Transverse Shear Stress Continuity Research Articles

Layerwise third-order shear deformation theory with transverse shear stress continuity for piezolaminated plates

Regarding laminated structures, an electromechanically coupled Finite Element (FE) model based on Layerwise Third-Order Shear Deformation (LW-TOSD) theory is proposed for static and dynamic analysis. LW-TOSD ensures the continuity of in-plane displacements and transverse shear stresses. The current LW-TOSD can be applied to arbitrary multi-layer laminated structures with only seven Degrees of Freedom (DOFs) for each element node and eliminates the use of the shear correction factors. Moreover, a shear penalty stiffness matrix is constructed to satisfy artificial constraints to optimize the structural shear strain. A dynamic finite element model is obtained based on LW-TOSD using the Hamilton’s principle. First, the accuracy of the current model is validated by comparing with literature and ABAQUS results. Then, this study carries out numerical investigations of piezolaminated structures for different width-to-thickness ratios, length-to-width ratios, penalty stiffness matrix, boundary conditions, electric fields and dynamics.

A compressible layerwise third-order shear deformation theory with transverse shear stress continuity for laminated sandwich plates

Sandwich structures with a soft core typically exhibit multiple deformation modes such as bending, twisting and compression when subjected to transverse loads. Compressive behavior of soft core significantly affects the structural response, which is usually neglected in conventional plate theories. Additionally, conventional plate theories do not take into account transverse shear stress continuity. In order to address this concern and precisely characterize the characteristics of arbitrary multilayer laminated structures, a novel electromechanical coupled finite element model is developed. This model is based on a compressible layerwise third-order shear deformation theory with a soft core. Notably, the proposed model theory requires only twelve degrees of freedom for each quadratic element node. The accuracy of the model is validated first. Afterwards, several numerical investigations are carried out for piezolaminated sandwich structures with different width-to-thickness ratios, reinforcement angle and skew angle. The results demonstrate the applicability of the current model in predicting the behavior of laminated sandwich plates.

Application of UPV-Instrument in Health Monitoring of Indian Rail Section Using AE Technique

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Buckling Analysis of Laminated Composite Plates under Thermal Conditions

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Higher-order vibration of thick composite and sandwich plates based on an alternative higher-order model

The transverse stretching vibration of thick sandwich plates, which is attributed to largely different stiffness at the adjacent layers, is a challenging issue, and efficient approach for such issue is less reported in the published literature. Thus, natural frequencies corresponding to stretching vibration modes are generally neglected in engineering design, which might impact structural safety as frequencies of the exciting force are close to transverse stretching vibration frequencies. This paper proposes an alternative higher-order model for dynamic analysis corresponding to the higher-order vibration modes. The proposed model is classified in the displacement-based equivalent single-layer theory, as the number of displacement parameters in the proposed model is independent of the layer number. The continuity of displacements and transverse shear stresses can be fulfilled at the interfaces between the adjacent layers of structures. To demonstrate the capability of the proposed model, typical examples are analyzed by utilizing the proposed model, the three-dimensional finite element method and the chosen higher-order models. By comparing with the exact three-dimensional elasticity solutions, it is found that the proposed model can yield more accurate natural frequencies corresponding to the higher-order displacement modes than the selected models. Moreover, the factors influencing reasonable prediction of the higher-order frequencies are investigated in detail, which can provide a reference for the accurate prediction of the higher-order frequencies.

New HSDT for free vibration analysis of elastically supported porous bidirectional functionally graded sandwich plate using collocation method

The present paper deals with the free vibration response of elastically supported porous bidirectional functionally graded soft-core sandwich rectangular plate with proposed new tangent shape function-based higher-order transverse shear deformation theory. The theory is accomplished to uphold the continuity of transverse shear stresses and zero shear stress conditions on both the extreme surfaces of a plate. The bidirectional gradation varies through the thickness ( z-axis) and length ( x-axis) directions. The effective material properties are calculated via the modified power-law. The governing differential equations (GDEs) of motion are acquired via the Hamilton principle. The inverse multi-quadric radial basis function-based Collocation method is implemented to discretize the GDEs obtained by strong form formulation. The current theory and method’s accuracy and efficacy are validated by comparing the present results with existing results in the literature. Studies show that the results of the proposed theory are in close agreement with 3D approach. Numerical results are obtained with a different scheme of sandwich layers for free vibration analysis. Effects of grading index, porosity fraction, two parameters foundation, different scheme, and the span to thickness ratio on free vibration response have been discussed. New results for free vibration response in porous media are presented with different skin materials.

