A bending-gradient model for thick plates, I : theory

International audienceThis is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff-Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner-Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner-Mindlin model. In part two (Lebee and Sab, 2010a), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner-Mindlin theory and to full 3D Pagano's exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity

Nonlinear analysis of laminated shells with alternating stiff/soft lay-up

This paper proposes a multi-layer formulation for the nonlinear analysis of laminated shells with an alternating stiff/soft lay-up. The zigzag variation of planar displacements is taken into account by adding to the Reissner–Mindlin formulation a specific set of zigzag function which is effective for the considered lay-up. Furthermore, a piecewise linear through-thickness distribution of the material transverse shear strain is assumed, which agrees well with the real distribution. The proposed lamination model with a low-order nonlinear strain–displacement relationship is incorporated within a co-rotational framework for geometric nonlinear analysis, thus upgrading the low-order local element formulation to large displacement analysis with relative ease. In addition, a local shell system is employed for direct definition of the additional zigzag displacement fields and associated parameters, which are thus excluded from the large displacement co-rotational transformations. The application of the proposed laminated shell modelling approach is illustrated in this paper for a 9-noded co-rotational shell element, which utilises the Mixed Interpolation of Tensorial Components (MITC) method in the local system for overcoming locking effects. Several linear and nonlinear numerical examples of multi-layer shell structures with alternating stiff/soft lay-ups are used to illustrate the effectiveness and efficiency of the proposed modelling approach

Static and Free Vibration Analyses of Composite Shells Based on Different Shell Theories

Equations of motion with required boundary conditions for doubly curved deep and thick composite shells are shown using two formulations. The first is based upon the formulation that was presented initially by Rath and Das (1973, J. Sound and Vib.) and followed by Reddy (1984, J. Engng. Mech. ASCE). In this formulation, plate stiffness parameters are used for thick shells, which reduced the equations to those applicable for shallow shells. This formulation is widely used but its accuracy has not been completely tested. The second formulation is based upon that of Qatu (1995, Compos. Press. Vessl. Indust.; 1999, Int. J. Solids Struct.). In this formulation, the stiffness parameters are calculated by using exact integration of the stress resultant equations. In addition, Qatu considered the radius of twist in his formulation. In both formulations, first order polynomials for in-plane displacements in the z-direction are utilized allowing for the inclusion of shear deformation and rotary inertia effects (first order shear deformation theory or FSDT). Also, FSDTQ has been modified in this dissertation using the radii of each laminate instead of using the radii of mid-plane in the moment of inertias and stress resultants equations. Exact static and free vibration solutions for isotropic and symmetric and anti-symmetric cross-ply cylindrical shells for different length-to-thickness and length-to-radius ratios are obtained using the above theories. Finally, the equations of motion are put together with the equations of stress resultants to arrive at a system of seventeen first-order differential equations. These equations are solved numerically with the aid of General Differential Quadrature (GDQ) method for isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with different boundary conditions using the above mentioned theories. Results obtained using all three theories (FSDT, FSDTQ and modified FSDTQ) are compared with the results available in literature and those obtained using a three-dimensional (3D) analysis. The latter (3D) is used here mainly to test the accuracy of the shell theories presented here

Static and Free Vibration Analyses of Composite Shells Based on Different Shell Theories

Equations of motion with required boundary conditions for doubly curved deep and thick composite shells are shown using two formulations. The first is based upon the formulation that was presented initially by Rath and Das (1973, J. Sound and Vib.) and followed by Reddy (1984, J. Engng. Mech. ASCE). In this formulation, plate stiffness parameters are used for thick shells, which reduced the equations to those applicable for shallow shells. This formulation is widely used but its accuracy has not been completely tested. The second formulation is based upon that of Qatu (1995, Compos. Press. Vessl. Indust.; 1999, Int. J. Solids Struct.). In this formulation, the stiffness parameters are calculated by using exact integration of the stress resultant equations. In addition, Qatu considered the radius of twist in his formulation. In both formulations, first order polynomials for in-plane displacements in the z-direction are utilized allowing for the inclusion of shear deformation and rotary inertia effects (first order shear deformation theory or FSDT). Also, FSDTQ has been modified in this dissertation using the radii of each laminate instead of using the radii of mid-plane in the moment of inertias and stress resultants equations. Exact static and free vibration solutions for isotropic and symmetric and anti-symmetric cross-ply cylindrical shells for different length-to-thickness and length-to-radius ratios are obtained using the above theories. Finally, the equations of motion are put together with the equations of stress resultants to arrive at a system of seventeen first-order differential equations. These equations are solved numerically with the aid of General Differential Quadrature (GDQ) method for isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with different boundary conditions using the above mentioned theories. Results obtained using all three theories (FSDT, FSDTQ and modified FSDTQ) are compared with the results available in literature and those obtained using a three-dimensional (3D) analysis. The latter (3D) is used here mainly to test the accuracy of the shell theories presented here

