computer science

INTRODUCTION

1

CHAPTER-I

INTRODUCTION

1.1 GENERAL

The general philosophy of earthquake resistant design for buildings is:

(i) to prevent non-structural damage in minor earthquakes which may occur

frequently in the service life of a structure; (ii) to prevent structural damage and

minimize non-structural damage in moderate earthquakes which may occasionally

occur; and (iii) to avoid collapse or serious damage in major earthquakes which may

rarely occur. However, codes only require buildings to be designed for one ultimate

force level. In effect, buildings are explicitly designed only for the third criterion. The

extensive damage and unprecedented economic losses caused by the 2001 Bhuj

Earthquake, have stimulated designers and owners to consider how the design

philosophy outlined above can be implemented to meet criteria (i) and (ii), and to

protect a building owner's economic investment.

The equivalent lateral force procedure for seismic design, as embodied in building

codes of the India[15] and most other countries, is based on implicit consideration of

inelastic structural response in the event of severe earthquakes. This approach has a

number of deficiencies as summarized by Younghui Roger Li [5]

(1) The internal forces determined from elastic analysis under code-specified static forces

are quite different from those produced during the inelastic earthquake response of the

structure.

(2) Although inelasticity may actually occur only at certain levels and in certain

locations, there is no way of determining those locations or the extent of inelasticity in

any of these locations through elastic analysis under code-prescribed static loads. As a

result, special ductility details must be provided in every structural member and every

connection. Also, there is no way to ascertain that the ductility provided through

conformance with prescribed detailing requirements will always suffice.

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(3) Elastic story drifts under code-specified forces, amplified by such multiplication

factors as may be prescribed, will be quite different from the actual inelastic story

drifts. Thus, keeping the amplified story drifts within prescribed limits may not

result in the intended damage control and safety against instability.

Information on the amount and distribution of internal forces and deformations

in yielding structures can be obtained through inelastic response history analyses of

structures subjected to earthquake motions. Over the past 20 years, several nonlinear

analysis programs have been developed. However, the applicability and accuracy of

those programs need to be evaluated before they can be used in routine design.

Structural acceleration records produced by earthquakes are one of the few

sources of quantitative information about the response of large structures to

damaging, or potentially damaging, earthquakes. They not only allow modern design

practices in earthquake engineering to be checked, but also they can play an important

role in research to improve these practices. As a consequence, considerable effort has

been made by earthquake engineers to record structural accelerations in addition to

ground accelerations during an earthquake. Over the last two decades, or so, these

efforts have been rewarded by the increasing number of sets of strong-motion

acceleration records which have been obtained in buildings. These data provide

excellent resources for calibrating the accuracy of those nonlinear programs.

1.2 SEISMIC ANALYSIS

The nonlinear time history analysis can be regarded as the most accurate method

of seismic demand prediction and performance evaluation of structures. However, this

method requires the selection of an appropriate set of ground motions and also a

numerical tool to handle the analysis of the data which is in many cases

computationally expensive. In this way, the nonlinear static analysis (pushover) can

simply be introduced as an effective alternative technique. In this method, structural

performance is evaluated using static non-linear analysis and estimation of the

strength and deformation capacities of the structure. The results are compared with

the demands at the corresponding performance levels. The use of the nonlinear static

analysis, named hereafter, pushover analysis, dates back to the 1970’s, but only after 3

gaining importance during the last 10-15 years have dedicated publications started to

appear on the subject. Initially, the majority concentrated on discussing the range of

applicability of the method and its advantages and disadvantages, compared to elastic

or non-linear dynamic procedures. Nonlinear static pushover analysis has some

limitations, such as the inability to include higher mode effects. The importance of

higher modes was discussed in the ATC-40 publication[3]. In an attempt to consider

higher mode effects, Paret and Sasaki et al[8] suggested the simple, yet efficient,

Multi-Mode Pushover procedure (MMP). This method comprises several pushover

analyses under forcing vectors representing the various modes deemed to be excited

in the dynamic response.

In 2001, Chopra and Goel[17] developed a method called multimode pushover

analysis, primarily for estimating inter-story drifts in framed structures. Later, the

ATC-55 project attempted to apply this procedure toward estimating story shears and

overturning moments, in addition to floor-displacements and inter-story drifts. The

project encountered some problems, pointing out reversals in the third mode pushover

of the three story steel moment-resistant frame building. This setback indicates that

increments in roof displacement are in a direction opposite to the base-shear, which

may happed depending on the mechanism that gets developed within the structure.

Recognizing that the roof displacement may not always be the best index as basis for

establishing the properties of so-called “Equivalent Single Degree of Freedom”

(ESDF) systems, Hernandez-Montes et al[16] developed an alternative index, known

as Energy-based Displacement. Subsequently, Tjhin et al[18] showed that using

energy-based displacement instead of roof displacement estimation to establish the

properties of the first mode ESDF system.

1.3 OBJECTIVE OF THE STUDY

The present work aims at the following objectives:

To study the effects of higher mode of a structure on its response.

To perform energy based pushover analysis (EBPOA).

To generate the Demand curve in energy based approach

Evaluation of performance point of the structure using the Energy based

Pushover analysis 4

1.4 SCOPE OF THE STUDY

The present work aims to demonstrate the effect of higher mode shapes on

performance of building and its importance in calculating the response of the same.

The procedure adopted in the present study is on the similar lines to what suggested

by Prof. Hernandez-Montes[16], Prof. Kotanidis[20], Prof. M.J Hashemi and Prof.

M. Mofid[22]. The building studied in this work is a reinforced concrete moment

resisting frame designed for gravity and seismic loads using response spectrum

analysis. The structure is evaluated in accordance with seismic code IS-1893:2002 [15]

using Conventional Pushover analysis and Energy based POA with the help of the

ETABS 9.7.4[26] and SAP2000 V.15.0[27] .

1.5 ORGANIZATION OF THESIS

Chapter 1 is the discussion of different seismic analysis procedures adopted

for the structures and a brief introduction to the Nonlinear static pushover analysis.

The scope and objective of the study has also been discussed.

Chapter 2 deals with the various literatures that have been published on the

energy based pushover analysis and seismic evaluation techniques.

Chapter 3 covers the complete study on performance point using pushover

analysis and energy based pushover analysis and the procedure adopted in the present

study and also discusses the need for performance based design.

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

consideration and the various building data surveys done to gather the information for

modeling of the structure.

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

pushover analysis method for the current building under study.

Chapter 6 details the discussions drawn based on the present work and the

scope for the further study.

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LITERATURE REVIEW

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CHAPTER-II

LITERATURE REVIEW

2.1 INTRODUCTION

The general philosophy for earthquake-resistant design of structures has

undergone some major changes in the past 15 years, following some of the most

devastating earthquakes all over the world. The prediction of the earthquake response

of a structure became more significant for the engineers to design the structures and

this became much easier with the availability of seismic data and software

enhancement.

Newer analysis methodologies are being proposed with focus on a realistic

characterization of seismic structural damage and its direct incorporation in the design

methodology. In addition, a major emphasis is given to the characterization of all the

uncertainties in the process of design. In general, these approaches are categorized

under performance-based seismic design (PBSD). The various ways of modeling

structural damage for PBSD lead to various design approaches. The most commonly

adopted approach for PBSD so far is the displacement-based design approach, where

the design criterion is set usually by a limit on the peak roof (inelastic) displacement,

the peak (inelastic) inter-story drift, or the peak ductility demand, etc.

Generally when the earthquake hits a structure, before actual failure of the

building, it passes both the linear and nonlinear stages. The linear analysis has been

the source of interest of the researchers as mentioned before, but the non-linearity of

the structures has now taken the stage. The strength of the structure in the non-linear

yield is significant for the structures and could be utilized with application of several

limit states for a monitored performance and damage.

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As mentioned in various codes and papers by number of researchers

[Hernandez-Montes[16],S. Chandrashekaran[13], Anubhab Roy[13],Li Younghai[5]],

nonlinear time history analysis (NL-THA) can be regarded as the most accurate

method of seismic demand prediction and performance evaluation of structures. In an

ideal world there would be no debate about the proper method of demand prediction

and performance evaluation at low performance levels other than inelastic time

history analysis that predicts with sufficient reliability the forces and cumulative

deformation (damage) demands in every element of the structural system is the final

solution. The implementation of this solution requires the availability of a set of

ground motion records (each with three components) that account for the uncertainties

and differences in severity, frequency characteristics, and duration due to rupture

characteristics and distances of the various faults that may cause motions at the site. It

requires further the capability to model adequately the cyclic load deformation

characteristics of all important elements of the three-dimensional soil-foundation

structure system, and the availability of efficient tools to implement the solution

process within the time and financial constraints imposed on an engineering office.

There is a need to work towards this final solution, but we also need to

recognize the limitations of today’s state of knowledge and practice, at this time none

of afore mentioned capabilities have been adequately developed and efficient tools for

implementation do not exist. Recognizing these limitations, the task is to perform an

evaluation process that is relatively simple, but captures the essential features that

significantly affect the performance goal. In this context, the accuracy of demand

prediction is desirable, but this is hardly possible since neither seismic input nor

capacities are known with accuracy.

The Nonlinear Static Analysis named here after Pushover Analysis (POA) had

been simply introduced as an effective alternative technique to the NL-THA. In this

method, structural performance is evaluated using static nonlinear analysis for

estimation of the strength and deformation capacities of the structure. The use of the

POA dates back to the 1970's, but only after gaining importance during the last 10-15

years have dedicated publications started to appear on the subject. Initially, the

majority concentrated on discussing the range of applicability of the method and its

advantages and disadvantages, compared to elastic or non-linear dynamic procedures. 9

2.2 NON-LINEAR STATIC PUSHOVER ANALYSIS

The ATC-40[3] document has been a source of great knowledge for researchers

working on the NSPs (Capacity Spectrum Method, Coefficient Method etc.). Excerpts

from the documents that would give a better picture of the pushover analysis have

been presented in the following paragraphs discussing the background and the

development of the performance based analysis.

The essence of virtually all seismic evaluation procedures is a comparison

between some measures of the "demand" that earthquakes place on a structure to a

measure-of the "capacity" of the building to resist. Traditional design procedures

characterize demand and capacity as forces. Base shear (total horizontal force at the

lowest level of the building) is the normal-parameter that is used for this purpose. The

engineer calculates the' base shear demand that would be generated by a given

earthquake, or intensity of ground motion, and compares this to the base shear-

capacity of the building. The capacity of the building is an estimate of a base shear

that would be "acceptable." If the building were subjected to a force equal to its base

shear capacity some deformation and yielding might occur in some structural

elements, but the building would not collapse or reach an otherwise undesirable

overall level of damage. If the demand generated by the earthquake is less than the

capacity then the design is deemed acceptable.

The first formal seismic design procedures recognized that the earthquake

accelerations would generate forces proportional to the weight of the building. With

the advancement of the understanding of the structural dynamics and empirical

knowledge of the actual behavior of the building, the basic procedure was modified to

reflect the fact that the demand generated by the earthquake accelerations was also a

function of the stiffness of the structure. Engineers also began to recognize the

inherently better behavior of some buildings over others. Consequently, they reduced

seismic demand based on the characteristics of the basic structural material and

system. The motivation to reduce seismic demand for design came because engineers

could not rationalize theoretically how structures resisted the forces generated by

earthquakes. This was partially the result of their fundamental assumption that

structures resisted loads linearly without yielding or permanent structural

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deformation. An important measure, the capacity of a structure, to resist seismic

demand is a property known as ductility. “Ductility is the ability to deform beyond

initial yielding without failing abruptly”. A glass rod would snap if it is attempted to

bend with force, where as a steel rod can be bend without breaking. This property is a

critical component of structural capacity.

Instead of comparing forces, nonlinear static procedures use displacements to

compare seismic demand with capacity of a structure. This approach includes

consideration of the ductility of the structure on an element by element basis. The

inelastic capacity of a building is then a measure of its ability to dissipate earthquake

energy. The current trend in seismic analysis is toward these simplified inelastic

procedures.

The NSP may be used for any structure and any Rehabilitation Objective, with the

following exceptions and limitations as summarized in ATC-40[3].

The NSP should not be used for structures in which higher mode effects are

significant, unless an LDP evaluation is also performed.

To determine the response, if higher modes are significant, a modal response

spectrum analysis should be performed for the structure using sufficient modes

to capture 90% mass participation and a second response spectrum analysis

should be performed considering only the first mode participation. Higher

mode effects should be considered significant if the shear in any story

calculated from the modal analysis considering all modes required to obtain

90% mass participation exceeds 130% of the corresponding story shear

resulting from the analysis considering only the first mode response. When an

LDP is performed to supplement an NSP for a structure with significant higher

mode affects, the acceptance criteria values for deformation-controlled actions

(m values) may be increased by a factor of 1.33.

