shallow foundation notes

7/29/2019 shallow foundation notes

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1Level 3 Shallow foundationPart 2EG3027 Lecture Notes (AI)

8 Shallow foundation (Part 2) - Settlements and Stress distribution

The term settlement is used to indicate the vertical downward movement of a structural

foundation, due to the compression and deformation of the ground below the foundation.Settlements may be caused by stresses which produce elastic deformation of the ground,

volume decrease of the ground underneath the loaded area, changed in pore water pressure,

plastic deformation combined with lateral expansion of the ground, ground failure or by

vibration in the case of loose granular materials.

When designing foundations, retaining walls etc. engineers need to check the construction with

respect to both the stress limit (no collapse) and serviceability (no deformation under working

conditions).

Uniform settlement is generally less dangerous than differential settlement.

If the Youngs modulus and Poissons ratio are known then we need to examine the

applicability of elasticity methods in soil mechanics and their use to calculate the vertical stress

and settlement increase with different surface loads applied (shallow foundation).

Elastic parameters

It is difficult to find an elastic stiffness modulus since it has been shown that there is a decrease

in the shear stiffness modulus G with an increase in shear strain obtained from the triaxial test.

(G=E/2(1+)) for an elastic material; Youngs modulus; Poissons ratio). This means that

soil is non-linear elastic material otherwise G would be constant with further increase in

shear strain. It is important to appreciate the stress paths when considering what value of G or

E to use. It has been shown that G is significantly different for the excavation phase and the

construction phase. If the stress-strain curve is non-linear there are two alternative definitions

of the effective stiffness modulus that can be used.

Figure 8.13


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Secant modulus - Line joining the origin and current stress-strain point (,) ratio of the

change in stress to the change in strain (from the datum point)

Tangent modulus - slope of the stress-strain curve at the point under consideration rate ofchange of stress with strain.

When standard formulas and methods regarding elasticity are used the foundation is assumedeither to be

Rigid in comparison with the soil (settlement is uniform) Perfectly flexible (stress distribution is uniform)

The one dimensional (1-D) consolidation we discussed earlier is a special case of the use of

elasticity to estimate soil settlement. In this case there is no opportunity for the load to be

transmitted into the surrounding ground. When a surface load is applied, an increase in total

stresses occurs immediately while an increase in effective stresses occurs with time (when the

pore water pressures return to their initial equilibrium).

The elasticity calculation can be used to estimate either:

The initial conditions (undrained)E, should be obtained from undrained tests and shouldbe defined in terms of total stresses.

The long term conditions (drained) - E, should be obtained from drained tests and should be

defined in terms of effective stresses.

When designing foundations, retaining walls and other structures engineers need to check theconstruction with respect to the stress limit and serviceability. Various types of load are

applied to soils. Depending on the geometry of the surface load the vertical stresses imposed

can be either uniform or not.


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Stress distribution

Aim: to find how the stresses are distributed within a soil mass and what are the resulting

deformations. The soil is assumed to be a semi infinite, homogenous, linear, isotropic elastic

material.

semi infinite - infinite below the surface; no boundaries apart from the surface

isotropic - the same properties in all directions

elastica linear stressstrain relationship

homogeneous - the same material properties throughout the region of interest

Stress Distribution Under a Concentrated (Point) Load

For a point load Q on an isotropic, homogenous, elastic, semi infinite medium the increase of

stress induced at any point within the medium may be determined by Boussinesqs theory

(1885). In polar coordinates, the components of this stress are:

Figure 8.14 Point load and vertical distribution of stress with depth and radial distance

5

3

2

3

R

Pzz

zR

R

R

zr

R

Pr

)21(3

23

2

2

zRR

Rz

RP

22

)21(

5

2

2

Pr3

R

zrz

0 zr


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By integrating Boussinesqs expression over any type of load acting on the area (e.g.

line load, uniform strip load, triangular strip load etc.) the stresses and strains can be

determined.

Figure 8.15 Different types of load imposed to the ground (Budhu, 2007)

For the case of a uniform surcharge of magnitude q acting over a circular area of

radius R at the surface of the half space, integration of Boussinesqs solution provides

the expression for the increase in vertical stress z at a depthz below the centreline.

(8.1)

This expression is shown on the graph below.


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Figure 8.16 Increase in vertical stress below the centreline of a uniform circular

surcharge

It is clear that the influence or intensity of the vertical stress below the foundation

induced by the foundation loading decreases both vertically and horizontally. Thismeans that at some point in the soil there will be almost no influence of the load at the

surface. The influence of the load will, however, be dependent on the shape of the

foundation. This can be represented in terms of pressure bulbs. The figure below

shows the pressure bulbs drawn for different foundation types where the stress is

represented as a fraction of the applied loading intensity, and is plotted against the

breadth of the foundation.

