Chapter VIII - INFLIBNETshodhganga.inflibnet.ac.in/bitstream/10603/7927/17/17_chapter 8.pdf · Chapter VIII 223 Chapter VIII Dynamic soil structure interaction analysis under earthquake

Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 223

Chapter VIII

Dynamic soil structure interaction analysis under

earthquake

8.1. Introduction

In the dynamic analysis of SSI problems two key aspects are to be considered are (i) The

method of dynamic analysis (ii) The modeling of interaction between structure and soil.

8.2. The method of dynamic analyses

The general dynamic methods that are used to analyze buildings subjected to earth quake are

as follows:

1. Quasi-static method.

2. Response spectrum analysis.

3. Push over analysis.

4. Modal analysis.

5. Time history analysis

The first two methods cannot be used in SSI analyses. However all have been explained

briefly here for clarity and completeness, highlighting their advantages and limitations:

8.2.1. Quasi-static method.

The most commonly employed method is the quasi-static method as it is the simplest,

requires only static analysis, and estimates the response of the structure for an ensemble of

earthquakes. In the traditional approach to earthquake engineering design the computations

are carried out on the basis of linear elastic static analysis. The nonlinear behaviour and

energy dissipation can be accounted for in a trivial manner by a force-based approach, where

the level of seismic loading is computed by analyzing the elastic response spectra on a single

degree of freedom system and the so – called R reduction factor method is then used to

introduce the ductility of materials (steel and concrete).


A more rational concept developed during this last decade, named displacement-based

procedure, turns towards the design based on the critical limit states of the structural

elements. In other words, defining the ultimate behaviour of materials, one has to deal with

maximum strains or curvatures, thus taking into account the structure ductility demand in a

more physical way.

The force-based design uses the elastic properties of the structure, the displacement-based

design characterizes it through the secant stiffness at maximum displacement. The evaluation

of vibration frequencies during the loading is more appropriate than for the purely elastic

analysis.

8.2.2. Response spectrum analysis.

The response spectrum method is identical to the quasistatic method except that it considers

more than just the fundamental mode of vibration. Most codes require that enough modes of

vibration are considered to account for 90% of the modal mass. Housner was instrumental in

wide spread acceptance of the concept of earth quake response spectrum introduced by Biot

in 1933. Response spectrum is the central concept of earth quake engineering.

There are significant computational advantages in using the response spectra method of

seismic analysis for prediction of displacements and member forces in structural systems.

The method involves the calculation of only the maximum values of the displacements and

member forces in each mode using smooth design spectra that are the average of several

earthquake motions.

(Wilkinson and Hiley) For the quasi-static method and the response spectrum method

the earthquake forces are divided by a behavior factor (also known as a structural response

factor or response modification coefficient). This factor accounts for the reserve strength of

the building after the formation of the first plastic hinge and allows a pseudo inelastic design

to be achieved without complicating the analysis. The only extra requirement to account for

inelastic behavior is for the designer to choose an appropriate building behaviour factor.

Typically, this is done by choosing a value from a table in a relevant earthquake code. This is

simple and reasonably effective but it is overly conservative. The various ductility factors


have been arrived at empirically based on past experience of structural behavior during

earthquakes and based on generalised analysis of simple models of various building types.

These classical procedures for design (force or displacement formulation) are based on the

study of an equivalent single degree of freedom system or in fact a simplified substitute

structure which is not capable to account for the load redistribution inside the structures due

to local nonlinearities. This is one of the major drawbacks preventing a realistic description

of the global and local behaviour of a structure up to failure.

Some codes have tried to improve the selection of the behaviour factor by introducing

Capacity design. The philosophy adopted in capacity design is best outlined by Park [5] ‘‘In

the capacity design of structures, appropriate regions of the primary lateral earthquake force

resisting structural system are chosen and suitably designed and detailed for adequate

strength for a severe earthquake. All other regions of the structural system and other possible

failure modes, are then provided with sufficient strength to ensure that the chosen means for

achieving ductility can be maintained throughout the post-elastic deformations that may

occur.’’ With certain limitations, this procedure could allow designers to choose whatever

behaviour factor they want as long as they then ensure that the elements and connections

have sufficient rotational capacity to redistribute the forces and hence achieve the ultimate

load.

However, because of the simplifications used in the analysis it is not a straightforward task to

relate the actual ductility of a structure to a building behaviour factor. The ductility of a

structure can be defined as the ratio of ultimate load to the load producing the first plastic

hinge and is generally referred to as the ductility factor, which is different from a behaviour

factor. Various researchers have tried to relate behaviour factors to ductility factors but there

is still no exact way to do this.

8.2.3. Push over analysis.

One remedy consists in using nonlinear finite element method to perform Push-over

analysis, i.e. a nonlinear static analysis under monotonic increasing lateral load. This load is

increased until a collapse mechanism is formed. The rotations of all the plastic hinges are

calculated and the joints detailed to ensure that these rotations can be achieved. It allows


determining the maximum carrying capacity of the structure in terms of forces,

displacements, ductility (deformation), and crack pattern and failure mode. The main

assumption of such a computation is that the response is related to an equivalent single

degree of freedom and thus, controlled by a single mode which justifies that the position and

amplitude of horizontal lateral loads are typically defined by an elastic analysis.

Though the finer aspects of structural modelling like the variation in structural form, the

influence of geometric nonlinearity and the nonlinear constitutive response of structural

materials under serviceability and extreme loading conditions can be taken care in above

methods the approximation in the forces or deformation nullifies such an effort being

8.2.4. Modal analysis

The modal analysis procedure is based on the fact that for certain forms of damping that are

reasonable models for many buildings, the response in each natural mode of vibration can be

computed independent of the others, and the modal responses can be combined to determine

the total response. Each mode responds with its own particular pattern of deformation, the

natural mode of vibration, with its own frequency, natural frequency of vibration and with its

own modal damping ratio. Each modal response can be computed by analysis of an SDF

system with properties chosen to be representative of particular mode. The modal analysis

procedure avoids simultaneous solution of coupled equations. This method is valid only for

the earth quake analysis of structures responding within their elastic range of behaviour. This

method, after a set of orthogonal vectors are evaluated, reduces the large set of global

equilibrium equations to a relatively small number of uncoupled second order differential

equations. The numerical solution of these equations involves greatly reduced computational

time. It has been shown that seismic motions excite only the lower frequencies of the

structure. Typically, earthquake ground accelerations are recorded at increments of 200

points per second. Therefore, the basic loading data does not contain information over 50

cycles per second. Hence, neglecting the higher frequencies and mode shapes of the system

normally does not introduce errors.

There are two major disadvantages of using this approach. First, the method produces a large

amount of output information that can require an enormous amount of computational effort


to conduct all possible design checks as a function of time. Second, the analysis must be

repeated for several different earthquake motions in order to assure that all the significant

modes are excited, since a response spectrum for one earthquake, in a specified direction, is

not a smooth function.

During the last couple of decades, it has been well recognized that the soil on which a

structure is constructed may interact dynamically with the structure during earthquakes,

especially when the soil is relatively soft and the structure is stiff. This kind of dynamic soil-

structure interaction can sometimes modify significantly the stresses and deflections of the

whole structural system from the values that could have been developed if the structure were

constructed on a rigid foundation. These interactions cause energy dissipation, and change

the natural modes of vibration of the structure such as natural frequencies and corresponding

mode shape. Because the dynamic – stiffness matrix is frequency dependent and complex,

the orthogonality condition is not satisfied for a soil- structure interaction system.

Consequently, the equation of motion cannot be uncoupled, and the classical mode

superposition method is not applicable to the soil-structure system.

8.2.5. Time history analysis

An alternative choice to perform the earthquake resistant design is by making use of the

nonlinear time- history analysis, assuming a physical description of materials and applying

transient loadings on the structure in terms of natural or simulated ground motion. The

evolution of Eigen modes concomitant to the stiffness degradation governed by local yield

criterion provides currently the most refined method of analysis for ultimate behavior of

concrete structure.

All the above mentioned dynamic methods can also be applied to soil structure interaction

problems with their advantages and limitations. Time history analysis yields more accurate

results and was not used by researcher’s earlier due to high requirement of computational

efforts. As the present generation of computers are very fast with large memory this

limitation is no more relevant and this method of analysis is adopted in the present work.

Time history analysis can be done either in frequency domain or time domain. For a long

time, modelling of dynamic soil structure-interaction was carried out in the Frequency


Domain, which restricted the analysis of the soil-structure system to be linear

[Aydinoglu(1992)]. The basic approach, used to solve the dynamic equilibrium equations in

the frequency domain, is to expand the external loads in terms of Fourier series or Fourier

integrals. The solution is in terms of complex numbers that cover the time span from -to .

Therefore, it is very effective for periodic types of loads such as mechanical vibrations,

acoustics, sea-waves and wind (Clough and Penzien). However, the use of the frequency

domain solution method for solving structures subjected to earthquake motions has the

following disadvantages:

1. The mathematics is difficult to understand and also the solutions are difficult to verify.

2. Earthquake loading is not periodic; therefore, it is necessary to select a long time period in

order that the solution from a finite length earthquake is completely damped out prior to the

application of the same earthquake at the start of the next period of loading.

3. For seismic type loading the method is not numerically efficient. The transformation of the

result from the frequency domain to the time domain, even with the use of Fast Fourier

Transformation methods, requires a significant amount of computational effort.

4. The method is restricted to the solution of linear structural systems.

5. The method has been used, without sufficient theoretical justification, for the approximate

nonlinear solution of site response problems and soil/structure interaction problems.

Typically, it is used in an iterative manner to create linear equations. The linear damping

terms are changed after each iteration in order to approximate the energy dissipation in the

soil. Hence, dynamic equilibrium, within the soil, is not satisfied.

8.3. Dynamic soil structure interaction

Two important characteristics that distinguish the dynamic soil-structure interaction system

from other general dynamic structural systems are the unbounded nature and the nonlinearity

of the soil medium. Another effect which is also taken into account concerns the interaction

of structure with the foundation, which provides both a more realistic boundary condition as

well as the damping model due to radiation effect.


Of the three widely used methods of SSI analyses (discussed in detail in Chapter I) viz., i)

Elimination method, ii) Sub-structure method, and iii) Direct method, Direct Method has

become more attractive to researchers and engineers because it can consider the nonlinearity

of the unbounded soil as long as sufficiently accurate transmitting boundaries can be

developed [wolf (1988)].

Therefore in the present work Direct Method of analysis by using Finite elements is applied

for non-linear dynamic analysis of soil-structure-problem in time domain.

The following numerical experiments are conducted in dynamic analysis in the course of

development of software

1. Dynamic analysis of three dimensional frame with fixed support under transient loads.

2. Dynamic analysis of three dimensional frame with fixed support under earth quake load.

3. Comparative Dynamic SSI analyses of following models, under transient loading, with

linear constitutive relations of soil:

a Three dimensional frame supported on raft foundation neglecting stiffness of infills.

b Three dimensional frame supported on raft foundation considering stiffness of infills.

4. Comparative Dynamic SSI analyses of following models, under earthquake loading, with

linear constitutive relations of soil:

a Three dimensional frame supported on raft foundation neglecting stiffness of walls.

b Three dimensional frame supported on raft foundation considering stiffness of walls.

5. Dynamic SSI analyses of Three dimensional frame supported on isolated footings under

earth quake load.

6. Non-linear Dynamic SSI analyses of three dimensional frame supported on raft

foundation.

8.4. Derivation of dynamic equations of equilibrium

In Dynamic problems the displacements, velocities, strains, stresses and loads are all time

dependent.

