Bending features of a sandwich panel with asymmetric structure under local loads

The bending characteristics of a composite panel with asymmetric layered structure under local surface loads are obtained. A refined version of the applied theory is developed using the analytical solution of the bending problem of a sandwich plate with arbitrary asymmetric structure under a point load. Local effects are investigated within the limits of a discrete model allowing for the specific character of elastic properties of a soft filler. The advantages of the solution are expressions of bending characteristics — layer curvatures, displacements, and stresses — in a closed form. It is shown that these characteristics can vary several times depending on the asymmetry parameters of the structure. Degeneration peculiarities of the solution, stemming from the slipping of layers or, otherwise, their rigid linking by the Kirchoff—Love hypothesis, as well as from account of the transverse shear and compression of the normal, are examined in line with the degeneration of geometric and physical parameters of the discrete model adopted. The results obtained are illustrated by curves and surfaces for the characteristics studied.Submitted for the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Institute of Polymer Mechanics, Latvian University, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 717–742, November–December, 1999.     

A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Dynamic programming in minimum-weight design of axisymmetric plates

The paper deals with minimum-weight design of axisymmetric plates. The design is subject to restrictions imposed by static equilibrium conditions and various yield criteria governing the failure of the material. Herein, the optimization problem is formulated as a dynamic programming problem and is solved. Such a formulation unifies the design procedure. The paper also discusses the dynamic programming approach.     

A system of steel-elastomer sandwich plates for strengthening orthotropic bridge decks

The use of a sandwich plate system (SPS) composed of two steel plates with a solid polymer (polyurethane) core has been introduced as a refurbishment procedure for steel decks of bridges, the so-called orthotropic decks consisting of a deckplate with longitudinal stiffeners and transverse crossbeams. Unfortunately, a great many of existing steel bridges still have structural members that do not comply with the recommendations given in design codes, and therefore damages have developed in them. For a satisfactory refurbishment of the bridges, the SPS technique fulfils all necessary requirements. To this end, both experimental and calculative investigations were carried out at RWTH Aachen to demonstrate the reinforcing and stiffening effect and to prove the suitability of the SPS-overlay technique for general use. The practical applicability of a SPS has been tested successfully in a pilot project for a German motorway bridge under severe traffic. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 271–282, March–April, 2007.     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

Bimodal optimal design of vibrating plates using theory and methods of nondifferentiable optimization

This paper is concerned with an optimal design problem of vibrating plates. The optimization problem consists in maximizing the smallest eigenvalue of the elliptic eigenvalue problem describing the free plate vibration. The thickness of the plate is the variable subject to optimization. The volume of the plate is constant and the thickness of the plate is bounded.In this paper, we consider the case where the smallest eigenvalue is multiple. This implies that the optimization problem is nondifferentiable. A necessary optimality condition is formulated. The finite-element method is employed as an approximation method. A nonsmooth optimization method is used to solve this optimization problem. Numerical examples are provided.This work was supported by the Polish Academy of Sciences and the Education Ministry of Japan. Lemarechal's implementation of his method was used for numerical computations.on leave from Systems Research Institute, Warsaw, Poland.     

A continuous model for investigation of physically nonlinear deformation of orthotropic sandwich plates

A phenomenological yield condition for quasi-brittle and plastic orthotropic materials with initial stresses is suggested. All components of the yield tensor are determined from experiments on uniaxial loading. The reliability estimates of the criterion suggested is discussed. For a plastic material without initial stresses, the given condition transforms into the Marin—Hu criterion. The defining equations of the deformation theory of plasticity with isotropic and “anisotropic” hardening, associated with the yield condition suggested, are obtained. These equations are used as the basis for a highly accurate nonclassical continuous model for nonlinear deformation of thick sandwich plates. The approximations with respect to the transverse coordinate take into account the flexural and nonflexural deformations in transverse shear and compression. The high-order approximations allow us to model the occurrence of layer delamination cracks by introducing thin nonrigid interlayers without violating the continuity concept of the theory. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. pp. 329–340, May–June, 2000.     

Bilateral estimates for solutions of boundary value problems in Kirchhoff plates

Dirichlet boundary value problems are studied for thin elastic plates on an elastic foundation within Kirchhoff's classical model. The aim is to construct dual problems that make it possible to obtain bilateral error estimates for approximate solutions. In the absence of an elastic foundation, the dual functionals are maximized on function sets whose elements satisfy certain differential restrictions. The theory is illustrated by means of a numerical example.     

On the optimal design of plates for a given deflection

The optimum design problem of an elastic plate for a given deflection is considered. The design variable is chosen to be the thickness of the plate. Using the principle of stationary mutual potential energy first introduced by Shield and Prager, a sufficient optimality condition (which makes the volume stationary) is derived for plates under bending caused by general loading conditions. As an example, the optimum profile of a simply supported circular plate under a given rotationally symmetric loading is obtained.     

