dynamics of structures volume 2 - applications chowdhury

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Dynamics of Structure andFoundation A Unied Approach

2. Applications

2009 Taylor & Francis Group, London, UK


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Dynamics of Structure andFoundation A Unied Approach

2. Applications

Indrajit ChowdhuryPetrofac International Ltd Sharjah, United Arab Emirates

Shambhu P. DasguptaDepartment of Civil Engineering Indian Institute of Technology Kharagpur, India



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Also available:Dynamics of Structure and Foundation A Unied Approach1. Fundamentals

Indrajit Chowdhury & Shambhu P. Dasgupta 2009, CRC Press/Balkema

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Contents

Preface xiii

1 Dynamic soil structure interaction 1

1.1 Introduction 11.1.1 The marriage of soil and structure 11.1.2 What does the interaction mean? 21.1.3 It is an expensive analysis do we need

to do it? 41.1.4 Different soil models and their coupling to

superstructure 61.2 Mathematical modeling of soil &structure 6

1.2.1 Lagrangian formulation for 2D frames orstick-models 6

1.2.2 What happens if the raft is f lexible? 141.3 A generalised model for dynamic soil structure

interaction 281.3.1 Dynamic response of a structure with multi

degree of freedom considering the underlyingsoil stiffness 28

1.3.2 Extension of the above theory to systemwith multi degree of freedom 29

1.3.3 Estimation of damping ratio for the soilstructure system 30

1.3.4 Formulation of damping ratio for single degreeof freedom 31

1.3.5 Extension of the above theory to systems withmulti-degree freedom 32

1.3.6 Some fal lacies in coupl ing of soi l and s tructure 401.3.7 What makes the structural response attenuate

or amplify? 41

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vi Contents

1.4 The art of modelling 421.4.1 Some modelling techniques 421.4.2 To sum it up 46

1.5 Geotechnical considerations for dynamic soil

structure interaction 461.5.1 What parameters do I look for in the soilreport? 47

1.6 Field tests 491.6.1 Block vibration test 491.6.2 Seismic cross hole test 501.6.3 How do I co-relate dynamic shear modulus

when I do not have data from the dynamicsoil tests? 51

1.7 Theoretical co-relation from other soil parameters 521.7.1 Co-relation for sandy and gravelly soil 521.7.2 Co-relation for saturated clay 58

1.8 Estimation of material damping of soil 611.8.1 Whitmans formula 611.8.2 Hardin formula 621.8.3 Ishibashi and Zhangs formula 63

1.9 All things said and done how do we estimate

the strain in soil, specially if the strain is large? 651.9.1 Estimation of strain in soil for machine

foundation 651.9.2 Estimation of soil strain for earthquake

analysis 701.9.3 What do we do if the soil is layered with

varying soil property? 771.9.4 Checklist of parameters to be looked in the

soil report 791.10 Epilogue 80

2 Analysis and design of machine foundations 83

2.1 Introduction 832.1.1 Case history #1 832.1.2 Case history #2 84

2.2 Different types of foundations 852.2.1 Block foundations resting on soil/piles 852.2.2 How does a block foundation supporting

rotating machines differ from a normalfoundation? 86

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Contents vii

2.2.3 Foundation for centrifugal or rotary typeof machine: Different theoretical methodsfor analysis of block foundation 88

2.2.4 Analytical methods 90

2.2.5 Approximate analysis to de-couple equationswith non-proportional damping 992.2.6 Alternative formulation of coupled equation

of motion for sliding and rocking mode 1052.3 Trick to by pass damping Magnification factor,

the key to the problem . . . 1132.4 Effect of embedment on foundation 117

2.4.1 Novak and Beredugos model 1192.4.2 Wolfs model 119

2.5 Foundation supported on piles 1192.5.1 Pile and soil modelled as f inite element 1212.5.2 Piles modelled as beams supported on elastic

springs 1232.5.3 Novaks (1974) model for equivalent spring

stiffness for piles 1242.5.4 Equivalent pile spr ings in ver ti cal d irect ion 1252.5.5 The group effect on the vertical spring

and damping value of the piles 1272.5.6 Effect of pile cap on the springand damping stiffness 128

2.5.7 Equivalent pile springs and dampingin the horizontal direction 129

2.5.8 Equivalent pile springs and dampingin rocking motion 130

2.5.9 Group effect for rotational motion 1312.5.10 Model for dynamic response of pile 138

2.5.11 Dynamic analysis of laterally loaded piles 1622.5.12 Partial ly embedded piles under rocking mode 1932.5.13 Group effect of pile 2012.5.14 Comparison of results 2032.5.15 Practical aspects of design of machine

foundations 2052.6 Special provisions of IS-code 213

2.6.1 Recommendations on vibration isolation 2132.6.2 Frequency separation 2132.6.3 Permissible amplitudes 2142.6.4 Permissible stresses 2142.6.5 Concrete and its placing 214

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viii Contents

2.6.6 Reinforcements 2142.6.7 Cover to concrete 215

2.7 Analysis and design of machine foundation underimpact loading 231

2.7.1 Introduction 2312.7.2 Mathematical model of a hammer foundat ion 2382.8 Design of hammer foundation 248

2.8.1 Design criteria for hammer foundation 2482.8.2 Discussion on the IS-code method of analysis 2522.8.3 Check l is t for analysis of hammer foundat ion 2532.8.4 Other techniques of analysis of Hammer

foundation 2532.9 Design of eccentrically loaded hammer foundation 268

2.9.1 Mathematical formulation of anvil placedeccentrically on a foundation 268

2.9.2 Damped equation of motion with eccentricanvil 270

2.10 Details of design 2712.10.1 Reinforcement detailing 2712.10.2 Construction procedure 271

2.11 Vibration measuring instruments 2722.11.1 Some background on vibration measuring

instruments and their application 2722.11.2 Response due to motion of the support 2722.11.3 Vibration pick-ups 272

2.12 Evaluation of friction damping fromenergy consideration 283

2.13 Vibration isolation 2842.13.1 Active isolation 2852.13.2 Passive isolation 287

2.13.3 Isolation by trench 2882.14 Machine foundation supported on frames 289

2.14.1 Introduction 2892.14.2 Different types of turbines and the

generation process . . . 2902.14.3 Layout planning 2922.14.4 Vibration analysis of turbine foundations 293

2.15 Dynamic soil-structure interaction model forvibration analysis of turbine foundation 305

2.16 Computer analysis of turbine foundation based onmulti degree of freedom 312

2.17 Analysis of turbine foundation 319

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Contents ix

2.17.1 The analysis 3192.17.2 Calculation of the eigen values 3202.17.3 So the ground rule is . . . 3212.17.4 Calculation of amplitude 321

2.17.5 Calcula tion of moments, shears and torsion 3212.17.6 Practical aspects of design of Turbine

foundation 3222.18 Design of turbine foundation 322

2.18.1 Check list for turbine foundation design 3222.18.2 Spring mounted turbine foundation 330

3 Analytical and design concepts for earthquakeengineering 389

3.1 Introduction 3893.1.1 Why do earthquakes happen in nature? 3903.1.2 Essential difference between systems

subjected to earthquake and vibrationfrom machine 391

3.1.3 Some history of major earthquakes aroundthe world 392

3.1.4 Intensity 3943.1.5 Effect of earthquake on soil-foundationsystem 395

3.1.6 Liquefaction analysis 3953.2 Earthquake analysis 412

3.2.1 Seismic coeff icient method 4123.2.2 Response spectrum method 4173.2.3 Dynamic analysis under earthquake loading 4243.2.4 How do we evaluate the earthquake force? 425

