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Addis Ababa University, Faculty of Technology, Department of Civil Engineering

Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 20

CHAPTER TWO BEARING CAPACITY OF SHALLOW FOUNDATIONS Table of Contents

2.0 Introduction ........................................................................................ 21

2.1 Bearing Failure Modes .......................................................................... 21

2.2 Ultimate Bearing Capacity Equations ...................................................... 22

2.2.1 Terzaghis Bearing Capacity equation .................................................. 22 2.2.2 Meyerhofs Bearing Capacity equation ................................................. 24 2.2.3 Hansens Bearing Capacity Equation.................................................... 25 2.2.4 A comparative summary of the three bearing capacity equations ............ 28

2.2.5 Effects of Groundwater Table on Bearing Capacity ................................ 29

2.2.6 Allowable bearing capacity and factor of safety .................................... 30

2.2.7 Eccentric Loads ................................................................................ 31

2.3 Field Tests .......................................................................................... 32

2.3.1 Plate Loading Test ............................................................................ 32

2.3.2 Standard Penetration Test (SPT) ......................................................... 33



2.0 Introduction

A foundation, often constructed from concrete, steel or wood, 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,

shallow and deep foundations. Shallow foundations are comprised of footings, while

deep foundations include piles that are used when the soil near the ground surface has

no enough strength to stand the applied loading. The ultimate bearing capacity, qu,

(in kPa) is the load that causes the shear failure of the soil underneath and adjacent

to the footing. In this chapter, we will discuss equations used to estimate the ultimate

bearing capacity of soils. When you complete this chapter you should be able to:

Calculate the bearing capacity of soils.

2.1 Bearing Failure Modes

Figure 2.1: Modes of bearing failures (a) General shear (b) Local shear and (c)

Punching shear.

Relative density of the soil and size of the foundation are among the major

factors that affect the mode of bearing failure likely to occur. The modes of bearing

failure are generally separated into three categories: The general shear failure (Fig.



1.1 a) is usually associated with soils of low compressibility such as dense sand and

stiff cohesive soils. In this case, if load is gradually applied to the foundation,

settlement will increase. At a certain point when the applied load per unit area equals to the ultimate load qu a sudden failure in the soil supporting the foundation will take place. The failure surface in the soil will extend to the ground surface and full

shear resistance of the soil is developed along the failure surface. Bulging of the soil

near the footing is usually apparent.

For the local shear failure (Fig. 1.1 b), which is common in sands and clays of

medium compaction, the failure surface will gradually extend outward from the

foundation but will not reach the ground surface as shown by the solid segment in Fig.

1.1 b. The shear resistance is fully developed over only part of the failure surface

(solid segment of the line). There is a certain degree of bulging of the soil.

In the case of punching shear failure, a condition common in loose and very

compressible soils, considerable vertical settlement may take place with the failure

surfaces restricted to vertical planes immediately adjacent to the sides of the

foundation; the ground surface may be dragged down. After the first yield has

occurred the load-settlement curve will be steep slightly, but remain fairly flat.

2.2 Ultimate Bearing Capacity Equations

2.2.1 Terzaghis Bearing Capacity equation

Many of the present day principles regarding bearing capacity equations appear

to have had their origin on a failure mechanism proposed by Prandtl in the early 1920s

(refer literature for Prandtls failure mechanism). Prandtl developed a bearing capacity

Figure 2.2: Failure mechanism for Terzhagis bearing capacity solution.

equation assuming a smooth (frictionless) footing and ignoring the weight of the soil

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



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, f soil for Terzhagis bearing capacity solution is shown in Fig. 2.2. Terzhagis ultimate bearing capacity equations are given as follows:

Strip (or long) footing: ggg NBDNNcq qcu 5.0' ++= (2.1)

Square footing: ggg NBDNNcq qcu 4.0'3.1 ++= (2.2)

Circular footing: ggg NBDNNcq qcu 3.0'3.1 ++= (2.3)

where Nc, Nq and Ng are called the bearing capacity factors and are obtained as

follows:

)2/'45(cos2 2'tan)'2/3(

f

ffp

+=

-eN q , )1('cot -= qc NN f ,

-= 1

'cos'tan 22

1

ff gg

pKN (2.4)

Figure 2.3: Terzhagis bearing capacity coefficients.

Figure 2.3 shows the variation of the bearing capacity factors provided by Terzhagi.

