5 stress distribution elastic settlement

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STRESS DISTRIBUTION IN SOIL

CE352A - FOUNDATION DESIGN

CE352A Dr. Rajesh Sathiyamoorthy, IIT Kanpur

Vertical Stress Increase due to Applied Load

Equation of static equilibrium

Theory of elasticity


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Equation of equilibrium written in terms of effective



stress


Strain – displacement relations




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Constitutive relations

The axial strains for an ideal, elastic, isotropic material in terms of the stress components are given by Hooke’s law



components are given by Hooke s law





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Compatibility conditions

In terms of strain

Plane Strain Condition



In terms of strain

In terms of stress

Considering weightless medium


g g

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Stress Function

For the plane strain condition, in order to




determine the stress at a given point due to a given load, the problem reduces to solving the equations of equilibrium together with the compatibility equation and the boundary conditions

• The usual method of solving these problems is to introduce a stress function


preferred to as Airy’s stress function.

• This equation should satisfy equilibrium equation and compatibility condition.

• The problem reduces to finding a function ϕ in terms of x and z such that it will satisfy fourth order differential equation and the boundary conditions

Stress Distribution ‐ Boussinesq approach

Boussinesq solution Theory of elasticity

• The soil medium is an elastic, homogeneous, isotropic, and semi‐infinite

Assumptions

medium, which extends infinitely in all directions from a level surface. (Homogeneity indicates identical properties at all points in identical directions, while isotropy indicates identical elastic properties in all directions at a point).

• The medium obeys Hooke’s law.• The self‐weight of the soil is ignored.• The soil is initially unstressed.• The change in volume of the soil upon application of the loads on to it is neglected.

• The top surface of the medium is free of shear stress and is subjected to only


• The top surface of the medium is free of shear stress and is subjected to only the point load at a specified location.

• Continuity of stress is considered to exist in the medium.• The stresses are distributed symmetrically with respect to Z‐axis.

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Stress distribution due to line load

The stress at any point inside a semi‐


Boussinesq solutionTheory of elasticity

The stress at any point inside a semiinfinite medium due to a line load of

intensity q per unit length can be given by a stress function


Stress distribution due to Vertical line load

Boussinesq solutionVertical stress caused by a

Vertical line load


Influence factor


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Vertical stress caused by a Horizontal line load

Influence factor

Boussinesq solution


Influence factor


Vertical stress caused by a Point load

Boussinesq solution



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Influence factor

Boussinesq solution




Boussinesq solution



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Boussinesq solution


Vertical stress distribution on a horizontal plane at depth zVertical stress distribution on a horizontal plane at depth z

P

P


Vertical stress caused by a Point loadBoussinesq solution


Vertical stress distribution along a vertical line at radial distance r

Variation of vertical stress due to point load

PP


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Vertical stress caused by a Point loadBoussinesq solution


Isobar / Pressure bulbP

Δσz= 0.1P per unit area (10% isobar)

Pressure at points inside the bulb are greater than that at a point on the surface of the bulb; and pressures at points outside the bulb are smaller than that value.


Vertical stress caused by a flexible strip load

Boussinesq solution


Stress increase due to vertical line load

Stress increase due to vertical strip load


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Vertical stress caused by a flexible strip load

Boussinesq solution


Influence factor



Vertical stress caused by a flexible rectangular loaded area

Boussinesq solution

Stress increase due to point load

Vertical stress at depth Z under a corner of a rectangular area


The factors m and n are interchangeable

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Boussinesq solution

Vertical stress at depth z under a corner of a rectangular area [FADUM chart]


Vertical stress at Point A



Boussinesq solution

I II

IV III

A

)( 3333 IVIIIIIIZ IIIIq

By principle of super positionI

A

M N

O

Q

RP

ST

)(

Zone I = MQAT, Zone II = PRATZone III = NQAS, Zone IV = ORAS


)( 3333 IVIIIIIIZ IIIIq

Since zone IV is deducted twice, its influence has to be added once

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Boussinesq solution Contours of equal vertical stress

Under strip area Under square area



Boussinesq solution


Approximate methods


Equivalent point load method 2V:1H Method

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Vertical stress below the center of a uniformly loaded circular area

Boussinesq solution

Stress increase due to point load

The total load on the elemental area P (shaded in the figure) is equal to qr dr dα.


Boussinesq solution


Vertical stress below the center of a uniformly loaded circular area


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Influence chart for Vertical stress(Newmark Influence chart)

Boussinesq solution

Newmark constructed an influence chart, based on the Boussinesq solution,enabling the vertical stress to be determined at any point below an area of anyenabling the vertical stress to be determined at any point below an area of any shape carrying a uniform pressure q.

Stress increase due to uniformly loaded circular area

R i h l d

• R/z and Δσz/q are non‐dimensional quantities• Using the values of R/z obtained from equation and for various pressure ratios, Newmark(1942) presented an influence chart.

• The chart consists of influence areas, the boundaries of which are two radial lines and


Rearranging the terms leads totwo circular arcs.



Boussinesq solution

There are 200 elementshence, the influence value is 0.005

Influence chart – construction procedure

• Influence chart can be constructed by drawing concentric circles.

• The radii of the circles are equal to the R/z values corresponding to σz/q = 0, 0.1, 0.2, . , 1.

• Note: For σz/q = 0, R/z = 0, and For σz/q = 1, R/z = ∞

• Nine concentric circles are drawn. • The unit length for plotting the circles is AB• The circles are divided by several equally spaced


• The circles are divided by several equally spaced radial lines.

• The influence value of the chart is given by 1/N, where N is equal to the number of elements in the chart.

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Boussinesq solution

Procedure for obtaining vertical pressure at any There are 200 elements

hence, the influence value is 0.005point below a loaded area


ELASTIC SETTLEMENT

CE352A - FOUNDATION DESIGN


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Contact pressure distributionContact pressure and settlements for a flexible foundations

Elastic/ Clayey soil Granular/ Sandy soil

Contact pressure and settlements for a rigid foundations

Elastic/ Clayey soil Granular/ Sandy soil

Clay


Settlement ‐ Theory of Elasticity

H

ze dzS0

Elastic settlement of a shallow foundation

dzE ysxs

H

zs

)(1

0

(From Hooke’s law)

where,Se = Elastic settlementq = Net applied pressure on the foundation B = Width of the foundation

For flexible foundation fs

se I

EqBS

21

For rigid foundation ),()( 93.0 centreflexibleeRigide SS (limited to Z = 4B)

BIE

qS fs

se

21 ;


Es = Average modulus of elasticity of soil (measured from Z = 0 to 4B)µs= Poisson’s ratio of the soilIf = Influence factor depends on rigidity and shape of the foundationH = Thickness of the soil layerΔσx, Δσy, Δσz, are the stress increase due to the net applied foundation load in the x, y and z directions resp.

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Settlement ‐ Theory of Elasticity

For settlement at corners of loaded area (µs = 0.5)


REFERENCES

• Bowles Foundation Analysis and Design• Das Shallow FoundationsDas Shallow Foundations• Das Principles of Foundation Engineering•Murthy Advanced Foundation Engineering• Poulos, Davis Pile Foundation Analysis and Design• Scott Foundation Analysis• Som, Das Theory and Practice of Foundation Design• Tomlinson Foundation Design and Construction• Varghese Foundation Engineering


• Winterkorn,Fang Foundation Engineering Handbook

5 stress distribution elastic settlement

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