3.2.5 Earthquake analysis of systems with multi-degree of freedom 431

3.2.6 Modal combination of forces 4443.3 Time history analysis under earthquake force 448

3.3.1 Earthquake analysis of tall chimneysand stack like structure 456

3.4 Analysis of concrete gravity dams 4813.4.1 Earthquake analysis of concrete dam 4813.4.2 A method for dynamic analysis

of concrete dam 4853.5 Analysis of earth dams and embankments 519

3.5.1 Dynamic ear thquake analysi s of earth dams 519

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x Contents

3.5.2 Mononobes method for analysis of earth dam 519

3.5.3 Gazetas method for earth dam analysis 5223.5.4 Makadisi and Seeds method for analysis of

earth dam 5233.5.5 Calculation of seismic force in dam andits stability 526

3.6 Analysis of earth retaining structures 5263.6.1 Earthquake analysis of earth retaining

structures 5263.6.2 Mononobes method of analysis of

retaining wall 5273.6.3 Seed and Whitmans method 5303.6.4 Arangos method 5303.6.5 Steedman and Zengs method 5323.6.6 Dynamic analysis of RCC retaining wall 5333.6.7 Dynamic analysis of cantilever and

counterfort retaining wall 5333.6.8 Some discussions on the above method 5443.6.9 Extension to the generic case of soil

at a slope i behind the wall 5443.6.10 Dynamic analysis of counterfort

retaining wall 5473.6.11 Soil sloped at an angle i with horizontal 560

3.7 Unyielding earth retaining structures 5713.7.1 Earthquake Analysis of rigid walls when

the soil does not yield 5713.7.2 Ostadans method 575

3.8 Earthquake analysis of water tanks 5773.8.1 Analysis of water tanks under earthquake

force 5773.8.2 Impulsive time period for non rigid walls 5813.8.3 Sloshing time period of the v ibra ting flu id 5833.8.4 Calculation of horizontal seismic force for

tank resting on ground 5833.8.5 Calculation of base shear for tanks resting

on ground 5843.8.6 Calculation of bending moment on the

tank wall resting on the ground 5843.8.7 Calculation of sloshing height 585

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Contents xi

3.9 Mathematical model for overhead tanks underearthquake 588

3.9.1 E arthquake Analysis for overhead tanks 5883.9.2 Hydrodynamic pressure on tank wall

and base 5923.9.3 Hydrodynamic pressure for circular tank 5923.9.4 Hydrodynamic pressure for rectangular

tank 5933.9.5 Effect of vertical ground acceleration 5933.9.6 Pressure due to inertia of the wall 5933.9.7 Maximum design dynamic pressure 594

3.10 Practical aspects of earthquake engineering 5983.10.1 Epilogue 603

References 605

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Preface

The monograph entitled Dynamics of Structure and Foundation - A UniedApproach consists of two volumes. While in Volume 1 we dealt with backgroundtheories and formulations that constitute the above subject, this second volume dealswith application of these theories to various aspects of civil engineering problems con-stituting topics related to dynamic soil-structure interaction, machine foundation andearthquake engineering.

If we have managed to stir the wrath of the professionals in Volume 1 with mazesof tensors, differential and integral equations, it is our strong conviction that in thispresent volume we will be able to considerably appease this fraternity for it constitutesof a number of applications that are innovative, easy to apply and solutions to many

practical problems that puts an engineer into considerable difculty and uncertaintiesin a design ofce.We start Volume 2 with the topic of Dynamic Soil Structure Interaction (DSSI). We

believe this topic would play a key role in future and more so with the distinct pos-sibility of construction of Nuclear power plants (especially in India) globally. A clearconcept on this topic would surely be essential for designing such plants. Though wehave dealt this topic only in terms of fundamental concepts, yet we feel that we havegiven sufcient details to eradicate the misnomer from which many engineers sufferthat DSSI is nothing but adding some springs to the boundary of a structure and then

doing the analysis through a computer.The geotechnical aspects that play an extremely important role in selecting the soil-

spring value, (that are highly inuenced by the strain range) have been dealt in quitedetail. We hope that this section will do away with some of the major blunders that wemake in DSSI analysis, and appreciate how the results thus obtained become unrealisticand questionable. We sincerely hope that engineers performing DSSI analysis, wouldstart paying sufcient attention to some of the key engineering parameters as furnishedin the soil report that are being habitually ignored in design ofces.

Second chapter consists of design and analysis of machine foundations (both blockand frame type). In our collective experience as a consultant and academician we haveseen signicant confusion on this topic as to who is responsible for this hapless orphan,structural or geotechnical engineers? While people from classical soil mechanics dis-owns it, as it involves the evaluation of eigen-values and vectors that are far away from



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xiv Preface

their traditional failure theories of foundation, structural engineers on the other handare equally reluctant to shoulder the guardianship for their inherent apathy towardswhat lies beneath the machine foundation. As such, a design involving machinefoundation throws the most challenging and interesting task in the domain of civil

engineering that requires multi-discipline knowledge and should be equally interest-ing to an engineer having structural or geotechnical background. The matrix analysisconcept that we have introduced herein is quite easy to follow and we hope wouldbridge the gap that is still prevalent in academics and practice alike.

We would be looking forward to have some feedback from hardened professionalswho are working in this area, as to how they feel about our representation which webelieve is quite novel and has tried to answer a number of problems that often becomeburning issues on which they have spent signicant time on clarifying either to theirClients or Project Management Consultants.

The last chapter of this volume deals with the most fearful force Mother Nature hascreated Earthquake. Earthquake engineering as a topic is so vast, complex anddiverse (and ever changing) that we concede that it did give us some uncomfortablemoments as to what should justiably constitute this chapter? Majority of the booksthat address this topic are far too focused on buildings and there are hardly any bookaround, that has addressed other specialized structures like chimneys, dams, retainingwalls, water tanks etc (except some very specialized literature). It should be realizedthat some of these structures are expensive, important and cannot be ignored whilebuilding an earthquake resistant infrastructure.

Buildings, we concede are the biggest casualties during an earthquake and aredirectly related to human life but damages to other structures as mentioned above canalso create havoc especially in the post earthquake relief scenario. The major focusbeing still thrust on buildings, we were also quite surprised to nd that there is stillmuch room for improvement in many of these structures, where technologies whichare as old as 60 years are still in use (for instance earthquake response of retainingwalls). We tried to improve upon many of them and believe that we have brought abouta number of innovative solutions that can be adapted in a design ofce environment

and can also be used as a basis for further research.While presenting the topic no demarcation is made between geotechnical and

structural earthquake engineering. For, as a seismic specialist our job is to minimizethe destruction of property and save human lives. Thus doing a structural design wecan perform the most sophisticated analysis and provide the most expensive detailingand our building still fails due to liquefaction killing people__no medals for doingan excellent structural design!, so if you do something do it in totality and not inisolation and this has been our major endeavour- that we have tried to communicateto you through this book.

Indrajit ChowdhuryShambhu P. Dasgupta



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Chapter 1

Dynamic soil structure interaction

1.1 INTRODUCTION

This chapter deals with some of the basic concepts of dynamic soil-structure interactionanalysis. At the advent of this chapter we expect you to have some background on

Static soil structure interaction Theory of Vibration/structural dynamics Basic theory of soil dynamics

Based on the above , we build herein the basic concepts of dynamic soil structureinteraction, which is slowly and surely gaining its importance in analytical procedurefor important structures.

1.1.1 The marriage of soil and structure

As was stated earlier in Chapter 4 (Vol. 1) even twenty years ago struc-tures and foundations were dealt in complete isolation where the structural andgeo-technical/foundation engineers hardly interacted 1 .

While the structural engineer was only bothered about the structural configuration

of the system in hand he hardly cared to know anything more about soil other than theallowable bearing capacity and its generic nature, provided of course the foundationdesign is within his scope of work. On the other hand the geotechnical engineer onlyremained focused on the inherent soil characteristics like ( c, , N c, N q , N , eo , Cc, Getc.) and recommending the type of foundation (like isolated footing, raft, pile etc.)or at best sizing and designing the same.

