Notes 2_Bearing Capacity(9)

Geotechnical Engineering 367 Dr Mohamed Shahin Curtin University Page 1

Bearing Capacity of Soil

(for Shallow Foundations)


Introduction Foundation is a structure that transmits loads to the underlying soils.

Shallow foundation is a foundation that has a ratio of embedment (or foundation) depth, Df , to footing breadth (or width), B, of less than or equal to 2.5, i.e. Df /B 2.5, otherwise it is a deep foundation. See Figure 1 for illustration of Df and B.

Shallow foundations are those comprised of pad footings (also called spread or isolated footings), strip footings, combined footings and rafts, whereas deep foundations are driven piles and drilled shafts.

The design of shallow foundations relies on the satisfactory fulfilment of the following two criteria: bearing capacity and settlement. The design requirement of bearing capacity ensures that there is an adequate protection against possible shear failure of the underlying soil, whereas the design requirement of settlement (total and differential) ensures that serviceability is accepted and structural damage is avoided. The bearing capacity will be covered in this semester, while settlement will be covered in Geotechnical Engineering 368.

Figure 1: An example of a shallow foundation

Df

B L


In order to design a shallow foundation for bearing capacity, the bearing pressure should be less than the bearing capacity by a factor of safety, usually between 2 to 3. The bearing pressure is the applied contact force per unit area along the bottom of the foundation, and the bearing capacity is the ultimate pressure at which the soil beneath the foundation fails in shear, or the maximum shear strength that the soil provides against external loads. There are two definitions for bearing pressure or bearing capacity (i.e. gross and net), and it is important to understand both definitions and decide which will be used for design.

Gross Bearing Pressure: The gross bearing pressure along the bottom of a shallow foundation is as follows:

where:

qapp-gross = gross bearing pressure;

Qcol = column load;

Wfooting = weight of footing;

Wsoil = weight of soil located immediately above the footing, if any;

A = base area of foundation; and

uD = footing uplift (buoyancy) at depth Df due to water pressure

Bearing Pressure versus Bearing Capacity

D

soilfootingcol

grossapp uA

WWQq

(1)


Virtually, all shallow foundations are made of reinforced concrete, so Wf is computed using a unit weight of concrete of 24 kN/m3.

The pore water pressure term, uD, accounts for the uplift pressures (buoyancy forces) that are present if a portion of the foundation is below the ground water table. If the ground water table is at a depth greater than Df, then uD = 0.

Net Bearing Pressure:

An alternative way to define bearing pressure is the net bearing pressure, qapp-net, which is the difference between the gross bearing pressure, qapp-gross, and the vertical effective overburden pressure, vo, at depth Df. In other words, it is the additional bearing pressure applied at the foundation level in excess of the existing own-weight of soil.

Qcol

Wfooting

Wsoil /2 Wsoil /2

Figure 2: Components of gross bearing pressure for a shallow foundation

Df


Gross Bearing Capacity: In order to compute the gross bearing capacity for shallow foundations, it is important to first study the different modes of failure beneath the foundations. The principal modes of failure may be defined as:

General shear failure (Figure 3a): This mode of failure is associated with dense cohesionless or stiff cohesive soils of

low compressibility.

Failure starts with a soil wedge underneath the footing, followed by a spiral slip surfaces which extend outward to the ground surface.

Failure is sudden and accompanied by a considerable bulging at the ground surface.

Local shear failure (Figure 3b): This failure mode occurs for footing resting on compressible soils of medium

compaction.

Failure starts with a soil wedge underneath the footing, followed by a spiral slip surfaces that do not extend to the ground surface.

Failure is not sudden and some bulging may occur.

Punching shear failure (Figure 3c): This failure mode occurs if the foundation is supported by a fairly loose

cohesionless or soft cohesive soils of high compressibility.

Only a soil wedge underneath the foundation is occurred and the failure surface does not extend to the ground surface.

Failure is accompanied by a considerable vertical movement and bulging at the ground surface is absent.


