Civil Engineering Formulas

Civil Engineering Formulas

RCC Formula

1. 𝑓𝑚 = 𝑓𝑐𝑘 + 1.65 𝜎 

2. 𝐸𝑐 = 5000 √𝑓𝑐𝑘 and 𝑓𝑐𝑟 = 0.7 √𝑓𝑐𝑘 

3. 𝜀𝑠𝑡 = 0.002 + 0.87 𝑓𝑦 /𝐸𝑠 

4. 𝐸𝑠 = 2 × 10^5 𝑀𝑃𝑎 , 𝑚 = 280/ 3𝜎𝑐𝑏𝑐 

5. 𝑏𝑓 = 𝑙0 /6 + 𝑏𝑤 + 6𝐷𝑓                          (for T-beams.) 

6. 𝑏𝑓 = 𝑙0 /12 + 𝑏𝑤 + 3𝐷𝑓                        (for L-beams.) 

7. (𝑙0 /(𝑙0 /𝑏 +4)) + 𝑏𝑤 ≤ 𝑏                       (for isolated T-beams.) 

8. (0.5𝑙0 /(𝑙0 /𝑏 +4)) + 𝑏𝑤 ≤                      (𝑏 for isolated L-beams.) 

9. 𝑥𝑢 /𝑑 = 0.87 𝑓𝑦. 𝐴𝑠𝑡 0.36 /𝑓𝑐𝑘 𝑏𝑑  and  𝑥𝑢,𝑚𝑎𝑥 /𝑑 = 0.0035 /(0.0055+ 0.87 𝑓𝑦 /𝐸𝑠 )

10. 𝑀𝑢 = 0.87 𝑓𝑦.𝐴𝑠𝑡 (𝑑 − 0.42 𝑥𝑢)           for 𝑥𝑢 ≤ 𝑥𝑢,max 

11. 𝑀𝑢 = 0.36 𝑓𝑐𝑘. 𝑏. 𝑥𝑢,𝑚𝑎𝑥(𝑑 − 0.42 𝑥𝑢,𝑚𝑎𝑥) 

12. 𝑥𝑢,𝑚𝑎𝑥 /𝑑 = 0.87 𝑓𝑦 /0.36 𝑓𝑐𝑘 ( 𝑃𝑡,𝑙𝑖𝑚 /100 ) 

13. 𝑀𝑢 = 0.87 𝑓𝑦 𝐴𝑠𝑡 (𝑑 − 𝑓𝑦 𝐴𝑠𝑡 /𝑓𝑐𝑘 𝑏 ) 

14. 𝑦𝑓 = 0.15 𝑥𝑢 + 0.65 𝐷𝑓                         when (𝐷𝑓 /𝑑) > 0.2 

15. 0.87 𝑓𝑦 (𝐴𝑠𝑡 − 𝐴𝑠𝑡,𝑙) = (𝑓𝑠𝑐 − 0.447 𝑓𝑐𝑘).𝐴𝑠𝑐 

16. 𝑀𝑢 = 𝑀𝑢,𝑙 + 0.87 𝑓𝑦. (𝐴𝑠𝑡 − 𝐴𝑠𝑡,𝑙) (𝑑 − 𝑑 ′ ) 

17. 𝐴𝑠𝑡 /𝑏𝑑 = 0.85 /𝑓𝑦 (minimum tension reinforcement) 

18. 𝐴𝑠𝑡,𝑚𝑖𝑛 = 0.12 % of 𝐴𝑔 for 𝐹𝑒-415 and 0.15 % of 𝐴𝑔 for 𝐹𝑒-250. (slabs) 

19. 𝐴𝑠𝑡,𝑚𝑎𝑥 = 4% 𝑜𝑓 𝐴𝑔 (Beams) ,𝐴𝑠𝑐,𝑚𝑎𝑥 = 4% 𝑜𝑓 𝐴𝑔 (Beams) 

20. ( 𝑙 /𝑑 ) 𝑚𝑎𝑥 = ( 𝑙 /𝑑 ) 𝑏𝑎𝑠𝑖𝑐 𝑘𝑡 𝑘c

Transportation Engineering 

1. 𝑦 = 2𝑥^2 /𝑛𝑊 (parabolic camber) 

