FOUNDATION ENGINEERING GATE ONLINE STUDY MATERIAL CIVILENGGFORALL

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Foundation Engineering Free Online CivilEnggForAll GATE Study Material

CONTENTS

1. SUBSURFACE INVESTIGATION

2. EARTH PRESSURE

3. STABILITY OF SLOPES

STABILITY OF INFINITE SLOPES
STABILITY OF FINITE SLOPES

4. FOUNDATIONS

TYPES OF FOUNDATIONS
FOUNDATION DESIGN REQUIREMENT

5. BEARING CAPACITY

6. PILE FOUNDATION

LOAD CARRYING CAPACITY OF PILES
STATIC FORMULAE
DYNAMIC FORMULAE
NEGATIVE SKIN FRICTION

SUB SURFACE INVESTIGATION

Drilling bore holes

Depending upon type of soil and purpose of exploration, different methods are used for drilling bore holes.

(1) Auger boring

(2) Wash boring

(3) Rotary boring

(4) Percussion Drilling

(5) Core drilling/ Core Boring.

Sampling – Types of soil samples based on methods of sampling

(a) Undisturbed sample Sample whose natural structure, properties and water content remain intact. It is used to test consolidation compressibility, shear, permeability and other engineering properties.

(b) Disturbed sample Sample whose natural structure gets partly or fully modified or destroyed. It is used to determine index properties.

Equipment used for sampling:

(1) Split spoon sampler (for distributed samples)

(2) Shelly tubes (for undistributed samples)

(3) Piston sampler (for undistributed samples)

Penetration Tests

(1) Standard Penetration Test (SPT)

Used to find relative density (ID), I value and C-value of soil

Split spoon sampler of 50.8 mm outer diameter and 38 mm inner diameter is used for the test.

It is driven into soil by a hammer of weight 65 kg and height of fall 75 cm

Number of blows required for penetrating the sampler 30 cm deep beyond seating drive is called penetration resistance or standard penetration test value (N value)

Correction applied

a. Overburden correction

N value of soil at shadow depth is underestimated and that at greater depths are overestimated due to increase in confining pressure with depth.

Overburden correction is applied for uniformity in N value

N_e (Effective value of N) = N₀[350/σ’+70]

Where N₀ is observed N Value and σ’ is effective overburden pressure

b. Dilatancy correction

This is applied after overburden correction

Used when N > 15

N_C = 15 + (N₀ – 15)/2

When N₀ ≤ 15, take N_e = N₀

(2) Cone Penetration Test (Dutch Cone test)

A standard sized cone is pushed to soil steadily under static pressure.
The cone itself has arrangement to measure cone resistance and side friction
The maximum value of resistance obtained is called ‘cone resistance’

Note

Both these tests are mainly used for cohesion less soil.
Cone resistance is approx. equal to 10 times penetration resistance.

(3) Dynamic cone test

Similar to that of SPT
A cone is pushed to soil with a hammer of weight 65 kg and height of free fall 75 cm.
Number of blows required for 30 cm penetration is called ‘dynamic cone resistance’.

Plate Load Test

It is conducted to determine ultimate bearing capacity, allowable bearing pressure or probable settlement.
Plate is usually square sized of side 30 cm (clayey sand soils) to 60 cm (dense sandy and gravel) and thickness of 2.5 cm.
The test pit width is taken 5 times the width of plate i.e., B = 5 Bp
A seating load of 7 kN/m² is applied initially and maximum load that is to be applied is taken 1.5 times the probable ultimate load or 3 times proposed allowable bearing pressure.
Load increments shall be approximately one-fifth of the estimated safe-bearing pressure calculated.
Settlements are measured by four independent dial gauges and average value is taken at each increment.
Load at which the settlement occurs in a rapid rate is taken as ultimate bearing capacity of the soil.
For each load increment, the settlement value is taken and we draw the Load-settlement curve.
If yield point is not specific in the graph, load corresponding to settlement equal to one-fifth of wide of plate is taken.
Since load test is of short duration, settlement due to consolidation cannot be predicted especially in cohesive soil. Settlement of foundation. For clayey soil, S_f/S_p = B_f/B_p

S corresponds to settlement and B for breadth or width f indicates footing and p indicates plate.