The NSP should not be used unless comprehensive knowledge of the structure

has been obtained.

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2.3 LITERATURE REVIEW

Force and displacement based POA and capacity spectrum method (CSM)

have been discussed extensively in various papers and articles by many authors.

Major criticisms can be summarized by T. Albanesi et al. [14].

The imposed force/displacement profile, although based on modal analyses or

other source of experience and observation, seems hardly capable of

simulating the behavior of the structure throughout the deformation history up

to collapse. The risk being that, by imposing a displacement profile, strain

localization may be underestimated and that by imposing a force profile, the

capability of the structure to redistribute forces and energy dissipation is

neglected.

The equivalent damping used to scale the displacement spectrum is obtained

from assumptions and equations containing a substantial degree of

approximation. The equivalent damping is generally found from the energy

dissipation, which in turn is obtained from the monotonic force displacement

response of the structure (capacity curve found from POA).

In an attempt to consider higher mode effects, Paret and Sasaki et al[8] suggested

the simple, yet efficient, Multi-Mode Pushover procedure (MMP). This method

comprises several pushover analyses under forcing vectors representing the various

modes deemed to be excited in the dynamic response. The results obtained from the

analysis of the structure in different modes are then combined using a suitable

combination rule (discussed in detail in chapter 4). The results obtained using this

method were even more closer to that of the time history analysis and were also

enhanced because of the consideration of the higher modes responses being

considered.

Chopra and Goel [17] developed a method called Multimode Pushover Analysis

(MPA), primarily for estimating inter-story drifts in framed structures when the higher

modes are considered. Later, the ATC-55 project attempted to apply this procedure

toward estimating story-shears and overturning moments, in addition to floor-

displacements and inter-story drifts. The project encountered some problems, pointing

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out reversals in the third mode pushover of a three-story steel moment-resistant frame

building. This setback indicates that increments in roof displacement are in a direction

opposite to the base-shear, which may happen depending on the mechanism that gets

developed within the structure that is the effect of various mode shapes.

Fig 2.1 Force profiles in proportion to the first three mode shapes,

Fig 2.2 Reversal in third mode pushover curve of a structure.

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Peng Pan, Makoto Ohsaki[24] proposed another non-linear MPA procedure.

In their paper “Nonlinear Multimodal Pushover Analysis Method for Spatial

Structures” the authors have discussed in detail the applicability and procedure of the

MPA methods. They pointed out some shortcomings of the MPA procedure as it is

not necessarily straightforward to extend the conventional pushover analysis method

to spatial structures primarily due to the following reasons

(1) Most of the existing methods are strongly dependent on the properties of

the regular frame and use base shear and roof displacement;

(2) Spatial structures commonly have multiple dominant vibration modes,

which are all indispensable for determining the structural response.

(3) The dominant vibration modes may significantly interact with each other

particularly when the responses increases to plastic range.

The MPA procedure has been conducted in two different ways, in the first one

the SRSS rule is applied to the response of the structure, where as in the 2nd

procedure, it is applied to the model, forces are first combined using multiple rules to

define the equivalent static forces, the forces are applied to the structure to obtain the

response for each combination and the maximum value among the multiple

combinations is taken as the final response.

Chandrashekaran and Anubhab Roy[13] highlighted the need of a

completely up-to-date and versatile method of a seismic analysis and design of

structures. They conducted a detailed dynamic analysis of a 10-storey RC frame

building using response spectrum method, based on Indian Standard code provisions

and base shear, storey-shear and storey-drifts were computed. The same structure was

also analyzed by Modal Pushover Analysis (MPA) to determine the structural

response for the same acceleration spectra used in the earlier case. The major focus of

study was to bring out the superiority of POA method over the conventional dynamic

analysis method recommended by the code. Comparison of the results obtained from

the above analyses procedures show that the response spectrum method

underestimates the response of the. It was also shown that modal participation of

higher modes contributes to better results of the response distribution along the height

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of the building. Also pushover curves were plotted to illustrate the displacement as a

function of base shear.

The Energy Based Pushover Analysis was first proposed in 1984 by Zahra

and Hall [1], they argued that the cumulative energy dissipated due to cyclic-plastic

deformations occurring in a structure (that is, the hysteretic energy) is directly related

to seismic damage in structures. The argument was in favor of considering the

hysteretic energy demand as design criterion is that it can directly account for the

cumulative nature of damage in the structure and the dynamic nature of earthquake.

The methodology was proposed but no further work was done due to intensity of

work involved and lesser resources available, hence the researchers never seemed to

be interested in this regard. With the development of the computer application and

availability of different software to perform difficult cyclic procedures, the

researchers turned to the EBPOA.

Recognizing that the roof displacement may not always is the best index as a

basis for establishing the properties of so called “Equivalent Single Degree of

Freedom" (ESDF) systems, Hernandez-Montes et al. [16] developed an alternative

index, known as an Energy-Based Displacement.

Enrique Hernández-Montes et al.[16] in their paper “An Energy Based

Formulation For First and Higher Mode POA”, have discussed that existing nonlinear

static (pushover) methods of analysis establish the capacity curve of a structure with

respect to the roof displacement. Rather than viewing pushover analyses from the

perspective of roof displacement, the energy absorbed (or the work done) in the

pushover analysis is considered in this procedure. Simple relations were established

for energy-based displacement that is equivalent to the spectral displacement obtained

by conventional pushover analysis methods, within the linear elastic domain.

Extensions to the nonlinear domain allow pushover curves to be established that

resemble traditional first mode pushover curves and which correct anomalies

observed in some higher mode pushover curves. The same was explained with

application of a modified Multimode Pushover Analysis procedure using Yield Point

Spectra.

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Subsequently, Tjhin et al.[18] showed that using energy-based displacement

instead of roof displacement, improves peak roof displacement estimation to establish

the properties of the first mode ESDF system. Meanwhile, experiences from recent

earthquakes show that the behavior of irregular buildings is significantly different

from that of regular ones. Accordingly, the UBC code started to differentiate between

irregular and regular buildings. Hence, parameters such as resistance ratio, stiffness,

mass and geometrical irregularities of a story, with respect to neighboring stories,

were considered. Also, as mentioned in the SEAOC code, another type of irregularity

is irregularities due to the difference in story elevation, which is sometimes extremely

essential for architectural reasons.

The necessity of an energy-based design procedure for future seismic design

guidelines has been emphasized by many researchers, including a few attempts at

providing a framework for such design procedures. Discussions of these efforts can be

found in Prasanth, T. Ghosh, S. and Collins [2008] [25]. The first significant step in

a hysteretic energy-based design is the estimation of hysteretic energy demand due to

the design ground motion scenario. With the computing facilities available today, this

estimation for a specific structure under a certain earthquake ground motion is not

difficult, although it is computation intensive. However, one has to apply this detailed

method, nonlinear response history analysis (NLRHA) of a multi-degree of freedom

(MDOF) model, for each individual structure separately, making this direct method

unsuitable for incorporating in a general purpose design methodology based on

hysteretic energy demand. Thus, it becomes necessary to use some approximate

method for estimating the energy demand that can be easily incorporated in seismic

design codes. Such a method will also be useful for the energy based performance

assessment/evaluation of existing structures and for the purpose of energy-based

design checking.

Prasanth et al. [2008][25]used a modal pushover analysis or MPA-based

[Chopra and Goel, 2002 [17]] approximate method to estimate the hysteretic energy

demand in a structure when it is subjected to an earthquake ground motion. Although

their method was limited only to symmetric-in-plan building structures, the results

obtained for three such framed structures subjected to various earthquake scenarios

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were satisfactory and the method was deemed suitable for adopting in energy-based

design and evaluation guidelines since it could use hysteretic energy response spectra.

Grigorios Manoukas et al.[23] in their paper “Static Pushover Analysis Based

on an Energy-Equivalent SDOF System: Application to Spatial Systems”, have

proposed an efficient Non-linear static procedure based on energy equivalence. The

energy equivalence of the structure is used as a margin to estimate the roof

displacement under the monotonically increasing loads those are characterized by the

modal shape of the structure (the fundamental mode shape). The work done in

displacing the structure is a better index to estimate the performance of the structure

and to determine the characteristics of the same. The proposed methodology was

demonstrated using a framed structure and the results obtained were compared with

that of Capacity spectrum method. Good results for the 2D and 3D framed buildings

can be obtained using the procedure and is applicable for the structures with regular or

irregular multistory planar frames. It is clear that the method's good approximation of

the roof displacements shown in this paper does not ensure analogous accuracy for

other quantities of interest, e.g. plastic rotations (Manoukas et al., 2006[23]). The

author concludes that the method seems to be a generalization and pointing out its

advantages is not yet possible. In order to obtain secure generalized conclusions

further investigations and research must be conducted.

In their paper “Energy-Based Modal Pushover Procedure for Asymmetric

Structures” Li et al. [5] concluded that Energy-based seismic evaluations of structures

give clear illustrations of seismic demands made upon structures. The paper

demonstrated a 3-dimensional energy-based modal pushover analysis (3D EMPA)

method with equivalent three degree of freedom (ETDOF) system and equal-

displacement rule. The lateral-rotation couple effect on structure’s asymmetric plan

was considered using the energy-based modal pushover analysis (EBMPA), which is

more robust than the traditional Modal Pushover Analysis (MPA) procedure. The

proposed procedure was validated against a nonlinear time history analysis of a 5-

storey structure. The results show that the method takes higher mode effects into

account and that the instability of capacity spectra, as indicated in the previous papers,

will be avoided. The maximum deformation obtained from EBMPA shows good

agreement with a NL-THA results of the original system. The proposed EBMPA 17

procedure appears to be reliable and effective for evaluating the seismic response of

asymmetric-plan structures.

Prof. M. Mofid [22] (Sharif University of Tehran) proposed a EBMPA

methodology with the help of a strong base provided by Hernandez Montes [16], a

study of the structural performance of the models, with irregularities (i.e., mass,

geometrical and irregularities due to the difference in elevation) in various aspects of

a building and it has been made clear that different types of the above-mentioned

irregularities in elevation do not have any significant effect on the Energy-Based

MPA method. Further the authors have concluded that, this method can be considered

as an accurate alternative technique for NL-THA, to fairly estimate the seismic

demands of structures. In their work the authors have performed the energy based

pushover analysis over a number of frames and models and with changes in different

parameters such as change in geometry, difference in the levels of masses at similar

storey, change in elevation etc,. of the structure and found that the the procedure

yielded better results which were close to the results obtained from the NL-THA. In

the concluding part, the authors of the paper suggested that the EBPOA method is the

best method available with can replace the NL-THA.

Chou and Chia-Ming Uang [12] proposed a procedure for evaluating seismic

energy demand of framed structures. The need of a better performance evaluation

method for the structures has been highlighted and consequently a method is

developed on the basis of the absorbed energy by the structure. Study on the energy

demand in multistory frames is limited. Fajfar et al. showed that the hysteretic energy

demand in an MDOF system cannot be evaluated reliably from an equivalent SDOF

system; the researchers attributed the problem to the higher mode effect. This effect

also made it difficult to predict the energy distribution along the height of building

structures. A recent study by Chopra and Goel [17] also showed that the seismic

storey drift demand along the building height can be estimated if more than one

equivalent SDOF systems are considered. The analysis was conducted on MDOF

systems with some well recorded earthquake data. The total energy absorbed by the

MDOF system was calculated and the same was distributed to different storey levels

using the procedure presented by Chou and Chia-Ming Uang [12]. Using this

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distribution of the energy different parameters and characteristics of the structure can

be studied.

2.4 APPRAISAL OF THE LITERATURE

Rigorous work has been done in making methods more effective and accurate

to use it as an alternative to non linear time history analysis. Literature papers indicate

that the consideration of higher modes gives result close to non linear time history

analysis. Literature papers conclude that consideration of energy dissipation due to

deformation gives a better result than the conventional pushover analysis. The

proposed study aims to determine the effect of higher modes on the response of the

building, and to determine the performance point of the building for different modes

using energy based approach.

2.5 SUMMARY

In this chapter the complete literature review of Energy Based Pushover

Analysis has been presented, from the proposal of the basic background of the method

to how it was improved and implemented by the researchers. Major contribution of

work has been made on comparison of the method with different types or methods of

dynamic analysis, for different types of structures. The consideration of energy is of

great significance and the procedure yields better results compared to the

conventional NSPs. The appraisal of the study has been discussed regarding the

effects of higher modes of the structure in the structural response.

