Geotechnical Formulas 5q2925

GEOTECHNICAL Index and Classification Properties of Soils Unit weight :

γ =

W V

Gs γw 1+ e Moist unit weight : γ = γdry ( 1 + ω ) (Gs + e) γ w Saturated unit weight: γsat = 1+ e Zero air void unit weight: Gsγ w γzav = 1+ e Moisture content (water content) Ww ω = Ws ωG s Degree of saturation : s = e e Porosity : n = 1+ e Air void ratio = n (1 – s)

Dry unit weight :

γd =

DRY

H1

MOIST

H2 G.W.T.

SATURATED

Phases of Soil

G.W.T. = ground water table where : W = total weight V = volume Gs = specific gravity e = void ratio γw = unit weight of water = 9.81 kN/m3 ( 1gram/cm3 )

H3

GEOTECHNICAL Relative density ( Density index ) Dr = Dr =

emax - e or emax - emin

Soil Characteristic

Liquidity Index

brittle solid plastic liquid

<0 <1 >1

1 / γd min - 1 / γd 1 / γd min - 1 / γd max

Shrinkage Ratio : S.R.

m2 e = in-situ void ratio S.R. = V2 ρw emax = void ratio in loosest condition Specific Gravity: Gs emin = void ratio in densest condition γd = dry unit weight in the field 1 γd(max) = dry unit weight in densest condition Gs = 1 SL γd(min) = dry unit weight in loosest condition SR 100 Shrinkage limit : S.L. (m1 - m2 ) (V1 - V2 ) S.L. = ρw in % m2 m2

m1 m2 V1 V2

= initial mass in saturated state = final mass in dry state = initial volume in saturated state = final volume in dry state

Plasticity Index : P.I. P.I. = L.L. - P.L. L.L. = liquid limit P.L. = plastic limit Liquidity Index : L.I. ω - P.L. L.I. = P.I. ω = in-situ moisture content P.L. = plastic limit P.I. = plasticity index Consistency Index : C.I. L.L. - ω C.I. = L.L. - P.I.

SOIL CLASSIFICATION U.S. Dept of Agriculture (USDA Method) Gravel

Sand

Silt

Clay

> 2 mm

2 to 0.05mm

0.05 to 0.002mm

< 0.002 mm

AASHTO Method Gravel

Sand

Silt

Clay

76.2 to 2mm

2 to 0.075mm

0.075 to 0.002mm

< 0.002 mm

Unified Soil Classification System Gravel Sand Fines (Silt & Clay) 76.2 to 4.75 mm

4.75 to 0.075 mm

< 0.075 mm

GEOTECHNICAL Particle Size Distribution Effective Size, D10 - is the diameter in the particle size distirbution curve corresponding to 10% finer. Uniformity coefficient, Cu D Cu = 60 D10

SOIL COMPACTION Compaction – is the densification of soils by the application of mechanical energy. It may also involve a modification of the water content as well as gradation of the soil. Some methods for determining density of soil in the field: 1. Sand Cone Method 2. Balloon Method 3. Oil Method

Coefficient of gradation or curvature, Cc (D30 ) 2 Cc = D60 • D10

D30 = particle diameter corresponding to 30% finer To determine the dry unit weight of D60 = particle diameter corresponding to 60% finer compaction in the field: dry unit weight excavated from the hole γd = Sorting Coefficient, So volume of the hole D75 So = D25 W - Wc Volume of hole, V = s D75 = particle diameter corresponding to 75% finer γd( sand ) D25 = particle diameter corresponding to 25% finer W = weight of sand to fill the hole and cone s

AASHTO Classification System

Wc = weight of sand to fill the cone only

Group Index, G.I. = (F – 35)[ 0.2 + 0.005(LL – 40)] + 0.01(F – 15)(PI – 10)

Relative Compaction, R γd( field) R= γd(max -lab)

Partial Group Index = 0.01(F – 15)(PI – 10)

γd(field) = compacted field dry unit weight γd(max-lab) = maximum dry unit weight determined in the laboratory by Proctor Test

F = % ing sieve # 200 LL = liquid limit PI = plasticity index Group index must be whole and positive, if the computed value is negative use zero. If the group classification is A-2-6 & A-2-7 attached Partial Group Index only.