Analysis of Laminated Anisotropic Plates and Shells Via a Modified Complementary Energy Principle Approach

The present work is concerned with the finite element structural analysis of laminated anisotropic plates and shells. New elements based on a modified complementary energy principle are proposed to improve the analysis of such composite structures. Third order deformation plate and shell models incorporating a convergence parameter are developed to govern the general displacement field. An eight-node isoparametric quadrilateral element with two independent cross-sectional rotations and three normal displacements is utilized to describe the displacement field. The present modified complementary energy formulation incorporates a number of in-plane strain functions of various orders. The corresponding in-plane stresses for each lamina are derived from the constitutive relations. The transverse stresses are then computed from the application of equilibrium equations. The element comprises an arbitrary number of lamina rigidly bonded together. The analysis technique employed, although using a higher order formulation, does not increase the number of variables associated with each lamina. Moreover, the use of a convergence parameter permits one to achieve excellent results for very thin as well as thick composite plates and shells. The static bending analysis of several example problems for various geometries, transverse loads and material properties is analyzed via a code written in MATLAB. The results are compared with those from technical theories, other finite element models and three-dimensional elasticity solutions available in the literature. It is demonstrated that marked improvements in the results for stress and displacement can be achieved by the use of the new modified complementary energy elements incorporating a convergence parameter

Nonlinear analysis of composite shells with application to glass structures

Laminated glass is a special composite material, which is characterised by an alternating stiff/soft lay-up owing to the significant stiffness mismatch between glass and PVB. This work is motivated by the need for an efficient and accurate nonlinear model for the analysis of laminated glass structures, which describes well the through-thickness variation of displacement fields and the transverse shear strains and enables large displacement analysis. An efficient lamination model is proposed for the analysis of laminated composites with an alternating stiff/soft lay-up, where the zigzag variation of planar displacements is taken into account by adding to the Reissner-Mindlin formulation a specific set of zigzag functions. Furthermore, a piecewise linear through-thickness distribution of the material transverse shear strain is assumed, which agrees well with the real distribution, yet it avoids layer coupling by not imposing continuity constraints on transverse shear stresses. Local formulations of curved multi-layer shell elements are established employing the proposed lamination model, which are framed within local co-rotational systems to allow large displacement analysis for small-strain problems. In order to eliminate the locking phenomenon for the shell elements, an assumed strain method is employed and improved, which readily addresses shear locking, membrane locking, and distortion locking for each constitutive layer. Furthermore, a local shell system is proposed for the direct definition of the additional zigzag displacement fields and associated parameters, which allows the additional displacement variables to be coupled directly between adjacent elements without being subject to the large displacement co-rotational transformations. The developed multi-layer shell elements are employed in this work for typical laminated glass problems, including double glazing systems for which a novel volume-pressure control algorithm is proposed. Several case studies are finally presented to illustrate the effectiveness and efficiency of the proposed modelling approach for the nonlinear analysis of glass structures.Open Acces

Hierarchical component-wise models for enhanced stress analysis and health monitoring of composites structures

L'abstract è presente nell'allegato / the abstract is in the attachmen

Nonlinear Formulations of a Four-Node Quasi-Conforming Shell Element

The quasi-conforming technique was introduced in the 1980’s to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming and nonconforming elements. While the linear formulation is well established, the nonlinear formulation based on the quasi-conforming technique that includes geometric and material nonlinearity is presented in this paper. The formulation is derived in the framework of an updated Lagrangian stress resultant, co-rotational approach. The geometric nonlinear formulation provides solutions to buckling and postbuckling behaviour while the material nonlinear formulation considers the spread of plasticity within the element while maintaining an explicit construction of element matrices. Aside from the elasto-plastic constitutive relation, formulations on laminate composites and reinforced concrete are also presented. The formulations of laminate composite and reinforced concrete material are present based on the layer concept, the material properties can vary throughout the thickness and across the surface of a shell element. The various failure criteria for laminate composite are included in the formulation which makes it possible to analyses the progressive failure of fibre and matrix. For the reinforced concrete material, the nonlinearities as a result of tensile cracking, tension stiffening between cracks, the nonlinear response of concrete in compression, and the yielding of the reinforcement are considered. The steel reinforcement is modeled as a bilinear material with strain hardening