The crux of this scenario was that nobody got the overall picture, while in realityunder static or dynamic loading the foundation and the structure do behave in tandem.

For theoretical background on these topics please consider Volume 1.1 Even today there are companies which has divisions like structural and civil engineering!! Where the

responsibility of the structural division is to design the superstructure considering it as fixed base frame,furnish the results (Axial load, Moments and Shear) and the column layout drawing to the civil divisionwho releases the foundation drawing based on this input data.



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2 Dynamics of Structure and Foundation: 2. Applications

In chapter 4 (Vol. 1), in the problem Example 1.3.1, we have shown how thesoil stiffness can affect the bending moment and shear forces of a bridge girder andignoring the same how we can arrive at a result which can be in significant variationto the reality.

Drawing a similar analogy one can infer that ignoring the soil stiffness in the overall response (and treating it as a fixed base problem) the dynamic response of structure(the natural frequencies, amplitude etc.) can be in significant variation to the realityin certain cases.

This aspect came to the attention of engineers while designing the reactor buildingof nuclear power plant for earthquake. Considering its huge mass and stiffness, thefundamental time period for the fixed base structure came around 0.15 sec whileconsidering the soil effect the time period increased to 0.5 second giving a completelydifferent response than the fixed base case.

With the above understanding that underlying soil signicantly affects the responseof a structure, research was focused on this topic way back in 1970, and under thepioneering effort of academicians and engineers, the two diverging domain of technol-ogy was brought under a nuptial bond of Dynamic soil structure interaction , wheresoil and structure where married off to a unied integrated domain. To our knowl-edge the first signicant structure where the dynamic effect of soil was considered inthe analysis in Industry in India was the 500 MW turbine foundations for Singrauliwhere the underlying soil was modeled as a frequency independent linear spring andthe whole system was analyzed in SAP IV (Ghosh et al. 1984).

1.1.2 What does the interaction mean?We have seen earlier that considering the soil as a deformable elastic medium thestiffness of soil gets coupled to the stiffness of the structure and changes it elasticproperty. Based on this the characteristic response of the system also gets modified.This we can consider as the local effect of soil.

On the other hand consider a case of a structure resting on a deep layer of softsoil underlain by rock. It will be observed that its response is completely differ-ent than the same system when it is located on soft soil which is of much shallowdepth or resting directly on rock 2 . Moreover the nature of foundation, (isolatedpad, raft, pile), if the foundation is resting or embedded in soil, layering of soil,type of structure etc. has profound inuence on the over all dynamic response of thesystem.

We had shown for static soil-structure interaction (Chapter 4 (Vol. 1)) case that thesoil can be modeled as equivalent springs or as finite elements and are coupled with thesuperstructure.

Thus for a simple beam resting on an elastic support can be modeled as shownin Figure 1.1.1 and an equivalent mathematical model for the same is shown inFigure 1.1.2 .

Based on matrix analysis of structure the element stiffness for this element may bewritten as

2 The reason for these effects we will discuss subsequently.



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Dynamic soil structure interaction 3

Node i iNode i Node jNode j

Soil Spring K iSoil Spring K i Soil Spring K jSoil Spring K j

Figure 1.1.1 Equivalent beam element connected to soil springs.

2 4

1 1 3 2

Figure 1.1.2 Mathematical model of the equivalent beam element.

[Kbeam ] =EIzL3

12 6L 0 12 6L 06L 4L2 0 6L 2L2 00 0

IxL2

2Iz(1 + )0 0 IxL

2

2Iz(1 + ) 012 6L 0 12 6L 06L 2L2 0 6L 4L2 0

0 0 IxL 22Iz(1 + )0 0 IxL2

2Iz(1 + )

(1.1.1)

and the displacement vector is given by

{} = T (1.1.2 )

When the soil springs are added to the nodes, the overall stiffness becomes

[Kbeam] =EIzL3

12 +L3KiiEIz

6L 0 12 6L 06L 4L2 0 6L 2L2 0

0 0 IxL 2

2Iz(1 + )0 0 IxL

2

2Iz(1 + )

12 6L 0 12 +L3KjjEIz

6L 0

6L 2L2 0 6L 4L2 0

0 0 IxL2

2Iz(1 + )0 0

IxL 2

2Iz(1 + )(1.1.3)



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where, [Kbeam ] = combined stiffness matrix for the beam and the spring; Kii = K jj =spring values of soil at node i and node j of the beam respectively.The above is a very convenient way of representing the elastic interaction behavior

of the underlying soil and can be very easily adapted in a commercially available finiteelement or structural analysis package.

1.1.3 It is an expensive analysis do we need to do it?

This is a common query comes to the mind of an engineer before starting of an analysis.Based on this fact an engineer do become apprehensive if his/her analysis would sufferfrom a cost over run or whether he/she will be able to finish the design within theallocated time frame.

If he is convinced that soil structure interaction do takes place and the structure isa crucial one 3 our recommendation would be its worth the effort rather than to be

sorry later. The additional engineering cost incurred is trivial compared to the riskand cost involved in case of a damage under an earthquake or a machine induced load.

Now the first question is for what soil condition does dynamic soil structureinteraction takes place?

Veletsos and Meek (1974) suggest that chances of dynamic soil structure interactioncan be significant for the expression

V sfh 20 (1.1.4)

where V s = shear wave velocity of the soil; f = fundamental frequency of the fixedbase structure; h = height of the structure.Let us now examine what does Equation (1.1.4) signifies?Knowing the time period T = 1/ f , the above expression can be rewritten as

V sT h 20 (1.1.5)

For a normal framed building considering the fixed base time period as (0.1 n), wheren is the number of stories and thus, we have

V snh 200 (1.1.6)

For a normal building the average ratio of h/ n (height : storey ratio) is about 3 to3.3 meter. Thus considering h/ n = 3, we have

V s

600 m / sec. (1.1.7)

3 Like Power House, Turbine foundations, Nuclear reactor Building, Main process piper rack, distillationcolumns, bridges, high rise building catering to large number of people etc.



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From which we conclude that for ordinary framed structure, when shear wavevelocity is less or equal to 600 meter/sec we can expect dynamic soil structureinteraction between the frame and the soil.

Incidentally, V s = 600 m/sec is the shear wave velocity which is associated with rock.Thus it can be concluded that for all other type of soil, framed structures will behavedifferently than a fixed base problem-unless and until it rests on rock. For Cantileverstructures like tall vessels, chimneys etc of uniform cross section fundamental timeperiod T is given by

T = 1.779 mh4EI (1.1.8)where, m = mass per unit length of the system; h = height of the structure; EI =flexural stiffness of the system.Substituting the above value in Equation (1.1.5) we have

V sT h 20; or

V s1.779 mh 4EI h 20; or, V s 11.24h EI m (1.1.9)Considering, I = Ar2 and m = A, where A = area of cross section; r = radius of gyration; = Mass density of the material, we have

V s 11.24 r

h E (1.1.10)Shear Wave Velocity for Soil-Structure interaction for

Chimneys

0.00

200.00

400.00

600.00

800.00

1000.00

1200.001400.00

1 0 0

1 2 5

1 5 0

1 7 5

2 0 0

2 2 5

2 5 0

2 7 5

3 0 0

Slenderness Ratio

S h e a r W a v e

v e l o c i t y ( m / s e c ) Shear Wave velocity

steel chimney

Shear Wave velocityconcrete chimney

Figure 1.1.3 Chart to assess soil-structure interaction for steel and concrete chimney.



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For steel structure the above can be taken as, V s 57580 / where = h/ r, theslenderness ratio of the structure.For concrete structure we have

V s 123970

(1.1.11)

Based on the above expressions one can very easily infer if soil structure interactionis significant or not.

The chart in Figure 1.1.3 shows limiting shear wave velocity below which soil-structure interaction could be significant for a steel and concrete chimney.