Based on this figure, Aysen (2002) proposed the following equation to obtain the



value of Kpg in the Ng equation:

)2/'60(tan)8.3'4'8( 022 fffg ++-=pK (2.5)

where 'f in the first term is in radians. In the undrained conditions (cu and 0=uf ):

1=qN , 71.5)1( 23 =+= pcN , 0=gN (2.6)

2.2.2 Meyerhofs Bearing Capacity equation

Meyerhof (1951) developed a bearing capacity equation by extending Terzhagis failure mechanism and taking into account the effects of footing shape, load

inclination and footing depth by adding the corresponding factors of s, d, and i. For a

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

gggggg disNBdisDNdisNcq qqqqccccu 5.0' ++= (2.7)

For vertical load, ic = iq = ig = 1

ggggg dsNBdsDNdsNcq qqqcccu 5.0' ++= (2.8)

Figure 2.4: Meyerhofs bearing capacity coefficients. The bearing capacity factors:



)2/'45(tan)'tanexp( 2 ffp +=qN , )1('cot -= qc NN f , )'4.1tan()1( fg -= qNN (2.9)

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

1=qN , 14.5)2( =+= pcN , 0=gN

The bearing capacity factors are graphically presented in Fig. 2.4. The shape,

inclination and depth factors are according to: Shape Depth Inclination

Any 'f LBKs pc 2.01+= B

DKd pc 2.01+= 2

0

0

901

-==

aqc ii

For 0'=f sq = s g= 1 dq = d g= 1 i g= 0

For 010'f LBKss pq 1.01+== g B

DKdd pq 1.01+== g 2

0

0

'1

-=

fa

gi

+=

2'45tan 2 fpK , a =angle of resultant measured from vertical axis.

when triaxial 'f is used for plane strain, adjust 'f to obtain 'triaxialff

-=

LB1.01.1'

For the eccentric load, the length and width of the footing rectangle are modified to:

L = L 2eL and B = B 2eB (2.9) where eL and eB represent the eccentricity along the appropriate directions.

2.2.3 Hansens Bearing Capacity Equation

Hansen (1961) extended Meyerhofs solutions by considering the effects of sloping ground surface and tilted base (Fig. 2.5) as well as modification of Ng and other

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

and load:

gggggggg gbidsNBgbidsDNgbidsNcq qqqqqqccccccu 5.0' ++= (2.10)

Equation 2.9 is sometimes referred to as the general bearing capacity equation. In the

special case of a horizontal ground surface,

ggggggg bidsNBbidsDNbidsNcq qqqqqcccccu 5.0' ++= (2.11)



Figure 2.5: Identification of items in Hansens bearing capacity equation. Figure 2.6 provides the relationships between Nc, Nq, and Ng and the 'f values, as proposed by Hansen.

Figure 2.6: Hansens bearing capacity coefficients. The bearing capacity factors Nc and Nq are identical with Meyerhofs factors. Ng is defined by:



fg tan)1(5.1 -= qNN (2.12)

Since failure can take place either along the long side or along the short side, Hansen

proposed two sets of shape, inclination and depth factors.

The shape factors are:

Bcc

qBc iL

BNN

s ,, 1 += , 'sin1 ,, f+= BqBq iLBs , 6.04.01

,

,, -=

B

LB i

iLBs

g

gg (2.13)

Lcc

qLc iB

LNN

s ,, 1 += , 'sin1 ,, f+= LqLq iBLs , 6.04.01 ,, -= LL iB

Ls gg (2.14)

For cu, fu=0 soil: BcBc iLBs ,, 2.0= , LcLc iB

Ls ,, 2.0= (2.15)

The inclination factors are:

11 ,

,, --

-=q

iqiqic N

iii ,

1

'cot5.01,

a

f

+

-=b

iiq AcV

Hi , 2

'cot7.01,

a

g f

+

-=b

ii AcV

Hi (2.16)

where the suffix i (in Eqn. 2.15) stands for B or L. 52 1 a . 52 2 a . A is the area of the footing base and cb is the cohesion mobilized in the footing-soil contact area. For

the tilted base:

2

'cot)4507.0(1

00

,

a

g fh

+-

-=b

ii AcV

Hi (2.17)

For cu, fu=0 soil: biic AcHi --= 15.05.0, (2.18)

In the above equations, B and L may be replaced by their effective values (B and L) expressed by Eqn. (2.9).

The depth factors are expressed in two sets:

For D/B 1 & D/L 1:

BDd Bc += 4.01, , B

Dd Bq -+= 2, )'sin1('tan21 ff (2.19)

LDd Lc += 4.01, , L

Dd Lq -+= 2, )'sin1('tan21 ff (2.20)

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



( )BDd Bc 1, tan4.01 -+= , )(tan)'sin1('tan21 12, BDd Bq --+= ff (2.21)

( )LDd Lc 1, tan4.01 -+= , )(tan)'sin1('tan21 12, LDd Lq --+= ff (2.22) For both sets: 1=gd (2.23)

For cu, fu soil: BDd Bc = 4.0, , L

Dd Lc = 4.0, (2.24)

For the sloping ground and tilted base, the ground factors gi and base factors bi are

proposed by the following equations. The angles b and h are at the same plane, either

parallel to B or L.

