chapter-2-bearing capacity of shallow foundation

Foundation Engineering Bearing Capacity of shallow Foundations

Arba Minch University/ Engineering Faculty/ Civil Eng’g Dep’t Lecture Notes - 17 -

CHAPTER TWO BEARING CAPACITY OF SHALLOW FOUNDATIONS Table of Contents

Page No.

2.0 Introduction................................................................................... - 18 -

2.1 Bearing Failure Modes ..................................................................... - 18 -

2.2 Ultimate Bearing Capacity Equations ................................................. - 19 -

2.2.1 Terzaghi’s Bearing Capacity equation ............................................ - 19 -

2.2.2 Meyerhof’s Bearing Capacity equation........................................... - 21 -

2.2.3 Hansen’s Bearing Capacity Equation ............................................. - 23 -

2.2.4 A comparative summary of the three bearing capacity equations...... - 26 -

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

2.2.6 Eccentric Loads .......................................................................... - 31 -

2.3 Field Tests .................................................................................. - 33 -

2.3.1 Plate Loading Test........................................................................ - 33 -

2.3.2 Standard Penetration Test (SPT) .................................................... - 34 -

2.4. Methods of Improving Bearing Capacity of soils……………………………………...-38

2.5. Bearing Capacity of footings on slopes……………………………………………….……-43

2.6. Foundations on Rocks…………………………………………………………………………….-

2.7. Bearing Capacity of footings on Layered soils……………………………………….-

2.8. Proportioning of footings…………………………………………………………………………..-45

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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:

ü 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 22

1

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 LBKs pc 2.01+=

BDKd 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 LBKss pq 1.01+== γ

BDKdd pq 1.01+== γ

2

0

0

'1

−=φα

γi

+=

2'45tan 2 φ

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

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

−=

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 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

LB

NN

s ,, 1 ⋅+= , 'sin1 ,, φ⋅+= BqBq iLBs , 6.04.01 ,, ≥−= BB i

LBs γγ (2.13)

Lcc

qLc i

BL

NN

s ,, 1 ⋅+= , 'sin1 ,, φ⋅+= LqLq iBLs , 6.04.01 ,, ≥−= LL i

BLs γγ (2.14)

For cu, φu=0 soil: BcBc iLBs ,, 2.0= , LcLc i

BLs ,, 2.0= (2.15)

The inclination factors are:

11 ,

,, −−

−=q

iqiqic N

iii ,

1

'cot5.01,

α

φ

+

−=b

iiq AcV

Hi , 2

'cot7.01,

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:

()ΤϕΕΤΘθ243.36 563.8497 3.96 19.8 ρεΩ∗ ν0 ΓΒΤ/Φ11 22.4688 Τφ0.5258 0 0 1 243.36 568.7698 Τµ 22.4688 ΤΛ()BDd Bc

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

, BDd Bq

−⋅−+= φφ (2.21)

()ΤϕΕΤΘθ245.16 527.8497 3.96 19.8 ρεΩ∗ ν0 ΓΒΤ/Φ11 22.4688 Τφ0.5261 0 0 1 245.16 532.7698 Τµ 22.4688 ΤΛ()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 , ()ΤϕΕΤΘθ401.64 347.7298 3.96 17.4 ρεΩ∗ νΒΤ/Φ11 15.8789 Τφ0.7446 0 0 1 401.64 351.2098 Τµ 15.8789 ΤΛ()5tan5.01 βγ −==gg q (2.25)

For cu, φu soil: 0

0

147β=cg (2.26)

0

0

1471 η−=cb , 'tan2 φη−=ebq , 'tan7.2 φηγ

−=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:





ü Shape, depth and inclination factors are not considered.

ü 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.

ü 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.





EXAMPLE 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.

EXAMPLE 2.5

A square footing is shown in figure below. Determine the safe gross load (Factor of

safety of 3.0) that the footing can carry.

Fig.E2.5 Square footing.

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:

FSqq 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

uqFSσ

= (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

PM

e yB = and

PMe 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:

+=

Be

BLP B61maxσ and

−=

Be

BLP B61minσ (3)

and along the y axis are:

+=

Be

BLP L61maxσ and

−=

Be

BLP 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 2.6

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.

1. DILATANCY CORRECTION: Silty fine sands and fine sands below the water

table develop pore pressure that is not easily dissipated. The pore pressure

increases the resistance of the soil and hence the penetration number (N)

Terzaghi and peck (1967) recommended the following correction in the case of





silty fine sands when the observed value of N exceeds 15.

