seismic performance evaluation of non-ductile rc frames

分类号： TU3 学号： M2016028

U D C ： 624 密级：无

硕士学位论文

Seismic Performance Evaluation of Non-ductile RC Frames Retrofitted with

BRB and SMA Braces

Ezechias MUGISHA

指导教师姓名：宋良龙讲师河海大学土木与交通学院

南京市鼓楼区西康路 1 号

申请学位级别：工学硕士专业名称：结构工程

论文提交日期： 2019 年 4 月论文答辩日期：2019 年 5 月 26 日

学位授予单位和日期：河海大学 2019 年 6 月 18 日

答辩委员会主席：朱召泉教授论文评阅人：伞冰冰副教授，

姚菲副教授，张勤副教授，唐永圣副教授

2019年 5月中国南京

分类号(中图法) TU3 UDC(DDC) 624 密级无

论文作者姓名 Ezechias MUGISHA 学号 M2016028 单位河海大学土木与交通学院

论文中文题名 BRB 和 SMA 支撑加固非延性 RC 框架的抗震性能评估

论文中文副题名无

论文英文题名 Seismic Performance Evaluation of Non-ductile RC Frames

Retrofitted with BRB and SMA Braces

论文语种英语论文摘要语种英

论文页数 68 论文字数 1.8 (万)

论文主题词：Non-ductile frame, Buckling restrained brace (BRB), Shape

memory alloy (SMA), Seismic performance, Fragility

申请学位级别工学硕士专业名称结构工程

研究方向新型抗震结构体系

指导教师姓名宋良龙讲师导师单位河海大学土木与交通学院

论文答辩日期 2019 年 5 月 26 日

4

Seismic Performance Evaluation of Non-ductile

RC Frames Retrofitted with BRB and SMA Braces

A thesis submitted to

Hohai University

In fulfillment of the requirement

for the degree of

Master of Engineering

By

Ezechias MUGISHA

(College of Civil and Transportation Engineering，Structural

Engineering)

Thesis Supervisor : Lecturer Lianglong SONG

May, 2019 Nanjing, P.R. China

学位论文独创性声明：

本人所呈交的学位论文是我个人在导师指导下进行的研究工作及取得的研

究成果。尽我所知，除了文中特别加以标注和致谢的地方外，论文中不包含其

他人已经发表或撰写过的研究成果。与我一同工作的同事对本研究所做的任何

贡献均已在论文中作了明确的说明并表示了谢意。如不实，本人负全部责任。

论文作者（签名）： 2019 年月日

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分内容的公布(包括刊登)授权河海大学研究生院办理。

论文作者（签名）： 2019年月日

ABSTRACT

VI

ABSTRACT

Around the world, a large percentage of reinforced concrete (RC) frame buildings constructed

prior to the implementation of modern building codes tend to exhibit non-ductile behavior under

earthquakes. Thus, there is a significant need of modern retrofit techniques for these non-ductile RC

frames in order to reduce the extensive loss and casualties during future earthquakes. This thesis

conducted seismic performance evaluation of an old non-ductile RC frame building before and after

retrofit with buckling restrained braces (BRBs) and shape memory alloy (SMA) braces. In detail,

this study includes contents as follows:

(1) A beam-column joint model was developed to address the deficiencies in the non-ductile

RC joints and was incorporated in the numerical model of a non-ductile RC frame using the Open

System for Earthquake Engineering Simulation (OpenSees). The joint and frame models are

validated using the results of an experimental joint test and a reference model.

(2) To evaluate seismic performance of the as-built and retrofitted frames, a series of dynamic

time history analyses using the Frequently Occurred Earthquake (FOE), Design Basis Earthquake

(DBE) and Maximum Considered Earthquake (MCE) ground motions were performed. Analyses

results show that the BRB and SMA bracing systems can both improve the performance of the

non-ductile RC frame, with reduced peak and residual story drifts. It is found that the SMA braced

frames are capable of achieving the same level of seismic response control as the BRB frame in

terms of peak story drifts. Furthermore, the SMA braced frames have much smaller residual story

drifts than the BRB frame.

(3) The performance of the as-built and retrofitted frames was studied using the probabilistic

evaluation approach. Incremental dynamic analysis was performed to obtain the structural demands,

and regression analyses were conducted to determine the relationship between earthquake intensity

and structural responses. Using fragility function and results of incremental dynamic analysis,

seismic fragility curves of various performance levels were derived. The seismic fragility analyses,

combined with seismic hazard analysis, are used to determine the annual probability of exceedance

for various performance levels.

Key words: Non-ductile frame; Buckling restrained brace (BRB); Shape memory alloy (SMA);

Seismic performance; Fragility

目录

TABLE OF CONTENTS

ABSTRACT ............................................................................................................................................................ VI

TABLE OF CONTENTS ........................................................................................................................................ VII

CHAPTER 1 INTRODUCTION ............................................................................................................................... 1

1.1 Background .................................................................................................................................................. 1

1.2 Literature Review ........................................................................................................................................ 2

1.2.1 Non-ductile RC Frames .................................................................................................................... 2

1.2.2 Retrofit Methods ............................................................................................................................... 6

1.2.2.1 Buckling Restrained Braces (BRB) ....................................................................................... 7

1.2.2.2 SMA Braces ......................................................................................................................... 10

1.2.2.3 Other Retrofit Methods ........................................................................................................ 13

1.3 Research Objectives ................................................................................................................................... 14

CHAPTER 2 DESCRIPTION AND MODELING OF FRAMES........................................................................... 16

2.1 Description of the Prototype Frame Building ............................................................................................ 16

2.2 Numerical Models ..................................................................................................................................... 18

2.2.1 Non-ductile Frame Model ............................................................................................................... 18

2.2.2 BRB Frame Model .......................................................................................................................... 24

2.2.3 SMA Frame Model ......................................................................................................................... 27

CHAPTER 3 PERFORMANCE EVALUATION OF AS-BUILT AND RETROFITTED FRAMES ..................... 31

3.1 Selection of Ground Motions ..................................................................................................................... 31

3.2 Performance Evaluation ............................................................................................................................. 34

CHAPTER 4 FRAGILITY ANALYSIS OF AS-BUILT AND RETROFITTED FRAMES ................................... 43

4.1 Seismic Fragility Theory ........................................................................................................................... 43

4.2 Regression Analysis of Structural Demands .............................................................................................. 46

4.3 Fragility Analysis ....................................................................................................................................... 49

CHAPTER 5 SUMMARY, CONCLUSIONS AND FUTURE RESEARCH .......................................................... 54

5.1 Summary and Conclusions ........................................................................................................................ 54

5.2 Future Research ......................................................................................................................................... 55

References ............................................................................................................................................................... 56

CHAPTER 1 INTRODUCTION

1


1.1 Background

Reinforced concrete (RC) frame structures designed according to the current seismic design

codes are generally perceived to behave ductile so that under moderate-to-strong earthquakes, the

life and safety of building are protected. However, many older RC buildings constructed before the

implementation of modern building codes were designed for gravity loads only and tend to exhibit

non-ductile behavior under seismic excitations, which leads to substantial casualties and economic

losses during earthquakes. As such, modern societies in seismically-active areas are dealing with a

fair amount of seismic deficient RC frame buildings. The problem is not limited to developing

nations where building codes development and implementation are lax but also equally critical in

developed nations such as United States as modern building codes for RC frame structures did not

appear until the mid-1970s [1-2].

With increased seismological and geological knowledge about geophysical behavior of faults

and realistic earthquake hazard modeling, it is certain that the seismic risks of these seismically

vulnerable frame structures cannot be neglected as extremely-low probability of occurrence. In

addition, these non-ductile buildings have higher seismic damage probability and expected repair

cost for the structural damage induced by earthquakes than their counterparts which were

constructed based on modern building codes [3-4]. Considerable damage of these non-ductile RC

buildings has been reported in past earthquakes, including the 1994 Northridge, the 1999 Kocaeli

and the 2008 Wenchuan earthquakes. The knowledge of the poor structural performance of these

non-ductile RC buildings has also been augmented by the field observations of the collapsed and

damaged RC buildings, such as the structural and soft-story collapse, initiated from beam-column

joint failures [5-7] shown in Figure 1.1.

Due to that these non-ductile RC frame buildings are highly vulnerable in moderate-to-strong

earthquakes, there is a significant need of modern retrofitting strategies and techniques to reduce

their seismic risks. Over the past several decades, a number of retrofitting schemes for these

non-ductile frames have been investigated, such as providing RC infilled walls [8-9], column and

beam jacketing [10-13] and adding new steel braces [14-16].

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(a) Kaiser-Permanente Clinic,

Northridge 1994

(b) Yingxiu school,

Wenchuan 2008

(c) Femina Hotel,

Padang, Indonesia 2009

Figure 1.1 Structural collapse and damage due to beam-column joint failures

In recent years, the use of energy dissipating elements, such as buckling restrained braces

(BRBs), has gained popularity as an alternative way to retrofit an existing structure [17-18], due to the

stable hysteretic behavior of these braces that result in large energy dissipation. Although BRB

frames are believed to have an enhanced capability of withstanding strong earthquakes, the potential

for large residual drifts in this system diminish the practical feasibility of repairing the building

after an earthquake more difficult and financially costly. Therefore, there is a need for developing a

system in which structural damage following an earthquake is minimized and the building can

return to its original position with reasonable efforts.

Using Shape Memory Alloy (SMA) materials in bracing systems can be an imminent

alternative, given the excellent re-centering and supplemental energy dissipation capabilities of

SMA materials [19]. SMAs are unique materials that have the ability to undergo large deformation

and return to a predetermined shape upon unloading or by heating. Hence, their incorporation in

retrofitting strategies can offer a way of minimizing losses associated with damage to structural

systems during an earthquake.

1.2 Literature Review

1.2.1 Non-ductile RC Frames

The seismic deficiencies of non-ductile RC frames have been widely recognized and well

documented in literature. Based on a comprehensive review of design codes and detailing manuals

from the past five decades and consultation with practicing structural engineers, the following

reinforcing details that are typical in non-ductile RC frames designed for gravity loads only are

known to be problematic [20]: (1) little or no transverse shear reinforcement is provided within the


3

beam-column joints; (2) bottom reinforcement in the beams is terminated within the beam–column

joints with a short embedment length; (3) columns have bending moment capacities that are close to

or less than those of the joining beams, promoting column-side sway or soft-story mechanisms; (4)

the longitudinal reinforcement ratio in columns is seldom more than 2%; (5) there is minimal

transverse reinforcement in columns to provide shear resistance and confinement; (6) lightly

confined lapped splices of column reinforcement often are placed in potential plastic hinge zones

just above the floor levels; and (7) construction joints are placed immediately below and above the

beam-column joints.

Extensive research programs were undertaken in the early 1990s to examine the impact of

these deficiencies on the seismic response of non-ductile RC frames. Experimental studies at

member or component scales under reversed cyclic loading and at system scales under seismic

excitations were carried out and supplemented by analytical studies. Thirty-four full-scale interior

and exterior beam-column joints, a two-story one-bay 1/6-scale frame, and a three-story three-bay

1/8-scale frame were tested by Beres et al. [20], Pessiki et al. [21], and El-Attar et al. [22]. While test

results have highlighted the non-seismic details (e.g. unreinforced joint and lightly reinforced

columns) were sources of damage and inelastic deformation, inherent resistance of the frames

prevented complete structural collapse under a moderate earthquake excitation (up to 0.3g peak

ground acceleration (PGA) ground motions). The review of El-Attar et al. [22] results implies that the

small scaling factors (1/6 and 1/8) have possibly suppressed shear and/or axial failure of the

beam-column joints and columns. The above research programs found that despite their flexibility,

non-ductile RC frames possess some inherent seismic strength. However, structural performance of

such frames is questionable in case of a major earthquake.

