Probabilistic Structural Design Research Articles

A validated approach for probabilistic structural integrity design code implementation

This paper provides a possible approach for incorporating probabilistic structural integrity assessment which is sufficiently simple to encourage its adoption by design engineers. The approach, called the “Three-Term Reference Damage Model (3T-RDM)”, is developed in the context of a particular benchmark problem and a proposed Design Chart where the probability of crack initiation by creep-fatigue in a widely used material in nuclear plants, namely 316H austenitic stainless steel, is estimated based on the R5V2/3 high temperature procedure. In principle, the 3T-RDM is not restricted to a particular failure mechanism, i.e., fracture, creep rupture, creep-fatigue crack initiation and growth are all within its scope, and non-metallic materials may also be addressed. To further validate the accuracy and generality of the proposed 3T-RDM and Design Chart, deterministic and Monte Carlo probabilistic structural integrity assessments are conducted for three EDF nuclear plant case-studies and the results are superimposed on the Design Chart obtained from the benchmark problem. The analysis indicates an excellent consistency between the results, justifying the adoption of the 3T-RDM for an intermediate level of probabilistic assessment in the nuclear sector.

Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques

Although more and more reliability-oriented sensitivity analysis (ROSA) techniques are now available, review and comparison articles of ROSA are absent. In civil engineering, many of the latest indices have never been used to analyse structural reliability for very small failure probability. This article aims to analyse and compare different sensitivity analysis (SA) techniques and discusses their strengths and weaknesses. For this purpose, eight selected sensitivity indices are first described and then applied in two different test cases. Four ROSA type indices are directly oriented on the failure probability or reliability index beta, and four other indices (of a different type) are oriented on the output of the limit state function. The case study and results correspond to cases under common engineering assumptions, where only two independent input variables with Gaussian distribution of the load action and the resistance are applied in the ultimate limit state. The last section of the article is dedicated to the analysis of the different results. Large differences between first-order sensitivity indices and very strong interaction effects obtained from ROSA are observed for very low values of failure probability. The obtained numerical results show that ROSA methods lack a common platform that clearly interprets the relationship of indices to their information value. This paper can help orientate in the selection of which sensitivity measure to use.

Risk-based decision making and the calibration of structural design codes – prospects and challenges

ABSTRACTTo date, our built environment is broadly developed and maintained on the basis of structural design standards. Most design standards contain simplified semi-probabilistic safety concepts that help daily structural engineering decision making using simple calculus. For few projects, the semi-probabilistic safety concept is replaced by advanced probabilistic methods that allow for reliability-based design. Neither include explicit consideration of optimality criteria. However, for the development and the calibration of the semi-probabilistic and probabilistic safety concepts it is important that societal preferences in regard to risk acceptance and the optimal use of natural resources are explicitly considered. In the present contribution, the development and calibration of semi-probabilistic and probabilistic structural design standards is represented as a formal decision problem allowing an optimal combination of reliability elements to be identified.

3-dimensional environmental contours based on a direct sampling method for structural reliability analysis of ships and offshore structures

ABSTRACTThis paper derives a simple way of calculating 3-dimensional environmental contours for structural reliability assessment and design of ships and offshore structures. The 3-dimensional contours are based on Monte Carlo sampling from the joint distribution of the input parameters for response calculations and are an extension of previously proposed 2-dimensional contours. The 3-dimensional contours offers a practical way of identifying multivariate joint extremes of relevant input variables in order to provide for analyses of extreme structural responses. The contour estimation method is applied to some simple example cases and is demonstrated to work well and give reasonable results. Hence, the 3-dimensional contours represent a practical tool for anyone involved in probabilistic structural reliability analysis and design of ships and offshore structures.

Efficient algorithm for probability-based design optimisation of complex structures and related issues

The probabilistic structural design optimisation problem can be carried out using either the conventional reliability index approach (RIA) or the performance measure approach (PMA). The direct approach for either RIA or PMA is to form a two-level optimisation by simply connecting algorithms of probabilistic calculation and design optimisation, which is computationally prohibitive for complex structures. The authors used the sequential approximate programming (SAP) strategy for RIA and extended to PMA. Examples show that SAP for both RIA and PMA reduce the total number of function evaluations notably, whereas PMA with SAP is more efficient and less dependent on probabilistic distributions and thus serves as a promising approach for complex structures. This study introduces the two SAP approaches in a unified way and shows that in these approaches, Taylor linear approximation is applied to both design variables and random variables. Two kinds of variables are treated in the same way from the mathematical viewpoint.

