9781305081567 Das Foundation Engineering 8e SI Chapter04

1 © 2016 Cengage Learning Engineering. All Rights Reserved.

Chapter 4Shallow Foundations:Ultimate Bearing Capacity

Principles of Foundation Engineering, SI, 8th edition Das

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© 2016 Cengage Learning Engineering. All Rights Reserved.

Introduction Shallow foundations must have two main characteristics: 1. Be safe against overall shear failure in the soil.

2. Cannot undergo excessive displacement or settlement.

Ultimate bearing capacity: The load per unit area of the foundation at which shear failure in soil occurs.

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Introduction This chapter discusses the following:

Development of the theoretical relationship for ultimate bearing capacity of shallow foundations subjected to centric vertical loading.

Effect of the location of water table and soil compressibility on ultimate bearing capacity.

Bearing capacity of shallow foundations subjected to vertical eccentric loading and eccentrically inclined loading.

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General Concept Consider a strip foundation with a width of B resting on the

surface of a dense sand or stiff cohesive soil.

If a load is gradually applied foundation, settlement will increase.

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General Concept Failure in the soil supporting the foundation will take place

at a certain point when load per unit area reaches a certain value.

The tipping point of this load per unit area is called the ultimate bearing capacity of the foundation ( ).

General shear failure is the term used for the sudden failure in the soil.

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General Concept For foundations resting on sand or clayey soil of medium

compaction, increasing the load will increase in settlement.

Failure surface in the soil will gradually extend outward from the foundation shown by the solid lines in the figure.

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General Concept When the load per unit area on the foundation equals ,

movement of the foundation will be accompanied by sudden jerks

is referred to as the first failure load

A considerable movement is then required for the failure surface in soil to extend to the ground surface

This is shown in the previous figure by the dashed lines

The load per unit area at which this happens is the ultimate bearing capacity ( ).

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General Concept If the foundation is supported by a fairly loose soil, the

load–settlement plot will be like this figure.

Here the failure surface in soil will not extend to the ground surface.

Beyond the ultimate failure load ( ) the load–settlement plot will be steep and practically linear. This type of failure in soil is called the punching shear failure.

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General Concept Relationship for the mode of bearing capacity failure of foundations

resting on sands.

= relative density of sand = depth of foundation measured from the ground surface = width of foundation = length of foundation for square foundations for circular foundations so

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General Concept This figure shows the settlement ( ) of the circular and

rectangular plates a sand at ultimate load.

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General Concept Foundations at a shallow depth ( ) show the

ultimate load occurring at a settlement of 4 to 10% of B.

This condition occurs with general shear failure in soil.

For local or punching shear failure, the ultimate load may occur at settlements of 15 to 25% of the width of the foundation (B).

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Terzaghi’s Bearing Capacity Theory According to Terzaghi, a foundation is shallow if its depth ( ) is less

than or equal to its width.

Later investigators suggested that foundations equal to 3 to 4 times their width be defined as shallow foundations.

The effect of soil above the bottom of the foundation may be assumed to be replaced by an equivalent surcharge .

( = unit weight of soil)

For a continuous or strip foundation the failure surface in soil at ultimate load may be assumed to be similar to that shown in the figure.

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Terzaghi’s Bearing Capacity Theory

The failure zone under the foundation can be separated into three parts: 1. The triangular zone ACD immediately under the foundation

2. The radial shear zones ADF and CDE, with the curves DE and DF being arcs of a logarithmic spiral

3. Two triangular Rankine passive zones AFH and CEG

(The angles CAD and ACD are assumed to be equal to the soil friction angle .)

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Terzaghi’s Bearing Capacity Theory The ultimate bearing capacity of the foundation can be obtained by

considering the equilibrium of the triangular wedge ACD from the previous figure and shown on a larger scale here.

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Terzaghi’s Bearing Capacity Theory If the load per unit area is applied to the foundation and general shear

failure occurs, the passive force will act on each of the faces of the soil wedge.

Consider that AD and CD are two walls that are pushing the soil wedges ADFH and CDEG, respectively, to cause passive failure.

