EVALUATION OF A SIMPLIFIED METHOD FOR THE DETERMINATION OF THE NON LINEAR SEISMIC RESPONSE OF RC FRAMES

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1 EVALUATIO OF A SIMPLIFIED METHOD FOR THE DETERMIATIO OF THE O LIEAR SEISMIC RESPOSE OF RC FRAMES 9 Misael REQUEA And A. Gustavo AYALA SUMMARY In this paper a siplified ethod is developed for the evaluation of the seisic behavior of building structures, in which the contribution to the response of higher odes of vibration is iportant. The steps are given for the application of the ethod. To validate results of the proposed ethod a 7-story reinforced concrete building is studied. The results obtained are copared with those corresponding to a nonlinear step by step dynaic analysis and show a good approxiation. ITRODUCTIO Current seisic design codes for structures are based on providing enough capacity so that they resist without collapse the seisic forces to which they will be subected during their useful life. In the definition of these forces it is generally accepted that structures can suffer daage when subected to the action of intense earthquaes, however, the actions for which the structures are designed are defined fro elastic analysis, which only considers in an approxiate way the daage that they ay experience. In general, it is not possible, through a linear analysis, to estiate the behavior of the structures when they undergo seisic forces that ae the behave nonlinearly, since the forces to which they are subected not only depend on the characteristics of the excitation but also on the dynaic properties of the structures in the linear and nonlinear range of behavior. Thus, in to now the real behavior of the structures, it is necessary to carry out nonlinear step by step dynaic analyses which have the inconvenience of being coplex and expensive. Recently, research has been carried out to approxiate using siplified ethods of evaluation the nonlinear behavior of building structures subected to earthquaes, e.g., Cailo (995). With these ethods it has been possible to evaluate the behavior of regular buildings, with the restriction that they do not include in their forulations the contribution to the response of higher odes of vibration, which is of iportance for structures of considerable height. Another aspect that these ethods ignore is that when subected to intense earthquaes, the lateral stiffness of the structure diinishes with increasing levels of base shear, which in turn causes changes in the distribution of the equivalent lateral forces to which the structure is subected. The ain obective of this wor is to develop and verify an approxiate ethod for the evaluation of the seisic behavior of fraes, based on the Capacity Spectru Method, CSM, as originally proposed by Freean et al (975). This ethod considers explicitly in its forulation the contribution of higher odes of vibration and the variation of the stiffness of the structure when inelastic effects occur during its earthquae response. This paper presents the different existing approaches to calculate the distributions of lateral static forces equivalent to the seisic ones. It describes in detail the evaluation ethod and an application exaple, which illustrates its practical iplications. The results obtained fro this exaple are copared with those corresponding to a nonlinear step by step analysis. Finally, the conclusions derived fro this wor are presented and recoendations are given stressing the fact that there are soe iportant aspects that deserve further research. Instituto de Ingeniería, UAM, APDO 7-, México D. F. 5,Mexico. Instituto de Ingeniería, UAM, APDO 7-, México D. F. 5,Mexico, eail: gayala@ercury.iingen.una.x