19

METHODOLOGY

20

CHAPTER-III

METHODOLOGY

3.0 GENERAL

The purpose of the pushover analysis is to evaluate the expected performance

of a structural system by estimating its strength and deformation demands in design

earthquakes by means of a static inelastic analysis, and comparing these demands to

available capacities at the performance levels of interest. The evaluation is based on

an assessment of performance parameters, like global drift, inter-storey drift, inelastic

element deformations (either absolute or normalized with respect to a yield value),

deformations between elements and connection forces. The inelastic static pushover

analysis can be viewed as a method for predicting seismic force and deformation

demands, which accounts in an approximate manner for the redistribution of internal

forces occurring when the structure is subjected to inertia forces that no longer can be

resisted within the elastic range of structural behavior. The pushover is expected to

provide information on many response characteristics that cannot be obtained from an

elastic static analysis. The following are examples of such response characteristics:

The realistic force demands on potentially brittle elements, such as axial

force demands on columns, force demands on brace connections, moment

demands on beam-to-column connections, shear force demands in deep

reinforced concrete spandrel beams, shear force demands in unreinforced

masonry wall piers, etc.

Estimates of the deformation demands for elements that have to deform

inelastically in order to dissipate the energy imparted to the structure by

ground motions.

Consequences of the strength deterioration of individual elements on the

behavior of the structural system.

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Identification of the critical regions in which the deformation demands

are expected to be high and that have to become the focus of thorough

detailing.

Identification of the strength discontinuities in plan or elevation that will

lead to changes in the dynamic characteristics in the inelastic range.

Estimates of the inter-story drifts that account for strength or stiffness

discontinuities and that may be used to control damage and to evaluate P-delta

effects.

Verification of the completeness and adequacy of load path, considering

all the elements of the structural system, all the connections, the stiff

nonstructural elements of significant strength, and the foundation system.

The last item is perhaps the most relevant one, provided the analytical model

incorporates all elements, whether structural or nonstructural, that contribute

significantly to lateral load distribution. For instance, load transfer across connections

between ductile elements can be checked with realistic forces; the effects of stiff

partial-height infill walls on shear forces in columns (short columns) can be

evaluated; and the maximum overturning moment in walls, which is often limited by

the uplift capacity of foundation elements, can be estimated.

Clearly, these benefits come at the cost of additional analysis effort, associated

with incorporating all important elements, modeling their inelastic load-deformation

characteristics, and executing incremental inelastic analysis, preferably with a three-

dimensional analytical model. At this time, with few exceptions, adequate analytical

tools for this purpose are either very cumbersome or not available, but several good

tools are under development, since the demand for the pushover analysis has been

established, primarily through the recent pre-publication of the FEMA-273[6]

document.

The use of pushover analysis methods for characterizing the predominant

mode of response of structures responding nonlinearly to earthquake ground motions

has become well established. Approximate and exact first mode analysis procedures

have been accepted in documents such as ATC-40 [3] and FEMA-356[10]. Recent

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proposals have focused on combined mode and multiple mode analysis procedures

[Chopra and Goel, 2002 [17]]. In each analysis method, lateral forces are applied

monotonically in a step-by-step static analysis. In the case of modal analysis

procedures, the applied lateral forces are proportional to the product of the mass and

mode shape amplitude at each level of the structure. In other cases, a non-modal shape

vectors may be used in place of the mode shape, where the shape vector may be an

inverted triangular shape or a rectangular shape. In still other cases, the lateral forces

may be proportional to a code lateral force distribution.

As we know in adaptive pushover analyses, the load patterns are modified as

the analysis progresses to reflect changes in structural properties that occur with the

development of nonlinearity in the structural components. In pushover analysis

procedures, the behaviour of the structure is characterized by a capacity curve. In

nearly all cases, the capacity curve is a plot of the base shear force versus the

displacement of the roof, as determined in the step-by-step pushover analysis. This

representation is convenient for use in design offices and is both meaningful to and

easily visualized by the engineer. The roof displacement is used in current procedures

not only as an index for the capacity curve, but also to establish the seismic demands

over the height of the structure at the estimated peak displacement, or performance

point. For example, for the conventional first mode pushover (e.g. ATC-40 [3]) and the

Multimode Pushover Analysis [Chopra and Goel, 2002 [17]] methods, the seismic

demands are determined throughout the structure based on the peak roof displacement

estimated in each of the modal pushover analyses. It is recognized that the roof

displacement was selected for this representation because it is convenient for use in

practice. In the linear elastic domain, the floor and roof displacements increase

proportionately when subjected to proportionately increasing lateral forces, as is done

in a typical pushover analysis. The capacity curve could just as easily be based on the

displacement at any floor, but the roof displacement has been preferred because it

emphasizes the overall response of the structure and provides better numerical

accuracy, particularly when higher modes are involved. While the roof displacement

is useful for characterizing the behavior of many buildings such as the moment

resistant frame of Fig.3.1 (a), it is not clear that the roof displacement is the most

23

meaningful index for other structures, such as the braced frames of Fig.3.l (b) and (c),

even for linear elastic behaviour under quasi-first mode lateral force patterns.

When nonlinear behaviour develops in the pushover analysis, the

displacements of the floors and roof will increase disproportionately with increasing

load, in general. The arbitrary choice to plot the base shear as a function of the roof

displacement introduces an arbitrariness to the inelastic portion of the capacity curve.

For systems with sharply defined yield points, disproportionate increases in

displacements over the height of the building, primarily affects the post-yield stiffness

of the capacity curve. Because small deviations in the post-yield stiffness of the

capacity curve of the "equivalent" single-degree-of-freedom (SDOF) system typically

have only minor effects on the dynamic response statistics, any departures from

theoretically ideal values can be difficult to discern in computational studies.

Where yielding is more gradual, disproportionate increases in the roof

displacement may, in addition, affect the effective yield strength that is determined for

the structure: when methods such as those described in ATC-40[3] [I997] are used.

(a) (b) (c)

Fig 3.1 Different type of frames

The arbitrariness in the choice of the roof displacement as the index used for

plotting the capacity curve is readily apparent in pushover curves obtained for the

second, third, or higher modes, as has been observed by some of the primary

contributors to multiple mode pushover analysis methods [Goel and Chopra, 2002 17]].

Roof displacements may increase at a decreasing rate or may even reverse, leading to

capacity curves that display unusual behaviour, as illustrated subsequently. A literal

24

interpretation of the capacity curves obtained in these cases would indicate that the

structure does not always absorb energy in a pushover analysis, but instead, may be a

source of energy for some inelastic regimes. Such an interpretation implies a violation

of the first law of thermodynamics, and points out the degree to which the use of the

roof displacement can be misleading. There is no doubt that external work is

consumed by the deformations of plastic hinges (and any changes in recoverable

strain energy) that take place in a monotonic pushover analysis. The notion that the

structure may be a source of energy is a consequence of the arbitrary choice to use the

roof displacement as the index (abscissa) of the capacity curve.

A more general characterization of the resistance developed by a multi-

degree-of-freedom (MDOF) structure under increasing lateral loads in a pushover

analysis by explicit consideration of the work done throughout the entire structure

gives better results of the response of the structure. This retains the utility of both first

mode and multiple mode methods of analysis while improving their theoretical basis.

Rather than relying on the displacements at an arbitrary location, absorbed energy (or

equivalently, external work) is used to characterize the resistance of the structure to

lateral forces. Simple relationships are developed to allow the resistance function to

be expressed in terms of the displacement of an equivalent SDOF system, thereby

providing an alternative means to generate the capacity curve of the equivalent SDOF

system. For the sake of simplicity, the resistance of the structure is determined with

respect to invariant modal force patterns; simple extensions can make the procedure

compatible with adaptive force patterns.

The resulting capacity curves appear to be generally compatible with those

obtained from traditional first mode pushover analyses, and thereby provide a stronger

theoretical underpinning for the use of conventional first mode pushover analyses.

The results obtained for higher mode pushovers avoid the problems associated with

the arbitrary choice of the roof displacement. However, the hypothesis that

independent modal pushover analyses may be combined in the domain of inelastic

response [Chopra and Goel, 2002 [17]] is not addressed and may require further study

to be adequately assessed.

25

3.2 DEFINATIONS

Acceptability (response) limits: Refers to specific limiting values for the

deformations and loadings, for deformation-controlled and force controlled

components respectively, which constitute criteria for acceptable seismic

performance.

Capacity: The expected ultimate strength (in flexure, shear, or axial loading) of a

structural component excluding the reduction (factors commonly used in design of

concrete members. The capacity usually refers to the strength at the yield point of the

element or structure's capacity curve. For deformation-controlled components,

capacity beyond the elastic limit generally includes the effects of strain hardening.

Capacity Curve: The plot of the total lateral force, V, on a structure, against the

lateral deflection, d, of the roof of the structure. This is often referred to as the

'pushover' curve.

Capacity Spectrum: The capacity curve transformed from shear force vs. roof

displacement (V vs. d) coordinates into spectral acceleration vs. spectral displacement

(Sa vs. Sd) coordinates.

Capacity Spectrum Method: 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.

Components or members: The local concrete members that comprise the major

structural elements of the building such as columns, beams, slabs, wall panels,

boundary members, joints, etc. Concrete frame building: A building with a

monolithically cast concrete structural framing system composed of horizontal and

vertical elements which support all vertical gravity loads and also provide resistance

to all lateral loads through bending of the framing elements. Deformation- Controlled

Refers to components, elements, actions, or systems which can, and are permitted to, 26

exceed their elastic limit in a ductile manner. Force or stress levels for these

components are of lesser importance than the amount or extent of deformation beyond

the yield point (see ductility demand).

Degradation: Refers to the loss of strength that a component or structure may suffer

when subjected to more than one cycle of deformation beyond its elastic limit.

Degrading components are generally referred to as being force-controlled, brittle, or

non-ductile. Some or all of their flexural, shear or axial loading must be redistributed

to other, more ductile, components in the structural system.

Demand: A representation of the earthquake ground motion or shaking that the

building is subjected to, in nonlinear static analysis procedures, demand is represented

by an estimation of the- displacements or deformations that the structure is expected

to undergo.

Demand spectrum: The reduced response spectrum used to represent the earthquake

ground motion in the capacity spectrum method.

Displacement-based: Refers to analysis procedures, such as the nonlinear static

analysis procedures recommended in this methodology, whose basis lies in estimating

the realistic, and generally inelastic, lateral displacements or deformations expected

due to actual earthquake, ground motion. Component forces are then determined

based on the deformations.

Ductility: The ability of a structural component, element, or system to undergo both

large deformations and/or several cycles of deformations beyond its yield point or

elastic limit and maintain its strength without significant degradation or abrupt failure.

These elements only experience a reduction in effective stiffness after yielding and

are generally referred to as being deformation controlled or ductile.

Elastic (linear) behaviour: Refers to the first segment of the bi-linear load-

deformation relationship plot of a component, element, or structure, between the

unloaded condition and the elastic limit or yield point. This segment is a straight line

whose slope represents the initial elastic stiffness of the component.

27

Energy Based Pushover Analysis: A type of Pushover analysis in which the energy

of absorbed by the structure during the pushing, is used as a scale to predict the

response of the structure.

Non-Linear Static procedure: The generic name for the group of simplified

nonlinear analysis methods central to this methodology characterized by: use of a

static pushover analysis to create a capacity curve representing the structure's

available lateral force resistance, a representation of the actual displacement demand

on the structure due to a specified level of seismic hazard, and verification of

acceptable performance by a comparison of the two.

Performance level: Refers to the level of the structure as defined for the measuring

the performance of the structure for a deformation, usually characterized for scaling

the damage in the structure.

Performance-based: Refers to a methodology in which structural criteria are

expressed in terms of achieving a performance objective. This is contrasted to a

conventional method in which structural criteria are defined by limits on member

forces resulting from a prescribed level of applied shear force.

Performance level: A limiting damage state or condition described by the physical

damage within the building, the threat to life safety of the building's occupants due to

the damage, and the post-earthquake serviceability of the building. A building

performance level is the combination of a structural, performance level and a

nonstructural performance level.

Performance objective: A desired level of seismic performance of the building

(performance level), generally described by specifying the maximum allowable (or

acceptable) structural and nonstructural damage, for a specified level of seismic

hazard.

Performance Point: The intersection of the capacity spectrum with the appropriate

demand spectrum ill the capacity spectrum method (the displacement at the

performance point is equivalent to the target displacement in the coefficient method).

ap, dp: coordinates of the performance point on the capacity spectrum, api, dpi

28

coordinates of successive iterations (i = 1, 2, etc.) of the performance point, ay, dy

coordinates of the effective yield point on the capacity spectrum.

Pushover Analysis: An incremental static analysis used to determine the force-

displacement relationship, or the capacity curve, for a structure or structural element.

The analysis involves applying horizontal loads, in a prescribed pattern, to a computer

model of the structure, incrementally; i.e. "pushing." the structure; and plotting the

total applied shear force and associated lateral displacement at each increment, until

the structure reaches a limit state or collapse condition.

