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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 06 Aug 2020 08:25:17 GMT2020-08-06T08:25:17ZON THE FORMULATION OF A TENSORIAL LAMINATE-LEVEL FAILURE CRITERION THROUGH INVARIANTS
http://hdl.handle.net/10985/12559
ON THE FORMULATION OF A TENSORIAL LAMINATE-LEVEL FAILURE CRITERION THROUGH INVARIANTS
CATAPANO, Anita; MONTEMURRO, Marco
Anisotropic materials, such as fibre-reinforced composite materials, are extensively used in many industrial fields thanks to their mechanical performances. The main characteristic of an anisotropic material is the dependency of physical properties upon the direction. Anisotropy influences strongly also the mechanical strength of a material, usually described by a failure criterion. We can separate the failure criteria into two distinct classes: the phenomenological/polynomial ones and the physically-based ones. The polynomial failure criteria are called in this way because the occurrence of the failure is checked through the computation of a scalar indicator, i.e. the failure index. In this framework, a unique scalar condition has to be verified, regardless the nature of the failure mechanism that is activated. It is noteworthy that all polynomial failure criteria are “ply-level failure criteria", thus, when utilised to analyse the failure of laminated structures, they are applied to each ply composing the structure in order to check the so-called first-ply-failure. However, this approach is not compatible with the methodologies often used for the preliminary design of composite structures. Indeed in this background, the number of design variables is economised by representing each laminate (composing the structure) as an equivalent homogeneous anisotropic plate characterised by a few number of parameters describing its overall mechanical response regardless to the nature of the stacking sequence. In order to include the failure mechanisms within the mathematical framework of preliminary design of composite structure an alternative approach is needed. To this purpose, a method to generalise the ply-level failure criterion of Tsai-Wu to the laminate-level (with the aim of introducing strength requirements at the macroscopic scale within the optimisation process of a composite structures) is proposed in this work.The Tsai-Wu criterion is formulated in the framework of the First-order Shear Deformation Theory (FSDT) in order to take into account the out-of-plane shear stress and strain components. Then, through the use of the polar method, the criterion is firstly formulated in terms of invariants. Finally it is extended, via a through-the-thickness homogenisation step, to the laminate-level in order to evaluate the strength of the entire laminate. The laminate-level failure criterion is thus expressed for a laminated plate modelled as an equivalent single layer having the same thickness of the laminate. Thanks to the polar representation a physical meaning of each tensor appearing in the laminate-level criterion is also given. The resulting criterion is then used in the framework of a strength optimisation problem in order to show the effectiveness of the proposed method. The multi-scale two-level optimisation strategy is used to firstly optimise the polar parameters of the tensors of the laminate-level criterion (macro-scale optimisation step) and then the lay-up design is carried out to find at least one stacking sequence satisfying the strength polar parameters provided by the first-step of the strategy (meso-scale optimisation step)
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/125592017-01-01T00:00:00ZCATAPANO, AnitaMONTEMURRO, MarcoAnisotropic materials, such as fibre-reinforced composite materials, are extensively used in many industrial fields thanks to their mechanical performances. The main characteristic of an anisotropic material is the dependency of physical properties upon the direction. Anisotropy influences strongly also the mechanical strength of a material, usually described by a failure criterion. We can separate the failure criteria into two distinct classes: the phenomenological/polynomial ones and the physically-based ones. The polynomial failure criteria are called in this way because the occurrence of the failure is checked through the computation of a scalar indicator, i.e. the failure index. In this framework, a unique scalar condition has to be verified, regardless the nature of the failure mechanism that is activated. It is noteworthy that all polynomial failure criteria are “ply-level failure criteria", thus, when utilised to analyse the failure of laminated structures, they are applied to each ply composing the structure in order to check the so-called first-ply-failure. However, this approach is not compatible with the methodologies often used for the preliminary design of composite structures. Indeed in this background, the number of design variables is economised by representing each laminate (composing the structure) as an equivalent homogeneous anisotropic plate characterised by a few number of parameters describing its overall mechanical response regardless