Nonlinear Finite Element Solution Research Articles

Development of Design Analysis Methods for Carbon Silicon Carbide Composite Structures

The stress—strain behavior at room temperature and at 1100° C (2000°F) is measured for two carbon fiber-reinforced silicon carbide (C/SiC) composite materials: a two dimensional (2D) plain-weave quasi-isotropic laminate and a 3D angle interlock woven composite. Previously developed micromechanics-based material models are calibrated by correlating the predicted material property values with the measured values. Four-point beam-bending subelement specimens are fabricated with these two fiber architectures and four-point bending tests are performed at room temperature and at 1100°C. Displacements and strains are measured at the mid-span of the beam and recorded as a function of load magnitude. The calibrated material models are used in concert with a nonlinear finite-element solution using ABAQUS to simulate the structural response of the two materials in the four-point beam bending tests. The structural response predicted by the nonlinear analysis method compared favorably with the measured response for both materials and both test temperatures. Results show that the material models scale-up fairly well from coupons to subcomponent level.

Nonlinear analysis of moderately thick reinforced concrete slabs at elevated temperatures using a rectangular layered plate element with Timoshenko beam functions

A simple displacement-based 4-node, 24-DOF rectangular layered plate element for nonlinear finite element analysis of thin-to-moderately-thick reinforced concrete slabs subjected to combined external and thermal loading is developed in this paper. Timoshenko’s composite beam functions are extended to form the basis of the bending strain of the layered plate element developed herein, so that the proposed model and approach not only provide a unified formulation for thin and moderately thick plates, but also avoid shear locking naturally. In addition to temperature-dependent material nonlinearity, geometric nonlinearity is accounted for to include the tension membrane effect, which has been identified as the major contributor to the load carrying capacity of RC slabs at elevated temperatures. Based on the proposed element model, the updated Lagrangian approach is employed to formulate the nonlinear finite element analysis and solution procedures for the nonlinear analysis of reinforced concrete slabs under constant external loading and thermal loading which are proposed in this paper. Comparison of the results from the proposed model and previous literatures for the numerical analysis of reinforced concrete slabs at ambient and elevated temperatures have demonstrated the effectiveness of the proposed model and solution procedures.

Cantilevered Plate Rayleigh-Ritz Trial Function Selection for von Kármán's Plate Equations

The accuracy and convergence characteristics of the classical Rayleigh-Ritz solution of the nonlinear von Karman plate equations are studied with respect to the types of cantilever plate in-plane trial functions used in the solution. The static deflection of the cantilever plate is computed for an applied static gravity loading. Four different in-plane trial function types are studied. In each of these cases the same out-of-plane trial functions are used. It is found that in two of the four cases good convergence and accuracy are achieved when compared to the solution from a nonlinear finite element model. The degree of satisfaction of the problem natural boundary conditions is also examined, and it is shown that for the two cases that show inadequate convergence characteristics this satisfaction is poor. It is noted in particular that for these two cases at points on the problems' free boundaries, the in-plane trial functions satisfy, either exactly or approximately, the linear in-plane natural boundary conditions. At these points the addition of inplane degrees of freedom to the solution will not contribute to the satisfaction of the nonlinear natural boundary condition. Thus for these two cases the convergence is poor as more in-plane trial functions are added to the solution. The change in the statically loaded plate natural frequencies are also computed. Similar to the static deflection results, convergence to the nonlinear finite element solution is slow when in-plane trial functions are used that approximately satisfy the linear in-plane boundary conditions.

Degradation investigation in a postbuckling composite stiffened fuselage panel

COCOMAT is a four-year project under the European Commission 6th Framework Programme that aims to exploit the large strength reserves of composite structures through a more accurate prediction of collapse. Accordingly, one of the COCOMAT work packages involves the design of test panels with a focus on investigating the progression of composite damage mechanisms. This paper presents the collaborative results of some of the partners for this task. Different design alternatives were investigated for fuselage-representative test panels. Non-linear structural analyses were performed using MSC.Nastran and ABAQUS/Standard. Numerical predictions were also made applying a stress-based adhesive degradation model, previously implemented into a material user subroutine for ABAQUS/Standard. Following this, a fracture mechanics analysis using MSC.Nastran was performed along all interfaces between the skin and stiffeners, to examine the stiffener disbonding behaviour of each design. On the basis of the structural and fracture mechanics analyses, a design was selected as being the most suitable for the experimental investigation within COCOMAT. Though the COCOMAT panels have yet to be manufactured and tested, experimental data on the structural performance and damage mechanisms were available from a separate project for a panel identical to the selected design. This data was compared to the structural, degradation and fracture mechanics predictions made using non-linear finite element solutions, and the application of the design within the COCOMAT project was discussed.