Target Displacement: In the displacement coefficient method. the target

displacement is the equivalent of the performance point in the capacity spectrum

method. The target displacement is calculated by use of a series of coefficients.

Yield (effective yield point): The point along the capacity spectrum where the

ultimate capacity is reached and the initial linear elastic force-deformation

relationship ends and effective stiffness begins to decrease. For larger elements or

entire structural systems composed of many components, the effective yield point (on

the bi-linear representation of the capacity spectrum) represents the point at which a'

sufficient number of individual components or elements have yielded and the global

structure begins to experience inelastic deformation.

3.3 MODAL ANALYSIS PROCEDURE

Keeping in mind that our main intention and aim is safe and economic design

of structures with speed, nonlinear static procedure have made analysis simpler but

some assumptions have to be made therein. A major assumption is that the response

of a nonlinear MDOF system with n-degrees of freedom can be expressed as

superposition of the responses of n appropriate SDOF systems just like in the linear

range. Of course, such an assumption violates the very logic of nonlinearity, as the

superposition principle is not valid in nonlinear systems. However, it must be thought

as a fundamental postulate, which constitutes the basis on which simplified pushover

procedures are built. Thus, each SDOF system corresponds to a vibration "mode" i

with "modal" vector φi (the quotation marks indicate the irregular use of the

29

superposition principle). The displacements vector ui and the inelastic resisting forces

vector Fsi are supposed to be proportional to φi and Mφi respectively, where M is the n

x n diagonal mass matrix. Furthermore, "modal" vectors φi are supposed to be

constant, despite the successive development of plastic hinges.

Taking into account the aforementioned assumptions and applying well-known

principles of structural dynamics, the following fundamental conclusions can be

derived.

a) the nonlinear response of a MDOF system with N degrees of freedom subjected to

an horizontal earthquake ground motion üg can be expressed as the sum of the

responses of N SDOF systems, each one corresponding to a vibration "mode" i,

having mass equal to the effective "modal" mass Μi* and displacement equal to

Di = uNi/ νi φNi ...1

where uNi, νi and φNi are the roof displacement parallel to the excitation direction, the

"modal" participation factor and the component of the vector φi corresponding to uNi

respectively.

b) the inelastic resisting force of each SDOF system is equal to the "modal" base

shear Vi parallel to the direction of excitation.

c) the external work of "modal" forces Fsi on the displacements dui = νi φi dDi in a

differential time interval dt is equal to the work of the resisting force (or the strain

energy) of the corresponding SDOF system on the displacement dDi.

It is worth noticing that Μi*, Vi , Fsi ,Di, uNi, νi, φi and φNi refer to the elastic

vibration mode i. However, those conclusions are derived on the basis of the

aforementioned assumptions and cannot be true all together when a pushover analysis

is conducted. Thus, the EBMPA uses the energy to determine modified resisting

forces the system.

3.3.1 Determination of eigen values and eigen vectors:

Let the shear stiffness of the ith storey is ki and the mass is mi subjected to an

external dynamic force fi(t) and the corresponding displacement xi(t). Assuming

damping in the system in small, so it may be ignored and the system is analyzed as

undamped system. Using D’Alemberts’s principle, the dynamic equilibrium equation

of mass at each floor is,

30

...2

The equilibrium equations can be expressed in matrix form as,

...3

where M and K are called mass and stiffness matrices respectively, which are

symmetrical. X”, X and F are called acceleration, displacement and force vectors

respectively, and all functions of time (t).

If the structure is allowed to freely vibrate with no external force (vector F is

equal to zero) and no damping in simple harmonic motion, then the system represents

undamped free vibration. In that case, displacement x can be defined at time t as,

x(t) = x sin (ωt+φ) ...4

where,

x = amplitude of vibration,

ω = natural circular frequency of vibration

φ = phase difference, which depends on the displacement and velocity at time t=0.

Differentiating s(t) twice with respect to time enables the relationship between

acceleration and displacement

Substitution, equation for free undamped vibration of the MDOF system becomes

K X = ω2M X ...5

Where ω2 is known as the eigen value or natural frequencies of the system.

this is known as eigen-value characteristic value problem.

From the relation that, natural time period, T = 2Π / ω ...6

X is known as an eigen-vector/modal vector or mode shape represented as,

{Ф} = { Ф1 , Ф2 , Ф3 , Ф4 … Фn}

31

After obtaining the eigen-values of the system, the next step is to calculate the

eigen-vectors (mode Shape) corresponding to each eigen-value. The characteristics

equation governs the undamped free vibration of the MDOF system. There are no

external forces acting on the system. The displacement and velocity at particular

storey level results from giving the initial displacement is perfectly arbitrary.

Therefore, one can determine the relative rather than absolute displacements (Bhatt,

2002). In this example eigen-vectors corresponding to eigen-values are obtained from

the co-factors of any row of the characteristics equation.

The shape of each mode of free vibration is unique but the amplitude of the

mode shape is undefined. The mode shape is usually normalized such that the largest

term in the vector is 1.0 or the sum of the squares of the terms in the vector is 1.0 or

the vectors are normalized so the generalized mass M* is 1.0. (Carr, 1994),

M* = { Фi} T [M]{ Фi } = 1.0 …7

Eigen-vectors for ω2 can be obtained by setting either of the xi=1.0, usually u1.

After obtaining the {X} matrix, we obtain the mode shape as,

{Фi} = [ X ]/ (XTMX)1/2 …8

Hence all the mode shapes are obtained.

3.3.2 Determination of the modal participation factors:

Using the eigen-vectors determined for the structure, modal participation

factors and effective masses for all the four modes can be calculated as:

Pk = Фik)/ ( Фik)2) …9

Where k is the number of the mode shape.

Determination of the modal mass:

The modal mass (Mk) of mode k is given by,32

Pk = Фik) / (Фik)2) …10

where,

g = Acceleration due to gravity

Фik = Mode shape coefficient at floor I in mode k, and

Wi = Seismic weight of floors i.

Using the data obtained above, the basic modal pushover analysis can be started.

3.4 MODAL COMBINATION RULES

The peak storey shear force (Vi) in ith storey due to all modes considered is

obtained by combining forces from all modes in accordance with modal combination

as per IS:1893-2002[15]. The combinations are usually achieved by using statistical

methods.

In general these modal maximum values will not occur simultaneously. To overcome

this difficulty, it is necessary to use an approximate method. The use of either of the

combination rules presented gives good results of the structural response in the

situations as highlighted.

3.4.1 Maximum Absolute Response (ABS)

The ABS for any system response quantity is obtained by assuming that the

maximum response in each mode occurs at the same instant of time. Thus the

maximum values of the response quantity are the sum of the maximum absolute value

of the response associated with each mode. Therefore using ABS, maximum storey

shear for all modes shall be obtained as,

…11

where the summation is for the closely-spaced modes only. The peak response

quantity due to the closely spaced modes (λ*) is then combined with those of the

remaining well- separated modes by the method of SRSS.

33

3.4.2 Square Root of the Sum of Squares (SRSS)

A more reasonable method of combining modal maxima for two-dimensional

structural system exhibiting well-separated vibration frequencies is the Square Root of

the Sum of Squares (SRSS). The peak response quantity (λ) due to all modes

considered shall be obtained as,

…12

where λk is the absolute value of quantity in mode ‘k’ and r is the number of modes

being considered.

3.4.3 Complete Quadratic Combination (CQC)

For three dimensional structural systems exhibiting closely spaced modes, the

peak response quantities shall be combined as per Complete Quadratic Combination

(CQC) method,

…13

where, r = number of modes being considered

λi = response quantity in mode i (including sign)

λj = response quantity in mode j (including sign)

ρij = cross modal coefficient,

…14

where, ζ = modal damping ratio (in fraction)

βij= frequency ratio ωj/ ωi,

34

ωi = circular frequency in ith mode, and

ωj = circular frequency in jth mode.

Here the terms λi and λj represent the response of the different modes of a certain

storey level.

Using the matrix notation the storey shears are worked out as V1,V2,V3,V4… Vn

respectively.

3.5 THEORETICAL BASIS OF PUSHOVER ANALYSIS

The differential equation of the dynamic response of a linear elastic multi-

degree of freedom structure subjected to a horizontal base excitation üg is:

…15

In the case of a multistory building, u is a vector of N components that

represents the lateral displacements of the floors relative to the base, and m, c and k

are the mass, damping and stiffness matrices of the structure. The vector 1 is a column

vector with each component equal to 1, s is a vector equal to ml and represents the

shape of the effective forces peff(t).

The displacement vector, u, can be decomposed into components expressed in

terms of the free vibration mode shapes (φn), where qn is the nth modal coordinate.

…16

The expression of the displacement vector in terms of the mode shapes [Eq.

(2)] allows the system of N coupled equations represented by Eq. (1) to be uncoupled

in terms of the modal coordinates. Substitution of Eq. (2) into Eq. (1) and application

of the properties of orthogonality of the free vibration mode shapes with respect to m,

c and k result in:

…17

35

where ζn, is the damping ratio, ωn, is the natural vibration frequency and Гn, is the

modal participation factor. A further simplification can be achieved by setting qn (t) =

Гn Dn (t), resulting in the following differential equation of motion for the SDOF

system:

…18

The solution of Eq. (4) for the Dn (t) corresponding to each mode is the basis

of modal response history analysis (MRHA), for which the vector u is given by:

…19

In order to address the forces that act on the structure for each modal response,

the equation of motion for the response of the SDOF system representing the nth

mode is presented. Displacements are proportional to the nth mode shape.

…20

Allowing Eq. (15) to be expressed as

…21

The effective force, peff (t) can be decomposed taking note of the orthogonality

of the mode shapes with respect to the mass matrix:

…22

Substituting Eq. (21) into Eq. (22), and multiplying both sides by φnT results in

…23

36

which indicates that only the sn, component of pee results in a non-zero response in

the nth mode, for which a, is equal to the participation factor Гn. Thus, sn and peff,n can

be expressed as:

…24

Thus, it is apparent that only peff,n causes response in the nth mode. An

equivalent static force can be associated with the nth mode displacement u n(t).

The equivalent static force f n is the statically applied force that results in a

displacement equal to u n(t):

where An( t ) is the pseudo-acceleration [A,( t )= ω n D n (t)]. For the elastic response,

any response quantity r(t) (e.g. displacements, internal element forces, or moments)

may be calculated as a combination of each of the modal responses rn(t) in MRHA:

…25

where rnst is the static response of quantity r due to the external force sn,.

The peak value of r for the nth mode is called rno. Thus, rno = rnst - An, where An

is the ordinate of the pseudo-acceleration design (or response) spectra corresponding

to the nth modal period. The peak value of the total response of the quantity r, given

by ro can be estimated according to combination rules such as CQC or SRSS. These

rules combine the peak values obtained for each mode.

Also of interest is the static force that produces the maximum displacements for each

mode, fno,

...26

37

where sn and sn* are vectors having different lengths but the same shape.

3.6 THE CONVENTIONAL PUSHOVER ANALYSIS METHOD

The ATC-40[3] document refers to several pushover methods involving various

quasi-first mode approaches and multiple mode approaches. The Multimode Pushover

Analysis (MPA) procedure of Chopra and Goel [2002] [17] combines quantities

determined in independent modal pushover analyses. The capacity curve determined

for the "equivalent" SDOF system for the first mode load pattern of the MPA

procedure is identical to that determined in the first mode, pushover method of ATC-

40[3]. Thus, the following discussion will refer to the nth mode pushover of the MPA

procedure, recognizing that the specialization for n = 1 applies equally to the ATC-

40[3] first mode pushover. The MPA procedure proposes to estimate peak dynamic

response quantities of inelastic structures based on a combination of possibly

nonlinear responses obtained independently for each mode. The structure is subjected

to a static force distributed over the height of the building according to sn*, with

amplitude increasing until the roof displacement equals or exceeds the maximum

displacement, urno expected in each mode (urno=ΓnφrnDn) where φrn is the component of

φn at the roof level. The peak modal responses rno, each determined in an independent

modal pushover analysis, are combined according to the SRSS method to obtain an

estimate of the peak value ro of the total response, where rno values may refer to floor

displacements, inter-storey drifts, storey shears, overturning moments, or other

response quantities.

When applied to structures responding inelastically, the method neglects the

influence of the modal forces sn on the response of other modes. That is, superposition

is assumed, or alternatively, interaction among the modes is neglected, just as in

elastic modal analysis.

An algebraic mapping is used in order to relate the capacity curves obtained in

modal pushover analyses to a common design or response spectrum for each of the

SDOF systems. To establish the algebraic mapping, the equations of motion for the

MDOF system are developed, and from these, the expressions for the SDOF system

may be extracted.

