Geotechnical Engineering / Foundation Design
1) Introduction:
1. Shallow foundation:
Df / B ≤ 1.
Df = depth of foundation below ground level.
B = width of foundation.
2. Deep foundations:
Df / B ≥1.
2) Engineering properties of the soil:
1. Volume of voids:
Vv = Vw + Va.
Vw = volume of water.
Va = volume of air.
2. Total volume of soil mass:
V = Vv + Vs.
Vs = volume of soil solids.
3. Weight of soil mass:
W = Ws + Ww.
Ws = weight of soil solids.
Ww = weight of water.
4. Void ratio:
e = Vv / Vs = n / (1-n).
5. Porosity:
n = Vv / V = e / (1+e).
6. Water content:
w = Ww / Ws.
7. Degree of saturation:
S = Vw / Vv.
8. Bulk unit weight:
γt = bulk / total unit weight = W / V = [(G +Se) / (1+e)]
γw = [G(1 +w) / (1+e)] γw = [( 1 + w) / (1+e)] γs .
G = specific gravity of soil solids.
γw = unit weight of water.
9. Unit weight of soil solids:
γs = Ws / Vs.
10. Unit weight of water:
γw = Ww / Vw.
10. Se = wG.
11. Saturated unit weight:
γsat = [(G +e) / (1+e)] γw.
As S = 100% = 1.
12. Submerged unit weight:
γsub = γsat - γw = [(G - 1) / (1+e)] γw.
13. Dry unit weight:
γdry = [ G / (1 + e) ] γw.
14. Uniformity coefficient:
Cu = D60 / D10.
D60 = grain diameter finer than 60%.
D10 = grain diameter finer than 10%.
15. Coefficient of permeability:
k = 100D102.
16. Plasticity index:
PI = LL - PL.
LL = Liquid limit.
PL = Plastic limit.
17. Liquidity index:
LI = ( w - wp) / (wL - wp).
w = normal water content of the soil.
wp = water content at plastic limit.
wL = water content at liquid limit.
18. Toughness index:
TI = If / PI.
If = flow index = slope of the curve for number of blows v/s water content by Casagrande´s method of determination of LL.
19. Darcy´s law for permeability:
ν = k i.
ν = superficial velocity.
k = coefficient of permeability of soil.
i = hydraulic gradient = h / L.
h = head loss between any two cross sections of flow.
L = straight distance between the cross sections.
20) Quick sand condition:
Seepage force = ic γw = submerged unit weight of soil. ic = critical hydraulic gradient = [ ( G - 1) / ( 1 + e) ].
21) Over consolidation ratio:
OCR = past effective pressure / present overburden pressure.
22) Relative Density:
Dr = [ ( emax - e) / (emax - emin) ].
emax = void ratio of the soil in the loosest state.
emin = void ratio of the soil in the densest state.
23) Terzaghi´s effective stress principle:
(P / A) = σ = (1 - a)u + σ ´.
P = load.
A = total area of soil mass.
σ = total stress at any point in the soil mass = σ ´ + u.
σ ´ = effective stress.
u = pore water pressure.
a = As / A.
As = contact area between soil grains.
24) Relative compaction:
RC = γdry,field / γd,max = A / [1 - Dr( 1 - A)].
A = γd,min / γd,max.
25) Empirical relationship between relative compaction and relative density:
Dr in % = (RC - 80) / 0.2.
26) Δe / (1 + e) = (ΔH / H).
Δe = change in void ratio.
ΔH = change in thickness of the sample.
H = initial thickness of the sample.
27) Shear strength of soils:
Total stress components:
s = c + σ tanΦ = c + σf.
Effective stress components:
s= c´ + σ ´ tanΦ ´ = c´ + σ ´f´.
s = shear strength of the soil.
C and c´ = cohesion of the soil.
σ´ = effective stress = σ - u.
u = pore water pressure.
Φ and Φ´ = angle of internal friction of the soil.
f , f´, tanΦ , tanΦ´ = coefficient of friction.
28) Direct shear test:
Normal stress:
σ = N / A.
