Mtech Dissertation - Analysis of Raft Foundation Using Soil-structure Interaction

ANALYSIS OF RAFT FOUNDATION USING SOIL-STRUCTURE

INTERACTION

A project report submitted in partial fulfillment of the

requirement of the degree of

Master of Technology (Civil – Geotechnical Engineering)

Submitted by

SHEVADE. B.S. M. Tech. (Civil – Geotechnical Engineering)

Guide

Prof. Dr. S.R. PATHAK

DEPARTMENT OF CIVIL ENGINEERING

PUNE INSTITUTE OF ENGINEERING & TECHNOLOGY

PUNE- 411005

2004 - 2005

ANALYSIS OF RAFT FOUNDATION USING SOIL-STRUCTURE

INTERACTION

A project report submitted in partial fulfillment of the

requirement of the degree of

Master of Technology

(Civil – Geotechnical Engineering)

Submitted by Shevade. B.S.

M. Tech. (Civil – Geotechnical Engineering)

Guide Prof. Dr. Mrs. S. R. Pathak

DEPARTMENT OF CIVIL ENGINEERING

PUNE INSTITUTE OF ENGINEERING & TECHNOLOGY

PUNE- 411005

2004 - 2005

ACKNOWLEDGEMENT

I cherish this opportunity of being able to express my deep

gratitude to my guide Prof. Dr. Mrs. S. R. Pathak for his constant guidance,

advice & encouragement during the preparation & presentation of this project

work.

I would like to thank the Head Of Civil Engineering Department

Prof. Dr. U. J. Kahalekar for constructive encouragement.

Last but not the least, I am extremely grateful to

Mr. G.L. RAUT for his guidance and support during the software analysis

using STAAD Pro-2004. I am highly obliged to the library staff for their kind

co-operation during the literature survey.

Balraj Suresh Shevade, Place- Pune

M.Tech ( Civil – Geotechnical Engg.) Date- 06.06.2005

Roll. No. - M0310G08

INDEX

Sr.No. Description PageNo. List of Figures. List of Tables.

Abstract. 1. Introduction.

General 1

Types of raft. 1

Analysis of raft. 2

2. Soil Structure Interaction. General. 3

Soil Structure Interaction. 3

Contact Pressure. 4

2.3.1. Contact Pressure by theory of 5

Elasticity.

2.3.2. Contact Pressure by theory of 7

Sub grade reaction.

2.4. Soil models used in Soil Structure 9

Interaction.

2.4.1. Numerical Models 11

2.4.2. Centrifuge Modeling. 12

2.4.3. Estimation of Modulus of Sub- 13

grade (k) value.

Sr.No. Description PageNo.

3. Methods of Analysis. 3.1. General. 16

3.2. Conventional Method. 17

3.2.1. Methodology. 17

3.3. Finite Difference Method. 18

3.3.1. Assumptions. 19

3.3.2. Finite Difference Plate Bending 19

Theory.

3.4. Finite Element Analysis. 22

3.4.1. Kirchoff’s plate theory. 23

3.4.2. Sub-structure Method. 23

3.4.3. Discretization 25

3.4.4. Displacement Models. 27

3.4.5. Variational Formulation. 29

3.4.6. Jacobian Operator. 32

3.4.7. Numerical Integration. 33

3.4.8. Boundary Conditions. 34

3.4.9. Global Stiffness Matrix 35

4. Analytical Work. 4.1. Conventional Method. 39

4.2. Finite Difference Method. 43

4.2.1. Without considering SSI 43

4.2.2. Considering SSI on Winkler’s 46

Soil model.

4.2.3. Considering SSI on Linear Elastic 51

Model.

Sr.No. Description PageNo. 4.3. Finite Element Method. 54

4.3.1. Discretization. 54

4.3.2. Nodal Degrees of Freedom and

Interpolation Function. 55

4.3.3. Jacobian and Numerical Integration 57

4.3.4. Boundary Conditions. 58

4.3.5. Global Stiffness Matrix. 58

4.3.6. Without considering SSI. 59

4.3.7. Considering SSI on Winkler’s 61

Soil model.

4.4. Analysis of raft foundation considering 64

SSI on Winkler’s model using STAAD

Pro 2004 software.

5. Results and Discussions. 5.1. Comparison of deflection values by 67

FDM and FEM without considering SSI. 5.2. Comparison of deflection values by 68

FDM and FEM without considering SSI.

5.3. Comparative study of deflection values 69

raft foundation with and without SSI.

5.4. Parametric study.

5.4.1. Conventional Method of raft 77

analysis.

5.4.2. FDM of raft analysis (Winkler’s 78

model).

Sr.No. Description PageNo. 5.4.3. FDM of raft analysis (LEM) 79

5.4.4. FEM of raft analysis (Winkler’s 80

model).

Conclusions. 83

Appendix –A. 85

Appendix –B. 86

Appendix –C. 91 References.

LIST OF FIGURES Fig.No. Description Page No.

2.1. Contact Pressure by Theory of Elasticity. 8

2.2. Contact Pressure by Theory of Sub grade 10

reaction.

2.3. Plate load test results. 14

3.1. Grid pattern and numbering system for FDM 20

3.2. Two dimensional discretization of raft for FE analysis 26

4.1. Plan of two bay two-storied structure. 36

4.2. Elevation of two bay two-storied structure. 36

4.3. Loads acting on raft. 37

4.4. Load transmission to supporting beams. 38

4.5. Grid pattern and numbering system for FDM 43

8m x 10m raft.

4.6. Contributing area for each node on raft grid. 47

4.7. Two dimensional discretization of raft for FE 55

analysis 8m x 10m raft, element size 2m x 2.5m.

4.8. Nodal Degrees of freedom. 55

5.1. Numbering system for FDM. 76

5.2. Numbering system for FEM. 76

LIST OF GRAPHS Fig.No. Description Page No.

4.1. Pressure Distribution along ABC, DEF and 40

GHI.

4.2. Deflection Profile along ABC, DEF and GHI. 40

4.3. Pressure Distribution along ADG, BEH and 41

CFI.

4.4. Deflection Profile along ADG, BEH and CFI. 41

4.5. Deflection Profile along17-10-4-12-20, 44

9-3-0-1-5 and 16-8-2-6-13 by FDM.

4.6. Deflection Profile along 18-10-3-8-15, 45

11-4-0-2-7 and 19-12-1-6-4 by FDM.


9-3-0-1-5 and 16-8-2-6-13 considering SSI

on Winkler’s model by FDM.

4.8. Contact Pressure along17-10-4-12-20, 49






4.10. Contact Pressure along18-10-3-8-15, 50





on LEM model by FDM.



on LEM model by FDM.


11-12-13-14-15 and 16-17-18-19 by FEM.


3-8-13-18-23 and 4-9-14-19-24 by FEM.


11-12-13-14-15 and 16-17-18-19 by FEM on

Winkler’s soil model.

4.16. Contact Pressure Distribution along 6-7-8-9-10, 61

11-12-13-14-15 and 16-17-18-19 by FEM on



3-8-13-18-23 and 4-9-14-19-24 by FEM on


4.18. Contact Pressure Distribution along 27-2-12-17-22, 62

3-8-13-18-23 and 4-9-14-19-24 by FEM on



11-12-13-14-15 and 16-17-18-19 by FEM using

STAAD Pro.


3-8-13-18-23 and 4-9-14-19-24 by FEM using

STAAD Pro.

5.1. Comparison of deflection with and without SSI 69

by FDM (along short span).


by FDM (along long span).


by FEM (along short span).


by FEM (along long span).

5.5. Comparison of deflection without SSI by 73

Conventional, FDM and FEM along short span.

5.6. Comparison of deflection with SSI by 74

FDM (Winkler’s model), FDM (LEM) and

FEM (Winkler’s Model) along short span.

5.7. Relation of L/B ratio and deflection obtained at 81

center by various methods of analysis.

5.8. Relation of L/B ratio and contact pressure 82

obtained at center by various methods of analysis.

LIST OF TABLES Fig.No. Description Page No. 4.1. Contact Pressure and deflections at various 39

locations on the raft by Conventional Method.

4.2. Deflection at various locations on raft without 44

SSI by FDM.

4.3. Deflection and contact pressures at various 48

locations on grid considering

SSI by FDM (Winkler’s model).

4.4. Deflection at various nodes on raft on grid 52

considering SSI by FDM(LEM).

4.5. Deflections at various nodes raft without SSI 58

by FEM.

4.6. Deflection and contact pressure distribution 60

at various nodes on the grid considering SSI

by FEM (Winkler’s model).

4.7. Deflection at various nodes on raft grid 64

considering SSI using STAAD Pro-2004.

5.1. Comparison by deflection values by FDM and 66

FEM without considering SSI.

5.2. Comparison by deflection values by FDM and 67

FEM considering SSI.

5.3. Comparative study of various methods of 75

analysis used without SSI and with SSI.

5.4. For various L/B ratio deflection and contact 77

pressures by Conventional Method.


pressures by FDM on Winkler’s model.

5.6. For various L/B ratio deflection by FDM on LEM. 79


pressures by FEM on Winkler’s model.

NOTATIONS

k - Modulus of sub grade reaction. Es - Modulus of elasticity of soil.

νs -Poisson’s ratio of soil. B – Width of the footing. Q - Resultant of all column loads.

A - Plan area of the raft.

x - Distance from the Y-axis to the point of application of the resultant.

y - Distance from the X-axis to the point of application of the resultant.

ex - eccentricity in the direction of X axis = B/2 -x.

ey - eccentricity in the direction of Y axis

B - width of raft.

L - length of the raft.

Iyy and Ixx - moment of inertia of plan area of raft with respect to Y

and X-axis respectively. qnet - Net intensity of pressure.

Cd - Shape and rigidity factor.

λx - width of rectangular area in X – direction.

λy - width of rectangular area in Y – direction.

D - Flexural rigidity of plate.

Ec - Modulus of Elasticity of concrete.

νc - Poisson’s ratio of concrete.

h -Thickness of plate.

w - deflection.

U - strain energy.

Wp - potential of applied loads.

X, Y, Z - body forces.

Tx , Ty , Tz - surface applied loads.

{u} - The displacements at any point within the element.

{q} - Displacements at nodes

[N] - Shape function obtained by isoparametric formulation.

{ε} - Vector of relevant strain components at an arbitrary point within the

finite element.

[B] - Strain displacement matrix.

V - volume of the body.

s1 - portion of body over which surface traction is specified.

[k] - Element stiffness matrix.

{Q} - Load matrix.

J - Jacobian operator or Jacobian Matrix.

αi,j - Weights.

ζi,ηj - sampling points.

ABREVATIONS LEM – Linear Elastic soil model.

FDM – Finite Difference Method.

FEM - Finite Element Method.

SSI – Soil Structure Interaction.

ABSTRACT

Raft foundations are large slabs supporting a number of columns and walls under

the entire structure. Raft slab is required when the allowable pressure is low or

where the columns are spaced so close that the individual footings overlap. Raft

foundations are useful in reducing the differential settlements and sustaining

large variations in loads on the individual columns. In conventional analysis of

raft foundation the reactive soil pressures due to the loads from the structure are

not considered. This reactive pressure is important as the raft is subjected to

bending due to loads from the structure and also from the reactive pressure

offered by the soil. These effects considerably alter the forces and the moments

in the structural members. This is where soil structure interaction comes into

play. The effect of soil immediately beneath and around the structure, on the

response of the structure when subjected to external loads is considered in soil

structure interaction. In this case, the soil and structure are considered as

components of one elastic system. During the analysis soil can be modeled using

various soil models such as Linear elastic soil model, Winkler’s soil model etc. In

the present work, analysis is carried out using Winkler’s soil model, the methods

of analysis being used are, Finite Difference Method and Finite Element Method.

The deflections obtained from these two methods by considering soil structure

interaction are compared with the Conventional analysis. It has been observed

that the deflection using soil structure interaction is considerably reduced than

those by Conventional method of analysis. Thus the moments acting on the raft

slab are significantly reduced. This dissertation work deals with a comparative

study of effect of soil structure interaction on raft foundation using Finite

Difference Method and Finite Element Method considering two soil models,

Winkler’s soil model and Linear Elastic soil model.

1

CHAPTER 1

INTRODUCTION

1.1. GENERAL:

A raft foundation is a large concrete slab used to interface one or

more columns in several lines with the base soil. It is a combined footing

that covers the entire area beneath the structure and supports the entire

load bearing columns and walls. Rafts are necessitated on account of

overlap of large individual footings under columns if they are closely

spaced. When the footing covers more than half the plan area, raft would

be adopted in preference to individual footings.