1.1.4 Different soil models and their couplingto superstructure

The various types of soil model that are used for comprehensive dynamic analysis areas follows:

1 Equivalent soil springs connected to foundations modeled as beams, plates, shelletc.,

2 Finite element models (mostly used in 2D problems),3 Mixed Finite element and Boundary element a concept which is slowly gaining

popularity.

Of all the options, spring elements connected to superstructure still remain the mostpopular model in design practices due to its simplicity and economy in terms of analysisespecially when the superstructure is modeled in 3-dimensions.

It is only in exceptional or very important cases that the Finite elements and Bound-ary elements are put in to use and that too is mostly restricted to 2 dimensional cases.

1.2 MATHEMATICAL MODELING OF SOIL & STRUCTURE

We present hereafter some techniques that are commonly adopted for coupling thesoil to a structural system.

1.2.1 Lagrangian formulation for 2D frames or stick-models

This formulation is one of the most powerful tool to couple the stiffness of soil to thesuperstructure-specially when one is using a stick model or a 2D model.

For the frame shown hereafter we formulate the coupled stiffness and mass matrixfor the soil structure system which can be effectively used for dynamic analysis.

In the system shown in Figure 1.2.1 , mf , J = mass and mass moment of inertiaof the foundation; m1 , J 1 = mass and mass moment of inertia of the 1st story; m2 , J 2 = mass and mass moment of inertia of the top story; Kx , K = translational androtational stiffness of the soil; and k1 , k2 = stiffness of the columns.



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y2

m2 J2

k 2y1 h2

m1, J1

k 1 h1

K x K m f , J

Figure 1.2.1 2D Mathematical model for soil structure interaction.

The equation for kinetic energy of the system may be written as

T =12

mf u2 +12 J 2 +

12

m1( u +h1 + y1)2 +12 J 1 2

+12

m2(u + (h1 +h2) + y2)2 +12 J 2 2

(1.2.1)

U =12

Kxu2 +12

K 2 +12

k1y21 +12

k2 (y2 y1)2 (1.2.2)

Considering the expression 4 , d dt T qi + U qi = 0, we have the free vibration

equation as

mf +m1 +m2 m1h1 +m2H m 1 m2m1h1 J +m1h

21 +m2H

2m1h1 m2H

m1 m1h1 m1 0m2 m2H 0 m2

u

y1y2

+

Kx 0 0 00 K 0 00 0 k1 +k2 k20 0 k2 k2

u

y1

y2

= 0 (1.2.3)

4 Refer Chapter 2 (Vol. 2) for further application of this formulation where we have derived a 2D soil-structure interaction model for a Turbine framed foundation.



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Figure 1.2.2 Typical finite element mesh with soil springs, for a exible raft.

where

J = J + J 1 + J 2 sum of all mass moment of inertia;H = h1 +h2 = the total height of the structure.

Above formulation can very well be used in cases the foundation is significantly rigidand can be modeled as rigid lumped mass having negligible internal deformation 5 .

However for cases where the foundation is more flexible one usually resorts to finiteelement modeling of the base raft which is connected to the soil springs as shown inFigure 1.2.2.

For the problem as shown above irrespective of the raft being modeled as a beamor a plate the soil stiffness is directly added to the diagonal element Kii of the globalstiffness matrix to arrive at the over all stiffness matrix of the system.

Before we proceed further we explain the above assembly by a conceptual problemhereafter.

Example 1.2.1

For the beam as shown in Figure 1.2.3 , compute the global stiffness matrix whensupported on a spring at its mid span. Take EI as the flexural stiffness of the beam. The spring support has stiffness @ K kN/m.

Solution:

For a beam having two degrees of freedom per node as shown in Figure 1.2.4 ,the element stiffness matrix is expressed as follows.

5 A classic example is a turbine frame foundation resting on a bottom raft whose thickness is usuallygreater than 2.0 meter.



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L L

K

Figure 1.2.3 Spring supported beam.

2

1 4 3

Figure 1.2.4 Two degrees of freedom of a beam element.

The element matrix for such case is given by

1 2 3 4

Kij =

12EI L3

6EI L2

12EI L3

6EI L2

6EI L2

4EI L

6EI L2

2EI L

12EI L3

6EI L2

12EI L3

6EI L2

6EI L2

2EI L

6EI L2

4EI L

Assembling the element matrix for the two beams we have

[K] g =

12EI L3

6EI L2

12EI L3

6EI L2 0 0 0 0

6EI L2

4EI L

6EI L2

2EI L

0 0 0 0

12EI L3

6EI L2

12EI L3 +

12EI L3

6EI L2 +

6EI L2

12EI L3

6EI L2

0 0

6EI L2

2EI L

6EI L2 +

6EI L2

4EI L +

4EI L

6EI L2

2EI L

0 0

0 0 12EI L3

6EI L2

12EI L3

6EI L2 0 0

0 0 6EI L2

2EI L

6EI L24EI L 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0



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As Left hand support is fixed hence we have to eliminate row and column 1and 2.

Similarly, as right hand support is hinged we have to eliminate row and column5 from the above when we have

[K] g =

24EI L3

0 6EI L2

0 8EI

L2EI L

6EI L2

2EI L

4EI L

with appropriate boundary conditions.

To use the spring support, the spring is now directly added to the diagonal

element of the global matrix.Thus the combined stiffness matrix is given by

[K ] g =

24EI L3 +Ks 0

6EI L2

0 8EI

L2EI L

6EI L2

2EI L

4EI L

The above is the normal practice adapted in global assemblage of soil springin a finite element assembly.

We further elaborate the phenomenon with a suitable practical numericalexample.

Example 1.2.2

Shown in Figure 1.2.5 is a bridge girder across a river is resting at points A and Bon rock abutments at ends, and resting on a pier at center of the girder (point C )

A 5.0 m C 5.0 m B

Water Level

Figure 1.2.5 Bridge girder across abutments.



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A C B

1 1 2 2 3 3 4 4 5

Figure 1.2.6 Idealisation of the bridge girder ignoring soil effect.

which is resting on the soil bed of the river. The exural stiffness of the girder is EI = 100,000 kN m2 . Area of girder is 5.0 m 2 . The dynamic shear modulusof soil is G = 2500 kN/m 2 . The bridge pier foundation has plan dimension of 6 m 6 m. Determine the natural frequencies of vibration of the girder consid-ering with and without soil effect. Unit weight of concrete = 25 kN/m 3. Mass

moment of inertia per meter run=

30 kN

sec2

m.

Solution:

The bridge girder can be mathematically represented by a continuous beam asshown in Figure 1.2.6. Here node 2 and 4 are at the center of beam.

Thus, for beam element 1, 2, 3, and 4, we have element stiffness matrix as

[Kij] =EI L3

12 6L

12 6L

6L 4L2 6L 2L212 6L 12 6L6L 2L2 6L 4L2

The unconstrained combined stiffness matrix as

[Kij

]

=EI L3

12 6L 12 6L 0 0 0 0 0 06L 4L2 6L 2L2 0 0 0 0 0 012 6L 24 0 12 6L 0 0 0 06L 2L2 0 8L2 6L 2L2 0 0 0 00 0 12 6L 24 0 12 6L 0 00 0 6L 2L2 0 8L2 6L 2L2 0 00 0 0 0 12 6L 24 0 12 6L0 0 0 0 6L 2L2 0 8L2 6L 2L20 0 0 0 0 0 12 6L 12 6L0 0 0 0 0 0 6L 2L2 6L 4L2

Substituting the values we have



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[K] =76800 96000 76800 96000 0 0 0 0 0 096000 160000 96000 80000 0 0 0 0 0 0

76800 96000 153600 0 76800 96000 0 0 0 0

96000 80000 0 320000

96000 80000 0 0 0 00 0 76800 96000 153600 0 76800 96000 0 00 0 96000 80000 0 320000 96000 80000 0 00 0 0 0 76800 96000 153600 0 76800 960000 0 0 0 96000 80000 0 320000 96000 800000 0 0 0 0 0 76800 96000 76800 960000 0 0 0 0 0 96000 80000 96000 160000