The displacement in an element is given by


-(8.1)---)(),,(

),,,(

),,,(

),,,(

),,,( )(etUzyxN

tzyxw

tzyxv

tzyxu

tzyxU

Where

U is the vector of displacements,

[N] is the matrix of shape functions and

)(eU is the vector of nodal displacements that is function of time t

The relation between the strains and nodal displacements is given by

-(8.2)---][ )( euB

And stresses as

-(8.3)---]][[][ )( eUBDD

By differentiating Eq.(1) with respect to time, the velocity field can be obtained as

-(8.4)---),,(

),,,(

),,,(

),,,(

),,,(ˆ )(eUzyxN

tzyxw

tzyxv

tzyxu

tzyxU

Where, )(eU is vector of nodal velocities.

To derive the dynamic equations of motion the Lagrange equations are used.

-(8.6)---bygivenfunctionLagrangianis

-(8.5)---0

pTLwhere

Q

R

Q

L

Q

L

dt

d


velocitynodalis

ntdisplacemenodalis U

functionndissipatio theisR

functionpotentialis

energykinetic theisT

U

p

-(8.7)---elementanofenergykineticThe)(

21)( dVUUT

T

V

e

e

functiondampingiswhere

-(8.9)---UR

as writtenbecanelementanoffunctionndissipatioThe

s velocitierelative the toalproportionforcesedissipativofexistence theassumingBy

density theis

eelementofs velocitieof vector theis

element theof volumetheV

-(8.8)---1

element theofenergyPotentialThe

)(

)()(

21(e)

(e)

)(21

e

eel

V

T

V

T

eV S

TT

p

dVU

U

Where

dVUdSUdV

By using equations (1), (2), (3), (7).(8) and (9) the expressions for T, p ,and R can be

written as

-(8.11)---)(..N..N

B

-(8.10)---N

1 V

~

S

~

~

~V1

~21

1

)(

~1 V

~21

1

)(

(e)(e)l

(e)

(e)

tPUdVdSU

UdVBDU

UdVNUTT

Tnel

e

Tl

TT

Tnel

e

Tnel

e

ePP

nel

e

TTnel

e

e


The expressions for T, p , and R can be written as

-(8.17)---..Nforcesbodybyproduced

forcesnodalelementofVector

-(8.16)---..Nforcessurfacebyproduced

forcesnodalelementofVector

-(8.15)---Nmatrixdampingelement

-(8.14)---Bmatrixstiffnesselement

-(8.13)---Nmatrixmassmasselement

followsasdefinedareintegralsinvolve whichmatricesThe

(e)

(e)l

(e)

(e)

(e)

V

~)(

S

~

)(

V

)(

V

)(

V

)(

dVP

dS

P

dVNC

dVBDK

dVNM

Te

B

lT

e

S

Te

Te

Te

The expressions for T, p ,and R can be written as

structure.offorcesnodaledconcentratof vector theisP

vector, velocitynodalglobalis

nt vector,displacemenodalglobal theis

Where

-(8.12)---N

~

~

~

~1 V

~21

1

)(

(e)

