Handling of Constraints in Finite-Element Response Sensitivity Analysis

In this paper, the direct differentiation method (DDM) for finite-element (FE) response sensitivity analysis is extended to linear and nonlinear FE models with multi-point constraints (MPCs). The analytical developments are provided for three different constraint handling methods, namely: (1) the transformation equation method; (2) the Lagrange multiplier method; and (3) the penalty function method. Two nonlinear benchmark applications are presented: (1) a two-dimensional soil-foundation-structure interaction system and (2) a three-dimensional, one-bay by one-bay, three-story reinforced concrete building with floor slabs modeled as rigid diaphragms, both subjected to seismic excitation. Time histories of response parameters and their sensitivities to material constitutive parameters are computed and discussed, with emphasis on the relative importance of these parameters in affecting the structural response. The DDM-based response sensitivity results are compared with corresponding forward finite difference analysis results, thus validating the formulation presented and its computer implementation. The developments presented in this paper close an important gap between FE response-only analysis and FE response sensitivity analysis through the DDM, extending the latter to applications requiring response sensitivities of FE models with MPCs. These applications include structural optimization, structural reliability analysis, and finite-element model updating.     

Parameter Sensitivity and Importance Measures in Nonlinear Finite Element Reliability Analysis

Finite element reliability methods allow the analyst to define material, load, and geometry parameters as random variables to represent uncertainties in these model parameters. Approximate probabilistic analysis methods produce estimates of the response variance/covariances, probabilities of exceeding specified structural performance thresholds, and parameter importance measures. A necessary ingredient for such analysis is consistent, efficient, and accurate algorithms for computing finite element response sensitivities. In this paper, unified response sensitivity equations with respect to material, load, and geometry parameters are developed for the time- and space-discretized finite element model. The sensitivities with respect to nodal coordinates and global shape parameters in the presence of material and geometric nonlinearities represent an extension of previous work. Practical computer implementation issues are emphasized. The equations are implemented in the comprehensive, open-source, object-oriented finite element software OpenSees. Importance measures from reliability analysis, employing the sensitivity results, are presented to enable the investigation of the relative importance of uncertainty in the parameters of a finite element model. Two example applications demonstrate that the variability in nodal coordinates of a structure can be a significant source of uncertainty along with that in key material and load parameters.     

Second-Order Sensitivities of Inelastic Finite-Element Response by Direct Differentiation

In this paper analytical equations are developed and implemented to obtain second-order derivatives of finite-element responses with respect to input parameters. The work extends previous work on first-order response sensitivity analysis. Of particular interest in this study is the computational feasibility of obtaining second-order response sensitivities. In the past, the straightforward finite difference approach has been available, but this approach suffers from serious efficiency and accuracy concerns. In this study it is demonstrated that analytical differentiation of the response algorithm and subsequent implementation on the computer provides second-order sensitivities at a significantly reduced cost. The sensitivity results are consistent with and have the same numerical precision as the ordinary response. The computational cost advantage of the direct differentiation approach increases as the problem size increases. Several novel implementation techniques are developed in this paper to optimize the computational efficiency. The derivations and implementations are demonstrated and verified with two finite-element analysis examples.     

Methods and Object-Oriented Software for FE Reliability and Sensitivity Analysis with Application to a Bridge Structure

This paper addresses the growing demand for finite-element software with capabilities to incorporate uncertainty in the input parameters. Reliability and response sensitivity algorithms are implemented in the general-purpose finite-element software OpenSees, which employs an object-oriented programming approach to achieve a sustainable software with focus on maintainability and extensibility. The product is a comprehensive and freely available library of software tools for finite-element reliability and response sensitivity analysis. A numerical example involving a detailed model of a highway bridge with inelastic material behavior and 320 random variables is presented to demonstrate features of the methodology and the software. Importance vectors are employed to rank the input parameters according to their relative influence on the structural reliability. The required response sensitivities are obtained by an extensive implementation of the direct differentiation method.     

Software Framework for Parameter Updating and Finite-Element Response Sensitivity Analysis

The finite-element software framework OpenSees is extended with parameter updating and response sensitivity capabilities to support client applications such as reliability, optimization, and system identification. Using software design patterns, member properties, applied loadings, and nodal coordinates can be identified and repeatedly updated in order to create customized finite-element model updating applications. Parameters are identified using a Chain of Responsibility software pattern, where objects in the finite-element model forward a parameterization request to component objects until the request is handled. All messages to identify and update parameters are passed through a Facade that decouples client applications from the finite-element domain of OpenSees. To support response sensitivity analysis, the Strategy design pattern facilitates multiple approaches to evaluate gradients of the structural response, whereas the Visitor pattern ensures that objects in the finite-element domain make the proper contributions to the equations that govern the response sensitivity. Examples demonstrate the software design and the steps taken by representative finite-element model updating and response sensitivity applications.     

