A Study on Soil Flexibilty on Internal Force Distribution of Buildings

ADDIS ABABA UNIVERSITY

SCHOOL OF GRADUATE STUDIES

A STUDY ON

THE INFLUENCE OF SOIL FLEXIBILITY

ON THE

INTERNAL FORCE DISTRIBUTION OF BUILDINGS

SUBJECTED TO LATERAL STATIC LOADS

By Amsalu Gashaye Birhan

January 2005

ADDIS ABABA UNIVERSITY

SCHOOL OF GRADUATE STUDIES

A STUDY ON


ON THE

INTERNAL FORCE DISTRIBUTION OF BUILDINGS

SUBJECTED TO LATERAL STATIC LOADS


Approved by the Board of Examiners

Dr. - Ing. Asrat Worku _________________

Advisor

Dr. Shiferaw Taye ________________

External Examiner

Dr Adil Zekaria ________________

Internal examiner

Prof. Alemayehu Tefera ________________

Chairman

A STUDY ON


ON THE

INTERNAL FORCE DISTRIBUTION OF BUILDINGS SUBJECTED

TO LATERAL STATIC LOADS

A thesis submitted to the School of Graduate studies of Addis Ababa University

in partial fulfillment of the Requirements of the Degree of Masters in Civil

Engineering.


-i-

ACKNOWLEDGEMENT

I express my special thanks to my advisor Dr.-Ing. Asrat Worku, Associate Professor at the

Department of Civil Engineering, Technology Faculty, Addis Ababa University for his

timely guidance and support throughout the work.

And most importantly, I must put on record my gratitude to my families and friends for their

support during the course of my work.

Last but not least, thanks to all those who have made their contributions towards the success

of this work.

-ii-

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT.............................................................................................................. i

List of Figures............................................................................................................................... iv

List of Tables ............................................................................................................................... vii

Abstract .......................................................................................................................................... 1

1. INTRODUTION........................................................................................................................ 2

1.1 Background ..................................................................................................................... 2

1.2 Objective .......................................................................................................................... 2

1.3 Methodology .................................................................................................................... 3

2. MODELS TO ACCOUNT FOR SOIL FLEXIBILITY ............................................................ 4

2.1 The Winkler Model......................................................................................................... 4

2.2 Simple Cone Model for the Half space.......................................................................... 5

3. SOIL STATIC STIFFNESS COEFFICIENTS ......................................................................... 8

3.1 Theoretical Background ................................................................................................. 8

3.2 Stiffness Coefficients for Foundations on the Surface of an Elastic Halfspace......... 8

3.2.1 Stiffness Coefficients for Rigid Circular Foundation on the Surface of Halfspace......10

3.2.2 Stiffness Coefficients for Rigid Strip Foundation on the Surface of Halfspace ...........12

3.2.3 Stiffness Coefficients for Rigid Rectangular Foundation on the Surface of Halfspace 12

3.2.4 Stiffness Coefficients for Arbitrarily Shaped Rigid Foundation on the Surface of Half-

space…………………..............................................................................................................13

3.2.5 Influence of Soil Layering on Stiffness Coefficients.....................................................15

3.2.6 Influence of Foundation Embedment on Stiffness Coefficients ....................................17

3.2.7 Influence of Flexibility of Foundation on Stiffness Coefficients ..................................20

3.2.8 Influence of Inhomogeneity and Anisotropy of Soil on Static Stiffness .......................21

4 MODELING OF THE BUILDING STRUCTURE ................................................................. 26

4.1 The Foundation Model ................................................................................................. 26

4.2 The Building Model ...................................................................................................... 27

5 PARAMETRIC STUDY .......................................................................................................... 28

5.1 The Foundation Models Employed ............................................................................. 28

5.2 The Soil Categories ....................................................................................................... 29

-iii-

5.3 The Structural Systems Studied .................................................................................. 29

5.4 Results of the Parametric Study ................................................................................ ..33

5.4.1 Structural System 1 ...................................................................................................33



6 CONCLUSIONS AND RECOMMENDATIONS ................................................................... 92

REFERENCES ............................................................................................................................ 94

DECLARATION......................................................................................................................... 95

APPENDIX A.............................................................................................................................. 96

-iv-

List of Figures

Page

Figure 2.1: Surface displacements of the Winkler model ………………………………….....4

Figure 2.2: The cone model for a half space……………………………………………….….5

Figure 2.3: FBD of the actual foundation and Substitute SDF model …………………….….6

Figure 2.4: Two dimensional model……………………...……………………………..…….7

Figure 3.1: Rigid foundation on the surface of an elastic half space excited by a vertical

harmonic load, free body diagrams, an SDF replacement model with the same load and a

mass less model excited by a harmonic action P applied on to the ground…………………...9

Figure 3.2: Definitions of geometric parameters ……………………...…………………….12

Figure 3.3: Definitions of parameters…………………………...…………………………...17

Figure 3.4: Static stiffnesses of cylindrical foundations with different d/D ratios (H/R=3,

D/R=1, ν=1/3)………………………...……………………………………………………...19

Figure 3.5: Static stiffnesses of rigid strip foundation on a homogenous halfspace (ν=0.25).23

Figure 4.1: Foundation model for planar analysis……………………………...……………26

Figure 4.2: The Building Model and the building model with the foundation model……….27

Figure 5.1: The Foundation models………………………………………………………….28

Figure 5.2: Floor Plan for Structural System1……………………………………………….30

Figure 5.3: Typical lateral force resisting systems in the Y direction……………...…….….31

Figure 5.4: Floor Plan for Structural System 2………………………………………………32

Figure 5.5: Floor Plan for Structural System 3………………………………………………33

Figure 5.6: Frame on Axis 4 (Fixed base & Soil C)-Bending Moment Diagram in kNm…..36

Figure 5.7: Frame on Axis4 (Flexible base, D=1m &Soil C) -BMD in kNm…………...….36

Figure 5.8: Plot showing difference in bending moment along the beam length………...…37

Figure 5.9: Frame on Axis 4 (Fixed base & Soil C) -Shear Force Diagram in kN..…………38

Figure 5.10: Frame on Axis 4 (Flexible base, D=1m& Soil C) -SFD in kN…….…..………38

Figure 5.11: Plot showing difference in shear force along the beam length………………...39

Figure 5.12: Frame on Axis 4 (Soil A & depth 3m) -Bending Moment Diagram in kNm…..40

Figure 5.13: Frame on Axis 4 (Soil B & depth 3m) -Bending Moment Diagram in kNm…..41

Figure 5.14: Frame on Axis 4 (Soil C & depth 3m) -Bending Moment Diagram in kNm….41

Figure 5.15: Frame on Axis 4 (Soil A &depth 3m) - Shear Force Diagram in kNm……..…42

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Figure 5.16: Frame on Axis 4 (Soil B &depth 3m) -Shear Force Diagram in kNm……..….43

Figure 5.17: Frame on Axis 4 (Soil C & depth 3m) -Shear Force Diagram in kNm…..……43

Figure 5.18: Beam and column labeling…………………………………………………..…45

Figure 5.19: Plot of difference in moment along the beam……...……………………..……45

Figure 5.20: Plot of difference in shear along a particular beam……………………….……46

Figure 5.21: Plot of difference in axial load along a particular column………………...…...48

Figure 5.22: Plot of difference in axial load at the foundation level……………...…………48

Figure 5.23: Plot of difference in moment at the foundation level……………………..……48

Figure 5.24: Frame on Axis 4 (Fixed base& Soil C)-Bending Moment Diagram in kNm….51

Figure 5.25: Frame on Axis4 (Flexible base, D=1m&Soil C) -BMD in kNm…………...…52

Figure 5.26: Plot showing difference in bending moment along the beam length…….……53

Figure 5.27: Overturning Moment in Shear wall SW2 (Soil C and D=1m) ………...………54

Figure 5.28: Frame on Axis 4 (Fixed base & Soil C) -Shear Force Diagram in kN…..……..55

Figure 5.29: Frame on Axis 4 (Flexible base, D=1m& Soil C) -SFD in kN…………..…….56

Figure 5.30: Plot showing difference in shear force along the beam length…………......…57

Figure 5.31: Shear force at the storey level for Shear wall SW2 (Soil C & D=1m)...………57

Figure 5.32: Frame on Axis 4 (Fixed base & Soil C) -Axial Force Diagram in kN……..…58

Figure 5.33: Frame on Axis 4 (spring base, D=1m& Soil C) -Axial Force Diagram in kN...59

Figure 5.34: Difference in Axial Force of Shear wall SW2 (Soil C & D=1m) ………..……60

Figure 5.35: Frame on Axis 4 (Soil A & depth 3m) -Bending Moment Diagram in kNm…61

Figure 5.36: Frame on Axis 4 (Soil B & depth 3m) - Bending Moment Diagram in kNm…62

Figure 5.37: Frame on Axis 4 (Soil C & depth 3m) - Bending Moment Diagram in kNm…63

Figure 5.38: Overturning moment for Shear wall SW2 (Depth D =3m) ……………………64

Figure 5.39: Storey shear for Shear wall SW2 (Depth D = 3m) ……………………….……65

Figure 5.40: Axial Force diagram of Shear wall SW2 (Depth D=3m) ……………...………66

Figure 5.41: Beam, column and Shear wall labeling……………………………………...…67

Figure 5.42: Plot of difference in moment along the beam length………………………..…67

Figure 5.43: Share of overturning moment along the height of SW2……………………..…68

Figure 5.44: Plot of difference in shear along the beam length…………….……………..…69

Figure 5.45: Shear force distribution along the height of SW2……….…………………..…69

Figure 5.46: Plot of difference in axial load along the column height………………..…..…70

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Figure 5.47: Axial force variation along the height of SW2…………………….….……..…71

Figure 5.48: Plot of differences in moment along the column height……………...……..…71

Figure 5.49: Plot of axial loads at the base………………………………………………..…72

Figure 5.50: Plot of bending moments at the base………………………….……………..…72

Figure 5.51: Frame on Axis 4 (Fixed base& Soil C)-Bending Moment Diagram in kNm….76

Figure 5.52: Frame on Axis4 (Flexible base, D=1m&Soil C) -BMD in kNm……….…..….77

Figure 5.53: Plot showing difference in bending moment along the beam length……….…78

Figure 5.54: Overturning Moment in Shear wall SW2 (Soil C and D=1m) ……………...…79

Figure 5.55: Plot showing difference in shear force along the beam length……………..…80

Figure 5.56 Shear force variation at storey level for Shear wall SW2 (Soil C & D=1m) …..80

Figure 5.57: Axial Force on Shear wall SW2 (Soil C & D=1m) …………..……………..…81

Figure 5.58: Share of overturning moment for Shear wall SW2 (Depth D=3m) ….……..…82

Figure 5.59: Storey shear for Shear wall SW2 (Depth D = 3m) ………...………………..…83

Figure 5.60: Axial Force diagram of Shear wall SW2 (Depth D=3m) ……………….…..…84

Figure 5.61: Beam, column and Shear wall labeling……………………….……………..…85

Figure 5.62: Plot of difference in moment along the beam length………………………..…86

Figure 5.63: Overturning moment along the height of SW2……………….……………..…86

Figure 5.64: Plot of shear forces along the beam length…………………………………..…87

Figure 5.65 Storey shear along the height of SW2………………………………………..…88

Figure 5.66: Plot of axial load along the column height…………………………………..…88

Figure 5.67: Axial force variation along the height of SW2………………..……………..…89

Figure 5.68: Plot of axial load at the base………………………………..………………..…90

Figure 5.69: Plot of bending moment at the base……………………..…………………..…90

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List of Tables

Page

Table 3.1: Values of shape-dependent correction factor for vertical static stiffnesses………..…14

Table 3.2: Static Stiffnesses of rigid circular foundations on a stratum-over-rigid base……...…15

Table 3.3: Static stiffnesses of circular foundations on a stratum-over-half space…………....…16

Table 3.4: Static stiffnesses of rigid embedded cylindrical foundations ‘welded’ in to a

Homogenous soil stratum over bed rock……………...…………………………………...…..…18

Table 3.5: Static Stiffnesses of rigid embedded strip foundations ‘welded’ into a homogenous

stratum over bedrock………………….……………………………………………...…...…..…19

Table 3.6: Static vertical stiffness of flexible circular mat on halfspace……………………....…20

Table 3.7: Static stiffness of rigid foundations on inhomogeneous and cross- anisotropic soils...22

Table 5.1: Calculated Static Springs for Different Cases of Structural System 1……..……....…34

Table 5.2: Summary of differences in Axial Force…………………………………………....…39

Table 5.3: Summary of Differences in Frame Bending Moments ……………………......…..…42

Table 5.4: Summary of Differences in Frame Shear Forces ……………...………….…...…..…44

Table 5.5: Influences of Soil Type on Column Axial Forces ……………...……..…….....…..…44

Table 5.6: Minimum Variation in Storey Drifts……………...……………...…………………..49

Table 5.7: Maximum Variation in Storey Drifts……………...…………………………...…..…49

Table 5.8: Calculated Static Springs for Different Cases of Structural System 2…….…...…..…50

Table 5.9: Summary of differences in Axial Force……………...………………………...…..…58

Table 5.10: Summary of Differences in Frame bending Moments...……………………...…..…63

Table 5.11: Percentage Difference in Shear Forces for Different Soil Types.......………...…..…64

Table 5.12: Percentage Difference in Axial Forces for Different Soil Types..……….…...…..…65

Table 5.13: Minimum variation in Storey Drifts………………..………………….……...…..…73

Table 5.14: Maximum Variation in Storey Drifts……………...…………..……………...…..…73

Table 5.15: Static Springs for Different Cases of Structural System 3……………......…………75

Table 5.16: Summary of differences in Axial Force……………...…………………..…...…..…81

Table 5.17: Summary of Differences in Frame Bending Moments..……………………...…..…82

Table 5.18: Percentage Difference in Shear Forces for Different Soil Types ..…………...…..…83

Table 5.19: Percentage Difference in Axial Forces for Different Soil Types ..…………...…..…84

-1-

Abstract

This work deals with the investigation of the influence of soil flexibility on the internal force

distributions of buildings subjected to lateral static loads. The soil flexibility, which usually is

neglected in the conventional analysis of buildings subjected to static lateral loads, has been

accounted for by introducing static elastic springs. Three structural systems have been

considered. They include Structural system 1 representing framed buildings and Structural

Systems 2 and 3 representing dual (frame-wall) systems. They have been selected in such a

manner that their properties fall within the range of applicability of the pseudo- static method

of analysis for lateral loads. Comparison was made between the internal force distributions

of the fixed-base structural system and the flexible-base structural system as obtained using

the structural analysis software-ETABS. The flexible-base system has been modeled to

account for the flexibility of the soil with all the influencing parameters like the foundation

shape and type, embedment depth and soil types. For the structural systems studied, it was

found that the soil flexibility has resulted in a significance difference in internal force

distribution specially in the bending moments around the supports of end spans of beams,

where the designs moments are already large in the conventional analysis. In the dual

systems, accounting for the flexibility of the soil has resulted in differences of bending

moments, shear forces and axial forces in the shear walls. Generally, the influence has been

found to be more pronounced in the dual systems. This observation has a direct practical

significance in that dual systems are the common types of structural systems employed in

buildings susceptible to lateral load effects.

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1. INTRODUTION

1.1 Background

Currently, almost all building structures are modeled as if they are perfectly fixed at their

foundations. This is equivalent to assuming that the flexibility of the foundation soil is too

negligible to influence the response of the structure to any loading it might be designed for.

But, in reality, the soil has some flexibility which can in some cases significantly influence

the distribution of the internal forces in the structure. Whereas the influence in some cases

could be on the liberal side of the conventional analysis, it could in other cases be on the

conservative side. This trend changes from systems to systems that it could even in some

cases be reversed. The trend can be ascertained through a systematic parametric study, in

which the type of the structural system, the soil type, foundation embedment depth and

foundation type are varied.

The flexibility of the soil can be accounted for by introducing static springs with coefficients

that can be easily obtained for practically all modes including translational, rocking and

torsional modes. These static stiffness coefficients are dependant on factors like type of the

soil, depth of foundation embedment, size and type of foundation and soil heterogeneity.

1.2 Objective

The objective of this work is to model and study the influence of soil behavior on the

response of buildings to equivalent static lateral forces due to earthquakes. Hence if the

internal forces obtained in the customary pseudo-static analysis of buildings are close to those

obtained in the analysis which considers soil flexibility, confidence can be laid up on the

usual pseudo-static analysis. However if there is any considerable change which is going to

influence the design of structural elements, then a recommendation can be made on how to

account for the flexibility of the soil in the structural models employed for the analysis of

buildings.

