THE USE OF DAMAGE FUNCTION IN PERFORMANCED-BASED SEISMIC DESIGN OF STRUCTURES

Transcription

1 3 th Word Conference on Earthquae Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 357 THE USE OF DAMAGE FUNCTION IN PERFORMANCED-BASED SEISMIC DESIGN OF STRUCTURES Chin-Hsiung Loh and Shu-Hsien Chao SUMMARY A design phiosoph, based on the energ concept and ied point spectra method, is deveoped for the performance based seismic design. A phsica based anatica mode capabe of describing the effects of pinching, stiffness degradation and strength degradation is used to construct the seismic design parameters and the seismic demand. Seismic design parameters, which were proposed to derive the seismic demand, wi be constructed for different hsteretic oops b using 75 earthquae records. Incorporate with the deveoped Yied Point Spectra and the proposed equivaent ductiit ratio the acceptabe design region of the structures can be determined for considering both safet and serviceabiit in design and rehabiitation. INTRODUCTION Performance based seismic design procedures and codes have been recommended to substitute traditiona design procedures and design codes. Differing from traditiona design codes, the characteristics and the advantages of the performance-based designs are isted beow.. Muti design performance eves. Damage reated design Index In order to perform it, the structura noninear behavior shoud be reaized to contro the structura behavior in different earthquae excitations. A arge amount of studies have been discussed about the structura noninear behavior. In genera, both the maximum structura dispacement and the structura Damage Index have been recommended to serve as the design indices in the performance based seismic design. Methods used to determine the sstem maximum dispacement have been discussed b severa researchers (Miranda, Kowas and Chopra ). The uses of the Damage Index in the performance based seismic design have aso been discussed (Bertero 996, ) based on the energ concept. Some parameters, such as the ied strength reduction factor and the dispacement modification factor etc., have been proposed and constructed to derive the information of the structura noninear behavior. In this stud, important seismic design parameters that can be used to contro and reaize the structura noninear behavior are seected and studied. In the past the reationship between the seismic design parameters and the structura hsteretic oops has not been discussed adaptive because the ac Professor, Department of Civi Engineering Nationa Taiwan Universit, Taipei, Taiwan Graduate Student, Department of Civi Engineering Nationa Taiwan Universit, Taiwan

2 of the anatica hsteretic mode, which can invove the stiffness, strength degradation effects and pinching effects. In this stud, an anatica noninear SDOF mode which can consider these effects is used to derive the seismic design parameters. The reationship of these parameters wi be discussed and suggested to revise the seismic code. In this stud the Yied Point Spectra Method is seected to determine the performance point of structures. B combining the seismic design parameters and the design phiosoph based on the Yied Point Spectra Method, a compete performance based seismic design phiosoph wi be constructed. The objectives of this stud are: a. Construct the seismic design parameters, which can be used to construct the seismic demand for the ineastic SDOF sstem. The infuence of different hsteretic oops on the seismic design parameters wi be discussed. b. Construct a design phiosoph for performance-based seismic design based on energ concepts and ied point spectrum, which can be used easi, convenient and quic for structura design and rehabiitation. SEISMIC DESIGN PARAMETERS For performance based seismic design, it is important to contro the structura behavior in the ineastic state during earthquae ground motions. In order to attain this goa, it is suggested to use seismic design parameters to reaize and contro the structura behavior in the ineastic state. These incude:. Dispacement Modification Factor,. Yied Strength Reduction Factor, 3. Seismic Energ Response Parameter, 4. Damage Index, 5. Equivaent Ductiit Ratio, These seismic design parameters can hep us to construct the seismic demand spectra for the ineastic sstems, and to reaize and contro the structura behavior in the ineastic state. A tota of 75 strong ground motion data were coected from hard site condition in Taiwan to generate the above-mentioned seismic design parameters. The predominant period of each ground motion is identified which wi be used as a normaized factor for period in deveoping the seismic demand spectra. Next, the noninear anatica SDOF mode, which can represent the degradation and pinching effects whie structures suffer strong ground motion, wi be introduced. Noninear Anatica SDOF Mode In this stud, the noninear SDOF hsteretic mode proposed b Mostaghe (999) was used for anasis. This mode can be impemented to predict the pinching and degrading hsteretic behavior of a structure. In this mode, the noninear contro parameter λ p (ranges from to ) contros the eve of pinching effects. If λ p the mode wi be equa to the biinear sstem which indicates no pinching effects. The noninear contro parameters λ and λ contro the veocit of stiffness and strength degradation. In summar, the noninear mode used in this stud has four noninear contro parameters to consider the pinching effects and the degradation of a sstem:. Post-ieding stiffness ratio α ( α, and when α indicates an eastic sstem),. Stiffness degradation contro parameter λ ( λ, and when λ indicates no degradation), 3. Strength degradation contro parameter λ ( λ, and when λ indicates no degradation), 4. Pinching contro parameter λ p ( λ p, and when λ p indicates no pinching effect),