38

The maximum displacement of the SDOF system is plotted on the abscissa of

the common capacity spectrum representation (hereafter referred to as the "common"

representation). The nth mode capacity curve is plotted with respect to one of the

lateral degrees of freedom, typically the roof displacement urn.

…27

Equation (27) gives the relation between the displacement of the capacity curve of the

“equivalent” SDOF system (Dn) and the roof displacement for the nth mode (urn).

Thus, the common representation for the nth mode pushover is obtained bya plotting

on the abscissa the following quantity:

…28

wehre urn is obtained from the pushover analysis with forces applied proportional to sn.

In Chopra and Goel's [17] description of the nth mode capacity curve (Vbn versus urn)

the term kn is used to represent the ratio of Vbn and urn in the elastic domain. Because

this may be misinterpreted as suggesting that the ratio is independent of the locations

where the displacements and shears are measured, we prefer to use the term krn, for

this ratio, resulting in Vbn = brn urn in the elastic domain.

The ordinate of the common representation is the restoring force of the unit mass

SDOF system, given by the third term of the left side of Eq. (19) which has maximum

value ωn2Dno. The equation for the base shear, Vbn, of the MDOF system is developed

below, in order to identify the values to be plotted on the ordinate of the common

representation.

…29

39

An expression for krn can be obtained from:

…30

The eigenproblem used to determine the undamped free vibration mode shapes

establishes that

Thus the value to plot on the ordinate of the common representation is obtained from

the equations above as :

…31

wehre αn is the modal mass coefficient for the nth mode.

3.7 NON LINEAR STATIC PUSHOVER ANALYSIS PROCEDURE IN SAP

2000[27]

Pushover analysis can be performed as either force controlled or displacement

controlled depending on the physical nature of the load and behavior expected from

the structure.

Force controlled option is useful when the load is known (such as gravity

loads) and the structure is expected to be able to support the load. Displacement

controlled procedure should be used when specified drifts are sought (such as in

seismic loading), where the magnitude of applied load is not known in advance, or

when the structure can be expected to lose the strength or become unstable. Following

40

are the general sequence of steps involved in performing non-linear static pushover

analysis using SAP 2000[27] in the present study:

1. A two or three dimensional model that represents the overall structural behaviour

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

for the cross sections.

2. Frame hinge properties are defined and assigned to the frame elements.

3. Gravity loads composed of dead loads and a specified portion of live loads are

applied to the structure model initially.

4. The non-linear static analysis cases to be used for pushover analysis are defined.

These cases include one or more pushover cases that start from previous cases loaded

with gravity and fixed loads.

5. Pushover analysis is set to run.

6. Lateral loads are increased until some members yield under the combined effect

of gravity and lateral loads.

7. Base shear and roof displacement is recorded in first yielding.

8. Analysis is repeated until the roof displacement reaches a certain level of

deformation or the structure becomes unstable for different mode shapes of the

structure.

The roof displacement is plotted with the base shear to get the global capacity

(pushover) curve of the structure.

Fig. 3.2 A typical pushover curve

3.8 GROUND MOTION AND RESPONSE SPECTRA

41

Response spectra provide a very handy tool for engineers to quantify the

demands of earthquake ground motion on the capacity of buildings to resist

earthquakes. Data on past earthquake ground motion is generally in the form of time-

history recordings obtained from instruments placed at various sites that are activated

by sensing the initial ground motion of an earthquake. The amplitudes of motion can

be expressed in terms of acceleration, velocity and displacement. The first data

reported from an earthquake record is generally the peak ground acceleration (PGA)

which expresses the tip of the maximum spike of the acceleration ground motion

(Fig. 3.3).

Fig. 3.3 Recorded ground motion

Although useful to express the relative intensity of the ground motion (i.e.,

small, moderate or large), the PGA does not give any information regarding the

frequency (or period) content that influences the amplification of building motion due

to the cyclic ground motion. In other words, tall buildings with long fundamental

periods of vibration will respond differently than short buildings with short periods of

vibration. Response spectra provide these characteristics. Picture a field of lollipop-

like structures of various heights and sizes stuck in the ground. The stick represents

the stiffness (K*) of the structure and the lump at the top represents the mass (M*).

The period of this idealized single-degree-of-freedom (SDOF) system is calculated by

the equation:

T = 2π (M * / K*)½ ...32

42

If the peak acceleration (Sa) of each of these SDOF systems, when subjected

to an earthquake ground motion, is calculated and plotted with the corresponding

period of vibration (T), the locus of points will form a response spectrum for the

subject ground motion. Thus, if the period of vibration is known, the maximum

acceleration can be determined from the plotted curve. When calculating response

spectra, a nominal percentage of critical damping is applied to represent viscous

damping of a linear-elastic system, typically five-percent. Response spectra can be

plotted in a variety of formats. A format commonly used in the 1960s was the

tripartite logarithmic plot, where the vertical scale is spectral velocity (Sv) and the

horizontal scale is T in seconds or frequency (f) in Hertz. On diagonal lines are

designated Sa and spectral displacement (Sd). An example is shown in Figure 3.3.

Mathematical relationships between the components of response spectra are

given in ATC 40 [3] by the following equations:

Sv = (T /2π ) Sa

Sa = (2π/ T) Sv

Sd = (T /2π ) Sv = Sa (T /2π )² …33

f = 1/T

By performing a time history analysis of a structure it is possible to determine the

peak acceleration, velocity and displacement of the structure’s response to a ground

motion. If such analysis are performed for a series of single degree of freedom

structures, each having a different period, T, and the peak response accelerations,

velocities and displacements are plotted vs. the period of the structures, the resulting

graphs are termed respectively acceleration, velocity and displacement response

spectra.

43

Fig. 3.4 Structural response to ground motion

Researchers commonly display response spectra on 3-axix plot known as tri-

partite plot in which peak response acceleration, velocity and displacement are all

plotted simultaneously against structural period. Researchers (Newmark and Hall,

1982) have found that response spectra for typical records can be enveloped by a plot

with three distinct ranges: a constant peak spectral acceleration (PSA). Aonstant peak

spectral velocity (PSV) and constant spectral displacement (PSD).

Fig.3.5 Tri-partite plot

Response spectra contained in the building code indicate the constant

acceleration and velocity ranges plotted in acceleration vs period domain. This is

convenient to the code design procedure which is based on forces (or strength) which

are proportional to acceleration.

44

Fig.3.6 Standard Format Response Spectrum

For nonlinear analysis, both force and deformation are important. Therefore,

spectra arte plotted in an acceleration vs displacement domains, which has been

termed ADRS (acceleration-displacement response). Period in those ADRS are

represented by a series of radial lines extending from the origin of the plot.

Fig 3.7 ADRS Format Response Spectrum

An elastic response spectrum, for each earthquake hazard level of interest at a

site, is based on the site seismic coefficients CA and CV, defined in the previous

sections. The seismic coefficient CA represents the effective peak acceleration (EPA)

of the ground. A factor of about 2.5 times CA represents the average value of peak

response of a 5 percent-damped short-period system in the acceleration domain. The

seismic coefficient CV represents 5 percent-damped response of a 1-second system and

45

when divided by period defines acceleration response in the velocity domain. Fig.

illustrates the construction of an elastic response spectrum.

Fig. 3.8 Elastic Response Spectrum

3.9 PROCEDURE TO CALCULATE PERFORMANCE LEVELS OF

BUILDINGS

A performance level describes a limiting which may be considered satisfactory

for a given building and a given ground motion. ATC-40 [3] describes standard

performance level for structural and non structural systems and several commonly

used combinations of structural and nonstructural levels. Combinations of structural

performance level and nonstructural performance level form a building performance

level to completely describe the desired limiting damage state for building.

The performance level of a building is determined based up on its function and

importance. Structures like hospital buildings, telecommunication centers,

transportation facilities etc are expected to have a performance level of operational or

immediate occupancy for an identified seismic hazard that can occur for the structure.

Meanwhile a residential building must have a performance level of damage control or

life safety. Temporary structures or unimportant buildings or structures came under

the performance level of structural stability or sometimes are not considered. The

46

force deformation relationship as well as the performance levels of structural element

is given in Fig. 3.16. As shown in Fig 3.9, five points labeled A, B, C, D, and E are

used to define the force deflection behavior of the hinge and three points labeled IO,

LS and CP are used to define the acceptance criteria for the hinge. (IO, LS and CP

stand for Immediate Occupancy, Life Safety and Collapse Prevention respectively.)

The values assigned to each of these points vary depending on the type of member as

well as many other parameters defined in the ATC-40[3] and FEMA-273[6] documents.

Fig. 3.9 Force deformation

The following are the step by step procedure for obtaining performance point of

building:

1. Develop the 5 percent damped response spectrum appropriate for the site.

2. Draw the 5 percent damped response spectrum and draw a family of reduced

spectra on the same chart. It is convenient if the spectra plotted correspond to

effective damping values (eff) ranging from 5 percent to the maximum value allowed

for the building's structural behavior type. The maximum eff for Type A construction

is 40 percent, Type B construction is 29 percent and Type C construction is 20

percent. Fig. 3.10 shows an example family of demand spectra.

47

Fig.3.10 Capacity Spectra Procedure After Step 2

3. Transform the capacity curve into a capacity spectrum using equations 34, 35, 36

and 37. Plot the capacity spectrum on the same chart as the family of demand spectra,

as shown in Fig 3.11.

…34

…35

…36

48

…37

Fig. 3.11 Capacity spectra Procedure after Step 3

4. Develop a bilinear representation of the capacity spectrum as illustrated in Fig.

3.12. The initial slope of the bilinear curve is equal to the initial stiffness of the

building. The post-yield segment of the bilinear representation should be run through

the capacity spectrum at a displacement equal to the spectral displacement of the 5

percent damped spectrum at the- initial pre-yield stiffness (equal displacement rule),

point a*, d*. The post-yield segment should then be rotated about this point to balance

the areas A1 and A2 as shown in Fig.3.12.

49

Fig. 3.12 Capacity Spectrum Procedure after Step 4

5. Calculate the effective damping for various displacements near the point a*, d*.

The slope of the post-yield segment of the bilinear representation of the capacity

spectrum is given by:

…38

For any point , , on the post-yield segment of the bilinear representation, the

slope is given by:

…39

Since the slope is constant, equations 38 and 39 can be equated:

50

…40

Solve equation 40 for api in terms of dpi. Call solved for in these terms of I

I ...41

This value can be substituted for api into equation 8-8 to obtain an expression for eff

that is in terms of only one unknown, dpi.

eff

Solve equation 51 for eff for a series of dpi values. Then substitute the expression

for o, that is, .

6. For each dpi value considered in step 5, plot the resulting dpi, eff point on the same

chart as the family of demand spectra and the capacity spectrum. Fig. 3.13 shows five

of these points.

7. As illustrated in Fig. 3.13, connect the points created in step 6, to form a line. ,The

intersection of this line with the capacity spectrum defines the performance point.

51

Fig.3.13 Capacity Spectrum Procedure after Step 7

3.10 ENERGY BASED APPROACH FOR THE PUSHOVER ANALYSIS:

At its core, the capacity curve of a structure represents the development of

resistance to lateral forces as a function of increasing lateral displacements. The

capacity curve has great value in characterizing the degree of nonlinearity that may

develop in a first or predominant "mode", recognizing, of course, that the onset of

nonlinearity causes changes in modal properties and invalidates modal superposition.

Because floor displacements over the height of the building generally increase

disproportionately as the response becomes increasingly nonlinear, one cannot

rigorously justify the use of the displacement at any one location for the abscissa of

the capacity curve, since the apparent post-yield stiffness of the capacity curve

depends on the location selected. As shown outright reversals in the capacity curve

may result in some cases. Rather than relying on the roof displacement the use of

energy absorbed by the structure in each modal pushover analysis to determine the

corresponding capacity curve of the equivalent SDOF system, recognizing that the

behavior of the MDOF system and its analogous SDOF system can be appreciated

from both conventional and energy-based perspectives. The energy-based formulation

developed below avoids the arbitrary selection of a single floor (or roof) location as

the parameter for representing the capacity curve, and may be used with single or

multimode analysis procedures.

52

The equation of motion is often expressed as the dynamic equilibrium of force

quantities [Eq. (15)], but can equivalently be expressed in terms of energy quantities.

The "absolute" energy form of Eq. (15) expressed in terms of the energy developed

from the time that the excitation starts, can be obtained by integrating Eq. [15] with

respect to displacement, as described by Usang and Bertero-1988 [2].

…42

where mi is the lumped mass associated with the ith story and it is the absolute (or

total) acceleration at the ith story, and fs is the restoring force.

