Lecture-2 Bearing Capacity of Soil

Objective of the chapter When you complete this chapter you should be

able to: Calculate the bearing capacity of soils. Differentiate the different methods of bearing capacity

equation. Determine the bearing capacity for eccentrically

loaded footings.

1.1 INTRODUCTION

A foundation, is a structure designed to transfer loads from a superstructure to the soil underneath the superstructure.

In general, foundations are categorized into two groups, namely,

1. Shallow foundations and

2. Deep foundations.

1.2 Some Basic Definition 1) Ultimate Bearing Capacity (qu) :

The ultimate bearing capacity is the gross pressure at the base of the foundation at which soil fails in shear. 2) Net ultimate Bearing Capacity (qnu) : It is the net increase in pressure at the base of foundation that cause shear failure of the soil. Thus, qnu = qu – γDf (overburden pressure) 3) Net Safe Bearing Capacity (qns) : It is the net soil pressure which can be safely applied to the soil considering only shear failure.

Thus, qns = qnu / [FS]

Cont’d…

4) Gross Safe Bearing Capacity (qs) : It is the maximum pressure which the soil can carry safely without shear failure. qs = qnu / FS + γ Df

5) Net Safe Settlement Pressure (qnp) :It is the net pressure which the soil can carry without exceeding allowable settlement.

6) Net Allowable Bearing Pressure (qna ): It is the net bearing pressure which can be used for design of foundation. Thus, qna = qns ; if qnp > qns

qna = qnp ; if qns > qnp

1.3 Modes of shear Failure

Vesic (1973) classified shear failure of soil under a foundation base into three categories depending on the type of soil & location of foundation.

a) General Shear Failure Most common type of shear failure; occurs in dens sand and

stiff clay soil under strip footing.b) Local Shear Failure Intermediate b/n general and punching

shear failure. This is common in sand and clays of medium compaction. failure surface will gradually extend outward from the foundation but will not reach the ground surface as shown by the solid segment

c) Punching Shear Failure Occurs in very loose sands weak clays.

Cont’d…

Bulge/heave

GENERAL

LOCAL

PUNCHING

Bearing capacity failure

2.1 Ultimate Bearing Capacity Equations1. Prandtl (1920) developed the equation by assuming

The footing as frictionless Ignoring the weight of the soil at the failure zone

For pure cohesive soil ( = 0)

For the footing at the surface

For the footing at a certain depth from the ground

surface

His assumptions are not true in practice and therefore the equation is never used in practical design, but it was a beginning.

uu cq 14.5

fuu Dcq 14.5

2.1.1 Terzaghi bearing capacity equation Terzhagi (1943) improved the Prandtl equation to include the roughness of the footing and the weight of

the failure zone. The failure mechanism in a c’, ’ soil is given in fig. below

Terzaghi’s Bearing Capacity equation is applicable for general shear failure

Terzhagi’s ultimate bearing capacity equations Strip (or long) footing Square footing: Circular footing:

Where Nc, Nq and N are called the bearing capacity factors and are obtained as follows

In the undrained conditions (cu and u = 0):

NBDNNcq qcu 5.0'

NBDNNcq qcu 4.0'3.1

NBDNNcq qcu 3.0'3.1

)2/'45(cos2 2

'tan)'2/3(

eN q

)1('cot qc NN

1

'cos'tan

221

pK

N

)2/'60(tan)8.3'4'8( 022 pK

1qN 71.5cN 0N

Terzaghi Bearing capacity coefficients

2.1.2 Meyerhof’s Bearing Capacity equation Meyerhof (1951) developed a bearing capacity equation by extending

Terzhagi’s failure mechanism and taking into account the effects of footing shape, load inclination and footing depth by adding the corresponding factors of s, d, & i.

For a rectangular footing of L by B (L > B) and inclined load:

For vertical load, ic = iq = i = 1.

The bearing capacity factors:

In the undrained conditions (cu and u= 0):

disNBdisDNdisNcq qqqqccccu 5.0'

)2/'45(tan)'tanexp( 2 qN )1('cot qc NN

)'4.1tan()1( qNN

1qN 71.5cN 0N

Meyerhof’s bearing capacity coefficients

The shape, inclination and depth factors

2.1.3 Hansen’s Bearing Capacity Equation Hansen (1961) extended Meyerhof’s solutions by

considering the effects of sloping ground surface and tilted base as well as modification of N and other factors.