2. 𝑆𝑆𝐷 = 𝑣𝑡 + 𝑣^2 /2𝑔(𝑓±𝑛) , 𝐼𝑆𝐷 = 2 × 𝑆𝑆𝐷 

3. 𝑂𝑆𝐷 = 𝑑1 + 𝑑2 + 𝑑3 , 𝑑1 = 𝑣𝑏 𝑡, 𝑑2 = 2𝑠 + 𝑣𝑏𝑇 , 𝑑3 = 𝑣𝑇 , 𝑇 = √4𝑠/𝑎 

4. 𝑣^2 /𝑅𝑔 = 𝑒 + 𝑓 , 𝑊𝑒 = 𝑛𝑙^2 /2𝑅 + 𝑉/ 9.5√𝑅 

5. 𝐿𝑠 = 𝑣^3 /𝐶𝑅 (comfort condition) 

6. 𝐿𝑠 = 𝑒𝑁(𝑊 + 𝑊𝑒 ) (rotated about inner edge) 

7. 𝐿𝑠 = 2.7 𝑉^2 /𝑅 for plain and rolling terrain, 𝐿𝑠 = 𝑉^2 /𝑅 for steep and mountainous terrain. 

8. IRC recommends Spiral as transition curve. 

9. 𝑆 = 𝐿𝑠 ^2 /24𝑅 

10. 𝑚 = 𝑅 (1 − cos 𝜃 /2 ), 𝜃 = 𝑆/𝑅 (single lane, 𝐿 > 𝑆) 

11. 𝑚 = 𝑅 (1 − cos 𝜃 /2 ) + (𝑆−𝐿) /2 sin 𝜃 /2 , 𝜃 = 𝐿/ 𝑅 (Single lane, 𝐿 < 𝑆) 

12. Curve resistance = 𝑇(1 − cos 𝛼) 

13. Grade compensation = (30+𝑅) /𝑅 or 75 /𝑅 whichever is less 

14. 𝐿 = 𝑁𝑠^2 /(√2ℎ+√2𝐻)^2 when 𝐿 > 𝑠 (Summit curve for SSD), ℎ = 0.15 𝑎𝑛𝑑 𝐻 = 1.2 

15. 𝐿 = 2𝑠 − (√2ℎ+√2𝐻)^2 /𝑁 when 𝐿 < 𝑠 (Summit curve for SSD), ℎ = 0.15 𝑎𝑛𝑑 𝐻 = 1.2 

16. 𝐿 = 𝑁𝑠^2 /(√2𝐻+√2𝐻)^2 when 𝐿 > 𝑠 (Summit curve for OSD), 𝐻 = 1.2 

17. 𝐿 = 2𝑠 − (√2𝐻+√2𝐻)^2 /𝑁 , 𝐿 < 𝑠 (Summit curve for OSD), 𝐻 = 1.2 

18. 𝐿 = 𝑁𝑠^2 /2ℎ+2𝑠 tan 𝛼  when 𝐿 > 𝑠 (valley curve), ℎ = 0.75 𝑎𝑛𝑑 𝛼 = 1° 

19. 𝐿 = 2𝑠 − (2ℎ+2𝑠 tan 𝛼) /𝑁 , 𝐿 < 𝑠 (valley curve), ℎ = 0.75 𝑎𝑛𝑑 𝛼 = 1° 

20. 𝐿 = 2 ( 𝑁𝑣^3 /𝐶 )^1/2 (valley curve comfort condition) 

21. 𝐶 = 80 /(75+𝑉) 

22. 𝐹𝑙𝑎𝑘𝑖𝑛𝑒𝑠𝑠 𝐼𝑛𝑑𝑒𝑥 = (𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑎𝑘𝑦 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑡ℎ𝑖𝑛𝑛𝑒𝑟 𝑡ℎ𝑎𝑛 0.6 𝑑𝑚 /𝑡𝑜𝑡𝑎𝑙 𝑤𝑒𝑖𝑔ℎ𝑡) × 100 

23. 𝐸𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛 𝐼𝑛𝑑𝑒𝑥 = (𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑎𝑘𝑦 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑙𝑜𝑛𝑔𝑒𝑟 𝑡ℎ𝑎𝑛 1.8 𝑑𝑚 /𝑡𝑜𝑡𝑎𝑙 𝑤𝑒𝑖𝑔ℎ𝑡) × 100 

24. Angularity number= 67 − 100( 𝑊 /𝐺 ) /𝐶 , C is weight water in the cylinder, W is weight of aggregate packed in the cylinder. 