Raft foundations are used to support storage water tanks, several

pieces of industrial equipment, silo clusters, chimneys, high rise buildings

etc. Raft foundations are used where the base soil has low bearing

capacity and the column loads are so large that more than 50% of the

area is covered by spread footings. It is most common to use raft

foundation for deep basement both to spread column loads to a more

uniform pressure distribution and provide the floor slab for the basement.

A particular advantage of raft for basement at or below GWT is to provide

a water barrier.

1.2. TYPES OF RAFT: The two basic structural forms of raft are,

1) Flat slab raft, and

2) Beam and Slab raft.

(Kurian,N, 1992)

A flat slab raft is a raft of uniform thickness supporting the columns

without the aid of beams. The flat slab type of raft is most suitable when

column loads are relatively light and spacing is relatively small and

uniform.

2

The beam slab raft consists of comparatively thinner slab

continuously spanning set of beams running through the column points in

both directions. Columns are normally located at the junction of these

beams. This type is suitable when bending stresses are high because of

large column spacing and unequal column loads.

1.3. ANALYSIS OF RAFT: Analysis of raft by Conventional Method is done by proportioning

the raft so that centroid of the area of contact is vertical load and soil

pressure is assumed to be uniform. Analysis by this method assumes raft

as a rigid beam.

(Horvath,J.S, 1983)

Two major limitations while analyzing raft foundation by Conventional

Method are -

(1) If eccentricities are absent i.e. ex = ey = 0, the reaction pressure will be

uniform and all points on the raft will deflect by same amount.

(2) If eccentricities are present the raft will rotate as a rigid body and there

will be differential vertical movement between points on raft.

This leads to uncertainty in analysis and over design of raft

foundation. To overcome these uncertainties raft is to be analyzed as

flexible plate. This assumption gives the clear view of contact pressure

distribution. Conventional Method does not consider this contact pressure

distribution during the analysis of raft foundation.

Finite Difference Method and Finite Element Method consider these

assumptions for the analysis of raft foundation. The contact pressure is

also considered during analysis, as the raft will be subjected to bending

due to loads coming from the columns as well as the loads due to the

contact pressures or the reactive pressures. This is the case where Soil-

Structure Interaction comes in picture. The present work deals with the

analysis of raft by all three methods discussed in the subsequent chapters

ahead.

3

CHAPTER 2

SOIL STRUCTURE INTERACTION

2.1. GENERAL:

Rafts are structural elements in contact with soil. When the loads

are transmitted to soil through foundation at the interface of soil and

foundation the reactive pressures are offered by soil to the foundation.

Due to these reactive pressures the footing is subjected to bending from

above loads i.e. loads from structure and from below due to soil reaction.

In conventional designs of raft footings, these reactive pressures

are not considered during design. This effect may considerably alter the

forces and moments in the structural members. Therefore, design must be

done by considering both loads from structures as well as reactive

pressures.

These reactive pressures are the contact pressures, and are

defined as, the reactive pressures offered by soil on foundation, at the

interface between soil and foundation, due to load transmission to soil

through foundation.

2.2. SOIL STRUCTURE INTERACTION: Design of raft footings by soil structure interaction approach

considers structural loads on the foundation and the soil reaction

produced by the loads on the foundation. The self weight of the foundation

and the contact pressure produced by it is not considered during the

calculations. Theoretically, the load on the foundation from structural loads

and that from soil reaction must be in static equilibrium. Therefore the soil

reaction takes any form consistent with the above loading conditions.

(Kurain,N,1981)

The actual distribution of soil reaction is the result of soil foundation

interaction and is determined by interactive analysis, involving the elastic

4

properties of both foundation and soil. Thus contact pressures are

statically indeterminate.

From the above discussions, we can define soil structure

interaction as, the effect of soil, immediately beneath and around the

structure on the response of the structure when subjected to external

loads is soil structure interaction.

When interactive analysis is considered, superstructure, foundation

and soil are considered as three components of one elastic system. The

interaction between the components of elastic system i.e. soil structure

system (superstructure, foundation and soil), under loads, depend on

interacting elastic effects on components of system. It is also seen that all

interacting systems are elastic and statically indeterminate.

2.3. CONTACT PRESSURE: The reactive pressures are the pressures offered by the soil on the

foundation at interface between the foundation and the soil against the

loads transmitted to the soil through foundation. They may also be called

as interface stresses. Loads transmitted from column to soil must not be

concentrated but have to be distributed uniformly. In this process of load

transmission soil is subjected to soil reaction i.e. contact pressure.

Structurally soil is subjected to bending due to load from structures and

also soil reaction acting from below. Actual distribution is result of soil

foundation interaction. This can be derived or determined by interaction

analysis involving both elastic properties of soil and foundation. Therefore

these contact pressures are statically indeterminate.

(Kurain,N,1981)

Theoretically contact pressures developing between interfaces

have two components –

1) Normal.

2) Tangential.

5

Tangential forces are not called upon to resist horizontal

components of applied load as they form equilibrium among themselves.

These tangential forces can be sustained if friction between soil and

foundation is fully mobilized. Its maximum value is limited to coefficient of

friction multiplied by normal reaction. (F = µN).

If foundation surface is too smooth no tangential component will

exist or when soil is of soft consistency.

Thus, foundation exerts pressure on soil, which is equal in

magnitude but opposite in direction of contact pressure. This is the

manner in which the superimposed load on foundation is felt by the soil as

it is transmitted through the medium of foundation.

Contact pressure is determined by two approaches –

1) Theory of Elasticity.

2) Theory of Sub grade reaction.

2.3.1. Contact pressure by theory of elasticity: Contact pressure is the result of elastic response of soil to applied

load. The best and rigorous approach to determine the magnitude and

distribution of contact pressure is from theory of elasticity. The extreme

cases that are considered are:

a. The flexibility or rigidity of the footing

b. The type of soil, and

c. The stage of loading.

Perfectly flexible footing is the one that cannot withstand any

bending moment and shear force. As it has little or no stiffness it can

undergo any amount of deflection. The flexural rigidity i.e. EI=0 which

means that it has thickness-approaching zero. A very thin membrane

represents the case of perfect flexibility.

Perfectly rigid footing is the one that can withstand enormous

bending moment and shear force with hardly any deflections. Footing

settles bodily or undergoes only rigid movements. Its flexural rigidity

6

approaches infinity. A very thick block represents the case of perfect

rigidity.

Types of soils considered are stiff clay and dry sand. The stages of

loading are the ones, which invoke the elastic response of the soil against

loading which invites ultimate response.

(1) Contact pressure under perfectly flexible footings:

The flexible footing cannot withstand any bending moment and

shear force, the loading on it must be such that reaction distribution does

not induce any moments or shear force. Therefore the reaction distribution

is identical.

Figure 2.1(a) shows, the soils considered are cohesionless soils and

Figure 2.1(b) shows, the soils as cohesive soils.

When footing is subjected to uniformly distributed load and rests on

cohesionless soil, Figure 2.1(a), the soil outside the footing is not under

pressure and has no strength. Therefore outer edge of the footing

undergoes large settlements, due to sudden loss of support felt at the

edges. Below the center of the footing the soil develops strength and

rigidity, and because of this the settlement is relatively smaller.

When footing subjected to uniformly distributed load and rests on

cohesive soils, Figure 2.1(b), the footing settlement is more at the center

and less at the edges.

(2) Contact pressure under perfectly rigid footing:

Consider a footing carrying concentrated load. The soil must be

perfectly isotropic, elastic half space. From cohesionless soil and cohesive

soils the latter satisfies the definition of elastic medium more closely, as

continuity of cohesive soils is good due to cohesion than that of

cohesionless soils.

As shown in figure 2.1(c), considering rigid footing on cohesionless

soils, the maximum intensity is at the center and minimum at the edges

7

approaching zero. The edge distribution approaches zero, due to the quick

yielding of sand at the edges as a result in the break of continuity in this

region. Settlement is uniform under rigid footings. The contact pressure

distribution is approximately parabolic for individual footing, while

ellipsoidal for mat foundation.

When we consider rigid footings on cohesive soils, the contact

pressure distribution shows less pressure at the center and more at the

edges. Settlement of rigid footing is uniform. The maximum bending

moment is induced at the center.

2.3.2. Contact pressure from the theory of sub grade reaction: Theory of sub grade reaction is based on Winkler’s assumption;

contact pressure (p) is proportional to the deflection (y) of the system.

(Kurain,N,1981)

In this assumption soil mass is replaced by a bed of closely spaced

elastic, identical and independent springs. Thus as stated above,

p α y

p = k y

k = Constant of proportionality = Modulus of sub grade reaction.

p and y are mutually dependent.

9

(1) Contact pressure under perfectly rigid footing:

As shown in figure 2.2(a), consider rigid footings, as seen earlier

the settlement of rigid footing is uniform and as the contact pressure is

directly proportional to settlement, the contact pressure is also uniform.

The settlement diagram and the contact pressure diagram are identical.

The magnitude of contact pressure is k times that of settlement. This

contact pressure can be determined from equations of equilibrium alone,

and hence the contact pressure under rigid footing by theory of sub grade

reaction is statically determinate.

(2) Contact pressure under perfectly flexible footing:

As shown in figure 2.2(b), consider flexible footing the maximum

settlement is at the center and so the contact pressure distribution. This is

due to soil structure interaction. This problem is statically indeterminate

due to the consideration of soil footing interaction.

2.4. SOIL MODELS USED IN SOIL STRUCTURE INTERACTION:

The behavior of soil must be defined initially to study soil structure

interaction by which further analysis part becomes less complicated. For

this purpose soils must be modeled. As discussed earlier that in any soil

structure interaction problem soil is considered as an elastic material the

models given herewith follow this rule.

(Chandrasekaran, V.S, 2001)

The soil structure interaction can be studied by,

1) Numerical modeling.

2) Centrifuge modeling.

11

2.4.1. Numerical Models: The numerical models give the relationship between the applied

forces and resulting displacement. These relationships are given by linear

functions, which are further used for analysis.

The numerical models used are:

1) Winkler’s model.

2) Elastic half space theory model.

1) Winkler’s model:

In this model soil mass is replaced by a bed of closely spaced

elastic, identical and independent springs. The shear resistance in soil is

neglected. The soil outside the loaded area does not undergo any

deflection.

This model is based on simple assumption that contact pressure is

proportional to deflection of elastic system.

p α y

p = k y

k = Constant of proportionality = Modulus of sub grade reaction.

p and y are mutually dependent. This mutual dependency is the essence

of interaction. If the structure in contact is vertical, contact pressure is

horizontal (kh) and if structure in contact is horizontal, contact pressure is

vertical (kv).

The value of k is dependent on material and dimensions of

foundations. From the above assumptions we can conclude that, the value

of k remains same whatever be the value of p and y.

The above assumptions are collectively referred as Winkler’s

model. It has been assumed that soil bed is considered as medium of

elastic, identical and independent springs. By elastic it ensures that there

is linear relationship between p and y. Identical ensures that the value of k

remains same whatever be the value of p and y may be. Independent

12

means that each spring deflects independently due to load coming on it,

without the interference of adjacent springs.

The value of modulus sub grade reaction can be determined

experimentally from load settlement diagram obtained plate load test.

2) Elastic half space model:

The elastic half space model for soil is superior to the Winkler’s

model, as the continuity present in the soil medium is accounted for in the

model. Also advantage of this model is its versatility in transferring

horizontal shear stresses beneath the foundation.

Soil is assumed to be homogenous, isotropic elastic and semi-

infinite. Displacement will not only occur in loaded area but also within

certain limited zones outside the loaded area.

2.4.2. Centrifuge modeling:

When scaled models are studied, it is difficult to simulate the body

forces in normal 1g fields. So to get near approximate field conditions

centrifuge technique is used.

In centrifuge technique the models are subjected to predetermined,

high acceleration levels to produce similarity conditions satisfactorily in

most situations.

The real full-scale structures are called as prototypes. The

miniatures of the prototypes, which satisfy the geometric similarities, are

called models. Thus a physical model involves a real object subjected to

forces and physical quantities such as resulting displacement and

stresses are measured. The physical measurements are made on model

and the corresponding quantities are predicted for prototype. For this

purpose two systems must be geometrically similar, and must be related

in following manner:

13

Rm = λ Rp

where,

Rm and Rp = same physical quantity pertaining to the model and the

prototype.

λ = proportionality constant.

When two systems behave similarly, knowledge of behavior of one

enables to determine the behavior of other.

Centrifuge is equipment in which models can be subjected to high

acceleration field. If model is placed at a radius r and if angular velocity is

w rad/sec, then radial acceleration is w2r. This can be visualized as the

unit weight of material and is increased by factor n = w2r/g

where, g is acceleration due to gravity.

Models in geotechnical engineering lack similitude because stress

levels in models do not match with those in prototypes. Therefore by

placing the model in the centrifuge and subjecting it to increased

acceleration field it is possible to obtain prototype stress levels in models.