Finite element-based structural models of non-ductile RC frames must incorporate the critical

details in predicting the response of such frames under earthquake loading. The need for such

models is becoming apparent with the move toward performance-based seismic engineering and the

use of seismic fragility assessment in decision-making [23]. Previous research [20-21, 24-28] on the

seismic performance of such frames has revealed that: (1) the first four problematic reinforcing

details listed previously are critical in making such frames vulnerable to seismic demands; (2) the

lack of adequate transverse reinforcement has only a marginal effect on performance (confinement

of the concrete core by transverse reinforcement can be taken into account through confined

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concrete models such as the modified Kent and Park model [29] in fiber models); and (3) previous

tests have not pointed to lapped splices and construction joints as a source of poor behavior. Of the

non-ductile reinforcing details that make these gravity load designed (GLD) RC frames vulnerable

to seismic demands (i.e., the first four of the non-ductile reinforcing details listed above), the latter

two are reflected explicitly in existing finite element platforms. However, additional modeling is

required to capture the inadequate joint shear capacity that results from a lack of transverse shear

reinforcement and the insufficient positive beam bar anchorage, in the finite element model.

Beam-column joint behavior is governed by shear and bond-slip in non-ductile RC frames. The

typical practice of providing little or no joint shear reinforcement leads to shear deformations in the

panel zone that may be substantial. This practice also leads to joint shear failure that can restrict the

utilization of the flexural capacities of the joining beams and columns. Moreover, the common

practice of terminating the beam bottom reinforcement within the joints makes the bottom

reinforcement prone to pullout under a seismic excitation. Insufficient beam bottom bar anchorage

precludes the formation of bond stresses necessary to develop the yield stress in the beam bottom

reinforcement. Thus, the positive beam moment capacity cannot be utilized. Hoffmann et al. [26] and

Kunnath et al. [27] modified the flexural capacities of the beams and columns of non-ductile RC

frames to model insufficient positive beam bar anchorage and inadequate joint shear capacity

implicitly. To account for insufficient positive beam bar anchorage, the pullout moment capacity of

the beam was approximated as the ratio of the embedment length to the required development

length per ACI 318-89 [30] multiplied by the yield moment of the section. This approximation

required that the yield strength of the discontinuous steel be reduced by the ratio of the actual to the

required anchorage length. To model inadequate joint shear capacity, the flexural capacities of the

beams and columns framing into the joint were reduced to a level that would induce shear failure of

the joint. The proposed procedure was utilized in inelastic dynamic time history analyses of typical

three-, six-, and nine-story non-ductile RC frames, which revealed that they are susceptible to

damage from joint shear failures and weak column-strong beam effects leading to soft-story

collapses. Alath and Kunnath [31] modeled the joint shear deformation with a rotational spring model

with degrading hysteresis. The finite size of the joint panel was taken into account by introducing

rigid links (see Figure. 1.2(a)). The envelope to the shear stress-strain relationship was determined

empirically, whereas the cyclic response was captured with a hysteretic model that was calibrated to


5

experimental cyclic response. The model was validated through a comparison of simulated and

experimental response of a typical non-ductile RC frame interior beam-column joint subassembly.

Lowes and Altoontash [32] proposed a 4-node 12-degree-of-freedom (DOF) joint element (see Figure.

1.2(b)) that explicitly represents three types of inelastic mechanisms of beam-column joints under

reversed cyclic loading. Eight zero-length translational springs simulate the bond-slip response of

beam and column longitudinal reinforcement; a panel zone component with a zero-length rotational

spring simulates the shear deformation of the joint; and four zero-length shear springs simulate the

interface-shear deformations. Because experimental research that reports bond-slip data of full-scale

frames or beam-column joint subassemblies is scarce, the envelope and cyclic response of the bar

stress versus slip deformation relationship were developed from tests of anchorage-zone specimens

and assumptions about the bond stress distribution within the joint. To define the envelope to the

shear stress-strain relationship of the panel zone, the modified-compression field theory (MCFT) [33]

was utilized. The cyclic response of the panel zone was modeled by a highly pinched hysteresis

relationship, deduced from experimental data provided by Stevens et al. [34]. A relatively stiff elastic

load-deformation response was assumed for the interface-shear components. Lowes et al. [35] later

attempted to model the interface-shear based on experimental data; this later effort also predicted a

stiff elastic response for the interface-shear.

(a) Scissors model (b) Panel zone model

Figure 1.2 Sample beam-column joint models

Rigid offset

Joint spring

Zero-length bond-slip spring

Zero-length shear spring

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Mitra and Lowes [36] subsequently evaluated the model proposed earlier by Lowes and

Altoontash [32] by comparing the simulated response with the experimental response of

beam-column joint subassemblies. The experimental data included specimens with at least a

minimal amount of transverse reinforcement in the panel zone, which is consistent with the

intended use of the model. Joints with no transverse reinforcement, a reinforcing detail typical in

non-ductile RC frames, were excluded from this study. Mitra and Lowes [36] noted that in joints with

low amounts of transverse reinforcement, shear is transferred primarily through a compression strut,

a mechanism, which is stronger and stiffer than predicted by the MCFT. Altoontash [37] adapted the

constitutive model developed for the translational bond-slip springs in Lowes and Altoontash [32] in

a fiber section analysis to derive the constitutive model for the member-end rotational springs, but

noted that detailed information on bond-slip response is needed. Furthermore, the development

length was assumed to be adequate to prevent complete pullout, which is not necessarily true for

bottom reinforcement in beams of non-ductile RC frames. The validation studies include RC

interior beam-column joint tests by Walker [38] and a two-story RC frame.

Based a review of the literature on models for simulating the RC beam-column joint response,

Celik[39] and Celik and Ellingwood [40] developed a joint model based on the experimental

determination of joint panel shear stress-strain relationship. The panel zone constitutive parameters

are defined to replicate the experimental joint shear stress-strain relationships, eliminating the need

for further calibration, while the effect of bond-slip is taken into account through a reduced

envelope for the joint shear stress-strain relationship. The application of the modeling scheme is

validated on two full-scale experimental RC beam-column joint test series. A fragility analysis of a

three-story non-ductile RC frame in which the proposed beam-column joint model is employed

demonstrates the importance of modeling shear and bond-slip behavior in joints when performing

seismic risk assessments of non-ductile frames.

1.2.2 Retrofit Methods

Seismic retrofit is the modification of existing structures to increase their resilience to seismic

activity. Following the availability of new seismic provisions and advanced materials, several

retrofitting strategies and techniques have been developed for seismic assessment, retrofit and

rehabilitation of existing structures. Retrofitting existing frame structures involves: (1) local

strengthening of structural elements by concrete jacketing or steel angles connected by transverse


7

plates); (2) increase global capacity by adding new strong structural elements such as cross braces

or new structural walls to absorb most of the seismic forces; (3) reduction of seismic demand by

using base isolation systems or any additional damping. Each retrofit technique shall consider the

effects of building modifications on the stiffness, strength, yield behavior, and deformability of the

upgraded structure [41].

1.2.2.1 Buckling Restrained Braces (BRB)

Existing framed structures may be suitably retrofitted by using diagonal braces, either

traditional steel or innovative. Braced systems exhibit high lateral stiffness and strength under

moderate-to-large magnitude earthquakes. The most common structural configurations for

lateral-resisting systems are concentrically brace frames (CBFs), which possess a lateral stiffness

significantly higher than that of unbraced frames, e.g. moment resisting frames. Nevertheless, due to

buckling of the metal compression members and material softening due to the Bauschinger effect,

the hysteretic behavior of CBFs with traditional steel braces is unreliable. Alternatively,

buckling-restrained braces (BRBs) may be employed as diagonal braces in seismic retrofitting of

steel and RC frames designed for gravity loads only.

Buckling Restrained Braces (BRBs) have proved to be beneficial in providing resistance

against earthquake ground motions while simultaneously enhancing the energy dissipation capacity

of both new and existing structures. However, their applicability and effectiveness in RC framed

structures is still uncertain. A relatively large number of different types of BRBs have been studied,

tested and proposed in the past few decades. Regardless of detailing differences, they all share the

same concept: prevent both global and local (cross-section) buckling in the braces and allow equal

tensile and compressive strength, and thus higher hysteretic energy dissipation. These elements

exhibit nearly ideal bilinear hysteretic characteristics, have large cumulative energy dissipation

capacities, and employ conventional design and construction for their incorporation into a structural

system.

A typical BRB consists of a steel profile encased in a circular or rectangular hollow section

steel profile, filled with concrete or mortar and an unbonding layer is placed at the contact surface

between the core plates and the filling concrete, thus this brace is called “unbonded brace”. The

main purpose of the concrete-filled tube is to prevent the buckling of the steel core. The steel

core-concrete interface usually consists of a slip surface to allow relative axial deformations

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between the steel core and the tube infill. The slip surface is achieved by placing a low friction

material between the infill material and the steel core. The transversal expansion of the brace under

compressive loads due to Poisson’s effect should be accommodated providing a gap between the

brace and the encasing material. In addition, the dissipative part of the brace, which is the zone

where yielding occurs, can be replaced by detaching it from the brace non-yielding segment, which

is retained (e.g., in the aftermath of a major seismic event).

Comparative studies, as well as completed construction projects, confirm the advantages of

buckling-restrained braced frame (BRBF) systems over conventional braced frames. In a study

presented by Sutcu et al. [42], a typical five-story RC school building in Turkey was selected for the

seismic evaluation. To assess the effectiveness of the proposed seismic retrofit scheme, nonlinear

time-history analysis using high-intensity seismic waves was performed on the RC frames. The

results indicated that the building retrofitted with BRBs shows relatively reduced story drift when

compared to the original building and the building with conventional braces.

Kim and Choi [43] presents an investigation on energy dissipation capacity and performance of

structures with BRB, and a design procedure to meet a given target displacement in the framework

of the capacity spectrum method was presented and investigated. It was observed that story-wise

distribution of BRB in proportion to the story drifts and story shears resulted in better structural

performance, and the equivalent damping ratios of SDOF structures with BRB generally increase as

the stiffness of BRB increases. The use of BRB devices is mostly restricted to steel framed

structures. In fact, the existing standards for BRBs, mainly in the United States or Japanese, and the

design procedures described in them are mostly applicable to steel buildings.

Although some studies performed the seismic assessment on the BRB retrofitted RC frames,

the problematic reinforcing details associated with non-ductile or pre-code buildings, such as the

inadequate joint shear capacity and the insufficient positive beam bar anchorage, were rarely

considered. Recently, Almeida et al. [44] studied a simplified method for predicting the response of

damped RC structures and employed in the preliminary design of a retrofitting scheme, with BRBs

in a three-story RC school building located in a high seismicity region of Portugal. The assessment

results varied according to the accelerograms and the behavior model RC cross-sections, which

demonstrated that it is difficult to predict the response of an RC structure retrofitted with BRBs,

when subjected to ground motion. However, it is worth mentioning that all beam-column joints


9

were assumed to be monolithic, while the columns were considered to be pinned at their base due to

poor reinforcement of the column-footing joints.

The research by Di Sarno and Manfredi [17] assess the seismic structural performance of a

typical RC framed school building retrofitted with BRBs. The frame was designed for gravity loads

only. The devices adopted in the study included a short-length BRB connected in series with a

traditional metallic brace, as shown in Figure 1.3. The code horizontal input spectra was defined

with reference to the 5% damped acceleration response spectra evaluated for the four limit states

compliant with the recent Italian code of practice. A set of seven code-compliant natural earthquake

records was selected and employed to perform inelastic response history analyses. It was found that

BRBs are effective to enhance the ductility and energy dissipation of the sample as-built structural

system, and the results of extensive nonlinear dynamic analyses showed that more than 60% of

input seismic energy was dissipated by the BRBs at ultimate limit states. Shear deformability of

beams and columns was included in the structural model. Panel zone strengths and deformations

were not considered.

Figure 1.3 Brace system with a short-length BRB connected with a traditional brace

Laboratory test data on BRB retrofitted frames are available in the published literature. The

majority of test were, however, conducted on steel frames, or concrete frames without proper

scenarios to study the effect on non-ductile failure modes or the retrofit effectiveness.

