Multiple Limit States Research Articles

Semi-Markov process-driven maintenance scheduling for Tainter gate system considering multiplelimit states

In extended periods of operation, Taiwan’s reservoir electromechanical systems increasingly require substantial maintenance. This research adopts the semi-Markov process, which accommodates non-exponential distribution of state durations, to formulate optimal maintenance strategies for Tainter gate systems that are noted for their prolonged dormancy and significant operational uncertainties. The methodology initiates with the estimation of failure probabilities across four condition states, analyzing deterioration through the Weibull distribution for both general and latent limit states. The general limit state accounts for time-induced deterioration and effects of dormancy using a Bayesian–Weibull first-order reliability method, while the latent limit state addresses activation failures. Employing the semi-Markov process, an annual transition matrix is computed and combined with failure probabilities to assess the Tainter gates’ system reliability. To identify the most efficient maintenance schedule, a genetic algorithm is applied, targeting the minimization of failure probabilities for both limit states below predefined thresholds and cost reduction. Numerical simulations validate the framework’s efficacy, demonstrating its potential to enhance maintenance planning objectivity and decrease dependence on subjective assessments. The findings highlight the predominance of the general limit state in dictating system failure and underscore the risk Tainter gates face during transition from dormancy to activation, emphasizing the need for thorough monitoring.

Material optimization of reinforced concrete structures in multiple limit states

AbstractA common limitation in current material optimization methods for reinforced concrete structures is that they often concentrate solely on either the serviceability limit state (SLS) or the ultimate limit state (ULS). In response, this paper introduces a novel approach to optimize the material layout in planar reinforced concrete structures in plane stress, accounting for both limit states simultaneously. The proposed method employs an iterative strategy, integrating criteria for serviceability and ultimate limit states while approximating the elastoplastic behavior of cracked reinforced concrete. Here, the criteria for the limit states are incorporated as limits on the strains. The methodology initiates with a finite element limit analysis for the preliminary layout. The minimum potential energy principle is utilized to determine the strains for the nonlinear material response, and subsequently, an iterative linearized method is applied for material optimization. The research showcases the successful application of this approach to simple examples and presents improved and optimized solutions that adhere to the limit criteria for more complex structural scenarios.

Reliability-Based Design Optimization for a Vertical-Type Breakwater with an Emphasis on Sliding, Overturn, and Collapse Failure

To promote the application of reliability-based design within the Korean coastal engineering community, the author conducted reliability analyses and optimized the design of a vertical-type breakwater, considering multiple limit states in the seas off of Pusan and Gunsan – two representative ports in Korea. In this process, rather than relying on design waves of a specific return period, the author intentionally avoided such constraints. Instead, the author characterized the uncertainties associated with wave force, lift force, and overturning moment – key factors significantly influencing the integrity of a vertical-type breakwater. This characterization was achieved by employing a probabilistic model derived from the frequency analysis results of long-term in-situ wave data. The limit state of the vertical-type breakwater encompassed sliding, overturning, and collapse failure, with the close interrelation between wave force, lift force, and moment described using the Nataf joint probability distribution. Simulation results indicate, as expected, that considering only sliding failure underestimates the failure probability. Furthermore, it was shown that the failure probability of vertical-type breakwaters cannot be consistently secured using design waves with a specific return period. In contrast, breakwaters optimally designed to meet the reliability index requirement of <i>β</i>-3.5 to 4 consistently achieve a consistent failure probability across all sea areas.

Reliability-based design optimization for seismic structures considering randomness associated with ground motions

When addressing the dynamic reliability analysis of structures, it becomes necessary to account for multiple limit state functions or their combinations. In scenarios where structures are subjected to random excitation, this can lead to intricate inter-dependencies among different limit states, and the computational workload can pose a substantial challenge in ensuring sufficient precision. Code-based design primarily ensures safety at the member level, while deterministic optimization fails to accommodate the inherent uncertainties associated with external excitation or the system as a whole. Therefore, in such cases, to address both the uncertainties in excitations and the presence of multiple limit states while mitigating computational challenges, equivalent extreme-value criteria are employed within the framework of the probability density evolution method to calculate the global reliability of the structure subjected to stochastic ground motions generated from the physically motivated stochastic ground motion model. Numerical optimization is subsequently conducted using genetic algorithms, aiming to minimize the cost of the superstructure while adhering to the design performance criteria related to the inter-story drift ratio and considering global reliability. Additionally, multi-objective optimization is carried out using NSGA-II, permitting the generation of multiple solutions, from which one can select the most suitable solution as needed. The numerical results illustrate the effectiveness of this technique in achieving an optimal balance between the cost of the structure and the consideration of global reliability, providing a comprehensive solution for dynamic reliability analysis and design optimization of structures under random excitations.

