Variation Of Transverse Shear Stresses Research Articles

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

On accurately capturing the through-thickness variation of transverse shear and normal stresses for composite beams using FSDT coupled with GPR

On accurately capturing the through-thickness variation of transverse shear and normal stresses for composite beams using FSDT coupled with GPR

Hygrothermal dynamic behavior of fiber metal laminated structure under dual-ellipse distribution heat source

Refined plate theory could account for parabolic variation of transverse shear stress through the thickness, and satisfy the zero traction boundary conditions on the top and bottom surfaces of the plate without introducing shear correction factor, of which all displacement variables can be depicted in functions of shear and bending components. Based on three-dimensional transient thermal conduction theory and the refined plate theory, an analytical solution of fiber metal laminated (FML) plate under hygrothermal environment is presented. In present work, distinct from other literatures, the separate variable method and Newton-Cotes method is applied to solve the three-dimensional transient temperature field (subjected to moving dual-ellipse distribution laser heat source) firstly. Secondly, the constitutive equations and geometrical equations are derived. Subsequently, the governing equations in hygrothermal environment are obtained. Finally, the Newmark method and Galerkin method are used to solve the governing equations of which in terms of displacement functions. The results show that fiber orientation affects the bending stiffness, the selection of (45°/ 0°) fiber orientation is more resistance to deformation and normal stress, and bending part is more sensitive to shear factor between layers.

Thermo-mechanical coupling analysis of laminated composite plates using wavelet finite element method

This paper proposes a novel high performance wavelet finite element formulation for thermo-mechanical coupling analysis of laminated composite plates. The Reddy’s higher-order shear deformation theory (HSDT) adopted for theoretical formulation represents parabolic variation of transverse shear stresses through thickness to take into account the effects of transverse shears and traction-free boundary conditions for laminated composite plates. The governing equations involving displacement field and temperature gradient field in term of the thickness variable are derived by implementing the principle of virtual work. Then the B-spline wavelet on the interval (BSWI) wavelet-based element is constructed by means of scaling functions as approximating functions instead of polynomial interpolations used in traditional finite element method (FEM). The convergence, accuracy and efficiency of proposed BSWI wavelet-based method are verified for mechanical and thermal analysis of laminated composite plates with fewer degrees of freedom. Then numerical examples concerning various length-to-thickness ratios, aspect ratios and stacking sequences are investigated for thermo-mechanical coupling analysis of symmetric and anti-symmetric laminated composite plates. These comparison examples demonstrate the accuracy and reliability of the proposed BSWI wavelet-based method comparing with the three-dimensional elasticity solutions and referential numerical solutions available in literatures.

Influence of porosity distribution on static and buckling responses of porous functionally graded plates

This article studies the effect of porosity distribution on static and buckling response of a functionally graded (FG) porous plate with all its four edges simply supported and subjected to a transverse load. The plate’s displacement field is approximated based on an inverse hyperbolic shear deformation theory (IHSDT) involving five variables. In this theory, inplane displacements vary as an inverse hyperbolic function through the plate thickness, so account for non-linear variation of transverse shear stresses and satisfies traction free boundary conditions on the surfaces of plate without the need of any shear correction factors. The governing equations of motion are derived using the principle of virtual work, and thus, the expressions of its strain energy and work done are obtained. Effective material properties of porous plate are assumed with an additional term of porosity in the direction of thickness. Power-law is used for continuous variation of material properties along the thickness direction. In this study, five types of porosity distribution functions are considered. The effect of the porosity parameter, the power-law exponent, side-thickness ratio, and aspect ratio on the static and buckling responses of the porous FG plate is evaluated. In problem-solving, the Navier solution technique is used to obtain the exact solutions. The non-dimensional deflections and critical buckling loads of FG porous plate are thus obtained. The results are compared with available results in the literature, and good agreement is found. Moreover, the results are presented to study the effects of various parameters on the dimensionless deflections, stresses, and critical uni-axial and bi-axial buckling loads.

