Analysis of laminated doubly-curved shells by alayerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations

In this paper, the static and free vibration analysis of laminated shells is performed by radial basis functions collocation, according to a sinusoidal shear deformation theory (SSDT). The SSDT theory accounts for through-the-thickness deformation, by considering a sinusoidal evolution of all displacements with the thickness coordinate. The equations of motion and the boundary conditions are obtained by the Carrera's Unified Formulation, and further interpolated by collocation with radial basis functions

Bisector and zero-macrospin co-rotational systems for shell elements

A principal issue in any co-rotational approach for large displacement analysis of plates and shells is associated with the specific choice of the local reference system in relation to the current deformed element configuration. Previous approaches utilised local co-rotational systems, which are invariant to nodal ordering, a characteristic that is deemed desirable on several fronts; however, the associated definitions of the local reference system suffered from a range of shortcomings, including undue complexity, dependence on the local element formulation and possibly an asymmetric tangent stiffness matrix. In this paper, new definitions of the local co-rotational system are proposed for quadrilateral and triangular shell elements, which achieve the invariance characteristic to the nodal ordering in a relatively simple manner and address the aforementioned shortcomings. The proposed definitions utilise only the nodal coordinates in the deformed configuration, where two alternative definitions, namely, bisector and zero-macrospin definitions, are presented for each of quadrilateral and triangular finite elements. In each case, the co-rotational transformations linking the local and global element entities are presented, highlighting the simplicity of the proposed approach. Several numerical examples are finally presented to demonstrate the effectiveness and relative accuracy of the alternative definitions proposed for the local co-rotational system

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

Finite Element Analysis of Free Vibration of the Delaminated Composite Plate with Variable Kinematic Multilayered Plate Elements

Composite laminates are prone to delamination. Implementation of delamination in the Carrera Unified Formulation frame work using nine noded quadrilateral MITC9 element is discussed in this article. MITC9 element is devoid of shear locking and membrane locking. Delaminated as well as healthy structure is analyzed for free mode vibration. The results from the present work are compared with the available experimental or/and research article or/and the three dimensional finite element simulations. The effect of different kinds and different percentages of area of delamination on the first three natural frequencies of the structure is discussed. The presence of open-mode delamination mode shape for large delaminations within the first three natural frequencies is discussed. Also, the switching of places between the second bending mode, with that of the first torsional mode frequency is discussed. Results obtained from different ordered theories are compared in the presence of delamination. Advantage of layerwise theories as compared to equivalent single layer theories for very large delaminations is stated. The effect of different kinds of delamination and their effect on the second bending and first torsional mode shape is discussed. (C) 2014 Elsevier Ltd. All rights reserved

A variable kinematic doubly-curved MITC9 shell element for the analysis of laminated composites

The present article considers the linear static analysis of composite shell structures with double-curvature geometry by means of a shell finite element with variable through-the-thickness kinematic. The refined models used are grouped in the Unified Formulation by Carrera (CUF) and they permit the distribution of displacements and stresses along the thickness of the multilayered shell to be accurately described. The shell element has nine nodes and the mixed interpolation of tensorial components (MITC) method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement (PVD) and the finite element method (FEM) is employed to solve them. Cross-ply spherical shells with simply-supported edges and subjected to bi-sinusoidal pressure are analyzed. Various laminations, thickness ratios, and curvature ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in the literature and the analytical solutions obtained using the CUF and the Navier's method. From the analysis, one can conclude that the shell element based on the CUF is very efficient and its use is mandatory with respect to the classical models in the study of composite structures. Finally, shells with different lamination, boundary conditions, and loads are also analyzed using high-order layer-wise theories in order to provide FEM benchmark solution