-3-

1.3 Methodology

In order to assess the effect of flexibility of soil on the internal force distribution, the

following scheme is followed. Analysis of the buildings is made for the usual case of analysis

which completely neglects the influence of soil flexibility and for the case which incorporates

the flexibility of the soil. Thus, three structural systems including G+5, G+10 and G+20

buildings, having symmetrical plan area but structural rigidity increased accordingly, are

used. On these systems the flexibility of the soil is accounted for by introducing static springs

during the pseudo-static method of analysis of the building using the structural software-

ETABS Nonlinear V-8.00.

The procedures used here include:

(1) analysis of the structural system where the foundation is assumed to be fixed at the

base as the usual practice

(2) analysis of the structural system where the flexibility of the soil is accounted for by

using springs with their respective static stiffness coefficients at the base

(3) then each of the above two cases are again treated by varying the parameters, which

are likely to influence the internal force distribution, including (a) depth of foundation

embedment (b) type of soil (c) structural system (d) type of foundation.

Finally, the results are systematically compared under given similar set of conditions, where

only one parameter is varied at a time. Conclusions are drawn and recommendations are

made following the results of the parametric study.

-4-

2. MODELS TO ACCOUNT FOR SOIL FLEXIBILITY

When subjected to loads, foundations of structures deform in a way that depend on the nature

and deformability of the supporting ground, the shape of the foundation, depth of

embedment of the foundation and the mode of the applied loads. Thus the key step in the

analysis of the system is to model the soil around the foundation, and estimating the stiffness

of flexibly supported foundation. Some common models used to account soil flexibility are

discussed below.

2.1 The Winkler Model

The idealized model of soil media proposed by Winkler (1867) assumes that the deflection,

w, of the soil medium at any point on the surface is directly proportional to the stress, q,

applied at that point and independent of stresses applied at other locations, i.e.

q(x, y) = k w(x, y) ……….……………………..……………...…………………..(2.1)

where, k is termed the modulus of subgrade reaction with units of stress per unit length

Physically, Winkler’s idealization of the soil medium consists of a system of mutually

independent spring elements with spring constant k. One important feature of this soil model

is that the displacement occurs immediately under the loaded area and outside this region the

displacements are zero (Figure 2.1). Also, it can be seen that, for the Winkler model, the

displacements of a loaded region will be constant whether the soil is subjected to an infinitely

rigid load or a uniform flexible load [14].

x

z

q (x,y)

(a)(b)

(c)

z

P

x

(d)

z

q

x

z

(c)

P

x

Figure 2.1: Surface displacements of the Winkler model due to (a) a non-uniform load,

(b) a concentrated load, (c) a rigid load, (d) a uniform load

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2.2 Simple Cone Model for the Half space

The model discussed hereunder is a rough approximation for a rigid circular foundation of

radius, Ro, at the surface of an elastic half space subjected to a vertical harmonic dynamic

loading. Even though the discussion is for a dynamic load, it can be applied for a static load

when the frequency, ω of the load is zero. Here the soil mass is going to be replaced by a

beam of increasing cross sectional area with depth i.e. a cone having area Ao = πRo2 at the

soil foundation interface as shown in Figure 2.2. Eventhough this model is not an accurate

representation of the halfspace; it provides an excellent indication of the important influence

of soil flexibility of the soil on the response of the foundation to the applied load.

z

R

Ro

dz

N+N'dz

h

α

dzA

A

(a)(b)

N

mwdzo ..

oP=Psinωt

Figure 2.2: The cone model for a half space

The equation of motion of the replacement beam is developed based on the differential

element in Figure 2.2(b). For the axial vibration considered

0..

' =− wmN ------------------------------------------------------------------------------- (2.2)

With the constitutive law of 'EAwN = and the variable cross sectional

areaoA

h

hzRA

2

22 )( +

== π , Equation 2.2 becomes 0)''''

(..

=+− wwA

AEw

ρ or

0)'2

''(2

..

=+

+− whz

wcw L -------------------------------------------------------------------------- (2.3)

Where ρ

EcL =

2 is the velocity of longitudinal waves

Equation 2.3, a partial differential equation with non constant coefficients, is transfer to an

equation with constant coefficients with the substitution

-6-

),(),( tzhz

htzw φ

+= ----------------------------------------------------------------------- (2.4)

This results in,

0''2

..

== φφ Lc -------------------------------------------------------------------------------- (2.5)

Equation 2.5 is the differential equation governing the propagation of longitudinal waves.

Because of the influence extent of the beam in Figure 2.2 and the harmonic nature of the

applied load, the solution in this particular case is

)(cos)(sin),( 21 tc

zAt

c

zAtz

LL

−+−= ϖϖφ -------------------------------------------- (2.6)

The normal force at the foundation soil interface is found from

),0('),0( twEAtN o= ----------------------------------------------------------------------- (2.7)

Using Equations 2.4 and 2.6, yields

)sincos()cossin(1

),0(' 2121 tAtAc

tAtAh

twL

ϖϖϖ

ϖϖ +++−−= --------------------- (2.8)

Equation 2.8 can be shown to be

),0(1

),0(1

),0('.

twc

twh

twL

−−= ---------------------------------------------------------- (2.9)

Substituting Equation 2.9 in 2.7, one obtains

),0(),0(),0(.

twc

EAtw

h

EAtN

L

oo −−= --------------------------------------------------- (2.10)

Consider the free body diagram of the foundation of Figure 2.2 as shown in Figure 2.3 below.

K C

N(0,t) w(0,t)

(a)

(b)

mw(0,t)f

..

P=Psinωto

P=Psinωto

mfmf

Figure 2.3: (a) FBD of the actual foundation in Figure 2.2 (b) Substitute SDF model

The equation of motion from the FBD is

tPtNtwm of ϖsin),0(),0(..

−=+ --------------------------------------------------------- (2.11)

Inserting equation 2.10 in 2.11, the equation of motion of the foundation takes the form:

-7-

tPtwh

EAtw

c

EAtwm o

o

L

of ϖsin),0(),0(),0(

...

=++ -------------------------------------- (2.12)

Now the soil foundation system of Figure 2.2 can be replaced by a classical SDF system as

shown in Figure 2.3 (b).

The equation of motion of the SDF model is

tPtKwtwCtwm of ϖsin),0(),0(),0(...

=++ --------------------------------------------- (2.13)

Comparison of equations 2.12 and 2.13 yields

..

L

oo

c

EACand

h

EAK == -------------------------------------------------- (2.14)

Equation 2.14 is an interesting result that shows that the vibrating foundation on top of the

conical beam of infinite height can be replaced by the same massless foundation supported by

a linear spring and a dash pot arranged in parallel i.e. classical SDF oscillator. This model can

also be used for other modes of vibration and thus the SDF model can be employed for the

whole half space under the various modes of oscillation irrespective of the foundation shape

[13].

For a two dimensional (planar) system subjected to a statically applied load (ω=0 , cL=0 and

C=0),we will thus have three springs in the three degrees of freedom under the foundation as

shown in Figure 2.4. The damping does not exist at ω=0.

Figure 2.4: Two dimensional model

In Figure 2.4, Kv, Kh, and Kr are static spring constants for the vertical, horizontal and

rocking modes respectively. A detailed discussion of the static stiffnesses is made in the next

chapter.

Similar extension can be made for a three dimensional system having six degrees of freedom

requiring six springs in the respective degrees of freedom under the foundation.

Kh

Kv Kr

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3. SOIL STATIC STIFFNESS COEFFICIENTS

3.1 Theoretical Background

A comprehensive analysis of waves in an elastic continua made by Lamb has indicated that

the existence of body and surface waves in elastic media play an important role in the

prediction of vibrations of foundations on soils. The energy imparted at the foundations due

to foundation vibration can be dissipated by the various forms and combinations of these

waves, which in this case propagate into the natural soil deposit outwards from the

foundations.

The vibrations of foundations could be easily studied by using simple, but appropriate,

models that capture the above indicated phenomenon in the vast mass of soil below the

foundation. Also the dynamic analysis of structures with due consideration of the interaction

of their foundations with the soil is possible by making use of the displacement-force ratios

called compliance functions. The inverse of these functions are called impedance functions.

The determination of the dynamic impedance functions, for the dynamic analysis of

structures with due consideration of interaction of their foundations, could be easily studied

by using simple, but appropriate models that capture the above indicated phenomena in the

vast mass of soil below the foundation. For instance the vibration of the half space can be

replaced by a mass-spring-dashpot SDF oscillator.

The dynamic impedance of an SDF oscillator may be expressed as a product of the static

spring constant Kst, and the complex function (k + iωc) which encompasses the dynamic

characteristic of the system and is called the dynamic part of the impedance. In this function,

k is the dynamic stiffness coefficient, c is the damping coefficient, ω is the circular frequency

of excitation, and i is the complex number given by 1−=i [4]. At zero frequency, there is

no accelerated motion yielding thus k=1, and c=0. The impedance coincides then with the

static stiffness Kst of the system. Thus the study of impedance and compliance functions in

soil dynamics could be used as a means to obtain the static stiffness at zero frequency.

3.2 Stiffness Coefficients for Foundations on the Surface of an Elastic Halfspace

In order to use a simple spring-dashpot-mass SDF model to replace a soil foundation system,

the equivalence of the two systems - the actual soil foundation system and the model

oscillator subjected to the same loading shall be justified. For this, consider the system in

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Figure 3.1 to explain the method with the help of the easy-to-visualize case of vertical

vibrations. Figure 3.1 portrays a rigid foundation block of mass mf, assumed to have a

vertical axis of symmetry z, passing through the centroid of the soil foundation contact

surface. The foundation is underlain by a deposit consisting of horizontal linearly deforming

soil layers, subjected to a vertical harmonic force Q(t) along the z axis , this foundation will

experience only a vertical harmonic displacement wz(t). The two free body diagrams are

sketched in the same figure and include the inertial (D’Alembert) forces. The foundation

“actions” on the soil generate equal and opposite “reactions”, distributed in some unknown

way across the interface and having unknown resultant P(t)[6].

Elastic half space: G,ν

wz(t)

wz(t)

K C

K C

(a)

(b)

(c)

(d)

(e)

Figure 3.1: (a) Rigid foundation on the surface of an elastic half space excited by a vertical

harmonic load, (b) and (c) Free body diagrams, (d) An SDF replacement model with the same

load, (e) A mass less model excited by a harmonic action P applied on to the ground

With the vertical displacement of the center of the loaded region denoted by wo(t), the

equation of motion of the actual rigid foundation in Figure 3.1(a) and 3.1(b) is:

)1.3.......(................................................................................)(..

ti

o

ti

oof eQePtwmϖϖ =+

The equation of motion of the SDF model of Figure 3.1(d) is given by

)2.3.......(............................................................)()()(...

ti

oooof eQtKwtwCtwmϖ=++

Comparison of the above two equations for equivalence indicates that

..

wm f Internal force

mf mf

ti

o eQQϖ= Applied force

P=Poeiωt

Resultant of

soil

reaction

ti

oeQQϖ=

ti

o eQQϖ=

(Resultant of

Foundation reaction)

P=Poeiωt

P=Poeiωt

mf

=

-10-

)3.3....(................................................................................)()(.

ti

ooo ePtKwtwCϖ=+

The equation of motion of a massless SDF model of Figure 3.1(e) is

)4.3......(................................................................................)()(.

ti

ooo ePtKwtwCϖ=+

Thus from equations (3.3) and (3.4) one can conclude that the half space subjected to a

harmonic vertical concentrated load on a rigid footing at the surface can indeed be replaced

by a simple massless oscillator SDF supported by a spring and a dashpot. The determination

of the dynamic impedance functions of an ‘associated’ rigid but massless foundation, as a

function of the excitation frequency, ω, is thus an essential step in the study of vibration of

foundations. Once the harmonic response of such a massless foundation has been determined,

the steady state response of the massive foundation or of any structure supported on it may be

evaluated using standard procedures.

It can be easily shown that the analogue models of Figure 3.1 are also applicable for all six

modes of vibration of a rigid foundation. However, the inherent coupling between the

horizontal translational motions with the associated rocking is neglected as it is commonly

done.

3.2.1 Stiffness Coefficients for Rigid Circular Foundation on the Surface of Halfspace

In the preceding section it is shown that knowledge of the dynamic compliance is necessary if

structural analysis is to be made incorporating the effect of soil-structure interaction. It is the

purpose of this section to present static stiffness coefficients of circular footings that are

derived from the dynamic compliances at zero frequency.

With the following basic assumptions the dynamic response of circular footings is determined

and the compliance functions and the corresponding static stiffnesses are established. [11]

a. The footing is modeled as a rigid disc with radius R resting on a homogenous, elastic

half space.

b. For the four cases of harmonic vibrations produced by torsional moment, a vertical

force, a rocking moment, and a horizontal force, it is assumed that the surface traction

outside the disc is zero. For vertical and rocking vibrations, the vertical displacement

under the disc is prescribed while the disc-foundation interface is taken to be

frictionless. For horizontal vibrations, the horizontal displacements are prescribed

under the disc while it is assumed that the contact is such that the normal component

of the surface traction is zero everywhere, whereas in considering the effect of

-11-

overturning moment , the horizontal or shearing component of the interface pressure

are assumed to be zero.

For each particular mode of harmonic excitation with frequency ω, the dynamic impedance is

defined as the ratio between the steady state force (moment) and the resulting displacements

(or rotations) at the base of the massless foundation. For example, the vertical impedance of a

foundation whose plan has a center of symmetry is defined by:

)(

)(

tw

tPK

z

v

v =

In which,

Pv(t) = Poeiωt

is the harmonic vertical force applied at the base of the disc

wz(t) = woeiωt

is the uniform harmonic settlement of the soil foundation interface.

Po= is the total soil reaction against the foundation; it is made up of normal

stresses against the basement (plus, in case of embedded foundation, the shear

stress along the vertical side wall)

Similarly one may define the torsional impedance, Kt, as the ratio of the torsional moment to

rotation; the horizontal impedances, Kh, as the horizontal forces to the displacements along

the principal axes of the base; and the rocking impedances, Kr, as the moments to rotations

around the same horizontal principal axes. However, since horizontal forces along the

principal axes produce rotations in addition to horizontal displacements, cross horizontal-

rotational impedances, Krh, may also be defined; they are usually negligibly small in case of

surface and very shallow foundations, but their effect may become appreciable for greater

depth of embedment.

These impedance functions can be written in the form K =Kst (k + iaoc), where Kst is the

corresponding static stiffnesses given by the following relations: [11]

( )

υ−=

1

4GRK

stv

( )

3

16 3GR

Kstt =

-------------------------------- (3.5)

( )υ−

=2

8GRK

sth

( ))1(3

8 3

υ−=

GRK

str

As the scope of this work is limited to static analysis (ω=0), the dynamic coefficients are not

presented herein.

-12-

3.2.2 Stiffness Coefficients for Rigid Strip Foundation on the Surface of Halfspace

When dealing with long and narrow foundations, the length of which is larger than their

width by a factor of 5 or greater, it is a common practice to idealize their shape as an

infinitely long strip. If, moreover, the dynamic loading is reasonably uniform along the

longitudinal direction, plane strain conditions prevail throughout and 2D analyses are

sufficient to obtain the response. The dynamic impedances (hence corresponding static

stiffnesses) of a rigid strip foundation on the surface of a homogeneous half space are given

in Reference [6].

The static vertical and horizontal stiffness of an infinite strip on a half space are zero. This is

at variance with the behavior of circular foundations whose static stiffnesses are non zero.

The zero static vertical and horizontal stiffnesses are due to the infinite displacements of a

strip loaded half space which in turn arise from the large depths of the corresponding ‘zones

of influence’. In other words, the static stresses induced by the strip surface loads decay

slowly with depth and, thus, cause appreciable straining of even remote soil elements;

accumulations of these strains yields infinite displacements and hence zero stiffnesses. On the

other hand, the stress and strain fields induced by moment loading are confined to the near

surface soil only, thereby producing small displacements and non zero static stiffnesses.

Here there are three possible modes of vibration (vertical, horizontal and rocking) as

compared to the four modes of a circular foundation. Apparently, torsional oscillations

involve out of plane motions and hence are impossible with strip footings. In general, the

dependence of the dynamic impedances on Poisons ratio is very similar for strip and circular

foundations.

