Optimal Design with Probabilistic Objective and Constraints

Significant challenges are associated with solving optimal structural design problems involving the failure probability in the objective and constraint functions. In this paper, we develop gradient-based optimization algorithms for estimating the solution of three classes of such problems in the case of continuous design variables. Our approach is based on a sequence of approximating design problems, which is constructed and then solved by a semiinfinite optimization algorithm. The construction consists of two steps: First, the failure probability terms in the objective function are replaced by auxiliary variables resulting in a simplified objective function. The auxiliary variables are determined automatically by the optimization algorithm. Second, the failure probability constraints are replaced by a parametrized first-order approximation. The parameter values are determined in an adaptive manner based on separate estimations of the failure probability. Any computational reliability method, including first-order reliability and second-order reliability methods and Monte Carlo simulation, can be used for this purpose. After repeatedly solving the approximating problem, an approximate solution of the original design problem is found, which satisfies the failure probability constraints at a precision level corresponding to the selected reliability method. The approach is illustrated by a series of examples involving optimal design and maintenance planning of a reinforced concrete bridge girder.     

Reliability Analysis Using Parabolic Failure Surface Approximation

In the second-order reliability method the failure surface is approximated by a general quadratic surface in the neighborhood of the design point. In this paper this general quadratic surface is further approximated by a parabolic surface. Several methods are proposed to obtain the probability content associated with this parabolic failure surface. It is assumed that the basic random variables are Gaussian. The proposed methods can be broadly grouped into: (1) nonasymptotic approximate methods, (2) exact methods, and (3) asymptotic distribution methods. Most of these methods result in a closed-form expression for the failure probability. For nonasymptotic approximations, a least-square approach and an optimal point expansion method using approximate probability density functions of a quadratic form in Gaussian random variables have been proposed. It is shown that such approximations give accurate results without significant numerical effort. Exact results, however, require greater numerical effort. The new asymptotic result is derived for the case when the number of random variables approaches infinity. Several numerical examples are provided to compare the proposed results with existing equivalent results and Monte Carlo simulations.     

New Multidimensional Visualization Technique for Limit-State Surfaces in Nonlinear Finite-Element Reliability Analysis

Structural reliability problems involving the use of advanced finite-element models of real-world structures are usually defined by limit-states expressed as functions (referred to as limit-state functions) of basic random variables used to characterize the pertinent sources of uncertainty. These limit-state functions define hyper-surfaces (referred to as limit-state surfaces) in the high-dimensional spaces of the basic random variables. The hyper-surface topology is of paramount interest, particularly in the failure domain regions with highest probability density. In fact, classical asymptotic reliability methods, such as the first- and second-order reliability method (FORM and SORM), are based on geometric approximations of the limit-state surfaces near the so-called design point(s) (DP). This paper presents a new efficient tool, the multidimensional visualization in the principal planes (MVPP) method, to study the topology of high-dimensional nonlinear limit-state surfaces (LSSs) near their DPs. The MVPP method allows the visualization, in particularly meaningful two-dimensional subspaces denoted as principal planes, of actual high-dimensional nonlinear limit-state surfaces that arise in both time-invariant and time-variant (mean out-crossing rate computation) structural reliability problems. The MVPP method provides, at a computational cost comparable with SORM, valuable insight into the suitability of FORM/SORM approximations of the failure probability for various reliability problems. Several application examples are presented to illustrate the developed MVPP methodology and the value of the information provided by visualization of the LSS.     

Large-sample analysis for a stochastic epidemic model and its parameter estimators

This paper considers the simplest stochastic model for the spread of an epidemic in a closed, homogeneously mixing population. Approximate methods are presented for calculating the probability distribution of the epidemic size (i.e. number of infected individuals). In fact, a functional central limit theorem and a large deviation principle for the epidemic size when the population increases are shown. These results enable us to both obtain a global approximation for the epidemic size and study asymptotic properties of other random variables depending on the complete history of the epidemic. As an application of our results, we derive two sequences of estimators for the contact rate and analyze their asymptotic behaviour.     

