100 RCC Numerical Questions and Answer for GATE Exam

This 100 Questions and answers for the gate examination will provide you good and effective way to solve the numerical for GATE EXAM.

1. Slab Design

1. Calculate the thickness of a one-way slab with a span of 3 m subjected to a total load of 6 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 6 kN/m², L = 3 m, h ≈ sqrt((8 * 6 * 3²) / 25) = 0.36 m.

2. Determine the required depth of a two-way slab of span 4 m x 4 m subjected to a live load of 5 kN/m² and a dead load of 4 kN/m². Assume f_y = 500 MPa and f_c = 30 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 9 kN/m², L = 4 m, d = (9 * 4²) / (8 * 30) = 0.50 m.

3. Calculate the area of steel required for a one-way slab of span 5 m and width 1.2 m subjected to a live load of 6 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Moment M = (w * L²) / 8. For w = 6 kN/m², L = 5 m, M = (6 * 5²) / 8 = 18.75 kNm.

4. Determine the effective depth for a slab of span 4 m with a total load of 7 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 7 kN/m², L = 4 m, d = (7 * 4²) / (8 * 25) = 0.56 m.

5. Calculate the thickness of a two-way slab of span 5 m x 5 m with a total load of 6 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 6 kN/m², L = 5 m, h ≈ sqrt((8 * 6 * 5²) / 25) = 0.47 m.

2. Beam Design

6. Find the bending moment capacity of a beam with a width of 250 mm, effective depth of 500 mm, and reinforcement area of 1500 mm². Assume f_y = 415 MPa and f_c = 25 MPa.

Bending moment capacity M = 0.87 * f_y * A_s * (d - a/2). For A_s = 1500 mm², d = 500 mm, M = 0.87 * 415 * 1500 * (500 - 0.42 * 500 / 2) = 192.2 kNm.

7. Determine the shear force in a simply supported beam of span 5 m subjected to a point load of 50 kN at the center.

Shear force at supports V_max = P / 2. For P = 50 kN, V_max = 50 / 2 = 25 kN.

8. Calculate the moment of inertia for a beam of dimensions 350 mm x 600 mm.

Moment of inertia I = (b * h³) / 12. For b = 350 mm, h = 600 mm, I = (350 * 600³) / 12 = 7.56 x 10¹⁰ mm⁴.

9. Find the effective depth of a beam with a width of 300 mm and subjected to a bending moment of 60 kNm. Assume f_y = 415 MPa and f_c = 30 MPa.

Effective depth d = M / (0.87 * f_y * b). For M = 60 kNm, b = 300 mm, d ≈ (60 * 10⁶) / (0.87 * 415 * 300) ≈ 435 mm.

10. Determine the maximum bending moment for a cantilever beam of length 2.5 m subjected to a uniformly distributed load of 4 kN/m.

Maximum bending moment M_max = (w * L²) / 2. For w = 4 kN/m, L = 2.5 m, M_max = (4 * 2.5²) / 2 = 12.5 kNm.

11. Calculate the shear force at a section 3 m from the fixed end of a cantilever beam of 5 m subjected to a point load of 20 kN at the free end.

Shear force V = P. For P = 20 kN, V = 20 kN.

12. Determine the required area of steel for a beam with a width of 300 mm and an effective depth of 600 mm subjected to a moment of 80 kNm. Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For M = 80 kNm, d = 600 mm, calculate A_s.

13. Calculate the maximum shear force for a simply supported beam of span 4 m with a central load of 30 kN.

Shear force at supports V_max = P / 2. For P = 30 kN, V_max = 30 / 2 = 15 kN.

14. Find the effective depth for a beam of span 7 m subjected to a uniformly distributed load of 10 kN/m. Assume f_y = 500 MPa and f_c = 30 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 10 kN/m, L = 7 m, d = (10 * 7²) / (8 * 30) = 0.54 m.

