Review on Recent developments in the performance-based seismic design of reinforced concrete structures

Structures 6 (2016) 119–133

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Review

Review on Recent developments in the performance-based seismic design of reinforced concrete structures Mohd. Zameeruddin 1, Keshav K. Sangle 2 Department of Structural Engineering, Veermata Jijabai Technological Institute, H. R Mahajan Road, Matunga, Mumbai 400 019, India

a r t i c l e

i n f o

Article history: Received 23 August 2015 Received in revised form 10 March 2016 Accepted 10 March 2016 Available online 15 March 2016 Keywords: Performance-based seismic design State of development Performance-based seismic evaluation Performance assessment procedures Example building

a b s t r a c t The performance-based seismic design was suggested to overcome the limitations of force-based seismic design methods used for addressing the inelastic behavior and cyclic loading effects in reinforced concrete structures. The suggested method permits buildings to be designed with a realistic and reliable understanding of the risk to life, occupancy, and economic loss that may result from future seismic events. The design has two primary goals: appropriately quantifying the uncertainties associated with the performance evaluation process and satisfactorily characterizing the associated structural damage for direct incorporation into the design or performance evaluation methodology. This study reviewed recent developments in performance-based seismic design by defining the performance objectives (levels), evaluation techniques, and assessment procedures. In addition, the current state-of-practice in performance-based seismic evaluations were compared. In contrast to the performance evaluation procedures of previous studies, in which the pros and cons were highlighted, the performance evaluation procedure of the present study was implemented for force-based design structures. A damage index that was expressed using nonlinear responses obtained in the output of the procedure was introduced. A possible integration of the damage value with the performance level was proposed. © 2016 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . State of development . . . . . . . . . . . . . . . . . Performance-based seismic evaluation . . . . . . . . . 3.1. Capacity spectrum method . . . . . . . . . . . 3.2. Displacement coefficient method . . . . . . . . 4. Performance assessment procedures . . . . . . . . . . 4.1. Damage indices for reinforced concrete structures 5. Example building . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . Abbreviations . . . . . . . . . . . . . . . . . . . . . . . Notations . . . . . . . . . . . . . . . . . . . . . . . . . Conflict of interest . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction

E-mail addresses: [email protected] (M. Zameeruddin), [email protected] (K.K. Sangle). 1 Tel.: +91 982 291 3231; fax: +91 246 222 2999. 2 Tel.: +91 982 112 9187.

The occurrence of recent earthquakes in countries across the world raises the need for a fundamental change in the present seismic design procedure [1]. Present seismic design codes are force-based, that is, forces and displacements within elastic limits are calculated, and their

http://dx.doi.org/10.1016/j.istruc.2016.03.001 2352-0124/© 2016 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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combination is used to design the structural and nonstructural components. Serviceability checks are applied using displacement limits and ductile detailing. Inelastic responses are calculated by applying the response reduction factor, which relates to force or displacement amplification; however, such an indirect approach causes misjudgment in the actual building response [2]. With an aim to communicate the safety-related decisions, the design practice is focused on the predictive method of assessing potential seismic performance, known as performance-based seismic design (PBSD). PBSD is a generalized design philosophy in which design criteria are expressed in terms of achieving stated performance objectives when the structure is subjected to the stated levels of seismic hazard [1]. PBSD permits the design and construction of buildings with a realistic and reliable understanding of the risk to life, occupancy, and economic loss that may occur because of future seismic events [3,4]. PBSD is an iterative process, which begins with the selection of performance objectives (that are defined by the owners, designers, and building officials), followed by the development of a preliminary design (considering stated set of performance objectives), an assessment of whether the design meets the performance objectives, and finally redesign and reassessment, if required, until the desired performance level is achieved [16]. Fig. 1 displays the flowchart representing key steps in the PBSD procedure. In contrast to the force-based design (FBD), PBSD provides a systematic methodology for assessing the performance capability of a building, system, or components. The methodology provides a framework for determining the levels of safety and property protection, and the cost acceptable to owner, designer, and building officials for the project according to the specific project requirement. The advantages of PBSD are as follows [17]: 1. Designing individual buildings with a higher level of confidence. 2. Designing individual buildings capable of meeting the stated set of performance objectives with lower construction cost. 3. Designing individual buildings to achieve higher performance than intended using present seismic codes.

4. Designing individual buildings, which are beyond the present seismic code-prescribed limits regarding the configuration, materials, and systems, to meet the performance intended by the present seismic codes. 5. Assessing the potential seismic performance of existing structures, and estimating the potential losses in the event of a seismic hazard. 6. Assessing the potential performance of the present seismic code requirement for new buildings to serve as the basis of improvement to code-based seismic design criteria. In PBSD, performance levels are described in terms of displacements and drifts. A structure's damage state can be related to performance levels. This idea has given a new design approach based on displacement, known as displacement-based seismic design (DBSD) of structures. The fundamental assumption in DBSD is that for an inelastic system, the strength is less important than the displacement [18,19]. DBSD has focused on three major weakness of FBD: (i) it is based on the assumption that initial stiffness determines the structural period and distribution of design forces among different structural elements because stiffness is dependent on strength of element, which remains unknown until the design completion; (ii) allocation of seismic forces among elements on the basis of initial stiffness is illogical for many structures because it is incorrectly assumed that different structural elements can be forced to yield simultaneously; and (iii) there is no unique force-reduction factor (R-based on ductility) for a given structure and material. The direct displacement-based seismic design is among several other attempts to improve the FBD. In the direct displacementbased design approach, the strength required at a designated plastic hinge to achieve a targeted design in terms of defined displacement objectives is evaluated. To ensure the formation of plastic hinges only where intended, the direct displacement-based design must be integrated with a capacity-based design [20]. PBSD and DBSD have been used many times interchangeably because both are performance-oriented design procedures. The assumption of DBSD was found to be oversimplification as the level of damage is

Step 1 Select performance objectives

Step 2 Develop preliminary design

Revise design and/ or objectives

Inputs from Owner, designer and building officials

Inputs from Owner, designer and contractors

Step 3 Access performance capability Comments Peer reviewers, building officials

No

Step 4 Does performance meets objective?

Yes

Construction Fig. 1. PBSD flow diagram [16].

M. Zameeruddin, K.K. Sangle / Structures 6 (2016) 119–133

influenced by several other factors, such as accumulation and distribution of damages, failure modes of elements and components, the number of cycles and duration of earthquakes, and acceleration levels as in the case of the secondary system [21]. To overcome these PBSD deficiencies, a correlation between the desired performance level and a corresponding level of damage or structural safety must be calibrated [22]. This study aimed to obtain an update review of the state of development of PBSD, performance-based seismic evaluation (PBSE) techniques, and performance assessment procedures. 2. State of development The interest in PBSD was initiated in the 1980s among engineers engaged in the practice of the seismic design and retrofit of existing buildings [3]. The roots of PBSD can be traced to the recommendations provided by the Structural Engineers Association of California (SEAOC, 1960) and the publications of the Portland cement association. The SEAOC publication introduced a lateral force equation V = ZKCW, accounting the seismic hazard (Z), structural systems performance (K), and structural dynamics (C as a function of period T), which was modified later by the introduction of various coefficients to relate with ductility and inelastic behavior. The Portland cement association publication introduced PBSE techniques, which cover the dynamics, inelastic response of structures, energy concepts, and pushover analysis [4–6]. The Applied Technical Council (ATC) 13 report [5], funded by the Federal Emergency Management Agency (FEMA), provided a methodology to estimate the probable repair costs for California buildings, which were damaged because of earthquakes. According to the expert opinions of practicing engineers, the methodology was further modified with an intention to apply it to a broader class of buildings rather than an individual structure. The Department of Defense [6,7], published the Tri-services manual for essential buildings (Navy, Army, and Air Force) addressing the issue of occupancy interruption. The Tri-services manual presented two basic analytical procedures: a linear methodology that uses inelastic demand ratios, and a capacity spectrum method (CSM) for inelastic analysis. The ATC 14 report [5–7], funded by the national science foundation, provided practicing engineers a standard methodology by which buildings confirming to certain model building types could be evaluated for seismic hazards. The need to accommodate multiple performance objectives, and their implementation in the design process, leads toward the development of first generation procedures for seismic evaluation and retrofit existing reinforced concrete buildings. Three documents are credited for laying the foundation for first generation procedures: SEAOC vision 2000 [8], ATC 40 [10], and FEMA 273 [11]. SEAOC vision 2000 developed the framework for procedures, which facilitates in designing structures for a predefined set of performance objectives, defined in terms of permanent and transient drift. The ATC 40 considers the performancebased design as a methodology in which design criteria are expressed in terms of achieving performance objectives, which are the desired levels of building performances defined in terms of acceptable structural and nonstructural damages for specified levels of seismic hazard. The ATC 40 puts forth a set of procedures for CSM-based seismic evaluation. In this procedure, the capacity spectrum of the structure is obtained using pushover analysis. FEMA 273 puts forth a straightforward method to estimate the peak displacement of a nonlinear system. The method involves multiplying the response of the structure elastic linear system by using coefficients, namely C1, C2, and C3. In this system, C1 represents the ratio of the peak displacement of the inelastic system to that of corresponding elastic systems with the same unyielding periods of vibration. C2 reveals the pinching effects in the load–displacement relationship, and C3 accounts for the P–Δ effect. ASCE 41 [13], provided a framework for PBSD of existing buildings with rehabilitation objectives, which can accommodate buildings of different types, addresses a variety of performance levels, and reflects the variation of seismic hazards across the United States and its territories.

