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Introduction
As per the conventional earthquake-resistant design philosophy, the structures are designed
for forces, which are much less than the expected design earthquake forces. Hence, when a
structure is struck with severe earthquake ground motion, it undergoes inelastic deformations.
Even though the structure may not collapse but the damages can be beyond repairs. In
reinforced cement concrete (RCC) structures, a structural system can be made ductile, by
providing reinforcing steel according to the IS:13920-1993 code. A sufficiently ductile
structural system undergoes large deformations in the inelastic region. In order to understand
the complete behaviour of structures, time history analysis of different Single Degree of
Freedom (SDOF) and Multi Degree of Freedom (MDOF) structures having non-linear
characteristics is required to be performed. The results of time history analysis, i.e. non-linear
analysis of these structures will help in understanding their true behavior. From the results, it
can be predicted, whether the structure will not collapse / partially collapse or totally
collapse.
In this chapter, the modeling of SDOF and MDOF structures having non-linear
characteristics for seismic response analysis is carried out. The push over analysis of the RCC
building is also presented.
7.2 Non-linear Force-Deformation Behavior
The structural systems which have linear inertia, damping and restoring forces, are analysed
by linear methods. Whenever, the structural system has any or all of the three reactive forces
(i.e. inertia, damping and stiffness) having non-linear variation with the response parameters,
namely displacement, velocity, and acceleration; a set of non-linear differential equations is
evolved. To obtain the response, these equations need be solved. The most common nonlinearity
is the stiffness and the damping non-linearity. The stiffness non-linearity comprises
of two types namely the geometric non-linearity and the material non-linearity.
For the material non-linearity, restoring action shows a hysteretic behavior under
cyclic loading. For the geometric non-linearity, no such hysteretic behavior is exhibited.
During unloading, the load deformation path follows that of the loading. Figure 7.1(a) shows
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the case of load deformation behavior of the non-hysteretic type. Figure 7.2(b) shows the
hysteretic behavior of a non-linear restoring force under cyclic loading (material nonlinearity).
Damping non-linearity may be encountered in dynamic problems associated with
structural control, offshore structures, and aerodynamics of structures. Most of the damping
non-linearities are of a non-hysteretic type. Most structures under earthquake excitation
undergo yielding. Hence, it is necessary to discuss material non-linearity exhibiting hysteretic
behavior.
For structural systems having linear behaviour (when subjected to weak ground
motions) of inertial forces, spring elastic forces and linear damping characteristics, linear
methods of analysis can be employed. Displacement, velocity and acceleration are important
response parameters of any structural system. When any or all of the reactive forces, viz.
inertia force / spring force or damping force has nonlinear variation with the response
parameters, the analysis involves non-linear differential equations. Solution of these
equations will give the response of the system. The popular method to obtain the response is
Newmark’s Beta method.
Pushover Analysis
Amongst the natural hazards, earthquakes have the potential for causing the greatest
damages. Since earthquake forces are random in nature & unpredictable, the engineering
tools need to be sharpened for analyzing structures under the action of these forces.
Earthquake loads are to be carefully modeled so as to assess the real behavior of structure
with a clear understanding that damage is expected but it should be regulated. In this context
pushover analysis which is an iterative procedure is looked upon as an alternative for the
conventional analysis procedures. Pushover analysis of multi-story RCC framed buildings
subjected to increasing lateral forces is carried out until the preset performance level (target
displacement) is reached. The promise of performance-based seismic engineering (PBSE) is
to produce structures with predictable seismic performance.
The recent advent of performance based design has brought the non linear static push
over analysis procedure to the forefront. Pushover analysis is a static non linear procedure in
which the magnitude of the structural loading along the lateral direction of the structure is
incrementally increased in accordance with a certain pre-defined pattern. It is generally
assumed that the behavior of the structure is controlled by its fundamental mode and the
predefined pattern is expressed either in terms of story shear or in terms of fundamental mode
shape.
With the increase in magnitude of lateral loading, the progressive non-linear behavior
of various structural elements is captured, and weak links and failure modes of the structure
are identified. In addition, pushover analysis is also used to ascertain the capability of a
structure to withstand a certain level of input motion defined in terms of a response spectrum.
Recently, modifications to push over procedures have also been proposed so as to capture
contribution of higher modes of vibration of structure, change in distribution of story shear
subsequent to yielding of structural members, etc. Push over procedure is gaining popularity
during the last few years as appropriate analytical tools are now available (SAP-2000,
ETABS).
Pushover analysis is of two types, (i) force controlled or (ii) displacement controlled.
In the force control, the total lateral force is applied to the structure in small increments. In
the displacement control, the displacement of the top storey of the structure is incremented
step by step, such that the required horizontal force pushes the structure laterally. The
distance through which the structure is pushed, is proportional to the fundamental horizontal
translational mode of the structure. In both types of pushover analysis, for each increment of
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the load or displacement, the stiffness matrix of the structure may have to be changed, once
the structure passes from the elastic state to the inelastic state. The displacement controlled
pushover analysis is generally preferred over the force controlled one because the analysis
could be carried out up to the desired level of the displacement