Stress distribution through soil

When a footing is loaded, a certain stress will be applied on the soil immediately below the footing.

The applied stress on the soil will decrease away from the footing location as the stress is distributed

over a larger area. Consider a small soil element as shown below and the stresses applied on the

element under plain strain condition.

Figure – Stress on a soil element under a strip footing.

Due to the stress applied on the soil from the foundation, the increase in the normal and the shear

stress in the soil mass under the foundation may lead to failure or cause deformation of the soil

causing settlement of the foundation. Therefore, it is very important to study the stress distribution

under a loaded area.

Even though the soil is neither elastic nor homogeneous, most of the stress distributions are obtained

assuming linear elastic and homogeneous soil medium below the foundation. Under the working

loads, the soil is stress well below the ultimate stress, as a factor of safety of about 3 against ultimate

stress in generally used. Therefore, under the working conditions, the assumption of linear elastic

behaviour may be somewhat accurate.

The most important stress component in the design of foundation is the vertical stress and hence the

distribution of vertical stress with depth is considered here.

F

X

Z

ZZ

XX

ZX

XZ

Vertical stress distribution in homogeneous isotropic soil medium

Approximate stress distribution

It is assumed that the stress within the soil mass is distributed under the footing so that a larger area is

involved in carrying the applied load with the depth and the boundary of the area involved in carrying

the applied load is linearly varying with the depth as shown in the following Figure.

As the concentrated force applied on the foundation is resisted by the soil pressure developed over a

(L + 2Z) x (B + 2Z) the stress developed at a depth Z, Z is given by the following relationship.

)2)(2( ZLZB

FZ

Generally, is taken as 0.5 with the tan = 0.5/1, = 26.5o

Boussinesq (1883) solved the problem of stress inside a semi-infinite mass due to a point load acting

on the surface.

L

Z 1

F = F/(BL)

z

B

L

B + 2Z

L + 2Z

The stress at point (x, y, z) due to a point load, Q, acting on the ground surface at the originate of a

rectangular coordinate system is given by Boussinesq as given by the following equations:

2

2

2

3

R

Qzz

[1]

323

2

5

2

)(

2

)(

1

3

21

2

3

R

z

zRR

xzR

zRRR

zxQx

[2]

323

2

5

2

)(

2

)(

1

3

21

2

3

R

z

zRR

xzR

zRRR

zyQy

[3]

235 )(

2

3

21

2

3

zRR

xyzR

R

xyzQxy

[4]

Where

222 zyxR

The practical application of the above equations is a difficult task. Therefore, other charts and Tables

are prepared to estimate of the stress increased due to a loaded area.

Considering a circular loaded area with radius r is loaded with a surface stress of qo, the stress

variation along the vertical axis through the centre may be estimated using the Equation proposed by

Boussinesq for the vertical stress due to a concentrated surface load, as shown in the following Figure.

Z

Y

X

y

x

z

Q

The loaded area is considered with small areas dA and the vertical stress at point D due to the

concentrated force qodA acting over the small area. Integrating the stress due to force acting over

small areas, the vertical stress increment is estimated, as given below as Equation [5].

2

32)/(1

10.1

zr

qo [5]

Newmarks Charts

The Boussinesq Equation for the vertical stress given in Equation [5] is re-arranged to develop the

Newmark’s charts. The Equation [5] is rearranged to give Equation [6].

113

2

o

v

q

q

z

r [6]

As the stress ratio (qv/qo) varies from 0.1 to 0.9, the ratio (r/z) varies as given in the following table.

qv/qo 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

r/z 0.0 0.270 0.400 0.518 0.637 0.766 0.918 1.110 1.387 1.908

Concentric circles of radius 0.27, 0.40 …1.908 are drawn and divided into 20 equal divisions by

drawing radial lines as shown in the following Figure. As the stress increment between any two

adjacent circles is 0.1, one small element is 0.1/20 = 0.005, which is called the influence value.

dq = qo dA

z

z

r

D

The use of the chart is explained using the following example.

Example

Estimate the stress at a depth of 5m below the center of the rectangular loaded area shown below. The

rectangular area is loaded with a distributed load of 200 kPa.

