Improved Direct Displacement-Based Design Procedure for Performance-Based Seismic Design of Structures

Transcription

1 Iproved Direct Displaceent-Based Design Procedure for Perforance-Based Seisic Design of Structures Rakesh K. Goel, California Polytechnic State University, San Luis Obispo, and Anil K. Chopra, University of California, Berkeley Abstract Direct displaceent-based design ruires a siplified procedure to estiate the seisic deforation of an inelastic SDF syste, representing the first (elastic) ode of vibration of the structure. This step is usually accoplished by analysis of an uivalent linear syste using elastic design spectra. In this paper, an ually siple procedure is developed that is based on the well-known concepts of inelastic design spectra. This procedure provides: (1) accurate values of displaceent and ductility deands, and (2) a structural design that satisfies the design criteria for allowable plastic rotation. In contrast, the existing procedure using elastic design spectra for uivalent linear systes is shown to underestiate significantly the displaceent and ductility deands. The existing procedure is shown to be deficient in yet another sense; the plastic rotation deand on structures designed by this procedure ay exceed the acceptable value of the plastic rotation, leaving an erroneous ipression that the allowable plastic rotation constraint has been satisfied. Introduction Direct displaceent-based design is being advocated as a ore rational and relevant approach to seisic design of structures copared to traditional strength-based design (Shibata and Sozen, 1976; Moehle, 1992; Kowalsky, Priestley, and MacRae, 1994). Displaceent-based design involves several steps (to be described later), one of which is to estiate the seisic deforation of an inelastic SDF syste representing the first (elastic) ode of vibration of the MDF syste. In soe of the present procedures, this step is accoplished by approxiate ethods in which the nonlinear syste is replaced by an uivalent linear syste (Shibata and Sozen, 1976; Priestley, Seible, and Calvi, 1996). The period and daping of this linear syste are deterined by the secant stiffness ethod (Jennings, 1968) or its variants, e.g., the substitute structure ethod (Shibata and Sozen, 1976). The purpose of this paper is to deonstrate application of inelastic design spectra to direct displaceent-based design of structures. The resulting design procedure is shown to produce a satisfactory structural design. In contrast, it is shown that the design produced by the procedure that uses elastic design spectra and uivalent linear systes does not necessarily satisfy the design criteria. In particular, it can leave an erroneous ipression that the allowable plastic rotation constraint has been satisfied. Note that this paper is a suary of a recent full-length anuscript accepted for publication (Chopra and Goel, 2000).

2 This presentation is intentionally restricted to structures idealized as SDF systes with bilinear force-deforation relations (Figure 1). The distraction of approxiations inherent in a one-ode representation of MDF systes and bilinear idealization of a pushover curve are thus avoided. h u y θ p Force f y f y ( 1+αµ α ) 1 k u y 1 1 ksec Deforation (a) (b) Figure 1. Idealized SDF syste with bilinear force-deforation relation. Displaceent-Based Design Using Elastic Design Spectra Equivalent Linear Syste. Needed in existing displaceent-based design procedures, the properties of the uivalent linear syste are suarized here. Consider an inelastic SDF syste with bilinear force-deforation relationship on initial loading (Figure 1b). The stiffness of the elastic branch is k and that of the yielding branch is αk. The yield strength and yield displaceent are denoted by f y and u y, respectively. If the peak (axiu absolute) deforation of the inelastic syste is u, the ductility factor µ = u / u y. For the bilinear syste of Figure 1b, the natural vibration period of the uivalent linear syste with stiffness ual to k sec, the secant stiffness, is αk u T = T n µ 1 + αµ α (1) where T n ( u u y ). is the natural vibration period of the syste vibrating within its linearly elastic range The ost coon ethod for defining uivalent viscous daping is to uate the energy dissipated in a vibration cycle of the inelastic syste and of the uivalent linear syste. Based on this concept, it can be shown that the uivalent viscous daping ratio is (Chopra and Goel, 1999): 2 (µ 1)( ζ = 1 α ) (2) π µ(1 + αµ α ) The total viscous daping of the uivalent linear syste is ζ Û = ζ + ζ where ζ is the viscous daping ratio of the bilinear syste vibrating within its linearly elastic range ( u u y ). (3)

