CHAPTER TWO BEARING CAPACITY OF SHALLOW · PDF fileMeyerhof (1951) developed a bearing capacity equation by extending Terzhagi’s failure mechanism and taking into account the effects

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 Terzaghi’s Bearing Capacity equation ................................................... 22

2.2.2 Meyerhof’s Bearing Capacity equation .................................................. 24

2.2.3 Hansen’s Bearing Capacity Equation .................................................... 25

2.2.4 A comparative summary of the three bearing capacity equations ............ 28

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

2.2.6 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 Terzaghi’s 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 Prandtl’s failure mechanism). Prandtl developed a bearing capacity

Figure 2.2: Failure mechanism for Terzhagi’s 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 Prandtl’s

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’, φ’ soil for

Terzhagi’s bearing capacity solution is shown in Fig. 2.2. Terzhagi’s ultimate bearing

capacity equations are given as follows:

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

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

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

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

follows:

)2/'45(cos2 2

'tan)'2/3(

φ

φφπ

+=

−eN q , )1('cot −= qc NN φ ,

−= 1

'cos'tan

221

φφ γ

γpK

N (2.4)

Figure 2.3: Terzhagi’s 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 Kpγ in the Nγ equation:

)2/'60(tan)8.3'4'8( 022 φφφγ ++−=pK (2.5)

Where 'φ in the first term is in radians. In the undrained conditions (cu and 0=uφ ):

1=qN , 71.5)1( 23 =+= πcN , 0=γN (2.6)

2.2.2 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, and i. For a

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

γγγγγγ disNBdisDNdisNcq qqqqccccu 5.0' ++= (2.7)

For vertical load, ic = iq = iγ = 1

γγγγγ dsNBdsDNdsNcq qqqcccu 5.0' ++= (2.8)

Figure 2.4: Meyerhof’s bearing capacity coefficients.



The bearing capacity factors:

)2/'45(tan)'tanexp( 2 φφπ +=qN , )1('cot −= qc NN φ , )'4.1tan()1( φγ −= qNN (2.9)

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

1=qN , 71.5)2( =+= πcN , 0=γN

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

inclination and depth factors are according to:

Shape Depth Inclination

Any 'φ L

BKs pc 2.01+=

B

DKd pc 2.01+=

2

0

0

901

−== α

qc ii

For 0'=φ sq = s γ= 1 dq = d γ= 1 i γ= 0

For 010'≥φ

L

BKss pq 1.01+== γ

B

DKdd pq 1.01+== γ

2

0

0

'1

−=

φα

γi

+=2

'45tan2 φ

pK , α =angle of resultant measured from vertical axis.

when triaxial 'φ is used for plane strain, adjust 'φ to obtain 'triaxialφφ

−=L

B1.01.1'

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 Hansen’s Bearing Capacity Equation

Hansen (1961) extended Meyerhof’s solutions by considering the effects of

sloping ground surface and tilted base (Fig. 2.5) as well as modification of Nγ and other

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

and load:

γγγγγγγγ 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,



γγγγγγγ bidsNBbidsDNbidsNcq qqqqqcccccu 5.0' ++= (2.11)

Figure 2.5: Identification of items in Hansen’s bearing capacity equation.

Figure 2.6 provides the relationships between Nc, Nq, and Nγ and the 'φ values, as

proposed by Hansen.

Figure 2.6: Hansen’s bearing capacity coefficients.

The bearing capacity factors Nc and Nq are identical with Meyerhof’s factors. Nγ is

defined by:



φγ 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 i

L

B

N

Ns ,, 1 ⋅+= , 'sin1 ,, φ⋅+= BqBq i

L

Bs , 6.04.01 ,, ≥−= BB i

L

Bs γγ (2.13)

Lcc

qLc i

B

L

N

Ns ,, 1 ⋅+= , 'sin1 ,, φ⋅+= LqLq i

B

Ls , 6.04.01 ,, ≥−= LL i

B

Ls γγ (2.14)

For cu, φu=0 soil: BcBc iL

Bs ,, 2.0= , LcLc i

B

Ls ,, 2.0= (2.15)

The inclination factors are:

1

1 ,,, −

−−=

q

iqiqic

N

iii ,

1

'cot

5.01,

α

φ

+−=

b

iiq

AcV

Hi ,

2

'cot

7.01,

α

γ φ

+−=

b

ii

AcV

Hi (2.16)

where the suffix i (in Eqn. 2.15) stands for B or L. 52 1 ≤≤ α . 52 2 ≤≤ α . 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