Centrifuge is a convenient way of providing artificial gravity resulting from

centripetal acceleration. Centrifuge modeling can be used to study the soil

structure interaction effects on various structures.

2.4.3. Estimation of Modulus of Sub grade (k) value:

The value of modulus of sub grade reaction can be determined

experimentally or as given by Vesic’s formula.

(Kurian, N, 1981)

Experimental determination of k value:

Modulus of sub grade reaction can be determined experimentally

from the results of plate bearing tests. These are normally plotted in

pressure – settlement diagram.

14

Fig.No.2.3. Plate load test results. (Kurian, N, 1981)

If the soil is linear as assumed by Winkler the slope of load

settlement diagram is the value of k. As per load settlement diagram k has

to be calculated by one of the following,

(1) Initial tangent modulus.

(2) Tangent modulus.

(3) Secant modulus.

Initial tangent modulus means slope of the tangent to load

settlement diagram at origin. Tangent modulus means slope of tangent to

curve.

Secant modulus gives more definite value k. Secant to a curve

means line joining the point on the curve to origin. Slope of secant to

curve is called secant modulus at a specified value of either load or

settlement. k value is normally taken as secant modulus corresponding to

settlement of 1.25mm.

Load or Pressure

Settlement

1.25mm

15

Vesic’s formula:

Vesic (1961) proposed the following relationship for computing the

value of k in analysis of raft,

k = 0.65(EsB4 / EbI)1/12(Es/1-νs2)

Es = Modulus of elasticity of soil.

EbI = Flexural rigidity of structure.

νs = Poisson’s ratio.

Since twelfth root of any term multiplied by 0.65 will approximately

be equal to 1, so for all practical purposes the Vesic’s equation reduces to,

k = Es / B (1-νs2) ------------------------(2.1)

He recommended that if a value of modulus of sub grade reaction

based equation (2.1) is used then the results of analysis on Winkler’s

model and elastic half space model would practically be same.

16

CHAPTER 3

METHODS OF ANALYSIS

3.1. GENERAL:

The analytical studies for solution of soil structure interaction

problems requires the consideration of deformational characteristics of soil

medium and the flexural behavior of the structure. By defining the stress

strain behavior of soil and the stiffness behavior of the structure, the soil

structure interaction problem is reduced to the determination of contact

pressure distribution at the soil structure interfaces. Once the contact

pressure distribution is computed, it is then possible to evaluate the

moments and forces in the structure and the stresses, strains and

deformations in the idealized supporting soil medium.

Methods used for analysis of foundation by soil structure

interaction approach are:

1) Finite Difference Method.

2) Finite Element Method.

In the present work, methods of analysis of raft foundation are

studied and a comparative study of raft, by considering soil – structure

interaction and without considering soil – structure interaction is carried

out using following methods,

1) Conventional Method.

2) Finite Difference Method.

3) Finite Element Method.

17

3.2. CONVENTIONAL METHOD: The analysis of raft foundation by Conventional method (Rigid

beam method) is one of the simplest method of analysis used in practice.

The basic assumption is that the mat or raft will move as a rigid body

when loads are applied. Raft is considered to be infinitely rigid compared

to soil. The self-weight of raft is directly taken by the soil. For example, the

theory of elasticity would predict vertical stresses of infinite magnitude

beneath the edges of a rigid body.

(Kurain, N, 1992)

The basic assumption is that the reaction pressures are distributed

linearly across the bottom of the mat. It is assumed that the resultant of

column loads and soil pressures coincide.

3.2.1 Methodology: Initially the column loads are calculated by the regular methods of

analysis of frames. The eccentricity if any is evaluated.

The contact pressure distribution is calculated by combined direct

bending stress formula,

xx

y

yy

x

IxQe

IyQe

AQq ±±= ----------(3.1)

If the resultant is not eccentric then the pressure distribution will be

uniform,

AQq = ----------(3.2)

where, q = Pressure intensity.

Q = Resultant of all column loads.

A = Plan area of the raft.

ex and ey = eccentricities in X and Y directions respectively.

x and y = co-ordinate locations where soil pressures are desired.

Iyy and Ixx = moment of inertia of plan area of raft with respect to Y

and X-axis respectively.

18

Iyy = BL3/12

Ixx = LB3/12

L and B = plan dimensions of the raft.

The maximum contact pressure distribution obtained must be less

than the safe bearing capacity of the soil. The slab is divided into strips

and each strip is considered as a rigid beam subjected to contact

pressures and column loads. The bending moment and shear force

diagrams are then obtained.

The settlement of the raft is obtained by,

s

2snetd

i E)νB(1qC∆ −

= --------------(3.3)

Where, qnet = Net intensity of pressure

= Average value taken along one line.

Cd = Shape and rigidity factor. νs = Poisson’s ratio of soil.

Es = Elastic modulus of soil.

Each strip is designed individually for the bending moments

calculated as before and the actual reinforcement provided must be twice

the area of steel obtained by conventional method, as per National

Building Code regulations.

3.3. FINITE DIFFERENCE METHOD:

Finite Difference method is numerical method used to calculate the

deflections and moments at various locations selected on the grid. The raft

is considered as flexible plate.

(Milovic, S.D, 1998)

It is assumed that the deflections of the plate are small compared to

the thickness of the plate. For the purpose of analysis the loads on the raft

are calculated from the frame analysis, and then the by using plate

bending equation are evaluated deflections of raft and the moments.

19

3.3.1. Assumptions: During the analysis of raft by finite difference method it is assumed that,

(Timoshenko, S.P and Krieger, S, 1959)

(1) Load acting on the plate is normal to the surface of the plate.

(2) Deflections of the plate are small compared to the thickness of the

plate.

(3) It is assumed that there are no horizontal shearing forces acting on the

plate.

(4) As there are no forces normal to the sides of element so any strain on

the middle plane occurring during bending is neglected.

(5) In addition to the moments Mx and My, twisting moments, Mxy, are

also considered in pure bending.

3.3.2. Finite Difference Plate Bending Equation: The basic plate bending equation is,

(Bowels, J.E, 1988)

Dq

yw

yxw

xw

=∂∂

+∂∂

∂+

∂∂

4

4

22

4

4

4

2 ------------(3.4)

where, q = Q/λx λy

Q = Column Load.

λx and λy = longer side and shorter side of the grid respectively.

D = Flexural rigidity of plate = Ech3/12 (1-νc2)

Ec = Modulus of Elasticity of concrete.

νc = Poisson’s ratio of concrete.

h = Thickness of plate. w = deflection.

The grid pattern and numbering system for finite difference method

is as shown in the figure3.1. This plate bending equation is then converted

to finite difference equation as below.

21

Referring the Figure 3.1. at point 0, we have,

[∂2w/∂x2] 0 = 1/λx2 [w1 – 2w0 + w3]

[∂2w/∂x2]1 = 1/λx2 [w5 – 2w1 + w0]

[∂2w/∂x2] 3 = 1/λx2 [w0 – 2w3 + w9].

[∂4w/∂x4]0 = 1/λx4{[∂2w/∂x2]1 - 2[∂2w/∂x2]0 - [∂2w/∂x2]3}

= 1/λx4 [w5 – 2w1 + w0 – 2(w1 – 2w0 + w2) + w0 – 2w3 + w9]

[∂4w/∂x4] 0 = 1/λx4 [6 w0 – 4(w1 + w3) + w5 + w9 ] --------------------(3.5)

Similar equation for ∂4w/∂y4 can be obtained as,

[∂4w/∂y4] 0= 1/λy4[ 6 w0 – 4(w2 + w4 ) + w7 + w11 ] -----------(3.6)

and for ∂4w/∂x2∂y2 at point 0,

[∂4w/∂x2∂y2] 0 = 1/ λx2λy2 [4 w0 – 2(w1 + w2 + w3 + w4 )

+ w6 + w8 + w10 + w12] ------(3.7)

Combining equations (3.5), (3.6), (3.7) and writing it in form of

equation (3.4), we have,

[ ] [ ]

[ ]Dqww)w4(w6w

λy1

)www2(w)www4(w8wλyλx1www4(w6w

λx1

1174204

1210864321022953104

=+++−+

+++++++−++++−

-----------(3.8)

From equation (3.8) we get equation in terms of unknown values

i.e. deflection at node 0. Similarly by evaluating equations for all the nodes

on the grid a set of simultaneous equations are obtained. These

simultaneous equations are solved by Gauss Elimination Method and

deflections at the prescribed points on the grid are calculated.

22

From the values of deflections obtained we can evaluate the

moments at the respective nodes are evaluated by using following

formulae,

Mx = - D [∂2w/∂x2 + ν∂2w/∂y2]

My = - D [∂2w/∂y2 + ν∂2w/∂x2] --------------(3.9)

Mxy = D (1 - ν)[∂2w/∂x ∂y]

3.4. FINITE ELEMENT ANALYSIS:

The basis of finite element method is representation of a body or a

structure by an assemblage of subdivisions called Finite Elements. These

elements are considered interconnected at joints, which are called nodes

or nodal points. Simple functions are chosen to approximate the

distribution or variation of actual displacement over each finite element.

Such assumed functions are called displacement function or displacement

models. So the final solution will yield the approximate displacement at

discrete locations in the body, at the nodal points.

(Desai. C.S and Abel. J.F, 2000)

The displacement models are expressed in terms of polynomials or

trigonometric functions. Since polynomials offer ease in mathematical

manipulations, they are employed commonly in Finite Element

applications.

A variational principle of mechanics, such as principle of minimum

Potential Energy, is usually employed to obtain the set of equilibrium

equations for each element.

23

3.4.1. Kirchoff’s Plate Theory: In case of raft foundation, raft is considered as plate resting on soil.

The plate follow Kirchoff’s plate theory, and accordingly certain

assumptions are made as follows:

(Bathe.K-J, 1997)

1) Structure is thin in one direction.

2) The stress through the thickness i.e. perpendicular to the mid surface

of the plate is zero.

3) Material particles that are originally on the straight line perpendicular to

the mid surface of the plate remain on the straight line during

deformation.

4) Shear deformations are neglected and the straight line remains

perpendicular to mid surface during deformation.

3.4.2. Sub-Structure Method: In soil structure interaction problem the basic unknowns to be

determined are raft settlements, forces in structure and raft and contact

pressure distribution at the raft-soil interface. The contact pressure

distribution will depend on structure-raft interaction and raft-soil

interaction. Therefore to evaluate the actual contact pressure distribution

full interactive analysis is carried out. This can be achieved precisely by

Finite Element Method. During the finite element analysis of the system,

structure, soil and raft are considered as elements of one single system.

(Hain. S.J and Lee. I.K, 1974)

The moments and shearing forces in the raft are sensitive to

column loads, which are in turn influenced by the settlement profile of the

raft. Therefore it is necessary to treat structure, raft and soil as a part of

one single system. The column loads are calculated by frame analysis.

The moments created by the column loads cannot be ignored as they

have great influence on settlement of raft foundation.

24

The substructure finite element analysis is considered here, as it is

the most efficient, flexible and effective technique. In this method stiffness

matrix of structure and supporting soil is incorporated into stiffness matrix

of raft.

Force displacement relation for structure and raft,

[P] = [K] [U] --------------- (3.10)

[K] = Raft stiffness matrix.

[ ]

=

i

b

UU

U , [ ]

=

i

b

PP

P

[Ub] = Boundary displacements common to super structure and soil.

[Ui] = interior displacements of super structure.

[Pb] and [Pi] = set of external forces.

=

i

b

i

b

iiib

bibb

PP

UU

KKKK

---------(3.11)

By partial inversion on above equation,

[Kbb - Kbi Kii-1 Kib] [Ub] = [Pb] – [Kbi Kii

-1] [Pi] -------(3.12)

This equation is free body equilibrium equation in matrix form for

the boundary nodes of superstructure (structure and raft).

[Kbb - Kbi Kii-1 Kib] = Boundary stiffness matrix.

[Kbb] = partition of structure raft stiffness matrix which refers to

boundary nodes.

[Kbi] = partition of structure raft stiffness matrix which refers to

boundary forces due to interior displacements.

[Kii] = partition of structure raft stiffness matrix which refers to

interior nodes.

[Kib] = partition of structure raft stiffness matrix which refers to

interior forces due to boundary nodes.

25

Force displacement relation for supporting medium i.e. soil,

[Ks] [δ] = [F] --------------- (3.13)

[Ks] = Supporting medium stiffness matrix.

[ ]

=

s

b

δU

δ , [ ]

=

s

b

FF

F

=

s

b

s

b

iisibs

bisbbs

FF

δU

KKKK

---------------- (3.14)

Combining equations (3.12) and (3.14).

