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1

Optimal form-finding of cable systems

Dzierżanowski G., Wójcik-Grząba I.

Archives of Civil Engineering

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2020

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Vol. 66, nr 3

305--321

PL

W artykule rozważane są zagadnienia związane z projektowaniem wstępnie napiętych konstrukcji cięgnowych. Głównym celem jest znalezienie konfiguracji kinematycznie niezmiennej o minimalnym ciężarze. W odróżnieniu od typowych zagadnień optymalizacji topologicznej celem zadania nie jest optymalizacja konstrukcji pod działaniem konkretnego obciążenia, chociaż zastosowana metoda rozwiązania jest podobna. Na początku formułujemy problem wariacyjny poszukiwania kształtu konstrukcji, a następnie przedstawiamy numeryczną procedurę iteracyjną służącą do znajdowania optymalnego położenia węzłów siatki cięgnowej.

EN

Tensile structures in general, achieve their load-carrying capability only after the process of initial form-finding. From the mechanical point of view, this process can be considered as a problem in statics. As cable systems are close siblings of trusses (cables, however, can carry tensile forces only), in our study we refer to equilibrium equation similar to those known from the theory of the latter. In particular, the paper regards designing pre-tensioned cable systems, with a goal to make them kinematically stable and such that the weight of so designed system is lowest possible. Unlike in typical topology optimization problems, our goal is not to optimize the structural layout against a particular applied load. However, our method uses much the same pattern. First, we formulate the variational problem of form-finding and next we describe the corresponding iterative numerical procedure for determining the optimum location of nodes of the cable system mesh. We base our study on the concept of force density which is a ratio of an axial force in cable segment to its length.

2

Zagadnienia kształtowania konstrukcji w ujęciu współczesnej mechaniki

Bołbotowski K., Czarnecki S., Czubacki R., Dzierżanowski G., Lewiński T., Łukasiak T., Sokół T.

Inżynieria i Budownictwo

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2017

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R. 73, nr 8

437--441

PL

Przedstawiono współczesne kierunki rozwoju teorii optymalizacji topologii konstrukcyjnej dźwigarów płaskich, przestrzennych oraz rusztów. Omówiono metody konstrukcji bazowych prowadzące do kratownicowych lub belkowych aproksymacji dźwigarów Michella, metodę strut and tie w wersji wspomaganej przez metodę konstrukcji bazowych, metodę interpolacji materiałowej oraz metodę projektowania materiału. Każda z tych metod łączy w sobie optymalizację kształtu i materiału.

EN

The paper concerns the contemporary trends of the theory of optimization of structural topology of planar and spatial girders as well as grillages. The following methods discussed are the ground structure method leading to truss-like or beam-like approximations of Michell structures, the strut and tie method supported by the ground structure method, the method of material interpolation schemes and the free material design method. Each of these methods combines the shape design and the material design.

3

Application of the isotropic material design and inverse homogenization in 3D printing

Dzierżanowski G.

Engineering Transactions

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2016

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Vol. 64, iss. 4

465--471

EN

Homogenization-based approach to structural optimization and Free Material Design (FMD) technique are discussed in case of isotropy. The problem is elaborated from the perspective provided by: (i) the theory of composites and (ii) Isotropic Material Design (IMD) – a variant of FMD. Results provided by IMD are interpreted in light of the Hashin-Shtrikman bounds on the effective isotropic properties of material-void mixtures. This in turn provides practical guidelines for 3D printing.

4

Topology optimization in structural mechanics

Lewiński T., Czarnecki S., Dzierżanowski G., Sokół T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2013

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Vol. 61, nr 1

23-37

EN

Optimization of structural topology, called briefly: topology optimization, is a relatively new branch of structural optimization. Its aim is to create optimal structures, instead of correcting the dimensions or changing the shapes of initial designs. For being able to create the structure, one should have a possibility to handle the members of zero stiffness or admit the material of singular constitutive properties, i.e. void. In the present paper, four fundamental problems of topology optimization are discussed: Michell’s structures, two-material layout problem in light of the relaxation by homogenization theory, optimal shape design and the free material design. Their features are disclosed by presenting results for selected problems concerning the same feasible domain, boundary conditions and applied loading. This discussion provides a short introduction into current topics of topology optimization.

