Transverse Shear Stress Continuity Research Articles

A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core

A new finite element (FE) model has been developed based on higher order zigzag theory (HOZT) for the static analysis of laminated sandwich beam with soft core. In this theory, the in-plane displacement variation is considered to be cubic for both the face sheets and the core. The transverse displacement is assumed to vary quadratically within the core while it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. The nodal field variables are chosen in an efficient manner to overcome the problem of continuity requirement of the derivatives of transverse displacements. A C0 quadratic beam finite element is implemented to model the HOZT for the present analysis. Numerical examples covering different features of laminated composite and sandwich beams are presented to illustrate the accuracy of the present model. Many new results are also presented which should be useful for future research.

Variable-Kinematics Approach for Linearized Buckling Analysis of Laminated Plates and Shells

This work deals with the linearized buckling analysis of laminated plates and shells. A variable-kinematics approach with hierarchical capabilities is considered to establish the accuracy of a large variety of classical and advanced plate/shell theories in order to evaluate buckling loads. So-called equivalent-single-layer as well as layerwise-variable descriptions are implemented. Interlaminar continuity of transverse shear and normal stresses are a priori fulfilled by referring to Reissner's mixed variational theorem. Stability equations are derived in compact form by referring to Carrera's unified formulation for the most general case of doubly curved shells. The eigenvalue problem is solved in the case of closed-form solutions related to simply supported boundary conditions and axially loaded multilayered plates/shells made of orthotropic layers. The casesofaxial constant strains and constant stresses are considered and compared to available three-dimensional and two-dimensional results. Results related to Love and Donnell approximations are implemented for comparison purposes. The accuracy of various approximations is established for significant multilayered plate and shell problems.

Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams

Large amplitude periodic forced vibration of curved beams under periodic excitation is investigated using a three-noded beam element. The element is based on the higher-order shear deformation theory satisfying interlayer continuity of displacements and transverse shear stress, and top-bottom conditions on the latter. The periodic responses are obtained using shooting technique coupled with Newmark time marching and arc length continuation algorithm developed. The second order governing differential equations of motion are solved without transforming to the first order differential equations thereby resulting in a computationally more efficient algorithm. The effects of excitation amplitude, support conditions and beam curvature on the frequency versus response amplitude relation are highlighted. The typical frequency response curves for isotropic and cross-ply laminated curved beams are presented. Phenomenon of strong modal interactions is observed.

Analysis of Laminated Sandwich Plates Based on an Improved Higher Order Zigzag Theory

A finite element model based on an improved higher order zigzag plate theory developed by the authors is refined in this study and applied to bending and vibration response of soft core sandwich plates. The theory satisfies interlayer transverse shear stress continuity including transverse shear stress free condition at the plate top and bottom surfaces and transverse normal compressibility of the core. The in-plane displacements vary cubically through the entire thickness, while transverse displacement is assumed to vary quadratically within the core. In order to have a better computational benefit, a C0 finite element formulation is adopted. This is refined through satisfaction of certain constrains variationally using a penalty function approach. The performance of the model is demonstrated by comparing the present results with 3D elasticity solutions and other available results.

Stochastic perturbation-based finite element for deflection statistics of soft core sandwich plate with random material properties

The effect of random variations in material properties of laminated sandwich plates on the transverse deflections is investigated. An improved higher-order plate model is proposed earlier by the authors, which satisfies the transverse shear stress continuity conditions at the layer interfaces including the zero transverse shear stress conditions at the plate top and bottom surfaces. The theory assumes the variation of in-plane displacements to be cubic with discontinuities in the transverse shear strains at the layer interfaces, while the transverse displacements varies quadratically across the core thickness, thereby including transverse normal deformation of the soft core. The core is considered to behave as a 3-D elastic medium. To obtain the second-order statistics of deflections of sandwich plate, a stochastic C 0 finite element (FE) based on the first-order perturbation technique is developed, where the lamina properties are considered as basic random variables while the other system properties are assumed to be deterministic. The performance of the improved stochastic laminated sandwich model is demonstrated through comparison of mean and standard deviations (SDs) of deflections obtained through independent Monte Carlo simulations and by comparison with results available in literature.

