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Shallow Foundation

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Introduction

Foundation is a structure made of concrete, steel or wood to transmit loads to the underlying soils. A geotechnical engineer must ensure that a foundation satisfies the following two stability conditions:(a) Ultimate Limit State - The foundation must not collapse or become

unstable under any conceivable loading. (Check the bearing capacity).(b) Serviceability Limit State - Settlement of the structure must be within

tolerable limits (check the ultimate settlement).

Shallow foundation is one in which the ratio of the embedment depth (Df) to the minimum plan dimension, which is usually the width(B), is Df/B ≤ 2.5.

Serviceability limit state is commonly adopted in the design of foundation. Serviceability limit state defines a limiting deformation or settlement of a foundation, which if exceeded, will impair the function of the structure that it supports.

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Ultimate net bearing capacity (qult minus surcharge of soil) is a pressure that capable to bring the soil to collapse or failure. In practice, we NEVER USE ULTIMATE NET BEARING CAPACITY TO DESIGN FOUNDATION. This is because of spurious soil variations, and uncertainties in soil test values and structural loads. We usually divide the ultimate net bearing capacity by a factor of safety (normally FS =3) and name it as allowable net bearing capacity, qallow(net).Gross allowable bearing capacity ( qallow ) = Ultimate bearing capacity / FS

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Assume a load apply on a dense and loose sand, respectively. The failure mechanisms for both cases will be as follows :

Footing

Failure plane: General shear failure

Footing

Failure plane: Local shear failure

Dense sand Loose sand

Load

Sett

lem

ent

Qult

Collapse, Pu = 5.14Bcu

Failure

Load

Sett

lem

ent

Qult

Failure

B

Modes of Bearing Capacity Failure in Soil1) General Shear Failure – Dense soils2) Local Shear Failure – Loose soils3) Punching Shear Failure – Very loose soils

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Failure plane: Punching shear failure

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General Equation of Ultimate Bearing Capacity, qult:qult = c Nc (rc sc dc ic bc) + 0.5B’ g1 N g (rg sg dg ig bg) + g2 Df Nq (rq sq dq iq bq)

Where g1 = effective unit weight of soil below foundation level (kN/m3) g2 = effective unit weight of soil below foundation level (kN/m3)N = soil bearing capacity factors

r = rigidity index reduction factors to compensate for a possible punching-local shear conditions = shape factorsd = depth factorsi = load inclination factorsb = base tilt factors

qult = qc + qg + qq

Ultimate bearing capacity of soil is the contribution of cohesion, unit weight of soil below base of footing and surcharge of soil

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Consideration of eccentric load :B’ = B -2eB

L’ = L – 2eL

Note: If the load acts at the center of the foundation, B’ = B and L’ = L.eB > B/6 or eL > L/6, then tension develop. Since tensile strength of soil is approximately zero, part of the footing will not transmit loads to soil. You should design the footing eB < B/6 and eL < L/6

B

L

Q

eL

eB

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Soil bearing capacity factors, N:Nq = tan2 (45 + f/2)(e) p tanf

Nc = (Nq – 1) cot f (for f > 0o)Nc = 5.14 (for f = 0o)Ng = 2(Nq + 1) tan f

Rigidity index reduction factors, r:Actual rigidity index,

Where, E = soil modulus of elasticity (Normal soil ranging from 500 – 180,000 kN/m2)v = Poisson’s ratio (Normal soil ranging from 0.1 – 0.5)

= Effective overburden pressure at a depth of Df + B/2 Critical rigidity index,

)tan)(1(2 vr cv

EI

245cot

'

'45.03.3exp

2

1)(

L

BI criticalr

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Continue on rigidity index reduction factors, r:In most of the cases, Ir > Ir(critical), which means general shear failure mode applies, thus rc , rq, rg = 1.

If Ir < Ir(critical), the rc , rq, rg can be obtained as below:

rq =

rqrq

rc

rc

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Shape factors, s:sc = 1 + (B’/L’)(Nq/Nc)sg = 1 – 0.4 (B’/L’)sq = 1 + (B’/L’) tan f

Depth factors, d:dg = 1

(when f > 0o)

(when f = 0o)

All the depth factors = 1 if the shear strength of the soil above the footing is low compared with the soil below the footing.

3.57

)'/(tan)sin1(tan21

12 BD

d fq

tan

1

c

qqc N

ddd

)3.57)(3(

)'/(tan1

1 BDd fc

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B

L

q

Q

Qh

Qv

m

hq cLBQ

Qi

cot''

1

1

cot''1

m

h

cLBQ

Qi

tan

1

c

qqc N

iii

''1

LBcN

mQi

c

hc

(when f > 0o)

(when f = 0o)

m = mL = (2+L’/B’) / (1+L’/B’) if the load inclined along L-axis.m = mB = (2+B’/L’) / (1+B’/L’) if the load inclined along B-axis.m =mq = mLcos2q + mBsin2q if the load inclined in a plane makes an angle q with the L-axis.