3 The strength degradation and stiffness degradation of the sstem are reated to the sstem absorbed hsteretic energ: Φ Φ, i+ i+ + ( ) E H, i+ u ( E u ) λ (a), i+, i+ u + u λ (b) H, i+ where i : ith time step, Φ : Stiffness degrading eve, stiffness, u : Initia sstem ied dispacement. If the vaue of degradation is faster. Φ : Strength degrading eve, : Initia sstem λ and λ are arger, the veocit of Combining these four noninear contro parameters adaptive, the hsteretic oop for different inds of noninear behavior of the reaistic structures can be modeed. B comparing the resuts of the noninear anatica mode and the experimenta resuts (ccic oading test of /-scae mode in NCREE Structura Lab.), both the hsteretic behavior of the fexure faiure mode (BMR Mode) and the infi wa-frame mode (W_F mode) can be generated b using the noninear anatica mode, as shown in Figure a and Figure b. Based on the proposed ineastic mode the seismic demands can be studied in a more reaistic wa. In this stud, the Biinear mode ( α ), the BMR mode and the W_F mode wi be used to constructed the seismic design parameters. The noninear contro parameters for different hsteretic mode are shown in Tabe. Based on the proposed ineastic mode the foowing three seismic demand factors are evauated: a. Dispacement Modification Factor C: The definition of the Dispacement Modification Factor is defined as the ratio between ineastic sstem maximum dispacement for specific ductiit eve (Sd ineastic ) and eastic sstem maximum dispacement (Sd eastic ). Through numerica stud it is found that the vaue of the Dispacement Modification Factor has highest vaue for the W_F mode because of the severe stiffness and strength degradation of the sstem mode. b. Yied Strength Reduction Factor Ru: The Yied Strength Reduction Factor for different hsteretic modes and sstem ductiit is studied. It is found that the vaue of the Yied Strength Reduction Factor has highest vaue for Biinear mode because of the higher vaue of the post ieding stiffness ratio. This factor has owest vaue for W_F mode because of the severe stiffness and strength degradation of the sstem mode. c. Seismic Energ Response Parameter γ: Fajfar (99) proposed using the parameter γ to consider the hsteretic energ absorbed b a sstem during earthquae ground motion excitations. The parameter γ is defined as: γ E / m H () ωsdineastic where E H : hsteretic energ absorbed b the sstem for a specific ductiit eve; m: sstem mass; ω: sstem natura frequenc of vibration; Sd ineastic : sstem maximum dispacement for a specific ductiit eve. Researches indicate that the damage caused b earthquae excitation is reated not on to a sstem maximum dispacement but aso to the ccic oad reversas resuting in ow cce fatigue. Low cce fatigue occurs when a sstem absorbs arge hsteretic energ during earthquae ground motion. To avoid ow cce fatigue the hsteretic energ absorbed b the structura sstem needs to be considered quantitative during earthquae excitation, and the parameter γ can refects the structura hsteretic energ quantitative. Figure a and Figure b show the parameter γ for different hsteretic mode and for sstem ductiit equa to and 6, respective. The vaue of the parameter γ cacuated b the Biinear mode has the highest vaue and cacuated b W_F mode has the owest vaue. The reationship between the sstem ductiit, period and the parameter γ is depend on the seected ineastic mode. Fortunate the vaue of the parameter γ is not wide ranging. From the resut of this anasis resuts (sstem ductiit equa