Buckling analysis of functionally graded plates using HSDT in conjunction with the stress function method

In this article, the mechanical buckling analysis of simply-supported functionally graded plates is carried out using a higher shear deformation theory (HSDT) in conjunction with the stress function method. The proposed formulation is variationally consistent, does not use a shear correction factor and gives rise to a variation of transverse shear stress such that the transverse shear stresses vary parabolically through the thickness satisfying the surface conditions without stress of shear. The properties of the plate are supposed to vary across the thickness according to a simple power law variation in terms of volume fraction of the constituents of the material. Numerical results are obtained to study the influences of the power law index and the geometric ratio on the critical buckling load.

Flexural behavior of porous functionally graded plates using a novel higher order theory

In this paper, the flexural response of functionally graded plates with porosities is investigated using a novel higher order shear deformation theory, which considers the influence of thickness stretching. This theory fulfills the nullity conditions at the top and bottom of the plate for the transverse shear stresses, thus avoids the need of a shear correction factor. The effective material properties are computed through the rule of mixtures. The principle of virtual displacements is employed to derive the equilibrium equations. The Navier’s method is adopted to obtain the solutions in closed form for simply supported boundary conditions. The accuracy and consistency of the developed theory are established with numerical results of perfect and porous functionally graded plates available in the open literature. The dimensionless transverse displacements and stresses have been reported. The effect of even, uneven and logarithmically-uneven porosity distributions with different porosity volume fraction, gradation index, side-to thickness ratios and aspect ratios are studied. The numerical results show that, the increase of volume fraction of porosity increases the dimensionless transverse deflections and axial stresses, and decreases the transverse shear stresses. No variation of transverse shear stresses observed for a completely ceramic and metallic plate for all kinds of porosity models. The provided numerical results can be used to evaluate various plate theories and also to compare the results of other analytical methods and finite element methods.

On the buckling analysis of functionally graded sandwich beams using a unified beam theory

In this paper, a unified beam theory is developed and applied to study the buckling response of two types of functionally graded sandwich beams. In the first type (Type A), the sandwich beam has a hardcore whereas in the second type (Type B), the sandwich beam has a softcore. In both the type of beams, face sheets are made up of functionally graded material. The material properties of face sheets are varied through the thickness according to the power-law distribution. A unified beam theory developed in the present study uses polynomial and non-polynomial type shape functions in-terms of thickness coordinate to account for the effect of shear deformation. The present theory is built upon classical beam theory and shows a realistic variation of transverse shear stresses through the thickness of the beam. The governing equations are deduced based on the principle of virtual work. Analytical solutions for simply supported sandwich beams subjected to axial force are presented. The critical buckling load factors of two types of FG sandwich beams are investigated. The numerical results are obtained for various power law coefficients and face-core-face thickness ratios. The validity of the present theory is proved by comparing the present results with various available solutions in the literature.

A meshless study of antisymmetric angle-ply laminates using high-order shear deformation theories

Composite laminated plates have found several applications in engineering field. For instance, aerospace and automotive industry use these structures in several components of their products due to their exceptional specific properties along the direction of the fibres. There are several two-dimensional plate models that aim to more accurately and more efficiently predict the kinematics of those structures. Unlike the Classical Plate Theory (CLPT), which can only be applied to thin laminates, the First-Order Shear Deformation Theory (FSDT), developed by Reissner and Mindlin, already accounts shear effects and, due to that, it can describe in a satisfactory way the kinematic of a generic thick laminated plate. Nevertheless, the FSDT considers linear shape functions to describe the in-plane displacements through the plate thickness, which results in constant shear strains. Thus, the traction boundary conditions on the top and bottom surface of plate are violated. Additionally, the FSDT needs shear correction factors, which are not easily obtained for a generic problem. Equivalent single layer (ESL) theories following high-order shear deformation theories (HSDTs) have been proposed and more efficiently applied to composite laminated plates. These theories allow to represent the nonlinear variation of transverse shear stresses through thickness of plate because they possess nonlinear shape functions interpolating the in-plane displacements. Thus, the kinematics of these structural elements can be better described and more accurate stress fields can be obtained for problems that cannot be handled with the FSDT.These structures have been analysed for several years using the Finite Element Method (FEM), which is the most popular numerical tool in structural analysis. However, several other advanced numerical techniques, such as meshless methods, can also handle this kind of problems and, in some cases, in a more efficient way. This work makes use of a recently developed meshless method – the Natural Neighbour Radial Point Interpolation Method (NNRPIM) (Belinha, 2014) – to study the bending of antisymmetric angle-ply composite laminates using distinct HSDTs.

