Analytical Fragility Functions Research Articles

Development of fragility functions of low-rise steel moment frame by artificial neural networks and identifying effective parameters using SHAP theory

Estimating analytical fragility functions requires high computational costs due to numerous incremental non-linear dynamic analyses. This study employs a soft computing approach to reduce this process and increase estimation accuracy. Utilized a fragility database of low-rise steel moment structures designed for different versions of Iranian seismic design code, and six feedforward Artificial Neural Networks (ANN) with various architectures were trained to estimate the fragility function’s parameters to determine the most desired architectures. The neural networks used the design parameters (frames story, code edition of design, ductility, soil type, near- and far-field earthquake, and seismic zone factor and importance factor) as inputs. For each of the fragility function’s parameters (Median PGA and βs of damages: slight, moderate, extensive, and complete) as outputs, an ANN was trained. The best architect of ANN was selected based on the accuracy of the models from different statistical indicators and the most balanced of the inputs’ importance on the outputs using SHapley Additive explanations (SHAP) analysis. To demonstrate the accuracy of the selected architect, the coefficient of determination was estimated for all database values, and the fragility functions of four randomly chosen frames were compared. The high R2 value in most cases and the closeness of the predicted fragility curved with the actual one indicates the reliability of the ANN method for predicting the fragility function.

Evaluation of the optimal ground motion intensity measure in the prediction of the seismic vulnerability of earth dams

The large majority of existing earth dams were designed with old standards, which often accounted for the effects of earthquakes in a simplified manner. As a result, seismic re-evaluation of these structures is now becoming increasingly important for dam owners. The seismic performance of earth dams can be evaluated at different levels of resolution, typically ranging from simple sliding block analysis, to the use of advanced numerical models. Seismic inputs for these models are typically chosen using code-based or site-specific probabilistic seismic hazard analysis (PSHA). In this study, we analyze the seismic vulnerability of two major earth dams located in Southern Italy, a seismically active area. The two dam systems have different geotechnical characteristics: one of them is founded on potentially liquefiable materials, whereas the other one is not susceptible to liquefaction. Our analyses are based on finite difference numerical simulations using both simplified and advanced constitutive models. Seismic vulnerability of both dams is analyzed by means of analytical fragility functions for various damage mechanisms and ground motion intensity measures (IMs). Such fragility functions are based on nonlinear deformation analyses within the multiple stripe analysis framework. We analyze efficiency and predictability of various IMs in predicting different damage levels and mechanisms. We show that the cumulative absolute velocity is overall the optimal IM in predicting vulnerability of both earth dams. We also analyze the relative importance of PSHA inputs (i.e. alternative ground motion and earthquake rupture forecast models; GMMs and ERFMs, respectively) and numerical models (simplified vs advanced) in the seismic analysis of earth dams by means of tornado diagrams. We show that the choice of GMMs and ERFMs is always more important than that of the constitutive model. However, for dams prone to liquefaction, the choice of the constitutive model has a higher importance than for non-liquefiable earth dams.

Seismic Fragility of Aging Elevated Water Tank with Smooth Bars Considering Soil Structure Interaction

The functionality of elevated water tanks is pivotal to assure after an earthquake as water supply is expected to be uninterrupted. Although elevated water tanks with deformed bars are widely studied, limited works exist for water tanks with smooth bars, although such tanks comprise a considerable fraction, even in the high seismic regions. To quantify the seismic vulnerability of aging elevated water tanks with smooth bars, we created analytical fragility functions for full, half, and empty reservoir conditions, considering fluid–structure and soil–structure interactions. The sum of findings reflects that soil flexibility and the amount of water present in the tank have a significant effect on overall seismic fragility, especially at higher damage states. The tanks are found to be most vulnerable when they are fully filled with water. The effect of soil flexibility is more pronounced at higher damage states. The difference between the fragility of flexible base and fixed base structures is found to increase with increasing ground motion intensity and it is the highest for the empty tank condition.

Seismic fragility assessment of bridges with as-built and retrofitted splice-deficient columns

A significant proportion of existing bridges in high seismic regions were constructed prior to the 1970s. As a result of poor reinforcement detailing, pre-1970s bridge columns are susceptible to lap-splice or shear failure in the plastic region. Given the high economic impact of retrofitting all pre-1970s reinforced concrete (RC) bridges, it is essential to identify the most vulnerable bridges for retrofit prioritisation. Analytical fragility functions are useful for quantifying the seismic vulnerability of existing bridge stock. However, the accuracy of these fragility functions relies on the adequacy of the adopted modelling approach. This paper presents a hinge-type modelling approach for capturing the seismic response of as-built splice-deficient and retrofitted RC bridge columns. Fragility analysis is carried out for typical seat and diaphragm abutment two-span bridges using the proposed hinge-type modelling approach. The results showed that the vulnerability of the bridges depends on the column failure mode and the limit state under consideration. Also, the common notion that the column is the most vulnerable component may not necessarily be true. The study underscored that retrofitting columns without retrofitting other components may not effectively mitigate the damage and associated risk.

