A comparison of deterministic, reliability-based and risk-based structural optimization under uncertainty

In this paper, the effects of uncertainty and expected costs of failure on optimum structural design are investigated, by comparing three distinct formulations of structural optimization problems. Deterministic Design Optimization (DDO) allows one the find the shape or configuration of a structure that is optimum in terms of mechanics, but the formulation grossly neglects parameter uncertainty and its effects on structural safety. Reliability-based Design Optimization (RBDO) has emerged as an alternative to properly model the safety-under-uncertainty part of the problem. With RBDO, one can ensure that a minimum (and measurable) level of safety is achieved by the optimum structure. However, results are dependent on the failure probabilities used as constraints in the analysis. Risk optimization (RO) increases the scope of the problem by addressing the compromising goals of economy and safety. This is accomplished by quantifying the monetary consequences of failure, as well as the costs associated with construction, operation and maintenance. RO yields the optimum topology and the optimum point of balance between economy and safety. Results are compared for some example problems. The broader RO solution is found first, and optimum results are used as constraints in DDO and RBDO. Results show that even when optimum safety coefficients are used as constraints in DDO, the formulation leads to configurations which respect these design constraints, reduce manufacturing costs but increase total expected costs (including expected costs of failure). When (optimum) system failure probability is used as a constraint in RBDO, this solution also reduces manufacturing costs but by increasing total expected costs. This happens when the costs associated with different failure modes are distinct. Hence, a general equivalence between the formulations cannot be established. Optimum structural design considering expected costs of failure cannot be controlled solely by safety factors nor by failure probability constraints, but will depend on actual structural configuration. (c) 2011 Elsevier Ltd. All rights reserved.

Reliability-based design optimization strategies based on FORM: a review

In deterministic optimization, the uncertainties of the structural system (i.e. dimension, model, material, loads, etc) are not explicitly taken into account. Hence, resulting optimal solutions may lead to reduced reliability levels. The objective of reliability based design optimization (RBDO) is to optimize structures guaranteeing that a minimum level of reliability, chosen a priori by the designer, is maintained. Since reliability analysis using the First Order Reliability Method (FORM) is an optimization procedure itself, RBDO (in its classical version) is a double-loop strategy: the reliability analysis (inner loop) and the structural optimization (outer loop). The coupling of these two loops leads to very high computational costs. To reduce the computational burden of RBDO based on FORM, several authors propose decoupling the structural optimization and the reliability analysis. These procedures may be divided in two groups: (i) serial single loop methods and (ii) unilevel methods. The basic idea of serial single loop methods is to decouple the two loops and solve them sequentially, until some convergence criterion is achieved. On the other hand, uni-level methods employ different strategies to obtain a single loop of optimization to solve the RBDO problem. This paper presents a review of such RBDO strategies. A comparison of the performance (computational cost) of the main strategies is presented for several variants of two benchmark problems from the literature and for a structure modeled using the finite element method.

Fuzzy Monte Carlo model for transmission power systems reliability based decision making

This thesis presents the Fuzzy Monte Carlo Model for Transmission Power Systems Reliability based studies (FMC-TRel) methodology, which is based on statistical failure and repair data of the transmission power system components and uses fuzzyprobabilistic modeling for system component outage parameters. Using statistical records allows developing the fuzzy membership functions of system component outage parameters. The proposed hybrid method of fuzzy set and Monte Carlo simulation based on the fuzzy-probabilistic models allows catching both randomness and fuzziness of component outage parameters. A network contingency analysis to identify any overloading or voltage violation in the network is performed once obtained the system states. This is followed by a remedial action algorithm, based on Optimal Power Flow, to reschedule generations and alleviate constraint violations and, at the same time, to avoid any load curtailment, if possible, or, otherwise, to minimize the total load curtailment, for the states identified by the contingency analysis. For the system states that cause load curtailment, an optimization approach is applied to reduce the probability of occurrence of these states while minimizing the costs to achieve that reduction. This methodology is of most importance for supporting the transmission system operator decision making, namely in the identification of critical components and in the planning of future investments in the transmission power system. A case study based on Reliability Test System (RTS) 1996 IEEE 24 Bus is presented to illustrate with detail the application of the proposed methodology.