Finite-Element Simulations of Full-Scale Modular-Block Reinforced Soil Retaining Walls under Earthquake Loading

A finite-element procedure was used to simulate the dynamic behavior of four full-scale reinforced soil retaining walls subjected to earthquake loading. The experiments were conducted at a maximum horizontal acceleration of over 0.8 g, with two walls subjected to only horizontal accelerations and two other walls under simultaneous horizontal and vertical accelerations. The analyzes were conducted using advanced soil and geosynthetic models that were capable of simulating behavior under both monotonic and cyclic loadings. The soil behavior was modeled using a unified general plasticity model, which was developed based on the critical state concept and that considered the stress level effects over a wide range of densities using a single set of parameters. The geosynthetic model was based on the bounding surface concept and it considered the S-shape load-strain behavior of polymeric geogrids. In this paper, the calibrations of the models and details of finite-element analysis are presented. The time response of horizontal and vertical accelerations obtained from the analyses, as well as wall deformations and tensile force in geogrids, were compared with the experimental results. The comparisons showed that the finite-element results rendered satisfactory agreement with the shake table test results.     

Multigrid Preconditioner for Unstructured Nonlinear 3D FE Models

Multigrid and multigrid-preconditioned conjugate-gradient solution techniques applicable for unstructured 3D finite-element models that may involve sharp discontinuities in material properties, multiple element types, and contact nonlinearities are developed. Their development is driven by the desire to efficiently solve models of rigid pavement systems that require explicit modeling of spatially varying and discontinuous material properties, bending elements meshed with solid elements, and separation between the slab and subgrade. General definitions for restriction and interpolation operators applicable to models composed of multiple, displacement-based isoparametric finite-element types are proposed. Related operations are used to generate coarse mesh element properties at integration points, allowing coarse-level coefficient matrices to be computed by a simple assembly of element stiffness matrices. The proposed strategy is shown to be effective on problems involving spatially varying material properties, even in the presence of large variations within coarse mesh elements. Techniques for solving problems with nodal contact nonlinearities using the proposed multigrid methods are also described. The performance of the multigrid methods is assessed for model problems incorporating irregular meshes and spatially varying material properties, and for a model of two rigid pavement slabs subjected to thermal and axle loading that incorporates nodal contact conditions and both solid and bending elements.     

Modeling and Simulation of Porous Elastoviscoplastic Material

Many materials exhibit elasto–visco–plastic behavior when subjected to loadings with certain strain rate. Examples include natural materials such as metals, clays, and soils and manmade materials such as some biomimic materials. Some voids may exist or be introduced in these materials. The effects of the voids on the material response are important in predicting the strength, reliability, and service life of structural systems containing these materials. This paper presents the results of applying a statistical micromechanical approach to describe the macroscopic behavior of elasto–visco–plastic materials containing many randomly dispersed spherical voids. Most existing micromechanics based models are only applicable to monotonic proportional loadings. The limitation is removed by integrating the material model into the framework of continuum plasticity. With the discrete integration algorithm and local return mapping algorithm, the proposed computation method is applicable to any loading and unloading histories and is ready for implementing into finite element analysis.     

Finite Element Response Sensitivity Analysis of Steel-Concrete Composite Beams with Deformable Shear Connection

The behavior of steel-concrete composite beams is strongly influenced by the type of shear connection between the steel beam and the concrete slab. For accurate analytical predictions, the structural model must account for the interlayer slip between these two components. In numerous engineering applications (e.g., in the fields of structural optimization, structural reliability analysis, and finite element model updating), accurate response sensitivity calculations are needed as much as the corresponding response simulation results. This paper focuses on a procedure for response sensitivity analysis of steel-concrete composite structures using displacement-based locking-free frame elements including deformable shear connection with fiber discretization of the cross section. Realistic cyclic uniaxial constitutive laws are adopted for the steel and concrete materials as well as for the shear connection. The finite element response sensitivity analysis is performed according to the direct differentiation method. The concrete and shear connection material models as well as the static condensation procedure at the element level are extended for response sensitivity computations. Two steel-concrete composite structures for which experimental test results are available in the literature are used as realistic testbeds for response and response sensitivity analysis. These benchmark structures consist of a nonsymmetric, two-span continuous beam subjected to monotonic loading and a frame subassemblage under cyclic loading. The new analytical derivations for response sensitivity calculations and their computer implementation are validated through forward finite difference analysis based on the two benchmark examples considered. Selected sensitivity analysis results are shown for validation purposes and for quantifying the effect and relative importance of the various material parameters in regards to the nonlinear monotonic and cyclic response of the testbed structures.     