3 u, c % 10% ζ = 2% 20% T, sec n Figure 2. Elastic deforation design spectru. Elastic Design Spectra. To ipleent the existing displaceent-based design procedure, an elastic design spectru is needed. For this purpose, a edian-plus-one-standard-deviation spectru constructed for peak values of the ground acceleration, velocity, and displaceents of u&& = 1g, u& = 122 c/s (48 in/s), and u = 91.4 c (36 in), respectively, by the procedure of go go go Newark and Hall (1982) is used. The displaceent (or deforation) design spectru needed in the displaceent-based design is deterined fro the standard pseudo-acceleration design spectru, using the well-known relationship between pseudo-acceleration A and deforation D: u = D = T n 2π A 2 Figure 2 shows such spectra for several values of the daping ratio. Step-By-Step Procedure. Adapted fro Priestley and Calvi (1997), a direct displaceent-based design procedure for bilinear SDF systes (Figures 1a and 1b) using elastic design spectra is outlined as a suence of steps: 1. Estiate the yield deforation u y for the syste. 2. Deterine acceptable plastic rotation θ p of the hinge at the base. (5) 3. Deterine design displaceent u u = u + hθ y p (6) and design ductility factor µ = u / u y. 4. Estiate the total uivalent viscous daping, Û ζ, for the design ductility factor fro Equations 2 and Enter the deforation design spectru for elastic systes with known u and Û ζ to read T (Figure 2). Deterine the secant stiffness

4 4π 2 k = 2 sec T (7) where is the ass of the syste. 6. Deterine the ruired yield strength f y fro Figure 1b: f = y ksec u 1 + αµ α 7. Estiate eber sizes and detail (reinforceent in R/C structures, connections in steel structures) to provide f. Calculate initial elastic stiffness k and u = f / k. y y y 8. Repeat steps 3 to 7 until a satisfactory solution is obtained. Exaple. Consider a portion of a long reinforced-concrete viaduct that is a part of a freeway. The total weight of the superstructure, 190 kn/, is supported on identical bents 9 high, uniforly spaced at Each bent consists of a single circular colun 1.5 in diaeter (Figure 3a). Using the design procedure described earlier, this syste is designed (longitudinal reinforceent of the colun) for the design earthquake defined by Figure 2 scaled to u&& go = 0.5g. (8) h w = 7517 kn k = 3EI h 3 (a) Figure 3. Exaple single-colun bent and idealized SDF syste. For the transverse ground otion, the viaduct can be idealized as an SDF syste (Figure 3b) with its lateral stiffness coputed fro k = 3EI / h 3, where E is the elastic odulus of concrete, I is the effective oent of inertia of the reinforced-concrete cross section, and h is the colun height. Based on the Aerican Concrete Institute design provisions ACI , the effective EI for circular coluns subjected to lateral load is given by EI = E I ( ρ γ 2 c g t E s / E c ) where I g is the second oent of inertia of the gross section, E c and E s are the elastic oduli of concrete and reinforcing steel, ρ t is the longitudinal reinforceent ratio, and γ is the ratio of the distances fro the center of the colun to the center of the outerost reinforcing bars and to the colun edge. The syste properties selected are: concrete strength = 27.6 MPa (4 ksi), steel strength = 413 MPa (60 ksi) and γ = 0.9. The ass of the idealized SDF syste is the tributary ass for one bent, i.e., the ass of 39.6 length of the superstructure, = w / g = ( N)/9.8 /s 2 = kg. The initial elastic vibration period of this syste is 1.82 s, which falls in the velocity-sensitive region of the design spectru. (b)