3.2.3 Stiffness Coefficients for Rigid Rectangular Foundation on the Surface of Halfspace

It has been known for some time that the static stiffness of a typical rectangular foundation,

Figure 3.2 can be approximated with reasonable accuracy by the corresponding stiffnesses of

an ‘equivalent’ circular foundation with the corresponding ‘equivalent’ radius.

x

y2L

2B

(a) (b)

Figure 3.2: Definitions of geometric parameters

Ro

-13-

For the translational modes in the three principal directions (x, y and z) the radius Ro of the

‘equivalent’ circular foundation is obtained by equating the areas of the contact surfaces;

hence:

ππ

BLARo

2.2== ---------------------------------------------------------------- (3.6)

For the rotational modes around the three principal axes, the ‘equivalent’ circular foundations

is set to have the same area moments of inertia around x, y and z, respectively, with those of

the actual foundation. Thus, the equivalent radii are

axisxtheaboutrockingforBLI

R x

xo −== 4

3

4

3

.164)(

ππ

axisytheaboutrockingforLBI

Ry

yo −== 4

3

4

3

.164)(

ππ------------- (3.7)

axiszthearoundtorsionforLBLBI

R z

to −+

== 4

22

4

6

).(..162)(

ππ

Recent parametric studies have confirmed the similar static behavior of rectangular and

‘equivalent’ circular foundations. For aspect ratios, L/B, less than 4 the ‘equivalent’

stiffnesses are in very good agreement with the actual ones. Typically, the error is within

10% and, hence, it is insignificant for all practical purposes. Even for aspect ratios as high as

8, the ‘equivalent’ circular foundations yield stiffnesses which are within 30% of the

corresponding stiffnesses of the actual rectangular foundation. This is by no means a large

error, in view, for example, of the uncertainty in estimating the soil modulus in practice. The

greater differences are observed between the actual and the ‘equivalent’ stiffness for torsion

(Kt) and for horizontal displacement in the y direction (Ky). For L/B=4, the error in Kt is

about 17% and in Ky about 22%. [6]

Nowadays, results are available for the complete dynamic impedance matrix and static

stiffness of rigid rectangular foundations with varying aspect ratios, L/B. The results of

several investigations for the static stiffnesses are presented in different publications [6].

3.2.4 Stiffness Coefficients for Arbitrarily Shaped Rigid Foundation on the Surface of

Halfspace

Only a few numerical results are available for impedance functions of foundations having

‘arbitrary’ geometries, i.e. plan shapes other than strip, circular or rectangular. One reason for

the lack of interest is that foundations of such arbitrary shape are not constructed very

-14-

frequently. Moreover, substantial computational effort must be expended to obtain solutions

for such foundation geometries.

Analytical expressions for the static stiffnesses of rigid foundations supported by an elastic

half space and having several different shapes are available. For example the following

equation can be used to determine the vertical static stiffness of arbitrarily-shaped foundation

on homogenous half space [6].

v

o

v JGR

Kυ−

=1

4 ----------------------------------------------------------- (3.8)

Where: foundationcircularequivalenttheofradiustheis

ARo ''

π=

A= the area of the soil footing contact surface.

Jv = shape dependant correction factor, numerical values of which have

been tabulated in Table 3.1for numerous plan shapes.

Table 3.1: Values of shape-dependent correction factor for vertical static stiffnesses [6]

Shape of foundation plan Jv Shape of foundation plan Jv

Circle

Regular hexagon

Semicircle

Equilateral triangle

Triangle with angles, 45o, 45

o,90

o

Triangle with angles, 30o, 60

o,90

o

Ellipse with a/b=2 (a =major axis)

Ellipse with a/b=3(b =minor axis)

Ellipse with a/b=4

Ellipse with a/b=6

1.00

1.01

1.05

1.07

1.10

1.12

1.03

1.07

1.13

1.21

Rhombus with an angle of 60 o



Rectangle with L/B = 2



1.07

1.14

1.27

1.03

1.13

1.23

Table 3.1 and 3.2 can be used for determining the vertical static stiffness of a variety of

foundations with very good accuracy. It can be noted that (a) the circular disc yields the

smallest stiffness of all footings with a given contact area (b) of all rigid footings with an n-

sided polygon-shaped plan of a given area, the regular n-sided polygon yields the smallest

stiffness (c) the correction factor depends primarily on the aspect ratio of the foundation,

being surprisingly insensitive to the details of each particular shape. By aspect ratio it is

somewhat loosely meant the ratio between largest and smallest critical foundation

dimensions.

-15-

3.2.5 Influence of Soil Layering on Stiffness Coefficients

a) Elastic Halfspace

When dealing with a deep and relatively uniform soil deposit, it makes engineering sense to

model it as a homogenous halfspace. This idealization primarily because of its simplicity has

been widely employed to determine stresses and deformations in soils, and its use in soil

dynamics has led to results in qualitative agreement with observation.

b) Homogenous Soil Stratum

Natural soil deposits very rarely have uniform properties within large depths from the loaded

surface. More typical is the presence of a stiffer material or even bed rock at a relatively

shallow depth. The response of a foundation on a soil stratum underlain by such a stiffer

medium can be substantially different from the response of an identical foundation resting on

a uniform halfspace. We can have two types of idealized soil profiles here: 1) a homogenous

soil stratum over a rigid base and, 2) a homogenous soil stratum over a flexible homogenous

halfspace.

1) Foundations on a Homogenous Soil Stratum Overlying a Rigid Base

For example the static stiffnesses of circular foundation on the surface of a stratum over a

rigid base are shown in Table 3.2.

Table 3.2: Static Stiffnesses of rigid circular foundations on a stratum-over-rigid base [6]

Type of loading Static Stiffness Range of Validity Soil Profile

Vertical

+

−=

H

RGRKv 28.11

1

4

υ

H/R>2

Horizontal

+

−=

H

RGRKh

2

11

2

8

υ

H/R>1

Rocking

+

−=

H

RGRK r

6

11

)1(3

8 3

υ

4>2H/R>2

Torsion 3

3

16GRK t =

H/R>1.25

H

G , ν

R

It is evident from the above simple and quite accurate formulae for the determination of the

static stiffnesses that existence of rigid base bed rock at a relatively shallow depth may

drastically increase the stiffnesses of a rigid surface foundation. The four expressions reduce

-16-

to the corresponding half space stiffnesses when H/R tends to infinity, but their values

increase with decreasing H/R.

Vertical stiffnesses are particularly sensitive to variation in the depth to bed rock. Horizontal

stiffnesses are also appreciably affected by H/R while the rotational stiffnesses (rocking and

torsion) are the least affected. In fact, for H/R>1.5 the response to torsional loads is

particularly independent of the layer thickness. An indication of the causes of this different

behavior of a circular footing to the four different types of loading can be obtained by

observing the depth of the ‘zones of influence’ (known as ‘pressure bulb’ ever since

Terzaghi) in each case.

2) Foundation on a Stratum Overlying a Flexible Halfspace

A more general model, of a stratum over a flexible half space needs the moduli ratio G1/G2 to

describe it in addition to H/R or H/B ratio. When G1/G2 tends to zero, the stratum on rigid

base is recovered, when it becomes equal to 1, the model reduces to a homogenous halfspace.

Thus the model here helps in bridging the gap between ‘halfspace’ and ‘stratum’ solutions

which are presented above.

Based on the results provided by Hadjian and Luco, Gazetas [6] has derived simple but

reasonably accurate formula for the static stiffnesses of a rigid circular disc, in terms of H/R

and G1/G2.Table 3.3 displays these formulae, which are valid for the usual case in which,

G1<G2, i.e. a halfspace stiffer than the overlying layer. Note that as the rigidity of the

supporting halfspace decreases, the static stiffnesses of the supporting half space decreases,

apparently due to increasing magnitude of strains in the halfspace.

Table 3.3: Static stiffnesses of circular foundations on a stratum-over-halfspace [6]

Type of loading Static Stiffness Range of Validity Soil Profile

Vertical

+

+

−

2

11

1

28.11

28.11

1

4

G

G

H

RH

R

RG

υ

1<H/R<5

Horizontal

+

+

−

2

11

1

2

11

2

11

2

8

G

G

H

RH

R

RG

υ

1 < H/R < 4

Rocking

+

+

−

2

11

3

1

6

11

6

11

)1(3

8

G

G

H

RH

R

RG

υ

0.75 < H/R < 2

G1

R

H

hG2

0 < G1/G2 < 1

-17-

3.2.6 Influence of Foundation Embedment on Stiffness Coefficients

Results for the response of foundations have been presented for circular, strip and rectangular

foundations embedded in a variety of idealized soil profiles, including the halfspace, stratum

over bed rock and stratum over halfspace. In each case, the new key dimensionless

parameter, in addition to the parameters controlling the response of surface foundations, is

the relative embedment, D/B or D/R.

H

dD

R

Figure 3.3: Definitions of parameters

Moreover, the assumed interface behavior at the contact between vertical sidewalls and

backfill is of critical importance. Most studies assume that walls and soil remain in full

contact during motion, as if they were welded at their interface. In reality, however, no tensile

stresses can be sustained between the two media, while the magnitude of developing shear

traction cannot violate Coulomb’s friction low. Hence, separation and sliding are likely to

occur between sidewalls and backfill, depending primarily on the mode of motion and the

nature and method of placement of the soil.

a) Embedded Circular Foundations

Based on the assumed interface behavior at the contact between vertical sidewalls and

backfill we have cylindrical ‘welded’ and imperfect contact between sidewalls and backfill.

‘Welded’ Cylindrical Foundations in a Homogeneous Stratum

Results are presented herein based on the work of Kausel and are strictly applicable to

foundations having infinitely rigid sidewalls and mat, which are all in perfect contact with the

soil. Moreover, the backfill must be of very good quality and have the same properties with

the soil beneath the mat. These are rather extreme conditions and thus, yield an upper bound

of the possible effect of embedment.

-18-

Table 3.4 displays five simple and sufficiently accurate formulae for the static stiffnesses of

cylindrical foundations, perfectly embedded in homogenous soil layer overlying bedrock. It is

evident that embedment increases the values of the static stiffnesses substantially. The

increase in D/R is especially beneficial to the two rotational modes, rocking and torsion; the

two translational modes, vertical and horizontal, are considerably less affected (factors of ½

and 2/3 for vertical and horizontal loading, as compared to 2 and 2.67 for rocking and torsion

respectively)

Table 3.4: Static stiffnesses of rigid embedded cylindrical foundations ‘welded’ in to a

Homogenous soil stratum over bed rock [6]

Type of loading Static Stiffness

Vertical

−−+

+

+

− HD

HD

R

D

R

D

H

RGR

/1

/28.085.01

2

1128.11

1

4

υ

Horizontal

+

+

+

− H

D

R

D

H

RGR

4

51

3

21

2

11

2

8

υ

Rocking

+

+

+

− H

D

R

D

H

RGR7.0121

6

11

)1(3

8 3

υ

Coupled

horizontal rocking

0.4Kh D.

Torsion

+

R

DGR 67.21

3

16 3

Profile

H

D

R

Range of validity:

D/R<2 and D/H< 0.5

In contrast, the effect of D/H is more visible in the vertical and horizontal modes,

appreciably less important in rocking, and negligible in torsion; this is consistent with the

expected depths of the corresponding ‘pressure bulbs’. With embedded foundation the cross-

coupling stiffness, Khr, can no longer be neglected, being approximately equal to 0.4Kh D,

where Kh is the horizontal stiffness.

Imperfect Contact between Sidewall and Backfill of Cylindrical Foundations in a

Homogeneous Stratum

The sensitivity of the static stiffness to variation in contact height over embedment ratio, d/D

is graphically displayed in the Figure 3.4 below. The effect is essentially independent of H/R

and D/R; hence only one curve is plotted for each mode. The effect of d/D is very significant

for rocking and torsional loading, substantial for horizontal loading and secondary for vertical

loading. Here it is assumed that no contact exists between sidewall and backfill near the

ground surface but that a perfect contact is effective over a height equal to d above the

basement.

-19-

H

dD

R

r

t

h

v

0.5 1

2

3

1

d/D

K(d

)/K

(0)

Figure 3.4:Static stiffnesses of cylindrical foundations with different d/D ratios (H/R=3, D/R=1, ν=1/3)

b) Embedded Strip Foundations

Jakub and Roesset, by utilizing the results of an extensive parametric study, developed simple

expressions for the static horizontal and rocking stiffnesses, which are given in Table 3.5

below. It is evident that the influence of embedment is much smaller for strip than it is for

circular foundations as the strip foundation has sidewalls along two sides only. Thus, per unit

length, the ratio of sidewall area to basement area is equal to 2D/2B=D/B. Where as, for a

circular foundation the ratio of the two areas is 2πRD/πR2=2(D/R)! This seems to imply that

the influence of D/R or D/B is proportional to the sidewall-over-basement area ratio.

Table 3.5: Static Stiffnesses of rigid embedded strip foundations ‘welded’ into a

homogenous stratum over bedrock[6]

Type of loading Static Stiffnesses Profile

Horizontal

+

+

+

− H

D

B

L

H

BG

3

41

3

1121

2

1.2

υ

L = length of the foundation

Rocking

+

+

+

− H

D

B

D

H

BGB

3

211

5

11

)1(2

2

υ

π

H

D

B

H/B>2

D/B<2/3

c) Embedded Rectangular Foundations in Halfspace

The sensitivity of the stiffnesses of rectangular foundations on D/B is not as strong as in the

case of circular foundations, but is quite stronger than that of a strip footing. Note that the

sidewall-basement area ratio in this case becomes equal to 4(B+L) D/ (2B.2D) = (1+B/L)

(D/B), which is in between the 1 and 2 times the embedment ratio of the strip and circular

foundations.

-20-

3.2.7 Influence of Flexibility of Foundation on Stiffness Coefficients

The in-plane (membrane) rigidity of shallow foundations is particularly infinitely large, when

compared to the deformability of soils. Hence, for horizontal and torsional loading most

foundations clearly qualify as ‘rigid’, and the results of the preceding sections are applicable.

However, in many practical situations, the foundation response to vertical and rocking

loading cannot be properly predicted without accounting for the finite out-of-plane (flexural)

rigidity of the shallow foundations [6].

A few studies have appeared lately on the behavior of flexible circular and rectangular plates

resting on a homogenous half space [16, 9].The additional dimensionless parameter which in

this case controls the foundation response is the relative flexural rigidity factor defined as

3

2)1(

−=

B

t

E

ERF f

s

fυ ……………………………………………………..…. (3.9)

where Ef, νf and t are respectively, the Young’s modulus, Poisson’s ratio and thickness of the

foundation raft.

In addition, the exact distribution of the applied loading influences appreciably the behavior,

especially of the flexible foundations.

An idea of how sensitive the static stiffnesses, Kv to changes in the relative rigidity factor,

RF, can be obtained from the results of Table 3.6. The vertical static stiffnesses of a circular

mat supported by a homogenous halfspace and loaded by either a uniformly or a parabolically

distributed load, are expressed in the form of an Equation 3.8. Here the ‘correction’ factor Jv,

which accounts for the mat flexural rigidity, is given as a function of RF. However,

additional parametric studies have to be made to draw more definitive conclusions [6].

Table 3.6: Static vertical stiffness of flexible circular mat on halfspace [4]

General Expression Jv(RF)

RF Uniform load Parabolic load

0.01 0.67 0.46

0.1 0.72 0.54

1 0.92 0.82

10 0.99 0.97

v

o

v JGR

Kυ−

=1

4RF

100 1.00 0.98

-21-

3.2.8 Influence of Inhomogeneity and Anisotropy of Soil on Static Stiffness

The results discussed so far have been based on the simplifying assumption that the soil can

be modeled as a homogenous, isotropic and linearly visco-elastic stratum or half space.

However, real soil strata frequently increase in rigidity with depth as a reflection of the

increase in overburden pressure, while in some other cases weathered crusts, in which rigidity

decreases with depth, overlay deposits of softer clay. Furthermore, laboratory tests show that

soils deform differently in the vertical and horizontal directions-a manifestation of

anisotropic fabric acquired during natural formation and subsequent loading. Finally, when

subjected to large enough stresses, soils respond as non linear and inelastic materials. This

section, thus presents characteristic results and important conclusions from a number of

recent studies aimed at assessing the influence of soil inhomogeneity, anisotropy and non

linearity on the static stiffness coefficients.

