Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction

A procedure is presented for the probabilistic analysis of the seismic soil-structure interaction problem. The procedure accounts for uncertainty in both the free-field input motion as well as in local site conditions, and structural parameters. Uncertain parameters are modeled using a probabilistic framework as stochastic processes. The site amplification effects are accounted for via a randomized relationship between the soil shear modulus and damping on the one hand, and the shear strain of the subgrade on the other hand, as well as by modeling the shear modulus at low strain level as randomly fluctuating with depth. The various random processes are represented by their respective Karhunen-Loève expansions, and the solution processes, consisting of the accelerations and generalized forces in the structure, are represented by their coordinates with respect to the polynomial chaos basis. These coordinates are then evaluated by a combination of weighted residuals and stratified sampling schemes. The expansion can be used to carry out very efficiently, extensive Monte Carlo simulations. The procedure is applied to the seismic analysis of a nuclear reactor facility.     

Probabilistic Settlement Analysis by Stochastic and Random Finite-Element Methods

The paper discusses finite element models for predicting the elastic settlement of a strip footing on a variable soil. The paper then goes on to compare results obtained in a probabilistic settlement analysis using a stochastic finite element method based on first order second moment approximations, with the random finite element method based on generation of random fields combined with Monte Carlo simulations. The paper highlights the deficiencies of probabilistic methods that are unable to properly account for spatial correlation.     

Latin Hypercube Sampling for Stochastic Finite Element Analysis

A Latin hypercube sampling method, including a reduction of spurious correlation in input data, is suggested for stochastic finite element analysis. This sampling procedure strongly improves the representation of stochastic design parameters compared to a standard Monte Carlo sampling. As the correlation control requires the number of realizations to be larger than the number of stochastic variables in the problem, a principal component analysis is employed to reduce the number of stochastic variables. In many cases, this considerably relaxes the restriction on the number of realizations. The method presented offers the same general applicability as the standard Monte Carlo sampling method but is superior in computational efficiency.     

A Study of Two Stochastic Search Methods for Structural Control

Many engineering tasks involve the search for good solutions among many possibilities. In most cases, tasks are too complex to be modeled completely and their solution spaces often contain local minima. Therefore, classical optimization techniques cannot, in general, be applied effectively. This paper studies two stochastic search methods, one well-established (simulated annealing) and one recently developed (probabilistic global search Lausanne), applied to structural shape control. Search results are applied to control the quasistatic displacement of a tensegrity structure with multiple objectives and interdependent actuator effects. The best method depends on the accuracy related to requirements defined by the objective function and the maximum number of evaluations that are allowed.     

Contribution of Nonlinear Finite-Element Analysis to Evaluation of Two Structural Concrete Failures

The failure of two reinforced concrete structures is recounted, one involving a warehouse structure and the other an offshore platform base structure. Design details and factors leading to the collapses are identified and discussed. The structures were subsequently analyzed using nonlinear finite-element analysis procedures, taking into account relevant second-order behavior models. The analyses provided an accurate assessment of the load capacities and failure modes observed, as well as meaningful insights into the underlying behavior mechanisms and factors leading to the failures. This paper supports the view that nonlinear analysis techniques have become useful everyday tools for design office applications, particularly in forensic work, and also gives evidence suggesting that errors in the design of modern structures can be potentially more catastrophic than in the past, and that advanced assessment techniques will assume increased importance as a result.     

Probabilistic Stability Analyses of Embankments Based on Finite-Element Method

Conventional limit equilibrium methods are commonly used to assess the stability of embankments. The finite-element method, as an alternative to limit equilibrium methods, is being increasingly used in the deterministic stability analysis of slopes or embankments. In this paper, a practical procedure for integrating the finite-element method and the limit equilibrium methods into probabilistic stability analysis for embankments is presented. The response surface method is adopted to approximate the performance function for the stability problems and the first-order reliability method is used to calculate the reliability index based on an intuitive expanding ellipsoid perspective. The advantages of the response surface method as a bridge between stand-alone numerical packages and spreadsheet-based reliability analysis via automatic constrained optimization are demonstrated and discussed through a hypothetical two-layer slope and an actual case of the James Bay Dykes. The results show the ease and successful implementation of the proposed procedure for reliability analysis of embankments.     

Galerkin Solution for Linear Stochastic Algebraic Equations

Algebraic equations with random coefficients, referred to as stochastic algebraic equations, are used extensively to solve approximately differential equations describing mechanics problems with uncertain material properties and applied loads. This paper (1) constructs optimal and suboptimal Galerkin solutions for linear stochastic algebraic equations, (2) reviews current procedures for deriving stochastic algebraic equations from stochastic differential equations and proposes alternative methods, (3) demonstrates the implementation of the proposed Galerkin method by numerical examples, and (4) calculates statistics of the displacement field for a plate on random elastic foundation. The optimal Galerkin solution coincides with the conditional expectation of the exact solution with respect to a σ-field coarser than the σ-field relative to which the exact solution is measurable, and is unbiased. Generally, suboptimal Galerkin solutions are biased but may provide approximations for the tails of the distribution of the exact solution that are superior to those by the optimal Galerkin solution.     