Figure 3: Principal modes of bearing capacity failure (a) General shear failure; (b)

local shear failure; (c) punching shear failure (modified after Das, 1963)

Settlement

Load

Failure

Settlement

Load

Failure

peak

General failure mode

Punching failure mode

Local failure mode

Settlement

Load

Failure

qult

qult

qult

qult

qult

qult


The gross bearing capacity, qult-gross, can be determined by considering one of the failure modes shown in Figure 3 and applying a limit equilibrium analysis to evaluate the stresses and strengths along the failure surface. For nearly all shallow foundations design problems, it is only necessary to check the general shear failure mode, and then conduct settlement analysis to verify that the foundation will not settle excessively. This settlement analysis implicitly protect against local and punching shear failures. Furthermore, assuming general shear failure mode is practical as in reality the ground conditions are always improved through compaction and soil stabilisation before placing the footings. In the sections that follow, we will study four different methods of calculating qult-gross.

Net Bearing Capacity: The net bearing capacity, qult-net, is the difference between the gross bearing

capacity, qult-gross, and the effective vertical overburden pressure, vo, at depth Df.

Both the gross and net bearing pressures and bearing capacities can be used for design of shallow foundations; however, in this course, we will use only the gross ones (i.e. qult-gross and qapp-gross ) as they are used by many codes. For simplicity, we will refer to qult-gross and qapp-gross as qult and qapp, respectively. It should be noted that for design of shallow foundations based on settlement, only the net bearing pressure should be used, as will be seen in Geotechnical Engineering 368.


For bearing capacity design, we want to see that the bearing pressure applied at the foundation level less than the ultimate bearing capacity by a factor of safety, thus, limiting the probability of failure. This factor of safety, FS, then can be calculated as follows:

For a safe design, FS obtained from Equation (2) should not be less than 2; however, if certain FS needs to be achieved (usually between 2 to 3), then the allowable bearing pressure, qall, is used as follows:

where: qall is the pressure that can be safely applied at the foundation level such that the shear failure is unlikely to occur. We then design the foundation so that the bearing pressure, qapp, does not exceed qall, i.e. qapp qall.

FS

qq ultall (3)

app

ult

q

qFS (2)


Terzaghis Method of Bearing Capacity

Terzaghi (1943) used the general failure mode of Figure 4 below and proposed the following equation for the bearing capacity of shallow foundations, qult:

Figure 4: General shear failure as proposed by Terzaghi (after Das, 1998)

where:

c = soil cohesion;

b = unit weight of soil below the foundation level;

B = footing breadth or width;

q = applied overburden vertical stress (surcharge) at the foundation level;

Nc, Nq & N = bearing capacity factors (solely dependent on soil friction angle )

sc & s = shape factors.

(4) sBNqNscNq bqccult 5.0

Cohesion

term

Surcharge

term Density

term


The bearing capacity factors Nc, Nq, N are given by the following equations:

The above equations are translated into a simple form in Table 1, in which Nc, Nq, N are obtained by knowing the friction angle . The shape factors sc & s can be obtained from Table 2.

1)2/45(cos2

cot2

tan)2/4/3(2

eNc

1)2/45(cos2 2

tan)2/4/3(2

eNq

tan1cos2

12

pKN

Table 1: Bearing capacity factors of Terzaghis method

Table 2: Shape factors of Terzaghis method

Footing Shape factor

Strip Square Circular Rectangular

Sc 1.0 1.3 1.3 )3.01(

L

B

S 1.0 0.8 0.6 )2.01(

L

B

It should be noted that Terzaghi equation for the ultimate bearing capacity did not consider the impact of the depth of foundation or inclined footing load.


Worked Example (1)

For the square footing shown in the figure below, use Terzaghis method to

determine the following (the water table is far below the ground surface):

(a) The magnitude of the allowable column load that can be applied for a factor

of safety FS = 3.

(b) If the footing is subjected to a column load of 450 kN, what would be the

factor of safety against bearing capacity failure.

[Answers: Qcol = 355 kN, FS = 2.4]

0.7 m c = 15 kPa

= 20o

= 18 kN/m3

0.3 m concrete = 25 kN/m3

0.25 0.25 m

1.5 1.5 m

Qcol


Meyerhof (1965) presented a general bearing capacity equation that takes into account the impact of the depth of foundation, Df, and load inclination to the vertical, , as follows:

idsBNidsqNidscNq bqqqqccccult 5.0

Nc, Nq, N are the bearing capacity factors and are given as:

2/45tan2tan eNq cot)1( qc NN )4.1tan()1( qNN

sc, sq, s = shape factors;

dc, dq, d = depth factors; and

ic, iq, i = load inclination factors.

