Linear Elastic Structures Research Articles

An enhanced finite volume method to model 2D linear elastic structures

This paper details the evaluation and enhancement of the vertex-centred finite volume method for the purpose of modelling linear elastic structures undergoing bending. A matrix-free edge-based finite volume procedure is discussed and compared with the traditional isoparametric finite element method via application to a number of test-cases. It is demonstrated that the standard finite volume approach exhibits similar disadvantages to the linear Q4 finite element formulation when modelling bending. An enhanced finite volume approach is proposed to circumvent this and a rigorous error analysis conducted. It is demonstrated that the developed finite volume method is superior to both standard finite volume and Q4 finite element methods, and provides a practical alternative to the analysis of bending-dominated solid mechanics problems.

Computational framework for effective force testing and a compensation technique for nonlinear actuator dynamics

SUMMARY This paper presents a development of a computational framework for effective force testing (EFT). The framework is intended to facilitate accurate computational simulations of EFTs that allow for performance evaluation of experimental tests, stability assessment of force feedback controllers, and others. To properly capture critical issues in EFT such as control–structure interaction and oil-column resonance, the computational framework incorporates nonlinear dynamics of servo hydraulic actuators including power electronics of a converter, electromagnetics of a flapper nozzle, hydrodynamics of hydraulic fluid, and kinematics of an actuator piston. The computational framework is verified through comparative studies with three different effective force tests that were previously conducted at the Johns Hopkins University. The comparative studies demonstrate that the computational framework is able to accurately simulate experimental EFTs of an single-degree-of-freedom (SDOF) linear elastic structure, an SDOF nonlinear inelastic structure, and an multi-degrees-of-freedom (MDOF) test structure. Using the developed computational framework, an investigation of nonlinear dynamics of servo hydraulic actuators is performed. Simulation results showed that the influence of the nonlinear actuator dynamics becomes significant as the level of force increases. A compensation technique that uses only force and valve command to reduce the effects of valve nonlinearities is developed and investigated using the computational framework. It was shown that the nonlinear valve dynamics compensation technique is effective at improving peak responses of force and reducing undesirable effects. The examples shown in this paper demonstrate that the computational framework facilitates further developments and applications of effective force test methods for seismic performance assessment of structural systems. Copyright © 2013 John Wiley & Sons, Ltd.

A newly developed qp-relaxation method for element connectivity parameterization to achieve stress-based topology optimization for geometrically nonlinear structures

The aim of this work is to present a novel computational approach to employ the stress-based topology optimization method (STOM) to minimize the volume subject to the locally defined stress constraints of a geometrically nonlinear structure in the framework of the element connectivity parameterization (ECP) method. Considering the locally defined stress constraints in topology optimization (TO) is a classic and challenging engineering problem, and successful optimization procedures have recently been developed using the density-based TO method for linear elastic structures. However, no study has yet considered the static failure constraint when using TO for a geometrically nonlinear structure. Therefore, the present study develops a novel computational approach for the STOM for a geometrically nonlinear structure. To successfully optimize a geometrically nonlinear structure, the unstable element issue must be properly addressed, in addition to the stress singularity issue, the existence of a large number of constraints, and the highly nonlinear behavior of the local stress constraints. To effectively resolve these issues, this research adopts the ECP method to interpolate and optimize the connectivities among solid finite elements. Furthermore, we find that a stress singularity issue linked to the local optima issue arises in the ECP method that is different from that of the density-based TO. By investigating the singularity behavior in detail, we develop a new qp-relaxation method that is suitable for the ECP method. To demonstrate the improved capability of the proposed ECP method with the modified qp-relaxation, several two-dimensional TO problems are solved.

Precise computation and error control of stress intensity factors and certain integral characteristics in anisotropic inhomogeneous materials

The numerical simulation of quasi-static crack propagation is closely related to the computation of characteristics such as stress intensity factors or energy release rates. In this work, ideas are proposed, how such quantities can be calculated precisely in linear elastic, anisotropic and inhomogeneous plane structures. Stress intensity factors and other local characteristics can be evaluated in terms of functionals depending on solutions of certain elasticity problems. The approach used here to calculate these functional values precisely with Galerkin finite elements is a dual-weighted-residual method for adaptive mesh refinement and a posteriori error control. Especially in structures under mixed-mode loadings contact of crack faces can occur. A numerical realization of mutual non-penetration conditions of inequality type for the crack faces and the effect of such constraints on stress intensity factors is shown. Numerical results are presented for anisotropic and functionally graded materials.