Improved Explicit Method for Structural Dynamics

An explicit method, which simultaneously has the most promising advantages of the explicit and implicit methods, is presented. It is shown that numerical properties of the proposed explicit method are exactly the same as those of the constant average acceleration method for linear elastic systems. However, for nonlinear systems, it has unconditional stability for an instantaneous stiffness softening system and conditional stability for an instantaneous stiffness hardening system. This conditional stability property is much better than that of the Newmark explicit method. Hence, the proposed explicit method is possible to have the most important property of unconditional stability for an implicit method. On the other hand, this method can be implemented as simply as an explicit method, and hence, possesses the most important property of explicit implementation for an explicit method. Apparently, the integration of these two most important properties of explicit and implicit methods will allow the proposed explicit method to be competitive with other integration methods for structural dynamics.     

Boundary Finite-Element Method Coupling Finite-Element Method for Steady-State Analyses of Dam-Reservoir Systems

The boundary finite-element method (BFEM) is extended for steady-state analyses of dam-reservoir system problems. In this study, the dam is assumed to be rigid and subjected to horizontal ground motions, and the liquid in the reservoir is assumed to be semiinfinite. The semiinfinite reservoir domain is partitioned into two subdomains: a near-field domain and a far-field domain. In it, the near-field domain is modeled by using the finite-element method (FEM), whereas the far-field domain is modeled by BFEM and is treated as a layered semiinfinite fluid domain. A BFEM/FEM coupling procedure is employed to solve the steady-state response of the reservoir. The coupling procedure is easy to implement and suitable for all frequencies, be it real or complex. The BFEM/FEM coupling procedure is validated in the frequency domain. Numerical results that are based on the present procedure are in good agreement with analytical and other available numerical solutions.     

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.     

Finite-Element Analysis and Vibration Testing of a Two-Span Masonry Arch Bridge

This paper presents the analytical modeling, modal testing, and finite-element model updating for a two-span masonry arch bridge. An Ottoman masonry arch bridge built in the 19th century and located at Camlihemsin, Rize, Turkey is selected as an example. Analytical modal analysis is performed on the developed 3D finite-element model of the bridge to obtain dynamic characteristics. The ambient vibration tests are conducted under natural excitation such as human walking. The operational modal analysis is carried out using peak picking method in the frequency domain and stochastic subspace identification method in the time domain, and dynamic characteristics (natural frequencies, mode shapes, and damping ratios) are determined experimentally. Finite-element model of the bridge is updated to minimize the differences between analytically and experimentally estimated dynamic characteristics by changing boundary conditions. At the end of the study, maximum differences in the natural frequencies are reduced on average from 18 to 7% and a good agreement is found between analytical and experimental dynamic characteristics after finite-element model updating.     

Development of Dynamic-Response-Based Objective Functions for Finite-Element Modeling of Bridges

The basic mechanism and procedures of finite-element (FE) bridge modeling and calibration are briefly presented. Different physical parameters of FE models are adjusted during the calibration process. Dynamic-response-based objective functions are carefully developed based on two powerful indices: the modal assurance criterion and frequency correlation trend line. The nominal bridge models are calibrated by minimizing the quantified difference between analytical results and experimental measurements. Using an existing calibration strategy, a nominal FE bridge model is optimized by minimizing this global dynamic-response-based objective function. The value of the objective function is reduced from 10.70 to 4.61%. The minimization of the objective function indicates the convergence of calibration and it is shown that the automated calibration becomes practical due to the formulation of the dynamic-response-based objective function.     

Domain-by-Domain Algorithm for Nonlinear Finite-Element Analysis of Structures

Parallel and distributed computers have been shown to provide the necessary computational power to solve large-scale engineering problems. However, in order for this computation style to be effectively used, efficient computational algorithms must be devised. In this work, a domain-by-domain algorithm is developed for the parallel solution of nonlinear structural dynamics problems. In the proposed algorithm, the original structure is partitioned into a number of subdomains. Each subdomain is solved independently and, therefore, concurrently using a traditional direct-solution method. Finally, the solution for the interface degrees of freedom between neighboring subdomains is obtained by enforcing compatibility and equilibrium using an iterative procedure. The nonlinear version of the algorithm involves two iterations: The nonlinear dynamic equilibrium iteration and the interface equilibrium and compatibility iteration. The integration of these two iterations is investigated and two strategies are developed. It is found that the strategy in which the two iterations are isolated is the most efficient. As a demonstration, the fully nonlinear transient analysis of a 20-story model building is carried out. Excellent accuracy in the solution and significant speed up values are obtained.     

