State-of-the-art

Previous research on the seismic risk of industrial facilities is quite limited. It mostly focuses on liquid storage (atmospheric pressure) tanks, whereas high-pressure vessels and piping have received considerably less attention. In an early work, O’Rourke & So (2000) produced vulnerability curves for a wide range of atmospheric tanks that vary in diameter, height and fill height. In Salzano et al. (2003) vulnerability curves were calculated for atmospheric tanks using the “probit analysis” method, while in Talaslidis et al. (2004) vulnerability curves were produced using both detailed 3D as well as simplified finite element (FE) models. Apart from storage tanks, Salzano et al. (2009) studied the risk assessment of pressure vessels in an effort to develop criteria for an early warning system in case of an earthquake. Also, Meskouris et al. (2009) presented an integrated methodology for the design of segments of an industrial structure that is sensitive to displacements, such as the piping system of a structure, which may lead to leakage of dangerous substances. The above efforts do not introduce a systematic framework for seismic risk assessment, but still propose useful approaches to investigate various parameters of the problem. Most of the previous research efforts use simplified numerical models to calculate the associated vulnerability which are not calibrated with robust detailed models. In the following, an overview is presented on the state-of-the-art relevant to the proposed integrated seismic risk assessment framework for industrial equipment structures. The discussion is structured according to the flowchart of the project methodology. 

Simulation and synthesis of seismic ground motions
Modelling of strong ground motions may take the form of synthetic time histories, or spectra (i.e response spectra). In general, there are two major approaches in modelling earthquake strong ground motion: the “stochastic (engineering) approach” and the “kinematic modelling approach”. According to the stochastic (engineering) approach, earthquake motion (acceleration) is modelled as Gaussian noise with a spectrum that is either empirical, or is based on a physical model, such as the Specific Barrier Model (SBM) (Papageorgiou & Aki 1983ab, Papageorgiou 2003) of the earthquake source. This approach is expedient, thus cost-effective, and has been extensively used in the past by engineers (using empirical spectra) and recently by seismologists (using spectra derived from physical models of the source). The intent of this approach to strong motion simulation is to capture the essential characteristics of high-frequency motion at an average site from an average earthquake of specified size. As a consequence, the artificially generated accelerograms do not represent a specific earthquake, but embody certain average properties of past earthquakes of a given magnitude. On the other hand, the kinematic approach developed by seismologists, involves the prediction of motions from a fault that has specific dimensions and orientation in a specified geologic setting. As such, this approach reflects more accurately the various wave propagation phenomena and is preferable for site-specific simulations (Aki & Richards 1980).

The seismic input required for the proposed project will be provided by the stochastic approach. This is the most expedient approach while at the same time is the most suitable for the needs of the present investigation. The stochastic approach is ideally suited for the generation/development of vulnerability curves that are significant components of the seismic risk assessment of structural systems. The team of the University of Patras has proposed and developed the Specific Barrier Model SBM for representing realistically the earthquake source (Papageorgiou & Aki 1983a, 1983b, Papageorgiou 2003), which was recently calibrated using a very extensive database (Halldorsson & Papageorgiou 2005) and was used to provide input for the research community of the Multidisciplinary Center for Earthquake Engineering Research (MCEER), USA. The SBM is ideal for modelling near-fault ground motions accounting properly for the very intense intermediate period near-fault pulses that have been observed in the vicinity of a rupturing fault. The latter was made possible by making use of a simple mathematical model for simulating such pulses that was proposed and developed by Mavroeidis & Papageorgiou (2003) and Mavroeidis et al. (2004). Furthermore, recently SBM was used successfully in modelling the ground motions from a number of moderate to large (Mw∼6.2-7.6) earthquake events (Halldorsson et al. 2009).

Numerical simulation of seismic response of industrial structures
Seismic analysis of industrial equipment structures such as liquid storage tanks, pressure vessels and industrial piping, is a challenging issue of Computational Structural Mechanics mainly because of their special characteristics and particularities. 

