Towards multiple hazard resilient bridges: a methodology for modeling frequent and infrequent time-varying loads Part I, Comprehensive reliability and partial failure probabilities

The current AASHTO load and resistance factor design (LRFD) guidelines are formulated based on bridge reliability,which interprets traditional design safety factors into more rigorously deduced factors based on the theory of probability.This is a major advancement in bridge design specifications.However,LRFD is only calibrated for dead and live loads.In cases when extreme loads are significant,they need to be individually assessed.Combining regular loads with extreme loads has been a major challenge,mainly because the extreme loads are time variables and cannot be directly combined with time invariant loads to formulate the probability of structural failure.To overcome these difficulties,this paper suggests a methodology of comprehensive reliability,by introducing the concept of partial failure probability to separate the loads so that each individual load combination under a certain condition can be approximated as time invariant.Based on these conditions,the extreme loads (also referred to as multiple hazard or MH loads) can be broken down into single effects.In Part Ⅱ of this paper,a further breakdown of these conditional occurrence probabilities into pure conditions is discussed by using a live truck and earthquake loads on a bridge as an example.There are three major steps in establishing load factors from MH load distributions:(1) formulate the failure probabilities;(2) normalize various load distributions;and (3) establish design limit state equations.This paper describes the formulation of the failure probabilities of single and combined loads.     

Bridge pier failure probabilities under combined hazard effects of scour, truck and earthquake. Part I: occurrence probabilities

In many regions of the world, a bridge will experience multiple extreme hazards during its expected service life. The current American Association of State Highway and Transportation Officials (AASHTO) load and resistance factor design (LRFD) specifications are formulated based on failure probabilities, which are fully calibrated for dead load and nonextreme live loads. Design against earthquake loads is established separately. Design against scour effect is also formulated separately by using the concept of capacity reduction (or increased scour depth). Furthermore, scour effect cannot be linked directly to an LRFD limit state equation, because the latter is formulated using force-based analysis. This paper (in two parts) presents a probability-based procedure to estimate the combined hazard effects on bridges due to truck, earthquake and scour, by treating the effect of scour as an equivalent load effect so that it can be included in reliability-based bridge failure calculations. In Part I of this series, the general principle of treating the scour depth as an equivalent load effect is presented. The individual and combined partial failure probabilities due to truck, earthquake and scour effects are described. To explain the method of including non-force-based natural hazards effects, two types of common scour failures are considered. In Part II, the corresponding bridge failure probability, the occurrence of scour as well as simultaneously having both truck load and equivalent scour load are quantitatively discussed.     

Towards multiple hazard resilient bridges: a methodology for modeling frequent and infrequent time-varying loads Part II, Examples for live and earthquake load effects

The current AASHTO load and resistance factor design (LRFD) guidelines are formulated based on bridge reliability,which interprets traditional design safety factors into more rigorously deduced factors based on the theory of probability.This is a major advancement in bridge design specifications.However,LRFD is only calibrated for dead and live loads.In cases when extreme loads are significant,they need to be individually assessed.Combining regular loads with extreme loads has been a major challenge,mainly because the extreme loads are time variable and cannot be directly combined with time invariant loads to formulate the probability of structural failure.To overcome these difficulties,this paper suggests a methodology of comprehensive reliability,by introducing the concept of partial failure probability to separate the loads so that each individual load combination under a certain condition can be approximated as time invariant.Based on these conditions,the extreme loads (also referred to as multiple hazard or MH loads) can be broken down into single effects.In this paper,a further breakdown of these conditional occurrence probabilities into pure conditions is discussed by using a live truck and earthquake loads on a bridge as an example.     

