Answers to Selected Problems

Chapter 1 Analysis of Stress

1.1. P = 4.27 kN

1.4. P = 32.5 kN, θ = 26.56°

1.6. P_all = 38.3 kN

1.8. σ_x′ = 156 MPa, τ_x′y′ = –131 MPa

1.10. σ_x′ = –13.15 MPa, τ_x′y′ = –15.67 MPa

1.11. F = 67.32 kN/m³

1.14. F_x = F_y = F_z = 0

1.21.

a. σ = 153.3 MPa,

b. σ = 23.1 MPa

1.26.

a. σ₁ = 121 MPa, σ = –71 MPa, τ_max = 96 MPa

b. σ₁ = 200 MPa, σ₂ = –50 MPa, τ_max = 125 MPa

1.31.

a. σ_x = 38.68 MPa, σ_y = 12.12 MPa, τ_xy = 7 MPa

b. σ₁ = 40.41 MPa, σ₂ = 10.39 MPa,

1.34. P = 3πpr²

1.37. τ_max = 48 MPa, θ_p = 29°

1.38. σ_w = 61.5 MPa, τ_w = 26.3 MPa

1.42.

a. σ = –0.237τ_o, τ = 0.347τ_o

b. σ₁ = 3.732τ_o, σ₂ = –0.268τ_o,

1.44. σ₁ = –σ₂ = 51.96 MPa,

1.46. σ₁ = 46.17 MPa, σ₂ = 13.83 MPa,

1.49.

a. σ_x = 100 MPa, τ_xy = –30 MPa

b. σ₁ = 110 MPa,

1.53. P = 1069 kN, p = 467 kPa

1.55.

a. σ_x = 186 MPa,

b. σ₁ = 188 MPa, τ_max = 101 MPa,

1.58. p = 494 kPa

1.62. σ₁ = 66.06 MPa, l₁ = 0.956, m₁ = 0.169, n₁ = 0.242

1.64. σ₁ = 24.747 MPa, σ₂ = 8.48 MPa, σ₃ = 2.773 MPa, l₁ = 0.647, m₁ = 0.396, n₁ = 0.652

1.69.

a. σ₁ = 12.049 MPa, σ₂ = –1.521 MPa, σ₃ = –4.528 MPa, l₁ = 0.618, m₁ = 0.533, n₁ = 0.577

1.73. σ = 52.25 MPa, τ = 36.56 MPa

1.75.

a. τ₁₃ = 8.288 MPa, τ₁₂ = 6.785 MPa, τ₂₃ = 1.503 MPa

1.79. σ_oct = 12 MPa, τ_oct = 9.31 MPa

1.82.

a. σ₁ = 108.3 MPa, σ₂ = 51.7 MPa, σ₃ = –50 MPa,

1.85. σ = –12.39 MPa, τ = 26.2 MPa, p_x = 16.81 MPa, p_y = –3.88 MPa, p_z = –23.30 MPa

Chapter 2 Strain and Material Properties

2.4. c₀ = 2a₀, c₁ = 2(a₁ + b₀)

2.9.

a. ε_x = 667 μ, ε_y = 750 μ, γ_xy = –1250 μ

b. ε₁ = 1335 μ, ε₂ = 82 μ,

2.11. ε₁ = 758 μ, ε₂ = –808 μ,

2.13. Δ_QB = 0.008 mm, Δ_AC = 0.016 mm

2.15.

a. γ_max = 200 μ, θ_s = 45°

b. ε_x = 350 μ, ε_y = 250 μ, γ_xy = –173 μ

2.17. ε₁ = –59 μ, ε₂ = –1141 μ,

2.21.

a. J₁ = –3 × 10^–4, J₂ = –44 × 10^–8, J₃ = 58 × 10^–12

b. ε_x′ = 385 μ

c. ε₁ = 598 μ, ε₂ = –126 μ, ε₃ = –772 μ

d. γ_max = 1370 μ

2.28.

a. ΔL = 5.98 mm,

b. Δd = –0.014 mm

2.33.