2 EQUIVALET LATERAL STATIC LOADS FOR THE SEISMIC AALYSIS OF BUILDIGS Most existent design regulations accept as a substitute to the dynaic seisic analysis of a structure a linear static analysis with lateral loads equivalent to the seisic ones. Recently, use of these equivalent loads has been extended to the evaluation of the nonlinear seisic behavior of buildings through a nonlinear static analysis in which the agnitude of the loads is onotonically increased until collapse. In this type of analysis, nown as pushover, the agnitude of these lateral loads is not iportant, but their distribution with height is, for which different investigators have proposed different ways to calculate their distribution. MODELS OF EQUIVALET LATERAL LOADS The ost coon distribution of equivalent lateral loads specified in seisic design codes is in the for of an inverted triangle, which approxiately corresponds to the distribution if only the contribution in the fundaental ode were considered. However, when structures of considerable height are analyzed, the contribution to the response of higher odes of vibration can be iportant, aing it necessary to carry out a dynaic or approxiate static analysis with distributions that explicitly consider this contribution. Load distributions with the contributions of only the fundaental ode (approach ). In buildings that respond doinantly in their fundaental ode, the distribution of equivalent static lateral loads is calculated using the equation: F i ii = V () where Fi, i and i are the respective lateral force, ass and floor ordinate of the fundaental ode corresponding to the level i, V is the base shear force and is the nuber of levels of the structure. This load distribution is siilar to that proposed in the Construction Code for the Federal District, DDF (993), aong other regulations to define the equivalent static seisic forces in buildings. Load distributions with the contributions of higher odes (approach ). A generalization of the previous pattern of loads, which includes the contribution of higher odes of vibration in the calculation of the distribution of the equivalent lateral loads, involves the use of a probabilistic approach of superposition of odal contributions. In this wor two forulations are investigated, one used by Freean et al (99), approach -A, and the other by Valles et al (99), approach -B. The forulation used by Freean et al (99) is based on the static ethod of seisic analysis, which includes the contribution of higher odes of vibration in the distribution of the lateral loads by eans of the cobination of the contributions of each considered ode with the rule of odal superposition of the square root of the su of squares, SRSS, Chopra (95). In this forulation the distribution of lateral loads depends on the pseudo spectral acceleration, Sa, corresponding to each ode, which in turn depends on the state of daage of the structure. The equation to calculate this distribution of the lateral loads is: F () = i = isai = = where is the odal shape of floor and ode and Sa is the pseudo spectral acceleration of ode. 9

3 An alternative to the distribution of eq, is that proposed by Valles et al (99) in which, the contribution of higher odes of vibration is included by accepting the existence of an equivalent fundaental ode, i, which is deterined through a cobination of vibration odes using the SRSS rule, i.e.: i ( ) iγ = (3) = where Γ is the participation factor of the ode defined as: Γ = The equation that defines the distribution of equivalent static loads is: i i Fi = V (5) Using a distribution of lateral loads in a pushover analysis, the capacity of the structure when subected to increental lateral forces ay be deterined. During this analysis in each load step, the base shear and the corresponding roof displaceent are recorded, and with these the curve called the capacity or pushover curve of the structure ay be plotted. It is possible to consider that when the stiffness of the structure changes during pushover, i.e. structural eleents reach axiu strength and plastic hinges appear, a new distribution of lateral loads ay be calculated with which it is possible to consider the state of deforation of the structure in the calculation of the lateral loads with which the pushover analysis is carried out. () DETERMIATIO OF THE OLIEAR SEISMIC BEHAVIOR OF PLAE FRAMES To deterine the nonlinear behavior of plane fraes, in this paper a siplified procedure is proposed for the calculation of the distribution of story displaceents, interstory drifts and distribution and agnitude of plastic hinges as the result of subecting the structure to a given seisic deand. The proposed ethod is a variation of the CSM, but differs in that to deterine the perforance point, the capacity curve of the structure is not copared with the response spectru of the excitation. Instead, to deterine the behavior of a frae subected to one or several earthquaes, the original structure represented by a ultiple degrees of freedo syste, MDFS, is transfored into one equivalent single degree of freedo syste (SDFS) using concepts of structural dynaics. The equivalent SDFS is then subected to one or several seisic records and its axiu displaceent is obtained. This displaceent corresponding to the spectral displaceent (Sd) of the fundaental ode is then transfored to the corresponding roof displaceent of the MDFS with which the perforance point of the structure to a given seisic deand is deterined. To include the contribution of higher odes of vibration to the distribution of equivalent static loads required by this ethod, the two forulations discussed above are evaluated. In approach A, fro the capacity curve resulting fro the pushover only the contribution of the fundaental ode is extracted, aing it possible to deterine the properties of the equivalent SDFS. In approach B, the behavior curve of SDFS corresponding to the equivalent ode, which includes the contribution of higher odes is directly obtained fro the capacity curve derived fro the pushover analysis. The steps involved in the procedure proposed in this investigation are as follows:. A static nonlinear pushover analysis of the structure represented by a MDFS is carried out and with the results of base shear and roof displaceent the capacity or pushover curve of the structure is constructed.. The capacity curve is approxiated with a bilineal for and this, in turn, is transfored to the space of Sa versus Sd by eans of the following equations: 3 9