Convergence control of the iterative procedure for performance-measure-based probabilistic structural design optimization

The advanced mean value (AMV) iterative scheme is commonly used to evaluate probabilistic constraints in the performance measure approach (PMA) for probabilistic structural design optimization (PSDO). However, the iterative procedure of PSDO may fail to converge. In this article, the chaotic dynamics theory is suggested to investigate and attack the non-convergence difficulties of PMA-based PSDO. Essentially, the AMV iterative formula forms a discrete dynamical system with control parameters. If the AMV iterative sequences present the numerical instabilities of periodic oscillation, bifurcation, and even chaos in some control parameter interval, then the outer optimization loop in PSDO cannot converge and acquire the correct optimal design. Furthermore, the stability transformation method (STM) of chaos feedback control is applied to perform the convergence control of AMV, in order to capture the desired fixed points in the whole control parameter interval. Meanwhile, PSDO is solved by the approaches of PMA two-level and PMA with the sequential approximate programming (SAP)—PMA with SAP. Numerical results of several examples illustrate that STM can smoothly overcome the convergence failure of PSDO resulting from the periodic oscillation, bifurcation, and chaotic solutions of AMV iterative procedure for evaluating the probabilistic constraints. Moreover, the probabilistic optimization with uniform random variables, which is widely recognized as a highly nonlinear and fairly difficult problem, can be attacked through introducing the strategy of chaos control. In addition, the approach of PMA with SAP combining with STM is quite effective and efficient.

Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD)

Probabilistic structural design deals with uncertainties in response (e.g. stresses) and capacity (e.g. failure stresses). The calculation of the structural response is typically expensive (e.g., finite element simulations), while the capacity is usually available from tests. Furthermore, the random variables that influence response and capacity are often disjoint. In previous work we have shown that this disjoint property can be used to reduce the cost of obtaining the probability of failure via Monte Carlo simulations. In this paper we propose to use this property for an approximate probabilistic optimization based on exact capacity and approximate response distributions (ECARD). In Approximate Probabilistic Optimization Using ECARD, the change in response distribution is approximated as the structure is re-designed while the capacity distribution is kept exact, thus significantly reducing the number of expensive response simulations. ECARD may be viewed as an extension of SORA (Sequential Optimization and Reliability Assessment), which proceeds with deterministic optimization iterations. In contrast, ECARD has probabilistic optimization iterations, but in each iteration, the response distribution is approximated so as not to require additional response calculations. The use of inexpensive probabilistic optimization allows easy incorporation of system reliability constraints and optimal allocation of risk between failure modes. The method is demonstrated using a beam problem and a ten-bar truss problem. The former allocates risk between two different failure modes, while the latter allocates risk between members. It is shown that ECARD provides most of the improvement from risk re-allocation that can be obtained from full probabilistic optimization.

Reliability-Based Aircraft Structural Design Pays, Even with Limited Statistical Data

Probabilistic structural design tends to apply higher safety factors to inexpensive or light-weight components, because it is a more efficient way to achieve a desired level of safety. We show that even with limited knowledge about stress probability distributions we can increase the safety of an airplane by following this paradigm. This is accomplished by a small perturbation of the deterministic design that maximizes safety for the same weight. The structural optimization for safety of a representative system composed of a wing, a horizontal tail and a vertical tail is used to demonstrate the paradigm. We find that moving small amount of material from the wing to the tails leads to substantially increased structural safety. Since aircraft companies often apply additional safety factors beyond those mandated by the Federal Aviation Administration (FAA), this opens the door to obtaining probabilistic design that satisfies also the FAA code based rules for deterministic design. We also find that probabilistic design is insensitive to errors committed while assessing the stress probability distribution of the deterministic design, which is the starting point of the probabilistic design. This suggests that using the deterministic design as the basis for the probabilistic design insulates the latter from the extreme sensitivity to statistical data that have been observed in the past. Finally, we find that for independent components subject to the same failure mode, the probabilities of failure at the probabilistic optimum are approximately proportional to the weight. So a component which is ten times lighter than another should be designed to be about 10 times safer.

Further study on efficiency of sequential approximate programming for probabilistic structural design optimization