Values of the bearing capacity factors do not have to be obtained using the

above equations and an alternative easy way is to use Table 3. Also, the

equations needed to obtain the values of the shape, depth and load

inclination factors are included in Table 4.

Meyerhofs Method of Bearing Capacity

(5)

Df

Q

P

T


o Nc Nq N

0 5.14 1.0 0.0

5 6.49 1.6 0.1

10 8.34 2.5 0.4

15 10.97 3.9 1.1

20 14.83 6.4 2.9

25 20.71 10.7 6.8

26 22.25 11.8 8.0

28 25.79 14.7 11.2

30 30.13 18.4 15.7

32 35.47 23.2 22.0

34 42.14 29.4 31.1

36 50.55 37.7 44.4

38 61.31 48.9 64.0

40 72.25 64.1 93.6

45 133.73 134.7 262.3

50 266.50 318.50 871.7

Table 3: Bearing capacity factors of Meyerhofs method

Table 4: Shape, depth and load inclination factors of Meyerhofs method

)2/45(tan2 N

For strip footings, B/L = 0.0

o = tan-1(T/P) (has no sign and measured from the vertical)

P = normal component of Q perpendicular to the footing base

T = shear component of Q parallel to the footing base

Factor Meyerhof

sc

L

BN2.01

sq

L

BN1.01 and 1 for = 0

s Same as sq

dc

B

DN

f

2.01

dq

B

DN

f

1.01 and 1 for = 0

d same as dq

ic

2

901

for any

iq same as ic for any

i

2

1

for > 0 and zero for = 0

T

Q P


Worked Example (2)

In the figure shown below, use Meyerhofs method to check the stability of the

footing against bearing capacity and sliding failures.

[Answers: FSbearing capacity = 1.23 and FSsliding = 0.5]

1.0 m c = 15 kPa

= 20o

soil = 18 kN/m3

1.5 1.5 m

Qapp-gross = 500 kN

35o


Hansens Method of Bearing Capacity

Hansen (1970) extended Meyerhofs work to include two additional factors to take care of the sloping of ground surface, , and tilted base, , as follows:

The bearing capacity factors Nc and Nq are the same as Meyerhof, but N is:

tan)1(5.1 qNN

sc, sq, s = shape factors;

dc, dq, d = depth factors;

ic, iq, i = load inclination factors;

gc, gq, g = ground slope factors; and

bc, bq, b = tilted base factors.

Values for the bearing capacity factors can be obtained directly from Table 5,

and equations to compute the values of the shape, depth, load inclination,

ground slope and tilted base factors are given in Table 6.

bgidsBNbgidsqNbgidscNq bqqqqqqccccccult 5.0 (6)

+

+ T + 90o

P

Df


Table 5: Bearing capacity factors of Hansens method

o Nc Nq N

0 5.14 1.0 0.0

5 6.49 1.6 0.1

10 8.34 2.5 0.4

15 10.97 3.9 1.2

20 14.83 6.4 2.9

25 20.71 10.7 6.8

26 22.25 11.8 7.9

28 25.79 14.7 10.9

30 30.13 18.4 15.1

32 35.47 23.2 20.8

34 42.14 29.4 28.7

36 50.55 37.7 40.0

38 61.31 48.9 56.1

40 72.25 64.1 79.4

45 133.73 134.7 200.5

50 266.50 318.50 567.4


Table 6: Shape, depth, load inclination, ground slope and tilted base factors of Hansens method

)2/45(tan2 N

tan/1cot



ca = adhesion on the base of footing

Af = contact area of footing

o = base inclination in degrees

= base inclination in radians, i.e.

180)(

rad

+

+

T + 90o

P

Df

Factor Hansen

sc

L

B

N

N

c

q1

tan1L

B

L

B4.01

sq

s

dc

B

D f4.01

B

D f2)sin1(tan21

1 for all

dq

d

ic 1

1

q

q

qN

ii for > 0 and

50

150

.

af cA

T.

for = 0

iq

5

cot

5.01

cAP

T

af

i

5

cot

7.01

cAP

T

af

gc

1471

gq

5)tan5.01(

g

same as gq

bc

1471

bq tan2e b tan7.2e


Worked Example (3)

Calculate the ultimate bearing capacity for the footing shown in the figure

below, using Hansens method. [Answer: qult = 358 kPa]

1.0 m

c = 15 kPa

= 20o

soil = 18 kN/m3 1.5 1.5 m

30o


Vesic (1963) used the same form of equation suggested by Hansen but he developed his own bearing capacity factor N as well as load inclination, ground slope and tilted base factors.

