The stiffest designs of elastic plates: Vector optimization for two loading conditions 



The stiffest designs of elastic plates: Vector optimization for two loading conditions, S. Czarnecki, T. Lewiński Comput. Methods Appl. Mech. Engrg. 200 (2011) 1708–1728 Read online>> The paper deals with optimal design of linearly elastic plates of the Kelvin moduli being distributed according to a given pattern. The case of two loading conditions is discussed. The optimal plate is characterized by the minimum value of the weighted sum of the compliances corresponding to the two kinds of loads. The problem is reduced to the equilibrium problem of a hyperelastic mixture of properties expressed in terms of two stress fields. The stressbased formulation (P) is rearranged to the displacement based form (P^{*}). The latter formulation turns out to be wellposed due to convexity of the relevant potential expressed in terms of strains. Due to monotonicity of the stress–strain relations the problem (P^{*}) is tractable by the finite element method, using special Newton's solvers. Exemplary numerical results are presented delivering layouts of variation of elastic characteristics for selected values of the weighting factors corresponding to two kinds of loadings.

On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight 



On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight, Tomasz Sokół , Tomasz Lewiński
Read online>> Two problems of minimum weight design of plane trusses are dealt with. The first problem concerns construction of the lightest fully stressed truss subject to three selfequilibrated forces applied at three given points. This problem has been solved analytically by H.S.Y. Chan in 1966. This analytical solution is rederived in the present paper. It compares favourably with new numerical solutions found here by the method developed recently by the first author. The solution to the three forces problem paves the way to halfanalytical as well as numerical solutions to the problem of minimum weight design of plane symmetric frameworks transmitting two symmetrically located vertical forces to two fixed supports lying along the line linking the points of application of the forces. 

Bounds on the effective isotropic moduli of thin elastic composite plates 



Bounds on the effective isotropic moduli of thin elastic composite plates, G. Dzierżanowski Arch. Mech., 62, 4, pp. 253–281, Warszawa 2010 Read online>> The main aim of this paper is to estimate the effective moduli of an isotropic elastic composite, analyzed within the framework of the KirchhoffLove theory of thin plates in bending. Results of calculations provide explicit functional correlations between the homogenized properties of a composite plate made of two isotropic materials, thus yielding more restrictive bounds on pairs of effective moduli than the classical (uncoupled) Hashin–Shtrikman–Walpole ones. Applying the staticgeometric analogy of Lurie and Goldenveizer, enables rewriting of these new bounds in the twodimensional elasticity (plane stress) setting, thus revealing a link to the formulae previously found by Gibiansky and Cherkaev. Consequently, simple crossproperty estimates are proposed for the plate subject to the simultaneous bending and inplane loads.

Sandwich plates of minimal compliance 



Sandwich plates of minimal compliance, S. Czarnecki, M. Kursa, T. Lewiński, Comput. Methods Appl. Mech. Engrg. 197 (2008) 4866–4881 Read online>> The subject of the paper is an optimal choice of material parameters characterizing the core layer of sandwichplates within the framework of the conventional plate theory in which the core layer is treated as soft in the inplane direction. The mathematical description is similar to the Hencky–Reissner model of plates with transverse shear deformation. Here, however, the bending stiffnesses and the transverse shear stiffnesses can be designed independently. The present paper deals only with optimal design of the core layer to make the plate compliance minimal. Two core materials are at our disposal, which leads to the illposed problem. To consider it one should relax this problem by admitting composite domains and characterize their overall properties by the homogenization formulae. The numerical approach is based on this relaxed formulation thus making it meshindependent. The equilibrium problem is solved by the DSG3 finite element method. The optimization results are found with using the convergent updating schemes of the COC method. 