For liquid storage tanks, the simulation of the interaction between the liquid and the deformable tank is a major challenge. Several numerical methodologies have been employed for the solution of the free-surface hydrodynamic problem in a moving container. In particular, the boundary element method has been employed for the solution of the liquid motion (Gedik & Ergüven 2003), whereas the finite element method has been shown to be more efficient in problems involving the deformation of the thin-walled container (Scharf (1990), Subhashbabu & Bhattacharyya (1996) and Karamanos et al. (2009)). It should be noted that such a rigorous approach would simulate not only the seismic action on the container wall, but also the deformation of the steel container. For the most common case of ground-supported tanks, the simulation should be able to account for elephant’s foot buckling, which is the major failure mode, as well as sloshing buckling at the top of the container, the uplifting of the tank, the failure of the base plate and its welded connection with the shell. These failure modes have not been examined in a rigorous manner, despite some simplified methodologies that have been proposed in the past (Malhotra (1995), Yamaguchi et al. (2006), Natsiavas & Babcock (1987)). A thorough description using an analytical methodology of the liquid motion within a container and its effects on the seismic response can be found in the recent work of Patkas & Karamanos (2007), which can be also used for benchmarking purposes. 

Furthermore, the capacity of liquid storage tanks, being thin walled structures, against buckling, strongly depends on the presence of initial imperfections, i.e. the deviation of their actual geometry with the respect to the ideal “perfect” cylindrical geometry, as well as on the spatial variation of their thickness and steel material properties. It is generally recognized, that the only way to account for the effect of these parameters on the response variability is to describe initial imperfections in a rational probabilistic context as stochastic fields (Schueller 2007). Various signal processing methodologies have been applied in the past for extracting necessary information from real data on initial imperfections when available. Given that the in most cases the processed data on initial imperfections resulted is strongly non-stationary power spectral densities, estimation methods were usually based on moving-window techniques (Schenk & Schueller 2002, Spanos et al. 2009, Papadopoulos & Papadrakakis 2005, Papadopoulos & Inglessis 2007, Papadopoulos et al. 2009a) in order to capture the evolutionary properties of the power spectra. The main disadvantage of these techniques is that they cannot be simultaneously accurate in both the frequency and the space (or time) domain. Recently, an improved method, known as the “method of separation” was proposed (Schillinger & Papadopoulos 2010, Schillinger et al. 2010, Broggi & Schueller 2011) which offers an simultaneously accurate resolution of the power spectral density both in frequency and space and has been proven to be accurate and efficient, especially when the spectrum has a narrow band frequency profile, as in the case of initial imperfections. However, actual measurements of initial imperfections are rarely available and therefore computationally expensive sensitivity analyses must be performed in order to identify extreme structural response and strength distributions (worst case scenarios). In order to reduce the computational effort required for the stochastic sensitivity analyses, efficient solution procedures have been proposed, which can reduce the required computational effort by more than one order of magnitude and making such procedures applicable to real-world problems (Papadopoulos & Papadrakakis 2004, 2005, Papadopoulos et al. 2009b).

As opposed to thin-walled “atmospheric” liquid storage tanks, the seismic analysis and design of industrial pressure vessels has received much less attention. Pressure vessels are comparatively thick-walled structures in order to resist high levels of internal pressure, they are not imperfection sensitive and can be categorized in: (a) spherical vessels, (b) horizontal-cylindrical vessels and (c) vertical-cylindrical vessels also referred as “stacks”. For cases (a) and (b), the interaction between the enclosed liquefied gas and the vessel walls is of special interest in order to determine the total hydrodynamic forces. The necessity for developing a methodology which accounts for this interaction has been demonstrated in a recent study (Wieschollek et al. 2011) and will constitute a significant part of the proposed research. Moreover, the principal failure mode under earthquake loading is the excessive distortion (or fracture) at the welded connections with attached piping (often referred to as “nozzles”). Previous publications have indicated the ability of pressure vessels to sustain deformations well-beyond the yield limit of their material. It should be noted though that according to current design practice, where the allowable stress design concept is used, quasi-elastic behaviour is assumed, despite the fact that the proposed value of behaviour (reduction) factor in ASCE 7 is equal to 2 or 3, which is associated with significant inelastic behaviour. The determination of a reliable behaviour factor value for the seismic analysis of industrial pressure vessels which takes into account the principal failure modes and the vessel’s ability to dissipate inelastic seismic energy is an open issue (Diamanti et al. 2011) and will be further investigated in the present project.

