**DUPIT FORMULA FOR TWO DIMENSIONAL FLOWS ON A horizontal impervious boundary,. Q= k(h1. 2-h2. 2)/2L. 38. Empirical coefficient of Permeability,.
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P E CivilExam.com Copyright © 200 8 – 201 2 Pecivilexam.com all rights reserved – Geotechnical F ormula 1 GEOTECHNICAL AND FOUNDATION FORMULA SHEET Table Cont en ts Page 1. IDENTIFICATION AND CLASSIFICATION OF SOIL AND ROCK 1 2. HYDRAULIC PROPERTIES OF SOIL AND ROCK 3 3. EFFECTIVE STRESS AND SEEPAGE PRESSURE 5 4. SEEPAGE OF WATER THROUGH SOI LS 5 5. COMPRESSIBILITY OF SOIL AND ROCK 6 6. STRENGTH OF SOIL AND ROCK 7 7. ENGINEERING GEOLOGY OF THE ROCKS AND SOIL 8 8. ENGINEERING SUBSURFACE INVESTIGATION 8 9. SHALLOW FOUNDATION FOOTING AND RAFT 10 10. DEEP FOUNDATION P ILES AND PIERS 11 11. RETAINING STRUCTURES 12

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P E CivilExam.com Copyright © 200 8 – 201 2 Pecivilexam.com all rights reserved – Geotechnical F ormula 2 IDENTIFICATION AND CLASSIFICATION OF SOIL AND ROCK 1. The Coefficient of uniformity, C u = D 60 /D 10 2. The Coefficient of Curvature, C z = (D 30 ) 2 / (D 60 x D 10 ) 3. Plasticity index, PI= LL Œ PL 4. Liquidity index, LI= (w – PL) /(LL – PL) 5. Activity index, AI= PI / (%> 0.002mm) , Clay contain greater than 40% 6. Activity index, AI= PI / (%>0.002mm – 5) , Clay contain less than 40% i f AI=>.75, low active clay; if AI= .75 to 1.25, normal active clay ; if < 1.25, active clay 7. Group index, GI= (F - 35) x [0.2+0.005 x (LL - 40)]+0.01 x(F - 15) x (PI - 10) , F IS % OF PASSING #200 8. VOLUME OF VOID, V v =V - V s ; 9. VOLUME OF SOLID SOIL, V s = W s / G s w 10. VOLUME OF SOIL, V =V s + V v 11. TOTAL WEIGHT, W =W w +W s 12. WEIGHT of SOIL, Ws=W/(1+w) 13. WATER CONTENT, w =W w / W s 14. BULK DENSITY, s w /(1 + e) = (G s + S r w /(1 + e) 15. SATURATED UNIT WEIGHT, sat = (G s w / 1+e; S r =1 16. DRY U NIT WEIGHT, d = W w / V=G s w 17. UNIT WEIGHT OF WATER, w = 62.4 PCF = 9.8KN/m 3 18. SUBMERGED UNIT WEIGHT, ™ = (G s Œ w sat - w = (G s w / 1+e 19. DEGREE OF SATURATION, S r = V w /V v = w G s /e 20. SPECIFIC GRAVIT Y, G s = W s /V s w 21. VOID RATIO, e = V v /V s = n/1 - n = [G s w - 1 22. VOID RATIO, e = w G s / S r ; WHERE FULLY SATURATED 23. SOIL, S r =1 POROSITY, n = V v /V 24. SPECIFIC VOLUME, v = 1 + e 25. AIR CONTENT, A = V a /V = (e - w G s ) / 1+e = n ( 1 - S r ) 26. RELATIVE DENSITY, D r =100 (e max Œ e) / (e max Œ e min ) D r min Œ d min Œ max )
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 3 27. Critical Hydraulic gradient, i c w =(G s - 1)/(1+e),Where, 28. Terminal velocity of particle, - w )D 2 /18µ s, D=dia, µ s=viscosity=.001 (SI unit) HYDRAULIC PROPERTIES OF SOIL AND ROCK 29. DISCHARGE VELOCITY , q =VA=kiA , discharge, v =ki ; k = Coefficient of permeability length o f flow path V= ki = q/A = q/Ta=Q/At, 30. VOLUME OF WATER, Q = Volume of water collected k = Coefficient of permeability i = Hydraulic gradient, h/L A = Cross - sectional area of sample t = Duration of time for collection of water L = Length of the sample For granular soil, 31. K=1 /e 2 For Horizontal flow 32. K=e 3 /1+e For vertical flow 33. Constant Head Permeability, 34. Falling Head Permeability, k = 2.303(aL/At) Log 10 (h o /h 1 ) a = cross - sectional area of standpipe h o = water level in the standpipe at start of the time h 1 = water level in the standpipe at end of the time 35. Equivalent Permeabili ty of Stratified Deposit. Equivalent Horizontal Permeability, K h(eq) = (k h1 x h 1 + k h2 x h 2 –.. k hn x h n )