Now imposing the boundary condition that vertical displacement are zero at1, 3, 5,6 we have

[K] =160000 96000 80000 0 0 0 0

96000 153600 0 96000 0 0 080000 0 320000 80000 0 0 0

0 96000 80000 320000 96000 80000 00 0 0 96000 153600 0 960000 0 0 80000 0 320000 800000 0 0 0 96000 80000 160000

Lumped mass at each node is given by Mii = 25 5 2.5/9.81 = 31.85kN sec2 /m.Mass moment of inertia at each node is given by J ii = 30 1.25 = 37.5.Thus combined mass matrix is given by

[M] =

37.5 0 0 0 0 0 0 0

0 31.85 0 0 0 0 0 00 0 65 0 0 0 0 00 0 0 31.85 0 0 0 00 0 0 0 37.5 0 0 00 0 0 0 0 31.85 0 00 0 0 0 0 0 65 00 0 0 0 0 0 0 37.5

6 We assume that since the bridge is supported on hard rock at ends, displacement at node 1 and 5are zero.



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A C B

1 2 3 4 5

Kz

Figure 1.2.7 Idealisation of the bridge girder considering soil effect.

Considering the equation[K] [M] 2 = 0 we have

MODE 1 2 3 4 5 6 7

Eigen value 692 1328 2684 4897 7448 7787 11722Natural 26.30 36.44 51.80 69.97 86.59926 88.24996 108.26855

frequency(rad/sec)

Considering the effect of soil we can construct the model as in Figure 1.2.7.

Here Kz =4Gr 01

where r0 = LxB , Here L = B = 6.0 mHere r0

= 3.38 m and for G

= 2500 kN/m 2 and

= 0.3 Kz

= 48285.71

kN/m.Now imposing the boundary condition that vertical amplitude at node 1 and

5 are zero (node 3 is not zero) we have[K] =

160000 96000 80000 0 0 0 0 0 96000 153600 0 76800 96000 0 0 0

80000 0 320000 96000 80000 0 0 00 76800 96000 201959.1 0 76800 96000 0

0 96000 80000 0 320000 96000 80000 00 0 0 76800 96000 153600 0 960000 0 0 96000 80000 0 320000 800000 0 0 0 0 96000 80000 160000

The Mass matrix remains same as derived earlier.Performing the eigen value solution we have

Modes 1 2 3 4 5 6 7 8

Eigen-values 75 692 2684 3045 7067 7448 9489 11722Natural 8.660 26.30 51.80 56.18 84.06 86.30 97.41 108.27

frequency(rad/sec)



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Having established the fact as to how soil affects the dynamic response let ussee further what different type of soil model is possible. For design office practicesspring values considered are usually based on Richart/Wolfs model which are effec-tively combined with structure as shown above to find out the overall response of asystem.

The example above, though it has been worked out based on beam the theory, it iseffective for any kind of structural elements like plates, shells, 8-nodded brick elementetc. Thus implementing the above in a general purpose Finite element package isquite straight forward. For raft modeled as beam with underlain spring, the essenceof arriving at individual springs at each node is same as shown in the case of staticanalysis based on influence zone 7 .

The only difference being that the nodal influence area is to be converted into anequivalent circular area to arrive at vertical spring values. The horizontal springs arebased on the full area and are divided equally at the end.

1.2.2 What happens if the raft is f lexible?

Methodology described in previous section is usually adapted when the raft is uncon-ditionally rigid. However there could be cases where the raft could be perfectly exibleor intermediate (i.e. somewhere between perfectly rigid and perfectly exible) whenthe calculation of spring values is different than what has been mentioned in thepreceding.

Before we get into this issue the obvious query would be what is the boundary

condition for raft rigidity in terms of dynamic loading?Unfortunately there is none, and the condition pertaining to static load still applies 8 .Thus as explained in Chapter 4 (Vol. 1), if L is the c / c distance between the columns,

then for

L 4 the raft will behave as rigid raft For L the raft will behave as flexible raft For all values between / 4 L , the slab behave in between rigid/flexible

in which = 4 kB4EcI , k = modulus of sub-grade reaction, (in kN/m 3); B = widthof raft in meter; Ec = modulus of elasticity of concrete (in kN/m 2); I = moment of inertia of the raft (in m 4).1.2.2.1 Calculation of spring constant for rigid raft

The rigidity of raft plays a significant role in the soil spring values connected to theplate elements as mentioned above.

7 Refer Example 4.6.1 in Chapter 4 (Vol. 1) for further details.8 This is not illogical for dynamic load can be conceived as a system under static equilibrium at a time t .

Thus condition of rigidity as explained in Chapter 4 (Vol. 1) should hold good.



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When the raft is rigid the gross spring value is obtained based on the full raftdimension and then are broken up into discrete values

kz =

KzA p

AG (1.2.4)

where, kz = value of discrete spring for the rigid finite element; Kz = value of grossspring considering the overall dimension of the raft; Ap = area of the finite elementplate, and AG = gross area of the raft.

1.2.2.2 Calculation of spring constant for flexible raft

When the raft is flexible an equivalent radius within which the load gets dispersed isfirst obtained from the formula

r0 = 0.8 t sEcG s

1 1 2c

13

(1.2.5)

The gross spring value is then obtained based on this equation. Finally the discretespring for the finite element is obtained as

kz = Kz A p r20

(1.2.6)

where, r0 = equivalent radius within which the load gets dispersed; Ec = dynamicmodulus of the concrete raft; G s = dynamic shear modulus of the soil; = Poissonsratio of soil; c = Poissons ratio of the raft, and t s = thickness of the raft.A suitable problem cited hereafter elaborates the above more clearly.

Example 1.2.3

A raft of dimension 30 m 15 m is resting on a soil having dynamic shearmodulus of 35000 kN/m 2 and Poissons ratio of soil = 0.4. Determine the soilsprings for plate elements of size 2.0 m 2.0 m for finite element analysisconsidering,

The raft as rigid Considering the raft as flexible.

The thickness of the raft is 1.8 m.



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Solution:

Considering the raft as rigid :

r0 = 30

15

= 11.968 meter;Kz =

4Gr 01 =

4 35000 11.960.6 = 2790666.67 kN/m

For finite element of size 2 m 2 m discrete spring value will be

kz = KzA pAG

kz = 2790666.67 2 230 15 =

24806 kN / m

Thus spring values at four nodes are 6201 kN/m i.e 1/4th of the abovecalculated value. When the raft is considered exible, we have:

r0 = 0.8 t sEcG s

1 1 2c

1/ 3

Here Ec = 3 108 kN/m2; c = 0.25 (say) ,

then r0 = 0.8 1.83 10835000

1 0.41 0.25 2

1/ 3

= 25.39 m

Thus Kz =4Gr 01 =

4 35000 25.390.6 = 5924333.333 kN/m

Thus for finite element of size 2 m 2 m the discrete spring value is

kz = 5924333.333 2 2

25.39 2 = 11701 kN/m

Thus spring values at four nodes are 2925 kN/mIt will be observed that the spring values vary considerably for the two different

approach.

1.2.2.3 What sin thou make in treating foundation& the structure separately?

Difficult to pass a sweeping judgment for depending on the situation, the sin could becardinal or even trivial.



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Based on a number of analysis carried out it can be stated that treating themin isolation can result in conservative design 9 or dangerously un-conservative, thusresulting in an unsafe structure which could be a danger to human life and property.

Having made the above statement a number of questions obviously come tomind10 like

1 How conservative or how susceptible the system can be ignoring the soil effect?2 Considering soil effect (specially for FEM analysis) makes the analysis more

laborious and time consuming thus more costly is it worth?3 My boss is a traditionalist and under project time pressure can I convince him

it is worth the effort.4 Before doing the detailed analysis itself can I come up with a quantitative value

based on which I can assess how far this effect will be (for good or worse) andthus convince my boss on the value addition to this effort?