N = normal load.
A = area of the failure plane = the cross sectional area of the shear box.
Shear stress:
s = Q / A.
Q = shear force.
Failure angle:
Φ = tan-1( s / σ).
29) Vane shear test:
Undrained shear strength:
s = c = T / [Π (d2h / 2) + d3 / 6)].
c = cohesion.
T = torque at failure.
d = diameter of the shear vane.
h = height of the shear vane.
30) Triaxial shear test:
Normal stress in the vertical direction at failure:
σ1 = σ3 + Δσ1.
σ3 = confining pressure.
Δσ1 = vertical stress.
31) Unconfined compression test:
Shear strength:
s = c = Su / 2.
Su = unconfined compression strength of the soil.
32) Sensitivity ratio of the soil:
Sr = Su,undisturbed / Su,remolded.
33) Split spoon sampler and standard penetration test:
Degree of disturbance of the soil sample:
AR = (Do2 - Di2) / Di2 in %.
AR = area ratio which is ratio of disturbed area to total area of soil.
Do = outside diameter of the sampling tube.
Di = inside diameter of the sampling tube.
34) Undrained shear strength or cohesion of clay:
cu = K N60.
K = constant = 3.5 - 6.5 kN/m2.
N60 = standard penetration number obtained from the field.
35) Cone penetration test:
Friction ratio:
Fr = frictional resistance / cone resistance = fc / qc.
3) Bearing capacity, settlement, stresses and lateral pressures in soils:
1. Prandtl´s theory for shallow foundations:
Ultimate bearing capacity:
qu = [ (c / tanΦ) + (1/2)( γB√Kp)] [KpeΠtanΦ - 1].
Kp = Rankine´s passive earth pressure coefficient = (1 + sinΦ) / (1- sinΦ).
2. Terzaghi´s theory for shallow foundations:
Ultimate bearing capacity of a continuous or strip footing:
qu = cNc + qNq + 0.5γBNγ.
q = γDf = surcharge at the foundation base level.
Nc, Nq, Nγ = non dimensional bearing capacity factors.

3. Terzaghi equation for local shear failure:
qu = c´Nc´ + qNq´ + 0.5γBNγ´.
c´ = (2 / 3) c.
Φ´ = tan-1[(2/3)tanΦ].
4. Ultimate bearing capacity of square foundation:
qu = 1.3cNc + qNq + 0.4γBNγ.
5. Ultimate bearing capacity of circular foundation:
qu = 1.3cNc + qNq + 0.3γBNγ.
6. Modified bearing capacity factors for smooth base:
Nc = (Nq -1) cotΦ.
Nq = tan2 [ 45 + (Φ / 2)] eΠtanΦ. Nγ = 2(Nq+ 1) tanΦ.
7. Allowable bearing capacity:
qall = qult / FS.
qult = ultimate bearing capacity.
FS = factor of safety.
8. Effect of ground water table:
q = γ (Df - D) + γsubD.
9. Ultimate bearing capacity at the base for deep foundations:
qp = cNpc + σoNpq + (γB Npγ / 2).
Npc, Npq, Npγ = Meyehof´s bearing capacity factors for deep foundations.
σo = normal stress on an equivalent free surface as defined by Meyerhof.
10. Equivalent value of N:
Nequivalent = 15 + [(1 / 2) (N - 15)].
11. Corrected value for penetration resistance:
N = N´[ 5 / ( 1.422p + 1 ) ].
N´ = actual blow count.
p = actual effective overburden pressure in kg / cm2.
12. qc = 4N.
qc = resistance in kg / cm2.
N = SPT values.
13. Settlement for clays:
s = (0.6qB) / E.
q = load intensity.
B = width.
E = modulus of elasticity of a soil.
14. Settlement for sand:
s2 = s1{[ B2( B1 + 0.3)] / [ B1( B2 + 0.3)]}2.
s2 = settlements of an area.
s1 = settlement of a plate.
B2 = area width.
B1 = width of the plate.