Generalized Stratified Sampling for Efficient Reliability Assessment of Structures against Natural Hazards

Performance-based engineering for natural hazards facilitates the design and appraisal of structures with rigorous evaluation of their uncertain structural behavior under potentially extreme stochastic loads expressed in terms of failure probabilities against stated criteria. As a result, efficient stochastic simulation schemes are central to computational frameworks that aim to estimate failure probabilities associated with multiple limit states using limited sample sets. In this work, a generalized stratified sampling scheme is proposed in which two phases of sampling are involved: the first is devoted to the generation of strata-wise samples and the estimation of strata probabilities, whereas the second phase aims at the estimation of strata-wise failure probabilities. Phase-I sampling enables the selection of a generalized stratification variable (i.e., not necessarily belonging to the input set of random variables) for which the probability distribution is not known a priori. To improve the efficiency, Markov Chain Monte Carlo Phase-I sampling is proposed when Monte Carlo simulation is deemed infeasible, and optimal Phase-II sampling is implemented based on user-specified target coefficients of variation for the limit states of interest. The expressions for these coefficients are derived with due regard to the sample correlations induced by the Markov chains and the uncertainty in the estimated strata probabilities. The proposed stochastic simulation scheme reaps the benefits of near-optimal stratified sampling for a broader choice of stratification variables in high-dimensional reliability problems with a mechanism to approximately control the accuracy of the estimators of multiple failure probabilities. The practicality and efficiency of the scheme are demonstrated using two examples involving the estimation of failure probabilities associated with highly nonlinear responses induced by wind and seismic excitations.

Robust reliability‐based design approach by inverse FORM with adaptive conjugate search algorithm

AbstractThe concept of reliability‐based design (RBD) of geotechnical works has seen great success in recent years, nevertheless, its application in real‐world engineering practice is still limited. This phenomenon could be mainly attributed to two aspects: the engineer's unfamiliarity of reliability algorithms even in their simplest forms, and the lack of direct connection between reliability algorithms and deterministic design tools. To promote the geotechnical RBD application, this study proposes an inverse first‐order reliability method (FORM) integrating adaptive conjugate direction search technique (RBD via conjugate FORM) to solve classical RBD problems where the number of (multiple) limit state functions is usually equal to the number of unknown design parameters. Notably, the RBD via conjugate FORM can readily back‐calculate those unknown design parameters subject to prescribed reliability index or probability of failure, by means of iterative evaluation of the parametric sensitivities. The proposed algorithm is first verified by several well‐documented examples. Comparative studies show that when the reliability problem to solve is highly nonlinear, the RBD results based on steepest descent search direction may fall in oscillation and non‐convergence, while the proposed RBD via conjugate FORM algorithm performs robustly and efficiently. It is then illustrated in combination with standalone numerical codes to conduct risk‐based optimization of unknown design parameters for two soil slope stability models. The first slope RBD model achieves designing the slope angle under specified reliability index, and the second slope RBD model is to optimize multiple design parameters of the stabilization pile, which can meet different slope reliability requirements under static and seismic loading conditions simultaneously.

Seismic capacity and performance of code-conforming single-story RC precast buildings considering multiple limit states and damage criteria

The seismic capacity and performance of code-conforming precast buildings is assessed in the study. The case study consists in single-story precast buildings designed according to Italian building code considering low to high seismicity sites and varying both geometry layout and soil conditions. The seismic behavior of the buildings is assessed through multiple-stripe analyses performed through nonlinear three-dimensional modeling. Fragility of the buildings is computed by considering several methods, and performance evaluation is based on capacity margin ratios, failure probabilities, and reliability indexes. Damage limitation, life safety prevention, and near collapse prevention performance levels are considered by accounting for multiple damage criteria and damage states, including both local and global approaches and both ductile and brittle failure of beam-to-column connections.The paper provides guidance for implementation of seismic capacity and performance evaluation of precast buildings according to latest advances, considering both European and US approaches. The study proves that the current code prescriptions might not supply adequate seismic performance, especially in the case of more critical soil conditions (i.e., C soil type compared to A soil type). A relatively critical performance was observed for damage limitation and life safety prevention performance levels, whereas collapse performance of the buildings was adequate. The extremely critical condition associated with brittle failure of the beam-to-column connection was also confirmed. The study stresses the need for an enhancement of current design requirements and highlights the need for further investigation into the adequacy of seismic performance of precast buildings and reliability of related code-conforming design.