0

0

1471b-=cg , ( )5tan5.01 bg -== gg q (2.25)

For cu, fu soil: 00

147b=cg (2.26)

0

0

1471h-=cb , 'tan2 fh-= ebq , 'tan7.2 fhg -= eb (2.27)

For cu, fu soil: 00

147h=cb (2.28)

2.2.4 A comparative summary of the three bearing capacity equations

Terzaghis equations were and are still widely used, perhaps because they are somewhat simpler than Meyerhofs and Hansens. Practitioners use Terzaghis equations for a very cohesive soil and D/B < 1. However, Terzaghis equations have the following major drawbacks:

Shape, depth and inclination factors are not considered. Terzaghis equations are suitable for a concentrically loaded horizontal

footing but are not suitable for eccentrically (for example, columns with

moment or titled forces) loaded footings that are very common in practice.

The equations are generally conservative than Meyerhofs and Hansens.

Currently, Meyerhofs and Hansens equations are more widely used than Terzaghis. Both are viewed as somewhat less conservative and applicable to more general conditions. Hansens is, however, used when the base is tilted or when the footing is on a slope and for D/B > 1.



EXAMPLE 2.1

Given the data in Fig. E2.1, determine the ultimate bearing capacity qu using:

a)Terzaghis, b) Meyerhofs and c) Hansens bearing capacity equations.

Figure E2.1: An isolated footing.

EAMPLE 2.2

Determine the ultimate bearing capacity of a square footing 1.5 m, at a depth of 1 m

in a soil c = 10 kPa, 'f =280, cu = 105 kPa, uf =0 andg = 19 kN/m3. Use Terzaghis, Meyerhofs and Hansens bearing capacity equations. Strategy It is a good policy to sketch a diagram illustrating the conditions given.

EAMPLE 2.3

A square footing 1.5 m is to be constructed in sand with c = 0, 'f =400. The thickness of the footing is 0.45 m and its top surface is level with the horizontal ground surface.

The footing is subjected to a central vertical force of 700 kN and a central horizontal

force (parallel to the sides) of 210 kN. Find the ultimate bearing capacity by a)

Meyerhofs and b) Hansens equations. (Note that Terzaghis equations are not applicable for inclined loads). The unit weight of the sand is 18 kN/m3.

2.2.5 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 Dg in the bearing capacity equations refers to the vertical stress of the soil above the base of the footing. The last term Bg refers to the vertical stress of a soil mass of thickness B, below the base of the footing. You

need to 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.



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

Bg is )(' zBz -+gg or )(' zBzsat -+gg . The later equation is used if the soil above the groundwater level is also saturated. The term Dg 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, then the term Dg is )(' zDz -+gg or )(' zDzsat -+gg . The latter equation is used if the soil above the

groundwater level is also saturated. The term Bg becomes B'g .

Figure E2.7: Groundwater within a) a depth B below base, b) embedment depth.

EAMPLE 2.4

Re-do example 2.3 assuming that the groundwater level is at the footing level (0.45 m

below the ground surface). The saturated unit weight is 21 kN/m3.

2.2.6 Allowable bearing capacity and factor of safety

The allowable bearing capacity, qa is calculated by dividing the ultimate bearing

capacity by a factor, called the factor of safety, FS. The FS is intended to compensate

for assumptions made in developing the bearing capacity equations, soil variability,

inaccurate soil data, and uncertainties of loads. The magnitude of FS applied to the

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

FSqq ua = (2.29)

Alternatively, if the maximum applied foundation stress max)( as is known and the dimension of the footing is also known then you can find a factor of safety by replacing

qa by max)( as in Eqn. (2.29):

max)( auqFS

s= (2.30)



2.2.7 Eccentric Loads

Meyerhof (1963) proposed an approximate method for loads that are located

off-centered (or eccentric loads).

Figure A1

He proposed that for a rectangular footing of width B and length L, the base area

should be modified with the following dimensions:

B = B 2eB and L =L - 2eL (2.31) Where B and L are the modified width and length, eB and eL are the eccentricities in the directions of the width and length, respectively. From your course in mechanics

you should recall that

PM

e yB = and PMe xL = (2.32)

where P is the vertical load, and My and Mx are the moments about the y and x axes,

respectively, as shown in Fig. A1.