For sandy or cohesionless soils, S_f = S_p[B_f(B_p+0.3/B_p(B_f+0.3))]²

Note: All values should be in meters.

Ultimate bearing capacity of foundation

For Clayey Soils, (q_u)_f = (q_u)_p

For Sandy Soils, (q_u)_f/(q_u)_p = B_f/B_p (Kindly, remember this equations carefully as these are very important for GATE Exam)

EARTH PRESSURE

Depending on the lateral pressure exerted by the soil on the retaining structure and based on movement of the retaining structure, there are three types of earth pressures.

Comparison of Earth Pressure at rest, Active Earth Pressure and Passive Earth Pressure

Earth Pressure at rest	Active Earth Pressure	Passive Earth Pressure
Retaining wall does not move. Elastic Equilibrium Horizontal earth pressure, k₀ = µ/1-µ ≈ 1-sinφ where µ is Poisson’s ratio σ_v = γ.z	Retaining wall moves away from backfill Plastic equilibrium σ_h = σ₃ σ_v = σ₁	Retaining wall moves towards back fill Plastic equilibrium σ_h = σ₁ σ_v = σ₃
	Failure plane inclined at (45+φ/2) to horizontal	Failure plane inclined at (45-φ/2) to horizontal
	At plastic equilibrium, σ₁ = σ₃.tan²(45+φ/2)+2C.tan(45+φ/2)

Active earth pressure has minimum value and passive earth pressure has maximum.

Tip – Either study the properties of active earth pressure or passive earth pressure and note the difference in the other one. It is better not to study both together and confuse values of σ_h and σ_v

What is Rankine’s Theory and what does it say?

RANKINE’S THEORY

Assumptions:

Soil dry, homogeneous, cohesionless, semi-infinite.
Ground surface is a horizontal or inclined plane.
Back of retaining wall is smooth and vertical.
Plastic equilibrium condition.
Wall yields about the base.

1. Dry cohesion less soil

Active Pressure	Passive Pressure
Active pressure, P_a = K_a.σ_v σ_v= rz K_a is coefficient of active earth pressure K_a= 1/tan²(45+φ/2) = tan²(45-φ/2) = 1-sinφ/1+sinφ For cohesive soil, φ=0 => K₀ = 1	Passive pressure, P_p = K_p.σ_v σ_v= rz K_p is coefficient of passive earth pressure K_p= 1/tan²(45-φ/2) = tan²(45+φ/2) = 1+sinφ/1-sinφ For cohesive soil, φ=0 => K_p = 1

Note:

K_a= 1/K_p
When φ = 0, K_a= K_p = 1
When z = 0, i.e., at top of the wall, P_a = P_p = 0

VARIATION OF LATERAL PRESSURE

Case I – Pressure distribution is triangular.

Resultant active earth pressure, P_a = 0.5 x K_a.γ.H x H

Note – Replacing K_a with K_p gives the passive earth pressure. The resultant is acting horizontally at height of H/3 from bottom.

Case II – Water table at the top of the retaining wall.

Pressure at any depth, P_a = K_a.γZ + γ_wZ

Case III – Water table between top and base of retaining wall.

If water is there on both sides of retaining wall, the effect of water need not be considered. Take P_aas in case I

Case IV – Inclined ground surface

Ground surface is inclined E with horizontal.

The resultant P_a will be inclined at the same angle β at height of H/3 from horizontal.

Case V – Backfill with uniform surcharge

At any depth,

P_a = K_a.γ.Z + K_a.q

At top (H=0) P_a = K_aq.q

At bottom (H=Z) P_a = K_aγH + K_aq

Note – There is no change in sign inside the root. There is square for cos b and cos f only inside the root.

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2. Cohesive Backfill

Active Pressure P_a = K_aσ_v– 2C√K_a = K_aγ_z– 2C√K_a

When Z=0, i.e., at top P_a = -2C√K_a

When Z=H at bottom, P_a = K_aγH – 2C√K_a

There is a point where tension becomes zero, it is called Tension Crack.