Compared to the traditional deterministic optimization based on safety factors, the probabilistic structural design optimization (PSDO) is considered to be a more rational design philosophy because of reasonable account of uncertainties in material properties, loading, boundary condition and geometry, and even mathematical representation of the system model. However, it is well known that the computation for PSDO can be prohibitive when the associated function evaluation is expensive. As a result, many approximate PSDO methods have been developed in recent literatures. In previous works, we developed two sequential approximate programming (SAP) strategies for PSDO based on reliability index approach (RIA) and performance measure approach (PMA). In PMA with SAP, a sequence of approximate programming of PSDO was formulated and solved before the final optimum was located. In each subprogramming, rather than relying on direct linear Taylor expansion of the probabilistic performance measure (PPM), we developed a formulation for approximate PPM at the current design point and used its linearization instead. The approximate PPM and its sensitivity were obtained by approximating the optimality conditions in the vicinity of the minimum performance target point (MPTP). The present paper further elaborates the SAP for PMA. In addition to detailed description of the algorithm, we present error analysis and show that in the ɛ-vicinity of optimum design and corresponding MPTP, the difference between the Taylor expansion of PPM and the linear expansion of approximate PPM is of higher order of ɛ. Four examples are optimized by six algorithms appearing in recent literatures for efficiency comparison. The effect of target reliability index and statistical distribution of random variables on the comparison is discussed. The third example shows that PMA with SAP performs well even for the problem for which reliability index calculation by first order reliability method (FORM) fails. Finally, the fourth example with 144 probabilistic constraints is shown to demonstrate the effectiveness of PMA with SAP. All example results illustrate that with the algorithm PMA with SAP, we get concurrent convergence of both design optimization and probabilistic performance measure calculation, which agrees well with the error analysis.

A sequential approximate programming strategy for performance-measure-based probabilistic structural design optimization

The evaluation of probabilistic constraints in the probabilistic structural design optimization (PSDO) problem can be carried out using either the conventional reliability index approach (RIA) or the more recently proposed performance measure approach (PMA). The latter is sometimes regarded as more efficient and stable with less dependence on probabilistic distribution types of random variables. Herein we apply PMA to evaluate probabilistic constraints and solve the PSDO using the sequential approximate programming (SAP) strategy. The sequential linear programming (SLP) approach in structural optimization achieves optimum design by solving a sequence of sub-programming problems iteratively. Each sub-programming consists of a linear objective subjected to a set of linear constraints, all based on approximation of the original objective and constraints at the current design. Implementing this approach to PSDO in a straightforward manner will require linear approximation of probabilistic performance measure and its sensitivities, thus implying large number of iterations and huge computational cost. In our new approach, rather than using the linear expansion of the probabilistic performance measure, we propose a formulation for an approximate probabilistic performance measure and its linearization. Obtained based on optimality conditions in the vicinity of the minimum performance target point (MPTP), the approximate measure and its sensitivity enables efficient sub-programming step. We update MPTP simultaneously at each step using iterative formula from the advanced mean-value (AMV) method and apply it as the initial estimate for the next step. As the sub-programming steps are no longer the linear approximation of the original problem, this is essentially a sequential approximate programming (SAP) approach. Through the application of this method to the most relevant examples frequently cited in similar studies, we compare its efficiency to other existing approaches and illustrate the concurrent convergence of both optimization and probabilistic performance measure calculation.

Advanced Load Effect Model for Probabilistic Structural Design

In probabilistic structural design some actions on structures can be well described by renewal processes with intermittencies. The expected number of renewals for a given time interval and the probability of “on“-state at an arbitrary point in time are of a main interest when estimating the structural reliability level related to the observed period. It appears that the expected number of renewals follows the Poisson distribution. The initial probability of “on”-state is derived assuming random initial conditions. Based on a two-state Markov process, the probability of “on”-state at an arbitrary point in time then proves to be a time-invariant quantity under random initial conditions. The results are numerically verified by Monte Carlo simulations. It is anticipated that the proposed load effect model will become a useful tool in probabilistic structural design. The aims of future research are outlined in the conclusions of the paper.

Equivalent strengths for reliability assessment of MEMS structures

Abstract The scattering and inconsistency of tested strength of brittle microelectromechanical systems (MEMS) materials imposes a critical obstacle for structural reliability assessment of MEMS devices. In this article, the nature of such a discrepancy and the effort to solve this issue are discussed. A method based on equal failure probability is proposed to map the material strength obtained from test specimens to the equivalent strength for MEMS structural design. This conversion can be classified into three types to deal with difference in size, geometry, and applied loadings manner and the possible approach to form a material strength database for brittle MEMS material is suggested. The weakest link theory and Weibull statistics are adapted for illustrating the proposed data reduction process. Several examples are provided to illustrate the possible applications of this work. Finally, an equivalent safety factor concept is proposed to promote probabilistic structural design and the perspective to achieve a MEMS material strength database is discussed.

A comparative study on reliability-index and target-performance-based probabilistic structural design optimization

Probabilistic structural design optimization enables designers and engineers to quantitatively take into account the uncertainties observed in the structural and environmental properties. In this paper, two approaches to determine the satisfaction of probabilistic constraints are discussed. One is the conventional reliability-index-based approach and the other is a more recently proposed target-performance-based approach. An algorithm, which detects and eliminates the excessive zigzagging iterations during the searches for the most probable failure point and the minimum performance target point, was incorporated. The number of iterations required by the two approaches was investigated in three examples: a cantilever beam, a three-bar truss and a ten-bar truss structure. Based on the results, the target-performance-based approach was found to be superior to the reliability-index-based one in view of both computational efficiency and numerical stability.