Uncertainty propagation in inverse reliability-based design of composite structures

An approach for the analysis of uncertainty propagation in reliability-based design optimization of composite laminate structures is presented. Using the Uniform Design Method (UDM), a set of design points is generated over a domain centered on the mean reference values of the random variables. A methodology based on inverse optimal design of composite structures to achieve a specified reliability level is proposed, and the corresponding maximum load is outlined as a function of ply angle. Using the generated UDM design points as input/output patterns, an Artificial Neural Network (ANN) is developed based on an evolutionary learning process. Then, a Monte Carlo simulation using ANN development is performed to simulate the behavior of the critical Tsai number, structural reliability index, and their relative sensitivities as a function of the ply angle of laminates. The results are generated for uniformly distributed random variables on a domain centered on mean values. The statistical analysis of the results enables the study of the variability of the reliability index and its sensitivity relative to the ply angle. Numerical examples showing the utility of the approach for robust design of angle-ply laminates are presented.

Improved ground motion intensity measures for reliability-based demand analysis of structures

In Performance-Based Earthquake Engineering (PBEE), evaluating the seismic performance (or seismic risk) of a structure at a designed site has gained major attention, especially in the past decade. One of the objectives in PBEE is to quantify the seismic reliability of a structure (due to the future random earthquakes) at a site. For that purpose, Probabilistic Seismic Demand Analysis (PSDA) is utilized as a tool to estimate the Mean Annual Frequency (MAF) of exceeding a specified value of a structural Engineering Demand Parameter (EDP). This dissertation focuses mainly on applying an average of a certain number of spectral acceleration ordinates in a certain interval of periods, Sa,avg (T1,…,Tn), as scalar ground motion Intensity Measure (IM) when assessing the seismic performance of inelastic structures. Since the interval of periods where computing Sa,avg is related to the more or less influence of higher vibration modes on the inelastic response, it is appropriate to speak about improved IMs. The results using these improved IMs are compared with a conventional elastic-based scalar IMs (e.g., pseudo spectral acceleration, Sa ( T(¹)), or peak ground acceleration, PGA) and the advanced inelastic-based scalar IM (i.e., inelastic spectral displacement, Sdi). The advantages of applying improved IMs are: (i ) "computability" of the seismic hazard according to traditional Probabilistic Seismic Hazard Analysis (PSHA), because ground motion prediction models are already available for Sa (Ti), and hence it is possibile to employ existing models to assess hazard in terms of Sa,avg, and (ii ) "efficiency" or smaller variability of structural response, which was minimized to assess the optimal range to compute Sa,avg. More work is needed to assess also "sufficiency" and "scaling robustness" desirable properties, which are disregarded in this dissertation. However, for ordinary records (i.e., with no pulse like effects), using the improved IMs is found to be more accurate than using the elastic- and inelastic-based IMs. For structural demands that are dominated by the first mode of vibration, using Sa,avg can be negligible relative to the conventionally-used Sa (T(¹)) and the advanced Sdi. For structural demands with sign.cant higher-mode contribution, an improved scalar IM that incorporates higher modes needs to be utilized. In order to fully understand the influence of the IM on the seismis risk, a simplified closed-form expression for the probability of exceeding a limit state capacity was chosen as a reliability measure under seismic excitations and implemented for Reinforced Concrete (RC) frame structures. This closed-form expression is partuclarly useful for seismic assessment and design of structures, taking into account the uncertainty in the generic variables, structural "demand" and "capacity" as well as the uncertainty in seismic excitations. The assumed framework employs nonlinear Incremental Dynamic Analysis (IDA) procedures in order to estimate variability in the response of the structure (demand) to seismic excitations, conditioned to IM. The estimation of the seismic risk using the simplified closed-form expression is affected by IM, because the final seismic risk is not constant, but with the same order of magnitude. Possible reasons concern the non-linear model assumed, or the insufficiency of the selected IM. Since it is impossibile to state what is the "real" probability of exceeding a limit state looking the total risk, the only way is represented by the optimization of the desirable properties of an IM.