New Point Estimates for Probability Moments

There are many areas of structural safety and structural dynamics in which it is often desirable to compute the first few statistical moments of a function of random variables. The usual approximation is by the Taylor expansion method. This approach requires the computation of derivatives. In order to avoid the computation of derivatives, point estimates for probability moments have been proposed. However, the accuracy is quite low, and sometimes, the estimating points may be outside the region in which the random variable is defined. In the present paper, new point estimates for probability moments are proposed, in which increasing the number of estimating points is easier because the estimating points are independent of the random variable in its original space and the use of high-order moments of the random variables is not required. By using this approximation, the practicability and accuracy of point estimates can be much improved.     

Least-Squares, Moment-Based, and Hybrid Polynomializations of Drag Forces

In order to carry out nonlinear dynamic analyses of fixed offshore structures in the frequency domain, polynomial approximations of the distributed drag term, u∣u∣, and inundation drag term, ηu∣u∣, of the Morison forces are studied. The methods of least-squares and moment-based approximations are considered for cases with and without current. Numerical results and analytical expressions of the polynomial coefficients are presented for the cubic approximations of u∣u∣ and quartic approximations of ηu∣u∣. The curve shapes, first four central moments, and probability density functions of the different approximations are evaluated and compared with the corresponding exact solutions. For the nonmonotonic inundation drag term with the current effect included, a hybrid polynomialization, based on the least-squares approximation for the odd-order polynomial coefficients and the moment-based approximation for the even-order coefficients, is proposed.     

Transverse Shear Stiffness of Composite Honeycomb Core with General Configuration

Based on the zeroth-order approximation of a two-scale asymptotic expansion, equivalent elastic shear coefficients of periodic structures can be evaluated via the solution of a local function τklij(y), and the homogenization process reduces to solving the local function τklij(y) by invoking local periodic boundary conditions. Then, effective transverse shear stiffness properties can be analytically predicted by reducing a local problem of a given unit cell into a 2D problem. In this paper, an analytical approach with a two-scale asymptotic homogenization technique is developed for evaluation of effective transverse shear stiffness of thin-walled honeycomb core structures with general configurations, and the governing 3D partial differential equations are solved with the assumptions of free warping constraints and constant variables through the core wall thickness. The explicit formulas for the effective transverse shear stiffness are presented for a general configuration of honeycomb core. A detailed study is given for three typical honeycomb cores consisting of sinusoidal, tubular, and hexagonal configurations, and their solutions are validated with existing equations and numerical analyses. The developed approach with certain modifications can be extended to other sandwich structures, and a summary of explicit solutions for the transverse shear stiffness of common honeycomb core configurations is provided. The lower bound solution provided in this study is a reliable approximation for engineering design and can be efficiently used for quick evaluation and optimization of general core configurations. The upper bound formula, based on the assumption of uniform shear deformation, is also given for comparison. Further, it is expected that with appropriate construction in the displacement field, the more accurate transverse stiffness can be analytically attained by taking into account the effect due to the face-sheet constraints.     

Approximate Solutions for Forchheimer Flow to a Well

An exact solution for transient Forchheimer flow to a well does not currently exist. However, this paper presents a set of approximate solutions, which can be used as a framework for verifying future numerical models that incorporate Forchheimer flow to wells. These include: a large time approximation derived using the method of matched asymptotic expansion; a Laplace transform approximation of the well-bore response, designed to work well when there is significant well-bore storage and flow is very turbulent; and a simple heuristic function for when flow is very turbulent and the well radius can be assumed infinitesimally small. All the approximations are compared to equivalent finite-difference solutions.     

Limit-State Surface Element Method: Application to Fatigue Reliability with NDE Inspections

A new method, named the limit-state surface element (LSSE) method, for problems in structural reliability is presented. The method is based on discretization of the limit-state surface into a finite number of elements. This allows for better representations of the failure surface in regions of high curvature than is possible with standard FORM techniques. The reliability is estimated as a sum of integrals over subdomains in the space of standard normal variables associated with the surface elements. Several benchmark problems are presented to demonstrate the accuracy and utility of the method, and a problem in fatigue reliability with NDE inspections is presented as a practical application of the method and a demonstration of the potential interrelations between QNDE and reliability assessment.     

