On the evaluation of probabilistic displacement spectra

Transcription

1 On the evaluation of probabilistic displacement spectra L. Decanini. L. Liberatore, F. Mollaioli Department of Structtwal and Geotechnical Engineering University of Rome "La Sapienza", Italy Abstract In this paper, on the basis of a large set of ground motion records. the result of a statistical analysis on the elastic displacement spectra and the comparison between different theoretical probabilistic distribution will be presented for a selected number of vibration periods. The following four theoretical distributions have been taken into account: Normal. Lognormal. Gamma and Gumbel type I. Since the best fit to the data was provided from the Lognormal distribution, in this paper lognormal displacement spectra will be defined, for different levels of probability of non-exceedance. for the estimation of the displacement demands to structures located on different local soil condition. at different distance from the causative fault, and for different levels of magn~tude. 1 Introduction In recognition of the need for new methodologies in the evaluation of seismic demands, alternative to the usual force-based methods, new approaches such as displacement-based and energy-based are in development varying in complexity and applicability. The former approach is based on the assumption that an adequate damage control can be achieved if deformation are controlled, while the latter is based on the premise that the damage potential of an earthquake is closely associated to the energy input to a structure and its energy dissipative capacities. However, since displacement and energy are strictly connected (Fajfar & Gaspersic [l], Teran-Gilmore [2], Decanini et al. [3]; Decanini & Mollaioli [4]) a comprehensive approach which include both procedures may be more adequate for a correct understanding of the behaviour observed during

2 4 Enrtlzyuake Resrstant Eng~neemg Structwes If1 severe earthquakes, and for the design of new structures and the retrofitting of existing ones. In this context, it is important to acquire new knowledge on the displacement spectral quantities and their correlation with the energy-based parameters in order to quantify the demand imposed on structural systems. A number of researchers have investigated in the past procedures to determine reliable estimates of elastic and inelastic displacement demands. More recently, this issue has gained a considerably growing attention (Priestley & Calvi, [5]; Bommer & Elnashai, [6]; Tolis & Faccioli, [7]). The analysis carried out in this paper is focused on the probabilistic aspect correlated with the displacement demand, defined as a function of three important features of the earthquake ground motion, named soil condition, magnitude and distance from the causative fault. 2 Elastic displacement spectra Dynamic analyses of 5% damped SDOF systems subjected to a large number of strong ground motion were computed to estimate the displacement demand. The records were grouped according to three parameters: magnitude (M), minimum distance from the surface projection of the fault rupture (Df) and soil site category. In this context elastic displacement spectra were obtained from 247 time histories (Decanini et al.. [g]) recorded on three different soil types (Sl, rock or stiff soil; S2, intermediate soil; S3, soft soil), at source-to-site distances up to 106 km, after earthquakes of magnitude varying between 5.4 and 7.1. The distribution of the recordings according to soil class and magnitude is indicated in Table <M< <M<7.1 Table 1. Distribution of strong motion records. Soil S Soil S Soil S The source-to-site distance subdivision considers four different intervals: 0<Df<5 km; %D612 km; 12<D630 km; Dp-30 km. The range D+O, which corresponds to the whole distance range, was also included. Such subdivision is based on the study of the influence of distance, magnitude and soil type on the input energy (Decanini & Mollaioli, [9]). Particularly it was found that input energy increases discontinuously with magnitude, resulting that energy does not vary significantly in certain magnitude ranges. This trend can be encountered also for displacement. In order to reduce at acceptable levels the influence of the recording instruments, elastic displacement spectra were derived in the period range 0-4 S. In this range, which includes a large number of structural systems, the adopted processing method does not affect significantly the displacement spectrum (Bommer & Elnashai, [6]; Tolis & Faccioli, [7]).