15. Boussinesq´s solutions for vertical concentrated load on the surface:
Vertical stress:
σz = KB ( P / Z2 ).
P = concentrated load.
Z = depth.
KB = ( 3 / 2Π ) [ 1 + ( r / z )2 ]-5/2.
r = cylindrically radial distance.
16. Total settlement of a footing on clay:
S = Si + Sc + Ss.
Si = immediate elastic settlement.
Sc = settlement due to primary consolidation.
Ss = settlement due to secondary consolidation.
17. Primary consolidation settlement by Terzaghi:
Sc = mv Δp H.
mv = coefficient of volume compressibility of the clayobtained by consolidation test.
Δp = vertical stress at the middle of the compressible layer due to load on footing.
H = thickness of the compressible clay.
Sc = [ Cc / ( 1 + e )] H log10 [ (po + Δp ) / po].
Cc = compression index determined by consolidation test.
e = void ratio.
po = vertical effective pressure due to soil overburden.
18. Rankine´s theory:
Total passive pressure:
Pp = γz [(1 + sinΦ) / (1 - sinΦ)] = γztan2[45 + (Φ / 2)] = Kpγz. γ = unit weight of soil.
Z = H2 / 2 = depth coordinate from top of the wall.
Kp = 1 / KA.
KA = active earth pressure coefficient.
19. For cohesive soils with horizontal backfill:

20. Lateral pressure at rest at any point:
σh = Koσv.
Ko = coefficient of earth pressure at rest.
σv = γz = vertical pressure at that point.
4) Rational design of shallow foundations:
1. Proportioning the size of the footing:
Area of footing:
A = Ll + d/ qa.
Ll + d = live load + dead load for the column which has the largest live load/dead load ratio.
qa = allowable bearing pressure.
Design pressure for all footings:
qd = Ls / A.
Ls = service load for the same column.
Area of other footings = service load / qd.
2. Stress on lower strata:
σz = (3Pz3) / (2ΠR5).
3. Bearing capacity of footings on slopes:
q = cNcq + (1 / 2)γBNγq.
4. Minimum depth of foundation required for the stability of adjoining soil in loose sand:
Df = ( q / γ) [ (1 - sinΦ) / (1 + sinΦ) ]2.
q = contact pressure.
γ = unit weight of soil.
Φ = angle of internal friction of soil.
5. Equilibrium in the vertical direction:
q = kw - T∇2w.
q = distributed vertical load applied.
k and T are the two parameters depending on foundation.
w = vertical deflection of the surface.
∇2 = Laplace operator.
6. Modulus of subgrade reaction:
ks = q / w = Cu.
q = bearing pressure at a point along the beam.
w = vertical displacement of the beam at a point.
Cu = coefficient of elastic uniform compression.
7. Mean settlement for a square plate:
wsp = [0.95(1 - νs2)qB] / Es.
νs = Poisson´s ratio of the soil.
q = average bearing pressure under plate.
B = side of the plate.
Es is the modulus of elasticity of the soil.
8. Mean settlement for a circular plate:
wsp = [0.85(1 - νs2)qB] / Es.
B = diameter of the plate.
9. Modulus of subgrade reaction for cohesionless soils:
ks = kt[(0.305 + B2) / 2B]2.
10. Modulus of subgrade reaction for cohesive soils:
ks = kt(0.305 / B).
5) Analysis of footings on elastic foundations:
1. Finite beams on elastic foundations:
Bending moment:
Mo = P / 4λ.
λ = [k / (4EI)]1/4.
6) Deep foundations:
1. Allowable pile load based on strength of the wooden pile:
Qall = Ap fw.
Ap = cross sectional area of the pile.
fw = allowable compressive strength of wood.
2. Resistance of soil:
R = 2E / ( S + C) in pounds.
E = driving energy.
S = pile penetration / blow in inches.
C = empirical constant.
3. Ultimate load carrying capacity of the pile by static equilibrium:
Qu = Qp + Qaf = Apqp + As Saf.
Qp = point/end bearing capacity.
Qaf = resistance due to adhesion and friction along the shaft of pile.