Effect of Soil Inhomogeneity

Among the numerous studies published for the vertical static problems, a prominent work is

that of Gibson and his co-workers [8], who studied the response to arbitrary surface loads of a

halfspace or stratum whose moduli increases linearly with depth, i.e. in the form G =Go +

~

m (z/R), where Go and ~

m are the moduli at the surface and at a one-radius (or one semi-

width). These studies revealed that for an incompressible medium, i.e. with Poisson’s ratio of

0.5, the stress distribution is hardly influenced by the degree of inhomogeneity; in the

particular case of zero surface modulus (Go=0) this distribution is identical with the

distribution in a homogeneous halfspace, regardless of foundation geometry. The surface

settlement on the other hand, being quite sensitive to the assumed soil profile, becomes

directly proportional to the applied normal pressure when Go=0, independent of the size and

shape of the loaded area and the thickness, H, of the soil layer on a rigid but frictionless base.

Thus, such a soil behaves like a Winkler medium rather than a homogenous halfspace, its

spring constant being simply equal to 2~

m /R. Expressions for the vertical static stiffnesses of

surface foundations of several shapes supported by such a deposit (frequently referred to as

‘Gibson soil’) are shown in Table 3.7.

This behavior remains qualitatively true when drained soil behavior is taking place (i.e.

ν<0.5). Thus, with increasing degree of inhomogeneity (e.g. increasing~

m ) normal and shear

stresses affect the soil at greater vertical and lesser horizontal distances, in agreement with

intuition that expects stiffer material to attract larger stresses. On the other hand, surface

-22-

displacements, being moderately sensitive to ν, do tend to become proportional to the applied

local pressure as ~

m increases. It is, thus, generally concluded that an inhomogeneous deposit

leads to more uniform stresses under rigid foundations than simple elastic theory

(homogenous halfspace) predicts.

Table 3.7: Static stiffness of rigid foundations on inhomogeneous and cross- anisotropic soils

[8, 7]

Type of loading Static stiffness Range of validity Soil profile

AB

m~

2

A= contact area

Undrained

loading

Cross-anisotropic

‘Gibson’ halfspace

obeying eqn. (3.11) , with

a modulus GVH=~

m (z/B)

Vertical, on

foundations of any

shape

−+

n

GEA

B

m VHV

41

~

A= contact area

Undrained

loading

General cross-anisotropic

‘Gibson’ halfspace not

obeying eqn. (3.11) , with

a modulus GVH=~

m (z/B)

Vertical, on rigid

strip ( )[ ] )6/(4/5.31

45BH

V

nBH

E

−+

5.25.0

41

≤≤

≤≤

n

BH

Horizontal, on rigid

strip 1.0)/(1.4

3/51

5

8

BHn

HBEV

−

+

5.25.0

61

≤≤

≤≤

n

BH

Shallow cross-anisotropic

undrained layer; soil

properties are uniform

throughout the layer and

they satisfy eqn.(3.11)

The static vertical, horizontal and rocking behavior of rigid strip foundation supported by a

halfspace or a stratum whose wave velocities increase linearly with depth, has been studied

by Gazetas [7]. Some results of that study presented here for a halfspace consisting of soil

with a constant mass density, a constant Poisson’s ratio, ν=0.5, constant hysteretic damping,

ξ=0.05, and an S-wave velocity varying with depth according to: )1(B

zVV os λ+= ; in which:

Vo= surface velocity; 2B= foundation width; and λ= the dimensionless rate of

H

-23-

inhomogeneity. Figure 3.5 portrays the dependence of λ of the normalized vertical, horizontal

and rocking stiffnesses. As one might expect, the vertical stiffness exhibits the largest

sensitivity to λ and the rocking stiffness the smallest- another manifestation of the difference

in the pressure bulbs of the three types of loading.

60

0.5 1.5 5 10

20

40v

r

h

λ

No

rmal

ized

sti

ffnes

s

Figure 3.5:Static stiffnesses of rigid strip foundation on a homogenous halfspace(ν=0.25)[7]

Effect of Soil Anisotropy

Numerous experimental studies have shown that most natural soils and rocks posses

anisotropic deformational characteristics. This anisotropy stems from the fact that soil fabric

is intimately related to the mechanical processes occurring during formation, which involves

anisotropic stress systems. Thus, for example, natural clay deposits formed by sedimentation

and, subsequently, one-dimensional consolidation over long periods of time acquire a fabric

that is characterized by particles or particle clusters oriented in a horizontal arrangement. This

preferred orientation makes the clay a cross-anisotropic material with a vertical axis of

symmetry. Similarly, fabric anisotropy in sands arises from the influence of gravity forces

and particle shape on the deposition process, while in rocks the anisotropy may result from

the anisotropy of forming minerals and micro or macro-fabric-features.

While an isotropic elastic material is characterized by only two independent elastic constants

(e.g. shear modulus and Poisson’s ratio), five parameters are needed to describe the stress-

strain relationships of an elastic cross-anisotropic material: a Young’s modulus EV in the

vertical direction; a Young’s modulus EH in the horizontal direction (EH=n EV); a Poisson’s

ratio νVH for the effect of vertical on horizontal strain; a Poisson’s ratio νHH for the effect of

horizontal on complementary horizontal strain; and a shear modulus GVH = GHV for distortion

-24-

in any vertical plane, i.e. any plane parallel to the vertical axis of material symmetry. The

condition of incompressibility, appropriate for undrained loading conditions, requires that:

21,5.0

nHHVH −== νν -------------------------------------------------- (3.10)

and, thus, reduces the number of independent material constants to three. Moreover, utilizing

the results of several experimental investigations, Gazetas [7] has recently shown that, in

many clays, the shear modulus GVH is closely related to the other four material constants.

Under undrained conditions, for example, with reasonable accuracy:

n

EG V

VH−

=4

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

Thus, the number of independent material constants reduces to two, under undrained

conditions, and to four, under drained conditions.

Table 3.7 offer some simple but fairly accurate formulae for the vertical static stiffness of

arbitrary-shape foundations on a cross-anisotropic, incompressible and inhomogeneous half

space (‘Gibson’ soil), and for the vertical and horizontal static stiffnesses on a homogenous

and incompressible cross-anisotropic shallow soil stratum on rigid base. Notice that on

anisotropic ‘Gibson’ halfspace obeying equation (3.11), the degree of anisotropy has no

influence on the vertical stiffness. In all other cases, however, the stiffnesses increase

substantially with n=EH/EV. In fact, for n→ 4 all stiffnesses tend to infinity, since the strain

energy of such a material is zero for all possible stress systems [8].

In general for most problems considered so far, the following groups of dimensionless

parameters which appreciably influence the static stiffnesses have been identified:

a) The ratio H/B of the top layer thicknesses, H, over a critical foundation plan dimension,

B; the latter may be interpreted as the radius, R, of a circular foundation or half the width

of a rectangular or strip foundation.

b) The embedment ratio D/B, where D is the depth from the surface to the horizontal soil

footing interface

c) The shape of the foundation plan: Circular, strip, rectangular, circular ring; in the last two

cases the plan geometry may be defined in terms of the length-to-width or ‘aspect’ ratio,

L/B, or the internal or external radii ratio, Ri/R, respectively.

-25-

d) The ratio G1/G2 of the shear moduli corresponding to the upper soil layer and the

underlying half space , respectively, this ratio may attain values ranging from 0, in the

case of a uniform stratum or rigid base , to 1, in case of a uniform half space.

e) The Poisson’s ratio(s) ν of the soil layer(s)

f) The relative flexural rigidity factor

3

2)1(

−=

B

t

E

ERF f

s

fυ where Ef, νf and t are

respectively, the Young’s modulus, Poisson’s ratio and thickness of the foundation raft;

RF ranges from ∞, for a perfectly rigid foundation , to 0, for an ideally flexible mat.

g) The factors n and m which express the degree of anisotropy and the rate of in-

homogeneity, respectively; n = EH / EV, where EH and EV are the horizontal and vertical

Young’s moduli of a cross anisotropic soil; while m, for a certain type of in-

homogeneity, describes the change of shear modulus from the surface to a depth equal to

B.

A comprehensive compilation of characteristic numerical results for static stiffnesses of

massless foundation, pertaining to all possible (translational and rotational) modes of

vibration are presented in Ref. [6].

-26-

4. MODELING OF THE BUILDING STRUCTURE

The building structure is considered to be composed of two basic different parts. Theses are

the soil-foundation system and the superstructure. The soil foundation system includes the

foundation itself and the soil around the foundation. As discussed in the previous chapters,

the force displacement ratio at ω=0 yields the required soil static stiffness coefficients.

Introducing the appropriate springs at the base of the superstructure model, would then enable

one to account for the soil flexibility in the static analysis of the structure. This approach is

commonly referred to as sub structure method.

4.1 The Foundation Model

As discussed in the previous chapters, foundation of structures, when subjected to static

loads, deform in a way that depends on: the nature and deformability of the supporting

ground, and the shape of the foundation, depth of embedment of the foundation and the

applied loads. Thus the key step in the analysis of the system is to model the soil around the

foundation, and estimate the static stiffnesses of flexibly supported foundation.

A rigid foundation on a flexible soil in general possesses six degrees of freedom- three

translational and three rotational. The foundation-soil system is replaced by a massless rigid

foundation of appropriate shape supported by springs along the degrees of freedoms

considered. Figure 4.1 below, for example, shows foundation model for a planar structure.

Figure 4.1: Foundation model for planar analysis

Kh

Kv

Kr

-27-

4.2 The Building Model

The general case of a framed building structure whose foundations are embedded in a

flexible soil layer of mass density ρ1 shear modulus of rigidity G1, and Poisson’s ratio υ1,

which overlies the half space of corresponding parameters ρ2, G2, υ2 is considered, Fig 4.2.

The building is assumed to be supported at the foundation level by springs (static springs) in

each of the six degrees of freedom. This model is then subjected to gravity and lateral loads.

The latter types of loads are generally of dynamic in nature like earthquake and wind loads.

These loads are, however, assumed to be quasi-static as it is the case in code provisions

which are applicable to quite a large class of building structures.

(a)

(b)H

Figure 4.2: a) The Building Model

b) The Building Model with the foundation model

G1, ρ1, ν1

G2, ρ2, ν2

-28-

5. PARAMETRIC STUDY

So far the background information on the soil stiffness coefficients and the modeling of the

system has been reviewed and systematically presented for the purpose of using it in this

work and also for future reference.

In this chapter, parametric studies are carried out to investigate the effect of soil flexibility on

the internal force distribution of systematically selected symmetrical building structures

founded on flexible soil formations.

Some of the major factors which are expected to influence the internal force distribution of

building structures include the nature and layering of the soil stratum, the shape and the

dimensions of the foundation, the embedment depth of the foundation and the type of the

structural system. The effect of these parameters is studied systematically. In order to avoid

unnecessary complications due to torsion that could possibly obscure the influences of the

major factors mentioned above, the buildings considered are symmetric with respect to both

rigidity and geometry. Furthermore, the height of the structure studied is limited to a height

of 20 stories. This is conformant with the range of applicability of the pseudo-static method

of analysis for lateral earthquake loads according to the current seismic codes including

EBCS 8 [12].

5.1 The Foundation Models Employed

The foundation models considered in the analysis of different cases are shown in Figure 5.1.

G,ρ,ν G,ρ,ν

G,ρ,ν

D

G,ρ,ν

D

Rigid formationRigid formation

(a) (b)

(c) (d)

Figure 5.1: The Foundation models: (a) Foundation on the surface of halfspace; (b)

Foundation embedded in half space; (c) Foundation on the surface of homogenous stratum

overlying bedrock; (c) Foundation embedded in homogenous stratum overlying bedrock

-29-

5.2 The Soil Categories

For practical reasons related to the provisions of the Ethiopian Building Code Standard 8, the

soil conditions considered are soil classes A, B, and C, which are described as follows [12]:

Subsoil Class A

1 Rock or other geological formation characterized by the shear wave velocity vs of

at least 800m/s , including at most 5m of weaker material at the surface.

2 Stiff deposits of sand, gravel or over consolidated clay, at least several meters

thick, characterized by a gradual increase of the mechanical properties with depth

and by vs-values of at least 400m/s at a depth of 10m.

Subsoil Class B

Deep deposits of medium dense sand, gravel or medium stiff clays with thickness

from several tens to many hundreds of meter, characterized by vs-values of at least

200m/s at a depth of 10m; increasing to at least 350m/s at a depth of 50m.

Subsoil Class C

1 Loose cohesionless soil deposit with or without some soft cohesive layers,

characterized by vs-values below 200m/s in the uppermost 20m.

2 Deposits with predominant soft-to-medium stiff cohesive soils, characterized

by vs-values below 200m/s in the uppermost 20m.

It is to be noted, however, that this categorization is too rough and has recently been replaced

by more refined ones including larger number of soil classes established in more rational

ways [10].

5.3 The Structural Systems Studied

For this parametric study three types of building structures are considered. The buildings are

assumed to cover the same plan area, but with different lateral force resisting systems

including frame and dual systems. The size and the number of columns and concrete shear

walls are increased realistically as the numbers of stories are increased. The details of each of

the three structural systems considered are discussed as follows.

-30-

a) Structural System 1

The first structural system is a ground-plus-five (G+5) regular reinforced concrete building,

whose plan is shown in Figure 5.2. The column size used is 400mmX400mm. It is assumed

that the columns are rigidly connected to the floor beams. The floor height is taken 3m

throughout. There are no concrete shear walls used for this building model. A typical frame

in the y-direction is also shown in Figure 5.3(a). However, the analysis is made on the three-

dimensional model of the building. The foundation type used is isolated foundations. Some of

the features of the building are:

� Size of beam - 250mm X 300mm

� Size of column - 400mm X 400mm

� Type of structural system - Framed (no shear walls)

� Foundation type used - Isolated footings

� Loads are given in Appendix A

x

5.0 5.0 5.0 5.0 5.0 5.0

5.0

5.0

5.0

5.0

5.0

5.0

5.0

5.0

D

C

B

A

1 2 3 4 5 6 7

E

y

Figure 5.2: Floor Plan for Structural System1

-31-

(a) (b) (c)

Figure 5.3: Typical lateral force resisting systems in the Y direction: (a) Structural

System 1 (G+5); (b) Structural System 2 (G +10); (c) Structural System 3 (G +20)

b) Structural System 2

The second structural system is a regular ground plus 10 (G+10) reinforced concrete building

whose plan is shown in Figure 5.4 where SW stands for concrete shear walls. The column

size used here is 650mmX650mm. The assumption that columns and concrete walls are

rigidly connected to the floor slabs applies. The floor height is taken 3m throughout.

Eventhough analysis is based on the three-dimensional model of the building; a typical frame

in the y-direction along with concrete shear wall is shown in Figure 5.3(b). Mat foundation is

employed for this structural system. The data used for the analysis of this building are

summarized below.



� Type of structural system - Dual system (Frames with shear walls)

� Shear wall thickness - 200mm

� Four Shear walls of length = 5 m

� Foundation type used - Mat foundation


-32-

y

x

5.0 5.0 5.0 5.0 5.0 5.0

5.0

5.0

5.0

5.0

SW

SW

(S

W2)

SW

SW

5.0

5.0

5.0

5.0

5.0

5.0 5.0 5.0 5.0 5.0 5.05.0

5.05.0C

A

B

1 2 3 4 5 6 7

E

D

Figure 5.4: Floor Plan for Structural System 2

c) Structural System 3

Structural System 3 is a regular ground plus 20 (G+20) reinforced concrete building whose

plan is shown in Figure 5.5 with shear walls labeled as SW. The column size used is

900mmX900mm. Once again it is assumed that the columns and the shear walls are rigidly

connected to the floor slabs. The floor height is taken 3m throughout. The typical frame

linked with concrete shear wall in the y-direction is shown in Figure 5.3(c). However, the

analysis is made on the three-dimensional building model. Mat foundation is employed for

this system. A summary of some features of the building are as follows.



� Type of structural system - Dual system (Frames with shear walls)

� Shear wall thickness - 200mm

� Six shear walls of length - 5m

� Foundation type used - Mat foundation


-33-

5.05.0

5.05.0

5.0

5.0

SW

5.0

5.0

5.0

5.0

5.0

5.0

5.0

5.0

SW

(S

W2

)

5.0

y

5.05.05.0

SW

5.0

5.0

x

5.05.05.0

SW

5.0

SW SW5.0 5.0

B

A

1 2 3 4 5 6 7

C

D

E

Figure 5.5: Floor Plan for Structural System 3

5.4 Results of the Parametric Study

This section defines the various cases studied and presents the results of the analysis. The

combination of the main parameters employed for each particular case is also specified.

The responses studied include the bending moments, shear forces, axial loads, storey drifts

and reactions at the foundation of the building due to lateral equivalent static earthquake

loads.

