INTRODUCTION

1

CHAPTER-I

INTRODUCTION

1.1. GENERALStarting from the very beginning of civilization, mankind has faced several threat

of extinction due to invasion of severe natural disasters. Earthquake is the most disastrous

among them due to its huge power of devastation and total unpredictability. Unlike other

natural catastrophes, earthquakes themselves do not kill people, rather the colossal loss of

human lives and properties occur due to the destruction of man-made structures. Building

structures are one of such creations of mankind, which collapse during severe

earthquakes, and cause direct loss of human lives. Numerous research works have been

directed worldwide in last few decades to investigate the cause of failure of different

types of buildings under severe seismic excitations. Massive destruction of high-rise as

well as low-rise buildings in recent devastating earthquake of Gujarat on 26th January,

2001 proves that also in developing counties like ours, such investigation is the need of

the hour.

Post-earthquake field surveys show that torsionally irregular buildings sustain

more damage compared to regular ones. Irregularities in stiffness and mass tend to

influence the capacity and demand. Plan irregularities cause non-uniform damage among

the structural elements of the same story due to non-uniform displacement demands [12].

Hence, seismic behavior of asymmetric building structures has become a topic of

worldwide active research since about last two decades. The previous studies were

limited to planar frames and symmetric buildings. Several research efforts have been

made to extend and apply the pushover analysis to unsymmetrical plan buildings whose

inelastic seismic responses are intricate. Kilar and Fajfar [30, 31], De Stefano and Rutenberg [15], Faella and Kilar [18], Moghadam and Tso [36, 37], Ayala et al. [5], Fujii et al. [25] and

Barros and Almeida [8] investigated on the application of pushover analysis for seismic

evaluation of unsymmetric-plan buildings. Recently, the modal pushover analysis (MPA)

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[13], the N2 method [21, 22] and a simplified seismic analysis [32] were extended to the

asymmetric-plan buildings.

Asymmetric building structures are almost unavoidable in modern construction due to

various types of functional and architectural requirements. The lateral-torsional coupling

due to eccentricity between centre of mass (CM) and centre of rigidity (CR) in

asymmetric building structures generates torsional vibration even under purely

translational ground shaking. During seismic shaking of the structural systems, inertia

force acts through the centre of mass while the resistive force acts through the centre of

rigidity as shown in Fig. 1.1. Due to this non-concurrency of lines of action of the inertia

force and the resistive force a time varying twisting moment is generated which causes

torsional vibration of the structure in addition to the lateral vibration. As an approximate

method, the early stage NSP has certain shortcomings on well considering the torsion

effects of the buildings. Starting in 1997, many researchers put significant efforts into

extending pushover analysis to asymmetric-plan buildings.

Figure 1.1: Generation of torsional moment in asymmetric structures during seismic excitation

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1.3 OBJECTIVES OF THE STUDY Performing pushover analysis of Symmetrical building.

Performing Non-linear time history analysis of Symmetrical building.

Evaluating N2 method for obtaining the target displacement for symmetrical building.

Performing pushover analysis of Asymmetrical building.

Performing Non-linear time history analysis of Asymmetrical building.

Performing response spectrum analysis of Asymmetrical building.

Evaluating N2 method for obtaining the target displacement for Asymmetrical building.

Evaluating Extended N2 method for obtaining the target displacement for Asymmetrical building.

1.4 ORGANIZATION OF THESIS

Chapter 1 This introductory chapter presents the background and motivation behind this

study.

Chapter 2 deals with brief report on the literature survey of nonlinear static procedures.

Chapter 3 covers the complete study on pushover analysis, N2 method and Non linear

time history analysis and the procedure to be adopted in the present analysis.

Chapter 4 completely takes care of the case study of a building under consideration.

Chapter 5 copes with the numerical study and presentation of results of pushover

analysis, N2 method, extended N2 method and time history analysis.

Chapter 6 Details the discussions drawn based on the present work and the scope for the

further study.

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1.5 SUMMARY

In this chapter, the importance of earthquake and the post disaster effects of it and

some light has been thrown on torsional effects occurring in asymmetrical buildings and

its effects. The objectives are also been discussed. Based on the objective of the present

study, research papers were collected and studied thoroughly. The review of research

papers is discussed in the next chapter named as literature review.

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LITERATURE REVIEW

CHAPTER-II

6

LITERATURE REVIEW

2.1. GENERALOne of the crucial steps of performance based design procedures is the

determination of the displacement demand of structures under the seismic actions defined

by either acceleration time history or design response spectrum methods. The most

accurate way to compute the displacement demands of a structure under a given seismic

action is to carry out a nonlinear response history analysis (RHA) of the detailed three

dimensional (3D) mathematical model of the structure. However, this is a tedious and

time consuming procedure. Hence, nonlinear static analysis procedures (NSP) [1-2, 4] attract

the attention of both practicing engineers and the research community in the sense that it

is more practical and faster to implement [10, 25–26]. Nevertheless, being an approximate

method, nonlinear static analysis has certain shortcomings and researchers have put

significant efforts into overcoming these shortcomings. However, most of the research on

nonlinear static analysis procedures has been limited to planar structures.

Seismic damage surveys and analyses conducted on modes of failure of building

structures during past severe earthquakes concluded that most vulnerable building

structures are those, which are asymmetric in nature. In this context, it is of significant

importance to be able to capture the variations of displacement demands within the same

story due to torsional effects for reliable damage assessment of structures with

asymmetric plans. As stated in the work by Chopra and Goel [13], current practice is

based on judgmental extension of NSP originally developed for planar analysis of

buildings to 3D analysis of structures with plan irregularities, which appear inaccurate in

capturing torsional effects. More recently, efforts have been made to address and

overcome the deficiencies of nonlinear static analysis procedures in estimating the

response of structures with plan irregularities [5-6, 13, 18-22, 32, 51].

2.2. Pushover methods for 3D plan asymmetric buildings

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The use of Nonlinear Static Procedures (NSPs) for the seismic assessment of plan

regular buildings and bridges is widespread nowadays. Their good performance in such

cases is widely supported by the extensive number of scientific studies described in the

previous pages. However, the applicability of NSPs on plan-irregular 3D buildings has so

far been the object of a limited number of papers. This limitation leads to a minor use of

these methods to assess current existing structures, the majority of which do tend to be

irregular in plan. The most important issue that controls the structural response of this

kind of structures is torsion. The aforementioned NSPs are not able to reproduce in a

correct manner the torsional response of plan irregular buildings; therefore one should be

cautious when using these methods to assess these structures.

In order to overcome the torsional problem in plan asymmetric buildings, some

researchers have proposed new pushover approaches. In 1997 Kilar and Fajfar [30] have

presented the use of a 3D model for the pushover analysis of plan irregular buildings.

They used an invariant force pattern with an inverted triangular shape at the centre of

mass of the floors. In this study the authors arrived at the conclusion that the torsional

rotation was strongly dependent on the orthogonal structural elements.

In 1998 Faella and Kilar [18] tested different location in plan to apply the lateral

forces of the pushover analysis of plan irregular buildings. Three eccentricities, measured

from the centre of mass location, were applied. In this study, the target displacement was

defined as the maximum response obtained from the nonlinear time-history analysis. The

torsional rotation was always underestimated, even when the eccentricity was maximum.

In the same year, De Stefano and Rutenberg [15] considered in their study the

interaction between walls and frames in a pushover analysis of 3D asymmetric

multistorey wall-frame structures. They applied the pushover forces considering the

design eccentricities prescribed in the Uniform Building Code [29]. The results obtained

were generally close to the time-history except at the flexible edges where the pushover

analysis overestimated the response.

In 2000, Azuhata et al. [7] included the torsional effects in the pushover analysis

by introducing two factors to the analysis: the strength modification factor and

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deformation amplification factor, introduced originally by Ozaki et al. [38]. Despite

leading to approximate results, the authors concluded that the method was conservative.

Moghadam and Tso [37] proposed in 2000 a 3D pushover procedure. In this

method, the conventional pushover is performed independently in each resisting element

using a planar analysis, and the target displacements are calculated considering the equal

displacement rule. In fact, the target displacements of each resisting element (planar

frames and walls) are calculated with an elastic response spectrum analysis of a 3D

model of the building. However, it is generally recognized that the equal displacement

rule may lead to small inelastic displacements in the case of: near-fault ground motions;

systems with low strength; soft soil conditions; hysteresis behaviour of the elements with

considerable pinching or stiffness and strength degradation. Therefore, the use of this

method may lead to not so accurate results.

In 2002 Ayala and Tavera [5] proposed a 3D pushover procedure where the load

vector was applied simultaneously in the two directions at the centre of mass of each

floor and it was constituted by forces and torques. This load vector also took into account

the contribution of higher modes.

In the same year, Penelis and Kappos [39, 40] proposed a 3D pushover analysis.

The load vector is applied at the centre of mass of the floors and it is constituted by

lateral forces and torques. These components of the vector are defined as the ones

necessary to produce the storey displacements and torsion profiles obtained from a multi-

modal response spectrum analysis. This is a single run method with an invariant load

vector. The equivalent SDOF capacity curve is obtained using the concept of Substitute

Structure proposed by Gulkan and Sozen [27]. The results show that further studies are

necessary to confirm the good performance of the method, namely for predicting local

responses.

In 2004 Chopra and Goel [13] extended the application of the aforementioned

Modal Pushover Analysis (MPA) to the case of plan asymmetric buildings. The method

was based on multi-run pushover analysis, where the load vectors in each run are

proportional to each 3D elastic mode of vibration of the structure. The load vectors are

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constituted by modal forces in the two translational directions and by torques. The total

seismic response was obtained combining the response due to each modal load. Some

drawbacks can however be mentioned for this method: since each run corresponding to

each mode is run independently, the yielding in one mode is not reflected in the others

and no interaction between modes in the nonlinear range is considered. A three

dimensional modal pushover analysis of buildings subjected to two ground motions was

also performed by Reyes and Chopra in 2011 [45]. In this work tall buildings were also

evaluated.

