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

Abstract

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 throughthickness 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 straindisplacement 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.

Effective modelling of structural glass with laminated shell elements

“…, which are orthogonal to the constant and linear modes and associated with the change of normals in the soft PVB layers [18]. The zigzag modes for a threeply LG are depicted in Figure 5 for an illustration.…”

“…Two efficient geometrically nonlinear laminated shell elements, Ln-MITC6 and Ln-MITC9, have been recently developed for the analysis of laminations with an alternating stiff/soft lay-up [18][19], where 'L' stands for 'layer', n is the number of layers, and 'MITC6'…”

“…Recently, triangular and quadrilateral laminated shell elements have been developed by the authors [18][19], both of which fulfil the requirements (i)-(iii). In the embedded lamination model, zigzag displacements and transverse shear strain distributions following an alternating stiff/soft lay-up with very large stiffness ratio, as encountered in LG, were proposed [18].…”

“…In the embedded lamination model, zigzag displacements and transverse shear strain distributions following an alternating stiff/soft lay-up with very large stiffness ratio, as encountered in LG, were proposed [18]. In order to eliminate locking phenomena which arise with conforming displacement-based shell elements, the Mixed Interpolation of Tensorial Components (MITC) method [20][21][22] was employed, which overcomes locking by offering a two-level approximation that samples and interpolates strain components in a covariant coordinate system at a selection of positions.…”

“…In order to eliminate locking phenomena which arise with conforming displacement-based shell elements, the Mixed Interpolation of Tensorial Components (MITC) method [20][21][22] was employed, which overcomes locking by offering a two-level approximation that samples and interpolates strain components in a covariant coordinate system at a selection of positions. The application of the MITC method to each constitutive layer of the 6-noded and 9-noded laminated shell elements has been shown to yield improved element performance [18][19]. Furthermore, a co-rotational approach has been developed [23][24] and adopted for geometric nonlinear analysis of the laminated shell elements, allowing the consideration of large displacement and buckling analysis.…”