## 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 * L

^{2}) / f_{c}). For w = 6 kN/m², L = 3 m, h ≈ sqrt((8 * 6 * 3^{2}) / 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 * L

^{2}) / (8 * f_{c}). For w = 9 kN/m², L = 4 m, d = (9 * 4^{2}) / (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 * L

^{2}) / 8. For w = 6 kN/m², L = 5 m, M = (6 * 5^{2}) / 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 * L

^{2}) / (8 * f_{c}). For w = 7 kN/m², L = 4 m, d = (7 * 4^{2}) / (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 * L

^{2}) / f_{c}). For w = 6 kN/m², L = 5 m, h ≈ sqrt((8 * 6 * 5^{2}) / 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 * h

^{3}) / 12. For b = 350 mm, h = 600 mm, I = (350 * 600^{3}) / 12 = 7.56 x 10^{10}mm^{4}.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 * 10^{6}) / (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 * L^{2}) / 2. For w = 4 kN/m, L = 2.5 m, M_{max}= (4 * 2.5^{2}) / 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 * L

^{2}) / (8 * f_{c}). For w = 10 kN/m, L = 7 m, d = (10 * 7^{2}) / (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 * h

^{3}) / 12. For b = 400 mm, h = 500 mm, I = (400 * 500^{3}) / 12 = 2.08 x 10^{10}mm^{4}.## 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 Î» = h / L

_{e}. For h = 4 m, L_{e}= 3.5 m, Î» = 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 / Î». For h = 6 m, Î» = 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 Î´ = (P * L

^{3}) / (48 * E * I). For P = 20 kN, L = 6 m, use E and I to calculate Î´.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 * L^{2}) / 2. For w = 10 kN/m, L = 4 m, M_{max}= (10 * 4^{2}) / 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 * h

^{3}) / 12. For b = 300 mm, h = 500 mm, I = (300 * 500^{3}) / 12 = 1.04 x 10^{10}mm^{4}.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 * L

^{2}) / (8 * f_{c}). For w = 5 kN/m², L = 4 m, d = (5 * 4^{2}) / (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 * 10^{6}) / (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 Î´ = (5 * w * L

^{4}) / (384 * E * I). For w = 8 kN/m, L = 5 m, use E and I to calculate Î´.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 * L

^{2}) / f_{c}). For w = 9 kN/m², L = 6 m, h ≈ sqrt((8 * 9 * 6^{2}) / 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 * h

^{3}) / 12. For b = 500 mm, h = 700 mm, I = (500 * 700^{3}) / 12 = 1.84 x 10^{11}mm^{4}.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 * L

^{2}) / f_{c}). For w = 8 kN/m², L = 3 m, h ≈ sqrt((8 * 8 * 3^{2}) / 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 * L

^{2}) / (8 * f_{c}). For w = 11 kN/m², L = 5 m, d = (11 * 5^{2}) / (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 * L

^{2}) / 8. For w = 5 kN/m², L = 4 m, M = (5 * 4^{2}) / 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 * L

^{2}) / (8 * f_{c}). For w = 7 kN/m², L = 4 m, d = (7 * 4^{2}) / (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 Î´ = (P * L

^{3}) / (48 * E * I). For P = 25 kN, L = 7 m, use E and I to calculate Î´.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 * L^{2}) / 8. For w = 15 kN/m, L = 5 m, M_{max}= (15 * 5^{2}) / 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 * h

^{3}) / 12. For b = 400 mm, h = 600 mm, I = (400 * 600^{3}) / 12 = 8.00 x 10^{10}mm^{4}.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 * L

^{2}) / (8 * f_{c}). For w = 8 kN/m², L = 3 m, d = (8 * 3^{2}) / (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 Î´ = (P * L

^{3}) / (48 * E * I). For P = 15 kN, L = 4 m, use E and I to calculate Î´.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 * L

^{4}) / (384 * E * Î´)). For w = 10 kN/m, L = 3 m, Î´ = L / 250, calculate d based on E and Î´.## 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 * L

^{2}) / (8 * f_{c}). For w = 10 kN/m², L = 5 m, d = (10 * 5^{2}) / (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 Î´ = (5 * w * L

^{4}) / (384 * E * I). For w = 12 kN/m, L = 6 m, use E and I to calculate Î´.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 * h

^{3}) / 12. For b = 250 mm, h = 400 mm, I = (250 * 400^{3}) / 12 = 1.067 x 10^{10}mm^{4}.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 * L

^{2}) / (8 * f_{c}). For w = 9 kN/m², L = 6 m, d = (9 * 6^{2}) / (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 * L

^{2}) / f_{c}). For w = 7 kN/m², L = 5 m, h ≈ sqrt((8 * 7 * 5^{2}) / 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 Î´ = (P * L

^{3}) / (48 * E * I). For P = 10 kN, L = 5 m, use E and I to calculate Î´.78. Find the moment of inertia for a beam with a cross-section of 300 mm x 600 mm.

Moment of inertia I = (b * h

^{3}) / 12. For b = 300 mm, h = 600 mm, I = (300 * 600^{3}) / 12 = 1.080 x 10^{10}mm^{4}.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 Î´ = (w * L

^{4}) / (8 * E * I). For w = 7 kN/m, L = 3 m, use E and I to calculate Î´.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 * L

^{2}) / f_{c}). For w = 10 kN/m², L = 7 m, h ≈ sqrt((8 * 10 * 7^{2}) / 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 * L

^{4}) / (384 * E * Î´)). For P = 30 kN, L = 5 m, Î´ = L / 250, calculate d based on E and Î´.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 * L

^{2}) / (8 * f_{c}). For w = 5 kN/m², L = 6 m, d = (5 * 6^{2}) / (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 * L

^{2}) / f_{c}). For w = 6 kN/m², L = 5 m, h ≈ sqrt((8 * 6 * 5^{2}) / 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 Î´ = (P * L

^{3}) / (3 * E * I). For P = 25 kN, L = 2 m, use E and I to calculate Î´.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 Î´ = (5 * w * L

^{4}) / (384 * E * I). For w = 8 kN/m, L = 3 m, use E and I to calculate Î´.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 * L

^{2}) / (8 * f_{c}). For w = 10 kN/m², L = 6 m, d = (10 * 6^{2}) / (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 * L

^{2}) / f_{c}). For w = 9 kN/m², L = 4 m, h ≈ sqrt((8 * 9 * 4^{2}) / 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 * L^{2}) / 2. For w = 10 kN/m, L = 5 m, M_{max}= (10 * 5^{2}) / 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.

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