Fajfar et al. [20, 21] proposed in 2005 an extended version of the N2 method for

plan asymmetric buildings. In this proposal the pushover analysis of the 3D model is

performed independently in each direction, the target displacement being calculated using

the original N2 method procedure. In order to take torsional effects into account, the

pushover results are amplified by torsional correction factors. These factors are computed

through an elastic response spectrum analysis and a pushover analysis. No de-

amplification of displacements due to torsion is considered by the method. In 2009

D’Ambrisi et al. [16] tested the Extended N2 method in an existing school, and in 2011

Koren and Kilar [33] tested the method in asymmetric base isolated buildings.

In 2008, an International Workshop on Nonlinear Static Methods for

Design/Assessment of 3D structures [43] took place in Lisbon, Portugal. Twelve famous

researchers on the topic from different countries were invited. Names like Anil Chopra,

Peter Fajfar, Helmut Krawinkler, Rakesh Goel, Stavros Anagnostopoulos, Andreas

Kappos, amongst others, contributed to the high quality of the presented papers, to the

lively and fruitful discussions that followed and to the substantial and objective

recommendations produced. At this international Workshop on 3D Pushover, several

contributions were made and important results were outlined, which were compiled in the

book [43]. The more relevant are described in the following:

• Kunnath examined one subset of issues related to the sensitivity of nonlinear

material models, element formulations, analysis assumptions, etc, in seismic

demand estimation in the inelastic range. It was clear that due attention and care

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should be paid to modelling issues (e.g. by means of accurately devised sensitivity

studies) prior to the actual comparison between different seismic assessment

methods.

• Goel presented a generalized pushover curve, defined as a relationship between the

scaling factor that is applied to the “modal” force distribution during the pushover

analysis versus displacement at any selected reference location. Since this

generalized pushover curve does not explicitly need the base shear, it can be

developed for three-dimensional structures for which modes excited during the

earthquake ground motion may induce little or no base shear.

• Savoia proposed a new procedure, termed Force/Torque Pushover (FTP) analysis,

to select storey force distributions for 3D pushover analysis of plan irregular RC

frame structures.

• Krawinkler demonstrated that pushover analysis is readily capable of accurately

revealing P-Delta (or deterioration) structural sensitivity, when the latter contributes

in a significant manner to seismic response. This needs, however, to be

complemented with appropriately derived rules for computation of the target

displacement, which should equally account for 2nd order effects and deterioration,

something that seems not to be regularly recognized by analysts. Caution was also

recommended in the use of the pushover analysis in an intensity range for which

results may become collapse sensitive, given the uncertainty introduced by record-

to-record variability, and the consequent effects on the results obtained.

• Moghadam calibrated, by means of an extensive parametric study considering 150

steel buildings, a modification factor that may be introduced within the scope of a

Displacement Coefficient Method application to improve results prediction when

the latter is applied to assess the seismic response of plan irregular structures.

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• Kappos introduced an improved version of the MPA procedure for application to

bridges. It seems to be a promising approach that yields more accurate results

compared to the ‘standard’ pushover.

• Bento carried out a study that aimed at scrutinizing the effectiveness with which

four commonly employed Nonlinear Static Procedures (CSM, N2, MPA, ACSM)

are able to reproduce the actual dynamic response of the well known SPEAR

building.

Other researchers have performed 3D pushover analysis on plan irregular

buildings, such as: Fujii et al. [23], Yu et al. [50], Erduran and Ryan [15], Stefano and

Pintucchi [47]. Despite the results obtained, definitive answers still need to be reached.

2.3. CRITICAL APPRAISAL OF LITERATURE REVIEW The use of Non linear static procedure methods (NSPs) in the

case of plan irregular structures has so far been studied by a limited number of authors. This fact limits the application of NSPs to assess irregular plan structures. Existing studies on this topic usually focus on the evaluation of a single NSP. In order to obtain useful elements of comparison between different methodologies, the performance of 2 commonly employed nonlinear static procedures (pushover analysis and N2 method) is evaluated and it is compared with nonlinear dynamic analysis (Time history) in this thesis.

2.4. SUMMARYThe present chapter deals with the various literatures that have been

published on pushover analysis, N2 method and Extended N2 method. The literature

available emphasizes on the evaluation of a single NSP. Hence the present study

aims at comparison between different methodologies (pushover analysis, N2 method and Time history) using SAP2000(48) as a tool.

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METHODOLOGY

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CHAPTER-III

METHODOLOGY

3.1 GENERALSeismic design philosophies: The seismic evaluation of structures has been

generally based on a Force-Based design philosophy, where the structural elements are

assessed in terms of stresses caused by the equivalent seismic forces. Therefore, the main

concern within this design philosophy is to give strength to the structure rather than

displacement capacity. Some procedures have been proposed in the past, such as the

Response Spectrum Analysis (RSA), which has been implemented in seismic codes all

over the world and is still commonly used by the majority of structural design engineers.

In this procedure, the structure is considered to have an elastic behavior. The periods and

the modes of vibration are calculated, and the response of the structure is computed

through the application of a response spectrum. The forces in the elements are divided by

a behavior factor in order to take into account the nonlinearity of the materials. A

complete description of the method can be found in [53].

More recently, Priestley [42] published a critical review on the drawbacks of this method.

The main fallacies pointed out are the following:

1) A response spectrum is obtained from an accelerogram by running this record in several single degree of freedom (SDOF) systems with different periods of vibration. The value of the response spectrum corresponding to a certain period is obtained taking the maximum response of the SDOF with that period. As a consequence the duration effects of the dynamic response are ignored, which may not be valid in the case of plastic responses;

2) The response spectrum analysis uses the combination of modal responses, so the final response is a combination of the response

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associated with each mode of vibration. Thus, this principle leads to internal forces that do not respect the equilibrium;

3) The stiffness degradation is not taken into account. The method considers a mean value of this parameter which may produce wrong estimations of internal forces;

4) The design forces obtained from the modal combination are reduced using a behavior factor, in order to take into account the ductility and over strength of the structure. The use of a single value to reduce the internal forces seems to be a rough solution to represent the nonlinearity of the materials. In fact, the higher modes may not be controlled by the same level of ductility of the fundamental mode. Therefore, using the same force reduction factor in all modes may underestimate the higher mode effects in terms of internal forces;

In recent years the need for changes in the existing seismic design methodology implemented in codes has been generally recognized. The structural engineering community has been creating a new generation of design and rehabilitation procedures based on a new philosophy of performance-based engineering concepts. It has become widely accepted that one should consider damage limitation as an explicit design consideration [35]. In fact, the damage and behavior of the structures during an earthquake is mainly governed by the inelastic deformation capacity of the ductile members. Therefore, the seismic evaluation of structures should be based on the deformations induced by the earthquake, instead of the element stresses caused by the computed equivalent seismic forces, as happens in the Force-Based philosophy. In recent years, several attempts have been made to introduce displacement-based methodologies in seismic engineering practice. These methodologies can be divided into two main groups: displacement-based design

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methods for the design of new structures [42, 43], and displacement-based evaluation methods for the seismic performance assessment of pre-designed or existing structures.

Two key elements of a performance-based procedure are demand and capacity. The demand represents the effect of the earthquake ground motion (it can be defined by means of a response spectrum or an accelerogram). The capacity of a structure represents its ability to resist the seismic demand. The performance depends on how the capacity is able to handle the demand. The structure must have the capacity to resist the demands of the earthquake such that its performance is compatible with the design objectives.

Within this context, nonlinear seismic analyses of structures are extremely important in order to correctly assess their seismic performance, Figure 3.1

Figure 3.1 – Use of inelastic analysis procedures to estimate inelastic forces anddeformations for given seismic ground motions and a nonlinear analysis model of

the building [2].

3.2 ANALYSIS PROCEDURES

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The analysis procedures can be divided into linear procedures (linear static and

linear dynamic) and non-linear procedures (non-linear static and non-linear dynamic).

3.2.1 Linear static analysisIn a linear static analysis procedure, the building is modeled as an equivalent

single degree of freedom (SDOF) system with a linear elastic stiffness and an equivalent

viscous damping. The seismic input is modeled by an equivalent lateral force with the

objective to produce the same stresses and strains as the earthquake it represents. Based

on an estimation of the first fundamental frequency of the building using empirical

relationships or Rayleigh’s method, the spectral acceleration is determined from the

appropriate response spectrum which, multiplied by the mass of the building, results in

the equivalent lateral force. The coefficients take into account not only issues like second

order effects, stiffness degradation, but also force reduction due to anticipated inelastic

behavior. The lateral force is then distributed over the height of the building and the

corresponding internal forces and displacements are determined using linear elastic

analysis.

These linear static procedures are used primarily for design purposes and are

incorporated in most codes. Their expenditure is rather small. However, their

applicability is restricted to regular buildings for which the first mode of vibration is

predominant.

3.2.2 Linear dynamic analysis In a linear dynamic analysis procedure the building is modeled as a multi-degree-

of-freedom (MDOF) system with a linear elastic stiffness matrix and an equivalent

viscous damping matrix.

The seismic input is modeled using either modal spectral analysis or time history

analysis. Modal spectral analysis assumes that the dynamic response of a building can be

found by considering the independent response of each natural mode of vibration using

linear elastic response spectra. Only the modes contributing significantly to the response

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need to be considered. The modal responses are combined using schemes such as the

square-root-sum-of-squares. Time-history analysis involves a time-step-by-step

evaluation of building response, using recorded earthquake acceleration data as base

motion input. In both cases the corresponding internal forces and displacements are

determined using again linear elastic analysis. The advantage of these linear dynamic

procedures with respect to linear static procedures is that higher modes can be considered

which makes them suitable for irregular buildings. However, again they are based on

linear elastic response and hence their applicability decreases with increasing non-linear

behavior which is approximated by global force reduction factors.

3.2.3 Non-linear static analysisIn a non-linear static analysis procedure, the building model incorporates directly

the non-linear force-deformation characteristics of individual components and elements

due to inelastic material response. Several methods exist (e.g. ATC 40[1] and FEMA

356(4)). They all have in common that the non-linear force-deformation characteristic of

the building is represented by a pushover curve, i.e. a curve of base shear verses top

displacement, obtained by subjecting the building model to monotonically increasing

lateral forces or increasing displacements, distributed over the height of the building in

correspondence to the first mode of vibration, until the building collapses. The maximum

displacements likely to be experienced during a given earthquake are determined using

either highly damped or inelastic response spectra. Clearly, the advantage of these

procedures with respect to the linear procedures is that they take into account directly the

effects of non-linear material response and hence, the calculated internal forces and

deformations will be more reasonable approximations of those expected during an

earthquake. However, only the first mode of vibration is considered and hence these

methods are not suitable for irregular buildings for which higher modes become

important.