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 4 36. Equivalent Horizontal Permeability, K v(eq) = h . . . ( h 1 /k v1 )+ ( h 2 /k v2 ).. .. ( h n / k vn ) 37. DUPIT FORMULA FOR TWO DIMENSIONAL FLOWS ON A horizontal impervious boundary, Q= k(h 1 2 - h 2 2 )/2L 38. Empirical coefficient of Permeability, k = CD 10 2, C =.4 to 1.5 , normally 1.0 C u < 5.0 Confined Aquifer 39. Fully Penetrating Coefficient of Permeability, k = [2.303 q Log 10 (r 1 /r 2 1 /h 2 ), 40. Partiall y Penetrating Coefficient of Permeability, k = [2.303 q Log 10 (r 1 /r 2 1 /h 2 )G, G = W/D [(1 +7 w W= Partially Penetrating depth r w = Radius of the well D= depth of aquifer Unconfined Aquifer 4 1. Fully Penetrating Coefficient of Permeability, k = [2.303 q Log 10 (r 1 /r 2 2 2 Œ h 1 2 ) 41 Partially Penetrating Coefficient of Permeability, k = [2.303 q Log 10 (R/r w - s) 2 - t 2 ] C= 1, nearly 1.0 s= length of un - penetrating de pth t= depth from draw - down to bottom r w = Radius of the well R= Radius of the draw - down cylinder
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 5 EFFECTIVE STRESS AND SEEPAGE PRESSURE No flow condition, 42. Total vertical pressure, p = H 0 w + sat 43. Pore water pressure, u w = H 0 w + w 44. Effective vertical pressure, - u w sat - w) = Z ™ z = certain depth of the soil Downward flow condition, 45. Pore water pressure, u w = z (H 0+ H s Œ w / H s 46. Total vertical pressure, w /H s H s = total depth of the soil, h= depth down 47. Effective vertical pressure, + w Upward flow condition, 48. Pore water pressure, u w = z (H 0+ H s w / H s 49. Effective vertical pressure, - w 50. Critical Hydraulic grad ient, i c w where, SEEPAGE OF WATER THROUGH SOILS Flow net in isotropic soil, 51. Total quantity of water flow under dam, sheet pile, q t =kH(N f /N d ) N f = number of flow channels in the net N d = number of equipotential drop H= Head difference Flow net in Anisotropic soil, 51. Total quantity of water flow under dam, sheet pile, q t = x . k z )h(N f /N d ) 52. Seepage line - free Surface , a= (d/cos ) - 2 /cos 2 - h 2 /sin 2 ) Heaving of soil at Exit Point 53. The pore water pressure at certain point A, u A w { z A+ d w +(rest of N d at point A / N d )h } Like, u A w { z A+ d w +(2 / 9 )h } (at tailwater side) z A= Depth of soil Point A to top of the soil (at tailwater side) d w = Depth of water from top of the soil to water level (at tailwater side)
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 6 Factor of safety for sheet pile against heave or boiling of the soil Where , i = Hydraulic gradient, h/L is too high. 54. Factor of safety, av w ) , sat - w ) x h, h=depth of heave soil pri sm/unit length pile. i av = N d at middle of heave soil prism /unit length pile. W™= Submerged weight of soil in the heave zone per unit width of sheet pile U= Uplift force due to seepage on the same volume of soil W™= D 2 sat - w )/2= D 2 /2, Where, D= is the depth of embedment into Permeable soil U= D 2 (i av w )/2 Block of heave soil = D/2 x D, max heave within D/2 from sheet pile