GEOTECHNICAL PERMEABILITY, SEEPAGE of WATER in SOIL Coefficient of Permeability, k ( Laboratory Test)

Constant Head Test : k =

QL Ah t

Q = volume of water collected A = area of cross section of soil specimen t = duration of water collection h i = hydraulic gradient = L Falling Head Test: k =

a L h1 ln A t h2

a = cross-sectional area of the stand pipe A = cross-sectional area of soil specimen t = duration of water collection h1 = initial head when t1 = 0 h2 = final head when t2 = t L = length of soil specimen h h2 i = hydraulic gradient = 1 L Rate of water flow in soil, Q Q=kiA k = coefficient of permeability i = hydraulic gradient A = cross-sectinal area of soil sample Discharge velocity, V V=ki Seepage velocity, Vs V Vs = n n = porosity

Absolute Permeability, K kη K= γw k = coefficient of permeability η = viscosity of water γw = unit weight of water Transmissibilty of Soil Stratum, T T=Kb K = average coefficient of permeability b = thickness of aquifer Equivalent Coefficient of Permeability in Layered Soil:

k1

H1

k2

H2

Equivalent Horizontal Coeffient, KH(eq) 1 KH (eq) = (k1H1 + k2H2 + … + knHn) H Equivalent Vertical Coefficient, KV(eq) H KV(eq) = H1 H2 H + + ... + n k k2 kn

GEOTECHNICAL Vertical Stresses in Soil

Compressibility of Soil Settlement – the total vertical deformation at the surface resulting from the load. When a soil deposit is loaded (by a structure or a man-made fill) deformation will occur. The movement may be downward with an increase in load or upward (called swelling) with a decrease in load.

A h1

Dry, γd B

G.W.T. Saturated, γsat

h2 C

1. Total stress at C : σc = γd h1 + γsat h2 2. Pore water pressure at C : uc = γw h2 3. Effective stress at C = Total stress – Pore pressure σc’ = σc − uc

S

A h1

Dry, γd B

G.W.T.

h

Components of Settlement:

Saturated, γsat

h2

S = S1 + S2 + S3

C 1. Total stress at C : σc = γd h1 + γsat h2 2. Pore water pressure at C : uc = γw h 3. Effective stress at C = Total stress – Pore pressure σc’ = σc − uc A h1

Primary Consolidation Settlement, S2 Normally Consolidated Clays Dry, γd

S2 =

B Moist, γ Degree of saturation, S

h2 C

S = total settlement S1 = immediate or distortion settlement S2 = primary consolidation settlement S3 = secondary consolidation settlement

zone of capillary rise G.W.T.

1. Total stress at C : σc = γd h1 + γ h2 σB = γd h1 2. Pore water pressure at C : uB = -S γw h2 uc = 0 3. Effective stress at C = Total stress – Pore pressure σc’ = σc − uc σB’ = σB − (−uB)

CcH P + ∆P log o 1 + e0 Po

Cc = compression index eo = in-situ void ratio Po = effective overburden pressure at the middle of the clay layer ∆P = average increase of stress on clay layer H = thickness of clay layer

GEOTECHNICAL Over – Consolidated Clays 1. when Po + ∆P < Pc S2 =

C sH P + ∆P log o 1 + e0 Po

Cs = swell index = ranges from 1/5 to 1/10 of Cc Pc = preconsolidation pressure 2.

when Po + ∆P > Pc

C sH P CH P + ∆P S2 = log c + c log o 1 + e0 Po 1 + e 0 Pc Secondary Consolidation Settlement, S3 S3 = C’a H log C’a = Ca =

Compression Index, Cc Cc = 0.009 (LL -10)

t2 t1

e1 - e 2 P log 2 P1

Cc =

Coefficient of Compressibility, av av =

e1 - e 2 P2 - P1

Coefficient of Volume Compressibility, mv

av 1 + e ave

mv =

Time Rate of Consolidation, t

open

open Hdr

open Tv =

Ca = secondary compression index ep = void ratio at the end of primary consolidation = eo - ∆e ∆e = Cc log