Bridge pier failure probabilities under combined hazard effects of scour, truck and earthquake. Part II: failure probabilities

In many regions of the world, a bridge will experience multiple extreme hazards during its expected service life. The current American Association of State Highway and Transportation Officials (AASHTO) load and resistance factor design (LRFD) specifications are formulated based on failure probabilities, which are fully calibrated for dead load and non-extreme live loads. Design against earthquake load effect is established separately. Design against scour effect is also formulated separately by using the concept of capacity reduction (or increased scour depth). Furthermore, scour effect cannot be linked directly to an LRFD limit state equation because the latter is formulated using force-based analysis. This paper (in two parts) presents a probability-based procedure to estimate the combined hazard effects on bridges due to truck, earthquake and scour, by treating the effect of scour as an equivalent load effect so that it can be included in reliability-based failure calculations. In Part I of this series, the general principle for treating the scour depth as an equivalent load effect is presented. In Part II, the corresponding bridge failure probability, the occurrence of scour as well as simultaneously having both truck load and equivalent scour load effect are quantitatively discussed. The key formulae of the conditional partial failure probabilities and the necessary conditions are established. In order to illustrate the methodology, an example of dead, truck, earthquake and scour effects on a simple bridge pile foundation is represented.     

基于Ferry Borges理论的地震与重卡车作用下桥梁内力的组合问题

目前的AASHTO LRFD桥梁规范主要考虑重力荷载和卡车等活荷载的组合情况,而没有在概率基础上考虑地震等极端荷载的组合问题.包括LRFD在内的很多桥梁抗震规范都是主要考虑地震的作用,甚至设计时不考虑其他荷载的作用,LRFD设计指导手册中在考虑地震等极端荷载时,也只是提到在特殊情况和桥梁比较长的情况下再考虑卡车的作用,...     

Uniform reliability safety format for seismic design of reinforced concrete structures

A safety format is proposed for the flexural design of reinforced concrete members for the combination of seismic and gravity loads, with load and resistance factors which depend on member type, on the value of the target theoretical probability of failure and on the ratio of the load effect due to gravity loads to that due to the nominal value of the seismic action, both obtained by elastic analysis. Safety factors are computed through an advanced Level II reliability procedure, using a limit state inequality between the member rotation ductility supply under monotonic loading and the peak rotation ductility and cyclic energy dissipation demands. Uncertainties considered are: for resistance, the uncertainty of failure under imposed cyclic deformations, and for action, the maximum peak ductility and energy dissipation demands in the structure's lifetime, as obtained through a series of non-linear dynamic analyses of multistorey buildings in 3D. using as input ensembles of bidirectional acceleration time-histories which describe probabilistically the extreme bidirectional seismic action in the structure's lifetime. Computed load and resistance factors are practically independent of the load-effects ratio. The load factor on the seismic action is found to be independent of member type and to increase with the theoretical probability of failure much faster than the elastic spectral value at the structure's fundamental period with probability of exceedance in the structure's lifetime. Simple rules for the dependence of the resistance modification factors on the theoretical failure probability are also derived. As for the computed values of the load factors the moment due to gravity loads is negligible in comparison to the factored seismic moment, a simplified safety checking inequality between the design flexural capacity and a reduced seismic moment is proposed, in which the ratio of the resistance to the load factor plays the role of a force reduction or effective behaviour factor for the member.     

Reliability analysis of offshore structures within a time varying environment

The analysis and design of offshore structures necessitates the consideration of environmental loads. Realistic modeling of the environmental loads is particularly important to ensure reliable performance of these structures. In this paper, structural reliability analysis of offshore structures subjected to a time varying environment is investigated. In this work, an extreme value statistical model for the wave height is adopted as a basis for the performance assessment of a jacket structure. Due to the changing environment, the model parameters are modeled to be time varying. To deal with this issue, two segmentation algorithms are proposed and applied to the observed data in order to derive piecewise stationary processes for a statistical analysis. The investigation includes the extreme value modeling of the wave height in the characterization of the sea load. The implementation of the segmentation algorithms in the original data eventually leads to approximations of the safety quality of the existing structure within different time interval. The computed result is developed to reflect the time varying effects in the failure probability of structures. The results are compared with the traditional extreme values approach in view of the accuracy and information content. The investigation is also extended to a case where the design of the structure ignores the time varying property.     