1) Introduction:
1. Shallow foundation:
Df / B ≤ 1.
Df = depth of foundation below ground level.
B = width of foundation.
2. Deep foundations:
Df / B ≥1.
2) Engineering properties of the soil:
1. Volume of voids:
Vv = Vw + Va.
Vw = volume of water.
Va = volume of air.
2. Total volume of soil mass:
V = Vv + Vs.
Vs = volume of soil solids.
3. Weight of soil mass:
W = Ws + Ww.
Ws = weight of soil solids.
Ww = weight of water.
4. Void ratio:
e = Vv / Vs = n / (1-n).
5. Porosity:
n = Vv / V = e / (1+e).
6. Water content:
w = Ww / Ws.
7. Degree of saturation:
S = Vw / Vv.
8. Bulk unit weight:
γt = bulk / total unit weight = W / V = [(G +Se) / (1+e)]
γw = [G(1 +w) / (1+e)] γw = [( 1 + w) / (1+e)] γs .
G = specific gravity of soil solids.
γw = unit weight of water.
9. Unit weight of soil solids:
γs = Ws / Vs.
10. Unit weight of water:
γw = Ww / Vw.
10. Se = wG.
11. Saturated unit weight:
γsat = [(G +e) / (1+e)] γw.
As S = 100% = 1.
12. Submerged unit weight:
γsub = γsat - γw = [(G - 1) / (1+e)] γw.
13. Dry unit weight:
γdry = [ G / (1 + e) ] γw.
14. Uniformity coefficient:
Cu = D60 / D10.
D60 = grain diameter finer than 60%.
D10 = grain diameter finer than 10%.
15. Coefficient of permeability:
k = 100D102.
16. Plasticity index:
PI = LL - PL.
LL = Liquid limit.
PL = Plastic limit.
17. Liquidity index:
LI = ( w - wp) / (wL - wp).
w = normal water content of the soil.
wp = water content at plastic limit.
wL = water content at liquid limit.
18. Toughness index:
TI = If / PI.
If = flow index = slope of the curve for number of blows v/s water content by Casagrande´s method of determination of LL.
19. Darcy´s law for permeability:
ν = k i.
ν = superficial velocity.
k = coefficient of permeability of soil.
i = hydraulic gradient = h / L.
h = head loss between any two cross sections of flow.
L = straight distance between the cross sections.
20) Quick sand condition:
Seepage force = ic γw = submerged unit weight of soil. ic = critical hydraulic gradient = [ ( G - 1) / ( 1 + e) ].
21) Over consolidation ratio:
OCR = past effective pressure / present overburden pressure.
22) Relative Density:
Dr = [ ( emax - e) / (emax - emin) ].
emax = void ratio of the soil in the loosest state.
emin = void ratio of the soil in the densest state.
23) Terzaghi´s effective stress principle:
(P / A) = σ = (1 - a)u + σ ´.
P = load.
A = total area of soil mass.
σ = total stress at any point in the soil mass = σ ´ + u.
σ ´ = effective stress.
u = pore water pressure.
a = As / A.
As = contact area between soil grains.
24) Relative compaction:
RC = γdry,field / γd,max = A / [1 - Dr( 1 - A)].
A = γd,min / γd,max.
25) Empirical relationship between relative compaction and relative density:
Dr in % = (RC - 80) / 0.2.
26) Δe / (1 + e) = (ΔH / H).
Δe = change in void ratio.
ΔH = change in thickness of the sample.
H = initial thickness of the sample.
27) Shear strength of soils:
Total stress components:
s = c + σ tanΦ = c + σf.
Effective stress components:
s= c´ + σ ´ tanΦ ´ = c´ + σ ´f´.
s = shear strength of the soil.
C and c´ = cohesion of the soil.
σ´ = effective stress = σ - u.
u = pore water pressure.
Φ and Φ´ = angle of internal friction of the soil.
f , f´, tanΦ , tanΦ´ = coefficient of friction.
28) Direct shear test:
Normal stress:
σ = N / A.