to the nature of the stacking sequence. In order to include the failure mechanisms within the mathematical framework of preliminary design of composite structure an alternative approach is needed. To this purpose, a method to generalise the ply-level failure criterion of Tsai-Wu to the laminate-level (with the aim of introducing strength requirements at the macroscopic scale within the optimisation process of a composite structures) is proposed in this work.The Tsai-Wu criterion is formulated in the framework of the First-order Shear Deformation Theory (FSDT) in order to take into account the out-of-plane shear stress and strain components. Then, through the use of the polar method, the criterion is firstly formulated in terms of invariants. Finally it is extended, via a through-the-thickness homogenisation step, to the laminate-level in order to evaluate the strength of the entire laminate. The laminate-level failure criterion is thus expressed for a laminated plate modelled as an equivalent single layer having the same thickness of the laminate. Thanks to the polar representation a physical meaning of each tensor appearing in the laminate-level criterion is also given. The resulting criterion is then used in the framework of a strength optimisation problem in order to show the effectiveness of the proposed method. The multi-scale two-level optimisation strategy is used to firstly optimise the polar parameters of the tensors of the laminate-level criterion (macro-scale optimisation step) and then the lay-up design is carried out to find at least one stacking sequence satisfying the strength polar parameters provided by the first-step of the strategy (meso-scale optimisation step)On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angle-tow laminates
http://hdl.handle.net/10985/11438
On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angle-tow laminates
MONTEMURRO, Marco; CATAPANO, Anita
In this work a multi-scale two-level (MS2L) optimisation strategy for optimising VAT composites is presented. In the framework of the MS2L methodology, the design problem is split and solved into two steps. At the first step the goal is to determine the optimum distribution of the laminate stiffness properties over the structure (macroscopic scale), while the second step aims at retrieving the optimum fibres-path in each layer meeting all the requirements provided by the problem at hand (mesoscopic scale). The MS2L strategy has been improved in order to integrate all types of requirements (mechanical, manufacturability, geometric, etc.) within the first-level problem.The proposed approach relies on: a) the polar formalism for describing the behaviour of the VAT laminate, b) the iso-geometric surfaces for describing the spatial variation of both the laminate stiffness properties (macro-scale) and the layers fibres-path (meso-scale) and c) an hybrid optimisation tool (genetic and gradient-based algorithms) to perform the solution search. The effectiveness of the MS2L strategy is proven through a numerical example on the maximisation of the first buckling factor of a VAT plate subject to both mechanical and manufacturability constraints.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/114382017-01-01T00:00:00ZMONTEMURRO, MarcoCATAPANO, AnitaIn this work a multi-scale two-level (MS2L) optimisation strategy for optimising VAT composites is presented. In the framework of the MS2L methodology, the design problem is split and solved into two steps. At the first step the goal is to determine the optimum distribution of the laminate stiffness properties over the structure (macroscopic scale), while the second step aims at retrieving the optimum fibres-path in each layer meeting all the requirements provided by the problem at hand (mesoscopic scale). The MS2L strategy has been improved in order to integrate all types of requirements (mechanical, manufacturability, geometric, etc.) within the first-level problem.The proposed approach relies on: a) the polar formalism for describing the behaviour of the VAT laminate, b) the iso-geometric surfaces for describing the spatial variation of both the laminate stiffness properties (macro-scale) and the layers fibres-path (meso-scale) and c) an hybrid optimisation tool (genetic and gradient-based algorithms) to perform the solution search. The effectiveness of the MS2L strategy is proven through a numerical example on the maximisation of the first buckling factor of a VAT plate subject to both mechanical and manufacturability constraints.A general multi-scale design strategy for the optimisation of variable stiffness composites
http://hdl.handle.net/10985/11437
A general multi-scale design strategy for the optimisation of variable stiffness composites
MONTEMURRO, Marco; CATAPANO, Anita
