ¼ 2n a3 þ a1ð Þ; T13 ¼ T23 ¼ 0; Ei3 ¼ 0; E11 ¼ 2 1=EYð Þ 1� n2

� �a3 � n 1þ nð Þa1

� �,

(e) Ti3 ¼ 0; E13 ¼ E23 ¼ 0; E33 ¼ �2 n=EYð Þ a3 þ a1ð Þ; E11 ¼ 2 1=EYð Þ a3 � na1ð Þ.5.66 (b) T11 ¼ 2ax1 þ 6x1x2; T22 ¼ 0; T12 ¼ �2ax2 � 3x22;

ðcÞ a ¼ �3c=2; ðdÞ tx1¼0 ¼ 3x2 x2 � cð Þe2; tx1¼b ¼ 3b 2x2 � cð Þe1 � 3x2 x2 � cð Þe2; tx2¼0 ¼ 0.

5.67 u1 ¼ Px21x22EYI

þ nPx326EYI

� Px326mI

þ P

2mI

h

2

2

x2 � c1x2 þ c3.

5.69 (a) T12 ¼ 2Am� lmcð Þcos h lmcf gsin h lmx2 þ sin h lmc lmx2 cos h lmx2ð Þ

sin h 2lmcþ 2lmc

� �sin lmx1.

5.72 ur ¼ 1þ nð ÞEY

�A

rþ 2B 1� 2nð Þr ln r � Br þ 2C 1� 2nð Þr

� �þ H sin yþ G cos y,

uy ¼ 1=EYð Þ 4Bry 1� nð Þ 1þ nð Þ½ � þ H cos y� G sin yþ Fr:

5.74 r; r�1; r3 and r ln r.

5.84 Trr ¼ � 1� 2nð ÞzR3

þ 3r2z

R5; Tyy ¼ � 1� 2nð Þz

R3; Trz ¼ 1� 2nð Þr

R3þ 3rz2

R5

,

Tzz ¼ 3z3

R5þ 1� 2nð Þ z

R3.

5.87 Txx ¼ � Fz

2p3x2z

R5� 1� 2nð Þz

R3þ 1� 2nð ÞR Rþ zð Þ �

1� 2nð ÞR Rþ zð Þ

1

R

x2

Rþ zð Þ þx2

R2

� �� �.

10 Answers to Problems

5.88 Tzz ¼ qoz3

r2o þ z2� �3=2 � qo.

5.101 C11 ¼ 1

DE2E3

1� n32n23ð Þ; C12 ¼ 1

DE2E3

n21 þ n31n23ð Þ; C13 ¼ 1

DE2E3

n31 þ n21n32ð Þ,

C23 ¼ 1

DE1E3

n32 þ n31n12ð Þ; D ¼ 1� 2n13n21n32 � n13n31 � n23n32 � n21n12½ �E1E2E3

.

5.112 Brr ¼ a=rð Þ2; Byy ¼ rcð Þ2; Bzz ¼ 1; Bry ¼ 0; Brz ¼ 0, Byz ¼ 0.

5.113 Brr ¼ l21; Bry ¼ Brz ¼ 0; Byy ¼ l21 þ rKð Þ2; Bzz ¼ l23; Bzy ¼ Byz ¼ l3rK.

CHAPTER 6

6.1 RB ¼ 5:1� 104N:

6.2 h ¼ 2:48m.

6.3 h2 ¼ r1h1 � r3h3ð Þ=r2.6.5 (b) Fx ¼ g 2r2L

� �. Fx is 2r/3 above the ground. Fy is 4r/3p left of the diameter.

6.6 p� pa ¼ rðgþ aÞh:6.8 h ¼ aℓ=g.

6.10 h1 � h2 ¼ o2ðr21 � r22Þ=ð2gÞ.6.12 (A) for n 6¼ 1; p n�1ð Þ=n ¼ p�1= n�1ð Þ

o po � fðn� 1Þ=ngrog z� zoð Þ½ �n= n�1ð Þ.

(B) for n ¼ 1, p ¼ po exp �rop�1=no g z� zoð Þ

h i.

6.14 (b) Tn ¼ mk � p; T2s ¼ 0, (c) any plane n1; 0; n3ð Þ and n1; n2; 0ð Þ.

6.16 (a) �Tnð Þ � p ¼ 44m=5; (b) Ts ¼ 8m=5.

6.20 (a) x2 ¼ a2; (b) x1 ¼ 1þ kX2t

1þ kX2toX1 and x2 ¼ X2.

