Answers to Selected Problems

Chapter 1 Analysis of Stress

1.1. P = 4.27 kN

1.4. P = 32.5 kN, θ = 26.56°

1.5. σ_x′ = −13.15 MPa, τ_{x′ y′} = −15.67 MPa

1.6. F = 67.32 kN/m³

1.9. F_x = F_y = F_z = 0

1.10. (a) σ₁ = 121 MPa, σ = −71 MPa, τ_max = 96 MPa

σ₁ = 200 MPa, σ₂ = −50 MPa, τ_max = 125 MPa

(b)

1.15. (a)σ_x = 38.68 MPa, σ_y = 12.12 MPa, τ_xy = 7MPa

(b)σ₁ = 40.41 MPa, σ₂ = 10.39 MPa,

1.18. P = 3πpr²

1.24. (a)σ = −0.237τ_o, τ = 0.347τ_o

(b)σ₁ = 3.732τ_o,σ₂ = −0.268τ_o,

1.26. σ₁ = −σ₂ = 51.96 MPa,

1.28. σ₁ = 46.17 MPa, σ₂ = 13.83 MPa,

1.30. (a) σ_x = 100 MPa, τ_xy = −30 MPa

(b) σ₁ = 110 MPa,

1.34. P = 1069 kN, p = 467 kPa

1.36. (a)σ_x = 186 MPa, (b)σ₁ = 188 MPa, τ_max = 101 MPa,

1.39. p = 494 kPa

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

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

1.50. (a) σ₁ = 12.049 MPa, σ₂ = −1.521 MPa, σ₃ = −4.528 MPa, l₁ = 0.618, m₁ = 0.534, n₁ = 0.577

1.54. σ = 52.25 MPa, τ = 36.56 MPa

1.56. (a) τ₁₃ = 8.288 MPa, τ₁₂ = 6.785 MPa, τ₂₃ = 1.503 MPa

1.60. σ_oct = 12 MPa, τ_oct = 9.31 MPa

1.62. (a) σ₁ = 108.3 MPa, σ₂ = 51.7 MPa, σ₃ = −50 MPa,

(b) σ_G = 56 MPa, τ_G = 38 MPa

1.66. (a) τ_max = 21 MPa, (b) σ_oct = 35 MPa, τ_oct = 17.15 MPa

1.67. 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 Stress and Stress–Strain Relations

2.4. c₀ = 2a₀, c₁ = 2(a₁ + b₀)

2.6. (a) ε_x = 667 µ, ε_y = 750 µ, γ_xy = −1250µ

(b) ε₁ = 1335 µ, ε₂ = 82 µ

2.8. ε₁ = 758 µ, ε₂ = −808 µ

2.10. Δ_OB = 0.008 mm, Δ_AC = 0.016 mm

2.12. (a) γ_max = 200µ, θ_s = 45°

(b) ε_x = 350 µ, ε_y = 250 µ, γ_xy = –173 µ

2.14. ε₁ = −59 µ, ε₂ = −1141 µ,

2.18. (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 = 1370µ

2.23. Δ_BD = 0.283/E m

2.26. ε_x = 522µ, ε_y = 678 µ, γ_xy = −1873 µ

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

2.33. (b)ΔV = −2250 mm³

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

2.39. ε_x = γ(a − x)/E, ε_y = −νε_x, σ_x = γ(a − x), γ_xy = τ_xy = σ_y = 0. Yes.

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

2.48. (a) U = 60.981T²a/πd⁴G, (b) U = 2.831 kN · m

2.54. U_oυ = 3.258 kPa, U_od = 38.01 kpa

2.56.

Chapter 3 Two-Dimensional Problems in Elasticity

3.1. (b)

3.2.

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

3.16.

3.19. Yes, ε₁ = 32.8µ,

3.20. σ_x = σ_y = EαT₁/(ν − 1), ε_z = 2ναT₁/(1 − ν) + 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

3.37. k = 3

3.42. (a) σ₁ = 4.29 MPa, (b) τ_max = 2.69 MPa,

(c) σ_oct = 1.07 MPa, τ_oct = 2.32 MPa

3.44. (a) σ_c = 1240 MPa, (b) σ_c = 1959 MPa

3.46. σ_c = 418 MPa, b = 0.038 mm

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

3.50. σ_c = 1014.7 MPa

Chapter 4 Failure Criteria

4.3. (a) σ_yp = 152.6 MPa, (b) σ_yp = 134.9 MPa

4.5. t = 8.45 mm

4.6. (a) d = 27.95 mm, (b) d = 36.8 mm

4.8. (a) T = 31.74 kN · m, (b) T = 26.05 kN · m

4.11. (a) R = 932 N, (b) R = 959 N

4.13. (a) p = 6.466 MPa, (b) p = 5.6 MPa

4.17. (b) σ₁ = 75 MPa, σ₂ = − 300 MPa

4.20. (a) τ = σ_u/2, (b)