For a rectangular footing of L by B (L > B) and inclined ground surface, base and load:

gbidsNBgbidsDNgbidsNcq qqqqqqccccccu 5.0'

In the special case of a horizontal ground surface,

The bearing capacity factors Nc and Nq are identical with Meyerhof’s factors. But N :-

Failure can take place either along the long side or along the short side of the footing so that Hansen proposed that

1. Shape factors is given as

For cu, u=0 soil

bidsNBbidsDNbidsNcq qqqqqcccccu 5.0'

tan)1(5.1 qNN

Bcc

qBc i

L

B

N

Ns ,, 1 'sin1 ,, BqBq i

L

Bs 6.04.01 ,, BB i

L

Bs

Lcc

qLc i

B

L

N

Ns ,, 1 'sin1 ,, LqLq i

B

Ls 6.04.01 ,, LL i

B

Ls

BcBc iL

Bs ,, 2.0 LcLc i

B

Ls ,, 2.0

The inclination factors are:

For the Tilted base:-

Where A = is the area of the footing base 0 = angle of inclination of the base of the footing. 1 and 2 are in the range of

1

1 ,,,

q

iqiqic

N

iii

1

'cot

5.01,

b

iiq

AcV

Hi

2

'cot

7.01,

b

ii

AcV

Hi

52 1 52 2

2

'cot

)4507.0(1

00

,

b

ii

AcV

Hi

Cont… H and V are horizontal and vertical component of the

total load. b = is the angle of friction between the base of footing

and soil. ca = is the adhesion between footing and soil.

For cu, u=0 soil:

In the above equations, B and L may be replaced by their effective values (B’ and L’) expressed as :

L’ = L – 2eL and B’ = B – 2eB 　

Where eL and eB represent the eccentricity along L and B directions 　　　　　　　　　

biic AcHi 15.05.0,

Cont…

The depth factors are expressed in two sets For D/B 1 & D/L 1:

For D/B > 1 & D/L > 1:

For both sets:

　　　　　　　　

BDd Bc 4.01, B

Dd Bq 2, )'sin1('tan21

LDd Lc 4.01, L

Dd Lq 2, )'sin1('tan21

BDd Bc

1, tan4.01

LDd Lc

1, tan4.01

)(tan)'sin1('tan21 12, B

Dd Bq

)(tan)'sin1('tan21 12, L

Dd Lq

1d

For cu, u=0 soil:

1. For the sloping ground and tilted base:-

The ground factors “gi“

The base factors “bi“

For cu, u soil

BDd Bc 4.0, L

Dd Lc 4.0,

0

0

1471 cg 5tan5.01 gg q

0

0

1471 cb 'tan2 ebq

'tan7.2

eb

0

0

147cg

0

0

147cb

2.2 A comparative summary of the three bearing capacity equations

1. Terzaghi’s equation is widely used, because it is some what simpler than Meyerhof’s and Hansen’s.

2. Practitioners use Terzaghi’s equations for a very cohesive soil and D/B < 1.

However, Terzaghi’s equations have the following major drawbacks:

Shape, depth and inclination factors are not considered. Terzaghi’s equations are suitable for a concentrically

loaded horizontal footing but are not suitable for eccentrically loaded footings that are very common in practice.

Cont… The equations are generally conservative than

Meyerhof’s and Hansen’s. That is why Currently, Meyerhof’s and Hansen’s

equations are more widely used than Terzaghi’s. Both are applicable to more general conditions.

Hansen’s is, however, used when the base is tilted or when the footing is on a slope and for D/B > 1.

2.3 Effects of Groundwater Table on Bearing Capacity

For all the bearing capacity equations, you will have to make Some adjustments for the groundwater condition.

The term “ D ” refers to the vertical stress of the soil above the base of the footing.

The term “B” refers to the vertical stress of a soil mass of thickness B, below the base of the footing. Here we have three condition for ground water effect ,so check which one of the three groundwater situations is applicable to your project.

Situation 1: Groundwater level at a depth B below the base of the footing.

In this case no modification of the bearing capacity equations is required.

Cont…

Situation 2:

Groundwater level within a depth B below the base of the footing.

If the groundwater level is at a depth z below the base, such that z < B. Then

The term or

The term remains unchanged.

Situation 3:

Groundwater level within the embedment depth. If the groundwater is at a depth z within the embedment such that z < D

)(' zBzB )(' zBzsat

D Soil is saturated above ground water level

Cont…

Then the term or

The term is changed in to .

)(' zBzB )(' zBzsat B B'

Situation 2 Situation 3

2.4 Allowable bearing capacity and factor of safety

The allowable bearing capacity “qa “ is calculated by dividing the ultimate bearing capacity by a factor.

The FS is intended to compensate for

1) Assumptions made in developing the bearing capacity equations.

2) Soil variability.

3) Inaccurate soil data, and

4) Uncertainties of loads. The magnitude of FS applied to the ultimate bearing

capacity may be between 2 and 3. The allowable bearing capacity is

FS

qq u

a

Cont…

If the maximum applied foundation stress is known and the dimension of the footing is also known. By replacing “qa“ with ()max then FS is given by:-

Meyerhof (1963) proposed an approximate method for loads that are located off-centered (or eccentric loads).