An efficient stratified sampling scheme for the simultaneous estimation of small failure probabilities in wind engineering applications

With the advent of performance-based engineering frameworks for extreme winds, greater efficiency in the estimation of failure probabilities associated with multiple limit states at ultimate load levels, including collapse, is required. To meet this demand, an optimal stratified sampling-based simulation scheme is proposed that shares several meritorious characteristics of standard Monte Carlo simulation (MCS), including the capability to deal with high-dimensional uncertain spaces and multiple limit state functions. The optimality of the scheme is governed by the allocation of MCS samples among the strata as well as the strata division itself. A constrained optimization problem is formulated to address efficient sample allocation which is an extension of the Neyman allocation. Details for the practical implementation of the scheme within wind engineering reliability assessment applications are discussed. It is observed that the proposed methodology can reduce the variance by several orders of magnitude compared to MCS, which is illustrated on a wind-excited steel structure for the limit states of member/system first yield and collapse. The applicability of Latin hypercube sampling within stratified sampling-based simulation and the resulting efficiency gains are also critically studied. The results show promise in the wider applicability of the proposed scheme in performance-based wind engineering.

Progressive collapse fragility of substandard and earthquake-resistant precast RC buildings

Several natural and man-made disasters have highlighted that precast reinforced concrete (RC) buildings often have insufficient levels of structural robustness, which is the last line of defence under extreme events. Even though robustness studies were originated by the progressive collapse of the Ronan Point precast RC building in 1968, such a structural feature of disaster resilience has been partially investigated in case of precast RC buildings to date. In this study, a precast RC frame structure representative of low-rise commercial buildings is analysed under different column loss scenarios, which can produce the partial or total collapse of the structural system. Two building classes are considered, namely, buildings designed only to gravity loads and buildings designed for earthquake resistance. Structural robustness is probabilistically assessed through a fragility analysis procedure, using three-dimensional fibre-based finite element models with nonlinear links simulating connections, large-displacement incremental dynamic analysis, and multiple performance limit states for damage assessment. The output of the study is threefold: (i) a detailed assessment of the effects of column loss on precast RC frame buildings, quantifying the major role of structural detailing and beam-column connections; (ii) a fragility-informed evaluation of beneficial effects of seismic detailing on structural robustness, which can drive designers in retrofitting of existing precast RC buildings and decision-makers towards prioritization-related issues; and (iii) the generation of typological fragility curves for progressive collapse risk assessment in both single- and multi-hazard environments. Progressive collapse fragility curves can be convolved with more classical fragility models describing the failure of single components (under, e.g., flow-type landslide impact, vehicle collision or blast) and hazard, to assess systemic structural risk.

An enhanced PDEM-based framework for reliability analysis of structures considering multiple failure modes and limit states

In this paper, an enhanced probability density evolution method (PDEM) framework considering multiple failure modes and limit states is proposed for reliability analysis of structures. Firstly, the PDEM principle and the enhanced mechanism are illustrated, and during the process three typical combination types (i.e., circle, triangle, square ways) are introduced. Secondly, two case studies are given to verify the effectiveness of the enhanced PDEM-based framework and the necessity to consider multiple limit states. The first example is a simply supported beam under two-point concentrated forces with two failure conditions (i.e., shear failure and flexural failure), and the second example is a 3-span-6-story reinforced concrete frame under seismic excitation with three failure conditions (i.e., maximum displacement failure, residual displacement failure and floor acceleration failure). Meanwhile, the Monte Carlo simulation (MCS) is also performed for both examples as a comparison and validation. Thirdly, parametric studies with related to two important aspects in the enhanced PDEM-based framework are primarily performed, including a modified equation of the target variable value via representative points incorporating the influence of individual quantile parameters (e.g., 16%, 50% and 84% quantile), as well as the other potential combination types in the enhanced PDEM-based framework (i.e., more than circle, triangle, square ways). In general, the paper provides a reference to perform the PDEM-based reliability assessment for multiple limit states and multiple failure patterns in the future. The enhanced framework presents less calculation burden and shows comparative calculation accuracy with the MCS. Meanwhile, the enhanced results are generally more conservative and commonly illustrate a lower reliability when compared with the single limit state, which can result in a more comprehensive decision and more robust strategy under the same condition in the practical engineering.

