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 requirement of the degree of

Master of Technology (Civil – Geotechnical Engineering) 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

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 obliged to the library staff for their kind co-operation during the literature survey.

Balraj Suresh Shevade, M.Tech ( Civil – Geotechnical Engg.) Roll. No. - M0310G08

Place- Pune Date- 06.06.2005

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.

3.

Description

PageNo.

Methods of Analysis. 3.1.

General.

16

3.2.

Conventional Method.

17

3.2.1. Methodology. 3.3.

17

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. 3.4.6. Jacobian Operator.

4.

29 32

3.4.7. Numerical Integration.

33

3.4.8. Boundary Conditions.

34

3.4.9. Global Stiffness Matrix

35

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 Model.

51

Sr.No.

Description 4.3.

PageNo.

Finite Element Method. 4.3.1. Discretization.

54 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 model).

78

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. 4.1.

Description Pressure Distribution along ABC, DEF and

Page No. 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. 4.7.


48

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



49


Contact Pressure along18-10-3-8-15,

50



52

9-3-0-1-5 and 16-8-2-6-13 considering SSI on LEM model by FDM. 4.12.

Deflection Profile along18-10-3-8-15, 11-4-0-2-7 and 19-12-1-6-4 considering SSI on LEM model by FDM.

53

4.13.


59

11-12-13-14-15 and 16-17-18-19 by FEM. 4.14.


59

3-8-13-18-23 and 4-9-14-19-24 by FEM. 4.15.


61

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 Winkler’s soil model. 4.17.


62

3-8-13-18-23 and 4-9-14-19-24 by FEM on Winkler’s soil model. 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 Winkler’s soil model. 4.19.


64

11-12-13-14-15 and 16-17-18-19 by FEM using STAAD Pro. 4.20.


65

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). 5.2.


70

by FDM (along long span). 5.3.


71

by FEM (along short span). 5.4.

Comparison of deflection with and without SSI by FEM (along long span).

72

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 obtained at center by various methods of analysis.

82

LIST OF TABLES Fig.No. 4.1.

Description

Page No.

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. 5.5.


78

pressures by FDM on Winkler’s model. 5.6.

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

79

5.7.


80

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,η j - 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.

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.

1

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. e x = 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 SoilStructure Interaction comes in picture. The present work deals with the analysis of raft by all three methods discussed in the subsequent chapters ahead.

2

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 3

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.

4

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

5

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

6

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, pαy p=ky k = Constant of proportionality = Modulus of sub grade reaction. p and y are mutually dependent.

7

(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.

9

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. pαy p=ky 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 (k h) and if structure in contact is horizontal, contact pressure is vertical (k v). 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

11

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 semiinfinite. 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:

12

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 w 2r. This can be visualized as the unit weight of material and is increased by factor n = w 2r/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.

13

Load or Pressure 1.25mm

Settlement

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.

14

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.

15

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.

16

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, q=

Q Qe x y ± A I yy

±

Qe y x Ixx

----------(3.1)

If the resultant is not eccentric then the pressure distribution will be uniform, q=

Q A

----------(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.

17

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, 2

∆ i =

C d qnet B(1 − ν s ) Es

--------------(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.

18

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)

∂4w ∂4w ∂4w q +2 2 2 + 4 = D ∂ x 4 ∂ x ∂ y ∂ y

------------(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 = E ch3/12 (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.

19

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(w 1 + 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, 1 [6w 0 λx 4

+

− 4(w 1 + w 3 + w 5 + w 9 ] +

1 [6w 0 λy 4

1

λx λy 2 2

− 4(w 2 + w 4 ) + w 7 + w 11 ] =

[8w 0 − 4(w 1 + w 2 + w 3 + w 4 ) + 2(w 6 + w 8 + w 10 + w 12 )]

q D

-----------(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.

21

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.

22

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.

23

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.

U  P  [U] =  b  , [P] =  b   Ui   Pi  [Ub] = Boundary displacements common to super structure and soil. [Ui] = interior displacements of super structure. [Pb] and [Pi] = set of external forces.

K bb K bi  Ub  Pb  K U  = P  K ii   i   ib  i

---------(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.

24

Force displacement relation for supporting medium i.e. soil, [Ks] [δ] = [F]

----------------------------- (3.13)

[Ks] = Supporting medium stiffness matrix.

