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Evaluation of the Seismic Behavior of the Reinforced Concrete Structures Based on the Probabilities by Nonlinear Static Analysis M. Javanpour MS.C. Structural .Civil Engineering, Islamic Azad University, Dezful Branch, Iran, E-mail: mohsen.javan1982@gmail.com ABSTRACT: This paper presents an analytical foundation for probability-based formats for seismic design and assessment of structures. These formats are designed to be suitable for code and guideline implementation. The framework rests on non-linear, static seismic analysis. The formats can be used to ensure that the structural seismic design can be expected to satisfy specified probabilistic performance objectives, and perhaps (more novel) that it does so with a desired, guaranteed degree of confidence. Performance objectives are presumed to be expressed as the annual probability of exceeding a structural performance level. Structural performance levels are in turn defined as specified structural parameters (e.g., ductility, strength, maximum drift ratio, etc.) reaching a structural limit state (e.g. onset of yield, collapse, etc.). The degree of confidence in meeting the specified performance objective may be quantified through the upper confidence bound on the (uncertain) probability. In order to make such statements, aleatory (random) uncertainty and epistemic (knowledge limited) uncertainty must be distinguished. Key words: Non-Linear, Probability, Performance, uncertainty, Seismic 1 GENERAL INSTRUCTIONS number 2800 offer simplified formulas to In the recent severe earthquake, it was observed that statistically analyze the structures’ equation which the existing buildings possessed high levels of of course these formulas are not held accountable to vulnerability. Considerable point is that even the real behavior of the structures. modern buildings have suffered severe losses Since the damage occurs in the plastic region, the including heavy seismic breakdown and in some vulnerability analysis should be done in this area. cases overall destruction. Therefore, to reduce the Modeling of reinforced concrete structures is a earthquake risks in urban areas and predict the significant step toward determination of damages vulnerability of the existing buildings due to resulted from earth quake. A structure model has to probable quakes of the future, achieving an be able to model all kinds of reinforced concrete appropriate instruction is of the most necessary tasks elements and present accurate analysis of their non- of the seismic engineering. linear dynamic performance. One of the commonest and most conventional Considering complexity of the factors affecting seismic systems applied in quake-riser regions is structure’s behavior under earthquake condition and reinforced concrete structure. The experience also variety of the existing buildings, it hasn’t been obtained from the earthquakes occurred over the past possible to present a definite method to investigate 25 years has illustrated that even if the reinforced the seismic vulnerability of the existing buildings so Concrete buildings are constructed based on the far. Design and construction regulations of the relevant To determine a method including the entire country, breakdown and even destruction will mentioned specifications is a complicated and happen in them. multidimensional task. On the one hand, capacity of Heavy damages observed in the quake-influenced the structural system has to be assessed according to buildings are derived from the severe seismic its resistance and transformability and on the other vibrations in the non-linear limit while the hand, through electing the probable impacts of earthquake regulations such as Iran’s regulation earthquake on the affected area in the form of intensity, frequency content and duration, the non- linear behavior of the structure should be predicted Table 1. Range of Each Parameter Used To Select as much as possible and subsequently on the basis of Records its results, the force or the acceleration resulting in damages of the elements and the whole structural system have to be calculated. Soil B,C,D usgs classification The main goal of this paper provide fundamental Magnitude 6---7.3 and