4.25. t = 9.27 mm

4.28. p = 11.11 MPa

4.31. t = 0.973 mm

4.35. P_max = 18.7 kN

4.38. 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

5.15. P = 15.6 kN

5.19. e = 4R/π

5.23. R = −13pL/32

5.25. υ = M_ox²(x − L)/4EIL, R_B = 3M_o/2L

5.32. (a) P = 11.436 kN, (b) (σ_θ)_B = −86.3 MPa

5.34. P = 1517 N

5.38. (a) σ_θ = −152.87P, (b) δ_p = 215.65P/E m

Chapter 6 Torsion of Bars

6.1. (a) τ_e > τ_c; (b) T_e > T_c

6.3. T = 256.5 kN · m

6.5. k = Gθ/2a²(b − 1)

6.8. θ_A = aT/2r⁴G, θ_B = 2θ_A

6.10.

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

6.19. (a) C = 2.1 × 10⁻⁷G, τ_max = 112,860T

6.20. θ = 0.1617 rad/m

6.22. θ = 2T/9Ga³t

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

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

Chapter 7 Numerical Methods

7.3. τ_B = 0.0107Gθ

7.6. υ(L) = 7PL³/32EI

7.10. υ_max = 0.01852pL³/EI

7.12. υ_B = −Pa³/16EI, θ_A = − Pa²/16EI

7.14. υ_max = 1.68182Ph⁴/EI, θ_max = −0.02131pL³/EI

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

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

Chapter 8 Axisymmetrically Loaded Members

8.2. p_i = 51.2 MPa

8.3. (a), (b) r_x = 27.12 mm

8.5. (a) p_i = 1.6p_o, (b) p_i = 1.16p_o

8.7. t = 0.59 m

8.11. (a) t = 0.825d_i, (b) Δd = 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

8.24. T = 101.79 N · m

8.26. ω = 36,726 rpm

8.30. (a) σ_θ,max = 108.68 MPa, (b) ω = 4300 rpm

8.34. (a) σ_θ,max = 554.6 MPa,

Chapter 9 Beams on Elastic Foundations

9.1. P = 51.18 kN

9.3.

9.5. υ_max = 3.375 mm σ_max = 103.02 MPa

9.8.

9.11. υ = – M_Lf₂(βx)/2β² EI

9.13. (a) υ = 10 mm; (b) υ_L = 15 mm, υ_R = 5 mm

9.14. υ_C = 0.186 mm

9.15. υ_C = 2.81 × 10⁻⁸ m, θ_E = 5 × 10⁻⁷ rad

Chapter 10 Energy Methods

10.3. υ_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.

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

10.18. R_Aυ = 3(λ + 1)pL₁/2 (3λ + 4), R_Ah = λpL²₁/4(3λ + 4)L₂, λ = E₂I₂L₁/E₁I₁L₂

10.21. δ_E = 41.5(P/AK)³

10.26. N_A = P/2π, M_A = PR/4

10.32. υ = Px²(3L − x)/6EI

10.33. υ = Pc²(L − c)²/4EIL

Chapter 11 Elastic Stability

11.2. (a) 6.25%, (b) P_all = 17.99 kN

11.5. (a) σ_cr = 34 MPa, (b) σ_all = 35.25 MPa

11.7. L_e = L

11.9. (a) ΔT = δ/2αL, (b) ΔT = (δ/2αL) + (π²I/4L² Aα)

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

11.13. σ_cr = 59.46 MPa

11.17. (a) σ_max = 93.71 MPa, υ_max = 150.6 mm

11.19. P = 0.89π²EI/L²

11.21. P_cr = 12EI/L²

11.23. P_cr = 9EI₁/4L²

11.24.

11.29. P_cr = 16EI₁/L²

Chapter 12 Plastic Behavior of Materials

12.2. α = 42.68°

12.5. σ_max = 3Mh/4I

12.8. M = 11ah²σ_yp/54

12.11. P = 46.18 kN

12.14. P_u = 9M_u/2L

12.17. (a) M_u = 2M_yp, (b) M_u = 16b(b³ − a³)M_yp/3π(b⁴ – a⁴)

12.22. t_o = 6.3 mm

12.27. (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 − υ)

13.6. (b) p_o = 12.05 kPa

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

13.10. n = 3.33

13.11.

13.15. σ_max = 2.48 MPa

13.16.

13.17. t = 0.791 mm

13.20.