U  [δ] =  b  δs 

Fb  , [F] =   Fs 

K s bb K s bi  Ub  Fb  K   =    s ib K s ii   δ s  Fs 

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

Combining equations (3.12) and (3.14).

K sbb + K bb − K biK ii −1K ib K sbi  Ub  Fb + Pb − K biK ii −1Pb    δ  =   K K F s   sib sii  s   

-----(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 soilstructure 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 platebending elements.

25

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 must include rigid body displacements of the element.

27

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

28

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) -------------( 3.18)

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

29

------------(3.19)

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) ------------( 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) ------------ (3.22)

{ε} = Vector of relevant strain components at an arbitrary point within the finite element. [B] = Strain displacement matrix.

30

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

[k] {q} = {Q}

-----------(3.24) -----------(3. 24)

---------------(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,

31

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 coordinates, we have,

∂Ni ∂Ni ∂x ∂Ni ∂y = + ∂ζ ∂x ∂ζ ∂y ∂ζ

---------------(3.26)

Similar relation can be devised for η co-ordinates.

∂Ni ∂Ni ∂x ∂Ni ∂y = + ∂η ∂x ∂η ∂y ∂η

---------------(3.27)

Writing the above equations in matrix form,

 ∂Ni /∂ζ   ∂x/∂ζ ∂y/∂ζ  ∂Ni /∂x  ∂N /∂η = ∂x/∂η ∂x/∂η ∂N /∂y   i   i  

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

 ∂Ni /∂ζ  ∂Ni /∂x  ∂N /∂η = J∂N /∂y   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

32

-----------------(3.30)

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,

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

0

------------------------(3.31)

where, αi,j = Weights.

ζi,η j = sampling points. During this procedure to change the intervals of integration from (a,b) to (-1,1).

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

0

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.

33

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.

34

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/m 3 Wall thickness = 230mm Unit weight of wall = 19.5 kN/m 3 Height of parapet wall = 1.5 m Floor Height = 3 m Grade of concrete – M20

Grade of steel – Fe415

0.5 Econcrete = 5700 (f ck ck)

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,

35

4m

45° 5m

Fig No.4.4. Load transmission to supporting beams.

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.

38

From the formula given in section 3.2.1, we have, q=

Q Qe x y ± A I yy

±

Qe y x Ixx

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, 2

∆ i =


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 Distribution

Deflections

kN/m2

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

39

The graphs below show the pressure distribution and deflection profile of the raft.

Points ADG n o i t 0 u b i 20 r t i i 40 4 6.47 3 D e 60 r 7 0.82 u 80 s 9 5.16 7 s 100 e r P

BEH

ABC CFI

DEF GHI

4 6 .4 7 3

4 6 .4 7 3

7 0 .8 2

7 0 .8 2

9 5 .1 6 7

9 5 .1 6 7

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

Points n ADG o i t 0 u b i r 5 5.667 t i i D10 11.604 e r 15 u 12.502 s s e 20 r P

BE H

ABC CFI

DEF GHI

7.527

5.667 11.604

15.415

12.502

17.529

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

40

Points AB C n 0 o i t u 20 b i r t i 40 i D e 60 r u s 80 s e r 100 P

DEF

ADG GHI

BEH CFI

46.473 46.473 70.82 70.82 95.167 95.167

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

Points A BC

n o i t 0 u b i r 5 t i i D10 e r u 15 s s e r 20 P

DEF

ADG GHI

BEH CFI

5.667 5.667 7.527 11.604 11.604 15.415

12.502 12.502 17.529

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

41

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. 0.5 Econcrete = 5700 (f ck = 25.5 x 10 6 kN/m2 ck)

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,

1 [6w 0 λx 4

+

− 4(w 1 + w 3 + w 5 + w 9 ] +

1 [6w 0 λy 4

1

λx λy 2 2

− 4(w 2 + w 4 ) + w 7 + w 11 ] =

[8w 0 − 4(w 1 + w 2 + w 3 + w 4 ) + 2(w 6 + w 8 + w 10 + w 12 )]

q D

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.