Depth of tension crack Z_c = 2Ctanα/γ = 2C/γ√K_a where K_a = 1/tan²α and α = 45+φ/2

Total Active Force

Before formation of tension crack, P_a = K_aγH²/2 – 2C√K_a.H

After formation of tension crack, P_a = K_aγH²/2 – 2C√K_a.H + 2C²/r

This acts at a height of one-third from base of bottom triangle, i.e. (H-Z_c)/3 from bottom (Height of lower triangle = H-Z_c)

Passive Pressure

P_P = K_p.σ_v + 2C√K_p

Total passive force = K_p_γH²/2 + 2C√K_p.H

Critical Height (H_c)

It is the depth upto which vertical excavation can be done on cohesive soil without any lateral support.

H_c = 2Z_c = 4C/γ√K_a = 4C/γ x tan²α (α = 45+φ/2)

For pure cohesive soil (clay), φ=0 => H_c = 4C/γ

Note

There is no tension formation in case of passive pressure as value of P_p is always positive for any value of Z.
Effect of cohesion is to reduce active pressure by a 2C√K_aand increase passive pressure by 2C√K_p

What is Coulomb’s Thery and what does it say?

COULOMB’S THEORY

The main difference between Rankine’s theory and coulomb’s wedge theory is that here the wall surface is assumed to be rough.

The inclination of resultant earth pressure to normal to the wall is taken as δ. It is the angle of friction between wall and the backfill.

For smooth walls, Angle of friction δ = φ/3

For concrete walls, Angle of friction δ = 2φ/3

For rough walls, Angle of friction δ = 3φ/4

For backfill subject to vibration, Angle of friction δ = 0

STABILITY OF SLOPES

Types of Slopes:

Infinite slope: slopes of greater extend having soil properties and soil compression constant for all identical depths below the top surface.
Finite slope: slopes having limited length like in earth dams, highway cuts, railway cuts etc.

Types of Slope Failures

Translational failure – Failure of slope occurs parallel to the slope at some depth below it. It is common in case of infinite slope.
Rotational failure – Failure surface is curved and is caused by rotation.

Stability of Infinite Slope

Infinite slope in C – φ Soil

Factor of Safety = Shear Strength/Shear Stress

F_c = S/τ
F_c = C+γZcos²i.tanφ)/γZcos(i).sin(i)

Where φ is angle of internal friction, i = angle of infinite slope

Infinite slope in dry sand

For dry sand, C = 0

Therefore, F = (0 + γZcos²i.tanφ)/ γZcos(i).sin(i) = tanφ/tan(i)

Stability Number (S_n)

Used forC – φ Soil, it is dimensionless quantity.

S_n = C/F_cγH = C_m/γH = C/γH_c x (tani – tanφ)cos²i

Reciprocal of stability number is called stability factor

Note

If the slope angle i is less than or equal to angle of internal friction φ, the slope is stable.
If i > φ , slope can be stable only upto a certain height called critical height (H_c)

H_c = C/r(tan i – tan φ)cos²C

Also, H_c = H x F_c where H is actual height of slope
Mobilised cohesion (C_m) is cohesion of soil when factor of safety (F_c) is applied.

C_m = C/F_c => F_c = C/C_m => F_c = H_c/H = C/C_m

For pure cohesion less soil, C=0 and S_n = 0
Maximum value of S_n = 0.261
For fully submerged soil, F_s = r_sub/r_sat x tanφ/tanβ, S = C/F_cγ’H (γ’= γ_sub)
For slope saturated by capillary water, S_n = C/F_cγ_satH

Stability of Finite Slope:

Types of failure in finite slope

Face failure
Toe failure
Base failure

Failure of finite slope is due to rotational failure.
Factor of safety against rotational failure.

F = Resisting moment/Driving moment

Depth Factor (D_f)

D_f = H+d/H = depth of hard stratum from top of slope/depth of slope

When,

D_f < 1, face failure (hard stratum inside slope)

D_f= 1, toe failure (i.e., d = 0)

D_f > 1, base failure (hard stratum below base of slope)

Methods of stability analysis of finite slope

φ_u = 0 analysis (for purely cohesive soil)
C – φ analysis (for cohesive soil)
Slip circle method.
Friction circle method.
Stability number method.
Bishop’s method

FOUNDATION

TYPES OF FOUNDATION

1. Based on depth

Shallow foundation – If the depth or width of the footing is less than or equal to its breadth, it is called shallow foundation. Failure is usually rotational bearing failure. Example – spread footing, Raft or mat footing.