Predicting Shrinkage Stress Field in Concrete Slab on Elastic Subgrade

Concrete is a material that changes volumetrically in response to moisture and temperature variations. Frequently, these volumetric changes are prevented by restraint from the surrounding structure, resulting in the development of tensile stresses. This paper provides a method for computing the stress and displacement fields that develop in response to this restraint by considering the concrete slab as a plate resting on an elastic foundation. The interface between the slab and the foundation is capable of simulating all cases between complete perfect bond and perfect compression∕zero tension bond to permit debonding. In addition, stress relaxation is considered in the concrete to account for the reduction in stress due to creep∕relaxation-related phenomena. For this reason, the stress-strain relationship and equilibrium equations have been considered in the rate or differential form. The history-dependent equilibrium equations are obtained by integrating the differential equations with respect to time. An example is presented to illustrate the favorable correlation that exists between the predicted behavior of the plate model and finite-element modeling.     

Stiffness Evaluation and Damage Detection of Ceramic Candle Filters

This paper presents a nondestructive evaluation method to identify the structural stiffness of ceramic candle filters. A ceramic candle filter is a hollow cylindrical structure made of a porous ceramic material used in advanced, coal-fired power generation systems. The candle filters need to sustain an extreme thermal and chemical environment over a great period of time to protect the gas turbine components from exposure to particulate matter. A total of 92 new candle filters and 29 used candle filters have been tested nondestructively using a dynamic characterization technique. All filters were subjected to an excitation force, and the response was picked up by an accelerometer in a free-free boundary condition. The frequency response function and vibration mode shapes of each filter were evaluated. Beam vibration equations and finite-element models were built to calculate the filter's dynamic response. Results indicate that the vibration signatures can be used as an index to quantify the structural properties of ceramic candle filters. The results also show estimations of the overall bending stiffness values for four different types of candle filters. The used filters show a trend of stiffness degradation, which was related to the filter's exposure time. Damage detection procedures using modal strain energy and finite-element simulation were studied for detection of a localized damage in the candle filter. The location and the size of the damaged section can be identified using the measured model strain energy.     

Sensitivity Statistics of 3D Structures under Parametric Uncertainty

Determination of sensitivity gradient is a major prerequisite for structural optimization, reliability assessment, and parameter identification. As the conventional deterministic sensitivity analysis cannot provide complete information, stochastic analysis is needed to tackle the uncertainties in structural parameters. This study focuses on the utility of the stochastic finite-element method for random response sensitivity analysis. The stochastic modeling of a random parameter is based on a commonly used 2D local averaging method generalized for a 3D case. The Choleski decomposition technique is then employed for digital simulation. The Neumann expansion based finite-element simulation method is extended for stochastic sensitivity analysis. This technique leads to a considerable saving of computational time. Example problems are used to compare the accuracy of this method to the direct Monte Carlo simulation and perturbation method in terms of varying stochasticity and efficiency in CPU time.     

Loading of a Gravel-Buried Steel Pipe Subjected to Rockfall

Increasing rockfall activity in the European Alps has raised the need for designing protection systems for Alpine infrastructure. This paper is concerned with protection of steel pipelines by a gravel overburden of height H. Rockfall-induced loading of such pipes is estimated by means of a three-dimensional, quasi-static, elasto-plastic finite-element (FE) model. Maximum impact forces F and corresponding penetration depths w are estimated based on dimensionless formulas, related to real scale impact tests onto gravel layers. The forces F are applied as surface loads onto the FE model, at a distance (H?w) from the pipe. Material behavior of gravel is represented by a cap model, which is based on pressure-independent linear elasticity and associated plasticity. Related material parameters are identified from acoustic and static material tests. The structural FE model is validated by comparing FE-predicted stresses in the pipe with stresses determined in a real-scale structural experiment. This is reasonable only because the real-scale test is independent of the experiments used for identification of the material parameters used as input for the structural FE model. Satisfactory FE predictions motivate use of the FE model for estimating the loading of the steel pipe in untested scenarios, concerning, e.g., different heights of overburden, or different impact intensities. These estimates show some efficiency of gravel protection systems for modest rockfall, with impact energies well below 3,500?kJ.     