Prepare the plan of the building so that the length AB given on the Newmark’s chart is equal to the

depth 5m. Therefore, the scale is AB = 5m. The plan of the building to the estimated scale is drawn on

a tracing paper. Then the plan view of the building (drawn on the tracing paper) is placed on the

Newmark’s chart so that the centre of the rectangle coincides with the centre of the Newmark’s chart.

Count the number of units (M) covered by the plan view of the foundation, the stress increment at 5m

below the ground surface may be estimated by:

MIqq o .

Where,

q - Increased intensity of soil pressure due to foundation loading at depth 5m,

M – Number of units covered by the plan view of the rectangle drawn to the scale mentioned above.

I – Influence factor based on the chart used (I = 0.005 for the chart given below)

Stress Isobars

Isobars of the vertical stress under footings are developed based on the Boussinesq equations given as

Equations [1] to [4]. Pressure isobars, developed to estimate the vertical stress under a square and a

strip footing, is given below.

Figure – Pressure Isobars based on the Boussinesq equation for square and strip footings.

Newmark developed the Equation [7] to estimate the pressure under a corner of a square footing

loaded with a pressure of qo.

1

1

1

2tan

12

4

1

VV

VMN

V

V

VV

VMNqq ov

[7]

Where

z

BM

z

LN 122 NMV 2

1 MNV

Equation [7] may be sexpressed in terms of a stress influence factor I as given by Equation [10]

Iqq ov

The values of stress influence factor I in terms of M and N are given in Tabular format.

Table – Stress influence factors beneath a corner of a rectangular loaded area; M=B/z; N=L/z, M and

N a interchabgable

Any internal point within a loaded area may be considered by dividing the loaded area into rectangles

with the point under consideration is a common corner and using the method of superposition to add

the stress caused by each rectangular area.

Example

A 12mx 16m rectangular raft foundation, shown below, is used to support a square water tower.

Estimate the soil pressure at 5m below the foundation level at points A, B, and C using the

Newmark’s charts.

Stress distribution through layered mediums

Most natural soil deposits consist of layers of different properties. If the difference in the stiffness is

not very high, the methods described above may be used to estimate the stress distribution through

such mediums. However, if the stiffness between layers differ significantly, the above distribution

methods developed for homogeneous mediums are not applicable. Therefore, different researchers

have developed methods to estimate the stress distribution through layered soil mediums.

Distribution of vertical stress through a two layer system

Burmister (1958) developed the chart given below to estimate the vertical stress distribution through a

two layer medium. The chart is developed to estimate the vertical stress below the centre line of a

circular loaded area and the thickness of the Layer I is assumed to be equal to the radius of the

circular loaded area, as shown in the following Figure. The stiffness ratio between the two layers is

the governing factor in determining the stress distribution among layers. It is clearly seen from the

charts that as the ratio E1/E2 increases, the load taken by the layer 1 becomes higher. When the

stiffness ratio E1/E2 decreases, the vertical stress distribution reaches to that given by the Boussinesq

equations.

A B C

A

16 m

12 m

4 m 2m

The stress distribution given by the chart for the two layer system is of particular importance in

foundation engineering, where shallow foundations are constructed in hard stiff filled layers, which

overlies soft compressible layers. In such situations, it is seen from the chart below that the stress

applied on the bottom layer significantly reduced if the stiffness ratio is high. Another branch of civil

engineering, where frequently layered mediums are encountered, is highway engineering. The road

subgrade and the pavement are of significantly different stiffness and hence, the assumption of

homogeneous medium is not applicable.

Figure – Subsurface condition assumed by Burmister.

1 = 0.5

E1

2 = 0.5

E2

h1 = b

h2 =

2b

E1= Elastic modulus of layer I

E2= Elastic modulus of layer II

Figure – Vertical stress distribution through a two layer system with the top layer thickness equal to

the radius of the loaded area.

Osterburg method

The influence diagram for the vertical stress at any depth Z directly beneath the vertical edge of the

semi-embankment can be estimated by using the chart proposed by Osterberg (1957). The chart

proposed by Osterburg is shown below.

Figure – Osterburg chart to estimate the vertical stress distribution under embankment loading

Example

Estimate the vertical stress increment at points A & B due to the weight of the embankment

as given below. The unit weight of the embankment material is 20 kN/m3.

10m 1 6m

1.5

6m 10m

5m A

B