15. Calculate the moment of inertia for a beam with dimensions 400 mm x 500 mm.

Moment of inertia I = (b * h³) / 12. For b = 400 mm, h = 500 mm, I = (400 * 500³) / 12 = 2.08 x 10¹⁰ mm⁴.

3. Column Design

16. Determine the axial load capacity of a column with a cross-sectional area of 4000 mm², effective height of 3 m, and concrete strength of 25 MPa. Assume a safety factor of 1.5.

Axial load capacity P = A * f_c / SF. For A = 4000 mm², f_c = 25 MPa, SF = 1.5, P = 4000 * 25 / 1.5 = 66.67 kN.

17. Calculate the slenderness ratio of a column with a height of 4 m and an effective length of 3.5 m.

Slenderness ratio λ = h / L_e. For h = 4 m, L_e = 3.5 m, λ = 4 / 3.5 = 1.14.

18. Find the required cross-sectional area of a column if the applied axial load is 200 kN and the concrete strength is 30 MPa. Assume a safety factor of 1.5.

Required area A = P / (f_c / SF). For P = 200 kN, f_c = 30 MPa, SF = 1.5, A = 200 / (30 / 1.5) = 10,000 mm².

19. Determine the maximum bending moment for a column subjected to a moment of 30 kNm and an axial load of 100 kN.

Maximum bending moment M_max = M + (P * e). For M = 30 kNm, P = 100 kN, e = 0 (no eccentricity), M_max = 30 kNm.

20. Calculate the effective length of a column with a height of 6 m and a slenderness ratio of 10.

Effective length L_e = h / λ. For h = 6 m, λ = 10, L_e = 6 / 10 = 0.6 m.

4. Design of Foundations

21. Determine the bearing capacity of soil if the load on a footing is 500 kN and the area of the footing is 2 m².

Bearing capacity q = P / A. For P = 500 kN, A = 2 m², q = 500 / 2 = 250 kN/m².

22. Calculate the settlement of a footing of size 1.5 m x 1.5 m with a load of 200 kN on a soil with a settlement modulus of 25 MPa.

Settlement S = P / (A * E_settlement). For P = 200 kN, A = 1.5 * 1.5 m², E_settlement = 25 MPa, S = 200 / (1.5 * 1.5 * 25) = 1.11 mm.

23. Find the depth of a strip footing if the width of the footing is 1.2 m and the allowable bearing capacity of soil is 200 kN/m². The load on the footing is 80 kN.

Depth d = P / (B * q). For P = 80 kN, B = 1.2 m, q = 200 kN/m², d = 80 / (1.2 * 200) = 0.33 m.

24. Calculate the size of a raft foundation if the total load on the foundation is 600 kN and the soil bearing capacity is 150 kN/m².

Size A = P / q. For P = 600 kN, q = 150 kN/m², A = 600 / 150 = 4 m².

25. Determine the factor of safety of a footing if the ultimate load is 500 kN and the allowable load is 400 kN.

Factor of safety FS = Ultimate Load / Allowable Load. For ultimate load = 500 kN, allowable load = 400 kN, FS = 500 / 400 = 1.25.

5. Structural Analysis

26. Calculate the deflection of a simply supported beam of span 6 m subjected to a point load of 20 kN at the center.

Deflection δ = (P * L³) / (48 * E * I). For P = 20 kN, L = 6 m, use E and I to calculate δ.

27. Find the maximum bending moment in a cantilever beam of length 4 m with a uniformly distributed load of 10 kN/m.

Maximum bending moment M_max = (w * L²) / 2. For w = 10 kN/m, L = 4 m, M_max = (10 * 4²) / 2 = 80 kNm.

28. Determine the shear force at the support of a simply supported beam of length 5 m with a point load of 25 kN at 2 m from one end.

Shear force V = P * (L - x) / L. For P = 25 kN, L = 5 m, x = 2 m, V = 25 * (5 - 2) / 5 = 15 kN.

29. Calculate the moment of inertia of a beam with a rectangular cross-section of width 300 mm and height 500 mm.

Moment of inertia I = (b * h³) / 12. For b = 300 mm, h = 500 mm, I = (300 * 500³) / 12 = 1.04 x 10¹⁰ mm⁴.