Improved Geometric Design of Bridge Asphalt Plug Joints

Asphalt plug joints (APJs) have several advantages over traditional bridge joints. They are easy and cheap to install and have good surface flatness. However, widespread application of APJs in bridges has been hindered by frequently observed premature failures. Detailed finite-element simulations are conducted to develop a better understanding of the parameters that influence APJ response under traffic and thermal loading conditions. The computational model employs a time and temperature dependent viscoplastic material model and is validated by comparing model results to previously published experimental data. The key parameters investigated are gap plate width, gap plate thickness, gap plate edge geometry, and geometry of the interface between pavement and APJ. The resulting information is synthesized into a proposed alternative APJ design that minimizes local demands deemed to be responsible for the observed early failures.     

Parallel Computing Techniques for Sensitivity Analysis in Optimum Structural Design

Among different activities of the optimum structural design using the gradient-based optimization approaches, design sensitivity analysis is the most time-consuming computational process. By introducing parallel computing techniques for sensitivity computation, significant speedup has been obtained in optimum structural design. Computation of design sensitivities is characteristically uncoupled, thus opening the door to parallelization. In this paper, two types of approaches viz. single-level and multilevel parallelisms are pursued for design sensitivities. The design sensitivities are computed using analytical and finite-difference methods. Numerical studies show that the performance of the parallel algorithms for design sensitivities on message passing systems is very good. Good speedups have been achieved in parallel multilevel sensitivity calculation. The parallel algorithms for design sensitivity analysis have been implemented on message passing parallel systems within the software platform of Parallel Computer Adaptive Language.     

Bayesian State Estimation Method for Nonlinear Systems and Its Application to Recorded Seismic Response

The focus of this paper is to demonstrate the application of a recently developed Bayesian state estimation method to the recorded seismic response of a building and to discuss the issue of model selection. The method, known as the particle filter, is based on stochastic simulation. Unlike the well-known extended Kalman filter, it is applicable to highly nonlinear systems with non-Gaussian uncertainties. The particle filter is applied to strong motion data recorded in the 1994 Northridge earthquake in a seven-story hotel whose structural system consists of nonductile reinforced-concrete moment frames, two of which were severely damaged during the earthquake. We address the issue of model selection. Two identification models are proposed: a time-varying linear model and a simplified time-varying nonlinear degradation model. The latter is derived from a nonlinear finite-element model of the building previously developed at Caltech. For the former model, the resulting performance is poor since the parameters need to vary significantly with time in order to capture the structural degradation of the building during the earthquake. The latter model performs better because it is able to characterize this degradation to a certain extent even with its parameters fixed. For this case study, the particle filter provides consistent state and parameter estimates, in contrast to the extended Kalman filter, which provides inconsistent estimates. It is concluded that for a state estimation procedure to be successful, at least two factors are essential: an appropriate estimation algorithm and a suitable identification model.     

Dynamic Field Test, System Identification, and Modal Validation of an RC Minaret: Preprocessing and Postprocessing the Wind-Induced Ambient Vibration Data

The investigation of dynamic response for civil engineering structures largely depends on a detailed understanding of their dynamic characteristics, such as the natural frequencies, mode shapes, and modal damping ratios. Dynamic characteristics of structures may be obtained numerically and experimentally. The finite-element method is widely used to model structural systems numerically. However, there are some uncertainties in numerical models. Material properties and boundary conditions may not be modeled correctly. There may be some microcracks in the structures, and these cracks may directly affect the modeling parameters. Modal testing gives correct uncertain modeling parameters that lead to better predictions of the dynamic behavior of a target structure. Therefore, dynamic behavior of special structures, such as minarets, should be determined with ambient vibration tests. The vibration test results may be used to update numerical models and to detect microcracks distributed along the structure. The operational modal analysis procedure consists of several phases. First, vibration tests are carried out, spectral functions are produced from raw measured acceleration records, dynamic characteristics are determined by analyzing processed spectral functions, and finally analytical models are calibrated or updated depending on experimental analysis results. In this study, an ambient vibration test is conducted on the minaret under natural excitations, such as wind effects and human movement. The dynamic response of the minaret is measured through an array of four trixial force-balanced accelerometers deployed along the whole length of the minaret. The raw measured data obtained from ambient vibration testing are analyzed with the SignalCAD program, which was developed in MATLAB. The employed system identification procedures are based on output-only measurements because the forcing functions are not available during ambient vibration tests. The ModalCAD program developed in MATLAB is used for dynamic characteristic identification. A three-dimensional model of the minaret is constructed, and its modal analysis is performed to obtain analytical frequencies and mode shapes by using the ANSYS finite-element program. The obtained system identification results have very good agreement, thus providing a reliable set of identified modal properties (natural frequencies, damping ratios, and mode shapes) of the structure, which can be used to calibrate finite-element models and as a baseline in health monitoring studies.     

