Chapter 1_Vertical Stresses

8/13/2019 Chapter 1_Vertical Stresses

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CHAPTER : VERTICAL STRESS ONSOILS

By

Siti Fatimah bt Sadikon
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Lesson Outcomes:

At the end of this chapter, students should be able to:-

Explain the total stress and effective stress analysis.

Solve total stress and effective stress problems.

Analyze the empirical analysis for point load, line load, strip load, circular

load and rectangular load by using Boussinesq Theory, Fadums Chart andNewmarks Chart.


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INTRODUCTION

When soils are subjected to external loads due to buildings,

embankments or excavations, the state of stress within the soilin the vicinity changes.

To study the stability or deformations of the surrounding soil,as a result of the external loads, it is often necessary to knowthe stresses within the soil mass fairly accurately.


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EFFECTIVE STRESS CONCEPT

In saturated soils, the normal stress () at any point within the soil massis shared by the soil grains and the water held within the pores.

The component of the normal stress acting on the soil grains, is calledeffective stress or intergranular stress, and is generally denoted by '.

The remainder, the normal stress acting on the pore water, is knows aspore water pressure or neutral stress, and is denoted by u.

Thus, the total stress at any point within the soil mass can be written as:tot = + u


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This applies to normal stresses in all directions at any pointwithin the soil mass.

In a dry soil, there is no pore water pressure and the total stressis the same as effective stress.

Water cannot carry any shear stress, and therefore the shearstress in a soil element is carried by the soil grains only.


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VERTICAL NORMAL STRESSES DUE TOOVERBURDEN

Unsaturated soil

a) Single layer

In a dry soil mass having a unit weight of , the

normal vertical stress at a depth of h is: = h

If there is a uniform surcharge qplaced at the ground level, this stress

becomes:

= h + q

GL

X

h

q


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b) Layered soil

In a soil mass with three different soillayers as shown in Figure above, thevertical normal stress at X is :

= 1h1 + 2h2 + 3h3

GL

h1 1

h2

h3

Soil 1, 1

Soil 2, 2

Soil 3, 3X


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Saturated soil

a) Single layer

Assume that the water table is at theground level.

The total vertical normal stress at X isgiven by:

= sath + q

The pore water pressure at this point issimply,

u = wh

Therefore, the effective vertical normal

stress is, = sath - wh + q

= h + q

where = sat w

GL

X

h1

q

h2

sat


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When the water table is at some depth below the groundlevel as shown in Figure above, the total and effectivevertical stresses and the pore water pressure can be writtenas:

' = h1 + (sath2 - wh2) + q

= h1 + h2 + q

When computing total vertical stress, use saturated unitweight for soil below the water table and bulk or dry unit

weight for soil above water table.


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b) Layered soil

At point A:

A = 1h1

At point B:B = 1h1 + (2 w)h2

At point X:

x = 1h1 + (2- w) h2 + (3- w)h3

= 1h1 + 2h2 + 3h3

GL

h1 1

h2

h3

Soil 1, 1

Soil 2, 2

Soil 3, 3X

A

B


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Soil response to stress

The stress-strain curve of a soil has features which are characteristic for different

material behavior. Soils show elastic, plastic and viscous deformation whenexposed to stresses.

OA: linear and recoverable

ABC: non-linear and irrecoverable

BCD: recoverable with hysteresisDE: continuous shearing

The relationship between a strain and stress is termed stiffness


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Elastic deformation

In linear-elastic behavior (OA) the stress-strain is a straight line and strains are fully

recovered on unloading, i.e. there is no hysteresis. The elastic parameters are thegradients of the appropriate stress-strain curves and are constant.