STRAIN–DISPLACEMENT EXPRESSIONS AND THEIR EFFECT IN DEFLECTION OF RECTANGULAR PLATE

Due to enormous application of thick plate and its relevance in engineering, various theories for plate analysis have been developed using linear strain–displacement expressions. It is proven from previous studies that results obtained using linear strain–displacement expressions may be unreliable for nonlinear stress and bending analyses. In the present paper, nonlinear strain– displacement expressions are employed for rectangular plates subjected to uniform distributed loads to suggest a more reliable refined plate theory that satisfies the continuity of all of the transverse stress components. This theory, which is based on traditional third-order shear deformation theory of plate is presented and applied in a bending analysis of rectangular thick plate. Governing equations and associated boundary conditions of the theory are obtained using the principle of variational calculus. From the formulated expression, the formula for calculation of the actual critical lateral imposed load, q𝑖𝑤, on the plate before deflection reaches the specified maximum specified limit and critical lateral imposed load, q𝑖𝑝, before plate reaches an elastic yield stress were obtained. By solving using the formulated expression, the effect of deflection and crack in a mild steel rectangular plates with opposite edge clamped and the other edge simply supported (CSCS) and simply supported at first and fourth edge and clamped at second edge and free of support at third edge (SCFS) were analysed and compared. This approach overcomes the challenges of the conventional practice in the structural analysis/design, which involves checking of deflection and shear; the process which is proved unreliable. In the result of CSCS plate, the positive value of the critical lateral imposed load, q𝑖𝑤(between 31.08735 N/mm to 155.4414 N/ mm) before deflection reaches the maximum specified limit and the critical lateral imposed load, q𝑖𝑝 (between 193.8246 N/mm to 193.8246 N/mm) before mild steel plate reaches the elastic yield stress, reveals that the plate neither failed in q𝑖𝑤 nor in q𝑖𝑝 for plate span (a) of 1000mm at allowable deflection, wa of 1mm to 5mm. Also, the positive value of critical lateral imposed load q𝑖𝑤 (between 16.23992 N/mm to 81.20424 N/mm) 𝑎𝑛𝑑 q𝑖𝑝(between 115.3523 N/ mm to 115.3523 N/mm) reveals that the plate neither fail in q𝑖𝑤 nor in q𝑖𝑝 for plate span (a) of 1000mm at allowable deflection, wa of 1mm to 5mm for SCFS. This means that the plate structure is safe. It is observed that the value of q𝑖𝑝 is constant at any value of wa for SCFS plate. This means that change in specified deflection limit does not affect the overall performance of SCFS rectangular plate unlike CSCS plate. Hence, it also reveals that the values of critical lateral imposed loads q𝑖𝑤 𝑎𝑛𝑑 q𝑖𝑝 decrease as the length-width ratio increases. This continues until failure occurs. This means that increase in plate width increases the chance of failure in a plate structure.It is concluded that the values of critical lateral load obtained by this theory gives realistic variation of transverse shear stress through the thickness of plate and satisfied the transverse flexibility of the rectangular plate’s condition while predicting the bending behaviour of isotropic thick rectangular plate. Therefore, using this theory it is possible to predict actual load that cause the bending behaviour of isotropic rectangular plate.

On the static deformation of FG sandwich beams curved in elevation using a new higher order beam theory

This article presents the static analysis of FG sandwich beams curved in elevation. Navier-type semi-analytical solutions are obtained based on polynomial type fifth order shear and normal deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are graded in z-direction according to the power-law distribution. The present theory accounts for a fifth-order distribution of axial displacement and fourth-order distribution of transverse displacement. The present theory considers the effect of thickness stretching and gives a realistic variation of transverse shear stress through the thickness of the beam. The governing equations are obtained within the framework of the principle of virtual work. Semi-analytical static solutions for the simply supported FG sandwich beams curved in elevation are obtained using Navier’s technique. The beam is subjected to uniformly distributed load. The non-dimensional numerical values for displacements and stresses are obtained for various power-law index and thickness of the core. The present results are found in good agreement with previously published results.