Values for the bearing capacity factors are given in Table 7 and equations

to obtain the values of the shape, depth, load inclination, ground slope and

tilted base factors are in Table 8.

Vesics Method of Bearing Capacity


Table 7: Bearing capacity factors of Vesics method

o Nc Nq N

0 5.14 1.0 0.0

5 6.49 1.6 0.4

10 8.34 2.5 1.2

15 10.97 3.9 2.6

20 14.83 6.4 5.4

25 20.71 10.7 10.9

26 22.25 11.8 12.5

28 25.79 14.7 16.7

30 30.13 18.4 22.4

32 35.47 23.2 30.2

34 42.14 29.4 41.0

36 50.55 37.7 56.2

38 61.31 48.9 77.9

40 72.25 64.1 109.4

45 133.73 134.7 271.3

50 266.50 318.50 762.84


Table 6: Shape, depth, load inclination, ground slope and tilted base factors of Vesics method

+

+

T + 90o

P

Df

L B

L B m m B

/ 1

/ 2 for T parallel to B

B L

B L m m L

/ 1

/ 2 for T parallel to L

)2/45(tan2 N

tan/1cot







180)(

rad

Factor Vesic

sc

same as Hansen sq

s

dc

same as Hansen

dq

d

ic 1

1

q

q

qN

ii for > 0 and

caf NcA

mT1 for = 0

iq

m

af cAP

T

cot1

i

1

cot1

m

af cAP

T

gc

1471

gq

2)tan1(

g

same as gq

bc )tan14.5(

21

bq 2)tan1(

b same as bq


Worked Example (4)

Calculate the ultimate bearing capacity for the footing shown in the figure below,

using Vesics method, and neglecting any applied shear to the footing base.

[Answer: qult = 481.5 kPa]

15o

1.0 m

c = 15 kPa

= 20o

soil = 18 kN/m3

1.4 m


Values of the bearing capacity factors for Meyerhof (M), Hansen (H) and Vesic (V)

Summary of Bearing Capacity Calculations

o Nc Nq N (M) N (H) N (V)

0 5.14 1.0 0.0 0.0 0.0

5 6.49 1.6 0.1 0.1 0.4

10 8.34 2.5 0.4 0.4 1.2

15 10.97 3.9 1.1 1.2 2.6

20 14.83 6.4 2.9 2.9 5.4

25 20.71 10.7 6.8 6.8 10.9

26 22.25 11.8 8.0 7.9 12.5

28 25.79 14.7 11.2 10.9 16.7

30 30.13 18.4 15.7 15.1 22.4

32 35.47 23.2 22.0 20.8 30.2

34 42.14 29.4 31.1 28.7 41.0

36 50.55 37.7 44.4 40.0 56.2

38 61.31 48.9 64.0 56.1 77.9

40 72.25 64.1 93.6 79.4 109.4

45 133.73 134.7 262.3 200.5 271.3

50 266.50 318.50 871.7 567.4 762.84


Factors of shape, depth of foundation, load inclination, ground slope and tilted base for Meyerhof, Hansen and Vesic

Factor Meyerhof Hansen Vesic

sc

L

BN2.01

L

B

N

N

c

q1

same as Hansen sq

L

BN1.01 and 1 for = 0

tan1L

B

s

Same as sq L

B4.01

dc

B

DN

f

2.01

B

D f4.01

same as Hansen

dq

B

DN

f

1.01 and 1 for = 0

B

D f2)sin1(tan21

d same as dq 1 for all

ic

2

901

for any

1

1

q

q

qN

ii for > 0 and

50

150

.

af cA

T.

for = 0 1

1

q

q

qN

ii for > 0 and

caf NcA

mT1 for = 0

iq

same as ic for any

5

cot

5.01

cAP

T

af

m

af cAP

T

cot1

i

2

1

for > 0 and zero for = 0

5

cot

7.01

cAP

T

af

1

cot1

m

af cAP

T

gc

N/A

1471

1471

gq

5)tan5.01( 2)tan1(

g

same as gq same as gq

bc

N/A

1471

)tan14.5(

21

bq tan2e 2)tan1(

b tan7.2e same as bq


L B

L B m m B

/ 1

/ 2 for T parallel to B

B L

B L m m L

/ 1

/ 2 for T parallel to L

)2/45(tan2 N

tan/1cot







180)(

rad

+

+ T + 90

o

P

Df


Which equation to use?