He modified base area or dimensions of the footing pad, as

B’ = B – 2eB and L’ =L - 2eL

Where eB and eL are eccentricity along B and L.

if the maximum applied foundation stress if the maximum applied foundation stress if the maximum applied foundation stress

max)( a

uqFS

Cont’d…..

Cont…

Where and Mx= moment in X-direction.

My= moment in Y-direction.

p = vertical load.

The maximum and minimum vertical stresses

Along the x axis are:

Along the y axis are:

Since the tensile strength of soils = 0, it should always be > 0. Therefore, eB < B/6 & eL < L/6.

The bearing capacity equations are modified for eccentric loads by replacing B with B’.


P

Me y

B P

Me x

L

B

e

BL

P B61max

B

e

BL

P B61min

B

e

BL

P L61max

B

e

BL

P L61min

Cont’d…..

Cont… The ultimate bearing capacity for footings with eccentricity,

using either the Meyerhof or Hansen equations, is found in either of two ways:

Method 1. Use either the Hansen bearing-capacity equation with the following adjustments:

a. Use B' in the BN term.

b. Use B' and L' in computing the shape factors.

c. Use actual B and L for all depth factors. The computed ultimate bearing capacity [qult] is then

reduced to an allowable value qa with an appropriate safety factor SF as

qa = qult / SF and Pa = qa B'L'


Cont…

Method 2. Use the Meyerhof general bearing-capacity

equation and a reduction factor Re used as:-

Where reduction factor :- For cohesive soil:-

For cohesion less soil and for 0 < e/B < 0.3. In practice the e/B ratio is seldom greater than 0.2.if it is equal

to 0.5 the foundation will be unstable or Area (A’)is at corner. Alternatively, use the Meyerhof equation with B' and L’

for shape and depth factors & B’ in the 0.5B'N term.


ecomputedultdesignult Rqq *)()(

B

eRe

21

B

eRe 1

2.6 Field Tests It is difficult to obtain undisturbed coarse-grained samples

for laboratory testing and one has to use results from field tests to determine the bearing capacity of shallow foundations. Some field tests are

1. Plate Loading TestThe plate loading test is carried out to estimate the bearing

capacity of single footings.

The plates (made of steel) dimensions are

25 mm thick and 150 mm to 762 mm in diameter.

In practice a circular plate of 300 mm is commonly used. square plates that are 300 mm×300 mm are also used.


Procedure Step – 1

Excavate bore hole with a minimum diameter 4BP [Bp = plate diameter] to a depth of D (D = depth of the proposed foundation).

Step – 2

Then place the plate at the center of the hole.

Step – 3

Apply initial setting load to make the proper contact of the plate with the ground.

Step – 4

set the dial gages reading as the initial readings


Step – 5

Apply the first load approximately equal to 1/5th of the anticipated ultimate load

Step – 6 Keep this load constant till the settlement

of the plate under this load ceases(where the settlement is about 0.3mm/hr, it is supposed that the settlement is almost stopped.)

If this situation is achieved before 1hr, the load has to kept applied for an hr. If this situation is achieved after 2hrs,then the load has to kept constantly applied for 2hrs

Step – 7

Record the settlement dial gage readings.

difference of initial & final reading gives settlement of the plate

Step – 8 change the next load=2*1/5*ultimate

load repeat as the pervious steps


Procedure plot a graph load with the corresponding settlement.

For tests in clay,

For tests in sandy soil,

Where

qu(F) & qu(P) are ultimate bearing capacity of foundation and plate, and BF and BP stand for width of foundation and plate


)()( PuFu qq

p

FPuFu

B

Bqq )()(

Under the same intensity of loading on plate & the actual foundation, the following r/ship used:-

For cohesion less soil

For cohesive soil

Draw back In case of non- homogeneous or stratified soil, which

affect the foundation but not the plate, the result of plate load test may be misleading

It has elaborate arrangement and take sufficient time for erecting & dismantling.

A plate load test is not recommended in soils which are not homogeneous at least to a depth equal to 1.5 to 2 times the width of the prototype foundation.

plate load test should not be relied on to determine the ultimate bearing capacity of sandy soil as the scale effects give very misleading result. However, when the test are carried on clay soils, the ultimate bearing capacity as determined by the test may be taken as equal to that of the foundation since the bearing capacity of clay is essentially independent of the footing size.

Example:-

The result of the two plate load tests on dense sand at a depth of 1.5m where:-

Test 1:-settlement 7mm Test 2:-settlement 9mm

The plate were 0.3m square and the applied load was 20KN. Determine the size of the squre foundation and load it would carry at the same depth, if the limit of settlement is 25mm.