Asymptotic Approach for Thermoelastic Analysis of Laminated Composite Plates

A thermoelastic model for analyzing laminated composite plates under both mechanical and thermal loadings is constructed by the variational asymptotic method. The original three-dimensional nonlinear thermoelasticity problem is formulated based on a set of intrinsic variables defined on the reference plane and for arbitrary deformation of the normal line. Then the variational asymptotic method is used to rigorously split the three-dimensional problem into two problems: A nonlinear, two-dimensional, plate analysis over the reference plane to obtain the global deformation and a linear analysis through the thickness to provide the two-dimensional generalized constitutive law and the recovering relations to approximate the original three-dimensional results. The nonuniqueness of asymptotic theory correct up to a certain order is used to cast the obtained asymptotically correct second-order free energy into a Reissner–Mindlin type model to account for transverse shear deformation. The present theory is implemented into the computer program, variational asymptotic plate and shell analysis (VAPAS). Results from VAPAS for several cases have been compared with the exact thermoelasticity solutions, classical lamination theory, and first-order shear-deformation theory to demonstrate the accuracy and power of the proposed theory.     

Simultaneous Determination of Critical Slip Surface and Reliability Index for Slopes

This paper presents a new method for applying reliability-based design approaches to slope stability analysis. In this method the soil properties are considered to be random variables. The factor of safety of the slope is found using Bishop’s simplified method for noncircular slip surfaces. By considering the variability of the soil properties, the probability of failure is determined from the reliability index (β). The minimization problem (determination of the lowest β value for the range of variables and possible slip surfaces considered) is solved using a genetic algorithm approach, which simultaneously locates the critical slip surface and determines the reliability index. The performance of the new method is compared to some existing reliability approaches when applied to case histories of slope failures from the geotechnical literature. The new approach is seen to provide reasonable and consistent estimates of the reliability index and critical slip surface.     

Vibration of Tensioned Beams with Intermediate Damper. II: Damper near a Support

Analytical solutions are used to investigate the free vibrations of tensioned beams with a viscous damper attached transversely near a support. This problem is of particular relevance for stay-cable vibration suppression, but no restrictions on the level of axial load are introduced, and the results are quite broadly applicable. Characteristic equations for both clamped and pinned supports are rearranged into forms suitable for numerical solution by fixed-point iteration, whereby the complex eigenfrequencies and corresponding damping ratios can be accurately computed within a few iterations. Explicit asymptotic approximations for the complex eigenfrequencies are also obtained, subject to restrictions on the closeness of the eigenfrequencies to their undamped values. These asymptotic approximations are expressed in the same “universal” form identified in previous studies. It is observed that the maximum attainable modal damping ratios and the corresponding optimal values of the damper coefficient can be significantly affected by bending stiffness and by the nature of the support conditions, and a nondimensional parameter grouping is identified that enables an assessment of when bending stiffness should be considered.     

Transformation of Domain Integrals in BEM for Thick Foundation Plates

In this paper, the domain integrals due to uniform load or self-weight that appear in the boundary element method (BEM) formulation for thick plates resting on elastic foundations are transformed to boundary integrals. The Reissner plate bending model is used to model the plate behavior, and the two-parameter Pasternak model is used to model the behavior of the foundation. The necessary particular solutions are derived, and the explict forms for the new boundary kernels are given. Two different collocation procedures are considered—external and boundary collocations. In the case of the boundary collocation and internal collocation (for computing internal functions) procedures, an additional free term is obtained, due to the discontinuity of the transformed kernels. The new boundary integrals are hypersingular integrals. However, it will be shown that these hypersingular terms vanish when integrated around a closed contour. Three numerical examples are presented with several parametric studies to demonstrate the accuracy of the present formulation.     

Efficient Algorithm for Computing Einstein Integrals

Analytical approximations to Einstein integrals are proposed. The approximations represented by two fast-converging series are valid for all values of their arguments. Accordingly, the algorithm can be easily incorporated into professional software like HEC-RAS or HEC-6 with minimum computational effort.     

Stochastic epidemics: the probability of extinction of an infectious disease at the end of a major outbreak

The aim of this study is to derive an asymptotic expression for the probability that an infectious disease will disappear from a population at the end of a major outbreak ('fade-out'). The study deals with a stochastic SIR-model. Local asymptotic expansions are constructed for the deterministic trajectories of the corresponding deterministic system, in particular for the deterministic trajectory starting in the saddle point. The analytical expression for the probability of extinction is derived by asymptotically solving a boundary value problem based on the Fokker-Planck equation for the stochastic system. The asymptotic results are compared with results obtained by random walk simulations.     