桁架结构基于地震可靠性分析的优化方法

本文把桁架结构地震可靠性分析和最优化设计方法结合起来，以结构的地震失效率概率为目标函数，给出一种考虑地震可靠性的桁架结构的优化方法。该方法能够解决线性桁架体系在平稳的随机地震地面运动激励下的优化问题，并在给定投资的条件下设计出了安全可靠的桁架结构。     

结构分灾抗震设计:概念和应用

详细论述了结构分灾抗震设计的产生背景、设计思想、优化模型和基本原则，指出结构分灾设计是在分析基于投资—效益准则的结构抗震设计模型的基础上，对工程实践中一些成功经验的提炼和概括而形成的设计方法，工程领域中一些现行设计方法和措施就是分灾设计的具体应用。当工程师们待处理的问题必须考虑高度不确定性因素时，将分灾设计作为一种可能选用的设计理念，将有助于工程师们实现设计创新。分灾设计符合基于性能的抗震设计思想，可以方便地实现基于性能的设计。     

Site-specific seismic design of damage tolerant structural systems using a novel concept

Designing more seismic load-tolerant structures is one of the major challenges of the world communities. It is due to the inability of the profession to predict future design earthquake time histories at a site compounded by the failure to appropriately incorporate uncertainties in other design variables and structural behavior just before failure. A site-specific method is proposed to generate a suite of ground excitation time histories and a novel risk estimation is developed considering major sources of nonlinearity and uncertainty. For wider application and acceptance, the risk evaluation procedure essentially consists of few deterministic time domain finite element analyses. The procedure is verified and showcased by estimating risks associated with three buildings designed by professional experts in the Los Angeles area satisfying the post-Northridge design criteria for the overall lateral deflection and inter-story drift. The accuracy of the estimated risk is verified using the Monte Carlo simulation technique. In all cases, the probabilities of collapse are found to satisfy the current code requirements. The spread in the reliability indexes for each building for both limit states cannot be overlooked, indicating the significance of the frequency contents of the time histories. The inter-story drift is found to be more critical than the overall lateral displacement. The reliability indexes for both limit states are similar only for few cases. The authors believe that they proposed an alternative to the classical random vibration and simulation approaches. The proposed site-specific seismic safety assessment procedure can be used by practicing engineers for routine applications.     

非均质软基上桶形基础承载性能有限元分析

作为海洋平台的基础部分,桶形基础不仅承受海洋平台结构及自身重量等竖向荷载的长期作用,而且往往还遭受波浪等所产生的水平荷载及力矩等其它荷载分量的作用。因此,确定软基上桶形基础在竖向荷载（V）、水平荷载（H）和力矩（M）等共同作用下的承载特性,建立其在复合加载模式下的破坏包络面,并进而依此评价海洋平台基础及地基的稳定性,是桶形基础设计与施工中的关键问题。在大型通用有限元分析软件ABAQUS平台上,采用基于Mises屈服准则的完全弹塑性本构模型,对横观各向异性软基上桶形基础的承载性能进行了三维弹塑性数值分析,探讨了软黏土不排水抗剪强度的各向异性对单个荷载和复合加载作用下桶形基础承载性能的影响。     

Adapting earthquake actions in Eurocode 8 for performance‐based seismic design

Performance‐based seismic design (PBSD) can be considered as the coupling of expected levels of ground motion with desired levels of structural performance, with the objective of achieving greater control over earthquake‐induced losses. Eurocode 8 (EC8) already envisages two design levels of motion, for no collapse and damage limitation performance targets, anchored to recommended return periods of 475 and 95 years, respectively. For PBSD the earthquake actions need to be presented in ways that are appropriate to the estimation of inelastic displacements, since these provide an effective control on damage at different limit states. The adequacy of current earthquake actions in EC8 are reviewed from this perspective and areas requiring additional development are identified. The implications of these representations of the seismic loads, in terms of mapping and zonation, are discussed. The current practice of defining the loading levels on the basis of the pre‐selected return periods is challenged, and ideas are discussed for calibrating the loading‐performance levels for design on the basis of quantitative earthquake loss estimation. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability analysis of slope with building loads