Stress analysis of laminated composite plates with a circular hole according to a single-layer higher-order model

This paper presents a single-layer higher-order model for predicting the stresses at curved free boundaries of laminated composite plates subjected to inplane loading. The proposed model can automatically satisfy the continuity of transverse shear stresses at interfaces. Based on both the proposed model and the methodology of the discrete Kirchhoff plate bending element, a triangular finite element is also presented. Numerical examples of curved free-edge problems have been taken into account to illustrate the performance of the present approach. Numerical results showed that the proposed model is capable of predict inplane and interlaminar stresses around the circular hole. In addition, distribution of free-edge stress through-the-thickness of a laminate is also presented, which are usually unavailable in literature.

Buckling of Sandwich Plates with Random Material Properties Using Improved Plate Model

An improved higher-order zigzag plate theory in a random environment is proposed and it is implemented in a stochastic finite element framework to study the buckling characteristic of sandwich plates with random material properties. The theory satisfies transverse shear stress continuity at all of the layer interfaces and the transverse shear-stress-free condition at the top and bottom surfaces. It also includes the effect of transverse deformability to the core. The through-thickness variation of in-plane displacements is assumed to be cubic, whereas transverse displacement varies quadratically within the core and it remains constant over the face sheets. Note that sandwich and composite structures are characterized by inherent uncertainties in their material properties. The effect of these uncertainties on the buckling characteristic of sandwich plates is studied by modeling the macromechanical material properties as basic random variables. A stochastic finite element method consisting of an efficient C° finite element in conjunction with a mean-centered first-order perturbation approach is developed, and the model is employed to evaluate the second-order statistics of the buckling loads. The published results are used to validate the deterministic part of the proposed approach, and its stochastic component is validated with an independent Monte Carlo simulation.

A higher-order displacement model for stress concentration problems in general lamination configurations

Abstract The higher-order displacement model has been proposed to analyze the stress concentration problems in angle-ply laminates under different loads. The proposed model a priori satisfies the transverse shear stress continuity conditions as well as free surface conditions. The total number of degrees of freedom in this model does not depend on the layer number of laminates. Thus, computations of the proposed model are simple and efficient. It is important that the proposed model is extended to solve the stress concentration problems in laminates with arbitrary stacking sequences. Based on the proposed model, a three-node triangular element has been presented. To circumvent the requirement of C 1 continuity at element interfaces, the refined element method has been employed. Numerical results show that the proposed model is suitable for the free-edge problems in the cross-ply laminates as well as the angle-ply laminates. After ascertaining the accuracy of present model, the model is extended to analyze the free-corner problems in laminates subjected to thermal loads. Numerical results imply that the analytical approach proposed by Becker et al. seem to underestimate the peak stresses in the vicinity of free corner in laminates.

Vibration characteristic of laminated sandwich plates with soft core based on an improved higher-order zigzag theory

This paper presents an improved higher order zigzag theory for vibration of laminated sandwich plates. It ensures continuity of transverse shear stresses at all the layer interfaces and transverse shear stress-free condition at the top and bottom surfaces apart from core compressibility. The through-thickness variation of in-plane displacements is assumed to be cubic, whereas transverse displacement varies quadratically across the core, which is modelled as a three-dimensional elastic continuum. An efficient C0 finite element is developed for the implementation of the plate theory. The model is validated using three-dimensional elasticity solutions and some other relevant results available in the literature.

A new shear deformation theory for laminated composite plates

Abstract In the present study, a new higher order shear deformable laminated composite plate theory is proposed. It is constructed from 3-D elasticity bending solutions by using an inverse method. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. It was observed that this theory gives most accurate results with respect to 3-D elasticity solutions for bending and stress analysis when compared with existing five degree of freedom shear deformation theories [Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51:745–52; Touratier M. An efficient standard plate theory. Int J Eng Sci 1991;29(8):901–16; Karama M, Afaq KS, Mistou S. Mechanical behaviour of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct 2003;40:1525–46]. All shear deformation theories predict the vibration and buckling results with reasonable accuracy, generally within %2 for investigated problems. Previous exponential shear deformation theory of Karama et al. (2003) can be found as a special case.

Vibration Analysis of Inclusion Reinforced Composite Square Plates under Various Boundary Conditions

The vibration behavior of E-glass epoxy cross-ply laminated plates containing unidirectionally aligned inclusions was studied under different boundary conditions by using first-order shear deformation theory (FSDT). The effective elastic modulus of composite was expressed by using the Mori—Tanaka mean field approach for unidirectionally aligned inclusions. The governing equations were obtained by means of the Hamilton's principle and solved by using the Navier-type solution and Ritz method. The continuity of transverse shear stress between layers of the symmetric cross-ply laminated plates was satisfied by the use of the shape function incorporated into the theory. The effects of the inclusion volume fraction, the inclusion aspect ratio, and the plate span to thickness ratio on the vibration behavior of the laminated plates were studied. It was found that the dimensionless frequency increases with increasing the aspect ratio of the reinforcements and approaches asymptotically to frequency of the continuous fibers.