Po + ∆P Po

t1 = time for completion of primary consolidation t2 = time after completion of primary consolidation Immediate Settlement, S1 S1 = Cs q B

(1 - µ 2 ) Es

Cs = shape and foundation rigidity factor q = pressure due to load B = width of foundation or diameter of circular foundation µ = Poisson’s ratio of soil Es = modulus of elasticity of soil

Hdr

H

Ca 1 + ep ∆e t log 2 t1

remolded clays

Cv t H 2 dr

Cv = coefficient of consolidation Tv = time factor Hdr = drainage distance of water Preconsolidation Pressure, Pc OCR =

Pc Po

OCR = overconsolidation ratio Pc = preconsolidation pressure Po = soil overburden pressure

closed

GEOTECHNICAL Bearing Capacity of Foundations

Case 2: the water table is located at the bottom of the foundation

Terzaghi’s Bearing-Capacity Ultimate Bearing Capacity, qult A. General Shear Failure (Dense sand and Stiffclay) 1. Square Footing

qult = 1.3c Nc + q Nq + 0.4 γ B Nγ

G.W.T.

B 2. Circular Footing

qult = 1.3c Nc + q Nq + 0.3 γ B Nγ c = cohesion q = effective pressure at the bottom of the footing B = width of footing or diameter of footing Nc Nq Nγ = bearing capacity factors

γ = γsat - γw Case 3: the water table is located so that d < B

Bearing Capacity Factors 1. see Tables 2. If no table available Nc = (Nq – 1) cot φ π tan φ

Nq = e

B d
2

tan (45 +

2

)

Modification of γ value in the 3rd term of the ultimate bearing capacity equation:

′

[

1 γ + γ (B - d) B dry γ ‘ = γsat - γw

γ=

Nγ = (Nq + 1) tan 1.4φ

]

Case 4: the water table is located so that d > B

Case 1: the water table is located above the bottom of the foundation

G.W.T.

B d>B G.W.T.

B

γ = γsat - γw

γ = γdry

GEOTECHNICAL Shear Strength of Soil Direct Shear Test Method

Unconfined Compression Test Method Normal Force

σ1

τ

Shear Force

σ τ φ

σ1

c

σ

τ

φ = angle of internal friction c = cohesion = 0 if normally consolidated clay σ = normal stress τ = shearing stress

c

σ

∆σ σ3

Tri-Axial Test Method

deviator stress

σ1

Unconsolidated - Undrained Test Method

τ

τ σ cu

σ

τ φ

c

σ

σ3 σ3

deviator stress

deviator stress

σ1

σ1 cu = undrained shear strength

σ3 = chamber confining pressure, cell pressure = minor principal stress σ1 = major principal stress ∆σ = deviator stress

GEOTECHNICAL Lateral Earth Pressure

σv

z

Pressure diagram due to water:

γs = unit weight of soil σh = k σv

σv = vertical pressure at depth z σh = lateral pressure at depth z k = coefficient of lateral earth pressure

Pressure diagram due to surcharge:

Lateral Earth Pressure on Retaining Wall with Horizontal Backfill At rest earth pressure coefficient, ko (normally consolidated soil) ko = 1 – sin φ φ = angle of internal friction Rankine active earth pressure, ka ka =

1 - sin φ 1 + sin φ

Rankine ive earth pressure, kp kp =

1 + sin φ 1 - sin φ

Pressure diagram due to effective unit weight of soil:

Pressure diagram due to cohesion of soil:

GEOTECHNICAL Braced Sheetings

Piles on Clay

Cuts in Sand

α method

Qall

End bearing capacity Qb = C Nc Atip H

assumed hinge

Frictional capcity Qf = Σ αL

Qf Pa= 0.65 γ ka H Cuts in Clay when

γΗ >4 C

C = cohesion of soil Nc = bearing capacity factor Atip = area of plie at the tip α = adhesion factor P = perimeter of pile L = length of pile

0.25 H

H

Design Load :

assumed hinge

Qall =

Qb + Qf F.S.