The present paper focuses on the development of a multi-scale design strategy for the optimisation of variable angle stiffness laminates. The main goal consists in proving that it is possible to design structures having complex shapes made of variable stiffness composites by taking into account, from the early stages of the design process, the constraints linked to the manufacturing process.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/114372016-01-01T00:00:00ZMONTEMURRO, MarcoCATAPANO, AnitaThe present paper focuses on the development of a multi-scale design strategy for the optimisation of variable angle stiffness laminates. The main goal consists in proving that it is possible to design structures having complex shapes made of variable stiffness composites by taking into account, from the early stages of the design process, the constraints linked to the manufacturing process.A multi-scale approach for the simultaneous shape and material optimisation of sandwich panels with cellular core
http://hdl.handle.net/10985/10674
A multi-scale approach for the simultaneous shape and material optimisation of sandwich panels with cellular core
MONTEMURRO, Marco; CATAPANO, Anita; DOROSZEWSKI, Dominique
This work deals with the problem of the optimum design of a sandwich panel made of carbon-epoxy skins and a metallic cellular core. The proposed design strategy is a multi-scale numerical optimisation procedure that does not make use of any simplifying hypothesis to obtain a true global optimum configuration of the system. To face the design of the sandwich structure at both meso and macro scales, a two-level optimisation strategy is employed: at the first level the goal is the determination of the optimum shape of the unit cell of the core (meso-scale) together with the material and geometric parameters of the laminated skins (macro-scale), while at the second level the objective is the design of the skins stacking sequence (skin meso-scale) meeting the geometrical and material parameters provided by the first-level problem. The two-level strategy is founded on the polar formalism for the description of the anisotropic behaviour of the laminates, on the NURBS basis functions for representing the shape of the unit cell and on the use of a genetic algorithm as optimisation tool to perform the solution search. To prove its effectiveness, the multi-scale strategy is applied to the least-weight design of a sandwich plate subject to constraints of different nature: on the positive-definiteness of the stiffness tensor of the core, on the admissible material properties of the laminated faces, on the local buckling load of the unit cell, on the global buckling load of the panel and geometrical as well as manufacturability constraints related to the fabrication process of the cellular core.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/106742016-01-01T00:00:00ZMONTEMURRO, MarcoCATAPANO, AnitaDOROSZEWSKI, DominiqueThis work deals with the problem of the optimum design of a sandwich panel made of carbon-epoxy skins and a metallic cellular core. The proposed design strategy is a multi-scale numerical optimisation procedure that does not make use of any simplifying hypothesis to obtain a true global optimum configuration of the system. To face the design of the sandwich structure at both meso and macro scales, a two-level optimisation strategy is employed: at the first level the goal is the determination of the optimum shape of the unit cell of the core (meso-scale) together with the material and geometric parameters of the laminated skins (macro-scale), while at the second level the objective is the design of the skins stacking sequence (skin meso-scale) meeting the geometrical and material parameters provided by the first-level problem. The two-level strategy is founded on the polar formalism for the description of the anisotropic behaviour of the laminates, on the NURBS basis functions for representing the shape of the unit cell and on the use of a genetic algorithm as optimisation tool to perform the solution search. To prove its effectiveness, the multi-scale strategy is applied to the least-weight design of a sandwich plate subject to constraints of different nature: on the positive-definiteness of the stiffness tensor of the core, on the admissible material properties of the laminated faces, on the local buckling load of the unit cell, on the global buckling load of the panel and geometrical as well as manufacturability constraints related to the fabrication process of the cellular core.A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part I: homogenisation of core properties
http://hdl.handle.net/10985/8498
A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part I: homogenisation of core properties
CATAPANO, Anita; MONTEMURRO, Marco