42

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 0 1 2 3 4 6 8 10 12

Deflection mm 21.3 9 7.4 9 7.4 1.8 1.8 1.8 1.8

Nodal Points '17--9-16 0

0

10-3-8

20

12-1-6

1.8

5

n o i t 10 c e l f 15 e D

4-0-2

9

1.8 7.4

9

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

21.3

25

Graph No. 4.5. Deflection profile along 17-10-4-12-20, 9-3-0-1-5 and 16-8-2-6-13.

44

Nodal Points '18-11-19 0

10-4-12

0

n 5 o i t 10 c e l 15 f e D 20

3-0-1

8-2-6

1.8 7.4

15-7-14'

1.8

0

7.4

9

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

21.3

25

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.


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,

45

1

λx 4

+

[6w 0 − 4(w 1 + w 3 + w 5 + w 9 ] +

1

λy

4

1

λx 2 λy 2

[6w 0 − 4(w 2 + w 4 ) + w 7 + w 11 ] =

Vesic

(1961)

[8w 0 − 4(w 1 + w 2 + w 3 + w 4 ) + 2(w 6 + w 8 + w 10 + w 12 )]

q Q + D λxλyD

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/m 3. 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. 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. Solving, these simultaneous equations by Gauss Elimination method deflections at various points on the grid are obtained.

46

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

Deflection mm 8.1 3.4 2.8 3.4 2.8 -1 -1 -1 -1

0 1 2 3 4 6 8 10 12

Contact Pressure kN/m2 20.769 8.717 7.179 8.717 7.179 -

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

Nodal Points -2 '17--9-16 0 n o i t c e l f e D

0

10-3-8 -1

2 4

3.4

4-0-2

2.8

20-5-13' 0 17-10-4-12-20

3.4

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

6 8

12-1-6 -1

8.1

10


48

Nodal Points '1-57--9-16 e 0 r u n s o 5 s i e t r u P b 10 t i r c t s a i 15 t D n o 20 C

0

10-3-8

4-0-2

0

12-1-6 0

8.717

7.179

20-5-13' 0 17-10-4-12-20

8.717

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

20.769

25

Graph No. 4.8. Contact Pressure Distribution along17-10-4-12-20, 9-3-0-1-5 and 16-8-2-6-13.

Nodal Points -2 '18-11-19 0 n o i t c e l f e D

2 4

10-4-1 -12

3-0-1

0

15-7-14' 0

2.8

3.4

2.8

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

6 8

8-2-6 -1

8.1

10


49

Nodal Points -5 '18-11-19 0

10-4-12

0

3-0-1

0

n 5 o i t c 10 e l f e D 15

7.179

8-2-6 0

8.717

15-7-14' 0

7.179

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

20

20.769

25

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.


νconcrete = 0.15 D = Ech3/12 (1-νc2) = 17385.29 Esoil = 40000 kN/m2

νsoil = 0.30.

50

From finite difference plate bending equation, section 3.3.2, the equation for deflection at 0 is, 1

λx

+

4

[6w 0 − 4(w 1 + w 3 + w 5 + w 9 ] +

1

λy 4

1

λx λy 2 2

[6w 0 − 4(w 2 + w 4 ) + w 7 + w 11 ] =

[8w 0 − 4(w 1 + w 2 + w 3 + w 4 ) + 2(w 6 + w 8 + w 10 + w 12 )]

q Q + D λxλyD

Where q = -k 0 w0 k = Spring stiffness at point 0. 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. 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. Solving, these simultaneous equations by Gauss Elimination method deflections at various points on the grid are obtained.

51

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

Node Number

Deflection mm 8.9 4.3 3.7 4.3 3.7 -1 -1 -1 -1

0 1 2 3 4 6 8 10 12

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

Nodal Points -2 '17--9-16 0 n o i t c e l f e D

0

10-3-8 -1

4-0-2

12-1-6 -1

2 4

10

0 17-10-4-12-20

4.3

3.7

4.3

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

6 8

20-5-13'

8.9


52

Nodal Points -2 '18-11-19 10-4-12 -1 0 0 n o i t c e l f e D

3-0-1

8-2-6 -1

15-7-14' 0

2 4

18-10-3-8-15 3.7

4.3

3.7

11-4-0-2-7 19-12-1-6-14

6 8

8.9

10


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.

53

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.

54

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,

 ∂ 2 w/∂x 2    [ε ] =  ∂ 2 w/∂y 2  ∂ 2 w/∂x∂y  = [B] [q]

--------------------------------------- ---------------(4.2) ---(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)

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

a 0  0 b  

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.