Buckling and Postbuckling of Delaminated Composite Plates

Chování laminátových desek namáhaných na tlak či na smyk může být výrazně ovlivněno přítomností delaminací, tedy oblastí, kde je porušena vazba mezi sousedními vrstvami. Cílem této práce je rozšířit znalosti o chování delaminovaných desek, a to především o chování desek s větším počtem delaminací a desek s delaminacemi libovolného tvaru, neboť taková podoba porušení laminátu více odpovídá poškození vznikajícího v důsledku nízkorychlostního dopadu cizího tělesa na laminátovou desku. Disertační práce se skládá ze tří hlavních částí. V první části jsou stručně nastíněny postupy využívané při analýze boulení delaminovaných desek a jsou diskutována omezení těchto analýz. Dále jsou v této části shrnuty hlavní poznatky o boulení delaminovaných desek. V druhé části práce je popsán výpočtový model použitý v rámci disertační práce pro analýzu boulení delaminovaných desek. Schopnost modelu předpovědět chování delaminovaných desek je pak dokumentována na několika ověřovacích úlohách. Třetí část disertační práce se skládá ze tří samostatných studií chování desek s několika delaminacemi eliptického či kruhového tvaru a jedné studie zabývající se možností náhrady obecného tvaru delaminace kruhem či elipsou. Je probírán vliv řady parametrů na chování delaminovaných desek, konkrétně vliv orientace vrstev laminátu a dále vliv počtu, tvaru, orientace a umístění delaminací. Na základě těchto studií jsou pak zformulována doporučení ohledně postupu při posuzování únosnosti delaminovaných konstrukcí.Buckling and postbuckling behaviour of a composite plate can be significantly influenced by the presence of delaminations, i.e. by regions where connectivity between layers is lost. Therefore, the aim of this work is to extend our knowledge about the buckling of delaminated plates, especially about the behaviour of plates with multiple delaminations and plates with delaminations which have irregular shape, because such damage more closely reflects the true nature of low-velocity impact induced damage in laminated plates. The thesis consists of three main parts. The first part comprises of the summary of analysis techniques used to investigate the behaviour of delaminated plates and presents the main results of the studies on the buckling of delaminated plates. A brief discussion of limits of computational analyses is also included. In the second part of the thesis, the computational model used to analyse the behaviour of delaminated plates is presented. The capability of the model to predict the postbuckling behaviour of delaminated plates is illustrated on several verification studies based on experimental and numerical analyses found in literature. Finally, in the third part, three studies on the buckling of plates with multiple delaminations and one study on the buckling response of plates with a delamination of an arbitrary shape are presented. Effects of several parameters on the buckling response such as ply orientation, number of delaminations, their shape and orientation, out-of-plane and in-plane position are discussed. Some implications of the studies for the design praxis are formulated.

An adaptive shell element for explicit dynamic analysis of failure in laminated composites Part 1: Adaptive kinematics and numerical implementation

To introduce more fibre-reinforced polymers in cars, the automotive industry is strongly dependent on efficient modelling tools to predict the correct energy absorption in crash simulations. In this context, an adaptive modelling technique shows great potential. However, as the critical energy absorption in a crash occurs over a very short period of time, and since the deformation behaviour is very complex, car crash simulations are usually performed using explicit dynamic finite element solvers. Therefore, any practical adaptive technique must be adapted to an explicit setting in a software available to the automotive companies. In this paper, we propose an adaptive method for explicit finite element analysis and describe its implementation in the commercial finite element solver LS-DYNA. The method allows for both so-called weak discontinuities (discontinuities in strain), which are crucial for accurate stress and intralaminar damage predictions, and strong discontinuities (discontinuities in displacements), needed for a proper representation of growing delamination cracks. In particular, we detail the implementation of the proposed method into LS-DYNA and also how we propose to remedy the non-physical oscillations arising from the implementation of the adaptive scheme in a explicit dynamic setting. The paper is concluded with numerical examples where we demonstrate the potential for the adaptive approach and also perform a detailed study on its accuracy and stability