This work deals with the problem of the optimum design of a sandwich panel. The design process is based on a general two-level optimisation strategy involving different scales: the meso-scale for both the unit cell of the core and the constitutive layer of the laminated skins and the macro-scale for the whole panel. Concerning the meso-scale of the honeycomb core, an appropriate model of the unit cell able to properly provide its effective elastic properties (to be used at the macro-scale) must be conceived. To this purpose, in this first paper, we present the numerical homogenisation technique as well as the related finite element model of the unit cell which makes use of solid elements instead of the usual shell ones. A numerical study to determine the effective properties of the honeycomb along with a comparison with existing models and a sensitive analysis in terms of the geometric parameters of the unit cell have been conducted. Numerical results show that shell-based models are no longer adapted to evaluate the core properties, mostly in the context of an optimisation procedure where the parameters of the unit cell can get values that go beyond the limits imposed by a 2D model.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10985/84982014-01-01T00:00:00ZCATAPANO, AnitaMONTEMURRO, MarcoThis work deals with the problem of the optimum design of a sandwich panel. The design process is based on a general two-level optimisation strategy involving different scales: the meso-scale for both the unit cell of the core and the constitutive layer of the laminated skins and the macro-scale for the whole panel. Concerning the meso-scale of the honeycomb core, an appropriate model of the unit cell able to properly provide its effective elastic properties (to be used at the macro-scale) must be conceived. To this purpose, in this first paper, we present the numerical homogenisation technique as well as the related finite element model of the unit cell which makes use of solid elements instead of the usual shell ones. A numerical study to determine the effective properties of the honeycomb along with a comparison with existing models and a sensitive analysis in terms of the geometric parameters of the unit cell have been conducted. Numerical results show that shell-based models are no longer adapted to evaluate the core properties, mostly in the context of an optimisation procedure where the parameters of the unit cell can get values that go beyond the limits imposed by a 2D model.Simultaneous shape and material optimization of sandwich panels with honeycomb core for additive manufacturing
http://hdl.handle.net/10985/9922
Simultaneous shape and material optimization of sandwich panels with honeycomb core for additive manufacturing
MONTEMURRO, Marco; CATAPANO, Anita; DOROSZEWSKI, Dominique
This works deals with the problem of the optimum design of a sandwich plate composed of CFRP faces and Al honeycomb core. The proposed design strategy is a multi-scale numerical optimization procedure that does not make use of any simplifying assumption to find a global optimum configuration of the system. The goal of such a procedure consists in simultaneously optimizing the shape of the unit cell of the honeycomb core (meso-scale) and the geometrical as well as the material parameters of the CFRP laminated skins (meso and macro scales). To prove its effectiveness, the multi-scale optimization strategy is applied to the problem of the least-weight design of a sandwich panel subject to constraints of different nature: on the positive-definiteness of the stiffness tensor of the core, on the admissible material properties of the laminated faces, on the local buckling load of the unit cell of the core, on the global buckling load of the panel and geometrical as well as manufacturability constraints linked to the fabrication process of the honeycomb core.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/99222015-01-01T00:00:00ZMONTEMURRO, MarcoCATAPANO, AnitaDOROSZEWSKI, DominiqueThis works deals with the problem of the optimum design of a sandwich plate composed of CFRP faces and Al honeycomb core. The proposed design strategy is a multi-scale numerical optimization procedure that does not make use of any simplifying assumption to find a global optimum configuration of the system. The goal of such a procedure consists in simultaneously optimizing the shape of the unit cell of the honeycomb core (meso-scale) and the geometrical as well as the material parameters of the CFRP laminated skins (meso and macro scales). To prove its effectiveness, the multi-scale optimization strategy is applied to the problem of the least-weight design of a sandwich panel subject to constraints of different nature: on the positive-definiteness of the stiffness tensor of the core, on the admissible material properties of the laminated faces, on the local buckling load of the unit cell of the core, on the global buckling load of the panel and geometrical as well as manufacturability constraints linked to the fabrication process of the honeycomb core.A new multi-scale design methodology for the optimisation of variable stiffness composites
http://hdl.handle.net/10985/12936
A new multi-scale design methodology for the optimisation of variable stiffness composites
MONTEMURRO, Marco; CATAPANO, Anita