novel method for calculating probability of the distance 594488.21"---1133858.3" vulnerability of structures using statistical PGA 28.58 ---212.035 in/sec² distribution is in addition to simply being more Number of point 1800 --- 5961 point efficient, more accurate performance of structures DT 0.005,0.01,0.02 sec likely to be able to offer us. (Being symmetric is one Duration 21.9---40 Sec of the important characteristics of this distribution which should be noted.) In this course, different distributions (such as beta, gamma, povason, Laplace, gamble etc.) were 3 Modal Pushover Analysis (MPA) examined but not applied due to some specific characteristics like being unsymmetrical. 3.1 Introduction Logistic distribution was additionally investigated and it was illustrated that this distribution is very The basics of this method were proposed by Chopra much like the Log-normal distribution from the & Geol [2], [3]. Applying the concept of one- degree aspect of statistic specifications. The presented -free structure as well as the desired seismogram mathematics papers are suggestive of this claim, too. allows the equivalent displacement of one-degree- In this paper, the attempts have been made to apply free structure to be achieved, by means of which the the logistic distribution simultaneously with the displacement of the main structure can be calculated. Log-normal distribution and to compare their results. In order to be able to achieve favorable results of a 3.2 Implication Equivalent One-Degree-Freedom structural model also was to check the results on the Structure nonlinear dynamic model to calculate the likely vulnerability of structures to be used. The dominant concept in the entire non-linear static analysis methods is the equivalent one-degree-free 2 STRUCTURAL CHARACTREISTIC Structure(Valley [10]). Indeed, the acceleration displacement response spectrum (ADRS) in the The applied model was a reinforced concrete considered mode is gained through the main bending frame which has been designed according to structure’s capacity curve divided by the modal the specific plasticity principles and by means of participation multiplied by the structure’s modal IDARC Non-linear Software. The mentioned frame possesses 8 storeys with 126 inch in height and also amount at roof point ( nn ) and the normal axis of four 197 inch openings. the main structure’s capacity curve divided by the To provide the necessary plasticity and meet the effective modal mass ( M n* ) as well. economical considerations, the percentage of the columns’ armatures and that of the posts were Objective of this process in the Capacity Spectrum limited to 1 – 3 and 1.7 percents, respectively. The Tables 1 contain the dimensions and kinds of the Method is the simultaneous drawing of demand and armatures used in the frame’s different parts. capacity parameters in one coordinate system as well In the current research, in order to do the Time as determining their crossing point as the structure’s history analysis, a series of Record were needed. So, performance point, although in fact obtained curve 30 reformed Record were selected. To make sure of indicates behavior of the equivalent one-degree-free being reformed, the entire records were controlled structure. In this bilinear curve, gradient of the first by Seismo Signal software. All the selected records part reveals behavior of the equivalent one-degree- belonged to California, United States and some controlling parameters, such as distance from the free structure and gradient of the second part, fault and largeness, were taken into account while represented usually as a multiple of the first part’s selecting records. gradient, is indicative of hardening after yield. Having this curve in hand, it is possible to determine un-linear behavior of the one-degree-free structure. Coordinate axes of structure curve are achieved by an un-linear incremental static analysis. First being changed into ADRS, they are transformed into the un-linear behavior of equivalent one-degree-free structure of the unit mass. It is essential to formulate the movement equation to calculate the structure’s response and then solve it. According to the curves shown in Fig. 1, the movement equation of the equivalent