Direct Inelastic Earthquake Design Using Secant Stiffness

Transcription

1 Direct Inelastic Earthquake Design Using Secant Stiffness Honggun Park and Taesung Eom ABSTRACT A new earthquake design method erforming iterative calculations with secant stiffness was develoed. Since basically the roosed design method uses linear analysis, it is convenient and stable in numerical analysis. At the same time, the roosed design method can accurately estimate the inelastic strength and ductility demands of the structural members through iterative calculations. In the resent study, the rocedure of the roosed design method was established, and a comuter rogram incororating the roosed method was develoed. Design examles using the roosed method were resented to verify its advantages. The roosed method, as an integrated analysis and design method, can directly address the earthquake design strategy intended by the engineer, such as limited ductility of member and the concet of strong column - weak beam. Through iterative calculations on a structural model with member sizes reliminarily assumed, the strength and ductility demands of each member can be determined so as to satisfy the given design strategy. As the result, structural safety and economical design can be achieved. EYWORDS: Earthquake Design, Secant Stiffness, Inelastic Analysis, Reinforced Concrete Honggun Park, Deartment of Architecture, Seoul National University, San56-1, Shinlim-dong, wanak-gu, Seoul, , South orea Taesung Eom, Deartment of Architecture, Seoul National University, San56-1, Shinlim-dong, wanak-gu, Seoul, , South orea

2 INTRODUCTION The equivalent static analysis/design method using linear elastic analysis and the resonse modification factor, which is the traditional earthquake design method, has serious disadvantages though it can be used conveniently in analysis and design. Although buildings of the same structural tye frequently show different resonses deending on their caacities of strength and ductility, the equivalent static method uses a common resonse modification factor. And the equivalent static method cannot accurately estimate the inelastic strength and deformation of each member because it uses linear elastic analysis. Recently, to overcome the disadvantages of the equivalent static method, a variety of earthquake analysis/design methods using nonlinear static analysis were develoed; the Caacity Sectrum ethod (CS, ATC 1996) and the Direct Dislacement-Based Design ethod (DDBD, Priestley 2). Unlike the conventional equivalent static method, the nonlinear static methods can estimate the inelastic seismic erformance of structures, and as the result, the structural safety can be secured against an earthquake. However, the existing nonlinear static methods still have several disadvantages in alication. The CS can be alied only if the analytical model describing the nonlinear behavior of each member has been established. This means that the CS can be used to evaluate the seismic erformance of existing structures or the structures already designed, but cannot be used as a direct design method to determine directly the strength and ductility demand. Therefore, this method has difficulty in directly addressing the earthquake design strategy intended by the structural engineer. Inconveniently, after the structure is designed arbitrarily, reeated evaluation and redesign on the structure are required to secure its structural safety and economical design. The DDBD simlifies an actual comlex structure to a substitute structure of single degree-of-freedom. The DDBD can determine the strength and ductility demand for the substitute structure. However, it has difficulty in determining the strength and ductility of each member consisting of the actual structure from those of the substitute structure. This is because a variety of actual structures with different local demands can be reresented by the same substitute structure. Therefore, ractically, DDBD is alicable to low-rise buildings and bridges, which have a comlete lastic failure mechanism, in a high-seismic zone. It is not aroriate for buildings in a low-seismic zone and for high-rise buildings with limited ductility demand instead of a comlete lastic mechanism. Due to such technical disadvantages, alication of the existing nonlinear static methods is limited, and the equivalent static method is still oular regardless of its technical inaccuracy. The urose of the resent study is to develo a new earthquake analysis/design method that can overcome the disadvantages of the existing methods: As a direct earthquake design method, the roosed method can directly determine the strength and ductility required in each member. Furthermore, it is alicable to structures with various seismic erformances including structures with a limited ductility demand. BASIC DESIGN CONCEPT The basic concet of the earthquake design method roosed in the resent study is to calculate the strength and ductility demands of structures and members resulting from their inelastic behavior by erforming linear analysis for secant stiffness instead of the traditional nonlinear analysis. Figure 1 (a) shows the deformed shae of a structure, and Figure 1 (b) shows the load-deflection curve and the moment-curvature curve at a lastic hinge of a member. Figure (b) shows the global and local