25. Penetration test unit is 1/10th mm. Weight used 100 grams. Temperature 25℃. 

26. 𝐺𝑎 = (𝑊1+𝑊2+𝑊3)/( 𝑊1/ 𝐺1 + 𝑊2/ 𝐺2 + 𝑊3 /𝐺3)  and 𝐺𝑡 = (𝑊𝑎+𝑊𝑏) /(𝑊𝑎 /𝐺𝑎 + 𝑊𝑏 /𝐺𝑏) 

27. 𝐺𝑚 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑚𝑜𝑢𝑙𝑑 /𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑜𝑢𝑙𝑑 

28. 𝑣𝑎 = 𝐺𝑡−𝐺𝑚 /𝐺𝑡 , 𝑣𝑏 = 𝑊𝑏 /𝑊𝑚 × 𝐺𝑚 /𝐺𝑏 and 𝑉𝑀𝐴 = 𝑣𝑎 + 𝑣𝑏 

29. 𝑉𝐹𝐵 = 𝑣𝑏 /𝑉𝑀𝐴 

30. Flow value units 1/4th mm 

31. 𝑁 = 365𝐴 [ (1+𝑟)^𝑛 −1 /𝑛 ]𝑉𝐷𝐹 × 𝐿𝐷𝐹 and 𝐴 = 𝑃(1 + 𝑟)^𝑥 

32. 𝐿𝐷𝐹 = 0.75 𝑓𝑜𝑟 𝑡𝑤𝑜 𝑙𝑎𝑛𝑒𝑠 𝑎𝑛𝑑 0.4 𝑓𝑜𝑟 𝑓𝑜𝑢𝑟 𝑙𝑎𝑛𝑒 (single carriageway) 

33. LDF=0.75 for two lanes and 0.60 for three lane and 0.45 for four lane (dual carriageway) 

34. 𝑉𝐷𝐹 = ( 𝑃/80)^4 , where P is in kN 

35. 𝐶𝐵𝑅 𝑎𝑡 2.5 𝑚𝑚 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛 𝑘𝑔/𝑐𝑚^2 /70 = 𝑙𝑜𝑎𝑑 𝑖𝑛 𝑘𝑔 /1370 

36. 𝐶𝐵𝑅 𝑎𝑡 5 𝑚𝑚 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛 𝑘𝑔/𝑐𝑚^2 /105 = 𝑙𝑜𝑎𝑑 𝑖𝑛 𝑘𝑔 /2055 

37. 𝐸𝑆𝑊𝐿: interpolate load for depth from line joining (𝑙𝑜𝑔𝑃, log 𝑧 /2 ) and (𝑙𝑜𝑔2𝑃, log 2𝑠). 

 38. Radius of relative stiffness 𝑙 = [ 𝐸ℎ^3 /12𝑘(1−𝜇^2) ] ^1/ 4 

39. 𝑘 = 𝑝 ( 𝑘𝑔 /𝑐𝑚^2 ) /0.125 𝑐𝑚 (modulus of subgrade reaction) 

40. 𝑘 = 𝐸 /1.18 𝑎 (𝑎 is rigid plate radius) 

41. 𝑘 = 𝐸 /1.5 𝑎 (𝑎 is flexible plate radius) 

42. 𝑘1𝑎1 = 𝑘2𝑎2 

43. 𝑅𝑖𝑔𝑖𝑑𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟 = 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 /𝑡𝑦𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (below 7 𝑘𝑔/𝑐𝑚2 contact pressure is more) 

44. 𝐿𝛼𝑡 = 𝛿/ 2 (expansion joint) 

45. 𝐿𝑐 /2 𝑏ℎ𝛾𝑐𝑓 = 𝑆𝑐 𝑏ℎ 

46. 𝑏ℎ𝛾𝑐𝑓 = 𝐴𝑠𝑡 𝑆𝑠 (tie bar area of steel per meter) 

47. (𝐿 /2) 𝜋∅𝜏𝑏𝑑 = (𝜋∅^2 /4) 𝑆𝑠 (length of tie bar) 

48. 𝑃(𝑥) = [(𝜆𝑡) ^(𝑥𝑒 −𝜆𝑡)] /𝑥! (Poisson distribution) 