A semi-analytical approach for two-dimensional rolling/sliding contact with applications to shakedown analysis

Knowledge of residual stresses is crucial in rolling contact fatigue, particularly when high cycle fatigue conditions must be evaluated. Furthermore, knowledge of residual stresses is a key factor in order to correctly understand the shakedown phenomenon in metallic materials. In this paper, an efficient algorithm to compute the residual stresses caused by two-dimensional rolling and sliding contact is proposed and compared with previous semi-analytical approaches. Plastic accumulation in the material can also be computed with this algorithm. The model takes into account full non-linear kinematical hardening, and is stable for relatively high values of plastic loading, showing a very good agreement with non-linear finite element solutions. However, the model has an important advantage when comparing with finite element analyse. In fact, the model is much more efficient than finite element analysis regarding computational time. Lastly, the model is applied to obtain the two-dimensional shakedown maps, for both pure rolling and rolling/sliding conditions.

An automated hybrid procedure for capturing mode-jumping in postbuckling composite stiffened structures

This paper presents a robust finite element procedure for modelling the behaviour of postbuckling structures undergoing mode-jumping. Current non-linear implicit finite element solution schemes, found in most finite element codes, are discussed and their shortcomings highlighted. A more effective strategy is presented which combines a quasi-static and a pseudo-transient routine for modelling this behaviour. The switching between these two schemes is fully automated and therefore eliminates the need for user intervention during the solution process. The quasi-static response is modelled using the arc-length constraint while the pseudo-transient routine uses a modified explicit dynamic routine, which is more computationally efficient than standard implicit and explicit dynamic schemes. The strategies for switching between the quasi-static and pseudo-transient routines are presented.

Design and analysis of a composite beam for infrastructure applications - Part I: Preliminary investigation in bending

This study aims to contribute to the development of a composite beam for use in civil engineering systems. Based on the limitations in existing concepts, a new beam design is proposed and its behaviour studied. Using the classical beam theory, the Timoshenko beam theory, the Timoshenko plate theory, as well as the transformed section approach, borrowed from reinforced concrete, a simplified analytical approach, which could be used in design, is developed to conduct first and second order analysis of the proposed beam in order to achieve a rational sizing of its section before a rigorous testing regime is carried out. Finally, to validate the analytical model and gain confidence in the design, the analytical and experimental results are compared to a rigorous nonlinear finite element solution. The analytical model agreed relatively well with the experiments and the FE analyses, giving confidence in the validity of the underlying assumptions.

Buckling analysis of tri-axial woven fabric composite structures subjected to bi-axial loading

The buckling behavior of several simple tri-axial woven composite structures is studied. These structures include basic tri-axial structure, modified tri-axial structure and enlarged tri-axial structure. The structure is subjected to bi-directional loading. Approximate analytical method using equivalent anisotropic plate theory is presented in this paper. Its confirmation is obtained by non-linear finite element solution. Approximate analytical method provides a simpler way to obtain the buckling load, while the non-linear finite element method gives the numerical results and detailed analysis of buckling behavior of the tri-axial structures. The non-linear finite element analysis results reveal that the structures investigated in the paper approximately have the same buckling behavior, but with different values of buckling load. When more X-crossovers are involved in the tri-axial structures, their buckling loads increase and tend to approach a certain value. The results show that techniques used for approximate analytical solution can be used to obtain the buckling load of a real life tri-axial structure. The sensitivity of the buckling behavior of the basic tri-axial structure to the change in the boundary conditions and to the imperfection due to initial configuration is investigated.