3 Earthquake Res~stant Engrnee~.rng ~Ytructures On the basis of the computed elastic displacement spectra, design displacement spectra were proposed (Decanini et al., [8]). Elastic design spectra have been defined so as to approach the mean plus one standard deviation through a simplified bilinear shape in the range 0+4 S, with the first line increasing from zero up to the maximum displacement 6h, and then remaining constant up to 4 S. The proposed maximum design displacements, which vary with soil condition, magnitude and distance from the fault, approach, as mentioned above, the mean plus one standard deviation, that corresponds to a probability of non-exceedance of 84% in case of normal distribution. If the assumption of normal distribution of the data is improper, 6D, correspond to a different probability of non-exceedance; in this context it seemed necessary to compare the proposed design displacement spectra with the probabilistic displacement spectra carried out in this paper. 3 Statistical and probabilistic characterization of elastic displacement The displacement demand at a given site and for a given structure vary from one ground motion to another. This variability may be encountered from one earthquake to another, from one site to another, for different location of structures related to the propagation of seismic waves, and so on. The resulting time histories are then characterized by different amplitudes, duration, frequency content, which are due to the acceleration pulses and their sequence within the record. As their distribution and amplitude is unknown a priori, the derived displacement demand may be considered as a random variable. There are different ways to deal with the uncertainties involved in the evaluation of the lateral deformation demand. If the estimation is founded on strong motion records, it is possible to specify directly the probability of nonexceedance of the displacement demands without recurring to random vibration techniques to compute probabilistic elastic spectra or to artificial signals, which can be generated so as to match an acceleration elastic spectrum, as, in both cases, the variability of individual acceleration pulses and their distribution within the time history can not be reproduced (Miranda, [10]). Otherwise, if the number of available records is not ample, it could be possible to resort to synthetic signals, which can be generated by means of advanced numerical models taking into account earthquake source mechanism, propagation of seismic waves, and local soil and geological conditions (Panza et al., [l l]). Anyway, in this paper, the probabilistic analysis is based on the response of SDOF systems subjected to 247 real strong motion records and consist in a preliminary investigation to determine the distribution that better fits the actual distribution and in the definition of probabilistic displacement spectra; this topics will be discussed in the two following paragraphs.

4 6 Earthquake Reszstaizt Engineering Structzwes Probability distribution of spectral displacements Two commonly adopted probability models in civil engineering practice are the Normal and the Lognormal: the former is one of the most used model in applied probability theory, mainly because it represent those variable which arise as the sum of a number of random effects, no one of which dominates the other; the latter is widely used mainly for its skewed shape, that makes the lognormal distribution suitable for many kinds of data. Another broadly adopted distribution is the Gamma, which is limited to positive values and skewed to the right; its shape depend on the parameter k, for kll the Gamma distribution is not bell shaped, but decreasing; particularly, for k=l it is equal to the Exponential distribution. Finally, Tipe I distribution of largest value (Gumbel) is expected to fit well some kinds of data since it has a positive skewness coefficient and un upper tail falling exponentially. The four theoretical probabilistic distribution adopted in this study are described by the following probability density functions: Normal 1 a G ;!:!!I] Lognormal 1 f (4 = exp---in-. sal, & Gamma Gumbel type I f (6) = a ex& a(6 - v) - e-a~6-v)]. (5) Where m, o, p, ol,, k, h, a and v are probability distributions parameters estimated directly from observed data and are defined as follows: m = mean; o = standard deviation; p = median; ol, = standard deviation of ln(6); k = (mlo)'; h = Wm; a = iz/&o ; v G m ~~ (Benjamin & Cornell [12]). The comparison between the observed values (computed elastic displacement) and the values corresponding to the considered theoretical distribution was made for eight periods of vibration: 0.5, l, 1 S, 2, 2.5, 3, 3.5, 4 S.

5 Earthquake Resistant Engineermg Strzictures The case relative to the maximum displacement demand, independently from the corresponding period, was also examined. For soil S1 and S3, the whole set of available records (independently from the distance), grouped according to the two magnitude intervals, was considered, while for soil S2, two different distance interval (D65 km; 12<Df<30 km) were also taken into account for both magnitude ranges. In order to verify the assumption that the data are distributed according one of the mentioned theoretical probability density function, method based on the goodness-of-fit plots has been adopted. Diagrams of expected values, as a function of experimental data, according to the examined distributions (Q-Q plot) and the corresponding deviations (detrended Q-Q plot), have been plotted. Generally, the percentage scatter of the data does not varies significantly with period, and does not present a clear trend. From the analysis of the goodness-of-fit plots it was found that the Gamma and the Lognormal distributions give the best fit to the real data. Particularly, for the latter the theoretical expected values differ significantly from the experimental values only for the upper bound of the considered intervals, as shown in Figure 1, where Lognonnal Q-Q plot, with the corresponding detrended Q-Q plot, are reported for a period equal to 2 S, for soil S3. Observed Value (cm) Observed Value (m) Figure 1: Lognormal Q-Q Plot (a) and detrended Lognormal Q-Q Plot (b); Soil S3, 6.5<M<7.1, T=2 S. The Gamma distribution can also provide a good fit to the observed data, especially in the few cases in which the histogram is not bell shaped; in fact in those cases the shape parameter k of the gamma distribution is less then 1, therefore the theoretical distribution present a decreasing shape, similar to that of the histogram. In addition to the goodness-of-fit plots, the Kolmogorov-Smimov test (K-S) was also used. In this paper, the K-S test was adopted as it has the advantage, to the respect of the Chi-square test, to be independent on the adopted class, and, consequently, to be valid for continuous distributions of data; moreover, unlike the Chi-square test, it works well for all sample size (Benjamin & Comell [12]). The K-S test considers the evaluation of a parameter D which represents the