56

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.

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

0

where, αi,j = Weights.

ζi,η j = 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.

57

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. Node Number

Deflection mm

13

24

14

12

18

8

12

12

8

8

19

5

17

4

7

4

9

4

58


7-12-17

8-13-18

9-14-19

0

0 4

5

10-15-20'

4 8

n o 10 i t c e 15 l f e 20 D

12

6-7-8-9-10

12

11-12-13-14-15 16-17-18-19-20 24

25 30


Nodal Points '2-3-4 0 5 n o i 10 t c 15 e l f e 20 D 25 30

7-8-9

12-13-14

17-18-19

0

22-23-24' 0

4 8

4 8 12 24

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

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

59

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). Node Number 13

Deflection mm 10.9

Contact Pressure KN/m2 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

60


8-13-18

9-14-19

10-15-20'

0

0 1.1

2 n o i t c e l f e D

7-12-17

4

1.1 4.03

5

6

6-7-8-9-10

5

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

8 10

10.9

12


Nodal Points '6-11-16 0 5 n o 10 i t c 15 e l f e 20 D

7-12-17

8-13-18

9-14-19

0

10-15-20' 0

2.82

12.82

2.82 10.33

12.82

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

25 30

27.94

Graph No. 4.16. Contact Pressure Distribution along 6-7-8-9-10, 11-12-13-14-15 and 16-17-18-19-20.

61


12-13-14

17-18-19

0 1.1

3.8

4

22-23-24' 0

1.1

2 n o i t c e l f e D

7-8-9

5

4.03

2-7-12-17-22

6

3-8-13-18-23

8

4-9-14-19-24

10

10.9

12


Nodal Points '2-3-4 0 5 n 10 o i t c 15 e l f e D 20

7-8-9

12-13-14

17-18-19

0

22-23-24' 0

2.82

2.82

9.74 12.82

10.33

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

25 30

27.94

Graph No. 4.18. Contact Pressure Distribution along 2-7-12-17-22, 3-8-13-18-23 and 4-9-14-19-24.

62

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 threedimensional 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.


νconcrete = 0.15 kplate = 5128.205 kN/m 3

63

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. Node Number

Deflection mm

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

Nodal Points '6-11-16 0 n o i t c e l f e D

2 4

7-12-17

8-13-18

9-14-19

0

10-15-20' 0

1.07 3.01

1.07 2.75

3.01

6-7-8-9-10 11-12-13-14-15

6

16-17-18-19-20

8 10

9.064


64

Nodal Points '2-3-4 0 2 n o i t c e l f e D

7-8-9

12-13-14

17-18-19

0

22-23-24' 0

1.07 2.75

1.07 3.01

2.75

4

2-7-12-17-22 3-8-13-18-23

6

4-9-14-19-24

8 9.064 10


65

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 C d 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 Nodes (FDM)

Nodes (FEM)

FDM

FEM

Deflection

Deflection

mm

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

66

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 SOILSTRUCTURE INTERACTION:

Table No. 5.2. Comparison of deflection values by FDM and FEM considering Soil Structure Interaction.

Method of

FDM

FEM

Analysis

Deflection

Deflection

mm

mm

Nodes (FDM)

Nodes (FEM)

Winkler’s

Linear Elastic

Winkler’s

Model

Model

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

67

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

68

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%.

9 0

3

0

Nodal Points 0

3.4

1

5 0

3.4

0

0

5 4.3

s n o 10 i t c e l f 15 e D

9

8.1 8.9

4.3 9

Without SSI Winklers Model LEM

20 21.3

25

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.

69

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%. Nodal Points 11 0 0 0 5 s n 10 o i t c e l f e 15 D

4

0

2

7

2.8

2.8

0 0

3.7

3.7 8.1

7.4

7.4

8.9

Without SSI Winklers Model LEM

20 25

21.3

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 number 13. When SSI is not considered the deflection at the center is 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.

70

Nodal Points 11 0 5 s n 10 o i t c 15 e l f e 20 D

25

12

13

14

15

0

0 5

5 Without SSI

12

10.9

12

With SSI (Winklers model)

24

30

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 number 13. When SSI is not considered the deflection at the center is 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.