4 to to 6) the upper and ower imits of the parameter γ are shown in Tabe. One can seect a higher vaue of γ to mae design conservative. Damage Index Damage Index has been proposed and used to identif the damage condition of the members and the structures. It was found that the damage of member and structure is reated to not on on the maximum dispacement of the members and the structures but aso on the absorbed hsteretic energ. As described above the sstem absorbed hsteretic energ, E, can be cacuated b using the predefined parameter γ: H H mγ ω Sdineastic γ u (3) E Combining this equation with the Damage Index proposed b Par and Ang (985), DI pa, the damage index can be rewritten as: γ DI pa + b (4) The vaue of b depended on the structura noninear characteristics. Different from Par & Ang damage mode, Bozorgnia and Betero () introduced two improved Damage Index equations as shown beow DI bb, [( α )( e ) ( ) ] + α ( E H E H ) (5) DI [( )( ) ( )] ( ). 5 bb, α e + α E H E H where u e u eastic /u maximum eastic portion of deformation / u and u e is for ineastic behavior, and u e is for the response remains eastic, α and α are constant depended on the stabiit of hsteretic behavior (simiar to the vaue b in Par-Ang damage mode). E H is the hsteretic energ capacit under monotonica increasing atera deformation. For the specia case of the EPP sstems the damage index can be expressed as: DI α + α E F u DI bb, bb, [( )( e ) ( )] ( H ) ( ) [( α)( e ) ( ) ] + αγ ( ) ( α )( e ) ( ) + α E H Fu [( α )( ) ( ) ] + α γ ( ) [ ] [( ) ( )] e Few characteristics of this improved Damage Index are isted beow: a. If the response remains eastic, both DI bb, and DI bb, wi equa to zero. b. Under monotonic atera deformation if the maximum deformation capacit (d ) is reached, then both DI bb, and DI bb, wi equa to one. c. If α and α the Damage Index is assumed to be on reated to the maximum pastic deformation. d. If α and α the Damage Index is assumed to be on reated to the hsteretic energ absorbed b the sstems. This Damage Index proposed b Bozorgnia and Bertero can caibrate some drawbacs of the Damage Index proposed b Par and Ang. In this mode it is necessar to define the vaue of α and α for different hsteretic mode. This ind of stud is beond our scope. The Damage Index proposed b Bracci et a. (993) is aso introduced. The definition of the Damage Index proposed b Bracci et a. is given b DI b D + D D D.5 (6)