Bowles (1996) suggested the various equations be used in the following

situations:

Use Best for

Terzaghi Very cohesive soil where D f / B 1 or for a quick estimate of q ult to compare with other methods

Hansen, Meyerhof, Vesic Any situation which applies, depending on used preference or familiarity with a particular method.

Hansen, Ves ic When base is tilted; when ground surface is sloped ; or when D f / B = 1.


Bearing Capacity for Footings on Layered Soils

The bearing capacity equations examined thus far have treated the soil beneath the footing as being a single homogeneous deposit (i.e. c, and are constant

with depth). In some instances, the subsoil beneath the footing may be stratified

into layers. One way of dealing with such situation is to use the weighted

average values of c, and based on the relative thicknesses of each stratum in

the zone between the bottom of the footing and a depth B below the bottom. The

weighted average values can be calculated as follows (see Figure 6):

ni

i

i

ni

i

ii

av

H

Hc

c

1

1

ni

i

i

ni

i

ii

av

H

H

1

1

ni

i

i

ni

i

ii

av

H

H

1

1

where:

ci = cohesion of layer i;

i = friction angle of layer i;

i = unit weight of soil for layer i;

Hi = thickness of layer i;

Hi = effective depth beneath the footing and is limited to B, as a mximum.

(7)


B

H1

H2

Hn

H B

c1, 1, 1

c2, 2, 2

cn, n, n

Figure 6: Footing on layered soils

GS


Effective versus Total Stresses for Bearing Capacity

If the water table is far below the foundation level, the bearing capacity should be computed using the effective shear strength parameters c and , and the

bulk unit weight of soil, . However, if the water table is located close to the

foundation, the shear strength parameters will be in accordance with whether

the design should be based on effective stress analysis (drained condition) or

total stress analysis (undrained condition). Also, some modifications are

necessary in the bearing capacity equations for the surcharge load q and unit

weight of soil, depending on the depth of water table below the ground surface.

Effective Stress Analysis: This analysis considers the drained condition where the excess pore water pressures, if any, that may be created due to loading has

time to dissipate. In this case, in the various bearing capacity equations, use the

effective shear strength parameters c and , and the values of q and b should

be computed taking into account the level of water table as per the following

three cases (see Figure 5).

Case I: the water table is located so that (0 D1 Df). The surcharge pressure q in the bearing capacity equations will take the following form:

2121221 )( DDDDDDDq wsatwsat


where:

= bulk unit weight of soil above the water table;

sat = saturated unit weight of soil below the water table;

= effective unit weight of soil; and

w = unit weight of water.

Also, the parameter b , in the last term ( b BN ) is replaced by .

Case II: the water table is located above the ground surface by a height hw. The surcharge pressure q in the bearing capacity equations will be:

Also, the parameter b , in the last term ( b BN ) is replaced by .

Case III: the water table is located below the base of footing so that (0 d B). The surcharge load q in the bearing capacity equation will take the form:

Also, the parameter b , in the last term ( b BN ) is replaced by the factor:

fDq

)'(' B

d

ffwsatfwwfsatww DDDhDhq ')()(


Figure 5: The three ground water cases that influence bearing capacity

Total Stress Analysis: This analysis considers the undrained condition where excess pore water pressures develop due to loading and will not have enough

time to dissipate. In this case, in the various bearing capacity equations, use the

undrained strength parameters cu (or su) and u = 0, and in calculation of the

surcharge load q, use the bulk unit weight above the water table and saturated

unit weight sat below the water table, depending on the location of the water

table between the ground surface and foundation level. It should be noted that

in this case, N = 0, thus the last term ( b BN ) is zero.

fDq

Df

B

D2

D1 WT

GS

Case I

sat

Case III

Df

B d

WT

GS

sat fDq

Df

B

WT

Case II

sat

GS hw


Worked Example (5)

Refer to Example (1) and use Terzaghis method to calculate the allowable column

load if the water table is located at:

(a) the ground surface;

(b) 0.5 m below the ground surface; and

(c) 2.0 m below the ground surface.