−+=

−+ −−

s

b1

iibibb

s

b

siisib

sbiibiibibbsbb

FPKKPF

δU

KKKKKKKK 1

-----(3.15)

This method does not make use of interface elements and so the

calculation part is reduced. This is most efficient method used in soil-

structure interaction problems.

3.4.3. Discretization:

The process of discretizing or subdividing a continuum is an

exercise of engineering judgment. The number, shape, size and

configuration of elements should be such that the original body is

simulated as closely as possible. The general objective of discretization is

to divide the body into elements sufficiently small so that the simple

displacement models can adequately approximate the true solution.

(Desai.C.S. and Abel.J.F, 2000)

The raft in this work is considered as two dimensional plane stress

problem and thus is discretized into two-dimensional rectangular plate-

bending elements.

27

3.4.4. Displacement Models: The basic philosophy of finite element method is piecewise

approximation. In this method, we approximate a solution to a complicated

problem is approximated by subdividing the region of interest and

representing the solution within each subdivision by a relatively simple

function. The simple functions, which approximate the displacements for

each element, are called displacement models, displacement functions or

interpolation functions.

(Desai, C.S. and Abel, J.F, 2000)

Polynomial is the most common form of displacement model that is

used in finite element formulation. It is easy to handle the mathematics of

polynomials in formulating the desired equations for various elements and

in performing digital computations. The use of polynomials permits us to

differentiate and integrate with relative ease. Also, the polynomial of

arbitrary order permits a recognizable approximation to the true solution. A

polynomial of infinite order corresponds to an exact solution, but, for

practical purpose the polynomials are limited to one of finite order.

Displacement is considered as polynomial function,

u = α1 + α 2 x + α 3 y + α 4 xy + α 5 x2 +--------etc.

Numerical solution must converge or tend to converge to the exact

solution of the problem. In finite element analysis, displacement

formulation gives upper bound to true stiffness of the structure i.e.

stiffness coefficients have higher values than the exact solution. Therefore

simulated structure deforms less than actual structure. So if the finite

element mesh is made finer is obtained exact solution.

The polynomial must satisfy certain convergence requirements,

(1) Displacement model must be continuous within the elements and

displacement model must be compatible between adjacent elements.

(2) Displacement models must include rigid body displacements of the

element.

28

(3) Displacement models must include constant strain states of the

element.

The formulation satisfying the first criteria is compatible or conforming.

Elements meeting both second and third criteria are complete. It should

ensured that the displacement model will allow continuous non-zero

derivatives of higher order appearing in potential energy functional. All

three conditions must be satisfied but practical results for elements that

satisfy only third criteria appear to converge acceptably.

The inter element compatibility must be satisfied and are imposed

not only on the displacement quantities but also on their derivatives. This

is to ensure that the plate remains continuous and does not kink.

Therefore at each node three conditions of continuity are imposed. The

three conditions of inter element compatibility are –

(1) Same isotropic displacement model is used in both the elements.

(2) For each element the displacement on the interface must depend only

on the nodal displacement occurring on that interface.

(3) Inter element nodal compatibility must be enforced.

Models satisfying compatibility for displacement does not

necessarily yield continuity for slopes or derivatives of displacement

across element interface. Continuity in slopes is achieved when they are

considered in the model as nodal degrees of freedom. So this condition

must be satisfied in terms of slopes if slope compatibility is to occur.

3.4.5. Variational Formulation:

Principle of virtual work is considered as the basis of the variational

formulations. The majority of theorems are called minimum principles

because the stationary value of the functional can be shown to be a

minimum.


29

The Total potential energy of an elastic body is defined as,

π = U + Wp ---------------(3.16)

where, U = strain energy

Wp = potential of applied loads.

Because the forces are assumed to remain constant during the

variation of the displacements, we can relate the variation of the work

done by the loads (W) and of potential of the loads can be related as

follows –

δW = - δ Wp -------------(3.17)

The principle of minimum potential energy is,

δπ = δ U + δWp

δπ = δU - δ W

δπ = 0 -------------(3.18)


The principle and its accompanying conditions can be stated as,

“Of all possible displacement configurations a body can assume which

satisfy compatibility and the constraints or kinematic boundary conditions,

the configurations satisfying equilibrium makes the potential energy

assumed a minimum value.”

The important point to note here is, a variation of displacements is

considered while forces and stresses are assumed constant. Moreover,

the resulting equations are equilibrium equations.

The total potential energy of a linearly elastic body can be

expressed as the sum of the internal work (strain energy due to internal

stresses) and the potential of the body forces surface tractions.

π = ∫∫∫v dU(u,v,w) - ∫∫∫ v (Xu + Yv + Zw) dV

- ∫∫ s1 ( Tx u + Ty v + Tz w) ds1 . ------------(3.19)

30

Where, s1 = surface of body along which surface applied loads are

prescribed.

dU(u,v,w) = strain energy per unit volume.

- ∫∫∫ v (Xu + Yv + Zw) dV - ∫∫ s1 ( Tx u + Ty v + Tz w) ds1 = work done by

constant external forces.

X, Y, Z = body forces.

Tx , Ty , Tz = surface applied loads.

dU = ½ { ε }T { σ} = ½ { ε }T [C] { ε }

π = ∫∫∫v [ ½ { ε }T [C] { ε } - 2{ u }T { X } ] dV - ∫∫ s1 { u }T{ Tx } ds1

-----------(3.20)

where, {u}T = {u, v, w}

{X} T = {X, Y, Z}

{Tx} T = {Tx, Ty, Tz}

Formula 3.20. for evaluation of stiffness matrix.

Evaluation of element stiffness matrix –

The displacement model is formulated in terms of

interpolation function,

Element displacement,

{u} = [N] {q} ------------(3.21)

{u} = The displacements at any point within the element.

{q} = Displacements at nodes

[N] = Shape function obtained by isoparametric formulation.

Element strains,

{ε} = [B] {q} ------------(3.22)

{ε} = Vector of relevant strain components at an arbitrary point

within the finite element.

[B] = Strain displacement matrix.

31

These strains are expressed in terms of some combination of

derivatives of the nodal displacement, {q}. Since nodal displacements are

functions of spatial co-ordinates these derivatives must be formed in terms

of matrix [N].

Further substituting these values in equation (3.20), we have,

π = ∫∫∫v [ ½ { q }T [B] T [C] { q } [B] - 2{ q }T [N] T [X] ]dV

- ∫∫ s1 { q }T[N] T{Tx} ds1 ------------(3.23)

V = volume of the body.

s1 = portion of body over which surface traction is specified.

Applying variational principle to the above equation,

{ δq }T { ∫∫∫v [ [B] T[C] [B] {q} - [N] T [X] ]dV

- ∫∫ s1 [N] T{Tx} ds1 }= 0 -----------(3.24)

[k] {q} = {Q} ---------------(3.25) [k] = Element stiffness matrix = ∫∫∫v [

[B] T[C] [B] dV

{Q} = Load matrix = ∫∫∫v [N] T [X] dV + ∫∫ s1 [N] T{Tx} ds1

From the equation 3.25. general formula for element characteristics

is obtained. The formulation of stiffness and load matrix can be obtained

by Numerical integration.

3.4.6. Jacobian Operator:

In finite element analysis, as given in equation (3.26.), [k] and [Q]

matrices are to be evaluated. These can be found out from [B] matrix,

which is defined in terms of derivatives of [N]. [N] Matrix is defined in

terms of local co-ordinates. Therefore it is necessary to devise some

means of expressing global derivatives in terms of local derivatives. Also,

32

elements of volume over which integration are carried out, needs to be

expressed in terms of local co-ordinates with appropriate change in limits

of integration.

(Zienkiewicz.O.C, 1997)

Consider two dimensional plate bending problem,

Let ζ and η be the local co-ordinates and x and y be the global co-

ordinates, we have,

ζy

yN

ζx

xN

ζN iii

∂∂

∂∂

+∂∂

∂∂

=∂∂ ---------------(3.26)

Similar relation can be devised for η co-ordinates.

ηy

yN

ηx

xN

ηN iii

∂∂

∂∂

+∂∂

∂∂

=∂∂ ---------------(3.27)

Writing the above equations in matrix form,

∂∂∂∂

∂∂∂∂∂∂∂∂

=

∂∂∂∂

y/Nx/N

ηx/ηx/ζy/ζx/

η/Nζ/N

i

i

i

i --------------------------(3.28)

∂∂∂∂

=

∂∂∂∂

y/Nx/N

Jη/Nζ/N

i

i

i

i --------------------------(3.29)

J is called Jacobian operator or Jacobian Matrix and is evaluated in

terms of local co-ordinates.

To transform the variables and the region with respect to which the

integration is made a standard process is used which involves the

determinant of J. Therefore integration made over element area becomes,

dx dy = det J dζ dη -----------------(3.30)

33

3.4.7. Numerical Integration: To calculate the element stiffness matrix to integrate the elements

of the matrix are to be integrated individually. There are two possibilities –

(1) Numerical Integration

(2) Explicit multiplication and term-by-term integration.

The second possibility is exhaustive and time taking so various

Numerical integration schemes are used. Gauss Quadrature scheme is

most commonly used.

(Bathe. K-J, 1997)

The basic integration schemes, such as, Trapezoidal rule,

Simpson’s rule use equally spaced sampling points. These methods are

effective when measurements of an unknown function to be integrated are

taken at certain intervals. But in Finite element methods the location and

values of sampling points as well as the weights are unknown, so a

numerical integration scheme, which optimizes both the sampling points

and the weights, is to be used. This can be done using Gauss Quadrature

rule.

The basic assumption for Gauss Quadrature rule is,

0∫a0∫b F (ζ, η) dζdη = Σ Σ αi,j F(ζi,ηj) ------------------------(3.31)

where, αi,j = Weights.

ζi,ηj = sampling points.

During this procedure to change the intervals of integration from

(a,b) to (-1,1).

0∫a0∫b F(ζ,η) dζdη = (ab/4) 1∫-11∫-1 F(ζ,η) dζdη

It is important to select the proper order of integration. If higher

order of integration is used all matrices will be evaluated accurately. If the

order of integration is too low the matrices evaluated are not accurate. So

it is important to devise the appropriate order of integration to reduce or

minimize the errors.

Order of integration = 2(p – m).

where, p = order of polynomial. m = order of differential.

34

3.4.8. Boundary Conditions: Finite Element problem is not completely specified unless boundary

conditions are prescribed. A loaded body or structure is free to experience

unlimited rigid body motion unless some supports or kinematic constraints

are imposed that will ensure the equilibrium of loads. These constraints

are Boundary Conditions.

(Desai.C.S. and Abel.J.F, 2000)

There are basically two types of boundary conditions,

(1) Geometric boundary conditions.

(2) Natural boundary conditions.

In finite element method only geometric boundary conditions are to

be specified, the natural boundary conditions are implicitly satisfied in the

solution procedure as long as we employ a suitable valid variational

principle is employed.

3.4.9. Global Stiffness and Load Matrix:

The direct stiffness method is employed universally for assembling

the algebraic equations in finite element application. The boundary

conditions prescribed or derived are applied to the element stiffness

matrices and then these reduced element stiffness matrices are

assembled together to obtain the global stiffness matrix.

(Bathe.K-J, 1997)

The individual stiffness and loads are added directly to locations in

overall matrices [k] and [Q], in conformity with the requirement of one to

one correspondence between the nodes of the element and those of

assemblage. The values of deflections are then obtained by solving the

equation by Gauss Elimination Method.

35

CHAPTER 4

ANALYTICAL WORK

Analysis of two bays – two-storied building is carried out. Raft footing of

size 8m x 10m is provided for this building.

The properties of structure and material used for construction are,

All beam sizes = 230 mm x 380 mm

All column sizes = 230 mm x 380 mm

Slab thickness = 125 mm

Unit weight of concrete = 25 kN/m3

Wall thickness = 230mm

Unit weight of wall = 19.5 kN/m3

Height of parapet wall = 1.5 m

Floor Height = 3 m

Grade of concrete – M20 Grade of steel – Fe415

Econcrete = 5700 (fck) 0.5 νconcrete = 0.15

Soil - Medium dense sand (sandy clay)

Esoil = 40000 kN/m2 νsoil = 0.30.

Frame load analysis:

The building frame is made of reinforced concrete. The openings in

the frame are filled with 230mm-thickness brick wall. Parapet wall is

constructed along the periphery of the structure on the top floor. The slabs

transmit loads to the beams. The slabs adopted in this problem are two

way slabs, so, the load transmitted on the beams is as shown on the

Figure 4.4,

38

The beams further transfer the loads to the columns and columns

to the raft. The loads transmitted to the raft in this problem are as shown in

the Figure 4.3.

4.1. CONVENTIONAL METHOD: The dimensions of the raft are 8m x 10m, with X and Y-axis as

shown in Figure 4.1. Taking moments of the applied loads along X-axis

and Y-axis, x and y are estimated.

x = Distance from the Y-axis to the point of application of the resultant.

y = Distance from the X-axis to the point of application of the resultant.

