Geotechnical engineering basic formulas

 A handy reference for use in geotechnical analysis and design Field  Given Below :

1. 𝐼𝑓=𝑊1−𝑤2log10𝑁2/𝑁1

2. 𝐼𝑡=𝐼𝑝𝐼𝑓

3. 𝐶𝑢=𝐷60𝐷10

4. 𝐶𝑐=𝐷302𝐷60𝐷10

5. 𝐼𝑃=0.73 (𝑤𝐿−20)

6. 𝐼𝑃=𝑤𝐿−𝑤𝑃

7. 𝐼𝑠=𝑤𝑃−𝑤𝑆

8. 𝐼𝐿=𝑤−𝑤𝑃𝐼𝑃

9. 𝐼𝑐=𝑤𝐿−𝑤𝐼𝑃

10. 𝐴=𝐼𝑝𝐹 where 𝐹 is clay fraction (Activity)

11. 𝑅𝐷=𝑒𝑚𝑎𝑥−𝑒𝑒𝑚a𝑥−𝑒𝑚𝑖𝑛=1/𝛾𝑑,𝑚𝑖𝑛−1/𝛾𝑑1/𝛾𝑑,𝑚𝑖𝑛−1/𝛾𝑑,𝑚𝑎𝑥

12. 𝑘1𝑘2=tan𝛼1tan𝛼2 (non homogeneous)

13. 𝑘=𝐶𝛾𝑤𝜇𝑒31+𝑒 𝑑2

14. 𝐾=𝑘𝜇𝛾𝑤 (absolute permeability)

15. 𝑘=𝑞𝜋 ln(𝑟2/𝑟1)𝑧22−𝑧12 (Permeability in unconfined aquifer)

16. 𝑘=𝑞2𝜋𝑏 ln(𝑟2/𝑟1)𝑧2−𝑧1 (Permeability in confined aquifer)

17. 𝑘ℎ=𝑘1𝐻1+𝑘2𝐻2𝐻1+𝐻2 (effective horizontal permeability in stratified soils)

18. 𝑘𝑣=𝐻1+𝐻2𝐻1𝑘1+𝐻2𝑘2 (effective vertical permeability in stratified soils)

19. 𝑘𝑒=√𝑘ℎ𝑘𝑣 (effective permeability)

20. 𝑘=𝑎𝐿𝐴𝑡lnℎ1ℎ2 (falling head permeability test)

21. 𝑘=𝑞𝐿𝐴ℎ (constant head permeability test)

22. 𝑞=𝑘𝑒 ℎ𝑁𝑓𝑁𝑑 (seepage discharge)

23. 𝜎𝑧=3𝑄2𝜋1𝑧2 (11+(𝑟𝑧)2 )52 (Boussinesq’s formula)

24. 𝜎𝑧=𝑐𝑄2𝜋1𝑧2 (1𝑐2+(𝑟𝑧)2 )32 where 𝑐=√1−2𝜇2−2𝜇 (Wesrwegaard’s formula)

25. 𝜎𝑧=2𝑞𝜋𝑧 (11+(𝑥𝑧)2)2 (line load)

26. 𝜎𝑧=𝑞𝜋(2𝜃+sin2𝜃) where 𝜃=tan−1𝑏𝑧 (stress under centre of strip load of width 2𝑏 )

27. 𝜎𝑧=𝑞𝜋(2𝜃+sin2𝜃sin2∅) where 2𝜃=𝛽1−𝛽2 𝑎n𝑑 2∅=𝛽1+𝛽2 ( strip eccentric point)

28. 𝜎𝑧=𝑞(1−cos3𝜃) where 𝜃=tan−1𝑅𝑧 (stress under centre of circular load)

29. sin∅=𝜎1−𝜎3𝜎1+𝜎3 (for cohesion less soils)

30. sin∅=(𝜎1−𝜎3)/2𝑐cot∅+(𝜎1+𝜎3)/2 (for cohesive soils)

31. 𝜎1=2𝑐tan𝛼+𝜎3tan2𝛼 where 𝛼=45+∅2

32. tan∅=𝜏𝜎 (shear box test for cohesion less soils)

33. 𝑇=𝑐 𝜋𝐷2(𝐻2+𝐷6) (if both top and bottom surfaces contributes)

