Finite Element Equilibrium Equations Research Articles

A nonlinear interval finite element method for elastic–plastic problems with spatially uncertain parameters

This paper proposes a nonlinear interval finite element method for elastic–plastic analysis of structures with spatially uncertain parameters. The spatially uncertain parameters are described by the interval field, and the variation bounds of the elastic–plastic structural responses can be calculated effectively. Quantified by the interval field, the spatially uncertain parameters are represented by the interval Karhunen–Loève (K-L) expansion, based on which the nonlinear interval finite element equilibrium equation is formulated. An interval iterative method is then presented to solve the equilibrium equation and obtain an outer solution of the variation bounds of structural responses such as displacement. In this method, the Newton-Raphson iterative method is used to transform the nonlinear problem into a linear one, and then the interval iterative method is introduced to solve the interval linear equations. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method.

Behavior analysis of energy piles in layered transversely isotropic saturated soils

This paper conducts theoretical research on energy piles in layered saturated transversely isotropic soils under different load forms. Firstly, the Timoshenko beam coupled axial force rod elements is used to simulate the pile elements, and the finite element equilibrium equation for energy single piles is derived accordingly. Then, the extended precise integration method (EPIM) is applied to obtain the fundamental solution of soil, which is regarded as the kernel function of the boundary element method to construct the boundary integral equation. Based on the theory of the coupled finite element method - boundary element method (FEM-BEM), and combined with the equilibrium conditions of force and displacement coordination at the pile-soil interface, the boundary integral equation of soil is coupled with the finite element equilibrium equation of the pile to establish the interaction equation. The calculation results of this method are compared with theoretical solutions and in-site field tests to verify the correctness of this theory. Finally, numerical examples are designed to explore the influence of the parameters of energy piles and soils on the behavior of energy piles in layered saturated transversely isotropic soils under vertical and horizontal loadings.

Comparative Studies between Frequency Domain Analysis and Time Domain Analysis on Free-Field One-Dimensional Shear Wave Propagation

In Korea, the underground silo structure for low- and intermediate-level radioactive waste disposal facilities has been constructed and operated since 2014. Large-scale earthquakes occurred in 2016 and 2017, respectively, in Gyeongju and Pohang areas near the underground silo structures, and interest in the stability of the underground silo increased significantly. In this paper, one-dimensional free-field analyses have been carried out before the three-dimensional silo dynamic analyses subjected to earthquake loadings. As an additional study, a new form of the finite element equilibrium equation is derived in terms of relative motions, which is essentially the same equation expressed in terms of total motions where the base shear force is applied to the earthquake load. The accuracy of conventional finite element solutions is evaluated by directly comparing them with closed-form solutions by frequency domain analysis such as SHAKE91.

An interval iterative method for response bounds analysis of structures with spatially uncertain parameters

Parameters with spatial uncertainties are very common in practical engineering, which could have a significant effect on structural performances. Different from the framework of probability theory, the interval field model describes the spatial uncertainty with upper and lower bounds rather than the probability distribution function or other statistical characteristics. By introducing the interval field model into finite element analysis and solving the resulting interval finite element equilibrium equation, the upper and lower bounds of structural responses can then be obtained. This paper proposes an interval iterative method for static displacement response analysis of structures with spatial uncertainties, effectively improving the overestimation existing in traditional interval analysis due to dependency problems and obtaining a compact interval envelope of the exact response bounds. Firstly, the spatially uncertain parameters described by the interval field are represented by the interval Karhunen–Loève (K-L) expansion, based on which the interval finite element equilibrium equation is formulated. Secondly, the interval finite element problem is decomposed into several subproblems, and an interval iterative method is developed for an envelope solution of the structural response bounds. Finally, the accuracy and efficiency of the proposed method are verified by several numerical examples.