The corrected penetration number, Nc

Nc=15+0.5(NR-15)

Where NR is the recorded value and Nc is the corrected value.

If NR=15, Nc=NR

2. OVERBURDEN PRESSURE CORRECTION: In granular soils, the overburden

pressure affects the penetration resistance. If two soils having the same

relative density but different confining pressures are tested, the one with a

higher confining pressure gives a higher penetration number. As the confining

pressure in cohesionless soils increases with the depth, the penetration

number for soils at shallow depths is underestimated and that at greater

depths is overestimated. For uniformity, the N-values obtained from field tests

under different effective overburden pressure are corrected to a standard

effective overburden pressure.

2;8.95'0

≤

= N

zN cc

σ (Liao and Whitman, 1985) (2.33)

kPa 24,2;1916log77.0 '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:

)(221

BDzcW +

+= (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 cor2512

= B ≤ 1.22 m (2.36)

dea kB

BNSq2305.0

258

+

= 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 cor2520

= B ≤ 1.22 m (2.36)

dea kB

BNSq2305.0

255.12

+

= cor B > 1.22 m (2.37)

In the above equations N is the statistical average value for the footing influence zone

of about 0.5B above footing base to at least 2B below. Weighted average using depth

increment X N may be preferable to an ordinary arithmetic average: that is

Nav=(?N*Zi)/(?Zi





For pile foundations there may be merit in the simple averaging of blow count N

for any stratum unless it is very thick- (thick being a relative term). Here it may be

better to subdivide the thick stratum into several “strata” and average the N count for

each subdivision.

If there are consistently low values of N below this zone, settlements may be

troublesome if N is not reduced somewhat to reflect this event.

It can be noted, in the above equations, that footing width is a significant parameter.

Obviously if the depth of influence is on the order of 2B a larger footing width will

affect the soil to a greater depth and strains integrated over a greater depth will

produce a larger settlement. This is taken in to account somewhat for mats, which are

considered by both Meyerhof and Bowles to obtain in the previous equations.

Both Meyerhof’s and Bowle’s equations are most viable and only reliable in

formations of sand, silty sand, and mixtures of silt, sand, and fine gravel. Thus, careful

scrutiny should be used in establishing a qall from SPT tests in fine-grained soils such

as silt and particularly clay, since silt and clay may be softened or stiffened with an

increase or decrease in the moisture content. Correspondingly, the SPT results may

vary in the same silt or clay formations if the moisture conditions change.

Practical Problem Encountered:

A one-story school building that was designed using a qall based on a very high

blow count (large N values) obtained during a dry season (and low water table). The

SPT information was used as a sole basis for determining qall. Gradually, but

cumulatively, over a three-year span from construction, cracks exceeding 5cm

developed in the on-grade concrete slab. A subsequent evaluation revealed a clay

stratum vulnerable to significant shrinkage and swelling from notable changes in the





water content.

In the same context, SPT numbers may be misleading if the formation should contain

large-size gravel. The large size gravel may wedge itself into a split-spoon sampler

thereby resulting in a large, misleading N value.

2.4 Methods Improving the Bearing Capacity of Soils

Significant increase in the bearing capacity of a soil can be achieved by altering the

soil properties ofφ, cohesion c, or densityρ. Usually an increase in density (or unit

weightγ) is accompanied by an increase in either φ or c or both (assuming the soil is

cohesive). Particle packing (compaction) always increases the density, with a

resulting decrease in void ratio, and reduces long term settlements. Particle packing

usually increases the stress-strain modulus so that any “immediate” settlements are

also reduced.

Methods of Soil Property Modification

Mechanical stabilization:

§ Stabilization is achieved by altering grain size gradation of the site soil.

§ Binder (material passing through No. 4 (0.425mm) sieve) is added for soil

dominated by gravel (from 75mm – 1mm). Where the soil is predominantly

cohesive, granular soil is imported and blended with the site soil.

§ It usually requires much more granular materials to stabilize cohesive deposits

than binder for cohesionless deposits and as result other stabilizing methodsare

usually used for clayey soils.





Compaction:

§ This method usually uses some kind of rolling equipment to achieve particle

packing for both cohesionless and cohesive soils and is usually the most

economical.

a

b





c

d





e

Preloading:

§ Used in combination with drainage, it is primarily taken to reduce future

settlement but may also be used to increase shear strength.

Drainage:

§ A method undertaken to remove soil water and to speed up settlements under

preloading.

Densification using vibratory equipment:

§ The method uses some type of vibrating probe, which is inserted into the soil

mass and withdrawn.