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3.2.4 Non-linear dynamic analysisIn a non-linear dynamic analysis procedure, the building model is similar to the

one used in non-linear static procedures incorporating directly the inelastic material

response using general finite elements. The main difference is that the seismic input is

modeled using a time-history analysis which involves time-step-by-time-step evaluation

of the building response. This is the most sophisticated analysis procedure for predicting

forces and displacements under seismic input. However, the calculated response can be

very sensitive to the characteristics of the individual ground motion used as seismic input,

therefore several time-history analysis are required using different ground motion

records. The major advantage of non-linear dynamic procedures is that it acts as a

research tool with the objective to simulate the behavior of a building structure in detail,

i.e. to describe the exact state of the art displacement profiles, the propagation of cracks,

the distribution of vertical and shear stresses, the shape of the hysteretic curves, etc.

3.3 PUSHOVER ANALYSIS

In Pushover analysis, a static horizontal force profile, usually proportional to the

design force profiles specified in the codes, is applied to the structure. The force profile is

then incremented in small steps and the structure is analyzed at each step. As the loads

are increased, the building undergoes yielding at a few locations. Every time such

yielding takes place, the structural properties are modified approximately to reflect the

yielding. The analysis is continued till the structure collapses, or the building reaches

certain level of lateral displacement. With the increase in the loads, non-linear responses

of the members are captured.

The pushover analysis can determine the lateral load verses deformation behavior

of the building corresponding to the incremental loads. Programs supporting pushover

analysis provide elegant visualization of the damage state for each load step and the

redistribution of the internal forces in the members. At each step, the base shear and the

roof displacement can be plotted to generate the capacity curve or pushover curve. It

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gives an idea of the lateral strength and the maximum inelastic drift the building can

sustain. For regular buildings it can also give a rough estimate of the lateral stiffness of

the building. Fig. 3.2 illustrates shows the way to plot the force deformation curve.

Figure 3.2 Inverted triangular loading for pushover analysis

Figure 3.3 Global capacity curve of structure

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Figure 3.4 Global capacity (pushover) curve

3.5 TYPES OF PUSHOVER ANALYSISPresently, there are two non-linear static analysis procedures available, one

termed as the Displacement Coefficient Method (DCM) included in the FEMA-356(4)

document and the other termed as the Capacity Spectrum Method (CSM) included in the

ATC-40(1) document. Both of these methods depend on the lateral load-deformation

variation obtained by using the non-linear static analysis under the gravity loading and

idealized lateral loading due to the seismic action. This analysis is generally called as the

pushover analysis.

3.5.1 Capacity spectrum method(3)

A nonlinear static analysis procedure that provides a graphical representation of

the expected seismic performance of the existing or retrofitted structure by the

intersection of the structure’s capacity spectrum with a response spectrum (demand

spectrum) representation of the earthquake’s displacement demand on the structure. The

intersection is the performance point, and the displacement coordinate, dp, of the

performance point is the estimated displacement demand on the structure for the specified

level of seismic hazard.

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3.5.2 Displacement coefficient method(29)

A nonlinear static analysis procedure that provides a numerical process for'

estimating the displacement demand on the structure, by using a bilinear representation of

the capacity curve and a series of modification factors, or coefficients, to calculate a

target displacement. The point on the capacity curve at the target displacement is the

equivalent of the performance point in the capacity spectrum method.

Various elastic (linear) and inelastic (non-linear) methods are available for the

analysis of existing concrete buildings. Elastic analysis methods available include static

lateral force procedures, dynamic lateral force procedures and elastic procedures using

demand capacity ratios. The most basic inelastic analysis method is the complete non-

linear time history analysis, which at this time is considered more complex and

impractical for general use. Available simplified non-linear analysis methods, referred to

as non-linear static analysis procedures, include the capacity spectrum method (CSM)

that uses the intersection of the capacity (pushover) curve and a reduced response

spectrum to estimate maximum displacement; the displacement coefficient method

(FEMA-273) that uses pushover analysis and a reunified version of the equal

displacement approximation to estimate maximum displacement; and the secant method

that uses a substitute structure and secant stiffness. A non-linear static procedure in

general focuses on the capacity spectrum method. This method has not been developed in

detail previously. It provides a particularly rigorous treatment of the reduction of seismic

demand for increasing displacement.

Although elastic analysis gives a good indication of the elastic capacity of

structures and indicates where first yielding will occur, it cannot predict failure

mechanisms and account for redistribution of forces during progressive yielding. Inelastic

analyses procedures help demonstrate how buildings really work by identifying modes of

failure and the potential for progressive collapse. The use of inelastic procedures for

design and evaluation is attempts to help engineers better understand how structures will

behave when subjected to major earthquakes, where it is assumed that the elastic capacity

of the structure will be exceeded. This resolves some of the uncertainties associated with

elastic procedures. The capacity spectrum method, a non-linear static procedure that

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provides a graphical representation of the global force-displacement capacity curve of the

structure (i.e., pushover) and compares it to the response spectra representations of the

earthquake demands, is a very useful tool in the evaluation and retrofit design of existing

concrete buildings. The graphical representation provides a clear picture of how a

building responds to earthquake ground motion, and it provides an immediate and clear

picture of how various retrofit strategies, such as adding stiffness or strength, will impact

the building's response to earthquake demands.

3.4.2 Pushover analysis in ATC-40(1) Seismic Evaluation and Retrofit of Concrete Buildings commonly referred to as

ATC-40(1) was developed by the Applied Technology Council (ATC) with funding from

the California Safety Commission. Although the procedures recommended in this

document are for concrete buildings, they are applicable to most building types.

ATC-40(1) recommends the following steps for the entire process of evaluation and

retrofit:

Initiation of a Project: Determine the primary goal and potential scope of the

project.

Selection of Qualified Professionals: Select engineering professionals with a

demonstrated experience in the analysis, design and retrofit of buildings in

seismically hazardous regions. Experience with PBSE and non-linear procedures

are also needed.

Performance Objective: Choose a performance objective from the options

provided for a specific level of seismic hazard.

Review of Building Conditions: Perform a site visit and review drawings.

Alternatives for Mitigation: Check to see if the non-linear procedure is

appropriate or relevant for the building under consideration.

Peer Review and Approval Process: Check with building officials and consider

other quality control measures appropriate to seismic evaluation and retrofit.

Detailed Investigations: Perform a nonlinear static analysis if appropriate.

Seismic Capacity: Determine the inelastic capacity curve also known to pushover

curve. Covert to capacity spectrum.

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Seismic Hazard: Obtain a site specific response spectrum for the chosen hazard

level and convert to spectral ordinates format.

Verify Performance: Obtain performance point as the intersection of the capacity

spectrum and the reduced seismic demand in spectral ordinates (ADRS) format.

Check all primary and secondary elements against acceptability limits based on

the global performance goal.

3.5 N2 method:

In recent years, a breakthrough of simplified inelastic analysis and performance

evaluation methods has occurred. Such methods combine the non-linear static (pushover)

analysis of a multi degree-of-freedom (MDOF) model and the response spectrum analysis

of an equivalent single-degree-of-freedom (SDOF) model. They can be used for a variety

of purposes such as design verification for new buildings and bridges, damage

assessment for existing structures, determination of basic structural characteristics in

direct displacement based design, and rapid evaluation of global structural response to

seismic ground motion of different intensities. An example is the N2 method, which has

been implemented into the final version of the Eurocode 8 standard (CEN 2003).

3.5.1 SUMMARY OF THE N2 METHOD:

The N2 method (N comes from Nonlinear analysis and 2 comes from 2

mathematical models) was developed at the University of Ljubljana in mid-eighties. The

formulation of the method in the acceleration displacement format enables the visual

interpretation of the procedure and of the relations between the basic quantities

controlling the seismic response. The basic version of the N2 method, limited to planar

structural models.

In the N2 method, first the pushover analysis of the MDOF model is performed.

Pushover curve is then transformed to the capacity diagram. The seismic demand for the

equivalent SDOF system with a period T can be determined as follows: Elastic demand in

terms of acceleration Sae and displacement Sde is determined from the elastic spectrum.

The inelastic acceleration demand Sa is equal to the yield acceleration Say, which

represents the acceleration capacity of the inelastic system. The strength reduction factor

24

due to ductility Rμ, which will be denoted in this paper as R (R ≡ Rμ), can be determined

as the ratio between the accelerations corresponding to the elastic and inelastic system.

The ductility demand μ is then calculated from inelastic spectra, which are defined by the

period dependent relation between reduction factor and ductility (R-μ-T relation), and the

inelastic displacement demand Sd is computed as Sd = (μ/R)Sde . The target

displacement, which represents the seismic demand of the MDOF model, is obtained as

Dt = ΓSd , where Γ is the transformation factor from the MDOF to the SDOF system. In

principle any R-μ-T relation can be used. A very simple and fairly accurate R-μ-T relation

is based on the equal displacement rule in the medium- and long-period range. This

relation is used in the basic variant of the N2 method. It has been implemented in

Eurocode 8 and is discussed below.

For many years, the ductility factor method has been used in seismic codes. The

basic assumption of this method is that the deformations of a structure produced by a

given ground motion are essentially the same, whether the structure responds elastically

or yields significantly. This assumption represents the “equal displacement rule”. Using

this rule, the ductility dependant reduction factor R is equal to ductility factor μ. The

simple chart in Fig.1 is essential for understanding of the concept of reduction factors and

of the ductility factor method. The educational value of the figure can be greatly

increased by using the AD format, introduced by Freeman. In AD format, Fig.3.5 (force

has to be divided by mass) can be combined with demand spectra (Fig.3.6.). Fig.3.6,

which enables a visualization of the basic variant of the N2 method, resembles to the

basic chart in capacity spectrum method. The main difference is in inelastic demand,

which is defined by an inelastic spectrum rather than by an equivalent highly damped

elastic spectrum. Inelastic spectrum in medium and long-period range in Fig.3.6 is based

on the equal displacement rule.

25

Figure 3.5. Basic diagram explaining the ductility factor method

Figure 3.6. Elastic and inelastic demand spectra versus capacity diagram.