Bending and free vibration analysis of symmetric and unsymmetric functionally graded CNT reinforced sandwich beams containing softcore

In the present work, bending and free vibration analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) sandwich beams are carried out using finite element-based higher-order zigzag theory. Face sheets are assumed to be made up of FG-CNTR composite, and the core is assumed to be made up of balsa wood (softcore). The present formulation also takes into account transverse normal stresses. The computational model incorporates transverse shear stress and transverse normal stress continuity condition at interfaces. Zero transverse shear stress condition at the bottom and top surfaces of the beam is also satisfied. The principle of minimum potential energy is employed for carrying out bending analysis, while Hamilton’s principle is adopted for free vibration analysis. The investigation is carried out for different gradation laws which govern the distribution of CNTs across the thickness of face sheets. The influence of the core’s thickness on stresses and displacements is also critically analyzed in the present work. It has been observed that the thickness of the core and CNT gradation law significantly affect the mechanical behavior of the sandwich FG-CNTRC beam.

Non-polynomial zigzag theories with C° finite element formulation for buckling analysis of laminated composite and sandwich plates

In this present work, non-polynomial zigzag theories (algebraic zigzag theory (AZT), exponential zigzag theory (EZT), hyperbolic zigzag theory (HZT), inverse hyperbolic zigzag theory (IZT), logarithmic zigzag theory (LZT) and trigonometric zigzag theory (TZT)) are performed for buckling response of laminated composite and sandwich plates. The present models assume parabolic variation of out – plane stresses through the depth of the plate and also accomplish the zero transverse shear stresses over the surface of the plate. Thus a need of shear correction factor is obviated. The present zigzag models able to meet the transverse shear stress continuity and zigzag form of in-plane displacement continuity at the plate interfaces. An efficient eight noded C° continuous isoparametric serendipity element is established and employed to examine the buckling analysis. Like FSDT, the considered mathematical model possesses similar number of variables and which decides the present models computationally more effective. Several numerical examples are carried out to study the effects of span to thickness ratio, ply orientation, lay-up number, modular ratio, loading condition and boundary condition on the buckling response. To ensure the capability of the proposed models, higher modes of buckling are obtained for laminated plates and sandwich plates. Further, the efficiency and superiority of the proposed models is ascertained by comparing it with 3 D elasticity solution and also with various existing shear deformation theories in the literature. Most remarkably, the present models are accurately estimates the buckling load parameter and they are insensitive of shear-locking.

Hygro-thermo-mechanical based bending analysis of symmetric and unsymmetric power-law, exponential and sigmoidal FG sandwich beams

In the present work, an attempt has been made to carry out the bending analysis of sandwich FGM beams under combined hygro-thermo-mechanical loadings. Temperature and moisture-dependent material properties are used during the present study. A comparative study has also been carried out between the beams made up of different homogenization rules. The study has been carried out using recently proposed finite element-based HOZT. The present model satisfies the inter-laminar transverse shear stress continuity condition at interfaces and the zero condition at the top and bottom surfaces of the beam. The data is presented in the form of tables for displacement and stresses and graphs representing stress distribution across the thickness of the beam. For the first time, upward displacement of the sandwich FGM beam has been reported in the present work when subjected to hygro-thermo-mechanical loading. Several new results are also reported in the present work, which will serve as a benchmark for future studies in a similar direction.

Investigation of layered composite plates under acoustic emission using an appropriate finite element model

Structural damages generate acoustic emission (AE) in the media and cause extensional and flexural acoustic waves. Often in structures like plates, flexural mode is predominant. In this study, the flexural mode AE waveforms due to simulated damage are studied for multi-layered composite plates. A generalized refined 2D plate theory, which satisfies the transverse shear stress continuity at the layer interfaces, is proposed here for modelling the plates. This formulation is implemented through finite element method (FEM) where a four-node rectangular element, that satisfies C1 continuity, is used. Plates, having different thickness ratios, are studied through numerical examples using the model. Results are validated wherever applicable and some new results are obtained. The results indicate that the proposed model can simulate the flexural waveforms realistically for ‘very thin’ to ‘moderately thick’ plates. It is also found that the present model is as accurate as 3D FEM but it possesses much better computational efficiency.

Bending analysis of laminated sandwich plates under hygrothermal loadings

Abstract In present work, hygrothermal based analysis of laminated composite and sandwich plates is carried out by using recently proposed finite element based higher-order zigzag theory. Present formulation satisfies transverse shear stress continuity condition at top and bottom surface of the plate. Also, the model satisfies transverse shear stress free condition at the bottom and top surface of the plate. A nine-noded C-0 Lagrangian finite element having eleven degrees of freedom per node is used during study. Proposed model is free from any kind of inclusion of penalty function. Effectiveness of the present model is carried out by comparing the present results with those available in the literature.

