Fragility Analysis of Space Reinforced Concrete Frame Structures with Structural Irregularity in Plan

With significant structural damages and losses observed during past destructive earthquakes (Anderson 1987; Ellingwood 1980; DesRoches et al. 2011; Elnashai et al. 2010a; Motosaka and Mitsuji 2012; Rosenblueth 1986; Rossetto and Peiris 2009), many researchers have made great efforts to evaluate the seismic vulnerability of various structures so that the extent of structural damage can be predicted in advance and further minimized (Calvi et al. 2006). In this regard, a seismic fragility curve has been widely used as a probabilistic indicator of structural safety against earthquake hazards (Moon et al. 2016). A seismic fragility curve, or seismic vulnerability curve, is expressed as conditional failure probabilities meeting a predefined damage state criteria for different levels of ground motion intensity (Ellingwood 2001); it shows how well a structure performs during earthquake events as a graphical representation of seismic risk (Rossetto and Elnashai 2003). Deriving appropriate and accurate fragility curves is one of the most critical tasks in seismic vulnerability assessment of structures because they significantly affect the overall assessment outcomes.

To derive analytical fragility curves, most previous research studies have used simplified structural representations, such as one-dimensional or two-dimensional models. When simplified analytical models can properly represent seismic behavior of target structures, they give valid seismic fragility curves with a substantial reduction in simulation time. In case of plan-regular space structures, two-dimensional analytical models can be used in deriving effective and efficient seismic vulnerability curves (Moon et al. 2016). However, in many cases, simplified models may be inadequate to assess seismic performance of structures and their seismic vulnerability. Especially for a space structures having a high degree of plan irregularity, it is not desirable to use a simple model in the analysis because it could not properly capture highly nonlinear structural behavior due to the lateral-torsional coupling effect; it is reported that overall structural performance with detailed or sophisticated models can be very different, both qualitatively and quantitatively, from that with simplified models (Anagnostopoulos et al. 2009, 2010; De Stefano and Pintucchi 2008). Thus, for such structures, more representative analytical models should be used in deriving their seismic fragility curves; otherwise, overall vulnerability assessments would be questionable.

In this study, three-dimensional representations are used to evaluate appropriate and accurate seismic performance of space reinforced concrete (RC) frame structures with structural irregularity in their plans. However, using a three-dimensional model in fragility analysis is very challenging because it inevitably causes a significant increase in computational time and cost. From a practical point of view, the use of a computationally expensive model is not feasible with existing approaches to seismic vulnerability analysis. In order to overcome the great challenge in using computationally demanding models, this study provides a novel approach that can derive seismic fragility curves efficiently even with three-dimensional analytical models for space structures. In the adopted approach, computational platforms of reliability and structural analysis are coupled with the development of a linking interface. This integrated computational framework enables quick calculation of structural failure probabilities, using the first order reliability method (FORM), by not evaluating the limit-state function explicitly. In deriving fragility curves, the adopted method is computationally very efficient when compared with the most widely-used Monte Carlo simulation method (Lee and Moon 2014), which makes possible to use a more realistic structural representation or a computational expensive model for vulnerability analysis of space structures. Unlike previous studies, this study investigates the seismic vulnerability of space RC frame structures having different degrees of plan irregularity by deriving more accurate fragility curves with the use of three-dimensional models. The uncertainty in concrete and steel strength is considered, and the influence of plan irregularity on seismic vulnerability is discussed.

In plan-irregular structures, center of mass does not coincide with center of stiffness. Earthquake-induced inertia forces act through center of mass, and reaction forces generated by lateral load-resisting members act through center of stiffness. The distance between the centers of mass and stiffness is defined as eccentricity, and it causes an additional torsional moment. The torsional response is coupled with the lateral response, leading to a considerable increase in deformation demand (Chandler and Hutchinson 1986). In general, plan-irregular structures are much more vulnerable to earthquake damage (Moon 2013). Evidently, considerable damages to such structures due to torsional vibration have been repeatedly observed during many past earthquakes (Anderson 1987; Ellingwood 1980; Durrani et al. 2005; Elnashai et al. 2010a; Rosenblueth 1986). Accordingly, there have been continuous research efforts to assess seismic vulnerability of plan-irregular structures.