30. Find the reaction at the supports of a simply supported beam with a span of 8 m subjected to a uniformly distributed load of 12 kN/m.

Reaction R = w * L / 2. For w = 12 kN/m, L = 8 m, R = 12 * 8 / 2 = 48 kN.

6. Mixed Problems

31. A one-way slab of span 4 m is subjected to a total load of 5 kN/m². Calculate the required depth of the slab.

Effective depth d = (w * L²) / (8 * f_c). For w = 5 kN/m², L = 4 m, d = (5 * 4²) / (8 * 25) = 0.40 m.

32. Determine the size of a rectangular column with a load capacity of 300 kN and a concrete strength of 20 MPa.

Size A = P / f_c. For P = 300 kN, f_c = 20 MPa, A = 300 / 20 = 15,000 mm².

33. Calculate the depth of a beam of width 250 mm and effective depth 500 mm subjected to a moment of 70 kNm. Assume f_y = 415 MPa and f_c = 25 MPa.

Effective depth d = M / (0.87 * f_y * b). For M = 70 kNm, b = 250 mm, d ≈ (70 * 10⁶) / (0.87 * 415 * 250) ≈ 550 mm.

34. Determine the maximum shear force in a cantilever beam of length 3 m subjected to a point load of 15 kN at the free end.

Maximum shear force V = P. For P = 15 kN, V = 15 kN.

35. Calculate the bending moment for a simply supported beam of length 6 m with a central load of 30 kN.

Bending moment M = (P * L) / 4. For P = 30 kN, L = 6 m, M = (30 * 6) / 4 = 45 kNm.

36. Find the maximum deflection of a simply supported beam of span 5 m subjected to a uniformly distributed load of 8 kN/m.

Maximum deflection δ = (5 * w * L⁴) / (384 * E * I). For w = 8 kN/m, L = 5 m, use E and I to calculate δ.

37. Determine the shear force at a section 2 m from the support of a simply supported beam of span 6 m with a point load of 40 kN at the center.

Shear force V = P / 2. For P = 40 kN, V = 40 / 2 = 20 kN.

38. Calculate the area of steel required for a slab of span 5 m with a total load of 7 kN/m². Assume f_y = 500 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). Calculate A_s based on the given load and dimensions.

39. Find the effective thickness of a slab of span 6 m subjected to a live load of 9 kN/m². Assume f_y = 500 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 9 kN/m², L = 6 m, h ≈ sqrt((8 * 9 * 6²) / 25) = 0.52 m.

40. Determine the load-carrying capacity of a column with a cross-sectional area of 5000 mm² and a concrete strength of 30 MPa.

Load-carrying capacity P = A * f_c. For A = 5000 mm², f_c = 30 MPa, P = 5000 * 30 = 150,000 kN.

41. Calculate the required area of steel for a beam with a width of 400 mm and an effective depth of 600 mm subjected to a moment of 100 kNm. Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For M = 100 kNm, d = 600 mm, A_s can be calculated.

42. Find the moment of inertia of a rectangular beam with a width of 500 mm and a height of 700 mm.

Moment of inertia I = (b * h³) / 12. For b = 500 mm, h = 700 mm, I = (500 * 700³) / 12 = 1.84 x 10¹¹ mm⁴.

43. Determine the maximum shear force for a simply supported beam of length 5 m with a central point load of 50 kN.

Maximum shear force V_max = P / 2. For P = 50 kN, V_max = 50 / 2 = 25 kN.

44. Calculate the thickness of a slab of span 3 m with a total load of 8 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 8 kN/m², L = 3 m, h ≈ sqrt((8 * 8 * 3²) / 25) = 0.37 m.

45. Find the bending moment capacity of a beam with a width of 300 mm, effective depth of 550 mm, and reinforcement area of 1200 mm². Assume f_y = 500 MPa and f_c = 30 MPa.