.' constddKvv

Bulk modulus

Poisson's ratio

.constddrara

Youngs modulus

.

constddG

Shear modulus

.constddEaaaa


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Typical values of elastic moduli E and

Typical E

Unweathered overconsolidated clays 20 ~ 50 MPa

Boulder clay 10 ~ 20 MPa

Keuper Marl (unweathered) >150 MPa

Keuper Marl (moderately weathered) 30 ~ 150 MPa

Weathered overconsolidated clays 3 ~ 10 MPa

Organic alluvial clays and peats 0.1 ~ 0.6 MPa

Normally consolidated clays 0.2 ~ 4 MPa

Steel 205 MPa

Concrete 30 MPa

Typical

Soil 0.25-0.4

Rock 0.3

Steel 0.28

Concrete 0.17


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Relationships between elastic moduli

In bodies of isotropic elastic material the three stiffness moduli E', K' and

G' and Poissons ratio (') are related as:

12

'EG

213

''

EK

Therefore the deformation behavior of an isotropic elastic material can be

described by only two material constants.


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Plastic deformation

Soils material behavior is often simplified as

elastic-perfectly plastic. During perfectly

plastic straining (AB), plastic strains continueindefinitely at constant stress. In a brittle

perfectly plastic material, the yield stress at

point A this is the same as the failure stress

at a point B.

With increasing stress the material

behavior goes over from elastic to plastic.

This transition is called yield (A). Plastic

strains (AB) are not recovered on

unloading (BC). Unloading (BC) and

reloading (CD) show a hysteresis. With

increasing strain (at constant stress) the

material eventually fails if brittle or flowsif ductile (E).

yield


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Elastic-plastic

GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com


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VERTICAL STRESS DUE TO VARIOUS TYPES OFLOADING

There are many types of loading such as point load, line load, circularload, triangular load, rectangular load and etc.

Factors affecting vertical stress: Size and shape of foundation.

Load distribution.

Contact pressure (surcharge load)

Modulus of elasticity.

Type of soil.

Rigid boundary.


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In estimating the soil stresses, 3 assumptions have to be made:

Soil mass in elastic medium and modulus of elasticity is constant.

The soil mass is homogeneous, isotropic and linear, where its constituent partsor elements are similar.

Soil mass is semi-infinite.

The calculation of vertical stress is using Boussinesq Equation.


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VERTICAL STRESS UNDER A POINT LOAD

Boussinesq Equation for vertical stress:

Q = surcharge load

z = depth

r = radius

It simplified using Boussinesq influence factor.

The values of influence factor, Ip for point load

given as in Table Ia and Ib.

The equation become:

Y

rz

Q (kN)


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Stress istribution iagram of stress intensity forPoint Loada) Isobar diagram

- An isobar is a contour connectingall points below the groundsurface of equal vertical stress.

- the vertical stress value of the

outer isobar contour will stateless value than inner isobarcontour.

b) Vertical stress distributionsdiagram in horizontal plane

c) Vertical stress distributions onvertical line.


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VERTICAL STRESS UNDER A LINE LOAD



The values of influence factor, IL for line loadgiven as in Table III.


z

r X

Q

(kN/m)


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VERTICAL STRESS UNDER A UNIFORMCIRCULAR LOAD


It simplified using Boussinesq influence

factor.

The values of influence factor, IC foruniformly circular load given as in TableII.


A

Q (kN/m2)

GL

z

r

a
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VERTICAL STRESS UNDER A UNIFORM STRIPLOAD


If x 0 or

If x = 0


The values of influence factor, Is for uniformstrip load given as in Table IV.


A

2b

Q

x

GL

z

B
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VERTICAL STRESS UNDER A UNIFORMTRIANGULAR LOADD


It simplified using Boussinesq influence

factor.

The values of influence factor, IT foruniform triangular load given as inTable V.


A

Q

C

X

Z


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VERTICAL STRESS UNDER A UNIFORMLYLOADED RECTANGULAR AREA


Where: m = B/z and n = L/z


The values of influence factor, Ir for uniform rectangular load given asin Table VI or using Fadums chart.


For using Table VI and Fadums chart, it can be used by the case ofdetermining the z right at below any one corner of the rectangle.

Besides, the vertical stress also could be determined by usingNewmarks chart.

QB

L

Z

A


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FADUMS CHART


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NEWMARKS CHART

A plan of the loaded area is drawnon a tracing paper to indicate the

scale.

The plan of the loaded area isplaced over the chart such thatthe point below which pressure isrequired coincides with the centreof the chart.

The vertical stress could be

determined as;

z = Q x influence value x number of area units covered

Documents

Chapter 1_Vertical Stresses