§ Densification is particularly useful in sand, silty sand, gravelly sand deposits with

Dr less than about 50 to 60 percent.

Use of in-situ reinforcement:

§ The treatment produces composite ground. A trial spacing is chosen and column

of material such as stone, sand, cement, or lime is inserted in the excavated soil

and rammed.

§ The drilled diameters usually range in between 600mm and 800mm and depth of

4m to 8m.





Grouting:

§ Injection of a viscous fluid to reduce the void ratio (and k) or to cement rock

cracks. Most commonly, the viscous fluid is a mix water and water or water and

lime, and/or with additives such as fine sand, bentonite clay, or fly ash.

Geotextiles:

§ Synthetic fabric that is sufficiently durable to last a reasonable length of time in

the hostile soil environment.

§ Because of their tensile strength, geotextiles are sometimes placed over weak

(poor bearing capacities) soils to form reinforcement. Generally, a layer of

controlled fill is placed over the geotextile, thereby creating a form of composite

that spans over the weak soil.

Chemical stabilization:

§ It involves use of chemical stabilizers (also termed chemical grouting). It is

seldom employed because of cost.

§ The more commonly used chemical agents are phosphoric acid, calcium chloride,

and sodium silicate (or water glass).

§ Various chemicals added to a soil may yield one but more likely a number of

changes in a soil formation: (i) reduce permeability of the soil (e.g. in dam

construction, excavation infiltration). (ii) Increase soil strength. (iii) Increase

bearing capacity (IV) decrease settlement. (v) Produce a stiffening of loose sand

formation and thus minimize undesirable effects, such as from vibrations.





2.5. Bearing Capacity of Footings on slopes:

Before construction of footings on sloping ground, the stability of the slope itself must

be investigated. Footings should not be constructed on slopes which are unstable.

They should also be avoided on slopes where slow creep of the superficial material

takes place. The stability of a stable slope may be endangered by the addition of

footings. Hence the stability of footings must be investigated both before and after

construction of footings.

Footings on sloping ground:

Ø Should have sufficient edge distance (minimum 2 to 3 ft) as protection against

erosion.

Ø Should be carried below the depth of frost penetration.

Ø Should be carried below the top (organic) soil, miscellaneous fill, abandoned

foundation, and debris.

The bearing capacity of footings on sloping ground may be determined by the

following equation (Meyerohf’s, 1957):

q=CNcq+0.5?BNq

Where Ncq and N?q vary with the slope of the ground, the relative position of the

footing and the angle of internal friction of the soil





Fig.2.9. Ultimate bearing capacity of continuous footings on slopes.





Example 2.7.

Figure E2.7. Shows a shallow strip footing on the top of a clay slope, Determine the allowable

bearing capacity of the foundation with a factor of safety of 4.0

Fig.E2.7 Strip Foundation on clay slope

2.8. Proportioning of footings:

2.8.1 Proportioning of footings using presumptive allowable soil pressures:

Through many years of practice, it has been possible to estimate the allowable soil

pressure for the different types of soils for uncomplicated soil conditions. Accordingly

different building codes give allowable average soil pressure sas.

After picking up the allowable soil pressure sas for a given soil, one may determine the

area and subsequently the proportions of a footing necessary to sustain a given load

or a combination of loads as in the figure …..

The allowable soil pressure, sas is given by:





Where

P=Load sustained by the footing.

A=a.b=area of footing.

a=Length of footing.

b=Width of footing.

The designer should fix the geometric shape (square, rectangle, circle) and the ratio

between a and b of the footing prior to the application of the above equation. Since all

other quantities in the above equation are known, one readily determines the area A

of the footing.

Figure: Proportioning of footings using presumptive value

2.8.2. Proportioning of footings using the soil strength parameters ? and C:

For cases where presumptive allowable soil pressures cannot be used, one should

determine the soil strength parameters ? and C. These parameters may be

approximated or determined from laboratory tests. If the nature of the project calls

for relatively accurate determination of ? and C, one should carry out a series of

triaxial tests on undisturbed soil samples taken from several points. Using the value of

? and C thus obtained, one can easily determine the area of the foundation in





question using bearing capacity equations (2.1-2.11).

Figure: Proportioning of footings using shear strength parameters of a soil

In applying the bearing capacity equations one should differentiate two states of

loading, namely, the initial or instantaneous loading condition and the final or

long-term loading condition.

In the Initial loading condition, the load is assumed to act instantaneously. At this

stage the pore water pressure in the soil does not have time to dissipate. This

situation corresponds to the quick or undrained test condition of the triaxial test. The

soil parameters are designated by ?u and Cu –in most cases ?u=0.