121

The commentary of this standard are based primarily on the second generation procedure FEMA 356 [12]. FEMA 356 presented an incremental improvement to the first generation procedure (FEMA 273) with respect to technical updates of the analytical requirements and acceptance criteria according to the information gained from using the procedure in engineering practice and case studies given in the FEMA 349 report. This standard requires the selection of rehabilitation objectives for a building that has been previously identified as requiring seismic rehabilitation. A seismic rehabilitation objective consists of one or more rehabilitation goals. Each goal consists of building performance and earthquake hazard levels. The building performance is described qualitatively in terms of the safety afforded to the building occupants during and after an earthquake, the cost feasibility of restoring the building to its pre-requisite condition, and the downtime to effect repairs. The earthquake hazard level corresponds to the mean return period. When a rehabilitation objective achieves the dual rehabilitation goal, it is termed as the basic safety objective. This standard recommended four analysis procedures to estimate seismic demands: The first two procedures are linear static and dynamic, which are both force-based; the third and the fourth are nonlinear static and dynamic procedures, which are displacement based. In the displacement-based procedure, a structure is subjected to targeted displacements to obtain capacity curve; the procedure is termed as pushover analysis [13]. The documents of both the first and second generation procedures are widely used throughout the profession. As these documents were referred to increasingly, the following shortcomings were identified [15]: 1. The current procedure predicts structural response and a demand on the basis of the global behavior of the structure; however, it evaluates the performance on the basis of damage sustained by individual components (i.e., poorest performing element), which questions the accuracy and reliability of available analytical procedures in predicting the actual building response. 2. Many of the acceptance criteria presented in the document are based on expert judgment drawn from practicing engineers, rather than laboratory test results or other substantiating evidence raising questions about the level of conservatism underlying the acceptance criteria. 3. The guidelines presented in the document are excessively conservative compared to those of the FBD. 4. The need of an alternative for communicating performance to stakeholders (owner, designer, and building officials) that is more meaningful and useful for decision-making purpose. To overcome the aforementioned shortcomings, FEMA contracted a long-term project with ATC, namely ATC 58 project [16], to prepare the next-generation PBSD guidelines for new and existing buildings. The ATC 58 recommended four subject areas for improvements: a. b. c. d.

primary performance metrics, discrete or continuous performance levels, levels of analysis, and risk communication concepts.

Primary performance metrics focus on the primary concerns of the stakeholders with respect to i) direct losses, including both the cost of damage and repairs; ii) downtime associated with the loss of using a building; iii) indirect loss associated with the loss of using a building; and iv) loss of life and casualties of the occupants and those in the immediate vicinity of the building. The first and second generation procedures have discrete performance levels, with explicit structural and nonstructural design requirements associated with them. ATC 58 considers the future possibility of continuum between discrete performance levels, as illustrated in Table 1 [16]. ATC 58 provided an improved deterministic language for describing the levels of analysis for evaluating alternate deterministic and probabilistic approaches, and identified five global approaches that

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Table 1 Recommended implications of performance levels with discrete levels overlay [16]. Performance level

Building usability

Damage description

Life safety

Reoccupation of the building is unlikely and it will need to be replaced Reoccupation of the building is delayed and repairs may be costly

Collapse prevention

Interrupted occupancy and Interrupted operations Continued occupancy and interrupted operations Continued occupancy and Continued operations

Significant or Substantial damage

Reoccupation of the building is almost immediate and the cost of repair is modest

Limited damage

The building can continue its operation almost immediately

Minimal to no damage

could be considered for the development of next-generation PBSD (Table 2). FEMA 445 [17] offers a step-by-step, task-oriented program that will develop next-generation PBSD procedures and guidelines for structural and nonstructural components, applicable to new and existing buildings. The technological framework developed under this program can be adopted for other extreme hazards, such as hurricane, wind, terrorist attack, and fire. Next-generation PBSD provides an analytical framework for estimating risks by using risk-evaluating software packages under practice by insurance industries, hazard mitigation agency, and others. The framework will be used to estimate the possibility of incurring of earthquake-induced damage in terms of direct loss (repair cost), casualties, and downtime for individual buildings under interest, whether new or existing. Under a contract with FEMA, national building sciences has developed loss estimation software, Hazards U.S. (HAZUS). This software enables an experienced engineer, with considerable expertise, to develop building-specific relationships between earthquake intensity, damage, and losses. The next-generation PBSD defines performance objectives as the statement of an acceptable risk of incurring casualties, direct economic loss (repair cost), and occupancy interruption (downtime) associated with the repair or replacement of damaged structural and nonstructural building elements at a specified level of seismic hazard. The performance objectives stated in next-generation PBSD are of three risk types: i) intensity-based, ii) scenario-based, and iii) timebased. The next-generation PBSD highlights the need for basic research on simulation techniques, risk communication, structural and nonstructural component testing, and ground motion hazard characterization, to complete the development of PBSD procedures [15–17]. Recent studies investigated the application of PBSD to a specific structural system such as concrete [9,14,24,35,46–50] and coupled-wall [51,52] with a discussion on achieving the PBSD performance level from seismic evaluation of force-based designed concrete structures. In DBSD for structural

Table 2 Improved deterministic language of describing the levels of analysis [16]. Analysis Communication language Level 1 Level 2 Level 3

Level 4 Level 5

Lowest level of analysis, including uncertainties in both demand and capacities, but not including consequences. Same as level 1, but includes consequences such as direct and indirect cost, downtime, life loss, and injuries. Similar to that used in SAC design procedures, and includes integration over the full range of seismicity impacting the site and the uncertainties in the demand and the capacities, but does not include uncertainties in consequences. Same as level 3 but includes consequences such as direct and indirect cost, downtime, life loss, and injuries. Most rigorous level, which includes a completely coupled analysis and the uncertainties in all aspects that impact the result of the analysis

system such as concrete [19,20] with a comparative discussions on a FBD and DBSD methodologies. 3. Performance-based seismic evaluation Performance-based seismic engineering evokes the need for highlevel analysis procedures [24]. Next-generation PBSD guidelines have introduced a comprehensive framework for linear and nonlinear analysis procedures [24–31]. The choice of using analytical procedures depends on several parameters, such as the importance of structure, the target performance level, the structural characteristics (e.g., regularity, complexity, frequencies, and mode shapes), and the modeling efforts involved. Nonlinear analysis procedures provide better perception of inelastic behavior and failure modes of structure during a severe seismic event [24]. Nonlinear analysis procedures appearing in literature include nonlinear static and dynamic analyses [15–17,24]. Nonlinear dynamic analysis (NRHA) is a powerful tool for studying a structural seismic response. A set of carefully selected ground motion records result in more accurate and reliable structural responses [27]. Despite the accuracy and efficiency of NRHA, engineers are attracted to nonlinear static procedures (NLSPs) because of NRHA's time consuming computational process. NLSP attempts to obtain essential inelastic dynamic response characteristics of the structure under monotonically increasing load (Pushover analysis) [25,26]. The key steps involved in NLSP are illustrated in Fig. 2. 3.1. Capacity spectrum method This method was first introduced in the 1970s as a rapid evaluation procedure for assessing the seismic vulnerability of buildings at the Puget Sound Naval Shipyard project. In the 1980s, this method was developed in a design verification procedure for the Tri-services manual, Seismic design guidelines for essential buildings [6,7]. The procedure compares the capacity of the structure (capacity spectrum) with the demands of the structure (demand spectrum). The graphical intersection of these two curves is termed as the performance point, which approximates the response of the structures. Fig. 3 describes the performance evaluation procedure [10]. ATC 40 [10], Seismic evaluation and retrofit of concrete buildings emphasizes the use of the CSM. The procedure involves determining the target displacement using the following equation:
 δt ¼ C 0 Sd T eq ; βeq