5 φα M α / α M D and D (7) ϕu M u / u M where Φ α current eve of curvature (dispacement) Φ u utimate curvature (dispacement) for monotonic oading M α current eve of moment (force) M u moment (force) at utimate monotonica deformation α unoading stiffness at current eve u unoading stiffness at utimate monotonica deformation M strength oss during ccic oading The advantage of the Damage Index proposed b Bracci et a. is that it considered the characteristics of the sstem hsteretic oop such as the strength degrading and the stiffness degrading effects direct in the form of the Damage Index. B combining the anatica noninear mode and the Damage Index proposed b Bracci et a., the Damage Index can be rewritten as beow (assume the vaue of the post ied stiffness ratio equa to zero) DI b D + D D D D D M M ( Φ ) ( Φ ) F, ( Φ ) F λγ + λ γ ( + λγ ) ( Φ ), where Φ : Stiffness degrading eve Φ, : Stiffness degrading eve for the utimate monotonic oading Φ : Strength degrading eve This Damage Index aso can caibrate the drawbacs of the Damage Index proposed b Par and Ang. Such as for the case of the eastic response, the vaue of the DI b wi be equa or smaer than (D constant and D ), and for the case of under monotonic atera deformation if the maximum deformation capacit (d ) is reached the vaue of the DI b wi be equa to (D and D ). (8) Based on the Damage Index DI b proposed b Bracci et a., the Damage Index spectra b using the BMR mode and the W_F mode can be generated (the effects of the stiffness degradation was negected, i.e. Φ Φ, ), with different eves of strength degrading, for utimate monotonica ductiit equa to 6, as shown in Figures 3a and 3b. It is found that the Damage Index proposed b Bracci et a. is not sensitive to the normaized structura period and is sensitive to the sstem ductiit. Besides, the vaue of the Damage Index is arger b using the W_F mode than using the BMR mode, because of the severe strength degrading in W_F mode. It shoud be noted the vaue of the Damage Index wi equa. when a sstem ductiit is equa or arger than the utimate ductiit obtained from monotonic oading. Equivaent Ductiit The concept of the Equivaent Ductiit proposed b Fajfar (99) can aso be used to cacuate the constraint of sstem ductiit for seismic design. From the definition of the Damage Index and the Damage Index Spectra, it is found that the capacit of the sstem ductiit can not reach the sstem utimate monotonica ductiit during the earthquae ground motions excitation. From this point of view

6 one can cacuate the constraint of the design sstem ductiit, which is caed the design Equivaent Ductiit, for a specific vaue of the sstem utimate monotonic ductiit b using the definition of the Damage Index during the earthquae ground motion. Based on this concept if the design sstem ductiit is ower than the design Equivaent Ductiit, the Damage Index of the structures during the earthquae ground motion excitation wi not exceed the constraint of the previous defined Damage Index. Before cacuating the design Equivaent Ductiit from the definition of Damage Index an important assumption must be made on parameter γ in advance. It is observed that the parameter γ is independent of the sstem ductiit and the structura period. This assumption is rationa because the vaue of the parameter γ is not wide ranging as shown in Tabe. If the vaue of the upper imit of the parameter γ is assumed to cacuate the design Equivaent Ductiit and is used to design, the ower vaue of the design Equivaent Ductiit wi be derived. A designer can use this equivaent ductiit to mae a conservative design. The design Equivaent Ductiit can be derived using different damage index mode and was shown beow a. Par & Ang Damage Mode: + 4DI pabγ eq, pa (9a) bγ b. Bozorgnia and Bertero Damage Mode (a specia case for EPP mode): [ ] ( α ) + ( α ) + 4α γ ( α ) + DI ( ) eq, bb α γ DI bb, ( ) + ( α ) e eq, bb (9b) [ + α ( γ ) ] c. Bracci er a. Damage Mode: e bb, ( DI b )( ) [ + DI b ( ) ] ( DI )( ) λ γ + + 4λγ eq, b (9c) b The Equivaent Ductiit cacuated from the definition of the Damage Index proposed b Bracci et a did not consider the effects of the stiffness degrading (set Φ and Φ, equa to ). Based on the definition of the design Equivaent Ductiit it is found that if the vaue of the sstem utimate ductiit from monotonic oading and the vaue of the parameter γ were pre-estimated, the vaue of the design Equivaent Ductiit can be derived. The vaue of b, α, α and λ are depend on the hsteretic oop of the sstem. Of course, for different hsteretic oops different vaues of the parameter γ, and different vaues of b, α, α and λ need to be determined to cacuate the design Equivaent Ductiit. B using the Damage Index proposed b Bracci et a the Equivaent Ductiit using different hsteretic mode, such as the BMR mode and the W_F mode which have a different eve of the strength degrading, can aso be obtained as shown in Tabe 3 for service eve (assume DI b.4) and Tabe 4 for ife safet eve (assume DI b.8). The Equivaent Ductiit for the BMR mode and the W_F mode can be potted with respect to the parameter γ for different sstem utimate monotonica ductiit, as shown in Figure 4a on service eve (DI b.4) and Figure 4b on the ife safet eve (DI b.8). From Figure 4 it is found that for the same vaue of the parameter γ, the design Equivaent Ductiit is smaer b using the W_F mode than using the BMR mode because in the W_F hsteretic mode more severe strength degradation was observed. It is a rationa resut because if the structures absorbed the same vaue of hsteretic energ, the W_F mode wi have arger damage than the BMR mode because of the oss of severe strength in the W_F mode. From Figure 4a it is found that the vaue of the Equivaent Ductiit