C

nel

e

TTnel

e

e

Q

Q

UdVNURR


nel

1e

)(

~

nel

1e

)(

~

nel

1e

)(

~

~~~21

~~~~~21

P

~~~21

structureofmatrixdampingGlobal

structursofmatrixStiffnessGlobal

structursofmatrixmassGlobal

PQQQ

e

e

e

T

TT

T

CC

KK

MM

Where

QCQR

K

QMQT

~

)(nel

1e

)(

~)())()(( vectorload totalGlobal)( tPtPtPtP C

eb

eS

By substituting equations (18) to (20) in (5) ,

(8.21)-----)()()()(~~~~~~~

tPtQKtQCtQM

This is the Dynamic equation of motion in which

nsacclerationodalglobalof vector theis~

Q

Irrespective of the kind of structure it is required to find global Mass matrix, damping

matrix and Stiffness matrix for dynamic analysis.

In the proposed work it is planned to attempt for a static case initially in which case

the equation (21) reduces to

----(8.18)

----(8.19)

----(8.20)


(8.22)-----~~~ staticstataic

PQK

Wherestatic~Q is displacement under the static load

static~P.

Equation (21) results in a set of ordinary differential equations where as equation (22)results

in a set of linear simultaneous equations.

8.5. Finite elements formulation

For the interaction of finite element static or dynamic analysis of a three dimensional rigid

jointed framed structure resting on soil, the following types of elements have been adopted.

1) Soil elements are modeled as eight nodded brick element with three translational

degrees of freedom per node.

2) Mat Foundation is modeled as plate element with five degrees of freedom per node

(three translational and two rotational dof).

3) Columns and beams modeled as one dimensional element with six degrees of

freedom per node (three translational and three rotational dof).

4) In filled walls modeled as plane stress element with two translational degrees of

freedom per node.

For Dynamic analysis following subroutines are developed

Subroutine esm Consistent mass matrix of hexahedron element

Subroutine bmass Consistent mass matrix of beam element

Subroutine wallm Consistent mass matrix of 2D plane stress element

Subroutine platem Consistent mass matrix of plate element.

Two computer programs LDYN and LDYNF have been developed, using FORTRAN, to

conduct linear dynamic soil-structure interaction analysis of a three dimensional frame


resting on raft foundation/ isolated footings. Programs NLDYN and NLDYNF have been

developed to conduct non-linear SSI analyses on structures resting of raft / isolated footings.

The developed programs satisfy the requirements of researchers and as well as requirements

of practicing engineers. Salient features of the programs developed are as follows:

1. The programs can be used to conduct linear / non-linear SSI analyses of structures

resting on raft / isolated footings for dynamic loads which may be transient, pulse or

seismic loads.

2. Time histories of kinematic responses like displacement, velocity and acceleration

along any global co-ordinate can be obtained.

3. Time history response of stress resultants in members of structures, stresses and

strains in soil can de recorded.

4. Maximum value of above mentioned responses and their time of occurrence can also

be recorded.

5. In non-linear analyses failure of soil element, its position and also the load at which it

fails can be obtained.

8.6. Technique of direct method of dynamic analysis

Direct methods model the entire soil-foundation-structure system in a single step

and are more robust than multistep methods, although they are also more computationally

demanding. Because of the non-linearities arising from modeling the frictional links at the

base, direct method of SFSI analysis is used in this study. Newmarks method is employed

to solve the nonlinear finite element equations of motion. Newmark developed a family of

time-stepping methods based on following equations.

UtUtUUU

UUUU

iiiii

iiii

t

tt

1

22

1

11

5.0

1


The parameters and define variation of acceleration over time step and determine the

stability and accuracy characteristics of the method. Typical selection of for is ½ and 1/6 <

<1/4 is satisfactory from all points of view, including that of accuracy. These two

equations, combined with equilibrium equations 21 at the end of time step provide basis for

computing ,1,1,1 and iii UUU at the time i+1from the known ,, and iii UUU at the time i.

Newmark’s method for linear systems:

1. Initial calculations

2.1m

kUUcpU 000

0

2.2 Select t

2.3

mt

ct

kk2

1ˆ

2.4 cmt

a

1

2.5 ctmb

1

22

1

2 Calculation for each time step, i

2.1 iubiuaipip ˆ

2.2k

pu i

i ˆˆ

2.3 iiii utuut

u

21

2.4 iiii uu

tu

tu

2

1112

2.5 iiiiiiiii UUUUUUUUU 111 ,,

3 Repetition of steps 2.1 to 2.5 for next time step.

For time integration the second-order accurate Newark-beta algorithm with

parameters 4/1and2/1 was selected. For linear problems, this method is

unconditionally stable and exhibits no numerical damping.

As mentioned in article 3.4 profile or skyline single array storage scheme is adopted to

save the memory space needed in the storage of the stiffness matrix. In dynamic analysis,


in addition to stiffness matrix, the other matrices stored in compact column matrix are

mass matrix, damping matrix, and tangent stiffness matrix, parameter “a”( defined in steps

1.3 in articles 8.4.1 and 8.4.2) and parameter “b”(defined in steps 1.4 in articles 8.4.1 and

8.4.2). Subroutines have been developed to obtain product of compact column matrices

and vectors which are used in step 2.1 of articles 8.4.1 and 8.4.2. An active column profile

(skyline) solution algorithm is employed in the equation solution module to solve the

equations efficiently.

8.7. Validation of dynamic analysis program for a simplestructure.

To verify the numerical correctness of the computer program developed based on the

proposed methodology, an example of a space frame structure is analyzed by both the

proposed program and a renowned commercial software package. Since the seismic analysis

capacity in the time domain is not yet available in most popularly used commercial software,

the analysis of the example structure is focused on the frequency response of the structure.

The developed program to conduct linear dynamic SSI analysis of structure-raft-soil system

was validated for a simple RCC structure shown in figure 8.1, for a rectangular pulse load of

1000N for 1.25 seconds in time step of .025 seconds. The frequency analysis performed by

using Nastran shows that the fundamental period of the structure is 0.75 s. The fundamental

period found by the developed program is 0.75.seconds, almost same as that from Nastran.

Since the value of a structure’s fundamental period is dependent on the modelling of the

distribution of the stiffness and masses of the structure, the comparison of the results

indicates that the proposed program is reliable. The close agreement of the computational

values also denotes that the numerical procedures used in the proposed model is trustworthy.

8.8. Non-interactive, linear Dynamic analysis for frame withoutand with stiffness of infills.

As results non-interactive analysis of frame is taken as reference to compare the results on

interactive analyses linear dynamic analysis is conducted on the frame shown in figure 8.2.

The following numerical experiments are conducted on frame with fixed support conditions:


(i) Dynamic analysis of three dimensional frame with fixed support under transient loads.

(ii) Dynamic analysis of three dimensional frame with fixed support under seismic load.

8.7.1. Dynamic analysis of three dimensional frames with fixed supportunder transient loads.

The horizontal response of the structure at the top of node “A1” with and without interaction

of infills is shown in figure 8.3. It may be observed from figure 8.3(a) that the period of

vibration is 0.8 for frame without the effect of infiills. But when the contribution of stiffness

of infills is considered the period is reduced to 0.075.

8.7.2. Dynamic analysis of three dimensional frames with fixed supportunder seismic load.

The same structure is subjected to Elcentrino earthquake load in longitudinal direction (X-

direction). The horizontal acceleration of Elcentrino earth quake is shown in figure 8.4.

Figure 8.5 shows response of the structure due to earth quake. The responses of the system in

undamped and damped analysis are close in the absence of stiffness of wall. For the damped

case, damping ratio of 5% is taken by appropriately evaluating the Raleighs damping

coefficients.

But when stiffness of wall is added too much of deviation is observed. The magnitudes of

responses increase further if mass of infills is not considered. Tables 8.1 to 8.6 give details of

peak responses of stress resultants in frame for undamped and damped analyses when

stiffness of wall is not considered. In general for stress resultants for beams in damped

analysis are about 80% of undamped analysis where as for columns it is about 10%.

8.9. Linear dynamic SSI analysis of Structure-raft-soil system.

The structure-raft-soil system under consideration is shown in figure 8.7 and the geometrical

details are given in table 8.7. The structure is mounted on flat soil of 20m thick and is

supported by a bed rock. FEM model is as shown in figure 8.8.

The soil is modeled with 33 X 21 X 7 layers in the longitudinal, transverse and vertical

directions respectively resulting in a total of 4851 brick elements. Mat foundation is modeled


by four nodded plate elements. The number of plate elements used is 135. The number of

beam elements in the longitudinal direction (X-direction) is 80, in transverse (Z-direction) it

is 72 and in vertical (y-direction) is 96.

For the size of soil mass considered (as in static analysis), the yielded the responses as shown

in figure 8.9 and 8.10 in restrained boundary condition of soil mass. The structure-raft-soil

system is in the state of beat (figure 8.9). It may be noted that maximum response in

structure-raft-soil system are approximately 10 times the maximum response in structure-

infill-raft-soil system.

In the analyzing SSI problems it is required to carefully model the unbounded nature of the

underlying semi-infinite soil medium. The unbounded nature of the soil medium requires

special Boundary Conditions (BC) that does not reflect seismic waves into the soil-structure

system. Various models of BC exist that enable the energy transmission. Various boundaries

that are used are transmitting, nonreflecting and silent boundaries. Among them, a type of

transmitting boundary called the plane strain boundary, rather widely employed and

described by Novak et al (1978) is used in this wok. This local boundary has proven to be

