Transportation engineering basic formulas

A handy reference for use in analysis and design of Transportation Engineering.

1. 𝑦=2𝑥2𝑛𝑊 (parabolic camber)

2. 𝑆𝑆𝐷 =𝑣𝑡+𝑣22𝑔(𝑓±𝑛) , 𝐼𝑆𝐷=2 ×𝑆𝑆𝐷

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

4. 𝑣2𝑅𝑔=𝑒+𝑓 , 𝑊𝑒=𝑛𝑙22𝑅+𝑉9.5√𝑅

5. 𝐿𝑠=𝑣3𝐶𝑅 (comfort condition)

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

7. 𝐿𝑠=2.7𝑉2𝑅 for plain and rolling terrain, 𝐿𝑠=𝑉2𝑅 for steep and mountainous terrain.

8. IRC recommends Spiral as transition curve.

9. 𝑆=𝐿𝑠224𝑅

10. 𝑚=𝑅(1−cos𝜃2), 𝜃=𝑆/𝑅 (single lane, 𝐿>𝑆)

11. 𝑚=𝑅(1−cos𝜃2)+𝑆−𝐿2sin𝜃2 , 𝜃=𝐿𝑅 (Single lane, 𝐿<𝑆)

12. Curve resistance =𝑇(1−cos𝛼)

13. Grade compensation =30+𝑅𝑅 or 75𝑅 whichever is less

14. 𝐿=𝑁𝑠2(√2ℎ+√2𝐻)2 when 𝐿>𝑠 (Summit curve for SSD), ℎ=0.15 𝑎𝑛𝑑 𝐻=1.2

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

16. 𝐿=𝑁𝑠2(√2𝐻+√2𝐻)2 when 𝐿>𝑠 (Summit curve for OSD), 𝐻=1.2

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

18. 𝐿=𝑁𝑠22ℎ+2𝑠tan𝛼 when 𝐿>𝑠 (valley curve), ℎ=0.75 𝑎𝑛𝑑 𝛼=1°

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

20. 𝐿=2(𝑁𝑣3𝐶)12 (valley curve comfort condition)

21. 𝐶=8075+𝑉

22. 𝐹𝑙𝑎𝑘𝑖𝑛𝑒𝑠𝑠 𝐼𝑛𝑑𝑒𝑥=𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑎𝑘𝑦 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑡ℎ𝑖𝑛𝑛𝑒𝑟 𝑡ℎ𝑎𝑛 0.6 𝑑𝑚𝑡𝑜𝑡𝑎𝑙 𝑤𝑒𝑖𝑔ℎ𝑡×100

23. 𝐸𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛 𝐼𝑛𝑑𝑒𝑥=𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑎𝑘𝑦 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑙𝑜𝑛𝑔𝑒𝑟 𝑡ℎ𝑎𝑛 1.8 𝑑𝑚𝑡𝑜𝑡𝑎𝑙 𝑤𝑒𝑖𝑔ℎ𝑡×100

24. Angularity number=67−100(𝑊𝐺)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+𝑊3𝑊1𝐺1+𝑊2𝐺2+𝑊3𝐺3 and 𝐺𝑡=𝑊𝑎+𝑊𝑏𝑊𝑎𝐺𝑎+𝑊𝑏𝐺𝑏

27. 𝐺𝑚=𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑚𝑜𝑢𝑙𝑑𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑜𝑢𝑙𝑑

28. 𝑣𝑎=𝐺𝑡−𝐺𝑚𝐺𝑡 , 𝑣𝑏=𝑊𝑏𝑊𝑚×𝐺𝑚𝐺𝑏 and 𝑉𝑀𝐴=𝑣𝑎+𝑣𝑏

29. 𝑉𝐹𝐵=𝑣𝑏𝑉𝑀𝐴

30. Flow value units 1/4th mm

31. 𝑁=365𝐴[(1+𝑟)𝑛−1𝑛 ]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=(𝑃80)4, where P is in kN

35. 𝐶𝐵𝑅 𝑎𝑡 2.5 𝑚𝑚=𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛 𝑘𝑔/𝑐𝑚270=𝑙𝑜𝑎𝑑 𝑖𝑛 𝑘𝑔1370

36. 𝐶𝐵𝑅 𝑎𝑡 5 𝑚𝑚=𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛 𝑘𝑔/𝑐𝑚2105=𝑙𝑜𝑎𝑑 𝑖𝑛 𝑘𝑔2055

37. 𝐸𝑆𝑊𝐿: interpolate load for depth from line joining (𝑙𝑜𝑔𝑃,log𝑧2) and (𝑙𝑜𝑔2𝑃,log2𝑠).