ð3:1Þ

where coefficient C0 is the fundamental mode participation factor and Sd(Teq,βeq) is the maximum displacement of linearly-elastic SDF system with equivalent time period Teq and effective damping βeq given by; T eq ¼ T 0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ ; 1 þ αμ−α

βeq ¼ β0 þ κ

1 ðμ−1Þð1−α Þ : π μ ð1 þ μα−α Þ

ð3:2Þ

To use CSM, converting the pushover curve, which is in terms of base shear and roof displacement, into the capacity spectrum format is essential. Capacity spectrum is a representation of the pushover curve in an acceleration–displacement response spectra (ADRS) format. ATC 40 defines three types of hysteretic behaviors: Type A denotes hysteretic behavior with stable, reasonably complete hysteretic loops; Type C represents either severely pinched loops or degraded loops; and Type B denotes a hysteretic behavior intermediate between Types A and C. ATC 40 specifies three iterative procedures, procedures A, B, and C, to estimate the earthquake-induced deformation demand. These procedures are based on the same underlying principles but differ in the implementation. Procedures A and B are transparent and amenable to programming, whereas procedure C is a purely graphical method and most suitable for a hand analysis [1,10,25,26].The ATC 40 procedures provide a realistic method for PBSE, but the results of the evaluation are frequently inconsistent with those obtained from NRHA [1,9,30].

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123

Δ

V

Obtaining capacity spectrum Represents the lateral displacement as a function of force applied to the structure in ADRS format

Sa

Δ

Obtaining demand Spectrum For a given structure and ground motion, displacement demand is estimated for maximum expected response of Sa

Sd

Obtaining performance point Intersection point of demand spectrum and capacity spectrum

Performance point

Sd Fig. 2. Nonlinear static procedures [10,24].

 2 Te ; δet ¼ sa ðT e Þ 2π

ð3:3Þ

Spectral acceleration (Sa)

where, Sa(Te) is the elastic acceleration response spectrum at period Te.

Elastic demand spectrum

Performance point Reduced demand spectrum

For calculating the target displacement for short and medium to long ranges of time, the following expressions are given: For short period range (T ⁎ b Tc); F

When my ≥Sa ðT e Þ, the response is elastic, and the target displacement is calculated as δ⁎t = δet⁎; δt = c0δt⁎; otherwise, the response is inelastic, and the target displacement is calculated as δt ¼

  δet Tc ≥δet ; 1 þ ðqu −1Þ qu Te

T0 Teff

Elastic demand spectrum

Performance point

Capacity spectrum

Capacity spectrum Spectral displacement (Sd) Fig. 3. Determination of performance point as per ATC 40 [10,27].

ð3:4Þ

where qu is the ratio of the product of the acceleration of the structure with unlimited elastic capacity, and the modal mass m⁎ to its yield force qu = Sa(Te)m⁎/fy⁎. For medium and long period ranges (T⁎ ≥Tc); The target displacement of an inelastic system is equal to that of an elastic structure; thus, δt⁎ = δet⁎. The displacement of the MDF system is always calculated as δt =c0δt⁎. In FEMA 440 [31], Improvement in nonlinear static seismic analysis procedures, the CSM procedure was renewed with an aim to obtain more accurate seismic evaluation. An improved CSM has more favorable procedures to estimate the equivalent period and equivalent damping. These associations were based on an optimization process, in which the error between the displacements predicted using the equivalent

Spectral acceleration (Sa)

A more straightforward method for determining the seismic demand, using inelastic strength and displacement spectrum obtained directly by NRHA of inelastic SDF, was proposed by Fajfar (1999). The approach represents the so-called N2 method formulated in the CSM format. The N2 method combines the pushover analysis of the MDF system with the response spectrum analysis of an equivalent SDF system. The N2 method visually interprets the procedure and relationship between the basic quantities controlling the seismic response. In the N2 method, N stands for nonlinear analysis and 2 stands for two mathematical methods. The method has been included in the Euro code 8 (EC8:2004). The method in its present form in EC8:2004 consists of the following steps [28,29]; (a) performing pushover analysis and obtaining capacity curve in Vb–D format, (b) converting the pushover curve of the MDF system to the capacity curve of an equivalent SDF system, and approximating the capacity curve with an idealized elasto–perfectly plastic relationship to obtain the period Te of the equivalent SDF system, (c) the target displacement is then calculated as

Reduced demand spectrum

Spectral displacement (Sd) Fig. 4. Determination of performance point by improved CSM [27].

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Base shear

124

Initial stiffness Ki

Vy

Capacity curve Post-yield stiffness Ke

0.6Vy

Effective stiffness Ke

y

Roof displacement

t

3.2. Displacement coefficient method

Fig. 5. Calculation of target displacement as FEMA 273 [11].

linear isolator and using NRHA was minimized. The procedure involves the determination of the target displacement as [30–34]:
 δt ¼ C 0 Sd T eff ; βeff

ð3:5Þ

where coefficient C0 is the fundamental mode of participation and Sd(Teff,βeff) is the maximum displacement of linearly-elastic SDF system for vast variety of cyclic behavior, viz. bilinear, stiffness degrading and in-cycle strength degrading. In the improved CSM, both effective period (Teff) and effective damping (βeff) expressions are discontinuous at two distinct ductility values of μ = 4 and 6. The formats of Teff and βeff are respectively;

T eff

β eff

i 8h > 0:2ðμ−1Þ2 þ 0:038ðμ−1Þ3 þ 1 þ T 0 > > > < ½0:28 þ 0:13ðμ−1Þ þ 1 þ T "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0# ) ¼ ( > ðμ−1Þ > > > 0:89 −1 þ 1 T 0 : 1 þ 0:05ðμ−2Þ 8 4:9ðμ−1Þ2 þ 1:1ðμ−1Þ3 þ β0 > > > < 14:0 þ 0:32ðμ−1Þ þ β " # 02 ! ¼ 0:64ðμ−1Þ−1 T eff > > > þ β0 : 19 T0 0:64ðμ−1Þ2

1:0 b μ b 4:0; 4:0≤μ ≤6:5

improved CSM does not follow the secant stiffness as in case ATC 40 CSM. As part of improvements, a numerical transformation was included to generate a modified ADRS that represents the correct acceleration values. Improved CSM specifies three iterative procedures: procedure A (Direct iteration), procedure B (Intersection with modified ADRS), and procedure C (Modified ADRS locus of possible performance points). In procedure A, the iteration converges directly on a performance point. In procedure B, the performance point is defined as the intersection of the capacity spectrum with modified acceleration displacement response. Procedure C uses the modified ADRS for multiple solutions and the corresponding ductility to generate a locus of the performance point. Fig. 4 illustrates the procedure of determination of the performance point in improved CSM [31–34].

