GEOTECHNICAL ENGINEERING FORMULAS

A handy reference for use in geotechnical analysis and design

TABLE OF CONTENTS Page

1. SOIL CLASSIFICATION ...............................................................................3 1.1 USCS: Unified Soil Classification System..........................................3

1.1.1 Relative Density of Cohesionless Soils: .....................................4 1.1.2 Fine Grained(Cohesive) Soil Charts using the USCS System:..4 1.1.3 Consistency of Fine Grained Soils: .............................................5

1.2 USDA Soil Classification System ........................................................5 1.3 AASHTO Soil Classification System:..................................................6

2. PHASE RELATIONSHIP EQUATIONS: .......................................................7 2.1 Shear Strength of Soils ........................................................................7 2.2 Bearing Capacity of Soils ....................................................................7

3. STRESSES IN SOILS ...................................................................................9 3.1 Various Loading Conditions:...............................................................9 ..........................................................................................................................9 ..........................................................................................................................9

4. SHALLOW FOUNDATIONS .......................................................................10 4.1 Conventional Footings.......................................................................10

4.11Geotechnical Analysis........................................................................10 4.12 Structural Design:..............................................................................10

4.2 Strap or Cantilever Footings: .................................................................11 4.3 Trapezoidal Footings: .............................................................................12

5. SOIL CONSOLIDATION EQUATIONS .......................................................14 5.1 Instant Settlement of footings: ..........................................................14 5.2 Primary Consolidation: ......................................................................14 5.3 Overconsolidated Soils ..........................................................................14 5.4 Time rate of settlement ..........................................................................15

5.41 Coefficient of consolidation..............................................................15 6. RETAINING STRUCTURES: ......................................................................16

6.1 Horizontal Stresses: Active, At Rest and Passive ................................16 6.2 Basement Wall with surcharge: .............................................................17 6.3 Braced Excavations: ...............................................................................17 6.4 Forces on Struts:.....................................................................................18 6.5 Cantilever Sheetpiles in Sand ................................................................20 6.6 Cantilever Sheetpiles in Clay .................................................................21 6.6 Anchored Sheetpiles in Sand (Also called Bulkheads) .......................22 6.7 Anchored Sheetpiles in Clay (Also called Bulkheads).........................24

7. PILE FOUNDATIONS .................................................................................25 8. Post Tensioned Slabs: ..............................................................................28 9. Asphalt Mix Design: ..................................................................................30 10. Concrete Mix Design:.............................................................................33

1. SOIL CLASSIFICATION

1.1 USCS: Unified Soil Classification System Coarse Grained soils have less than 50% passing the # 200 sieve:

Symbol Passing the #200

Cu= 30

60

D

D

Cc =

6010

2

30)(

DD

D

Soil Description

GW < 5% 4 or higher 1 to 3 Well graded gravel GP < 5% Less than 4 1 to 3 Poorly graded gravel GW-GM 5 to12% 4 or higher 1 to 3 but with

15%sand Clayey Gravel with sand GM-GC >12% N/A N/A,12% N/A N/A,15%sand Clayey Silt with sand

SW < 5% 6 or higher 1 to 3 Well graded sand SP < 5% Less than 6 1 to 3 Poorly graded sand SM >12% N/A N/A Silty Sand or Sandy Silt SC >12% N/A N/A Clayey Sand or Sandy

Clay SC-SM >12% N/A N/A Silty Clay with Sand

Where:

Cu = Uniformity Coefficient; gives the range of grain sizes in a given sample. Higher Cu means well graded. Cz = Coefficient of Curvature is a measure of the smoothness of the gradation curve. Usually less than 3. D10, D3, & D60 are the grain size diameter corresponding to 10%, 30% and 60% passing screen.

1.1.1 Relative Density of Cohesionless Soils: SPT or N value Relative Density % Relative Density

0 3 Very loose 0 15

4 10 Loose 15 35

11 30 Medium dense 35 65

31 50 Dense 65 -85

> 50 Very dense 85 - 100

1.1.2 Fine Grained(Cohesive) Soil Charts using the USCS System:

1.1.3 Consistency of Fine Grained Soils:

SPT or N value Cohesion, C or Su Consistency

< 2 < 500 psf Very soft

2 4 500 1000 psf Soft

5 8 1000 2000 psf Firm

9 15 2000 4000 psf Stiff

16-30 4000 8000 psf Very stiff

>30 > 8000 psf Hard

1.2 USDA Soil Classification System

The percent SAND,SILT,and CLAY lines are drawn and their intersection gives the soil classification.