3 erformance oints defined with strength and maximum deformation resulting from the inelastic behavior of the structure and the member. Here, as shown in Figure 1 (c), the same erformance oint can be obtained by carrying out a linear analysis for the secant stiffness corresonding to the erformance oint. This is ossible because if the rofile of the earthquake load does not change, only one strength exists for the same deformation even though the loading aths may be different. For this rincile to be valid, the rofile of earthquake load should not change during inelastic behavior, and each member should not be unloaded. These assumtions are generally acceted in the conventional nonlinear static methods. Conversely, if an arbitrary secant stiffness is used at each lastic hinge of the members and linear analysis using the secant stiffness is erformed for a given earthquake load, a erformance oint can be calculated. If each member is designed so that its inelastic behavior asses the local erformance oint of the member, the same erformance oint can be obtained from the conventional nonlinear static analysis using inelastic member model. This result means that the linear analysis using the secant stiffness has the same effect as the conventional nonlinear analysis. s :Secant Stiffness ember A Structure ember A Structure ember A (a) Deformed shae Load-Deflection Curve oment-rotation Curve (b) Nonlinear analysis Load-Deflection Curve oment-rotation Curve (c) Elastic analysis using secant stiffness Figure 1. Nonlinear static analysis vs. equivalent linear elastic analysis using secant stiffness If the erformance oint is fixed to a secific value, as is done when the seismic erformance of existing structures is evaluated, the linear analysis using arbitrary secant stiffness cannot obtain the same erformance oint as the nonlinear analysis does. However, when a new structure is designed, as shown in Figure 2, a variety of the erformance oints for a given earthquake load can be selected according to the design strategy intended by the engineer. Therefore, a erformance oint can be determined by erforming the linear analysis for an arbitrary secant stiffness. As mentioned, if each member is designed so as to satisfy the strength and deformation demands calculated by the linear analysis using the secant stiffness, the erformance oint resulting from the conventional nonlinear analysis is the same as that determined by the linear analysis. This result indicates that in structural design, a erformance oint and the related strength and deformation demands of each member can be determined by linear analysis for secant stiffness. Figure 2. arious design alternatives for the same earthquake load

4 Though basically the secant stiffness can be assumed arbitrarily in a structural design, the secant stiffness at the lastic hinge of each member should be aroriately selected to secure structural safety and economical design. For the urose, the boundary of allowable secant stiffness must be established. Figure 3 shows the admissible zone of the erformance oint (, ), and the secant stiffness that can be acceted generally. The conditions for the admissible zone are as follows. 1) The secant stiffness should be less than the elastic stiffness. 2) The strength required to resist earthquake load should be not less than that required for gravity load. 3) A member designed with seismic details according to current earthquake design rovisions has a secific deformability u. Therefore, deformation of each member at the erformance oint should be less than its deformability. Elastic Stiffness Admissible Zone (, ) Secant Stiffness my Possible Performance Point my (a) Admissible zone for erformance oints u Actual behavior (, ) my ( y, y) my (b) Actual behavior of designed member corresonding to erformance oint Figure 3. Determination of inelastic strength and deformation of member using secant stiffness The shaded area in Figure 3 (a) indicates the admissible zone where the erformance oint can exist. In the reliminary analysis of a structure, the arbitrary secant stiffness is tried because it is not known if the erformance oint of each member belongs to the admissible zone. As the result of the analysis, if the erformance oint resulting from the analysis does not belong to the admissible zone, the secant stiffness is modified, and the analyses are reeated until at all the lastic hinges the erformance oints belong to the admissible zone. When the erformance oint of a member is u

