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).
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 224
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 225
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 226
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 227
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 228
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.
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 229
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 230
-(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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 231
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 232
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 233
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)
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 234
(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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 235
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 236
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,
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 237
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:
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 238
(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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 239
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 240
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.
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 241
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.
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 242
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 243
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 244
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 245
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.
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 246
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.
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 247
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 248
Table 8.2 Maximum response values of Stress resultants (moments) inbeams(X-direction) in Dynamic Non-interactive analysis
Un-damped analysis Damped analysisDamped
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 249
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
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 -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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 250
Table 8.4: Maximum response values of Stress resultants (moments)in beams(Z-direction) in Dynamic Non-interactive analysis
Un-damped analysis Damped analysisDamped
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 251
Table 8.5: Maximum response values of Stress resultants (translatory) inColumns in Dynamic Non-interactive analysis
Un-damped analysis Damped analysisDamped
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 252
Table 8.6: Maximum response values of Stress resultants (moments) in Columns in Dynamic Non-interactive analysis
Un-damped analysis Damped analysisDamped
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 253
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 254
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 255
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
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)
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 256
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
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)
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 257
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
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)
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter VIII 258
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
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)
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter-IV
260
Figure 8.2: Frame with details of node and the degree of freedom for which time histories are plotted.
Rectangular pulse load
1000 KN
1.25 S
21
A
BC
Y
X
Z
Horizontal DOF for which responses are shown
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter-IV
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
Rectangular pulse load
1000 k N
1.25 S
Rectangular pulse load
1000k N
1.25 S
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter-IV
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
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter-IV
263
X (Longitudinaldirection)
Plan
Figure 8.8: Fem model for frame- raft-soil interaction analysis
Z ( Transversedirection)
L
B
X (Longitudinaldirection)
Y (Verticaldirection)
Raft discretization
A B
AC
A
2
A1
A
Elevation
Z ( Transversedirection)
Figure 8.7: Frame-raft-Soil system
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter-IV
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
Rectangular pulse load
1000 kN
1.25 S
Rectangular pulse load
1000 kN
1.25 S
Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditionsChapter-IV
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
Rectangular pulse load
1000kN
1.25 S
Rectangular pulse load
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
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 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
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 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
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
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
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 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
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 Z - d i r e c t i o n
hapter IX
X (Longitudinaldirection)
X (Longitudinaldirection)
Plan
Elevation
A BA
CA
2A
1A
B
X (Longitudinaldirection)
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
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 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
T i m e i n s e c o n d sA
cc
ler
ati
on
in
m/ s
ec
2
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
- 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 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