very simple to use as well as sufficiently accurate for many practical applications. It is better

suited than the very popular viscous boundary because it has a stiffness (in-phase) part.

The following numerical experiments are conducted on frame with restrained boundary

conditions at the end of soil mass:

1. Linear dynamic SSI analysis of Structure-raft-soil system under rectangular pulse

load for different boundary conditions.

2. Linear dynamic SSI analysis of Structure-raft-soil system under seismic load.

8.8.1. Linear dynamic SSI analysis of Structure-raft-soil system underrectangular pulse load for different boundary conditions.

Comparative dynamic SSI analyses are performed on structure-raft-soil system under

rectangular pulse load for three boundary condition of the soil mass, viz., 1) free boundary

conditions, 2) Restrained boundary conditions and 3) transmitting boundary for various sizes


of soil mass to converge on the size of soil mass that has to be considered in dynamic

analysis. Refer figure 8.11.

In the figure 8.12 the time history of horizontal displacement at top node of frame in non-

interactive undamped analysis are compared with the responses that are obtained with

different boundary condition at the edge of soil mass. And response for different stiffness of

soil is also plotted for comparison. It may be observed that with increase in stiffness of soil

the peak response also increase.

In figure 8.13 the response of the frame-raft-soil system with transmission boundary is

compared with response of frame-raft-soil system having restrained and free boundaries

conditions of the soil mass. It may be observed that the transmitting boundary adopted is

effective in its purpose. Figures 8.14 and 8.15 give the horizontal response at the top floor

node above the ground node “A1”neglecting and considering wall effect respectively.

8.8.2.Linear dynamic SSI analysis of Structure-raft-soil system under seismicload.

The frame-raft-soil system shown in figure 6.7 is analyzed for seismic load subjecting it to El

Centrino earthquake’s ground motion in longitudinal direction (x-direction). Two numerical

experiments are conducted.

In the first experiment in the SSI analysis the effect of wall stiffness is neglected and

following results are plotted.

1. The time history of the horizontal displacement at the node of top floor which is

directly above node “A1” is plotted in figure 8.16 for two cases; undamped and

damped. For the damped case, damping ratio of 5% is taken by appropriately

evaluating the Raleigh’s damping coefficients.

2. The time histories, of longitudinal acceleration, at top node are plotted in figure

8.17. The magnification of acceleration at the top node is considerable.

3. The time history of longitudinal acceleration at node “A1” are plotted in figure 8.18

in which SSI effect on it are apparent. For comparison the acceleration of the earth

quake is also plotted.


4. Though seismic acceleration is applied in longitudinal direction accelerations are

induced in transverse direction and vertical direction clearly indicating three

dimensional effect of SSI. These accelerations are plotted in figures 8.19 and 8.20

In second experiment SSI analysis of structure-Infill-raft-soil system is done considering soft

storey in ground floor. The response of horizontal displacement the node mentioned above is

plotted in the figure 8.21. The acceleration along X,Y and Z direction at the node A1 are

shown in figures 8.23, 8.24 and 8.25 respectively.

Table 8.9 and 8.10 give comparison of stress resultants in soft storey of structure-raft-soil

system and structure-infill-raft-soil system. It may be observed that infills increase axial

forces in interior columns and reduce bending moments in most of the columns.

8.10. Linear dynamic SSI analysis of Structure-isolated footings-soil system.

The structure-isolated footings-soil system under consideration is shown in figure 8.7 except

that it is supported on isolated footings and the geometrical details are given in table 8.8. The

structure is mounted on flat soil of 20m thick and is supported by a bed rock. Finite element

formulation in the SSI analysis of the frame - isolated footings -soil system is shown in figure

8.26. The soil is modelled with 33 X 21 X 7 layers in the longitudinal, transverse and vertical

directions respectively resulting in 4851 brick elements. Each footing is modelled by four

plate elements. The number of plate elements used is 96. The number of beam elements in

the longitudinal direction (X-direction) is 80, in transverse (Z-direction) it is 72 and in

vertical (y- direction) is 96.

Linear SSI analysis is performed with a damping co-efficient of 5% under the seismic load

and some of the results are as follows

1. The time history of the horizontal displacement at the node of top floor which is

directly above node “A1” is plotted in figure 8.27. For comparison time history of

horizontal displacement at the same node obtained from structure-raft-soil interaction

analysis is also plotted.


2. The time history of longitudinal acceleration at node “A1” are plotted in figure

8.28 in which SSI effect on it are apparent.

3. The time histories, of longitudinal acceleration, at top node are plotted in figure

8.29.

4. Though seismic acceleration is applied in longitudinal direction accelerations are

induced in transverse direction and vertical direction clearly indicating three

dimensional effect of SSI. These accelerations are plotted in figures 8.30 and 8.3.

Tables 8.9 and 8.10 give comparison of stress resultants in soft storey of structure-raft-soil

system and structure-infill-raft-soil system. It may be observed that infills increase axial

forces in interior columns and reduce bending moments in most of the columns.

8.11. Conclusions on linear dynamic SSI analyses

Following conclusions are drawn from linear dynamic SSI analyses of frame-raft-soil and

frame-isolated footings-soil systems:

1. The extent of the soil mass to be considered for dynamic analysis is six times the

length and breadth of building. This limit is found to give satisfactory results in

dynamic analysis of SSI problems with the transmitting boundaries proposed by

Novak et al.(1978).

2. The frame and infill interaction is found to reduce displacement and acceleration in

the direction of applied ground motion.

3. The SSI reduced the magnitude of ground acceleration at foundation level. The

magnitude of horizontal motion and acceleration increased with the height of floors.

4. The effect of infill is found to reduce the magnitude of acceleration and horizontal

response in the direction of applied ground movement.

5. In structure-infill-raft-soil system the maximum stress resultants occurred in soft

storey. These maximum values were less than the maximum stress resultants obtained

from structure-raft-soil system. The reason for this may be attributed to conclusion in

point number 4. This indicates that critical values for design are not soft storey stress


resultants in structure-infill-raft-system but maximum stress resultants in structure-

raft-soil system.

6. The spatial effect of three dimensional dynamic SSI is clearly observed from the fact

that accelerations were induced in Y and Z directions though seismic excitation was

applied only in X-direction.

7. It was also observed that the vertical inertial forces induced due to horizontal

acceleration balance for the whole frame.

8. Though the frame and soil mass chosen were same in the case of frame-raft-soil and

frame-isolated footing-soil were same, the effects of size and shape of foundation on

dynamic SSI analyses are clearly visible in the modification of ground acceleration to

different extents (upto ten folds) in the later case. How ever response of horizontal

displacement at top storey was almost same.

9. The maximum stress resultants in frame supported on isolated footings are more than

the maximum stress resultants obtained in frame supported on raft foundation.

8.12. Non-Linear dynamic SSI analysis of Structure-raft-soilsystem.

In most of devastating earthquakes, soil and structures appear as large deformations, which

get into the non-linear phase. Through seismological observation of a reinforced concrete

structure founded on piles in Los Angeles, Sivanovic (2000) considered the non-linear

property of soil to be one of the most significant factors influencing the seismic response of a

structure. In 1980, Roesset indicated that the second element that controls the veracity and

rationality of the analysis of SSI is the non-linearity of the soil. However, because of the

complexity and time- consuming calculation of non-linear phenomena, there has been little

work associated with non-linear property in this subject.

In the present work it is proposed to conduct studies on dynamic analysis of structure-raft-

soil system subjected to seismic load to estimate the realistic time histories of responses, and

maximum force quantities in the structural members, accounting for their three dimensional


behavior and nonlinear constitutive relations of soil. The non-linear model used in present

analysis is hypo elastic model.

8.12.1. Implementation of Hypo-elastic model in dynamic analysis

The loading and unloading, and reloading in hypoelastic model processes by using sign of

trace of the product of the deviatoric stress tensor Sij and increment of the deviatoric strain

tensor dEij . ijijD dES . Refer figure 5.1

Loading and unloading are defined as follows:

Loading: 0dESij , K=P’/( iV/ )

Unloading: 0dESij , K=P’/( iV/ )

Treatment in the computer program for loading and unloading are done as below:

During loading, the bulk modulus is given by K=P’/( iV/ )

During unloading, the bulk modulus is given by K=P’/( iV/ )

For details, articles 5.4 and 5.5 may be referred.

Prior to the dynamic analysis the structure-foundation-soil system is subjected to

static vertical loads for which the structure is designed. This is done to get the stste of

stresses in soil as the constitutive relations of hypoeleastic model depend on state of stress

and strains, and their increments.

8.12.2. Newmark’s method for non-linear systems:

1. Initial calculations

1.1.m

kUUcpU 000

0

1.2. Select t

1.3. cmt

a

1


1.4. ctmb

1

22

1

2. Calculation for each time step, i

2.1. iubiuaipip ˆ

2.2. Determine tangent stiffness matrix ki

2.3.

mt

ct

kk i 2

1ˆ

2.4. Solve for iU from iP and ik

2.5. iiii utuut

u

21

2.6. iiii uu

tu

tu

2

1112

2.7. iiiiiiiii UUUUUUUUU 111 ,,

3. Repetition of steps 2.1 to 2.5 for next time step