Conjugate Bayesian updating of analytical fragility functions using dynamic analysis with application to corroded bridges

Analytical fragility functions provide a way to quantify the risk of a structure. One method to construct seismic fragility curves is to perform a series of nonlinear dynamic analyses of the structure. As structural inspection data is increasing, updating fragility functions based on these measurements provides a way to translate collected data into risk-based decision-making support. However, the high computational cost in running and re-running analyses over the full finite-element model can be prohibitive. This paper presents a new methodology based on conjugate Bayesian updating to efficiently and accurately update analytical fragility curves. The reduction in computational cost from both reducing the number of analyses required and simplifying the structural complexity are investigated. The method is applied to update calculations of seismic bridge fragilities accounting for varying levels of measured corrosion. Results comparing updated fragility curves obtained from using the proposed approach versus using the full set of dynamic analyses show that the proposed method achieves accurate, stable, and more quickly converging fragility calculations. The number of structural analyses required is reduced by at least 60%, and the use of component-level analyses reduces computation time by 97% and more than an order of magnitude compared to existing methods.

A mechanics-based method towards risk assessment of RC buildings under tsunami and flow-type hazards

The study presents a simple mechanics-based method to develop analytical fragility functions for the risk assessment of Reinforced Concrete (RC) frame buildings subjected to tsunami or other flow-type hazards (i.e., flood, mud flows, flow-type landslides, etc.). A harmonized framework for the analysis of building portfolios to flow-type hazards is first illustrated. The mechanics-based method for the performance assessment of RC frames with breakaway infill walls subjected to tsunami and flow-type loads, called SAFETI (Structural Assessment to Flows and Extreme Tsunami Inundation), is then presented. For the large-scale risk analysis, a damage levels classification is adopted for both structural and non-structural (i.e., infill walls) components, based on the HAZUS approach. The flow depth is assumed as Intensity Measure. Damage levels computed through SAFETI are validated with ones obtained from a refined finite-element modelling in OpenSees for three case-study RC frames subjected to tsunami. Finally, an application of the SAFETI method is presented to show the feasibility of the mechanics-based procedure to derive analytical fragility curves for a simulated case-study building portfolio subjected to tsunami, flow-type landslides, and flood. The SAFETI method appears a robust solution to perform a consistent fragility assessment at regional scale for tsunami and flow-type hazards, explicitly accounting for the building-to-building variability with limited modelling and computational efforts.

Empirical seismic fragility assessment and optimal risk mitigation of building contents

Building contents proved challenging in performance-based earthquake engineering frameworks because of the data scarcity. Despite the content seismic losses in the past two decades, the overall effectiveness of restrainers to contents has not been thoroughly investigated. This paper aims to provide a detailed analysis for assessing content damage, loss, and protection to better understand content risk and performance. In this paper, analytical fragility functions for rigid block-type contents are developed based on nonlinear time history analysis of four dynamic models to represent freestanding and anchored building contents with elastic-brittle restrainers considering multiple failure modes. The effectiveness of restrainers is quantified by the improvement of the fragility median. A simplified expression is proposed to estimate content lognormal fragility parameters based on a wide range of variables' correlation between contents' characteristics and restraint strength. The proposed simplified expression is adopted in FEMA P-58 loss assessment framework through a case study of a 4-story light-frame wood office building to estimate content dollar loss. To further investigate content risk mitigation, we used a quantitative loss assessment for multiple mitigation scenarios by anchoring different content types such as heavy, expensive, electrical, furniture, or glassware. It is found that elastic-brittle restrainers are more effective for rigid systems than flexible systems, and decision-makers should design restrainers based on a combination of block-like content and restrainer characteristics for efficient protection and life safety. For the case study, anchoring expensive components was the optimal mitigation scenario, which resulted in a 74% reduction in the average annual losses compared to freestanding contents.