N = normal load.
A = area of the failure plane = the cross sectional area of the shear box.
Shear stress:
s = Q / A.
Q = shear force.
Failure angle:
Φ = tan-1( s / σ).
29) Vane shear test:
Undrained shear strength:
s = c = T / [Π (d2h / 2) + d3 / 6)].
c = cohesion.
T = torque at failure.
d = diameter of the shear vane.
h = height of the shear vane.
30) Triaxial shear test:
Normal stress in the vertical direction at failure:
σ1 = σ3 + Δσ1.
σ3 = confining pressure.
Δσ1 = vertical stress.
31) Unconfined compression test:
Shear strength:
s = c = Su / 2.
Su = unconfined compression strength of the soil.
32) Sensitivity ratio of the soil:
Sr = Su,undisturbed / Su,remolded.
33) Split spoon sampler and standard penetration test:
Degree of disturbance of the soil sample:
AR = (Do2 - Di2) / Di2 in %.
AR = area ratio which is ratio of disturbed area to total area of soil.
Do = outside diameter of the sampling tube.
Di = inside diameter of the sampling tube.
34) Undrained shear strength or cohesion of clay:
cu = K N60.
K = constant = 3.5 - 6.5 kN/m2.
N60 = standard penetration number obtained from the field.
35) Cone penetration test:
Friction ratio:
Fr = frictional resistance / cone resistance = fc / qc.
3) Bearing capacity, settlement, stresses and lateral pressures in soils:
1. Prandtl´s theory for shallow foundations:
Ultimate bearing capacity:
qu = [ (c / tanΦ) + (1/2)( γB√Kp)] [KpeΠtanΦ - 1].
Kp = Rankine´s passive earth pressure coefficient = (1 + sinΦ) / (1- sinΦ).
2. Terzaghi´s theory for shallow foundations:
Ultimate bearing capacity of a continuous or strip footing:
qu = cNc + qNq + 0.5γBNγ.
q = γDf = surcharge at the foundation base level.
Nc, Nq, Nγ = non dimensional bearing capacity factors.

3. Terzaghi equation for local shear failure:
qu = c´Nc´ + qNq´ + 0.5γBNγ´.
c´ = (2 / 3) c.
Φ´ = tan-1[(2/3)tanΦ].
4. Ultimate bearing capacity of square foundation:
qu = 1.3cNc + qNq + 0.4γBNγ.
5. Ultimate bearing capacity of circular foundation:
qu = 1.3cNc + qNq + 0.3γBNγ.
6. Modified bearing capacity factors for smooth base:
Nc = (Nq -1) cotΦ.
Nq = tan2 [ 45 + (Φ / 2)] eΠtanΦ. Nγ = 2(Nq+ 1) tanΦ.
7. Allowable bearing capacity:
qall = qult / FS.
qult = ultimate bearing capacity.
FS = factor of safety.
8. Effect of ground water table:
q = γ (Df - D) + γsubD.
9. Ultimate bearing capacity at the base for deep foundations:
qp = cNpc + σoNpq + (γB Npγ / 2).
Npc, Npq, Npγ = Meyehof´s bearing capacity factors for deep foundations.
σo = normal stress on an equivalent free surface as defined by Meyerhof.
10. Equivalent value of N:
Nequivalent = 15 + [(1 / 2) (N - 15)].
11. Corrected value for penetration resistance:
N = N´[ 5 / ( 1.422p + 1 ) ].
N´ = actual blow count.
p = actual effective overburden pressure in kg / cm2.
12. qc = 4N.
qc = resistance in kg / cm2.
N = SPT values.
13. Settlement for clays:
s = (0.6qB) / E.
q = load intensity.
B = width.
E = modulus of elasticity of a soil.
14. Settlement for sand:
s2 = s1{[ B2( B1 + 0.3)] / [ B1( B2 + 0.3)]}2.
s2 = settlements of an area.
s1 = settlement of a plate.
B2 = area width.
B1 = width of the plate.