7 for W_F mode in service eve (DI b.4) is not sensitive to the vaue of the sstem utimate ductiit from monotonic oading. It means that no matter what the vaue of the sstem utimate ductiit is seected the wa frame structure wi be damaged as shown in the figure and un-repairabe when the sstem ductiit exceeds a constant vaue during the earthquae ground motion excitation. GENERATE ACCEPTABLE DESIGN REGION USING YIELD POINT SPECTRA The Yied Points Spectra Method, proposed b Mar Aschheim (), can be used to design the sstem stiffness, structura period and design strength direct. This method can aso be used to design muti performance objectives incuding the demand for service eve and the ife safet eve. It is suggested to use the Yied Point Spectra Method in conjunction with the concept of equivaent ductiit ratio to construct the performance based design procedure. Based on the deveoped ineastic ied strength demand spectra and ineastic ied dispacement spectra ( u ineastic Sd ineastic ), respective, the Yied Point Spectra can be constructed b potting the ied strength coefficient demand spectra C with respect to the ied dispacement demand spectra in ADRS format. For seismic design, the Yied Point Spectra can hep us to construct the Acceptabe Design Region and gives a reference to a designer on how to determine the design structura ied strength, the design structura period and the design sstem ductiit for specific seismic constraints such as the constraints of the Damage Index and the constraint of the sstem maximum dispacement to be used in the performance based seismic design. To determine the Acceptabe Design Region in the Yied Point Spectra it is necessar to specif some constraint vaues, such as the deformation constraint and/or damage eve constraint. Three different tpes of acceptabe design region are described as foows: (a) Constraint on maximum dispacement of structure As shown in Figure 5, the combination of the Yied Point Spectra to the ied point of four different inds of structure, and the structura response, incuding the sstem ductiit and sstem maximum dispacement during the earthquae ground motion excitations is demonstrated. From this figure it is found that sstem (TT ) and sstem (TT ) have the same vaue of the ied dispacement but have different vaue of the sstem ductiit and sstem maximum dispacement ( u, i i, i,), and sstem 4 (TT 4 ) has arger vaues of the ied dispacement and its sstem ductiit is smaer than. which means the sstem 4 (TT 4 ) wi remain eastic during the earthquae ground motion. (b) Constraint on design equivaent ductiit For a specific constraint vaue of the Damage Index of the structures one can use the deveoped design Equivaent Ductiit as mentioned before to constrain the sstem ductiit, and usua the seected design sstem ductiit must be ower than the design Equivaent Ductiit. From the use of the Yied Point Spectra, as shown in Figure 6, it is nown that the position of the ied point of the structures must be on the right hand side of the bac ine for eq, as shown in the figure of Acceptabe Design Region. (c) Combining constraints of Damage Index and maximum dispacement As shown in Figure 7, both Damage Index and maximum dispacement are considered as constraints in the ied point spectrum. From this figure it is found that if the position of the ied point of the structure is ocated in the Acceptabe Design Region then the sstem ductiit ratio wi not exceed the vaue of the Equivaent Ductiit eq. It means that the vaue of the Damage Index during the earthquae ground motion wi not exceed the vaue of the constrained Damage Index, and the vaue of the maximum dispacement wi not exceed the vaue of the constrained S d i. (d) Combining constraints of two performance eves Suppose two performance objectives (consider both ife safet and serviceabiit) were used in the design exampe, as shown in Tabe 5. The Acceptabe Design Region for both performance eves incuding the service eve and the ife safet eve is shown in either Figure 8a or Figure 8b. The vaue of the T ser, the structura period for the

