A 2D equivalent single-layer formulation for the effect of transverse shear on laminated plates with curvilinear fibres

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a b s t r a c tThe flexural response of laminated composite and sandwich beams is analysed using the notion of modelling the transverse shear mechanics with an analogous mechanical system of springs in series combined with a system of springs in parallel. In this manner a zig-zag function is derived that accounts for the geometric and constitutive heterogeneity of the multi-layered beam similar to the zig-zag function in the refined zig-zag theory (RZT) developed by Tessler et al. (2007). Based on this insight a new equivalent single layer formulation is developed using the principle of virtual displacements. The theory overcomes the problem in the RZT framework of modelling laminates with Externally Weak Layers but is restricted to laminates with zero B-matrix terms. Second, the RZT zig-zag function is implemented in a third-order theory based on the Hellinger-Reissner mixed variational framework. The advantage of the Hellinger-Reissner formulation is that both in-plane and transverse stress fields are captured to within 1% of Pagano's 3D elasticity solution without the need for additional stress recovery steps, even for highly heterogeneous laminates. A variant of the Hellinger-Reissner formulation with Murakami's zig-zag function increases the percentage error by an order of magnitude for highly heterogeneous laminates. Corresponding formulations using the Reissner Mixed Variational Theory (RMVT) show that the independent model assumptions for transverse shear stresses in this theory may be highly inaccurate when the number of layers exceeds three. As a result, the RMVT formulations require extra post-processing steps to accurately capture the transverse stresses. Finally, the relative influence of the zig-zag effect on different laminates is quantified using two non-dimensional parameters.

Section: Hellinger-reissner Zig-zag Theorymentioning

confidence: 99%

On displacement-based and mixed-variational equivalent single layer theories for modelling highly heterogeneous laminated beams

Groh

Weaver

2015

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“…40:1, taking the shorter inplane dimension which drives the buckling mode shape (refer to Section 2 ), sandwich panels are typically used in regions where enhanced structural efficiency, in terms of bending and buckling, is required due to large distances between supported boundaries. Furthermore, transverse shear effects become significant in composite sandwich panels at much lower thickness to characteristic length ratios than monolithic composites due to the high ratio of in-plane stiffness to transverse shear stiffness present in sandwich panels ( Groh et al, 2013 ). The critical buckling load was determined for the case of uniform end-shortening with longitudinal edges free to expand and all edges simply-supported.…”

Section: Parametric Studymentioning

confidence: 99%

“…However, recent advancements of automated fibre placement (AFP) and tape laying technologies have led to the possibility of steering fibres in the plane of a ply, thus creating variable-stiffness laminates and significantly increasing the design space available to engineers. In recent times, performance benefits of VS laminates have been shown for: the buckling and post-buckling of plates ( Groh et al, 2013;IJsselmuiden et al, 2010;Olmedo and Gürdal, 1993;Raju et al, 2015;Weaver et al, 2009;Wu et al, 2013;2012b ), shells ( White et al, 2014 ) and stiffened panels ( Coburn and Weaver, 2015;Coburn et al, 2014a;2014b ); the stress distribution around discontinuities ( Hyer and Lee, 1991;Khani et al, 2011 ); * Corresponding author. Tel.…”

Section: Introductionmentioning

confidence: 98%

Buckling analysis, design and optimisation of variable-stiffness sandwich panels

Coburn

Weaver

2016

a b s t r a c tIn recent years variable-stiffness (VS) technology has been shown to offer significant potential weight savings and/or performance gains for both monolithic and stiffened plate structures when buckling is a driving consideration. As yet, little work has been reported on VS sandwich structures. As such, a semianalytical model is developed based on the Ritz energy method for the buckling of sandwich panels with fibre-steered VS face-sheets. The model captures both global and shear crimping instabilities and is shown to explain both types of buckling responses observed and the mode switching between them. Quantitative agreement with detailed three-dimensional finite element analysis was found to be within 13% . Subsequent parametric and optimisation studies, which were performed for many practical geometries using the developed model, reveal that, whilst VS sandwich panels show a significant improvement in global buckling performance, they suffer a reduction in shear crimping performance when compared to their straight-fibre counterparts. This behaviour is found to be due to the VS face-sheets creating a pre-buckling load redistribution where regions locally exceed the critical shear crimping load and induce the short wavelength instability at a reduced panel level load. For VS sandwich panels with modest to low transverse shear moduli of the core, shear crimping can become the critical mode diminishing performance benefits relative to straight-fibre configurations. Cores with sufficient rigidity, thus preventing shear crimping, showed improvements in critical buckling load in the order of 80% when using VS, however this improvement reduces to a negligible amount with decreasing core transverse shear moduli. The transverse shear flexibility and load redistribution are thus two key parameters that must be considered carefully in the design of sandwich panels, in order to exploit the benefits of VS fully in this novel structural configuration.

“…It has been previously shown that DQM is an efficient strategy for solving the stretching and bending problems [22,23] of composite laminates.…”

Section: Differential Quadrature Methodsmentioning

confidence: 99%

Higher-order beam model for stress predictions in curved beams made from anisotropic materials

Thurnherr

Groh

Ermanni

2016

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A higher-order beam model for analyzing the flexural response of curved multilayered beams with constant curvature and arbitrary constant thickness is developed. The new model is derived from the Hellinger-Reissner mixed variational statement and predicts inherently equilibrated 3D stresses from an equivalent single-layer model. As a starting assumption, the hoop stress is formulated as a series of higher-order stress resultants multiplied by Legendre polynomials. The governing equations are derived in a generalized manner such that the modeling order can be adjusted and is not defined a priori. Hence, the highest order Legendre polynomial determines the modeling order. The through-thickness shear and normal stresses are derived by integrating the generalized hoop stress in Cauchy's polar equilibrium equations. As a result, all stress fields are based on the same set of variables, thereby considerably reducing the computational effort. The three stress fields, and two displacements in the radial and hoop directions are used in the Hellinger-Reissner functional to derive a new set of stress-displacement relations. The enforcement of the classical membrane and bending equilibrium equations of curved beams in the Hellinger-Reissner functional guarantees that all interlaminar and surface traction equilibrium conditions are satisfied exactly. A validation study of a composite laminate using a high-fidelity 3D finite element model shows that the stresses are captured very accurately by the present model, but with much less computational effort than the finite element model. As a result, the developed model can provide rapid and accurate insights into the expected damage onset behavior of curved laminates.