In the Final loading or long term loading condition, the load is assumed to act

gradually as construction progresses, thus giving the pore water pressure in the soil

ample time to dissipate. Here the situation corresponds to the slow or drained test





condition of the triaxial test. The soil parameters in this case are designated by ?’ and

C’.

When one compares the respective magnitudes of the soil parameters; one finds that

Cu is much bigger than C’ and ?u-if not equal to zero- is much less than ?’.

Example 2.8:

Determine the Dimensions of a square footing necessary to sustain an axial column

load of 850KN as shown in the figure below, given that Df=2m, γ=19.1 KN/m3,if

a) An allowable presumptive bearing pressure of 150KN/m2 is used.

b) Cu=40KN/m2;C’=7.5KN/m2;?’=22.50.

Figure: Proportioning of a square footing.





Example 2.9:

A Rectangular mat foundation measuring 10m X 20m is to be placed at a depth of

3.50m below ground level. The subsurface profile comprises of multi layer soil

deposits, the details of which are shown in figure below. Determine safe bearing

capacity of the soil by adopting suitable factor of safety. Use Meyerhof’s bearing

capacity equation. Assume that the foundation carries a concentrically applied vertical

load.





Example 2.10:

At a site for a proposed building, SPT tests were conducted in a borehole at a depth

interval of 0.75m. The results of blow counts (N) observed at different depths below

ground level are given in the table below. At this site the soil in general is fine sand

with an average bulk unit weight of 17.0KN/m3 and saturated unit weight of 21KN/m3.

The ground water table is located at a depth of 3m below ground level. A rectangular

footing of size 3.0m X 4.0m is to be placed at a depth of 2.25m. Determine the

allowable bearing capacity of the footing for an allowable settlement of 50mm.

Table: Measured SPT blow counts.

Depth, m 0.75 1.50 2.25 3.00 3.75 4.50 5.25 6.00 6.75 7.50 8.25 9.00 9.75

Recorded, N 9 12 15 14 21 18 22 24 19 21 25 20 16





Example 2.11:

A Building is to be constructed over a site that has the soil stratification shown in

Figure below.

A. Determine the area of a square footing that can safely transfer the load

from the superstructure without shear failure, i.e. bearing capacity failure.

B. Determine the corresponding total settlement for the footing area

proportioned above. Check if the load can be transferred without excessive

settlement. Is an isolated footing the right choice for this condition? Why?

Use the following data:-

§ The load from the superstructure; P=2645KN.

§ The footing is to be placed at a depth of 2.0m below the ground

surface.

§ The allowable total settlement is 75mm.

§ Maximum center-to-center spacing between columns is 5.0m.

§ Assume the foundation to be a rigid foundation.

§ Ground water table exists at a depth of 5.0m below the ground

surface. Use Meyerhof’s Bearing Capacity equation. Use F.S=3.0.

Figure: Subsurface profile

CLAY SOIL

C=32 KN/m2, Φ =22o , Sr=80%, Gs=2.70, γ=17 KN/m3

E=25Mpa, υ=0.50, eo=0.80, Cc=0.21

10.0m

G.L

Rock Extended to a great depth





Example 2.12:

A square footing is shown in figure below. Determine the safe gross allowable load

(factor of safety=3) that the footing can carry. Use any two bearing capacity

equations.

Figure: Square footing.





Example 2.13:

A square footing is shown in figure below. Determine the safe gross allowable load

(factor of safety=3) that the footing can carry. Use Terzaghi bearing capacity

equations for general shear failure.

Given: ρsat=1980KN/m3, ∅ =25,ρ=1800Kg/m3 ,C=23.94KN/m2, B=1.8m, Df=1.2m,

h=2m





Example 2.14:

In the figure shown below, this shows a shallow strip foundation on the top a slope.

Given:

Slope (Sand)

ß=15o

C=0 KN/m2

∅ =40o

γ=15KN/m3

Foundation:

B=0.75m

D=1.5m

Estimate the allowable bearing capacity. Use factor of safety of 4.





Example 2.15:

A square footing of 4m width is shown in the figure below. The footing is subjected to

an eccentric load. For the following cases, Determine the gross allowable load that the

footing could carry. Use Meyerhof’s bearing capacity procedure and safety of

factors=4.

Given: Df=3m, x=y=0.5m, ∅ =25,ρ=1800Kg/m3 ,C=23.94KN/m2



chapter-2-bearing capacity of shallow foundation

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