Geotechnical engineering basic formulas

Civil Study October 18, 2021

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

1. 𝐼𝑓=π‘Š1βˆ’π‘€2log10𝑁2/𝑁1

2. 𝐼𝑑=𝐼𝑝𝐼𝑓

3. 𝐢𝑒=𝐷60𝐷10

4. 𝐢𝑐=𝐷302𝐷60𝐷10

5. 𝐼𝑃=0.73 (π‘€πΏβˆ’20)

6. 𝐼𝑃=π‘€πΏβˆ’π‘€π‘ƒ

7. 𝐼𝑠=π‘€π‘ƒβˆ’π‘€π‘†

8. 𝐼𝐿=π‘€βˆ’π‘€π‘ƒπΌπ‘ƒ

9. 𝐼𝑐=π‘€πΏβˆ’π‘€πΌπ‘ƒ

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

11. 𝑅𝐷=π‘’π‘šπ‘Žπ‘₯βˆ’π‘’π‘’π‘šaπ‘₯βˆ’π‘’π‘šπ‘–π‘›=1/𝛾𝑑,π‘šπ‘–π‘›βˆ’1/𝛾𝑑1/𝛾𝑑,π‘šπ‘–π‘›βˆ’1/𝛾𝑑,π‘šπ‘Žπ‘₯

12. π‘˜1π‘˜2=tan𝛼1tan𝛼2 (non homogeneous)

13. π‘˜=πΆπ›Ύπ‘€πœ‡π‘’31+𝑒 𝑑2

14. 𝐾=π‘˜πœ‡π›Ύπ‘€ (absolute permeability)

15. π‘˜=π‘žπœ‹ ln(π‘Ÿ2/π‘Ÿ1)𝑧22βˆ’π‘§12 (Permeability in unconfined aquifer)

16. π‘˜=π‘ž2πœ‹π‘ ln(π‘Ÿ2/π‘Ÿ1)𝑧2βˆ’π‘§1 (Permeability in confined aquifer)

17. π‘˜β„Ž=π‘˜1𝐻1+π‘˜2𝐻2𝐻1+𝐻2 (effective horizontal permeability in stratified soils)

18. π‘˜π‘£=𝐻1+𝐻2𝐻1π‘˜1+𝐻2π‘˜2 (effective vertical permeability in stratified soils)

19. π‘˜π‘’=βˆšπ‘˜β„Žπ‘˜π‘£ (effective permeability)

20. π‘˜=π‘ŽπΏπ΄π‘‘lnβ„Ž1β„Ž2 (falling head permeability test)

21. π‘˜=π‘žπΏπ΄β„Ž (constant head permeability test)

22. π‘ž=π‘˜π‘’ β„Žπ‘π‘“π‘π‘‘ (seepage discharge)

23. πœŽπ‘§=3𝑄2πœ‹1𝑧2 (11+(π‘Ÿπ‘§)2 )52 (Boussinesq’s formula)

24. πœŽπ‘§=𝑐𝑄2πœ‹1𝑧2 (1𝑐2+(π‘Ÿπ‘§)2 )32 where 𝑐=√1βˆ’2πœ‡2βˆ’2πœ‡ (Wesrwegaard’s formula)

25. πœŽπ‘§=2π‘žπœ‹π‘§ (11+(π‘₯𝑧)2)2 (line load)

26. πœŽπ‘§=π‘žπœ‹(2πœƒ+sin2πœƒ) where πœƒ=tanβˆ’1𝑏𝑧 (stress under centre of strip load of width 2𝑏 )

27. πœŽπ‘§=π‘žπœ‹(2πœƒ+sin2πœƒsin2βˆ…) where 2πœƒ=𝛽1βˆ’π›½2 π‘Žn𝑑 2βˆ…=𝛽1+𝛽2 ( strip eccentric point)

28. πœŽπ‘§=π‘ž(1βˆ’cos3πœƒ) where πœƒ=tanβˆ’1𝑅𝑧 (stress under centre of circular load)

29. sinβˆ…=𝜎1βˆ’πœŽ3𝜎1+𝜎3 (for cohesion less soils)

30. sinβˆ…=(𝜎1βˆ’πœŽ3)/2𝑐cotβˆ…+(𝜎1+𝜎3)/2 (for cohesive soils)

31. 𝜎1=2𝑐tan𝛼+𝜎3tan2𝛼 where 𝛼=45+βˆ…2

32. tanβˆ…=𝜏𝜎 (shear box test for cohesion less soils)

33. 𝑇=𝑐 πœ‹π·2(𝐻2+𝐷6) (if both top and bottom surfaces contributes)