5 What is the risk in terms of cost and safety if I do not do this analysis?The questions are surely pertinent and not always very easy to answer. However with

a little bit of intelligent analysis it is not difficult to come up with a logical conclusionon this issue.

We try to explain . . .The obvious answer is it essentially could modify the natural frequency/time period

of the system11 .What needs to be evaluated is what is the effect of this modified time period

on the system compared to, if the soil is ignored (i.e. it is considered a fixed base

problem).The two classes of problems under which dynamic soil structure interaction plays asignificant role are

Systems subjected to vibration from machines like block foundations (machinefoundations for pumps, compressors, gas turbines etc), frame foundations (turbinefoundations, compressor foundations, boiler feed pump foundations)

Structures subjected to earthquake.For the machine foundation source of disturbance is the machine mounted on the

system the dynamic waves generated are transferred from the machine via structureto the surrounding soil-which is an infinite elastic half space.

While for earthquake the source of disturbance is the ground itself where elasticwaves generate within the soil mass due to the tectonic movement/rupture of the rockmass (geologically known as faults).

It is obvious that soil will affect these two classes of problem in different ways.For instance a machine supported on a frame- the frame is usually made signi-

cantly stiff to ensure stress induced in it are not signicant and are generally made

9 For big projects which could mean a cost over run.10 SpeciallyforfreshmannewtothetopicwhohasgotaleadengineerandadepartmentalHEADtoanswerto.11 We say the word could as because the extent of modification will depend upon the shear wave velocity

of the soil. We had shown previously the boundary limits within which it can have a significant effect.



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over tuned for medium or low frequency machine when considered as a fixed basedproblem.

But in reality considering the soil effect, the foundation may actually be under tunedor even hover near the resonance zone when the underlying soil participates in thevibration process. Thus the amplitude of vibration could significantly vary than thecalculated one.

Generically, considering the soil stiffness will make the system more flexible thena fixed base problem and it can be intuitively deduced that though the stress mightremain within the acceptable level the amplitude of vibration will be more and couldwell exceed the acceptable limit which might have secondary damaging effect to themachine and its appurtenances.

For earthquake the effect is quite different. In this case the structure resting on thesite can be visualized as a body resting on an infinite elastic space (similar to a shipfloating in sea). Due to rupture in the fault as waves dissipate in all direction the soilmass starts vibrating at its own fundamental frequency known as the free field timeperiod of the site.

In such case the earthquake acts as an electronic lter and tries to excite the super-structure resting on it to its own fundamental frequency and suppressing or eveneliminating other modal frequencies 12 . Thus if the fixed base frequency of the struc-ture matches the fundamental frequency of the soil strata on which it is resting, theyare in resonance and catastrophe could well be a reality.

Before dwelling into the mathematical aspect of it we further substantiate the abovestatement by some real life facts and observations.

Dowrick (2003) reports that in the Mexico earthquake in 1957 extensive damageoccurred to the buildings that were tall and were found to be resting on alluvium soilof depth > 1000 m. In 1967, the Caracas earthquake showed identical result where thetall structures underwent extensive damage and those were resting on deep alluviumsoil overlying bedrock. In 1970 earthquake at Gediz in Turkey a part of a factorywas demolished in a town about 140 Km from the epicenter while no other build-ings in the town underwent any damage! Subsequent investigation revealed that thefundamental period of the building matched the free field time period of the site . TheCaracas earthquake as cited earlier also showed a distinctive pattern where mediumrise buildings (59 storeys) underwent extensive damage where depth to bedrock wasless than 100 m, while buildings over 14 stories were damaged where the depth tobedrock was greater than 150 meters.

Let us see why such thing happened and how does it substantiate the free field timeperiod phenomenon as stated earlier.

The free field time period of a site is given by the equation

T n =4H

(2n 1)V s(1.2.7)

12 It can be visualized as a giant hand trying to shake a small body resting on it. Since the body is muchweaker to the giant it tries to follow the same phase of vibration as the soil medium.



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0

20

40

60

80

100

120

0 0 .

1 5 0 . 3

0 . 4 5 0 .

6 0 .

7 5 0 . 9

1 . 0 5 1 .

2 1 .

3 5 1 . 5

Depth of soil/Shear wave Velocity

N u m b e r o f S t o r i e s

n for RCC framen for steel frame

Figure 1.2.8 Limiting value of storeys for frames.

where, T = time period of the free field soil (i.e. without the structure); H = depth of soil over bedrock 13 ; n = number of mode; and V s = shear wave velocity of the soil.Thus based on the explanation above it can be argued that if the fixed base frequencyof structure is in the close proximity of the free field time period of the site the structuremay be subjected to significant excitation.

The above statement can be extended to a very interesting hypothesis.If we equate the free field time period of the site to the fixed base time period of

the structure we can arrive at some limiting design parameters which can result insignificant dynamic amplification and which should be avoided at the very out set of planning of the structure.

For instance as per IS-1893 RCC moment resisting frames with no infill brick work,the fundamental time period is given by

T = 0.075 h0.75 (1.2.8)Thus equating it to fundamental free field time period of the site we have

0.075 h0.75 =4H V s

, which gives h =160H 3V s

4/ 3(1.2.9)

Considering 1 floor is of height 3.3 m, we can further simplify the equation to

n = 0.303160H 3V s

4/ 3

(1.2.10)

13 Here bedrock is perceived as that level where the shear wave velocity of soil is greater or equal to600m/sec.



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The curves shown in Figure 1.2.8 give limiting stories for RCC and steel frames forwhich resonance can occur in a structure during an earthquake as per IS-1893 14 forvarious values of H / V s.

Let us now probe the problem a bit more based on a suitable numerical problem.

Example 1.2.4

A particular site has been found to consist of 100 m soil overlying bedrockwhen the shear wave velocity of the soil is 222.22 m/sec. Find the limitingnumber of stories of height 3.3 meter for an RCC frame for which resonancecan occur. What would be resonance story if the depth of the overlying soft soil isonly 30 m.

Solution:Based on above data H / V s = 100 / 222.2 = 0.45 when H = 100 m.As per the chart as shown above the limiting story for which resonance canoccur is 18.

Thus for a 18 storied building resonance can very well occur and the strategywould be to build the building at least ( )25% away i.e. either it should be 23storied or more or 14 storied or less.

When the depth of soil is only 30 m, H / V s = 30222.2 = 0.135.Based on the above chart the limiting story height is roughly 4-storey only.Thus to avoid resonance the building should be either more than 5-storey or lessthan 3-storey.

The above problem well explains the phenomenon as to what happened in theMexico and Turkey earthquakes and perhaps challenges the myth quite prevalent inmany design offices that for one or two storied building earthquake is not important and can well be ignored.

It is evident from the above problem that the response depends on the depth of soilon which it is resting and depending on the free field time period the response caneither amplify or attenuate. It can well affect even a one storied building.

The chart in Figure 1.2.9 shows limiting story height of buildings with infill brickpanels and all other type of frames as per IS 1893 for different width of buildingvarying from 10 meter to 50 m 15 .

The above theory is though explained in terms of building, can very well be adaptedfor any class of structure for which it is possible to establish the fundamental timeperiod expression.

14 In this case time period for steel frame is considered as T = 0.085( h)0.75 as per IS-1893.15 Time periodof thefixed base structureconsidered as T =0.09 h /(d )0.5 as per, IndianStandards Institution(1984, 2002). Indian Standard Criteria for Earthquake Resistant Design of Structures, IS: 1893(Part 1), ISI, New Delhi, India.