The maximum and minimum vertical stresses along the x axis are:

+=

Be

BLP B61maxs and

-=

Be

BLP B61mins (2.33)

and along the y axis are:

+=

Be

BLP L61maxs and

-=

Be

BLP L61mins (2.34)

Since the tensile strength of soils is approximately zero, mins should always be

greater than zero. Therefore, eB & eL should always be less than B/6 & L/6, respectively.

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



For the case where there are moments about both x and y directions:

++=

Le

Be

BLP LB 661maxs and

--=

Le

Be

BLP LB 661mins (2.35)

EXAMPLE 2.5

A footing 2 m square is located at a depth of 1 m below the ground surface in a deep

deposit of compacted sand, 'f =300, c=0, and satg =18 kN/m3. The footing is subjected to a vertical load of 500 kN and a moment about the Y-axis of 125 kNm. The ground water table is 5 m below the ground surface. Use Meyerhofs bearing capacity equation and calculate the factor of safety. Assume the soil above the ground

water is also saturated.

2.3 Field Tests

Often, it is difficult to obtain undisturbed samples of especially coarse-grained

soils for laboratory testing and one has to use results from field tests to determine the

bearing capacity of shallow foundations. Some of the most common methods used for

field tests are briefly described below.

2.3.1 Plate Loading Test

Tests on full sized footings are desirable but expensive. The alternative is to

carry out plate loading tests. The plate loading test is carried out to estimate the

bearing capacity of single footings. The plates that are used in the field are usually

made of steel and are 25 mm thick and 150 mm to 762 mm in diameter. A circular

plate of 300 mm is commonly used in practice. Occasionally, square plates that are

300 mm300 mm are also used.

To conduct a plate load test, a hole is excavated (Fig. 2.8) with a minimum

diameter 4BP (BP = diameter of the test plate) to a depth of D (D = depth of the

proposed foundation). The plate is placed at the center of the hole. Load is applied to

the plate in increments of 10% to 20% of the estimated ultimate load. Each load

increment is held until settlement ceases. The final settlement at the end of each

loading increment is recorded. The test should be conducted until the soil fails, or at

least until the plate has gone through 25 mm of settlement.



Figure 2.8: Plate Loading Test

For tests in clay,

)()( PuFu qq = (2.36)

where qu(F) & qu(P) are ultimate bearing capacity of foundation and plate, respectively.

Eqn. (2.31) implies that the bearing capacity in clays is independent of plate size.

For tests in sandy soil,

p

FPuFu B

Bqq )()( = (2.37)

where BF and BP stand for width of foundation and plate, respectively.

There are several problems associated with the plate load test. The test is

reliable if the soil layer is thick and homogeneous, local conditions such as a pocket of

weak soil near the surface of plate can affect the test results but these may have no

significant effect on the real footing, the correlation between plate load results and

real footing is problematic, and performance of the test is generally difficult.

2.3.2 Standard Penetration Test (SPT)

The Standard Penetration Test (SPT) is used to determine the allowable bearing

capacity of cohesionless coarse-grained soils such as sands. The test procedure for

SPT has been introduced in Chapter 1. The N values obtained from SPT are usually

corrected for various effects such as overburden pressure and energy transfer. The

following are two of the most commonly used methods in practice for correcting the N

values.

2;8.95'0

= N

zN cc s

(Liao and Whitman, 1985) (2.38)

kPa 24,2;1916log77.0 ' 0'0

10 >

= zN

zN cc ss

(Peck et al., 1974) (2.39)

where cN is a correction factor for overburden pressure, and '0zs is the effective



overburden pressure in kPa. A further correction factor is imposed on N values if the

groundwater level is within a depth B below the base of the footing. The groundwater

correction factor is:

)(221

BDzcW +

+= (2.40)

where z is the depth to the groundwater table, and D and B are the footing depth and

width. If the depth of the groundwater table is beyond B from the footing base cW = 1.

The corrected N value is:

NccN WN=cor Meyerhof (1956, 1974) proposed the following equations to determine the allowable

bearing capacity qa from SPT values.

dea kNSq cor2512

= B 1.22 m (2.41)

dea kBBNSq

2305.0258

+= cor B > 1.22 m (2.42)

where Se is the elastic settlement of the layer in mm and kd = 1 + 0.33D/B 1.33. In practice, each value of N is a soil layer up to a depth B below the footing base is

corrected and an average value of Ncor is used in Eqn. (2.37).

Bowles (1996) modified Meyerhofs equations by 50% increase in the allowable bearing capacity. Bowless equations are:

dea kNSq cor2520

= B 1.22 m (2.43)

dea kBBNSq

2305.025

5.12

+= cor B > 1.22 m (2.44)

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