A semi-analytical interval method for response bounds analysis of structures with spatially uncertain loads

This paper proposes a semi-analytical interval method for static response bounds analysis of structures subjected to spatially uncertain loads. In the investigated problem, the external loads applied to the structure are spatially uncertain but bounded, which are quantified by an interval field model using upper and lower bounds. By introducing the interval field and its series expansion form into finite element analysis, the interval finite element equilibrium equations are then formulated, where the force vector is an interval vector. Solving the interval equations by the proposed method, a semi-analytical solution of the upper and lower bounds of structural static responses such as displacement and stress can be obtained efficiently. Finally, the effectiveness and accuracy of the proposed method are verified by three numerical examples.

Integrated Digital Design and Manufacturing for 3D Printing Technologies

Abstract A process is proposed and outlined in which optimal structural design, design post-processing, manufacturing process planning, and subsequent manufacturing may be done algorithmically in a digital design and manufacturing platform. Customized algorithms are developed to define the structural design problem, mesh the design domain and solve the finite element equilibrium equations, perform robust structural topology optimization, optimal build orientation determination for additive manufacturing, support structure location requirements, adaptive slicing of the finite element mesh, and control of a custom made Digital Light Processing (DLP), stereolithography-based 3D Printer.

Performance analysis of geometrically nonlinear static deformations of laminated composite plates using PFRC/AFC materials

The present work focuses on geometrically nonlinear static analysis of laminated smart composite plates integrated with the patches of Active fiber composites (AFC) and piezoelectric fiber-reinforced composite (PFRC) materials acting as the distributed actuators has been compared using generalized energy based finite element models. To derive this coupled electromechanical finite element model the Von Karman type nonlinear strain displacement relations and a simple first-order shear deformation theory (FSDT) are used. Eight-node diso-parametric serendipity elements are used for discretization of the overall plate integrated with AFC/PFRC patch material .Picard’s direct iteration method with under relaxation is used to solve the nonlinear finite element equilibrium equations. The performance of both AFC and PFRC patches has been investigated and compared for symmetric cross-ply laminated composite plate substrate.

Static behavior of thermally loaded multilayered Magneto-Electro-Elastic beam

The present article examines the static response of multilayered magneto-electro-elastic (MEE) beam in thermal environment through finite element (FE) methods. On the basis of the minimum total potential energy principle and the coupled constitutive equations of MEE material, the FE equilibrium equations of cantilever MEE beam is derived. Maxwell\'s equations are considered to establish the relation between electric field and electric potential; magnetic field and magnetic potential. A simple condensation approach is employed to solve the global FE equilibrium equations. Further, numerical evaluations are made to examine the influence of different in-plane and through-thickness temperature distributions on the multiphysics response of MEE beam. A parametric study is performed to evaluate the effect of stacking sequence and different temperature profiles on the direct and derived quantities of MEE beam. It is believed that the results presented in this article serve as a benchmark for accurate design and analysis of the MEE smart structures in thermal applications.

BFree Finite Element Approach for Saturated Porous Media: Consolidation

TheBfree finite element approach is applied to the governing equations describing the consolidation process in saturated poroelastic medium with intrinsically incompressible solid and fluid phases. Under this approach, where Voigt notation is avoided, the finite element equilibrium equations and the linearization of the coupled governing equations are fully derived using tensor algebra. In order to assess theBfree approach for the consolidation equations, direct comparison with analytical solution of the response of a homogeneous and isotropic water-saturated poroelastic finite column under harmonic load is presented. The results illustrate the capability of this finite element approach of reproducing accurately the response of quasistatic phenomena in a saturated porous medium.

Uncertain temperature field prediction of heat conduction problem with fuzzy parameters

This paper proposes a first-order fuzzy perturbation finite element method (FFPFEM) and a modified fuzzy perturbation finite element method (MFPFEM) for the uncertain temperature field prediction with fuzzy parameters in material properties, external loads and boundary conditions. Fuzzy variables are used to represent the subjective uncertainties associated with the expert opinions. Based on the α-cut method, the finite element equilibrium equation is equivalently transformed into groups of interval equations. The interval matrices and vectors in both methods are expanded by the first-order Taylor series. First-order Neumann series is employed in FFPFEM for the inversion of interval matrix, while some high-order terms of the Neumann series are retained in MFPFEM. Based on the fuzzy decomposition theorem, the fuzzy solutions to the original uncertain heat conduction problem are eventually derived. By comparing the results with traditional Monte Carlo simulation and vertex method, two numerical examples evidence the feasibility and effectiveness of the proposed methods at predicting the fuzzy temperature field in engineering.