In deriving seismic fragility curves for vulnerability assessment, it is very important to use an appropriate representation for a target structure that can reflect important characteristics of structural behavior. This is because final assessment outcomes would be unreliable and misleading if a selected analytical model could not represent real seismic behavior. For space structures with high irregularity, three-dimensional analytical models are more proper structural representations than one-dimensional or two-dimensional models to estimate the complex dynamic behavior from the coupled lateral-torsional responses. However, there is an actual challenge associated with the simulation of a three-dimensional model in vulnerability analysis. Each structural analysis with a three-dimensional model will take much more time, and correspondingly, overall computational time of the fragility analysis will increase drastically with many iterations of the time-consuming earthquake simulation. Unfortunately, the use of such a computationally expansive model is not readily available with existing methods of seismic fragility curve derivation. This study provides a novel way to derive fragility curves very efficiently even with a three-dimensional analytical model.

The seismic fragility curve is defined as the relationship between ground motion intensity and failure probability when a structure reaches or exceeds a certain response level (Jeong and Elnashai 2007). The failure probability can be calculated either by simulation or analytically. Accordingly, existing methods can be grouped into two categories: a simulation-based method and an analytical function–based method. The simulation-based method, such as Monte Carlo simulation, is conceptually straightforward. The failure probability is determined by conducting a series of numerical simulations and counting failure cases out of total cases (Melchers 1999). This method can easily handle any type of structural analysis technique and can give good accuracy with enough samples. However, a simulation-based method can be extremely time-consuming when each structural analysis is pricey or an expected level of failure probability is relatively low. In the analytical function–based method, the probability of failure is expressed as a conditional probability of reaching a specified limit-state condition given a certain level of hazard intensity, where the limit-state function is defined as the difference between structural capacity (supply) and seismic response (demand). This method is viable only when the structural capacity and seismic response can be expressed as analytical functions of selected random variables, so that the limit-state function has a closed-form or explicit expression. Although this method requires extensive statistical knowledge and mathematical techniques, it can give an exact solution for the failure likelihood. However, the use of an analytical function–based method is limited to very simple structures.

This study provides a different method to derive seismic fragility curves for plan-irregular space structures. The reason is that existing methods are not practically applicable when considering the fact that more realistic and complicated models are wanted for space structures with high structural irregularities. This new method can handle computational demanding models in fragility analysis, and it lies between the simulation-based method and analytical function–based method. In the adapted method, the failure probability is analytically computed without constructing a closed-form expression of the limit-state function. The seismic supply and demand are separately estimated through the structural analysis, and the limit-state function is numerically calculated as supply minus demand. This method necessitates the coupling of reliability analysis and structural analysis so as to evaluate the limit-state function implicitly and to obtain the failure probability numerically. This method does not involve complicated or cumbersome calculations of analytical functions, and it can be generally applied to any structural system.

To tackle the computational challenge, the adopted method uses the first-order reliability method. FORM is known to be an efficient, yet reliable, method to estimate the failure probability. In FORM, an event, often called “failure,” is initially defined with selected random variables, and the limit state of the failure is expressed by a function of $\mathbf{x}$ in the original space, where $\mathbf{x}$ is the column vector of random variables. Then, the random variables in the original space are transformed into corresponding standard normal space using a one-to-one mapping transformation matrix, and the limit-state function is redefined by a function of $\mathbf{u}$, where $\mathbf{u}$ is the column vector of standard normal variables. Afterward, failure surface defined by the limit-state function is linearly approximated with the first-order Taylor series expansion, and a reliability index $\beta$ is computed as the minimum distance from the origin to the approximated limit-state function in the standard normal space. Finally, the failure probability is calculated aswhere $\mathrm{\varphi }(·)$ = cumulative distribution function of the standard normal distribution. Fig. 1 shows the linearized limit-state function for the failure domain in the two-dimensional standard normal space and the minimum distance from the origin to the approximated linear function. In FORM, linearization of the failure surface makes it possible to deliver an estimate of the failure probability cost effectively. Further details about FORM are available elsewhere (Der Kiureghian 2005; Rackwitz and Flessler 1978).

The adopted method requires the coupling of reliability analysis and structural analysis. For this, finite-element reliability using MATLAB (FERUM) and ZEUS-NL are selected as computational tools for the reliability and structural analysis. FERUM, provided by the University of California at Berkeley, is one of the popular reliability analysis software packages. It offers various reliability analysis methods, including FORM (Haukaas 2003). FERUM has been widely used in solving a variety of structural reliability problems. ZEUS-NL is an advanced structural analysis software package that is specially developed for earthquake engineering applications by the Mid-America Earthquake (MAE) Center (Elnashai et al. 2010b). This software adopts a fiber-based element modeling approach so that it can consider the material inelasticity spread within the member cross-section as well as along the member length. ZEUS-NL has been extensively used for nonlinear static and dynamic analysis of various structural systems (Jeong et al. 2012). Because source codes for both software packages are open to the public, necessary modifications can be made to develop an integrated platform of FERUM and ZEUS-NL.