6 8 Eai-thquake Resistant Engineeving Strucrwes Ill maximum absolute deviation between the actual (observed), F*, and hypothesized cumulative distribution function, F : The comparison of this coefficient with the critical one, for the desired significance level a, is the basis for the acceptance or rejection of the hypothesis that the theoretical distribution could represent the real distribution. In Tables 2, 3 the K-S parameter is tabulated for soil S 1 and S3, respectively, for magnitude in the range between 6.5 and 7.1; the critical value of D is also reported for two different level of significance: a=o. 1 and a=o.o5. In Figures 2, 3 the correspondent cumulative distribution functions are reported for a period equal to 2.5 seconds. Table 2. Kolmogorov-Smimov parameter: Soil S1, 6.5<M<7.1. Table 3. Kolmogorov-Smirnov parameter: Soil S3,6.5<M<7.1. In general, the K-S test confirmed that the Lognormal distribution can supply a good fit to the actual data, even taught, in some case the Gamma distribution can be suitable as well. Anyway, on the basis of the results derived for each class of soil, magnitude and distance, it seems reasonable to resort to the Lognormal distribution in order to represent the experimental data.

7 Eczvthyuake Resistnnt Engirzeevq Stv~lctwes "f x real Figure 2:Cumulative Density Function: Soil S 1, 6.5<M<7.1, T=2.5 S (cm) Figure 3:Cumulative Density Function: Soil S3, 6.5<M<7.1, T=2.5 S. 3.2 Probabilistic elastic displacement spectra On the basis of the Lognormal distribution, displacement spectra having 50, 70, 80, 84, 90, 95 constant probability of non-exceedance were calculated for each soil condition and each range of magnitude and distance. Such spectra were then compared with the design displacement spectra, with the mean and mean+ l o spectra, and the spectra obtained as envelope of the maximum displacement values for each period. Generally, the mean spectra correspond to percentiles higher than 50%, and for 5.4<M<6.2 they reach a 70% percentile. For highest percentiles (90-95%) the cumulative probability densities show small increases with displacement, therefore displacement with percentile of 90-95% can be greater then maximum dispalcements. The mean + l o spectra (fractile 0.84 of the normal distribution) do not show significant deviations f?om the spectra corresponding to the 84% probability of non-exceedance, with reference to the Lognormal distribution, for magnitude between 6.5 and 7.1; while for the lowest magnitude range the deviation is

8 10 Earthquake Resistant Engineering Structz~res Ill generally about 15-20%, but reaches the 27% for soil S1 in the third interval of distance for T=3.6 S, and the 36% for soil S2 in the first interval of distance for T = 2.9 S. In Table 4 the maximum spectral displacements, grouped for soil condition, range of magnitude and distance from the fault, are reported for 50, 70, 80, 90% probability of non-exceedance together with maximum proposed design displacements, 6D,,. In Figures 4a, 4b probabilistic spectra are reported, together with the proposed design spectra, for soil S1 and S3, in the higher range of magnitude ( ), for source-to-site distances less then 5 km. Table 4. Maximum spectral displacement (cm) for probability of non-exceedance of 50, 70, 84, 90% and maximum design spectral displacement. -a- Dr>30 km DpO km ( From the reported values of maximum spectral displacement it is evident the great influence of the choice of a certain probability of non-exceedance, especially for high values of the fractile: the ratio between the displacement demand corresponding to a fractile of 0.9 and those corresponding to a fractile of 0.84 can be, in some cases, greater then 1.4. About the comparison between the proposed displacement spectra and the constant percentile spectra it is worthwhile to point out that the increasing branch of the design displacement spectra corresponds to fiactiles variable with the period of vibration, but always higher than 0.70, while the constant branch generally corresponds to fractiles between 0.80 and 0.90 for each soil condition. In conclusion, due to the fact that the bilinear shape of the design spectrum corresponds to a simplified shape of the actual displacement spectrum, it was not possible to associate a single fiactile in the whole period range. Anyway, it is possible to assert that the maximum spectral displacements (constant branch of