Deep foundation – If the depth of footing is more than its breadth, it is called deep foundation. Example – Pile foundation D > B

2. Based on its structure

Isolated footing – It is constructed under base of the wall or below a column.

Combined footing – Two or more footing joined together. Used if the base of footing overlap or is very close to each other.

Strip footing – Length of foundation is much more than its width.

Mat or raft foundation – The whole footing or column is on a single layer of foundation.

Pile foundation – Piles are inserted into the soil where there is no hard stratum near the surface.

(a) End bearing pile – The pile which directly transmits the load to the hard stratum at a greater depth.

(b) Friction pile – Pile which carry the load by friction b/w surface of pile and surrounding soil.

(c) Compaction pile – Piles which are inserted into cohesion-less soil to increase shear strength of soil. Example – Sand pile.

3. Based on Material

Concrete foundation
Steel foundation
Wooden foundation
Bamboo foundation
Sand pile
Brick foundation

Foundation design requirement

Foundation is designed to satisfy two main requirements

No shear failure
No excessive settlement

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SHEAR FAILURE

TYPES OF SHEAR FAILURE

General shear failure (GSF)

– failure is well defined

– occurs in dense soils

– heave on both sides

Punching shear failure (PSF)

– no well defined slip line

– No heaving

– occurs in loose sand

Local shear failure (LSF)

– characteristics of both GSF and PSF

– Shows vertical shear and slight building

– Slight bulging/ heaving

– Occurs in cohesive soil for shear strength C_u < 50 kN/m²

Criteria for GSF and LSF

Criteria	GSF	LSF
Φ	> 36⁰	< 28⁰
Relative Density	> 70%	< 20%
N	> 30	< 5
e	< 0.55	> 0.75
Failure Strains	< 5%	10 – 20%

Excessive settlement

– Allowable settlement depends on type of soil and foundation

– Allowable settlements

Foundation → Soil ↓	Isolated Footing	Raft Footing
Sand, hard clay, (cohesion less soil)	50 mm	75 mm
Plastic clay (cohesive soil)	75 mm	100 mm

Total settlement

Total settlement = immediate settlement + consolidation settlements Immediate settlements S_i, occurs in highly permeable soils

S_i = qB(1-µ²/E_s)I

Where,

q – Net intensity of contact pressure (kN/m2)
B – Least lateral dimension of footing or loaded area (m)
E_s – Modulus of elasticity (kN/m²)
I – Influence factor
µ – Poisson’s ratio value of influence factor depends on the type, shape and rigidity of the footing. (The value will be given in the question paper itself during the exam). The value ranges from 0.8 to 1.7.

Consolidated Settlement (S_c)

S_c = C x C_i/1+C₀ x H log (σ₀+Δσ/σ₀)

Where,

C – Correction factor
H – thickness of compressible layer
C_i– Compression index
σ_o – effective overburden pressure
e_o – initial void ratio
Δσ – vertical stress on footing

Bearing capacity of Soil

Various terms to express the bearing capacity/ bearing pressure of soil.

Gross pressure (q)

Total bearing pressure at the base of the footing due to weight of the structure, self weight of footing and earth fill. In some cases earth fill is not provided. Then take that load as zero.

Net pressure (q_n)

It is the pressure in excess of that which existed before construction

q_n = Gross pressure – Original overburden pressure that acted on the unit area = q – γD

where, γ = unit weight of soil, D = depth of foundation

Ultimate bearing capacity (q_u) – Maximum gross pressure that the soil can withstand before it fails in shear

Net ultimate bearing capacity (q_nu)

Maximum net intensity of press. the soil can bear before it fails in shear

q_nu = q_n – γD

Net safe bearing capacity (q_ns)

Net safe pressure that the soil can safely withstand without any risk of shear failure

q_ns = q_nu/F

where F – factor of safety, usually taken between 2.5 and 3.

