Fragility Surfaces as Measure of Seismic Performance

Fragility surfaces give the probability of failure for structural or non-structural systems as a function of two parameters, the earthquake magnitude and epicentral distance. These parameters provide a unique characterization of the ground motion. In contrast, the current characterization of the ground motion by a single parameter, e.g., the Peak Ground Acceleration (PGA), can be unsatisfactory since events with the same PGA can cause very different damages. In order to demonstrate the methodology, fragility surfaces are calculated for a 22,000 gallon water tank located at the 20 floor of a hospital in New York City. The analysis is based on the linear random vibration theory, Monte Carlo simulation and nonlinear dynamic analysis. Introduction Earthquakes are one of the most catastrophic events on earth. They are capable of causing loss of lives, damage to buildings and systems, and interruption of essential services such as electricity, gas, and water supply. Failure and movement of non-structural systems and components can cause much more damage than the failure of the structural systems themselves. It is crucial to evaluate the vulnerability of structural and non-structural systems to earthquakes, as this helps in predicting realistic economic losses. It also helps with planning emergency and recovery efforts. Most importantly, it helps in identifying the most vulnerable systems and components and determining the need for retrofitting and strengthening of such systems. It has been a common trend to express the fragility of structures to earthquakes as a relationship between the ground motion severity and structural damage. Fragility curves and Potential Damage Matrices (PDM) are the most common format of this relationship. Both methods describe the conditional probability of exceeding different levels of damage at given levels of ground motion severity. As the names could suggest, fragility curves express the data in a graphical format while PDM express it numerically. Professor, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853 Graduate Research Assistant, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853 Properties of both system and soil are embedded in the fragility curves and PDM. However the seismic hazard at the site is not reflected. Fragility curves and/or PDM can be integrated with seismic hazard information to give idea about the potential damage to the system. Fragility Curves Fragility curves give the conditional probability of exceeding different levels of damage usually referred to as limit states at different intensities of ground motion. Every curve gives this conditional probability for a specific limit state over a range of ground motion intensity. It has been proven that the uncertainty in the ground motion and soil properties at the site can be much more than the uncertainty in the structural performance of the system. Accordingly, many researchers have assumed deterministic parameters of the system. Some others considered variation in the system parameters in addition to the ground motion and soil properties. Uniform and lognormal distributions are among the most common models used with system parameters. Types of Fragility Curves Fragility curves can be developed for one specific system or for a class of systems. Based on the method used to generate them, fragility curves are classified as either analytical or empirical. Analytical fragility curves are generated using results of numerical simulations of the system under artificial or historical earthquake records. Empirical fragility curves are based on experimental results or damage data collected from the field after earthquakes. In some cases, opinions of experts and personal judges can be the basis for empirical fragility curves. Fragility curves have been developed for a great variety of structural systems over the last few years. Examples include RC frame/wall systems (Hwang 1990 and 1994 and Singhal 1996), RC structural walls (Sasani 2001), steel frames (Dimova 2000), and RC bridges. Unrestrained equipment and piping systems are examples of non-structural systems for which fragility curves have been developed (Chong 2000 and Mostafa 1999). The main challenges for analytical methods are generating artificial ground motions that are consistent with the site and relating numerical results of the simulation to predefined levels of damage. Scarcity of available data is the main challenge for the empirical methods. Characterization of Ground Motion Intensity It is of great importance to select a suitable indicator that describes the intensity or strength of ground motion against which fragility curves are plotted. The main properties of ground motion, including amplitude, frequency content, and duration of strong motion, are supposed to be reflected in this parameter. Any two or more ground motions having the same value of this parameter are supposed to have the same potential ability to cause damage. Different parameters have been considered by researchers to characterize ground motion. Peak Ground Acceleration (PGA) is the most common parameter used (Hwang 1990 and 1994a). Elastic response spectral ordinates Sa (spectral acceleration) and Sv (spectral velocity) at various frequencies have been used (Dimova 2000, Singhal 1996, and Sasani 2001), as have qualitative parameters such as the Modified Mercali Intensity (MMI) (Dumova-Jovanoska 2000). The main advantage of PGA and response spectral ordinates is the ease of estimation. MMI has the advantage over other parameters in that it has been recorded for old events for which other parameters cannot be calculated. However, for artificial records, it is not visible how to estimate such quantitative parameter for the generated motions. It has long been recognized, however, that PGA has poor correlation with both actual observed and theoretically computed damage in structures. In addition, elastic response-based measures, Sa or Sv are incapable of describing the ability of ground motion to cause damage to the structure through the various sequences and magnitudes of nonlinear response cycles. In order to