A method for optimally designing laminated plates subject to fatigue loads for minimum weight using a cumulative damage constraint

The purpose of this paper is to study shear locking-free parametric earthquake analysis of thick and thin plates resting on Winkler foundation using Mindlin’s theory, to determine the effects of the thickness/span ratio, the aspect ratio and the boundary conditions on the linear responses of thick and thin plates subjected to earthquake excitations. In the analysis, finite element method is used for spatial integration and the Newmark-β method is used for the time integration. Finite element formulation of the equations of the thick plate theory is derived by using higher order displacement shape functions. A computer program using finite element method is coded in C++ to analyze the plates clamped or simply supported along all four edges. In the analysis, 8-noded finite element is used. Graphs are presented that should help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that 8-noded finite element can be effectively used in the earthquake analysis of thick plates. It is also concluded that, in general, the changes in the thickness/span ratio are more effective on the maximum responses considered in this study than the changes in the aspect ratio.

A methodology to design symmetrically laminated fibre-reinforced structures under transverse loads for minimum weight, with manufacturing uncertainty in the ply angle, is described. The ply angle and the ply thickness are the design variables, and the Tsai-Wu failure criteria is the design constraint implemented. It is assumed that the probability of any tolerance value occurring within the tolerance band, compared with any other, is equal, and thus the approach is a worst-case scenario approach. The finite element method, based on Mindlin plate and shell theory, is implemented, and thus effects like bending-twisting coupling are accounted for. The Golden Section method is used as the search algorithm, but the methodology is flexible enough to allow any appropriate finite element formulation, search algorithm and failure criterion to be substituted. In order to demonstrate the procedure, laminated plates with varying aspect ratios and boundary conditions are optimally designed and compared.

Optimization of FRP composites against impact induced failure using island model parallel genetic algorithm

Automated Fiber Placement (AFP) machines can manufacture composite panels with curvilinear fibers. In this article, the critical buckling load of grid-stiffened curvilinear fiber composite panels is maximized using a genetic algorithm. The skin is composed of layers in which the fiber orientation varies along one spatial direction. The design variables are the fiber orientation of the panel for each layer and the stiffener layout. Manufacturing constraints in terms of maximum curvature allowable by the AFP machine are imposed for both skin and stiffener fibers. The effect of manufacturing-induced gaps in the laminates is also incorporated. The finite element method is used to perform the buckling analyses. The panels are subjected to in-plane compressive and shear loads under several boundary conditions. Optimization results show that the percentage difference in the buckling load between curvilinear and straight fiber panels depends on the load case and boundary conditions.

The sandwich composites fuselages appear to be a promising choice for the future aircrafts because of their structural efficiency and functional integration advantages. However, the design of sandwich composites is more complex than other structures because of many involved variables. In this paper, the fuselage is designed as a sandwich composites cylinder, and its structural optimization using the finite element method (FEM) is outlined to obtain the minimum weight. The constraints include structural stability and the composites failure criteria. In order to get a verification baseline for the FEM analysis, the stability of sandwich structures is studied and the optimal design is performed based on the analytical formulae. Then, the predicted buckling loads and the optimization results obtained from a FEM model are compared with that from the analytical formulas, and a good agreement is achieved. A detailed parametric optimal design for the sandwich composites cylinder is conducted. The optimization method used here includes two steps: the minimization of the layer thickness followed by tailoring of the fiber orientation. The factors comprise layer number, fiber orientation, core thickness, frame dimension and spacing. Results show that the two-step optimization is an effective method for the sandwich composites and the foam sandwich cylinder with core thickness of 5 mm and frame pitch of 0.5 m exhibits the minimum weight.

An exact three dimensional solution for bending of thick rectangular FGM plate

The paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior. The nonlinear critical load is determined by finding the load level at which the Hessian of the potential energy ceases to be positive definite. A recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm. Sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables. NUMBER of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements, element stresses, system stability, etc.1'2 The optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables. The optimality criterion was derived by, differentiating the Lagrangian with respect to the design variables. The displacements were assumed to be small and, in the finite element analysis, linear equilibrium equations were solved to determine the response of the structure to the ap- plied loads. In the case of system stability the constraints were defined (see Refs. 3-6) by the associated linear eigenvalue problem. This definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections. The correct procedure for these structures would be to analyze the structure* by using nonlinear equilibrium equations. This will be particularly true for large space structures (LSS). In this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior. In the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode. This design tends to become im- perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub- stantially. The imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal (see Ref. 7). A real structure has geometric im- perfections and these structures tend to have nonlinear

A nine-node isoparametric plate element in conjunction with first-order shear deformation theory is used for free vibration analysis of rectangular plates with central cutouts. Both thick and thin plate problems are solved for various aspect ratios and boundary conditions. In this article, primary focus is given to the effect of rotary inertia on natural frequencies of perforated rectangular plates. It is found that rotary inertia has significant effect on thick plates, while for thin plates the rotary inertia term can be ignored. It is seen that the numerical convergence is very rapid and based on comparison with experimental and analytical data from literature, it is proposed that the present formulation is capable of yielding highly accurate results. Finally, some new numerical solutions are provided here, which may serve as benchmark for future research on similar problems.