Reliability-based evaluation of design code provisions for circular concrete-filled steel columns

This paper presents an investigation of design code provisions for steel-concrete composite columns. The study covers the national building codes of United States, Canada and Brazil, and the transnational EUROCODE. The study is based on experimental results of 93 axially loaded concrete-filled tubular steel columns. This includes 36 unpublished, full scale experimental results by the authors and 57 results from the literature. The error of resistance models is determined by comparing experimental results for ultimate loads with code-predicted column resistances. Regression analysis is used to describe the variation of model error with column slenderness and to describe model uncertainty. The paper shows that Canadian and European codes are able to predict mean column resistance, since resistance models of these codes present detailed formulations for concrete confinement by a steel tube. ANSI/AISC and Brazilian codes have limited allowance for concrete confinement, and become very conservative for short columns. Reliability analysis is used to evaluate the safety level of code provisions. Reliability analysis includes model error and other random problem parameters like steel and concrete strengths, and dead and live loads. Design code provisions are evaluated in terms of sufficient and uniform reliability criteria. Results show that the four design codes studied provide uniform reliability, with the Canadian code being best in achieving this goal. This is a result of a well balanced code, both in terms of load combinations and resistance model. The European code is less successful in providing uniform reliability, a consequence of the partial factors used in load combinations. The paper also shows that reliability indexes of columns designed according to European code can be as low as 2.2, which is quite below target reliability levels of EUROCODE. (C) 2009 Elsevier Ltd. All rights reserved.

An approach for reliability-based robust design optimisation of angle-ply composites

Variations of manufacturing process parameters and environmental aspects may affect the quality and performance of composite materials, which consequently affects their structural behaviour. Reliability-based design optimisation (RBDO) and robust design optimisation (RDO) searches for safe structural systems with minimal variability of response when subjected to uncertainties in material design parameters. An approach that simultaneously considers reliability and robustness is proposed in this paper. Depending on a given reliability index imposed on composite structures, a trade-off is established between the performance targets and robustness. Robustness is expressed in terms of the coefficient of variation of the constrained structural response weighted by its nominal value. The Pareto normed front is built and the nearest point to the origin is estimated as the best solution of the bi-objective optimisation problem.

A reliability-based measure of robustness for concrete structures subjected to corrosion

This work is a contribution to the definition and assessment of structural robustness. Special emphasis is given to reliability of reinforced concrete structures under corrosion of longitudinal reinforcement. On this communication several authors’ proposals in order to define and measure structural robustness are analyzed and discussed. The probabilistic based robustness index is defined, considering the reliability index decreasing for all possible damage levels. Damage is considered as the corrosion level of the longitudinal reinforcement in terms of rebar weight loss. Damage produces changes in both cross sectional area of rebar and bond strength. The proposed methodology is illustrated by means of an application example. In order to consider the impact of reinforcement corrosion on failure probability growth, an advanced methodology based on the strong discontinuities approach and an isotropic continuum damage model for concrete is adopted. The methodology consist on a two-step analysis: on the first step an analysis of the cross section is performed in order to capture phenomena such as expansion of the reinforcement due to the corrosion products accumulation and damage and cracking in the reinforcement surrounding concrete; on the second step a 2D deteriorated structural model is built with the results obtained on the first step of the analysis. The referred methodology combined with a Monte Carlo simulation is then used to compute the failure probability and the reliability index of the structure for different corrosion levels. Finally, structural robustness is assessed using the proposed probabilistic index.