For time integration the second-order accurate Newark-beta algorithm with

parameters 4/1and2/1 was selected. For linear problems, this method is

unconditionally stable and exhibits no numerical damping.

However, instability can develop in the presence of non-linearities, which can be

circumvented by reducing the time step and/or adding (stiffness) damping. In general, the time

step should not exceed 10% of the period of the highest relevant frequency in the structure;

otherwise, large phase errors will occur. The phenomenon usually associated with a time step

that is too large is strong oscillatory accelerations leading to numerical instability. To

circumvent this problem the analysis has been repeated with significantly different time steps

and compared the responses. After several trials, the time step was fixed respectively for the

chosen problems.


8.12.3. Results and discussion.

Non-linear dynamic SSI analysis is conducted on structure-raft-soil system subjected

ElCentrino seismic excitation. The analysis was done for time step of 0.01 seconds. As the

digitized data is available in the time interval.0.04 seconds, linear variation of acceleration is

assumed in the time interval of 0.04 seconds to obtain acceleration for every time step of 0.01

seconds. Instability is observed at the end of 4.77 seconds. To check the stability of the hypo

elastic model, non-linear dynamic is conducted for 50% and 25% of El Centrino seismic

excitation. The response of the system for 100%, 50% and 25% seismic excitation is shown in

figure 8.32 which is true to soil nonlinearity. It is observed that with the reduction in magnitude

of seismic acceleration the structure had stability for longer time. Figures 8.33, 8.34 and 8.35

give ground acceleration in X, Y and Z direction respectively.

The maximum values of stress resultants in non-linear SSI analysis are compared with the

maximum values of stress resultants of linear SSI analysis in tables 8.12 and 8.13. Table 8.12

gives member end forces. Table 8.13 gives member end moments.

8.13. Conclusions on non-linear dynamic SSI analyses

Following are conclusions from comparison of linear and non-linear dynamic analyses of

frame-raft-soil system:

1. End forces in beams and shear forces (in X-direction) in columns are lesser in non-

linear SSI analysis when compared to end forces in linear SSI analysis.

2. Where as axial forces and shear forces in lateral directions (Z-direction) increase in

non-linear analysis with respect to linear analysis.

3. One more interesting point needs to be observed in non-linear SSI analysis is the axial

forces in exterior columns are lesser than axial forces in interior columns which is

exactly reverse trend that is found in linear SSI analysis.


Table 8.1: Maximum response values of Stress resultants (translatory) inbeams(X-direction) in Dynamic Non-interactive undamped analysis

Un-damped analysis Damped analysisDamped

VsUndamped

Mem

ber

No

X-c

oord

inat

e

Y-c

oord

inat

e

Z-c

oord

inat

ePo

siti

on o

fm

emeb

er

Fx1 Fy1 Fz1 Fx2 Fy2 Fz2

Fx2/F

x 1

Fx2/F

x 1

Fx2/F

x 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

64 23.5 40 -114.46 -820.32 11.81 -102.97 -727.24 -9.41 0.900 0.887 -0.7971

69 23.5 40 -49.81 -626.48 8.83 -44.89 -555.82 -7.04 - -0.797 0.887

69 23.5 40 0 -655.32 9.37 0 -581.19 -7.46 0.901 0.887 -0.7972

74 23.5 40 20.37 -663.89 -18.87 18.36 -596.02 -16.3 - -0.798 0.888

3 74 23.5 40 1.65 -541.24 -14.09 1.1 -486.09 -12.17 - 0.887 -0.796

64 27 40 0 -550.33 -14.94 0 -494.16 -12.9 - -0.796 0.8876

69 27 40 6.82 -406.64 -23.96 5.86 -368.95 -21.14 0.901 0.898 0.864

69 27 40 4.71 -349.87 -17.89 -3.72 -317.61 -15.79 - 0.864 0.8987

74 27 40 0 -348.12 -18.97 0 -315.94 -16.74 0.667 0.898 0.864

8 74 27 40 36.57 149.64 -28.48 33.15 -130.15 -24.16 - 0.864 0.898

64 30.5 40 17.99 136.84 -21.26 16.2 -121.77 -18.04 - 0.898 0.86311

69 30.5 40 0 127.63 -22.54 0 -111.74 -19.12 - 0.864 0.898

69 30.5 40 -115.94 -869.99 9.74 -102.42 -779.64 -7.75 0.859 0.907 0.88212

74 30.5 40 -50.8 -664.1 7.95 -44.77 -595.64 -6.33 - 0.882 0.908

13 74 30.5 40 0 -694.8 8.15 0 -622.93 -6.48 -0.790 0.908 0.883

64 34 40 33.9 -705.58 -15.52 30.08 -641.86 -13.4 - 0.883 0.90816

69 34 40 4.61 -573.22 -12.66 4.08 -521.87 -10.93 - 0.908 0.882

69 34 40 0 -583.78 -12.97 0 -531.28 -11.19 - 0.882 0.90817

74 34 40 21.57 -416.92 -19.7 19.55 -377.28 -17.37 0.906 -0.870 0.848

18 74 34 40

Ext

erio

r B

eam

s

-2.87 -360.03 -16.07 2.03 -325.82 -14.18 -1.000 0.848 -0.893

64 23.5 45 0 -357.75 -16.47 0 -323.74 -14.52 0.901 -0.890 0.84921

69 23.5 45 54.03 164.55 -23.41 49.2 130.57 -19.86 - 0.849 0.909

69 23.5 45 25.71 148.82 -19.09 23.42 -119.97 -16.2 - -0.875 0.84822

74 23.5 45 0 139.54 -19.56 0 111.21 -16.59 - 0.848 -0.900

23 74 23.5 45 -114.46 -820.32 11.81 -102.97 -727.24 -9.41 0.883 0.896 -0.796

64 27 45 -49.81 -626.48 8.83 -44.89 -555.82 -7.04 - -0.795 0.89626

69 27 45 0 -655.32 9.37 0 -581.19 -7.46 0.881299 0.896913 -0.79623

69 27 45 20.37 -663.89 -18.87 18.36 -596.02 -16.3 - -0.79554 0.89728827

74 27 45 1.65 -541.24 -14.09 1.1 -486.09 -12.17 - 0.89656 -0.79509

28 74 27 45 0 -550.33 -14.94 0 -494.16 -12.9 - -0.79451 0.896811

64 30.5 45 6.82 -406.64 -23.96 5.86 -368.95 -21.14 0.887316 0.909691 0.86340231

69 30.5 45 4.71 -349.87 -17.89 -3.72 -317.61 -15.79 - 0.863231 0.90982

69 30.5 45 0 -348.12 -18.97 0 -315.94 -16.74 0.885033 0.910418 0.86334932

74 30.5 45 36.57 149.64 -28.48 33.15 -130.15 -24.16 - 0.863564 0.910772

33 74 30.5 45 17.99 136.84 -21.26 16.2 -121.77 -18.04 - 0.910069 0.86276

64 34 45 0 127.63 -22.54 0 -111.74 -19.12 - 0.863314 0.91031236

69 34 45 -115.94 -869.99 9.74 -102.42 -779.64 -7.75 0.906351 0.904922 0.881726

69 34 45 -50.8 -664.1 7.95 -44.77 -595.64 -6.33 - 0.881661 0.90512737

74 34 45 0 -694.8 8.15 0 -622.93 -6.48 -0.70732 0.90498 0.88239

38 74 34 45

Inte

rior

Bea

ms

33.9 -705.58 -15.52 30.08 -641.86 -13.4 - 0.881777 0.90549


Table 8.2 Maximum response values of Stress resultants (moments) inbeams(X-direction) in Dynamic Non-interactive analysis


VsUndamped

Mem

ber

No

X-c

oord

inat

e

Y-c

oord

inat

e

Z-c

oord

inat

e

Posi

tion

of m

emeb

er

Mx1 My1 Mz1 Mx2 My2 Mz2

Mx 2

/ Mx 1

My 2

/ My 1

Mz 2

/ Mz 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

64 23.5 40 0 -32.6 -2273.19 0 25.97 -2015.66 - -0.797 0.8871

69 23.5 40 0 -26.45 -1840.92 0 21.09 -1633.04 - -0.797 0.887

69 23.5 40 0 -21.58 -1541.83 0 17.22 -1368.51 - -0.798 0.8882

74 23.5 40 0 -22.55 -1603.1 0 17.97 -1423.09 - -0.797 0.888

3 74 23.5 40 0 -23.44 -1642.47 0 18.66 -1457.14 - -0.796 0.887

64 27 40 0 52.06 -1823.89 0 44.96 -1637.96 - 0.864 0.8986

69 27 40 0 42.27 -1508.09 0 36.52 -1354.62 - 0.864 0.898

69 27 40 0 34.48 -1343.42 0 29.79 -1207 - 0.864 0.8987

74 27 40 0 35.97 -1375.26 0 31.07 -1235.94 - 0.864 0.899

8 74 27 40 0 37.35 -1380 0 32.26 -1239.57 - 0.864 0.898

64 30.5 40 0 66.1 -1123.48 0 58.33 -1019.77 - 0.882 0.90811

69 30.5 40 0 53.68 -922.2 0 47.39 -837.49 - 0.883 0.908

69 30.5 40 0 43.78 -873.81 0 38.65 -793.68 - 0.883 0.908

74 30.5 40 0 45.68 -888.03 0 40.31 -806.89 - 0.882 0.909

13 74 30.5 40 0 47.43 -874.46 0 41.85 -794.02 - 0.882 0.908

64 34 40 0.01 78.57 426.87 -0.01 66.65 -381.21 -1.000 0.848 -0.89316

69 34 40 -0.01 63.83 -310.01 0.01 54.15 -282.04 -1.000 0.848 0.910

69 34 40 0 52.03 -342.38 0 44.15 -311.26 - 0.849 0.90917

74 34 40 0 54.28 -340.71 0 46.05 -310.1 - 0.848 0.910

18 74 34 40

Ext

erio

r B

eam

s

0 56.35 314.9 0 47.81 -283.51 - 0.848 -0.900

64 23.5 45 0 -26.08 -2410.57 0 20.73 -2160.25 - -0.795 0.89621

69 23.5 45 0 -22.64 -1951.9 0 18.01 -1750.43 - -0.795 0.897

69 23.5 45 0 -19.71 -1634.08 0 15.68 -1466.24 - -0.796 0.89722

74 23.5 45 0 -20.06 -1698.92 0 15.95 -1524.46 - -0.795 0.897

23 74 23.5 45 0 -20.39 -1741.17 0 16.2 -1561.5 - -0.795 0.897

64 27 45 0 41.53 -1939.01 0 35.85 -1764.15 - 0.863 0.91026

69 27 45 0 36.06 -1601.39 0 31.14 -1457.67 - 0.864 0.910

69 27 45 0 31.37 -1421.97 0 27.09 -1295.09 - 0.864 0.91127

74 27 45 0 31.92 -1456.64 0 27.56 -1326.77 - 0.863 0.911

28 74 27 45 0 32.41 -1463.63 0 27.98 -1332.36 - 0.863 0.910

64 30.5 45 0 52.73 -1151.44 0 46.49 -1042.2 - 0.882 0.90531

69 30.5 45 0 45.79 -945.68 0 40.38 -856.68 - 0.882 0.906

69 30.5 45 0 39.84 -899.38 0 35.13 -814.38 - 0.882 0.90532

74 30.5 45 0 40.53 -913.26 0 35.74 -827.24 - 0.882 0.906

33 74 30.5 45 0 41.17 -898.53 0 36.29 -813.52 - 0.881 0.905

64 34 45 0.01 62.64 469.59 0 53.14 -373.56 0.000 0.848 -0.79636

69 34 45 -0.01 54.41 340.64 0 46.15 -276.32 0 0.84819 -0.81118

69 34 45 0 47.32 369.44 0 40.14 -306.93 - 0.848 -0.83137

74 34 45 0 48.14 362.26 0 40.84 -305.42 - 0.848359 -0.8431

38 74 34 45

Inte

rior

Bea

ms

0 48.89 344.69 0 41.47 -278.49 - 0.848 -0.808


Table 8.3:Maximum response values of Stress resultants (translatory) in beams(Z-direction) in Dynamic Non-interactive analysis

Non-interactiveAnalsysis

Uncoupled VsNon-interactive

Coupled VsNon-nteractive

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

of

mem

eber


Fx2/F

x 1

Fx2/F

x 1

Fx2/F

x 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)64 23.5 40 -12.1 3.53 -13.01 9.57 3.25 10.59 -0.791 0.921 -0.814

8164 23.5 45 -0.19 30.97 4.64 0.16 -25.23 3.96 -0.842 -0.815 0.853

82 64 23.5 45 -22.13 2.5 0 17.47 2.5 0 -0.789 1.000 -

64 27 40 0 -9.88 2.66 0 7.85 2.54 - -0.795 0.95584

64 27 45 18.88 4.41 21.39 16.35 3.73 18.56 0.866 0.846 0.868

85 64 27 45 0.18 -50.96 6.76 0.14 -44.22 5.13 0.778 0.868 0.759

64 30.5 40 34.42 2.5 0 29.8 2.5 0 0.866 1.000 -87

64 30.5 45 0 15.73 3.15 0 13.58 2.87 - 0.863 0.911

88 64 30.5 45 24.24 5.13 27.14 21.2 4.1 24.15 0.875 0.799 0.890

64 34 40 -0.19 -64.67 8.6 0.12 -57.57 6.06 -0.632 0.890 0.70590

64 34 45 44.09 2.5 0 38.57 2.5 0 0.875 1.000 -

91 64 34 45

Ext

erio

r Bea

ms

0 19.97 3.49 0 17.6 3.08 - 0.881 0.883

69 23.5 40 26.74 4.38 32.41 22.82 3.66 27.52 0.853 0.836 0.84993

69 23.5 45 0.13 -77.23 6.37 -0.08 -65.59 4.72 -0.615 0.849 0.741

94 69 23.5 45 48.83 2.5 0 41.67 2.5 0 0.853 1.000 -

69 27 40 0 23.71 4.08 0 20.12 3.46 - 0.849 0.84896

69 27 45 3.05 2.7 -18.79 -2.41 2.67 15.16 -0.790 0.989 -0.807

97 69 27 45 0.1 46.77 2.59 0.08 -37.73 2.51 0.800 -0.807 0.969

69 30.5 40 4.9 2.5 0 -3.86 2.5 0 -0.788 1.000 -99

69 30.5 45 0 -6.44 2.19 0 5.13 2.16 - -0.797 0.986

100 69 30.5 45 -4.77 2.84 30.5 -4.13 2.76 26.41 0.866 0.972 0.866

69 34 40 0.1 -75.89 2.92 0.08 -65.73 2.73 0.800 0.866 0.935102

69 34 45 -7.62 2.5 0 -6.59 2.5 0 0.865 1.000 -

103 69 34 45

Inte

rior B

eam

s

0 10.26 2.29 0 8.86 2.21 - 0.864 0.965

74 23.5 40 -6.11 2.99 38.71 -5.34 2.83 34.33 0.874 0.946 0.887105

74 23.5 45 -0.11 -96.34 3.3 0.07 -85.44 2.91 -0.636 0.887 0.882