34. 𝑇=𝑐 πœ‹π·2(𝐻2+𝐷12) (if only bottom surface contribute)

35. 𝑆𝑖=π‘ž 𝐡 1βˆ’πœ‡2𝐸 𝐼𝑓 (immediate settlement )

36. 𝑆𝑓=𝑆𝑝(𝐡𝑓𝐡𝑝 𝐡𝑝+0.3𝐡𝑓+0.3)2 (settlement of footing based on plate settlement)

37. Δ𝑒=𝐡(Ξ”πœŽπ‘)+𝐴𝐡 (Ξ”πœŽπ‘‘) (Skempton’s pore pressure parameters)

38. π‘ž=π‘šπ‘ƒ+π‘˜ is stress path equation where βˆ…=tanβˆ’1π‘š and 𝑐=π‘˜/cosβˆ…

39. 𝐢𝑐=Δ𝑒log10𝜎0+Ξ”πœŽπœŽ0

40. 𝐢𝑐=0.009 (π‘€πΏβˆ’10) (for normally consolidated soil)

41. 𝐢𝑐=0.007 (π‘€πΏβˆ’10) (for over consolidated soil)

42. Δ𝑒1+𝑒0=Δ𝐻𝐻

43. π‘šπ‘£=Δ𝐻/π»Ξ”πœŽ0

44. 𝑐𝑣=π‘˜π›Ύπ‘€π‘šπ‘£

45. 𝑇𝑣=𝑐𝑣𝑑𝑑2

46. 𝑇𝑣=πœ‹4π‘ˆ2 when π‘ˆβ‰€0.6

47. 𝑇𝑣=βˆ’0.933 log10(1βˆ’π‘ˆ)βˆ’0.085 when π‘ˆ>0.6

48. 𝑆𝑓=𝐢𝑐𝐻1+𝑒0 log10𝜎0+Ξ”πœŽπœŽ0

49. 𝑆𝑓=πΆπ‘Ÿπ»1+𝑒0 log10πœŽπ‘πœŽ0+𝐢𝑐𝐻1+𝑒0 log10𝜎0+Ξ”πœŽπœŽπ‘

50. π΄π‘Ÿ=𝐷02βˆ’π·π‘–2𝐷𝑖2

51. 𝑆𝑛=𝑐𝑒𝐹𝛾𝐻

52. π‘žπ‘’=𝑐𝑁𝑐+π‘ž π‘π‘ž+0.5 𝛾 𝐡 𝑁𝛾 (Terzaghi’s strip)

53. π‘žπ‘’=1.3 𝑐𝑁𝑐+π‘ž π‘π‘ž+0.4 𝛾 𝐡 𝑁𝛾 (Terzaghi’s square)

54. π‘žπ‘’=1.3 𝑐𝑁𝑐+π‘ž π‘π‘ž+0.3 𝛾 𝐡 𝑁𝛾 (Terzaghi’s circle)

55. π‘žπ‘’=(1+0.3𝐡𝐿) 𝑐𝑁𝑐+π‘ž π‘π‘ž+(1βˆ’0.2𝐡𝐿)0.5 𝛾 𝐡 𝑁𝛾 (Terzaghi’s rectangle)

56. π‘žπ‘’=𝑐𝑁𝑐𝑆𝑐𝑑𝑐𝑖𝑐+π‘ž π‘π‘žπ‘†π‘žπ‘‘π‘žπ‘–π‘ž+0.5 𝛾 𝐡′ 𝑁𝛾 𝑆𝛾𝑑𝛾𝑖𝛾 (Meyerhof)

𝐡′=π΅βˆ’2𝑒π‘₯ and 𝐿′=πΏβˆ’2 𝑒𝑦

57. π‘žπ‘›π‘’=𝑐𝑁𝑐 (Skempton

𝑁𝑐=5(1+0.2𝐷𝑓𝐡)(1+0.2𝐡𝐿)

Limiting value of 𝐷𝑓/𝐡 𝑖𝑠 2.5

58. 𝑄𝑒=π‘Šβ„Ž πœ‚β„Žπ‘†+𝐢 (ENR) where 𝐢=2.54 π‘π‘š π‘“π‘œπ‘Ÿ π‘‘π‘Ÿπ‘œπ‘ β„Žπ‘Žπ‘šπ‘šπ‘’π‘Ÿ π‘Žn𝑑 0.254 π‘π‘š π‘“π‘œπ‘Ÿ π‘ π‘‘π‘’π‘Žπ‘š β„Žπ‘Žπ‘šπ‘šπ‘’π‘Ÿ