In Fig.3.6 the quantities relevant for the seismic response of an ideal elasto-plastic SDOF

system can be visualized. Seismic demand is expressed in terms of accelerations and

displacements, which are the basic quantities controlling the seismic response. Demand is

compared with the capacity of the structure expressed by the same quantities. Fig.3.6

helps to understand the relations between the basic quantities and to appreciate the effects

of changes of parameters. The intersection of the radial line corresponding to the elastic

period of the idealised bilinear system T with the elastic demand spectrum Sae defines the

26

acceleration demand (strength) required for elastic behaviour, and the corresponding

elastic displacement demand Sde. The yield acceleration Say represents both the

acceleration demand and capacity of the inelastic system. The reduction factor R is equal

to the ratio between the accelerations corresponding to elastic (Sae) and inelastic systems

(Say). If the elastic period T is larger than or equal to TC, which is the characteristic

period of ground motion, the equal displacement rule applies and the inelastic

displacement demand Sd is equal to the elastic displacement demand Sde. From triangles

in Figs.3.5 and 3.6 it follows that the ductility demand μ is equal to R. Fig.3.6. also

demonstrates that the displacements Sdd obtained from elastic analysis with reduced

seismic forces, corresponding to design acceleration Sad, have to be multiplied by the

total reduction factor, which is the product of the ductility dependent factor R and the

overstrength factor, defined as Say/Sad. The intersection of the capacity diagram and the

demand spectrum, called also performance point, provides an estimate of the inelastic

acceleration and displacement demand, as in the capacity spectrum method. This feature

allows the extension of the visualisation to more complex cases, in which different

relations between elastic and inelastic quantities and different idealisations of capacity

diagrams are used. Fig.3.6. can be used for both traditional force-based design as well as

for the increasingly popular deformation-controlled (or displacement-based) design. In

these two approaches, different quantities are chosen at the beginning. Let us assume that

the approximate mass is known. The usual force-based design typically starts by

assuming the stiffness (which defines the period) and the approximate global ductility

capacity. The seismic forces (defining the strength) are then determined, and finally

displacement demand is calculated. In direct displacement based design, the starting

points are typically displacement and/or ductility demands. The quantities to be

determined are stiffness and strength. The third possibility is a performance evaluation

procedure, in which the strength and the stiffness (period) of the structure being analysed

are known, whereas the displacement and ductility demands are calculated. Note that, in

all cases, the strength corresponds to the actual strength and not to the design base shear

according to seismic codes, which is in all practical cases less than the actual strength.

Note also that stiffness and strength are usually related quantities. All approaches can be

easily visualized with the help of Fig.3.6. The relations apply to SDOF systems.

27

However, they can be used also for a large class of MDOF systems, which can be

adequately represented by equivalent SDOF systems.

3.6 Limitations of N2 method:

The N2 method is, like any approximate method, subject to several limitations.

Applications of this methods are time being, restricted to the planar analysis of structures.[] This method is restricted to symmetrical structures.

3.7 Extended N2 method for Plan Asymmetric Buildings:

The original N2 method is, like other simplified non-linear methods, restricted to

2D analysis. In order to extend the applicability of the method to plan-asymmetric

buildings, which require a 3D structural model, a procedure based on pushover analysis

of a 3D building model was proposed in (Fajfar 2002) and implemented in (Fajfar et al.

2002).

3.8 SUMMARY OF THE EXTENDED N2 METHOD:

Fajfar and his co-workers worked on the seismic behavior of plan-asymmetric

buildings and observed that the torsional effects are mostly pronounced in the elastic

range and early stages of plastic behavior and tend to decrease with an increase in the

plastic deformations [21, 22]. Hence, the amplifications in the displacement demands due

to torsional effects computed from elastic dynamic analysis can be used as a rough and

conservative estimate both in the elastic and inelastic range. Based on this observation,

Fajfar et al. [21] developed the extension of the N2 method to plan asymmetric buildings.

The fundamental steps of the procedure are summarized below:

The N2 method was included in Eurocode 8 [11] as the recommended nonlinear static

procedure.

The fundamental steps of the procedure are summarized below:

1. Pushover analysis of the 3D mathematical model of the building is performed

independently in two horizontal directions and the target roof displacement for each

horizontal direction is computed using the N2 method [21].

28

2. A linear response spectrum analysis of the 3D model is carried out independently in

two horizontal directions and the results are combined using the SRSS rule.

3. The correction factors to be applied to the relevant results of pushover analysis are

determined. The correction factor is defined as the ratio between the normalized roof

displacements obtained by elastic modal analysis and by pushover analysis. The

normalized roof displacement is the ratio of the roof displacement at an arbitrary

location to the roof displacement at the center of mass (CM). If the normalized roof

displacement obtained from elastic response spectrum analysis is less than 1.0, it is

taken as 1.0.

4. All the relevant quantities obtained by pushover analysis are multiplied with the

appropriate correction factors. Examples for the relevant quantities may be the

deformation demand of the ductile elements and the force demand of the brittle

elements.

3.9 Time History Analysis

It is an analysis of dynamic response of the structure at each increment of time,

when its base is subjected to specific ground motion history. This means the method

requires site specific ground motion studies. Dynamic analysis using either Time History

method or response spectrum, method shall be performed for following buildings.

a) Regular buildings : These buildings having height greater than 40 m In Zone IV and

V, and those greater than 90 m height in Zone II and III.

b) Irregular buildings: All buildings higher the 12 m in Zone IV and V , and those

greater than 40m in height Zone II and III.

However, for irregular buildings having height less than 40 m in Zone II and III,

even though not mandatory, dynamic analysis is recommended. Thus, in general,

dynamic analysis shall be performed for building in Zone IV and V while for buildings

having height less than 40m in Zone II and III seismic coefficient method which is

simple in application.

29

3.9.1 OVERVIEW OF TIME HISTORY ANALYSIS

Time-history analysis is used to determine the dynamic response of a structure to

arbitrary loading. The dynamic equilibrium equations to be solved are given by:

................... 3. 3

Where K is the stiffness matrix; C is the damping matrix; M is the diagonal mass

matrix; and are the displacements, velocities, and accelerations of the

structure; and r is the applied load. If the load includes ground acceleration, the

displacements, velocities, and accelerations are relative to this ground motion.

Any number of time-history Load Cases can be defined. Each time-history case can differ

in the load applied and in the type of analysis to be performed.

There are several options that determine the type of time-history analysis to be

performed:

Linear vs. Non linear.

Modal vs. Direct-integration: These are two different solution methods, each with

advantages and disadvantages. Under ideal circumstances, both methods should

yield the same results to a given problem.

Transient vs. Periodic: Transient analysis considers the applied load as a one-time

event, with a beginning and end. Periodic analysis considers the load to repeat in

definitely, with all transient response damped out. Periodic analysis is only

available for linear modal time-history analysis.

Initial Conditions

The initial conditions describe the state of the structure at the beginning of a time-

history case. These include:

Displacements and velocities

Internal forces and stresses

Internal state variables for non linear elements

Energy values for the structure

External loads

30

The accelerations are not considered initial conditions, but are computed from the

equilibrium equation. For linear transient analyses, zero initial conditions are always

assumed. For periodic analyses, the program automatically adjusts the initial conditions

at the start of the analysis to be equal to the conditions at the end of the analysis.

Time Steps

Time-history analysis is per formed at discrete time steps. Number of output time

steps are specified with parameter n step and the size of the time steps with parameter dt.

The time span over which the analysis is carried out is given by n step·dt. For periodic

analysis, the period of the cyclic loading function is assumed to be equal to this time

span. Responses are calculated at the end of each dt time increment, resulting in nstep+1

values for each output response quantity.

Response is also calculated, but not saved, at every time step of the input time

functions in order to accurately capture the full effect of the loading. These time steps are

called load steps. For modal time-history analysis, this has little effect on efficiency.

For direct-integration time-history analysis, this may cause the stiffness matrix to be re-

solved if the load step size keeps changing. For example, if the output time step is 0.01

and the input time step is 0.005, the program will use a constant internal time- step of

0.005. However, if the input time step is 0.075, then the input and output steps are out of

synchrony, and the loads steps will be: 0.075, 0.025, 0.05, 0.05,0.025, 0.075, and so on.

For this reason, it is usually advisable to choose an output time step that evenly divides,

or is evenly divided by, the input time steps.

3.9.2 CLASSIFICATION

Time-history analysis is a step-by-step anal sis of the dynamical response of a

structure to a specified loading that may vary with time. The analysis may be linear or

non-linear.

Linear Time History analysis:

A Linear Time History analysis overcomes all the disadvantages of modal response

spectrum analysis, provided non-linear behaviour is not involved. This method requires

greater computational efforts for calculating response quantities at discrete times. One

interesting advantage of such procedure is that the relative signs of response quantities

31

are preserved in the response histories. This is important when interaction effects are

considered in design among stress resultants.

Non-Linear Time History analysis:

A seismically deficient building will be subjected to inelastic action during

earthquake motion. The Non-Linear T.H.A off the building under strong ground motion

brings out the regions of weakness and ductility demand of the structure. This is the most

rational method available for assessing building performance. There are computer

programs available to perform this type of analysis. However, there are complexities with

regard to biaxial non linear response of the columns, modelling of joints behaviour,

interaction of flexural and shear strength and modelling of degrading characteristics of

member. The methodology is used to ascertain deficiency and post elastic response under

strong ground shaking.

Modal time history analysis:Modal superposition provides a highly efficient and accurate procedure for

performing time-history analysis. Closed-form integration of the modal equations is used

to compute the response, assuming linear variation of the time functions between the

input data time points.

Therefore, numerical instability problems are never encountered, and the time

increment may be any sampling value that is deemed fine enough to capture the

maximum response values. One-tenth of the time period of the highest mode is usually

recommended; however, a larger value may give an equally accurate sampling if the

contribution of the higher modes is small.

If all of the spatial load vectors pi, are used as starting load vectors for Ritz-vector

analysis, then the Ritz vectors will always produce more accurate results than if the same

number of eigenvectors is used. Since the Ritz-vector algorithm is faster than the

eigenvector algorithm, the former is recommended for time-history analyses.

To determine if the Modes calculated by the program are adequate to represent the

time history response to the applied load, these things should be checked.