6.22 (a) x21 þ x22 ¼ a21 þ a22, (b) x21 þ x22 ¼ X21 þ X2

2, time history:

x1 ¼ X2 sin otþ X1 cos ot; x2 ¼ X2 cos ot� X1 sin ot.

6.23 (a) y ¼ yo, (b) y ¼ Y, time history: r2 ¼ R2 þ Qt=ðpÞ.6.26 v ¼ ða=2mÞðx2d � x22Þ þ vox2=d; Q ¼ ad3=ð12mÞ þ vod

2=ð2dÞ.6.27 mv ¼ rg sin y d � x2=2ð Þx2.

6.29 m1vtð Þ ¼ �a

x222� b

2

m2 � m1m1 þ m2

x2 � b2

m1m1 þ m2

� �.

6.32 (b) v ¼ 1

4mdp

dzr2 þ a2 � b2ð Þ

ln b=að Þ ln r þ b2ln a� a2ln bð Þln b=að Þ

� �.

Answers to Problems 11

6.34 A ¼ ba2b2=f2ða2 þ b2Þg; B ¼ �A.

6.36 wave length ¼ffiffiffiffiffiffi2p

p=103 ¼ 2:51� 10�3m.

6.38 (b) A ¼ Qm=ð2pÞ.

6.40 Y ¼ � m3

12ka2vodþ a

2m

d � 2x2ð Þ

� �4þ Cx2 þ D.

D ¼ Yo þ m3

12ka2vodþ ad2m

4

.

6.42 Y ¼ � mB2

kr2þ C ln r þ D; C ¼ r2o � r2i

r2i r2o

mB2

k

�ln

riro

:

6.44 (b) T11 ¼ �pþ 2mk; T22 ¼ �p� 2mk; T33 ¼ �p; T12 ¼ T13 ¼ T23 ¼ 0;

(c) a1 ¼ k2x1; a2 ¼ k2x2; a3 ¼ 0, (d) p ¼ �ðr=2Þðv21 þ v22Þ þ po, (f) F ¼ 4mk2;(h) the nonslip boundary condition at x2 ¼ 0 is not satisfied for a viscous fluid.

6.46 Ans. B ¼ � 1=mð Þ @p=@x1ð Þx2e3.

6.48 (c)dy

dx

’¼constant

dy

dx

c¼constant

¼ �1: (d) B ¼ � @2c@y2

þ @2c@x2

ez.

6.49@2c@y2

þ @2c@x2

¼ 0.

6.51 Q ¼ A1A2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 p1 � p2ð Þ � rgh½ �= r A2

1 � A22

� �� �q:

CHAPTER 7

7.1

ðv � ndS ¼

ðdiv vdV ¼ 16.

7.3

ðdiv vdV ¼

ðv � ndS ¼ 64p.

7.9 (b) m ¼ 3roe�a t�toð ÞA; dm=dt ¼ �3aroe

�a t�toð ÞA.

7.11 (b) m ¼ kAro ln 3; dm=dt ¼ 0.

7.14 (a)dP

dt¼ � 9

2a2Aroe

�a t�toð Þe1; (b) 9Aroa2e�a t�toð Þ, (c) F ¼ 9

2Aroa

2e�a t�toð Þe1.

7.15 (a) P ¼ 2rokaAe1; (b) 2kroAa2; (c) F ¼ 2kroAa

2e1.

7.17 ðrC2pr6o=3Þe1; ðprC2r6o=4Þe1.

7.19 gx ¼ xd2x

dt2þ dx

dt

2

.

12 Answers to Problems

7.21 Force from water to the bend is Fw ¼ 1100e1 � 282e2N:

7.22 rAv2o e1.

7.24 Force on the vane is Fvane ¼ rA vo � vð Þ2 1� cos yð Þe1 � sin ye2½ �.

CHAPTER 8

8.3 S ¼ Geþ �dedt

; S=eo ¼ GHðtÞ þ �dðtÞ.

8.4 (b)S

eo¼ �2oG

ð� þ �oÞ2e

�Gt�þ�o þ ��o

ð� þ �oÞdðtÞ.

8.5 S12 ¼ mvoð1� e�t=lÞ.

8.8 G 0 ¼ð1

l¼o

l2o2H lð Þl 1þ l2o2� � dl; G00 ¼

ð1l¼o

loH lð Þl 1þ l2o2� � dl .

8.11 (b) Ct½ � ¼1 0 0

0 1 0

0 0 1

24

35þ

0 k 0

k 0 0

0 0 0

24

35 t� tð Þ þ

2k2 0 0

0 0 0

0 0 0

24

35 t� tð Þ2

2; k dv=dx1.