Eccetricities are then obtained as,

ex = eccentricity in the direction of X axis = B/2 -x.

ey = eccentricity in the direction of Y axis = L/2 -y.

where, B = width of raft.

L = length of the raft.

45°

Fig No.4.4. Load transmission to supporting beams.

4m

5m

39

From the formula given in section 3.2.1, we have,

xx

y

yy

x

IxQe

IyQe

AQq ±±=

Choose the cartesian co-ordinates (0,0) at the center of the raft and

decide the sign conventions in above equation. Putting the values of x and

y as per the co-ordinates, contact pressure at each point where columns

are located on the raft are estimated

Deflections beneath each column points are obtained by formula

given in section 3.2.1,

s

2snetd

i E)νB(1qC∆ −

=

The value of Cd is obtained from the table given in Appendix-A.

The results obtained from the above calculations are shown

in the Table 4.1.below,

Table No. 4.1. Pressure Distribution and Deflections at various locations on raft.

Locations Pressure DistributionkN/m2

Deflections mm

A 95.167 11.604

B 95.167 15.415

C 95.167 11.604

D 70.820 12.502

E 70.820 17.529

F 70.820 12.502

G 46.473 5.667

H 46.473 7.527

I 46.473 5.667

40

The graphs below show the pressure distribution and deflection

profile of the raft.

95.167 95.167 95.16770.82 70.82 70.8246.473 46.473 46.473

020406080

100

ADG BEH CFI

Points

Pres

sure

Diit

ribut

ion

ABCDEFGHI

Graph No. 4.1. Pressure Distribution along ABC, DEF and GHI.

11.60415.415

11.60412.502

17.529

12.502

5.667 7.527 5.667

0

5

10

15

20

ADG BEH CFI

Points

Pres

sure

Diit

ribut

ion

ABCDEFGHI

Graph No. 4.2. Deflection profile along ABC, DEF and GHI.

41

95.167

70.82

46.473

95.167

70.82

46.473

95.167

70.82

46.473

0

20

40

60

80

100

ABC DEF GHI

PointsPr

essu

re D

iitrib

utio

n

ADGBEHCFI

Graph No. 4.3. Pressure Distribution along ADG, BEH and CFI.

11.604 12.502

5.667

15.41517.529

7.52711.604 12.502

5.667

0

5

10

15

20

ABC DEF GHI

Points

Pres

sure

Diit

ribut

ion

ADGBEHCFI

Graph No. 4.4. Deflection profile along ADG, BEH and CFI.

42

4.2. FINITE DIFFERENCE METHOD:

4.2.1. Without considering Soil-Structure Interaction: When soil structure interaction is not considered. The grid pattern is

selected by dividing the raft into rectangular areas as shown in the Figure

4.5.

L = 10m.

B = 8m.

λx = 2 m.

λy = 2.5 m.

Econcrete = 5700 (fck) 0.5 = 25.5 x 106 kN/m2

νconcrete = 0.15

D = Ech3/12 (1-νc2) = 17385.29

From finite difference plate bending equation, section 3.3.2, we can

write equation for deflection at 0 as,

[ ] [ ]

[ ]Dqww)w4(w6w

λy1


λx1

1174204

1210864321022953104

=+++−+

+++++++−++++−

Where q = Q/ λx λy

Q = Column load at 0 = 1694.813kN

λx = width of rectangular area in X – direction = 2 m.

λy = width of rectangular area in Y – direction =2.5 m.

So we can find the algebraic equation for deflections at point 0.

Similarly we can find the algebraic equations at various points on the grid.

Solving, these simultaneous equations by Gauss Elimination

method we can obtain deflections at various points on the grid.

44

The graphs below show the deflection profiles along the raft length and widths.

Table No. 4.2. Deflections at various locations on raft without SSI.

Node

Number Deflection

mm 0 21.31 9 2 7.4 3 9 4 7.4 6 1.8 8 1.8 10 1.8 12 1.8

0

9

21.3

9

001.8

7.4

1.800

5

10

15

20

25

'17--9-16 10-3-8 4-0-2 12-1-6 20-5-13'

Nodal Points

Def

lect

ion 17-10-4-12-20

9-3-0-1-516-8-2-6-13

Graph No. 4.5. Deflection profile along 17-10-4-12-20, 9-3-0-1-5 and

16-8-2-6-13.

45

0

7.4

21.3

7.4

00 1.8

9

1.8 005

10152025

'18-11-19 10-4-12 3-0-1 8-2-6 15-7-14'

Nodal PointsD

efle

ctio

n 18-10-3-8-1511-4-0-2-719-12-1-6-14

Graph No. 4.6. Deflection profile along 18-10-3-8-15,11-4-0-2-7 and

19-12-1-6-14.

4.2.2. Considering Soil-Structure Interaction on Winkler’s Soil model: When soil structure interaction is considered using Winkler’s soil model.

The grid pattern is selected by dividing the raft into rectangular areas as

shown in the figure 4.5.

L = 10m.

B = 8m.

λx = 2 m.

λy = 2.5 m.


νconcrete = 0.15

D = Ech3/12 (1-νc2) = 17385.29

Esoil = 40000 kN/m2 νsoil = 0.30.

From finite difference plate bending equation, section 3.3.2, the

equation for deflection at 0 is,

46

[ ] [ ]

[ ]λxλyD

QDqww)w4(w6w

λy1


λx1

1174204

1210864321022953104

+=+++−+

+++++++−++++−

Vesic (1961) proposed the following relationship for

computing the value of k in analysis of raft,

k = Es / B (1-νs2)

where, q = -k0 w0

k0 = Spring stiffness at point 0 = 5128.205 kN/m3.




As it is assumed that spring is present beneath each node so

stiffness of spring beneath each node depends on the contributing area of

the grid.

For point 0, the contributing area = 4 x (2/2) x (2.5/2) = 5 sqm.

Therefore,

k0 = Spring stiffness at point 0 = 5128.205 x 5 =25641.03 kN/m.

The contributing area for each node is obtained as shown in the figure 4.6.

So the algebraic equation for deflections at point 0 is found out.

Similarly the algebraic equations at various points on the grid are

obtained.


method deflections at various points on the grid are obtained.

48

Table No. 4.3. Deflection and Contact Pressure at various nodes on the grid considering SSI (Winkler’s Model).

Node Number Deflection

mm Contact Pressure

kN/m2 0 8.1 20.769 1 3.4 8.717 2 2.8 7.179 3 3.4 8.717 4 2.8 7.179 6 -1 - 8 -1 -

10 -1 - 12 -1 -

The graphs below show the deflection profiles along the raft length

and widths.

0

3.4

8.1

3.4

00-1

2.8

-10

-202468

10

'17--9-16 10-3-8 4-0-2 12-1-6 20-5-13'

Nodal Points

Def

lect

ion 17-10-4-12-20

9-3-0-1-516-8-2-6-13


16-8-2-6-13.

49

0

8.717

20.769

8.717

00 0

7.179

0 0

-5

0

5

10

15

20

25

'17--9-16 10-3-8 4-0-2 12-1-6 20-5-13'

Nodal PointsC

onta

ct P

ress

ure

Dis

trib

utio

n

17-10-4-12-209-3-0-1-516-8-2-6-13

Graph No. 4.8. Contact Pressure Distribution along17-10-4-12-20,

9-3-0-1-5 and 16-8-2-6-13.

0

2.8

8.1

2.8

00-1

3.4

-10

-2

0

2

4

6

8

10

'18-11-19 10-4-12 3-0-1 8-2-6 15-7-14'

Nodal Points

Def

lect

ion 18-10-3-8-15

11-4-0-2-719-12-1-6-14


19-12-1-6-14.

50

0

7.179

20.769

7.179

00 0

8.717

0 0

-5

0

5

10

15

20

25

'18-11-19 10-4-12 3-0-1 8-2-6 15-7-14'

Nodal PointsD

efle

ctio

n 18-10-3-8-1511-4-0-2-719-12-1-6-14

Graph No. 4.10. Contact Pressure Distribution along18-10-3-8-15,

11-4-0-2-7 and 19-12-1-6-14. 4.2.3. Considering Soil-Structure Interaction on Linear Elastic Soil model:

When soil structure interaction is considered using linear elastic soil

model, the grid pattern is selected by dividing the raft into rectangular

areas as shown in the figure 4.5.

L = 10m.

B = 8m.

λx = 2 m.

λy = 2.5 m.


νconcrete = 0.15

D = Ech3/12 (1-νc2) = 17385.29

Esoil = 40000 kN/m2 νsoil = 0.30.

51

From finite difference plate bending equation, section 3.3.2, the

equation for deflection at 0 is,

[ ] [ ]

[ ]λxλyD

QDqww)w4(w6w

λy1


λx1

1174204

1210864321022953104

+=+++−+

+++++++−++++−

Where q = -k0 w0

k = Spring stiffness at point 0.




The assumptions made for linear elastic soil model are,

(1) The foundation has the properties of semi-infinite elastic

body.

(2) Plate rests on sub grade without friction i.e. smooth base.

From dynamic tests k value is,

k = Es / 2(1-νs2).

Here the k value does not depend on contributing area and remains

same for all the nodes on the grid.

So the algebraic equation for deflections at point 0 is found out.

Similarly the algebraic equations at various points on the grid are

obtained.


method deflections at various points on the grid are obtained.

52

Table No. 4.4. Deflection at various nodes on the grid considering SSI on Linear Elastic Soil Model.

The graphs below show the deflection profiles along the raft

length and widths.

0

4.3

8.9

4.3

00-1

3.7

-10

-202468

10

'17--9-16 10-3-8 4-0-2 12-1-6 20-5-13'

Nodal Points

Def

lect

ion 17-10-4-12-20

9-3-0-1-516-8-2-6-13

Graph No. 4.11. Deflection profile along 17-10-4-12-20, 9-3-0-1-5 and 16-8-2-6-13.

Node Number Deflection mm

0 8.9 1 4.3 2 3.7 3 4.3 4 3.7 6 -1 8 -1 10 -1 12 -1

53

0

3.7

8.9

3.7

00-1

4.3

-10

-2

0

2

4

6

8

10

'18-11-19 10-4-12 3-0-1 8-2-6 15-7-14'

Nodal Points

Def

lect

ion 18-10-3-8-15

11-4-0-2-719-12-1-6-14


19-12-1-6-14.

4.3. FINITE ELEMENT METHOD: For analyzing raft foundation, the problem is considered as plane

stress problem, as the thickness in Z – direction is very small. Raft is

assumed as plate resting on soil. The plate follows Kirchoff’s plate theory

and the assumptions made are valid for raft foundation. Sub-structure

method of finite element analysis is used for solving the problem.

4.3.1. Discretization: Raft is divided into rectangular elements each of width (a= 2m) and

length (b= 2.5m). The discretization is done is similar to that of finite

difference method, so as to compare the deflections obtained by both

methods. Total raft is now divided into 16 elements.

The discretization is as shown in the figure 4.7.

54

4.3.2. Nodal Degrees of freedom and Interpolation function: Many researchers found that solving the problem with 3 degrees of

freedom at each node may solve the problem but the inter element

compatibility criteria is not satisfied. So introduction of additional degree of

freedom is required. (Desai.C.S. and Abel.J.F, 2000)

So Bogner et al, 1965, gave the formulation of interpolation function

with 4 DOF at each node.

DOF at each node,

w, θx = ∂w/∂x, θy = ∂w/∂y, θxy = ∂2w/∂x ∂y,

giving total 16 DOF for each element.

Cubic Hermitian polynomial is used as interpolation model.

Nx1 = 1- 3ζ2 + 2 ζ3. Nx2 = ζ2 (3 - 2 ζ).

Nx3 = aζ (ζ -1) 2.

Nx4 = aζ2 (ζ - 1). ζ = x/a

0<ζ < 1.

∂ /∂x = (1/a) ∂ /∂ζ.

Ny1 = 1- 3η2 + 2 η3. Ny2 = η2(3 - 2 η).

Ny3 = aη (η -1) 2.

Ny4 = aη2 (η - 1).

η = y/b

0<η < 1.

∂ /∂y = (1/b) ∂ /∂η.

56

By using shape functions we have defined, the displacement model

for plate bending element is,

w = Nx1 Ny1 w1 + Nx2 Ny1 w2 + Nx2 Ny2 w3 + Nx1 Ny2 w4

+ Nx3 Ny1 θx1 + Nx4 Ny1 θx2 + Nx4 Ny2 θx3 + Nx3 Ny2 θx4

+ Nx1 Ny3 θy1 + Nx2 Ny3 θy2 + Nx2 Ny4 θy3 + Nx1 Ny4 θy4

+ Nx3 Ny3 θxy1 + Nx4 Ny3 θxy2 + Nx4 Ny4 θxy3 + Nx3 Ny4 θxy4.