1.3 AASHTO Soil Classification System:

CLASSIFICATION OF HIGHWAY SUBGRADE MATERIALS(With suggested subgroups)

General Classification

Group Classification

Sieve Analysis,Percent Passing:

No. 10No. 40No. 200

Characteristics offraction passing # 40:

Liquid LimitPlasticity IndexGroup Index

Usual Types of Significicant Constituent

Materials

General Rating as Subgrade

A-2

A-1-a A-1-b A-2-4 A-2-5 A-2-6 A-2-7

0-60

Stone Fragments, Gravel and Sand

A-3

51-1000-10

N.P.0

Fine Sand

0-50

Granular Materials (35% or less passing No. 200)

0-35 0-35 0-35 0-350-300-15

0-500-25

A-1

0-100-40 41+

0-100-4011+

41+11+

0 0-4

Silty or Clayey Gravel and Sand

Excellent to Good

Silt-Clay Materials (More than 35% passing #200)

A-4 A-5 A-6A-7

A-7-5A-7-6

36-100 36-100 36-100 36-100

0-400-10

41+0-10

0-4011+

41+11+

Silty Soils Clayey Soils

Fair to Poor

0-8 0-12 0-16 0-20

Cohesive soils classification in AASHTO System:

2. PHASE RELATIONSHIP EQUATIONS: Dry Unit Weight, d

Bulk or Wet or Total Unit Weight, m or w or t or

Saturated Unit Weight, s or sat

2.1 Shear Strength of Soils

2.2 Bearing Capacity of Soils Hansen B.C. Factors:

Terzaghi B.C. Factors

Note:If Df/B > 1, terzaghis B.C. factors do not apply. Use Hansens B.C. factors. For example, if depth of footing (Df) is 3 ft but footing width (B) is 2.75 ft.

3. STRESSES IN SOILS 3.1 Various Loading Conditions:

Strip

4. SHALLOW FOUNDATIONS

4.1 Conventional Footings

4.11Geotechnical Analysis qall = Q / Bx1 for Continuous Footings qall = Q / BxL for Rectangular Footings qall = Q / BxB for Square Footings qall < qu / 3 from Bearing Capacity Calculations e < B/6, where e=eccentricity Df > 1.0 ft minimum Df > frost depth Df > setback distance for footings on slope Df > scour depth Df > high moisture variations depth(expansive soils)

4.12 Structural Design: Given: A Continuous footing with m = 100 pcf, Df = 5 ft, qall = 4,000 psf, D.L=22 k/ft, L.L.=12 k/ft, fc=3 ksi, fy= 60 ksi. Design the footings using the ACI code:

Layer 1

Layer 2

Layer 33B

2B

1B

BDf

q all

GWT

2

1 Cc___1+Eo

C=

Q

4.2 Strap or Cantilever Footings: Strap Footing with varying beam thickness

Strap Footings with constant beam thickness

4.3 Trapezoidal Footings:

5. SOIL CONSOLIDATION EQUATIONS 5.1 Instant Settlement of footings:

or

5.2 Primary Consolidation: or

5.3 Overconsolidated Soils

or

or

5.4 Time rate of settlement (i=immediate, c=consolidation, & s=secondary)

5.41 Coefficient of consolidation, Cv:

6. RETAINING STRUCTURES:

6.1 Horizontal Stresses: Active, At Rest and Passive

6.2 Basement Wall with surcharge:

6.3 Braced Excavations:

6.4 Forces on Struts:

Bottom Heave Calculations:

6.5 Cantilever Sheetpiles in Sand

6.6 Cantilever Sheetpiles in Clay

6.6 Anchored Sheetpiles in Sand (Also called Bulkheads)

6.7 Anchored Sheetpiles in Clay (Also called Bulkheads)

7. PILE FOUNDATIONS 7.1 Single Piles Equations:

7.2 Group capacity of piles:

Example:

7.3 Settlement of Group Piles:

8. Post Tensioned Slabs: Edge Lift:

Center Lift:

The Structural Engineer also needs Kv (given in immediate settlement section), effective PI(pp 138 of Geotechnical DVD book) and other climatic constants that are from building codes(given).