34. 𝑇=𝑐 𝜋𝐷2(𝐻2+𝐷12) (if only bottom surface contribute)

35. 𝑆𝑖=𝑞 𝐵 1−𝜇2𝐸 𝐼𝑓 (immediate settlement )

36. 𝑆𝑓=𝑆𝑝(𝐵𝑓𝐵𝑝 𝐵𝑝+0.3𝐵𝑓+0.3)2 (settlement of footing based on plate settlement)

37. Δ𝑢=𝐵(Δ𝜎𝑐)+𝐴𝐵 (Δ𝜎𝑑) (Skempton’s pore pressure parameters)

38. 𝑞=𝑚𝑃+𝑘 is stress path equation where ∅=tan−1𝑚 and 𝑐=𝑘/cos∅

39. 𝐶𝑐=Δ𝑒log10𝜎0+Δ𝜎𝜎0

40. 𝐶𝑐=0.009 (𝑤𝐿−10) (for normally consolidated soil)

41. 𝐶𝑐=0.007 (𝑤𝐿−10) (for over consolidated soil)

42. Δ𝑒1+𝑒0=Δ𝐻𝐻

43. 𝑚𝑣=Δ𝐻/𝐻Δ𝜎0

44. 𝑐𝑣=𝑘𝛾𝑤𝑚𝑣

45. 𝑇𝑣=𝑐𝑣𝑡𝑑2

46. 𝑇𝑣=𝜋4𝑈2 when 𝑈≤0.6

47. 𝑇𝑣=−0.933 log10(1−𝑈)−0.085 when 𝑈>0.6

48. 𝑆𝑓=𝐶𝑐𝐻1+𝑒0 log10𝜎0+Δ𝜎𝜎0

49. 𝑆𝑓=𝐶𝑟𝐻1+𝑒0 log10𝜎𝑐𝜎0+𝐶𝑐𝐻1+𝑒0 log10𝜎0+Δ𝜎𝜎𝑐

50. 𝐴𝑟=𝐷02−𝐷𝑖2𝐷𝑖2

51. 𝑆𝑛=𝑐𝑢𝐹𝛾𝐻

52. 𝑞𝑢=𝑐𝑁𝑐+𝑞 𝑁𝑞+0.5 𝛾 𝐵 𝑁𝛾 (Terzaghi’s strip)

53. 𝑞𝑢=1.3 𝑐𝑁𝑐+𝑞 𝑁𝑞+0.4 𝛾 𝐵 𝑁𝛾 (Terzaghi’s square)

54. 𝑞𝑢=1.3 𝑐𝑁𝑐+𝑞 𝑁𝑞+0.3 𝛾 𝐵 𝑁𝛾 (Terzaghi’s circle)

55. 𝑞𝑢=(1+0.3𝐵𝐿) 𝑐𝑁𝑐+𝑞 𝑁𝑞+(1−0.2𝐵𝐿)0.5 𝛾 𝐵 𝑁𝛾 (Terzaghi’s rectangle)

56. 𝑞𝑢=𝑐𝑁𝑐𝑆𝑐𝑑𝑐𝑖𝑐+𝑞 𝑁𝑞𝑆𝑞𝑑𝑞𝑖𝑞+0.5 𝛾 𝐵′ 𝑁𝛾 𝑆𝛾𝑑𝛾𝑖𝛾 (Meyerhof)

𝐵′=𝐵−2𝑒𝑥 and 𝐿′=𝐿−2 𝑒𝑦

57. 𝑞𝑛𝑢=𝑐𝑁𝑐 (Skempton

𝑁𝑐=5(1+0.2𝐷𝑓𝐵)(1+0.2𝐵𝐿)

Limiting value of 𝐷𝑓/𝐵 𝑖𝑠 2.5

58. 𝑄𝑢=𝑊ℎ 𝜂ℎ𝑆+𝐶 (ENR) where 𝐶=2.54 𝑐𝑚 𝑓𝑜𝑟 𝑑𝑟𝑜𝑝 ℎ𝑎𝑚𝑚𝑒𝑟 𝑎n𝑑 0.254 𝑐𝑚 𝑓𝑜𝑟 𝑠𝑡𝑒𝑎𝑚 ℎ𝑎𝑚𝑚𝑒𝑟