Solution:- The soil is dense sand(i.e cohesion less soil)

From the given data average settlement under plate is 8mm and Sf=25mm

25=8((B(0.3+0.3))/(0.3(B+0.3)))2

Solve the above relation we get;

B=2.2836m

B=2.3m

pressure under plate load=20/(0.3*0.3)=222KN/m2

Since it’s dense sand assume that

2. Standard Penetration Test (SPT) It is used to determine the allowable bearing capacity of cohesion less

coarse-grained soils [sands].

Step 1 Driven a standard split spoon sampler into the ground by dropping a hammer of mass 64 kg from 760 mm high.

The sampler is driven 150 mm into the soil at the bottom of a borehole.

Step 2 Then count the number of blows (N) required to drive it an additional 300 mm.

Step 3 Correct “N“ values obtained from SPT for various effects such as overburden pressure and energy transfer.


Standard Penetration Test (SPT)Commonly methods used in practice for correcting the N values.

Where [’z0 is the effective overburden pressure (kPa).

CN=Correction factor for overburden pressure

The groundwater correction factor is:

Where [“z” is the depth to the groundwater table, and

D & B are the footing depth and width.


2;8.95

'0

N

zN cc

kPa 24,2;1916

log77.0 '0'

010

zN

zN cc

)(22

1

BD

zcW

Nc

Standard Penetration Test (SPT)The corrected standard penetration number will be

Meyerhof (1956, 1974) proposed allowable bearing capacity qa equation as follows.

Where

Se is the elastic settlement of the layer in mm and

kd = 1 + 0.33D/B 1.33.


NccN WNcor

mBkNSq dea 22.125

12cor

mBkB

BNSq dea 22.1

305.0

25

82

cor

Se is the elastic settlement of the layer in mm and kd = 1 + 0.33D/B

1.33.

Standard Penetration Test (SPT)Bowles (1996) modified Meyerhof’s equations by 50%

increase in the allowable bearing capacity.

Bowles’s equations are:


Se is the elastic settlement of the layer in mm and kd = 1 + 0.33D/B

1.33.

mBkB

BNSq dea 22.1

305.0

25

5.122

cor

mBkNSq dea 22.125

20cor

The observed value is affected by the following factor Hammer efficiency bore hole diameter Sampler Length of the rod used

Example:- A 4m square foundation is to be constructed at a depth of

2m below ground level in a medium dense sand of unit weight 19KN/m3. the water table is located 4m below the ground level. The values obtained from a SPT were as follows

Making any appropriate corrections to allow for the effect of the overburden & the GWT, estimate the allowable load on the foundation if the immediate settlement is limited to 25mm.

Depth,Z (m) 2 2.8 3.6 4.4 5.2 6.0 6.8

N values 12 13 15 15 18 21 25

Solution:-

Depth,m N value U(KN/M2) CN N*CN

2 12 38 0 38 1.31 15.72

2.8 13 53.2 0 53.2 1.2 15.6

3.6 15 68.4 0 68.4 1.11 16.65

4.4 15 83.6 4 79.6 1.0715 16.07

5.2 18 98.8 12 86.8 1.034 18.612

6.0 21 114.0 20 94.0 1.008 21.168

6.8 25 129.2 28 101.2 0.98 24.5

=16.04

0z '0z

kPa 24,2;1916

log77.0 '0'

010

zN

zN cc

YwhpU

and

UhY zzz

*

,* 0'00

Cw=1/2 + 4/(2(2+4))=0.833 and

Ncor=16.04*0.833=13.3667

Finally; 4m>1.22m

Kd=1+0.33*2/4=1.165<=1.33, Kd=1.165

qa = 8/25(25*13.3667*((4+0.305)/4)2*1.165

=144.3KN/m2

)(22

1

BD

zcW

NccN WNcor

mBkB

BNSq dea 22.1

305.0

25

82

cor

Using Bowel’s equation we get

qa= 225.4688KN/m2

mBkB

BNSq dea 22.1

305.0

25

5.122

cor

Summary of chapter two Define ultimate bearing capacity

of soils under foundation structure.

What is factor of safety regarding bearing capacity of the soils?

Define and sketch failure modes of bearing capacity of soils.

Discuss the reason of failure in bearing capacity of foundations.

What are the factors affects ultimate bearing capacity?

Cont’d… List ultimate bearing capacity

equations and discuss their drawn backs, and comparisons among them.

Make modification for groundwater conditions on bearing capacity.

What is the advantage of field test method of determination of bearing capacity and list the methods used for bearing capacity determination with some sketch and procedure.

chapter -3

Lateral Earth pressure

Documents

Lecture-2 Bearing Capacity of Soil