5 The step-by-step procedure described earlier in this section is now ipleented as follows: 1. An initial estiate of u y = 4.5 c. 2. The plastic rotation acceptable at the base of the colun is θ p = 0.02 radians. 3. The design displaceent given by Equation 6 is u = u + hθ = = 22.5 c y p and the design ductility factor is µ = u u y = 22.5/ 4.5 = For α = 5% and µ = 5, Equations 2 and 3 give ζ Û = 45%. 5. The deforation design spectru for elastic systes is constructed for Û ζ = 45%. Corresponding to u = 22.5 c this spectru gives T = 2.81 s and k sec is coputed by Equation 7, k sec = (2π / 2.81) = N/ = kn/c. 6. The yield strength is given by Equation 8, f y = ( ) /( ) = kn. 7. The circular colun is then designed using ACI for axial load due to superstructure weight of 7517 kn plus colun self weight of 375 kn and the bending oent due to lateral force = f y : M = hf y = 6472 kn-. For the resulting colun design, ρ t = 1.19%, flexural strength = 7395 kn-, and lateral strength = kn. For ρ t = 1.19%, EI = kn 2, k = 91.3 kn/c, and u y = f y k = 821.7/91.3 = 9 c. 8. Since the yield deforation coputed in Step 7 differs significantly fro the initial estiate of u y = 4.5 c, iterations are necessary. The procedure converged after three iterations giving a colun design with ρ t = 1.3%. This colun has an initial stiffness, k = kn/c and lateral yield strength, f y = kn. The deforation deand u = 26.8 c. Due to reason of brevity, results of interediate iterations are not presented here; they are available in Chopra and Goel (2000). Displaceent-Based Design Using Inelastic Design Spectra Presented next is a direct displaceent-based design procedure that uses the well-known constant-ductility design spectra instead of the elastic design spectra for uivalent linear systes. Inelastic Design Spectru. A constant-ductility spectru for an elastoplastic hysteretic syste is a plot of Ay versus the initial elastic period T n for selected values of µ. The pseudoacceleration Ay is related to the yield strength f y by f y = (A y / g)w where w is the weight of the syste. The yield strength reduction factor is given by Ry = f o / f y = A / Ay where f o = (A / g)w is the iniu yield strength ruired for the structure to reain elastic during the earthquake; A is the pseudo-acceleration ordinate of the elastic design spectru at (T n,ζ).

6 A constant-ductility design spectru is established by dividing the elastic design spectru by appropriate ductility-dependent factors that depend on T n ; detailed procedure is described in Chopra (1995, Chapter 7). The Newark and Hall (1982) recoendations for the reduction factor, R y, are 2µ 1 in the acceleration-sensitive region, and µ in the velocity- and displaceent-sensitive regions. In recent years, several other recoendations for the reduction factor have been developed (Krawinkler and Nassar, 1992; Vidic, Fajfar, and Fischinger, 1994; Miranda and Bertero, 1994). The peak deforation u of the inelastic syste is given by: 2 2 (9) µ T n 1 T n u = Ay = µ 2π 2π A R y Coputed by using Equation 9 and R y µ relations of Newark and Hall (1982), the deforation design spectru is shown in Figure 4. Step-By-Step Procedure. The first three steps of this procedure are identical to those in the previously-described displaceent-based design procedure, and steps 4 to 8 are replaced by the following steps based on the deforation design spectra for inelastic systes (Figure 4). 4. Enter Figure 4 with known u and µ to read T n. Deterine the initial elastic stiffness: 4π 2 (10) k = 2 T n 5. Deterine the ruired yield strength f y = k u y (11) 6. Estiate eber sizes and detail (reinforceent in R/C structures, connections in steel structures, etc.) to provide the strength deterined fro Equation 11. For the resulting design of the structure, calculate the initial elastic stiffness k and yield deforation u y = f y / k. 7. Repeat steps 3 to 6 until a satisfactory solution is obtained. u, c µ = T, sec Figure 4. Inelastic deforation design spectra.