71

Nodal Points 23 0 5 s n 10 o i t c 15 e l f e 20 D

25

18

13

8

3

0

0 3.8

4.03 8

8 10.9

Without SSI With SSI (Winklers Model)

24

30

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.

72

Conventional FDM Nodal Points 9 0

3

0

FEM 1

5

0

0

5

s n 10 o i t c 15 e l f e 20 D

12.5

9 12 15

25

17.52 21.3 24

9 12 15

12.5

30

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.

73

Nodal Points 9 0 00 4 6 8 10 12

0

1

0

2 0 n o i t c e l f e D

3

5 0 0

3.4 3.01 4.3 5

3.4 3.01 4.3 8.1 8.9

0 0

5

9.064

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

10.9

Graph No. 5.6. Comparison of deflections with SSI by FDM (Winkler’s model), FDM (LEM) and FEM (Winkler’s model). (Along short span).

74

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

Conventional Conventional

Finite Difference Method

Finite Element

Software

Method

STAAD Pro

Method Nodes

Nodes

Without

Winkler’s

(FDM)

(FEM)

SSI

Model

0(Center)

13(Center)

17.52

21.3

8.1

1

14

-

9

2

18

-

3

12

4

Without

Winkler’s

Winkler’s

SSI

Model

Model

8.9

24

10.9

9.064

3.4

4.3

12

5

3.01

7.4

2.8

3.7

8

4.03

2.75

-

9

3.4

4.3

12

5

3.01

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 1 .07

11

3

15.41

0

0

0

0

0

0

12

9

-

1.8

-1

-1

4

1.1

1.07 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

75

LEM

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 C d 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

1

1.25

1.5

1.75

2.0

Ratio Nodes

Defl

CP

Defl 2

CP

Defl

CP

mm

KN/m

2

mm

KN/m

Defl 2

KN/m

2

Defl

CP

mm

KN/m

2

mm

KN/m

A

23.05 23.05

102.37 102.37

11.60 11.60

95.16 95.16

9.41 9.41

92.63 92.63

8.02

92.13 92.13

8.01 8.01

92.65 92.65

B

23. 23.05

102.37

15.4 5.41

95.16

12.5 2.50

92. 92.63

10.6 0.66

92.13

10.3 0.32

92. 92.65

C

23.05 23.05

102.37 102.37

11.60 11.60

95.16 95.16

9.41 9.41

92.63 92.63

8.02

92.13 92.13

8.01 8.01

92.65 92.65

D

16.4 16.48 8

73.2 73.20 0

12.5 12.50 0

70.8 70.81 1

10.4 10.47 7

71.1 71.18 8

9.1 9.16 6

72. 72.66 66

9.50 9.50

74.6 74.64 4

E

16. 16.48

73.20

17.5 7.52

70.81

14.6 4.68

71. 71.18

12.8 2.84

72.66

12.9 2.90

74. 74.64

F

16.4 16.48 8

73.2 73.20 0

12.5 12.50 0

70.8 70.81 1

10.4 10.47 7

71.1 71.18 8

9.1 9.16 6

72. 72.66 66

9.50 9.50

74.6 74.64 4

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 5.66

46.47

5.05

49.73

4.63

53.19

4.89

56.63

77

mm

CP

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

1.0

1.25

1.50

1.75

2.0

Ratio Nodes

Defl

CP

Defl

CP

Defl

CP

Defl

CP

Defl

CP

mm

KN/m

mm

KN/m

2

mm

KN/m

2

mm

KN/m

2

mm

KN/m

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

-

2

Table 5.5, shows the values of deflections and contact pressure 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.

78

2

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

Deflection

Deflection

Deflection

Deflection

mm

mm

mm

mm

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.

79

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 Nodes

1.0

1.25

Defl

CP

mm

KN/m

13(center)

1.50

Defl

CP

2

mm

KN/m

11.95

32.82

14

3.9

18

1.75

Defl

CP

2

mm

KN/m

10.9

27.94

10.71

5

4

10.98

12

3.9

8

2.0

Defl

CP

Defl

CP

2

mm

KN/m

2

mm

KN/m

9.76

23.83

8.4

19.78

6.7

15.33

12.82

5.4

13.18

4.5

10.59

3.9

8.92

4.03

10.33

3.94

9.621

3.15

7.417

2.5

5.72

10.71

5

12.82

5.1

12.45

4.5

10.59

3.9

8.92

4

10.98 1 0.98

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 2.82

1.2 1.2

2.93

1.6

3.76

1.4 1.4

3.20

Table 5.7, shows the values of deflections and contact pressure 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.

