Constraining paleo PGA values by numerical analysis of overturned columns

Transcription

1 EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2010; 39: Published online 1 September 2009 in Wiley InterScience ( SHORT COMMUNICATION Constraining paleo PGA values by numerical analysis of overturned columns Gony Yagoda-Biran 1 and Yossef H. Hatzor 1,2,, 1 Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 2 Department of Structural Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel SUMMARY The estimated peak ground acceleration (PGA) for a specific region cannot always be determined reliably from compilation and statistical analyses of instrumental data acquired because of the relatively short time window available for such an approach, typically up to 100 years of instrumentation. We propose a complimentary approach for estimating PGA in a specific region, through back analysis of seismically driven column collapse in historic monuments, using the numerical discrete element discontinuous deformation analysis (DDA) method. Preliminary threshold paleo PGA values thus obtained constrain the lower bound of PGA estimates using information from a much broader time window, in the case study presented here of approximately 1200 years. Copyright q 2009 John Wiley & Sons, Ltd. Received 24 December 2008; Revised 5 July 2009; Accepted 13 July 2009 KEY WORDS: numerical analysis; discrete element methods; DDA; PGA; paleo-seismicity; historical monuments 1. INTRODUCTION The conventional method for assessing seismic risk in specific regions is through statistical analysis of empirical relationships between earthquake magnitude, epicenter distance, peak ground acceleration (PGA), and response spectra at a given site, as expressed in seismic building codes. Statistical analyses of existing data from earthquakes can provide suggested attenuation relationships for the regions at hand, on the basis of recorded data (see [1]). In some regions, however, recorded data are very scarce due to lack of instrumentation and seismic network infrastructure, and in all regions Correspondence to: Yossef H. Hatzor, Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. hatzor@bgu.ac.il Contract/grant sponsor: National Steering Committee for Earthquake Preparedness; contract/grant number: Copyright q 2009 John Wiley & Sons, Ltd.

2 464 G. YAGODA-BIRAN AND Y. H. HATZOR of the world recorded data never date back to before the late 19th century so that the data obtained are hardly representative of the true seismicity of the region over historic times. To overcome the limitation of narrow time window for instrumental data acquisition, alternative methods to estimate PGA in specific regions have been proposed. Brune [2], for example, introduced a method to assess upper bounds for PGA by performing a pseudo-static stability analysis of precariously balanced rocks in order to constrain earthquake intensities in Nevada and California. Similarly, a pseudo-static limit equilibrium analysis may be applied in the back analysis of recent or historic landslides, provided that the geometry of the slide (slip surface and original topography) and the available shear strength along the sliding surface are known [3]. Alternatively, the Newmark approach may be employed for the same purpose (e.g. [4]). With the advent of numerical tools over the past two decades it has become possible to apply a fully dynamic analysis in the time domain both for continuous as well as discontinuous media using the finite element (e.g. [5]), finite difference (e.g. [6]), and the distinct element [7] methods. Recently Kamai and Hatzor [8] introduced an approach to asses PGA values from inversion of mapped block displacement data in historic masonry structures. Their work focused on back analysis of key-stone displacements in ancient Roman and Byzantine arches. The advantage of using a fully dynamic approach is that it provides, in addition to PGA, constraints on the frequency and duration of the seismic ground motion that must have caused the preserved damage in the studied monuments. Furthermore, with a fully dynamic approach the response of the structure to real earthquake records can be studied as well. In this paper we continue with analysis of masonry structures as a means to asses historic PGA, but this time we focus on the dynamic rocking motion which ultimately culminates in toppling of massive stone columns found in a destructed Byzantine church along the eastern margins of the Sea of Galilee at the Susita site. A two-dimensional formulation of the discontinuous deformation analysis (DDA) method [9] is used here for this purpose. The seismic stability of columns in historic monuments has been studied extensively over the past several decades. Housner [10] was the first to investigate the response of a rocking rigid block as an inverted pendulum. More recently, the problem of a rocking column under harmonic base excitations as well as real earthquake excitations has been investigated analytically by Makris and Roussos [11] and numerically by Psycharis et al. [12] for a two-dimensional formulation. The response of a column to base excitations in three dimensions has been studied experimentally by Mouzakis et al. [13] using shaking table experiments, and numerically by Papantonopoulos et al. [14] and Psycharis et al. [15] using the distinct element code 3-DEC [7, 16]. An attempt to solve the very complicated problem of a wobbling motion of a cylindrical drum over a planar surface using a Lagrangian approach was recently proposed by Stefanou et al. [17]. DDA is an implicit discrete element method proposed originally by Shi [9] to provide a tool useful for investigating the kinematics of blocky rock masses. DDA models a discontinuous material as a system of individually deformable blocks that move independently with minimal amount of interpenetration. The formulation is based on dynamic equilibrium that considers the kinematics of individual blocks as well as friction along block interfaces. The displacement and deformation of the discrete blocks are the result of the accumulation of small time steps. The equilibrium equations are derived by minimizing the total potential energy of the block system with respect to the displacement at block center. In this study we use the original two-dimensional version of DDA in which each block i in the general block system has six degrees of freedom, and the resulting displacement components (u,v) of an arbitrary point (x, y) in the X and Y directions are derived using a first-order approximation. The algebraic equation for the increase in displacement is solved for each time increment by substituting the appropriate terms for acceleration and velocity