That enough Modes have been computed

That the Modes cover an adequate frequency range

32

That the dynamic load (mass) participation mass ratios are adequate for the Load

Patterns and/or Acceleration Loads being applied.

That the modes shapes adequately represent all desired deformations.

Direct-Integration Time-History Analysis:Direct integration for the solution of the differential equations of motion is available

in SAP2000 for linear and nonlinear time history analysis. For linear analysis, modal

superposition can be used instead with greater efficiency than the direct integration

method. When using direct integration for nonlinear transient analysis, all types of

nonlinearities (material, geometric) are considered in the algorithm.

A nonlinear direct-integration time-history analysis can be initiated from zero

initial conditions (unloaded structure) or continued from a nonlinear static analysis

(pushover) or another direct-integration time-history nonlinear analysis. The geometric

nonlinearity is taken as the previous analysis case. For the analysis of undamaged

building structures, nonlinear time history analysis is conducted including only the effects

of gravity loads.

While modal superposition is usually more accurate and efficient, direct-

integration does offer the following advantages for linear problems:

Full damping that couples the modes can be considered

Impact and wave propagation problems that might excite a large number of modes

may be more efficiently solved by direct integration

For non linear problems, direct integration also allows consideration of more types of

nonlinearity that does modal superposition.

Direct integration results are extremely sensitive to time-step size in a way that is not true

for modal super position. Therefore the direct-integration analyses should always be run

with decreasing time-step sizes until the step size is small enough that results are no

longer affected by it. In particular, stiff and localized response quantities should be

checked. For example, much smaller time step may be required to get accurate results for

the axial force in a stiff member than for the lateral displacement at the top of a structure.

3.7.3 Damping

33

For the damping calculations, there are three options in SAP2000. These are ‘direct

specification’, ‘specifying modal damping by period’ and ‘specifying damping by

frequency’ options. In ‘direct specification’ option, the damping values are entered

considering mass and stiffness proportional coefficients. In ‘specify modal damping by

period’ option, the damping values with the first and second periods are assigned

Using these values, the program calculates the mass proportional and stiffness

proportional coefficients. The ‘specify modal damping by frequency’ has the same

interface but this time frequency values instead of periods are assigned. In the analyses of

the analytical models ‘specify modal damping by period’ option is used.

In direct-integration time-history analysis, the damping in the structure is modelled

using a full damping matrix. Unlike modal damping, this allows coupling between the

modes to be considered. Direct-integration damping has three different sources, which

are described in the following. Damping from these sources is added together.

Proportional Damping from the Load Case:

For each direct-integration time-history Load Case, proportional damping has

coefficients that apply to the structure as a whole. The damping matrix is calculated as a

linear combination of the stiffness matrix scaled by specifying a coefficient and the mass

matrix scaled by specifying a second coefficient.

These two coefficients may be specified directly, or they may be computed by

specifying equivalent fractions of critical modal damping at two different periods or

frequencies.

Stiffness proportional damping is linearly proportional to frequency. It is related

to the deformations within the structure. Stiffness proportional damping may excessively

damp out high frequency components.

Mass proportional damping is linearly proportional to period. It is related to the motion of

the structure, as if the structure is moving through a viscous fluid. Mass proportional

damping may excessively damp out long period components.

Due to the limitations of the response spectrum analysis procedure to

approximate the dynamic nonlinear response of a complex three-dimensional structural

systems, nonlinear time-history analysis is strongly recommended instead. Nonlinear

time history analysis accounts for the nonlinearities or strength degradation of different

34

elements of the building, as well as the load pattern or ground motion intensity and

characteristics used during a nonlinear dynamic analysis.

The loading in a time history analysis is foundation displacement or ground

motion acceleration, not externally applied loads at the joints or members of the structure.

The design displacements are not established using a target displacement, but instead are

determined directly through dynamic analysis using suites of ground motion records.

Inertial forces are produced in the structure when the structure suddenly deforms due to

ground motion and internal forces are produced in the structural members. For complex

three-dimensional structures such as curved structures, the direction of the earthquake

that produces the maximum stresses, in a particular member or at a specified point, is not

apparent. For all building types time history analysis must therefore be performed using

several different earthquake motions at various input angles to assure that all the

significant modes are excited and the critical earthquake direction is captured, producing

the peak response and estimating accurately the seismic demand on the structure.

Another approach is to use a larger suite of earthquake ground motion records of

three components at one angle of input. Since seismic motions can excite the higher

frequencies of the structure, neglecting higher modes of the building system could

introduce a significant error in the dynamic analysis results.

The main disadvantage of the time history analysis method is the high

computational and analytical effort required and the large amount of output information

produced. During the analysis, the capacity of the main building components is evaluated

as a function of time, based on the nonlinear behaviour determined for the elements and

materials.

This evaluation is carried out for several input ground motions applied at different

angles, and the response of the structure is recorded at every time step. Despite these

challenges, the evaluation of the capacity using the THA method at each time step

produces superior results, since it allows for redistribution of internal forces within the

structure. Each member is therefore not designed for maximum peak values, as required

by the response spectrum method, but for the actual forces produced in the structure

during dynamic excitation.

35

The recent development of computer hardware has allowed to reduce the required

computational time and made it more practical to run many time history analyses for

complex building structures. In addition, the seismic demand can be estimated through

statistical approximations, using the mean and standard deviation values of joint

displacements and element forces to determine the peak response expected for the

structure.

3.10 Earthquake Ground Motion:

The earthquake ground motion of sufficient strength that affect human and their

environment (strong ground motion) are of interest for earthquake resistant design. The

strong motions are measured by accelerographs and its recorded history is time history of

acceleration (accelerogram).The temporal evolution of accelerogram is composed of

three parts viz. Rise, strong motion and decay. The motion strongly depends on source

parameters such as fault shape, its area, and maximum fault dislocation, complexity of

slipping process and the distance of fault from the ground surface. The elastic properties

of the material through which the generated seismic waves travel also influence the

strong motion. The effect of ground shaking is mostly dependent on duration on strong

motion part.

3.10.1 Input Ground Motions:Input ground motions should be selected and scaled so as to accurately represent

the specific hazard of interest. As outlined in ASCE 7, the ground motions should reflect

the characteristics of the dominant earthquake source at the building site, such as fault

mechanism, distance to the fault, site conditions, and characteristic earthquake

magnitude. Recent studies have further shown that the shape of the ground motion

response spectra is an important factor in choosing and scaling ground motions,

particularly for higher intensity motions (Baker and Cornell 2006). While a

comprehensive discussion on the selection and scaling of ground motion records is

beyond the scope of this Technical Brief, the following are some of the issues to

consider.

3.10.2 Target hazard spectra or scenario:

36

While the earthquake hazard is a continuum, building codes typically define

specific ground motion hazard levels for specific performance checks. Generally, the

hazard is defined in terms of response spectral accelerations with a specified mean annual

frequency of expedience, although other definitions are possible including scenario

earthquakes, e.g., an earthquake with a specified magnitude and distance from the site, or

deterministic bounds on ground motion intensities.

3.10.3 Source of ground motions

For building assessment and design, the input earthquake ground motions can

either be (1) actual recorded ground motions from past earthquakes, (2) spectrally

matched ground motions that are created by manipulating the frequency content and

intensity of recorded ground motions to match a specific hazard spectrum, or (3)

artificially simulated motions. Opinions differ as to which types are most appropriate.

Recorded ground motions are generally scaled to match the hazard spectrum at one or

more periods. For example, ASCE 7 specifies rules for scaling the ground motions based

on their spectral acceleration values for periods between 0.2T to 1.5T, where T is the

fundamental period of vibration of the structure. When structures are expected to

respond in multiple modes, such as in tall buildings, spectral matching may be more

appropriate, since scaling of actual recorded motions to a uniform hazard spectrum may

bias the analysis results to either overestimate the response at short periods or

underestimate it at longer periods.

3.10.4 Number of ground motions:

Given the inherent variability in earthquake ground motions, design standards

typically require analyses for multiple ground motions to provide statistically robust

measures of the demands. For example, ASCE 7 requires analyses for at least seven

ground motions (or ground motion pairs for three-dimensional analyses) to determine

mean values of demand parameters for design. In concept, it is possible to obtain reliable

mean values with fewer records, such as through the use of spectrally matched records,

but there is currently no consensus on methods to do so. Moreover, while one could

calculate additional statistics besides the mean, e.g., the standard deviation of the demand

37

parameters, the reliability of such statistics is questionable when based on only seven

ground motions. This is especially true when spectrally matched records are used, where

the natural variability in the ground motions is suppressed.

3.11 SUMMARY

In this chapter the basic theory, seismic methods and different analysis procedures

were discussed .The code based procedure for seismic analysis, methods for seismic

evaluation, overview and different types of time history analysis, earthquake ground

motion.

38

CASE STUDY

39

CHAPTER-IV

CASE STUDY

4.0 GENERAL

Two different buildings of similar height but with significantly different

characteristics were selected to assess the performance using Non linear dynamic

Analysis (Time history) and Non linear static Analysis methods. One building is

symmetrical with 5 bays in X direction and 3 bays in Y direction; and another is

Asymmetric in plan with 5 bays in X direction and 3 bays in Y direction, as shown in fig.

4.1 and 4.2. The preliminary building data required for analysis assumed are presented in

the following table.

Table 4.1 Assumed Preliminary data required for the Analysis of the frameSl. No Variable Data

1 Type of structure Moment Resisting Frame2 Number of Stories 6 stories3 Floor height 3 m4 Imposed Load 5 kN/m

5 Materials Concrete M20 and Grade of Steel Fe-415

6 Size of Columns 450 x 450 mm

7 Size of Beams 300 x 300 mm

8 Specific weight of RCC 25 kN/m3

9 Specific weight of infill 20 kN/m3

10 Type of soil Rock

40

4.1 STRUCTURAL SYSTEMS OF THE BUILDING:

The foundation system is isolated footings with a depth of the footing 1.5 m and

sizes of the footings are 1.5 m x 1.5 m. The column and beam dimensions are detailed in

Table 4.

In the calculation of dead loads, the unit weights of the materials have been taken

according to the IS-875 (Part 1) and the seismic load considerations according to IS-

1893 (Part 1): 2002[20].

Figure 4.1 Asymmetrical Plan and isometric view of a building

41

Figure 4.2. Symmetrical Plan and isometric view of a building

The building is a six storey building located in zone II. Tables 4.0 and Table 4.2

present a summary of the building parameters.