Bending moment capacity M = 0.87 * f_y * A_s * (d - a/2). For A_s = 1200 mm², d = 550 mm, M = 0.87 * 500 * 1200 * (550 - 0.42 * 550 / 2) = 230.3 kNm.

46. Determine the effective depth for a two-way slab of span 5 m x 5 m subjected to a live load of 6 kN/m² and a dead load of 5 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 11 kN/m², L = 5 m, d = (11 * 5²) / (8 * 25) = 0.55 m.

47. Calculate the area of steel required for a one-way slab of span 4 m and width 1 m subjected to a live load of 5 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Moment M = (w * L²) / 8. For w = 5 kN/m², L = 4 m, M = (5 * 4²) / 8 = 10 kNm.

48. Determine the maximum shear force for a cantilever beam of length 2.5 m subjected to a uniformly distributed load of 6 kN/m.

Maximum shear force V = w * L. For w = 6 kN/m, L = 2.5 m, V = 6 * 2.5 = 15 kN.

49. Calculate the required depth of a two-way slab of span 4 m x 4 m with a total load of 7 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 7 kN/m², L = 4 m, d = (7 * 4²) / (8 * 25) = 0.56 m.

50. Determine the load-carrying capacity of a column of cross-sectional area 6000 mm² and concrete strength 25 MPa. Assume a safety factor of 1.5.

Load-carrying capacity P = A * f_c / SF. For A = 6000 mm², f_c = 25 MPa, SF = 1.5, P = 6000 * 25 / 1.5 = 100,000 kN.

51. Calculate the deflection of a simply supported beam of span 7 m subjected to a central load of 25 kN.

Deflection δ = (P * L³) / (48 * E * I). For P = 25 kN, L = 7 m, use E and I to calculate δ.

52. Find the maximum bending moment for a simply supported beam of span 5 m subjected to a uniformly distributed load of 15 kN/m.

Maximum bending moment M_max = (w * L²) / 8. For w = 15 kN/m, L = 5 m, M_max = (15 * 5²) / 8 = 93.75 kNm.

53. Determine the shear force at the support of a cantilever beam of length 3 m subjected to a point load of 20 kN at the free end.

Shear force V = P. For P = 20 kN, V = 20 kN.

54. Calculate the moment of inertia for a beam with a rectangular cross-section of width 400 mm and height 600 mm.

Moment of inertia I = (b * h³) / 12. For b = 400 mm, h = 600 mm, I = (400 * 600³) / 12 = 8.00 x 10¹⁰ mm⁴.

55. Find the reaction at the supports of a simply supported beam of span 10 m subjected to a uniformly distributed load of 10 kN/m.

Reaction R = w * L / 2. For w = 10 kN/m, L = 10 m, R = 10 * 10 / 2 = 50 kN.

56. Calculate the required area of steel for a beam of width 250 mm and effective depth 400 mm subjected to a moment of 40 kNm. Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For M = 40 kNm, d = 400 mm, A_s can be calculated.

57. Determine the effective depth of a one-way slab of span 3 m and a load of 8 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 8 kN/m², L = 3 m, d = (8 * 3²) / (8 * 25) = 0.36 m.

58. Calculate the bending moment capacity of a beam with a width of 400 mm, an effective depth of 500 mm, and a reinforcement area of 1500 mm². Assume f_y = 415 MPa and f_c = 25 MPa.

Bending moment capacity M = 0.87 * f_y * A_s * (d - a/2). For A_s = 1500 mm², d = 500 mm, M = 0.87 * 415 * 1500 * (500 - 0.42 * 500 / 2) = 328.0 kNm.

59. Find the maximum deflection of a simply supported beam of length 4 m subjected to a central load of 15 kN.

Deflection δ = (P * L³) / (48 * E * I). For P = 15 kN, L = 4 m, use E and I to calculate δ.

60. Determine the depth of a cantilever beam of length 3 m subjected to a uniformly distributed load of 10 kN/m. Assume a maximum deflection limit of 1/250 of the span.