Reliability-Based Optimal Design of Series Structural Systems

A robust approach for approximately solving reliability-based optimal design problems, for series structural systems, is developed. The approach reformulates the problems by replacing reliability terms with deterministic functions. The reformulated problems can be solved by existing semiinfinite optimization algorithms, producing solutions that are identical to those of the original problems, when the limit-state functions are affine, or when first-order reliability approximations are used. An important advantage of the approach is that the required reliability and optimization calculations are completely decoupled, allowing flexibility in the choice of the optimization algorithm and the reliability method. Three sets of examples demonstrate applications of the approach.     

Reliability-based Calibration of Partial Safety Coefficients for Fiber-Reinforced Plastic

The scope of this paper is to present a possible methodology for the calibration of partial safety factors for the design of strengthening measures of reinforced-concrete (RC) members using fiber-reinforced plastic (FRP). The methodology is general and can be used for any type of strengthening measure, e.g., in flexure, shear, ductility, or for the design of anchorage zones. The approach considers the problem of strengthening an RC member from a current unsafe situation, where all involved quantities are known from assessment, though only in probabilistic terms, to a target safe one, of which only the desired reliability is known. All relevant random variables are attributed a predefined probability distribution, based on a statistical survey separately conducted on geometrical and mechanical characteristics of old-style components. A first-order reliability method based optimization procedure is used to seek the solution of such a problem so that the target reliability is attained with the optimal FRP quantity, the appropriate collapse mechanism of the strengthened member, and the FRP design strength, with the associated partial safety factor. From Monte Carlo design simulations, the partial safety factor is probabilistically characterized, thus allowing one to select an appropriate fractile value for it.     

Domain Decomposition Method for Calculating the Failure Probability of Linear Dynamic Systems Subjected to Gaussian Stochastic Loads

In this paper the problem of calculating the probability of failure of linear dynamic systems subjected to random vibrations is considered. This is a very important and challenging problem in structural reliability. The failure domain in this case can be described as a union of linear failure domains whose boundaries are hyperplanes. Each linear limit state function can be completely described by its own design point, which can be analytically determined, allowing for an exact analytical calculation of the corresponding failure probability. The difficulty in calculating the overall failure probability arises from the overlapping of the different linear failure domains, the degree of which is unknown and needs to be determined. A novel robust reliability methodology, referred to as the domain decomposition method (DDM), is proposed to calculate the probability that the response of a linear system exceeds specified target thresholds. It exploits the special structure of the failure domain, given by the union of a large number of linear failure regions, to obtain an extremely efficient and highly accurate estimate of the failure probability. The number of dynamic analyses to be performed in order to determine the failure probability is as low as the number of independent random excitations driving the system. Furthermore, calculating the reliability of the same structure under different performance objectives does not require any additional dynamic analyses. Two numerical examples are given demonstrating the proposed method, both of which show that the method offers dramatic improvement over standard Monte Carlo simulations, while a comparison with the ISEE algorithm shows that the DDM is at least as efficient as the ISEE.     

Simulation Model to Forecast Project Completion Time

Project cost is most sensitive to its schedule. The construction project environment comprising dynamic, uncertain, but predictable, variables such as weather, space congestion, workmen absenteeism, etc., is changing continuously, affecting activity durations. The reliability of project duration forecast can be enhanced by an explicit analysis to determine the variation in activity durations caused by the dynamic variables. A computer model is used to simulate the expected occurrence of the uncertainty variables. From the information that is collected normally for a progress update of the tactical plan and by simulating the project environment, the combined impact of the uncertainty variables is predicted for each progress period. By incorporating the combined impact in the duration estimates of each activity, the new activity duration distribution is generated. From these activity duration distributions, the probability of achieving the original project completion time and of completing the project at any other time is computed.     

Reliability-Based Optimum Design of Glulam Cable-Stayed Footbridges

This work presents a procedure for finding the reliability-based optimum design of cable-stayed bridges. The minimization problem is stated as the minimization of stresses, displacements, reliability, and bridge cost. A finite-element approach is used for structural analysis. It includes a direct analytic sensitivity analysis module, which provides the structural behavior responses to changes in the design variables. An equivalent multicriteria approach is used to solve the nondifferential, nonlinear optimization problem, turning the original problem into sequential minimization of unconstrained convex scalar functions, from which a Pareto optimum is obtained. Examples are given illustrating the procedure.