In both the "absolute" and "relative" energy formulations of the equation of motion,

the absorbed energy, Ea is

…43

which is the third term of Eq. (42). The absorbed energy is composed of the

recoverable elastic strain energy and non-recoverable energy associated with energy

dissipated by the hysteretic response of the structural components. The static force

associated with the nth mode is fn (t). The restoring force is assumed to be equal to

sum of the modal components fn (t). Following this assumption, the restoring force f,

can be represented in terms of its modal components:

…44

Due to the orthogonality of modes with respect to k (φiT . k. φ j 0 for i= j and

0 otherwise) the force fn does work only for displacements in the nth mode. The work

done by this force on the other modal displacements is zero. In the elastic domain, the

absorbed energy associated with the static force fn going through an elastic

displacement from 0 to un may be computed by substituting Eq. (22) for fn and Eq. (5)

for un:

53

…45

The corresponding base shear associated with the nth mode pushover is:

…46

Substituting Eq. (24) into Eq. (23), we obtain

…47

Equation (47) can be interpreted graphically as the area beneath the curve in a plot of

Vb,n with respect to Dn in the elastic domain (Fig. 3.2). Therefore, we define the

energy-based displacement, De,n to be equal to 2En/Vb,n in order to assure that De,n = Dn

in the elastic domain.

Fig. 3.14 Extension of De,n definition to elastic and inelastic domain.

More generally, for both the elastic and inelastic response, the work done by Vb,n in a

differential displacement dDe,n is dEn:

…48

54

which is necessarily equal to the work done by the static force f, in a differential

displacement of the structure in this mode. Using an incremental formulation, the

terms ΔEn and Vbn can be computed for each step in the pushover analysis. Then, the

corresponding increment in the energy-based displacement, ΔDe,n may be calculated

as:

…49

The value of De,n corresponding to the base shear is determined by summation.

Equation (27) is consistent with Eq. (25) in the elastic domain. The possible influence

of changes in the deformed shape from static forces associated with modes other than

the nth mode is neglected in this formulation, because orthogonality of the load vector

and the elastic mode shapes is assumed, as described earlier. As with conventional

pushover approaches, the mapping for the ordinate of the common representation can

be obtained by solving Eq. (24) for the term ωn2Dn(t):

…50

In this case, the values plotted on the ordinate of determined as before, as:

…51

3.11 ENERY-BASED PUSHOVER ANALYSIS PROCEDURE

In the energy-based pushover approach, the capacity curve associated with each

modal pushover is determined based on the work done in the analysis. The work is

computed incrementally typically for each step in the pushover analysis.

55

Analysis Procedure: Step-by-step implementation of the Energy-Based POA

procedure is presented.

1. Compute natural frequencies, ωn, and modes, φn, for linear-elastic vibration of

the structure. The conventional modal analysis has to be carried out for calculating

the mass stiffness matrices. Later the determinant of the |k – m * ωn2|= 0, for a trivial

solution, where n is the number of modes. Sequentially the modal frequency vectors

are established by one of the values in the vector as 1 and solving for the other values

of the vector.

[K – M * ωi2] * Фin = 0

2. Define the force distribution: Sn = m.Фn. The force distribution over the structure

is according to the mode shape of the structure and will be as defined by the mode

shapes for higher modes also.

3. Apply the force distribution characterized in step 2 incrementally, and record the

base-shears and associated story-displacements. The structure should be pushed just

beyond the expected targeted roof displacement, in the selected mode.

4. Employing the energy-based pushover approach presented by Hernandez-Montes et

al. [16], the capacity curve associated with each modal pushover analysis is determined

based on the work done/energy absorbed in displacing the structure. The work is

computed incrementally typically for each step in the pushover analysis. The

increment in the energy-based displacement of the nth mode ESDOF system, ΔDe, n, is

obtained as:

expression gives the response in the elastic domain of the structure.

ΔDe,n = ΔEn x Vb,n expression gives the response for an incremental step of the POA.

where En = increment of work done by the lateral forces acting through the

displacement increment associated with one step of the nth mode pushover analysis

and Vb,n = base-shear at that step of the pushover analysis, which is equal to the sum

of the lateral forces at that step. The incremental displacements, ΔDen, are

56

accumulated (summed) to obtain the displacement Den, of the ESDOF system at any

given step in the modal pushover analysis.

5. Idealize the pushover curve as a bilinear curve using the ATC-40[3] procedure.

6. Analyze the equivalent SDOF system of each mode through the POA method.

Subsequently, obtain the plastic moments of the members, for the given displacement

De, n.

7. Energy shapes (ψ1, ψ2, ψ3…) as shown in Fig. 3.14 are obtained by normalizing the

cumulative rotation on each floor by the second floor’s rotation. The first-mode

energy Ea1 is then distributed to each floor level and the base of the frame with the

chosen energy shape ψk

From these values of energy distribution vectors, the energy distribution for each and

every storey level is calculated selecting the appropriate energy shape vector as

suggested by C.C Chou and C.M Uang [9].

Fig. 3.15 Frame subjected to lateral forces fim and pushover curve of a single

floor i.

57

Fig. 3.16 Shape of the energy distribution vectors.

8. This procedure is repeated for each mode separately and finally, the desired

responses are extracted from pushover database values at the target displacement.

However, in this process, each energy-based displacement obtained from each record

is a target displacement, separately for Energy-Based POA. The desired responses are

extracted from pushover database values at every target displacement. This method is

more accurate, although huge amounts of calculation are needed.

9. Determine the Demand curve for each mode as describes earlier and plot the energy

based capacity spectrum and demand curve in a same graph for the respective modes,

and the intersection of this demand curve with the energy based capacity spectrum

defines the performance point.

58

Fig. 3.17 Flow chart showing the step by step procedure to obtain performance point using Energy Based Pushover Analysis

3.12 SUMMARY

In this chapter the basics and background of the methods have been presented.

The basis of the procedure is the energy, the theoretical formulations of the equations

and insight of the procedures, for the calculations, have been given for better

understanding of the methodology with diagrams and appropriate examples and for

59

convenience step wise procedure have been provided in the form of a chart. The

Pushover analysis has to be run through a convenient computing tool capable of

performing NSPs. In this case study ETABS 9.7.4 [26] and SAP V.15.0[27] (CSI Ltd)

non linear analysis engine has been utilized and the procedure regarding the

application has been explained in detail with appropriate diagrams wherever

necessary.

60

CASE STUDY

61

CHAPTER-IV

CASE STUDY

4.1 GENERAL

The building considered is having plan 4x3 bays of lengths 5m, 3m, 3m and 5m in X-direction and 4m, 3m and 4m in Y-direction of reinforced concrete ordinary moment resisting space frame of 7 storey configuration. Here stiffness of the infill is neglected in order to account the nonlinear behavior of seismic demands. The bottom storey height is 2.5m and remaining stories are 3m in height. The building has been analyzed by EBPOA method.

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 frame

Sl.no Variable Data

1 Type of structure Moment Resisting Frame

2 Number of Stories 7

3 Bottom storey height 2.5m

3 Floor height 3m

4 Live Load 5.0 kN/m

5 Dead load 10.0 kN/m

6 Materials Concrete (M30) and Reinforced with HYSD bars (Fe415)

7 Size of Columns C1-500x500 mm C2-600x400 mm

8 Size of Beams 300x400 mm

9 Depth of slab 120mm thick

10 Specific weight of RCC 25 kN/m3

11 Zone II12 Importance Factor 1

13 Response Reduction Factor 3

14 Type of soil Medium

62

4.2 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 4.1.

Fig. 4.1 Geometry of the structure

4.3 GENERAL DATA OF BUILDING

The building model is located in zone II. Tables 4.1 and Table 4.2 present a

summary of the building parameters.

63

Table 4.1 General data collection and condition assessment of building

Sl.No

.Description Information Remarks

1 Building height 20.5m Above ground level

2 Open ground storey Yes ----

3 Special hazards None ----

4 Type of building RegularIS 1893:2002[15]

Clause 7.1

5 Horizontal floor systemBeams and

Slabs----

6 Software usedETABS 9.7.4[26]

SAP V.15.0[27]----

4.4 ANALYSIS USING RESPONSE SPECTRUM METHOD

Analysis methods are broadly classified as linear static, linear dynamic, nonlinear

static and nonlinear dynamic methods. In these the first two methods are suitable when

the structural loads are small and no point, the load will reach to collapse load and are

differs in obtaining the level of forces and their distribution along the height of the

structure. Whereas the non- linear static and non-linear dynamic analysis are the

improved methods over linear approach. During earthquake loads the structural loading

will reach to collapse load and the material stresses will be above yield stresses. So in that

case material nonlinearity and geometrical nonlinearity should be incorporated into the

analysis to get better results. These methods also provide information on the strength,

deformation and ductility of the structures as well as distribution of demands.

64

Linear dynamic analysis of the building models is performed using ETABS

9.7.4[26]. The lateral loads generated by ETABS 9.7.4 correspond to the seismic zone II

and 5% damped response spectrum given in IS 1893-2002[15] (Part 1). The fundamental

natural period values are calculated by ETABS 9.7.4 [26], by solving the eigenvalue

problem of the model. Thus, the total earthquake load generated and its distribution along

the height corresponds to the mass and stiffness distribution as modeled by ETABS 9.7.4 [26].The response spectrum for the structure is defined in the SAP V.15.0 [27] as Spec 1

(Spectrum) and the response spectrum analysis data is defined in the load cases defined.

The following Fig 4.2 show the details of the Response history analysis case. The

response spectrum analysis is run and the members are designed for the analysis results

obtained, by IS-456:2000[11] design specifications. The SAP V.15.0[27] has predefined

member reinforcement details, these are overwritten by selecting the Reset all Concrete

Overwrites button.

Fig. 4.2 Response Spectrum Load Case Data

4.5 PUSHOVER ANALYSIS FOR DIFFERENT MODES

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 behaviour of the structure. Lateral load

patterns involved in determining pushover curve of the building structure should

65

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. SAP V.15.0[27] 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

FEMA-273[6]. 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[6] 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 [27] as a default method,

which is incorporated from the ATC-40 [3] document.

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

of the structure. The SAP V.15.0 [27] package gives the choice to the user for selecting or

defining the type of hinges required for analysis.

66

Fig. 4.3 Hinge assignments at the ends of the beams and columns.

Default hinge definitions according to the FEMA-356[10] 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. 4.4 Performance levels according to hinge states.

67

Force-displacement or moment-rotation curve for a hinge definition used in SAP2000 [2] is

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

by the following points as:

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 behaviour 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

68

this study, only two types of hinges are used to simulate the plastic hinge formation

through the non-linear behaviour of the structure. The first is the coupled 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 4.5. The top displacement of the

structure is set to 4% 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 ie., 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[11] and then pushover cases are run on the designed structure, for the analysis

results.

Figure 4.5 Pushover Analysis Load Case Data

69

4.6 SUMMARY

The geometrical and physical aspects of the structure under consideration have

been presented with the help of plan and elevation drawings of the structure. The sections

assigned to the structure have been highlighted and further information regarding the

structure has been tabulated including the details of the Dead / Imposed loads. Details of

the response spectrum analysis according to IS-1893:2002[15] that is used to analyze and

design the structure for dynamic loading has been presented. Further the definition of the

modal pushover cases and hinge properties assigned to the structural members have been

discussed in detail.

70

RESULTS

71

CHAPTER-V

RESULTS AND DISCUSSIONS

5.1 GENERAL

The Energy-Based POA was implemented for a symmetrical 3D R.C frame

subjected to the selected ground motion. To estimate seismic demands, the contributions

of higher modes were included in the analysis of the building. The selected pushover

curves and Energy based pushover curve are illustrated in Fig 5.1

5.2 RESULTS AND DISCUSSION

The following table contains the results of the push over analysis and energy

based pushover analysis carried out for the structure as discussed earlier. The result are of

different pushover cases and each pushover case is for specific mode and direction i.e

PUSH-1Y represents pushover in mode 1 and in Y-direction. The values tabulated are of

the critical modes of the structure, which have been considered in the analysis.

Table 5.1 Results of the POA and EBPOA of the structure.

Case:

POA EBPOA

Max

Base Shear (kN)

Max Displacement

(m)

Max

Base Shear (kN)

Max Displacement

(m)

PUSH-1Y 1635.9364 0.2831 1635.9364 0.21479

PUSH-2X 1860.1984 0.2945 1860.1984 0.223278

PUSH-3Y 3828.0293 -0.0357 3828.0293 0.02966

PUSH-4Y 3693.9202 0.0051 3693.9202 0.016502

72

PUSH-5X 4647.293 -0.0365 4647.293 0.025029

PUSH-6X 4448.1318 -0.0014 4448.1318 0.01206

The results for EBPOA are obtained from the equations proposed by Hernandez

Montes [16], while the results for POA have been obtained directly from the ETAB 9.7.4 [21]

output file. The use of the roof displacement in the conventional POA approach leads to

an apparent stiffening in the post – yield response, while the energy-based approach

shows monotonic softening with increasing displacement. Because each plastic hinge has

a bilinear moment-rotation relationship the apparent stiffening after initial yielding is not

physically reasonable, but instead must be viewed as a consequence of the arbitrary

choice to index the capacity curve by displacement that increases disproportionately in

the nonlinear regime.