Aniello et al.[45] carried out numerical modeling of two full-scale experimental test results on

the lateral load-displacement response of a RC structure seismically retrofitted by buckling

restrained braces of type “only steel” BRBs. They adopted a simplified approach to model the

inelastic response of the tested structures. In particular, BRBs were schematized as truss element

using a bilinear force-deformation relationship, which had been characterized by selecting an

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appropriate value of the post-elastic stiffness. Their proposed simplified analytical back-analysis

confirmed the feasibility of this simple modeling scheme for design purposes.

1.2.2.2 SMA Braces

Shape memory alloys include a class of metallic alloys that display several characteristics not

present in traditional civil engineering materials. At the macroscopic level, SMAs feature two

unique properties: the shape memory effect (SME) and the superelastic effect (SE), which can

recover their initial pre-deformed shape after unloading or by heating, as shown in Figure 1.4. The

SE is related to the ability to recover large deformations after the removal of the load, while the

SME refers to the ability to regain the original shape through heating [46]. As a result of these

properties, in recent years SMAs have attracted significant attention from the scientific community

with several applications being developed in the civil engineering, automotive, and biomedical

fields [47-50].

Figure 1.4 Idealized stress-strain curve for shape memory effect, and superelasticity effect

The ability of SMAs to recover their shape is related to the reversible martensitic phase

transformation that results from a solid-to-solid diffusionless process between a crystallographically

more-ordered phase, austenite, and a crystallographically less-ordered phase, martensite. The

austenitic phase tends to be stable at low stresses and high temperatures, while the martensitic phase

is stable at high stresses and at low temperatures. The SME occurs when the ambient temperature is

below the martensite finish temperature of the material resulting in the material being in its twinned

strain

Stress

strain

Stress

Temperature

Heating

'Shapememory'effect

Superelasticity

cooling


11

martensite phase. Upon uniaxial deformation, the twinned martensite undergoes a detwinning

process along with the accumulation of residual deformation. This deformation can only be

recovered by heating the material above the austenite finish temperature causing a phase

transformation to the high temperature austenite phase and a recovery of the residual deformation.

During cooling, the material returns to its original, more stable, low temperature twinned martensite

phase. Alternatively, the SE occurs when the ambient temperature is above the austenite finish

temperature of the material resulting in the SMA being in its austenite phase upon deformation. For

this case, the SMA undergoes a martensitic phase transformation to stress-induced detwinned

martensite upon reaching the forward transformation stress. Since the stress-induced martensite is

only stable due to the applied load, upon unloading, the material reverts back to its original

austenite phase with little or no residual deformation and without requiring the application of heat.

The martensite start, martensite finish, austenite start, and austenite finish temperatures can be

altered through the manufacturing and processing of the SMA to provide either a SME specimen or

a SE specimen for a desired temperature range [46].

Previous studies on the behavior and applicability of different SMA materials classed NiTi as

the most appropriate shape memory alloy due to its excellent superelasticity, large recoverable

strains and excellent corrosion resistance. It is stable and presents a superior thermo-mechanic

performance [51]. Details regarding modeling, design and testing of devices with NiTi or Fe-Mn-Si

and their potential application in earthquake engineering are discussed in [49, 52]. More recently, Qiu

and Zhu [53], Casciati and Faravelli [54] studied CuAlBe alloy towards its exploitation in passive

control devices. Monocrystalline CuAlBe-based SMA has an extraordinary strain recovery up to 19%

and a wide operating temperature from room temperature to -40°C [54]. CuAlBe alloy are available

at a reasonable cost and shows superelastic behavior at room temperature in tension cyclic tests [55].

Tanaka [56] studied a novel Ferrous monocrystalline SMA (e.g. FeNiCuAlTaB alloy) which

maintains superelasticity down to -62°C and is able to recover deformed shape up to a strain

slightly greater than 13%.

Given the distinct and unique properties of SMAs, many researchers have studied their

mechanical properties, constitutive modeling and their structural applications to explore the

applicability of shape memory alloy devices in earthquake engineering. In the study by McCormick

[57] on the use of SMA in seismic vibration control of structures, it was found that larger diameter

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12

bars perform as well as wire specimens under dynamic loading. The authors also studied the

dynamic performance of conventional steel buckling-allowed cross-bracing system and a

superelastic SMA cross-bracing system. They reported a reduction in residual drifts for the SMA

equipped system. Several researchers have investigated the contribution of modeling of superelastic

SMA materials for seismic applications [58-59]. Besides, there have been many studies to develop

design methods to implement SMA recentering bracing systems for seismic upgrade of structures

[60-62]. It was found that SMA with greater hysteretic parameters results in a more economical design,

even if SMA braced frame retrofitted with different hysteretic parameters of braces can

satisfactorily achieve the same performance objectives [63]. In reinforced concrete frames, the option

of incorporating superelastic shape memory alloys in reinforcement for seismic hazard mitigation

has been explored by researchers. A better deformability was achieved when SMA bars are used in

beams [64] and SMA RC frames are able to recover most of their post-yield deformation, even after a

strong earthquake [65]. A detailed review on the mechanical properties, behavior and application can

be found in the report by [66].

The use of BRB systems has gained popularity in the research community and building

practioners, considering their ability to reduce transient deformations in structures. The distinctive

recentering capabilities of SMAs and undergoing strain hardening raised the possibility of much

attention turning to SMA bracing systems for seismic retrofit. There are available literature on

numerical studies performed on steel structures concentrically braced frames equipped with SMA

braces compared to similar ones ( same unretrofitted frames) with buckling restrained braces [67-68].

It is observed that SMA elements are good alternatives for an effective dynamic response of

structures to seismic excitations as their implementation can reduce the roof displacement and peak

inter-story drift compared to the buckling restrained braced frames.

The re-centering capability of the SMA-based bracing system can be applied to recover the

undeformed shape of RC frames. Donatello et al. [69] reported the results of an extensive program of

shaking table tests on 1/4-scale three-dimensional R/C frames, conducted to evaluate the

effectiveness of passive control bracing systems for the seismic retrofit of R/C frames designed for

gravity loads only. Two alternative retrofitting strategies were investigated, one based on the

hysteretic behavior of steel elements and the other on the super-elastic properties of Shape Memory

Alloys (SMA). During dynamic analysis of the bare frame, some cracks were observed in the beams


13

and joints of the first two stories, caused by the slippage of the beam reinforcement at the joints.

The steel-based braces exhibited high energy dissipation but no re-centering capability. Their

effectiveness in limiting the maximum and the residual inter-story drift is conditioned upon their

stiffness, which must be higher or much higher than the RC structure, in order to limit the inelastic

deformations of this latter. The supplemental re-centering capability that was exhibited by SMA

based braces allowed high inelastic deformation in the RC members to be accepted, without

compromising the collapse safety of the structural system nor its seismic resistance at the end of the

earthquake. Such a property induces a pseudo-elastic behavior of the structure even if perfect hinges

are generated at the joints, only conditioned upon the capacity of joints and columns to transmit

axial forces.

1.2.2.3 Other Retrofit Methods

Han et al. [70] performed seismic risk analysis was for an old non-ductile RC frame building

before and after retrofit with base isolation. Various sources of uncertainty such as structural

uncertainties, ground motions uncertainties and modeling uncertainties are discussed and

propagated in the analysis procedure. A sensitivity study was also conducted to determine which

structural parameters have the most significant impact on both the seismic demands of the

un-retrofitted and base isolated building. A suite of recorded ground motions was utilized to

investigate the influence of considering aftershocks on the performance of these types of buildings.

The study revealed that base isolation can greatly reduce the seismic risk for higher damage levels,

as one would expect.

Tarfan et al. [71] presented a probabilistic seismic assessment of non-ductile RC frames

retrofitted by pre-tensioned aramid fiber reinforced polymers (AFRP). Three RC buildings with

different heights are designed according to older construction practice and the poorly detailed

columns of each model are then retrofitted using pre-tensioned AFRP belts. The numerical finite

element models are developed in OpenSees using concentrated plastic hinge models that can

capture shear weakness of original columns and deterioration of beams’ stiffness and strength.

Incremental dynamic and nonlinear static analyses are performed to quantify structures’

performance in terms of both global- and component-level metrics. The structures’ global response

is evaluated using fragility curves, mean annual frequency of collapse, and collapse margin ratios.

Furthermore, statistical analyses are performed to obtain median inter-story drift distribution along

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the height of the structure, structural members’ ductility and dissipated energy under three different

seismic hazard levels. The results indicate that retrofitting by pre-tensioned AFRP improves the

structure’s global-level ductility and reduces the collapse probability significantly. Moreover, it can

prevent weak story formation by engaging a larger number of stories in collapse mechanism. From

a component-level perspective, pre-tensioned AFRP increases columns’ ductility and dissipated

energy and enhances their performance, particularly at near-collapse-limit states.

Pampanin et al. [72] introduced a diagonal metallic haunch system at the beam–column

connections in existing RC frames designed before the introduction of modern seismic-oriented

design codes in the mid 1970s, to protect the joint panel zone from extensive damage and brittle

shear mechanisms, while inverting the hierarchy of strength within the beam–column subassemblies

and forming a plastic hinge in the beam. A complete step-by-step design procedure is suggested for

the proposed retrofit strategy to achieve the desired reversal of strength hierarchy. Analytical

formulations of the internal force flow at the beam-column-joint level are derived for the retrofitted

joints. Results from an experimental program carried out to validate the concept and the design

procedure are also presented. The experimental results demonstrated the effectiveness of the

proposed solution for upgrading non‐seismically designed RC frames and also confirmed the

applicability of the proposed design procedure and of the analytical derivations.

Günay et al. [73] presented the results of an investigation on the efficacy of using rocking spines

of strengthened infill walls as a retrofit measure for non-ductile reinforced concrete (RC) frames

with unreinforced masonry (URM) infill walls. The study examines the effects of spines of

strengthened URM infill walls on the behavior of the RC frame, with particular emphasis on

whether spines could reduce the tendency to form a soft story mechanism. A nine story frame with

five bays is selected to represent complex multi-story behavior, where the collapse of stiff infill

walls may lead to the formation of a soft story mechanism. The effect of the proposed retrofit is

investigated through nonlinear static and dynamic analyses. Fragility relationships are obtained for

the frames using pseudo-acceleration corresponding to the first mode as the intensity measure and

maximum story drift ratio as the response variable. Analyses show that infill retrofit with rocking

spines provides significant improvement in the seismic performance of non-ductile RC frames.

1.3 Research Objectives

The main objective of this research is to assess the seismic performance of non-ductile RC


15

frames retrofitted with BRB and SMA braces. This study focus on the following specific objectives:

Numerical models of the as-built non-ductile RC frame and retrofitted frames with BRB and SMA

braces are developed and validated.

Based on the nonlinear time-history analysis, the maximum and residual drift demands of the

as-built non-ductile RC frame and retrofitted frames with BRB and SMA braces are assessed at

various intensity levels of ground excitations.

The effect of hysteretic variety of SMAs on the seismic behavior of retrofitted frames with SMA

braces is studied.

The frame response parameters, such as maximum story drift and residual deformation, are

converted to performance levels. A seismic fragility assessment is then developed to measure the

probability of failure of the frames to meet different performance limits under the possible future

seismic events.

Finally, based on the analysis results of the as-built and retrofitted frames, conclusions

regarding the effectiveness of BRB and SMA bracing systems are made.

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CHAPTER 2 DESCRIPTION AND MODELING OF FRAMES

Seismic performance evaluation of a building inventory in a region requires the selection of

sample buildings that are representative of design and construction practices in the region and the

finite element simulations of those buildings. Accordingly, this chapter presents the description of

the prototype non-ductile frame building designed for gravity loads only; and develops finite

element structural models of the as-built and retrofitted frames.