a. ε₁ = 1222 μ, ε₂ = 58 μ

b. γ_max = 1164 μ,

c. (γ_max)t = 1222 μ

2.36. Δ_BD = 0.283/E m

2.39. ε_x = 522 μ, ε_y = 678 μ, γ_xy = –1873 μ

2.41. σ_x = 72 MPa, σ_y = 88 MPa, σ_z = 40 MPa, τ_xy = 16 MPa, τ_yz = 64 MPa, τ_xz = 0

2.46. (b) ΔV = –2250 mm³

2.48. σ₃: σ₂: σ₁ = 1 : 1.086:1.171, σ₁ = 139.947 MPa, σ₂ = 129.757 MPa, σ₃ = 119.513 MPa

2.52. ε_x = γ(a – x)/E, ε_y = –vε_x, σ_x = γ(a – x), γ_xy = τ_xy = σ_y = 0. Yes.

2.55. U₁ = P²L/2EA, U₂ = 5U₁/8, U₃ = 5U₁/12

2.61.

a. U = 60.981T²a/πd⁴G,

b. U = 2.831 kN · m

2.67. U_ov = 3.258 kPa, U_od = 38.01 kPa

2.69.

Chapter 3 Problems in Elasticity

3.1.

3.2.

3.14. All conditions, except on edge x = L, are satisfied.

3.16.

3.19. Yes, ε₁ = 32.8 μ,

3.20. σ_x = σ_y = EαT₁/(v – 1), ε_z = 2vαT₁/(1 – v) + αT₁

3.24. P_x = –161.3 kN

3.31.

a. (σ_x)_elast. = P/0.512L, (σ_x)_elem. = P/0.536L

b. (σ_x)_elast. = P/1.48L, (σ_x)_elem. = P/3.464L

3.33.

a. (σ)_elast. = 19.43F/L, (σ_x)_elem. = 20.89F/L

(τ_xy)_elast. = 5.21F/L, (τ_xy)_elem. = 2.8F/L

3.38. r = 6.67 mm, d = 26.7 mm

3.42. T = 153.5 N · m

3.47.

a. σ₁ = 36.5 MPa,

b. τ_max = 16.96 MPa,

c. σ_oct = 11.3 MPa, τ_oct = 17.85 MPa

3.49.

a. σ_c = 1233 MPa,

b. σ_c = 1959 MPa

3.51. σ_c = 418 MPa, b = 0.038 mm

3.53. a = 2.994 mm, b = 1.605 mm, σ_c = 505.2 MPa

3.58. σ_c = 1014.7 MPa

Chapter 4 Failure Criteria

4.1. P_all = 707 kN

4.3.

a. Yes,

b. No

4.6.

a. σ_yp = 152.6 MPa,

b. σ_yp = 134.9 MPa

4.8. t = 8.45 mm

4.9.

a. d = 27.95 mm,

b. d = 36.8 mm

4.11.

a. T = 31.74 kN · m,

b. T = 26.05 kN · m

4.14.

a. R = 932 N,

b. R = 959 N

4.16.

a. p = 6.466 MPa,

b. p = 5.6 MPa

4.22. (b) σ₁ = 75 MPa, σ₂ = –300 MPa

4.25.

a. τ = σ_u/2,

b.

4.30. t = 9.27 mm

4.36. p = 11.11 MPa

4.39. t = 0.973 mm

4.43. P_max = 18.7 kN

4.46. τ_max = 274.3 MPa, φ_max = 4.76°

Chapter 5 Bending of Beams

5.4. M_o = 266.8 N · m

5.6.

a. P = 3.6 kN,

b. P = 3.36 kN

5.9.

a. φ = –12.17°,

b. σ_A = 136.5 MPa

5.10. (b) σ_x = px³/Lth²,

(c) (σ_x)_elast. = 0.998(σ_x)_elem.