Passive force should be inclined at an angle (angle of wall friction) to the perpendicular drawn to the wedge faces (AD and CD).

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Terzaghi’s Bearing Capacity Theory For equilibrium we have the equation

= weight of soil wedge ACD =

= cohesive force acting along each face, AD and CD, that is equal to the unit cohesion times the length of each face =

Thus,

or

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Terzaghi’s Bearing Capacity Theory The passive pressure is the sum of the contribution of the weight of soil

( ), cohesion ( ), and surcharge ( ).

The following figure shows the distribution of passive pressure from each of these components on the wedge face CD.

are earth pressure coefficients that are functions of the soil friction angle ( ).

Taking these figures into consideration we can now write the equation

' 2 ' ' '1( tan ) ( tan ) ( tan )

2 p c qP b K c b K q b K

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Terzaghi’s Bearing Capacity Theory

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Terzaghi’s Bearing Capacity Theory Combining Equations

and

We now can write the equation

= bearing capacity factors

' 2 ' ' '1( tan ) ( tan ) ( tan )

2 p c qP b K c b K q b K

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Terzaghi’s Bearing Capacity Theory Since are very tedious to calculate, Terzaghi

created the following relations:

If and

Where

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Terzaghi’s Bearing Capacity Theory If and then

where

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Terzaghi’s Bearing Capacity Theory If and

Then

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Terzaghi’s Bearing Capacity Theory Variations on bearing capacity factors are given below.

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Terzaghi’s Bearing Capacity Theory To estimate the ultimate bearing capacity of square and

circular foundations, use the following equations.

Square foundation

Circular foundation ( )

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Factor of Safety Calculating allowable load-bearing capacity of shallow foundations

requires the applying a factor of safety (FS) to the gross ultimate bearing capacity.

Some engineers prefer Net stress increase on soil = Net Ultimate Bearing Capacity

FS

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Factor of Safety Net ultimate bearing capacity: The ultimate pressure per unit area of

the foundation that can be supported by the soil in excess of the pressure caused by the surrounding soil at the foundation level.

If the difference between the unit weight of concrete used in the foundation and the unit weight of soil surrounding is assumed to be negligible, then .

= net ultimate bearing capacity

So (here the factor of safety should be at least 3)

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Modification of Bearing Capacity Equations for Water Table

If the water table is close to the foundation, some modifications to the previous bearing capacity equations must be made.

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Case I. If the water table is located so that , the factor in the bearing capacity equations takes the form

= effective surcharge =

= saturated unit weight of soil = unit weight of water

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Case II. For a water table located so that

In this case, the factor in the last term of the bearing capacity equations must be replaced by the factor

Based on the assumption that there is no seepage force in the soil.

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Case III.

When the water table is located so that the water will have no effect on the ultimate bearing capacity.

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The General Bearing Capacity Equation Previous equations do not address the case of rectangular foundations.

They also do not consider shearing resistance along the failure surface in soil above the bottom of the foundation.

Also, the load on the foundation may be inclined.

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The General Bearing Capacity Equation To account for all those shortcomings, Meyerhof suggested the following

equation:

=cohesion =effective stress at the lever of the bottom of the foundation =unit weight of soil =width of foundation (=diameter for a circular foundation) =shape factors =depth factors =load inclination factors

=bearing capacity factors

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Bearing Capacity Factors The angle shown in the figure below is closer to than .

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Bearing Capacity FactorsIf the previous changes are accepted, then the following equations should be employed:

Relations between can be found on the following table.

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Shape, Depth, and Inclination Factors Commonly used shape, depth, and inclination factors are

given in the following tables.

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Shape, Depth, and Inclination Factors

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Other Solutions for Bearing Capacity ( ), Shape, and Depth Factors Other equations for bearing capacity factors

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Other Solutions for Bearing Capacity ( ), Shape, and Depth Factors

Variations of with soil friction angle ( )

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Other Solutions for Bearing Capacity ( ), Shape, and Depth Factors

Variations of with soil friction angle ( )

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Shape and Depth Factors Shape and depth factors proposed by Meyerhof

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Shape and Depth Factors Zhu and Michalowski shape factors based on the

elastoplastic model of soil and finite element analysis.