3 CONSTRAINING PALEO PGA VALUES 465 provided by a time integration formulation similar to Newmark direct integration method with parameters β=0.5 andγ=1.0, into the general equation of motion [18, 19]. The result is a system of equations for solving the dynamic problem which, after collecting terms on both sides, typically expressed as: ˆKD= ˆF, where K ij is a 6 6 coefficient sub-matrix, D i is a 6 1 deformation matrix of block i and F i is a 6 1 loading matrix of block i. Sub-matrices [K ii ] depend on the material properties of block i and sub-matrices [Kij] are defined by the contacts between blocks i and j. The validity and accuracy of DDA have been studied extensively over the past decade and a comprehensive review is presented by MacLaughlin and Doolin [20]. DDA validations pertaining to the specific goals of this research are reported by Kamai and Hatzor [8]. Three user-specified, numeric control parameters are required in DDA: The normal contact spring stiffness (g0), the time step size (g1), and the assumed maximum displacement per time step ratio (g2). The possible range for these control parameters in relation to the size of the problem is discussed by Shi [9], and optimal values for dynamic applications are reported by Tsesarsky et al. [21]. 2. DYNAMIC ROCKING OF A FREE STANDING COLUMN Makris and Roussos [11] studied the problem of dynamic rocking of a column subjected to a sinusoidal input acceleration and their analytical solution is briefly reviewed below for the free body diagram shown in Figure 1(A). The centers of rotation of the free standing column can be either0or0. Assuming there is no vertical base acceleration, the equations of motion are [11]: I 0 θ+mgrsin( α θ) = müg R cos( α θ), θ 0 (1) I 0 θ+mgrsin(α θ) = müg R cos(α θ), θ>0 (2) where I 0 is the mass moment of inertia, m the block mass, and ü g the ground acceleration. All the geometrical parameters are defined in Figure 1. Inserting the definition of I 0 into (1) and (2), introducing the parameter p (p = 3g/4R), linearizing the equations due to a tall and slender geometry of the column (small α), using a sinusoidal input ground acceleration in the form of ü g (t)=a p sin(ω p t +ψ) (where a p, ω p and ψ are the amplitude, frequency, and phase when rocking initiates, respectively,) and integrating the equations yields [11]: 1 θ(t) = A 1 sinh(pt)+ A 2 cosh(pt) α+ 1+(ω 2 p /p2 ) 1 θ(t) = A 3 sinh(pt)+ A 4 cosh(pt)+α+ 1+(ω 2 p /p2 ) a p g sin(ω pt +ψ), θ 0 (3) a p g sin(ω pt +ψ), θ>0 (4) where the coefficients of integration as well as the complete solution are presented by Makris and Roussos [11]. The dynamic equations for the angular velocity are [11]: ω p θ(t) = pa 1 cosh(pt)+ pa 2 sinh(pt)+ 1+(ω 2 p /p2 ) ω p θ(t) = pa 3 cosh(pt)+ pa 4 sinh(pt)+ 1+(ω 2 p /p2 ) a p g cos(ω pt +ψ), θ 0 (5) a p g cos(ω pt +ψ), θ>0 (6)