9 Earthquake Resistant Engineering Structzres the design spectrum) are included in a limited range of eactile values, between 0.80 and Figure 4: ~robabilistic displacement spectra 6.5<M<7.1; ~65krn; soil S1 (a); soil S2 (b). Finally, it seems important to remark the noticeable influence of soil condition, magnitude and distance fiom the fault on the displacement demand. Particularly, it was found that there is a large difference between near-fault and other distance ranges in terms of maximum displacements. 4 Conclusions Probabilistic elastic spectra computed in this study provide a rational estimation of displacement demand, since they allow to estimate the displacement corresponding to target probabilities of non-exceedance, corresponding to a certain theoretical distribution, for different soil condition, different level of magnitude and distance fi-om the fault. Providing that a suitable theoretical distribution is found, the adoption of a theoretical distribution to compute response spectra seems to be more adequate then defining them directly fi-orn the actual distribution; in fact, the use of a theoretical distribution permits to eliminate singularity in the actual response spectra. In this study the Lognormal distribution was found to supply a good fit to the observed displacement. Therefore probabilistic spectra with constant probability of non-exceedance of 50, 70, 80, 84, 90, 95%, according to the Lognormal distribution, were computed. The values of computed maximum spectral displacement highlighted the great influence of the choice of a certain probability of non-exceedance, especially for high values of the fiactile; this is a direct consequence of the cumulative density function shape, that for high values of the fi-actile can present a very low slope.

10 12. Eartlzqzinke Reszstant Enpneer~ng Strz~rures II! The comparison between the computed probabilistic spectra and the design displacement spectra displayed that, even though the design displacement spectra have a simplified bilinear shape and are defined so as to approach the mean plus one standard deviation, the design displacements correspond, with reference to the Lognormal distribution, to fractiles in the range of 0.8e0.9 in the constant branch of design spectra, and to fractiles larger then 0.7 in the first branch. References Fajfar, P., Gaspersic, P. The N2 method for the seismic damage analysis of RC buildings. Earthquake Engineering and Structural Dynamics, 25, , Teran-Gilmore, A. Performance-Based Earthquake-Resistant Design of Framed Buildings using Energy Concepts. Ph.D. Dissertation, Department of Civil Engineering, University of Cal$ornia at Berkeley, Decanini L., Mollaioli, F., Saragoni, R. Energy and displacement demands Imposed by near-source ground motions. Proc. of the 12Ih World Conference on Earthquake Engineering, January 29'h-~ebruary 4Ih 2000, New Zealand, paper 1136/6/A, Decanini, L., Mollaioli, F. An energy-based methodology for the assessment of the seismic demand. Soil Dynamics and Earthquake Engineering, 21, 2, pp , Priestley M.J.N., Calvi G.M. Concepts and procedures for direct displacement-based design and assessment, Seismic Design Methodologies for the Next Generation of Codes, Fajfar & Krawinkler (eds), Bakema, Rotterdam, Bommer, J.J., Elnashai, A.S. Displacement spectra for seismic design. Journal of Earthquake Engineering, Vol. 3, No. 1, pp. 1-32, Imperial College Press, Tolis, S.V., Faccioli, E. Displacement design spectra. Journal of Earthquake Engineering, Vol. 3, No l, pp , Decanini, L., Liberatore, L., Mollaioli, F. Definizione di Spettri di Spostamento in Funzione di Parametri Caratteristici del Moto del Suolo. Atti del 9' Convegno Nazionale "L' Ingegneria Sismica in Italia", Torino, Italy, September Decanini, L. D., Mollaioli, F. Formulation of Elastic Earthquake Input Energy Spectra. Earthquake Engineering and Structural Dynamics, 27, [l01 Miranda E. Probabilistic site-dependent non-linear spectra, Earthquake Engineering and Structural Dynamics, 22, , [l l] Panza, G.F., Romanelli, F. and Vaccari, F. Seismic wave propagation in laterally heterogeneous anelastic media: theory and applications to the seismic zonation. Advances in Geophysics, Academic press, 43, 1-95, [l21 Benjamin, J.R., Comell, C.A.. Probability, Statistic, and Decision for Civil Engineers. McGraw-Hill Book Company, 1970.