Structural Condition Assessment of Sewer Pipelines

The need of immediate supportive measures for sustainability of municipal infrastructures calls for better understanding of the behavior of various infrastructure network systems and their components. This paper presents a study which uses artificial neural networks to investigate the importance and influence of certain characteristics of sewer pipes upon their structural performance, expressed in terms of condition rating. In this study, back propagation and probabilistic neural network (NN) models were developed and validated. The data used in the development of these models were provided by the municipality of Pierrefonds, Quebec. It comprised of parameters related to sewer pipelines, pipe diameter, buried depth/cover, bedding material, pipe material, pipeline length, age, and closed circuit television (CCTV) based structural condition rating. The first six parameters are the independent variables of the models whereas CCTV based condition rating for these pipes is the dependent variable (i.e., the output of the models). The developed NN models were used to rank the parameters, in order of their importance/influence on pipe condition. It was found that, among the studied parameters, material attributes have highest influence on pipe structural condition, respectively, followed by the geometric and physical attribute group. Sensitivity analysis was then performed to simulate the structural condition of a pipe at a range of values of each input parameters. Results of sensitivity analysis describe the nature and degree of the influence of each parameter on pipe structural condition. The developed models are expected to benefit academics and practitioners (municipal engineers, consultants, and contractors) to prioritize inspection and rehabilitation plans for existing sewer mains.     

Explicit Finite-Element Analysis of 2024-T3/T351 Aluminum Material under Impact Loading for Airplane Engine Containment and Fragment Shielding

Uncontained aircraft engine failure can cause catastrophic damaging effects to aircraft systems if not addressed in the aircraft design. Mitigating the damaging effects of uncontained engine failure and improving the numerical modeling capability of these uncontained engine events are crucial. In this paper, high strain rate material behavior of one of the most extensively used materials in the aircraft industry is simulated and the results are compared against ballistic impact tests. Ballistic limits are evaluated by utilizing explicit finite-element (FE) simulations based on the corresponding ballistic impact experiments performed at different material thicknesses. LS-DYNA is used as a nonlinear explicit dynamics FE code for the simulations. A Johnson–Cook material model with different sets of parameters is employed as a thermo-viscoplastic material model coupled with a nonlinear equation of state and an accumulated damage evaluation algorithm for the numerical simulations. Predictive performance of the numerical models is discussed in terms of material characterization efforts, material model parameters, mesh sensitivities, and effects of stress triaxiality. It is shown that mesh refinement does not necessarily provide better results for ballistic limit simulations without considering and calibrating these interrelated factors. Moreover, it is shown that current models that can only fit a specific function for damage evaluation as a function of stress triaxiality are not always successful in predicting failure, especially if the state of stress changes significantly.     

Adjoint Sensitivity Analysis for Shallow-Water Wave Control

An adjoint sensitivity method based on the shallow-water equations is developed for water wave control in river and estuarine systems. The method is used to compute the gradient of a user-defined objective function in the N-dimensional parameter space consisting of system control settings with just one solution of the basic problem and one solution of the associated adjoint problem. Characteristic equations are derived for the adjoint problem and a new formalism is proposed for the sensitivity of shallow-water flow to boundary changes in depth and discharge. New adjoint boundary conditions are developed for river and estuarine forecasting models with open-water inflow and outflow sections. This gives rise to new expressions for sensitivities at these sections. Characteristic analysis of the adjoint and basic problems shows that sensitivities propagate in the reverse time direction along the characteristic paths of the basic problem. The Riemann variables of the adjoint problem are shown to precisely describe the sensitivity of the objective function to changes in depth and discharge at system boundaries. The method is extended to two space dimensions by bicharacteristic analysis.