4 466 G. YAGODA-BIRAN AND Y. H. HATZOR (A) (B) Figure 1. Validation of the dynamic column rocking problem: (A) Sign convention and (B) DDA (open circles) vs analytical (solid line) solutions. The solution for the dynamic rocking of a column subjected to an input loading function of a half-sine pulse is obtained in two stages: (1) Instantaneous response dynamic motion that takes place simultaneously with application of the input acceleration function: ü g (t)=a p sin(ω p t +ψ) from t =0to0.5s, where here ω is 2π, and the phase angle (ψ) is ψ=sin 1 (αg/a p ), (2) Consequent motion rocking oscillations after pulse termination from t =0.5s and onwards. Naturally when the pulse terminates the input acceleration diminishes (ü g (t)=0, hence a p =0) and the coefficients of integration are updated for changing rotation angle and angular velocity. Furthermore, following each impact (@θ=0), the angular velocity and the coefficients of integration are recalculated as well. The analytical and DDA solutions for column width and height of b =0.2m and h =0.6m are presented in Figure 1(B). In the upper panel of Figure 1(B) results obtained for amplitude a p =5.43m/s 2 (0.5535g), a value slightly lower than required for overturning, are shown and hence only column rocking is obtained. In the lower panel of Figure 1(B) results obtained with an

5 CONSTRAINING PALEO PGA VALUES 467 amplitude of a p =5.44m/s 2 (0.5545g), the minimum value required for overturning, are shown and indeed column overturning is obtained 1.5s after pulse termination. In the DDA model the column rests on a fixed base and is subjected to dynamic input at its centroid. The friction angle along the interface is set to 89 to avoid sliding, as in the analytical solution which ignores sliding. The numerical control parameters used here are: energy dissipation coefficient=1, maximum displacement ratio=0.0075, time step size=0.0025s, normal contact spring stiffness= N/m, E =3GPa, and ν=0.25. A remarkably good agreement between the analytical and numerical solutions is indicated in Figure 1(B) as can be seen from the plotted numerical error that after initial perturbations rapidly decreases below 1%. 3. THE CASE OF THE BYZANTINE CATHEDRAL AT SUSITA The archeological site of Susita is located at the top of a hill capped by a basalt plateau, 350 m above and about 1 km east of the Sea of Galilee, a rhomb-shaped graben formed due to a left segmentation of the sinister Dead Sea fault system [22, 23] (see Figure 2(A)). Seismicity in the Sea of Galilee area is moderate [24, 25], with an estimated M 6 earthquake reoccurrence time interval in the order of 10 2 years increasing to 10 3 years for M 7 earthquakes [26]. The last strong event (M 6.2) was recorded near Jericho in 1927 [27], about 100 km south of Susita. In 1973 a M 4.5 earthquake occurred few km south of Susita site [28]. According to the national seismic building code [29] a horizontal PGA of 0.3g is estimated for this region with 10% probability of exceedance for every 50 years. Susita, Antiochia-Hippos by its Greek name, was founded during the Hellenistic period after 200 BC [30] and is believed to have been destroyed during the large earthquake of 749 AD [31]. The church columns are monolithic, consisting of red and grey granites transported most likely from the Aswan region in Egypt [30], as well as of some white and green marbles. One row S N 0.6 m 4.7 m 0.75 m 1 meter (A) (B) (C) Figure 2. (A) Location map for Susita; (B) The toppled columns preserved at the site; and (C) The DDA model for a characteristic Susita column.