A new C0 layerwise wavelet finite element formulation for the static and free vibration analysis of composite plates

In this paper, a new C0 layerwise wavelet finite element is proposed for the static and free vibration analysis of composite plates. The refined zigzag theory is adopted to introduce the zigzag effects in multilayered plate structures by using piecewise linear C0 continuous functions. Then the layerwise wavelet-based BSWI element is derived based on the higher-order plate theory by means of two-dimensional BSWI scaling functions. The proposed model satisfies the conditions of transverse shear stress continuity at the layer interfaces as well as yields to the stress-free boundary conditions on the surface of plate without a shear correction factor. What’s more, the layerwise wavelet-based BSWI element also possesses the advantages of high convergence, high accuracy and reliability with fewer degrees of freedom on account of the excellent approximation property of BSWI. The accuracy and effectiveness of proposed layerwise wavelet-based BSWI element is assessed for static and free vibration analysis of laminated composite and sandwich plates with available 3D elasticity solutions, finite element solutions and other referential solutions in published literatures.

Free vibration analysis of laminated sandwich plates under thermal loading

In present work, free vibration analysis of laminated sandwich plates is carried out under thermal loading. C-0 finite element based higher-order zigzag theory is used during analysis. Third-order variation of in-plane displacement field is assumed while for transverse displacement field, quadratic variation is used for core region and is taken constant for the face layers. Present model is free from any kind of penalty requirements. Also, the proposed model satisfies transverse shear stress continuity condition at interface and same is zero at top and bottom faces. Present results are compared with those available in literature. New results for sandwich plates are proposed in the present work which will serve as benchmark for future studies.

Exact solution considering layerwise mechanics for laminated composite and sandwich curved beams of deep curvatures

In this work, a third-order efficient layerwise theory (ELWT) has been developed for laminated composite and sandwich curved beams of deep curvatures. The circumferential displacement is assumed to have global third-order variation in thickness (radial) coordinate with a linear layerwise variation. The number of independent variables is reduced to 3 by imposing the continuity of displacement and transverse shear stress at interfaces and shear free conditions on the outer and inner surfaces. Equations of the motion are derived using Hamilton’s principle. Navier type analytical solution is obtained for simply supported ends. Results for static deflection, stresses, and natural frequencies are presented for laminated composites and sandwich curved beams of different radii of curvature and thicknesses. The importance of inclusion of the deepness terms (1+z/R) in the formulation for the static and free vibration responses are discussed by comparing with the exact 2D elasticity solution. It is shown that the results predicted by the present ELWT are more accurate than equivalent single layer theories having same number of variables. For sandwich curved beams, the predictions of third-order theory (TOT) are extremely poor in comparison with ELWT. The results presented in the paper will be useful for assessing the accuracy of other simplified 1D theories.

Nonpolynomial Zigzag Theories for Random Static Analysis of Laminated-Composite and Sandwich Plates

The present paper aims to study the random static behavior of laminated composite and sandwich plates using nonpolynomial zigzag theories. The nonpolynomial zigzag theories are proposed earlier by the authors, which satisfy the transverse shear-stress continuity conditions at the layer interfaces including the traction-free boundary conditions on the top and bottom surfaces of the plate. These theories incorporate the realistic nonlinear distribution of transverse shear stresses across the thickness of the plate. A probabilistic procedure is developed using a stochastic finite element method that is in the conjunction of finite element method with a mean-centered first-order perturbation technique, to obtain the second-order statistics of deflections of the laminate. The problem is analyzed considering the material properties of laminated composite and sandwich structures to be random in nature while the other system properties are assumed to be deterministic. The stochastic results in terms of mean as well as standard deviation of the responses have been obtained for various combinations of geometric parameters, stacking sequences, span–thickness ratios, and boundary conditions and compared with those existing in the literature and Monte Carlo simulation approach.

A refined quasi-3D zigzag beam theory for free vibration and stability analysis of multilayered composite beams subjected to thermomechanical loading

A refined four-unknown quasi-3D zigzag beam theory is developed to model the free vibration and buckling behaviors of multilayered composite beams subjected to axial mechanical loading (e.g., distributed load and terminal force) and uniform temperature variation. Types of the composite beams considered include laminated composite beams, sandwich beams with composite face sheets, and fiber metal laminates. The proposed theory accounts for not only thickness stretching but also interlaminar continuity of transverse shear stresses and displacements. Associated eigenvalue problems for various boundary conditions are derived using the Ritz method. Accuracy and effectiveness of the theoretical predictions are verified by comparison with existing results and present finite element simulations. The theory is employed to quantify the effects of axial distributed load/terminal force and temperature variation on free vibration and buckling for different boundary conditions, geometric parameters and material properties. The present theory could produce sufficiently accurate predictions of natural frequencies and buckling capacities of multilayered beams at a very low computational cost.

