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Stress energy minimization as a tool in the material layout design of shallow shells |
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 G. Dzierżanowski, Stress energy minimization as a tool in the material layout design of shallow shells,
International Journal of Solids and Structures 49 (2012), pp. 1343-1354
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The present research deals with the compliance minimization problem of an elastic thin shallow shell subjected to simultaneous in-plane and bending loads. In this context, our goal is to lay out a given amount of material in the volume of a shell assuming that the distribution in the direction transversal to its middle surface S is homogeneous. The discussion hence reduces to the question of finding the optimal material arrangement on S. Similar problems were solved in the framework of two dimensional elasticity or Kirchhoff plate theory and the present research attempts to generalize these results. Following the pattern emerging from the above mentioned considerations, our research starts from the minimum compliance problem of a structure made of two elastic materials whose volumetric fractions are fixed. |
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The stiffest designs of elastic plates: Vector optimization for two loading conditions |
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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
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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 stress-based formulation (P) is rearranged to the displacement-
based form (P*). The latter formulation turns out to be well-posed 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.
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On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight |
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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
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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 self-equilibrated forces applied at three given points. This problem has been solved analytically by H.S.Y. Chan in 1966. This analytical solution is re-derived 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 half-analytical 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. |
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Bounds on the effective isotropic moduli of thin elastic composite plates |
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Bounds on the effective isotropic moduli of thin elastic composite plates, G. Dzierżanowski
Arch. Mech., 62, 4, pp. 253–281, Warszawa 2010
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The main aim of this paper is to estimate the effective moduli of an isotropic elastic composite, analyzed within the framework of the Kirchhoff-Love 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 static-geometric analogy of
Lurie and Goldenveizer, enables rewriting of these new bounds in the two-dimensional elasticity (plane stress) setting, thus revealing a link to the formulae previously found by Gibiansky and Cherkaev. Consequently, simple cross-property estimates are proposed for the plate subject to the simultaneous bending and in-plane loads.
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Sandwich plates of minimal compliance |
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Sandwich plates of minimal compliance, S. Czarnecki, M. Kursa, T. Lewiński,
Comput. Methods Appl. Mech. Engrg. 197 (2008) 4866–4881
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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 in-plane 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 ill-posed 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 mesh-independent. 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. |
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