Geotechnical engineering basic formulas
Geotechnical engineering basic formulas

59. 𝑄𝑒=π‘Šβ„Žπœ‚β„Žπœ‚π‘π‘†+𝐢2 (Hiley)

where 𝐢=𝐢1+𝐢2+𝐢3

𝐢1=9.05𝑅𝐴 with dolley and 𝐢1=1.77𝑅𝐴 without dolley and 𝐢2=0.657𝑅𝐿𝐴 𝐢3=3.55𝑅𝐴 𝐿=πΏπ‘’π‘›π‘”π‘‘β„Ž π‘œπ‘“ 𝑃𝑖𝑙𝑒 𝑖𝑛 π‘š 𝑅=𝑃𝑖𝑙𝑒 π‘π‘Žπ‘π‘Žπ‘π‘–t𝑦 𝑖𝑛 π‘‘π‘œπ‘›π‘›π‘’π‘ =0.1𝑄 𝐴=π‘π‘Ÿπ‘œπ‘ π‘  π‘ π‘’π‘π‘‘π‘–π‘œπ‘› π‘Žπ‘Ÿπ‘’π‘Ž π‘œπ‘“ 𝑝𝑖𝑙𝑒 𝑖𝑛 π‘π‘š2

πœ‚π‘=π‘Š+𝑒2π‘ƒπ‘Š+𝑃 when π‘Š>𝑃

πœ‚π‘=π‘Š+𝑒2π‘ƒπ‘Š+π‘ƒβˆ’(π‘Šβˆ’π‘’π‘ƒπΈ+𝑃)2 when π‘Š<𝑃𝑒

60. 𝑄𝑒=π‘Šβ„Žπœ‚β„Žπ‘†+𝑆02 (Danish) 𝑆0=√2πœ‚β„Žπ‘Šβ„ŽπΏπ΄πΈ

61. 𝑄𝑒=𝐴𝑝𝑐𝑁𝑐+𝐴𝑠𝛼 𝑐 (clays)

62. 𝑄𝑒=𝐴𝑝𝑐𝑁𝑐+𝐴𝑠 πœ†(πœŽΜ…+2𝑐) (clays)

63. 𝑄𝑒=π΄π‘πœŽΜ… π‘π‘ž+π΄π‘ πœŽΜ… π‘˜tan𝛿 (sands) πœŽΜ… π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’ π‘’π‘π‘‘π‘œ 15 𝑑 π‘‘π‘’π‘π‘‘β„Ž

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

65. π‘π‘Ž=π‘˜π‘ŽπœŽΜ…βˆ’2π‘βˆšπ‘˜π‘Ž+𝑒

66. 𝑝𝑝=π‘˜π‘πœŽΜ…+2π‘βˆšπ‘˜π‘+𝑒 

67. π‘˜π‘Ž=1βˆ’sinβˆ…1+sinβˆ… and π‘˜π‘=1+sinβˆ…1βˆ’sinβˆ…

 68. 𝐻𝑐=2π‘π›ΎβˆšπΎπ‘Ž and unsupported vertical cut =2𝐻𝑐

69. π‘˜π‘Ž=sin2(𝛽+βˆ…)sin2𝛽sin(π›½βˆ’π›Ώ) (1+√sin(βˆ…+𝛿)sin(βˆ…βˆ’π‘–)sin(π›½βˆ’π›Ώ)sin(𝛽+𝑖))2 (Coulomb’s active )

70. π‘˜π‘=sin2(π›½βˆ’βˆ…)sin2𝛽sin(𝛽+𝛿) (1βˆ’βˆšsin(βˆ…+𝛿)sin(βˆ…+𝑖)sin(𝛽+𝛿)sin(𝛽+𝑖))2 (Coulomb’s passive )

71. π‘˜π‘Ž=cosπ›½βˆ’βˆšcos2π›½βˆ’cos2βˆ…cos𝛽+√cos2π›½βˆ’cos2βˆ… and π‘ƒπ‘Ž=π‘˜π‘Žπ›Ύβ„Ž22cos𝛽 (Inclined backfill)

72. π‘˜π‘=cos𝛽+√cos2π›½βˆ’cos2βˆ…cosπ›½βˆ’βˆšcos2π›½βˆ’cos2βˆ… and 𝑃𝑝=π‘˜π‘π›Ύβ„Ž22cos𝛽 (Inclined backfill)

73. 𝑁𝑐=15+12(π‘βˆ’15) π‘€β„Žπ‘’π‘› 𝑁>15 π‘Žπ‘›π‘‘ 𝑁𝑐=𝑁 π‘€β„Žπ‘’π‘› 𝑁≀15 (dilatancy)

74. 𝑖𝑐=πΊβˆ’11+𝑒 (Quick sand condition)

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