Depth d = sqrt((5 * w * L⁴) / (384 * E * δ)). For w = 10 kN/m, L = 3 m, δ = L / 250, calculate d based on E and δ.

7. Reinforced Concrete Design

61. Determine the bending moment capacity of a reinforced concrete beam with a width of 300 mm, effective depth of 500 mm, and a reinforcement area of 1500 mm². Assume f_y = 415 MPa and f_c = 25 MPa.

Bending moment capacity M = 0.87 * f_y * A_s * (d - a/2). For A_s = 1500 mm², d = 500 mm, M = 0.87 * 415 * 1500 * (500 - 0.42 * 500 / 2) = 328.0 kNm.

62. Calculate the required area of steel for a rectangular column of dimensions 400 mm x 600 mm subjected to an axial load of 250 kN. Assume f_c = 30 MPa and a safety factor of 1.5.

Required area A_s = P / (f_c / SF). For P = 250 kN, f_c = 30 MPa, SF = 1.5, A_s = 250 / (30 / 1.5) = 12,500 mm².

63. Find the effective depth required for a one-way slab of span 5 m subjected to a total load of 10 kN/m². Assume f_c = 25 MPa and f_y = 415 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 10 kN/m², L = 5 m, d = (10 * 5²) / (8 * 25) = 0.50 m.

64. Determine the maximum bending moment for a simply supported beam of length 7 m with a central point load of 50 kN.

Maximum bending moment M_max = (P * L) / 4. For P = 50 kN, L = 7 m, M_max = (50 * 7) / 4 = 87.5 kNm.

65. Calculate the shear force at a section 2 m from the support of a simply supported beam of span 6 m subjected to a point load of 40 kN at the center.

Shear force V = P / 2. For P = 40 kN, V = 40 / 2 = 20 kN.

66. Find the area of steel required for a beam of width 250 mm and effective depth 400 mm subjected to a moment of 50 kNm. Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For M = 50 kNm, d = 400 mm, A_s = M / (0.87 * 415 * (400 - 0.42 * 400 / 2)).

67. Determine the maximum deflection of a simply supported beam of span 6 m subjected to a uniformly distributed load of 12 kN/m.

Maximum deflection δ = (5 * w * L⁴) / (384 * E * I). For w = 12 kN/m, L = 6 m, use E and I to calculate δ.

68. Calculate the moment of inertia for a rectangular beam with a width of 250 mm and a height of 400 mm.

Moment of inertia I = (b * h³) / 12. For b = 250 mm, h = 400 mm, I = (250 * 400³) / 12 = 1.067 x 10¹⁰ mm⁴.

69. Find the bending moment capacity of a beam with a width of 300 mm, effective depth of 600 mm, and reinforcement area of 1000 mm². Assume f_y = 500 MPa and f_c = 30 MPa.

Bending moment capacity M = 0.87 * f_y * A_s * (d - a/2). For A_s = 1000 mm², d = 600 mm, M = 0.87 * 500 * 1000 * (600 - 0.42 * 600 / 2) = 208.5 kNm.

70. Calculate the size of a column if the applied axial load is 150 kN and the concrete strength is 20 MPa. Assume a safety factor of 1.5.

Required area A = P / (f_c / SF). For P = 150 kN, f_c = 20 MPa, SF = 1.5, A = 150 / (20 / 1.5) = 11,250 mm².

8. Advanced Topics

71. Determine the shear force at a section 3 m from the support of a simply supported beam of span 10 m subjected to a point load of 60 kN at 5 m from one end.

Shear force V = P * (L - x) / L. For P = 60 kN, L = 10 m, x = 5 m, V = 60 * (10 - 5) / 10 = 30 kN.

72. Find the effective depth required for a two-way slab of span 6 m x 6 m subjected to a total load of 9 kN/m². Assume f_c = 25 MPa and f_y = 415 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 9 kN/m², L = 6 m, d = (9 * 6²) / (8 * 25) = 0.54 m.