explore the visibility of using PGA as a ground motion parameter, a group of 120 artificial records have been generated using the stochastic model described below. The epicentral distance was kept constant for all records and equal to 50 km. Six values of the magnitude were considered, ranging from 5 to 6 at increments equal to 0.2. Twenty samples have been generated for each magnitude value. The response of a linear SDOF system under each record was calculated and the maximum displacement, velocity, and acceleration were obtained for each case. The system has a frequency of 14 rad./sec and a modal damping ratio of 5%. Figure 1 shows the maximum response of the system against PGA for 20 samples with the same magnitude and epicentral distance. The figure and the values of the correlation coefficient reflect the lack of correlation between PGA and the maximum response of the system. For such simple system, PGA failed to represent the ability of the motion to affect the system. It is very unlikely for it to do the job with more complicated nonlinear systems. PGA has been shown to be a misleading and inadequate parameter to characterize the ground motion (Sewell 1989). It seems to be almost impossible to fully characterize the ground motion with one parameter. Fragility Surfaces Similar to fragility curves, fragility surfaces give the conditional probability of exceeding different limit states given the occurrence of earthquake of certain intensity. The only difference is that the ground motion intensity is characterized with two parameters, magnitude, M, and epicentral distance, D, instead of one parameter as the case is with fragility curves. Fragility surfaces are plotted against these two parameters. Figure 1. Maximum response vs. PGA Limit states represent different levels of damage and are characterized using one or more damage indices, DIi. The fragility surface corresponding to the limit state i is defined as follows: Fi (x, y) = P [DI > DIi | Mw = x, D = y] (1) where DI is the damage index and DIi is its value corresponding to the i th limit state. Probabilistic Model for Ground Motion The input parameters for this model are magnitude, epicentral distance, and soil properties. The model has been developed at the university at Buffalo and is based on the Specific Barrier model (Papagiorgio 1983a and 1983b). The ground motion acceleration, A(t), is modeled by a nonstationary stochastic process ) ( ~ ) ( ) ( t A t w t A ⋅ = (2) where t is the time, w(t) is a deterministic amplitude modulator envelope function and ) ( ~ t A is a stationary Gaussian process with a one-sided power spectral density g(ω). Numerical Example An example of a steel water tank located at the 20 floor of a hospital in New York City is considered to demonstrate the methodology of developing fragility surfaces for non-structural systems. The effect of adding bracing elements to the system on its seismic performance is explored by comparing fragility surfaces of the system with and without the bracing elements. Problem Definition The hospital consists of three sections: the plaza, the middle tower (core), and the main tower. The plaza is seven stories high while the two towers are twenty stories high. Figure 2 shows a photo for the plaza and the middle tower of the hospital and Figure 3 shows the location of the water tank on a schematic plan of the hospital. The hospital is a steel frame building with full and partial moment resistance connections. Most floors in the building are made of two-way concrete slabs. The building is supported by pile foundations. The water tank has dimensions of 20.3x16.8x4.8 ft. It is supported at its corners by four vertical legs each 6.83 feet high and a cross-section W 8 X 28. This representation is postulated because of lack of detailed information on the secondary systems of the hospital. The ground excitation is assumed to be a non-stationary stochastic process based on the Specific Barrier model. It is also assumed that the ground acceleration has only one component in the x direction (Fig. 3). The ground excitation depends on two parameters: magnitude and the epicentral distance of the earthquake event. The fragility surfaces are plotted against these two parameters. Fragility surfaces are generated for the water tank with three limit states. The limit states considered are DI > DIi, where DI is drift of the water tank legs in the x direction and DIi takes the values of 0.5, 2 and 10% of the height of the legs. Assumptions The primary structure remains linearly elastic during ground motion. Neither material nor geometrical nonlinearity is considered. The first and the fourth modes of the primary structure are dominant and are translation modes in the x direction. Therefore the four points supporting the water tank can be assumed to have equal displacements in x direction and zero rotations and translation in the y and z directions. The body of the tank is modeled as a rigid body. All the deformations are assumed to be concentrated in its four legs. The tank is assumed to be full of water and closed, so that the problem of water sloshing is not considered. The interaction between the secondary system and the primary structure is ignored. Only the effects of the primary system are considered while the possible feedback from the secondary system on the primary structure is ignored. The uncertainty in the primary-secondary system parameters is not considered. The only source of randomness is the seismic input. Only the first 20 modes of the primary structure are considered in the modal analysis. Solution Technique The primary structure is analyzed separately neglecting the interaction with the secondary system. The ground motion is filtered through the primary structure and the response at the attachment points is calculated and used as the input to the secondary system. The secondary system is analyzed under this input. Many samples of the ground excitation are used to generate the fragility surfaces of the system using Monte Carlo simulation. Figure 4 summarizes the main steps of the methodology. Figure 3. Schematic plan of the hospital Figure 2. Photo of the plaza and the middle tower X Y