The coefficients of the static springs are determined using assumed mass density of

1800Kg/m3 and Poisson’s ratio of 0.35 for all soil types whereas the shear wave velocity is

taken as 400m/s,200m/s, 100m/s for soil type A, B, and C respectively. The difference in the

soil categories is thus mainly represented by the magnitudes of the shear wave velocities

assigned to them in accordance with code specifications.

5.4.1 Structural System 1

a) Cases Considered for the Study of Structural System 1

In order to study the effect of flexibility of soil on the responses of Structural System 1, the

following cases are considered.

-34-

Case 1.1: The first case considered is the analysis of the G+5 building, where the structure is

assumed fixed at its base.

Case 1.2: This is a case in which the analysis is of the G+5 building, where the foundation is

placed at a depth of 1m below the surface of an elastic half space.

Case 1.3: This is the analysis of the G+5 building, where the foundation is placed at a depth

of 3m below the surface of an elastic half space.

Case 1.4: This case is the analysis of the G+5 building, where the foundation is placed at a

depth of 5m below the surface of an elastic half space.

Case 1.5: This one is the analysis of the G+5 building, where the foundation is placed at a

depth of 10m below the surface of an elastic half space.

Values of the static springs for each case can be appropriately calculated as discussed in the

previous chapters of this work. However, better results, accounting for all the factors

influencing the static stiffnesses, are obtained on the basis of tables provided by Gazetas [6],

and are summarized below in Table 5.1. In this table Khx is the lateral static spring constant

along the x axis in units of KN/m, Khy is the lateral static spring constant along the y axis in

KN/m, Kv is the vertical static spring constant in KN/m, Krx is the rocking static spring

constant about x axis in KNm, Kry is the rocking static spring constant about y axis in KNm

and Kt is the torsional static spring constant in KNm. The left-right direction on the plane of

the paper is considered as the x-direction.

Table 5.1: Calculated Static Springs for Different Cases of Structural System 1

Static Stiffnesses (X106)

Cases of Soil A Cases of Soil B Cases of Soil C

1.2 1.3 1.4 1.5 1.2 1.3 1.4 1.5 1.2 1.3 1.4 1.5

Khx 4.28 4.63 4.87 5.31 1.54 1.64 1.71 1.83 0.386 0.41 0.427 0.458

Khy 4.28 4.63 4.87 5.31 1.54 1.64 1.71 1.83 0.386 0.41 0.427 0.458

Kv 3.91 4.46 5.01 6.39 1.54 1.67 1.80 2.12 0.385 0.42 0.449 0.53

Krx 7.81 8.01 8.13 8.30 8.16 8.25 8.29 8.37 2.04 2.06 2.07 2.09

Kry 9.18 9.64 9.98 10.6 9.99 10.1 10.2 10.5 2.50 2.53 2.56 2.62

Kt 15.2 15.2 15.2 15.2 15.6 15.6 15.6 15.6 3.89 3.89 3.89 3.89

In Table 5.1, the vertical static stiffness is more influenced by increase in embedment depth

than the other static stiffnesses and the embedment depth has practically no influence on

-35-

torsional static stiffness. Except in the torsional mode, the static stiffnesses increase with the

embedment depth. Moreover, the soil type has an effect on stiffness values in such a way that

static stiffnesses generally decrease from Soil Type A through Soil type B to Soil Type C.

The above static springs are then assigned at the base in the respective degrees of freedom in

the analysis of the building to obtain the internal forces. Three-dimensional analysis of the

building is made using the structural software, ETABS Non linear V 8.00, which has features

to incorporate the soil flexibility by introducing static springs. The results of the analysis are

discussed next by comparing all cases of this system.

b) Discussion on the Analysis Results of Structural System 1

For the purpose of discussion, the internal forces obtained from the 3D analysis for a

systematically chosen frame are considered. In order to show the effects of the various

parameters on the internal force distribution, the frame on Axis 4 of Figure 5.2 is chosen and

the discussions are presented below in sections (i), (ii) and (iii).

i) Effect of embedment

In the previous discussion (5.4.1(a)), we saw that an increase in embedment depth has

increased the values of static stiffnesses. Thus one can expect that the deeper the foundation

is placed, the lesser the effect of flexibility of soil on the internal force distribution. The

general effect of embedment depth on the internal force distribution in this structural system

can be observed in Figures 5.6 to 5.11 for Soil Type C. This soil is chosen for it is expected

to show the influence clearly, than the other soil types because of its high deformability. The

results provided here for the frame on Axis 4 are for fixed-base and flexible-base cases,

where the foundations are placed at a depth of 1m below the surface of the soil. The results of

embedded foundations with flexible base at a depth of 3m, 5m and 10m were found to lie in

between the above two extreme cases.

The embedment depth has influenced the bending moments, which for the fixed-base and the

flexible-base cases are given in Figures 5.6 and 5.7 respectively, for the frame on Axis 4.

-36-

Figure 5.6: Frame on Axis 4 (Fixed base & Soil C)-Bending Moment Diagram in KNm

Figure 5.7: Frame on Axis4 (Flexible base, D=1m &Soil C) -BMD in KNm

-37-

From Figures 5.6 and 5.7, the following differences in bending moment are observed. In

beams located at the end bays (i.e. between Axis A&B or Axis D&E), average difference of

about 38.6 % at the supports and 0.35% around middle span are observed. In beams located

at the interior of the frame (i.e. between Axis B&D), an average difference is about 9.2 % at

the support and 0.42% around the middle spans.

Particularly, to show the relative difference in bending moments between the results of the

fixed-base and the flexible-base system(extreme case of embedment depth of 1m), the beams

located between Axis-A and Axis-B are chosen and the plot showing this difference along the

length of the beam for all the stories in Soil Type C is shown in Figure 5.8. The choice is

based on the fact that pronounced differences were observed in these spans than elsewhere.

Difference in BM along the length -70

-60

-50

-40

-30

-20

-10

0

10

20

30

40

0 1 2 3 4 5Length(m)

Dif

fere

nce(K

Nm

)

storey 7 storey 6storey 5storey 4storey 3storey 2storey 1

Figure 5.8: Plot showing difference in bending moment along the beam length

As indicated in Figure 5.8, significant differences are observed at the supports. The

differences in bending moments at the spans are relatively lower and especially at the mid

span the difference is almost zero. The deviations become larger when one goes down from

the upper stories to the lowest.

Effect of embedment depth on the shear force distributions can be seen on the shear force

diagrams for the two extreme cases of fixed-base and flexible-base systems at a depth of 1m

below the surface of Soil Type C as provided in Figures 5.9 and 5.10.

-38-

Figure 5.9: Frame on Axis 4 (Fixed base & Soil C) -Shear Force Diagram in KN

Figure 5.10: Frame on Axis 4 (Flexible base, D=1m& Soil C) -SFD in KN

-39-

In the beams located at the end bays (i.e. between Axis A&B or Axis D&E), average

differences of about 15.8 % around the supports and 0.7% around mid-spans are observed. In

beams located at the interior of the frame (i.e. between Axis B&D), an average difference of

about 2.75 % at the supports and 0.05% around mid-spans can be seen.

For the same reason cited above for the bending moments, spans between Axis-A and Axis-B

are chosen and the plot showing this difference along the length of the beam is shown in

Figure 5.11.

Difference in SF along the length

-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

0 1 2 3 4 5Length (m)

Dif

fere

nce(K

N)

storey7storey6

storey 5storey 4storey 3

storey 2Storey 1

Figure 5.11: Plot showing difference in shear force along the beam length

Here also, differences in the shear forces are observed along the length of the beam but

around the mid-span where the shear changes its direction the differences are too small.

Moreover, the quantitative differences are seen to increase as one goes down from the top

story to the lowest storey.

The axial forces in the columns for the fixed-base and flexible-base systems were also seen to

be affected by the embedment depth. The observed differences at the lowest storey are given

in Table 5.2. The plot showing the entire distribution of the axial forces is omitted for brevity

reason. The maximum differences are observed in the external columns. The differences

within a column increase as one goes down from the roof to the foundation. Generally

speaking, the effects on the column axial forces are not as pronounced as in the beam shears

and bending moments.

-40-

Table 5.2: Summary of differences in Axial Force

Column Axis Difference (%)

A 13.0

B 10.0

C 3.0

D 8.0

E 11.0

ii) Effect of Soil Type

In section 5.4.1(a) we saw that the static stiffness coefficients were found to change when the

soil type is changed keeping the other parameters the same. The static stiffnesses of relatively

softer soil were less than that of relatively firm soil. The effect of the soil type on the internal

force distribution of structural system under consideration can be seen in Figures 5.12 to

Figure 5.17 for the case of foundations placed at a depth of 3m.This depth is selected as it is

around a representative embedment depth for shallow foundations.

The bending moments are influenced by the soil type as shown in Figure 5.12, 5.13 and 5.14.

Figure 5.12: Frame on Axis 4 (Soil A & depth 3m) -Bending Moment Diagram in KNm

-41-

Figure 5.13: Frame on Axis 4 (Soil B & depth 3m) -Bending Moment Diagram in KNm

Figure 5.14: Frame on Axis 4 (Soil C & depth 3m) -Bending Moment Diagram in KNm

-42-

In Figures 5.12, 5.13 and 5.14, eventhough the trend in the variation of bending moment from

one soil type to another is not predictable, it can be seen that there is a general increase in

bending moments as we go from Soil Type A through B to C except around Axis C. Average

differences as summarized in Table 5.3 are noteworthy.

Table 5.3: Summary of Differences in Frame Bending Moments

Difference b/n Soil Type Average Difference (%)

A & B 12.8

B & C 22.2

A & C 24.9

The shear force diagrams of Figures 5.15 to 5.17 for the three soil types at a depth of 3m

show the effect of the soil types on the shear forces.

Figure 5.15: Frame on Axis 4 (Soil A &depth 3m) - Shear Force Diagram in KNm

-43-

Figure 5.16: Frame on Axis 4 (Soil B &depth 3m) -Shear Force Diagram in KNm

Figure 5.17: Frame on Axis 4 (Soil C & depth 3m) -Shear Force Diagram in KNm

-44-

Eventhough the trend in the variation of shear force from one soil type to the other soil type is

not predictable, the average differences as summarized in Table 5.4 are quite notable.

Table 5.4: Summary of Differences in Frame Shear Forces


A & B 8.3

B & C 17.7

A &C 19.7

The influence of the soil type on the column axial forces has also been studied in a similar

manner as that of the bending moments and shear forces. And the trend in the variation of the

axial force from one soil type to another is summarized in Table 5.5. A trend in increase of

the axial force is seen when the soil type is varied from Soil Type A through Soil Type B to

Soil Type C in the external columns (Axis A-4 & E-4). However, a decrease in the axial force

is observed in the internal columns (Axis B-4, C-4 & D-4).

Table 5.5: Influences of Soil Type on Column Axial Forces

Average difference (%) between Soil Type

Column Axis A & B B& C A & C

A-4 2.7 7.2 9.7

B-4 2.1 6.0 8.0

C-4 0.3 3.2 3.5

D-4 1.7 4.8 6.4

E-4 3.2 7.3 10.3

(iii) Comparison of Fixed-Base and Flexible-Base Systems

The usual analysis of building structures based on a fixed-base system results in internal

forces which are different from those obtained by the analysis that incorporates the flexibility

of the soil. This difference was observed in section (i) above. Besides, better and closer look

is made by considering the influences of all the parameters affecting the internal force

distribution on particular elements (i.e. beams and columns). For this purpose, Beam B13 and

Column C16 as shown in Figure 5.18 are chosen. This choice is made for the reason that

larger differences are observed on the edge members as discussed in the previous sections. A

series of plots provided below will show the differences in internal forces (shear, moment,

-45-

axial load) between analysis with fixed-base and that with flexible-base along the length of

that particular element.

Figure 5.18: Beam and column labeling

For Beam 13, the bending moment diagrams of the fixed-base and the flexible-base along the

length of the beam is plotted for Soil Types A, B & C, embedment depths of 1m, 3m, 5m,

10m and all stories. Finally four curves, two for fixed base and two for flexible base, showing

extreme differences are shown in Figure 5.19.

Variation in BM Along the Length of the Beam-120

-100

-80

-60

-40

-20

0

20

40

60

0 1 2 3 4 5

Length(m)

Mo

men

t(K

Nm

)

storey1,soilC,Fixed base D=1

storey7,soilA,Fixed base D=1

storey7,soilA,Flexible base D=1

storey1,soilC,Flexible base D=1

Figure 5.19: Plot of difference in moment along the beam

From the above plot, maximum difference at the support and negligible difference around the

mid span are observed. Also, moments for the flexible-base are greater in the left half span of

-46-

the beam and are less in the right half. This shows that accounting for flexibility of the soil

may lead to conservative design at one section and unsafe design at another section even

within a single span.

Similarly for this beam, the shear force diagrams of the fixed-base and the flexible-base

along the length of the beam are drawn for Soil Types A, B & C, embedment depths of 1m,

3m, 5m, 10m and all stories. Figure 5.20 shows extreme differences in shear forces.

Variation in Shear Force Along the Length of the Beam-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

0 1 2 3 4 5

Length(m)

Sh

ear(

KN

m)





Figure 5.20: Plot of difference in shear of a particular beam

The shear in the flexible-base system is greater than those of the fixed-base system for about

the half span and vice versa for the other half span of the beam.

In a similar manner, the axial force diagrams of the fixed-base and the flexible-base along the

height of the Column 16 are plotted for Soil Types A, B & C, embedment depths of 1m, 3m,

5m, and 10m and all stories. Finally four curves showing extreme differences are shown in

Figure 5.21.

-47-

Variation in Axial Force Along the Height of the Column

0

1

2

3

-2400 -1900 -1400 -900 -400

Axial Force(KN)

He

igh

t(m

)

storey7, soil A,Fixed base D=1

storey 2, soil C, Fixed base D=1

storey7, soil A,Flexible base D=1

storey 2, soil C, Flexible base D=1

Figure 5.21: Plot of difference in axial load of a particular column

Figure 5.21 shows that the axial force differences between the fixed-base system and flexible-

base system are negligible as two curves for a single storey are almost overlapping.

In the same manner as that of the shear forces, the bending moment diagrams for Column 16

have shown that there is a difference in the values of the bending moments between the fixed-

base and the flexible-base system. This difference is found to increase as we move down the

stories.

The difference in reactions at the base of the fixed-base and flexible base can also be an

indication of the influence of soil flexibility. Hence, reactions under columns located along

Axis 4 of Figure 5.2 are considered. The axial forces of the fixed-base and the flexible-base

systems at the base of columns located on Axis 4 (Col 16, Col 17, Col 18, Col 19, and Col

20) are plotted for Soil Types A, B & C, and embedment depths of 1m, 3m, 5m, and 10m.

Finally curves showing greater differences are shown in Figure 5.22.

-48-

Variation in Axial Load at the Base of Columns

2250

2750

3250

3750

4250

16 17 18 19 20Col

Axia

l F

orc

e(K

N)

soil A, Fixed base D=10

soil C, Fixed base D=1

soil A, Flexible base D=10

soil C, Flexible base D=1

Col Col Col Col

Figure 5.22: Plot of difference in axial load at the foundation level

In the above plot the maximum percentage difference is found to be 10%. Perhaps the overall

variation in axial loads is not significant.

The corresponding bending moments at the bases of columns on Axis 4 are plotted in Figure

5.23 for fixed-base and flexible-base.

Variation in Bending Moment at the Base

70

90

110

130

150

170

190

16 17 18 19 20Col

Mo

men

t(K

N)





Col Col Col Col

Figure 5.23: Plot of difference in moment at the foundation level

The difference in bending moments between the fixed-base and flexible base systems for soil

type A is negligible. However a maximum difference of 20% was observed along Axis 4.

Hence it can be seen that the observed difference in bending moment at the bases may affect

the foundation design.

-49-

The storey drifts along the direction of the application of lateral loads are also considered to

study the variation between fixed-base and flexible-base systems. Here the difference is

considered for Soil Types A, B & C, embedment depths of 1m, 3m, 5m, and 10m. The

smallest difference is observed in Soil type A and Table 5.6 shows this. The largest

difference is observed in Soil type C as shown in Table 5.7.