The seismic analysis and design of piping systems is also of particular importance and has not received the necessary attention. Previous research has been directed towards: (a) the strength of piping components (e.g. straight pipes, elbows, Tee-junctions, reducers, nozzles etc) under strong cyclic loading, and (b) the response of the entire piping system under seismic loading, considering its support system (pipe rack). For item (a) the previous research efforts have focused on the strength of the piping components, either experimentally or numerically, and most of this research is summarized in the Welding Research Council report (Slagis 1997). Of particular importance are the degradation of the piping component with repeated loading in the plastic range under strain-controlled conditions, and the material “ratcheting” under stress-controlled conditions, which can both lead to catastrophic results (Rahman & Hassan 2009, Varelis et al. 2011). In the case of item (b), research focuses on the interaction between the pipes and the supporting system, the effects of dissipative behaviour of the supporting structure and the development of a reliable value of the behaviour factor for seismic design (Hoffmeister et al. 2011, Paolacci et al. 2011).

Performance-based Engineering framework for industrial facilities
The state-of-the-art in earthquake engineering is based on the performance-based design concept and adopts rigorous approaches for direct consideration of the effect of uncertainties. Performance-Based Earthquake Engineering (PBEE) and seismic risk assessment combine computational tools and reliability-assessment procedures for calculating exceedance probabilities for a wide range of limit-states. PBEE aims at performing structural designs that can reliably display the required performance for both frequent and rare seismic events, typically remaining operational for the first and sustaining controlled damages for the latter. This is usually quantified via specific damage limitation requirements for a number of performance levels, akin to the limit-states defined in EC8 (EN 1998-1), that correspond to ground motion intensities with a prescribed mean annual frequency (MAF) of exceedance. For example, for a reinforced-concrete portal frame the interstorey drift (indicating damage) has to be less than 0.5% for a “frequent” earthquake with a 10% probability of being exceeded in 10 years (MAF) and less than 2% for a “rare” earthquake of 10% probability in 50 years. It is obvious, that PBEE is closely tied to advanced nonlinear static and dynamic methods of analysis, and also requires a more rigorous specification of seismic ground motions compared to the uniform hazard spectrum of seismic design codes. For ordinary civil engineering structures, PBEE is introduced in US design codes and guidelines (e.g. ATC-40, Vision 2000, ASCE-41), while in Europe Eurocode 8 (EC8) provides some general recommendations. In Greece, the recent Retrofitting Code (ΚΑΝΕΠΕ) provides a detailed framework for reinforced-concrete structures partly based on the recommendations of the ASCE-41 document.

Although PBEE has reached a mature stage in civil engineering structures, there are no provisions for industrial equipment structures. Their geometry, toxic or inflammable contents, and intrinsic failure modes make the problem substantially different from buildings or bridges where current provisions apply. In order to establish PBEE provisions for industrial equipment structures, a number of issues, still open to research, need to be addressed. Depending on the type of industrial structure and its importance relative to the whole facility, we need to define the acceptable performance levels and the corresponding levels of capacity that the structure should be designed to. Demand and capacity should be measured with appropriate parameters (e.g. stresses, strains) at critical locations, in accordance to the different damage (or failure) modes of the structure. For the proper seismic assessment, it is essential to provide recommendations on modelling and also on how the uncertainties, on seismic input and structural properties, will be considered in the analysis. Finally, a complete PBEE framework should consider how the engineer and the owner of the facility would interpret the findings of the analyses, in order to make decisions about its safety.

Reliability assessment and calculation of vulnerability curves
Vulnerability (or fragility) curves were initially developed for the reliability analysis of nuclear plants in an effort to separate the structural analysis part from the hazard analysis performed by engineering seismologists. Vulnerability curves are a useful tool for the seismic risk assessment of a facility, since they provide the conditional probability that a limit-state is exceeded as function of seismic intensity, thus combining seismic hazard with structural analysis. 