4 S a Sd V α W = () PF = (7) i where is the roof displaceent of the building, W is total weight of the structure and PF i is the odal participation factor for level i and ode, i.e.: and PF i = α is the base shear participation factor for ode, defined as: α = (9) 3. The ass corresponding to the fundaental ode,, is obtained by transforing the total ass of the MDFS with the chosen approach for distribution of loads. The equation required by approaches and - A is: and, by approach -B: i= i i i = () = () i= i i. The odal base shear (V) versus Sd curve is deterined by ultiplying the odal ass by Sa obtained in step. This curve is the behavior curve of the SDFS fro which one can calculate the elastic and inelastic stiffness of the syste, as well as its yielding force. In the case of approach A, fro the original capacity curve considering the contribution of higher odes, the contribution of only the fundaental ode required to calculate the V versus Sd curve needs to be extracted. To do this the original curve is approxiated with straight line segents, so that the points corresponding to first yielding and to other interediate states of interest in the behavior of the structure ay be defined, fig. To deterine the line segents of the reduced capacity curve corresponding only to the contribution of the first ode, curve O-A'-B'-C'-D ' in fig, it is necessary to transfor each of the line segents of the original curve O-A-B-C-D, with the following equations: () (ode ) ( odes ) α (ode ) ( odes ) αv = () V = (3) (ode ) (ode ) where and V are the roof displaceent and base shear corresponding to the ( odes ) ( odes ) contribution of the fundaental ode and and V are the corresponding roof displaceent and base shear including the contribution of all odes,. 5. A nonlinear analysis of the SDFS subected to a given seisic deand is carried out, obtaining the axiu Sd, which represents the perforance point of the SDFS.. The axiu Sd, obtained in step 5, is transfored fro the SDFS to the MDFS considering the evaluation approach used Approach 9

5 Approach -A Approach -B Sd PF = () S PF (ode ) = d (5) ( odes ) (ode ) = β () S PF (equi. ode ) = d (7) where is the roof displaceent neglecting the contribution of higher odes, β is the ratio of the roof displaceent with the contribution of odes to that corresponding to the fundaental ode, and (equi. ode ) is the roof displaceent considering the contribution of the equivalent fundaental ode Figure. Idealized capacity curves. 7. A new pushover analysis of the MDFS is carried out to the axiu roof displaceent obtained in step, fro which the lateral displaceents of the floors, the interstory drifts and the distribution and intensity of the plastic hinges of the structure can be deterined. MODEL STUDIED The structure studied is a regular square shape plan office building, located in area III of the seisic zoning of the valley of Mexico. The structure consists of four reinforced concrete fraes in each direction with three bays each long. The building has 7 floors with a story height of 3., except the first one which is. Bea sections are 35x9 c in all floors. Coluns are of types with sections, x c fro story to 7, 9x9 c fro story to, 75x75 c fro story to and x c for the last three floors. Slab thicness in all floors is c. The evaluation ethod was applied to an interior frae. COMPARISO AD AALYSIS OF THE RESULTS To validate the evaluation ethod, the changes in the odal shapes when inelastic effects in the frae occurred under increasing lateral loads during pushover were deterined. Fig a shows the changes in the fundaental ode as the daage increases in the structure. Liewise, figs b and c illustrate the changes in the second and third odes correspondingly. Taing into account the two considered approaches, the distributions of lateral loads in the frae with increasing values of base shear were deterined, fig 3a. In fig 3b the variation of the distribution of the lateral loads of approach is presented as daage in the structure occurs. Using the evaluation ethod with the considered approaches of distribution of equivalent lateral loads, the seisic perforances of the investigated frae were deterined. For approach the pushover analysis of the structure was carried out and the corresponding capacity curve deterined, fig. This capacity curve was approxiated with a bilineal for and then transfored, into the Sa versus Sd, curve, i.e., the behavior curve of the SDFS, fig 5. For the equivalent SDFS the odal ass was calculated with eq and a fraction of critical daping of.5 was assigned. Using as seisic deand the SCT-EW record of the 95 Michoacán earthquae, the axiu response of the equivalent SDFS was deterined and, in an inverse process, the corresponding 5 9