Use a saturated unit weight of soil, sat = 20 kN/m3, and comment on the results

[Answers: 313 kN, 325 kN, 349 kN]


Footings with Eccentric Loading

Most foundations are built so that the vertical load acts through the centroid, thus producing a fairly uniform distribution of bearing pressure underneath the

foundation. However, sometimes it becomes necessary to accommodate eccentric

loading which results from loads applied somewhere other than the footings

centroid or from applied moments (see Figure 7), such as those resulting at the

footings of a tall building from wind loads or earthquakes.

Eccentric loadings produce a non-uniform bearing pressure distribution underneath footings, and also change the bearing capacity calculations. When a

footing is subjected to an eccentric load, it tilts towards the side of the eccentricity

and the bearing pressure increases on that side and decreases on the opposite side.

When the vertical applied load reaches its ultimate value, there will be a failure of

the supporting soil on the side of eccentricity. It is then important to know how

bearing pressure and bearing capacity change with eccentric loading and how

they can be determined.

In Figure 7(a), the eccentricity of the bearing pressure is equal to:

(8) AuWWQ

M

Q

Me

Dsoilfcolapp


In Figure 7b, the eccentricity is equal to:

where:

e = eccentricity of bearing pressure distribution;

e1 = eccentricity of the column load;

M = applied moment at the footing centroid;

Qapp = applied force at the footing centroid;

Qcol = column load;

Wf = footing load;

Wsoil = weight of soli above footing;

uD = footing uplift (buoyancy) due to water pressure at depth Df; and

A = base area of foundation.

Figure 7: Examples of footings with eccentricity

GS

M

(a)

Qapp

GS

M

(b)

Qcol Qapp

e1

AuWWQ

eQ

Q

Me

Dsoilfcol

col

app 1 (9)


Bearing Pressure for Eccentric Foudations

One-Way Eccentricity:

For a footing with one-way eccentricity in the B direction, i.e. eB, the possible

bearing pressure distributions will be as shown in Figure 8. If eB < B/6, the

bearing pressure distribution is trapezoidal and the minimum and maximum

bearing pressure, qapp-min and qapp-max are as follows:

if eB = B/6, then qapp-min = 0 and the bearing pressure distribution is triangular, as

shown in Figure 8. Therefore, so long as eB B/6, there will be some bearing

pressure contact along the entire base area of the foundation. However, if

eB > B/6, part of the bearing pressure will be in tension and one side of

the foundation will lift off the ground, which should not be allowed.

If the eccentricity is in the L direction, substitute L by B and eL for eB in Eqns

(11) and (12), respectively.

)6

1)((minB

eu

A

WWQq BD

soilfcol

app

(11)

)6

1)((maxB

eu

A

WWQq BD

soilfcol

app

(12)


Figure 8: Possible bearing pressure distributions for a footing with one-way eccentricity in B direction

B

GS

Qapp

ML

qapp-max

qapp-min

qapp-max

qapp-max

qapp-min

qapp-min

L

B

Qapp

ML

B

GS

Qapp

ML


Two-Way Eccentricity:

In Figure 9 below, the footing is applied to two-way eccentricity and the

minimum and maximum bearing pressures are as follows:

where: eB and eL are the eccentricities in the B and L directions , respectively,

and are obtained as follows: eB = ML /Qapp and eL = MB /Qapp.

Figure 9: Footing with two-way eccentricity

B

L

eB

eL

ML

MB

Qapp

)66

1)((minL

e

B

eu

A

WWQq LBD

soilfcol

app

(13)

)66

1)((maxL

e

B

eu

A

WWQq LBD

soilfcol

app

(14)


Bearing Capacity for Eccentric Foundations Meyerhof (1963) suggested a method that is generally referred to as the

effective area method to obtain the bearing capacity of footings subjected to

eccentricity. The design of footing subjected to eccentricity is the same as

described earlier using any of the ultimate bearing capacity methods but with

some modifications to consider for the effective dimensions of footing, as

explained below.