The integrated computational platform, referred to as FERUM-ZEUS, is established with the development of linking interface between the FERUM and ZEUS-NL. The linking interface makes possible the automatic exchange of information between two different analysis tools during fragility analysis, and it enables implicit calculation of failure probability and efficient derivation of fragility curves. Fig. 2 presents the computational platform of FERUM-ZEUS. In this platform, FERUM provides ZEUS-NL with deterministic input values for selected random variables based on their distribution types, and ZEUS-NL generates an analytical model with the determined input parameters. Then, ZEUS-NL determines the structural capacity (supply) from a push-over analysis and estimates the seismic response (demand) from an inelastic response-history analysis. The analysis outcomes are sent back to FERUM, and FERUM evaluates the limits-state function numerically as supply minus demand. FERUM repeatedly calls ZEUS-NL until the failure probability is found with FORM. The integrated platform is developed in MATLAB, and the process of fragility curve derivation is automated. This platform can handle computationally expensive models in seismic vulnerability analysis, and it can derive fragility curves very efficiently with FORM. Additionally, in order to save computational time for the cases when an expected failure probability is too low or high, an algorithm is added to assign a probability zero or one depending on the calculated limit-state function value.

This study aims to derive more accurate and appropriate seismic fragility curves for space RC frame structures with different degrees of plan irregularity with their three-dimensional models and investigate the effect of structural irregularity on their seismic vulnerability. Instead of simplified models, three-dimensional analytical models are adopted to take into account true nonlinear coupled lateral-torsional responses. To address the significant computational challenge associated with the use of three-dimensional models, this study establishes a computational framework that integrates structural and reliability analysis. FERUM and ZEUS-NL are selected as the reliability and structural analysis tools, and a linking interface is provided that enables automatic interchange of the necessary data between two analysis tools. FORM is used to estimate the failure probabilities. With the adopted framework, seismic vulnerability of various space RC frame structures, with and without plan irregularity, is investigated. Five different models of RC frame structure are studied, with varying plan irregularities from 0 to 10% with a 2.5% increment. A total of 15 ground motions are used, and they are categorized into three groups based on the ratio of PGA to PGV. Uncertainties in structural capacity and earthquake demand are both considered. Three limit states are defined, serviceability, damage control, and collapse prevention. The corresponding values of interstory drift ratio for each limit state are obtained from a series of adaptive pushover analyses. Under the integrated framework, seismic fragility curves of space RC fame structures with different degrees of plan irregularity are successfully derived with their three-dimensional models on a standard personal computer. The lognormal cumulative probability distribution is assumed for the fragility curves, and statistical parameters are provided after conducting extensive regression analysis.

The proposed approach of deriving seismic fragility curves makes it practically possible to use computationally expensive models in the analysis, and it produces fragility curves very efficiently. From the derived fragility curves, it is observed that seismic vulnerability is considerably affected by the structural irregularity, and the PGA/PGV ratio of ground motions has a noteworthy effect on the fragility curves. The numerical results clearly indicate that space RC frame structures become much vulnerable to earthquake damage as the plan irregularity increases. This agrees well with many of the previous research results and the actual damage observed in past earthquakes. Structural irregularity is the one of the major causes of the failure or collapse of structures, so considerable attention should be paid when conducting seismic vulnerability analysis. The main contributions of this study are as follows: (1) it establishes an integrated computational framework for efficient fragility analysis, (2) it derives more representative fragility curves with the use of three-dimensional analytical models, (3) it provides functional forms of seismic fragility curves for space RC frame structures with varying plan irregularity, and (4) it investigates the effect of plan irregularity on seismic vulnerability with more realistic fragility curves. The proposed approach is expected to be very useful when more accurate seismic vulnerability for complex structures is required. This study delivers seismic fragility curves for typical low-rise space RC frame structures with varying plan irregularity, but the general application may be limited because seismic performance could be very different depending on the structural configuration and the damage state definition.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2015R1C1A1A01055825).