Topology Optimization of a Car Seat Frame

The optimization of consumer products through the use of numerical simulations has become a key factor to a continuously increasing requirement for time and cost efficiency, for quality improvement and materials saving, in many manufacturing areas such as automotive, aerospace, building, packaging and electronic industries. Environmental protection, fuel economy and safety specifications are today major concerns in automotive industry. Part of the overall strategy is a lower weight car which means increased performance, reduction of fuel consumption achieving a lower vehicle exhaust emissions to the environment. At the same time, the occupant safety must be continuously improved, as the safety specifications are more demanding. To meet these requirements, a call for new lightweight’s concepts and crash structures was raised, using lighter and/or stronger materials. In the last decade we have assisted to a development and application of high-strength steels and aluminum alloys in the manufacture of automotive structures. This paper presents a structural design problem of a car seat frame, aiming the desired weight reduction while satisfying a set of performance constraints. The numerical model of the seat frame has been developed and the numerical results were validated against experimental data obtained during static loading tests. Using the developed computational model, an optimized topology of the linear elastic structure has been determined, reaching a significant weight reduction.

Two-scale approach to predict multi-site cracking potential in 3-D structures using the generalized finite element method

This research utilized the novel computational framework of the generalized finite element method (GFEM) to predict the potential for crack propagation in concrete slabs. A two-scale approach, using the global–local concept within the GFEM framework (GFEMg–l), is applied to multi-site cracking problems (MSC), where different crack geometries are placed simultaneously at different positions in a three-dimensional airfield slab loaded by new generation aircraft gears. The GFEMg–l approach efficiently simulated multiple cracks without discretization in the global mesh, but only in the local domain. The GFEMg–l enrichment functions allow the displacements of the local problem to be represented in the global domain through enrichment functions from the local problems rather than explicitly modeling each crack discretely in the global domain. The main contribution of this work was extension of the GFEMg–l approach to a class of three-dimensional MSC problems involving realistic boundaries conditions and existence of multiple cracks spanning different orders of magnitude in size (scales) within the domain. For the linear elastic structure, bottom- and surface-initiated cracks with small dimensions were considered in conjunction with a larger macro-crack. Unlike traditional numerical methods, the proposed GFEMg–l made it possible to tackle this class of problems by avoiding refined crack front meshes in the global domain as well as numerical round-off errors. Furthermore, the two-scale approach significantly reduces the computational cost for large-scale 3-D MSC problems.

On the application of intervening variables for stochastic finite element analysis

This paper investigates the application of first order Taylor series expansion considering intervening variables for estimating second order statistics for problems of stochastic finite element analysis. In particular, two types of intervening variables are considered. The focus is on studying the response of linear elastic static structures whose properties are characterized by means of log-normal random fields. Results obtained indicate that the use of intervening variables can lead to improved estimation of second order statistics when compared with traditional approaches based on Taylor series while numerical efforts remain similar.

Influence of mesh size and truncation order on reconstruction of impact force on composite elastic beams

Abstract Reconstructing impact forces can improve considerably structural health monitoring as the extent of damage can be better made out. In this work an inverse problem formulation to identify the pressure generated by a non punctual impact is investigated. Considering the case of linear elastic layered composite structures, reconstruction of impact pressure is performed through a finite element model of the structure and impulse response functions between the impact zone and sensors placed at known positions. Assuming that the pressure is uniform, reconstruction is carried out by regularized deconvolution based on generalized singular value decomposition. The infl uence of mesh size, modal truncation and time sampling on the reconstructed pressure is discussed.

Shape optimal design of structures with dual boundary elements

This paper is concerned with the numerical implementation of the dual boundary element method for shape optimal design of two-dimensional linear elastic structures. The design objective is to minimize the structural compliance, subject to an area constraint. Sensitivities of objective and constraint functions, derived by means of Lagrangean approach and the material derivative concept with an adjoint variable technique, are computed through analytical expressions that arise from optimality conditions. The dual boundary element method, used for the discretization of the state problem, applies the stress equation for collocation on the design boundary and the displacement equation for collocation on other boundaries. The use of the stress boundary integral equation, discretized with discontinuous quadratic elements, allows an efficient and accurate computation of stresses on the design boundary. The perturbation field is described with linear continuous elements, in which the position of each node is defined by a design variable. Continuity along the design boundary is assured by forcing the end points of each discontinuous boundary element to be coincident with a design node. The optimization problem is solved by the modified method of feasible directions available in the program ADS. Examples of a plate with a hole are analyzed with the present method, for different loading conditions. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of shape optimal design of structures.