2.1 Description of the Prototype Frame Building

A 3-story, 3-bay non-ductile RC prototype frame building is chosen for the purpose of this

study. The building was studied by Celik and Ellingwood [2] and it has been used by many

researchers for performance evaluation of the pre-1970 RC frame office buildings in United States

[2] [40] [74]. The building was designed with gravity load and wind loads to mirror the lack of any

consideration of seismic actions including the inadequate transverse reinforcement, weak columns,

short embedment length, and poor beam-column joint confinement. Figures 2.1 and 2.2 show the

typical symmetric floor plan, together with elevation of the 3-story frame building. The beams and

columns section properties were identical over the 3 story-heights, given that gravity loads

governed the design. The beam and column reinforcement layouts are shown in Figure 2.3.

Specified concrete compressive strength of 28 MPa and the design steel yield strength of 280 MPa

were employed in the design.

Figure 2.1 Plan view and elevation of the prototype buildings

3

×5.49

mm

3×

5.4

86 m

4×5.486 m

3

×.49

mm


17

Figure 2.2 Elevation view of a typical interior frame

36

58

36

58

36

58

5,486 5,486 5,486

5486

1

2#6

2#5 1#5

2#5 2#6

2#6

2#5

Beam reinforcement details

1

2

2

3

3

4

4

5

5

6

6

6

6

5

5

4

4

3

3

229

2#6

2#5

2#6

152

305

229

2#6

2#6

152

305

229

2#6

1#52#6

152

305

229

2#6

1#5

2#6

152

305

229

2#6

2#6

152

305

229

2#6

152

305

2#5

SECTION 1-1 SECTION 2-2 SECTION 3-3

SECTION 4-4 SECTION 5-5 SECTION 6-6

2#52#5

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Figure 2.1 Reinforcement layout of the interior frame

2.2 Numerical Models

2.2.1 Non-ductile Frame Model

The non-ductile RC frames are characterized by poor reinforcement details. A proper modeling

requires a beam-column joint model that captures the absence of joint reinforcement to simulate the

shear and bond slip in the joints of a non-ductile frame. Celik and Ellingwood [40] developed a

beam-column joint model to capture the deficiency in joint behavior that results from inadequacy in

joint reinforcement details. They also validated their model using two full-scale experimental RC

beam-column joint test series. The approach to incorporate the joint model in the non-ductile RC

frames is described subsequently.

The adopted method requires a constitutive relationship for the panel zone for each

beam-column joint in the frame, as shown in Figure 2.2. The coordinates of the four key points that

define the panel zone backbone for a general beam-column joint correspond to joint shear cracking,

reinforcement yielding, joint shear strength/adjoining beam or column capacity, and residual joint

strength, respectively.

The first point corresponds to the shear cracking of the panel zone. The ordinate of the first

point (the joint cracking shear stress) is given by:

(��̅�)�� = 3.5�1 + 0.002 �

��

�� (2.1)

where Nu is the axial load and Ajh is the joint area. Joint shear stresses corresponding to the shear

cracking of the panel zone were reported to be in the range of 0.21-0.69��MPa, increasing with

higher axial loads.

152

305

229SECTION 6-6

2#6

1#5

2#6# 3 @ 152 mm

# 3 @ 152 mm

305

305

Internal Column

381

381

External Column

4#6

1372

8#8


19

Figure 2.2 The constitutive model of panel zone

In case that the shear failure of the joint does not occur before the adjoining beams/columns

reach their ultimate capacity, then the second and the third points on the backbone curve are given

by the yield and ultimate moment capacities of the adjoining beams and columns. The yield and

ultimate moment capacities of each frame member element were determined by section analysis.

The positive yield moment capacities of the beams were scaled by a bond-slip factor a = 0.5.

The positive ultimate capacities of the beams then were set equal to the scaled positive yield

capacities of the beams.

The moments transferred through the joint (i.e., the rotational spring) when the adjoining

beams/columns reach their yield and ultimate capacities are given below for the scissor model:

��

�,��,�

= �� (��

�)�,� + (��)�,�

��,�(��

� )� + (�� )�,�

�� (2.2)

for interior joints, and

��

��,�

= �� (��

�)�,� + (��)�,�

��,�(��

� )�

�� (2.3)

for the positive backbone, and

��

��,�

= �� (��

�)�,� + (��)�,�

��,(��

� )�,�

�� (2.4)

for the negative backbone of exterior joints, where �� = 1 −��

�� and �� = 1 −

��

�� ( subscripts C,

Strain

Stress

(γcr, τcr)

(γy, τy)(γmax, τmax)

(γres, τres)

(γcr, τcr)

(γy, τy)(γmax, τmax)

(γres, τres)

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IB and EB refer to column, interior beam, and exterior beam; and superscripts y and u refer to yield

and ultimate, and B and T refer to bottom and top, respectively).

The ordinate of the fourth point was assumed equal to that of the first point on the backbone,

as previous experimental research has revealed that strength degradation occurs once the peak point

is attained on the backbone curves of beam-column joints that are typical of non-ductile

construction.

The abscissas of the four key points were based on the available experimental data [40]. The

following joint shear strains were utilized as the abscissas of the four key points: 0.0005, 0.005,

0.02, and 0.08.

The conversion of joint shear stress into moments transferred through the rotational spring was

done with the following equations.

�� = ��

1

� (2.5)

for interior and exterior joints , and

�� = ��

1

�� (2.6)

for interior and exterior top floor joints, where

� =��⁄��

��−

�

��, �� =

��⁄��

��−

�

��, � = 1 − ℎ� ��⁄ − �� ⁄ , where Lb is the total length of

the left and right beams; bj is the width of the joint panel; jd is internal moment arm; Lc is the total

length of the top and bottom columns and hj is the height of the joint.

To validate the joint model described above for finite element analysis of non-ductile RC

frames, an experimental program tested by Pantelides et al. [75] involving an RC beam-column test

specimen that has no transverse shear reinforcement in the panel zone are considered. The finite

element analysis of the beam-column specimen was performed using OpenSees [76]. OpenSees is an

open-source computational platform that can account for geometric and material nonlinearities. The

built-in fiber approach to element/section modeling, which makes use of the nonlinear uniaxial

constitutive models of concrete and steel, enables the modeling of the spread of inelasticity across

the section depth and along the member length. Figure 2.5 shows the structural models developed

for analysis by OpenSees. Smaller elements were required near joint regions where significant

inelastic actions may occur. Concrete and steel properties reported in the test programs [75], rather

than specified minimum design values, were incorporated into the finite element models. Cover and


21

core concrete properties were calculated using the modified Kent and Park model, while steel

properties were represented through a bilinear steel model with strain hardening in the range of

1-2%, based on material tests. Concrete cover loss was taken into account in all cases. The

beam-column panel zone was modeled using the scissors model with a rotational spring. The

moment-rotation of the spring was determined by the Figure 2.3 and was modeled using the

Pinching4 material in OpenSees.

Figure 2.4 The OpenSees model for the specimen tested by Pantelides et al. [75]

The comparison of the experimental and numerical force-drift responses are shown in Figure

2.6. It is observed that the envelopes to numerical force-drift responses of the specimen showed

good agreement with the experimental data. The discrepancies that exist between the simulated and

experimental envelopes are attributed to how the backbone of joint shear stress-strain relationships

was defined, as the performance points corresponding to particular damage states do not necessarily

replicate the experimental shear stress-strain envelope. The degradation in backbone curves was

captured. The numerical cyclic responses were also in good agreement with the experimental data.

Hysteretic energy dissipated during the experimental loading cycles was represented well and the

pinching point was captured quite well.

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-10 -8 -6 -4 -2 0 2 4 6 8 10

-50

-40

-30

-20

-10

0

10

20

30

40

50

Late

ral Load, kip

s

Drift,%

(a) Experimental force-drift response (b) Numerical force-drift response

Figure 2.6 Validation of the beam-column joint model

Finite element model of the 3-story non-ductile RC frame was developed based the

beam-column joint model using OpenSees. Figure 2.7 shows the structural model of the three-story

frame. The 3-story building frame was modeled as a two-dimensional frame model, where

displacement based beam-column elements with 2 integration points were used to model frame

members and smaller elements were provided near beam-column joint regions where significant

inelastic actions may occur. The two-dimensional finite element model of the frame developed in

OpenSees cannot account for out-of-plane behavior or torsional effects caused by earthquakes, but

such a model is much less time consuming for analysis and can yield sufficiently accurate results

without interference from torsional action or bi-axial interaction.

The fiber approach to element/section modeling, which makesuse of the nonlinear uniaxial

constitutive models of concrete and steel, enabled the modeling of the spread of inelasticity across

the section depth and along the member length. Concrete compressive strength and steel yield

strength were increased by 25% from their nominal values to account for the conservatism in

nominal material strength with respect to in-situ strength and the increase in strength that occurs

under dynamic loading [77]. Cover and core concrete properties were calculated using the modified

Kent and Park model (concrete01 material in opensees), whereas steel properties were represented

through a bilinear steel model with 0.5% strain hardening. The increase in concrete strength due to

confinement was examined carefully and was found to be only marginal, varying in the range of 2–

4% in locations where minimal transverse reinforcement was provided in columns and beams.


23

Figure 2.7 Finite element model for the 3-story frame

The nonlinear displacement-based beam-column element of OpenSees is a more reliable

choice for modeling the beams and columns. In contrast with the force-based element, each beam or

column member must be modeled by multiple elements for better accuracy and capturing more

realistic results. Each frame member is modeled by 6 or more displacement-based elements with 2

integration points for a good capture of the steel reinforcement distribution and member

deformation, giving the model the ability to achieve a good behavior and accurate result similarly to

a single fore-based nonlinear element with five integrations points.

The effective width of the slab was defined according to ACI Standard 318 and the slab bars

within this width were considered, except in the region close to the columns where the bottom bars

lacked sufficient anchorage. A distributed gravity loading of 40,000 N/m was applied to the beam

spans, whereas a dynamic mass of 15,700 N/m/s2 was lumped at each beam–column joint for

dynamic analysis, based on the load combination 1.0DL+0.25LL.

The hysteretic damping of the frame is automatically calculated when the materials went

into nonlinear range in the analyses, and damping ratio of 5% is adopted for viscous damping which

is only valid when the models behaved elastically. An eigen value analysis was performed and the

fundamental period of the 3-story frame was found to be 1.07s, with a mass participation factor of

87.4% for the first mode and 10.2% for the second mode.

The analytical model of the un-retrofitted frame was validated through comparison with the

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reference model by Ruilong et al. [74]. In addition to the same fundamental period which implies

comparability for the linear behavior, the similarity in nonlinear behavior of the model in this study

and the reference model was also validated through the comparison of the story drift distribution

along the building height of the model in this study and the reference model using the same

earthquake records (ground motion No.215 and No.231 in the PEER-NGA database). As shown in

figure 2.8, the analytical model used herein was felt to adequately reproduce the seismic response of

the reference model and of the prototype non-ductile RC frame building.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1

2

3

Sto

ry

Story drift [%]

Reference model Numerical model in this study

Figure 2.8 Comparison of the story drift distribution along the building height of the model in this

study and the reference model using the same earthquake records

2.2.2 BRB Frame Model

In this study, the buckling restrained braces (BRBs) are placed in a diagonal configuration, as

shown in Figure 2.9. Each brace system includes two short-length BRB elements connected in

series with an elastic brace element exhibiting adequate over-strength. For low ductility RC frames,

this configuration leads to dissipative braces with short length and stiff elastic braces. The designed

dissipative braces yield for displacements significantly smaller than the ultimate displacement of the

bare frame, can undergo large plastic deformations, and can dissipate a large amount of energy

within the range of deformation compatible with the frame.

The stiffness and ductility of the resulting dissipative brace system can be obtained through

the following equation:

�� =��

�� , µ�� =

��µ��

�� (2.7)

where �� is axial elastic stiffness of the brace; �� is the ductility of the brace; ��, �� and ��


25

are the stiffness, the yield force and ductility capacity of the dissipative brace system, respectively.

�� is the stiffness of the elastic (non-dissipative) component.