ð3:6Þ

FEMA 273, NEHRP guidelines for the seismic rehabilitation of existing building provides a simple method for estimating the target displacement δt. The target displacement refers to the displacement of a characteristic node on the top of a structure, typically on the roof, during a seismic event. It does not require converting the capacity curve into its corresponding spectral coordinates. The target displacement helps in estimating maximum inelastic deformation demands on a building, which is defined by δt ¼ C 0 C 1 C 2 C 3 Sa

μ N 6:5

ð3:7Þ

Te ¼ TL

sffiffiffiffiffiffi Ki ; Ke

ð3:9Þ

where Ki represents the initial stiffness and Ke is the effective stiffness of the building obtained by idealizing the pushover curve as a bilinear relationship. The bilinear curve has postelastic stiffness αKe, as shown in Fig. 5. A line segment in the idealized base shear–displacement curve is located using an iterative graphical procedure that approximately balances the area above and below the curve. This nonlinear relationship facilitates in calculating the effective lateral stiffness Ke and effective yield strength Vy of the structure.

Base Shear

because Teff and βeff are functions of μ that is unknown a priori from the idealization of pushover curves. The effective period of the

ð3:8Þ

The effective fundamental period of the building in the direction under consideration is computed by modifying the fundamental vibration period by using the elastic dynamic analysis as follows:

μN6:5

1:0 b μ b 4:0; 4:0 ≤ μ ≤ 6:5

T e2 g: 4π 2

Vd

p-Δ Ke 1Ke

Vy

e

Ke

0.6 Vy Ke

2Ke

Δy

Δd

Fig. 6. Idealized force–displacement curves in ASCE-41 [13].

Displacement

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Table 3 Summary of the available PBSE techniques [14–35]. PBSE techniques

Advantages

Disadvantages

ATC 40 (CSM)

1. It is a simple graphical procedure for estimating load–deformation characteristics for predicting earthquake damages and survivability of structures. 2. The performance point obtained facilitates in assessing the structural damages from different scales of ground motion. 3. Is based on the assumption that the response is controlled using the fundamental mode, and the mode shape remains unchanged after the structure yields.

N2 method

1. The method provides a visual interpretation of the capacity method with a sound physical basis of an inelastic demand spectra. 2. The lateral load distribution is directly related to the assumed displacement shape. 3. The bilinear idealization is defined in order for the areas under the actual and idealized force–deformation curves to be equal. 4. Graphical presentation can be formulated in the acceleration–displacement (AD) format 1. The effective linear parameters are functions of the initial period, damping, and ductility demand. 2. The method determines the equivalent linear parameters (i.e., effective period Teff and effective damping βeff) by using a statistical analysis that minimizes the extreme differences between the maximum response of an actual inelastic SDF system and its equivalent linear counterpart.

1. No physical principle justifies the existence of a stable relationship between the hysteretic energy dissipation of the maximum excursion and equivalent viscous damping, particularly for highly inelastic systems. 2. The procedure provides a relationship between ductility and damping by using different hysteretic behavior types. For nondegrading structures, such relationships underestimate the deformation demands, whereas for degrading structures, the relationship leads to overestimation. 3. The period associated with a performance point does not completely clarify the dynamic response of structures. 1. An equivalent displacement rule is a successful approach for structures on firm soil sites with a fundamental period in the medium or the long period range, whereas for soft soils, the approach has been found to be unsatisfactory. 2. The inelastic demand spectra used in this method is found to be inappropriate for near-fault ground motions, hysteretic behavior with significant pinching/stiffness/strength deterioration effects.

FEMA 440 (CSM)

FEMA 273 (DCM)

ASCE 41 (DCM)

1. It is a straight forward method of estimating the peak displacement of a nonlinear system. The procedure uses displacement modification factors, which are more accurate than an elastic spectra with equivalent viscous damping. 2. It does not require converting the capacity curve to spectral coordinates. 3. Two sets of lateral load distributions are used for the pushover analysis: The first set consists of vertical distribution proportional to elastic first mode shape; and the second set encompasses mass proportional uniform load distribution.

- lateral load patterns, - assumed displacement shape, and - MDF to SDF equivalence coefficients for top displacement and base shear. 1. The method has provided significant improvements in FEMA 356 DCM by 1. The influence of post-yield stiffness ratio (α), which reduces the modifying the coefficient used for estimating target displacements. deformation demands, is not addressed. 2. The method has introduced coefficients representing the influence of 2. The method fails to capture the variation in actual inelastic deformation different soil site classes, which disregards the overestimation of demands that are influenced by the seismological features of ground displacements. motion. 3. The method addresses the effects of strength and stiffness degradation. 4. The method has imposed a limitation on the lateral strength to avoid dynamic instability (P–Δ).

The modification factor C0 relates the elastic response of a SDF system to the elastic displacement of an MDF system at the control node, which is considered as the first mode participation factor. The coefficient C1 reflects the ratio of the peak displacements of the inelastic system to that of the corresponding elastic system with the same unyielding period of vibrations. The coefficient C1 is given by; 8 > T e ≥T < 1:0; 1:0 þ ðR−1ÞT s =T e ; Te b T : C1 ¼ > R : T e b 0:1T s 1:5;

ð3:10Þ

Ratio of elastic strength to yield strength is ðSa =g Þ

: R¼ V y C 0 =W

C3 ¼

: 1:0 þ

jα jðR−1Þ3=2 Te

The major drawback of FEMA 273 DCM is that such a simple product formulation may not reflect the three distinct failure effects of the actual nonlinear behavior of structure [9]. ASCE 41, Seismic rehabilitation of existing buildings has adopted an improved DCM procedure documented in FEMA 440. The improvement includes revisions of the coefficients used to calculate pseudo-lateral force and target displacement to reflect the current understanding of Assembling building performance model It includes definition of structural, nonstructural components and occupancy

ð3:11Þ

The coefficient C2 reveals a pinching effect in the load–displacement relation. The coefficient C3 considers dynamic P–Δ effects as 8 < 1:0

1. Both Teff and βeff expressions are discontinuous at two distinct ductility values (μ = 4 and 6.5); thus, they should be used for the values of μ less than 10–12. 2. The effective period expression does not follow the second stiffness approach. Therefore, graphical solutions become more complicated. 3. Numerical analysis techniques must be carried out for convergence in the results. 1. Iteration is required for the bilinear idealization. 2. The initial stiffness and yielding points depend on the target displacement. The resulting curve is bilinear with post-yield stiffness. The positive post-yield stiffness does not affect the target displacement. 3. The MDF to SDF transformation is theoretically inconsistent. 4. The procedure allows partial or no association between

α≥0 αb0

ð3:12Þ

where α is the ratio of post-yield stiffness to the effective elastic stiffness.

Analyze building Response using NRHA or NLSP

Define earthquake hazards Address the earthquake performance associated with ground shaking

Develop collapse fragility From structural and nonstructural fragility and damage functions.

Calculate performance Perform loss calculations. Use performance assessment computing tool (PACT). Fig. 7. Flowchart for performance assessment methodology [17].

126

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Table 4 Summary of all available DIs with their parameter values [1,4,5,35–45]. Damage Index A. Vibration response-based DIs 1. Local DIs Newmark and Rosenblueth (1971) 1.1.1.1. Banon H, Biggs J. M, Irvine H. M (1981) Park (1986) Powell and Allahabadi (1988) Lybas and Sozen (1977) Banon H, Biggs J. M, Irvine H. M (1981) Roufaiel and Meyer (1987)

Bracci J. M, Reinhorn, Mander J. B, Kunnath S.K (1989) 2. Cumulative DIs Banon and Veneziano (1982)

Description

Formulation

θm −θy θm ¼1þ (4.1) θy θy ϕm −ϕy ϕ (4.2) μ r ðϕÞ ¼ m ¼ 1 þ ϕy ϕy δm δm −δy DI is defined in terms of the ductility factor expressed as a ¼1þ (4.3) μ r ðδÞ ¼ function of characteristic member displacements (δ) δy δy μ m −1 DI is either based on displacement or ductility demand and (4.4) DI ¼ capacity under monotonic conditions. μ u −1 Ko DI is defined as the ratio of initial stiffness to maximum elastic DI ¼ (4.5) Km stiffness. M u ϕm Flexural damage ratio is defined in terms of stiffness DI ¼ (4.6) M m ϕu degradation. θy θm Modified flexural damage ratio is defined in terms of increment − in flexibility before and after a failure Mm My DI ¼ (4.7) θy θu − Mu My ϕ −M u =K m Based on final residual curvature φu DI ¼ m (4.8) ϕu −M u =K u DI is defined in terms of the ductility factor expressed as a function of rotation (θ) DI is defined in terms of the ductility factor expressed as a function of curvature (φ)