COMPRESSIBILITY OF SOIL AND ROCK Vertical stress under Foundation Vertical pressure on each layer, 5 5. t m b ) /6 t, m, b are the increase in pressure at top, middle, bottom 56. Avarage Vertical pressure, av A B C ) /6 A, A, C are the pressure at LAYER Time rate Consolidation , Settlement 57. compression inde x,C c = 0.009(LL - 10) 58. swell index, C s = 1/5 to 1/6 59. Settlement, S = H (1+ e 0 ) , For One - dimensional consolidation 60. Settlement, S= C c H [log(p 0 + 0 ] / (1+ e 0 ), For p 0= p c, normal consolidated clay p 0= Effecti ve overberden pressure p c = Preconsolidation pressure 61. Pre - consolidation pressure, P c = .5q u /(.11+.0037 PI)
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 8 ENGINEERING GEOLOGY OF THE ROCKS AND SOIL 74. Earthquake , Lateral force, V=ZIKCSW Where, Z = zone factor, I=intensity =1 , 1.5 for Hospital K=0.67 , Space Frame K=0.80, Frame / shear wall K=1, Shear wall B ox K =1.33 C=1/ ( 15 S=1 or 1.5 for Rock foundation W= Total Building dead load plus 25% floor live load. ENGINEERING SUBSURFACE INVESTIGATION Field Vane Shear Test 75. Torque, T=px= C u (d 2 h/2)+(d 3 /6) Cu= 1.7 - 0.54 (PI) where C Correction factor , PI Plasticity index of the soil. Standard Penetration test, 76. Corrected N - value, N 1 (60)= N x C e x C l x C s x C d x C N C N = v ) where P 100 kPa or 2.0 ksf or 1 tsf, or 1 kg/cm 2 where ( N1 )60 =Normalized SPT blow count, for 60% rod - ene rgy ratio and 100 kPa (1 kg/cm 2 ; 1 tsf, 2 ksf) N= Field SPT blow count, from 6 to 18 inches C e = Correction for hammer release system energy C l = Correction for rod length C s = Correction for sampler type C d = Correction for bore hole diameter C N = Corre ction for effective overburden pressure Static - Cone Penetration Test A rod, having an enlarged cone - shaped tip of 1.4 inches diameter, is pushed into theground at the rate of 2 to 4 feet per minute of the soils encountered. An empirical relationship betw een normalized cone resistance, normalized friction ratio, and soil identification is . 77. q c1e = q c v ' ) c 78. … c1e v ') s 79. R … =100 (…s/q c ) v '= Vertical effective stress (1 atm, 1 tsf, or 100 kPa) q c 1e = Normalized cone resistance qc = Measured cone resistance (1 atm, 1 tsf, or 100 kPa)
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 9 c= Cone resistance stress exponent f s 1e = Normalized sleeve friction fs = Measured sleeve friction (1 atm, 1 tsf, or 100 kPa) R … = Friction ratio, percent. Estimating Relative Density and Friction Angle from SPT Data Presented empirical relationships that can be reasonably approximated by a straight line for N - values up to 50 blows per foot (0.3 m): 80. For coarse - grained sands: '= 30° N/3 81. For fine - grained sands: ' = 28° N/4 Estimating Unconfi ned Compressive Strength from CPT Data 82. Su=( qc - total) /Nk where Su = Untrained cohesive strength qc = Measured CPT cone resistance total l = In situ total overburden stress Nk = Empirical untrained strength - bearing factor. This equation is applicable for most sedimentary, non - sensitive clays. Estimating Drained Friction Angle from CPT Data There are two methods for estimating the drained friction angle of clean sands: An empirical correlation that indicates 83. '= 28° + 12.4 lo g(q c 1e) Where , the