= [N] [q]. -----------(4.1)

After obtaining the shape functions we can obtain the element

strain displacement matrix as,

[ ]

∂∂∂∂∂∂∂

=yxw/

yw/xw/

ε2

22

22

= [B] [q] -----------------------------(4.2)

The element stiffness matrix can be further obtained,

[k] = Element stiffness matrix = ∫∫∫v [B] T[C] [B] dV.

{Q} = Load matrix = ∫∫∫v [N] T [X] dV + ∫∫ s1 [N] T{Tx} ds1.

For each element the element stiffness matrix and load matrix are

as given in the form below,

[k] {q} = {Q} -------------------------------(4.3)

4.3.3. Jacobain and Numerical Integration: The Jacobian can be obtained as discussed in the section 3.4.6,

J =

b

a0

0

Det J = ab.

The order of numerical integration is selected as in section 3.4.7, Order of integration = 2(p – m).

If, p = order of polynomial = 3.

m = order of differential = 2.

Order of integration = 2(3 – 2) = 2.

57

In this problem the order of integration is 2 x 2. Obtaining the

values of weights and sampling points from the table given in Appendix-C

we can obtain the element stiffness matrix.

0∫a0∫b F(ζ,η) dζdη = Σ Σ αi,j F(ζi,ηj)

where, αi,j = Weights.

ζi,ηj = sampling points.

4.3.4. Boundary Conditions: We need to specify only geometric boundary conditions; the natural

boundary conditions are implicitly satisfied in the solution procedure as

long as we employ a suitable valid variational principle.

The boundary conditions are as below,

w = 0, θx = 0, θy = 0, θxy = 0.

So these conditions are applied at nodes on the boundary

(1,2,3,4,5,6,10,11,15,16,20,21,22,23,24,25). These boundary conditions

are applied during the derivation of element stiffness matrices.

4.3.5. Global Stiffness Matrix: The direct stiffness method is employed universally for assembling

the algebraic equations in finite element application. The boundary

conditions prescribed or derived are applied to the element stiffness

matrices and then these reduced element stiffness matrices are

assembled together to obtain the global stiffness matrix. The individual

stiffness and loads are added directly to locations in overall matrices [k]

and [Q], in conformity with the requirement of one to one correspondence

between the nodes of the element and those of assemblage. The values

of deflections are then obtained by solving the equation by Gauss

Elimination Method.

58

4.3.6. Without considering soil-structure interaction: The deflections obtained by solving the problem by finite element

method are shown below.

Table No. 4.5. Deflection at various nodes on the grid without considering soil structure interaction by FEM.


13 24

14 12

18 8

12 12

8 8

19 5

17 4

7 4

9 4

59

0

12

24

12

004

84

00

5

10

15

20

25

30

'6-11-16 7-12-17 8-13-18 9-14-19 10-15-20'

Nodal PointsD

efle

ctio

n 6-7-8-9-1011-12-13-14-1516-17-18-19-20

Graph No. 4.13. Deflection profile along 6-7-8-9-10, 11-12-13-14-15

and 16-17-18-19-20.

0

8

24

8

004

12

400

51015202530

'2-3-4 7-8-9 12-13-14 17-18-19 22-23-24'

Nodal Points

Def

lect

ion 2-7-12-17-22

3-8-13-18-234-9-14-19-24

Graph No. 4.14. Deflection profile along 27-2-12-17-22,3-8-13-18-23 and 4-9-14-19-24.

60

4.3.7. Considering Soil-Structure Interaction on Winkler’s Soil Model: It is assumed that spring is present beneath each node so stiffness

of spring beneath each node depends on the contributing area of the grid.

The spring stiffness is estimated as obtained for FDM and then added to

the element stiffness matrix. So we can obtain raft-soil stiffness matrix.

The values of deflections are then obtained by solving the equation by

Gauss Elimination Method.

The deflections and contact pressure obtained are shown below,

Table No. 4.6. Deflection and Contact Pressure at various nodes on the grid considering SSI by FEM(Winkler’s model).


Contact Pressure KN/m2

13 10.9 27.94

14 5 12.82

18 4.03 10.33

12 5 12.82

8 3.8 9.74

19 1.1 2.82

17 1.1 2.82

7 1.1 2.82

9 1.1 2.82

61

0

5

10.9

5

001.1

4.03

1.100

2

4

6

8

10

12

'6-11-16 7-12-17 8-13-18 9-14-19 10-15-20'

Nodal PointsD

efle

ctio

n 6-7-8-9-1011-12-13-14-1516-17-18-19-20


16-17-18-19-20.

0

12.82

27.94

12.82

002.82

10.33

2.8200

5

10

15

20

25

30

'6-11-16 7-12-17 8-13-18 9-14-19 10-15-20'

Nodal Points

Def

lect

ion 6-7-8-9-10

11-12-13-14-1516-17-18-19-20

Graph No. 4.16. Contact Pressure Distribution along 6-7-8-9-10,

11-12-13-14-15 and 16-17-18-19-20.

62

0

3.8

10.9

4.03

001.1

5

1.100

2

4

6

8

10

12

'2-3-4 7-8-9 12-13-14 17-18-19 22-23-24'

Nodal PointsD

efle

ctio

n 2-7-12-17-223-8-13-18-234-9-14-19-24

Graph No. 4.17. Deflection profile along 2-7-12-17-22, 3-8-13-18-23

and 4-9-14-19-24.

0

9.74

27.94

10.33

002.82

12.82

2.8200

5

10

15

20

25

30

'2-3-4 7-8-9 12-13-14 17-18-19 22-23-24'

Nodal Points

Def

lect

ion 2-7-12-17-22

3-8-13-18-234-9-14-19-24

Graph No. 4.18. Contact Pressure Distribution along 2-7-12-17-22,

3-8-13-18-23 and 4-9-14-19-24.

63

4.4. Analysis of Raft foundation considering Soil-Structure Interaction on Winkler’s Soil Model using STAAD Pro-2004 software:

STAAD Pro-2004 software is analysis and design software for

structures. The problem considered in the present work is two dimensional

plane stress problem. STAAD Pro-2004 is used to consider three-

dimensional problem of the same. Here superstructure, raft and soil are

considered as three components of one elastic system.

The beams and columns are considered as single line elements

and each beam and column is considered as individual element. Raft is

considered as three-dimensional plate, which follows Kirchoff’s plate

theory, which is thin in one direction, and the deflections of raft are small

compared to the plate thickness. The deflections are obtained on the

middle plane of the plate. The soil is modeled as Winkler’s soil model. It is

assumed that beneath each node of the plate a spring is present.

As discussed above the model is generated and the value of the

spring stiffness is entered in the foundation menu bar.

The properties incorporated are,

L = 10m.

B = 8m.

λx = 2 m.

λy = 2.5 m.


νconcrete = 0.15

kplate = 5128.205 kN/m3

64

The results obtained are shown in the table below,

Table No. 4.7. Deflection at various nodes on the raft grid

considering soil structure interaction using STAAD Pro-2004.


13 9.064

14 3.01

18 2.75

12 3.01

8 2.75

19 1.07

17 1.07

7 1.07

9 1.07

0

3.01

9.064

3.01

001.07

2.751.07

00

2

4

6

8

10

'6-11-16 7-12-17 8-13-18 9-14-19 10-15-20'

Nodal Points

Def

lect

ion 6-7-8-9-10

11-12-13-14-1516-17-18-19-20

Graph No. 4.19. Deflection profile along 6-7-8-9-10, 11-12-13-14-15 and 16-17-18-19-20.

65

0

2.75

9.064

2.75

001.07

3.01

1.0700

2

4

6

8

10

'2-3-4 7-8-9 12-13-14 17-18-19 22-23-24'

Nodal Points

Def

lect

ion 2-7-12-17-22

3-8-13-18-234-9-14-19-24

Graph No. 4.20. Deflection profile along 2-7-12-17-22,

3-8-13-18-23 and 4-9-14-19-24.

66

CHAPTER 5

RESULTS AND DISCUSSIONS

This chapter discusses the comparison of values of deflections

obtained by various methods used for analysis. Also the comparison of

values of deflection by considering soil structure interaction and without

soil structure interaction is done. Comparing contact pressure and

deflection values for different L/B ratios parametric study of raft foundation

is carried out. Parametric study enlightens the effect of L/B ratio on the

deflection values when soil structure interaction is considered and soil is

modeled as Winkler’s soil model. Also when Conventional Method is used

the effect of L/B ratio is important as Cd i.e. shape and rigidity factor,

affects the value of deflection.

5.1. COMPARISON OF DEFLECTION VALUES BY FINITE DIFFERENCE

METHOD AND FINITE ELEMENT METHOD WITHOUT CONSIDERING SOIL-STRUCTURE INTERACTION:

Table No. 5.1. Comparison of Deflection values by FDM and FEM without considering Soil Structure Interaction.

Method of Analysis FDM FEM

Nodes (FDM) Nodes (FEM) Deflection mm

Deflection mm

0 13 21.3 24

1 14 9 12

2 18 7.4 8

3 12 9 12

4 8 7.4 8

6 19 1.8 5

8 17 1.8 4

10 7 1.8 4

12 9 1.8 4

67

From the values given in the Table 5.1, it has been observed that

the values of deflections obtained by Finite Element Method are more than

those obtained by Finite Difference Method. At central node the value of

deflection obtained by FDM is 21.3mm and that obtained by FEM is

24mm. At central node the value of deflection obtained by FDM is 11.25%

more than that obtained by FDM. This change in value of deflection is due

to the consideration of more degrees of freedom in FEM. In FDM only one

DOF is considered i.e. deflection (w), but in FEM at each node four DOF

are considered (w,θx, θy, θxy).

5.2. COMPARISON OF DEFLECTION VALUES BY FINITE DIFFERENCE METHOD AND FINITE ELEMENT METHOD WITH CONSIDERING SOIL-STRUCTURE INTERACTION:

Table No. 5.2. Comparison of deflection values by FDM and FEM

considering Soil Structure Interaction.

Method of Analysis

FDM Deflection

mm

FEM Deflection

mm

Nodes (FDM) Nodes (FEM) Winkler’s Model

Linear Elastic Model

Winkler’s Model

0 13 8.1 8.9 10.9

1 14 3.4 4.3 5

2 18 2.8 3.7 4.03

3 12 3.4 4.3 5

4 8 2.8 3.7 3.8

6 19 -1 -1 1.1

8 17 -1 -1 1.1

10 7 -1 -1 1.1

12 9 -1 -1 1.1

68

From the values given in the Table 5.2, it has been observed that

the values of deflections obtained by Finite Element Method are more than

those obtained by Finite Difference Method. Also when soil model

changes the value of deflection changes. This is due to the change in

formulation of calculating the value of ‘k’ for Winkler’s model and for Linear

Elastic Model. As during the calculation of ‘k’ value for Winkler’s soil model

the effective area of the grid comes into consideration and the value of

spring stiffness at each node is different, but while calculating ‘k’ value for

Linear Elastic Model it remains same throughout.

At central node (Node no. 0) the value of deflection obtained by

Linear Elastic Model is 8.9mm and the value of deflection at central node

(Node no. 0) obtained on Winkler’s soil model is 8.1mm. At central node

(Node no. 0) the value of deflection obtained by Linear Elastic Model is

8.98% more than that obtained on Winkler’s soil model.

At central node (Node no. 0) the value of deflection obtained by FEM on

Winkler’s soil model is 10.90mm.

Now if Winkler’s soil model is considered and analysis is done by

FDM and FEM it is observed that the difference in deflection at central

node is 25.68%.

5.3. COMPARITIVE STUDY OF DEFLECTION (mm) VALUES FOR RAFT FOUNDATION WITH AND WITHOUT SSI:

Graph 5.1, shows the values of deflections obtained by FDM

without SSI and with SSI along short span. The central node is node

number 0. When SSI is not considered the deflection at the center is

21.3mm and considering SSI on Winkler’s soil model is 8.1mm. If soil is

modeled as LEM the deflection at center is 8.9mm. Similarly for adjacent

nodes (Node no. 3and 1), when SSI is not considered the deflection is

69

9mm and considering SSI on Winkler’s soil model the deflection is 3.4mm.

If soil is modeled as LEM the deflection is 4.3mm.

At central node the deflection is reduced by approximately 60%

when soil is modeled as Winkler’s soil model. When soil is modeled as

LEM the deflections are reduced by approximately 55%.

0

9

21.3

9

0

03.4

8.1

3.40

0

4.3

8.9

4.3

00

5

10

15

20

25

9 3 0 1 5Nodal Points

Def

lect

ions

Without SSIWinklers ModelLEM

Graph No. 5.1. Comparison of deflections with and without SSI by FDM (Along short span).