Figure 2.9 The non-ductile RC frame retrofitted with BRB brace systems

For the purpose of this study, the non-dissipative part was assumed to be rigid to transfer all

the loading directly to the dissipative part. Eq. (2.7) gives K� = K� and µ�� = µ��.

A simplified approach was used to obtain the design capacities of the bracing system.

Details of the detail approach can be found in Freddi et al. [78]. The method followed to design the

BRB bracing system is based on pushover analysis of the existing frame under a distribution of

forces corresponding to its first vibration mode. The structure is pushed up to an ultimate

displacement du corresponding to the design damage level, corresponding to the limit strain

demands in the most critical fibers (the extreme compression fibers) of the core of a RC section set

to 0.003 [79]. The corresponding value of the base shear is denoted as ��. The BRB bracing system

is assumed to behave as an elastic-perfectly plastic system, with base shear capacity ��, ductility

capacity μdu and with the same ultimate displacement of the bare frame. In this study, the ductility

capacity of the whole brace μdu is assumed equal to 12.

It is noteworthy that the value of �� is a design choice and depends on the ratio α, which is

measured by between the base shear capacity of the bracing system �� and that of the bare frame

��. For a given value of ��

� , the stiffness of the bracing system at the first story is as follows:

3×5.486m

3×3.6

6m

BRB

Rigid element

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��

� = ��

�µ��

�� (2.8)

in which δ1 denotes the story drift at the first story, normalized with respect to the top floor

displacement according to the first model shape. The shear �� and stiffness ��

� of the dissipating

bracing system at each story are determined by the following relations:

�� = ��

��, �� = ��

� �� (2.9)

where �� and �� denote, respectively, the shear force and stiffness at each story, normalized with

respect to the base shear and base stiffness according the first mode of the bare frame. The stiffness

distribution adopted for the BRB bracing system at each story ensures that the first modal shape of

the bare frame remains unvaried after the retrofit. This avoids drastic changes to the internal action

distribution in the frame, at least in the range of the elastic behavior. Additionally, the chosen

strength distribution of the BRB bracing system aims at obtaining simultaneous yielding of the

devices at all the stories and, thus, a global ductility of the bracing system coinciding with the

ductility of the single braces. Given �� and ��

� ,the braces properties at each story can be

determined on the basis of the number of braces and on geometrical considerations, and by applying

Equation (2.8). Finally, given the braces properties, the braces dimensions (length and area of the

dissipative devices and of the elastic braces) can be derived. Tables 2.1 and 2.2 show the

mechanical and geometric properties of the BRB elements, for a design ratio value α of 2.

Table 2.1 Mechanical properties of the BRB elements at each story

Story No. Yield force Fd [kN] Initial axial stiffness Kd [kN/m]

1 481 101709

2 400 75478

3 229 68969

Table 2.2 Geometric properties of BRB elements at each story

Story No. Area [mm2] Length [m]

1 1463 1.015

2 1.25E+03 1.25

3 7.20E+02 0.785


27

The behavior of BRBs can be described by an elasto-plastic constitutive law. In this study,

the braces behavior is modeled by the Giuffré-Menegotto-Pinto Model (referred to as Steel02 in

OpenSees). The strain-hardening ratio for kinematic hardening is set as 0.01, with a smooth

transition from elastic to post-elastic branches, whereas the isotropic hardening is disregarded.

2.2.3 SMA Frame Models

The hysteretic shape of SMAs determines the seismic response of structures that use SMA-based

braces. The mechanical properties of SMAs are sensitive to many parameters, including the

chemical composition, processing method, ambient temperature, and alloy composition. Although

loading rate also plays an important role in determining the hysteresis, the mechanical properties of

SMAs were reported to be insensitive to loading rates that characterized by the typical frequency

range of seismic loadings, mainly due to the heat exchange rate becomes saturated when SMAs are

subjected to high-frequency loadings. To address the effect due to the variation in the SMA

hysteresis, the current study selects three types of SMAs that consist of different alloy compositions

(e.g. Ni-Ti, CuAlBe, and FeNiCuAlTaB-based SMAs).

To make a quantitative comparison between the selected SMAs in terms of hysteretic

properties, the cyclic behavior of SMAs at constant temperature is simplified as shown in Figure

2.10. The stress-strain relationship of SMAs is defined by the parameters that include σMs (the stress

causing the start of martensite transformation), σMf (stress corresponding to the finish of martensite

transformation), σAs (stress triggering the start of austenite transformation), σAf (stress featuring the

finish of austenite transformation), austenite modulus ESMA, strains associated with the start and

finish of martensite transformation, εMs and εMf; and the superelastic plateau strain length εL. The

parameter, γ, is introduced to measure the ratio of martensite stiffness to austenite stiffness. Because

SMAs gain strength again as the martensite transformation is finished, γ is also called the hardening

behavior parameter. Table 2.3 lists the material parameters, clearly demonstrating the significant

difference in hysteretic properties between various SMAs. The CuAlBe-based SMA can maintain

self-centering capability with a recoverable strain higher than 19% and reduce structural permanent

deformation even after extremely strong earthquakes [53]. At large strain cycles, it exhibits an

increase in energy dissipation (an increased equivalent damping ration at high strains) and can

achieve a maximum recoverable strain of 23% [53]. Therefore, only superelastic strain was exploited

for the CuAlBe-based SMA, assuming a single stage loading and unloading path and corresponding

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phase transformations. The FeNiCuAlTaB-based SMA can recover strains to approximately

15%and the hardening behavior was also not considered. Consequently, the hysteresis curves of

FeNiCuAlTaB- and CuAlBe-based SMAs are assumed to be constant without being affected by the

strain magnitude. In particular, NiTi-based SMA has the lowest ductility; thus, this study takes into

account its hardening behavior with the ratio of martensite stiffness to austenite stiffness equal to

3%, and there are residual strains for strain demands greater than its recoverable strain. It is further

assumed that no strength degradation occurs during cycling loading. The SelfCentering uniaxial

material in OpenSees [76] is used to model the flag-shaped behavior of SMA elements.

Figure 2.10 Simplified hysteresis behavior of shape memory alloys at constant temperature

Table 2.1 Mechanical properties of SMA materials

Alloy type �� [MPa] �� [MPa] �� [MPa] �� [MPa] �� [GPa] εL [%] γ

NiTi 500 529.4 250 220.6 58.9 5 0.3

CuAlBe 170 261.8 214.2 122 17 17.515 -

FeNiCuAlTaB 753.6 1086.4 528.8 196 50.2 18.8 -

In similarity with the BRB bracing system, the SMA bracing system include two

superelastic SMA bar segments connected to the frame through connective segments in order to

reduce the length of SMA required. This connective segment is assumed to be a rigid element.

About designing of the SMA braces, cross section and length of the SMA elements should

be determined in such a way that the BRB and SMA braced frames be comparable. For this purpose,

superelastic SMA braces were designed to provide the same yielding strength, Fd and the same

σMs

σMf

σAs

σAf

ESMA

ESMA

εL

Stress, σ γESMA

Strain, ε


29

initial axial stiffness, Kd, as the BRB braces. Thus, the area of the SMA brace can be determined by

the following equation

�� =��

�� (2.10)

and the length of the SMA brace is computed by

�� =��

�� (2.11)

Since the required geometric properties (i.e. cross-sectional area and element length) of the

superelastic SMA braces can be calculated easily. It was also assumed that the SMA elements are

made of a number of large diameter superelastic bars able to undergo compressive loads without

buckling. Table 2.2 shows the geometric properties of the SMA braces for the 3 story frame.

Table 2.3 Geometric properties of the SMA braces for the 3 story frame.

Story NiTi

CuAlBe

FeNiCuAlTaB

Area [mm2] Length [mm]



1 457 781

303 442

1345 663

2 801 1250

531 707

2355 1061

3 963 1115

639 631

2832 947

It is noted that the impact of ambient temperature change on the hysteretic properties of

SMAs is not considered, by assuming all the structures are located in an indoor environment with

relatively stable room temperature. Usually, a higher temperature raises the phase transformation

stresses of SMAs, and thus increases the strength capacity of the SMA braced frames, which is thus

deemed to be beneficial to the seismic resistant capacity of the SMA braced frames; whereas a

lower temperature decreases the phase transformation stresses and even impair the superelasticity of

SMAs, which may probably lead to negative influence on the seismic performance of the SMA

braced frames. Thus, the temperature effect on the seismic behavior of the corresponding SMA

braced frames needs further evaluation in future.

2.3 Summary

The adopted gravity-load designed RC building was described and detailed. A beam-column joint

model was presented and validated with experimental results available in the literature. The

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beam-column joint model was found appropriate to simulate the seismic response seismic of the

non-ductile RC frames. The panel zone model was used to develop a 2-D numerical model of the

prototype building, which in turn was compared to a similar model in the literature. A BRB

equipped frame model was developed adding, in diagonal configuration, BRBs in the As-built

frame model with design strength and stiffness following the first mode shape of the As-built

building. Then, BRB were replaced by SMA braces designed with same yield strength and first

elastic stiffness as the BRBs. Both frames were found to have the same fundamental period, to

allow a proper comparison.

CHAPTER 3 PERFORMANCE EVALUATION OF AS-BUILT AND RETROFITTED FRAMES

31

CHAPTER 3 PERFORMANCE EVALUATION OF AS-BUILT AND

RETROFITTED FRAMES

To assess the effect of using different bracing systems on the performance of non-ductile RC frames,

a comparative study of the frame model with BRB bracing systems and those with SMA bracing

systems was conducted by subjecting them to a representative set of ground motion records. In this

chapter, the results of non-linear dynamic analysis are presented and a detailed discussion is

elaborated to discuss the drift responses of the frame structures and the energy dissipation capacity

of the bracing systems.

3.1 Selection of Ground Motions

A considerable number of dynamic analyses using different ground motion records are often

needed to derive meaningful conclusions about the seismic behavior of nonlinear structures. The

results from these dynamic analyses depend on the characteristics of the ground motions as well as

the dynamic characteristics of the structure. Ground motion characteristics depend on many factors,

including the seismicity of the region, magnitude, epicentral distance, fault rupture mechanism,

fault rupture direction, and site soil profile.

A total of seven far-fault ground motion records are used from the Pacific Earthquake

Engineering Research Center database for nonlinear time-history analyses of the structures in this

study. Table 3.1 summarizes the characteristics of selected earthquake ground motions. The

selection criteria of the ground motions were as follows: (1) magnitude of the earthquake Mw ⩾6.5;

(2) closest distance to the fault rupture 10<Rrup<100 km; and (3) recording station site was

classified as a stiff soil site, similar to the site conditions used in the design of the prototype

non-ductile RC frame building structures. These ground motions were found to have different peak

ground accelerations (PGA), the first-mode spectral acceleration at 5% damping ratio, Sa(T1,5%)

and durations. Figure 3.1 shows the acceleration time histories of earthquake ground motions. Note

that all the nonlinear time-history analyses are performed with additional zero acceleration values

padded to all ground motion records to allow the free vibration decay of the structure in order to

adequately capture the residual deformation.