Fragility-based framework for optimal damper placement in low-rise moment-frame buildings using machine learning and genetic algorithm

Commonly, seismic fragility is used as an assessment tool rather than a design constraint for existing building structures. In addition, the degree of seismic fragility safety margin after retrofitting existing structures is usually unknown during the early retrofit design stage. The current study proposes a framework that allows the designer to assign the preferred seismic fragility margin as a design variable before retrofit of a structure using soft computing techniques. In addition, it provides the design damping ratio to achieve the optimal damper capacity distribution required for retrofit. To this end, a machine learning model (MLM) is used to construct safety margin curve (SMC) graphs for the structural system considering different limit states to obtain the design damping ratio. This MLM is trained with a wide range of structural systems subjected to various levels of ground motion intensities for predicting seismic responses of the system. The main role of the MLM and SMC is to provide a reasonable approximate range for damper capacity in each story to reduce the computational time required for the optimization process. After that, a genetic algorithm (GA) optimization process is implemented to obtain the optimum capacities of the energy dissipation devices (EDDs) required for retrofitting the structure using the design damping ratio obtained from the SMC. Displacement-dependent EDDs are used in the current study because of their effectiveness in dissipating seismic energy. The EDDs are arranged such that the seismic demands are controlled to achieve the required basic seismic safety objective maintaining the optimal EDD capacity for each story. The reliability and robustness of the framework are investigated using various case-study examples and validated by comparing the results with those obtained from nonlinear time history (NLTH) and nonlinear static methods. The proposed framework is validated for low-rise moment-resisting frame buildings that can be represented using a single degree of freedom (SDOF) system. The framework is expected to enable researchers and engineers to effectively implement the required level of seismic safety in retrofitted structures with less computational cost and allows the designer to select the most suitable retrofit scheme considering multiple limit state criteria checks.

Sensitivity analysis of seismic performance of portfolio of bridge supports for multiple qualitative limit states

The seismic performance of bridges is of primary importance for the durability and functionality of these structures. Several numerical methods have arisen in order to evaluate this performance and one is generally led to define the most significant input parameters in the seismic response. Thus, the sensitivity analyses are practical tools to achieve this objective. In this study, a sensitivity study was carried out on the seismic performance of 88 supports of a road network located in a low to moderate seismic zone. A preliminary visual inspection of these structures was carried out and showed the degree of advanced degradation that the structure had undergone during its lifetime. This visual inspection was completed by a set of non-destructive and destructive tests in order to characterize the geometric and mechanical parameters of the materials that compose these supports. The data collected is directly inserted into the finite element model on OpenSees, which describes the nonlinear fibered behavior with distributed plasticity of these supports. The seismic capacity is thus described through the nonlinear static analysis which provides the shear force curves at the base as a function of the displacement at the head of the pile. This curve will be idealized into a bilinear curve in order to integrate it into the equivalent linear system composed of both the supports and the elastomers that serve as seismic isolators. The sensitivity study is carried out on the basis of the analysis of variance, which evaluates the part of the variance of the input parameters in the output responses that characterize the seismic behavior of these structures. The input parameters are vertical loads, geometric and material properties, and output responses are the deck and top pier displacements, in addition to the shear force at the base of the supports and the distortion of the elastomers. The results of the sensitivity study allowed to prioritize the most significant parameters in the seismic responses for different limit states. The most significant parameters are the ratio of longitudinal and transverse reinforcement, followed by the diameter of the piles and the applied vertical loads.