Graph 5.2, shows the values of deflections obtained by FDM

without SSI and with SSI along long span. When SSI is not considered the

deflection at the center is 21.3mm and considering SSI on Winkler’s soil

model is 8.1mm. If soil is modeled as LEM the deflection at center is

8.9mm. Similarly for adjacent nodes (Node no. 3and 1), when SSI is not

considered the deflection is 7.4mm and considering SSI on Winkler’s soil

model the deflection is 2.8mm. If soil is modeled as LEM the deflection is

3.7mm.

70

At adjacent nodes the deflections are reduced by approximately

60% when soil is modeled as Winkler’s soil model. When soil is modeled

as LEM the deflections are reduced by approximately 50%.

0

7.4

21.3

7.4

00

2.8

8.1

2.8

00

3.7

8.9

3.7

00

5

10

15

20

25

11 4 0 2 7

Nodal Points

Def

lect

ions Without SSI

Winklers ModelLEM

Graph No. 5.2. Comparison of deflections with and without SSI by FDM (Along long span).

Graph 5.3, shows the values of deflections obtained by FEM

without SSI and with SSI along short span. The central node is node


24.0mm and considering SSI on Winkler’s soil model is 10.90mm.

Similarly for adjacent nodes (Node no. 12 and 14), when SSI is not

considered the deflection is 12mm and considering SSI on Winkler’s soil

model the deflection is 5mm.

At central node the deflection is reduced by approximately 55%

when soil is modeled as Winkler’s soil model.

71

0

12

24

12

00

5

10.9

5

00

5

10

15

20

25

30

11 12 13 14 15

Nodal Points

Def

lect

ions

Without SSI With SSI (Winklers model)

Graph No. 5.3. Comparison of deflections with and without SSI by FEM (Along short span).

Graph 5.4, shows the values of deflections obtained by FEM

without SSI and with SSI along long span. The central node is node


24.0mm and considering SSI on Winkler’s soil model is 10.90mm.

Similarly for adjacent nodes (Node no. 18 and 8), when SSI is not

considered the deflection is 8mm and considering SSI on Winkler’s soil

model the deflection is 4.03mm.

At adjacent nodes the deflections are reduced by approximately

50% when soil is modeled as Winkler’s soil model.

72

0

8

24

8

004.03

10.9

3.800

5

10

15

20

25

30

23 18 13 8 3

Nodal Points

Def

lect

ions

Without SSIWith SSI (Winklers Model)

Graph No. 5.4. Comparison of deflections with and without SSI by FEM (Along long span).

Graph 5.5, shows the comparison between the different methods of

analysis used for raft foundation when soil structure interaction is not

considered. The deflection at the center when conventional method of

analysis is used is 17.52mm. The deflection obtained at the center by

FDM is 21.3mm and that obtained by FEM is 24mm.

The deflection at the center obtained by Conventional method

(Rigid raft) is approximately 27% less than that obtained by Finite Element

method (Flexible raft). The deflection at the center obtained by

Conventional method (Rigid raft) is approximately 17.74% less than that

obtained by Finite Difference method (Flexible raft). The deflection at the

center obtained by Finite Difference method is approximately 11.25% less

than that obtained by Finite Element method.

73

12.515

17.5215

12.5

0

9

21.3

9

00

12

24

12

005

1015202530

9 3 0 1 5

Nodal Points

Def

lect

ions

Conventional FDMFEM

Graph No. 5.5. Comparison of deflections without SSI by Conventional Method, FDM and FEM. (Along short span).

Graph 5.6, shows the comparison between various methods of

analysis used for raft foundation when soil structure interaction is

considered. Also, the results obtained by using STAAD Pro- 2004

software are shown in the graph 5.6. At central node the deflection

obtained by FDM on Winkler’s soil model is 8.10mm and that obtained on

LEM is 8.90mm. At central node the deflection obtained by FEM on

Winkler’s soil model is 10.90mm and that obtained by using STAAD Pro

software is 9.064mm.

At the adjacent nodes the deflections obtained by FDM on

Winkler’s soil model is 3.40mm and that obtained on LEM is 4.30mm. At

the adjacent nodes the deflections obtained by FEM on Winkler’s soil

model is 5mm and that obtained by using STAAD Pro software is 3.01mm.

74

0

3.4

8.1

3.4

00

4.3

8.9

4.3

00

5

10.9

5

00

3.01

9.064

3.01

002468

1012

9 3 0 1 5

Nodal PointsD

efle

ctio

n FDM (Winkler's model)FDM (LEM)FEM (Winkler's model)FEM (STAAD Pro)

Graph No. 5.6. Comparison of deflections with SSI by FDM (Winkler’s

model), FDM (LEM) and FEM (Winkler’s model). (Along short span).

75

Table No. 5.3. Comparative study of various methods of analysis used without considering SSI and with SSI.

Method of Analysis. Conventional Method

Finite Difference Method Finite Element Method

Software STAAD Pro

Nodes (FDM)

Nodes (FEM)

Without SSI

Winkler’s Model

LEM Without SSI

Winkler’s Model

Winkler’s Model

0(Center) 13(Center) 17.52 21.3 8.1 8.9 24 10.9 9.064

1 14 - 9 3.4 4.3 12 5 3.01

2 18 - 7.4 2.8 3.7 8 4.03 2.75

3 12 - 9 3.4 4.3 12 5 3.01

4 8 - 7.4 2.8 3.7 8 3.8 2.75

5 15 12.50 0 0 0 0 0 0

6 19 - 1.8 -1 -1 5 1.1 1.07

7 23 7.52 0 0 0 0 0 0

8 17 - 1.8 -1 -1 4 1.1 1.07

9 11 12.50 0 0 0 0 0 0

10 7 - 1.8 -1 -1 4 1.1 1.07

11 3 15.41 0 0 0 0 0 0

12 9 - 1.8 -1 -1 4 1.1 1.07

13 20 - 0 0 0 0 0 0

14 24 - 0 0 0 0 0 0

15 22 - 0 0 0 0 0 0

16 16 - 0 0 0 0 0 0

17 6 - 0 0 0 0 0 0

18 2 - 0 0 0 0 0 0

19 4 - 0 0 0 0 0 0

20 10 - 0 0 0 0 0 0

21 25 5.66 0 0 0 0 0 0

22 21 5.66 0 0 0 0 0 0

23 1 11.60 0 0 0 0 0 0

24 5 11.60 0 0 0 0 0 0

77

5.4. PARAMATRIC STUDY: The deflections that are obtained by any method of analysis as

discussed in previous sections depend largely on L/B ratio. When

deflections for Conventional method are to be obtained, it depends on Cd

i.e. shape and rigidity factor which in turn depends on L/B ratio and shape

of the raft.

When method of analysis used is Finite Difference Method

considering soil structure interaction, deflection value depends on

modulus of sub grade reaction (k). The k value changes with the shape of

the contributing area of the raft.

For rectangular footings,

k = kplate (2/3) [1+(B/2L)]

Finite Element Method by considering soil structure interaction uses

the same concept for analysis of raft foundation. So parametric study is

done to observe the deflections for various L/B ratios.

5.4.1. Conventional Method of raft analysis:

Table No.5.4. For various L/B ratios the deflections and contact pressure.

L/B Ratio

1 1.25 1.5 1.75 2.0

Nodes

Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

A 23.05 102.37 11.60 95.16 9.41 92.63 8.02 92.13 8.01 92.65

B 23.05 102.37 15.41 95.16 12.50 92.63 10.66 92.13 10.32 92.65

C 23.05 102.37 11.60 95.16 9.41 92.63 8.02 92.13 8.01 92.65

D 16.48 73.20 12.50 70.81 10.47 71.18 9.16 72.66 9.50 74.64

E 16.48 73.20 17.52 70.81 14.68 71.18 12.84 72.66 12.90 74.64

F 16.48 73.20 12.50 70.81 10.47 71.18 9.16 72.66 9.50 74.64

G 9.91 44.02 5.66 46.47 5.05 49.73 4.63 53.19 4.89 56.63

H 9.91 44.02 7.52 46.47 6.71 49.73 6.15 53.19 6.31 56.63

I 9.91 44.02 5.66 46.47 5.05 49.73 4.63 53.19 4.89 56.63

78

Table 5.4, shows the values of deflections and contact pressure

obtained by Conventional Method for L/B ratio varying from 1 to 2. It is

observed that the values of deflection decrease with increase in L/B ratio.

This trend continues upto L/B ratio 1.75 and for L/B ratio this trend

changes. Similar trend is followed by contact pressure.

5.4.2. Finite Difference Method of raft analysis (Winkler’s Model): Table No.5.5. Deflection values by Finite difference method on Winkler’s

soil model for various L/B ratios. L/B

Ratio 1.0 1.25 1.50 1.75 2.0

Nodes Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

0 9.3 25.54 8.1 20.76 6.9 16.85 5.6 13.18 4.5 10.30

1 2.8 7.692 3.4 8.717 3.5 8.54 3.5 8.24 3.4 7.78

2 2.8 7.692 2.8 7.179 2.7 6.59 2.6 6.12 2.4 5.50

3 2.8 7.692 3.4 8.717 3.5 8.54 3.5 8.24 3.4 7.78

4 2.8 7.692 2.8 7.179 2.7 6.59 2.6 6.12 2.4 5.50

6 -1 - -1 - -1 - 0 - 0 -

8 -1 - -1 - -1 - 0 - 0 -

10 -1 - -1 - -1 - 0 - 0 -

12 -1 - -1 - -1 - 0 - 0 -


obtained by Finite Difference Method on Winkler’s model for L/B ratio

varying from 1 to 2. It is observed that the values of deflection at central

nodes (Node no. 0) decreases with increase in L/B ratio. Similar trend is

followed by contact pressure. But for adjacent nodes the deflections tend

to increase initially for L/B ratio 1 but approximately remain same for

further increase in L/B ratio.

79

5.4.3. Finite Difference Method of raft analysis (Linear Elastic Model):

Table No.5.6. Deflection values by Finite difference method on Linear Elastic soil model for various L/B ratios.

L/B Ratio 1.0 1.25 1.50 1.75 2.0

Nodes Deflection mm

Deflection mm

Deflection mm

Deflection mm

Deflection mm

0 10.9 8.9 7.3 5.8 4.6

1 3.4 4.3 4.4 4.1 4.1

2 3.4 3.7 3.1 2.9 2.9

3 3.4 4.3 4.4 4.1 4.1

4 3.4 3.7 3.1 2.9 2.9

6 -1 -1 -1 0 0

8 -1 -1 -1 0 0

10 -1 -1 -1 0 0

12 -1 -1 -1 0 0

Table 5.5, shows the values of deflections obtained by Finite

Difference Method on LEM model for L/B ratio varying from 1 to 2. It is

observed that the values of deflection decrease with increase in L/B ratio.

But for adjacent nodes the deflections tend to increase initially for L/B ratio

1 but approximately remain same for further increase in L/B ratio.

80

5.4.4. Finite Element Method of raft analysis (Winkler’s Model): Table No.5.7. Deflection values by Finite element method on Winkler’s soil

model for various L/B ratios. L/B Ratio 1.0 1.25 1.50 1.75 2.0

Nodes Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

Defl mm

CP KN/m2

13(center) 11.95 32.82 10.9 27.94 9.76 23.83 8.4 19.78 6.7 15.33

14 3.9 10.71 5 12.82 5.4 13.18 4.5 10.59 3.9 8.92

18 4 10.98 4.03 10.33 3.94 9.621 3.15 7.417 2.5 5.72

12 3.9 10.71 5 12.82 5.1 12.45 4.5 10.59 3.9 8.92

8 4 10.98 3.8 10.33 3.9 9.52 2.9 6.82 2.7 5.72

19 1.9 5.219 1.1 2.82 1 2.442 1.58 3.72 1.35 3.09

17 1.9 5.219 1.1 2.82 1.2 2.93 1.6 3.76 1.4 3.20

7 1.9 5.219 1.1 2.82 1 2.442 1.58 3.72 1.35 3.09

9 1.9 5.219 1.1 2.82 1.2 2.93 1.6 3.76 1.4 3.20


obtained by Finite Element Method on Winkler’s model for L/B ratio

varying from 1 to 2. It is observed that the values of deflection at central

nodes (Node no. 13) decreases with increase in L/B ratio. Similar trend is

observed for contact pressure distribution. But for adjacent nodes (Node

no. 12,14 etc) the deflections tend to increase initially for L/B ratio 1 but

approximately remain same for further increase in L/B ratio.

81

From the graph 5.7, it is observed that the deflection obtained at

the center decreases as the L/B ratio increases. This comparison is done

by keeping the value of L same throughout and by changing the value of

B.