one-degree-free structure is presented as (1) formula: Fig. 1: Properties of the nth-‘mode’ inelastic SDF system from the pushover curve. Fsn ( Dn , Dn ) D 2 n Dn u g (t ) Ln (1) In MPA method, the seismic response of each mode, from the push of building to the target displacement Where, Dn is the displacement response of the of that mode, is determined by uniform distribution of modal lateral force S n m n .Since the equivalent one-degree-free structure, Dn and maximum response of building is obtained through D n show the first and second derivatives of the the combination of each mode’s seismic response selected Dn , respectively, n is the structure’s with the appropriate modal combination law, the effects of higher modes would be examined. This frequency at the nth mode as well as time history of method is directly applied to estimate the Fsn the seismic acceleration. Can be obtained deformation demand (such as the roof displacement Ln and the relative displacement); however, additional through the division of normal axis of capacity curve considerations are needed in order to calculate the (base shear of the main structure) by the effective rotational plastic hinge and elements force. The principal assumptions of this method are un- modal mass ( M n* ) and u g is the ground motion. coupling and super position of the modal responses It is concluded form Fig .1 that stiffness of the in the building possessing non-rubber system, i.e. structure is a function of displacement’s amount and the principal assumptions of non-linear static direction. Through numerically solving the Eq. (1) method. or direct modeling of one-degree-free structure with MPA method allows the seismic demand evaluation the selected un-linear behavior, maximum to be achieved in two stages (Donald [5]): displacement of the equivalent one-degree-free a) Execution of several one-modal pushover structure Dn can be calculated. Thereafter, by means analyses for different modes so as to of Eq. (2), maximum displacement of the main determine the matching modal response in structure’s end is obtained: the final displacement level. b) Then, evaluation of the structure’s final u r ,n nr ,n Dn response through combining responses of (2) several modes conforming to appropriate modal combination law. Where, r ,n is the deformation of the structure’s end at the considered mode and Dn is maximum 4 CALCULATION OF HAZARD CURVE displacement of the equivalent one-degree-free PARAMETERS structure. Indeed, if authors suppose Capacity Spectrum Method (CSM) as a method based on spectrum analysis, MPA method can be considered as a method based on time history analysis. It is This section has dealt with the process of notable that for both mentioned methods, by calculating the structure’s probability of exceeding a application of one of the modal combination drift demand value by means of the logistic methods, the results obtained can be generalized to distribution in order to be able to easily assess other modes. probability of structure vulnerability. (DOE [4], Luco[8]). expression for the limit state frequency into (conditional) frequencies of exceeding the limit state for a given spectral acceleration (the adopted IM), and composing the results by integration over all spectral acceleration values: H LS .P S a S a ,c P S a S a ,c / S a y ..g S a ( y ).dy P y S a ,c . dH Sa ( y ) (7) Where S a represents the IM-based demand, S a ,c represent the limit state capacity also expressed in Fig .2: A typical hazard curve for spectral spectral acceleration terms, and represents the acceleration. It corresponds to a damping ratio equal seismicity rate. We have used Eq. (8) in order to to and a structural fundamental period of 0.95 express the PDF of spectral acceleration in terms of seconds % 5 the increment in the spectral acceleration hazard. Coordinate of two end points considered hazard zone in relation Eq. (3) and the following system of P y S a y y g Sa ( y ) lim equations can be solved. y y P S a y y P S a x dG Sa ( y ) dR Sa ( y ) lim H 1 (Sa ) 0 (Sa )1 y y dy dy (8) H 2 (Sa ) 0 (Sa ) 2 (3) Where g Sa ( y ) is probability density function (PDF) at spectral acceleration value x, and, RSa ( y ) is the Eq. (3) on both sides of the divided complementary