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Table 3.1 Characteristics of selected earthquake records

ID Record Event Year Epicentral

distance (km)

Peak ground

acceleration(g)

Scale factor

DBE

1 IV79qkp Imperial valley 1979 23.6 0.309 1.95

2 LP89g04 Loma Prieta 1989 16.1 0.212 1.65

3 LP89hch Loma Prieta 1989 28.2 0.247 1.49

4 LP89hda Loma Prieta 1989 25.8 0.279 1.52

5 NR94cnp Northridge 1994 15.8 0.42 1.69

6 SH87icc Superstition

hills 1987 13.3 0.172 1.3

7 NR94cen Northridge 1994 30.9 0.322 2.07

0 10 20 30 40-0.4

-0.2

0.0

0.2

0.4

Acc

ele

ratio

n [g]

Time [sec]

0 10 20 30 40-0.4

-0.2

0.0

0.2

0.4A

cce

lera

tion [g]

Time [s]

(a) EQ1 (b) EQ2

0 10 20 30 40-0.4

-0.2

0.0

0.2

0.4

Acce

lera

tio

n [

g]

Time [s]

0 10 20 30 40-0.4

-0.2

0.0

0.2

0.4

Acce

lera

tio

n [

g]

Time [s]

(c) EQ3 (d) EQ4


33

0 10 20 30 40-0.4

-0.2

0.0

0.2

0.4

Acce

lera

tio

n [

g]

Time [s]

0 10 20 30 40-0.4

-0.2

0.0

0.2

0.4

Acce

lera

tion

[g

]

Time [s]

(e) EQ5 (f) EQ6

0 10 20 30 40-0.4

-0.2

0.0

0.2

0.4

Accele

ration [g]

Time [s]

(g) EQ7

Figure 3.1 Acceleration time histories of earthquake ground motions

The selected ground motions are linearly scaled to three seismic hazard levels, i.e., the

Frequently Occurred Earthquake (FOE), the Design Basis Earthquake (DBE) and the Maximum

Considered Earthquake (MCE), corresponding to 50%, 10% and 2% probabilities of exceedance in

50 years, respectively. As a result, the average pseudo-acceleration spectra approximately match the

target spectra over the period range of interest in this study. Figure 3.2(a) shows the mean response

spectra of the seven scaled earthquake records and their corresponding ASCE 7-10 [81] design response

spectra at a 5% damping level. Figure 3.2 (b) shows the individual earthquake response spectra and their

mean spectra for the DBE seismic hazard levels, respectively.

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0 1 2 30

1

2

3

Sp

ect

ral a

ccele

ratio

n [g]

Period [s]

FOE Design spectrum DBE Design spectrum MCE Design spectrum Average FOE Spectrum Average DBE Spectrum Average MCE Spectrum

0 1 2 3

0

1

2

3

Spect

ral a

ccele

ration [g]

Period [sec]

DBE Design Spectrum Average DBE spectrum Individual Response spectrum

(a) Design and mean spectra (b) Individual spectra scaled to DBE

Figure 3. 2 Design and earthquake response spectra scaled to FOE, DBE and MCE hazard levels

3.2 Performance Evaluation

Prior to dynamic time-history analyses, modal analyses are conducted to determine the

fundamental periods of the frames. The fundamental periods of the as-built non-ductile frame is

1.07s; while the fundamental period for the BRB and SMA braced frames is 0.476s. Note that frame

models resulting from the use of three SMA materials have the same fundamental period as the

BRB frame and are expected to exhibit similar behavior when subjected to low intensity ground

motions.

Figure 3.2 compares the roof drift time histories of the bare frame and the retrofitted frames

under the EQ1 earthquake ground motion scaled to the DBE and MCE. Under the DBE-level

ground motions, the peak story drift of the bare frame is 2.54%, while the peak story drifts for the

BRB, NiTi, Fe and Cu based SMA frames are 0.70%, 1.14%, 1.05%, 1.26%, respectively. Under the

MCE-level ground motions, the peak story drifts of the bare and BRB frame are 2.9% and 1.7%,

respectively; while the peak story drifts of NiTi, Fe and Cu based SMA frames are 1.91%, 1.84%

and 1.91 %, respectively. Note that the results of the roof drift time history showed much noticeable

residual deformation for the BRB frame, 0.17% at DBE and 0.4% at MCE, while the SMA braces

brought the frame back to its initial position or at least with negligible residual deformations.


35

0 10 20 30 40 50 60

-3

-2

-1

0

1

2

3R

oo

f D

rift

[%

]

Time [sec]

NiTi FeNiCuAlTaB CuAlBe BRB BARE

0 10 20 30 40 50 60

-3

-2

-1

0

1

2

3

Ro

of

Drift

[%

]

Time [sec]

NiTi-SMA FeNiCuAlTaB-SMA

CuAlBe-SMA BRB BARE

(1) DBE (2) MCE

Figure 3.3 Roof drift time histories of the bare and retrofitted frames under the EQ1 earthquake

ground motion

The force-deformation behavior of the bracing systems, as illustrated in Figure 3.4, shows

that buckling restrained braces are able to dissipate a significant amount of seismic energy. The

SMA braces exhibited a recentering capability, with the frame dissipating energy through the

flag-shaped hysteresis behavior. The brace with narrower hysteretic shape tends to exhibit larger

deformation. Despite the higher drift demands on the SMA retrofitted frames, the SMA frames

experienced very small permanent deformation (0.21%, 0.07% and 0.12% for the Ni-, Fe- and

Cu-SMA based frames, respectively) at MCE hazard level, as they came back to their initial

position after the earthquake, in contrary to BRB frame, which presents a residual story drift of

0.6%. In the case considered here (EQ1) the RC frame benefitted from the recentering capability of

the SMA braces can reduce the structural damage, even at such high simulated seismic event.

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

-600

-400

-200

0

200

400

600

Axi

al

Fo

rce

[k

N]

Axial deformation [m]

BRB NiTi FeNiCuAlTaB CuAlBe

Figure 3.4 Typical axial force-deformation relationships of BRB and SMA braces

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The three different SMA materials used in this research possess different characteristic

cyclic hysteresis loop. As illustrated in Figure 3.4, the Cu-SMA braces possess the narrowest

hysteresis flag, implying that they dissipated less amount of seismic energy compared to the other

SMA braces (Ni- and Fe-based SMA bracing systems), which have wider hysteretic loops. Note that

the three SMA-braced frames experienced relatively comparable peak inter-story drift ratio (PIDR),

e.g. 2.90%, 2.75%, and 3.23% at MCE level, for Ni-, Fe- and Cu-based SMA frames, respectively.

However, a relatively higher PIDR was observed for Cu-SMA retrofitted frame. It can be concluded

that the maximum story drift can be reduced by employing SMA materials with wider hysteresis

loops.

As it was corroborated in previous paragraphs, to each of the 7 ground motion record was

associated 3 scaling factors (for FOE, DBE and MCE levels) to much the mean response spectrum

of the selected ground motion with design response spectrum selected for the purpose of this

research. Therefore, for each level of seismic excitation, each ground motion record was scaled and

applied to all the 5 frame models. Figure 3.4-6 show the peak inter-story drift ratio (PIDR) under

FOE, DBE and MCE ground motions respectively. For the three levels of ground excitations, after

post-processing the results, the observed average values of PIDR for the As-built (Bare), BRB,

Ni-SMA, Fe-SMA, and Cu-SMA are presented Table 3. . Referring to the life safety level as defined

in ASCE 7, all the retrofitted frames were able to sustain the seismic excitation at DBE without

significant structural damage as the average PIDR was kept under 2% for all the retrofit methods.

EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7 --0.0

0.5

1.0

1.5

2.0

Earthquake Ground Motion

Bare Average BRB Average NiTi-SMA Average FeNiCuAlTaB-SMA Average CuAlBe-SMA Average

PID

R [

%]

BARE BRB NiTi-SMA FeNiCuAlTaB-SMA CuAlBe-SMA

Figure 3.4 Peak inter-story drift ratio (PIDR) at FOE seismic intensity


37

EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ70

1

2

3

4

5 Bare Average BRB Average NiTi Average FeNiCuAlTaB Average CuAlBe Average

PID

R [

%]


BARE BRB NiTi FeNiCuAlTaB CuAlBe

Figure 3.5 Peak inter-story drift ratio (PIDR) at DBE seismic Intensity

EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ70

2

4

6

8 Bare Average BRB Average NiTi Average FeNiCuAlTaB Average CuAlBe Average

PID

R [%

]



Figure 3.6 Peak inter-story drift ratio (PIDR) at MCE seismic Hazard

Table 3. 2 Mean values of PIDR at FOE, DBE and MCE hazard levels

Hazard level Bare BRB Ni-SMA Fe-SMA Cu-SMA

FOE 1.10% 0.31% 0.38% 0.37% 0.41%

DBE 2.86% 1.01% 1.38% 1.29% 1.49%

MCE 5.23% 2.29% 2.35% 2.24% 2.71%

In similarity with the single record case study analyzed above, the BRB frame exhibited

relatively lower PIDR compared to the three SMA equipped frames. At the FOE level of seismic

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excitations, BRB and the three SMA frames presented relatively similar behavior as expected,

referring to the fact that they possess the same initial elastic stiffness. Under that level of seismic

hazard, the deformations endured by bracing elements were mostly in the elastic range, and it is

enough to limit the PIDR of the frame to less than 0.5%. From the seismic performance of the frame

models at DBE level of seismic loading, it was observed that the use of BRB braces resulted in a

reduction in PIDR of 65% from that of the bare frame. Under the same level of seismic loading, the

SMA frames experienced reductions of PIDR by 52%, 55% and 48% for Ni-, Fe- and Cu-SMA

frames, respectively.

Another observation lies on the behavior of SMA frames with different SMA materials.

From Figure 3.4-6, the three frames showed relatively comparable performance, suggesting that the

initial elastic stiffness and the recentering capability dominated the overall frame structural behavior.

However, the distinctive response pattern observed in DBE analysis results can be related to their

different hysteretic properties. The three materials possess different parameters, those being the

postyield stiffness ratio, hysteresis width, and the ratio of martensitic stiffness to austenite stiffness.

The noticeable order is deemed to follow the materials characteristic hysteresis width, specifically.

Therefore, despite the relatively higher postyield stiffness of CuAlBe based braces, the Cu-based

frame exhibited higher transient deformations, given the lower damping capabilities of the material.

Cu-based SMA frame experienced higher PIDR for almost all the seven records and for both under

DBE and MCE level of seismic excitations. This could lead to the conclusion that the performance

of SMA braced non-ductile RC frame in terms of PIDR is strongly related to the energy dissipation

capacity and the area of hysteresis loop of the SMA braces. It is worth mentioning that CuAlBe

SMA would present higher damping at large strains, a behavior not considered in this study, given

the non-uniform nature of loops it develops in high strain cyclic loading [53].

After a seismic event, the damage of structures caused by the ground motions is assessed

and quantified to help on deciding the reparability, associated cost, and reusability of the structure.

The assessment involves both the comfort of the users and, most importantly, their life safety,

especially from imminent or probable aftershock. The residual deformation is commonly used in the

earthquake engineering community as a tool for the damage quantification (damage index) after a

seismic event [82] to link the structures transient deformations to structural damage. It this research,

residual deformation is used as a measure of evaluating the effect of the structures transient


39

deformation on the structures’ damage at the end of a ground motion loading scenario as adopted by

previous researchers.

As illustrated in Figure 3. below, the BRB frame presented an average residual drift close

to that of the bare frame under DBE excitations, being 0.156% and 0.158% in average, respectively

for the bare and BRB frames. SMA frames, on the other side, experienced 68%, 64% and 58%

reduction in residual deformations for Ni-, Fe- and Cu-based SMA frames, respectively, under the

DBE excitations. Figure 3.8 presents residual deformations under MCE ground motions. It can be

seen that the residual drift for the three SMA frames were under 0.5% for almost all the scenarios

simulated, with the mean values being 0.2%, 0.061% and 0.08%, respectively for NiTi, Fe and

Cu-based SMA frames. Moreover, the BRB frame has a mean residual drift close to 0.5%. Thus, the

SMA retrofit system was able to limit the frames’ damage, even at high seismic intensities, as it can

be seen from Figure 3.8 where the BRB frame presented relatively higher permanent deformation

for records with high seismic intensity than SMA frames. Additionally, note that the residual

deformation was kept below acceptable margins, for FOE to MCE increasing ground motion

intensity. Therefore, superelastic SMA bracing system was able to keep its performance unvaried,

even at high levels of seismic excitation. That observation suggests that the SMA can prevent the

failure of non-ductile RC frames even at unexpectedly high seismic event.

EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ70.0

0.1

0.2

0.3

0.4


Bare Average BRB Average NiTi Average FeNiCuAlTaB Average CuAlBe Average

Ma

x -

Re

sid

ua

l ID

R [

%]


Figure 3. 7 Residual IDR at DBE seismic Intensity

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EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ70.0

0.5

1.0

1.5

2.0

2.5

3.0

Bare Average BRB Average NiTi Average FeNiCuAlTaB Average CuAlBe Average

Resi

dual

IDR

[%

]



Figure 3.8 Residual IDR at MCE seismic Intensity

The effect of the retrofitting systems on damage distribution of the non-ductile RC frame

was also investigated. The behavior of the bare frame shows the damage localization and tends to

fail due to the weak story mechanism, in accordance with the fact that it is a weak column strong

beam design. Figure 3.9 illustrates a comparison of average of peak story drift distribution along

each story level for different frame models under the seven ground motions. The Bare and BRB

frames showed relatively higher maximum story drift demand at the first story caused by an

accelerated damage concentration at this story level. In contrast, SMA frame models exhibited

deformation demand distributed overall frame stories with smaller first story drift demand. Figure

3.10 shows the vertical distribution of the average residual drift of each story under DBE and MCE

excitations. It was observed that structural damage was concentrated at the first story. The BRB

frame kept the same pattern of residual story drift distribution as the bare frame, while residual

deformation for the first story was reduced close to zero for all the SMA frames, with relatively

uniform vertical damage distribution.


41

0 1 2 3 4 50

1

2

3

Flo

or

leve

l

Peak Story Drift [%]

BARE BRB NiTi-SMA

FeNiCuAlTaB-SMA CuAlBe-SMA

(a) DBE intensity level

0 1 2 3 4 50

1

2

3

Flo

or

level

Peak Story Drift [%]


(b) MCE intensity level

Figure 3.9 Average peak story drift distribution along the building height

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

Flo

or

leve

l

Residual Story Drift [%]


0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

Flo

or

leve

l

Residual Story Drift [%]


(a) DBE intensity level (b) MCE intensity level

Figure 3.10 Average residual story drift distribution along the building height

In general, as shown in Figure 3.10, the damage distribution for different retrofit response

mechanisms highlight the benefits of using SMA braces toward the avoidance of localization of

damage concentration in non-ductile RC frames subjected to high seismic excitations. Furthermore,

SMA frame models showed relatively similar damage distribution. The Cu-SMA and Fe-SMA

frames provides almost identical damage distribution in terms of residual deformation, despite the

fact that Fe-based SMA braces dissipates larger amount of seismic energy (given their wider

hysteretic loops) compared to the Cu-based SMA braces (given its narrow hysteretic loops).

3.3 Summary

A set of 7 natural far-fault ground motion records were selected form PEER- NGA database,

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and they were linearly scaled to three seismic hazard levels: FOE, DBE and MCE. Additional zero

acceleration values were added at the end of each ground motion to allow the free vibration decay

of the structure order to adequately capture the residual deformation. Nonlinear Time History

Analysis was performed on the As-built frame, the BRB frame and 3 SMA frame models. The

results of record EQ1 showed that the residual roof drift for the SMA frame was much less than the

residual roof drift of the BRB frame, 0.4, 0.21, 0.07 and 0.12 % respectively for BRB, Ni-, Fe- and

Cu- SMA frames at MCE. SMA and BRB frames exhibited comparable results in terms of PID,

while the energy dissipation capacity of SMAs influenced the PIDR results. As-built and BRB

frame shows the damage localization and would develop weak story mechanism, while the SMA

frame models exhibited deformation demands with a relatively even distribution along overall

frame stories. The residual deformation served as a tool for the damage quantification. SMA retrofit

system was able to limit the frames’ damage, even at high seismic intensities,

CHAPTER 4 FRAGILITY ANALYSIS OF AS-BUILT AND RETROFITTED FRAMES

43

CHAPTER 4 FRAGILITY ANALYSIS OF AS-BUILT AND

RETROFITTED FRAMES

Seismic fragility assessment deals with the probabilistic estimation of the safety and

performance of buildings and other civil infrastructure under uncertain future seismic events. It

should quantitatively treat and propagate all sources of uncertainties involved in order to obtain

reliable seismic fragility assessments. In seismic performance and risk assessment of structures,

both the randomness and uncertainty of the ground motion intensity demand, structural demand,

and structural capacity need to be considered. This chapter lays out the framework for seismic

fragility assessment; explains the concept of a seismic fragility, which is a key ingredient of the

framework; presents a probabilistic demand model that facilitates the derivation of fragilities

through simulation-based reliability analysis; and describes how the core elements involved in this

process are determined: seismic intensity, structural demand, and limit states-damage assessment;

followed by the fragility formulation adopted in this study.

4.1 Seismic Fragility Theory

For a specific seismic event, seismic fragilities express the capability of a structure in terms of

probability to withstand the seismic loading. It quantifies the probability that the seismic demand (D)

placed on the structure is greater than the capacity (C) of the structure, conditioned on a chosen

intensity measure (IM) that represents the level of seismic loading. The following expression

represents the fragility formulation concept.

�� = �(� > �|��) (4.1)

The seismic fragility modeling requires the selection of a proper seismic intensity measure

and appropriate engineering demand parameters (EDPs) to monitor the seismic response and to

evaluate the performance of the structures. Monotonically scalable ground motion intensities

present an apparent way of scaling them. Common examples of scalable IMs are the Peak Ground

Acceleration (PGA), Peak Ground Velocity, the 5% damped Spectral Acceleration at the structure’s

first mode period (Sa(T1,5%)), etc. EDPs can be deduced from the output of the corresponding

non-linear dynamic analysis.

The seismic fragility is commonly expressed by a lognormal cumulative distribution function:

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�(� < �|�� = �) = 1 − � �

ln �Ĉ

��

�� (4.2)

where �[] is the standard Gaussian distribution function; Ĉ is the median value of the structural

capacity at certain performance levels; �� is the median median value of structural demand; �� is

the coefficient representing aleatory and epistemic uncertainties. An estimate of �� is defined by:

�� = ��|�� + ��

� + �� (4.3)

with ��|�� and �� describe aleatoric uncertainties of structural demand and structural capacity,

respectively, and �� describe the modeling (epistemic) uncertainty.

The relation between the structural demand and the seismic intensity of earthquake, which is

commonly referred as the probabilistic seismic demand model (PSDM), can be established using

the power-law function suggested by Cornel et al.[83]

�� = � ∗ �� (4.4)

where a and b are regression coefficients. Therefore, a linear relationship can be obtained between

the median demand and the seismic intensity in the log-log space. By making a log-log linear

regression analysis between the structural demand and seismic intensity from the nonlinear dynamic

analysis results of N earthquake ground motions, the values of coefficients a and b can be obtained

and the dispersion value ��|�� can be calculated.

After that the dispersion ��|�� is estimated, the fragility can be written as the folowing:

�(� < �|�� = �) = 1 − �

⎣⎢⎢⎡ln�Ĉ� − ln(a. IM�)

��|�� + ��

� + ��

⎦⎥⎥⎤

(4.5)

In this study, a multi-record incremental dynamic analysis (IDA) is performed. It involves

subjecting the structural systems to a set of selected ground motions with increasing intensity values.

The 5% damped spectral acceleration at the structure’s first-mode period (Sa(T1,5%)) is adopted as

the seismic Intensity Measure ( IM) given its efficiency and hazard computability [84].

The limit states selected for fragility analysis can be identified by different levels of an

engineering demand parameter (EDP). Researchers have utilized displacement-based, energy-based

and hybrid measures to quantify damage; however, there has been little consistency on the most

appropriate measure. ASCE/SEI 41-06 proposes the use of the peak inter-story drift ratio (PIDR) to

evaluate building performance and damage levels of structural components. In this study, to


45

evaluate the post-earthquake seismic performance of the structures, the residual inter-story drift

ratio (RIDR) is also used to fully quantify the performance level of the structures.

The structural capacities are defined by the limit states that correspond to three widely used

performance levels [immediate occupancy (IO), life safety (LS), and collapse prevention (CP)] in

the earthquake community. These performance levels are adopted so that the results are consistent

with, and can be compared to, work published previously by other investigators. The IO level is

described by the limit below which the structure can be occupied safely without significant repair,

and is defined by the value of PIDR beyond which the frame enters the inelastic range. This value is

determined from a nonlinear static pushover analysis performed using a lateral force pattern that is

based on the first mode shape. The LS level occurs at a deformation at which “significant” damage

has been sustained, but at some margin below incipient collapse. Because this limit is hard to

quantify in terms of inter-story drift, the intermediate level is identified herein as the inter-story drift

at which significant structural damage (SD) has occurred, associated with a PIDR of 2%. Finally,

the CP level is defined by the point of incipient collapse of the frame due to either severe strength

degradation of members or significant P-delta effects resulting from excessive lateral deformations.

The CP limit is calculated from statistical analyses of incremental dynamic analysis (IDA) results,

and its median is defined by the median PIDR prior to failure to converge in the IDA.

The uncertainty in capacity, ��, depends on the limit state considered. At the IO and SD levels,

�� must capture the uncertainty with respect to the assumption that these performance levels can,

in fact, be defined by the elastic limit and a PIDR of 2%, respectively. It is assumed that ��=0.25

for both IO and SD levels. For the CP level, �� is set equal to the logarithmic standard deviation of

structural capacity calculated using the IDA results. Modeling uncertainty, ��, is assumed to be

0.20, based on the assumption that the modeling process yields an estimate of building frame

response that, with 90% confidence, is within 30% of the actual value. Table 2 summarizes all

parameters used in the fragility formulation.

To assess the possible benefits of reduced residual drift ratio of the SMA braced frames, two

performance levels associated with residual displacements are proposed in this study, i.e.,

“Re-centering” and “Repairable” performance levels, with the corresponding maximum allowable

residual drift of 0.2% and 0.5%, respectively. The “Re-centering” performance level is a damage

state where the structure can return close to its original vertical alignment with minimal residual

drift. Considering that so far there is no established criterion for the “Re-centering” performance

HOHAI UNIVERSITY

46

level, the residual drift of 0.2% was adopted in this study, which is based on the acceptable

out-of-plumb construction tolerance. The construction tolerance, typically found in the building

codes, set limits for deviation from the designed column and beam lines to ensure that the

performance of the structure is almost “as designed”. The “Repairable” performance level is a

damage state where the structure suffered certain damage and it may be financially more feasible to

repair the structure than to demolish it. The 0.5% residual drift limit for the “Repairable”

performance level was determined based on a survey of damaged structures during past earthquakes.

It was found that when direct and indirect repair costs were considered along with losses due to

building closure during the repair period, it was no longer practically suitable for repair from an

economic perspective when residual drifts were greater than 0.05% rad.

4.2 Regression Analysis of Structural Demands

To develop fragility curves by performing IDAs, the mean spectral acceleration of the ground

motion records is scaled to target IM values. The maximum and residual drift recorded from IDA

are incorporated into probabilistic seismic demand model (PSDM) and linear regression of demand

(EDP)-intensity measure (IM) pairs in the log-transformed space. This linear regression model is

used to determine the slope, intercept, and dispersion of the EDP-IM relationship. Figure 4.1-4.2

depict the seismic demand in terms of PIDR and RIDR of the bare and retrofitted frame models,.