,

α

γ φη

+−−=

b

ii

AcV

Hi (2.17)

For cu, φu=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 φφ (2.19)

LDd Lc ⋅+= 4.01, , L

Dd Lq ⋅−+= 2, )'sin1('tan21 φφ (2.20)

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

( )BDd Bc

1, tan4.01 −⋅+= , )(tan)'sin1('tan21 12

, BDd Bq

−⋅−+= φφ (2.21)



( )LDd Lc

1, tan4.01 −⋅+= , )(tan)'sin1('tan21 12

, LDd Lq

−⋅−+= φφ (2.22)

For both sets: 1=γd (2.23)

For cu, φu 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 β and η are at the same plane, either

parallel to B or L.

0

0

1471 β−=cg , ( )5tan5.01 βγ −== ggq (2.25)

For cu, φu soil: 0

0

147β=cg (2.26)

0

0

1471 η−=cb ,

'tan2 φη−= ebq , 'tan7.2 φη

γ−= eb (2.27)

For cu, φu soil: 0

0

147η=cb (2.28)

2.2.4 A comparative summary of the three bearing capacity equations

Terzaghi’s equations were and are still widely used, perhaps because they are

somewhat simpler than Meyerhof’s and Hansen’s. 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 (for example, columns with

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

� The equations are generally conservative than Meyerhof’s and Hansen’s.

Currently, Meyerhof’s and Hansen’s equations are more widely used than

Terzaghi’s. Both are viewed as somewhat less conservative and 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.



EXAMPLE 2.1

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

a)Terzaghi’s, b) Meyerhof’s and c) Hansen’s 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, 'φ =280, cu = 105 kPa, uφ =0 andγ = 19 kN/m3. Use Terzaghi’s,

Meyerhof’s and Hansen’s 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, 'φ =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)

Meyerhof’s and b) Hansen’s equations. (Note that Terzaghi’s equations are not

applicable for inclined loads). The unit weight of the sand is 18 kN/m3.

2.2.1 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 Dγ in the bearing capacity equations refers

to the vertical stress of the soil above the base of the footing. The last term Bγ 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

Bγ is )(' zBz −+γγ or )(' zBzsat −+γγ . The later equation is used if the soil above

the groundwater level is also saturated. The term Dγ 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 Dγ is

)(' zDz −+γγ or )(' zDzsat −+γγ . The latter equation is used if the soil above the

groundwater level is also saturated. The term Bγ becomes B'γ .

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.

2.2.5 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:

FS

qq u

a = (2.29)

Alternatively, if the maximum applied foundation stress max)( aσ is known and the

dimension of the footing is also known then you can find a factor of safety by replacing

qa by max)( aσ in Eqn. (2.29):

max)( a

uqFS

σ= (2.30)



2.2.6 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 (1)

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

P

Me y

B = and P

Me x

L = (2)

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:

+=B

e

BL

P B61maxσ and

−=B

e

BL

P B61minσ (3)

and along the y axis are:

+=B

e

BL

P L61maxσ and

−=B

e

BL

P L61minσ (4)

Since the tensile strength of soils is approximately zero, minσ 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’.



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, 'φ =300, c’=0, and satγ =18 kN/m3. The footing is

subjected to a vertical load of 500 kN and a moment about the Y-axis of 125 kN・m.

The ground water table is 5 m below the ground surface. Use Meyerhof’s 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 mm×300 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 = (2.31)

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 )()( = (2.32)

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

σ (Liao and Whitman, 1985) (2.33)

kPa 24,2;1916

log77.0 '0'

010 >≤

= zN

zN cc σ

σ (Peck et al., 1974) (2.34)

where cN is a correction factor for overburden pressure, and '0zσ 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:

)(22

1

BD

zcW +

+= (2.35)

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 cor25

12= B ≤ 1.22 m (2.36)

dea kB

BNSq

2305.0

25

8

+= cor B > 1.22 m (2.37)

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 Meyerhof’s equations by 50% increase in the allowable

bearing capacity. Bowles’s equations are:

dea kNSq cor25

20= B ≤ 1.22 m (2.36)

dea kB

BNSq

2305.0

25

5.12

+= cor B > 1.22 m (2.37)

Documents

CHAPTER TWO BEARING CAPACITY OF SHALLOW · PDF fileMeyerhof (1951) developed a bearing capacity equation by extending Terzhagi’s failure mechanism and taking into account the effects