Gross safe bearing capacity (q_gs) It is the maximum gross bearing pressure the soil can withstand without shear failure.

q_gs = q_ns + γD = q_nu +γD

Net safe settlement pressure (q_np)

It is the maximum net intensity of loading that can be allowed on the soil without the settlement of soil exceeding the permissible value

Net allowable bearing pressure (q_na)

It is the maximum net intensity of pressure that can be imposed on soil with no risk of shear failure as well as possibility of excessive settlement. This value is used to design foundation.

q_na = q_ns or q_np whichever is less

Area of foundation

Design criteria for design of foundation

q_n ≤ q_na

Q/A ≤ q_na (back filled)

Q/A – γD ≤ q_na (no backfill)

In case of no back fill, Q/A ≤ γD + q_na

It is clear that in case of no backfill, bearing capacity is considerably enhanced.

If Q/A = γD, then q_na = 0 This is the principle of floating raft foundation where the pressure due to superstructure is equal to overburden pressure, thereby reducing net pressure to zero

Rankine’s Analysis for minimum depth of foundation Depth of foundation,

D = q/γ x [1-sinφ/1+sinφ]²

Where φ – effective angle of shearing, q – intensity of loading, γ – density of soil solids

Terzaghi’s bearing capacity theory

Assumptions

Footing is shallow (D_f < B), continuous
Base of footing is rough
GSF occurs

Effect of water table on bearing capacity

Case 1: Water table (WT) is at or above ground level

use r’ instead of r in the equation.

q_u = CN_c + γDN_q + 0.5Bγ’N_γ

Case 2: When WT is above base of the footing

q_u = CN_c + γDN_qRW₁ + 0.5BγN_γ

R_W1 = 0.5[1 + Z_w2/D]

Where Z_w1 = Depth of WT below ground level

and Z_w2 = Depth of WT below base of the footing, B.

Case 3: WT is below base of footing at depth less than width of footing, B

q_u= CN_c + γDN_q + 0.5BγN_γ.R_wz

R_w2 = 0.5[1 + Z_w2/B]

Z_w2 = Depth of the WT below base of the footing, B.

Case 4: WT below base of footing, at depth more than width of footing, B.

NO Correction is required in this case.

Points to be noted:

Value of R_w1 and R_w2 ranges from 0.5 (where Z_w1 or Z_w = 0) to 1 (where Z_w1 or Z_w = 1)
Note that in equation of R_w1 it is Z_w1 divided by D and for R_w2 it is R_w2 divided by B.

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Correction of Bearing Capacity

Clayey Soils –

Size of footing or plate does not affect the bearing capacity. The bearing capacity found out during plate load test is taken for the design of foundation in clayey soil.

q_p ≈ q_f

Sandy Soils –

Bearing capacity is directly proportional to width or side of the footing/plate

q α B where q_f is Bearing capacity of footing

q_f/q_p = B_f/B_p where q_pis Bearing capacity of plate in plate load test

Skempton’s Method

Skempton’s method is applicable only to cohesive soil.

Bearing capacity factor N_c increases with ratio D_f/B

Strip footing

When D = 0, N_c = 5.14 (On Surface). This is the minimum value

When D/B < 2.5, N_c = 5 [1 + 0.2D/B] = 5 + D/B

When D/B > 2.5, N_c = 5 x 1.5 = 7.5, this is the maximum value

Rectangular Footing

N_c = 5[1 + 0.2D/B][1 + 0.2B/L]

Minimum value of N_c is 6.2 and maximum value is 9

PILE FOUNDATION

Classification of Piles

Based on load transfer

End bearing pile – The load of the superstructure (Q_u) is transferred to the hard strata through the pile tips (Q_b)
Friction pile or floating pile – Load is transferred by skin friction between soil and surfaced pile used when hard strata is not available at reasonable depth.
Combined end bearing and friction pile – Load transferred by combination of end bearing and friction force.

Based on function

Compaction pile – For compaction of loose granular soil and to increase the bearing capacity. Eg: Sand piles
Tension piles – To anchor down structures which are subjected to uplift force.
Anchor piles – To provide anchorage against horizontal pull from sheet pile walls.
Fender piles and dolphins – They are sheet piles used to protect water front structures against impact from ships and other floating bodies.
Batter piles – These piles are provided to resist lateral or inclined forces due to moving or berthing of ships.

Based on method of installation

Driven cast-in- situ piles – The casing is driven and pile is cast-in-situ.
Precast driven piles – Pile is precast and is driven into soil
Bored cast-in-situ – Bores are made on the ground and piles are cast in situ.