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C1 continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.

Buckling of laminated sandwich plates with soft core based on an improved higher order zigzag theory

An improved higher order zigzag theory is presented and it is applied to study the buckling of laminated sandwich plates. The present theory satisfies the conditions of transverse shear stress continuity at all the layer interfaces including transverse shear stress free conditions at the top and bottom surfaces of the plate. The variation of in-plane displacements through thickness direction is assumed to be cubic for both the face sheets and the core, while transverse displacement is assumed to vary quadratically within the core but it remains constant over the face sheets. The core is modeled as a three-dimensional elastic continuum. An efficient C 0 finite element is proposed for the implementation of the improved plate theory. The accuracy and range of applicability of the present formulation are established by comparing the present results with 3D elasticity solutions and other results available in literature.

An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams

Abstract Several displacement-based theories are assessed by analyzing the free vibration and the buckling behaviors of laminated beams with arbitrary layouts as well as soft-core sandwich beams. The equations governing the dynamic response of laminated structures are derived by using Hamilton’s principle. However, equations of equilibrium for buckling problems are given by employing the principle of virtual displacements. Moreover, using Navier’s technique and solving the eigenvalue equations, analytical solutions based on the global–local higher-order theory used in this paper are first presented in present study. At the same time, the effect of the order number of higher-order shear deformation as well as interlaminar continuity of transverse shear stress on the global response of both laminated beams and soft-core sandwiches has been also studied. Numerical results show that by increasing the order number of in-plane and transverse displacement components, the global higher-order theories can reasonably predict the natural frequencies and the critical loads of laminated beams with arbitrary layouts and soft-core sandwich beams whereas these global higher-order theories are still less accurate compared to the global–local higher-order theory and the zig-zag theory used herein.

A global-local higher order theory for multilayered shells and the analysis of laminated cylindrical shell panels

Based on the global-local superposition technique proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212.], a global-local higher order laminated shell model is proposed for predicting both displacement and stress distributions through the thickness of laminated shells. This shell model satisfies transverse shear stress continuity conditions at interfaces as well as free surface conditions of transverse shear stresses. The merit of this model is that transverse shear stresses can be accurately predicted directly from constitutive equations without smoothing techniques. Cylindrical bending of laminated and sandwich shell panels is chosen to assess the present model wherein the results from several 2D laminated shell models and three-dimensional elasticity solution are available for comparison. In addition, thermal bending and thermal expansion of laminated cylindrical shell panels are also considered in this paper.

Benchmark 3D solution and assessment of a zigzag theory for free vibration of hybrid plates under initial electrothermomechanical stresses

A benchmark three-dimensional (3D) exact piezothermoelasticity solution is presented for free vibration of simply supported cross-ply symmetrically laminated hybrid rectangular plates under initial in-plane electrothermomechanical stresses. The solution, based on second Piola stress tensor and nonlinear Lagrangian strains, accounts for the effect of initial transverse normal strain and the two-way electromechanical coupling. Benchmark results are presented for the natural frequencies under initial stresses and the modal distributions of displacements and stresses across the thickness, against which the accuracy of various 2D plate theories for hybrid plates can be assessed. A recently developed coupled 2D zigzag theory for hybrid plates is extended for the vibration under initial stresses. The theory uses a piecewise linear approximation for the thermal and potential fields across sublayers, and combines a global third-order variation with a piecewise linear (zigzag) variation across the layers for the inplane displacements. Using the conditions of continuity of transverse shear stresses at the layer interfaces and the shear traction-free conditions at the top and bottom, the number of primary displacement variables is reduced to five, as in the corresponding smeared third-order theory (TOT). The zigzag theory and the smeared TOT are assessed in comparison with the exact 3D solution for the natural frequencies and mode shapes of simply-supported rectangular plates under initial stresses. The zigzag theory results show excellent agreement with the 3D results when the initial thermal transverse normal strain is neglected, whereas the TOT may yield very inaccurate results depending upon the plate layup.