One-Way Eccentricity:

For footings with one way eccentricity, the effective footing dimensions are as

follows (see Figure 10):

B = effective width = B 2eB (for eccentricity in B direction)

or

L = effective length = L 2eL (for eccentricity in L direction)

In the bearing capacity equations, to obtain the shape factors (sc, sq and s), use the effective width B or effective length L instead of B or L. Also, use B in the

last term of the bearing capacity equations ( bBN ). However, to determine

the depth factors (dc, dq and d), use B & L and do not replace them by B & L.


Figure 10: Effective dimensions for footings with one-way eccentricity in B and L directions

B = B 2eB

B

B

L

eB

L = L 2eL

B

L

eL

L

It should be noted that if eccentricity was in the direction of the footing length, as the case in the left of Figure 8, the effective length L would be equal to L

2eL and the effective width would be equal to B. L and B should be compared

and the smaller of the two dimensions should be used as the effective width of

footing and the other dimension should be used as the effective length.


L

BBLBBB

2

)(

2

)( 111'

(15)

where, B1 and L1 can be determined from charts established by Higher and

Andres (1985) and reproduced in Figures 12 and 13, respectively. The effective

length is still the original length L.

Two-Way Eccentricity:

When a foundation is subjected to two-way eccentricity, its effective area (i.e.

the area affected by the load) will be similar to the dashed area depicted in

Figure 11. In this case, the effective width of foundation B will be such

that:

As with the one way eccentricity, in the bearing capacity equations, use the effective width B instead of B to obtain the shape factors (sc, sq and s). Also,

use B in the last term of the bearing capacity equation ( bBN ). However,

to determine the depth factors (dc, dq and d), use the original width B and do

not replace it by B.


Figure 11: Effective area related to a two-way eccentricity

B1

B

L

L1

eB

eL

Figure 12: Higher and Andres charts for the determination of B1

Figure 13: Higher and Andres charts for the determination of L1


For footings with eccentricity, the bearing capacity, qult, can then be compared with the maximum bearing pressure, qapp-max, and the factor of

safety can then be obtained as follows:

max

app

ult

q

qFS (16)


Worked Example (6)

Determine the factor of safety against the bearing capacity failure using Meyerhofs method for the footing shown in the figure below. The water

table is far below the foundation level, the depth of foundation is 1.5 m from

the ground surface and the soil has the following properties: = 18 kN/m3,

c = 15 kPa and = 20o. [Answer: FS = 4.3]

3 m

2 m

Qapp = 450 kN

MB= 90 kN.m

ML= 67.5 kN.m


Use of SPT for Allowable Bearing Pressure

Due to the difficulty in obtaining undisturbed samples for testing in the laboratory, especially for granular soils, many foundation design methods have

focussed on correlations with the in-situ tests such as the standard penetration test

(SPT), cone penetration test (CPT), dilatometer test, pressuremeter test and plate

load test. The methods that use the SPT results are popular and widely used,

thus, two of the available SPT methods will be covered in our course, including

the Meyerhofs method (1974) and Terzaghi & Pecks method (1948). These

methods rely on the calculation of the allowable bearing pressure for a maximum

settlement of 25 mm, and thus do not rely on the calculation of the ultimate

bearing capacity and its factor of safety, as will be described later.

Before talking about the SPT design methods, a description of the SPT procedure is first given according to the Australian Standards, as follows:

1. A vertical hole of at least 65 mm diameter is drilled to the depth at which the test

is to be conducted.

2. A split spoon sampler (Figure 14a) is inserted into the hole via steel rods.

3. A (63.5 1) kg hammer, as shown in Figure 14b, is raised a distance of 760 15

mm using a self-tripping mechanism and allowed to fall freely due to lifting

winch inertia.


Figure 14: Standard penetration test: (a) spoon sampler; (b) test procedure

(a)

(b)


4. The process is repeated until the sampler penetrates the soil for a total

distance of 450 mm.

5. The number of hammer blows required for each 150 mm interval is recorded.

6. The blow counts for the last 300 mm of penetration are summed and the

number of blows of the standard penetration test (N) is computed, noting that

the blow counts for the first 150 mm are not used for computing N, as this soil

is assumed to be disturbed by the drilling process.

7. The process is repeated at another depth and so on, as required.

The blow counts obtained from the SPT, as measured previously, may be affected by the overburden pressure and the ground water table. Peck et al.

(1974) presented the following equation to correct N for overburden pressure:

where, q = effective overburden pressure in (kPa) at the depth where N is

measured.