EFFECT OF L/B RATIO

23.05

17.5214.68

12.84 12.9

9.38.1 6.9

5.6 4.5

10.98.9

7.35.8

4.6

11.95 10.9 9.768.4

6.7

0

5

10

15

20

25

1 1.25 1.5 1.75 2

L/B Ratio

Def

lect

ion

at c

ente

r

Conventional MethodFDM (Winklers)FDM (LEM)FEM (Winklers)

Graph No. 5.7. Relation of L/B ratio and deflection obtained at the center

by various methods of analysis.

From the graph 5.8, it is observed that the contact pressure

obtained at the center decreases as the L/B ratio increases. This

comparison is done by keeping the value of L same throughout and by

changing the value of B.

82

EFFECT OF L/B RATIO

102.3795.16 92.63 92.13 92.65

25.54 20.76 16.85 13.18 10.3

32.82 27.94 23.83 19.78 15.33

0

20

40

60

80

100

120

1 1.25 1.5 1.75 2

L/B Ratio

Con

tact

Pre

ssur

e at

cen

ter

Conventional MethodFDM (Winklers)FEM (Winklers)

Graph No. 5.8. Relation of L/B ratio and contact pressure obtained at the center by various methods of analysis.

83

CONCLUSIONS

1) Analysis of raft foundation by Conventional Method, considering

raft as rigid body, is found to give conservative values than those

obtained by FDM and FEM. The values obtained during analysis

may result over designing of the raft.

2) Using Finite Difference Method, by considering SSI on Winkler’s

model, the deflection at the center of the raft has been observed to

reduce by 60% as compared with that obtained without SSI.

Similarly reduction in deflection with Linear Elastic Model is

of the order of 53% as that of without SSI.

3) Using Finite Element Method, by considering SSI on Winkler’s

model, then the deflection at the center of the raft has been

observed to reduce by approximately 55% as compared with that

obtained without SSI.

4) Contact pressure obtained by Conventional Method has been

observed to be about 70% and 60% more than that obtained by

Finite Difference Method and Finite Element Method respectively.

5) Difference in deflections at the center of the raft by Finite Element

Method using Winkler’s model with two dimensional plate element

and three-dimensional plate element (STAAD Pro-2004) is

approximately same. Similarly for adjacent nodes the increase is

about 35%.

6) The deflection obtained at the center decreases as the L/B ratio

increases. This comparison is done by keeping the value of L same

throughout and by changing the value of B. Similarly, the contact

84

pressure obtained at the center decreases as the L/B ratio

increases.

Thus it can be concluded that raft should be designed by

considering Soil Structure Interaction using either Finite Difference

Method or Finite Element Method. However, Finite Element Method

(two or three dimensional plate element) is more preferable as it

takes into account four degrees of freedom and therefore results

are more realistic.

85

APPENDIX – A

In conventional method the deflections are calculated by the formula,

s

2snetd

i E)νB(1qC∆ −

=

Where, qnet = Net intensity of pressure

Cd = Shape and rigidity factor. νs = Poisson’s ratio of soil.

Es = Elastic modulus of soil.

The value of shape and rigidity factor Cd depends on shape of loaded area,

position of point for which settlement is to be estimated and stratification in

foundation soils. When the subsoil is uniform to infinite depth, values of Cd are to

be obtained from the following table given below, by Nayak, N.V, 2001.

Table No. A-1. Values of Cd for calculating settlement of points on loaded area at surface.

Shape Values of Cd for point at

Center Corner Middle of

short side

Middle of long

side

Average

Circle 1.00 0.64 0.64 0.64 0.85

Circle (Rigid) 0.79 0.79 0.79 0.79 0.79

Square 1.12 0.56 0.76 0.76 0.95

Square

(Rigid)

0.99 0.99 0.99 0.99 0.99

Rectangle

Length/width

1.5 1.36 0.67 0.89 0.97 1.15

2 1.52 0.76 0.98 1.12 1.30

3 1.78 0.88 1.11 1.35 1.52

5 2.10 1.05 1.27 1.68 1.83

10 2.53 1.26 1.49 2.12 2.25

100 4.00 2.00 2.20 3.60 3.70

1000 5.47 2.75 2.94 5.03 5.15

10000 6.90 3.50 3.70 6.50 6.60

86

APPENDIX – B

B-1. DERIVATION FOR FLEXURAL RIGIDITY OF PLATE:

Considering a long rectangular plate subjected to transverse load.

The deflected portion of the plate is assumed to be cylindrical with axis of

cylinder parallel to the plate.

Elemental strip of unit thickness is considered. Plate is considered

of uniform thickness ‘h’. XY plane is the middle of plate before bending.

Let positive direction of Z – axis is downward. Now for elemental strip

width of plate is ‘l’. Therefore strip is considered as bar have length ‘l’ and

depth ‘h’. It is assumed that cross section of bar remains plane during

bending. Therefore it undergoes only rotation with respect to neutral axis.

l

Unit width

w

X

Y

Fig No. B-1. Cylindrical Bending of Plates. Ref – Timoshinko,S.P, and Krieger,S.W.(1959)

M M h/2

Fig No. B-2. Section of Plate Bending. (Timoshinko,S.P, and Krieger,S.W. 1959)

87

Curvature of deflection = d2w/ dx2

w = Deflection in Z – direction.

Deflection is assumed to be small in as compared to length of the bar.

ε x at a distance z from middle surface = - z d2w/ dx2

ε x = σx/ E - ν σy/ E ------------------------(1)

Lateral strain in Y direction must be zero in order to maintain continuity

during plate bending.

ε y = σy/ E - ν σx/ E = 0

σy = ν σx -----------------------(2)

ε x = (1 - ν2)σx/ E

σx = E ε x / (1 - ν2) = - [ Ez / (1 - ν2)] d2w/ dx2 --------( 3)

Bending Moment,

M = -h/2h/2∫ z σx dz = --h/2

h/2∫ [ Ez2 / (1 - ν2)] d2w/ dx2. dz

= --h/2h/2∫ [ Ez2 / (1 - ν2)] d2w/ dx2. dz

= - [Eh3/12(1 - ν2)] d2w/ dx2 ------------------(4)

D = Flexural rigidity of plate = Eh3/12(1 - ν2) -----------------(5)

The basic differential equation is,

M = - D d2w/ dx2 --------------------(6)

88

B-2. DIFFERENTIAL EQUATION OF THE DEFLECTION SURFACE: Assumptions made are,

1) Load acting on plate is normal to its surface.

2) Deflections are small in comparison with thickness of plate.

3) At boundary edges of plate are free to move in plane of plate.

4) From the assumptions we can neglect any strain in middle plane of

plate during bending.

Take co-ordinate axis X-Y in middle plane of plate and Z- axis

perpendicular to that plane. In addition to moments Mx and My, twisting

moments Mxy are considered in pure bending. There are vertical shearing

forces acting on sides of element.

Magnitude of these shearing forces,

Qx = -h/2 h/2∫ ζxz dz.

Qy = -h/2 h/2∫ ζyz dz.

dx dy

Mx +(∂Mx/∂x)dx

Mxy +(∂Mxy/∂x)dx

Qx +(∂Qx/∂x)dx

Mxy +(∂Mxy/∂y)dy

My Qy

Fig No. B-3. Moments and Shear forces acting on Plates. (Timoshinko,S.P, and Krieger,S.W,1959)

89

Moments and shearing forces are functions of X and Y. While

discussing conditions of equilibrium we take into consideration the small

change by quantities dx and dy.

Distributed load over the plate is considered as qdxdy.

∂Qx dx - ∂Mxy dy + qdxdy = 0

∂x ∂y

∂Qx - ∂Mxy + q = 0 --------------------------------(7)

∂x ∂y

Taking moments @ X axis,

∂Mxy dxdy - ∂My dxdy + Qydxdy = 0

∂x ∂y

∂Mxy - ∂My + Qy = 0 ------------------------------(8)

∂x ∂y

Similarly,

∂Mxy + ∂Mx - Qx = 0 ------------------------------(9)

∂y ∂x

Obtaining values of Qx and Qy from equation (g) and (h) and

substituting in equation (i)

∂2Mx + ∂2Myx + ∂2My - ∂2Mxy = - q

∂x2 ∂x∂y ∂y2 ∂x∂y

Myx = - Mxy

90

∂2Mx - 2 ∂2Myx + ∂2My = - q --------------------------(10)

∂x2 ∂x∂y ∂y2

Mx = -D [∂2w/∂x2+ ν ∂2w/∂y2]

My = -D [∂2w/∂y2+ ν ∂2w/∂x2]

Myx = - Mxy = D (1 - ν2) ∂2w/∂xy

∂4w - 2 ∂4w + ∂4w = - q ------------------------(11)

∂x4 ∂x2∂y2 ∂y4 D

This is basic plate bending equation.

91

APPENDIX – C

But in Finite element methods the location and values of sampling points

as well as the weights are unknown, so a numerical integration scheme, which

optimizes both the sampling points and the weights, is to be used. This can be

done using Gauss Quadrature rule.

The basic assumption for Gauss Quadrature rule is,

0∫a0∫b F (ζ, η) dζdη = Σ Σ αi,j F(ζi,ηj)

where, αi, j = Weights.

ζi,ηj = sampling points. While performing numerical integration the values of sampling points and weights

are given in the table given below,

Table No. C-2. Values of Sampling points and weights for different number of sampling points.

Bathe, K.J, 1997. Number of sampling points Sampling points Weights

1 0.000000000000000 2.000000000000000 2 +0.577350269189626 1.000000000000000

3 +0.774596669241483 0.555555555555556

0.000000000000000 0.888888888888889

4 +0.861136311594053 0.347854845137454

+0.39981043584856 0.652145154862546

5 +0.906179845938664 0.236926885056189

0.538469310105683 0.478628670499366

0.000000000000000 0.568888888888889

6 +0.932469514203152 0.171324492379170

+0.661209386466265 0.360761573048139

+0.238619186083197 0.467913934572691

93

REFERENCES 1) King, G.W.J (1977), “An introduction to superstructure/raft/soil interaction”, International Symposium on Soil Structure Interaction, pp-453-466. 2) Hain,S.J, Lee, I.K (1974), “Rational analysis of raft foundation", American Society of Civil Engineers, Journal of Geotechnical Engineering, Vol- 100 No.GT7, pp-843 - 860. 3) Chandrasekaran,V.S (2001), “Numerical and Centrifuge Modeling in Soil Structure Interaction”, Indian Geotechnical Journal, Vol 31 No 1, pp – 1-60 4) Lee, I.K (1977), “Interaction analysis of raft and raft pile system”, International Symposium on Soil Structure Interaction, pp-513-520. 5) Milovic, S.D(1998),”A Comparison between observed and calculated large settlements of raft foundation”, Canadian Geotechnical Journal, Vol- 35, pp-251 - 263. 6) Horvath,J.S. (1983), “New Sub grade model applied to mat foundation", American Society of Civil Engineers, Journal of Geotechnical Engineering, Vol- 109 No.GT7, pp-1567 - 1587.

7) Ungureanu,N, Ciongradi,I, and Strat,l,(1977) “Framed structure foundation

beams soil interaction”, International Symposium on Soil Structure Interaction,

pp-101-108.

8) Devaikar, D.M (1977), “Contact pressure distribution under rigid footings”, International Symposium on Soil Structure Interaction, pp-231-236. 9) Jagdish,R and Sharda Bai,H (1977), “Design of combined footings on elastic foundation”, International Symposium on Soil Structure Interaction, pp-271-278. 10) Chen, W and Snitbhan,N (1977), “Analytical studies for solution of soil structure interaction problems”, International Symposium on Soil Structure Interaction, pp-557-575.

94

BOOKS REFERRED

1) Kurian,N (1981), “Modern Foundation – Introduction to Advanced Techniques”, Tata Mcgraw Hill Publishing Company, New Delhi. 2) Kurian,N (1992), “Design of Foundation Ststems”, Narosa Publishing House, New Delhi. 3) Timoshenko,S.P and Krieger,S (1959), “Theory of Plates and Shells”, Tata Mcgraw Hill Publishing Company, New Delhi. 4) Desai,C.S and Abel,J.F (2000), “ Introduction to the Finite Element Method”, CBS Publishers and Distributers, New Delhi. 5) Bowels, J.E, (1988), “ Foundation Analysis and Design”, Tata Mcgraw Hill Publishing Company, New Delhi. 6) Bathe, K-J, (1997), “ Finite Element Procedures”, Princeton Hall of India Pvt. Ltd, New Delhi. 7) Zienkiewicz, O.C,(1997), “ The Finite Element Method”, Tata Mcgraw Hill Publishing Company, New Delhi.

Documents

Mtech Dissertation - Analysis of Raft Foundation Using Soil-structure Interaction