5 determined as shown in Figure 3 (a), the actual inelastic behavior of the member will be as shown in Figure 3(b). The boundaries reresenting the allowable minimum strength and maximum deformation can be established arbitrarily according to the earthquake design strategy intended by the engineer. For examle, if the concet of strong column weak beam is intended to be introduced, develoment of lastic hinges in the column members should be restrained by increasing the boundary of the lowest strength. Also, if the detail of lateral confinement for ductility cannot be used, deformation of the lastic hinge can be restrained by decreasing the boundary of the maximum deformation. As such, if the strategy of earthquake design intended by the engineer is alied to establish the admissible zone, the erformance oint can be determined so as to satisfy the design concet. PROCEDURE OF DIRECT INELASTIC EARTHQUAE DESIGN The rocedure of the Direct Inelastic Earthquake Design (DIED) roosed in the resent study can be summarized as follows: 1) After assuming sizes of the members, erform linear analysis for the gravity load, and establish the boundary of the minimum strength at each member. Here, the boundary of the minimum strength can be determined by the strategy of earthquake design such as the minimum flexural strength secified in the current design rovisions, as well as by the gravity load. 2) Secify the maximum rotation at otential lastic hinges of each member, according to the ductility detail alied. Refer to either existing design rovisions and manuals such as FEA-273(BSSL, 1997) or exerimental results. 3) odel the otential lastic hinges of each member. Generally, locate the otential lastic hinges at both ends of the member. If a conventional comuter rogram for linear elastic analysis is used, the lastic hinges can be modeled as elements searate from the main element. In the element of lastic hinge, secant stiffness is used to resent the inelastic behavior (Figure 4). Elastic Beam-Column Inelastic rotational srings using secant stiffness Figure 4. Beam-column element with lastic hinges at two ends 4) Assume a secant stiffness at each otential lastic hinge. The secant stiffness should be less than the elastic stiffness e and greater than the minimum stiffness u corresonding to the boundary of the maximum rotation u (Figure 5). Here, e and u are calculated as e my mu = and u my u = (1) where my = yield rotation, u = maximum rotation, the yield rotation, and my = minimum moment corresonding to mu = minimum moment corresonding to the maximum rotation.

6 According to Priestley (2), y (= my ) can be aroximately estimated, regardless of the amount of reinforcement. ε y h 1 y = αst = αstεy h 2 2 (2) where h = deth of member, ε y = yield strain of longitudinal re-bars, α ST = modification factor according to member tye and shae of cross-section: α ST = 2.35 for columns with rectangular section, 2.12 for columns with round section, 2. for walls with rectangular cross-section, and 1.7 for beams. my e s (, ) u u mu Figure 5 Admissible range of secant stiffness 5) Calculate the earthquake load in accordance with the earthquake design code. Perform linear analysis using the assumed secant stiffness for the earthquake load. 6) If, at each lastic hinge, the local erformance oint does not belong to the admissible zone, the secant stiffness is modified. Though the secant stiffness can be modified arbitrarily, the following method was roosed in the resent study. As shown in Figure 6, the secant stiffness is modified for the four cases classified by the locations of the erformance oint: For i < my, s i+ 1 = e (3a) my + ( i my ) For my i < u and i < my + ( i my), si+ 1 = (3b) i For my i u and i my + ( i my), s i+ 1 = s i (3c) i For i > u, si+ 1 (3d) At each lastic hinge, stiffness, and u my i = current rotation, i = stiffness of the boundary of the minimum moment. u = current moment, s i, s i + 1 = secant