Analytical model for analysis and design of V-shaped thermal microactuators

An analytical solution of the thermoelastic bending/buckling problem of thermal microactuators is presented. V-shaped beam actuators are modeled using the theory of beam-column buckling. Axial (longitudinal) deformations including first-order nonlinear strain-displacement relations and thermal strains are included. The resulting nonlinear transcendental equations for the reaction forces are solved numerically and the solutions are compared with a nonlinear finite element (FE) model. A test actuator has also been fabricated and characterized. The obtained accuracy of the prediction is within 1.1% of the nonlinear FE solution and agrees well with the experimental data. A corresponding one-dimensional (1-D) heat transfer model has also been developed and validated against experimental i-V measurements at various temperatures. The developed analytical models are then used to analyze maximum stress and the heat transfer paths. It has been confirmed that the heat flux toward the substrate is a dominant heat dissipation route in sacrificially released devices.

A novel sensorless control strategy of doubly fed induction motor and its examination with the physical modeling of machines

Available sensorless speed control strategy of doubly fed induction machines based on rotor flux linkage fails around the synchronous speed due to the inaccurate estimation of the rotor flux linkage. In order to solve this problem, an excitation-current-based speed/position estimation strategy is proposed. The corresponding speed regulation system is constructed. Using the nonlinear transient finite element solutions of the doubly fed induction machine, a physical phase variable motor model is developed. The speed regulation system is verified using the proposed machine model. Simulation results show that the novel sensorless speed control strategy works properly in the whole speed zone and provides good speed regulation performance.

Localized Buckling of a Bilinear Elastic Ring under External Pressure

A fully nonlinear finite element analysis for prediction of localization in moderately thick imperfect rings under applied hydrostatic pressure is presented. The present nonlinear finite element solution methodology includes all the nonlinear terms in the kinematic equations and utilizes the total Lagrangian formulation in the constitutive equations and incremental equilibrium equations. A curved six-node element, based on an assumed quadratic displacement field (in the circumferential coordinate), employs a two-dimensional hypothesis, known as linear displacement distribution through thickness theory, to capture the effect of the transverse shear/normal (especially, shear) deformation behavior. The driving factor behind this analysis is to determine the onset of localization arising out of the bilinear material behavior of the ring with modal imperfection. Numerical results suggest that material bilinearity is primarily responsible for the appearance of a limit or localization (peak pressure) point on the postbuckling equilibrium path of an imperfect ring.

Topology optimization design of crushed 2D-frames for desired energy absorption history

The present work deals with topology optimization for obtaining a desired energy absorption history of a crushed structure. The optimized energy absorbing structures are used to improve the crashworthiness of transportation vehicles. The ground structure consists of rectangular 2D-beam elements with plastic hinges. The elements can undergo large rotations, so the analysis accommodates geometric nonlinearities. A quasi-static nonlinear finite element solution is obtained with an implicit backward Euler algorithm, and the analytical sensitivities are computed by the direct differentiation method.

The finite element method for thin-walled members—basic principles

The application of the finite element method to thin-walled structures often requires non-linear analysis. Whereas in linear finite element analyses errors are easily made, this is even more so in the non-linear analyses. This paper focuses on possible sources of error in linear and non-linear finite element solutions, and gives suggestions how to check and prevent these errors.

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part III: estimation of crack opening area

Evaluation of the crack opening area (COA) plays a central role in the evaluation of the critical crack length for a detectable leak for piping systems. Simplified evaluation methods for the COA for a circumferential through-wall crack in a pipe subjected to axial and bending loading or their combination is reviewed in this paper. Elastic solutions are compared and recommendations are given. Plastic solutions by the reference stress method are compared with nonlinear finite element solutions. The reference stress method tends to overestimate the COA for medium or large crack angles. Considerable improvement is achieved by making empirical modifications to the limit load expressions used in the calculation of the reference stress.

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part II: J-integral estimation

Evaluation of the J-integral plays a central part in evaluation of the critical crack length for unstable fracture for piping systems. Simplified evaluation methods for the J-integral for a circumferential through-wall crack in pipes subjected to axial and bending loading or their combination is reviewed in this paper. Use of the LBB.ENG2 method and a similar approach based on the η-factor concept were found to result in significant underestimation of the J-integral for small and medium crack angles. On the other hand, the reference stress method based on the solutions for stress intensity factor and limit load recommended in the companion paper (Part I) provides solutions which agree well with the available non-linear finite-element solutions and can be utilized as a powerful tool for J-integral evaluation for arbitrary materials, not restricted to simple power-law hardening.