6 468 G. YAGODA-BIRAN AND Y. H. HATZOR of collapsed columns that originally supported the roof of the church can clearly be seen today where the collapsed columns rest parallel on the ground surface, all pointing in the same direction (Figure 2(B)). Two assumptions are made in the numerical approach: (1) we consider dynamic excitation of free standing columns, the results thus obtained pertain to free standing columns and not to the entire structure that probably included an architrave and a roof; (2) the dynamic analysis is restricted to two dimensions while the actual columns are three-dimensional. The DDA mesh used in this research for a typical Susita column is presented in Figure 2(C). The input material properties for the modeled column are: E =40GPa, ν=0.18, ρ=2700kg/m 3,and friction angle between column base and pedestal =45. The input numerical control parameters, except for the contact spring stiffens value, are as in the validation study. To find the optimal contact spring stiffness value, the numerical control parameter for which the DDA solution is most sensitive, an input peak acceleration value sufficiently small so as to avoid column toppling was applied for the modeled column geometry, in both analytical and numerical solutions. The numeric contact spring stiffness value was repeatedly adjusted until a good agreement was obtained between the DDA and analytical solutions for strictly rocking motion with no sliding or toppling, due to the low level of dynamic excitation used for this calibration exercise. The optimal contact spring stiffness value thus obtained was N/m; this is the value that was used in all further analyses for the dynamic response of Susita columns with DDA. Once optimal numerical control parameters are established, forward DDA modeling is performed where for each modeled input frequency the amplitude is increased until column toppling is obtained. The required PGA values for toppling obtained with DDA for one and three cycles of sinusoidal input functions are presented in Figure 3. Under a single loading cycle the required PGA for overturning clearly increases with input frequency (Figure 3(A)). The required PGA for overturning also increases with frequency for three loading cycles, but at a smaller rate (Figure 3B), and attains a steady state value of 1g for 3 Hz frequencies and above. Results of previous studies [32] suggest that for 1 Hz frequencies or below the required PGA for overturning is independent of the number of loading cycles, as the column will overturn during the first cycle or soon after it. For higher frequencies, however, three loading cycles seem to better represent earthquake thresholds [32]. It may seem more appropriate therefore to adopt the numerical results obtained for three cycles (Figure 3(B)) if only sinusoidal input motions are considered. The range of threshold PGA values thus obtained is 0.2g<Threshold PGA<1g (A) PGA, g 1 input loading cycle Column Toppling Frequency, Hz 4 Safe 5 PGA, g (B) 3 input loading cycles Column Toppling Frequency, Hz 4 Safe 5 Figure 3. Required PGA for overturning the modeled Susita column under one (A) and three (B) input loading cycles. Solid triangles stable columns, solid diamonds overturned columns.

7 CONSTRAINING PALEO PGA VALUES DISCUSSION Threshold paleo PGA values obtained with DDA for sinusoidal input functions clearly exhibit frequency dependency. A possible way to constrain the wide threshold PGA range obtained for sinusoidal input functions would be by assuming that the columns failed when rocking at their resonance frequency. However, as already shown by Housner [10] in his pioneering work, a resonance frequency for a free standing column does not exist. Consider for example the top panel of Figure 1(B) where the plotted ratio θ/α presents half periods that are getting shorter and shorter with time due to energetic damping and application of coefficient of restitution upon impact in the analytical solution (see [11]). A shorter period with time implies increasing frequencies over time and hence the resonance frequency of the motion cannot be assumed to be constant throughout the analysis. Furthermore, the initial half period or frequency of the motion depends upon the magnitude of the input. Therefore, this approach for constraining threshold PGA values is restricted as it provides a relatively wide range of possible paleo PGA values, in the case study analyzed here between 0.2 1g. A possible way to overcome the restrictions imposed by applying sinusoidal input functions would be to subject the modeled column to real accelerograms with realistic frequency content. For the problem at hand, ground motion records acquired during strong earthquakes triggered in tectonic settings similar to those as in the Dead Sea rift (DSR) system, namely where strike slip rather than normal or reverse faulting takes place, would provide the most relevant results. Possible candidates are the San Andreas Fault system in California, as well as strong earthquakes recorded along the DSR system. A characteristic earthquake record for the DSR system would have been ideal for such an approach, yet such a record does not exist in Israel due to the relatively low seismic intensity of the rift system over the time period during which ground motion data have been acquired instrumentally (past 50 years). Furthermore, even if a design or characteristic earthquake record for the DSR system did exist, the actual ground motions at any specific site would largely depend upon the magnitude of the event, the distance between the site and the source, and specific site conditions. The second best choice would be to subject the studied structure to the strongest earthquake record generated by the DSR and measured instrumentally during the past 50 years, the 1995, M L =7.2 event with epicenter in the Red Sea (Nueiba) measured at the city of Eilat at a distance of 60 km from the source [33]. Threshold PGA values for a typical Susita column obtained with such an approach are presented in Table I. As our analysis is restricted for now to two dimensions, the selected accelerograms are applied twice for each earthquake, where in each simulation a different horizontal component (E-W or N-S) is aligned with the horizontal axis of the modeled column. In all simulations the vertical component is applied as well, so as to allow also for vertical motions in the analysis. Five earthquake records are used as input, the important parameters of which are listed in Table I, including publication source, magnitude, distance between epicenter to measurement site, predominant frequency of the record obtained by us using a MATLAB V. 6.5 FFT algorithm to compute the discrete Fourier transform of the published earthquake records, the peak horizontal PGA, and the required PGA for toppling obtained by us with DDA. The modeled time histories are applied to numerical loading points located at the centroid of the column and the pedestal (see Figure 2(C)), while the foundation block remains fixed. Each earthquake record is used twice as explained above, resulting in 10 different simulations. To determine the threshold PGA required for column overturning under the modeled earthquake the input records are up-scaled or down-scaled (by multiplying the entire record by a scalar)