Load Inclination factors, i:Q: Vertical force to footingQh: Horizontal force along footing

m= (mL2 + mB2)0.5 Use B and L ,not B’ and L’

Inclined loads are produced when footing is loaded with both a vertical V and a horizontal Hi components of loading.Normally footing is subject to combination of wind loads Hi and gravity loads V.

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Base tilt factors, b:

bq = bg

bc = 1 – 0.0067a

2

tan3.57

1

b

tan

1

c

qqc N

bbb (when f > 0o)

(when f = 0o)

For a long strip footing (L much greater than B) with a vertical centric load:qult = c Nc + 0.5B g1 N g + g2 Df Nq

a

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4.2 Principle of Shallow Foundation Design: Spread Footings

Ultimate Net Bearing Capacitty, qnet:qult = c Nc (rc sc dc ic bc) + 0.5B g1 N g (rg sg dg ig bg) + g2 Df Nq (rq sq dq iq bq)

If both side of Df is different, use the smaller Df

Ultimate Net Bearing Capacity, qnet

Net Allowable Bearing Capacity, qallow:qallow (net) = qnet/FS ; FS is in the range from 2 to 3, with FS = 3 is most often used.

qnet = qult qnet = qult - g2 Df

g2

g1

Df

Gross Allowable Bearing Capacityqallow = qult / FS

qnet = qult - g2 Df

Factor or Safety (SF)= qult / Stress created by external loading to the actual dimension of footing

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Remember, we must use GROSS allowable bearing capacity, qall to design the foundation. Our responsibility is to ensure that the stress from the structure, s is lower than the allowable bearing capacity (i.e. s < qallow).

qallow

P (From structural analysis)

qallow

s = P/A

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Load Bearing Capacity of Shallow Foundation Based on Plate Bearing Test

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4.3 Principle of Shallow Foundation Design: Mat Footings

A mat foundation is used when:- Spread or individual footings cover over 50% of the foundation area.- Pockets of soft soils are present.- The structure is sensitive to differential settlement.

Consider the case below:

If all the loads act on a mat foundation are compensated by embedment (i.e. q = g/Df), we name it as fully compensated raft. This kind of foundation is deemed as a conservative design.

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4.4 Bearing Capacity for Layered SoilsIn practice, there are three common cases of layered soils:a)Soft clay over stiff clay: Shallow foundation in soft clay should be avoided. We may either replace the soft clay with compacted fill or consider the alternative of deep foundation.b)Thinly stratified soils: Use the shear strength properties of the weakest soil layers in the calculation of bearing capacity.c)Stiff clay over soft clay: First, we have to ensure that the stiff clay layer is thick enough to be considered as this category:

If the height of stiff soil below the footing to the top of soft clay, H is smaller than Hcr (i.e. H < Hcr), then this case is valid. Otherwise, we will assume the whole soil profile as soft clay.

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Secondly, the bearing capacity for this case should consider the smaller value of 2 mechanisms:i)The failure plane only develop in stiff clay, thus we only calculate the bearing capacity in stiff clay and ignored the existence of soft clay.

ii)The failure plane is extended into the soft clay layer. For this case, we assume an imaginary footing with dimensions (B + H, L + H) sitting directly on the soft clay layer.

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4.5 SettlementSettlement is practically impossible to be prevented. However, as an geotechnical engineer, we have to ensure that the settlement of foundation has not exceed the serviceability limit state. There are many descriptions of serviceability limit state depending on the function of the structure. Foundation settlement can be divided in 3 basic types: uniform settlement, non-uniform settlement and tilt or distortion.

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Distortion is caused by differential settlement. It is quantified by d/l, where d is the maximum differential settlement and l is the length over which the settlement occurs. The limit of distortion can be calculated from:

Soil type Values Spread Footing Mat Footing

Clay R 22,500 30,000

Pmax 75 mm 100 mm

Sand R 15,000 18,000

Pmax 50 mm 60 mm

lRP

max

The maximum tolerable settlement is as below:

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Immediate settlement:Immediate settlement is also known as elastic settlement. To determine the immediate settlement of shallow foundations, we need to take into account the effects of soil stiffness, embedment, and soil-wall resistance. The immediate settlement can be calculated as follows:

Where P = total vertical loads, Eu = undrained elastic modulus of soil, L is one-half the length of foundation, vu is poissons’ ratio of soil, ms, memb, and mwall are the soil stiffness, embedment, and soil-wall interface factors, respectively.

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Where Ab is the area of the foundation base and Aw is the area of the wall in contact with the embedded portion of the footing.

We should not consider the reduction of settlement resulting from the wall factor if we are unsure whether the full wall resistance is mobilized. In this case, mwall = 1.

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Primary consolidation Settlement: (Skempton-Bjerrum method)Calculation of primary consolidation for shallow foundation, we need to take into account the effect of lateral stress and strain. As the result, a factor, mSB is applied to the one-dimensional primary consolidation:

where mSB can be referred to Figure 1= Induced vertical stress referred to Table 1

= Modulus of volume compressibility= Thickness of clay layer

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Figure 1Hc : Depth of clay layer B: Equivalent diameter of circular A : Porewater coefficient footing =2(area of base/3.1415)0.5

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Table 1

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