normalized tip resistance, q c 1e , measured in MPa, Estimating Pre - consolidation Pressure 84. Effective overburden pressure, P™ c = .5q u /(.11+.0037 PI); C u =.5q u Estimation of Liquefaction Potential 85. cyc = 0.65 a max v rd /g cyc= Uniform cyc lic shear stress a max = Peak ground surface acceleration g= Acceleration of gravity v =Total vertical stress rd= Stress reduction factor (see Figure 7.25). The Cyclic Stress Ratio is defined as 86. CSR = cyc v0 , cyc / Earth quake shear st ress
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 10 SHALLOW FOUNDATION FOOTING AND RAFT 87. Ultimate Bearing Capacity, q d = cN c f N q For Continuous footing C= Cohesion D f= Depth of foundation B = Width of foundation N ,N c, N q = Bearing capacity factor 8 8. Bearing Capacity, q dr = 1.2cN c f N q For Circular footing on hard soil 89. Bearing Capacity, q dr = 1.2cN c f N q For Square (BxB) footing on hard soil 90. Bearing Capacity, q ult = cN cq For Continuous footing with inc lined load Continuous Footing at top of slope and on a slope (Case - I and Case - II) 91. Bearing Capacity, q ult = cN cq For Continuous footing with water level d o >= B 92. Bearing Capacity, q ult = cN cq sub N For Continuous footing with water level at GL Using 0.4B for squre and 0.6R for circular footing instate of 0.5B Bearing Capacity of Cohesive Soils Single Cohesive Layer. 93. The ultimate bearing capacity of cohesive soils, q d = cN c f q d(net) = cN c For a continuous footing, for Df / B<= 4 Nc= 5.14 +[( D… / B) / 0.37 +0.35 ( D… / B)] For a circular or square footing, for Df / B<= 4 Nc= 6.2 +[( D… / B) / 0.45 +0.24 ( D… / B)] For a rectangular footing, Nc=( 0.84 +0.16 B/ L) Nc (square)
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P E CivilExam.com Copyright © 200 8 - 201 2 Pecivilexam.com all rights reserved - Geotechnical F ormula 11 DEEP FOUNDATION PILE S AND PIERS Ultimate vertical load capacity of pile or pier 94. Qult= Qb+ Qs - Wp = 9c u A p u p L Where, Qult= Ultimate vertical load capacity of pile or pier Qb= Component of load capacity due to bearing capacity at pile or pier base Qs= Component of load capacity due to side friction p=perimeter L=Length Other method Load capacity at pile or pier base 95. Q b = A b ( cNc + t ' Nq _ 0.5 B b' N ) Where A b = Area of pile or pier base c= Soil cohesion t '= Effective vertical stress at pile or pier base B= Base diameter b'= Effective unit weight of soil in the failure zone beneath base Nc, Nq , N Bearing capacity factors. Page - 8.4 Fig - 8.4 The load capacity due to skin friction on the shaft of the pile 96. Qs= t ' K h c Where, t '= Effective overburden pressure K hc = Ratio of horizontal to vertical pressure Œ pile in compression P= Perimeter or circumference of pile, For circular pile, P= L= length of the pile. Carrying Capacity of a Single Pile or Pier in Granular Soil 97. Qult= Ab t ' Nq + t ' K hc c=0, N =0 Carrying Capacity of a Single Pile or Pier in Cohesive Soil 98. Qb - ult= A b c N c Where, c=.5q u , N q =0 and Skin Frictio n factor for Driven Piles. 99. Qs - ult = c u PL, ca / cu= 1.0, cu 0.25 tsf ca / cu= 1.25 - cu , 0.25< cu< 0.75 tsf ca / cu= 0.5, cu> 0.75 tsf Settlement of Pile Groups Pile Group in Granular Soil .

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