Geotechnical engineering basic formulas

59. 𝑄𝑢=𝑊ℎ𝜂ℎ𝜂𝑏𝑆+𝐶2 (Hiley)

where 𝐶=𝐶1+𝐶2+𝐶3

𝐶1=9.05𝑅𝐴 with dolley and 𝐶1=1.77𝑅𝐴 without dolley and 𝐶2=0.657𝑅𝐿𝐴 𝐶3=3.55𝑅𝐴 𝐿=𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑃𝑖𝑙𝑒 𝑖𝑛 𝑚 𝑅=𝑃𝑖𝑙𝑒 𝑐𝑎𝑝𝑎𝑐𝑖t𝑦 𝑖𝑛 𝑡𝑜𝑛𝑛𝑒𝑠=0.1𝑄 𝐴=𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑖𝑙𝑒 𝑖𝑛 𝑐𝑚2

𝜂𝑏=𝑊+𝑒2𝑃𝑊+𝑃 when 𝑊>𝑃

𝜂𝑏=𝑊+𝑒2𝑃𝑊+𝑃−(𝑊−𝑒𝑃𝐸+𝑃)2 when 𝑊<𝑃𝑒

60. 𝑄𝑢=𝑊ℎ𝜂ℎ𝑆+𝑆02 (Danish) 𝑆0=√2𝜂ℎ𝑊ℎ𝐿𝐴𝐸

61. 𝑄𝑢=𝐴𝑝𝑐𝑁𝑐+𝐴𝑠𝛼 𝑐 (clays)

62. 𝑄𝑢=𝐴𝑝𝑐𝑁𝑐+𝐴𝑠 𝜆(𝜎̅+2𝑐) (clays)

63. 𝑄𝑢=𝐴𝑝𝜎̅ 𝑁𝑞+𝐴𝑠𝜎̅ 𝑘tan𝛿 (sands) 𝜎̅ 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑢𝑝𝑡𝑜 15 𝑑 𝑑𝑒𝑝𝑡ℎ

64. 𝑄𝑢=𝑁(𝐴𝑝𝑐𝑁𝑐+𝐴𝑠𝛼 𝑐) or 𝑄𝑢=(𝐴𝑔𝑝𝑐𝑁𝑐+𝐴𝑔𝑠 𝑐) (Group)

65. 𝑝𝑎=𝑘𝑎𝜎̅−2𝑐√𝑘𝑎+𝑢

66. 𝑝𝑝=𝑘𝑝𝜎̅+2𝑐√𝑘𝑝+𝑢 

67. 𝑘𝑎=1−sin∅1+sin∅ and 𝑘𝑝=1+sin∅1−sin∅

 68. 𝐻𝑐=2𝑐𝛾√𝐾𝑎 and unsupported vertical cut =2𝐻𝑐

69. 𝑘𝑎=sin2(𝛽+∅)sin2𝛽sin(𝛽−𝛿) (1+√sin(∅+𝛿)sin(∅−𝑖)sin(𝛽−𝛿)sin(𝛽+𝑖))2 (Coulomb’s active )

70. 𝑘𝑝=sin2(𝛽−∅)sin2𝛽sin(𝛽+𝛿) (1−√sin(∅+𝛿)sin(∅+𝑖)sin(𝛽+𝛿)sin(𝛽+𝑖))2 (Coulomb’s passive )

71. 𝑘𝑎=cos𝛽−√cos2𝛽−cos2∅cos𝛽+√cos2𝛽−cos2∅ and 𝑃𝑎=𝑘𝑎𝛾ℎ22cos𝛽 (Inclined backfill)

72. 𝑘𝑝=cos𝛽+√cos2𝛽−cos2∅cos𝛽−√cos2𝛽−cos2∅ and 𝑃𝑝=𝑘𝑝𝛾ℎ22cos𝛽 (Inclined backfill)

73. 𝑁𝑐=15+12(𝑁−15) 𝑤ℎ𝑒𝑛 𝑁>15 𝑎𝑛𝑑 𝑁𝑐=𝑁 𝑤ℎ𝑒𝑛 𝑁≤15 (dilatancy)

74. 𝑖𝑐=𝐺−11+𝑒 (Quick sand condition)