Large deflection orthotropic plate approach to develop ultimate strength formulations for stiffened panels under combined biaxial compression/tension and lateral pressure

This paper uses the large deflection orthotropic plate approach to develop the ultimate strength formulations for steel stiffened panels under combined biaxial compression/tension and lateral pressure loads, considering the overall (grillage) buckling collapse mode. The object panel has a number of one-sided small stiffeners in either one or both orthogonal directions. The stiffened panel is then modeled as an equivalent orthotropic plate, for which the various elastic constants characterizing structural orthotropy are determined in a consistent systematic manner using classical theory of elasticity. The panel edges are considered to be simply supported. The influence of initial deflections is taken into account. The membrane stress distribution inside the panel under combined uniaxial loading (in either longitudinal or transverse direction) and lateral pressure is analyzed by solving the nonlinear governing differential equations of large deflection orthotropic plate theory. It is presumed that the panel collapses when the most highly stressed boundary location yields, resulting in closed-form expressions for the ultimate strength of the stiffened panel. Based on the insights previously developed through numerical studies, the panel ultimate strength interaction formulation between biaxial loads, with lateral pressure regarded as a secondary load component is then proposed as a relevant combination of the two sets of panel ultimate strength formulations, i.e. one for combined longitudinal axial load and lateral pressure and the other for combined transverse axial load and lateral pressure. The validity of the proposed ultimate strength formulations is verified by a comparison with nonlinear finite element and other numerical solutions.

The buckling of plastic oblate hemi-ellipsoidal dome shells under external hydrostatic pressure

The paper presents a theoretical and an experimental investigation into the buckling of seven oblate hemi-ellipsoidal dome shells under external hydrostatic pressure. Four of the shells were made in glass reinforced plastic and three were made from a thermosetting plastic called solid urethane plastic. All the vessels were tested to destruction. The theoretical study was made with the aid of a non-linear finite element solution, where both geometrical and material non-linearity were allowed for. Good agreement was found between experiment and theory for all the vessels. The very oblate domes failed axisymmetrically. Theoretical convergence was good for the more oblate domes but it was not as good as for the less oblate domes. This may have been because the less oblate domes did not fail in a classical axisymmetric manner as was expected. This work is of much importance in ocean engineering.

Lateral Buckling Behavior of Pneumatically Stiffened, Reinforced Composite Beams in Bending

The use of inflatable structures has often been proposed for aerospace applications. The advantages of such structures include low weight, easy assembly, and storage. The aim of this study is to investigate the effects of finite inflation on the subsequent buckling response of a carbon-fiber-reinforced composite beam (RCB) in bending. A finite element solution for three incremental beam configurations is developed. The first lateral buckling mode is used to determine the structural efficacy of the inflated beam. For each beam, the bending moment required to induce lateral buckling was determined. A nonlinear material and geometrically nonlinear finite element solution is obtained for the inflation load. A Rik method solution was then obtained to examine the postbuckling response of the RCBs. In addition, an eigenvalue extraction was performed to study the intrinsic influence of air pressure on the structural stability of RCBs. Finally, experimental data were obtained to verify the theoretical results. It is found that a simple eigenvalue extraction provides a conservative estimate of the critical buckling moment for preliminary design purposes. Furthermore, there is good correlation between theory and experiment and there exists noticeable interaction between the beam and pressurized air prior to final buckling for this type of reinforced composite beam.

Analysis and design of down-aisle pallet rack structures

This paper presents an efficient approach to the analysis and design of unbraced pallet rack structures subjected to horizontal and vertical loads. The structures are analysed by considering an equivalent free-sway column and solving the differential equations of flexure, including the P– Δ effect. Initial imperfections within the frame are allowed. Results of the analysis are compared with a traditional non-linear finite element solution of the same problem. The analysis algorithm is then incorporated into a design program, which performs code checks against global buckling, deformation and internal member forces.

An anisotropic elastoplastic constitutive model for large strain analysis of fiber reinforced composite materials

In this work a generalized anisotropic elastoplastic constitutive model for the large strain analysis of fiber-reinforced composite materials in the frame of the mixing theory and the finite element method is presented. The isotropic equivalent formulation proposed assumes the existence of a fictitious isotropic space where a mapped fictitious problem is solved. Both real anisotropic and fictitious spaces are related by means of linear fourth-order transformation tensors that contain the complete information about the real anisotropic material. Details of the numerical implementation of the model into a non-linear or large strain finite element solution scheme are provided. Application examples showing the performance of the model for analysis of fiber reinforced composite materials are given.