μ r ðθÞ ¼

n

Based on the normalized cumulative rotation DI ¼

Stephens and Yao (1987)

Based on the cumulative displacement ductility

∑ ϕim −ϕy i¼1 n

"

DI ¼ ∑ i¼1

Wang and Shah (1987)

Force-based DI defining strength decay

Jeang and Iwan (1988)

Force-based DI accounting the effects of combining cycles with various amplitudes Based on energy dissipated by a structure until its collapse

Gosain N. K, Brown R. H., Jirsa J. O. (1977) Hwang and Sonbner (1984) 3. Combined DIs Banon and Veneziano (1982)

Park and Ang (1985) Niu and Ren, (1996) Mehanny and Deierlien (2001)

Based on energy dissipated by the structure until its collapse, accounting change in stiffness DI is expressed as a linear combination of maximum displacement, failure displacements, and hysteretic energy dissipation DI is expressed as linear combination of maximum plastic displacement and plastic dissipated energy Similar to Park–Ang DI but formulated with different constants Based on ductility by considering cumulative effects and loading history of a structure

ϕu þ

Δd Δd f

(4.9) #1−br (4.10)

b = 0.77 (recommended) Fy DI ¼ 1− (4.11) Fm   n ni μ si DI ¼ ∑ (4.12) C i¼1   n F i di (4.13) DI ¼ ∑ i¼1 F y dy   2 n K i di (4.14) DI ¼ ∑ 2 i¼1 K e dy 0:38 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 u     !2 u dm 2Eh DI ¼ t þ (4.15) dy −1 F y dy

DI ¼

dm ∫dE þ βe (4.16) F y du du

θm E β þ αð Þ (4.17) Eu θu !β  α nþ þ þ ∑ Dþ ¼ θ j þ θ j p θ p FHC;i currentPHC

DI ¼

i¼1

α

nþ

ðθþ pu Þ þ

!β

; DI ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ γ − γ ðDþ θ Þ þ ðDθ Þ (4.18)

∑ θþ p jFHC;i i¼1

α = 1, β = 1.5, γ = 6 Columbo and Negro (2005)

Based on the ratio between the initial and the reduced resistance capacity of a structure

1 " ð1− θm Þ =β1  0:5 1− ; tanh β 2 ∫dE −π θu Eu ! DI ¼ 1− ∫dE  ; exp −β3 E

!! # (4.19)

u

H. J. Jaing, L. Z. Chen, and Q. Chen (2011) 4. Global DI Roufaiel and Mayer (1987) Di Pasquale and Cakmak (1988)

Eliminating the nonconvergence problem of existing Park–Ang model Strength-based global DIs Based on the ratio of the final and initial period of an equivalent SDF system of a structure Maximum softening DI Plastic DI Final softening DI

Park, Ang, Wen, (1985)

Hysteretic energy weighted average

α = 1, β1 = 0.1, β2 = 2.4, β3 = 1 γ = 0.80 μ ∫dE δm δu ;μ ¼ (4.20) DI ¼ ð1−βÞ m þ β ; μm ¼ f y dy ðμ u −1Þ μu δy u δy DI ¼ GDP

dm −dy (4.21) du −dy

Ta (4. 22) Tm Ta 2 DI ¼ 1−ð Þ (4.23) Tm Ta 2 DI ¼ 1−ð Þ (4.24) Td

DI ¼ 1−

M. Zameeruddin, K.K. Sangle / Structures 6 (2016) 119–133

127

Table 4 (continued) Damage Index

Description

Formulation N

Dstorey ¼

∑ Di E i i¼1 N

∑ Ei i¼1 N

(4.25)

∑

Dglobal ¼

storey;i¼1

Dstorey;i Estorey;i

N

∑

storey;i¼1

Bracci, (1989)

Gravity load weighted average

Estorey;i

N

Dstorey ¼

∑ W i Dbþ1 i i¼1 N

∑ W i Dbi i¼1 N

Dglobal ¼

(4.26)

∑

W storey;i Dbþ1 storey;i

∑

W storey;i Dbstorey;i

storey;i¼1 N storey;i¼1

B. Strength—parameter-based DIs Allemag R. J, and Brown D. L. (1982)

Lieven NAJ, and Ewins D. L. (1988)

2 Modal assurance criterion, is applied to correlate the two sets of jϕT ϕ j DI ¼ ðϕT ϕ utÞðϕdtT ϕ Þ (4.27) ut ut dt dt mode shape, that is, damaged and undamaged. The value ranges from 0 to 1. When two sets fit each other, the value is closer to 1, and value 0 implies no correlation. h i2 Coordinate modal assurance criterion, Examine the changes in DI¼ ∑ ðϕnj Þ ðϕnj Þ d u mode shape caused by damage is a better approach for n h i (4.28) combining data of various modes to obtain a single parameter ∑ ðϕnj Þ2d ∑ ðϕnj Þ2u

n

Pandey A. K, Biswas M., and Samman M. M. (1991)

Modal Flexibility DI, this method involves comparison of the flexibility matrices obtained from the two sets of mode shapes

n

N

∑ DI ¼ 1−

ϕ2ij

2 i¼1 ω j N

∑

ϕ2 ij

(4.29)

2 i¼1 ω j

Pandey A. K, and Biswas M. (1994)

Stiffness DI, this index represents a percentage reduction in storey stiffness before and after damage (expressed in terms of floor mass, modal frequency, and mode shape). The value 0 implies no damage, and 1 indicates collapse.

DI ¼

N m ϕ i ij ω2 ∑  j i¼1 Δϕij (4.30) 1− N mϕ i ij 2 ωj∑ i¼1 Δϕij N ϕ ij ω2 ∑  j i¼1 Δϕij (4.31) 1− N ϕ ij ω2j ∑ Δϕ i¼1 ij

Jer-Fu Wang, Chi-Chang Lin, and Shih-Min Yen Approximate stiffness DI, For most buildings, the floor mass (2007) distribution is generally uniform; thus, the approximate value of the Stiffness DI can be represented as

DI ¼

Ghobarah A., Abou-Elfath H., and Biddah A. (1999)

DI ¼ 1−

Stiffness-based DI, This index represents the change in the stiffness of a structure by performing pushover analysis on the structure twice: one before subjecting the structure to the earthquake, and one after subjecting the structure to ground motion

building behavior during earthquakes [31]. The pseudo-lateral force in a given horizontal direction for a building is determined using Eq. (3.13) and this load is used to design the vertical elements of lateral load resisting system [13]. V ¼ C 1 C 2 C m Sa W:

ð3:13Þ

The target displacement δt at each floor is calculated as [13];

δt ¼ C 0 C 1 C 2 Sa

Te2 g: 4π2

ð3:14Þ

Coefficient C0. C0 represents modification factor to relate spectral displacement of an equivalent SDF system to roof displacement of the building. Improved DCM modifies coefficients C1 and C2 to improve the expected elasto-plastic oscillator deformation estimates from their elastic counterparts (C1), and modify these estimations for the cyclic degradation (C2). This procedure eliminates coefficient C3, which in the former version accounts for P–Δ effects and establishes a limit on the lateral strength to avoid dynamic instability [34]. Coefficient C1.