Table 5.6: Minimum Variation in Storey Drifts

Story

Value of Max drift

in Fixed base (m)

Value of Max drift in

Flexible base at 10m(m)

Difference b/n Fixed base and

flexible base at D=10m (m)

STORY7 4E-05 4.20E-05 2E-06

STORY6 8E-05 8.20E-05 2E-06

STORY5 1E-04 1.02E-04 2E-06

STORY4 1E-04 1.02E-04 2E-06

STORY3 2E-04 2.02E-04 2E-06

STORY2 4E-04 4.04E-04 4E-06

STORY1 7E-04 7.10E-05 1E-05

Table 5.7: Maximum Variation in Storey Drifts

Story

Value of Max drift

in Fixed base (m)

Value of Max drift in

Flexible base at D=1m (m)


flexible base at D=1m (m)

STORY7 6E-05 9.0E-05 3E-05

STORY6 1E-04 1.3E-04 3E-05

STORY5 2E-04 2.3E-04 3E-05

STORY4 2E-04 2.3E-04 3E-05

STORY3 2E-04 2.3E-04 3E-05

STORY2 2E-04 2.4E-04 4E-05

STORY1 8E-05 1.6E-04 8E-05

From these tables it can be seen that the storey drifts are too small but the difference due to

soil flexibility is relatively significant. A difference of up to 100% is observed.

So far, for Structural System 1, differences in the internal force distributions and storey drifts

were observed between the fixed-base and the flexible-base systems. The flexibility of the

soil has influenced the internal forces around the supports significantly. However its

influence around the span is too small. This influence in bending moments is greatest while in

the axial loads is the least. The influences in the internal forces increase as we decrease the

embedment depth keeping the other parameters constant. Similarly, keeping the other

parameters constant these influences increase as the soil type is changed from relatively firm

soil type to a softer soil type. These influences are even larger for taller buildings as can be

seen in the Structural Systems 2 and 3 presented next.

-50-



The effect of flexibility of soil on the responses of Structural System 2 is studied by

considering similar cases as that of Structural System 1. Thus, we will have five cases which

are identical to that of Structural System 1 but G+10 building is considered instead of G+5.

Static springs for each case are calculated in exactly the same manner as that of Structural

System 1. Global static stiffnesses are first calculated for the mat foundation considering its

respective dimensions. Static stiffnesses at the base of each column and shear wall are

determined according to the tributary area. The calculated static stiffnesses are given in Table

5.8. Translational stiffnesses are given in kN/m whereas the rotational stiffnesses are in kNm.

Table 5.8: Calculated Static Springs for Different Cases of Structural System 2

Static Stiffnesses (X106)


2.2 2.3 2.4 2..5 2.2 2.3 2.4 2..5 2.2 2.3 2.4 2..5

Khx 0.669 0.690 0.705 0.733 0.167 0.173 0.176 0.183 0.042 0.043 0.044 0.046

Khy 0.681 0.704 0.719 0.747 0.170 0.176 0.180 0.187 0.043 0.044 0.045 0.047

Kv 0.788 0.801 0.815 0.848 0.197 0.200 0.204 0.212 0.049 0.050 0.051 0.053

Krx 78.1 78.2 78.2 78.2 19.5 19.5 19.5 19.6 4.9 4.9 4.9 4.9

Kry 163 163 163 163 40.8 40.9 40.9 40.9 10.2 10.2 10.2 10.2

Kt 174 174 174 174 43.4 43.4 43.4 43.4 10.9 10.9 10.9 10.9

Then, the above static springs are assigned at the base in the respective degrees of freedom in

the three dimensional analysis of the building to obtain the internal forces. The results of the

analysis are discussed in the following sections.


The internal forces obtained from the 3D analysis for a systematically chosen frame are

considered to discuss the results. To show the effects of the various parameters on the internal

force distribution, the frame on Axis 4 and the shear wall SW2 of Figure 5.4 are chosen and

the discussions are presented below in sections (i) to (iii).

-51-


For the same reason as that of Structural System 1, the general effect of embedment depth on

the internal force distribution for Soil Type C is considered in Figures 5.24 to 5.34. The

results provided here for the frame on Axis 4 and shear wall SW2 are those with fixed-base

and spring-base founded at a depth of 1m below the surface of the soil. All the results of

embedded foundations with flexible base at a depth of 3m, 5m and 10m fall in between the

above two extreme cases.

In order to assess the effect of the embedment depth on the bending moments, the other

parameters are kept constant. The bending moment diagrams for Soil Type C and embedment

depth of 1m are shown in Figure 5.24 and 5.25 for the fixed-base and flexible-base systems.

Figure 5.24: Frame on Axis 4 (Fixed base& Soil C)-Bending Moment Diagram in kNm

-52-

Figure 5.25: Frame on Axis4 (Flexible base, D=1m&Soil C) -BMD in kNm

From the plots in Figures 5.24 and 5.25, the following differences in bending moment are

observed. In beams located at the end bays (i.e. between Axis A&B or Axis D&E), average

difference of about 57.3 % at the supports and 20.6% around middle span are observed. In

beams located at the interior of the frame (i.e. between Axis B&D), an average difference is

about 48.5 % at the support and 10.1% around the middle spans. The differences observed in

-53-

some of the beams between Axis A & B are more than 100%. It is particularly to be noted

that these significant differences are observed in locations where the moments in the fixed-

base model are also maximum. In these locations, the bending moments in the flexibly

supported system are consistently larger than those in the fixed-base system. This is an

indication that the conventional design at these critical sections may be on the unsafe side.

To show the relative differences in bending moments between results of the fixed-base and

flexible-base system(extreme case of embedment depth of 1m), the beams in between Axis-A

and Axis-B are chosen and the plot showing this difference along the length of the beam for

all the stories in Soil Type C is shown in Figure 5.26. The choice is based on the fact that

pronounced differences are observed in these spans than elsewhere. Variation in Bending Moment Along the Beam Length Variation in Bending Moment Along the Beam Length Variation in Bending Moment Along the Beam Length Variation in Bending Moment Along the Beam Length-80

-60

-40

-20

0

20

40

60

0 1 2 3 4 5Length (m)

Dif

fere

nc

e (

KN

m)

storey 12, D=1

storey 8, D=1

storey 4, D=1

storey 2, D=1


The maximum difference is seen to be around the supports and negligible difference is

observed around the mid span of the beam. One can see that these differences are greater than

that of Structural System 1 by about 25.5%.

For the shear wall SW2, the overturning moments obtained by analyzing the building fixed

and flexible at the base are shown in Figure 5.27.

-54-

Variation of Overturning Moment Along the Shear w all Height

0

3

6

9

12

15

18

21

24

27

30

33

-1.5-0.7500.751.52.253

Moment(KNm) x103

Sto

rey H

eig

ht(m

)

Fixed base D=1

Flexible base, D=1

Figure 5.27: Overturning Moment in Shear wall SW2 (Soil C and D=1m)

The moments given are for the total shear wall length (5m). The overturning moment in the

flexible-base is larger than that of the fixed base for about the middle one-third of the wall

height. However, a significant reduction is observed around the bottom of the shear wall.

Since the latter is governing in design, one can easily note the potential saving in material if

the base is properly modeled.

The shear force diagrams for the fixed-base and flexible-base systems are shown in Figure

5.28 and 5.29 to observe the effect of embedment depth in the values of the shear forces.

-55-

Figure 5.28: Frame on Axis 4 (Fixed base & Soil C) -Shear Force Diagram in KN

-56-

Figure 5.29: Frame on Axis 4 (Flexible base, D=1m& Soil C) -SFD in KN

In Figures 5.28 and 5.29 we can easily observe differences in shear forces. In beams located

at the end bays (i.e. between Axis A&B or Axis D&E) average differences of about 40.5 %

around the supports are observed. In beams located at the interior of the frame (i.e. between

Axis B&D) an average difference of about 28.8 % at the supports can be seen.

Especially, to show the relative difference in shear force between results obtained using the

fixed-base and flexible-base models (extreme case of embedment depth of 1m), beam spans

in-between Axis-A and Axis-B are chosen and the plot showing these differences along the

length of the beam is shown in Figure 5.30. This choice is based on the fact that pronounced

differences are observed in these spans than elsewhere.

-57-

Variation in Shear Force Along the Beam Length

-40

-30

-20

-10

0

10

20

30

40

0 1 2 3 4 5Length (m)

Dif

fere

nc

e(K

N)

storey 12, D=1

storey 8, D=1

storey 4, D=1

storey 2,D=1


It can be said that the maximum differences are shown at the supports and these differences

are grater than the differences seen in Figure 5.11 of Structural System 1.

For the shear wall SW2, the shear forces carried by the shear wall at storey levels are

obtained and their variation along the height of the wall is shown in Figure 5.31.

Variation of Storey shear shared by the Shear w all

0

3

6

9

12

15

18

21

24

27

30

33

-500050010001500

Shear force (KN)

Sto

rey H

eig

ht(

m)

Fixed base, D=1

Flexible base, D=1

Figure 5.31: Shear force at the storey level for Shear wall SW2 (Soil C & D=1m)

-58-

In this figure, the shear force diagram shown for fixed-base is significantly greater than that

of the flexible-base system almost in the total height of the shear wall. Particularly, the

difference around the base is rather high.

Once again, the axial force diagrams for the two extreme cases of fixed-base and flexible-

base systems at a depth of 1m below the surface of soil C are shown in Figures 5.32 and 5.33

to study the effect of embedment depth on the member axial forces.

Figure 5.32: Frame on Axis 4 (Fixed base & Soil C) -Axial Force Diagram in KN

-59-

Figure 5.33: Frame on Axis 4 (spring base, D=1m& Soil C) -Axial Force Diagram in kN

From Figure 5.33 and 5.34 we see that the axial forces in the flexible-base system are greater

than those of the fixed-base system in the exterior columns and vice versa in the interior

columns. The differences in axial forces at the lowest storey (of coarse it is approximately the

same in the other stories) are summarized in Table 5.9.



A 33.0

B 27.0

C 25.0

D 23.1

E 28.7

-60-

The maximum differences are observed on the external columns as compared to the interior

columns.

For the shear wall SW2, the differences in axial forces as obtained are shown in Figure 5.34.

Variation of Axial Force Along the Shear w all Height

0

3

6

9

12

15

18

21

24

27

30

33

-6000-5000-4000-3000-2000-10000

Axial force(KN)

Sto

rey H

eig

ht(

m)

Fixed base, D=1

Flexible base, D=1

Figure 5.34: Difference in Axial Force of Shear wall SW2 (Soil C & D=1m)

In Figure 5.34, the axial forces in the flexible base are larger than that obtained in fixed base

through out the wall height.


Different soil types have shown different internal force distributions in Structural System 1 as

discussed in section 5.4.1(b).The influence of soil type on the internal force distribution of

Structural System 2 is shown in Figures 5.35 to 5.40 for the case of foundations placed at a

depth of 3m.

-61-

The influences of soil type on the distributions of bending moments in the frames can be seen

by comparing Figures 5.35, 5.36 and 5.37.

Figure 5.35: Frame on Axis 4 (Soil A & depth 3m) -Bending Moment Diagram in kNm

-62-

Figure 5.36: Frame on Axis 4 (Soil B & depth 3m) - Bending Moment Diagram in kNm

-63-

Figure 5.37: Frame on Axis 4 (Soil C & depth 3m) - Bending Moment Diagram in kNm

In Figures 5.35 to 5.37, there is a general trend of increase in bending moments as we go

from Soil Type A through B to C. The average differences in the three soil types are

summarized in Table 5.10.

Table 5.10: Summary of Differences in Frame Bending moments


A & B 26.6

B & C 39.8

A & C 51.1

The difference between the relatively firm soil (Soil Type A) and soft soil (Soil Type C) is

maximum as expected. These percentage differences are greater than those of Structural

System 1 (Table 5.3)

-64-

The overturning moments carried by the shear wall SW2 and obtained by analyzing the

flexible-base building with an embedment depth of 3m in the three soil types is shown in

Figure 5.38.

Variation of Overturning Moment

Along the Shear wall Height

0

3

6

9

12

15

18

21

24

27

30

33

36

-150

0

-100

0

-50005001000150020002500

Moment(KNm)

Sto

rey

Heig

ht(

m)

Soil A

Soil B

Soil C

Figure 5.38: Overturning moment for Shear wall SW2 (Depth D =3m)

The bending moments resisted by the wall increases as the soil becomes softer. However, the

moment around the base is smallest for the softest soil.

The shear force distribution for the three soil types has been studied in order to observe

influence of soil types on the shear forces. Eventhough the trend in the variation of shear

force from one soil type to another does not show a clear trend, the average differences

summarized in Table 5.11 are observed.

Table 5.11: Percentage Difference in Shear Forces for Different Soil Types

Difference b/n Soil Type Average difference (%)

A & B 18.6

B& C 33.7

A&C 43.0

Here the largest difference is observed between the relatively softer soil and firm soil (soil

types A & C). Note also that shear force differences are less than the bending moment

differences of Table 5.10. However, they are greater when compared with the shear force

differences of Structural System 1 (Table 5.4).

-65-

For the shear wall SW2, the storey shear carried by the wall as obtained by analyzing the

building flexible at the base for the three soil types is shown in Figure 5.39.

Variation of Shear Force Along the Shear

wall Height

0

3

6

9

12

15

18

21

24

27

30

33

36

-50005001000

Shear(KN)

Sto

rey H

eig

ht(

m)

soil A

Soil B

Soil C

Figure 5.39: Storey shear for Shear wall SW2 (Depth D = 3m)

In the above figure, the shear forces shown are for the whole span of the wall (5m) and it can

be seen that there is a little difference along the wall height except at the top and base.

However, it is to be recalled that the flexibly-supported system exhibited consistently much

smaller shear forces than the fixed-base system (see Figure 5.31).

The axial force distributions for the three soil types in the frame have shown increase in axial

force from Soil Type A through Soil Type B to Soil Type C in externally located columns

(Axis A & E). And a decrease in axial force is seen from Soil Type A through Soil Type B to

Soil Type C in internally located columns (Axis B, C & D). Table 5.12 shows average

percentage differences in axial force among the three soil types.

Table 5.12: Percentage Difference in Axial Forces for Different Soil Types



A 8.8 15.4 22.9

B 5.9 13.5 18.6

C 3.3 12.2 15.1

D 5.1 10.9 15.4

E 11.4 19.1 28.3

-66-

It can be seen that larger differences are observed in the external columns than in the internal

columns. Also, when we compare Tables 5.5 and 5.12, we see that the difference is more for

Structural System 2.

Differences in axial forces of the flexible-base system for the three soil types are shown in

Figure 5.40 for the shear wall SW2.

Variation of Axial Force Along the Shear

wall Height

0

3

6

9

12

15

18

21

24

27

30

33

36

-6000-5000-4000-3000-2000-10000

Axial Force(KN)

Sto

rey H

eig

ht(

m)

soil A

Soil B

Soil C

Figure 5.40: Axial Force diagram of Shear wall SW2 (Depth D=3m)

In Figure 5.40, the axial forces shown are for the whole span of the wall (5m) and it can be

seen that there is a consistent trend of increase in axial force when the soil type becomes

softer and softer.

(iii) Comparison of Fixed-Base and Flexible-Base Systems

In Structural System 1 we saw differences in internal force distribution between analyses of

building structures as fixed-base and flexible-base. This difference was again observed in

section 5.4.2 (b-i). However, particular elements (i.e. beams, columns and shear wall) are

chosen to closely observe the influences of all the parameters affecting the internal force

distribution together. Beam B13, Column C16 and Shear wall SW2 as shown in Figure 5.41

are chosen for the study. This choice is made for the reason that larger differences are

observed on these elements along the selected Axis 4 as discussed in the previous sections

and to make a parallel comparison with Structural System 1. The differences in internal

-67-

forces (shear, moment, axial load) between analysis with fixed-base and that with flexible-

base along the length of the given element are displayed in the following plots and a brief

discussion is made.

Figure 5.41: Beam, column and Shear wall labeling

Difference in bending moment for Beam 13 between fixed-base and flexible-base along the

length of the beam is determined and a plot is made for Soil Types A, B & C, embedment

depths of 1m, 3m, 5m, 10m and all the stories. Finally four curves showing extreme

differences are shown in Figure 5.42.


-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

0 1 2 3 4 5

Length(m)

Mo

me

nt(

KN

m)


storey 12,soilA,Fixed base D=1

storey 12,soilA,Flexible base D=1

storey 2,soilC,Flexible base D=1

Figure 5.42: Plot of difference in moment along the beam length

-68-

There is a larger difference in bending moment at the supports of the beam and when we

compare Figure 5.42 and 5.19 it is clear that the quantitative variation is more in structural

system 2. Hence, the probability that sections designed on the basis of fixed-base analysis

become either conservative or unsafe is greater for Structural System 2.

Variation in bending moment for shear wall SW2 between fixed-base and flexible-base along

the height of the wall is plotted for Soil Types A, B & C, embedment depth of 3m in Figure

5.43.