Fig 5.1 Pushover curve for different modes

73

Fig 5.2 Energy base pushover curve for different modes

5.2.1 CAPACITY CURVES

The pushover curves are obtained using the procedure given by Prof. Hernandez

Montes [16].The pushover analysis for different modes is performed and the capacity

curves are plotted from the analysis results obtained for POA and Energy-Based Pushover

analysis. Fig. 5.1 shows the capacity curves of conventional pushover and Energy based

pushover for different modes of the structure. It is seen that the energy based curve for the

first mode response nearly coincides with the conventional first mode pushover capacity

curve. The pushover curve is converted into Capacity spectrum curve using procedure

given in ATC-40[3]. For simplicity of calculation and understanding purpose the capacity

curve is converted into bilinear capacity spectrum. Fig. 5.9 shows capacity spectrum

curves and Fig. 5.10 shows bilinear curve of first mode.

74

Fig. 5.3 Push-over curve for mode 1

Fig. 5.4 Push-over curve for mode 2

75

Fig. 5.5 Push-over curve for mode 3

Fig. 5.6 Push-over curve for mode 4

76

Fig. 5.7 Push-over curve for mode 5

Fig. 5.8 Push-over curve for mode 6

77

Fig. 5.9 Capacity spectrum curve for EBPOA for mode 1

Fig.5.10 Bilinear capacity spectrum for mode 1

78

5.2.2 RESPONSE SPECTRA AND DEMAND CURVES

Response spectra is plotted for zone factor-0.10 and effective peak acceleration

(EPA) of the ground is 0.24*g as per ATC-40[3] as shown in Fig 5.11 . It is then converted

into Acceleration Displacement Response Spectra (ADRS) format i.e. spectral

acceleration v/s spectral displacement as shown in Fig. 5.12. And then a series of family

of response spectra is plotted corresponding to different effective damping values (eff)

starting from 5% to maximum of 29% for type B structure. Demand curve is generated

using procedure given in ATC-40[3]. The energy capacity and demand curves at yield of

the frame are shown in Fig. 5.14. Using the intersection points of the demand and

capacity curves as the maximum displacement values, the roof displacement and drift

demands could be determined.

Fig. 5.11 Response Spectrum (Standard Format)

79

Fig. 5.12 Response Spectrum (ADRS Format)

Fig. 5.13 Family of Reduced Reponse Spectra

80

Fig. 5.14 Demand curve and Capacity curve for mode 1

Fig. 5.15 Demand curve and Capacity curve for Mode 2

81

Fig. 5.16 Demand curve and Capacity curve for Mode 3

Fig. 5.17 Demand curve and Capacity curve for Mode 4

82

Fig. 5.18 Demand curve and Capacity curve for Mode 5

Fig. 5.19 Demand curve and Capacity curve for Mode 6

83

Table 5.2 Performance of building for different modes

Case Sd Sa DisplacementBase shear

Performance Level

POA-mode1 0.1025 0.1422 0.13298 1537.48 C.P

POA-mode2 0.0840 0.1653 0.10935 1768.84 L.S

EBPOA- mode 1 0.0948 0.1465 0.12300 1579.21 C.P

EBPOA- mode2 0.0779 0.1745 0.10135 1866.49 L.S

EBPOA- mode 3 0.0055 0.5094 0.00027 92.57 I.O

EBPOA- mode 4 0.0119 0.4727 0.00017 49.54 I.O

EBPOA- mode 5 0.0031 0.41630 0.0001 75.93 I.O

EBPOA- mode 6 0.0089 0.52019 0.0001 53.25 I.O

Table 5.3 Comparision of POA and EBPOA for mode 1 and 2

Mode 1 Mode 2 D(m) B.S (kN) D(m) B.S (kN)

POA 0.1329 1537.48 0.1093 1768.84EBPOA 0.1230 1579.21 0.1013 1866.49

Difference(%) 7.50 2.71 7.31 5.52

In the table 5.2, the performance of the building in terms of spectral acceleration (Sa)

and spectral displacement (Sd); displacement and base shear for EBPOA procedure are

shown for different modes and the same was shown for the first two modes of normal

POA. Table 5.3 shows the comparison in displacement and base shear at performance

point for mode 1 and mode 2. The results obtain from POA and EBPOA are nearly same .

The comparisons of the results are shown in bar charts in the Fig. 5.20 and Fig. 5.21. The

accuracy of this method can be claimed by the fact that, in both the cases, the building

performance is same for same modes i.e. in the first mode is collapse prevention (C.P),

and in the second mode is life safety (L.S). For the higher modes the performance point is

obtained using EBPOA which is not possible in conventional POA because of mode

reversal.

84

Fig. 5.20 Displacements at performance level for mode 1 and 2 using POA and EBPOA

Fig. 5.21 Base Shear at performance level for mode 1 and 2 of POA and EBPOA

5.3 SUMMARY

In this chapter, an insight to the results has been presented. The results of significant modes of the structure have been considered. The non-linear responses of the structure have been highlighted in the form of graphs and possible reasoning has been given for the response of the structure. The calculation details for the responses of the structure have been shown in the Appendix-I. From the study it has been observed that the results EBPOA is close to the POA for the first two modes. The POA procedure overestimates the storey drifts and underestimates the responses base shear of the structure. Overall, the performance level of building falls under same category (CP for mode 1 and LS for mode 2 and IO for other higher modes) for the two methods.

85

CONCLUSIONS

86

CHAPTER-VI

CONCLUSIONS

6.1 GENERAL

The general NSPs show the application of the force profile, and the response of

the structure is calculated accordingly, mostly without consideration of the energy or

work done in the system. The adaptive procedure should be based on the equations of

motion and if this is the case, adaptive procedures depend entirely upon the input forces.

Hence the response is associated with the energy stored in the structure.

The capacity curve determined in a pushover analysis reveals the rate at which a

structure develops resistance to lateral forces of a given pattern; this resistance has been

viewed conventionally from the point of view of roof displacement or alternatively from

the perspective of the energy absorbed in the lateral load analysis. Though roof

displacement remains a useful index for the first mode response of many structures but it

ceases to increase proportionately. However, for some structures, the roof displacement

may be a poor index, even for elastic response. For yielding systems, disproportionate

increases in displacements affect the slope of the capacity curve. The response of the

SDOF systems with negative post-yield stiffness is known to be sensitive to the post-yield

stiffness, and therefore, estimates made using analogous SDOF systems may be sensitive

to the displacement index used for the capacity curve. For higher mode pushover

analyses, the roof displacements may reverse, leading to the reversal of modes.

The energy absorbed by the MDOF structure in the pushover analysis has been

used to derive an energy-based displacement that characterizes the work done by the

equivalent SDOF system. Thus, in contrast to the conventional view of pushover

analysis, pushover analysis has been equivalently viewed in terms of the work done (or

absorbed energy). The data associated with roof displacements has been used to

determine the energy-based displacement, De,n. The capacity curve of the equivalent

SDOF system was then obtained using conventional transformations of the base shear

together with the energy-based displacements, as illustrated schematically in the previous

87

chapters. The energy-based capacity curve was defined to match the capacity curves

obtained using conventional approaches for the first and higher mode analyses in the

elastic and inelastic domains. The energy-based capacity curves are able to correct the

anomalies observed for the higher mode pushover curves. It is concluded that the energy-

based formulation provides a stronger theoretical basis for establishing the capacity

curves of the first and higher mode equivalent SDOF systems.

6.2 CONCLUSIONS

The Energy-based POA has been applied to a 7 storey moment resisting 3D plane

frame, designed according to the Indian Standards (IS-456: 2000[11]) for seismic loading

using response spectrum analysis. The performance of the structure is then evaluated for

the dynamic loading using the Energy Based Pushover Analysis, considering the effects

of the higher modes. The frame is subjected to dead, live and dynamic loading as

specified in IS-1893:2002[15] Codes of Practice and designed according to

IS-456:2000[11].

Based on the study of the structural performance of the model, it is concluded that

From the table 5.2 it is concluded that the performance level of the building

improves as higher modes are considered.

From the Fig. 5.2 it is proved that the EBPOA over comes “mode reversal”

limitation of conventional pushover analysis (Fig 5.1).

From the Fig. 5.2 and Fig. 5.3 we can conclude that EBPOA shows a gradual

softening of the structure rather than a quick failure.

The target displacement is decreased by 7.5% and base shear is increased by 2.7%

in the first mode, and target displacement decreased by 7.3% and base shear is

increased by 5.52% in the second mode using EBPOA ( table 5.3)

From the table 5.3, we can say that the performance point obtained by the energy

based push over analysis is close to the conventional POA for the first and second

mode, therefore this method of Energy based approach can be applied to higher

modes.

88

6.3 SCOPE FOR FURTHUR STUDY

As the various researchers are getting attracted towards the NSP’s, the scope of the

studies under the particular topic can be stretched to wide horizons.

The application of method with different structural members for a better

performance of the structure can serve as a good topic for a research program.

The method can be tested and applied for different zones of earthquake.

A comprehensive study of the performance of a base-isolated structure subjected

to seismic loads and the effects of different modes can be examined.

89

References1. Zahrah, T.F. and Hall, W.J. (1984). “Earthquake Energy Absorption in SDOF

structures,” Journal of Structural Engineering, ASCE, Vol. 110, No. 8, pp. 1757–

1772

2. Usang, C.M. & Bertero, V.V. (1988), Use of energy as a design criterion in

earthquake resistant design. Report no. UCB/EERC-88/18; Earthquake

Engineering Research Center, University of California at Berkeley

3. Applied Technology Council, ATC-40 (1996), ‘Seismic evaluation and retrofit

of concrete buildings’, Vol.1 and 2, California.

4. Paret, T.F., Sasaki, K.K., Elibeck, D.H. and Freeman, S.A. (1996),

“Approximate inelastic procedures to identify failure mechanism from higher

mode effects", Proceedings of the Eleventh World Conference on Earthquake

Engineering, Paper 966, Pergamon, Elsevier Science Ltd, Acapulco, Mexico.

5. Younghui Roger Li. (1996), “ Nonlinear Time History and Pushover Analysis

For Seismic Design And Evaluation”, University of Texas, Austin

6. FEMA 273 (1997) ‘NEHRP Guidelines for the seismic rehabilitation of

buildings’, Building Seismic Safety Council, Washington, D.C

7. Sasaki, F., Freeman, S. and Paret, T. (1998), “Multi-Mode Pushover Procedure

(MMP), a method to identify the effect of higher modes in a pushover analysis",

Proc. 6th U.S. National Conference on Earthquake Engineering, Seattle, CD-

ROM, EERI, Oakland.

8. Ashraf Habibullah, S.E.1, and Stephen Pyle (Winter, 1998), “Practical Three

Dimensional Nonlinear Static Pushover Analysis”, S.E.2 (Published in Structure

Magazine).

9. Chou C-C, Uang C-M.,“Establishing absorbed energy spectra-An attenuation

approach. Earthquake Engineering and Structural Dynamics 2000”; 29(10):1441–

1455.

90

10. FEMA 356 (2000) ‘Pre-standard and commentary for the seismic rehabilitation of

buildings’, ASCE for the Federal Emergency Management Agency, Washington,

D.C.

11. IS:456 (2000), Indian Standard for Plain and Reinforced Concrete Code of

Practice, Bureau of Indian Standards, New Delhi.

12. Chung-Che Chou and Chia-Ming Uang (2001), “A procedure for evaluating

seismic energy demand of framed structures”, Department of Structural

Engineering; University of California; San Diego; La Jolla; CA 92093; U.S.A.

13. S. Chandrashekaran, Anubhab Roy (2001), “Seismic evaluation of multi storey

r c frame building using modal pushover analysis”, Non-linear Dynamics, 2006,

Springer, pp. 329-342.

14. T. Albanesi, S. Binodi and M. Petranjali (2002), “An energy based approach”,

University “G.D’Annuzio” of Chieti, Prisco, Progettazione, Riabilitazione e

Controllo delle strtture.

15. IS:1893 (2002) Indian Standard Criteria for Earthquake Resistant Design of

Structures, Bureau of Indian Standards, New Delhi 110002

16. Hernandez-Montes, E., Kwon. O.S. and Aschheim, M. (2004), “An energy-

based formulation for first and multiplemode nonlinear static (pushover)

analyses", J. Earthquake Eng., 8(1), pp. 69-88.