Based on material of pile

Conevete pile
Jimber pile
Steel pile
Composite pile

Points to be noted

All driven piles comes under displacement piles i.e., they cause lateral displacement of soil when installed.
All bored piles are non displacement piles

LOAD CARRYING CAPACITY OF PILE

STATIC FORMULAE

This formula is based on strength properties of soil. The load transferred by a pile is equal to sum of that transferred by end bearing and skin friction.

Ultimate load capacity of pile,

Q_u = Q_b + Q_s

Where,

Q_b= end bearing resistance/ capacity
Q_s = skin friction capacity or side friction capacity

Q_b = A_bf_b

Q_s = A_sf_s

Where

A_b = base section area of pile
A_s = Surface area of pile (2πr.D)
f_b, f_s = bearing capacity and skin friction of pile

Load Capacity of pile in sand

Q_u= Q_b + Q_s = A_bf_b + A_sf_s

For piles in sand,

f_b = σv’.N_q , where N_q = bearing capacity factor

σv’ = γ’L if L<D_c where L = Length of pile

σv’ = γ’D_c if L≥D_c where D_c = Critical depth

i.e., σv’ = γ’.(L or D_c whichever is lesser)

D_c = 10xd for loose sand where d=diameter of pile

= 20xd

f_s = K.σ_a’.tanδ

K = Lateral earth pressure coefficient

σ_a’ = Average vertical pressure for the depth

δ = Angled friction between pile and soil

Therefore, Q_u = A_bσV’N_q = A_s.k. σ_a’.tanδ

Load Capacity of pile in clay

Q_u = A_bf_b + A_sf_s

f_b = CN_c

where C = Cohesion of soil at pile tip = shear strength of soil at pile tip, N_c=9

f_s = αC where α = Shear mobilization factor or adhesion factor

Q_u = A_b.C.N_c + A_s.α.C

Safe load capacity of pile = Q_u/F

Factor of Safety, F = 2.5 for static formulae

Group action of piles

Efficient of group pile, η_g = Q_gx100/n.Q_u

η_gcan be greater than or less than 100%. It is usually greater than 100% for loose and medium dense sands and less than 100% for dense sands.

Spacing of Piles in group pile

For end bearing piles, spacing = 2.5.d where d = diameter of pile.

For friction piles, spacing = 3 × d

Failure of group pile

a. Individual failure

Group capacity of pile group, Q_g = n x Q_u

Where n is number of piles, Q_u is ultimate load capacity of each pile

b. Block failure

Group capacity, Q_B = A_Bf_B + A_sf_s

A_B = L’ x B’

A_s = 2(L’ + B’) x L

L’ = (n-1)s + d

Where n is number of piles on that side, d is diameter of pile, L is length of pile

Note : Group capacity for calculations is taken the smaller value from these values of group capacity.

DYNAMIC FORMULAE

This formula is based on energy input to drive the pile.

Engineering news formulae : Q_s = Whη_h/F(S+C)

Where

W – Weight of hammer in kg.

h – height of drop

η_h – efficiency of pile hammer (Usually taken as 1)

F – factor of safety (usually 6)

S – penetration of pile/ hammer blow

C ≈ 2.5 (C=2.4 for drop hammer and 2.53 for singe and double acting hammers)

Where

P = weight of pile
e = coefficient of restitution
e = 0 for timber pile, e = 0.5 for RCC and steel pile

Note: This equation is applicable only to friction pile.

For end bearing P is reduced to half.

Where C = effective elastic compression = sum of temporary elastic compression of dolly and pecking, pile and ground = C₁ + C₂ + C₃

Point to remember – Dynamic formula is test suited for coarse grained soil and not for deep and silt underwater.

Pile Load Test

Used to determine capacity of individual pile.
Maximum load on the pile should not be less than 2.5 times estimated load and for working pile it should not be less than 1.5 times.
The load settlement curve is drawn, and least of the following is taken as safe load
2/3rd of final load which given total settlement 12 mm.
2/3rd load which gives net settlement of 6 mm.
1/2 of load at which total settlement is 1/10 of pile diameter.

Negative Skin Friction (Downward drag)