7 The graphical ipleentation of Step 4 in the odified design procedure ay be attractive for its siilarity to the previous procedure. However, the graphical feature is not essential and the Step 4 can be ipleented nuerically; fro Equation (9) T n = 2π u R y A µ where R y and µ are related, for exaple, by relations of Newark and Hall (1982). Because this relation depends on T n, iteration ay be necessary to deterine T n fro Equation 12. Exaple. The step-by-step procedure described in this section is now ipleented for the syste designed previously by the displaceent-based design procedure using the elastic design spectra. 1. An initial estiate of u y = 4.5 c. 2. The plastic rotation acceptable at the base of the colun is θ p = 0.02 radians. 3. The design displaceent given by Equation 6 is u = u y + hθ p = = 22.5 c and the design ductility factor is µ = u u y = 22.5/ 4.5 = The deforation design spectru for inelastic systes is constructed for µ = 5. Corresponding to u = 22.5 c, this spectru gives T n = 1.01 s and k is coputed by Equation 10, k = (2π /1.01) = N/ = kn/c. 5. The yield strength is given by Equation 11, f y = ku y = = 1344 kn. 6. The circular colun is then designed using ACI for axial load due to superstructure weight of 7517 kn plus colun self weight of 375 kn and the bending oent due to lateral force = f y : M = hf y = kn-. For the resulting colun design, ρ t = 3.62%, flexural strength = kn-, and lateral strength = 1441 kn. For ρ t = 3.62%, EI = kn 2, k = kn/c, and u y = f y k = 1441/174.4 = 8.27 c. 7. Since the yield deforation coputed in Step 6 differs significantly fro the initial estiate of u y = 4.5 c, iterations are necessary. The procedure converged after five iterations giving a colun design with ρ t = 5.5%. This colun has an initial stiffness, k = kn/c and lateral yield strength, f y = 1907 kn. The deforation deand u = 26.0 c. Due to reason of brevity, results of interediate iterations are not presented here; they are available in Chopra and Goel (2000). Evaluation of Exaple Designs The colun design resulting fro both procedures is evaluated in this section. Whether a design is satisfactory will be judged by calculating the deforation deand and plastic rotation deand iposed by the design earthquake. These deands can be coputed for a syste with known properties (initial elastic stiffness, k, ass,, and yield-strength f y ) by the following procedure: (12)

8 1. Calculate the initial elastic period, T n, fro the known ass,, and the initial elastic stiffness, k. 2. Deterine the pseudo-acceleration A fro the elastic design spectru; the elastic design force, f o = ( A/ g)w. 3. Calculate the yield-strength reduction factor, R = f / f, in which f is coputed in Step 2 y o y o and f y is known yield-strength of the designed syste. 4. Deterine the ductility deand µ using the R y µ T n relations. 5. Calculate u fro Equation 9, and θ fro Equation 6, where u = f / k and f is known p y y y yield-strength of the syste. The deforation and plastic rotation deands are coputed by this procedure for the exaple syste and copared with those estiated by the two afore-entioned design procedures. Structural Design Using Elastic Design Spectra. In designing the structure by using the elastic design spectra for uivalent linear systes, the deforation of the designed structure was estiated to be 26.8 c. However, when the designed structure is analyzed, the deforation deand is 39.7 c. Which of the two values is ore accurate? Clearly it is the latter value because it coes fro inelastic design spectra which are based on nonlinear response history analyses of inelastic systes considering a wide range of syste paraeters and any ground otions (Krawinkler and Nassar, 1992; Vidic, Fajfar, and Fischinger, 1994). In contrast, the forer value coes fro an approxiate procedure based on uivalent linear systes, a procedure that is known to be inaccurate (Chopra and Goel, 2000). Thus the design procedure has underestiated the deforation deand by 32.6%. The displaceent-based design procedure based on elastic design spectra for uivalent linear systes has additional deficiencies. Although the structure was designed for an acceptable value of the plastic rotation θ p = 0.02 radians, the plastic rotation deand = radians, 72% ore than the acceptable value. Thus the design procedure leaves an erroneous ipression that the allowable plastic rotation constraint has been satisfied. Structural Design Using Inelastic Design Spectra. In designing the structure by the procedure based on inelastic design spectra, the deforation deand for the designed structure was estiated to be 26.0 c. When the designed structure is analyzed, the deforation deand is 25.9 c. Clearly the design procedure has estiated the deforation deand consistent with that predicted by well-established concepts of inelastic design spectra. Furtherore, the plastic rotation deand of radians is essentially identical to the acceptable value of 0.02 radians that was iposed on the design, and the proposed procedure has produced a satisfactory design. Note that for the exaples considered, the displaceent-based design procedure using inelastic design spectra leads to a structure with ore longitudinal reinforceent and thus higher strength copared to the design based on elastic design spectra for uivalent linear systes. A stronger colun is necessary to satisfy the selected design criteria.