In hill areas, landslides are frequent and hazardous. In developing hill areas, many multistoreyed r.c.c. farmed buildings are constructed on hill slope. The building loads are transferred to the hill slope terrain at the foundation level, which may cause hill slope failure. Therefore the stability of hill slope with building loads needs to be checked. In this paper, a procedure has been developed to find the factor of safety against sliding failure of slope considering building loads transferred to the slope. Earthquake forces can also be considered in the analysis. Different types of soils in the slope can be considered. A computer program has been developed based on the formulation presented in the paper and is validated by solving few examples. Stability of slope with differently configured buildings have been studied. Provisions to be made for stepped foundations on hill slopes has been highlighted here.     

DUAL-LEVEL SEISMIC DESIGN: A RELIABILITY-BASED METHODOLOGY

The seismic design provisions of most building codes in the United States specify ground motion parameters for various regions of the country and provide simple formulae to determine a distribution of lateral forces for which the structure should be designed. Although the code provisions are very simple to use, they oversimplify a complex problem and are based on many implicit assumptions which many designers may not appreciate. Furthermore, the reliability of the final design is not easily determined. This paper describes a reliability-based seismic design procedure for building structures. It is a performance-based design procedure which requires the designer to verify that a particular structural design satisfies displacement-based performance criteria. An equivalent system methodology and uniform hazard spectra are used to evaluate structural performance. The performance criteria are expressed in probabilistic terms, and deterministic design-checking equations are derived from these criteria. The design-checking equations incorporate design factors (analogous to load and resistance factors) which account for the uncertainty in the seismic hazard, the uncertainty in predicting site soil effects, and the approximate nature of the simplified models of the structure. The alternative procedure should enable designers to achieve code-specified target performance objectives for moderate and severe levels of earthquake excitation.     

Analytical evaluation of structural component limit state probabilities

Probabilistic seismic assessment requires extensive computational effort resulting from variability both in input ground motions and mechanical properties. Nonetheless, such methodologies are of considerable importance, namely for pre-earthquake disaster planning or development of retrofitting programs. A framework for the analytical definition of closed form expressions for exceedance probabilities of structural component limit states, defined by limit values of structural response parameters, is proposed herein. The definition of these expressions is based on the probabilistic representation of the ground motion intensity and on the establishment of suitable expressions characterizing the evolution of structural demand with increasing earthquake intensity. Distinction is made between deformation-based and force-based structural parameters in the definition of such relations. Within the proposed framework, the limit states are defined by single deterministic thresholds of structural response quantities at the component level, as defined in structural codes. Different approaches are also discussed to account for the randomness of the mechanical properties and ground motion input within the proposed methodology. An application of the assessment of different limit state probabilities of members from a reinforced concrete building is presented, for which limit states and limit state capacities are defined according to the upcoming Part 3 of the Eurocode 8. Although the presented application only deals with member chord rotation and shear force limit state probabilities, the proposed methodology can be generalized to other deformation-based and force-based structural parameters.     

Effects of two alternative representations of ground‐motion uncertainty on probabilistic seismic demand assessment of structures

A probabilistic representation of the entire ground‐motion time history can be constructed based on a stochastic model that depends on seismic source parameters. An advanced stochastic simulation scheme known as Subset Simulation can then be used to efficiently compute the small failure probabilities corresponding to structural limit states. Alternatively, the uncertainty in the ground motion can be represented by adopting a parameter (or a vector of parameters) known as the intensity measure (IM) that captures the dominant features of the ground shaking. Structural performance assessment based on this representation can be broken down into two parts, namely, the structure‐specific part requiring performance assessment for a given value of the IM, and the site‐specific part requiring estimation of the likelihood that ground shaking with a given value of the IM takes place. The effect of these two alternative representations of ground‐motion uncertainty on probabilistic structural response is investigated for two hazard cases. In the first case, these two approaches are compared for a scenario earthquake event with a given magnitude and distance. In the second case, they are compared using a probabilistic seismic hazard analysis to take into account the potential of the surrounding faults to produce events with a range of possible magnitudes and distances. The two approaches are compared on the basis of the probabilistic response of an existing reinforced‐concrete frame structure, which is known to have suffered shear failure in its columns during the 1994 Northridge Earthquake in Los Angeles, California. Copyright © 2007 John Wiley & Sons, Ltd.     