Performance of variable curvature sliding isolators in base‐isolated benchmark building

AbstractThe performances of variable curvature sliding isolators like variable frequency pendulum isolator (VFPI) and variable curvature friction pendulum system (VCFPS) installed in the base‐isolated benchmark building subjected to bi‐directionally acting seven strong earthquakes have been studied. The shear type base‐isolated benchmark building is modelled as three‐dimensional linear elastic structure having three degrees‐of‐freedom at each floor level. Time domain dynamic analysis of the building has been carried out with the help of constant average acceleration Newmark‐Beta method and non‐linear isolation forces has been taken care by fourth‐order Runge‐Kutta method. The force‐displacement responses of the VFPI and VCFPS are studied under parametric variations of their key characteristics for the comparative performance evaluation. The time history variations of response characteristics and peak response evaluation criteria are also investigated for overall comparison of their performances. The performance of VFPI and VCFPS are observed both in uniform and hybrid isolation systems. The force‐displacement responses of both VFPI and VCFPS subjected to strong near‐field earthquakes show excessively large isolator displacements at higher initial radii of curvature of sliding surface. The large isolator displacements of VFPI and VCFPS can be restrained efficiently by addition of viscous fluid dampers in comparison to the increase in coefficient of friction of isolators. Copyright © 2010 John Wiley & Sons, Ltd.

The interaction of elasticity and rocking in flexible structures allowed to uplift

SUMMARYNumerous structures uplift under the influence of strong ground motion. Although many researchers have investigated the effects of base uplift on very stiff (ideally rigid) structures, the rocking response of flexible structures has received less attention. Related practical analysis methods treat these structures with simplified ‘equivalent’ oscillators without directly addressing the interaction between elasticity and rocking. This paper addresses the fundamental dynamics of flexible rocking structures. The nonlinear equations of motion, derived using a Lagrangian formulation for large rotations, are presented for an idealized structural model. Particular attention is devoted to the transition between successive phases; a physically consistent classical impact framework is utilized alongside an energy approach. The fundamental dynamic properties of the flexible rocking system are compared with those of similar linear elastic oscillators and rigid rocking structures, revealing the distinct characteristics of flexible rocking structures. In particular, parametric analysis is performed to quantify the effect of elasticity on uplift, overturning instability, and harmonic response, from which an uplifted resonance emerges. The contribution of stability and strength to the collapse of flexible rocking structures is discussed. Copyright © 2012 John Wiley & Sons, Ltd.

Torsional Responses under Bidirectional Seismic Excitations: Effect of Instantaneous Load Eccentricities

AbstractTorsional response for one-way or two-way asymmetric linear/nonlinear structural systems under uni- and bidirectional seismic excitations increases the maximum displacement demand on structural systems. In this study, we consider instantaneous load eccentricities caused by the motion of the center of mass. The eccentricities, which are time dependent, exist even for linear elastic two-way symmetric structures under seismic excitations, as the relative displacement between the center of mass and center of stiffness may not necessarily be negligible. The quantification of this second-order effect on structural response is not available in the literature. Equations are developed and some numerical results are obtained by incorporating this second-order effect for linear elastic structures under bidirectional excitations to quantify the importance of this phenomenon. The results obtained by considering more than 100 ground motion records suggest that, on average, an underestimation of seismic displace...

Response Spectral Characteristics and Reliabilities of Linear Structures to Both Intensity and Frequency Content Time-Varying Earthquake Loads

AbstractA numerical method for estimating the useful spectral characteristics of responses of classically damped linear elastic structures to the Yeh-Wen model of fully nonstationary random earthquake loads is developed. The derivation of the method is carried out in three steps. First, the stationary Gaussian white noise process in the space of a frequency modulation function is transformed to the uniformly modulated Gaussian white noise process in the time space. Second, the numerical computation procedure for the unit impulse response functions in the time domain of the Clough-Penzien filters modified in the space of a frequency modulation function is proposed. Finally, the numerical solutions for the useful spectral characteristics of responses of linear structures to the Yeh-Wen model are given, from which the first-passage reliability problem can be solved by using Vanmarcke’s approximation. Numerical investigation of a simple oscillator shows the effect of the frequency modulation functions on the ...

Large‐scale real‐time hybrid simulation involving multiple experimental substructures and adaptive actuator delay compensation