9 Conclusions Direct displaceent-based design ruires a siplified procedure to estiate the seisic deforation of an inelastic SDF syste, representing the first (elastic) ode of vibration of the structure, an MDF syste. A siplified procedure that uses the well-known inelastic design spectra has been presented in this paper. With the aid of exaples, it has been deonstrated that the procedure (1) provides displaceent estiates consistent with those predicted by the wellestablished concepts of inelastic design spectra, and (2) produces a structural design that satisfies the design criteria for acceptable plastic rotation. The displaceent-based design procedure proposed by several researchers in recent years uses elastic design spectra for uivalent linear systes based on the secant stiffness ethod or its variations like the substitute structure ethod. In this paper, it is deonstrated that the deforation and ductility factor that are estiated in designing the structure by this procedure are uch saller than the deforation and ductility deands deterined by nonlinear analysis of the syste using inelastic design spectra. Furtherore, it has been shown that the plastic rotation deand on structures designed by this procedure ay exceed the acceptable value of the plastic rotation. Thus, the design procedure leaves an erroneous ipression that the allowable plastic rotation constraint has been satisfied. Acknowledgents This research investigation is supported by the National Science Foundation under Grant CMS The authors are grateful for this support. References Chopra, A. K. (1995). Dynaics of structures: Theory and applications to earthquake engineering. Prentice Hall, Upper Saddle River, NJ. Chopra, A. K. and Goel, R. K. (2000). Direct displaceent-based design: use of inelastic design spectra versus elastic design spectra. Accepted for Publication, Earthquake Spectra. Chopra, A. K. and Goel, R. K. (2000). Evaluation of a NSP to estiate seisic deforation: SDF systes. Journal of Structural Engineering, ASCE, Vol. 126, No. 4, pp Chopra, A. K. and Goel, R. K. (1999). Capacity-deand-diagra ethods for estiating seisic deforation of inelastic structures: SDF systes. Report No. PEER-1999/02, Pacific Earthquake Engineering Research Center, University of California, Berkeley, April. Jennings, P. C. (1968). Equivalent viscous daping for yielding structures. Journal of the Engineering Mechanics Division, ASCE, Vol. 94, No. EM1, pp Kowalsky, M., Priestley, M. J. N., and MacRae, G. A. (1994). Displaceent-based design of RC bridge coluns. Proceedings, 2 nd International Workshop on the Seisic Design of Bridges, Queenstown, New Zealand, Vol. 1, pp Krawinkler, H., and Nassar, A. A. (1992). Seisic design based on ductility and cuulative daage deands and capacities. Nonlinear Seisic Analysis and Design of Reinforced Concrete Buildings, P. Fajfar and H. Krawinkler, Eds., Elsevier Applied Science, New York, 1992.

10 Miranda, E., and Bertero, V. V. (1994). Evaluation of strength reduction factors for earthquakeresistant design. Earthquake Spectra, Vol. 10, No. 2, pp Moehle, J. P. (1992). Displaceent-based design of R/C structures subjected to earthquakes. Earthquake Spectra, Vol. 8, No. 3, pp Newark, N. M., and Hall, W. J. (1982). Earthquake Spectra and Design. Earthquake Engineering Research Institute, Berkeley, CA. Preistley, M. J. N., Seible, F., and Calvi, G. M. (1996). Seisic design and retrofit of bridges. John Wiley & Sons, New York, NY. Priestley, M. J. N., and Calvi, G. M.. (1997). Concepts and procedures for direct displaceentbased design. Seisic Design Methodologies for the Next Generation of Codes, Fajfar and Krawinkler (eds), Balkea, Rotterda, pp Shibata, A., and Sozen, M. A. (1976). Substitute structure ethod for seisic design in R/C. Journal of the Structural Division, ASCE, Vol. 102, No. ST1, pp Vidic, T., Fajfar, P., and Fischinger, M. (1994). Consistent inelastic design spectra: strength and displaceent. Earthquake Engineering and Structural Dynaics, Vol. 23, No. 5, pp