Table 4.2 General data collection and condition assessment of buildingSl.No. Description Information Remarks1 Building height 18 m Above the ground2 Number of storeys above ground 6 ----3 Number of basements below ground 0 ----4 Open ground storey No ----5 Special hazards None ----6 Falling hazards Parapet wall ----

7 Type of building Regular frame IS 1893:2002[20]

Clause 7.18 Horizontal floor system Beams ----9 Software used SAP2000 v15 ----

4.2 ANALYSIS USING EQUIVALENT STATIC ANALYSIS METHOD AND

DESIGN ACCORDING TO IS- 1893: 2002[20]

The load cases for the structure is defined in the SAP 2000 [21] as Dead load, Live

load and Earthquake loads in x and y directions according to IS1893:2002. The lumped

mass is given on each floor using floor diaphragm with Dead load and 25% of live load.

Following fig 4.3 & 4.4 show the details of the analysis case. The static analysis is run

42

and the members are designed for the analysis results obtained, by IS-456:2000 design

specifications.

Figure. 4.3. Load cases defined for analysis.

43

Figure.4.4 Details of mass source and analysis cases to rum

44

Figure 4.5 Reinforcement details of beams and columns

4.3 PUSHOVER ANALYSIS CASES

Pushover analysis provides important features of structural response, such as the

initial stiffness of the structure, total strength and yield displacement. In addition, it

provides reasonable estimates for the post peak behavior of the structure. Lateral load

patterns involved in determining pushover curve of the building structure should

represent characteristics of inertia forces developed in the building under the input

ground motion excitation. Fixed load patterns suggested by seismic codes are usually

sufficient for the determination of the envelopes of the building inertia forces. These load

patterns have invariant distribution through the height of the building but gradually

increase until a target value of roof displacement is reached. The displacement at an

ultimate state of the building, when a global mechanism exists, is set as the target

displacement for comparison purposes. SAP2000 [] software is used to perform the

pushover analysis of buildings using displacement control strategy, where gravity loads

(set as nonlinear) of each building are applied prior to the pushover analysis.

Fixed lateral load patterns used to push the buildings are chosen such that they

represent the common patterns recommended by the seismic regulation provisions of

45

FEMA-273[2]. Additionally, a fixed pattern based on the first mode of vibration for the

building considering an ultimate deformed configuration is investigated.

The common lateral load patterns of FEMA-273[2] are as follows:

The uniform load pattern (ULP)

The equivalent lateral force pattern (ELP)

The first mode load pattern (FLP)

Deformed shape of the building (DLP) (Adaptive Pushover Analysis)

The DLP is used for performing the analysis by the SAP2000 [21] as a default

method, which is incorporated from the ATC-40 [1] document.

For defining the POA case we start with the assignment of the hinges to the members of

the structure. The SAP2000 [21] package gives the choice to the user for selecting or

defining the type of hinges required for analysis.

Figure. 4.6 Frame hinge properties for a typical member.

46

Figure. 4.7 Hinge assignments at the ends of the beams and columns.

Default hinge definitions according to the FEMA-356 guidelines have been provided at

the ends, where the formation of the potential plastic hinges is more probable for beams

and columns with degree of freedom as M3 and the shear value for the hinge is taken

from the Dead load case. The hinges are set so that they drop the load after reaching the

point E of the performance level.

Fig. 4.8. Performance levels according to hinge states.

Force-displacement or moment-rotation curve for a hinge definition used in SAP2000 [21]

is referred to as a plastic deformation curve. The plastic deformation curve is

characterized by the following points as:

47

Point A represents the origin.

Point B represents the yielding state. No deformation occurs in the hinge up to

point B, regardless of the deformation value specified for point B. The

displacement (rotation) at point B will be subtracted from the deformations at

points C, D, and E. Only the plastic deformation beyond point B will be exhibited

by the hinge.

Point C represents the ultimate capacity for pushover analysis.

Point D represents the residual strength for pushover analysis.

Point E represents total failure. Beyond point E the hinge will drop load down to

point F directly below point E on the horizontal axis. If the users do not want the

hinge to fail this way, a large value for the deformation at point E can be

specified.

The user may specify additional deformation measures at point’s immediate

occupancy, life safety, and collapse prevention. These are informational measures that are

reported in the analysis results and used for performance-based design. They do not have

any effect on the behavior of the structure. Prior to reaching point B, the deformation is

linear and occurs in the frame element itself, not in the hinge. Plastic deformation beyond

point B occurs in the hinge in addition to any elastic deformation that may occur in the

element. When the hinge unloads elastically, it does so without any plastic deformation,

i.e., the unloading path is parallel to line A-B. Curve scaling permits that the force-

displacement (moment-rotation) curve of the hinge can be defined by entering

normalized values and specify the required scale factor. Often, the normalized values are

based on the yield force (moment) and yield displacement (rotation), so that the

normalized values for point B on the curve would be (1,1). Any deformation given from

A to B is not used. This means that the scale factor on deformation is actually used to

scale the plastic deformation from point B to C, C to D, and D to E. However, it may still

be convenient to use the yield deformation for scaling. When default hinge properties are

used, the program automatically uses the yield values for scaling. These values are

calculated based on the frame section properties and the yield stress provided for the

element material. In this study, only two types of hinges are used to simulate the plastic

hinge formation through the non-linear behavior of the structure. The first is the coupled

48

axial and moment hinge which is assigned to the column elements. The hinge properties

of this type are created based on the interaction surface that represents where yielding

first occurs for different combinations of axial force, minor moment and major moment

acting on the section. The second type is the moment hinge which is assigned to the beam

elements. The hinge properties of this type can be considered as a special case of the first

type.

After assignment of the hinges, the pushover cases are defined under the

conjugate monitored displacement. The Dead loads case is assigned to non-linear static

case, and the pushover case data is given as shown in the Fig 4.9. The top displacement

of the structure is set to 1% of the total height of the structure at one of the top storey

nodes. The degree of freedom (U1) for the structure is in the direction of the application

of the pushover loads i.e., along the x-axis.

The RSM and Load cases are set to run at first, the structure is then designed according to

IS-456: 2000 and then pushover cases are run on the designed structure, for the analysis

results.

Figure 4.9 Pushover Analysis Case of 1st mode

49

4.4 Response Spectra Analysis:

To perform the extended N2 method, Pushover analysis and response spectrum

analysis of a building has to be carried out. From the results of these two analysis “β”

correction factor is calculated. Response spectra analysis is performed using the chamba

earthquake spectra, for which accelerogram has to be first converted to spectra, for this

purpose Seismospect software[] is used. In the figure 4.10- 4.14 showing accelerogram is

being converted to spectra and uploaded to SAP2000. Response spectra analysis is

performed and results are obtained from it.

Figure 4.10 chamba earthquake data into seismospect

50

Figure 4.11 seismospect converted accelerogram into response spectra.

Figure 4.12 Response spectra of chamba earthquake in SAP2000

51

Figure 4.13 performing response spectra analysis of building

Figure 4.14 Deformed shape of building after response spectrum analysis

52

4.5 Time History Analysis cases:4.5.1 Modeling:

4.5.1.1 Natural Earthquake Records

The behaviour of multi-story buildings during an earthquake event depends on the

Characteristics of the earthquake ground motion such as the amplitude of its peak ground

acceleration (PGA), as well as the duration and frequency content of the earthquake.

Also, the type of foundation and the soil characteristics at site have important influence

on the behaviour of the building. In addition, the building characteristics including the

stiffness, damping and modal properties play a significant role in the way the building

responds to the earthquake excitation.

A main factor in earthquake-resistant design is the amplitude of the pseudo

acceleration which corresponds to the fundamental period of the building using the

response spectrum curve for the earthquake ground motion. If the pseudo-acceleration

corresponding to the fundamental period of the building is high, then significant inertia

forces will be developed in the building during the earthquake event and vice-versa.

Hence, the response spectra of a number of earthquakes need to be investigated for

adequate evaluation of the building behavior under different seismic actions. The

structural performance of the reinforced concrete and steel buildings in this study is

estimated based on considering natural ground motion records chamba. These records

were obtained from the strong motion database of the Earthquake Station chamba in

Himachal Pradesh, Fig show the acceleration time history for Earthquake.

4.5.2. Chamba Earthquake Data

Fig4.15: Acceleration time-history of chamba Earthquake

53

4.5.3 Sample collection of Acceleration data points -.579E-02 -.134E-01 0.962E-02 0.959E-02 0.331E-01 -.918E-02 0.126E-01 0.476E-01-.454E-01 -.944E-01 -.136E-01 0.490E-01 0.979E-01 -.283E-01 -.134E+00 -.141E-010.320E-01 0.394E-01 0.198E-01 -.364E-01 0.301E-02 0.430E-01 -.614E-01 -.330E-010.115E+00 0.870E-01 -.587E-01 -.121E+00 -.282E-01 0.831E-01 0.117E+00 0.306E-01-.116E+00 -.425E-01 0.419E-01 -.330E-01 -.254E-01 0.543E-01 0.120E+00 0.468E-01-.128E+00 -.117E+00 0.409E-01 0.366E-01 -.303E-01 -.690E-01 -.779E-02 0.421E-010.320E-01 0.120E-01 0.203E-01 0.192E-01 0.204E-01 -.393E-01 -.830E-03 0.809E-010.205E-01 -.753E-01 -.352E-01 0.436E-01 0.817E-01 0.531E-01 -.476E-02 -.428E-01-.455E-01 -.163E-01 -.167E-02 0.262E-01 0.270E-03 -.202E-01 -.631E-01 -.859E-01-.155E-01 0.163E-01 0.723E-01 0.633E-01 0.161E-01 -.114E-01 -.840E-02 -.858E-02-.238E-02 -.242E-02 -.196E-01 0.486E-02 -.146E-01 -.544E-01 0.286E-01 0.790E-01

4.6 Modeling of building using time history analysis in SAP

Collecting of Earth Quake data (Chamba).

Defining Time History Function

Import the data of ground accelerations.

Display graph

Defining time history case.

Following are the general sequence of steps involved in performing linear time history

analysis using SAP2000 in the present study:

A two or three dimensional model that represents the overall structural behavior is

created. For reinforced concrete elements the appropriate reinforcement is provided for

the cross sections.