A simple four-unknown refined theory for bending analysis of functionally graded plates

In the present paper, a refined trigonometric higher-order plate theory is simply derived, which satisfies the free surface conditions. Moreover, the number of unknowns of this theory is the least one comparing with other shear theories. The effects of transverse shear strains as well as the transverse normal strain are taken into account. The number of unknown functions involved in the present theory is only four as against six or more in case of other shear and normal deformation theories. The bending response of FG rectangular plates is presented. A comparison with the corresponding results is made to check the accuracy and efficiency of the present theory. Additional results for all displacements and stresses are investigated through-the-thickness of the FG rectangular plate.     

The traction initial-boundary value problem for bending of thermoelastic plates with cracks

The initial-boundary value problem for bending of a thermoelastic plate weakened by a crack, with Neumann-type boundary conditions along the edges of the crack, is studied, and its unique solvability in spaces of distributions is proved by means of a combination of the Laplace transformation and variational methods.     

Design of induction motors using a mixed-variable approach

In this paper we are concerned with the problem of optimally designing three-phase induction motors. This problem can be formulated as a mixed variable programming problem. Two different solution strategies have been used to solve this problem. The first one consists in solving the continuous nonlinear optimization problem obtained by suitably relaxing the discrete variables. On the opposite, the second strategy tries to manage directly the discrete variables by alternating a continuous search phase and a discrete search phase. The comparison between the numerical results obtained with the above two strategies clearly shows the fruitfulness of taking directly into account the presence of both continuous and discrete variables.This work was supported by CNR/MIUR Research Program “Metodi e sistemi di supporto alle decisioni”, Rome, Italy.     

Regularization of optimal design problems for bars and plates,part 1

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control variables. A linear functional of the solutions to the equilibrium boundary-value problem is minimized under additional restrictions upon the control variables.Special variations of the control within a narrow strip provide a necessary condition for a strong local minimum (Weierstrass test). This necessary condition can be used for the detailed analysis of the following problems: bar of extremal torsional rigidity; optimal distribution of isotropic material with variable shear modulus within a plate; and optimal orientation of principal axes of elasticity in an orthotropic plate. For all of these cases, the stationary solutions violate the Weierstrass test and therefore are not optimal. This is because, in the course of the approximation of the optimal solution, the curve dividing zones occupied by materials with different rigidities displays rapid oscillations sweeping over a two-dimensional region. Within this region, the material behaves as a composite medium assembled of materials of the initial class.     

Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms

A variant of the refined geometric nonlinear theory is suggested for nonshallow shells with a transversely soft core of medium thickness with regard to modifications of metric characteristics across the core thickness. The kinematic relations for the core are derived by sequential integration of the initial three-dimensional equations of elasticity theory along the transverse coordinate. The equations are preliminarily simplified by the assumption that the tangential stress components are equal to zero. With the example of sandwich plates, it is shown that these equations allow us to investigate synphasic, antiphasic, mixed flexural, and mixed flexural-shear buckling forms of load-bearing layers and the core depending on the precritical stress-strain state. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 95–108, January–February, 2000.     

An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates

This paper presents an alternative alpha finite element method using triangular meshes (AαFEM) for static, free vibration and buckling analyses of laminated composite plates. In the AαFEM, an assumed strain field is carefully constructed by combining compatible strains and additional strains with an adjustable parameter α which can produce an effectively softer stiffness formulation compared to the linear triangular element. The stiffness matrices are obtained based on the strain smoothing technique over the smoothing domains and the constant strains on triangular sub-domains associated with the nodes of the elements. The discrete shear gap (DSG) method is incorporated into the AαFEM to eliminate transverse shear locking and an improved triangular element termed as AαDSG3 is proposed. Several numerical examples are then given to demonstrate the effectiveness of the AαDSG3.     

A new model for nonlinear elastic plates with rapidly varying thickness,ii: the effect of the behavior of the forces when the thickness approaches zero

We consider a family of nonlinear elastic plates with rapidly varying thickness under the assumption that the three-dimensional constitutive equation is linear with respect to the "full" strain tensor (St. Venant-Kirchhoff material). The main goal of this paper is to shown that the limit problem, when the mean plate thickness converges to zero, may be a ill posed problem if the forces do not behave in an appropriate manner     

Optimal Shape Design Problems for a Class of Systems Described by Parabolic Hemivariational Inequality

Optimal shape design problems for systems governed by a parabolic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.     

Design of Finite Element Tools for Coupled Surface and Volume Meshes

Many problems with underlying variational structure involve a coupling of volume with surface effects.A straight-forward approach in a finite element discretiza- tion is to make use of the surface triangulation that is naturally induced by the volume triangulation.In an adaptive method one wants to facilitate"matching"local mesh modifications,i.e.,local refinement and/or coarsening,of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA.We also present several important applications of the mesh coupling.     

Optimal Shape Design Problems for a Class of Systems Described by Hemivariational Inequalities

Optimal shape design problems for systems governed by an elliptic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.