Optimality-criteria-based probabilistic structural design

The possibility of extending absolute minimum weight designs for sandwich beam systems or fibre-reinforced beams and plates is considered when both the extreme loading applied to the system and the beam bending resistance properties are described in probabilistic rather than deterministic terms. It is shown that such an extension is possible provided certain plausible assumptions are made about the probabilistic modelling of the system. Examples considered include simple and continuous beams under loading of uncertain magnitude, direction or location. It is shown that the classical (deterministic) optimal solutions remain valid under reasonable assumptions. The extension of the probabilistic solutions to a wider class of examples is discussed.

Structural Design of a Silicon Micro-Turbo-Generator

The probabilistic structural analysis and design of a silicon micro-turbo-generator rotor are presented. This rotor was designed to have a tip speed of 500 m/s. Three-dimensional finite element analysis, fracture strength characterization of single crystal silicon,and structural failure probability calculations were performed. The results show that the design of micro-turbine-generator rotor, although highly stressed, is feasible. However, it is important to note that this feasibility is very dependent on the etching process achieving a high surface quality. The overall approach and tests employed are equally applicable to other highly stressed microelectromechanical systems.

A mixed design approach for probabilistic structural durability

In this paper, a mixed approach for probabilistic structural durability design of mechanical systems is proposed. In this approach, a deterministic design optimization that considers structural crack initiation and crack propagation lives at critical points of the structural component as design constraints is performed first. After an optimal design is obtained, a reliability analysis is performed to ascertain if the deterministic optimal design is reliable. If the probability of the failure of the deterministic optimal design is found to be acceptable, a reliability-based design approach that employs a set of interactive design steps, such as trade-off analysis and what-if study, is used to obtain a near-optimal design that is reliable with an affordable computational cost. A 3-D tracked vehicle roadarm is employed to demonstrate the feasibility of the proposed approach.

Application of Dynamic Reliability Theory to Probabilistic Structural Design.

Recent research related to probability-based structural design has become more active, especially in the field of the load and resistance factor design (LRFD) technique. This paper also describes some contributions to the improvement of the LRFD method. In this paper, a rational and analytical method to determine the required code format and load factors, which were determined experimentally in many former LRFD studies, considering the dynamic characteristics of a structure is newly proposed on the basis of the dynamic reliability technique with the load combination theory. The usefulness of this proposed method is described in an application to the selection problem of the required code format and load factors against a typical structure that is flexible and excited due to multiple random loads.

1質点系RC構造物の衝撃非線形時刻歴応答の統計的変動について

In order to obtain any response displacement x_n(t) within the nonlinear range, a new equivalent linearization method which uses repeatedly the equivalent linear-type spring modulus _<eq>k(x_n) (n=1, 2, ・・・) in terms of energy correspond to x_n(t), was devised. And by applying the perturbation method to the above method, a nonlinear random vibration method of analysis was proposed on condition that the variance of _<eq>k(x_n) which is a function of the x_n(t) is to be obtained by convergence calculations. In accordance with the proposed method, the formulations of a lumped-mass-system, where the acting loads as well as the structural properties are random variables, were endeavored and the applicability of the proposed method was examined. Futhermore, regarding a RC structure modelled trilinearly, a numerical example where almost all the structural properties are random variables, was shown. Major conclusions are summarized as follows. a) The proposed method of analysis can more accurately evaluate nonlinear random response of structures from elastic range to large deformation range, since the perturbation method and the conventional equivalent linearization method are not used in case of the linearization of the nonlinear equation of motion. b) In case that almost all the structural properties are random variables, the variance of the response dis-placement increases rapidly in the region of its maximum value. c) The proposed method can be useful means for probabilistic structural design to impulsive dynamic response, though there still remains some problems to be solved.

Forewords to “Probabilistic Structural Design and Analysis” and “Limitations of FEM in Analysis of Composite Structures”

Forewords to “Probabilistic Structural Design and Analysis” and “Limitations of FEM in Analysis of Composite Structures”

Probabilistic optimal design of framed structures

Probabilistic structural design is a degree of difficulty greater than deterministic design and effectively is only tractable using a computer. This paper outlines one approach to the design of framed structures subject to random loading. In particular stochastic approximation is used iteratively to improve the value of the optimality criterion, leading to a guaranteed optimal solution in the limit. Two framed structures are used to illustrate the computations which are performed in conjunction with a random number generator to simulate the probabilistic aspects of the loading. The approach is independent of the probability model assumed for the loading. It is straightforward to implement and has good solution convergence characteristics.