15. Boussinesq´s solutions for vertical concentrated load on the surface:
Vertical stress:
σz = KB ( P / Z2 ).
P = concentrated load.
Z = depth.
KB = ( 3 / 2Π ) [ 1 + ( r / z )2 ]-5/2.
r = cylindrically radial distance.
16. Total settlement of a footing on clay:
S = Si + Sc + Ss.
Si = immediate elastic settlement.
Sc = settlement due to primary consolidation.
Ss = settlement due to secondary consolidation.
17. Primary consolidation settlement by Terzaghi:
Sc = mv Δp H.
mv = coefficient of volume compressibility of the clayobtained by consolidation test.
Δp = vertical stress at the middle of the compressible layer due to load on footing.
H = thickness of the compressible clay.
Sc = [ Cc / ( 1 + e )] H log10 [ (po + Δp ) / po].
Cc = compression index determined by consolidation test.
e = void ratio.
po = vertical effective pressure due to soil overburden.
18. Rankine´s theory:
Total passive pressure:
Pp = γz [(1 + sinΦ) / (1 - sinΦ)] = γztan2[45 + (Φ / 2)] = Kpγz. γ = unit weight of soil.
Z = H2 / 2 = depth coordinate from top of the wall.
Kp = 1 / KA.
KA = active earth pressure coefficient.
19. For cohesive soils with horizontal backfill:

20. Lateral pressure at rest at any point:
σh = Koσv.
Ko = coefficient of earth pressure at rest.
σv = γz = vertical pressure at that point.
4) Rational design of shallow foundations:
1. Proportioning the size of the footing:
Area of footing:
A = Ll + d/ qa.
Ll + d = live load + dead load for the column which has the largest live load/dead load ratio.
qa = allowable bearing pressure.
Design pressure for all footings:
qd = Ls / A.
Ls = service load for the same column.
Area of other footings = service load / qd.
2. Stress on lower strata:
σz = (3Pz3) / (2ΠR5).
3. Bearing capacity of footings on slopes:
q = cNcq + (1 / 2)γBNγq.
4. Minimum depth of foundation required for the stability of adjoining soil in loose sand:
Df = ( q / γ) [ (1 - sinΦ) / (1 + sinΦ) ]2.
q = contact pressure.
γ = unit weight of soil.
Φ = angle of internal friction of soil.
5. Equilibrium in the vertical direction:
q = kw - T∇2w.
q = distributed vertical load applied.
k and T are the two parameters depending on foundation.
w = vertical deflection of the surface.
∇2 = Laplace operator.
6. Modulus of subgrade reaction:
ks = q / w = Cu.
q = bearing pressure at a point along the beam.
w = vertical displacement of the beam at a point.
Cu = coefficient of elastic uniform compression.
7. Mean settlement for a square plate:
wsp = [0.95(1 - νs2)qB] / Es.
νs = Poisson´s ratio of the soil.
q = average bearing pressure under plate.
B = side of the plate.
Es is the modulus of elasticity of the soil.
8. Mean settlement for a circular plate:
wsp = [0.85(1 - νs2)qB] / Es.
B = diameter of the plate.
9. Modulus of subgrade reaction for cohesionless soils:
ks = kt[(0.305 + B2) / 2B]2.
10. Modulus of subgrade reaction for cohesive soils:
ks = kt(0.305 / B).
5) Analysis of footings on elastic foundations:
1. Finite beams on elastic foundations:
Bending moment:
Mo = P / 4λ.
λ = [k / (4EI)]1/4.
6) Deep foundations:
1. Allowable pile load based on strength of the wooden pile:
Qall = Ap fw.
Ap = cross sectional area of the pile.
fw = allowable compressive strength of wood.
2. Resistance of soil:
R = 2E / ( S + C) in pounds.
E = driving energy.
S = pile penetration / blow in inches.
C = empirical constant.
3. Ultimate load carrying capacity of the pile by static equilibrium:
Qu = Qp + Qaf = Apqp + As Saf.
Qp = point/end bearing capacity.
Qaf = resistance due to adhesion and friction along the shaft of pile.
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