K final K initial



(4.32)

C1 represents modification factor to relate expected maximum inelastic displacement calculated for linear elastic response. 8 1:0; > > > > < 1:0 þ R−1 ; C1 ¼ aT 2e > > > R−1 > : 1:0 þ ; 0:04a

T e N1:0s 0:2s b T e ≤1:0s

ð3:15Þ

T e ≤0:2s

where a represents the influence of different soil site classes. The various recommended values for a are as follows: 130 for soil site classes A and B, 90 for soil site class C, and 60 for soil site classes D, E, and F. R is strength ratio computed using Eqs. (3.17)–(3.18). In FEMA 356, the coefficient C1 had to be limited (capped) for short-period structures. The modified coefficient C1 disregards the influence of the post-yield stiffness on the peak displacements of a short-period structure [31]. Coefficient C2. Coefficient C2 accounts for changes in the peak SDF displacements, produced by departures from the elasto-plastic models, because of either severe stiffness degrading or severe strength degrading hysteretic behavior, which was neglected in FEMA 356. C2 ¼

8 < 1:0;

  1 R−1 2 : 1:0 þ 800 T e

T e N0:7s T e ≤0:7s

ð3:16Þ

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M. Zameeruddin, K.K. Sangle / Structures 6 (2016) 119–133

Table 5 Material properties considered in the design of example MRF [53,54]. Material property

Concrete M 25 grade

Steel Fe 415 grade

Weight per unit volume (kN/m3) Mass per unit volume (kN/m3) Modulus of elasticity (kN/m2) Characteristic strength (kN/m2) Minimum tensile strength (kN/m2) Expected yield strength (kN/m2) Expected tensile strength(kN/m2)

25 2.548 25E + 06 25,000 (for 28 days) − − −

76.97 7.849 2E + 08 415,000 (yield) 485,800 456,500 533,500

Cm represents effective mass factor to account for higher mode mass participation effects obtained from structural configuration, the value equals to 1, when the fundamental period is less than 1. Improved DCM excludes coefficient C3, but imposes a limitation on the lateral strength to avoid dynamic instability by imposing a maximum limit on R as follows; The strength ratio shall be calculated as; R¼

Sa Cm: V y =W

ð3:17Þ

For the buildings with negative post-yield stiffness the maximum strength Rmax shall be calculated as; R max ¼

Δd jα e j−h ; þ Δy y 4

h ¼ 1:0 þ 0:15 ln ðT e Þ;

ð3:18Þ

where Δd is the deformation corresponding to the peak strength, Δy is the yield deformation, and αe is the effective negative post-yield slope given by α e ¼ α P−Δ þ λðα 2 −α P−Δ Þ:

ð3:19Þ

The nonlinear force–displacement association between base shear and displacement of the control node is replaced by an idealized association to calculate the effective lateral stiffness Ke and yield strength Vy of the building, as illustrated in Fig. 6. The first line segment in an idealized force–displacement relation begins at the origin and has a slope equal to the lateral stiffness Ke. Ke is the secant stiffness measured at the base shear force equal to 60% of the effective yield strength. The second line segment represents a positive post-yield slope (α1Ke), determined by point (Vd, Δd) and a point at the intersection with the first line segment such that the area above and below the actual curve is approximately

balanced. The point (Vd, Δd) shall be the point on the actual force– displacement curve that lies either at the calculated target displacement or at the displacement corresponding to maximum base shear, whichever is less. The third line segment represents the negative post-yield slope (α2Ke), determined by the point at the end of the positive post yield slope (Vd, Δd) and the point at which base shear degrades to 60% of the yield strength. The negative slope ratio caused by P–Δ effects is given as αP-Δ; and λ is the near-field effect factor, given as 0.8 for S1 ≥ 0.6, and 0.2 for S1 ≤ 0.6 (S1 is defined as the 1-second spectral acceleration for the maximum-considered earthquake) [13,25,26,28–34]. All the aforementioned procedures are based on monotonically increasing predefined load patterns until some displacement is achieved. However, such a simplified procedure, based on invariant load patterns, is inadequate for predicting elastic seismic demands of buildings when modes higher than the first mode contribute to the response and inelastic effects alter the height-wise distribution of inertia forces. Many scientists have determined loading vectors, derived from mode shapes, to overcome the drawbacks of the earlier procedures. The procedures attempt to account for the higher mode's contribution, and use elastic modal combination rules when using invariant loading vectors. The modal pushover analysis, modified-modal pushover analysis, and upper bound pushover analysis procedures are examples of such an approach. The adaptive pushover analysis is another enhanced pushover method, in which load factors are progressively updated to consider the change in the system modal attributes during the inelastic phase. A more recently developed procedure is the adaptive modal combination, in which a set of adaptive mode shape-based inertia force patterns are applied to the structure. Although all the aforementioned procedures predict reasonable level of capacity, demands, and performance evaluation of the structure, they represent only the onset of collapse mechanisms and fail to quantify the associated damage to the structure [33]. A comparative summarization of all the aforementioned performance evaluation procedures is presented in Table 3. A literature survey reveals that the results of the improved CSM are closer to those obtained using NRHA. A higher mode contribution by using modified-modal pushover analysis provides storey-drift estimates that are generally much closer to the mean NRHA estimates. The upper bound pushover analysis procedure estimates lead to significant underestimation of storey–drift demands and member rotations at the lower level, and to overestimation at the upper storeys [26–35]. Thus, for the development of the next-generation PBSD procedure, a unique performance evaluation procedure must be developed to obtain accuracy in the predicted response and its conservation into acceptance criteria.

Fig. 8. Cross-sectional and rebar details, plan, and elevation of example MRF.

M. Zameeruddin, K.K. Sangle / Structures 6 (2016) 119–133

4. Performance assessment procedures The current methods for practicing performance evaluation are capacity-demand (ATC 40) and linear and nonlinear static or dynamic methods (FEMA 273, 356, and ASCE 41). These procedures are initiated by selecting design criteria in the form of one or more performance objectives. Performance objectives indicate the statement of acceptable damages to the structural and nonstructural components for specified levels of seismic hazard. These damages correspond to the failure caused by transient or permanent drifts. These procedures provide information related to collapse mechanism but fail to quantify associated damages. The next-generation PBSD demands a measure of buildings' performance in terms of potential for casualties, repairs or replacement, and downtime resulting from earthquake-induced damages. This can be obtained through seismic performance assessment, which requires relating a damage indicator (or damage index; DI) with discrete performance levels. In addition, performance objectives are to be defined as statements of acceptable risk of incurring different levels of damage, and the consequential losses that occur because of this damage. Fig. 7 illustrates the individual steps in the performance assessment process [17]. 4.1. Damage indices for reinforced concrete structures The concept of damage and damageability in a structural design is of high importance [36]. The response of structures to a seismic hazard can be expressed in terms of damages experienced by structural components or the structure as a whole, which are termed as seismic DIs [39]. Damages to an RC structure are generally related to the failure of its components due to the crushing of concrete. This phenomenon initiates because of the spalling of concrete cover due to shrinkage, hydration, and carbonation and of the crushing of the confined core (due to buckling, fracture of longitudinal bars, and loss of anchorage), which are not easy to define even under predominantly flexural conditions [36]. Thus, any damage variable defined should preferably refer to physically measurable parameters known as engineering demand parameters (EDPs). These EDPs include stress, strain, displacement, curvature, deformation, base shear, strength, stiffness, and dissipated energy. DI represents a phenomenon of damage involving various combinations of these EDPs [36]. Thus, a DI indicates “a quantity with zero value when no damage occurs, and a value of 1 (or 100%) when failure or collapse occurs, with intermediate values giving some measures of the degree of partial damage” [36–41]. The damage quantification can be broadly categorized into the following [35]: 1. Empirical DIs. 2. Analytical DIs. a. Strength-based DIs. b. Vibration response-based DIs.