17. A.K Chopra and Goel (2004), “A modal Pushover analysis procedure for

Seismic analysis of the buildings”.

18. Tjhin, T., Aschheim, M. and Hernandez-Montes (2005), “Estimates of Peak

Roof Displacement Using Equivalent Single Degree of Freedom Systems", J.

Struct. Eng., 131(3), pp. 517-522.

19. Siddhartha Ghosh, and Kevin R. Collins (2006), “Merging energy-based design

criteria and reliability-based methods: Exploring a new concept”, Department of

Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai

400076, India Department of Civil Engineering, Lawrence Technological

University, Southfield, MI 48075, U.S.A.

20. C. Kotanidis and I.N. Doudoumis (2006), “Energy-Based Approach Of Static

Pushover Analysis”, Dipl. Civil Engineer, Dept. of Civil Engineering , Aristotle

University of Thessaloniki, Thessaloniki, Greece Professor, Dept. of Civil

Engineering , Aristotle University of Thessaloniki, Thessaloniki, Greece.

91

21. Sutat Leelataviwat , Winai Saewon , Subhash C. Goel, “An Energy Based

Method For Seismic Evaluation Of Structures”, the 14th World Conference on

Earthquake Engineering October 2008, Beijing, China

22. M.J. Hashemi and M. Mofid (2010), “Evaluation of Energy-Based Modal

Pushover Analysis in reinforced Concrete Frames with Elevation Irregularity”.

23. Grigorios Manoukas, Asimina Athanatopoulou and Ioannis, Avramidis

(2011), “Static Pushover Analysis Based on an Energy-Equivalent SDOF System:

Application to Spatial Systems”, CED, Aristotle University, Thessaloniki, Greece.

24. Peng Pan, Makoto Ohsaki (2008), “A Multi-mode Pushover Analysis method

for seismic assessment”, Department of Civil Engineering, Birmingham

University.

25. Prasanth, T., Ghosh, S. and Collins, K.R. (2008). “Estimation of Hysteretic

Energy Demand using Concepts of Modal Pushover Analysis,” Earthquake

Engineering and Structural Dynamics, Vol. 37, No. 6, pp. 975–990.

26. ETABS 9.7.4, Structural Analysis Program, Computers and Structures, Inc.,

Berkeley, CA.

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Berkeley, CA.

92

APPENDIX-I

93

APPENDIX-I

The energy absorbed by the structure at different steps in the analysis is calculated.

The target roof displacement is divided into required number of smaller displacements at

which the pushover forces and the roof displacement of the structure are obtained from

the output of the analysis program. The energy absorbed in each of these steps is added

up to get the total energy absorbed during the process of pushing the structure.

Table I.1: Detailed calculation of the energy absorbed by the structure for 7 steps of increment of the lateral pushover forces on the structure in the 1st mode of pushover

analysis.

S T E P 0

storey massnormalize

ØB.S step

0B.S step

1 F0 F1 U0 U1 ▲W7 183.938 0.233 0 646.533 0 150.652 0 0.0105 0.79096 201.524 0.217 0 646.533 0 140.634 0 0.0098 0.68915 201.524 0.191 0 646.533 0 124.066 0 0.0087 0.53964 201.524 0.156 0 646.533 0 101.333 0 0.0071 0.35973 201.524 0.113 0 646.533 0 73.592 0 0.0051 0.18762 201.524 0.066 0 646.533 0 42.768 0 0.003 0.06411 198.593 0.02 0 646.533 0 13.485 0 0.0009 0.006

Σ▲W = 2.6373Total

mass = 1390.157 1 = 646.5335 ▲Uen = 0.0081Uen = 0.0081

S T E P 1storey B.S step1 B.S step2 F1 F2 U1 U2 ▲W

7 693.3893 879.8298 150.6523 204.0401 0.0105 0.0158 0.9399356 693.3893 879.8298 140.6345 191.0083 0.0098 0.0149 0.8456895 693.3893 879.8298 124.0666 169.0405 0.0087 0.0133 0.674146

94

4 693.3893 879.8298 101.3339 138.509 0.0071 0.0111 0.4796863 693.3893 879.8298 73.59231 100.5307 0.0051 0.0081 0.2611852 693.3893 879.8298 42.76831 58.45674 0.003 0.0046 0.080981 693.3893 879.8298 13.4855 18.24446 0.0009 0.0013 0.006346

Σ▲W = 3.287967= 879.8298 ▲Uen = 0.00418

Uen = 0.012338

S T E P 2

storey B.S step2B.S step

3 F2 F3 U2 U3 ▲W

7 879.8298 1014.712 204.0401 235.3205 0.0158 0.022 1.362018

6 879.8298 1014.712 191.0083 220.2909 0.0149 0.0209 1.233898

5 879.8298 1014.712 169.0405 194.9553 0.0133 0.019 1.037388

4 879.8298 1014.712 138.509 159.7431 0.0111 0.016 0.730718

3 879.8298 1014.712 100.5307 115.9426 0.0081 0.0117 0.389652

2 879.8298 1014.712 58.45674 67.41846 0.0046 0.0065 0.119581

1 879.8298 1014.712 18.24446 21.04143 0.0013 0.0019 0.011786Σ▲W = 4.88504

= 1014.712 ▲Uen = 0.005157

Uen = 0.017495

s t e p 3storey B.S step3 B.S step F3 F4 U3 U4 ▲W

7 1014.712 1446.601 235.3205 335.4791 0.022 0.084 17.694796 1014.712 1446.601 220.2909 314.0525 0.0209 0.0809 16.03035 1014.712 1446.601 194.9553 277.9334 0.019 0.0735 12.886224 1014.712 1446.601 159.7431 227.734 0.016 0.0599 8.5051223 1014.712 1446.601 115.9426 165.2908 0.0117 0.0415 4.1903772 1014.712 1446.601 67.41846 96.11353 0.0065 0.0216 1.2346671 1014.712 1446.601 21.04143 29.99722 0.0019 0.0058 0.099525

Σ▲W = 60.641= 1446.601 ▲Uen = 0.049275

Uen = 0.066771

95

s t e p 4storey B.S step4 B.S step5 F4 F5 U4 U5 ▲W

7 1446.601 1505.242 335.4791 349.0786 0.084 0.1005 5.6476016 1446.601 1505.242 314.0525 326.7835 0.0809 0.0966 5.0305625 1446.601 1505.242 277.9334 289.2002 0.0735 0.0873 3.9132224 1446.601 1505.242 227.734 236.9658 0.0599 0.071 2.5790843 1446.601 1505.242 165.2908 171.9913 0.0415 0.0492 1.2985362 1446.601 1505.242 96.11353 100.0098 0.0216 0.0261 0.4412771 1446.601 1505.242 29.99722 31.21323 0.0058 0.0074 0.048968

Σ▲W = 18.95925

= 1505.242 ▲Uen = 0.012846Uen = 0.079616

s t e p 5storey B.S step5 B.S step6 F5 F6 U5 U6 ▲W

7 1505.242 1579.242 349.0786 366.2398 0.1005 0.1589 20.88736 1505.242 1579.242 326.7835 342.8486 0.0966 0.1515 18.38145 1505.242 1579.242 289.2002 303.4177 0.0873 0.1363 14.519144 1505.242 1579.242 236.9658 248.6154 0.071 0.1121 9.9786923 1505.242 1579.242 171.9913 180.4466 0.0492 0.0808 5.5685192 1505.242 1579.242 100.0098 104.9264 0.0261 0.0469 2.1313361 1505.242 1579.242 31.21323 32.74772 0.0074 0.0169 0.303815

Σ▲W = 71.7702= 1579.242 ▲Uen = 0.046536

Uen = 0.126153

s t e p 6storey B.S step6 B.S step7 F6 F7 U6 U7 ▲W

7 1579.242 1621.558 366.2398 376.0533 0.1589 0.2509 34.145486 1579.242 1621.558 342.8486 352.0353 0.1515 0.2307 27.51745 1579.242 1621.558 303.4177 311.5478 0.1363 0.2022 20.263114 1579.242 1621.558 248.6154 255.2771 0.1121 0.1644 13.176793 1579.242 1621.558 180.4466 185.2817 0.0808 0.1193 7.0402712 1579.242 1621.558 104.9264 107.7379 0.0469 0.0717 2.6370371 1579.242 1621.558 32.74772 33.6252 0.0169 0.028 0.36837

Σ▲W = 105.1485= 1621.558 ▲Uen = 0.065701

Uen = 0.191854

s t e p 7storey B.S step7 B.S step8 F7 F8 U7 U8 ▲W

7 1621.558 1635.936 376.0533 379.3877 0.2509 0.2831 12.16266 1621.558 1635.936 352.0353 355.1567 0.2307 0.2583 9.75925

96

5 1621.558 1635.936 311.5478 314.3103 0.2022 0.2252 7.1973684 1621.558 1635.936 255.2771 257.5406 0.1644 0.1826 4.666643 1621.558 1635.936 185.2817 186.9246 0.1193 0.1328 2.5123932 1621.558 1635.936 107.7379 108.6932 0.0717 0.0803 0.9306541 1621.558 1635.936 33.6252 33.92335 0.028 0.0318 0.128342

Σ▲W = 37.35725= 1635.936 ▲Uen = 0.022936

Uen = 0.21479

Table I.2 Detailed calculation of conversion of pushover curve to capacity spectrum curve

PUSH 1 Y STEP 0 1 2 3 4 5 6 7 8Uen 0 0.0081 0.0123 0.0174 0.0667 0.0796 0.1261 0.1918 0.2147B.S 0 646.53 879.82 1014.71 1446.6 1505.24 1579.24 1621.55 1635.93M 1390.157 1390.15 1390.15 1390.15 1390.15 1390.15 1390.15 1390.15 1390.15 0.7899 0.7899 0.7899 0.7899 0.7899 0.7899 0.7899 0.7899 0.7899

P.F 5.5638 5.5638 5.5638 5.5638 5.5638 5.5638 5.5638 5.5638 5.5638Øroof 0.2330 0.2330 0.2330 0.2330 0.2330 0.2330 0.2330 0.2330 0.2330

Sa 0 0.0607 0.0816 0.0941 0.1342 0.1397 0.1466 0.1505 0.1518Sd 0 0.0062 0.0095 0.0134 0.0515 0.0614 0.0973 0.1479 0.1656

The area under the actual capacity curve is calculated and then bilinear curve is

created so that its area under the curve and its initial stiffness is equal to that of actual

capacity curve. Trial and error procedure is adopted, initially assuming any point to be a

yield point and then changing the point to reduce the error.

Table I.3 Detailed calculation of conversion of capacity spectrum curve to bilinear curve

97

PUSH 1YSTEP 0 1 2 3 4 5 6 7 8

Sd 0 0.0062 0.0095 0.0134 0.0515 0.0614 0.0973 0.1479 0.1656 Total

Sa 0 0.0600 0.0816 0.0941 0.1342 0.1397 0.1466 0.1505 0.1518 area

AREA 0 0.0001 0.0002 0.0003 0.0043 0.0013 0.0051 0.0075 0.0026 0.02180

ITERATION NO: Sa Sd A,cs Sa,y 0.6XSa,y Sd,0.6 Ki Sd,y αi,n Ai,b ERR

1 0.1518 0.1657 0.0218 0.0817 0.049 0.0051 9.537 0.0086 0.0468 0.0187 14.3

2 0.1518 0.1657 0.0218 0.0953 0.0571 0.006 9.537 0.01 0.0381 0.0197 9.61

3 0.1518 0.1657 0.0218 0.1054 0.0632 0.0066 9.537 0.0111 0.0315 0.0205 6.13

4 0.1518 0.1657 0.0218 0.1123 0.0673 0.0071 9.537 0.0118 0.0269 0.0210 3.775 0.1518 0.1657 0.0218 0.1167 0.0700 0.0073 9.537 0.0122 0.0240 0.0213 2.266 0.1518 0.1657 0.0218 0.1194 0.0716 0.0075 9.537 0.0125 0.0222 0.0215 1.337 0.1518 0.1657 0.0218 0.121 0.0726 0.0076 9.537 0.0127 0.0211 0.0216 0.788 0.1518 0.1657 0.0218 0.1219 0.0731 0.0077 9.537 0.0128 0.0205 0.0217 0.459 0.1518 0.1657 0.0218 0.1225 0.0735 0.0077 9.537 0.0128 0.0201 0.0218 0.21

10 0.1518 0.1657 0.0218 0.1228 0.0736 0.0077 9.537 0.0129 0.0199 0.0218 0.1111 0.1518 0.1657 0.0218 0.123 0.0738 0.0077 9.537 0.0129 0.0198 0.0218 0.0612 0.1518 0.1657 0.0218 0.1231 0.0738 0.0077 9.537 0.0129 0.0197 0.0218 0.01

O.K

98