A new formulation for the stochastic control of systems with bounded state variables: An application to a single reservoir system

A new formulation is presented for the analysis of reservoir systems synthesizing concepts from the traditional stochastic theory of reservoir storage, moments analysis and reliability programming. The analysis is based on the development of the first and second moments for the stochastic storage state variable. These expressions include terms for the failure probabilities (probabilities of spill or deficit) and consider the storage bounds explicitly. Using this analysis, expected values of the storage state, variances of storage, optimal release policies and failure probabilities — useful information in the context of reservoir operations and design, can be obtained from a nonlinear programming solution. The solutions developed from studies of single reservoir operations on both an annual and monthly basis, compare favorably with those obtained from simulation. The presentation herein is directed to both traditional reservoir storage theorists who are interested in the design of a reservoir and modern reservoir analysts who are interested in the long term operation of reservoirs.     

钢筋混凝土建筑结构的抗地震破坏倒塌能力评估研究

在地震作用下钢筋混凝土建筑结构出现破坏倒塌为地震灾害中的关键,有效评估建筑结构抗地震破坏倒塌能力是建筑结构设计的前提,也是当前建筑结构提高抗震性能与加固的依据。提出变形指标极值、失效判断标准以及钢筋混凝土建筑结构倒塌极限状态判断标准,据此获取倒塌储备系数、倒塌易损性、结构整体超强系数、结构整体延性系数等评估标准。采用Pushover分析法选择相应地震波。依据梁柱线刚比对建筑结构抗倒塌能力的影响,以及柱端弯矩增加系数对建筑结构抗地震破坏倒塌能力的影响,对建筑结构易损性进行分析。结果表明:等跨建筑结构抗地震破坏倒塌能力更强;建筑结构底层是薄弱层,COF值越高,结构越容易倒塌。     

Reliability-based design of R/C water tank structures under seismic action

Based on a reliability level 2 method, a procedure is proposed to design reinforced concrete structures for elevated tanks subjected to seismic action, with a specified probability of failure in a 50-year design life. To evaluate the probability of failure the ultimate limit state is obtained when the top column displacement demanded by the earthquake, a random variable, reaches the allowable displacement, which is here treated as deterministic. The seismic action is characterized probabilistically by the power spectral density function of the ground acceleration, which is obtained from a design spectrum. The strength and ductility of an annular column section of confined reinforced concrete for cyclic loads are evaluated with design aids. Design charts are made for a given tank capacity and specified seismic zone that allow one to choose different combinations of strength, stiffness and ductility for the same tolerable probability of failure. A step by step method is suggested for the design of the annular column section, choosing finally the most convenient design. The advantage of this methodology is shown through a numerical example.     

Reliability-based load and resistance factor design approach for external seismic stability of reinforced soil walls

Load and resistance factor design (LRFD) approach for the design of reinforced soil walls is presented to produce designs with consistent and uniform levels of risk for the whole range of design applications. The evaluation of load and resistance factors for the reinforced soil walls based on reliability theory is presented. A first order reliability method (FORM) is used to determine appropriate ranges for the values of the load and resistance factors. Using pseudo-static limit equilibrium method, analysis is conducted to evaluate the external stability of reinforced soil walls subjected to earthquake loading. The potential failure mechanisms considered in the analysis are sliding failure, eccentricity failure of resultant force (or overturning failure) and bearing capacity failure. The proposed procedure includes the variability associated with reinforced backfill, retained backfill, foundation soil, horizontal seismic acceleration and surcharge load acting on the wall. Partial factors needed to maintain the stability against three modes of failure by targeting component reliability index of 3.0 are obtained for various values of coefficients of variation (COV) of friction angle of backfill and foundation soil, distributed dead load surcharge, cohesion of the foundation soil and horizontal seismic acceleration. A comparative study between LRFD and allowable stress design (ASD) is also presented with a design example.