5.13. p = 3.88 kN/m

5.14. P = 9320 N

5.16. (a) σ_A = 35.3 MPa,

(b) σ_B = 23.5 MPa

(b) r_Z = –205.48 m

5.19.

a. τ_max/σ_max = h/L,

b. p_all = 10.34 kN/m

5.20. P = 15.6 kN

5.24. e = 4R/π

5.28. R = –13pL/32

5.30. v = M_ox²(x – L)/4EIL, R_B = 3M_o/2L

5.34. P = 1.614 kN

5.35.

a. σ_max = –177 MPa,

b. σ_max = –176.2 MPa

5.45.

a. σ_θ = –182.3P,

b. δ_p = 215.65P/E m

Chapter 6 Torsion of Prismatic Bars

6.2. D = 60.9 mm

6.6. (τ_max)_B = 130.4 MPa

6.13. T = 1.571 kN · m

6.14.

a. τ_e > τ_c;

b. T_e > T_c

6.16. T = 256.5 kN · m

6.18. k = Gθ/2a²(b –1)

6.21. θ_A = aT/2r⁴G, θ_B = 2θ_A

6.23.

6.26. τ_max = 76.8 MPa, θ = 0.192 rad/m

6.32.

a. C = 2.1 × 10^–7G, τ_max = 112,860T

6.33. θ = 0.1617 rad/m

6.35. θ = 2T/9Ga³t

6.37. τ_max = 5.279 MPa, θ = 0.0131 rad/m

6.39. τ₂ = τ₄ = τ_max = 50.88 MPa, θ = 0.01914 rad/m

Chapter 7 Numerical Methods

7.3. τ_B = 0.0107Gθ

7.6. v(L) = 7PL³/32EI

7.10. v_max = 0.01852pL³/EI

7.12. v_B = –Pa³/16EI, θ_A = –Pa²/16EI

7.14. v_max = 1.68182ph⁴/EI, θ_max = –0.02131pL³/EI

7.15. R_A = R_B = P/2, M = PL/8

7.18. (b) u₂ = 2PL/11AE, u₃ = –3PL/11AE

(c) R₁ = 2P/11, R₄ = 9P/11

7.19. (b) u₂ = PL/9AE, u₃ = PL/18AE

(c) R₁ = 2P/3, R₄ = P/3

7.24. (c) u₂ = 0.9 mm, v₂ = –3.02 mm

(d) R_1x = 4.5 kN, R_1y = 6 kN, R_3y = –4.5 kN

(e) F₁₂ = –7.5 kN, F₂₃ = 36 kN

7.27. v₂ = –5pL⁴/384EI

7.29. v₂ = –5pL³/48EI, θ₂ = PL²/8EI

7.32. (c) v₃ = –7PL³/12EI, θ₃ = –3PL²/4EI, θ₂ = –PL²/4EI

(d) R₁ = –3P/2, R₂ = 5P/2, M₁ = PL/2, M₂ = –PL

7.35. (c) u₂ = 1.0 mm, v₂ = –3.9 mm

(d) R_1x = 7.63 kN, R_1y = 10.18 kN, R_3x = –7.5 kN, R_3y = 0

7.42. {σ_x, σ_y, τ_xy}_a = {66.46, 6.65, –92.12} MPa

Chapter 8 Axisymmetrically Loaded Members

8.2. pi = 51.2 MPa

8.3.

a.

b. r_x = 27.12 mm

8.5.

a. pi = 1.6p_o,

b. pi = 1.16p_o

8.7. t = 0.59 m

8.11.

a. t = 0.825d_i,

b. Δ_d = 0.0074 mm

8.13.

a.

b.

8.16. T = 5.073 kN · m

8.21. Δd_s = 0.23δ₀ m

8.23. σ_θ,max = 1.95E_bE_s(T₂ – T₁)/(E_s + 3E_b)10⁵

8.26.

a. pi = 155.6 MPa,

b. pi = 179.3 MPa

8.30. Shaft: σ_θ = σ_r = –49.4 MPa

Cylinder: σ_θ,max = 63.8 MPa, σ_r,max = 49.4 MPa

8.32. T = 101.79 N · m

8.36. ω = 36,726 rpm

8.40.

a. σ_θ,max = 108.68 MPa,

b. ω = 4300 rpm

8.44.

a. σ_θ,max = 554.6 MPa

Chapter 9 Beams on Elastic Foundations

9.1. P = 51.18 kN

9.3.