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0

5

10

15

20

25

30

35

40

0 0 . 0

8 0 . 1

5 0 . 2

3 0 . 3 0 . 3 8

0 . 4 5

0 . 5 3 0 . 6 0 . 6

8 0 . 7

5

Depth of soil/Shear wave velocity

N u m b e r o f s t o r y

n for d=10m

n for d=15m

n for d=20m

n for d=25mn for d=30m

n for d=35m

n for d=40m

n for d=45m

n for d=50m

Figure 1.2.9 Limiting story for building with inll brick panel.

Having assessed the resonance criteria and making sure at planning stage that thetwo periods do not match one would still like to quantify the combined time periodof the overall soil structure system and assess whether there is any amplification or

attenuation of the earthquake force.Before plunging into detailed analysis based on FEM or otherwise it would be usefulto have a rough estimate as to how much the underlying soil affects the overall response.

Veletsos and Meek (1974) has given a very useful expression based on which it ispossible to estimate the modified time period of a structure, and is given by

T = T 1 +k

Kx1 +

Kxh2

K (1.2.11)

where T = modified time period of the structure due to the soil stiffness, T = timeperiod of the fixed base structure, k = stiffnessof the fixed base structure @ 4

2W gT 2 ,

Kx , K = horizontal and rotational spring constant of the soil (IS-1893), h = effectiveheight or inertial centroid of the system, and, W = total weight of the structure.Based on the above expression one can immediately arrive at a rough estimate asto how strong could be soil response at the very outset of a design. We elaborate theabove based on two suitable problems hereafter.

Example 1.2.5

An RCC Chimney 150 meter in height has a uniform cross section area of Ac =8.5 m 2 and moment of inertia I = 92.5m 4 . Evaluate the base moment and



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shear under earthquake considering the problem as fixed base as well as thesoil effect. The structure is located in zone IV as per IS 1893. The structureis supported on raft of diameter 18 meter. The soil has a dynamic shear wavevelocity of 120 m/sec and unit weight of 19 kN/m 3 . Consider 5% dampingfor the analysis. 16 The grade of concrete used is M30 having dynamic Econc =3.12 108 kN/m 2 .

Solution:

Height of the structure = 150 m; Area of shell = 8.5 m2Weight of chimney = 150 8.5 25 = 31875 kN (unit weight of conc. =25 kN/m 3)Radius of gyration of the chimney =

I / A =

92.5 / 8.5 = 3.298 m

Thus slenderness ratio H / r = 1503.298 = 45.4.As per IS 1893 CT = 82.8.As per IS 1893 time period of a fixed base chimney is given by, T =CT 3.13 WH EcAc .where, W =weight of chimney in N; Ec =Dynamic Youngs modulus of conc.@ 3

108 kN/m 2

Thus, T =82.83.13 31875 1503.12 108 8.5 = 1.13 sec

For 5% damping referring to chart in IS-1893 we have Sa / g = 0.10.Thus the horizontal seismic coefficient is given by, h = IF o

Sa g

.

Here (Soil foundation factor) = 1.0 for chimney resting on raft, I = 1.5Importance factor, F o = Zone factor @ 0.25 for zone IV.This gives

h = 1.0 1.5 0.25 0.10 = 0.0375The Bending moment and shear force are given by,

M

= hW H 0.6

x

H

12

+0.4

x

H

4 and V

= Cv hW

5x

3H

2

3

x

H

2

16 In this case it is presumed that reader has some idea of how to use the code IS-1893 or is at leastfamiliar with it.



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Here, Cv = a coefficient depends on the slenderness ratio and as per the presentproblem is 1.47 as per IS 1893; H = height of c.g. of the structure above base@ 75 meter for the problem; x = distance from the top.Substituting the appropriate values, we have

M = 0.0375 31875 75 [0.6 (1.0 )12 +0.4 (1.0 )4] = 89648 kN/m

V = 1.47 0.0375 31875 [(5/ 3) (2/ 3)] = 1757 kN.

Considering the soil effect we have the dynamic shear modulus of soil, G = v2sOr G = (19 / 9.81 ) 120 120 = 27890 kN/m 2 .With radius of raft = 9.0 m, Kx =

8GR2

, = Poissons ratio of the soilconsidered as 0.35,

Kx =8 27890 9

2 0.35 = 1217018.2 kN/m

And K =8GR 3

3(1 )which gives, K =

8 27890 933(1 0.35 ) =

83412554 kN/m

The xed base stiffness of chimney is given by

k =4 2W gT 2 =

4 2 318759.81 1.13 2 =

100458 kN/m

Substituting the above numerical values in Veletsos equation we have

T = T 1 +k

Kx1 +

Kxh2

K

T = 1.13 1 + 1004581217018.2 1 + 1217018.2 75283412554 = 3.2 secAs per IS 1893 for T = 3.2 sec, Sa / g = 0.05, which gives

h =0.0375

0.10 0.05 = 0.01875 (By proportion)

The base moment and shear are given by

M =896480.0375 0.01875 = 44824 kN m;



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and V =1757

0.0375 0.01875 = 878.5 kN

The results are compared hereafter 17

Case Moment Shear Remarks

Without soil 89648 1757 Reduction in moment and shear by 34%With soil 44824 878.5

The problem shows a clear attenuation of the response.

We show another example hereafter.

Example 1.2.6

Shown in Figure 1.2.10 is a horizontal vessel having empty weight of 340 kNand operating weight of 850 kN is placed on two isolated footing of dimension8.5m 3 m. The center to center distance between the two foundations is 5.5meter. The center line of vessel is at height ( H f ) of 4.5 meters from the bottom of the foundation. Thickness of the foundation slab is 0.3 meter. The RCC pedestal

is of width 1.0 meter, length 6 meter having height of 3.45 meter. The shear wavevelocity of the soil is 200 m/sec having Poissons ratio of 0.3. Allowable bearingcapacity of the foundation is 150 kN/m 2 . Calculate the design seismic momentconsidering the effect of soil and without it, if the site is in zone III as per IS-1893.Consider soil density @ 18 kN/m 3 and unit weight of concrete as 25 kN/m 3?

Solution:

Plan are of footing = 8.5 3 = 25.5 m2

Equivalent circular radius = Af = 25.5 = 2.849 mMoment of inertia of the foundation about X -axis

112

BL3 =1

1238.53 =

153.5313 m 4

Moment of inertia of the foundation about Y -axis 112

LB3 =1

128.5 33 =

19.125 m 4

17 Without an elaborate analysis it could be an effective calculation to convince the boss thatyou can save some money and the worth of a dynamic soil-structure interaction analysis.



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yp

y p H f

H p

D s

W p

L p

Y

Lf X

B f B f

Ls

Figure 1.2.10 A horizontal vessel.

Equivalent circular radius about X axis =12

64I xx

0.25

= 3.739183 m

Equivalent circular radius about Y axis =12

64I yy

0.25

= 2.221 m

Mass density of soil ( ) =18

9.81 = 1.835 kN/m3

Dynamic shear modulus = v2s = 1.835 200 2 = 73400 kN/m 2



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Lateral spring in X and Y direction =32Gr 0(1 )

7 8 = 1018306 kN/m

Rocking spring about X axis

=

8Gr 3x

3(1 ) = 14627886 kN/m

Rocking spring about Y axis =8Gr 3y

3(1 ) = 3063462 kN/m

Moment of Inertia of the pedestal about X axis =1

12 1 63 = 18 m4

Moment of Inertia of the pedestal about X axis

=1

12 6

13

= 0.5 m4

Structural stiffness of pedestal about X axis =3EI xL3 =

3 3.2 108 183.45 3 =

3.95 108 kN/m

Structural stiffness of pedestal about Y axis =3EI yL3 =

3 3.2 108 0.53.45 3 =

1.10 107 kN/m

Contributing mass for the vessel empty case =340

2 9.81 = 17.33 kN-sec 2 /m

Contributing mass for the vessel operating case =850

2 9.81 =43.323 kN-sec 2 /mContributing uniformly distribute load for the pedestal = 25.5 25/9.81 =64.98 kN/mThe mathematical model for the pedestal thus constitute of a beam element

(pedestal) having a mass lumped at its tip (mass contribution from the vessel) isshown in Figure 1.2.11.