0

3

6

9

12

15

18

21

24

27

30

33

36

-1500-5005001500250035004500

Moment(KNm)

Sto

rey H

eig

ht(

m)

Soil A,Fixed base,D=3

Soil A,Flexible base,D=3

Soil B, Fixed base,D=3

Soil B,Flexible base D=3

Soil C, Fixed base,D=3

Soil C, Flexible base,D=3

Figure 5.43: Share of overturning moment along the height of SW2

As can be seen in Figure 5.43, the overturning moment of flexible-base is different from that

of fixed-base. The variation is larger around the foundation level and in the middle 1/3rd

of

the wall height.

Four curves showing extreme differences in shear forces for Beam 13 between fixed-base and

flexible-base system along the length of the beam is shown in Figure 5.44.

-69-


-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

0 1 2 3 4 5

Length(m)

Sh

ear(

KN

m)





Figure 5.44: Plot of difference in shear along the beam length

Comparing Figure 5.44 and 5.20, it is clear that larger difference is observed in Structural

System 2 and variation in Soil Type C is larger. The difference in soil type A is smaller. Also

the difference decreases as one goes up the stories.

Shear force variation in the shear wall SW2 between fixed-base and flexible-base along the

height of the wall is plotted for Soil Types A, B & C, embedment depth of 3m in Figure 5.45.

Variation of Shear Force Along the Shear w all Height

0

3

6

9

12

15

18

21

24

27

30

33

36

-500-2500250500750100012501500

Shear(KN)

Sto

rey H

eig

ht(

m)



Soil B, Fixed base,D=3Soil B,Flexible base D=3



Figure 5.45: Shear force distribution along the height of SW2

-70-

The storey shear of flexible-base system is smaller than that of fixed-base system. The

variation is larger around the foundation level.

The influence of soil flexibility on Column 16 is shown by determining the axial force

difference between fixed-base and flexible-base system along the height of the column. A

plot is made for Soil Types A, B & C, embedment depths of 1m, 3m, 5m, 10m and all the

stories. Finally four curves showing extreme differences are shown in Figure 5.46.


0

1

2

3

-2400 -1900 -1400 -900 -400

Axial Force(KN)

He

igh

t(m

)

storey12, soil A,Fixed base D=1


storey 12, soil A,Flexible base D=1


Figure 5.46: Plot of difference in axial load along the column height

Here also comparing Figure 5.46 and 5.21, it is clear that larger difference is observed in

structural system 2 and variation in Soil Type C is larger while in Soil Type A it is smaller.

The difference increases as one goes down the stories.

Axial force variation in the shear wall SW2 for the fixed-base and flexible-base systems

along the height of the wall is plotted for Soil Types A, B & C, embedment depth of 3m in

Figure 5.47.

-71-

Variation of Axial Load Along the Shear w all Height

0

3

6

9

12

15

18

21

24

27

30

33

36

-6000-5000-4000-3000-2000-10000

Axial Load(KN)

Sto

rey H

eig

ht(

m)

Soil A,Fixed base,D=3Soil A,Flexible base,D=3Soil B, Fixed base,D=3Soil B,Flexible base D=3Soil C, Fixed base,D=3Soil C, Flexible base,D=3

Figure 5.47: Axial force variation along the height of SW2

As shown in Figure 5.47 the axial force obtained in flexible base is greater than that of fixed-

base. The axial loads in the fixed-base case are almost equal for the three soil types.

Bending moment differences in Column 16 for fixed-base and flexible-base systems along

the height of the column are determined and a plot is made for Soil Types A, B & C,

embedment depths of 1m, 3m, 5m, 10m and all the stories together. Then four curves

showing extreme differences are shown in Figure 5.48.

Variation in Bending moment along the Height of the Column

0

1

2

3

-140 -100 -60 -20 20 60 100 140 180

Moment(KNm)

He

igh

t(m

)

storey 12, soil A,Fixed base D=1




Figure 5.48: Plot of differences in moment along the column height

-72-

The differences are significant when we move down the storey. For instance, in the figure the

bending moments are almost doubled for the flexible-base system at some locations.

In structural System 1 we have seen that accounting for flexibility of soil resulted in

differences in the reactions at the base. Similarly, this effect on Structural System 2 can be

studied by observing the different internal forces at the base. Thus, the axial forces for fixed-

base and flexible-base at the base of columns located on Axis 4 (Col 16, Col 17, Col 18, Col

19, and Col 20) are plotted for Soil Types A, B & C, embedment depths of 1m, 3m, 5m, and

10m. Four curves showing extreme differences in axial loads are shown in Figure 5.49. A

similar plot is made for the bending moments at the base in Figure 5.50.


1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

16 17 18 19 20Col

Ax

ial

Fo

rce(K

N)

soil A, Fixed base D=10soil C, Fixed base D=1soil A, Fixed base D=10soil C, Fixed base D=1

Col Col Col Col

Figure 5.49: Plot of axial loads at the base


0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

16 17 18 19 20Col

Mo

men

t(K

N)





Col Col Col Col

Figure 5.50: Plot of bending moments at the base

-73-

The above two figures show that there is a significant difference in bending moments and

axial forces of fixed base and flexible base. It is also clear that these variations are larger

when we compare with Structural System 1. These changes in the axial loads and bending

moments may indicate that there might be a change in foundation design.

Moreover, a difference in storey drifts along the direction of the application of lateral loads is

seen between analysis accounting soil flexibility and that of fixed base. A smaller difference

is observed in Soil type A as given in Table 5.13 and the larger difference is observed in Soil

type C and Table 5.14 shows this.

Table 5.13: Minimum variation in Storey Drifts

Story

Value of Max drift

in Fixed base (in m)


flexible base at D=1m (in m)

STORY12 3.40E-05 2.60E-05

STORY11 3.60E-05 2.70E-05

STORY10 3.70E-05 2.80E-05

STORY9 3.80E-05 2.80E-05

STORY8 3.80E-05 2.90E-05

STORY7 3.70E-05 3.00E-05

STORY6 3.60E-05 3.10E-05

STORY5 3.30E-05 3.20E-05

STORY4 2.90E-05 3.50E-05

STORY3 2.50E-05 3.80E-05

STORY2 1.80E-05 3.60E-05

STORY1 1.30E-05 3.20E-05

Table 5.14: Maximum Variation in Storey Drifts

Story

Value of Max drift

in Fixed base (in m)


flexible base at D=1m (in m)

STORY12 5.10E-05 1.49E-04

STORY11 5.40E-05 1.51E-04

STORY10 5.60E-05 1.52E-04

STORY9 5.70E-05 1.54E-04

STORY8 5.70E-05 1.56E-04

STORY7 5.60E-05 1.60E-04

STORY6 5.30E-05 1.64E-04

STORY5 4.90E-05 1.69E-04

STORY4 4.40E-05 1.76E-04

STORY3 3.70E-05 1.87E-04

STORY2 2.60E-05 1.80E-04

STORY1 1.70E-05 1.90E-04

-74-

Storey drifts are small but these values are found to be larger for flexible-base than fixed-base

system. The values for flexible-base system are even doubled that of fixed-base system.

In this section of the work a general difference in bending moments, shear forces and axial

forces between fixed-base systems and their corresponding flexible-base systems were

observed.

Similar to that of Structural System 1, the flexibility of the soil has influenced the internal

forces around the supports significantly; however its influence around the span was too small.

The influence on bending moments is greatest while on the axial loads the influence is the

least. The influences in the internal forces increase as we decrease the embedment depth

keeping the other parameters constant. Similarly, keeping the other parameters constant these

influences increase as the soil type is changed from relatively firm soil type to a softer soil

type. The differences in internal forces observed in Structural System 2 are greater than that

of Structural System 1. This increase in difference has shown that an increase in storey height

has an increasing influence in the distribution of internal forces.

The bending moments at around the base of the shear wall for the flexible-base system are

significantly smaller than that of the fixed-base system. However, higher bending moments

were observed at around the middle one-third of the height of the wall for the flexible-base

system than the fixed-base system. The shear forces in the flexible-base system are less than

that of the fixed-base system almost in the total height of the wall. However, the axial loads

in the flexible-base system are greater than those of the fixed-base system throughout the wall

height.

The axial loads and the bending moments at the foundation level for the fixed-base and

flexible-base system have shown some differences. The axial load difference is not that much

significant. However, moderate bending moment differences are observed. These differences

are highly pronounced in Structural System 2 than in Structural system 1.

The effects of soil flexibility in Structural System 3 are studied in the following sections.

-75-



Similar to Structural System 1, five cases are considered in order to study the influence of

flexibility of soil on the responses of Structural System 3 which is a G+20 building. Once

again, for the mat foundation employed global static springs for each case are appropriately

calculated and then springs at each column and wall are calculated based on the tributary area

as given in Table 5.15. Translational stiffnesses are in kN/m and rotational stiffnesses are in

kNm.

Table 5.15: Static Springs for Different Cases of Structural System 3

Static stiffnesses (X106)


3.2 3.3 3.4 3.5 3.2 3.3 3.4 3.5 3.2 3.3 3.4 3.5

Khx 0.746 0.770 0.787 0.817 0.186 0.193 0.197 0.204 0.047 0.048 0.049 0.051

Khy 0.750 0.774 0.791 0.822 0.188 0.194 0.198 0.205 0.047 0.048 0.049 0.051

Kv 0.823 0.836 0.850 0.885 0.206 0.209 0.213 0.221 0.051 0.052 0.053 0.055

Krx 85.1 85.3 85.4 85.6 21.3 21.3 21.4 21.4 5.32 5.33 5.34 5.35

Kry 180 181 181 181 45.1 45.1 45.2 45.2 11.3 11.3 11.3 11.3

Kt 201 201 201 201 50.3 50.3 50.3 50.3 12.6 12.6 12.6 12.6

Thus the above static springs are assigned at the base in the respective degrees of freedom in

the analysis of the building to obtain the internal forces. Three-dimensional analysis of the

building is made and the results of the analysis are discussed next by comparing all cases of

this system among themselves and with the other structural systems.


In order to show the effects of the various parameters on the internal force distribution the

frame on Axis 4 and the shear wall SW2 of Figure 5.5 are chosen and the discussions are

presented below in sections (i), (ii) and (iii).

-76-


The effect of embedment depth on the internal force distribution for Soil Type C is

considered in Figures 5.51 to 5.59. Results provided are for the frame on Axis 4 and shear

wall SW2 from models with fixed-base and spring-base founded at a depth of 1m below the

surface of the soil. All the internal forces of embedded foundations with flexible base at a

depth of 3m, 5m and 10m lie in between the above two extreme cases.

Influence of soil flexibility on the bending moments can be observed in Figure 5.51 and 5.52

respectively.

Figure 5.51: Frame on Axis 4 (Fixed base& Soil C)-Bending Moment Diagram in kNm

-77-

Figure 5.52: Frame on Axis4 (Flexible base, D=1m&Soil C) -BMD in kNm

The following differences in bending moment are observed in Figures 5.51 and 5.52. In

beams located at the end bays (i.e. between axis A&B or axis D&E), average difference of

about 50.5 % at the supports and 21.8% around middle span are observed. In beams located

at the interior of the frame (i.e. between axis B&D), an average difference is about 38.2 % at

the support and 1.1% around the middle spans.

Beams in between Axis-A and Axis-B are considered in particular to see the variation of the

moment differences along the beam length between fixed-base and flexible-base systems

-78-

(extreme case of embedment depth of 1m). The plot showing this difference along the length

of the beam for all the stories in Soil Type C is shown in Figure 5.53.

Variation in BM Along the Beam Length-120

-100

-80

-60

-40

-20

0

20

40

60

80

0 1 2 3 4 5Length(m)

Dif

fere

nc

e(K

Nm

)

storey22,soilC, D=1

storey18,soilC, D=1

storey10,soilC, D=1

storey2,soilC, D=1


The maximum difference is seen to be around the supports and negligible difference is

observed around the mid span of the beam. We can see that these differences are on the

average greater than that of Structural System 2 (Figure 5.26).

For the shear wall SW2, the overturning moments obtained by analyzing the building fixed

and flexible at the base are shown in Figure 5.54. From this figure, the overturning moment

in the flexible-base is larger than that of the fixed-base system in almost the whole height of

the wall. Unlike structural system 2, there was no reduction in bending moments around the

base for flexible-base system.

-79-


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

-10123456

Moment(KNm) x103

Sto

rey H

eig

ht(

m)

Fixed base D=1

Flexible base, D=1

Figure 5.54: Overturning Moment in Shear wall SW2 (Soil C and D=1m)

The shear forces for the two extreme cases of fixed-base and flexible-base systems at a depth

of 1m below the surface of soil C are assessed and the following observations are made. In

beams located at the end bays (i.e. between axis A&B or axis D&E) an average differences is

about 41.5 % around the supports. In beams located at the interior of the frame (i.e. between

axis B&D) an average difference is about 33.5 % at the supports.

The shear force difference between results obtained using fixed-base and flexible-base

(extreme case of embedment depth of 1m), for beam spans in-between Axis-A and Axis-B

are plotted along the length of the beam as shown in Figure 5.55. As usual, the maximum

differences are observed at the supports. These differences are greater when compared with

Figure 5.30 of Structural System 2.

-80-

Variation in Shear Force Along the Length of the Beam

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

0 1 2 3 4 5Length(m)

Dif

fere

nce

(KN

)

storey 22, D=1

storey 18, D=1

storey 10, D=1

storey 2,D=1


For the shear wall SW2, the shear force carried by the shear wall at storey level and obtained

by analyzing the building fixed and flexible at the base is shown in Figure 5.56.

Variation of Storey shear shared by the Shear w all

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

-900-2005001200

Shear force (KN)

Sto

rey H

eig

ht(

m)

Fixed base, D=1

Flexible base, D=1

Figure 5.56 Shear force variation at storey level for Shear wall SW2 (Soil C & D=1m)

In the figure above the shear force differences between fixed-base and flexible-base are small

for the top 17 stories or so but greater difference is observed in the lower three stories. Even

the direction of the shear force has changed for that of flexible-base at around the lowest

stories.

-81-

Once again, the axial force variation has been studied for the two extreme cases of fixed-base

and flexible-base systems at a depth of 1m below the surface of soil C. The differences in

axial forces at the lowest storey are summarized in Table 5.16.



A 33.2

B 39.9

C 40.0

D 38.3

E 35.1

Contrary to the other two structural systems considered above, the maximum difference is

found in the interior columns. Besides, the percentage differences are greater than that of

Structural System 2.

For the shear wall SW2, the axial forces as obtained by analyzing the building fixed and

flexible at the base are given in Figure 5.57.

Variation of Axial Force Along the Shear w all Height

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

-9.5-8.5-7.5-6.5-5.5-4.5-3.5-2.5-1.5-0.5

Axial force(KN) x 103

Sto

rey H

eig

ht(m

)

Fixed base, D=1

Flexible base, D=1

Figure 5.57: Axial Force on Shear wall SW2 (Soil C & D=1m)

-82-

In Figure 5.57, the axial forces in the flexible base are larger than that obtained in fixed base

throughout the wall height. Eventhough the magnitude is different this trend is similar to that

of Structural System 2. The axial forces given on the figure are for the whole length (5m) of

the wall.


Different soil types have showed different internal force distributions in structural systems 1

and 2 as discussed previously. The influence of soil type on the internal force distribution of

Structural System 3 is studied for the case of foundations placed at a depth of 3m and the

summary of the results obtained are presented briefly.

There is a general increase in bending moments as we go from Soil Type A through B to C.

The average differences in the three soil types are summarized in Table 5.17.

Table 5.17: Summary of Differences in Frame Bending Moments


A & B 23.8

B & C 15.8

A & C 32.2

The overturning moments carried by the shear wall SW2 and obtained by analyzing the

building flexible at the base for the three soil types is shown in Figure 5.58.

Variation of Overturning Moment Along

the Shear wall Height

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

-800200120022003200420052006200

Moment(KNm)

Sto

rey H

eig

ht(

m)

Soil A

Soil B

Soil C

Figure 5.58: Share of overturning moment for Shear wall SW2 (Depth D=3m)

-83-

In Figure 5.58, the bending moments shown are for the whole span of the wall (5m) and it is

clear that the share of the overturning moment on the wall increases as the soil becomes

softer and softer. Larger difference is found around the base.

The trend in the variation of shear forces from one soil type to another is not predictable.

However, average differences as summarized in Table 5.18 are observed.

Table 5.18: Percentage Difference between different soil types

Difference b/n Soil Type Average difference (%)

A & B 25.2

B& C 13.6

A&C 31.9

As expected, a larger difference is observed between a relatively softer soil and firm soil (soil

types A & C).