Determination of Capacities of Eccentric Stiffeners Part 1: Experimental Studies

Forty column specimens were experimentally tested with an effort to identify the effective stiffener capacities of eccentric stiffeners when used within moment connections of beams connecting to column flanges. Three test methods were performed described as (1) single tension with load pulling away from the column specimen, (2) single compression with load applied toward the column specimen, and (3) double compression with load applied toward the column specimen and with a reaction plate directly opposite from the applied load. The sizes were selected with a range of slenderness ratios for the web and flange and to study multiple concentrated load limit states using AISC Specification Section J10 (AISC, 2016b). For each column size and test method, four column specimens were tested: (1) without stiffeners, (2) with stiffeners concentric with the applied load, (3) with stiffeners at a low eccentricity of 2 in. or 3 in., and (4) with stiffeners at a high eccentricity of 4 in. or 6 in. For column specimens tested without stiffeners and for compression tests, the maximum loads compared favorably to those predicted for web local crippling and were significantly higher than those predicted for web compression buckling (double compression tests) and web local yielding. For single tension tests, the maximum load always exceeded that predicted for the limit states of flange bending and web local yielding.

Life-cycle cost analysis of pile-supported wharves under multi-hazard condition: aging and shaking

Investigating the economic consequences of the natural hazards on the infrastructures is a crucial task in any risk-based decision-making process. Life-cycle cost (LCC) analysis is an important tool in both design (to choose the most economic configuration) and analysis (to estimate the future cost of ownership) stages. The pile-supported wharves, as one of the main parts in a marine harbor system, may experience significant deterioration (i.e. strength reduction) and seismic shaking during their life time. In this paper, damage cost in multiple limit states of a deteriorated pile-supported wharf due to chloride corrosion is calculated. A precise finite element model is developed to account for structural aging and the simultaneous seismic shaking. Aging-dependent seismic fragility functions are first developed using incremental dynamic analysis. Next, the LCC analysis is conducted considering the crane damage, as well as, inspection and maintenance costs. The results are calculated for three seismic hazard levels. The findings confirm significance of corrosion on LCC of pile-supported wharf. It is observed that corrosion may increase the LCC of the structure and system (i.e. structure and cranes) by 15% and 8%, respectively. Finally, a set of analytical formulations are proposed for performance index as a function of age, damage state, and seismic hazard level.

Cross‐level fragility analysis of modularized suspended buildings based on experimentally validated numerical models

SummarySuspended buildings typically have a core as the primary and suspended floors as the secondary structures. These configurations offer visual transparency, smaller vertical components, and seismic attenuation via the primary–secondary structure interaction. Such attenuation is further enhanced by the modularization of the suspended segment which allows large drifts but prevents them from causing damage. Previously conducted shake‐table tests have confirmed these features. However, how the component performance contributes to system performance has not been quantitated. To address this gap, fragility analyses are conducted for 10‐story experimentally validated models with optimized supplemental dampers and inter‐module stiffness. Multiple limit state functions are proposed to provide a full account of damage sources. Additionally, a mapping rule from the component‐level to the system‐level limit states is developed which captures the influence of damage distribution on system‐level limit states. Results for the uncontrolled suspended building indicate that for the PGV of 0.5 m/s, the failure probabilities of the repairable and life safety limit states are 97% and 83%, respectively. These probabilities are 92% and 27% for the frame structure with viscous dampers, 58% and 5% for the passive‐controlled modularized suspended building system (MSBS), and 45% and 3% for MSBS with optimal vertical distributions of modularized secondary structure.

Failure probability assessment and prediction of corroded pipeline under earthquake by introducing in-line inspection data

Considering the transient ground deformation (TGD) and permanent ground deformation (PGD), uncertain modeling is proposed by multiple limit states with time-dependent corrosion growth and seismic damage, as well as the rehabilitation. In this work, the developed corrosion growth model implemented the correlated 3D stochastic growth for one-defect. It then embedded in the combination model involved in corroded defect and seismic loading. Further, the Bayesian method and Markov Chain Monte Carlo (MCMC) simulation were used to obtain the updating of failure probability by introducing rehabilitation and in-line inspection (ILI) data. Failure probability was performed to study this difference using Monte Carlo simulation (MCS) to compare with the effect of the different parameters. A numerical example was investigated proposed models, and the influence of assessment rehabilitation was also discussed. The results indicate the potential impact of seismic damage for the corroded pipeline is significant, which can optimize the frame reliability-based anti-seismic for pipe infrastructure.