Qall

λ method

Pa= γH – 4C Cuts in Clay when

End bearing capacity Qb = C Nc Atip Frictional capcity Qf = PLλ (Qv + 2C )

γΗ <4 C

0.25 H

H

Qb

assumed hinge

Qv =

Qf

area of Pv diagram L

Qb

0.25 H Pa= 0.3 γH ka = Rankine active pressure coefficient C = cohesion of clay γ = unit weight of soil H = depth of cut

Design Load :

Qall =

Qb + Qf F.S.

L

Pv diagram

GEOTECHNICAL Piles on Sand

Drilled Piles on Clay

Qall

Qall Ds

1.5 m

Dc L1

Qf

L Qf Ds

Qb

Pv diagram

L2 Db

End bearing capacity Qb = Pv Nq Atip

Qb

Frictional capcity Qf = P ( area of Pv diagram ) K µ Design Load :

Qall =

Qb + Qf F.S.

Pv = vertical soil pressure at the tip Nq = bearing capacity factor K = coef of lateral bet pile and sand earth pressure factor µ = tan θ θ = angle of friction bet pile and sand Critical Depth, Dc i. Dc = 20 x diameter of pile for dense sand ii. Dc = 10 x diameter of pile for loose sand Note : The pressure below the critical depth, Dc is assumed to be uniform.

End bearing capacity Qb = qp Atip qp = 6C [ 1 + 0.2

L1 + L2 ] but not greater than CNc Db

Db = bell diameter Ds = shaft diameter

Frictional capcity Qf = Σ α C P L α = adhesion factor P = perimeter of pile L = effective length of pile that resist friction Design Load :

Qall =

Qb + Qf F.S.

GEOTECHNICAL Drilled Piles on Sand Qall

Group of Piles Efficiency of Group of Piles, Eg

D

Qf

L

Converse – Labarre Equation: Eg = 1 -

Qb

End bearing capacity Qb = qp Atip

a. Nvalue < 50 qp = 57.5 Nvalue < 2900 kPa b. Nvalue > 50 P qp = 0.59[ Nvalue ( a )]0.8 Pvb Pa = 100 kPa Pvb = effective vertical pressure at base elevation

θ [(n - 1) m + (m - 1) n] 90 m n

Bowles : Eg =

2 ( m + n - 2) S + 4D πD m n

m = number of rows of piles n = number of piles in a row D tan θ = S D = diameter of pile S = spacing of piles center to center

Settlement of Piles Frictional capcity : Qf = Σ β Pv P L a. Nvalue ≥ 15 β = 1.5 – n (z)0.5 b. Nvalue < 15 N β = value [ 1.5 – n (z)0.5 ] 15 Pv = effective vertical overburden pressure at depth z. z = height from ground surface to mid-height of a given layer n = 0.245 P = perimeter of pile L = length of pile

Se = Se1 + Se2 + Se3 Elastic Settlement of Pile, Se1 (Qwp + ξ Qws ) L Se1 = Ap Ep

Qwp = load carried at the pile point under working load condition Qws = load carried by frictional resistance under working load condition Ap = cross sectional area of pile L = length of pile

GEOTECHNICAL Settlement of Pile caused by the Load at the Pile Tip , Se2 Se2 =

qwp =

qwp D Es Qwp

Settlement of Group of Piles

(1 - µs2 ) Iwp

Ap

Es = modulus of elasticity of soil µs = Poisson’s ratio of soil Iwp = influence factor Settlement of Pile caused by the Load transmitted along of the pile shaft , Se3 Se3 =

Qws D (1 - µs2 ) Iws PLEs

Iws = 2 + 0.35

L D

P = perimeter of pile L = length of pile Iws = influence factor D = diameter of pile

S= Σ

Cc H P + ΔP log o 1 + eo Po

Cc = compression index H = thickness of clay layer eo = initial void ratio ∆P = average increase in pressure on clay Po = effective overburden pressure at the midheight of the clay layer