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Case Studies on Ultimate Bearing Capacity Corn Silo bearing capacity failure

Load per unit area foundation when failure occurred

' 12 u c cs cd ci q qs qd qi s d iq c N F F F qN F F F BN F F F

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Case Studies on Ultimate Bearing Capacity

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Case Studies on Ultimate Bearing Capacity Corn Silo bearing capacity failure

FS= / applied load per unit area FS=181.8= 1.14 160

This factor of safety is too low and approximately equals one, for which failure occurred for the silo.

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Load Tests on Small Foundations in Soft Bangkok Clay Load tests of five small square foundations on soft clay.

According to the figure is about 35 kN/m2 for depths between zero and 1.5 m.

is approximately equal to 24k N/m2 for depths varying from 1.5 to 8 m.

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Load Tests on Small Foundations in Soft Bangkok Clay

The figure shows the load-settlement plots obtained from the bearing-capacity tests on all five foundations.

The ultimate loads are shown and can be determined from the graph.

The ultimate load is defined as the point where the load-settlement plot becomes practically linear.

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Effect of Soil Compressibility Vesic proposed the following equation to account for

change in failure due to soil compressibility:

are soil compressibility factors.

' 12 u c cs cd cc q qs qd qc s d cq c N F F F qN F F F BN F F F

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Effect of Soil Compressibility Calculating

1. Determine rigidity index at a soil depth approximately below the bottom of the foundation.

= shear modulus of soil

= effective overburden pressure at depth of

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Effect of Soil Compressibility

2. Calculate critical rigidity index

Variations of with in the following table

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Effect of Soil Compressibility

3. If then

If then

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Effect of Soil Compressibility 3. Cont’d

For use

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Eccentrically Loaded Foundations In several instances, as with the base of a retaining wall,

foundations are subjected to moments in addition to the vertical load.

In this situation the distribution of pressure by the foundation on the soil is not uniform.

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Eccentrically Loaded Foundations The nominal distribution of pressure is

= total vertical load

= moment on the foundation

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Eccentrically Loaded Foundations Using the equation

We get and

min

6(1 )

Q eq

BL B

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Eccentrically Loaded Foundations When the eccentricity becomes , then is zero.

When , will be negative and tension will develop.

Soil cannot take any tension, so there will be a separation between the foundation and the underlying soil.

The value of

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Eccentrically Loaded Foundations The figure shows the nature of failure surface in soil for a

surface strip foundation subjected to an eccentric load.

The factor of safety for such type of loading against bearing capacity failure is

= Ultimate load carrying capacity

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Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity Effective Area Method

Used for determining the ultimate load that the soil can support and the factor of safety against bearing capacity failure.

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Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity Step 1. Determine the effective dimensions of the

foundation.

= effective width = = effective length =

If the eccentricity were in the direction of the length of the foundation, the value of would be equal to . The value of would equal .

The smaller of the two dimensions is the effective width of the foundation.

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Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity

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Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity Step 2. Use the equation above to determine ultimate bearing

capacity.

Use relationships in Table 4.3 to determine

(use the effective width and length dimensions)

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Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity Step 3. The total ultimate load that the foundation can

sustain is

= effective area

' ' ' '{( ( )( )}uQ A q u B L

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Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity Step 4. The factor of safety against bearing capacity failure is

is the ultimate bearing capacity of a foundation of width with a centric load.

The actual distribution of soil reaction at ultimate load will be of the type shown in Figure to follow.

is the average load per unit area of the foundation. Thus

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Prakash and Saran Theory Analysis of the problem of ultimate bearing capacity of

eccentrically and vertically loaded continuous (strip) foundations.

Uses the one-sided failure surface in soil.

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Prakash and Saran Theory The ultimate load per unit length of a continuous

foundation is determined by the equation

= bearing capacity factors under eccentric loading.

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Prakash and Saran Theory The variations of with soil angle are given in the following figures.