Effects of Nonsymmetrical Prestressing Forces on Torsion in Prestressed Thin-Walled Flexural Members

AbstractTorsional resistance of a single prestressing tendon was derived on the basis of the theorem of virtual work to control the torsional rotation by nonequal prestressing forces on both sides of the webs of a flexural member. The energy formulation revealed that the vertical component of the prestressing force has a significant role in resisting the torsional rotation, whereas the axial component is responsible for the warping-related stiffness. The most influential location of the prestressing tendon was the point where the value of the warping function is maximized. The resulting finite-element equilibrium equations yielded nodal forces at the degrees of freedom corresponding to torsion and warping. Through numerical applications, it was suggested that the conventional solution method of beam finite-element formulation on the basis of the transformed cross section can be improved by adopting the nodal forces derived in this paper. Further, it was shown that the torsional rotation can be controlled ...

Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity

We propose an a posteriori estimator of the error of hyper-reduced predictions for elastoviscoplastic problems. For a given fixed mesh, this error estimator aims to forecast the validity domain in the parameter space, of hyper-reduction approximations. This error estimator evaluates if the simulation outputs generated by the hyper-reduced model represent a convenient approximation of the outputs that the finite element simulation would have predicted. We do not account for the approximation error related to the finite element approximation upon which the hyper-reduction approximation is introduced. We restrict our attention to generalized standard materials. Upon use of incremental variational principles, we propose an error in constitutive relation. This error is split into three terms including a tailored norm of the hyper-reduction approximation error. This error norm is defined by using the convexity of an incremental potential introduced to state the constitutive equations. The second term of the a posteriori error is related to the stress recovery technique that generates stresses fulfilling the finite element equilibrium equations. The last term is a coupling term between the hyper-reduction approximation error at each time step and the errors committed before this time step. Unfortunately, this last term prevents error certification. In this paper, we restrict our attention to outputs extracted by a Lipschitz function of the displacements. In the proposed numerical examples, we show very good preliminary results in predicting the validity domain of hyper-reduction approximations. The average computational time of the predictions obtained by hyper reduction, is accelerated by a factor of 6 compared to that of finite element simulations. This speed-up incorporates the computational time devoted to the error estimation. The numerical implementation of the proposed error estimator is straightforward. It does not require the computation of the incremental potential. In the numerical results, the estimated validity domain of hyper-reduced approximations is inside the reference validity domain. This paper is a first attempt for a posteriori error estimation of hyper-reduction approximations.

Spatial Embedded Slip Model for Analyzing Time-Relative Coupling Effects of Creep and Prestress on PC Bridges

This paper presents, for analysis of prestressed concrete (PC) bridges, a spatial embedded slip model that is based on a three-dimensional isoparametric element, truss element, and nonthickness bond element, to simulate concrete, tendon, and the interface of the two, respectively. The bond element is embedded into the slip model via the virtual nodes introduced at the intersection points of tendon and concrete. Based on the displacement-based finite-element framework and the constitutive relations of each component, the elastic finite-element equilibrium equation for the slip model is deduced according to the virtual work principle. The quasilinear regression method is used to fit the creep coefficient in the Chinese bridge design code. Then, based on the elastic equilibrium equation and theoretical incremental formula of creep analysis, the equilibrium equation for analyzing time-relative coupling effects of creep and prestress is derived and the finite-element program is developed. The proposed model allows tendon to go through concrete in any pattern, without consideration for layout and direction of tendon and therefore with convenience in a finite-element mesh. The freedoms of the virtual nodes can be reduced in the formation of the equilibrium equation. Verified by a numerical example of a simply supported beam with a rectangular section, the embedded slip model is proved to support accurate linear and creep computation within an elastic range of materials.