In this work a multi-scale two-level (MS2L) optimisation strategy for optimising VAT composites is presented. In the framework of the MS2L methodology, the design problem is split and solved into two steps. At the first step the goal is to determine the optimum distribution of the laminate stiffness properties over the structure (macroscopic scale), while the second step aims at retrieving the optimum fibres path in each layer meeting all the requirements provided by the problem at hand (mesoscopic scale). The MS2L strategy has been improved in order to integrate all types of requirements (mechanical, manufacturability, geometric, etc.) within the first-level problem. The proposed approach relies on: (a) the polar formalism for describing the behaviour of the VAT laminate, (b) the iso-geometric surfaces for describing the spatial variation of both the laminate stiffness properties (macro-scale) and the layers fibres-path (meso-scale) and (c) an hybrid optimisation tool (genetic and gradient-based algorithms) to perform the solution search. The effectiveness of the MS2L strategy is proven through a numerical example on the maximisation of the first buckling factor of a VAT plate subject to both mechanical and manufacturability constraints.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/129362017-01-01T00:00:00ZMONTEMURRO, MarcoCATAPANO, AnitaIn this work a multi-scale two-level (MS2L) optimisation strategy for optimising VAT composites is presented. In the framework of the MS2L methodology, the design problem is split and solved into two steps. At the first step the goal is to determine the optimum distribution of the laminate stiffness properties over the structure (macroscopic scale), while the second step aims at retrieving the optimum fibres path in each layer meeting all the requirements provided by the problem at hand (mesoscopic scale). The MS2L strategy has been improved in order to integrate all types of requirements (mechanical, manufacturability, geometric, etc.) within the first-level problem. The proposed approach relies on: (a) the polar formalism for describing the behaviour of the VAT laminate, (b) the iso-geometric surfaces for describing the spatial variation of both the laminate stiffness properties (macro-scale) and the layers fibres-path (meso-scale) and (c) an hybrid optimisation tool (genetic and gradient-based algorithms) to perform the solution search. The effectiveness of the MS2L strategy is proven through a numerical example on the maximisation of the first buckling factor of a VAT plate subject to both mechanical and manufacturability constraints.Architecture and materials selection in multi-materials design
http://hdl.handle.net/10985/8514
Architecture and materials selection in multi-materials design
BARACCHINI, Paul; GUILLEBAUD, Claire; KROMM, François-Xavier; CATAPANO, Anita; MONTEMURRO, Marco; WARGNIER, Hervé
The design process involving both the architecture and the materials represents an hard task mainly due to the high number of potential configurations, thus requiring firstly the development of new and more rigorous approaches but also the development of new tools. To this purpose, we present in this work a new strategy for the design of architectured materials. Such a strategy relies on one hand on the construction of some databases for the selection of both geometrical patterns and materials, and on the other hand on the use of well-known analytical models to describe the physical behaviour of the multi-material. In order to prove its effectiveness, we apply our strategy to the problem of the least-weight design of a multilayer plate that has to meet thermal, electrical and mechanical requirements. Moreover, we use a genetic algorithm, as a numerical tool, to perform the solution search for our problem. Numerical results show that we can obtain optimum configurations characterised by a weight saving up to 59% keeping the same (or even superior) thermal, electrical and stiffness properties than those of a monolithic reference plate.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10985/85142014-01-01T00:00:00ZBARACCHINI, PaulGUILLEBAUD, ClaireKROMM, François-XavierCATAPANO, AnitaMONTEMURRO, MarcoWARGNIER, HervéThe design process involving both the architecture and the materials represents an hard task mainly due to the high number of potential configurations, thus requiring firstly the development of new and more rigorous approaches but also the development of new tools. To this purpose, we present in this work a new strategy for the design of architectured materials. Such a strategy relies on one hand on the construction of some databases for the selection of both geometrical patterns and materials, and on the other hand on the use of well-known analytical models to describe the physical behaviour of the multi-material. In order to prove its effectiveness, we apply our strategy to the problem of the least-weight design of a multilayer plate that has to meet thermal, electrical and mechanical requirements. Moreover, we use a genetic algorithm, as a numerical tool, to perform the solution search for our problem. Numerical results show that we can obtain optimum configurations characterised by a weight saving up to 59% keeping the same (or even superior) thermal, electrical and stiffness properties than those of a monolithic reference plate.A new paradigm for the optimum design of variable angle tow laminates
http://hdl.handle.net/10985/11387
A new paradigm for the optimum design of variable angle tow laminates
MONTEMURRO, Marco; CATAPANO, Anita