Failure analysis of a laminated composite beam subjected to uniformly distributed load and thermal load is studied for different boundary conditions and fiber orientation angles, based on first ply failure load. Three different boundary conditions are studied: simply supported, fixed-fixed and fixed-free. The strength ratio is computed and compared for different failure theories. The effect of fiber orientation angle and aspect ratio on the strength ratio based on first ply failure load is presented in the paper. The strength ratio and transverse deflection are determined for Graphite/Epoxy and Glass/Epoxy composite and their hybrid combinations to find out the optimum hybrid composite beam with minimum weight, deflection and cost. The problem is solved in MATLAB platform. The mode of failure of the composite beam is determined by using maximum stress theory.

The mid-frequency range finite element (MFR-FE) method was developed by Christian Soize in the early 1980s, and it is well described in Soize’s book with Roger Ohayon [Structural Acoustics and Vibration, (Academic, New York, 1998), Chap. 7]. The method combines finite element equations and the frequency translation property of the Fourier transform to solve acoustics and vibration problems in the mid-frequency range, at frequencies higher than possible with conventional finite element codes. In an effort to understand the method fully, an implementation of MFR-FE was developed in MATLAB [Mathworks Inc., Natick, MA], a natural environment for prototyping algorithms involving matrices and signal processing. The FE formulation was built upon one already prepared by Kwon and Bang [The Finite Element Method Using MATLAB, (CRC, Boco Raton, FL, 1997)], and initial results were recently described [Paper AIAA–99-1855, 5th AIAA/CEAS Aeroacoustics Conference]. The present talk will provide an overview of MFR-FEs and give example results, extending those previously shown. One advantage of the method is that it can be used with other improvements in FE technology, such as hp adaptive formulations. [Work sponsored by NASA Langley Research Center Structural Acoustics Branch.]

Optimal design of clamped laminated composite plates

SummaryIn composite plate shear walls‐concrete encased (C‐PSW/CE), the headed studs are used to anchor concrete panels on both sides of the infill steel plate and make them resist lateral load together. The finite element models of C‐PSW/CE are developed and validated by testing three C‐PSW/CE specimens under cyclic loading. The responses of the components in C‐PSW/CE under cyclic loading are simulated by finite element method. The effects of the number of headed studs, stud diameter, concrete panel thickness, infill steel plate thickness, and wall aspect ratio on the shear force of headed studs are studied. The equation is proposed to evaluate the shear force demand of headed studs in C‐PSW/CE under cyclic loading using the numerical results. The shear force demand of headed studs increases with decreasing the number of headed studs and wall aspect ratio, increasing the stud diameter, and increasing the thicknesses of concrete panel and infill steel plate. Compared with the shear force demand of headed studs in C‐PSW/CE under monotonic loading, it is recommended that the shear force demand of headed studs in design of C‐PSW/CE in high seismic regions adopts the larger of shear force demands of headed studs under monotonic and cyclic loadings.

Experimental and numerical analysis of plates quasi-statically loaded by a rectangular indenter

Numerous engineering applications exist for the piezoelectric effect, which results from the electromechanical coupling between electrical and mechanical fields. In-plane vibrations of piezoelectric plates’ resonance frequencies and associated mode shapes have been thoroughly investigated. However, analytical solutions for in-plane-dominated vibrations of thick piezoelectric circular plates are limited. In this paper, higher-order plate theories for the in-plane-dominated vibration characteristics of piezoelectric circular thick plates under fully clamped and completely free boundary conditions are presented. The resonant frequencies and associated mode shapes were investigated based on two higher-order plate theories: second-order shear deformation plate theory and third-order shear deformation plate theory, as well as simplified third-order linear piezoelectric theory. Hamilton's principle was applied to derive equations of motion and boundary conditions. In the theoretical analysis, the resonant frequencies, associated mode shapes and distribution of electric displacements for various radius-to-thickness ratios were calculated. The numerical results obtained by the finite element method were compared with those obtained from theoretical analysis. Excellent agreement was found between the theoretical and numerical results for the thick piezoelectric circular plates.

In this paper an examination of methods for vibration analysis of moderately thick rectangular plates has been presented. First, the state-of-the art in thick plate vibration theories and analysis methods is described and it is followed with basic equations of the original Mindlin theory, which represents a starting point for the development of all other mathematical models. Then, the problem of analytical solving of equilibrium equations is considered based on the modified Mindlin theory of thick plate vibrations, which has been published in the literature recently. Further, energy methods that can be applied to arbitrary boundary conditions are discussed and outline of the assumed mode method is presented. Finally, in the context of numerical methods a new quadrilateral finite element, based on the above mentioned advanced thick plate theory, is included. It should be emphasized that it doesn’t suffers of shear locking problem associated with finite elements, due to natural relation among bending and shear polynomials, and moreover, it gives very accurate results. Application of the presented methods is illustrated by several numerical examples which include natural vibration analyses of rectangular plates with various thicknesses and different combinations of boundary conditions (simply supported, clamped, free and elastically restrained). Comparisons of natural frequencies and mode shapes with results available in the relevant literature and with those obtained by the commercial finite element software are also provided.