Reliability analysis methodologies are applied to calculate the conditional probabilities of various performance levels. Vulnerability curves can be calculated either by Nonlinear Static Procedures also referred as pushover methods (Mander & Basoz 1999, Shinozuka et al. 2000, Moschonas et al. 2009), or via Incremental Dynamic Analysis (IDA) (Vamvatsikos & Fragiadakis 2011). The first class of methods are an inexpensive way to obtain mean response estimates and thus require assumptions on the type and dispersion of the response distribution (usually lognormal). On the other hand, IDA uses nonlinear dynamic analyses under a suite of ground motion records that is scaled to multiple intensity levels to obtain response estimates for the full range of performance levels (from initial linear elastic upto failure). IDA allows a direct consideration of uncertainties due to seismic records (record-to-record variability, or aleatory uncertainty), which simplifies significantly the probabilistic seismic assessment process. Moreover, IDA procedures can be easily extended to account for other sources of uncertainty (e.g. geometry, material, or epistemic uncertainty) when combined with Monte Carlo Simulation (MCS) (Dolsek 2009, Liel et al. 2009, Vamvatsikos & Fragiadakis 2011).

Point estimate methods such as Rosenblueth’s 2K+1 method and FOSM (first-order, second-moment method) are simpler methods with less computational demands compared to MCS. These methods can be easily adopted to estimate the first moments of the response and therefore of the vulnerability. They require only 2K+1 simulations, where K is the number of random variables of the problem. Response surface methodologies are also computationally efficient since they approximate the failure surface through simulations at a number of points. A robust response surface representation however is not always a straightforward task. For this reason, meta-models such as Neural Networks (NN) have been used to obtain large populations of response surface points with minimum computational cost. This is accomplished by substituting the real FEM model with a trained NN in order to trace the failure (or limit-state) surface for a given performance level (Papadrakakis et al. 1996, Lagaros & Fragiadakis 2007). These procedures have the advantage that they require orders of magnitude less computing effort with respect to MCS. This becomes more important when the cost of a single simulation is high, as in the case of IDA.

Seismic risk assessment of industrial structures
Past research on seismic risk assessment of industrial structures does not go beyond the calculation of vulnerability curves and is not based on an integrated framework that considers all aspects of the application of the PBEE framework on such facilities. Seismic risk can be expressed as the mean annual frequency (MAF) of a limit-state being exceeded. The MAF is the reciprocal of the return period that a limit-state is exceeded and therefore allows to express the risk in terms that can be understood by non-engineers, thus simplifying the decision-making process. The most common approach to calculate the MAF is through convolving the vulnerability with the hazard curve. Therefore, the MAF can be calculated using the total probability theorem:

where λLS is the MAF of limit-state violation, λIM is the MAF of the chosen seismic intensity measure (IM) or simply the hazard curve and P(.) represents the conditional probability of limit-state exceedance, or the vulnerability. Accurate assessment using Eq. (1) requires cutting-edge research both on the vulnerability and the seismic hazard side. Most importantly, the choice of the IM becomes pivotal due to the implicit assumption that structural response can be conditioned solely on the IM, becoming independent of any seismological parameters. The sensitivity of MAF estimates to different IMs is an open issue and will be investigated in this project with respect to the specific industrial installation.

FEMA-350 (2000) was the first attempt to directly introduce probabilistic analysis concepts in seismic codes and guidelines for the approximate estimation of the MAF. This document is directed to the design of new steel structures and its theoretical framework was proposed by Cornell et al. 2002. FEMA-350 provides simplified expressions in order to estimate limit-state MAFs and, once the system’s capacity curve is known, it is capable to produce reasonable MAF estimates. 

A more rigorous approach to calculate the risk is by using models of synthetic ground motions. Using a reliable procedure for the generation of synthetic ground motions, the statistics of the response can be directly estimated with MCS using samples that contain ground motions covering a wide spectrum of magnitude (Mw) and distance (R) pairs. Similar procedures have been adopted in the past by Collins et al. (1996) and Wen & Wu (2001), who used seismological data from the Los Angeles area. The advantage of this procedure is that it can be applied directly to complicated linear or nonlinear problems, taking into consideration uncertainties in the system properties. More efficient variations of this concept are often based on importance sampling methods, where the ground motions are chosen based on appropriate Mw-R scenarios identified via disaggregation (Wen 2001).