B free

The linearized strain matrix and the matrix of material elasticities — commonly referred to as the B and D matrices — have been traditionally presented as convenient operators that enable the matrix formulation and finite element implementation of field theories in continuum mechanics. In this article it is argued that the opposite is true: neither matrix is necessary, they both complicate the finite element formulation of field theories and, moreover, they do not necessarily lead to the most efficient implementations. Examples are provided to back this statement in the context of infinitesimal and finite strain solid mechanics. For these problems, the finite element equilibrium equations and their linearization are fully derived using a formulation which is simpler, more compact, often computationally cheaper … and free of both B and D matrices.

Large deflection analysis of laminated composite plates using layerwise displacement model

In this paper the geometrically nonlinear continuum plate finite element model, hitherto not reported in the literature, is developed using the total Lagrange formulation. With the layerwise displacement field of Reddy, nonlinear Green-Lagrange small strain large displacements relations (in the von Karman sense) and linear elastic orthotropic material properties for each lamina, the 3D elasticity equations are reduced to 2D problem and the nonlinear equilibrium integral form is obtained. By performing the linearization on nonlinear integral form and then the discretization on linearized integral form, tangent stiffness matrix is obtained with less manipulation and in more consistent form, compared to the one obtained using laminated element approach. Symmetric tangent stiffness matrixes, together with internal force vector are then utilized in Newton Raphson\'s method for the numerical solution of nonlinear incremental finite element equilibrium equations. Despite of its complex layer dependent numerical nature, the present model has no shear locking problems, compared to ESL (Equivalent Single Layer) models, or aspect ratio problems, as the 3D finite element may have when analyzing thin plate behavior. The originally coded MATLAB computer program for the finite element solution is used to verify the accuracy of the numerical model, by calculating nonlinear response of plates with different mechanical properties, which are isotropic, orthotropic and anisotropic (cross ply and angle ply), different plate thickness, different boundary conditions and different load direction(unloading/loading). The obtained results are compared with available results from the literature and the linear solutions from the author\'s previous papers.

Active Control of Geometrically Non-linear Transient Response of Smart Laminated Composite Plate Integrated with AFC Actuator and PVDF Sensor

A coupled electromechanical finite element formulation for active control of geometrically non-linear transient response of laminated composite plate is studied. First-order shear deformation theory and von Karman type non-linear strain displacements are used. The plate is discritized using eight-noded quadratic isoparametric elements with five mechanical degrees of freedom and one electrical degree of freedom per node. Newton-Raphson iterative method in association with Newmark time integration method is used to solve the non-linear finite element equilibrium equation. Negative velocity feedback control algorithm is used to control the dynamic response of the smart laminated composite plate. AFC layer poled in fiber direction acting as distributed actuator and PVDF layer poled in thickness direction acting as sensor are considered. This study involves two types of actuator sensor arrangements: (1) the substrate is sandwiched between AFC actuator and PVDF sensor, called non-collocated arrangement and (2) AFC actuator and PVDF sensor are placed on top of the substrate, called collocated arrangement. The effect of piezoelectric fiber orientation in actuator layer on vibration control for both cross-ply and angle-ply laminates are examined.