Two extreme shapes are used a uniformly loaded circular foundation and a strip

foundation. For a uniformly loaded rectangular the distribution is somewhere in

between.


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Figure 8.17 Contours of increase in vertical stress (a) below a circular footing of

diameter B (b) below a strip footing of width B (Powrie, 2002)

Uniform circular load Uniform strip load

0.2q 0.1q 0.2q 0.1q

Max depth of pressure bulb

Max half-breadth of pressure bulb

It has been found that Boussinesqs solution can be used with a fair degree of

confidence when looking at the vertical stress distribution if there is a general increase

in stiffness with depth and if there are several distinct layers of different types of soil

present. The reason for this is that the stiffness of the material does not really feature

in the expressions for the increases in stress.


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Deviations from Boussinesqs solution would occur in cases where:

- the applied load is high enough to cause yielding- if there is inclusion of lens of soil of higher or lower stiffness which are not

continuous

- if a stiff layer on the surface overlies more compressive soil, since the stifflayer will redistribute the load to some extend.

Newmarks Chart

For determining the vertical stresses at any point beneath any shaped areas, which are

uniformly loaded (q), a chart of the type shown in Fig 8.18 may be used. It is

essentially a plan view of the surface divided into n elements. Bearing in mind the

expression for vertical stress beneath the centre of a uniformly loaded circular

foundation (see Equation 8.1), if a particular depth is chosen for z then a series of

concentric circles can be drawn.

The chart is divided by 9 concentric circles into 10 annuli, each of which issubdivided into N= 20 segments giving 200 elements in total. Loading an element

with a uniform stress of q will result in an increase in vertical stress of q/200 or

0.005q at depth z below the centre of the chart.

The radii of the concentric circles are calculated using Eq. 8.1

Table 8.1


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To construct a chart, a scale must be chosen for z (z = 40mm is usual for an A4 sheet).

Application of the chart is as follows:

- draw a plan view of the loaded area on the chart to the appropriate scale ( foundby setting z)

- place the plan view so that it is positioned with the point of interest (this can beeither below or to the side of the loaded areas)

- count the number of elements covered by the loaded area- the vertical stress at the depth z and beneath the pointx is given by

(8.2)

where N is number of squares covered in each case.

Construction of Chart

Choose a convenient dimension for z (say z = 20 metres), the radii of the circles are

then, 5.4, 8.0, 10.4, 12.8 metres etc (see Table 8.1 established). Establish a scale (say

1:500) and draw the circles. Select a suitable number of rays (20 is the usual figure)

and construct them at equally spaced intervals. The resulting diagram is shown in

Figure 8.18; on it is drawn a vertical line, AB, representing x to the scale used (AB =

500

metres20= 40 mm). With N = 20, the influence factor is

1/200 = 0.005.

Example:Increase in vertical stress using Newmarks chart


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Figure 8.18Newmarks chart


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Uniformly loaded rectangular area - Fadums chart

Although the Newmarks chart is very flexible and appropriate for any type of loading

area it can be time consuming, in particular when the soil is divided into several

layers. Since in practice foundations are mainly rectangular, Fadums chart can beused in which increases in vertical stress at the depth z below the corner of a

uniformly loaded rectangular surcharge of length L and breath B may be calculated.

The chart gives the values for an influence factor I in terms of m= L/Z and n= B/Z.The increase in vertical stress at the depth Z below the corner of a uniform surcharge

of dimensions L and B is

(8.3)

A very good feature of this method is that the principle of superposition applies. This

means that the chart in Figure 8.19 can be used for points both inside and outside theloaded area.

Figure 8.19 Influence factors for calculating the vertical stress increase under the

corner of a rectangle. (Powrie, 2002)


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Example


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Estimation of settlements due to surface loads and foundations

There are a number of charts and tables available for the calculation of surface

settlements. Soil is very often heterogenic with an increase in stiffness of the soil with

depth. This characteristic is significant to the settlement, which means that the charts

available may have limited use.

Terzaghi, however, suggested that the settlement could be calculated by simply

assuming that the soil movements are mainly vertical so that relationship / is

governed by the 1-D modulus E0. If the soil to be considered has a variation in

stiffness with depth this can be taken into account by dividing the soil into a number

of layers following the procedure:

Identify a number of different layers within the soil, where each can becharacterised by a uniform 1-D modulus E0and uniform increase in vertical

stress v. (Note that even if only one layer is present this method can still be

used in terms of accounting for an increase of stiffness with the depth)

Use a Newmark chart and calculate the increase in stress v for each layer atthe point of interest (where settlement is required). It may be a middle point in

the middle of the layer and it can be an average value of v taken at the top

and bottom of the layer.