Eccentric low-velocity impact on fiber-metal laminates under in-plane loading using unified zigzag theory

This paper investigates the eccentric low-velocity impact of Fiber metal laminates (FMLs) subjected to spherical projectile using a unified Zig-Zag plate theory. The presented zig-zag plate theory enforces transverse shear stress continuity through the thickness and can be reduced to conventional plate theories using appropriate shape function. The governing equations and suitable boundary conditions are obtained using the principle of minimum total potenital energy. Runge-Kutta method is employed to solve initial value problem resulted by the method of Ritz. The present model is validated by comparison and good agreement between its results and those of reports in open literature. Influence of various specifications of impact phenomenon such as laminate thickness, projectile radius, projectile velocity, in-plane load and eccentricity parameter is examined on deflection and contact force time history. The obtained results indicate that continuity of transverse shear stress is required to achieve accurate contact force even for moderately thin FMLs.

Static analysis of highly anisotropic laminated beam using unified zig-zag theory subjected to mechanical and thermal loading

In the present study, static behavior of short hybrid laminate beams was investigated using a unified zig-zag theory (ZZT) containing various beam theories as special cases. This theory satisfies transverse shear stresses continuity in the interface of layers via piece-wise continuous arbitrary shape functions. The principle of virtual work was employed to derive unified equilibrium equations and suitable boundary conditions. The present theory obviates the need for stress recovery for continuous transverse stresses. A general solution was presented to analyse high transversely anisotropic laminates under several kinds of transverse loads (general lateral, sinusoidal and point load) and non-linear thermal loads. The validity of this model is demonstrated by comparison of its predictions and good agreement with published results in literature. Numerical examples were given to investigate the impact of the transverse anisotropy on displacement, strain and stress fields through the thickness. The results show that the piece-wise continuous exponential and sinusoidal shape functions provide more accurate transverse stress distribution in comparison with other shape functions. In addition, the results show that the continuity of transverse shear stress through the thickness plays an important role in analysing transversely anisotropic laminated beams. A comparison of present ZZT and existing exact elasticity solutions shows that the current theory is simple and efficient.

Stability of laminated composite and sandwich beams by a Reddy-type higher-order zig-zag theory

ABSTRACTTransverse shear stresses have a significant influence on the buckling analysis of laminated structures. Thus, the models violating interlaminar continuity of transverse shear stress will encounter difficulty for the buckling analysis. The postprocessing method by integrating three-dimensional equilibrium equation is also invalid, as the transverse shear stresses obtained from the postprocessing method are unable to be involved in the strain energy. In this paper, a Reddy-type higher-order zig-zag theory is firstly proposed, in which the in-plane displacement field is obtained by superposing the developed zig-zag function on the displacement field of Reddy's theory. To predict accurately the buckling behaviors of laminated composite and sandwich beams, a new equilibrium approach in conjunction with the Hu-Washizu (HW) mixed variational principle has been developed to produce the improved transverse shear stresses. Moreover, the accurate transverse shear stresses can be involved in the strain energy which can actively impact the accuracy of buckling stresses. To assess the performance of the proposed method, the critical loads of the laminated composite and sandwich beams with various configurations have been analyzed. Agreement between the present results and the solutions of layerwise are very good, and the proposed model only includes the four displacement parameters which can illustrate the accuracy and effectiveness of the present model. In addition, new results using all the models considered in this paper have been presented which can serve as a reference for future investigations.

A layerwise C0-type higher order shear deformation theory for laminated composite and sandwich plates

A novel layerwise C0-type higher order shear deformation theory (layerwise C0-type HSDT) for the analysis of laminated composite and sandwich plates is proposed. A C0-type HSDT is used in each lamina layer and the continuity of in-plane displacements and transverse shear stresses at inner-laminar layer is consolidated. The present layerwise theory retains only seven variables without increasing the number of variables when the number of lamina layers are intensified. The shear stresses through the plate thickness derived from the constitutive equation of the present theory have the same shape as those calculated from the equilibrium equation. In addition, the artificial constraints are added in the principle of virtual displacements (PVD) and are certainly fulfilled through a penalty approach. In this paper, two C0-continuity numerical methods, such as the Finite Element Method (FEM) and Bézier isogeometric element (BIEM) are utilized to solve a discrete system of equations derived from the PVD. Several numerical examples with various geometries, aspect ratios, stiffness ratios, and boundary conditions are investigated and compared with the 3D elasticity solution, the analytical, as well as, numerical solutions based on various plate theories.