Seismic fragility analysis of RC bridges in high seismic regions under horizontal and simultaneous horizontal and vertical excitations

Although Himalayan highways generally pass close to the active seismic faults, near field excitation effects are not considered in designing bridges. The effect of strong vertical shaking is known to cause several anomalous and remarkable consequences in bridges. To this end, we present a comparative seismic vulnerability analysis of representative RC bridge from Nepal Himalaya, considering horizontal only and simultaneous horizontal and vertical excitations. Fragility functions for longitudinal and transverse directions considering horizontal only and simultaneous horizontal and vertical excitations are constructed for representative RC bridge from Asian Highway-2 (AH-2) in Nepal that generally follows the same alignment as that of one of the very active seismic faults in the region. Analytical fragility functions are also compared with the existing empirical fragility functions. The results depict that the horizontal response parameters, such as displacement and displacement ductility, are not considerably affected by the vertical ground motion, but the effect is found to be increasingly significant for higher damage states. The sum of observations highlights that the transverse direction of the bridge will be more prominent while considering vertical excitation effects together with the horizontal excitation rather than the horizontal excitation only.

A comparative analysis of empirical and analytical tsunami fragility functions for buildings in Tumaco, Colombia

The Pacific coast of Colombia has witnessed several tsunami events in the last century. However, the physical damage to structures at the time was not recorded, even when these events have brought overwhelming destruction to coastal communities. This lack of data increases uncertainties when assessing future tsunami impacts. Nonetheless, other regions of the world, which have been also affected by tsunami, have collected damage information that has been used to build empirical tsunami fragility functions. Based on a demand parameter, such as the tsunami inundation depth, a fragility function expresses the probability of building damage. Hence, regions with no recent tsunami damage cannot develop empirical fragility functions since post-disaster damage survey data is not available. One alternative for these regions is to develop analytical fragility functions out of computational nonlinear structural analysis. Reinforced concrete buildings and palafitte houses are selected to perform a comparative analysis of collapse damage assessment expressed in empirical and analytical fragility functions.

A comparative study of empirical and analytical fragility functions for the assessment of tsunami building damage in Tumaco, Colombia

ABSTRACT A fragility function shows the building damage likelihood based on a demand parameter such as the tsunami flow depth. While empirical fragility functions have been developed by using postdisaster damage survey data, regions with no recent tsunami damage cannot develop these estimations. Alternatively, analytical fragility functions could be developed to fill this gap. Along the Colombian Pacific coast, several tsunami events have occurred; however, the aftermath resulting from the structure physical damage was not recorded. Fortunately, other regions of the have collected tsunami damage information, which has been analyzed and represented in terms of empirical fragility functions. This paper provides a comparative study of empirical and analytical tsunami fragility functions. It is focused on the analytical fragility functions developed for typical structures in Tumaco, Colombia and on empirical fragility curves taken from damage survey data of recent tsunami events. The differences in the assessments of the damage produced by the two methods in the case of a two-story reinforced concrete building in Tumaco is discussed herein. As a result of the comparison, it was shown that the structures in Tumaco have higher collapse probabilities at lower flow depths than structures in other places along the Pacific and Indian Oceans.

Seismic fragility of buried steel natural gas pipelines due to axial compression at geotechnical discontinuities

This paper presents an extended set of numerical fragility functions for the structural assessment of buried steel natural gas (NG) pipelines subjected to axial compression caused by transient seismic ground deformations. The study focuses on NG pipelines crossing sites with a vertical geotechnical discontinuity, where high compression straining of a buried pipeline is expected to occur under seismic transient ground deformations. A de-coupled numerical framework is developed for this purpose, which includes a 3D finite element model of the pipe–trench system employed to evaluate rigorously the soil–pipe interaction effects on the pipeline axial response in a quasi-static manner. One-dimensional soil response analyses are used to determine critical ground deformation patterns at the vicinity of the geotechnical discontinuity, caused by the ground shaking. A comprehensive parametric analysis is performed by implementing the proposed analytical framework for an ensemble of 40 recorded earthquake ground motions. Crucial parameters that affect the seismic response and therefore the seismic vulnerability of buried steel NG pipelines namely, the diameter, wall thickness, burial depth and internal pressure of the pipeline, the backfill compaction level, the pipe–soil interface friction characteristics, the soil deposits characteristics, as well as initial geometric imperfections of the walls of the pipeline, are systematically considered. The analytical fragility functions are developed in terms of peak ground velocity at the ground surface, for four performance limit states, considering all the associated uncertainties. The study contributes towards a reliable quantitative risk assessment of buried steel NG pipelines, crossing similar sites, subjected to seismically-induced transient ground deformations.

Fragility Functions for RC Shear Wall Buildings in Australia

This research investigates the development of analytical fragility functions for reinforced concrete shear wall buildings in Australia. A building stock for the city of Melbourne is used in conducting an assessment of these types of structures. The assessment uses the best information available for selecting the building parameters applicable to the low-to-moderate seismic region, site soil class, expected earthquake ground motions, and site response. The capacity spectrum method is used to derive vulnerability functions for low-, mid-, and high-rise reinforced concrete shear wall buildings. Although there is a paucity of earthquake damage data available in Australia, some comparisons are made using the results from the fragility functions derived here to the damage data from the Newcastle earthquake in 1989.