9. Asphalt Mix Design:

AC Mix design Formulas:

When weighing in Water:

When weighing in Air:

Open Graded Mixtures:

10. Concrete Mix Design:

Fineness modulus:

Yield:

Relative Yield:

Modulus of Rupture:

= (7.5fc) or

Conversion Factor Table http://vertex42.com/edu/kinematics.html Copyright 2005 Jon Wittwer

Multiply by To Getinch 2.54 cmThis can also be written as: 1 inch = 2.54 cmacre 43,560 ft2amperehr (Ah) 3,600 coulomb (C)ngstrm () 1x10-10 matm (atmosphere) 1.01325 baratm, std 76.0 cm of Hgatm, std 760 mm of Hg at 0Catm, std 33.90 ft of wateratm, std 29.92 in of Hg at 30Fatm, std 14.696 lbf/in2 abs (psia)atm, std 101.325 kPaatm, std 1.013x105 Paatm, std 1.03323 kgf / cm2atm, std 14.696 psiabar 0.9869 atm, stdbar 1x105 PaBtu 778.169 ftlbfBtu 1055.056 JBtu 5.40395 psiaft3

Btu 2.928x10-4 kWhBtu 1x10-5 thermBtu / hr 1.055056 kJ / hrBtu / hr 0.216 ftlbf / secBtu / hr 3.929x10-4 hpBtu / hr 0.2931 WBtu / lbm 2.326* kJ / kgBtu / lbm 25,037 ft2 / s2Btu / lbmR 4.1868 kJ / kgKBtu / lbmF 4.1868 kJ / kgCBtu / lbmolR 4.1868 kJ / kmolKcal (g-calorie) 3.968x10-3 Btucal 1.560x10-6 hphrcal (IT calorie) 4.1868 JCalorie (Cal) 4.1868 kJcal / sec 4.1868 W (watt)cm (centimeter) 0.03281 ftcm 0.3937 incP (centipoise) 0.001 PaseccSt (centistokes) 1x10-6 m2 / secdegree /180 radiandyne 10 N (micronewton)eV (electronvolt) 1.602x10-19 Jerg 1x10-7 Jft (feet) 0.3048* mft 30.48 cmft2 2.2957x10-5 acreft2 144 in2

ft2 0.09290304* m2

ft3 7.481 gal (U.S.)ft3 0.02832 m3

ft3 28.317 Lft3 / lbm 0.062428 m3 / kgftlbf 1.285x10-3 Btuftlbf 1.35582 Jftlbf 3.766x10-7 kWhftlbf 1.35582 Nmftlbf 0.324 calorie (g-cal)ftlbf / sec 1.818x10-3 hpft / s2 0.3048* m / s2

U.S. gallon (gal) 0.13368 ft3gal 3.7854 Lgal 3.7854x10-3 m3

gal 231 in3gal (U.K.) 1.201 gal (U.S.)gal (U.K.) 277.4 in3gal / min (gpm) 0.002228 ft3 / secgamma (,) 1x10-9 tesla (T)gauss 1x10-4 Tgram (g) 2.205x10-3 lbmg / cm3 1 1 kg / Lg / cm3 1000 kg / m3

g / cm3 62.428 lbm / ft3

g / cm3 1.940 slug / ft3

g / cm3 0.036127 lbm / in3

hectare 1x104 m2hectare 2.47104 acreshp (horsepower) 42.41 Btu / minhp 0.7068 Btu / sec

hp 2544.5 Btu / hrhp 745.70 W (watt)hp 0.74570 kWhp 33,000 ftlbf / minhp 550 ftlbf / sechphr 2544 Btuhphr 1.98x106 ftlbfhphr 2.68x106 Jin 2.54* cmin of Hg 0.0334 atmin of Hg 13.60 in of waterin of Hg 3.387 kPain of water 0.0736 in of Hgin of water 0.0361 lbf / in2 (psi)in of water 0.002458 atmJ (joule) 9.4782x10-4 BtuJ 6.2415x1018 eVJ 0.73756 ftlbfJ 1 NmJ 1x107 ergsJ / s 1 Wkg (kilogram) 2.2046226 lbm (pound mass)kg 0.068522 slugkg 1x10-3 metric tonkg / m3 0.062428 lbm / ft3kgf 9.80665 newton (N)kip 1000 lbfkip 4448 NkJ 1 1 kPam3kJ 1000 NmkJ 0.94782 BtukJ 737.56 ftlbfkJ / kg 1000 m2 / s2kJ / kg 0.42992 Btu / lbmkJ / kgK 0.23885 Btu / lbmRkJ / kgC 1 kJ / kgKkJ / kgC 1 J / gCkJ / kgC 0.23885 Btu / lbmF kJ / kgC 0.23885 Btu / lbmRkm 3280.8 ftkm 0.6214 mikm/hr 0.6214 mi / hr (mph)km/hr 0.2778 m/skm/hr 0.9113 ft/skPa (kilopascal) 9.8693x10-3 atmkPa 0.14504 lbf / in2 (psi)kW 3412.14 Btu / hrkW 0.9478 Btu / seckW 737.56 lbfft / seckW 1.341 hpkWh (kW-hour) 3412.14 BtukWh 1.341 hphrkWh 3600 kJL (liter) 0.03531 ft3L 61.02 in3L 0.2642 gal (U.S.)L 0.001 m3