Delamination Modeling and Detection in Composite Structures

Composite laminated structures are prone to delamination. Rotorcraft flexbeams, apart from many other aerospace primary load carrying members are made up of composite laminated structures. A delaminated primary load carrying member can lead to catastrophic failure of the system of which it is a part. Delamination modeling and detection in composite laminated structures are challenging areas of ongoing research worldwide. Existing literature falls short of addressing effects of widthwise partial delamination on the modal characteristics of beams. To address this issue, a new partial delamination model for composite beams is proposed and implemented using the finite element method. Homogenized cross-sectional stiffness of the delaminated beam is obtained by the proposed analytical technique, including extension-bending, extension-twist and torsion-bending coupling terms, and hence can be used with an existing finite element method. A two-noded C1-type Timoshenko beam element with four degrees of freedom per node for dynamic analysis of beams is implemented. The results for different delamination scenarios and beams subjected to different boundary conditions are validated with available experimental results in the literature and/or with a 3-D finite element simulation using COMSOL. Results of the first torsional mode frequency for the partially delaminated beam are validated with the COMSOL results. The key point of the current work is that even partial delamination in long structures can be analyzed using a 1-D beam model, rather than using computationally more demanding 3-D or 2-D models. Rotor craft flexbeams are prone to delaminations, which in most realistic situations are partial along both the length and the width. However, the effect of partial delamination on the modal characteristics of the beam is not studied by researchers to the best of the author’s knowledge. Addressing this issue, a rotorcraft flexbeam is analysed here in the presence of delamination. A set of nonlinear governing equations for the rotating flexbeam are developed in hybrid basis. The flexbeam model developed has axial stretch, transverse displacement and flexural rotation in flapwise direction and twist as its degrees of freedom. The nonlinear governing differential equations are linearised and solved for eigenvalues and eigenvectors using a finite element method. The effects of angular speed and delamination size and location on the flexbeam modes are analysed. The results obtained using the proposed model are validated with the COMSOL 3-D finite element simulations. Next, the issue of delamination detection in beams is addressed. Mode shape curvature and Katz fractal dimension are used to detect the presence of partial delaminations in a beam. The effects of boundary conditions and location of delamination on the fractal dimension curve are studied. Usage of higher mode shape data for detection of delamination in beams is evaluated. Limitations of the Katz fractal dimension curve for delamination detection are enumerated. It is shown that fractal dimension measure and mode shape curvature can be used to detect the presence of partial delamination in beams. It is found that the torsional mode shape is best suited for partial delamination detection in beams. Apart from beams, Shell-and plate-like structures are also extensively used in aerospace structures. The modeling of multilayered plates is introduced herein with the intention to model delaminations in 2D. Carrera Unified Formulation(CUF)plate model, developed using variational formulations, is used to derive the stiffness matrices and to apply, the Principle of Virtual Displacement(PVD) and the Reissner Mixed Variational Theorem (RMVT). It is known that FEM implementation for plates leads to the phenomenon of numerical locking: the so-called membrane and shear locking effects. A well-known remedy for addressing locking is the use of the Mixed Interpolated Tensorial Components(MITC) technique. A strategy similar to MITC approach in the RMVT formulation is used to construct an advanced locking-free finite element to treat the multilayered plates. Composite laminated plates are prone to delamination. Implementation of delamination in the CUF frame work using nine-noded quadrilateral MITC9 elements is discussed. MITC9 elements are devoid of shear locking and membrane locking. Delaminated structures, as well as the corresponding healthy structures, are analysed for free vibration modes. The results from the present work are compared with those from available experimental or/and theoretical research articles or/and the 3-D finite element simulations. The effects of different kinds and different percentages of interfacial area of delaminations on the first three natural frequencies of the structure are discussed. The presence of the open-mode or breathing mode delamination mode shape for large delaminations within the first three natural frequencies is discussed. Also, the switching of the places between the second bending mode and the first torsional mode frequencies is discussed. Results obtained from different ordered theories are compared in the presence of delamination. Advantage of layer wise theory as compared to equivalent single layer theories for very large delaminations is stated. The effects of different kinds of delamination and its effect on the second bending and first torsional mode shapes are discussed

Virtual Element Methods for hyperbolic problems on polygonal meshes

In the present paper we develop the Virtual Element Method for hyperbolic problems on polygonal meshes, considering the linear wave equations as our model problem. After presenting the semi-discrete scheme, we derive the convergence estimates in H^1 semi-norm and L^2 norm. Moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the Newmark method and the Bathe method. Finally we show the practical behaviour of the proposed method through a large array of numerical tests

A hierarchic optimisation approach towards locking-free shell finite elements

A hierarchic optimisation approach is presented for relieving inaccuracies in conforming shell elements arising from locking phenomena. This approach introduces two sets of strain modes: (i) objective strain modes, defined in the physical coordinate system, and (ii) corrective strain modes, representing conforming strains enhanced with hierarchic strain modes. This leads to two alternative families of element, objective and corrective, both arising from minimising the difference between objective and corrective strains. Importantly, the proposed approach not only alleviates shear and membrane locking, but it also addresses locking arising from element distortion. The application of the proposed optimisation approach is demonstrated for a 9-noded quadrilateral Lagrangian shell element, where the membrane, bending and transverse shear strains are separately optimised, all within a local co-rotational framework that extends the element application to geometric nonlinear analysis. Several numerical examples, including cases with geometric and material nonlinearity, are finally presented to illustrate the effectiveness of the optimised 9-noded shell element in relieving the various sources of locking