73. Calculate the bending moment capacity of a reinforced concrete column of dimensions 500 mm x 500 mm with a concrete strength of 30 MPa and a safety factor of 1.5.

Bending moment capacity M = A * f_c / SF. For A = 500 mm x 500 mm, f_c = 30 MPa, SF = 1.5, M = 500 * 500 * 30 / 1.5 = 5,000,000 kN.mm.

74. Determine the required area of steel for a one-way slab of span 4 m subjected to a total load of 6 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For w = 6 kN/m², L = 4 m, A_s = M / (0.87 * 415 * (d - a/2)).

75. Calculate the thickness of a slab if the span is 5 m and the total load is 7 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 7 kN/m², L = 5 m, h ≈ sqrt((8 * 7 * 5²) / 25) = 0.45 m.

76. Determine the maximum bending moment for a cantilever beam of length 4 m subjected to a point load of 20 kN at the free end.

Maximum bending moment M_max = P * L. For P = 20 kN, L = 4 m, M_max = 20 * 4 = 80 kNm.

77. Calculate the deflection of a simply supported beam of length 5 m subjected to a central point load of 10 kN.

Deflection δ = (P * L³) / (48 * E * I). For P = 10 kN, L = 5 m, use E and I to calculate δ.

78. Find the moment of inertia for a beam with a cross-section of 300 mm x 600 mm.

Moment of inertia I = (b * h³) / 12. For b = 300 mm, h = 600 mm, I = (300 * 600³) / 12 = 1.080 x 10¹⁰ mm⁴.

79. Determine the load-carrying capacity of a column of cross-sectional area 4000 mm² with concrete strength of 25 MPa.

Load-carrying capacity P = A * f_c. For A = 4000 mm², f_c = 25 MPa, P = 4000 * 25 = 100,000 kN.

80. Calculate the area of steel required for a beam with a width of 300 mm and effective depth of 450 mm subjected to a moment of 60 kNm. Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For M = 60 kNm, d = 450 mm, A_s = M / (0.87 * 415 * (450 - 0.42 * 450 / 2)).

81. Find the maximum deflection of a cantilever beam of span 3 m subjected to a uniformly distributed load of 7 kN/m.

Maximum deflection δ = (w * L⁴) / (8 * E * I). For w = 7 kN/m, L = 3 m, use E and I to calculate δ.

82. Determine the shear force at the support of a simply supported beam of length 8 m with a point load of 40 kN placed at 2 m from one end.

Shear force V = P * (L - x) / L. For P = 40 kN, L = 8 m, x = 2 m, V = 40 * (8 - 2) / 8 = 30 kN.

83. Calculate the required area of steel for a two-way slab with a span of 4 m x 4 m and a total load of 8 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For w = 8 kN/m², L = 4 m, A_s = M / (0.87 * 415 * (d - a/2)).

84. Determine the effective thickness of a slab of span 7 m subjected to a total load of 10 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 10 kN/m², L = 7 m, h ≈ sqrt((8 * 10 * 7²) / 25) = 0.60 m.

85. Calculate the required depth of a beam if subjected to a central load of 30 kN over a span of 5 m. Assume a maximum deflection limit of L/250.

Required depth d = sqrt((5 * P * L⁴) / (384 * E * δ)). For P = 30 kN, L = 5 m, δ = L / 250, calculate d based on E and δ.

86. Find the bending moment capacity of a beam with a width of 250 mm, an effective depth of 450 mm, and a reinforcement area of 1200 mm². Assume f_y = 415 MPa and f_c = 25 MPa.

Bending moment capacity M = 0.87 * f_y * A_s * (d - a/2). For A_s = 1200 mm², d = 450 mm, M = 0.87 * 415 * 1200 * (450 - 0.42 * 450 / 2) = 226.8 kNm.

87. Calculate the shear force at the mid-span of a simply supported beam of length 4 m with a uniformly distributed load of 12 kN/m.

Shear force V = w * L / 2. For w = 12 kN/m, L = 4 m, V = 12 * 4 / 2 = 24 kN.