8 service eve, is determined b the constraint vaue of the maximum dispacement (Sd ser ) of the structures for the service eve. The vaue of the T saf, and T saf,, the structura period for the ife safet eve, is determined b the constraint vaues of the maximum dispacement of the structures for the service eve Sd saf and the vaue of the Sd saf eq, respective. The main difference between these two figures is the reative vaue among T ser, T saf, and T saf,. For Figure 8a the vaue of the T ser is smaer than the vaue of thet saf, and T saf,. For Figure 8b the vaue of the T ser is between the vaue of the T saf, and T saf,. If the position of the ied point of the structures is ocated in the Acceptabe Design Region as shown in these two figures the structures wi satisf the design performance objectives as shown in Tabe 5. SEISMIC DESIGN USING YIELD POINT SPECTRUM Based on the above discussion on the Damage Index, the design Equivaent Ductiit and the Yied Point Spectra, a design phiosoph for the performance-based seismic design can be introduced. The design procedure is isted as foows:. Define design performance objectives quantitative: Based on the initia estimation on the characteristics of the structure such as the materia of the structures, the structura tpe and height defined, the design performance objectives must be determined quantitative. It is suggested that at east two performance eves were seected to design a structure: the service eve and the ife safet eve. Then determine the constraints of the inter-stor drift index (IDI) and the Damage Index (DM) for both serviceabiit and ife safet.. Transform MDOF sstem to SDOF sstem: Transform the constraints of the inter-stor drift index (IDI) for MDOF to the constraints of the maximum dispacement (S d ) for SDOF. The process of transformation wi depend on the initia estimation on the characteristic of the structures. 3. Construct the seismic eastic spectra and the seismic demand spectra: The seismic eastic demand spectra wi depend on the site conditions and the characteristics of the structures (such as the damping ratio of the structure) as discussed before. The Seismic Demand Parameters, incuding the parameter C, Ru, γ, are dependent on the characteristics of the structure (such as the noninear behavior and the sstem damping ratio of the structures) and the site condition. 4. Cacuate the design Equivaent Ductiit: The vaue of the design Equivaent Ductiit wi depend on the parameter γ, the constraint vaue of the Damage Index and the vaue of the utimate ductiit of monotonic oading which must be estimated in advance from the characteristics of the structures. 5. Use the Yied Point Spectra method to determine the Acceptabe Design Region. 6. Determine design strength, period, sstem ductiit of the structure: B using the Acceptabe Design Region of the Yied Point Spectra, the design structura ied strength, the design structura period, and the design sstem ductiit during the earthquae ground motion excitation can be determined. Chec the position of the ied point of the structure in the Yied Point Spectra to find if the design resut satisf the design performance objectives. SUMMARY AND CCONCLUSIONS A design phiosoph, based on the energ concept, has been deveoped for performance based seismic design in this stud. B combining the use of the Yied Point Spectra and a of the seismic demands, incuding seismic strength demand, seismic dispacement demand and seismic energ demand for ineastic SDOF sstems, the Acceptabe Design Region of the Yied Point Spectra can be derived which provides design information considering muti performance eves. Some significant concusions of this stud are summarized as foows:

9 . Seismic Energ Response Parameter γ, which can refect the amount of the hsteretic energ absorbed b the structures, can be used to cacuate the Damage Index of the structures during the earthquae ground motion excitation and the Equivaent Ductiit. The variation of the vaue of the parameter γ is not wide-ranging and it is convenient to estimate the vaue of the Damage Index of the structures during the earthquae ground motion excitations and the vaue of the Equivaent Ductiit.. Advantages and Disadvantages of the Damage Index proposed b different researches are discussed. From the definition of the Damage Index the definition of the Equivaent Ductiit can be derived. Using the concept of the Equivaent Ductiit, the Damage Index of the structures can be imited b the constraint of the sstem ductiit during the earthquae ground motion excitations. 3. Based on the energ concept, a design phiosoph for performance based seismic design was proposed in this stud. Seismic demands for different performance eves, incuding seismic strength demand, seismic dispacement demand and seismic energ demand for ineastic SDOF sstems, can be constructed and iustrated b the Yied Point Spectra. Using the Yied Point Spectra the Acceptabe Design Region can be constructed and iustrated according to the constraint of the Damage Index and the maximum sstem dispacement, and based on it a designer can seect and derive the design structura strength, design structura period and design sstem ductiit to mae a design. For rehabiitation combining the Yied Point of the structure to the Yied Point Spectra can chec the performance eve of the structure. The design phiosoph proposed in this stud wi not need the iteration steps because a the information about the seismic demands for ineastic SDOF sstems can be shown and constructed in the Yied Point Spectra. ACKNOWLEDGEMENTS This research was supported b Nationa Science Counci under Grant No. NSC 9--E--7. REFERENCES [] Aschheim, Mar and Bac, Edgar F., Yied Point Spectra for Seismic Design and Rehabiitation, Earthquae Spectra, Vo.6, No., Ma, [] Bertero, R. D., V. V. Bertero and A. Teran-Gimore, Performance-based Earthquae-Resistant Design Based on Comprehensive Design Phiosoph and Energ Concepts, Proceedings of WCEE, Paper No. 6, 996. [3] Bertero, R. D. and V. V. Bertero, Performance-based Seismic Engineering: The Need for A Reiabe conceptua Comprehensive Approach, Earthquae Engineering and Structura Dnamics, 3,, [4] Bozorgnia, Y. and Bertero, V.V., Estimation of Damage Potentia of Recorded Earthquae Ground Motion Using Structura Damage Indices, Proceedings of ECEE, Paper ref. No.475,. [5] Bracci, J.M., Reinhorn, A.M., Mander, J.B. and Kunnath, S.K. Deterministic Mode for Seismic Damage Evauation of Concrete Structures, Technica Report NCEER-89-33, Nationa Center for Earthquae Engineering Research, State Universit of New Yor Buffao NY,. [6] Chopra, Ani K. and Raesh K. Goe Direct Dispacement Based Design: Use of Ineastic vs. Eastic Design Spectra, Earthquae Spectra, Voume 7, No., Februar a. [7] Fajfar, P., Equivaent ductiit factor, taing into account ow-cce fatigue Earthquae Engineering and Structura Dnamics, Vo.,,99, [8] Miranda, Eduardo and Jorge Ruiz-Garcia Evauation of approximate methods to estimate maximum ineastic dispacement demands, Earthquae Engineering and Structures Dnamics, Voume 3,,, [9] Mostaghe, Naser Anatica Description of Pinching, Degrading Hsteretic Sstems, ASCE, J. of Engineering Mechanics, 5(),, 999, 8-4.

10 [] Par, Y. J. and Ang, A. H-S. Mechanistic Seismic Damage Mode for Reinforced Concrete, J. of the Structura Division, ASCE, (ST4), 985. [] Prieste, M.J.N. and Kowas, M.J. Direct Dispacement Based Design of Concrete Buidings, Buetin of the New Zeaand Societ for Earthquae Engineering, Voume 33, No. 4, December Tabe : Noninear contro parameters for different hsteretic mode α λ λ λ p ζ Biinear mode BMR mode W_F mode Tabe : Suggested vaue on the Parameter γ for different hsteretic mode Upper imit of the parameter γ Lower imit of the parameter γ EPP mode..6 Biinear mode.3.8 BMR mode..6 W_F mode.8.5 Tabe 3: Estimated equivaent ductiit for BMR and W_F modes (DI b.4 for service eve) λ.(bmr mode), DI b.4 mon \ λ λ.5 (W_F mode), DI b.4 mon \ λ