7 = s i+ 1 e s i s i + 1 s i e my my ( i, i) my u my u (a) i < my (b) my < i < u e = s i+ 1 s i Admissible Zone e s i + 1 s i my my my u my (d) u i (c) my < i < < u Figure 6 Strategy for modifying secant stiffness at lastic hinge u 7) Reeat the analysis using the modified secant stiffness until the erformance oints belong to the admissible zone at all the lastic hinges. 8) Perform the strength and ductility design so as to satisfy the demands resulting from the analysis. PERFORANCE-BASED DESIGN Since the magnitude and rofile of the earthquake load are rescribed by the earthquake design code and are assumed to be unchanged, the roosed method described above is a traditional force-based design even though it is able to calculate the inelastic deformation of structure. However, in reality, the erformance oint reresenting the strength and deformation demands varies with the characteristics of the inelastic behavior of the structure. Therefore, for rational earthquake design, the earthquake load itself should be modified according to the characteristics of the inelastic behavior redicted by the secant stiffness. Though, there are various methods for modifying the inelastic demands, in the resent study, the method rovided by the DDBD (Priestley) was adoted. According to the method, the inelastic strength demand d varies with inelastic deformation and energy dissiation caacity of the structure. The resent study focused on the develoment of a erformance-based design rocess of the Direct Inelastic Earthquake Design (DIED) satisfying the inelastic strength demand modified during the roosed design rocess. The inertia force and deformation of a structure subject to earthquake is determined by the dynamic characteristics of the structure, such as effective eriod, ductility, and energy dissiation caacity. Therefore, for a structure with a secific deformability, a strength demand is determined by the deformability, and the structure should be designed to satisfy the strength demand. For an earthquake load arbitrarily assumed, a deformation is redicted using the DIED. Subsequently, the strength demand d corresonding to the deformation can be estimated by the method rovided in the DDBD or CS. If the caacity is not equal to the demand d, the virtual

8 erformance oint (, ) of the current strength and deformation is not the actual one. Therefore, the earthquake load should be corrected so that the caacity is equal to the demand. The rocedure to obtain the erformance oint can be summarized as follows. 1) Assuming an earthquake load, calculate the inelastic deformation using the DIED (Figure 7 Ste 1). Here, the estimated inelastic deformation should be less than the allowable drift secified by the earthquake design code alied. 2) Using the DDBD or the CS, calculate the strength demand d corresonding to the inelastic deformation (Figure 7 Ste 2). 3) Check if = d. Otherwise, reeat stes 1) through 3), for the modified earthquake load. ( ), d ste 2 erformance oint Caacity Curve (, ) Figure 7. Determination of erformance oints Demand Curve ste 1: virtual erformance oint Figure 7 shows the rocedure of the erformance-based design using the DIED. The erformance, calculated for the earthquake load arbitrarily assumed is a virtual one, which can oint ( ) become the actual erformance oint. The curve connecting the virtual erformance oints (, ) is the Caacity Curve reresenting the seismic erformance of the structure designed. On the other hand, the curve connecting the strength demands calculated for the current deformation is the Demand Curve. Here, it should be noted that the definition of the Caacity Curve of the roosed method is different from that of the CS. In the CS, the Caacity Curve describes the conventional inelastic load-deflection curve of a structure. To the contrary, the Caacity Curve of the roosed method (DIED) resents the variations in the seismic erformance of the structure designed differently using various secant stiffnesses. The intersection between the Caacity Curve and the Demand Curve is the actual erformance oint intended to solve. The corresonding strength and dislacement are the design earthquake load and the target dislacement. Generally, as the trial earthquake load increases, the resulting dislacement increases. As the estimated dislacement increases, the strength demand decreases. Therefore, to obtain the erformance oint, the trial load should be increased if < d and decreased if > d. The erformance-based design shown in Figure 7 is alicable to various erformance levels: Immediate Occuancy (IO); Life safety (LS); and Collase Prevention (CP). FEA 273 secifies the allowable maximum rotation of beams and columns for the three erformance levels. If the allowable maximum deformation aroriate for a given erformance level is used to establish the admissible P P