5

Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part I. Solving the local optimization problem

Dzierżanowski G., Lewiński T.

Archives of Mechanics

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2012

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Vol. 64, nr 1

21-21

EN

The paper deals with compliance minimization of a transversely homogeneous plate, subjected to the in-plane and transverse loadings acting simultaneously. The set of design variables includes the eigenstates of Hooke’s tensor whose eigenvalues, i.e. Kelvin moduli fields, are assumed to be fixed on the middle plane of the plate, but no isoperimetric condition is imposed. The optimization task reduces to an equilibrium problem of an effective hyperelastic plate. The effective potential is explicitly expressed in terms of the invariants of both the strain fields involved.

6

Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part II. The effective boundary value problem and exemplary solutions

Dzierżanowski G., Lewiński T.

Archives of Mechanics

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2012

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Vol. 64, nr 2

137-137

EN

The compliance minimization of transversely homogeneous plates with predefined Kelvin moduli leads to the equilibrium problem of an effective hyperelastic plate with the hyperelastic potential expressed explicitly in terms of both the membrane and bending strain measures, as derived in Part I of the present paper. The aim of this second part of the paper is to show convexity of this potential and, consequently, uniqueness of solutions of the minimum compliance problem considered. Theoretical results are illustrated by numerically calculated optimal trajectories of the eigenstate corresponding to the largest Kelvin modulus.

7

Bounds on the effective isotropic moduli of thin elastic composite plates

Dzierżanowski G.

Archives of Mechanics

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2010

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Vol. 62, nr 4

253-281

EN

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.

8

Layout optimization of two isotropic materials in elastic shells

Dzierżanowski G., Lewiński T

Journal of Theoretical and Applied Mechanics

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2003

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Vol. 41 nr 3

459-472

EN

The two-phase layout problem within the plate theory was solved by Gibiansky and Cherkaev in 1984. The same problem in the plane stress formulation was solved by the same authors in eventually cleared up by Allaire and Kohn in 1993. In the thin shell theory both these formulations are coupled, which is clearly seen in the homogenization formulae found by Lewiński and Telega in 1988, Telega and Lewiński in 1998, and a general setting of the layout problem presented in the book by the same authors. The aim of the present paper is to set this problem within the Mushtari-Donnell-Vlasov approximation. The main result of the present examination is the lower bound of the complementary energy found by using the translation method. The translation matrix involves off-diagonal components, which leads to the effective complementary potential of a specific coupled form, expressible in terms of invariants of the stress and couple results.

PL

Zagadnienie optymalnego rozkładu dwóch materiałów izotropowych w sprężystych plytach cienkich rozwiązali Gibianskij i Czerkajew w roku 1984. Minimalizacji podlegała podatność płyty. Analogiczne zadanie dotyczące teorii tarcz rozwiązali ci sami autorzy w 1987r. Sformułowanie to uzupełnili i uściślili Allaire i Kohn w roku 1993. W zadaniu dotyczącym powłok cienkich oba te sformułowania są ze sobą sprzężone, co jasno jest widoczne w formułach homogenizacji znalezionych w pracach Lewińskiego i Telegi z roku 1988 oraz pracach Telegi i Lewińskiego z roku 1998; ogólne, niejawne sformułowanie tego zadania optymalizacji omówiono w książce tych samych autorów. Celem niniejszej pracy jest sformułowanie tego zadania w sposób jawny w zakresie technicznej teorii powłok Musztariego-Donnella-Własowa. W pracy wyprowadzamy w sposób jawny dolne oszacowanie energii komplementarnej z wykorzystaniem metody translacji. Macierz traslacji zawiera tutaj składniki pozadiagonalne. Ta postać macierzy translacji prowadzi do zastępczego potencjału o specyficznej postaci sprzężonej, wyrażalnej za pomocą niezmienników sił wewnętrznych w powłoce.