80

2

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 25 23.05 r e t n e c t a n o i t c e l f e D

20 17.52 15

Conventional Method 14.68

11.95 10.9 10 9.3 5

12.9

12.84 10.9 8.9 8.1

9.76 7.3 6.9

8.4 5.8 5.6

FDM (Winklers) FDM (LEM) FEM (Winklers)

6.7 4.6 4.5

0 1

1 .2 5

1 .5

1.75

2

L/B Ratio

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.

81

EFFECT OF L/B RATIO 120

r e t 102.37 n 100 e c t a 80 e r u s 60 s e r 32.82 P 40 t c a t n 20 25.54 o C

95.16

92.13

92.65 Conventional Method FDM (Winklers)

27.94

20.76

0

1

92.63

1 .2 5

FEM (Winklers) 23.83

19.78

16.85

13.18

1 .5

1 .7 5

15.33 10.3 2

L/B Ratio

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

82

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

83

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.

84

APPENDIX – A

In conventional method the deflections are calculated by the formula, 2

∆ i =


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 C d for calculating settlement of points on loaded area at surface. Shape

Values of Cd for point at Center

Corner

Middle of

Middle of long

short side

side

Average

Circle

1.00

0.64 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

0.99

0.99

0.99

0.99

0.99

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 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

(Rigid) Rectangle Length/width

85

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.

l

X Unit width

w Y

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

M

h/2

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

86

Curvature of deflection = d 2w/ 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 d 2w/ dx2 ε x = σx/ E - ν σy/ E

----------------------------------- ------------(1) (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/2h/2∫ [ Ez2 / (1 = --h/2h/2∫ [ Ez2 / (1 = - [Eh3/12(1 -

ν2)] d2w/ dx2. dz

ν2)] d2w/ dx2. dz

ν2)] d2w/ dx2

------------------(4)

ν2)

-----------------(5)

D = Flexural rigidity of plate = Eh 3/12(1 The basic differential equation is, M = - D d2w/ dx2

--------------------(6)

87

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 M x and My, twisting moments M xy are considered in pure bending. There are vertical shearing forces acting on sides of element.

dx d

Mx +(∂Mx/∂x)dx Mxy +(∂Mxy/∂y)dy

Mxy +(∂Mxy/∂x)dx

My

Qx +(∂Qx/∂x)dx

Qy

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

Magnitude of these shearing forces, Qx = -h/2 h/2∫ ζxz dz. Qy = -h/2 h/2∫ ζyz dz.

88

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 ∂x

------------------------------------ --------------------(7) -------(7)

∂y

Taking moments @ X axis,

∂Mxy dxdy - ∂My dxdy + Qydxdy = 0 ∂x

∂y

∂Mxy - ∂My + Qy = 0 ∂x

---------------------------------------- ----------------(8) ---(8)

∂y

Similarly,

∂Mxy + ∂Mx - Qx = 0 ∂y

---------------------------------------- ----------------(9) ---(9)

∂x

Obtaining values of

Q x 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

89

∂2Mx - 2 ∂2Myx + ∂2My = - q ∂x2

∂x∂y

------------------------------------ --------------(10) -(10)

∂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 ∂x4

∂x2∂y2

∂y4

D

This is basic plate bending equation.

90

------------------------(11 -------------- ----------(11))

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,

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

0

where, αi, j = Weights.

ζi,η j = 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 1 2

Sampling points 0.000000000000000 +0.577350269189626

Weights 2.000000000000000 1.000000000000000

3

+0.774596669241483

0.555555555555556

0.000000000000000

0.888888888888889

+0.861136311594053

0.347854845137454

+0.39981043584856

0.652145154862546

+0.906179845938664

0.236926885056189

0.538469310105683

0.478628670499366

0.000000000000000

0.568888888888889

+0.932469514203152

0.171324492379170

+0.661209386466265

0.360761573048139

+0.238619186083197

0.467913934572691

4

5

6

91

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, Vol109 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.

93

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.

94

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

Recommend Documents