Beams and columns are modelled as linear finete elements with lumped plasticity

at start and end of each element. Beams and columns are modelled by 3d frame elements.

While modelling the beams and columns, the important properties to be assigned are

cross sections, reinforcement details and types of materials used. Plinth beams should

54

also be modelled as frame element. Columns at the end of foundation can be modelled by

considering the degree of fixity provided by the foundation. All the beam column joints

are considered as rigid.

4.7 UPLOAD ACCELEROGRAMS TO SAP2000

Define the accelerogram in SAP2000

An accelerogram is basically the time-history of the acceleration experienced by

the ground in a given direction during a seismic event

We need to input the accelerogram as a generic function defined in SAP2000

starting from a TXT file

Once the mechanical model of the structural system under investigation has been

created, SELECT from the menu bar:

DEFINE -> FUNCTIONS -> TIME HISTORIES. 

Define the accelerogram in SAP2000

• The accelerogram must be uploaded to SAP2000 as FUNCTION FROM FILE

, an option which can be selected from the drop-down list

Define the accelerogram in SAP2000. We can select:

Name of the function (e.g. Chamba earthquake)

Location of the file by using the button BROWSE.

Number of lines to skip (1 for chamba database)

Number of points per line (8 for chamba database)

55

Fig 4.16: Define the time history function from file.

Fig 4.17: Defining the function name and displaying function graph:

Define the accelerogram in SAP2000

By clicking on DISPLAY GRAPH we can visually check the waveform.

Define Loads Cases

56

A load case defines how loads are to be applied to the structure, and how the

structural response is to be calculated. An unlimited number of named load cases of any

type can be defined in SAP2000. The load case(s) is selected when the model is analyzed.

Note: Load patterns by themselves do not create any response (deflections, stresses, and

so forth). A load case must be defined to apply the load.

Before defining load cases, define any load patterns or functions that will be needed:

For seismic analysis, the ground-acceleration loads are already built-in and do not

need to be defined. For all other loads, use the Define menu > Load Patterns

command to define the loads to be applied.

For response-spectrum analysis, use the Define menu > Response Spectrum

Functions command to create the necessary input acceleration functions.

For time-history analysis, use the Define menu > Time History Functions

command to input the time functions

Define the accelerogram in SAP2000

• In order to be used in a seismic simulation, the waveform previously defined

must be properly assigned to an ANALYSIS CASE

• To do this, SELECT from the menu bar:

• DEFINE -> ANALYSIS CASES...

Define the accelerogram in SAP2000

• The analysis case in presence of accelerogram is a TIME HISTORY , which

can be selected from a drop-down list after clicking on the ADD NEW

CASE... button

Define the accelerogram in SAP2000

• The appropriate analysis in presence of accelerogram is a TIME HISTORY

case, which can be selected from the drop-down list

57

Define the accelerogram in SAP2000

• This leads to a more complicated windows, in which few parameters must be

input

Define the accelerogram in SAP2000

In particular:

• The load type is an acceleration of the ground (first drop-down list)

• For a planar model, the ground shaking is generally assumed to happen in the

X direction, i.e. Load name U1 (second drop-down list)

• The records from the ground motion are given in units of g, so the scale factor

would be 9.81 m/s 2 or 981 cm/s 2 , depending on the units selected in the

model of SAP2000

• The number of time steps should be given by the duration of the accelerogram

divided by the sampling time: For example: 40 s / 0.01 s = 4,000 steps

Fig 4.18: Defining the modal case window

58

Fig 4.19: Showing the run analysis window.

RUN THE ANALYSIS AND DISPLAY THE RESULTS

Run the dynamic analysis

Once the accelerogram has been defined in SAP2000, it is possible to run the time-

history analysis as any other type of analysis.

Run the dynamic analysis

o Once the analyses are complete, results from tables and diagrams

Display the results

Analysis type Linear: In a linear time history analysis, all objects behave

linearly. Only the linear properties assigned to link elements are considered in a linear

time history analysis.

Modal damping Use the Modal Damping form to specify or modify the modal

damping. For convenience, specify that a damping applies to all modes and then, if

desired, overwrite the damping for any mode(s). The following three options are

available:

Damping for All Modes: Enter the damping for all modes in this form. This is a percent

critical damping. For example a damping that is 5% 0f critical damping is entered as

0.05.

59

Number of output time steps: The number of output time steps is the number of equally

spaced steps at which the output results are reported. Do not confuse this with the number

of time steps in input time history function. The number of output time steps can be

different from the number of time steps in input time history function. The number of

output time steps times the output time step size is equal to the length of time over which

output results are reported.

Output time step size: The output time step size is the time in seconds between each of

the equally spaced output time steps. Do not confuse this with the time step size in your

input time history function. The number of output time step size can be different from the

input time step size in your input time history function. The number of output time steps

time the output time step size is equal to the length of time over which output results are

reported.

4.8 Load Assignments

Load: The Load may either be a defined static load case, acc dir 1, acc dir 2 or acc dir 3.

The three accelerations (acc dir 1, acc dir 2 and acc dir 3) are ground accelerations in the

local axes directions of the time history. Positive acc dir 3 corresponds to the positive

global Z direction always. See the discussion of the Angle item in this area for

information about acc dir 1 and acc dir 2. When you specify one of these three ground

accelerations your input function defines how the ground acceleration varies with time.

Function: In this the function name display which is given in the time history function.

So we select the function according to our use.

Scale factor: The Scale Factor item is used as a multiplier on the input function values.

The units for the scale factor depend on the type of load specified in the Load drop-down

box. If the load is specified as a ground acceleration (that is, acc dir 1, acc dir 2 or acc dir

3), this scale factor has units of Length/seconds2. If the load is a static load case, this

scale factor is unit less. The scale factor can be any positive or negative number, or zero.

60

Arrival time: The arrival time is the time that a particular load assignment starts.

Assume that you want to apply the same ground acceleration that lasts 30 seconds to your

building in global X and global Y directions. Further assume that you want the ground

acceleration in the global Y direction to start 10 seconds after the ground acceleration in

the global X direction begins. In that case you could specify an arrival time of 0 for the

load assignment for the global X direction shaking and arrival time of 10 for the load

assignment for the global Y direction shaking.

The arrival time can be zero or any positive or negative time. The time history analysis

for a given time history case always starts at time zero. Thus if you specify a negative

arrival time for a load assignment, any portion of its associated input function that occurs

before time zero is ignored. Before we run analysis we set the parameters to run dynamic

analysis.

4.9 Summary:

In this chapter the building description and plan of the building was discussed.

The structural systems of the building and general data collected, geological and

geotechnical data collected were presented in this chapter. Modeling of the building using

SAP2000 was presented and steps for uploading the accelerogram were presented.

61

RESULTS AND DISCUSSION

62

CHAPTER-V

RESULTS AND DISCUSSION

5.0 GENERAL

Results obtained from analysis and design investigations are the Lateral

displacements; Pushover Curves and comparing base shears.

5.1 RESULTS

For ease, results are briefly described and classified into three categories i.e. pushover

analysis results, N2 method results and Time history results.

5.2 PUSHOVER ANALYSIS RESULTS

Both the building models are analyzed using non-linear static (pushover) analysis. At

first, the pushover analysis is done for the gravity loads (DL+0.25LL) incrementing under

load control. The lateral pushover analysis (in X & Y-direction) is followed after the

gravity pushover, under displacement control. The building is pushed in lateral directions

until the formation of collapse mechanism. The capacity curve (base shear versus roof

displacement) is obtained from the analysis.

5.3 Pushover analysis results of Symmetrical building:

Figure 5.1 shows a capacity curve (pushover curve) of a symmetrical structure in

X-direction, the following curve is obtained under displacement control for in Mode 2.

Initially monitored displacement given to the structure is 0.72m (5 % of structure height).

It has been observed that the maximum base shear 1132 KN is obtained at a displacement

of 0.52 m and this is further applied for calculating target displacement using N2 method.

63

Figure 5.1 Pushover Curve for Symmetrical structure in X direction

Figure 5.2 shows a capacity curve (pushover curve) of a symmetrical structure in

Y-direction, the following curve is obtained under displacement control in Mode 1.

Initially monitored displacement given to the structure is 0.72m (5 % of structure height).

It has been observed that the maximum base shear 1073 KN is obtained at a displacement

of 0.53 m and this is further applied for calculating target displacement using N2 method.

Figure 5.2 Pushover Curve for Symmetrical structure in Y direction

64

5.4 Pushover Results of Asymmetrical building:

Figure 5.3 shows a capacity curve (pushover curve) of a Asymmetrical structure

in X-direction, the following curve is obtained under displacement control in Mode 2.

Initially monitored displacement given to the structure is 0.72 m (5 % of structure

height). It has been observed that the maximum base shear 1124 KN is obtained at a

displacement of 0.49 m but it also observed that the structure is failing at earlier

displacement of 0.57 m instead of 0.72 m. So, with this observation we can say that the

failure occur in Asymmetrical structure earlier when compared to Symmetrical structures.

This type of earlier failure in Asymmetrical structures is due to lateral force with

torsional force effect.

Figure 5.3 Pushover Curve for Asymmetrical structure in X direction

Figure 5.4 shows a capacity curve (pushover curve) of a Asymmetrical structure

in Y-direction, the following curve is obtained under displacement control in Mode 1.

Initially monitored displacement given to the structure is 0.72 m (5 % of structure

height). It has been observed that the maximum base shear 1062 KN is obtained at a

displacement of 0.52 m but it also observed that the structure is failing at earlier

displacement of 0.68 m instead of 0.72 m. So, with this observation we can say that the

failure occur in Asymmetrical structure are critical when compared to Symmetrical

65

structures. As it has been seen that building plan in asymmetrical in X-direction and

Symmetrical in Y-direction, for this symmetry in y direction the displacement at failure

and monitored displacement are very near to each other.

Figure 5.4 Pushover Curve for Asymmetrical structure in Y direction

5.5 Comparison of pushover for Symmetrical and Asymmetrical buildings

0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 7.00E+00 8.00E+000

200

400

600

800

1000

1200

Asymmetry -Xsymmetry-X

Fig. 5.5: Comparison of pushover curves of both buildings in X direction

66

Figure 5.5 shows capacity curve (pushover curve) for Symmetrical structure in X-

direction with maximum base shear of 1132KN at displacement of 0.52 m and

Asymmetrical structure in X-direction with maximum base shear of 1124 KN at

displacement of 0.49 m respectively. The variation of base shear and displacement is due

to Asymmetry in X-direction in both structures.