8.12 (b) Ct½ � ¼e�2k t�tð Þ 0 0

0 e2k t�tð Þ 0

0 0 1

24

35 ¼

I½ � þ t� tð Þ�2k 0 0

0 2k 0

0 0 0

24

35þ

4k2 0 0

0 4k2 0

0 0 0

24

35 t� tð Þ2

2þ

�8k3 0 0

0 8k3 0

0 0 0

24

35 t� tð Þ3

3!þ . . . :

8.13 (b) Ct½ � ¼e2k t�tð Þ 0 0

0 e2k t�tð Þ 0

0 0 e�4k t�tð Þ

24

35 ¼ I½ � þ t� tð Þ

2k 0 0

0 2k 0

0 0 �4k

24

35

þ4k2 0 0

0 4k2 0

0 0 16k2

24

35 t� tð Þ2

2þ

8k3 0 0

0 8k3 0

0 0 �64k3

24

35 t� tð Þ3

3!þ . . . :

8.14 (a) x 0 ¼ x1 cos h k t� tð Þ þ x2 sin h k t� tð Þ; x 0 ¼ x1 sin h k t� tð Þ þ x2 cos h k t� tð Þ,

(b) Ct½ � ¼cos h2fk t� tð Þg þ sin h2fk t� tð Þg sin hf2k t� tð Þg 0

sin hf2k t� tð Þg sin h2fk t� tð Þg þ cos h2fk t� tð Þg 0

0 0 1

264

375

¼ I½ � þ0 2k 0

2k 0 0

0 0 0

24

35 t� tð Þ þ

4k2 0 0

0 4k2 0

0 0 0

24

35 t� tð Þ2

2þ

0 8k3 0

8k3 0 0

0 0 0

24

35 t� tð Þ3

6þ . . . :

Answers to Problems 13

8.20 (a) A1½ � ¼0 k rð Þ 0

k rð Þ 0 0

0 0 0

24

35; k ¼ dv

dr� v rð Þ

r

.

(b) rA1ð Þrry ¼ � 2k

r; rA1ð Þryy ¼ 0; rA1ð Þrzy ¼ 0; rA1ð Þyry ¼ 0;

rA1ð Þyyy ¼2k

r; rA1ð Þyzy ¼ 0; rA1ð Þzry ¼ 0; rA1ð Þzyy ¼ 0; rA1ð Þzzy ¼ 0.

(c) A2 ¼2k2 0 0

0 0 0

0 0 0

24

35.

8.23 m S12k

¼ �ð10

sf2 sð Þd; s1 ¼ S11 � S22 ¼ �k2ð10

s2f2 sð Þds; s2 ¼ S22 � S33 ¼ 0.

8.27 (a) Corotational stress rate is: T∘h i

¼ mk2 0

0 �mk2

� �;

(c) T∘ ¼ mk2

cos 2otð Þ sin 2otð Þsin 2otð Þ �cos 2otð Þ

� �; (d) T

∘*

h i¼ Q½ � T∘

h iQ½ �T .

8.28 (b) The corotational derivative of T: rk v22 � v21� �

I.

8.30 T½ � ¼ �p I½ � þ f1 k2=4; 0� � 0 k=2 0

k=2 0 0

0 0 0

24

35þ f2 k2=4; 0

� � k2=4 0 0

0 k2=4 0

0 0 0

24

35.

8.36 S12 ¼ mkAðkÞ ; s1 S11� S22 ¼ 2lmk2

AðkÞ ; s2 ¼ S22� S33 ¼�lmk2 1þ að ÞAðkÞ ; AðkÞ ¼ 1þ 1� a2

� �lkð Þ2

h i.

8.37 � kð Þ ¼ S12=k ¼ m; s1 ¼ T11 � T22 ¼ 2mk2 l1 � l2ð Þ; s1 ¼ T22 � T33 ¼ 0.

8.38 � kð Þ ¼ S12k

¼ m 1þ l2mok2ð Þ

1þ l1mok2ð Þ ; s1 ¼ T11 � T22 ¼ 2mk2 l1 � l2ð Þ1þ l1mok2ð Þ ; s2 ¼ T22 � T33 ¼ 0.

8.40 (a) Szr ¼ t kð Þ; Szz � Srr ¼ s1 kð Þ; Srr � Syy ¼ s2 kð Þ; Szy ¼ Sry ¼ 0.

14 Answers to Problems