11 Tabe 4: Estimated equivaent ductiit for BMR and W_F modes (DI b.8 for safet eve) λ.(bmr mode), DI b.8 mon λ.5 (W_F mode), DI b.8 mo Tabe 5: Conceptua vaue of damage index, sstem ductiit and maximum dispacement for service eve and ife safet eve Damage Index Sstem Ductiit Maximum Dispacement Service Leve < Sd ser Life Safet Leve <.6 < eq (Cacuated b Damage Index) Sd saf

12 (a) BMR Mode Anatica Mode (b) RC-Wa frame mode α. λ. λ. λ.3 p α. λ.5 λ.5 λ p. - - Dispacement (cm) Dispacement (cm) Figure : Comparison between anatica (eft) and experimenta (right) ineastic Hsteretic mode: (a) fexure faiure mode,(b) Wa-frame mode, Force ( ton ) Force ( ton ) Experimenta Resuts Simuation K.56 ton/cm A β.888 γ -.86 n Error 3.4% WF_ (EXP.) A s.5, S.6 r.5, p. Z s.5,δυ. Z m u6 u.8.8 γ.6.4. Ductiit ratio. EPP BMR WF Biinear T/TG γ.6.4. Ductiit ratio T/TG Figure : Comparison on γ-spectrum using different hsteretic modes; (a) for ductiit ratio., (b) for ductiit ratio 4..

13 . BMR mode,μ6. W_F mode,μ6 DIb (a) (b) μeq μeq u u u3 u4 u5 u T/TG DIb Figure 3: Damage Index Spectra DI b ( mom 6): (a) for the BMR mode ( λ.), and (b) for the W_F mode ( λ.5) with 3 BMR mode(λ.),dib.4 U6 U8 U U γ BMR mode(λ.),dib.8 U6 U8 U U γ Figure 4: Pot of Equivaent Ductiit with respect to γ-vaue for different ductiit ratio of monotonic oading; (a) for case of DI b.4, (b) for case of DI b.8. μeq μeq u u u3 u4 u5 u T/TG W F mode(λ.5),dib.4 U6 U8 U U γ WF mode(λ.5),dib.8 U6 U8 U U γ

14 T T Pea Dispacement Yied Strength Coefficient, C T 3 T 4 u, u, u, 4u, u, Yied Dispacement Figure 5: Pot of ied point spectrum and the sstem capacit curve (four different structura sstems with period of T, T, T 3, and T 4 ). Yied Strength Coefficient, C Unacceptabe Design Region Acceptabe Design Region T eq Yied Dispacement Figure 6: Identified acceptabe design region using equivaent ductiit as constraint in Yied Point Spectrum. Yied Strength Coefficient, C Acceptabe Design Region T T eq Sd eq Sd Yied Dispacement Figure 7: Identified acceptabe design region using the constraint of the sstem equivaent ductiit and the maximum dispacement in Yied Point Spectrum.

15 Yied Strength Coefficient, C Service Demand Acceptabe Design Region T saf, T saf, T ser Life Safet Demand eq Sd ser Sdsaf Sd saf Yied Dispacement eq Figure 8a: Pot of acceptabe design region in ied point spectrum considering both service eve and ife safet eve (T saf, is in between T saf, and T ser ). Yied Strength Coefficient, C Service Demand Acceptabe Design Region T saf, T ser T saf, Life Safet Demand eq Figure 8b: Pot of acceptabe design region in ied point spectrum considering both service eve and ife safet eve (T ser is in between T saf, and T saf, ). eq Sd ser Sdsaf Sd saf Yied Dispacement