49. 𝑃(ℎ ≥ 𝑡) = 𝑒^−𝜆𝑡 

50. 𝑘 = 𝑘𝑗 (1 − 𝑣 /𝑣𝑓 ) and 𝑣 = 𝑣𝑓 (1 − 𝑘 /𝑘𝑗 ) 

51. 𝑞 = 𝑘𝑣 and 𝑘 = 1000 /𝑠𝑝𝑎𝑐𝑖𝑛𝑔 

52. 𝑃𝐻𝐹 = 𝑝𝑒𝑎𝑘 ℎ𝑜𝑢𝑟 𝑓𝑙𝑜𝑤 /4 ×𝑝𝑒𝑎𝑘 15 𝑚𝑖𝑛 𝑓𝑙𝑜𝑤   or    𝑃𝐻𝐹 = 30𝑡ℎ ℎ𝑜𝑢𝑟𝑙𝑦 𝑣𝑜𝑙𝑢𝑚𝑒 /𝐴𝐴𝐷𝑇 

53. 𝑇𝑖𝑚𝑒 𝑚𝑒𝑎𝑛 𝑠𝑝𝑒𝑒𝑑 = 𝑣1+𝑣2+⋯+𝑣𝑛 /𝑛 

54. 𝑆𝑝𝑎𝑐𝑒 𝑚𝑒𝑎𝑛 𝑠𝑝𝑒𝑒𝑑 = 𝑑 /(𝑡1+𝑡2+⋯+𝑡𝑛 /𝑛) 

55. 𝑞 = (𝑛𝑎+𝑛𝑦 )/(𝑡𝑎+𝑡𝑤) , 𝑡̅= 𝑡𝑤 − 𝑛𝑦 /𝑞 

56. Safe speed limit is 85th percentile speed 

57. Geometric design is based on 98th percentile speed. 

58. Road side facilities are based on 30th highest hourly volume. 

59. 𝐶0 = (1.5𝐿+5) /(1−𝑌0) 

60. 𝑄 = 280𝑤 [(1+ 𝑒 /𝑤 )(1− 𝑝 /3 )]/ (1+ 𝑤 /𝑙 ) 

61. 𝑇𝑟 = 𝑇𝑎 + (𝑇𝑚−𝑇𝑎)/ 3 

62. 𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 = 7 % 𝑓𝑜𝑟 300 𝑚 

63. 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑜𝑓 𝑎𝑖𝑟𝑝𝑜𝑟𝑡, 𝑇𝑠 = 15 − (6.5 /1000) ℎ 

64. Temperature correction = 1% 𝑝𝑒𝑟 1℃ 𝑟𝑖𝑠𝑒 𝑜𝑓 𝑇𝑟 𝑎𝑏𝑜𝑣𝑒 𝑇𝑠 

65. Gradient correction = 20% 𝑝𝑒𝑟 1% 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡. 

66. 𝑇𝑢𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑑𝑖𝑢𝑠 𝑅 = 𝑉^2 /125𝑓 

67. 𝑇𝑢𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑑𝑖𝑢𝑠 𝑅 = [(0.388𝑊^2)/ (𝑇 /2) −𝑆] where 𝑠 = 6 + 𝑡𝑟𝑒𝑎𝑑 /2 

68. Turning radius for subsonic aircraft is 120 m and for supersonic it is 180 m 

69. Grade compensation for BG is 0.04%, for MG is 0.03% and for NG is 0.02% per degree of curve. 

70. 𝐷 = 1720/ 𝑅 

71. 𝑉𝑒𝑟𝑠𝑖𝑛𝑒 𝑜𝑓 𝑐𝑢𝑟𝑣𝑒 = 𝑐^2 /8𝑅 

72. 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑐𝑎𝑛𝑡 = (𝑉^2 /127𝑅) × 𝐺 

73. Theoretical cant = 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑐𝑎𝑛𝑡 + 𝑐𝑎𝑛𝑡 𝑑𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 

74. Widening of gauge in cm, 𝑑 = 13(𝐵+𝐿)^2 /𝑅 where B is wheel base in m, lap of flange in m, 𝐿 = 0.02√ℎ^2 + 𝐷ℎ , h is depth of wheel flange below rail top level, D dia of wheel in cm. 

75. 𝐿𝑠 = 3.28 𝑣^3 /𝑅 (Transition curve) 

76. 𝑦 = 𝑥^3 /6𝐿𝑅 (Transition curve) 

77. Usually adopted transition curve for railways is cubic parabola