cumulative distribution function (CCDF) at S a y . H 1 (Sa ) (Sa )1 H (Sa ) (Sa )1 1 (4) We assume that the spectral acceleration capacity is H 2 (Sa ) (Sa )2 H 2 (Sa ) (Sa )2 a logistic variable with the following statistical With logarithms on both sides of the Eq. (4) parameters: H 1 (Sa ) median (S a , c ) Sa ,c ln (5) (9) 2 H (Sa ) ln(S a ,c ) S a ,c (Sa )1 ln (Sa ) 2 We can observe that the first term in the integral P[ S a S a ,c ] can be also interpreted as the CDF of And form there the spectral acceleration capacity at S a y : H 1 (Sa ) GS a ,c (Y ) P Y S a ,c H 1 (Sa ) 0 (Sa )1 0 (6) (10) (Sa )1 Since S a ,c is assumed to be a logistic variable, the 5 Annual Frequency of Exceeding a Limit State, the IM-based approach corresponding CDF can be expressed in terms of the standardized logistic CDF: In this section we are going to derive the annual frequency of exceeding a limit state, H Ls , by following an IM-based approach. The total probability theorem (TPT) is used to decompose the y 2 1 ln y ln S a ,c ln 2 S a ,c 1 S a ,c , . y S a ,c L G S a ,c ( y ) P (11) 2 S a ,c H LS .P S a S a ,c 0 (cos( ) isin( )) (17) . .y .dy In order to be able to integrate Eq. (7), we use y . S (1 cos( ) isin( )) 2 a ,c integration by parts and transform the equation into the following form: We transform the integrand into a complete square term and take all the constant terms outside of the y integrand: ln S a ,c H LS .P S a S a ,c L .dH ( y ) sa 9 2 S a ,c Sa ,c 2 , ln Sa ,c 4 (12) y 1 ln y Sa ,c 2 ln Sa ,c ln ( )2 S a ,c 2 Sa ,c d L .H ( y ) sa S a ,c cos( ) i sin( ) , cos( ) i sin( ) H LS .P S a S a ,c The derivative of the standard Logistic CDF can be 0 . . 1 . cos( ) i sin( ) .dy (18) calculated as: 0 y .Sa ,c 1 cos( ) i sin( ) 2 y The term inside the integral is itself the derivative ln d S a ,c 1 .l ln y ln S a ,c for a standard Logistic CDF: L (13) dy S a ,c y . S a ,c S a ,c H LS .P S a S a ,c 0 . . d .L cos( ) i sin( ) .dy (19) 0 dy 1 cos( ) i sin( ) 2 After the derivative of the Logistic CDF in Eq. (13) is substituted in Eq. (12), and the hazard term is Noting that the integral is equal to unity, the limit replaced by the power-law approximation from Eq. state probability can be derived as: (14): (14) H LS P S a S a ,c 0 . . (20) 0 . Sa ,c . cos( ) i sin( ) H LS .P S a S a ,c We can observe the power-law term outside the 0 ln y ln S a ,c (15) y .S .l S exponential is equal to the frequency of exceeding .y .dy (i.e., hazard) a spectral acceleration equal to the a ,c a ,c median spectral acceleration capacity: If we substitute the expression for the Logistic PDF H LS P S a S a ,c H Sa (Sa ,c ). cos( ) i sin( ) in Eq. (16) into the Eq. (15): (21) u e l (u ) (16) (1 e u ) 2 We can argue that H S a (S a ,c ) is a first-order Annual Frequency of Exceeding a Limit State considering the uncertainty approximation to the limit state probability and the Beta (UH) H (Sa) Med H (Sa) Exp (RC) Exp (UC) H (LS) term 2 .S2a ,c / 2 , cos() i sin() is a Logistic Distribution magnifying factor that accounts for the sensitivity of 0.5 0.002177 0.00227 1.04571 1.00297 0.002381 the limit state probability to the randomness in the Log-Normal Distribution spectral acceleration capacity. 0.5 0.002177 0.00247 1.0395 1.0027 0.00257 6 RESULTS AND DISCUSSION Presented in this section the results on the structure Table 4. Annual Frequency of Exceeding a Limit State, the IM-based Mpa desired and calculated probability of structural approach with using Interpolation the Main Hazard Curve vulnerability using two probability distribution MPA Drift – Structural statistical parameters Logistic and Log-normal pay. K0 K Median C Beta C Note: Two methods have been used to deliver results. 