Dynamic Response of Closed-Loop System with Uncertain Parameters Using Interval Finite-Element Method

Using the interval finite-element method, the vibration control problem of structures with interval parameters is discussed, which is approximated by a deterministic one. Based on the first-order Taylor expansion, a method to solve the interval dynamic response of the closed-loop system is presented. The expressions of the interval stiffness and interval mass matrix are developed directly with the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of dynamic responses is developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. The present method is applied to a vibration system to illustrate the application. The effect of the different levels of uncertainties of interval parameters on responses is discussed. The comparison of the present method with the classical random perturbation is given, and the numerical results show that the present method is valid when the parameter uncertainties are small compared with the corresponding mean values.     

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.     

Modal Testing, Finite-Element Model Updating, and Dynamic Analysis of an Arch Type Steel Footbridge

This paper describes an arch type steel footbridge, its analytical modeling, modal testing, finite-element model updating, and dynamic analysis. A modern steel footbridge which has an arch type structural system and is located on the Karadeniz coast road in Trabzon, Turkey is selected as an application. An analytical modal analysis is performed on the developed three-dimensional finite-element model of footbridge to provide analytical frequencies and mode shapes. Field ambient vibration tests on the footbridge deck under natural excitation such as human walking and traffic loads are conducted. The output-only modal parameter identification is carried out by using peak picking of the average normalized power spectral densities in the frequency domain and stochastic subspace identification in the time domain, and dynamic characteristics such as natural frequencies, mode shapes, and damping ratios are determined. The finite-element model of the footbridge is updated to minimize the differences between analytically and experimentally estimated modal properties by changing some uncertain modeling parameters such as material properties. Dynamic analyses of the footbridge before and after finite-element model updating are performed using the 1992 Erzincan earthquake record. At the end of the study, maximum differences in the natural frequencies are reduced from 22 to only 5% and good agreement is found between analytical and experimental dynamic characteristics such as natural frequencies and mode shapes by model updating. Also, maximum displacements and principal stresses before and after model updating are compared with each other.     

Stability of Shallow Tunnel Using Rigid Blocks and Finite-Element Upper Bound Solutions

Two methods of upper bound solutions were used to calculate the needed support pressure σT for shallow tunnel stability in cohesive-frictional ground. One uses a proposed rigid blocks failure mechanism for shallow rectangular tunnel of upper bound solution, and another uses a finite-element upper bound solution with linear program. In the rigid blocks failure mechanism, the velocity discontinuities in the side wall of the shallow rectangular tunnel occurred in both radial and tangent directions, so as to increase its flexibility. The main formulas for calculating support pressure σT were deduced for the above two methods; thereafter, the corresponding calculating programs were compiled. The needed support pressure σT for shallow tunnel stability and its parametric influences were obtained. The results show that both methods can be used effectively to analyze the stability problems of shallow rectangular tunnel using upper bound solutions.     

Multiple-Porosity Contaminant Transport by Finite-Element Method

An exponential finite-element model for multiple-porosity contaminant transport in soils is proposed. The model combines three compartments for dissolved contaminants: a primary compartment of diffusion–advection transport with nonequilibrium sorption, a secondary compartment with diffusion in rectangular or spherical soil blocks, and a tertiary compartment for immobile solutions within the primary compartment. Hence the finite-element model can be used to solve four types of mass-transfer problems which include: (1) intact soils, (2) intact soils with multiple sources of nonequilibrium partitioning, (3) soils with a network of regularly spaced fissures, and (4) structured soils. Hitherto, mobile/immobile compartments, fissured soils, and nonequilibrium sorption have been treated separately or in pairs. A Laplace transform is applied to the governing equations to remove the time derivative. A Galerkin residual statement is written and a finite-element method is developed. Both polynomial and exponential finite elements are implemented. The solution is inverted to the time domain numerically. The method is validated by comparison to analytical and boundary element predictions. Exponential elements perform particularly well, speeding up convergence significantly. The scope of the method is illustrated by analyzing contamination from a set of four waste repositories buried in fissured clay.     

Bridge Structural Condition Assessment Using Systematically Validated Finite-Element Model

The structural condition assessment of highway bridges is largely based on visual observations described by subjective indices, and it is necessary to develop a methodology for an accurate and reliable condition assessment of aging and damaged structures. This paper presents a method using a systematically validated finite-element model for the quantitative condition assessment of a damaged reinforced concrete bridge deck structure, including damage location and extent, residual stiffness evaluation, and load-carrying capacity assessment. In a trial of the method in a cracked bridge beam, the residual stiffness distribution was determined by model updating, thereby locating the damage in the structure. Furthermore, the damage extent was identified through a defined damage index and the residual load-carrying capacity was estimated.