The time period of such fixed base model is given by (Paz 1991)

T = 2 (M +0.25 mb )Kmb

M

Figure 1.2.11 Mathematical model for the pedestal.



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and the modified time period considering soil effect is given by

T = T 1 +k

Kx1 +

Kxh2

K

The time periods and the corresponding Sa / g values as per IS-1893 for 5%damping are as show hereafter.

Time period Time period Time period Time period (vessel empty ) (vessel empty ) (operating ) (operating )about X about Y about X about

Sl no Case direction direction direction Y direction

1 Without soil 0.0018 0.011 0.0024 0.0152 With soil effect 0.0579 0.1057 0.0771 0.1408

Corresponding Sa / g value is given by

Sa/g Sa/g Sa/g Sa/g (vessel empty ) (vessel empty ) (operating ) (operating )about about about about

Sl no Case X direction Y direction X direction Y direction


Base shear as per IS 1893 considering Importance factor as 1.0 for vessel emptycase and 1.25 for vessel in operation case we have

Shear Shear Shear Shear (vessel empty ) (vessel empty ) (operating ) (operating )about about about about


1 Without soil 21.25 25.5 26.5625 34.53125

2 With soil effect 42.5 42.5 53.125 53.125

The moment at the foundation level is given by

Moment Moment(vessel (vessel Moment Momentempty ) empty ) (operating ) (operating )about about about about



This case clearly shows an amplification of force considering the soil effect.



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Having established a basis of how to evaluate the coupled soil-structure interactionunder dynamic loading we now extend the above theory to system with multi degreeof freedom where the theory can be very well be adapted as a powerful tool for adetailed yet economic dynamic analysis.

1.3 A GENERALISED MODEL FOR DYNAMIC SOIL STRUCTUREINTERACTION

In this section we present a generalised model for dynamic soil-structure interaction.Though the model is developed based on 3D frames can also be adapted for a threedimensional Finite Element analysis.

1.3.1 Dynamic response of a structure with multi degreeof freedom consider ing the underlying so il s ti ffness

We had shown earlier that for a single degree of freedom system the modified timeperiod of a structure considering the soil effect is given by

T = T 1 +k

Kx1 +

Kxh2

K (1.3.1)

Both ATC (1982) and FEMA has adapted this formula for practical design office usage(Veletsos & Meek 1974, Jennings & Bielek 1973). The nomenclatures of the formulaare as explained earlier. Now squaring both sides of the above equation we have

T 2 = T 2 1 +k

Kx +kh

2

K (1.3.2)

Considering the expression T = 2 we have

42

2 =4

2

21 + 4

2m

T 2Kx + 42mh

2

T 2K or 4

2

2 =4

2

21 + m

2

Kx + 2mh

2

K

(1.3.3)

Simplifying and expanding the above we have

1

2 =1

2 +mKx +

mh2

K which can be further modified to

12 =

12 + 12x +

12

(1.3.4)

which gives the modified natural frequency relation for a system with single degree of freedom. This formulation has also been shown in, Kramer, S. (2004).



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Now considering generically = k/ m we havemke =

mk +

mKx +

mh2

K , or,

1ke =

1k +

1Kx +

h2

K (1.3.5)

where ke = equivalent stiffness of the soil structure system having single degree of freedom.We shall extend the above basis to multi degree of freedom hereafter ( Chowdhury

and Dasgupta 2002 ).

1.3.2 Extension of the above theory to system wi th multi degree of freedom

A 3-D frame shown in Figure 1.3.1, is considered for the presentation of the proposedmethod. The frame structure has n degrees-of freedom and subjected to soil reactionsin the form of translational and rotational springs.

For a system having n degrees of freedom the above equation can be written in theform

[M]nn[Ke]nn =

[M]nn[K]nn +

[M]nnKx +

[M]nn h2

nnK

(1.3.6)

Y

X O

Z

Kx

K -- mass points

Figure 1.3.1 A 3-D Frame having multi-degree-of freedom with representative foundation spring.



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Here, [Ke] = equivalent stiffness matrix of the soil structure system of order n, [M] =a diagonal mass matrix of order n having masses lumped at the element diagonals,[h2] = radius vectors of the lumped masses to the center of the foundation springsof order n, Kx , K = translation and rotation spring stiffness of the total foundationsystem represented by a unique value.

Taking out the common factor [ M], we have

[I ][Ke] =

[I ][K] +

[I ]Kx +

h2

K (1.3.7)

where, [I ] = identity matrix of order n having its diagonal element as 1.or [I ][Ke]1 = [I ][K]1 + [I / Kx] + [h2 / K ]

[ Fe] = [F ] + [F x] + [F ] (1.3.8)

where [F ] = Flexibility matrix of the system with suffixes as mentioned earlier forstiffness matrices.Once the flexibility matrix of the equivalent soil structure system is known the

stiffness matrix may be obtained from the expression

[Ke] = [Fe]1 (1.3.9)

Now knowing the modified stiffness matrix the eigen solution may be done basedon the usual procedure of

[Ke][ ] = [ e][M][ ]. (1.3.10)

1.3.3 Estimation of damping ratio for the soilstructure system

While calculating the damping ratio, the normal process is to guess a damping ratiofor the structure like 25%, and consider the same damping ratio for all the mode andobtain the value of Sa / g value for the particular structure per mode corresponding tothe time period based on the curves given in IS-1893.

The basis of assuming this damping ratio is purely judgmental and is dependenton either the experience of the engineer, recommendation of codes, or based on fieldobservations on the performance of similar structure under previous earthquakes.

When the effect of soil is neglected it is possible to obtain the material damping ratioof the structure depending on what constitute the material like steel, RCC etc.

However when the whole system is resting on soil an analyst is usually faced withthe following stumbling blocks for which clear solution is still eluding us specially formodal analysis in time domain.



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The difficulties encountered can be summarised as follows

The damping matrix of the coupled soil-structure system becomes non-proportional for which the damping matrix does not de-couple based onorthogonal transformation.

As the damping ratio of the structure and the soil could be widely varying itbecomes difficult to assess a common damping ratio which would affect the soilas well as structure.

Even after elaborate FEM modelling of the soil, the damping ratio contributionper mode still remains guess estimation at the best.We present hereafter a method by which one can estimate approximately the contri-

bution of combined soil structure system under earthquake for various modes, withoutresorting to an elaborate modelling of the soil itself.

We only estimate the contribution of the soil damping to the structural system whoseresponse we are interested in. The estimation is surely approximate but at least givesa rational mathematical basis to arrive at some realistic damping value rather thanguessing a damping value at the outset and presuming that it remain same for eachmode, specially for coupled soil structure system where widely varying damping for thefoundation and structure makes it difficult for the analyst to arrive at unified rationalvalue applicable to the system.

1.3.4 Formulation of damping ratio for single degreeof freedom

Neglecting the higher order, the material damping ratio for a soil structure systemhaving single degree of freedom is given (Kramer 2004) by

2 =

2 + x2x +

2

(1.3.11)

where, =

damping ratio of the equivalent soil structure system;

= damping ratio

of the fixed base structure; x = horizontal damping ratio of the soil, where x = 0.288 Bxand Bx =

(78 )mg 32(1 ) sr3x, where m = total mass of the structure and foundation; g =

acceleration due to gravity; = Poissons ratio of the soil; s = mass density of thesoil; rx = Equivalent circular radius in horizontal mode; = damping ratio of thesoil in rocking mode x = 0.15(1+B ) B and B =

0.375 (1) J g sr5

; and J = mas

dynamics of structures volume 2 - applications chowdhury

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