An application of reliability-based robustness assessment of steel moment resisting frame structures under post-mainshock cascading events

The paper presented herein proposes a reliability-based framework for quantifying the structural robustness considering the occurrence of a major earthquake (mainshock) and subsequent cascading hazard events, such as aftershocks that are triggered by the mainshock. These events can significantly increase the probability of failure of buildings, especially for structures that are damaged during the mainshock. The application of the proposed framework is exemplified through three numerical case studies. The case studies correspond to three SAC steel moment frame buildings of 3-, 9-, and 20- stories, which were designed to pre-Northridge codes and standards. Twodimensional nonlinear finite element models of the buildings are developed using the Open System for Earthquake Engineering Simulation framework (OpenSees), using a finite-length plastic hinge beam model and a bilinear constitutive law with deterioration, and are subjected to multiple mainshock-aftershock seismic sequences. For the three buildings analyzed herein, it is shown that the structural reliability under a single seismic event can be significantly different from that under a sequence of seismic events. The reliability-based robustness indicator used shows that the structural robustness is influenced by the extent by which a structure can distribute damage.

Artificial-Intelligence-Based Optimization of the Management of Snow Removal Assets and Resources Final Report, October 2002

Geographic information systems (GIS) and artificial intelligence (AI) techniques were used to develop an intelligent snow removal asset management system (SRAMS). The system has been evaluated through a case study examining snow removal from the roads in Black Hawk County, Iowa, for which the Iowa Department of Transportation (Iowa DOT) is responsible. The SRAMS is comprised of an expert system that contains the logical rules and expertise of the Iowa DOT’s snow removal experts in Black Hawk County, and a geographic information system to access and manage road data. The system is implemented on a mid-range PC by integrating MapObjects 2.1 (a GIS package), Visual Rule Studio 2.2 (an AI shell), and Visual Basic 6.0 (a programming tool). The system could efficiently be used to generate prioritized snowplowing routes in visual format, to optimize the allocation of assets for plowing, and to track materials (e.g., salt and sand). A test of the system reveals an improvement in snowplowing time by 1.9 percent for moderate snowfall and 9.7 percent for snowstorm conditions over the current manual system.

GA-based Optimization of a Fourth-order Sigma-delta Modulator for WLAN

Over-sampling sigma-delta analogue-to-digital converters (ADCs) are one of the key building blocks of state of the art wireless transceivers. In the sigma-delta modulator design the scaling coefficients determine the overall signal-to-noise ratio. Therefore, selecting the optimum value of the coefficient is very important. To this end, this paper addresses the design of a fourthorder multi-bit sigma-delta modulator for Wireless Local Area Networks (WLAN) receiver with feed-forward path and the optimum coefficients are selected using genetic algorithm (GA)- based search method. In particular, the proposed converter makes use of low-distortion swing suppression SDM architecture which is highly suitable for low oversampling ratios to attain high linearity over a wide bandwidth. The focus of this paper is the identification of the best coefficients suitable for the proposed topology as well as the optimization of a set of system parameters in order to achieve the desired signal-to-noise ratio. GA-based search engine is a stochastic search method which can find the optimum solution within the given constraints.

Differential-Evolution-Based Optimization of the Dynamic Response for Parallel Operation of Inverters With No Controller Interconnection

In this paper, the use of differential evolution ( DE), a global search technique inspired by evolutionary theory, to find the parameters that are required to achieve optimum dynamic response of parallel operation of inverters with no interconnection among the controllers is proposed. Basically, in order to reach such a goal, the system is modeled in a certain way that the slopes of P-omega and Q-V curves are the parameters to be tuned. Such parameters, when properly tuned, result in system's eigenvalues located in positions that assure the system's stability and oscillation-free dynamic response with minimum settling time. This paper describes the modeling approach and provides an overview of the motivation for the optimization and a description of the DE technique. Simulation and experimental results are also presented, and they show the viability of the proposed method.