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Prakash and Saran Theory

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Prakash and Saran Theory For rectangular foundations the ultimate load can be given

as

= Shape Factors

'( ) ( ) ( ) ( ) ( ) ( )

1[ ]

2 u c e cs e q e qs e e s eQ BL c N F qN F BN F

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Reduction Factor Method (For Granular Soil) Stability analysis of eccentrically loaded continuous foundations

supported by a layer of sand using the method of slices.

= Reduction Factor =

= average ultimate bearing capacity of eccentrically loaded continuous foundations

= ultimate bearing capacity of centrally loaded continuous foundations.

= functions of the embedment ratio found on the next slide.

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Reduction Factor Method (For Granular Soil)

Combining the equations on the previous slide we get

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Reduction Factor Method (For Granular Soil) Based on lab tests

The ultimate load per unit length of the foundation can then be given as

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Bearing Capacity—Two-Way Eccentricity Consider a foundation is subjected to a vertical ultimate

load and a moment as shown in Figures. For this case, the components of the moment about the x- and y-axes can be determined as and , respectively.

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Bearing Capacity—Two-Way Eccentricity This condition is equivalent to a load placed

eccentrically on the foundation with and .

= effective area =

Bx e Ly e

yB

u

Me

Q

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Bearing Capacity—Two-Way Eccentricity The terms can be found using the table below.

Use effective and width and length instead of and

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Bearing Capacity—Two-Way Eccentricity The terms can be found using the table below.

Use effective and width and length instead of and

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Bearing Capacity—Two-Way Eccentricity To determine do not replace with .

In determining , and , there are five possibilities.

Case I: and

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Bearing Capacity—Two-Way Eccentricity

Case I: and

The effective area in this condition is shown in the figure

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Bearing Capacity—Two-Way Eccentricity For Case I, the following equations apply

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Bearing Capacity—Two-Way Eccentricity Case II:

and

The effective area is show in the figure below

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Bearing Capacity—Two-Way Eccentricity For Case II, the following equations apply

(use whichever L value is larger)

(use whichever L value is larger)

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Bearing Capacity—Two-Way Eccentricity Case III:

and

The effective area is shown in the figure below.

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Bearing Capacity—Two-Way Eccentricity For Case III, the following equations apply

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Bearing Capacity—Two-Way Eccentricity The magnitudes for and can be found from the figure

below.

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Bearing Capacity—Two-Way Eccentricity Case IV: and

Effective area can be determined by the figure below

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Bearing Capacity—Two-Way Eccentricity The ratio can be determined using the upward

sloping lines in the figure below.

The ratio can be determined from the downward sloping lines in the figure below.

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Bearing Capacity—Two-Way Eccentricity For Case IV, the following equations apply.

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Bearing Capacity—Two-Way Eccentricity Case V: In the case of circular foundations under eccentric loading, the

eccentricity is always one way.

The effective area and effective width are determined from the table below.

The values from the table allow us to apply the equation .

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Bearing Capacity of a Continuous Foundation Subjected to Eccentrically Inclined Loading

Shallow continuous foundations are at times subjected to eccentrically inclined loads.

The figure shows two possible modes of load application.

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Bearing Capacity of a Continuous Foundation Subjected to Eccentrically Inclined Loading

In the figure the line of load application of the foundation is inclined toward the center line of the foundation.

This is referred to as partially compensated by Perloff and Baron.

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Bearing Capacity of a Continuous Foundation Subjected to Eccentrically Inclined Loading The line load application on the foundation can be inclined

away from the center line of the foundation.

This is called the reinforced case by Perloff and Baron and is shown in the figure below.

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Partially Compensated Case Start with the equation

For a continuous foundation and can be determined from the tables on the following slide.

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Partially Compensated Case Depth and inclination factors

Bearing capacity factors

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Partially Compensated Case After determining the value for we can apply the

equation

It has been proposed to use a reduction factor to estimate for granular soil

= Reduction factor

= ultimate bearing capacity of the foundation with centric vertical loading

' ' 'u u

u(ei)

(q )(B )(1) q (B 2e)Q

cos cos

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Partially Compensated Case The reduction factor is determined by the equation

Combining the previous equations we get

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Reinforced Case (Granular Soil) Use equation

9781305081567 Das Foundation Engineering 8e SI Chapter04

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