A new multiscale formulation for the electromechanical behavior of nanomaterials

We present a new multiscale, finite deformation, electromechanical formulation to capture the response of surface-dominated nanomaterials to externally applied electric fields. To do so, we develop and discretize a total energy that combines both mechanical and electrostatic terms, where the mechanical potential energy is derived from any standard interatomic atomistic potential, and where the electrostatic potential energy is derived using a Gaussian-dipole approach. By utilizing Cauchy–Born kinematics, we derive both the bulk and surface electrostatic Piola–Kirchhoff stresses that are required to evaluate the resulting electromechanical finite element equilibrium equations, where the surface Piola–Kirchhoff stress enables us to capture the non-bulk electric field-driven polarization of atoms near the surfaces of nanomaterials. Because we minimize a total energy, the present formulation has distinct advantages as compared to previous approaches, where in particular, only one governing equation is required to be solved. This is in contrast to previous approaches which require either the staggered or monolithic solution of both the mechanical and electrostatic equations, along with coupling terms that link the two domains. The present approach thus leads to a significant reduction in computational expense both in terms of fewer equations to solve and also in eliminating the need to remesh either the mechanical or electrostatic domains due to being based on a total Lagrangian formulation. Though the approach can apply to three-dimensional cases, we concentrate in this paper on the one-dimensional case. We first derive the necessary formulas, then give numerical examples to validate the proposed approach in comparison to fully atomistic electromechanical calculations.

A topological optimization method for flexible multi-body dynamic system using epsilon algorithm

In a flexible multi-body dynamic system the typical topological optimization method for structures cannot be directly applied, as the stiffness varies with position. In this paper, the topological optimization of the flexible multi-body dynamic system is converted into structural optimization using the equivalent static load method. First, the actual boundary conditions of the control system and the approximate stiffness curve of the mechanism are obtained from a flexible multi-body dynamical simulation. Second, the finite element models are built using the absolute nodal coordination for different positions according to the stiffness curve. For efficiency, the static reanalysis method is utilized to solve these finite element equilibrium equations. Specifically, the finite element equilibrium equations of key points in the stiffness curve are fully solved as the initial solution, and the following equilibrium equations are solved using a reanalysis method with an error controlled epsilon algorithm. In order to identify the efficiency of the elements, a non-dimensional measurement is introduced. Finally, an improved evolutional structural optimization (ESO) method is used to solve the optimization problem. The presented method is applied to the optimal design of a die bonder. The numerical results show that the presented method is practical and efficient when optimizing the design of the mechanism.

Torsional Stiffness of Prestressing Tendons in Double-T Beams

When a prestressed double-T beam is subjected to torsion, a pair of prestressing tendons resists torsional rotation because of the restoring action of the displaced prestressing tendons. A comprehensive formulation to account for the torsional restoring action of double-T beams is presented, based on Vlasov’s hypothesis of considering warping displacement in an open-section. The deformation energies of prestressing tendons and reinforcing bars are calculated based on the deformed geometry to obtain the total potential energy. A two-noded beam element with seven degrees of freedom per node approximates an axial displacement, two translations, two flexural, and one torsional rotations, and a warping displacement to derive the finite-element equilibrium equations by minimizing the potential energy function. The role of prestressing forces of the tendons on the torsional resistance and the limitations of the traditional transformed section approach are addressed when it is applied to torsional problems. As a ...

Geometrically Nonlinear Finite Element Analysis of Composite Plates Integrated with the Patches of Piezoelectric Fiber-reinforced Composite

Geometrically nonlinear static analysis of laminated smart composite plates integrated with the patches of piezoelectric fiber-reinforced composite (PFRC) material acting as the distributed actuators has been carried out by a generalized energy based finite element model. The Von Karman type nonlinear strain displacement relations and a simple first-order shear deformation theory are used for deriving this coupled electromechanical finite element model. Eight-noded isoparametric serendipity elements are used for discretizing the overall plates. The nonlinear finite element equilibrium equations are solved by Picard's direct iteration method under relaxation. The performance of the PFRC patches has been investigated for both symmetric and antisymmetric cross-ply and antisymmetric angle-ply laminated composite plate substrates with different boundary conditions. Particular emphasis has been placed on investigating the effect of variation of piezoelectric fiber orientation on the actuating capability of the PFRC patches for counteracting the nonlinear deformations of the smart composite plates. The results suggest the potential use of the patches of PFRC material for distributed control of nonlinear deformations of smart composite structures.