Developing Seismic Risk Prediction Functions for Structures

The paper presents the development of a nonlinear static displacement-based methodology for seismic risk assessment and loss estimation of stone masonry building stock of Pakistan. Experimental investigation of one-third scaled model, tested on shake table, is performed in order to obtain lateral strength and drift limits for stone masonry and develop damage scale for performance-based assessment. Prototype buildings are designed respecting the existing building stock and investigated through nonlinear static and dynamic time history analyses. Nonlinear static mechanical models, for both global and local vulnerabilities, are developed for the considered typology which are used to derive analytical structure-dependent fragility functions considering expected sources of uncertainties explicitly in contrary to the conventional procedures. Furthermore, seismic risk assessment is performed for different scenario earthquakes and presented in terms of structure-independent fragility functions to estimate the mean damage ratio, the repair cost as a fraction of replacement cost, and casualties, with the dispersion being quantified, given source-to-site distance and magnitude for an earthquake event. The methodology is tested for seismic risk assessment of the considered typology in recent 2005 Kashmir earthquake, which is reasonably predicted. Future development of the methodology is required with additional experimental tests on rubble stone masonry material in order to increase confidence in future applications.

A general framework for the estimation of analytical fragility functions based on multivariate probability distributions

A fragility curve is a function that expresses the probability of failure of a structure or component as a function of the intensity of external aggression. This paper proposes a general framework for the development of analytical fragility functions from data based on the copula approach. Such a model allows for any kinds of marginal distributions and dependence structures so that it can be applied to various types of fragility data, analytical or empirical. The fragility function is then derived from the joint distribution of intensity and damage measures. The Bayesian information criterion is used to select the most plausible model among the candidate joint distributions, given the data. The practical implementation of the methodology is illustrated by an analytical test case and by the evaluation of seismic fragility curves for a reinforced concrete building. Several candidate marginal distributions, in agreement with the nature and the physical properties of the variables (e.g. common intensity and damage measures take only positive values) are evaluated. In particular, seismic intensity measures are lognormal random variables according to seismological models. This paper is focused on bivariate distributions but the case of vector valued intensity measures can be treated accordingly.

A proposed methodology for deriving tsunami fragility functions for buildings using optimum intensity measures

Tsunami fragility curves are statistical models which form a key component of tsunami risk models, as they provide a probabilistic link between a tsunami intensity measure (TIM) and building damage. Existing studies apply different TIMs (e.g. depth, velocity, force etc.) with conflicting recommendations of which to use. This paper presents a rigorous methodology using advanced statistical methods for the selection of the optimal TIM for fragility function derivation for any given dataset. This methodology is demonstrated using a unique, detailed, disaggregated damage dataset from the 2011 Great East Japan earthquake and tsunami (total 67,125 buildings), identifying the optimum TIM for describing observed damage for the case study locations. This paper first presents the proposed methodology, which is broken into three steps: (1) exploratory analysis, (2) statistical model selection and trend analysis and (3) comparison and selection of TIMs. The case study dataset is then presented, and the methodology is then applied to this dataset. In Step 1, exploratory analysis on the case study dataset suggests that fragility curves should be constructed for the sub-categories of engineered (RC and steel) and non-engineered (wood and masonry) construction materials. It is shown that the exclusion of buildings of unknown construction material (common practice in existing studies) may introduce bias in the results; hence, these buildings are estimated as engineered or non-engineered through use of multiple imputation (MI) techniques. In Step 2, a sensitivity analysis of several statistical methods for fragility curve derivation is conducted in order to select multiple statistical models with which to conduct further exploratory analysis and the TIM comparison (to draw conclusions which are non-model-specific). Methods of data aggregation and ordinary least squares parameter estimation (both used in existing studies) are rejected as they are quantitatively shown to reduce fragility curve accuracy and increase uncertainty. Partially ordered probit models and generalised additive models (GAMs) are selected for the TIM comparison of Step 3. In Step 3, fragility curves are then constructed for a number of TIMs, obtained from numerical simulation of the tsunami inundation of the 2011 GEJE. These fragility curves are compared using K-fold cross-validation (KFCV), and it is found that for the case study dataset a force-based measure that considers different flow regimes (indicated by Froude number) proves the most efficient TIM. It is recommended that the methodology proposed in this paper be applied for defining future fragility functions based on optimum TIMs. With the introduction of several concepts novel to the field of fragility assessment (MI, GAMs, KFCV for model optimisation and comparison), this study has significant implications for the future generation of empirical and analytical fragility functions.