Empirical and theoretical approaches have been used to yield various estimates of structural damage [42]. The empirical damage models are based on statistics of observed structural damage following seismic events. Empirical systems cannot predict the reserve strength and response characteristics of a structure for a specified degree of damage because i) the systems do not comply with the mechanics of materials regarding inelastic cyclic deformation; ii) future earthquakes may have different intensities, duration, and frequency content; iii) a present code modification according to the post-earthquake experiences may change damage statistics; and iv) statistical analysis comprises the shifted dynamic characteristics of structures due to repairs and damages resulting from past earthquakes. The analytical damage models may involve various degrees of complexities caused

129

Table 6 Summary of designed column section for example MRF. Storey

External column

All floors

Interior column

size (mm)

Rebar's (mm2)

size (mm)

Rebar's (mm2)

380 × 380

1155

450 × 450

1620

by structure characteristics and the seismic response. Analytical damage models are broadly divided into structural parameter and vibration response-based DIs. The structural parameter-based DIs depend on the geometry of structural elements, such as column and wall area and their general material properties. In the absence of the field observation of damaged structures because of seismic loads, calibration is performed using nonlinear dynamic analysis. Vibration response-based indices use the structural response measurements for a single excitation event, and calculate damage-related physical factors, such as peak acceleration, peak velocity, and energy [43]. An updated review of all available DIs in related literature is presented in Table 4. The first damage model was developed on the basis of the ductility concept. Ductility-based damage models were expressed as functions of member rotation, curvature, and characteristic displacement, as defined in Eqs. (4.1)–(4.4). Because of easy quantification, these indices are used to assess the performance but fail to account for the effect of strength and stiffness degradation under the cyclic loads. The need for structural safety under cyclic loading against plastic incursions leads to the development of DIs expressed in terms of kinematic or cyclic ductility as measures of collapse (Eqs. (4.5)–(4.8)). The kinematic or cyclic ductility DI face difficulties related to the difference between the characteristics of expected earthquake and earthquake used in their calibration (such as intensity, duration, and frequency content), but are strongly supported because of their prediction characteristic that is typically similar to that of the ductility DIs [36–39]. During cyclic loading, energy dissipates in the structure accounting only for ductility, which is not an accurate measure of damage. This concept results in the development of cumulative DIs, which are expressed as functions of plastic deformation and absorbed hysteretic energy (Eqs. (4.9)–(4.14)). Cumulative DIs have proven to be simple measures for structural degradation during seismic events but were found dependent on the duration and intensity of an earthquake, and they failed to represent the complex behavior of concrete [36–39]. The concept of assessing damage according to the combined effects of strength, ductility, and energy dissipation leads to the development of combined DIs, as defined in Eqs. (4.14)–(4.20). Among all the combined DIs, the Park–Ang DI is widely supported by scientists as it was found to be consistent with the observed damage statistics for both concrete and steel structures. DIs defined in Eqs. (4.21)–(4.27) are used to measure damages to the entire structure and its characteristics. They inform about the global damage expressed as a function of the distribution and severity of local damage [33–39]. DIs defined in Eqs. (4.27)–(4.31) are based on the modal frequency, mode shape, or both. The damage is always accompanied by reduction in stiffness and modal frequency; however, determining the damage location only by observing the changes of modal frequencies is extremely Table 7 Summary of designed beam section for example MRF. Storey

Dimension (mm)

Rebar's (mm2) Bay 1

1st 2nd 3rd 4th

300 × 300

Bay 2

Bay 3

Top

Bot.

Top

Top

Bot.

Top

Top

Bot.

Top

600 670 552 330

261 261 261 261

604 651 519 360

605 668 545 372

261 261 261 261

605 668 545 372

604 651 519 360

261 261 261 261

600 670 552 330

M. Zameeruddin, K.K. Sangle / Structures 6 (2016) 119–133

V/W

130

Table 8 Results of modal analysis and lateral load profile for the example MRF.

Storey level

Storey height (m)

Mass × 103 (kgs)

Lateral distribution of base shear (V) (kN)

0.18 0.16 0.14

Loading diagram

0.12 0.1 0.08

12

28.63

36.00

3rd floor

9

28.63

24.35

2nd floor

6

28.63

10.82

1st floor

3

28.63

2.70

Roof

0.06 0.04 0.02 0 0

Difficult. DIs, which account for changes in mode shape, were used but were found to have low sensitivity to damage. Later DIs, which considered both modal frequencies and mode shapes, to detect the occurrence and location of damage were proposed. These DIs involved a tedious process of evaluating flexibility or stiffness matrices for every incremental time step. Next, a simple method of comparing the changes in stiffness before and after an earthquake was suggested (Eq. (4.32)), which was easy in quantifying damages to RC structures [43–45]. 5. Example building

0.01

M/My

Rotation/ SF

CP LS D

E

A

A B C D E

0.025

rotation capacity, labeled immediate occupancy (IO), life safety (LS), and collapse prevention (CP) are shown in Fig. 9. Fig. 10 shows the pushover curve obtained by performance evaluation of example MRF. Nonlinear responses determined at the performance point for all PBSE procedures are tabulated in Tables 9 and 10. The results obtained from pushover analysis for FEMA 440 (Improved CSM), show a variation of 3.30% in displacement with respect to ATC 40 (CSM); whereas, ASCE 41 (Improved DCM) shows a variation of 13.60% in displacement with respect to the FEMA 356 (DCM). A comparison between next-generation procedures, that is, ASCE 41 (Improved DCM) and FEMA 440 (Improved CSM) shows variations of 42.48% in displacements. Whereas; a comparison between ATC 40 (CSM) and FEMA 356 (DCM) shows variation of 48.60% in displacement (Table 9 and Fig. 11). A time-history analysis is known as the most accurate and reliable analysis method. NRHA was employed for the example MRF by using EL-Centro (1940) and Northridge (1994), with PGA 0.319 and 0.968, respectively. Data was obtained from the PEER page (http://peer.berkeley. edu). Obtained maximum displacements are 0.089 m for EL-Centro (1940) and 0.098 m for Northridge (1994) earthquakes. When the time-history results are used as reference points for comparison of responses obtained to study the PBSE procedure, it may be concluded that next-generation procedure results are consistent compared to those of the first and second generation procedures. Hence, for a practicing engineer, selecting a reliable performance evaluation procedure is tedious. At this stage, it is clear that for strong conclusive statements, an extensive analysis is required. The present study attempts to highlight the possible gray areas for future research. H and L, represent higher and lower percentages respectively. The performance evaluation procedures in PBSD are capable of performing nonlinear static analysis, and are widely used in research. However, they have not found wide use in damage prediction. In addition, such procedures may need to be modified to include structure response quantities, needed to compute damage parameters, in their

Acceptance criteria

IO

0.02

Fig. 10. Pushover curve for example MRF.

C B

0.015

Δ/H

Force

The response and damage states of moment resisting reinforced concrete frame (MRF) of an example medium-rise building were evaluated. The example MRF has 3-bay and 4-stories representing a mediumrise building. The MRF was subjected to displacement-controlled nonlinear static analysis (pushover). The response of the frame was evaluated according to the guidelines mentioned in the first, second, and next generation PBSE procedures. SAP 2000V 17.0 was used for analytical modeling of the example structures [57]. The MRF has a bay width of 3 m and the height of each storey is 3 m. The MRF was designed as per the guidelines mentioned in IS 456:2000 (rev) [53], IS 1893:2002 (part 1) [55], and IS 13920:1996 [56]. The frame was subjected to lateral loads, which were obtained by following the guidelines of IS 1893:2002 (part 1) [54], for seismic zone V (zone factor, z = 0.36) and importance factor 1, located on the hard soil strata. The various material properties, which were used for designing structural members, are presented in Table 5. Fig. 8 describes plan, and elevation of the example MRF. Tables 6 and 7 show the reinforcement details of reinforced concrete sections. For the present example MRF, a dead load of 16 kN/m and live load of 9 kN/m for all floors were assigned. The MRF was subjected to IS 1893 specified lateral load pattern to obtain a pushover curve by using SAP 2000 V 17.0 [50], a finite element-based computing tool. The details of the modal analysis and lateral load profile are presented in Table 8. To develop a moment–rotation curve of a default plastic hinge, a stress– strain relationship corresponding to FEMA 356-integrated in software was used. Beams and column elements were modeled as nonlinear frame elements by assigning concentrated M3 and P-M3 plastic hinges respectively, at both ends. The acceptance criteria for the ultimate

0.005

0 1 1.1 0.20 0.20

0 0 0.015 0.015 0.015

0 0 0.025 0.025 0.05

Plastic deformation /SF 3E-03 0.01 3E-03 0.01 0.012 0.02 0.015 0.025 0.015 0.025

Deformation Fig. 9. Force–deformation relationship of a typical plastic hinge.

Displacement

M. Zameeruddin, K.K. Sangle / Structures 6 (2016) 119–133 Table 9 Comparison of first, second, next-generation PBSD procedures and NRHA. Sr. no.