Assuming that the strains are 1 dimensional, estimate the settlement of eachlayer from:

E0 = v/v ; v = /h (8.4)

which then gives:

= vh/ E0 (8.5)

where h is the thickness of the layer.

For each point sum up the settlements calculated for all the layers.

E0 is defined as the ratio of the change in vertical effective stress v and the change

in vertical strain v during one dimensional loading and unloading. This can beobtained directly from an oedometer test stage or it can be obtained from a

conventional triaxial compression test.

Hooks law for an isotropic elastic material in terms of the undrainedparameters is:

1 = (1/Eu)(1- u2 - u3)

2 = (1/Eu)(2- u1 - u3)

3 = (1/Eu)(3- u1 - u2)


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If written for effective stresses:

1=v; 2= 3= h

v = (1/E)(v- 2h) (8.6)

h = (1/E)((1-)h-v) (8.7)

In 1D compression, h = 0 horizontal strain increment

(1-)h=v

h= [/(1-)] v

(8.8)

substituting (8.6) into (8.4)

v = (v/E)[(1-22)/(1-)] (8.9)

E0= v/ v= E (1-)/[ (1+)(1-2)] (8.10)

For undrained conditions Eu and ushould replace E and . If we do that and set

u= 0.5, the undrained 1-D modulus will become infinite which indicates that no

volume change can occur without drainage of pore water pressure.

Since the 1-D vertical strain must develop since horizontal strain is equal to zero only

long term or ultimate settlement can be found using the 1-D modulus.


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Settlements from elastic theory

There are formulas and charts available for settlements, which occur from various

patterns of surface load.

Vertical settlement under an area carrying uniform pressure q on the surface of a semi

infinite homogeneous, isotropic mass can be expressed through Eq 8.11.

sIE

qBs

21 (8.11)

Where EYoungs modulus, Bthe shorter dimension from the width and length, Isis an influence factor. These factors are given in Table 8.2 under the centre and corner

(rectangle)/edge (circle) for both flexible and rigid loaded area.

Shape of area Is(flexible) Is (rigid)

Centre Corner Average Average

Square L/B=1 1.12 0.56 0.95 0.82

Rectangle L/B=2 1.52 0.76 1.30 1.20

Rectangle L/B=5 2.10 1.05 1.83 1.70

Rectangle L/B=10 2.54 1.27 2.25 2.10

Rectangle L/B=100 4.01 2.01 3.69 3.47

Circle 1.00 0.64 0.85 0.79

Table 8.2 Influence factors for calculating the settlement below the corner of a

uniformly loaded rectangular area at the surface of a homogeneous, isotropic elastichalf space

In most cases the soil deposit will be of limited thickness and will be underlain by a

hard stratum (e.g.bedrock). For this case the proposed averaged vertical displacement

under a flexible loaded area carrying a uniform pressure q is given:

E

qBs

10 (8.12)

Where 0 depends on the depth of embedment and 1 on the layer of thickness and the

shape of the loaded area. Values of the coefficients 0 and 1 for Poissons ratio equal

to 0.5 are shown in Figure 8.20.


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Figure 8.20 Coefficient 0 and 1 for vertical displacements (after Christian and Carrier

1978)

For fine grained soils consolidation settlements are obtained (see consolidation lecture)

and are added to the immediate settlements using elastic theory. For coarse grained soils

empirical methods are used based on CPT data.

For fine grained soilsfrom consolidation theory

ci sss (8.13)

where si is an immediate settlement, under undrained conditions and sc is consolidation

settlement. Immediate settlement is zero if the lateral strain is zero whilst it will exist if

the lateral strain is not zero as is assumed in the 1D method.

In 1 D method consolidation settlement is equal to total settlements with an introduction

of a settlement coefficient.

oedcc ss (8.14)


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Where

)1( AAs (8.15)

Where A is pore water coefficient that can be explained experimentally (triaxial tests)

from u1=A1 where 1 is a change in deviatoric stress and u1 pore pressure at the

application of load.

For the case of strip foundation As = 0.866A +0.211 should be used instead of A.

Values of the settlement coefficient s for circular and strip footing in terms of A and

H/B (ratio of layer thickness to breath of footing (H/B) are given in Figure 8.21.

Figure 8.21 Settlement coefficients (Scott 1963)

Examples

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