L / s 2.119 ft3 / min (cfm)L / s 15.85 gal / min (gpm)lbf (pound force) 32.174 lbmft / s2lbf 4.44822 Nlbf 32.17 poundalslbf / in2 (psi) 0.06805 atmlbf / in2 2.307 ft waterlbf / in2 2.036 in Hglbf / in2 6894.757 Palbm 0.45359237* kglbm 0.031081 sluglbm / in3 1728 lbm / ft3

lbm / ft3 0.016018 g / cm3

lbm / ft3 16.018 kg / m3m (meter) 3.28083 ftm 1.0926 yardm 39.370 inm2 1550 in2

m2 10.764 ft2

m3 1x106 cm3 (cc)m3 35.315 ft3

m3 264.17 gal (U.S.)m3 1000 Lm3 / kg 16.02 ft3 / lbmm / s 196.8 ft / min

m / s 3.60 km / hm / s 3.2808 ft / sm / s 2.237 mi / h (mph)m / s2 3.2808 ft / s2metric ton 1000 kgmil 0.001 inmi (mile) 5280 ftmi 1.6093 kmmi2 (square mile) 640 acresmph (mile/hour) 1.6093 km / hrmph 88.0 ft / min (fpm)mph 1.467 ft / smph 0.4470 m / smicron 1x10-6 mmm of Hg 1.316x10-3 atmmm of Hg 0.1333 kPamm of water 9.678x10-5 atmN (newton) 1 kgm / s2N 1x105 dyneN (microN) 0.1 dyneN 0.22481 lbfNm 0.7376 ftlbfNm 1 JPa (pascal) 1 N / m2Pa 1.4504x10-4 lbf / in2 (psia)Pa 0.020886 lbf / ft2

Pa 9.869x10-6 atmPas 10 poisepsi (pounds per square inch) --- see lbf / in2radian 180/ degreeshort ton 2000 lbmshort ton 907.1847 kgslug 32.174 lbmslug 14.5939 kgslug / ft3 0.5154 g / cm3

stokes 1x10-4 m2 / stherm 1x105 Btuton of refrigeration 200 Btu / minW (watt) 3.4121 Btu / hrW 0.7376 ftlbf / secW 1.341x10-3 hpW 1 J / sW / cm2 1x104 W / m2

W / cm3 1x106 W / m3

W / m2 0.3171 Btu / (hft2)W / m3 0.09665 Btu / (hft3)W / mC 1 W / mKW / mC 0.57782 Btu / (hftF)W / (m2C) 1 W / (m2K)W / (m2C) 0.17612 Btu / (hft2F)weber / m2 10,000 gauss

* The exact conversion between metric and English.

TEMPERATURET(K) = T(C) + 273.15T(R) = T(F) + 459.67T(F) = 1.8 T(C) + 32

SOME IMPORTANT CONSTANTSAtomic Mass Unit (u) = 1.66054x10-27 kgAvogadros number (NA) = 6.02213x1023 particles/molBoltzmanns constant (kB)= 1.38065x10-23 J / Kelectron charge (e) = -1.6022x10-19 Celectron mass (me) = 9.10939x10-31 kgproton mass (mp) = 1.6726x10-27 kgGas Constant (R) = 8314 J / kmolKGravitational Constant (G)= 6.672x10-11 Nm2 / kg2Gravity (mean) = 9.8067 (9.81) m / s2Plancks constant (h) = 6.6260x10-34 JsSpeed of Light (c) = 2.99792458x108 m/s (exact)

SI PREFIXESyocto (10-24), zepto (10-21), atto (10-18), femto (10-15), pico (10-12), nano (10-9), micro (10-6), milli (10-3), centi (10-2), deci (10-1), deka (101), hecto (102), kilo (103), mega (106), giga (109), tera (1012), peta (1015), exa (1018), zetta (1021), yotta (1024)

A

B

C

D

E

F

G

H

J

K

L

M

T

N

P

RS

W