88. Determine the effective depth of a one-way slab of span 6 m and a total load of 5 kN/m². Assume f_c = 25 MPa and f_y = 415 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 5 kN/m², L = 6 m, d = (5 * 6²) / (8 * 25) = 0.36 m.

89. Calculate the thickness of a two-way slab of span 5 m x 5 m subjected to a total load of 6 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 6 kN/m², L = 5 m, h ≈ sqrt((8 * 6 * 5²) / 25) = 0.46 m.

90. Determine the load-carrying capacity of a column of cross-sectional area 5000 mm² and concrete strength 20 MPa. Assume a safety factor of 1.5.

Load-carrying capacity P = A * f_c / SF. For A = 5000 mm², f_c = 20 MPa, SF = 1.5, P = 5000 * 20 / 1.5 = 66,667 kN.

91. Calculate the deflection of a cantilever beam of span 2 m subjected to a point load of 25 kN at the free end.

Deflection δ = (P * L³) / (3 * E * I). For P = 25 kN, L = 2 m, use E and I to calculate δ.

92. Find the maximum shear force for a simply supported beam of length 4 m with a central load of 20 kN.

Maximum shear force V_max = P / 2. For P = 20 kN, V_max = 20 / 2 = 10 kN.

93. Determine the required reinforcement area for a beam of width 350 mm and effective depth 500 mm subjected to a moment of 70 kNm. Assume f_y = 415 MPa and f_c = 25 MPa.

Required area A_s = M / (0.87 * f_y * (d - a/2)). For M = 70 kNm, d = 500 mm, A_s = M / (0.87 * 415 * (500 - 0.42 * 500 / 2)).

94. Calculate the bending moment capacity of a beam with a width of 350 mm, an effective depth of 450 mm, and a reinforcement area of 1000 mm². Assume f_y = 415 MPa and f_c = 25 MPa.

Bending moment capacity M = 0.87 * f_y * A_s * (d - a/2). For A_s = 1000 mm², d = 450 mm, M = 0.87 * 415 * 1000 * (450 - 0.42 * 450 / 2) = 163.5 kNm.

95. Find the deflection of a simply supported beam of span 3 m subjected to a uniformly distributed load of 8 kN/m.

Deflection δ = (5 * w * L⁴) / (384 * E * I). For w = 8 kN/m, L = 3 m, use E and I to calculate δ.

96. Determine the effective depth of a slab subjected to a total load of 10 kN/m² and a span of 6 m. Assume f_c = 25 MPa and f_y = 415 MPa.

Effective depth d = (w * L²) / (8 * f_c). For w = 10 kN/m², L = 6 m, d = (10 * 6²) / (8 * 25) = 0.54 m.

97. Calculate the shear force at a section 4 m from the support of a simply supported beam of span 8 m subjected to a point load of 50 kN placed at 3 m from one end.

Shear force V = P * (L - x) / L. For P = 50 kN, L = 8 m, x = 3 m, V = 50 * (8 - 3) / 8 = 31.25 kN.

98. Find the thickness of a slab for a span of 4 m subjected to a total load of 9 kN/m². Assume f_y = 415 MPa and f_c = 25 MPa.

Effective thickness h = sqrt((8 * w * L²) / f_c). For w = 9 kN/m², L = 4 m, h ≈ sqrt((8 * 9 * 4²) / 25) = 0.40 m.

99. Calculate the maximum bending moment for a cantilever beam of length 5 m subjected to a uniformly distributed load of 10 kN/m.

Maximum bending moment M_max = (w * L²) / 2. For w = 10 kN/m, L = 5 m, M_max = (10 * 5²) / 2 = 125 kNm.

100. Determine the shear force at a section 2 m from the support of a simply supported beam of length 6 m subjected to a point load of 30 kN placed at 4 m from one end.

Shear force V = P * (L - x) / L. For P = 30 kN, L = 6 m, x = 4 m, V = 30 * (6 - 4) / 6 = 10 kN.