The aleatoric uncertainty of structural demands can be calculated using the following equation

��|�� = �1

� − 2��ln(��) − ln �� ∗ ��

��

�

��

(4.6)

-6 -5 -4 -3 -2 -1 0 1-4

-3

-2

-1

0

1

2

3

ln(P

IDR

)

ln(Sa_T1)

-4 -3 -2 -1 0 1-4

-3

-2

-1

0

1

2

3

ln(P

IDR

)

ln(Sa_T1) (a) Bare frame (b) BRB frame


47

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0-4

-3

-2

-1

0

1

2

ln(P

IDR

)

ln(Sa_T1)

-4 -3 -2 -1 0 1-4

-3

-2

-1

0

1

2

3

ln(P

IDR

)

ln(Sa_T1)

(c) Fe-based SMA frame (d) Cu-based SMA frame

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0-4

-3

-2

-1

0

1

2

ln(P

IDR

)

ln(Sa_T1)

(e) NiTi-based SMA frame

Figure 4.1 Peak inter-story drift ratio (PIDR) vs. spectral acceleration relationships

-6 -5 -4 -3 -2 -1 0 1-12

-10

-8

-6

-4

-2

0

2

ln(R

es_

Drift)

ln(Sa_T1)

-4 -3 -2 -1 0 1-10

-8

-6

-4

-2

0

2

ln(R

es_

Drift

)

ln(Sa_T1) (a) Bare frame (b) BRB frame

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-4 -3 -2 -1 0 1-10

-8

-6

-4

-2ln

(Re

s_D

rift)

ln(Sa_T1)

-4 -3 -2 -1 0 1-10

-8

-6

-4

-2

0

ln(R

es_

Drift)

ln(Sa_T1)

(c) Fe-based SMA frame (d) Cu-based SMA frame

-4 -3 -2 -1 0 1-10

-8

-6

-4

-2

0

ln(R

es_

Drift

)

ln(Sa_T1) (e) NiTi-based SMA frame

Figure 4.2 Residual inter-story drift ratio (RIDR) vs. spectral acceleration relationships

Table 4.1 Summary of parameters used in fragility analysis for PIDR

Parameter

RC-Ni RC-Fe RC-Cu BRB BARE

Demand

a

1.1 1.15 1.23 1.04 4.7

b

1.18 1.12 1.24 1.16 1

��|��

0.28 0.27 0.28 0.29 0.36

Capacity

Ĉ IO 1

LS 2

CP 4

��

0.25

Modeling

��

0.2


49

Table 4.2 Summary of parameters used in fragility analysis for RIDR

Parameter

RC-Ni RC-Fe RC-Cu BRB BARE

Demand

a 0.024 0.024 0.641 0.136 0.3935

b 1.86 1.862 1.982 2.549 1.766

��|�� 0.67 0.626 0.641 0.881 0.8

Capacity

Ĉ Recentering 0.2

Repairable 0.5

��

0.25

Modeling

��

0.2

4.3 Fragility Analysis

The fragility curves of different retrofit methods and levels of limit state are illustrated in

Figure 4. and 4.4. Both the BRB and all the SMA retrofitted frames possess comparable fragilities

considering PIDR as the EDP. Probabilities of exceeding the immediate occupancy limit state are

nearly identical for all the retrofitted frames. It can be observed from Figure 4. that the BRB

frame performed better than all the other frames in terms of peak drift limit states. At Life Safety

limit state, BRB frame has less chance of the drift demand on the frame exceeding the frame’s

capacity, conditioned on the seismic intensity measure given its large energy dissipation capacity.

NiTi braces were able to reduce the PIDR demand the RC frame, which can be associated to the

hardening of NiTi material. Thus NiTi-based SMA frame is expected to show relatively lower drift

at collapse limit state. The recentering capacity appears to compensate the energy dissipation

capacity at large deformation demands. It was observed that the transient peak story drift demand is

much correlated with the energy dissipation capacity of the system. Cu-based SMA material

exhibits the highest fragilities given its narrow hysteresis loops. The fragility curves obtained when

residual drift is the EDP show that the SMA based bracing system can protect the non-ductile frame

even at high seismic event, and BRB equipped frame tend to develop residual deformation after

yielding.

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50

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

Pro

babili

ty o

f exc

ed

ence

Sa_T1 [g]

BRB Cu Fe Ni BARE

0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

Pro

ba

bili

ty o

f e

xce

dence (

BR

B)

Sa_T1 [g]

BRB Cu Fe Ni BARE

LS

(a) IO limit state (b) LS limit state

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Pro

babili

ty o

f exc

edence

Sa_T1 [g]

BRB Cu Fe Ni BARE

(c) CP limit state

Figure 4. 3 Fragility curves of as-built and retrofitted frames for IO, LS and CP limit states

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

Pro

ba

bili

ty o

f e

xce

de

nce

Sa_T1 [g]

BRB Cu Fe NiTi BARE

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

Pro

ba

bili

ty o

f e

xce

de

nce

Sa_T1 [g]

BRB Cu Fe NiTi BARE

(a) Recentering limit state (b) Repairable limit state

Figure 4. 4 Fragility curves of the frames for Recentering and Repairable limit states


51

A seismic hazard curve expresses the annual probability that the seismic intensity of

earthquakes exceeds a particular intensity. Given the fragility function formulated previously, the

probability that the demand exceeds the structure capacity can be defined by integrating the fragility

and seismic hazard as expressed as Eq. (4.7).

�(� < �) = � �(� < �|�� = �) ∗ ��(�) (4.7)

where the �(� < �|�� = �) is the seismic fragility as calculated previously, and �(�) = seismic

hazard which provides the annual probability that the Intensisity measure SI exceeds x and can be

expressed using a power function as shown in following equation [83] when Sa(T1) is used as SI

�(�) = �(�� ≥ �) = �� ∗ �� (4.8)

where �� and � represents constants that vary by regions and site conditions. The limit state

probability analysis of the nonductile RC frame retrofitted with different bracing systems will

comparatively identify the reduction in fragility and limit state probability associated with the

replacement of one system by another, accounting for the mentioned uncertainty. The hazard curve

described by equation (9) is a straight line in log-log plot. With Equations (4.3) to (4.8), Eq. (4.7)

can be presented as

�(� < �) = �(��

�)��

2��

�|��2 + �

�2 + �

�2 �� (4.9)

�� = (�/�)�/� (4.10)

The assumed building site location is San Francisco, CA. the site condition is site class D.

Uniform Hazard spectra can then be constructed with site coefficient according to ASCE 7-10. The

Sa values corresponding to T1=0.473s are 3.68g, 2.61g and 1.96g , respectively for 2, 5 and 10%

probability of exceedance in 50 years. The hazard curve for building site is shown in Figure 4.5

with a power function fitting the annual probability of exceedance for different spectral

accelerations (Sa), where the constants in Equation (9) are �� = 4.04 × 10�� and k= 2.74

(�� = 0.476 s).

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52

1.0 1.5 2.0 2.5 3.00.000

0.001

0.002

0.003

An

nu

al P

rob

abili

ty o

f E

xce

ed

an

ce

, H

(Sa

)

Sa [g]

H(Sa)= ko Sa-k

ko = 4.04X10-3

k = 2.74

Figure 4.5 Annual Seismic Hazard curve for the retrofitted frame

The annual probabilities of exceedance of peak drift demand for the four cases of the

retrofitted building at different performance limit states are calculated using Eq. (4.9) and results are

illustrated in Figure 4.6. The Fe-based SMA frame exhibits the lowest annual probability of the drift

demand exceeding the capacity, whereas the Cu-based SMA frame has the highest among the four

retrofitted frames. The BRB frame will likely not meet the performance objectives compared to

NiTi- and Fe-based SMA frames. The Cu-based SMA braces possess the lowest energy dissipation

capability and Fe-based SMA braces possess the highest energy dissipation capability. It can be

deduced that probability of the peak transient drift demand exceeding the capacity limits is related

to both the recentering and energy dissipation mechanisms and the presence of both would benefit

the non-ductile RC frame.

Fe Cu Ni BRB0.00E+00

1.25E-02

2.50E-02

Annua

l pro

ba

bili

ty o

f exceeda

nce

IO


2.50E-03

5.00E-03

Annual p

robabili

ty o

f exceed

ance

LS


5.00E-04

1.00E-03

Ann

ua

l pro

bab

ility

of

exceeda

nce

CP

Figure 4.6 Annual probability of exceedance for different retrofit methods


53

4.4 Summary

In this chapter, a review of the basic concepts of fragility theory is presented and its core

elements involved in the process are determined: seismic intensity measure, engineering demand

parameters, and limit states-damage assessment parameters. Uncertainties parameters adopted in the

study are also presented. An IDA was performed and the maximum and residual drift recorded are

incorporated into probabilistic seismic demand model (PSDM). A linear regression of demand

(EDP)-intensity measure (IM) pairs in the log-transformed space is used to determine the slope,

intercept, and dispersion of the EDP-IM relationship. The fragility curves of different retrofit

methods and levels of limit state are illustrated. The fragility curves obtained when residual drift is

the EDP show that the SMA based bracing system can protect the non-ductile frame even at high

seismic event, and BRB equipped frame tend to develop residual deformation after yielding.

The annual probabilities of exceedance of peak drift demand for the four cases of the

retrofitted building at different performance limit states were calculated. It was observed that the

probability of the peak transient drift demand exceeding the capacity limits is related to both the

recentering and energy dissipation mechanisms and the presence of both would benefit the

non-ductile RC frame.

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CHAPTER 5 SUMMARY, CONCLUSIONS AND FUTURE

RESEARCH

5.1 Summary and Conclusions

This thesis presented a comparative study of the seismic performance of a non-ductile RC

frame building designed for gravity load only, with the retrofitted models of this frame containing

the buckling restrained braces (BRBs) and shape memory alloy (SMA) braces. Nonlinear time

history analyses were carried out to compare the maximum and residual story drifts of the as-built

and retrofitted frames and the braces’ demand. Finally, fragility analyses were used to estimate the

probabilities of exceeding certain limits of structural response, given increasing intensities of

ground motions. The effects of BRBs and SMA braces on residual deformation were discussed as

well as the effect of the variation of ground motion intensities. The general conclusions drawn from

the results of this study can be summarized as follows:

(1) The developed beam-column joint model for the non-ductile RC frame building was

sufficiently accurate in predicting the experimental beam-column joint responses for simulating the

seismic response of the non-ductile RC frames for purposes of fragility assessment and

performance-based earthquake engineering.

(2) When the frames are subjected to the Frequently Occurred Earthquake (FOE) and Design

Basis Earthquake (DBE) ground motions, the non-ductile RC frame retrofitted with BRB exhibit

slightly lower peak story drift demands compared to the non-ductile RC frame upgraded with SMA

braces. In general, the results of the nonlinear time-history analysis show that the SMA braced

frames can achieve a seismic response level comparable to that of the BRB frame while having

significantly reduced residual drifts. It was also observed that the retrofitted frame with higher

energy dissipation capacity would experience smaller peak story drifts.

(3) Despite the noticeable variety in the hysteretic properties of SMA materials, the non-ductile

RC frames with different SMAs can successfully achieve the same performance targets. All the

SMA-based non-ductile RC frames exhibit small residual deformations even after very strong

earthquakes, which clearly demonstrates their superior seismic performance.

(4) Statistical analyses of seismic demands reveal that the maximum drift ratios of the frames

CHAPTER 5 SUMMARY, CONCLUSIONS AND FUTURE RESEARCH

55

are approximately linearly correlated with the seismic intensity measure. However, the residual drift

demands increases non-linearly with the increase of seismic intensity measure. In addition, the

dispersion of residual drift demand is larger than that of maximum drift demand.

(5) Analyses of the fragility curves reveal that the effectiveness of a retrofit technique in

mitigating probable damage can be measured using fragility curves for a given damage state of

interest. The non-ductile RC frames with SMA braces experienced less seismic vulnerability than

the BRB frame in terms of the performance limit states based on the residual story drifts.

5.2 Future Research

The following areas are recommended for future research work:

(1) The seismic performance of non-ductile frames with BRB and SMA braces subjected to

mainshock-aftershock sequences should be evaluated.

(2) Proper or improved SMA material models should be developed for better simulation of the

complex hysteretic loops and the residual strain developed at large strains in SMA based elements.

(3) The effect of vertical component of earthquakes should be incorporated in the performance

assessment procedure to study the behavior of BRB and SMA based frames under the vertical

excitations.

(4) Further research work should expand the possibility of combining the BRB and SMA for

an efficient re-centering and energy dissipation bracing system toward an effective upgrade method

for non-ductile RC frames.

(5) This study should be expanded to high rise non-ductile RC frames to study the effect of

story numbers.

56

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