106 74 23.5 45 -9.75 2.5 0 -8.53 2.5 0 0.875 1.000 -

74 27 40 0 13.03 2.35 0 11.49 2.25 - 0.882 0.957108

74 27 45 -6.92 2.86 46.13 -5.88 2.73 39.16 0.850 0.955 0.849

109 74 27 45 0.06 -114.79 2.92 -0.03 -97.45 2.62 -0.500 0.849 0.897

74 30.5 40 -11.01 2.5 0 -9.36 2.5 0 0.850 1.000 -111

74 30.5 45 0 15.48 2.46 0 13.13 2.32 - 0.848 0.943

112 74 30.5 45 -0.55 2.52 -18.01 0.43 2.51 14.55 -0.782 0.996 -0.808

74 34 40 0.11 44.52 2.13 0.09 -35.97 2.12 0.818 -0.808 0.995114

74 34 45 -0.76 2.5 0 0.59 2.5 0 -0.776 1.000 -

115 74 34 45

Inte

rior B

eam

s

0 -6.8 2.09 0 5.4 2.09 - -0.794 1.000


Table 8.4: Maximum response values of Stress resultants (moments)in beams(Z-direction) in Dynamic Non-interactive analysis


VsUndamped

Mem

ber

No

X-c

oord

inat

e

Y-c

oord

inat

e

Z-c

oord

inat

e

Posi

tion

of m

emeb

erMx1 My1 Mz1 Mx2 My2 Mz2

Mx 2

/ Mx 1

My 2

/ My 1

Mz 2

/ Mz 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)64 23.5 40 -0.19 30.97 4.64 0.16 -25.23 3.96 -0.842 -0.815 0.853

8164 23.5 45 0.19 34.09 -4.72 -0.16 -27.73 -4.24 -0.842 -0.813 0.898

82 64 23.5 45 0 -9.88 2.66 0 7.85 2.54 - -0.795 0.955

64 27 40 0.18 -50.96 6.76 0.14 -44.22 5.13 0.778 0.868 0.75984

64 27 45 -0.18 -55.98 -6.76 -0.14 -48.57 -5.66 0.778 0.868 0.837

85 64 27 45 0 15.73 3.15 0 13.58 2.87 - 0.863 0.911

64 30.5 40 -0.19 -64.67 8.6 0.12 -57.57 6.06 -0.632 0.890 0.70587

64 30.5 45 0.19 -71.04 -8.52 -0.12 -63.21 -6.63 -0.632 0.890 0.778

88 64 30.5 45 0 19.97 3.49 0 17.6 3.08 - 0.881 0.883

64 34 40 0.13 -77.23 6.37 -0.08 -65.59 4.72 -0.615 0.849 0.74190

64 34 45 -0.13 -84.81 -7.13 0.08 -72.02 -5.53 -0.615 0.849 0.776

91 64 34 45

Ext

erio

r Bea

ms

0 23.71 4.08 0 20.12 3.46 - 0.849 0.848

69 23.5 40 0.1 46.77 2.59 0.08 -37.73 2.51 0.800 -0.807 0.96993

69 23.5 45 -0.1 47.18 -2.58 -0.08 -38.08 -2.45 0.800 -0.807 0.950

94 69 23.5 45 0 -6.44 2.19 0 5.13 2.16 - -0.797 0.986

69 27 40 0.1 -75.89 2.92 0.08 -65.73 2.73 0.800 0.866 0.93596

69 27 45 -0.1 -76.59 -2.97 -0.08 -66.34 -2.65 0.800 0.866 0.892

97 69 27 45 0 10.26 2.29 0 8.86 2.21 - 0.864 0.965

69 30.5 40 -0.11 -96.34 3.3 0.07 -85.44 2.91 -0.636 0.887 0.88299

69 30.5 45 0.11 -97.23 -3.33 -0.07 -86.24 -2.85 -0.636 0.887 0.856

100 69 30.5 45 0 13.03 2.35 0 11.49 2.25 - 0.882 0.957

69 34 40 0.06 -114.79 2.92 -0.03 -97.45 2.62 -0.500 0.849 0.897102

69 34 45 -0.06 -115.86 -3.06 0.03 -98.36 -2.68 -0.500 0.849 0.876

103 69 34 45

Inte

rior B

eam

s

0 15.48 2.46 0 13.13 2.32 - 0.848 0.943

74 23.5 40 0.11 44.52 2.13 0.09 -35.97 2.12 0.818 -0.808 0.995105

74 23.5 45 -0.11 45.54 -2.12 -0.09 -36.79 -2.12 0.818 -0.808 1.000

106 74 23.5 45 0 -6.8 2.09 0 5.4 2.09 - -0.794 1.000

74 27 40 0.11 -72.32 2.13 0.08 -62.66 2.12 0.727 0.866 0.995108

74 27 45 -0.11 -73.97 -2.13 -0.08 -64.08 -2.12 0.727 0.866 0.995

109 74 27 45 0 10.81 2.09 0 9.33 2.09 - 0.863 1.000

74 30.5 40 -0.11 -91.82 2.14 0.07 -81.46 2.12 -0.636 0.887 0.991111

74 30.5 45 0.11 -93.91 -2.14 -0.07 -83.32 -2.12 -0.636 0.887 0.991

112 74 30.5 45 0 13.73 2.09 0 12.1 2.09 - 0.881 1.000

74 34 40 0.07 -109.41 2.12 -0.04 -92.89 2.11 -0.571 0.849 0.995114

74 34 45 -0.07 -111.91 -2.12 0.04 -95.01 -2.11 -0.571 0.849 0.995

115 74 34 45

Inte

rior B

eam

s

0 16.3 2.1 0 13.83 2.09 - 0.848 0.995


Table 8.5: Maximum response values of Stress resultants (translatory) inColumns in Dynamic Non-interactive analysis


VsUndamped

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

ofm

emeb

erFx1 Fy1 Fz1 Fx2 Fy2 Fz2

Fx2/F

x1 Fx

2/Fx

1 Fx2/F

x1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)153 64 20 40 -2045.41 -846.71 -0.53 -1843.62 -752.4 0.48 0.90 0.89 -0.91

154 69 20 40 381.67 -1049.49 0.12 343.45 -932.14 -0.1 0.90 0.89 -0.83

155 74 20 40 -24.54 -1024.01 -0.02 -21.79 -909.67 -0.01 0.89 0.89 0.50

159 64 23.5 40 -1221.5 -619.91 -1.03 -1111.93 -547.43 0.83 0.91 0.88 -0.81

160 69 23.5 40 188.77 -965.93 0.19 171.59 -855.86 -0.16 0.91 0.89 -0.84

161 74 23.5 40 4.5 -926.11 0.02 3.85 -820.53 0.01 0.86 0.89 0.50

165 64 27 40 -561.69 -442.83 -1.62 -510.35 -401.26 1.13 0.91 0.91 -0.70

166 69 27 40 -68.89 -682.92 0.31 61 -619.66 -0.21 -0.89 0.91 -0.68

167 74 27 40 13.44 -665.82 -0.02 11.77 -604.15 0.01 0.88 0.91 -0.50

171 64 30.5 40 151.72 205.39 -2.28 -135.41 -181.74 1.52 -0.89 -0.88 -0.67

172 69 30.5 40 -13.89 -345.98 -0.44 -9.43 -314.68 -0.28 0.68 0.91 0.64

173 74 30.5 40

Ext

erio

r col

umns

11.27 -343.68 -0.02 10.04 -312.64 0.01 0.89 0.91 -0.50

177 64 20 45 -2143.88 -902.98 -0.37 -1936.98 -796.64 0.34 0.90 0.88 -0.92

178 69 20 45 400.54 -1117.93 0.08 361.18 -985.79 -0.07 0.90 0.88 -0.88

179 74 20 45 -26.08 -1090.83 -0.01 -23.24 -961.93 0.01 0.89 0.88 -1.00

183 64 23.5 45 -1270.48 -663.04 -0.83 -1154.14 -601.2 0.67 0.91 0.91 -0.81

184 69 23.5 45 194.43 -1026.65 0.15 177.02 -932.65 -0.13 0.91 0.91 -0.87

185 74 23.5 45 4.63 -984.54 0.01 4.09 -894.34 0.01 0.88 0.91 1.00

189 64 27 45 580.45 -464.61 -1.32 -508.5 -423.77 0.92 -0.88 0.91 -0.70

190 69 27 45 -75.74 -716.72 0.25 -59.21 -654.26 -0.17 0.78 0.91 -0.68

191 74 27 45 15.01 -698.45 -0.01 13.41 -637.62 0.01 0.89 0.91 -1.00

195 64 30.5 45 165.51 227.96 -1.71 -131.65 179.52 1.15 -0.80 0.79 -0.67

196 69 30.5 45 -16.66 379.57 -0.33 -12.31 -308.99 -0.21 0.74 -0.81 0.64

197 74 30.5 45

Inte

rior C

olum

ns

12.56 375.58 -0.02 11.27 -307.19 0.01 0.90 -0.82 -0.50


Table 8.6: Maximum response values of Stress resultants (moments) in Columns in Dynamic Non-interactive analysis


VsUndamped

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

ofm

emeb

erMx1 My1 Mz1 Mx2 My2 Mz2

Mx 2

/

Mx 1

My 2

/

My 1

Mz 2

/

Mz 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)153 64 20 40 0.13 0.84 -1730.94 -0.1 -0.72 -1537.76 -0.77 -0.86 0.89

154 69 20 40 0.07 -0.19 -1968.61 -0.05 0.16 -1748.44 -0.71 -0.84 0.89

155 74 20 40 0.08 0.03 -1939.36 -0.06 0.02 -1722.65 -0.75 0.67 0.89

159 64 23.5 40 -0.13 1.64 -1037.95 -0.11 -1.35 -918.73 0.85 -0.82 0.89

160 69 23.5 40 -0.07 0.31 -1674.76 -0.06 0.26 -1485.15 0.86 0.84 0.89

161 74 23.5 40 -0.08 -0.03 -1597.17 -0.06 -0.02 -1416.43 0.75 0.67 0.89

165 64 27 40 -0.11 2.76 -706.5 0.08 -1.94 -639.35 -0.73 -0.70 0.90

166 69 27 40 -0.06 -0.53 -1156.22 0.04 0.36 -1048.69 -0.67 -0.68 0.91

167 74 27 40 -0.07 0.03 -1121.68 0.05 -0.02 -1017.33 -0.71 -0.67 0.91

171 64 30.5 40 -0.11 3.63 296.84 0.07 -2.44 -253.83 -0.64 -0.67 -0.86

172 69 30.5 40 -0.06 0.69 563.63 0.03 0.45 -508.79 -0.50 0.65 -0.90

173 74 30.5 40

Ext

erio

r col

umns

-0.07 0.04 554.58 0.04 -0.02 -501.37 -0.57 -0.50 -0.90

177 64 20 45 0.1 0.52 -1844.38 -0.08 -0.47 -1626.54 -0.80 -0.90 0.88

178 69 20 45 0.06 -0.11 -2096.27 -0.05 0.1 -1848.18 -0.83 -0.91 0.88

179 74 20 45 0.07 0.02 -2065.13 -0.05 -0.01 -1820.78 -0.71 -0.50 0.88

183 64 23.5 45 -0.1 -1.32 -1116.03 -0.08 -1.08 -1010.94 0.80 0.82 0.91

184 69 23.5 45 -0.06 0.25 -1782.23 -0.05 0.2 -1618.73 0.83 0.80 0.91

185 74 23.5 45 -0.07 -0.02 -1700.86 -0.06 -0.02 -1544.56 0.86 1.00 0.91

189 64 27 45 -0.08 2.21 -730.11 0.06 -1.56 -667.66 -0.75 -0.71 0.91

190 69 27 45 -0.06 -0.42 -1207.91 0.04 0.29 -1103.49 -0.67 -0.69 0.91

191 74 27 45 -0.06 0.02 -1170.23 0.04 -0.02 -1069.33 -0.67 -1.00 0.91

195 64 30.5 45 -0.09 2.84 333.56 0.05 -1.91 260.88 -0.56 -0.67 0.78

196 69 30.5 45 -0.06 -0.54 620.64 0.03 0.35 -498.66 -0.50 -0.65 -0.80

197 74 30.5 45

Inte

rior C

olum

ns

-0.06 0.03 609.95 0.03 -0.02 -491.6 -0.50 -0.67 -0.81


Table 8.7: Details of SSI modelSl no Structure Component Details

No. of storey 5No. of bays 5X3Storey height 3.5mBay width 5mBeam size 0.3X0.6mColumn size 0.4X0.4

1 Frame

Raft size 25.0X15.0X0.42 Soil Soil mass 153.0mX95.0mX20.0m3 Elastic modulus of soil 1.33X107 N/m2

4 Poissons ratio of soil 0.455 Bulk modulus of concrete 6.1X106 N/m2

6 Elastic modulus of concrete 1.4X1010 N/m2

Table 8.8 Details of the structure modeled

Sl no Structure Component Details

No. of storey 5

No. of bays 5X3

Storey height 3.5m

Bay width 5m

Beam size 0.3X0.6m

Column size 0.4X0.4 m

1 Frame

Footing size 2.0 X 2.0 X0.4 m

2 soil Soil mass 153.0 X 95.0 X 20.0 m

3 Elastic modulus of soil 1.33X107 N/m2

4 Poisson’s ratio of soil 0.45

5 Bulk modulus of concrete 6.1X106 N/m2

6 Elastic modulus of concrete 1.4X1010 N/m2


Table 8.9: Stress resultants in structural members of soft storey in frame-raft-soilDynamic interaction analysis

Experiment 1 (without effectof infills)

Experiment 2 (with effect ofinfills)

Experiment 2/Experiment 1

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

of

mem

eber

Fx1 Fy1 MZ1 Fx2 Fy2 MZ2

Fx3/F

x 1

Fy3/F

y 1

Mz 2

/ Mz 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

1 64 23.5 40 23.9 121.48 334.5 -21.03 61.52 165.89 -0.880 0.506 0.496

1 69 23.5 40 20.72 103.78 260.41 21.03 60.42 -130.94 1.015 0.582 -0.503

2 69 23.5 40 15.57 102.67 254.02 -12.52 45.61 107.26 -0.804 0.444 0.422

2 74 23.5 40 13.4 85.79 246.81 12.52 44.81 -110.35 0.934 0.522 -0.447

3 74 23.5 40 0 107.59 264.81 -0.46 48.84 117.94 - 0.454 0.445

21 64 23.5 45 24.42 132.73 363.87 -23.74 66.47 -185.13 -0.972 0.501 -0.509

21 69 23.5 45 18.37 108.9 287.26 23.74 69.79 -151.33 1.292 0.641 -0.527

22 69 23.5 45 14.94 106.68 262.05 -13.38 48.79 -116.64 -0.896 0.457 -0.445

22 74 23.5 45 11.11 84.86 258.92 13.38 51.2 -126.86 1.204 0.603 -0.490

23 74 23.5 45

Bea

ms

in X

-dire

ctio

n

0 112.97 278.25 -0.48 52.41 -129.1 - 0.464 -0.464

81 64 23.5 40 6.04 4.69 7.36 -4.78 2.77 3.06 -0.791 0.591 0.416

81 64 23.5 45 6.96 5.11 3.67 4.78 2.78 -2.47 0.687 0.544 -0.673

82 64 23.5 45 8.27 2.5 4.63 -7.44 2.57 2.32 -0.900 1.028 0.501

93 69 23.5 40 3.03 4.68 6.92 -2.86 2.59 2.36 -0.944 0.553 0.341

93 69 23.5 45 3.3 4.46 3.98 2.86 2.6 -2.29 0.867 0.583 -0.575

94 69 23.5 45 4.71 2.5 4 -4.77 2.55 2.23 -1.013 1.020 0.558

105 74 23.5 40 1.29 3.48 4.32 -1.13 2.54 2.22 -0.876 0.730 0.514

105 74 23.5 45 1.4 3.39 0.58 1.13 2.54 -2.18 0.807 0.749 -3.759

106 74 23.5 45

Bea

ms

in Z

-dire

ctio

n

2.16 2.5 2.94 -1.84 2.52 2.16 -0.852 1.008 0.735

153 64 20 40 344.98 130.62 277.91 203.18 137.38 252.77 0.589 1.052 0.910

153 64 23.5 40 262.18 116.18 179.27 -203.18 -137.38 228.07 -0.775 -1.182 1.272

154 69 20 40 16.04 162.3 314.18 -72.58 158.3 283.82 -4.525 0.975 0.903

154 69 23.5 40 19.09 143.86 253.85 72.58 -158.3 270.23 3.802 -1.100 1.065

155 74 20 40 13.25 158.13 304.53 -27.57 155.97 281.03 -2.081 0.986 0.923

155 74 23.5 40 10.52 140.13 248.93 27.57 -155.97 264.85 2.621 -1.113 1.064

177 64 20 45 405.75 148 312.89 254.3 -157.66 -290.87 0.627 -1.065 -0.930

177 64 23.5 45 288.42 129.39 205.09 -254.3 157.66 -260.96 -0.882 1.218 -1.272

178 69 20 45 31.91 180.94 350.08 -88.9 -178.64 -319.2 -2.786 -0.987 -0.912

178 69 23.5 45 40.97 156.99 283.2 88.9 178.64 -306.03 2.170 1.138 -1.081