38. Radius of relative stiffness 𝑙=[𝐸ℎ312𝑘(1−𝜇2)]14

39. 𝑘=𝑝 (𝑘𝑔𝑐𝑚2)0.125 𝑐𝑚 (modulus of subgrade reaction)

40. 𝑘=𝐸1.18 𝑎 (𝑎 is rigid plate radius)

41. 𝑘=𝐸1.5 𝑎 (𝑎 is flexible plate radius)

42. 𝑘1𝑎1=𝑘2𝑎2

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

44. 𝐿𝛼𝑡=𝛿2 (expansion joint)

45. 𝐿𝑐2𝑏ℎ𝛾𝑐𝑓=𝑆𝑐 𝑏ℎ

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

47. 𝐿2𝜋∅𝜏𝑏𝑑=𝜋∅24𝑆𝑠 (length of tie bar)

48. 𝑃(𝑥)=(𝜆𝑡)𝑥𝑒−𝜆𝑡𝑥! (Poisson distribution)

49. 𝑃(ℎ≥𝑡)=𝑒−𝜆𝑡

50. 𝑘=𝑘𝑗(1−𝑣𝑣𝑓) and 𝑣=𝑣𝑓(1−𝑘𝑘𝑗)

51. 𝑞=𝑘𝑣 and 𝑘=1000𝑠𝑝𝑎𝑐𝑖𝑛𝑔

52. 𝑃𝐻𝐹=𝑝𝑒𝑎𝑘 ℎ𝑜𝑢𝑟 𝑓𝑙𝑜𝑤4 ×𝑝𝑒𝑎𝑘 15 𝑚𝑖𝑛 𝑓𝑙𝑜𝑤 or 𝑃𝐻𝐹=30𝑡ℎ ℎ𝑜𝑢𝑟𝑙𝑦 𝑣𝑜𝑙𝑢𝑚𝑒𝐴𝐴𝐷𝑇

53. 𝑇𝑖𝑚𝑒 𝑚𝑒𝑎𝑛 𝑠𝑝𝑒𝑒𝑑=𝑣1+𝑣2+⋯+𝑣𝑛𝑛

54. 𝑆𝑝𝑎𝑐𝑒 𝑚𝑒𝑎𝑛 𝑠𝑝𝑒𝑒𝑑=𝑑𝑡1+𝑡2+⋯+𝑡𝑛𝑛

55. 𝑞=𝑛𝑎+𝑛𝑦𝑡𝑎+𝑡𝑤, 𝑡̅=𝑡𝑤−𝑛𝑦𝑞

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+51−𝑌0

60. 𝑄=280𝑤 (1+𝑒𝑤)(1−𝑝3)(1+𝑤𝑙)

Transportation engineering basic formulas

61. 𝑇𝑟=𝑇𝑎+𝑇𝑚−𝑇𝑎3

62. 𝐸𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛=7 % 𝑓𝑜𝑟 300 𝑚

63. 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑜𝑓 𝑎𝑖𝑟𝑝𝑜𝑟𝑡,𝑇𝑠=15−6.51000ℎ

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

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

66. 𝑇𝑢𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑑𝑖𝑢𝑠 𝑅=𝑉2125𝑓

67. 𝑇𝑢𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑑𝑖𝑢𝑠 𝑅=0.388𝑊2𝑇2−𝑆 where 𝑠=6+𝑡𝑟𝑒𝑎𝑑2

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=1720𝑅 

71. 𝑉𝑒𝑟𝑠𝑖𝑛𝑒 𝑜𝑓 𝑐𝑢𝑟𝑣𝑒=𝑐28𝑅

72. 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑐𝑎𝑛𝑡=𝑉2127𝑅×𝐺

73. Theoretical cant =𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑐𝑎𝑛𝑡+𝑐𝑎𝑛𝑡 𝑑𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦

74. Widening of gauge in cm, 𝑑 =13(𝐵+𝐿)2𝑅 where B is wheel base in m, lap of flange in m, 𝐿=0.02√ℎ2+𝐷ℎ , h is depth of wheel flange below rail top level, D dia of wheel in cm.

75. 𝐿𝑠=3.28𝑣3𝑅 (Transition curve)

76. 𝑦=𝑥36𝐿𝑅 (Transition curve)

77. Usually adopted transition curve for railways is cubic parabola.