9 zone shown in Figure 3(a), the structure for the erformance level can be designed directly using the roosed method. DESIGN EXAPLES In the resent study, a comuter rogram for earthquake design of reinforced concrete structure was develoed to carry out the Direct Inelastic Earthquake Design using secant stiffness. The develoed comuter rogram uses flexural elements with otential lastic hinges at two ends to describe the inelastic behavior of the lastic hinges. Since the roosed method requires a simle algorithm of iterative calculation rather than a comlex algorithm as in a conventional nonlinear analysis, the comuter rogram can be develoed easily by modifying a conventional comuter rogram for linear elastic analysis. 3@6 mm Gravity loads(unfactored): wd = 32.7 kn/m w = 14.7 kn/m L Figure 8. Configuration and load rofiles for seismic design of 8-story frame Concrete Elastic modulus(e c ) Table 1. aterial roerties Comressive strength(f c ) 8@3 mm Column Size: nd 1-2 story: 6 6 th 3-5 story: 5 5 th 6-8 story: 4 4 Beam Size: th 1-8 story: 3 45 Yield strength(f y ) steel Yield strain( ε y ) 23.5 GPa 27 Pa 4 Pa.2 Zone factor(a) Soil factor(s) Table 2. Seismic roerties Imortance factor(i) Resonse modification factor(r) Natural eriod(t) The rototye structure designed was the ordinary reinforced concrete moment frame, as shown in Figure 8. Table 1 resents the characteristics of the material used, and Table 2 resents the coefficients for calculating the earthquake load in accordance with the Standard Design Loads for Buildings (SDLB) secifying the earthquake load in orea. Using the coefficients, the base shear is calculated as

10 = where C S ( T ) AIC = W (4) R In orea, moderate seismic zone, the area coefficient A is at most.11, but to highlight the advantages of the roosed method, A =.4 was used in this study. The dead load D and live load L alied at each floor were 6 kn and 27 kn, resectively. The sizes of the members that were initially assumed are shown in Figure 8. The total weight of the building was 48 kn. The base shear calculated by the Equivalent Static ethod of the SDLB was 648 kn. The base shear was distributed linearly along the height of the building (Figure 8). Conventional Equivalent Static ethod Conventional linear elastic analysis was erformed for the structure subject to the equivalent static design load, and strength demands for the members were obtained. The inelastic dislacement of the building calculated using the rincile of equal dislacement adoted in SDLB was 765 mm. The structure designed was analyzed by DRAIN-2DX to verify the validity of the equivalent static method. Figure 9 shows the locations and the magnitude of the lastic rotations calculated by the inelastic analysis. As shown in the figure, although the structure was designed as to satisfy the strength demand calculated by the equivalent static method, the actual inelastic behavior was not desirable: The lastic hinges were develoed at the columns and excessive rotations occurred at the lastic hinges of the lower stories. Since the conventional equivalent static method using linear elastic analysis cannot estimate accurately the locations and magnitude of the inelastic deformation, the soft-story and local failure of the members cannot be revented, and structural safety of the building against earthquake cannot be secured. Base Shear(kN) beam failure column failure =.25 = To Dislacement(mm) max =.88 Figure 9. Traditional elastic design for equivalent static seismic load Direct Inelastic Earthquake Design Using Secant Stiffness Force-Based Design First, the roosed method was alied for the force-based design. The earthquake load alied to the frame was calculated by the equivalent static method, using Eq. 4. The DIED was erformed for the structure subject to the earthquake load, shown in Figure 8. For the structural analysis using secant stiffness, the following assumtions and design strategies were alied.