179 74 20 45 12.12 176.09 339.21 -38.87 -175.9 -315.96 -3.207 -0.999 -0.931

179 74 23.5 45

Col

umns

9.74 153.21 277.11 38.87 175.9 -299.67 3.991 1.148 -1.081


Table 8.10 Comparison of maximum end forces in Ground floor structural members inlinear dynamic analysis of frame-isolated footings-soil and frame-raft-soil system

Structure on Footing Structure on raft Footing Vs Raft

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

of

mem

eber


Fx2/F

x 1

Fx2/F

x 1

Fx2/F

x 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

1 64 23.5 40 38.8 113.2 4.49 23.9 121.48 2.8 1.623 0.932 1.6041 69 23.5 40 -38.8 -108.2 -4.49 20.72 103.78 2.48 -1.873 -1.043 -1.8102 69 23.5 40 28.43 84.19 3.35 15.57 102.67 2.09 1.826 0.820 1.6032 74 23.5 40 -28.43 -79.19 -3.35 13.4 85.79 1.84 -2.122 -0.923 -1.8213 74 23.5 40 0 86.74 3.55 0 107.59 2.22 - 0.806 1.599

21 64 23.5 45 40.34 125.31 3.68 24.42 132.73 2.3 1.652 0.944 1.60021 69 23.5 45 -40.34 -120.31 -3.68 18.37 108.9 2.03 -2.196 -1.105 -1.81322 69 23.5 45 30.15 89.89 3.01 14.94 106.68 1.87 2.018 0.843 1.61022 74 23.5 45 -30.15 -84.89 -3.01 11.11 84.86 1.65 -2.714 -1.000 -1.82423 74 23.5 45

Bea

ms

in X

-dire

ctio

n

0 89.58 3.08 0 112.97 1.92 - 0.793 1.60481 64 23.5 40 -52.52 23.53 -5.19 6.04 4.69 2.87 -8.695 5.017 -1.80881 64 23.5 45 52.52 24.48 5.19 6.96 5.11 3.27 7.546 4.791 1.58782 64 23.5 45 72.02 2.5 0 8.27 2.5 0 8.709 1.000 -93 69 23.5 40 31.67 15.9 -7.35 3.03 4.68 4.05 10.452 3.397 -1.81593 69 23.5 45 -31.67 16.91 7.35 3.3 4.46 4.6 -9.597 3.791 1.59894 69 23.5 45 41.72 2.5 0 4.71 2.5 0 8.858 1.000 -

105 74 23.5 40 12.54 7.64 -7.06 1.29 3.48 3.89 9.721 2.195 -1.815105 74 23.5 45 -12.54 8.03 7.06 1.4 3.39 4.43 -8.957 2.369 1.594106 74 23.5 45

Bea

ms

in Z

-dire

ctio

n

16.79 2.5 0 2.16 2.5 0 7.773 1.000 -153 64 20 40 299.64 129.44 47.83 344.98 130.62 3.15 0.869 0.991 15.184153 64 23.5 40 -299.64 -129.44 -47.83 262.18 116.18 4.52 -1.143 -1.114 -10.582154 69 20 40 70.51 142.42 -29.87 16.04 162.3 1.21 4.396 0.878 -24.686154 69 23.5 40 -70.51 -142.42 29.87 19.09 143.86 1.12 -3.694 -0.990 26.670155 74 20 40 -32.84 132.7 -11.15 13.25 158.13 0.33 -2.478 0.839 -33.788155 74 23.5 40 32.84 -132.7 11.15 10.52 140.13 0.37 3.122 -0.947 30.135177 64 20 45 326.64 137.26 20.92 405.75 148 0.82 0.805 0.927 25.512177 64 23.5 45 -326.64 -137.26 -20.92 288.42 129.39 0.86 -1.133 -1.061 -24.326178 69 20 45 -49.73 153.1 12.7 31.91 180.94 0.25 -1.558 0.846 50.800178 69 23.5 45 49.73 -153.1 -12.7 40.97 156.99 0.27 1.214 -0.975 -47.037179 74 20 45 20.08 139.9 4.76 12.12 176.09 0.13 1.657 0.794 36.615179 74 23.5 45

Col

umns

-20.08 -139.9 -4.76 9.74 153.21 0.15 -2.062 -0.913 -31.733


Table 8.11 Comparison of maximum end moments in Ground floor structural membersin dynamic analysis of frame-isolated footings-soil and frame-raft-soil system

Structure on Footing Structure on raft Footing Vs Raft

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

of

mem

eber


Mx 2

/ Mx 1

My 2

/ My 1

Mz 2

/ Mz 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

1 64 23.5 40 -0.03 -12.37 310.93 0.01 6.82 334.5 - -1.81 0.931 69 23.5 40 0.03 -10.07 242.58 0.02 5.56 260.41 - -1.81 0.932 69 23.5 40 0.03 -8.2 205.4 0.01 4.51 254.02 - -1.82 0.812 74 23.5 40 -0.03 -8.56 205.43 0.01 4.71 246.81 - -1.82 0.833 74 23.5 40 -0.03 -8.87 212.67 0.01 4.9 264.81 - -1.81 0.80

21 64 23.5 45 -0.01 -9.85 344.35 0 5.42 363.87 - -1.82 0.9521 69 23.5 45 0.01 -8.57 269.7 0.01 4.72 287.26 - -1.82 0.9422 69 23.5 45 0.01 -7.45 214.41 0 4.1 262.05 - -1.82 0.8222 74 23.5 45 -0.01 -7.58 222.56 0 4.17 258.92 - -1.82 0.8623 74 23.5 45

Bea

ms

in X

-dire

ctio

n

0.01 -7.69 219.78 0.01 4.24 278.25 - -1.81 0.7981 64 23.5 40 0.05 12.38 63.66 0.04 7.81 7.36 - 1.59 8.6581 64 23.5 45 -0.05 13.6 -46.37 0.04 8.54 3.67 - 1.59 -12.6382 64 23.5 45 0 -3.75 9.54 0 2.05 4.63 - -1.83 2.0693 69 23.5 40 0.03 18.27 42.48 0.02 11.46 6.92 - 1.59 6.1493 69 23.5 45 -0.03 18.48 -32.05 0.02 11.57 3.98 - 1.60 -8.0594 69 23.5 45 0 -2.46 7.74 0 1.34 4 - -1.84 1.94

105 74 23.5 40 -0.04 17.43 17.44 0.03 10.94 4.32 - 1.59 4.04105 74 23.5 45 0.04 17.87 -13.76 0.03 11.19 0.58 - 1.60 -23.72106 74 23.5 45

Bea

ms

in Z

-dire

ctio

n

0 -2.6 4.28 0 1.41 2.94 - -1.84 1.46153 64 20 40 -0.21 -87.64 261.8 0.18 13.35 277.91 - -6.56 0.94153 64 23.5 40 0.21 -79.78 191.23 0.18 2.85 179.27 - -27.99 1.07154 69 20 40 -0.12 54.82 266.15 0.1 4.2 314.18 - 13.05 0.85154 69 23.5 40 0.12 49.71 232.33 0.1 0.57 253.85 - 87.21 0.92155 74 20 40 -0.13 20.56 250.96 0.11 1.5 304.53 - 13.71 0.82155 74 23.5 40 0.13 18.48 214.98 0.11 0.36 248.93 - 51.33 0.86177 64 20 45 -0.16 -35.59 278.15 0.14 2.6 312.89 - -13.69 0.89177 64 23.5 45 0.16 -37.61 202.25 0.14 1.56 205.09 - -24.11 0.99178 69 20 45 -0.11 -21.3 287.51 0.09 1.61 350.08 - -13.23 0.82178 69 23.5 45 0.11 -23.15 248.35 0.09 0.85 283.2 - -27.24 0.88179 74 20 45 -0.11 -7.93 268.14 0.1 0.8 339.21 - -9.91 0.79179 74 23.5 45

Col

umns

0.11 -8.72 221.51 0.09 0.35 277.11 - -24.91 0.80


Table 8.12: Comparison of maximum end forces in Ground floor structural members innon-linear and linear dynamic analysis of frame-raft-soil system

Non-linear analysis Linear analysis Non-linear Vs Linear

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

of

mem

eber


Fx2/F

x 1

Fx2/F

x 1

Fx2/F

x 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

1 64 23.5 40 -4.85 -16.17 -0.71 23.9 121.48 2.8 -0.203 -0.133 -0.2541 69 23.5 40 4.85 21.17 0.71 20.72 103.78 2.48 0.234 0.204 0.2862 69 23.5 40 -4.65 -8.75 -0.53 15.57 102.67 2.09 -0.299 -0.085 -0.2542 74 23.5 40 4.65 13.75 0.53 13.4 85.79 1.84 0.347 0.160 0.2883 74 23.5 40 -4.18 -9.48 -0.56 0 107.59 2.22 - -0.088 -0.252

21 64 23.5 45 -3.4 -16.53 -0.58 24.42 132.73 2.3 -0.139 -0.125 -0.25221 69 23.5 45 3.4 21.53 0.58 18.37 108.9 2.03 0.185 0.198 0.28622 69 23.5 45 -3.48 9.22 -0.47 14.94 106.68 1.87 -0.233 0.086 -0.25122 74 23.5 45 3.48 14.02 0.47 11.11 84.86 1.65 0.313 0.165 0.28523 74 23.5 45

Bea

ms

in X

-dire

ctio

n

-2.92 -9.79 -0.48 0 112.97 1.92 - -0.087 -0.25081 64 23.5 40 -3.65 2.58 0.82 6.04 4.69 2.87 -0.604 0.550 0.28681 64 23.5 45 3.65 6.23 -0.82 6.96 5.11 3.27 0.524 1.219 -0.25182 64 23.5 45 -4.2 2.5 0 8.27 2.5 0 -0.508 1.000 -93 69 23.5 40 -2.23 2.57 1.16 3.03 4.68 4.05 -0.736 0.549 0.28693 69 23.5 45 2.23 6.23 -1.16 3.3 4.46 4.6 0.676 1.397 -0.25294 69 23.5 45 -2.98 2.5 0 4.71 2.5 0 -0.633 1.000 -

105 74 23.5 40 -2.35 2.58 1.11 1.29 3.48 3.89 -1.822 0.741 0.285105 74 23.5 45 2.35 6.21 -1.11 1.4 3.39 4.43 1.679 1.832 -0.251106 74 23.5 45

Bea

ms

in Z

-dire

ctio

n

-2.97 2.5 0 2.16 2.5 0 -1.375 1.000 -153 64 20 40 596.51 18.52 -11.09 344.98 130.62 3.15 1.729 0.142 -3.521153 64 23.5 40 -596.51 -18.52 11.09 262.18 116.18 4.52 -2.275 -0.159 2.454154 69 20 40 904.39 -16.88 -12.02 16.04 162.3 1.21 56.383 -0.104 -9.934154 69 23.5 40 -904.39 16.88 12.02 19.09 143.86 1.12 -47.375 0.117 10.732155 74 20 40 890.18 -16.07 -11.72 13.25 158.13 0.33 67.183 -0.102 -35.515155 74 23.5 40 -890.18 16.07 11.72 10.52 140.13 0.37 -84.618 0.115 31.676177 64 20 45 918.91 20.32 0.32 405.75 148 0.82 2.265 0.137 0.390177 64 23.5 45 -918.91 -20.32 -0.32 288.42 129.39 0.86 -3.186 -0.157 -0.372178 69 20 45 1229.87 -16.08 0.58 31.91 180.94 0.25 38.542 -0.089 2.320178 69 23.5 45 1229.87 16.08 -0.58 40.97 156.99 0.27 30.019 0.102 -2.148179 74 20 45 1212.29 -15.02 0.56 12.12 176.09 0.13 100.024 -0.085 4.308179 74 23.5 45

Col

umns

1212.29 15.02 -0.56 9.74 153.21 0.15 124.465 0.098 -3.733


Table 8.13: Comparison of maximum end moments in Ground floor structural membersin non-linear and linear dynamic analysis of frame-raft-soil system

Non-linear analysis Linear analysis Non-linear Vs Linear

Mem

ber N

o

X-c

oord

inat

e

Y-c

oord

inat

e

Z-co

ordi

nate

Pos

ition

of

mem

eber


Mx 2

/ Mx 1

My 2

/ My 1

Mz 2

/ Mz 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

1 64 23.5 40 0 1.97 -56.32 0.01 6.82 334.5 - 0.289 -0.1681 69 23.5 40 0 1.6 -37.04 0.02 5.56 260.41 - 0.288 -0.1422 69 23.5 40 0 1.29 -25.32 0.01 4.51 254.02 - 0.286 -0.1002 74 23.5 40 0 1.34 -30.94 0.01 4.71 246.81 - 0.285 -0.1253 74 23.5 40 0 1.4 -28.07 0.01 4.9 264.81 - 0.286 -0.106

21 64 23.5 45 0 1.56 -56.83 0 5.42 363.87 - 0.288 -0.15621 69 23.5 45 0 1.36 -38.31 0.01 4.72 287.26 - 0.288 -0.13322 69 23.5 45 0 1.17 -26.02 0 4.1 262.05 - 0.285 -0.09922 74 23.5 45 0 1.19 -31.61 0 4.17 258.92 - 0.285 -0.12223 74 23.5 45

Bea

ms

in X

-dire

ctio

n

0 1.21 -28.88 0.01 4.24 278.25 - 0.285 -0.10481 64 23.5 40 -0.01 -1.95 -15.12 0.04 7.81 7.36 - -0.250 -2.05481 64 23.5 45 0.01 -2.14 -3.53 0.04 8.54 3.67 - -0.251 -0.96282 64 23.5 45 0 0.59 2.61 0 2.05 4.63 - 0.288 0.56493 69 23.5 40 0 -2.88 -14.78 0.02 11.46 6.92 - -0.251 -2.13693 69 23.5 45 0 -2.91 -3.85 0.02 11.57 3.98 - -0.252 -0.96794 69 23.5 45 0 0.38 2.6 0 1.34 4 - 0.284 0.650

105 74 23.5 40 0 -2.74 -14.81 0.03 10.94 4.32 - -0.250 -3.428105 74 23.5 45 0 -2.8 -3.74 0.03 11.19 0.58 - -0.250 -6.448106 74 23.5 45

Bea

ms

in Z

-dire

ctio

n

0 0.4 2.58 0 1.41 2.94 - 0.284 0.878153 64 20 40 0.01 17.79 32.62 0.18 13.35 277.91 - 1.333 0.117153 64 23.5 40 -0.01 21.04 32.23 0.18 2.85 179.27 - 7.382 0.180154 69 20 40 0.01 20.38 -31.58 0.1 4.2 314.18 - 4.852 -0.101154 69 23.5 40 -0.01 21.69 -27.5 0.1 0.57 253.85 - 38.053 -0.108155 74 20 40 0.01 19.58 -30.53 0.11 1.5 304.53 - 13.053 -0.100155 74 23.5 40 -0.01 21.46 -25.71 0.11 0.36 248.93 - 59.611 -0.103177 64 20 45 0.01 -0.43 36.82 0.14 2.6 312.89 - -0.165 0.118177 64 23.5 45 -0.01 -0.68 34.29 0.14 1.56 205.09 - -0.436 0.167178 69 20 45 0.01 -1.06 -30.5 0.09 1.61 350.08 - -0.658 -0.087178 69 23.5 45 -0.01 -0.97 -25.89 0.09 0.85 283.2 - -1.141 -0.091179 74 20 45 0.01 -1.04 -28.8 0.1 0.8 339.21 - -1.300 -0.085179 74 23.5 45

Col

umns

-0.01 -0.92 -23.88 0.09 0.35 277.11 - -2.629 -0.086

Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter-IV

259

Rectangular pulse load

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5 6 7 8

Undamped Damped

1000 KN

1.25 S

Figure 8.1: Response of structure ( test problem) subjected to rectangular pulse load

5 m

5 m

3 m


260

Figure 8.2: Frame with details of node and the degree of freedom for which time histories are plotted.


1000 KN

1.25 S

21

A

BC

Y

X

Z

Horizontal DOF for which responses are shown


261

3.125, -0.008912302

2.325, -0.007825552

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7

undamped damped

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

1.5 2.5 3.5 4.5 5.5 6.5 7.5

undamped damped

Figure 8.3: Time histories of horizontal displacement at top node of fixedframe (a) without interaction infills under rectangular pulse load

(b) with interaction of infills

(a) without interaction ofinfills


1000 k N

1.25 S


1000k N