SUMMARYReal‐time hybrid simulation provides a viable method to experimentally evaluate the performance of structural systems subjected to earthquakes. The structural system is divided into substructures, where part of the system is modeled by experimental substructures, whereas the remaining part is modeled analytically. The displacements in a real‐time hybrid simulation are imposed by servo‐hydraulic actuators to the experimental substructures. Actuator delay compensation has been shown by numerous researchers to vitally achieve reliable real‐time hybrid simulation results. Several studies have been performed on servo‐hydraulic actuator delay compensation involving single experimental substructure with single actuator. Research on real‐time hybrid simulation involving multiple experimental substructures, however, is limited. The effect of actuator delay during a real‐time hybrid simulation with multiple experimental substructures presents challenges. The restoring forces from experimental substructures may be coupled to two or more degrees of freedom (DOF) of the structural system, and the delay in each actuator must be adequately compensated. This paper first presents a stability analysis of actuator delay for real‐time hybrid simulation of a multiple‐DOF linear elastic structure to illustrate the effect of coupled DOFs on the stability of the simulation. An adaptive compensation method then proposed for the stable and accurate control of multiple actuators for a real‐time hybrid simulation. Real‐time hybrid simulation of a two‐story four‐bay steel moment‐resisting frame with large‐scale magneto‐rheological dampers in passive‐on mode subjected to the design basis earthquake is used to experimentally demonstrate the effectiveness of the compensation method in minimizing actuator delay in multiple experimental substructures. Copyright © 2011 John Wiley & Sons, Ltd.

Gauss-Newton and Inverse Gauss-Newton Methods for Coefficient Identification in Linear Elastic Systems

The "inverse problem" of determining parameter distributions in linear elastic structures has been explored widely in the literature. In the present article we discuss this problem in the context of a particular formulation of linear elastic systems, dividing the associated inverse problems into two classes which we call Case 1 and Case 2. In the first case the elastic parameters can be obtained by solving a certain set of linear algebraic equations, typically poorly conditioned. In the second case the corresponding problem involves nonlinear equations which usually must be solved by approximation methods, including the Gauss-Newton method for overdetermined systems. Here we discuss the application of this method and a related, empirically more stable, method which we call the inverse Gauss-Newton method. Convergence theorems are established and computational results for sample problems are presented.

Energy Distribution in Impulsively Excited Structures

This paper presents a novel approach to the study of shock in elastic structures. An energy storage property for linear elastic non-dissipative structures is demonstrated, then extrapolated also to damped systems. This property relates locations of energy accumulation along the structure to the form of its stiffness and mass matrices. This result can be profitable used at a design stage of complex structures undergoing shock excitations. On the theoretical ground, the obtained results disclose indeed general properties of energy distribution in dynamical systems with implications on energy localization as well as on energy equipartitioning. Finally, these findings open a new point of view on energy transmission in structures, with potential application in the context of Statistical Energy Analysis.

ROBUSTNESS ANALYSIS OF STRUCTURES UNDER COMPLIANCE CONSTRAINT

The main problem addressed in this paper is to evaluate robustness of a linear elastic structure with an uncertain static external load. We consider the compliance constraint as a performance requirement and use the robustness function as a qualitative measure of robustness. The robustness function is defined as the largest level of uncertainty in the external load, up to which the compliance constraint is satisfied. We show that the robustness function can be computed by solving a semidefinite programming problem. Numerical examples illustrate that robust preference between two given structural designs depends on the critical performance.

INFLUENCE OF HIGH INITIAL ISOLATOR STIFFNESS ON THE SEISMIC RESPONSE OF A BASE-ISOLATED BENCHMARK BUILDING

The influence of high initial isolator stiffness on the response of a base-isolated benchmark building is investigated. The base-isolated building is modeled as a three-dimensional linear-elastic structure having three degrees-of-freedom at each floor level. The time-history analysis of this building is carried out by solving the governing equations of motion using Newmark-beta method along with an iterative predictor–corrector approach. The force–deformation behavior of the isolation system is modeled by a bilinear law, which can be effectively used to model all isolation systems in practice. Three near-field earthquakes with bidirectional ground motions are considered. Structural response parameters such as absolute top floor acceleration, base shear, and base displacement are chosen for investigating the effects of high initial isolator stiffness. It was observed that the high initial isolator stiffness of the isolation system excites the higher modes in the base-isolated building and increases the top floor acceleration. Such a phenomenon can be detrimental to the sensitive instruments placed in the isolated structure. On the other hand, both the base displacement and base shear reduce marginally due to increase in the initial isolator stiffness. Further, the influences of high initial isolator stiffness are found to dependent on the period and characteristic strengths of the base isolation system.

Minmax topology optimization

We describe a systematic approach for the robust optimal design of linear elastic structures subjected to unknown loading using minmax and topology optimization methods. Assuming only the loading region and norm, we distribute a given amount of material in the design domain to minimize the principal compliance, i.e. the maximum compliance that is produced by the worst-case loading scenario. We evaluate the principal compliance directly by satisfying the optimality conditions which take the form of a Steklov eigenvalue problem and thus we eliminate the need of an iterative nested optimization. To generate a well-posed topology optimization problem we use relaxation which requires homogenization theory. Examples are provided to demonstrate our algorithm.

Analysis of implicit HHT‐α integration algorithm for real‐time hybrid simulation

SUMMARYReal‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 John Wiley & Sons, Ltd.