Analysis procedures

Displacement (m)

% variation with NRHA-1

% variation with NRHA-II

1 2 3 4 5 6

ATC 40 (CSM) FEMA 440 (CSM) FEMA 356 (DCM) ASCE 41 (DCM) NRHA- I (EL-Centro) NRHA-II (Northridge)

0.088 0.091 0.153 0.177 0.089 0.098

1.55 (L) 1.77 (H) 41.58 (H) 49.50 (H)

10.20 (H) 7.14 (H) 35.95 (H) 44.63 (H)

0.2 0.1

0.177

0.153 0.091

0.088

0.089

0.098

0 ATC 40 (CSM)

FEMA 440 FEMA 356 (CSM) (DCM)

ASCE 41 (DCM)

NRHA -I NRHA-II (EL-Centro) (Northridge)

Fig. 11. Comparison of PBSE and NRHA procedures.

output. To correlate damage variables and performance levels for present procedures, a DI is introduced, which accounts for the change in stiffness at every pushover step, namely DIs (static). DIs, is expressed as a simple expression; DI s ¼ 1−

131

Kj ; K 0p

ð5:1Þ

where Kop represents the stiffness at an operational level, that is, the formation of a first hinge or at the first crack and Kj is the stiffness at the considered performance level. DIs represents a possible extension of the DI by Ghorbah et al. (1999) for evaluating the damage value at different performance levels once the capacity curve is obtained. The other DI presented in Table 4 requires knowledge of nonlinear dynamic analyses, and hence has not been evaluated for the present example MRF. The various values of DIs were computed corresponding to the formation of hinges at various performance levels. Table 10 illustrates the computation of DIs with reference to hinge mechanism, accounting the loss in stiffness for every pushover step in NLSP. In this study, we compared the current state-of-practice in PBSE for reinforced concrete structures. In contrast to various other publications, in which investigations were carried out to study pros and cons of NLSPs for one or more structural design, we implemented all PBSE for the FBD structure, and introduced a DI from the nonlinear responses obtained in their output. Relevant literature contains two basic procedures, which can be used to compute structural DI. The first procedure involves demand versus capacity, and the second procedure is based on degradation. The demand versus capacity procedure is based on the estimation of some demand on the structure, substructure or member, and corresponding capacity (supply). The degradation procedure is based on the estimation of a property for a structure, substructure, or member in its undamaged state and a corresponding estimation in its damaged state. The demand versus capacity procedure is best suited for the NLSP procedure, in which computed demand will increase monotonically as the structure is damaged and the damage parameter can directly equal this demand. The change in stiffness at every pushover step was used to represent the damage value, and integrated with various performance levels defined in PBSD. 6. Conclusions PBSD is based on structural analysis and is a revolutionary development in seismic resistance design theory. PBSD addresses multiple

performance objectives and the associated earthquake hazard levels. With the development of a soft computing tool, modeling a nonlinear behavior of structural elements has become easier. Current PBSD procedures have recommended NRHA and NLSP for seismic performance evaluation. The simple computational efforts involved in NLSP have made it favorable among the engineers engaged in the design of earthquake resistant structures. The performance levels defined in first generation procedures (SEAOC vision 2000, ATC 40, and FEMA 273) and second generation procedures (ASCE 41) use an interstorey drift ratio and plastic rotations to establish building performance levels such as; immediate occupancy (IO), life-safety (LS) and collapse prevention (CP). While these measures provide information's on the deformation of elements and displacement profile at critical states, they are inadequate in themselves to provide an assessment of the state of damage or proximity to collapse. The nextgeneration PBSD (FEMA 445) procedure conveys the need to express the performance objectives regarding the primary concerns of stakeholders (viz., repair cost, casualties, and downtime). Therefore, DIs must be integrated within the performance evaluation procedure. The current evaluation procedures (CSM and DCM) are unique and simple in evaluating the performance. However, a performance evaluation of the present example MRF by using these procedures demonstrates differences in the results when available procedures are compared to each other. Hence, efforts are required to determine the accuracy in predicting the actual building response, and conservatism in the acceptance criteria. The collapse mechanism is clearly represented in the formation of plastic hinges; however, the associated damage to the structure is not quantified. Many scientists have proposed a DI by using EDPs, which may include stress, strain, drift, strength, change in stiffness, and hysteretic energy. Until recently, these damage variables were not included in performance evaluation procedures individually or in a group. All such performance evaluation procedures may need to be modified to include the structural response quantities required to compute damage parameters in their output. If these damage variables are integrated with the performance evaluation procedure, performance evaluation and damage assessment can be performed in a single analysis, thus facilitating design procedures and saving time. As a preliminary attempt, a corelationship was proposed between the performance levels and the associated DIs. DIs is based on the changes in stiffness associated with the formation of hinges, and accounts for changes in stiffness referring to the transfer hinge from one state of a performance level to another. The overall effort of this study was to provide an updated review of developments in PBSD.

Table 10 Computation of DIs with reference to hinge mechanism. Step

Displ. (m)

Base force (kN)

Different performance levels A to B

B to IO

IO to LS

LS to CP

CP to C

C to D

D to E

NE

1 19 44 56

0.0035 0.096 0.216 0.273

37.79 211.18 210.23 209.7

55 28 28 28

1 26 0 0

0 2 26 6

0 0 2 20

0 0 0 0

0 0 0 2

0 0 0 0

0 0 0 0

Stiffness kN/m

DIS

Remarks

10,714.77 2191.49 971.67 767.44

0 0.79 0.90 0.92

Appearance of first crack First hinge formation in IO-LS First hinge formation in LS-CP First hinge formation in C-D

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M. Zameeruddin, K.K. Sangle / Structures 6 (2016) 119–133

Abbreviations

Conflict of interest

ASCE ATC ADRS CSM DCM DBSD EDPs FBD FEMA PBSD PBSE MRF SEAOC SDF MDF NRHA NLSP

The authors of this paper hereby declare that no conflict of interest related to scientific research, direct or indirect financial relationships, employment, and personal beliefs exists.

American Society of Civil Engineers Applied Technological Council acceleration demand response spectrum capacity spectrum method displacement coefficient method displacement-bases seismic design engineering demand parameters force-based design Federal Emergency Management Agency performance-based seismic design performance based seismic evaluation Moment resisting frame Structural Engineers Association of California single degree of freedom multi-degree of freedom nonlinear response history analysis nonlinear static pushover analysis

Notations b C Di dm, dy dE Ei, Eh

counts severity of the damage elements constant value local damage index at location i maximum displacement and yield displacement, respectively incremental dissipated hysteric energy dissipated energy and dissipated hysteretic energy, respectively maximum force during previous cycle, and failure force durFm, Fy ing loading cycle, respectively g acceleration due to gravity Ko, Km, Ku, Ke initial, maximum, ultimate, and elastic bending stiffness's, respectively elastic stiffness of the building Ki ultimate bending moment resulting from pushover analysis Mu number of hysteretic cycles and number of cycles with inelasn, ni tic deformation, respectively R ratio of elastic and yield strengths response spectrum acceleration at effective fundamental Sa period and damping ratio of the building under consideration Ta, Tm, Td natural period at initial stage, maximum softening, and final softening, respectively initial period of vibration of a nonlinear system T0 characteristic period of ground motion Tc total shear force VT W total building weight α post-yield stiffness ratio parameter representing the cyclic loading βe μ maximum displacement ductility ratio κ adjustment factor for approximate account of changes in hysteretic behavior of reinforced concrete structure. ζ equivalent ductility ratios maximum and ultimate curvatures, respectively ϕm, ϕu ductility under monotonic loading, and ductility attained durμu, μm ing seismic response, respectively. roof displacement δmax incremental increase of positive displacements Δd+ incremental decrease of negative displacements Δd− recommended 10% of floor height Δdf value of Δd+ for a cyclic load that leads to failure Δddf elastic stiffness of the building Ki effective stiffness of the building obtained by idealizing the Ke pushover curve as a bilinear relationship.

Acknowledgments The authors of this paper acknowledge the research contributions of all the citations under reference and the support by faculty members of Veermata Jijabai Technological Institute, Mumbai, India and by the Chairman, MGM College of Engineering, Nanded, India.

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