0 1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

Asymmetry -Ysymmetry -Y

Fig. 5.6: Comparison of pushover curves of both buildings in Y direction

Figure 5.6 shows capacity curve (pushover curve) for Symmetrical structure in Y-

direction with maximum base shear of 1073KN at displacement of 0.53 m and

Asymmetrical structure in X-direction with maximum base shear of 1062 KN at

displacement of 0.52 m respectively. The small variation of base shear and displacement

is due to symmetry in Y-direction in both structures.

5.6 Conventional pushover analysis results of symmetrical building:

Both the building models are analyzed using non-linear static (pushover) analysis.

At first, the pushover analysis is done for the gravity loads (DL+0.25LL) incrementally

under load control. The Conventional pushover analysis (in X & Y-direction) is followed

after the gravity pushover, under full load pattern. The building is pushed in lateral

directions until the formation of collapse mechanism. The capacity curve (base shear

versus roof displacement) is obtained from the analysis.

67

Figure 5.7 Conventional Pushover Curve for Symmetrical structure in X direction

Figure 5.7 shows a conventional pushover curve for Symmetrical structure in X-

direction. To obtain curve joint 7 displacement is obtained against base shear of structure.

As we can see from figure maximum base shear is 847 KN with a displacement of 0.0947

m.

Figure 5.8 Conventional Pushover Curve for Symmetrical structure in Y direction

68

Figure 5.8 shows a conventional pushover curve for Symmetrical structure in Y-

direction. To obtain curve joint 7 displacement is obtained against base shear of structure.

As we can see from figure maximum base shear is 778 KN with a displacement of 0.0930

m.

5.7 Conventional pushover results of Asymmetrical building:

Figure 5.9 Conventional Pushover Curve for Asymmetrical structure in X direction

Figure 5.9 shows a conventional pushover curve for Asymmetrical structure in X-

direction. To obtain curve joint 169 (centre of mass) displacement is obtained against

base shear of structure. As we can see from figure maximum base shear is 819 KN with a

displacement of 0.096 m.

69

Figure 5.10 Conventional Pushover Curve for Asymmetrical structure in Y direction

Figure 5.10 shows a conventional pushover curve for Asymmetrical structure in

X- direction. To obtain curve joint 169 (centre of mass) displacement is obtained against

base shear of structure. As we can see from figure maximum base shear is 787 KN with a

displacement of 0.096 m.

0 0.02 0.04 0.06 0.08 0.1 0.120

100

200

300

400

500

600

700

800

900

Assymmetrical in X di-rection

Symmetrical in X direction

Fig. 5.11: Comparison of conventional pushover curves of both buildings in X

direction

70

Figure 5.11 shows comparison of conventional pushover curves of Symmetrical

and Asymmetrical Structure. Two pushover curves are overlapped with each other; in this

case it has been observed that variation in base shear of Asymmetrical building with 819

KN and Symmetrical building with 847 KN. And variation is due to Asymmetrical in

plan in X-direction.

0.00E+00 2.00E-02 4.00E-02 6.00E-02 8.00E-02 1.00E-01 1.20E-010

100

200

300

400

500

600

700

800

900

Assym in Y directionsymmetrical in Y direction

Fig. 5.12: Comparison of conventional pushover curves of both buildings in Y

direction

Figure 5.12 shows comparison of conventional pushover curves of Symmetrical

and Asymmetrical Structure. Two conventional pushover curves are overlapped with

each other; in this case it has been observed that there is very small variation in base

shear due to symmetry in Y-direction. The observed base shear for Asymmetrical

structure is 787 KN and Symmetrical structure is 778 KN with a difference percentage of

1%.

5.8 N2 method Results:

The capacity curves obtained from the pushover analysis of the Asymmetrical and

symmetrical structure, under invariant force distributions proportional to the first and

second mode shapes are presented in figures (Fig.5.1 to Fig.5.4) The capacity curve of

each mode was then converted to the force-deformation relationship of an equivalent

71

single-degree-of-freedom (SDOF) system (D* vs. F*) in figure 5.14. The single-degree-

of-freedom (SDOF) curve is converted to bilinear curve shown in figure 5.15. Now this

curve is passed over a demand curve which is obtained from IS1893-2000 and from this

overlapping of two curves target displacement is calculated. with the new target

displacement pushover curve is obtained.

0 0.05 0.1 0.15 0.2 0.25 0.30

200

400

600

800

1000

1200

pushover curve

Figure 5.13 Pushover Curve for Asymmetrical structure in X direction

0 0.05 0.1 0.15 0.2 0.250

100

200

300

400

500

600

700

800

900

1000D* vs F*

D* vs F*

Figure 5.14 Pushover Curve for Asymmetrical structure of Equivalent SDOF

72

0 0.05 0.1 0.15 0.2 0.250

100

200

300

400

500

600

700

800

900

1000 bilenar curve

bilenar curve

Figure 5.15 Bilinear Pushover Curve for Asymmetrical structure of Equivalent SDOF

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

bilinear pushover curve

Response spectra

Figure 5.16 Demand spectra for ground motions and Capacity diagram

73

5.9 Extended N2 method:

Extended N2 method is carried out to calculate the correction factor for

Asymmetrical Building. βis called correction factor.

For each combination and for each frame the following normalized displacement is

computed:

η MRSA , i= µ MRS A ,iµ MRSA ,CM =1.06

Where µMRSA ,i is the “i” frame top displacement and µMRSA ,CM is the CM

top Displacement; consequently,η MRSA , i are computed for each frame and their

maximum is assumed as the referenceη MRSA , i;

For model the NLSA with the maximum target displacement is considered; consequently,

maximum normalised displacements are obtained for each frame:

η NLSA ,i= µ NLSA ,iµNLSA , CM = 1.02

Where µNLSA , i is the “i” frame top displacement and µNLSA , CM is the CM

top displacement;

For each frame, 4 correction factors are computed:

β=η MRSA , iη NLSA , i

=1.03

Where η MRSA , i and η MRSA , i are the reference value are considered;

74

5.10 Time history Analysis Results

Base shear-versus-roof displacements curve is an important parameter to

understand the difference between two buildings with different level of asymmetry.

Results of Chamba ground motion data are presented separately for convenience. Figs.

5.17 and 5.19 represents time vs. base shear for chamba ground motion, Figs. 5.18 and

5.20 presents the time vs. roof displacement for chamba earthquake ground motion.

Fig. 5.17: Time-versus- Base shear data for Chamba Earth Quake ground motion for

symmetrical building

75

Fig. 5.18 : Time-versus- roof displacement data for Chamba Earth Quake ground motion

for Symmetrical building

Fig. 5.19: Time-versus- Base shear data for Chamba Earth Quake ground motion for

Asymmetrical building

76

Fig. 5.20: Time-versus- roof displacement data for Chamba Earth Quake ground motion

for Asymmetrical building

Table 5.1 showing the values of base shear and roof displacement of Asymmetrical

and symmetrical building

Base Shear(KN)

Roof Displacement(m)

Asymmetrical Building 842 0.015

Symmetrical Building 775 0.0046

5.11 Comparison of results of Pushover, N2 method and Time history analysis

77

Asymmetrical Symmetrical0

100200300400500600700800900

1000842

775800 804916 945.52557

Time history analysis of Asymmetrical and symmetrical buildingn2 methodconventional pushover analysis

Figure. 5.21: comparison of Base shear of Time history, Conventional pushover and

N2 method

1% 2% 4%0

200

400

600

800

1000

12001104 1136

842 842

Pushover analysis Time History Analysis

Fig. 5.22: Comparing base shears of Time history analysis and pushover analysis of

Asymmetrical building

78

5.3 Summary

This chapter copes with the numerical study and presentation of results of

pushover analysis and N2 method for both the buildings under study; the results are

compared and are represented in the form of graphs. The results were studied and based

on the study, the conclusions were drawn. The detail conclusions for the present study are

given in the next chapter

79

CONCLUSIONS

80

CHAPTER-VI

CONCLUSIONS

6.1 GENERALThe use of Nonlinear Static Procedures (NSPs) for the seismic

assessment of plan irregular buildings is challenging. The most common pushover-based approaches have led to adequate results in regular buildings, and hence, there is a need to improve and verify the validity of such methods on the assessment of irregular structures.

Fajfar and his team have developed the Extended N2 method [4, 8] from orignal N2 method, which is able to capture the torsional behaviour of plan-asymmetric buildings. This procedure is based on the application of correction factors to the pushover results obtained with the N2 method. The correction factors depend on a dynamic elastic analysis and on a pushover analysis.

The main objective of this chapter is to assess the seismic behaviour of the buildings, evaluating the accuracy of the pushover analysis, orignal N2 method and Extended N2 method with non linear dynamic analysis (Time history analysis).6.2 CONCLUSIONSBased on the study of the structural performance of the model,

It is observed from figure (5.5) that the Asymmetrical building collapse earlier than the symmetrical building

It is been observed from the figure (5.21) that conventional pushover analysis over estimates the base shear. Whereas N2

81

method is predicting the base shear which is nearly equal to non linear dynamic analysis.

Figure (5.22) shows comparison of base shear of pushover analysis with 1% and 2% monitored displacement of building height with non linear dynamic analysis of Asymmetrical building. It is observed that pushover analysis is over estimating the base shear with 1% and 2% of monitored displacement of building height. For Asymmetrical cases judgment through pushover analysis is not appropriate.

The Extended N2 method produces results in a good fashion when compare to the nonlinear dynamic results for all the buildings analyzed and through the seismic intensity tested. This method shows, for these case studies, a much better performance in estimating the torsional behavior of the buildings than the conventional pushover analysis and original N2 method.

6.3 SCOPE FOR FURTHER STUDY

In the present study, the analysis has been carried out for the G+5 storey

buildings. This study can further be extended for tall buildings.

In the present study asymmetric plan in one direction is considered, this study can

be further extended to asymmetric plan in both directions.

In the present analysis for symmetrical elevation carried out and it can be further

extended for asymmetrical elevation.

N2 method can be further studied and verify for higher modes of building.

82

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