Analytical Fragility Functions for Horizontally Curved Steel I-Girder Highway Bridges

Horizontally curved bridges were investigated following a statistical evaluation of typical details commonly used in the United States. Both seismically and non-seismically designed bridges are considered where the primary differences are in column confinement, type of bearings and abutment support length. Columns and bearings were found to be the most seismically vulnerable components for both categories. Central angle was identified as an important factor that increases the demand on various components, particularly columns. Furthermore, larger angles lead to increased deformations at the supports which adversely affect the seismic vulnerability. Consistent with the fragility curves that account for the central angle explicitly, a second set of system fragility curves were introduced for cases when central angle is not specified such as the case in the National Bridge Inventory. Comparison of fragility parameters to those suggested by HAZUS-MH highlighted the need for revisions to account for current design practices and central angle.

Assessment of Seismic Performance of High-Rise Thin RC Wall Buildings in Lima, Peru Using Fragility Functions

The actual behavior of thin RC wall high-rise buildings during an earthquake in Lima, Peru, and the associated seismic loss is unknown. This type of building was assessed done using analytical fragility functions. The numerical model was based on full-scale tests done in Lima, Peru. Nonlinear dynamic response analysis was performed using records simulated for Lima. The damage ratio was estimated for four damage states and fragility functions were obtained assuming that the damage ratio followed log-normal distributions. Seismic performance was evaluated by considering the probability of different damage states for three seismic hazard levels. It was found that highrise buildings present a low probability of collapse in severe earthquakes.

Development of empirical and analytical fragility functions using kernel smoothing methods

SummaryFragility functions that define the probabilistic relationship between structural damage and ground motion intensity are an integral part of performance‐based earthquake engineering or seismic risk analysis. This paper introduces three approaches based on kernel smoothing methods for developing analytical and empirical fragility functions. A kernel assigns a weight to each data that is inversely related to the distance between the data value and the input of the fragility function of interest. The kernel smoothing methods are, therefore, non‐parametric forms of data interpolation. These methods enable the implicit treatment of uncertainty in either or both of ground motion intensity and structural damage without making any assumption about the shape of the resulting fragility functions. They are particularly beneficial for sparse, noisy, or non‐homogeneous data sets. For illustration purposes, two types of data are considered. The first is a set of numerically simulated responses for a four‐story steel moment‐resisting frame, and the second is a set of field observations collected after the 2010 Haiti earthquake. The results demonstrate that these methods can develop continuous representations of fragility functions without specifying their functional forms and treat sparse data sets more efficiently than conventional data binning and parametric curve fitting methods. Moreover, various uncertainty analyses are conducted to address the issues of over‐fitting, bias, and confidence intervals. Copyright © 2014 John Wiley & Sons, Ltd.

Seismic risk assessment for a reinforced concrete frame designed according to Chinese codes

The analytical formulations of two types of seismic fragility functions, namely seismic demand fragility and seismic damage fragility, are derived from the general definition of fragility function and usually adopted assumptions in seismic risk assessment. Using suitable intensity measure (IM) and damage measure (DM), the well-known Cornell's IM- and displacement-based formulations for seismic risk are revisited from the viewpoint of the analytical fragility functions. It is found that the recently widely used formulations using engineering demand parameters (EDPs) as well as DM-based approaches are two specific cases of the general IM-based risk equation, depending on the chosen fragility parameters. To apply Cornell's formulations to assess the seismic performance of Chinese code-conforming buildings and to investigate the effects of the derived fragility parameters on the seismic performance, a five-storey reinforced concrete (RC) frame designed according to the Chinese codes has been used as a case study. The results of this example study demonstrate that the capacity randomness and the selection of earthquake IMs all have obvious influences on seismic fragility and risk. It is also found that the computed failure probabilities for different limit states in 50 years for the example frame all satisfy the probabilistic safety requirements by the Chinese code for seismic design of reinforced concrete buildings.

Developing Seismic Fragility Function of Structures by Stochastic Approach

Fragility function of structures is the major requirement of seismic loss estimation which is widely used in the seismic risk management. In this paper, firstly, a comprehensive and simplifies stochastic methods are presented for development of analytical fragility functions. Secondly, the effect of damage threshold uncertainty on fragility functions is estimated. It is shown that the results of the method are almost comparable with the result of previous studies and the effect of uncertainty of damage state on the deviation of fragility function in the lower intensity of ground motion is high which gradually decreases.