9.5. v_max = 3.375 mm, σ_max = 103.02 MPa

9.6. P = 41.7 kN

9.8. σ_max = 196.3 kN

9.11.

9.14. v = –M_Lf2(βx)/2β²EI

9.16.

a. v = 10 mm;

b. v_L = 15 mm, v_R = 5 mm

9.17. v_C = 0.186 mm

9.18. v_C = 2.81 × 10^–8 m, θ_E = 5 × 10^–7 rad

Chapter 10 Applications of Energy Methods

10.3. v_p = (11Pc₁a⁴/12E) + (7Pc₂a³/3E) + (Pa³/3EI₂)

10.6. δ_A = 3PL³/16EI, θ_A = 5PL²/16EI

10.9. δ_D = Fab²/8EI

10.11.

10.14. δ_D = 8.657WL/AE

10.17. v_c = pL⁴/96EI

10.18. N_BD = 0.826P, N_AB = –0.131P, N_BC = 0.22P

10.23. R_A = –3M_o/2L, R_B = 3M_o/2L, M_A = M_o/2

10.27. R_A = 4p_oL/10, M_A = p_oL²/15, R_B = p_oL/10

10.29.

10.32. δ_E = 41.5(P/AK)³

10.37. N_A = P/2π, M_A = PR/4

10.43. v = Px²(3L – x)/6EI

10.44. v = Pc²(L – c)²/4EIL

Chapter 11 Stability of Columns

11.1. F = 357.5 kN

11.3. F_all = 12.93 kN

11.5. σ_cr = 183.2 MPa

11.9. L = 4.625 m

11.12.

a. 6.25%,

b. P_all = 17.99 kN

11.15.

a. σ_cr = 34 MPa,

b. σ_all = 35.25 MPa

11.17. L_e = L

11.19.

a. ΔT = δ/2αL,

b. ΔT = (δ/2αL) + (π²I/4L²Aα)

11.21. Bar BC fails as a column; P_cr = 1297 N

11.23. σ_cr = 59.46 MPa

11.31.

a. σ_max = 93.71 MPa, v_max = 150.6 mm

11.33. P = 0.89π²EI/L²

11.35. P_cr = 12EI/L²

11.37. P_cr = 9EI₁/4L²

11.38.

11.43. P_cr = 16EI₁/L²

Chapter 12 Plastic Behavior of Materials

12.2. α = 42.68°

12.5. σ_max = 3Mh/4I

12.7.

a. P_max = 471 kN,

b. (δ_AB)_p = 0, (δ_BC)_p = 7.66 mm

12.9. P = 189.8 kN

12.13. M = 11ah²σ_yp/54

12.15.

a. M_yp = 15.36 kN · m, M = 21.12 kN · m

12.18. f = 1.7

12.19. f = 1.4

12.22. P = 46.18 kN

12.25. P_u = 9M_u/2L

12.28.

a. M_u = 2M_yp,

b. M_u = 16b(b³ – a³)M_yp/3π(b⁴ – a⁴)

12.36. t_o = 6.3 mm

12.41.

a. p_u = 25.53 MPa,

b. p_u = 29.48 MPa

Chapter 13 Plates and Shells

13.2. ε_max = 1250 μ, σ_max = 274.7 MPa

13.4. (b) w = M_a(x² – y²)/2D(1 – v)

13.6. (b) po = 12.05 kPa

13.8. w = M_o(a² – r²)/2D(1 + v), σ_r,max = σ_θ,max = 6M_o/t²

13.10. n = 3.33

13.11.

13.12. d = 3.85 mm, M_max = 320 N

13.16.

a. t = 18.26 mm,

b. w_max = 0.336 mm

13.21. p = 1.2 MPa

13.23.

a. t = 11.11 mm,

b. t = 15.6 mm,

c. t = 17.1 mm

13.29. σ_max = 2.48 MPa

13.30.

13.31. t = 0.791 mm

13.34.