8 470 G. YAGODA-BIRAN AND Y. H. HATZOR Table I. Threshold paleoseismic horizontal PGA obtained with DDA for toppling a characteristic Susita column using different earthquake records from similar tectonic settings. Predominant PGA in original Threshold Earthquake record Epicenter frequency (Hz) record (g) PGA (g) used for DDA Magnitude distance (km) EW NS EW NS EW NS Nueiba, Red Sea, (1995), M L = Fill response [33] Nueiba, Red Sea, (1995), M L = Rock response [34] Loma Prieta (1989), M L = Yerba Buena [35] SF Bay Area [36] M w = Imperial Valley, (1940), El Centro [35] M L = until column toppling is obtained with DDA, thus preserving the original frequency contents of the modeled earthquakes. The resulting threshold paleo PGA values thus obtained are shown in the two right columns in Table I. Interestingly, even though the predominant frequencies for the modeled earthquake records vary from 0.45 to 2.2 Hz, the threshold PGA boundaries remain within a relatively narrow range of g. The paleo PGA values thus obtained are much better constrained than those that would have been obtained with application of sinusoidal input functions. 5. SUMMARY AND CONCLUSIONS A new method to estimate paleo PGA, through back analysis of seismically driven failures in historic masonry structures using a numerical discrete elements method (DDA), is proposed. The DDA method is shown here to be adequate for such a task, as accurate results are obtained when compared with existing analytical solutions for the dynamic rocking of a column excited by a sinusoidal input function. DDA has significant advantages over the existing analytical solution as it allows for: (1) sliding at column base, (2) application of vertical motions, (3) application of true earthquake records with realistic frequency contents, and (4) modeling much more complex geometries involving multiple blocks for which analytical solutions do not exist. Modeling column response by studying sinusoidal input motions provides estimated PGA values that are highly dependent upon the input frequency. However, subjecting the columns to true acceleration time histories narrows the possible range of threshold PGA quite substantially. ACKNOWLEDGEMENTS The authors thank the Ministry of National Infrastructure of Israel for partial support of this research through a grant from the National Steering Committee for Earthquake Preparedness for the study of the seismic risk in the vicinity of the Sea of Galilee through contract Drs Rivka Amit and Oded Katz of Israel Geological Survey are thanked for their collaboration in the joint research program. Dr Gen-hua Shi is thanked for modifying the original DDA code for the purposes of this research. Dr Yuli Zaslavsky from the Geophysical Institute of Israel is thanked for providing the deconvoluted record of the Nueiba earthquake for rock response. Dr Yuri Karinski from the national building research

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