Vibration of porous functionally graded plates with geometric discontinuities and partial supports

Vibrational study of the porous functionally graded plate with geometric discontinuities and partial supports has been presented in the present paper. The kinematics of functionally graded plate is based on the refined exponential shear deformation theory. The displacement field has been refined by dividing the in-plane and out of the plane displacements into bending and shear components. The theory accounts for the nonlinear transverse shear stress variation along with the thickness with only four unknowns. The closed-form solution (Navier’s solution), as well as FEM-based solution, have been used for the vibration analysis of functionally graded plate. The geometric discontinuities have been incorporated in terms of a circular cut-out of different sizes at the center of the plate. Modified rule of mixtures, modified sigmoid law, and trigonometric law have been used to compute the effective material properties of the functionally graded plate. A C0 continuous iso-parametric FEM formulation has been used to attain the results in the case of FEM solution, and the efficacy of the present solution is demonstrated by comparing the results with the available literature. The results reflect that the porosity inclusion, circular cut-out, and position of the boundary constraints have a notable influence on the fundamental frequency of the functionally graded plate. It is also concluded that after a specific radius of circular cut-out, the vibration response of functionally graded plate exhibits nonlinearity in nature.

Analytical solutions of bending and free vibration of moderately thick micro-plate via two-variable strain gradient theory

In this paper, the strain gradient theory (SGT) and two-variable refined plate theory are implemented simultaneously to analyze bending and free vibration of micro-plate. The proposed theory is called two-variable strain gradient theory (TV-SGT). The TV-SGT, which can be used for both thin and moderately thick micro-plates, predicts parabolic variation of transverse shear stresses across the plate thickness and does not need the shear correction factor. The governing equations are obtained using a variational approach and solved analytically to find the deflection of simply supported micro-plate under uniformly distributed loading and natural frequency of this plate. Thereby, the effects of length scale parameters and dimensions on the bending deflection and natural frequency of the micro-plate are investigated. Results show that, strengthen micro-plate has higher shear rigidity. Moreover, it is inferred that TV-SGT is well defined to analyze the micro-plate.

Static analysis of functionally graded composite beams curved in elevation using higher order shear and normal deformation theory

Abstract In the present study, higher order shear and normal deformation theory is presented for static analysis of functionally graded composite beams curved in elevation. The material properties are varying according to the power-law distribution. Governing equations and boundary conditions in the present theory are obtained using the principle of virtual work. Navier’s solution technique is taken for the solution of simply supported beams curved in elevation. Stresses and deflections of functionally graded composite beams curved in elevation are obtained. Present theory gives parabolic variation of transverse shear stress through the thickness of functionally graded beams and also satisfies zero traction boundary conditions on the top and bottom surfaces of beam.

Static analysis of functionally graded sandwich plates with porosities

In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded sandwich plate with new definition of porosity distribution taking into account composition and the scheme of the sandwich plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Material properties of FGM layers are assumed to vary continuously across the plate thickness according to either power-law or sigmoid function in terms of the volume fractions of the constituents. The face layers are considered to be FG across each face thickness while the core is made of a ceramic homogeneous layer. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simply supported rectangular plate subjected to sinusoidal loading has been obtained by using the Navier method. Numerical results are presented to show the effect of the material distribution, the sandwich plate geometry and the porosity on the deflections and stresses of FG sandwich plates. The validity of the present theory is investigated by comparing some of the present results with other published results.

Dual-axis buckling of laminated composite skew hyperbolic paraboloids with openings

The dual-axis buckling of Laminated composite skew hyperbolic paraboloid with cutouts subjected to the in-plane biaxial and the shear load is investigated for various boundary conditions using the present mathematical model. Variation of transverse shear stresses is represented by a second-order function across the thickness, and the cross-curvature effect is also included via strain relations. The transverse shear stress-free condition at the shell top and bottom surfaces is also satisfied. This mathematical model (having a realistic second-order distribution of transverse shear strains across the thickness of shell) requires unknown parameters only at the reference plane. For generality in the present analysis, nine-node curved isoparametric element is used. So far, no solution exists in the literature for dual-axis buckling problem of laminated composite skew hyperbolic paraboloids with cutouts. As no result is available for the present problem, the present model is compared with suitable published results and then it is extended to analyze biaxial and shear buckling of laminated composite skew hyperbolic paraboloids. A C0 finite element coding in FORTRAN is developed to generate many new results for different boundary conditions, skew angles, lamination schemes, etc.

Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams

A simple modified exponential shear deformation theory (ESDT) is developed and applied for the bending, buckling, and free vibration analyses of functionally graded beams with different boundary conditions. The properties of functionally graded material are assumed to vary through the thickness direction according to power law (P-FGM) and exponential law (E-FGM). The present theory is different from existing theories, because in the present theory, the transverse displacement consists of bending and shear components to understand the contribution of transverse displacement due to bending and due to shear separately. The developed theory accounts for higher order variation of transverse shear stress through the thickness of the beam, and satisfies the traction-free conditions on the top and bottom surfaces of the beam. The theory appropriately represents the strain energy of shear deformation without using shear coefficient. Equations of motion and associated boundary conditions are derived from Hamilton’s principle. Closed-form solutions for various boundary conditions are obtained, and the numerical results are compared with those available in the literature. The present study contributes some new results on the P-FGM and E-FGM beams for the reference of future research in this area.

Numerical nonlinear frequency analysis of pre-damaged curved layered composite structure using higher-order finite element method

The influence of the internal debonding on the structural stiffness and the nonlinear modal characteristics of the layered structure are examined extensively in the current research article. For the investigation purpose, the shell frequency responses are obtained numerically for both the linear and the nonlinear cases via a generic type of mathematical formulation using the Equivalent Single Layer (ESL) theory in the framework of two kinematic models. The current formulation not only includes the influence of the transverse shear deformations but also satisfies the parabolic variation of transverse shear stress through the thickness. Additionally, the geometrical nonlinear distortion modeled via Green–Lagrange strain–displacement relations. Further, the internal debonding between the adjacent layers are modeled using sub-laminate approach and the displacement continuity between segments (laminate and delaminate) have been established through the intermittent continuity conditions. The nonlinear system governing equation of the vibrated structure is obtained via Hamilton’s principle and converted to set of nonlinear algebraic equations through the isoparametric finite element (FE) steps. The desired responses are solved numerically with the help of robust (direct iterative method) technique and compared with available results to demonstrate the solution accuracy. Subsequently, an adequate number of examples are solved for the delaminated structure using the current higher-order nonlinear models and the influential parameters discussed in detail.

A refined finite element method for bending analysis of laminated plates integrated with piezoelectric fiber-reinforced composite actuators

This research presents a finite element formulation based on four-variable refined plate theory for bending analysis of cross-ply and angle-ply laminated composite plates integrated with a piezoelectric fiber-reinforced composite actuator under electromechanical loading. The four-variable refined plate theory is a simple and efficient higher-order shear deformation theory, which predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies zero traction conditions on the plate free surfaces. The weak form of governing equations is derived using the principle of minimum potential energy, and a 4-node non-conforming rectangular plate element with 8 degrees of freedom per node is introduced for discretizing the domain. Several benchmark problems are solved by the developed MATLAB code and the obtained results are compared with those from exact and other numerical solutions, showing good agreement.

Efficient 2D thermo-mechanical analysis of composites and sandwich laminates

ABSTRACTThe static behavior of composites and sandwich plates in thermo-mechanical environment is investigated by a two dimensional (2D) FE model. An efficient higher-order zig-zag theory (HOZT) considering actual through-thickness temperature profile and a least square error (LSE) method to accurately predict the inter-laminar stresses is implemented in this model. The in-plane displacement field is obtained by superposing a cubically varying global displacement field on a zig-zag displacement field having different slopes at each layer. This plate theory represents parabolic through thickness variation of transverse shear stresses, which satisfy the inter-laminar continuity at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. In the present 2D finite element (FE) model, the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C1 continuity associated with the above plate theory (HOZT). The accurate through-thickness distribution of temperature is obtained by using a linear zig-zag thermal lamination theory proposed by the authors by using the thermal conduction properties of different constituent layers in the thickness direction. The LSE method is applied at the postprocessing stage to accurately calculate the inter-laminar stresses by the 3D equilibrium equations of the plate problem, after in-plane stresses are calculated. The proposed combined FE model (HOZT+LSE) is implemented to analyze the static behavior of laminated composites and sandwich plates subjected to thermo-mechanical loadings. Many new results are also presented that should be useful for the future reference.