1.25 S


262

- 0 . 4

- 0 . 3

- 0 . 2

- 0 . 1

0

0 . 1

0 . 2

0 . 3

0 . 4

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2

T im e in s e c o n d s

Gro

un

d a

cc

ele

rat i

on

in

te

r ms

of

'g'

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0 1 2 3 4 5 6 7 8 9 10 11 12

Tim e in seconds

Dis

pla

cem

ent

in m

eter

s

undam ped dam ped eq-pulse/5000

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 1 2 3 4 5 6 7 8 9 10 11 12Time in seconds

Dis

pla

cem

ents

in

met

ers

undamped damped eq-pulse/10 without wall mass

Figure 8.4: Acceleration due to earthquake

Figure 8.5: Time histories of horizontal displacement at top node of fixed frame neglectingstiffness of infills

Figure 8.6: Time histories of horizontal displacement at top node of fixed frame consideringstiffness of infills


263

X (Longitudinaldirection)

Plan

Figure 8.8: Fem model for frame- raft-soil interaction analysis

Z ( Transversedirection)

L

B


Y (Verticaldirection)

Raft discretization

A B

AC

A

2

A1

A

Elevation

Z ( Transversedirection)

Figure 8.7: Frame-raft-Soil system


264

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.040 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Time in seconds

Dis

pla

cem

ent

in

met

ers

undamped Damped

-0.006

-0.004

-0.002

0

0.002

0.004

0.0060 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Times in seconds

Dis

pla

cem

ents

in

met

er

Undamped Damped

Figure 8.9: Time histories of horizontal displacement at top node of frame indynamic SSI analysis of frame-raft-soil system neglecting stiffness of infills

Figure 8.10: Time histories of horizontal displacement at top node of frame indynamic SSI analysis of frame-raft-soil system neglecting stiffness of infills


1000 kN

1.25 S


1000 kN

1.25 S


265

Figure 8.11: Different boundary conditions for soil mass.(a) Unrestrained (b)Restrained (c) Transmitting boundary

hapter IX

-0.06

-0.04

-0.02

0

0.02

0.04

0.060 1 2 3 4 5 6 7 8

Time in seconds

Dis

plac

emen

ts in

met

ers

Non-interactive ends fixedends free stiff soil (E=33 MPa)very stiff(E=330 MPa) Undamped with transmitting boundary

Figure 8.12: Time history of horizontal displacement at top of node A1 for different boundaryconditions at the edge of soil mass and stiffness of soil in structure-raft-soil system

Figure 8.13: Time history of horizontal displacement at top of node A1 for differentboundary conditions at the edge of soil mass in structure-raft-soil system

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Time in seconds

Dis

plac

emen

ts i

n m

eter

s

Damped analysis with transmitting boundary Damped anlysis with soil boundary restrained

Damped analysis with soil boundary unstrained


1000kN

1.25 S


1000k N

1.25 S

266

hapter IX

hapter IX

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time in seconds

Dis

plac

emen

ts in

met

ers

Undamped with transmitting boundary Damped with transmitting boundary

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Time in secondsD

ispl

acem

ents

in m

eter

s

Undamped with transmitting boundary Damped with transmitting boundaryFigure 8.14: Time history of horizontal displacement at top floor node above A1 with

transmitting boundary in structure-raft-soil system.

Figure 8.15: Time history of horizontal displacement at top floor node above A1with transmitting boundary in structure-infill-raft-soil system.

267

hapter IX

hapter IX

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12

Tim e in seconds

Acc

eler

ati

on

in

m/s

ec2

Acceleration due to earth quake Accleration at top floor

Figure 8.17: Time histories of horizontal acceleration at top node of frame in dynamic SSIanalysis of frame-raft-soil system, with transmitting boundaries, under El Centrino seismic load.

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0 2 4 6 8 10 12 14 16 18 20 22 24

Time in seconds

Dis

pla

cem

ents

in

met

ers

Undamped Damped el-centrino earth quake/10

Figure 8.16: Time histories of horizontal displacement at top node of frame in dynamic SSIanalysis of frame-raft-soil system, with transmitting boundaries, under El Centrino seismicload.

268

hapter IX

Figure 8.18: Time histories of horizontal acceleration in X-direction at node A1 of frame in dynamicSSI analysis of frame-raft-soil system, with transmitting boundaries, under El Centrino seismic load.

- 3

- 2

- 1

0

1

2

3

0 2 4 6 8 1 0 1 2

T i m e i n s e c o n d s

Ac

cle

ra

tio

n i

n m

/ se

c2

G r o u n d a c c e l e r a t i o n i n X - d i r e c t i o n E l C e n t r i n o s e i s m i c a c c e l e r a t i o n i n X - d i r e c t i o n

- 0 . 0 5

- 0 . 0 4

- 0 . 0 3

- 0 . 0 2

- 0 . 0 1

0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5

0 2 4 6 8 1 0 1 2


Ac

cle

ra

tio

n i

n m

/ se

c2

G r o u n d a c c e l e r a t i o n Z - d i r e c t i o n

- 3

- 2

- 1

0

1

2

3

0 2 4 6 8 1 0 1 2


Ac

cle

ra

tio

n i

n m

/ se

c2

G r o u n d a c c e l e r a t i o n Y - d i r e c t i o n E l C e n t r i n o g r o u n d a c c e l e r a t i o n i n X - d i r e c t i o n

Figure 8.19: Time histories of horizontal acceleration in Z-direction at node A1 of frame in dynamicSSI analysis of frame-raft-soil system, with transmitting boundaries, under El Centrino seismic load.

Figure 8.20: Time histories of horizontal acceleration in Y-direction at node A1 of frame in dynamicSSI analysis of frame-raft-soil system, with transmitting boundaries, under El Centrino seismic load.

269

hapter IX

-0 .0 4

-0 .0 3

-0 .0 2

-0 .0 1

0 .0 0

0 .0 1

0 .0 2

0 .0 3

0 .0 4

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4

T im e in seco n d s

Dis

pla

ce

me

nt

in m

ete

rs

U n d a m p ed D a m p ed e l-cen tr in o ea r th q u a ke /1 0

Figure 8.21: Time histories of horizontal displacement at top node of frame in dynamic SSIanalysis of frame-infill-raft-soil system considering with transmitting boundaries, subjected toEl Centrino seismic load.

-5

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12Time in seconds

Acc

eler

atio

n in

m/s

ec2

Acceleration at top floor Acceleration due to earthquake

Figure 8.22: Time histories of horizontal acceleration at top node of frame in dynamic SSIanalysis of frame-infill-raft-soil system considering with transmitting boundaries, subjectedto El Centrino seismic load.

hapter IX

- 3

- 2

- 1

0

1

2

3

0 2 4 6 8 1 0 1 2

T i m e i n s e c o n d sA

cc

ler

ati

on

in

m/ s

ec

2


Figure 8.23: Time histories of horizontal acceleration in X-direction at node A1 of frame in dynamic SSIanalysis of frame-infill-raft-soil system, with transmitting boundaries, under El Centrino seismic load.

Figure 8.24: Time histories of horizontal acceleration in Z-direction at node A1 of frame in dynamic SSIanalysis of frame-infill-raft-soil system, with transmitting boundaries, under El Centrino seismic load.

Figure 8.25: Time histories of horizontal acceleration in Y-direction at node A1 of frame in dynamic SSIanalysis of frame-infill-raft-soil system, with transmitting boundaries, under El Centrino seismic load.

- 3

- 2

- 1

0

1

2

3

0 2 4 6 8 1 0 1 2


Ac

cle

ra

tio

n i

n m

/ se

c2

G r o u n d a c c e l e r a t i o n i n Y - d i r e c t i o n E l C e n t r i n o s e i s m i c a c c e l e r a t i o n i n X - d i r e c t i o n

- 0 . 0 5

- 0 . 0 4

- 0 . 0 3

- 0 . 0 2

- 0 . 0 1

0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5

0 2 4 6 8 1 0 1 2


Ac

cle

ra

tio

n i

n m

/ se

c2


hapter IX



Plan

Elevation

A BA

CA

2A

1A

B


Y (Vertical direction)

LZ (Transverse direction)

Figure 8.26: FEM modeling of structure-isolated footings-soil system

hapter IX

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0 2 4 6 8 10 12 14 16 18 20 22 24

Tim e in seconds

Dis

pla

cem

ents

in

met

ers

Structure on raft foundations Structure on isolated footings

Figure 8.27: Comparison of time histories of horizontal displacement at top node of frame, indynamic SSI analysis of frame-raft-soil system and frame-isolated footings-soil system withtransmitting boundaries subjected to El Centrino seismic load

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20 22 24

T im e in seconds

Acc

eler

ati

on

in

m/s

ec2

A cce le ra tion due to earth quake A cc le ra tion due in S S I ana lys is

Figure 8.28: Time histories of horizontal acceleration at top node of frame in dynamic SSIanalysis of frame-isolated footings-soil system with transmitting boundaries, subjected to ElCentrino seismic load.

hapter IX

- 0 . 5

- 0 . 4

- 0 . 3

- 0 . 2

- 0 . 1

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 2 4 6 8 1 0 1 2


Ac

cle

ra

tio

n i

n m

/ se

c2


- 3

- 2

- 1

0

1

2

3

0 2 4 6 8 1 0 1 2

T i m e i n s e c o n d sA

cc

ler

ati

on

in

m/ s

ec

2


- 3

- 2

- 1

0

1

2

3

0 2 4 6 8 1 0 1 2


Ac

cle

ra

tio

n i

n m

/ se

c2

G r o u n d a c c e l e r a t i o n i n Y - d i r e c t i o n E l C e n t r i n o s e i s m i c a c c e l e r a t i o n i n X - d i r e c t i o n

Figure 8.29: Time histories of horizontal acceleration (X-direction) at node “A1” of frame indynamic SSI analysis of frame-isolated footings-soil system with transmitting boundaries,subjected to El Centrino seismic load.

Figure 8.30: Time histories of horizontal acceleration (Z-direction) at node “A1” of frame indynamic SSI analysis of frame-isolated footings-soil system with transmitting boundaries,subjected to El Centrino seismic load.

Figure 8.31: Time histories of vertical acceleration (Y-direction) at node “A1” of frame indynamic SSI analysis of frame-isolated footings-soil system with transmitting boundaries,subjected to El Centrino seismic load.

hapter IX

-0.0010

-0.0005

0.0000

0.0005

0.0010

0 1 2 3 4 5 6 7 8 9 10

Time in seconds

Dis

plac

emen

t in

met

ers

!00% 50% 25%

Figure 8.32: Time histories of horizontal displacement at top node of frame in non-linear dynamicSSI analysis of frame--raft-soil system considering with transmitting boundaries, subjected to ElCentrino seismic load.

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0 0.5 1 1.5 2 2.5 3 3.5

Time in seconds

Acc

eler

atio

n in

m/s

ec2

Acceleration in X-direction

Figure 8.33: Time histories of horizontal acceleration (X-direction) at node “A1” of frame indynamic SSI analysis of frame-raft-soil system with transmitting boundaries, subjected to ElCentrino seismic load.

276

hapter IX

hapter IX

hapter IX

Documents

Chapter VIII - INFLIBNETshodhganga.inflibnet.ac.in/bitstream/10603/7927/17/17_chapter 8.pdf · Chapter VIII 223 Chapter VIII Dynamic soil structure interaction analysis under earthquake