The corrected N value is then can be calculated as:

; CN 2

'

2000log77.0

qCN (17)

NCN Ncorrected (18)


Meyerhofs method does not require correction of measured SPT blow counts for water table, as he suggested that the influence of the water table would be

implicitly incorporated in the measured SPT results.

The method, however, requires correction for overburden using Pecks method described earlier. Meyerhof (1974) provided relationships for the calculation of

the allowable bearing pressure using the average blow counts corrected for

overburden pressure, Ncorrected-average, within an influence zone equal to the

breadth of footing, B, below the depth of foundation, as follows:

d

averagecorrectede

all kNS

q25

12

(for B < 1.22 m)

Meyerhofs SPT Method of Bearing Pressure

d

averagecorrectede

all kB

BNSq

2305.0

25

8

(for B > 1.22 m)

where:

Se = elastic settlement of footing in (mm)

)/33.01( BDk fd

Df = depth of foundation in (m); and B = breadth of footing in (m)

(19)

(20)


Terzaghi and Peck (1948) proposed the determination of the allowable bearing pressure of a footing having a width B using the average blow counts

corrected for overburden pressure, Ncorrected-average, and the chart shown in

Figure 15. When using the chart, Ncorrected-average is obtained within an

influence zone equal to B below the depth of foundation and the settlement is

assumed to be 25 mm.

Figure 15: Allowable bearing pressure from the SPT (after Terzaghi and Peck, 1948)

Terzaghi & Pecks SPT Method of Bearing Pressure


The method also proposed a correction factor for the allowable bearing pressure obtained from Figure 15 to account for the effect of water table, as follows:

where,

Cw = correction factor for water table;

Dw = depth of water table below ground surface;

Df = depth of footing embedment; and

B = width of footing.

Thus, the allowable bearing pressure after correction for water table becomes:

BD

DC

f

ww 1

2

1

; Cw 1 (21)

)15(Figureallwall qCq (22)


Worked Example (7)

The SPT results at various depths in a soil are shown in the table below. Determine the allowable column load for a square footing 2 m wide located at

0.5 m below the ground surface. The thickness of footing is 0.5 m and the

unit weight of concrete is 25 kN/m3. The tolerable settlement is 25 mm and

the ground water table is at 1.0 m below the ground surface. The unit weights

of soil are = 19 kN/m3 and sat = 22 kN/m

3. Use both Meyerhofs method, as

well as Terzaghi and Pecks method. [Answers: 1974 kN and 1390 kN]

Depth (m) 0.6 1.2 2.0 3.0 4.2

SPT-N 25 33 28 31 41


References:

Bowles, J. E. (1996). Foundation analysis and design, McGraw-Hill, N.Y. Das, B. (1998). Principles of geotechnical engineering, PWS Publishing

Company, Boston, MA.

Hansen, J. B. (1970). A revised and extended formula for bearing capacity Danish Geotechnical Institute, Copenham, Bulletin (28), 5-11.

Highter, W. H., and Andres, J. C. (1985). Dimensioning footings subjected to eccentric loads Journal of Geotechncial Engineering, ASCE, 11(GT5), 659-665.

Meyerhof, G. G. (1963). Shallow foundations Journal of Soil Mechanics and Foundation Division, ASCE, 91(SM2), 21-31.

Meyerhof, G. G. (1974). General report: outside Europe Proceedings of the 1st European Symposium on Penetration Testing, Stockholm, 40-48.

Peck, R. B., Hanson, W. E., and Thornburn, T. H. (1974). Foundation engineering, Wiley, N.Y.

Terzaghi, K. (1943). Theoretical soil mechanics, Wiley & Sons, N.Y. Terzaghi, K., and Peck, R. (1948). Soil Mechanics in engineering practice,

Chapman and Hall, London, John Wiley, N.Y.

Vesic, A. S. (1963). Bearing capacity of deep foundation in sand Highway Research Record, No. 39, Washington D.C.

Vesic, A. S. (1973). Analysis of ultimate loads of shallow foundations Journal of Soil Mechanics and Foundation Engineering Division, ASCE, 99(SM1), 45-73.

Notes 2_Bearing Capacity(9)

Documents

net bearing pressure

additional bearing pressure

bearing capacity of

gross bearing pressure

ultimate pressure

foundation depth

shallow foundation df

introduction foundation