6 axiu roof displaceent. This displaceent was.3. The sae procedure was carried out using approaches -A and -B. For approach -A the capacity curve corresponding to the fundaental ode base shear was used, fig, and the axiu roof displaceent was.3. For approach -B the capacity curve of the frae and its bilineal idealization is presented in fig 7. The corresponding roof displaceent was Fundaental Modal Shape T=.75 T=.3 T=.9 T=3.53 T= Second Mode Shape T =.597 T =.55 T =.779 T =.977 T = Third Mode Shape T=.339 T=.357 T=.397 T=.5 T=.5 (a) (b) (c) Figure. Variation odal shapes with change of stiffness. To copare, a nonlinear dynaic step by step analysis was carried out using the sae seisic deand giving a axiu roof displaceent of.37. Fig shows the coparison of the lateral displaceents and fig 9 that of the interstory drifts. Finally, figs a to c show the distributions and sizes of plastic rotations of hinges as obtained with approaches, -A and the nonlinear dynaic analysis, respectively. COCLUSIOS In this paper an approxiate ethod of seisic evaluation of tall building fraes was presented and tested. The results obtained considering only the contribution of the fundaental ode and its variation with the state of deforation of the frae approxiate well those obtained fro the nonlinear step to step dynaic analysis. Also, the analyzed frae, being regular, is defored basically in its first ode, even when the frae presents considerable daage when plastic hinges occur in the structural eleents. The lateral displaceents, the distortions and the distribution and intensity of the plastic hinges obtained taing into account the contribution of higher odes approxiate very well those fro the nonlinear step by step dynaic analysis. 5 5 Lateral Loads (ton) Criterio Criterio Lateral Loads (ton) T =.75 T =.3 T =.9 T = 3.53 T = 5. (a) (b) Figure 3. Coparison of the distribution of lateral loads 9

7 Base Shear (ton) Roof Displaceent () Capacity Curve. Idealization Sa (g) Spectral Displaceent () Figure. Capacity curves corresponding to approach Figure 5. Behavior curve Sa vs Sd (approach of evaluation. of evaluation). Fro the results obtained when applying the siplified ethod of evaluation to the 7-story frae one ay conclude: ) The stiffness degradation of the structure influences ore the fundaental ode shape than those of higher odes. ) The distribution of lateral loads varies as plastic hinges are fored and/or increased in the structural eleents. 3). The effect of higher odes is iportant when the frae behaves in the nonlinear range. ). In the deterination of the seisic behavior, it is very iportant to consider the changes in the odes of vibration. ACKOWLEDGEMETS We acnowledge the ational Council for Science and Technology, COACyT for the scholarship of the first author and the sponsorship of the proect Developent of analytic and experiental ethods for the evaluation of the seisic behavior of structures fro which this wor was derived 9 Base Shear Roof Displaceent () Capacity Curve st ode Idealization Base Shear (ton) Roof Displaceent () Capacity Curve Idealization Figure. Capacity curves corresponding to approach -A of evaluation. Figure 7. Capacity curve corresponding to approach -B of evaluation. REFERECES Cailo, L. (995), Approxiate Procedure for the Deterination of the Seisic Capacity of Buildings, (in Spanish) aster thesis, DEPFI, UAM, Mexico. Chopra A. (995), Dynaics of Structures Theory and Applications to Earthquae Engineering, Prentice Hall, Englewood Cliffs, J. DDF, (993) Construction Code for the Federal District, (in Spanish) Mexico. 7 9

8 Freean, S., icoletti, J. P. and Tyrrell, J. V. (975), Evaluation of Existing Buildings for Seisic Ris. A Case Study of aval Puget Sound Shipyard, Breerton, Washington, Proceedings of the st US ational Conference on Earthquae Engineering, EERI, Ann Arbor, MI. Freean, S., Sasai, K., and Paret, T. (99), Multi-Mode Pushover Procedure (MMP)-A Method to Identify the Effects of Higher Modes in a Pushover Analysis, Proceedings of the th ational Conference on Earthquae Engineering, EERI, Seattle, WA. Valles, R., Reinhorn, A., Kunnath, S., Li, C. and Madan A. (99), IDARCD version.: A Coputer Progra for the Inelastic Analysis of Buildings, Technical Report CEER-9-, ational Center for Earthquae Engineering Research, Buffalo, Y Lateral Displaceents () Approach -A Step by Step Approach -B Approach... Interstorey D rift () Approach Approach -A Approach -B Step by Step Figure. Coparison of floor displaceents. Figure 9. Coparison of interstory drifts. (a) (b) (c) Figure. Distribution and intensity of plastic hinges, a) approach, b) approach and c) dynaic analysis. 9

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