Evaluation of wind directionality on wind load effects and assessment of system reliability of wind-excited structures

This study re-evaluate wind directionality effect on various structural responses having distinct directionality characteristics using a relatively simple but sufficiently accurate approach with closed-form estimation of response with a given mean recurrence interval. The approach is established on the basis that the directional yearly maximum responses observed at different hours can be assumed to be mutually independent at the distribution upper tail region, even when the directional yearly maximum wind speeds can be correlated to some extent. This study also evaluates the performance of AIJ procedure and traditional sector-by-sector approach through comprehensive case studies. Furthermore, this study presents an analysis framework for assessing overall reliability/risk of a wind-exited structure associated with multiple limit state responses. The limit state responses can be considered to be mutually independent especially in the distribution upper tail region and when the wind directionality is further considered. Therefore, the failure probability associated with multiple limit state responses can be estimated as the sum of those associated with each response at each wind direction. The characteristics of failure probability influenced by wind directionality and structural orientation are investigated. This study also highlights the sensitivity of failure probability estimation to assumed probability distribution model of extreme response.

EEK-SYS: System reliability analysis through estimation error-guided adaptive Kriging approximation of multiple limit state surfaces

In order to approximate the multiple limit state functions for different failure events, the active learning Kriging model proposed for component reliability analysis has been extended to system reliability analysis. Meanwhile, many efficient sampling strategies have been applied to reduce the high computational burden. However, these strategies meet a challenge in wasting some training points and terminating the training process inappropriately, since they do not directly relate to the estimation error of system failure probability. To address the challenge, this work proposes an estimation error-guided adaptive Kriging method. As Kriging prediction may be inaccurate before being well trained, the predicted system failure probability may deviate from the true result. To quantify this estimation error, the true number of failure points is approximated by adding the number of predicted failure points and the number of wrongly classified points. Since it is impossible to learn the exact number of wrongly classified points, its confidence interval is derived based on the probability of making wrong state classification. Subsequently, the refinement of Kriging is achieved by using the probability to identify new points and using the estimation error to determine the termination, which has been demonstrated by three different cases.

Highly efficient Bayesian updating using metamodels: An adaptive Kriging-based approach

Bayesian updating offers a powerful tool for probabilistic calibration and uncertainty quantification of models as new observations become available. By reformulating Bayesian updating into a structural reliability problem via introducing an auxiliary random variable, the state-of-the-art Bayesian updating with structural reliability method (BUS) has showcased large potential to achieve higher accuracy and efficiency compared with conventional approaches based on Markov Chain Monte Carlo simulations. However, BUS faces a number of limitations. The transformed reliability problem often involves a very rare event especially when the number of observations increases. This along with the fact that conventional reliability analysis techniques are not efficient, and often not capable of accurately estimating the probability of rare events, unavoidably lead to a very large number of evaluations of the likelihood function and simultaneously insufficient accuracy of the derived posterior distributions. To overcome these limitations, we propose Simple Rejection Sampling with Multiple Auxiliary Random Variables (SRS-MARV), where the limit state function in BUS is decomposed into a system reliability problem with multiple limit state functions. The main advantage of this approach is that the acceptance rate of each decomposed limit state function is significantly improved, which facilitates effective integration of adaptive Kriging-based reliability analysis into SRS-MARV. Moreover, a new stopping criterion is proposed for efficient, adaptive training of the Kriging model. The proposed method called BUAK is shown to be highly computationally efficient and accurate based on results of comprehensive investigations for three diverse benchmark problems. Compared to the state-of-the-art methods, BUAK substantially reduces the computational demand by one to three orders of magnitude, therefore, facilitating the application of Bayesian updating to computationally very intensive models.

Estimation of multiple limit state responses with various mean recurrence intervals considering directionality effects

Assessment of system reliability of structures against wind loading requires evaluation of the joint probability distribution of multiple limit state responses with consideration of wind and aerodynamic directionality effects. While the directionality effect has been extensively addressed in literature on the estimation of probabilistic single limit state response, it is not the case for multiple limit state responses. This study revisits estimation of extreme responses with various mean recurrence intervals, with a focus on adequate definition of directionality factor. A new framework is then presented for estimation of joint probability distribution of multiple extreme responses with consideration of directionality effect. This new framework can be considered as an extension of existing efforts on directionality effect for single limit state response to multiple ones, thus help in better assessing system reliability of structures against wind loading. The proposed framework is applied to responses of claddings on a large-span saddle type roof.