11 1) In the linear analysis for the gravity load, the effective stiffness of beams and columns were.5 and.7 times the stiffness of the gross section, resectively, as roosed by ATC-4 and FEA-273. However, since the beams were T-shaed, their effective stiffness was increased to two times the effective stiffness of the rectangular cross-section. 2) Linear elastic analysis was erformed for the gravity load 1.4D+ 1.7L. The boundary of the minimum strength at the otential lastic hinges was generally defined as the flexural moments resulting from the analysis for the gravity load. However, when necessary, the minimum strength was modified according to current design code. If the flexural moment was less than that corresonding to the minimum reinforcement ratio secified in the design code, the latter was assigned to the boundary of the minimum strength. Also, when the flexural moment was less than half of the flexural moment for the reversed earthquake load, the latter was assigned to the boundary. 3) The concet of strong column weak beam was used as the rincial design strategy. Therefore, the otential lastic hinges were located only at the beam ends and at the bottom of the column of the 1 st story. 4) As resented in Table 3, the allowable maximum rotations corresonding to the three erformance levels were secified at the otential lastic hinges. Table 3. Limitation on lastic rotation at lastic hinges (radians, FEA-273) ember Performance level IO LS CP Column (1 st story) Beams (all stories) ) The yield stiffness was assumed as 5 ercent of the elastic stiffness 6) The rotational sring element added to idealize the lastic hinge was modeled as the rigid lastic element resenting only lastic deformation excluding elastic deformation. In fact, though the magnitude of earthquake load should be changed deending on the erformance level alied, in this study, the same earthquake load was used to investigate variations of the strength and ductility demands with the allowable maximum rotation of lastic hinges. Table 4 resents the analytical results for the three erformance levels: IO, LS, and CP. The results show that though the earthquake load alied is uniform for all the erformance levels, the structural deformation tends to increase as the allowable deformation of members increases. Figure 1 shows the deformed shae of the structure and the locations and sizes of the lastic hinges for the erformance-level CP. In the figure, and resent normalized values of inelastic strength and lastic rotation, resectively. The lastic hinges at the two ends of the members are asymmetric, which occurred due to the effect of the gravity load. The analytical results show that unlike the conventional equivalent static method, the roosed method can accurately estimate the inelastic strength and deformation for a given earthquake load. e.

12 my u = = u my Rotation Ratio Rotation Ratio Figure 1. Proosed design method alied for force-based design Table 4. Results of force-based design Performance level IO LS CP (mm) (kn) 857 oment Ratio 2 st Columns of 1 story 1 Columns.5 1 Direct Inelastic Earthquake Design Using Secant Stiffness Performance-Based Design Next, the roosed method was alied to a erformance-based design. Here, the DDBD was used to evaluate the strength demand corresonding to a given inelastic deformation. The same design strategy, as had been used for the force-based design, was used in this analysis and design. As mentioned, in the erformance-based design, iterative calculations were erformed until the assumed strength (earthquake load) was equal to the strength demand. Figure 11 shows the rocedure seeking the final erformance oint. As the allowable maximum rotation of each member increases, the demand curve reresenting the inelastic strength demand decrease, and at the same time, the caacity curve is early softened. As the result, the design earthquake load of the erformance oint decreases. In short, as the ductility caacity of members is enhanced, the strength demand decreases (Table 5). The roosed method can directly address this trend in earthquake design of structures. Figure 12 shows the deformed shae of the structure and the locations and sizes of the lastic hinges for the erformance-level CP. As shown in the figure, the roosed method can estimate the strength and deformation demands at all the lastic hinges, and as the result, the ductility design to satisfy the deformation demand as well as the strength design can be carried out. As shown in the figure, lastic hinges were not develoed in the columns excet for the bottom of the 1 st story column because the roosed method adoting the strategy of strong column - weak beam fixed the secant stiffness of the columns to the elastic stiffness so that lastic hinges were not develoed in the oment Ratio 2 1 Beams.5 1

13 columns. Therefore, the roosed method securely restrained the develoment of the soft story. As shown in Figure 12, lastic hinges were develoed in all the beams along the building height. This develoment indicates that energy dissiation is maximized by sreading the lastic hinges. The maximized energy dissiation can be accomlished by the roosed design method adoting the rational strategy of earthquake design. Figure 13 shows the numerical results of the DRAIN-2DX for the structure designed by the Direct Inelastic Earthquake Design using secant stiffness. Figure 13 (a) shows the relationshis of the base shear and the to dislacement. Though the two aths are different from each other, the same erformance oint was obtained by the DRAIN-2DX using the conventional nonlinear analysis and the roosed method using the secant stiffness. Figure 13 (b) comares the lastic deformation occurring at the lastic hinges. The two results are exactly the same. Base Shear(kN) 1/R Demand 15 Caacity.3 1 (.2 P, P) erformance oint 5.1 virtual erformance oint. 2 4 Dislacement(mm) Base Shear(kN) Demand Caacity (, ) 2 4 Dislacement(mm) (a) Immediate Occuancy (IO) (b) Life Safety (LS) (c) Collase Prevention (CP) Figure 11. Determination of erformance oints P P 1/R Base Shear(kN) Demand Caacity (, ) 2 4 Dislacement(mm) P P 1/R my 4 3 oment Ratio 2 1 st Columns of 1 story Columns.5 1 Rotation Ratio 3 Figure 12. Proosed design method alied for erformance-based design oment Ratio = = u u my Beams.5 1 Rotation Ratio