The storey shear carried by the wall SW2 as obtained by analyzing the building flexible at the

base for the three soil types is shown in Figure 5.59.

Variation of Shear Force Along the Shear

wall Height

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

-600-350-100150400650900

Shear(KN)

Sto

rey H

eig

ht(

m)

soil A

Soil B

Soil C

Figure 5.59: Storey shear for Shear wall SW2 (Depth D = 3m)

-84-

In Figure 5.59, the shear force shown are for the whole span of the wall (5m) and it can be

seen that there is an increase in shear force as we go from Soil Type A through B to Soil

Type C. The differences among the different soil types of this structural system are greater

than those of Structural System 2 (Figure 5.39).

An average percentage difference in the axial forces for the three soil types and foundations

placed at a depth of 3m are summarized in Table 5.19.

Table 5.19: Percentage Difference between different soil types



A 9.0 3.1 11.8

B 12.0 5.9 17.2

C 13.2 6.7 19.0

D 10.6 4.8 14.9

E 10.4 4.1 14.1

In this structural system it is seen that larger differences are observed in the internal columns

than external columns.

Axial forces as obtained by analyzing the building flexible at the base for the three soil types

are shown in Figure 5.60 for the shear wall SW2.

Variation of Axial Force Along the Shear

wall Height

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

-9.6-8.6-7.6-6.6-5.6-4.6-3.6-2.6-1.6-0.6

Axial Force(KN) x 103

Sto

rey H

eig

ht(

m)

soil A

Soil B

Soil C

Figure 5.60: Axial Force diagram of Shear wall SW2 (Depth D=3m)

-85-

In Figure 5.60, the axial force shown are for the whole span of the wall (5m) and it can be

seen that there is an increase in axial force when the soil type becomes softer and softer.

(iii) Comparison of Fixed-Base and Flexible-Base System

By now we are sure that accounting flexibility of soil in the analysis of structures has

changed the internal force distribution. This change can also be confirmed by observing

internal force variations in particular elements (i.e. beams, columns and shear walls). Beam

B13, Column C16 and Shear wall SW2 as shown in Figure 5.61 are chosen for this study. A

series of plots provided next will show the differences in internal forces (shear, moment, axial

load) between analysis with fixed base and that with flexible base along the length of that

particular element.

Figure 5.61: Beam, column and Shear wall labeling

The bending moments on Beam 13 for fixed-base and flexible-base along the length of the

beam are determined and a plot is made for Soil Types A, B & C, embedment depths of 1m,

3m, 5m, 10m and all the stories. Four curves showing extreme differences are shown in

Figure 5.62.

SW2

-86-


-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2 3 4 5

Length(m)

Mo

men

t(K

Nm

)


storey 22,soilA,Fixed base D=1

storey 22,soilA,Flexible base D=1

storey 2,soilC,Flexible base D=1

Figure 5.62: Plot of difference in moment along the beam length

There is a larger difference in bending moment at the supports of the beam. The differences

in bending moment in this figure are greater than that of structural System 2 as shown in

Figure 5.42.

Variation in bending moment for shear wall SW2 for fixed-base and flexible-base along the

height of the wall is plotted for Soil Types A, B & C, embedment depth of 3m in Figure 5.63.


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

-1.501.534.56

Moment(KNm) x 103

Sto

rey H

eig

ht(

m)



Soil B, Fixed base,D=3

Soil B,Flexible base D=3



Figure 5.63: Overturning moment along the height of SW2

-87-

As can be seen in the above figure, the overturning moment of flexible base is different from

that of fixed base. The variation is greater around the lower stories and in the other parts of

the wall height the difference is not significant.

Similarly, shear forces for fixed-base and flexible-base along the length of the beam are

studied and a plot is made for Soil Types A, B & C, embedment depths of 1m, 3m, 5m, 10m

and all the stories. Figure 5.64 shows extreme shear force diagrams which are expected to

display greater differences in shear force values.


-150

-130

-110

-90

-70

-50

-30

-10

10

30

50

70

90

110

0 1 2 3 4 5

Length(m)

Sh

ear(

KN

m)





Figure 5.64: Plot of shear forces along the beam length

This figure shows that the variation in soil type C is greater and the difference in soil type A

is smaller. Also the difference decreases as one goes up the stories. The differences observed

here are greater than that of Structural System 2 (Figure 5.44).

Shear force variation in the shear wall SW2 for the fixed-base and flexible-base systems

along the height of the wall is plotted for Soil Types A, B & C, embedment depth of 3m in

Figure 5.65.

-88-

Variation of Shear Force Along the Shear w all Height

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

-600-1004009001400

Shear(KN)

Sto

rey

Heig

ht(

m)


Figure 5.65 Storey shear along the height of SW2

As shown in Figure 5.65 the storey shear of flexible-base is different from that of fixed-base

system. The variation is larger around the foundation level. Moreover, change in the direction

of the shear force is also observed at the base. Soil Type C shows a pronounced difference.

The influence of soil flexibility on Column 16 is shown by plotting the axial forces for fixed-

base and flexible-base along the height of the column and the plot is made for Soil Types A,

B & C, embedment depths of 1m, 3m, 5m, 10m and all the stories. Finally four curves

showing extreme differences are shown in Figure 5.66.


0

1

2

3

-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0

Axial Force(KN)

He

igh

t(m

) storey22, soil A,Fixed base D=1




Figure 5.66: Plot of axial load along the column height

-89-

Comparing Figure 5.66 and 5.46, it is clear that larger difference is observed in Structural

System 3 and variation in soil type C is larger while the difference in soil type A is smaller.

Axial force variation in the shear wall SW2 for fixed-base and flexible-base along the height

of the wall is plotted for Soil Types A, B & C, embedment depth of 3m in Figure 5.67.

Variation of Axial Load Along the Shear w all Height

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

-10-8-6-4-20

Axial Load(KN) x 103

Sto

rey H

eig

ht(

m)


Figure 5.67: Axial force variation along the height of SW2

As shown in Figure 5.67 the axial force obtained in flexible base is greater than that of fixed

base. The axial loads in the fixed base case are almost equal for the three soil types and the

three curves have overlapped.

The reactions at the foundation level are assessed for the fixed-base and flexible-base systems

of Structural System 3. At the base of columns located on Axis 4 (Col 16, Col 17, Col 18, Col

19, and Col 20), a plot for Soil Types A, B & C, embedment depths of 1m, 3m, 5m, and 10m

is made. Four curves showing maximum differences are shown in Figure 6.68. A similar plot

is made for the bending moments at the base in Figure 5.69.

-90-


4000

5000

6000

7000

8000

9000

10000

11000

12000

16 17 18 19 20Col

Axia

l F

orc

e(K

N)

soil A, Fixed base D=10soil C, Fixed base D=1soil A, Fixed base D=10soil C, Fixed base D=1

Col Col Col Col

Figure 5.68: Plot of axial load at the base


0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

16 17 18 19 20Col

Mo

me

nt(

KN

)





Col Col Col Col

Figure 5.69: Plot of bending moment at the base

In Figure 5.68 and 5.69, there is a significant difference in bending moments and axial forces

of fixed base and flexible base. It is also clear that these variations are larger when we

compare with Structural Systems 1 and 2.

-91-

Moreover, a difference in storey drifts along the direction of the application of lateral loads is

seen between analysis accounting soil flexibility and that of fixed base. A smaller difference

is observed in Soil type A and larger difference is observed in Soil type C.

Similar to that of Structural System 1 and Structural System 2, the flexibility of the soil has

influenced the internal forces around the supports significantly; however its influence around

the span was too small. The influence on bending moments is greatest while on the axial

loads the influence is the least. The influences in the internal forces increase as we decrease

the embedment depth keeping the other parameters constant. Similarly, keeping the other

parameters constant these influences increase as the soil type is changed from relatively firm

soil type to a softer soil type. The differences in internal forces observed in Structural System

3 are greater than that of Structural System 1 and Structural System 2. This increase in

difference has showed that an increase in storey height has an influence in the distribution of

internal forces.

For Structural System 3, the bending moments in the shear wall of the flexible-base system

are greater than that of the fixed-base system in the larger portion of the wall height and even

around the base. But in Structural System 2, the values of the bending moments around the

foundation level were less than that of the corresponding fixed-base system.

Similar to Structural System 2, the shears in the flexible-base system of Structural System 3

are less than that of the fixed-base system in almost the total height of the wall except that

around the base the shear force in the Structural System 3 has changed its direction with a

sharp increase in magnitude. However, the axial loads in the flexible-base system are greater

than those of the fixed-base system through out the wall height.

The axial loads and the bending moments at the foundation level for the fixed-base and

flexible-base system are different. The axial load difference is not that much significant,

however, moderate bending moment differences are observed. These differences are highly

pronounced in Structural System 3 and are least in Structural system 1.

-92-

6. CONCLUSIONS AND RECOMMENDATIONS

The parametric study made for the three structural systems has indicated that the bending

moments and shear forces in the beams and the axial loads and bending moments in the

columns for the fixed-base system are different from their corresponding values of the

flexible-base system. In particular, the following observations were made.

In Structural System 1, representing framed buildings:

• The flexibility of the soil has significantly influenced the internal forces around the

supports of beams. Its influence around the spans is too small. The influence is more

significant in beams located at the end bays.

• Bending moments in some beams of the flexible-base system were found to be greater

than those of the fixed-base system in a portion of a span and vice versa in the

remaining portion of that same span. This implies sections of beams (even within a

span) designed for the internal forces obtained using the usual fixed-base model can

be either on the conservative or the unsafe side.

• Accounting for the flexibility of soils in the analysis has greatest influence in bending

moment than the shear forces and axial loads. The influence on the axial forces is the

least.

• Extreme differences of up to 110% are observed in the bending moments of beams

around the end supports between flexible-base system and the fixed-base system. In

these locations the average deviation is about 38%.

• For a given embedment depth of the foundation, the internal forces are most

influenced when the soil is of Soil Type C as expected. The least influence is

observed in Soil Type A, which is the competent type of soil considered in this study.

• The influences in the internal forces increase as we decrease the embedment depth

keeping the other parameters constant as expected.

As studied on the basis of Structural System 2 and 3 representing buildings supported by dual

(frame-wall) structural systems the following were observed.

• On the frames, the flexibility of soil has influenced the internal forces in a trend similar

to Structural System 1 and hence all the above observations hold true. However, the

magnitude of influence grows when we move from Structural System 1 through

Structural System 2 to Structural System 3. This implies that within the range of

-93-

building height studied the differences in internal forces between the flexible-base and

fixed-base system increase with increasing height of the building.

• In the dual systems, the bending moments in the shear walls of the flexible-base system

are greater than that of the fixed-base system in a larger portion of the walls’ height. But

the bending moments around the foundation level are less than that of the corresponding

fixed-base system in Structural system 2 which is in contrast to that of Structural

System3.

• In both Structural Systems 2 and 3, the shear forces in the flexible-base system are less

than that of the fixed-base system in almost the entire height of the walls except that

around base, where the shear forces in the Structural System 3 changed their direction

with a sharp increase in magnitude. However, the axial loads in the flexible-base system

are greater than those of the fixed-base system throughout the wall height.

• Differences have also been observed in the axial loads and the bending moments at the

foundation level for the fixed-base and flexible-base systems. The differences in the

axial load are not, however that much significant. Modest differences are observed in

bending moments. These differences are highly pronounced in Structural System 3 and

are least in Structural system 1.This has a practical significance in that the difference

could force the designer to the extent of considering a change in the type of foundation.

On the basis of the above observation in regular structural systems, it can be recommended

that for such systems the soil flexibility be accounted for by simply introducing elastic static

springs, whose coefficients can easily be retrieved from the available and relatively recent

literature. The additional effort involved is practically insignificant that ignoring the soil

flexibility is difficult to justify.

However, another study on non regular structural systems and systems having lateral force

resisting elements other than simple rectangular shear walls shall be made to draw a more

general recommendation.

-94-

REFERENCES

1. Abdulwasi Usmail, The Effect of Soil-Structure Interaction on the Dynamic Response

of Symmetrical Reinforced concrete Buildings, Addis Ababa, 2004

2. Borodachev N. M., Determination of the Settlement on Rigid Plates, Soil Mechanics

and Foundation Engineering (USSR), 1964, 1, 210

3. ETABS Users Manual, Computers and Structures, Inc, Version 8.00,1995 University

Avenue, Berkley, California USA 94704

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Soils and Foundations, 1981g, 21, No.4, 109

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pp.553-593

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the Engineering Mechanics Division, ASCE, Vol. 97, No. EM5, Proc. Paper 8416,

Oct., 19971, pp.1381-1395

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Resistance, Ethiopian Building Code Standard-8, Addis Ababa, 1995

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Hall, Inc., Englewood Cliffs, New Jersey, 1970

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Scientific Publishing Co., 1979

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the Soil Mechanics and Foundations Division, ASCE, Vol.97, No.SM9, Sep., 1971,

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-95-

DECLARATION

This thesis is my original work. It has not been presented for a degree in any other university

and that all sources of material used for the thesis have been duly acknowledged.

-96-

APPENDIX A

1. Gravity Loads on Frames Located in Different Axes

1 & 7 2 & 6 3,4 & 5

Part Beam

Factored

Load

(KN/m)

Factored

Load

(KN/m)

Factored

Load

(KN/m)

A-B 24.4 41.2 45.92

B-C 23.4 44.1 42.2

C-D 23.4 44.1 42.2

D-E 24.4 41.2 45.92

A-B 38.7 62.5 67.25

B-C 37.8 65.4 63.53

C-D 37.8 65.4 63.53

D-E 38.7 62.5 67.25

A-B 23.8 23.8 23.77

B-C 23.8 23.8 23.77

C-D 23.8 23.8 23.77

D-E 23.8 23.8 23.77

A & E B & D C

Part Beam

Factored

Load

(KN/m)

Factored

Load

(KN/m)

Factored

Load

(KN/m)

1-2,6-7 24.41 48.4 45.92

2-3,3-4,4-5,5-6,6-7 30.61 44.1 42.2

1-2,6-7 38.74 69.7 67.25

2-3,3-4,4-5,5-6,6-7 44.94 65.4 63.53

1-2,6-7 23.77 23.8 23.77

2-3,3-4,4-5,5-6,6-7 23.77 23.8 23.77

Axes

Roof

Typical floor

Ground floor

Axes

Roof

Typical floor

Ground floor

2. Lateral Loads on Frames Located in Different Axes

Structural System 1

Soil Type A B C

Fi Fi Fi

Roof 264.1 316.9 396.1

5th

floor 336.4 403.7 504.6

4th

floor 277.0 332.4 415.5

3rd

floor 217.7 261.2 326.5

2nd

floor 158.3 190.0 237.5

1st floor 98.9 118.7 148.4

Ground floor 41.1 49.3 61.6

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Structural System 2

Soil Type A B C

Fi Fi Fi

Roof 301.8 362.2 452.7

10th

floor 302.0 362.4 453.0

9th floor 273.7 328.4 410.6

8th floor 245.4 294.5 368.1

7th floor 217.1 260.5 325.6

6th floor 188.8 226.5 283.1

5th floor 160.4 192.5 240.7

4th floor 132.1 158.6 198.2

3rd

floor 103.8 124.6 155.7

2nd

floor 75.5 90.6 113.3

1st floor 47.2 56.6 70.8


Structural System 3

Soil Type A B C

Fi Fi Fi

Roof 448.8 538.6 673.3

20th floor 260.1 312.1 390.1

19th

floor 247.5 297.0 371.3

18th floor 234.9 281.9 352.4

17th floor 222.3 266.8 333.5

16th

floor 209.8 251.7 314.6

15th floor 197.2 236.6 295.7

14th floor 184.6 221.5 276.9

13th

floor 172.0 206.4 258.0

12th floor 159.4 191.3 239.1

11th floor 146.8 176.2 220.2

10th

floor 134.2 161.1 201.4

9th

floor 121.7 146.0 182.5

8th

floor 109.1 130.9 163.6

7th

floor 96.5 115.8 144.7

6th

floor 83.9 100.7 125.9

5th

floor 71.3 85.6 107.0

4th

floor 58.7 70.5 88.1

3rd

floor 46.1 55.4 69.2

2nd

floor 33.6 40.3 50.3

1st floor 21.0 25.2 31.5


A Study on Soil Flexibilty on Internal Force Distribution of Buildings

Documents

influence of soil flexibility

lateral static loads

amsalu gashaye birhan

internal examiner

asrat worku

special thanks

winkler model

list of figures