In this work the authors propose a new paradigm for the optimum design of variable angle tow (VAT) composites. They propose a generalisation of a multi-scale two-level (MS2L) optimisation strategy already employed to solve optimisation problems of anisotropic structures characterised by a constant stiffness distribution. In the framework of the MS2L methodology, the design problem is split into two sub-problems. At the first step of the strategy the goal is to determine the optimum distribution of the laminate stiffness properties over the structure, while the second step aims at retrieving the optimum fibres-path in each layer meeting all the requirements provided by the problem at hand. The MS2L strategy relies on: a) the polar formalism for describing the behaviour of the VAT laminate, b) the iso-geometric surfaces for describing the spatial variation of the stiffness properties and c) an hybrid optimisation tool (genetic and gradient-based algorithms) to perform the solution search. The effectiveness of the MS2L strategy is proven through a numerical example on the maximisation of the first buckling factor of a VAT plate subject to both mechanical and manufacturability constraints.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/113872016-01-01T00:00:00ZMONTEMURRO, MarcoCATAPANO, AnitaIn this work the authors propose a new paradigm for the optimum design of variable angle tow (VAT) composites. They propose a generalisation of a multi-scale two-level (MS2L) optimisation strategy already employed to solve optimisation problems of anisotropic structures characterised by a constant stiffness distribution. In the framework of the MS2L methodology, the design problem is split into two sub-problems. At the first step of the strategy the goal is to determine the optimum distribution of the laminate stiffness properties over the structure, while the second step aims at retrieving the optimum fibres-path in each layer meeting all the requirements provided by the problem at hand. The MS2L strategy relies on: a) the polar formalism for describing the behaviour of the VAT laminate, b) the iso-geometric surfaces for describing the spatial variation of the stiffness properties and c) an hybrid optimisation tool (genetic and gradient-based algorithms) to perform the solution search. The effectiveness of the MS2L strategy is proven through a numerical example on the maximisation of the first buckling factor of a VAT plate subject to both mechanical and manufacturability constraints.A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part II: the optimisation strategy
http://hdl.handle.net/10985/8493
A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part II: the optimisation strategy
CATAPANO, Anita; MONTEMURRO, Marco
This work deals with the problem of the optimum design of a sandwich panel. The design strategy that we propose is a numerical optimisation procedure that does not make any simplifying assumption to obtain a true global optimum configuration of the system. To face the design of the sandwich structure at both meso and macro scales, we use a two-level optimisation strategy: at the first level we determine the optimal geometry of the unit cell of the core together with the material and geometric parameters of the laminated skins, while at the second level we determine the optimal skins lay-up giving the geometrical and material parameters issued from the first level. The two-level strategy relies both on the use of the polar formalism for the description of the anisotropic behaviour of the laminates and on the use of a genetic algorithm as optimisation tool to perform the solution search. To prove its effectiveness, we apply our strategy to the least-weight design of a sandwich plate, satisfying several constraints: on the first buckling load, on the positive-definiteness of the stiffness tensor of the core, on the ratio between skins and core thickness and on the admissible moduli for the laminated skins.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10985/84932014-01-01T00:00:00ZCATAPANO, AnitaMONTEMURRO, MarcoThis work deals with the problem of the optimum design of a sandwich panel. The design strategy that we propose is a numerical optimisation procedure that does not make any simplifying assumption to obtain a true global optimum configuration of the system. To face the design of the sandwich structure at both meso and macro scales, we use a two-level optimisation strategy: at the first level we determine the optimal geometry of the unit cell of the core together with the material and geometric parameters of the laminated skins, while at the second level we determine the optimal skins lay-up giving the geometrical and material parameters issued from the first level. The two-level strategy relies both on the use of the polar formalism for the description of the anisotropic behaviour of the laminates and on the use of a genetic algorithm as optimisation tool to perform the solution search. To prove its effectiveness, we apply our strategy to the least-weight design of a sandwich plate, satisfying several constraints: on the first buckling load, on the positive-definiteness of the stiffness tensor of the core, on the ratio between skins and core thickness and on the admissible moduli for the laminated skins.