0.0015 1.0442 0.7 0.424 a. Using Power-Law Method Annual Frequency of Exceeding a Limit State b. Using the Interpolation Main Hazard Curve H (Sa) EXP H (LS) Logistic Distribution Table 2. Annual Frequency of Exceeding a Limit State, the IM-based 0.000933 1.0179 0.00095 approach with using Power-Law Method Log-Normal Distribution MPA Drift – Structural statistical parameter 0.000933 1.0397 0.00097 K0 K Median C Beta C 0.0015 1.0442 0.7 0.424 Table 5. Annual Frequency of Exceeding a Limit State considering the uncertainty, the IM-based approach with using Interpolation the Main Hazard Curve Annual Frequency of Exceeding a Limit State MPA Drift – Structural statistical parameters considering the uncertainty H (Sa) EXP H (LS) Logistic Distribution K0 K Median C Beta (RC) Beta (UC) n 0.002177 1.0179 0.002216 0.0015 1.0442 0.7 0.424 0.1278 30 Log-Normal Distribution 0.002177 1.0397 0.002263 Annual Frequency of Exceeding a Limit State considering the uncertainty Table 3. Annual Frequency of Exceeding a Limit State considering Beta (UH) H (Sa) Med H (Sa) Exp (RC) Exp (UC) H (LS) the uncertainty, the IM-based approach with using Power-Law Method Logistic Distribution 0.5 0.00933 0.00973 1.04571 1.00297 0.010205 Log-Normal Distribution MPA Drift – Structural statistical parameters considering the uncertainty 0.5 0.00933 0.01057 1.0397 1.003 0.01102 K0 K Median C Beta (RC) Beta (UC) n 0.0015 1.0442 0.7 0.424 0.1278 30 7 CONCLUSION of Energy Facilities”, DOE-STD-1020-94, U. S. Dept. of Energy, Washington, D. C. This paper was a formula to calculate the probability 5. Donald E.Grierson ,yanglin Gong , Lei Xu of structural vulnerability (Annual Frequency of (2006) “ Optimal performance based seismic Exceeding a Limit State, the IM-based) using design using modal pushover analysis “ Logistic probability distribution is obtained and its journal of Earthquake Engineering results with the results of Log-normal probability ,Vol.10,No.1(2006) 73-96 distribution to be compared. 6. FEMA 350 (2000). “Recommended seismic According to investigations conducted by the author design criteria for new steel moment-frame determined that the log normal probability distribution, buildings.” Report No. FEMA-350, SAC the probability of the vulnerability of structures to be Joint Venture, Federal Emergency very conservative calculation. According to the research Management Agency, Washington, DC. was clear that the likely vulnerability of structures using the Log normal distribution approximately 20% of the 7. FEMA 352 (2000). “Recommended post- structures more vulnerable than the Logistic distribution earthquake evaluation and repair criteria for shows (Table. 2, 3, 4, 5). welded steel moment-frame buildings.” But according to studies carried out suggest the author to Report No. FEMA-352, SAC Joint Venture, calculate the probability distribution Logistic Federal Emergency Management Agency, vulnerability of structures to be used because in a Washington, DC. certain specified probability distribution Logistic 8. Luco, N.; Cornell, C. A. (1998) “Seismic more accurate than the Log-normal distribution drift demands for two SMRF structures with offers. The fact considers less risk, but designing brittle connections”, Structural Engineering structures in discussion of economic problems and World Wide 1998, Elsevier Science Ltd., desirable structural safety with regard to reasonable Oxford, England, 1998, Paper T158-3. risk of the most essential things. Therefore it is 9. Valley, M.T., Harris, J.R.(1998).” suggested to calculate the probability of structural Application of model techniques in a vulnerability of the Logistic distribution probability pushover analysis”.6th US National distribution simple, symmetrical, and the likelihood Conference on Earthquake Engineering. and ability to accurately calculate the probability of occurrence and the response it has caused (Bader taking some reasonable risks) instead of Log normal probability distribution use. REFRENCES 1. Banon, H., Cornell, C.A., Crouse, C.B., Marshall, P., Nadim, F., and Younan, A. H., (2001) “ISO seismic design guidelines for offshore platforms”, OMAE2001/S&R-2114, Proceedings, 20th OMAE, Rio de Janiero, June, 200l 2. Chopra,A.K.,Goel , R.K.(2001).” A modal pushover analysis procedure for estimating seismic demand for building:theory and preliminary evaluation “.[ Report No.PEER- 2001/03].Pacific Earthquake Engineering Research Center,University of California , Berkeley 3. Chopra,A.K.,Goel , R.K.(2002).” A modal pushover analysis procedure for estimating seismic demand for building:theory and preliminary evaluation “.J. Earthquake Engineering and Structural Dynamics,31,561-582. 4. DOE (1994). “Natural phenomena hazards design and evaluation criteria for Department

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