14 Table 5 Results of erformance-based design Performance level IO LS CP (mm) (kn) Base Shear(kN) (a) Load-dislacement curve (b) Inelastic deformations Figure 13. erification of roosed method for erformance criterion of collase revention SUARY AND CONCLUSIONS DRAIN-2DX Proosed method 2 4 To Dislacement(mm).1.2 Rotation(Proosed ethod) Existing nonlinear static methods for earthquake design have several disadvantages and inconvenience when alied to comlex structures such as multi-story buildings: The Caacity Sectrum ethod, an evaluation method used for existing structures and reliminary designed structures, cannot be used as a direct design method, which can directly determine the strength and deformation demands of a structure and members; and the Direct Dislacement-Based Design method cannot be used for buildings with limited ductility demand because it requires a comlete lastic failure mechanism. In the resent study, a new earthquake design method, which can be used as a direct design method, was develoed. The roosed method (Direct Inelastic Earthquake Design) uses a new numerical method erforming iterative calculations for secant stiffness, and it can directly determine the inelastic strength and deformation demands with only the member sizes and load conditions used in the conventional elastic analysis. The roosed method uses a simle algorithm so it is convenient to use in numerical calculations. At the same time, it can analyze the inelastic behavior of structure through iterative calculations. In the resent study, the rocedure of the roosed method was established, and a comuter rogram erforming integrated analysis and design was develoed. Design examles using the roosed method were resented to verify its advantages. The advantages of the roosed method can be summarized as 1) The roosed method can directly calculate the inelastic strength and deformation demands with only the member sizes and load conditions. 2) The roosed method does not require a comlete lastic failure mechanism of structure. Therefore, it is alicable to structures with limited ductility demand such as high-rise buildings and buildings in low and moderate seismic zones. 3) The roosed method can directly address the design strategy intended by the engineers such as limited ductility of a member and the concet of strong column weak beam. As the result, structural safety and economical design can be accomlished. Rotation(DRAIN-2DX).2.1

15 4) Earthquake design based on member ductility is ossible. As the result, the soft-story can be revented and energy dissiation can be maximized by sreading the lastic hinges along the building height. 5) Since the inelastic numerical method using secant stiffness is versatile in alication, it is alicable to existing earthquake design methods using elastic, inelastic, or lastic analysis. ACNOWLEDGEENT This research was financially suorted by the orea Earthquake Engineering Research Center, and the authors are grateful to the authorities for their suort. REFERENCES ATC, 1996, Seismic evaluation and retrofit of concrete buildings, ATC-4, Alied Technology Council, Redwood City, California Building Seismic Safety Council, 1997, NEHRP guidelines for the seismic rehabilitation of buildings, FEA-273, Federal Emergency anagement Council, Washington, D.C Shibata, A. and Sozen,. A., 1976, Substitute-Structure ethod for Seismic Design ethod in R/C, Journal of the Structural Engineering, ASCE, ol. 12, No. ST1, January Priestley,.J.N, 2, Performance Based Seismic Design, Paer No. 2831, 12 th World Conference on Earthquake Engineering (WCEE) Allahabadi, R. and Powell, G. H., 1988, DRAIN-2DX User Guide, Earthquake Engineering Research Center, Reort No. UCB/EERC-88/6 Prakash,., Powell, G. H., and Cambell, S., 1993, DRAIN-2DX Base Program Descrition and User Guide ersion 1.1 Powell, G. H., 1993, DRAIN-2DX Element Descrition and User Guide for Element Tye1, Tye2, Tye4, Tye6, Tye9, and Tye15 ersion 1.1, University of California, Reort No. UCB/SE-93/17