Effects of higher-order global–local shear deformations on bending, vibration and buckling of multilayered plates

Abstract In this paper, the effects of higher-order shear deformations on bending, natural frequencies and buckling loads of multilayered plates are studied. Based on the power series expansion of displacement components along thickness coordinate z and the double-superposition hypothesis proposed by Li and Liu [Li XiaoYu, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212], the m th-order global–local theory which satisfies the free surface conditions and the geometric and transverse shear stress continuity conditions at interfaces is presented. A refined four-node quadrilateral element is proposed for the numerical modeling of the global–local theory. In the process of finite element implementation, both the first and the second derivatives of transverse displacement w appear in the strain components. Transverse displacement functions satisfying C 0 and C 1 weak-continuity between elements are derived based on the refined element methods. Numerical results show that the transverse shear stress of multilayered plates, the natural frequency and the buckling loads of soft-core sandwich plates predicted by the proposed models of higher-order shear deformation are more accurate in comparison with the results based on the third-order global local theory and other lower order theories.

Comparison of Various Shear Deformation Theories for Bending, Buckling, and Vibration of Rectangular Symmetric Cross-ply Plate with Simply Supported Edges

In the present study, unified shear deformation theory which was proposed by Soldatos, K.P. and Timarci, T. (1993). A Unified Formulation of Laminated Composite, Shear Deformable Five-degrees-of-freedom Cylindrical Shells on the Basis of a Unified Shear Deformable Shell Theory, Compos. Struct., 25(1–4): 165–171 is used to analyze simply supported symmetric cross-ply rectangular plates for deflections, stresses, natural frequencies, and buckling loads. This theory enables the selection of different in-plane displacement components to represent shear deformation. Exponential shear deformation theory, which is proposed by Karama et al., 40(6) (2003). Mechanical Behavior of Laminated Composite Beam by New Multi-layered Laminated Composite Structures Model with Transverse Shear Stress Continuity, Int J Solids Struct., 40: 1525–1546, is used for the first time to analyze the problem considered. Results which are found by exponential theory are compared with those obtained by using the parabolic shear deformation theory of Reddy, J.N., 51(4) (1984). A Simple Higher-order Theory for Laminated Composite Plates, J Appl Mech., 51: 745–752, the trigonometric shear deformation theory of Touratier, M., 29(8) (1991). An Efficient Standard Plate Theory, Int. Jnl. Engng. Sci., 29(8): 901–916, the hyperbolic shear deformation theory of Soldatos, K.P., 94(3–4) (1992). A Transverse Shear Deformation Theory for Homogeneous Monoclinic Plates, Acta Mech., 94: 1995–220 and with the available three-dimensional elasticity solutions. The study shows that while the transverse displacement and the stresses are best predicted by the exponential shear deformation theory, the parabolic shear deformation and the hyperbolic shear deformation theories yield more accurate predictions for the natural frequencies and the buckling loads.

Spline finite strip analysis of composite plates based on higher-order zigzag composite plate theory

A spline finite strip method is developed for static analysis of composite plates using the higher-order zigzag composite plate theory presented by Cho and Parmerter. The theory uses the same number of unknowns as Reddy’s higher-order shear deformation theory and Mindlin’s first-order shear deformation theory, but satisfies the transverse shear stress continuity conditions at the interface between layers as well as the shear free surface conditions, leading to an improved accuracy, especially for the multilayer asymmetric laminates. Using this theory, the transverse shear stresses can be calculated directly and correctly from the constitutive equations, and an artificial shear correction factor is not needed. Results are compared with solutions based on three-dimensional elasticity theory and other shear deformation theories.

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

A recently developed coupled third-order zigzag theory for the statics of piezoelectric hybrid cross-ply plates is extended to dynamics. The theory combines a third-order zigzag approximation for the in-plane displacements and a sub-layerwise linear approximation for the electric potential, considering all components of the electric field. The nonuniform variation of the transverse displacement due to the piezoelectric field is accounted for. The conditions for the absence of shear traction at the top and bottom surfaces and continuity of transverse shear stresses in the presence of electromechanical loading are satisfied exactly, thereby reducing the number of displacement variables to five, which is the same as in a first- or third-order equivalent single-layer theory. The governing equations of motion are derived from the extended Hamilton's principle. The theory is assessed by comparing the Navier solutions for the free and forced harmonic vibration response of simply supported plates with the exact three-dimensional piezoelasticity solutions. Comparisons for hybrid test, composite and sandwich plates establish that the present theory is quite accurate for the dynamic response of moderately thick plates.