Distribution Of Transverse Shear Stresses Research Articles

A modified quasi-3D theory and mixed beam element method for static behaviour analysis of functionally graded beams

This paper develops a modified quasi-3D theory and the mixed beam element model for static analysis of functionally graded beams. The modified quasi-3D theory encompasses two innovations. Firstly, the accurate distribution of transverse shear stress is derived from the differential equilibrium equation that describes the relationship between stresses, rather than the traditional one derived from geometric relations and constitutive equations. Secondly, the effect of distributed loads associated with transverse stretching deformation, a factor typically overlooked in the existing quasi-3D theory-based beam models, is involved. In the development of beam finite element model, the mixed variational principle is firstly utilized to formulate the model of a quasi-3D theory-based beam element, where generalized displacements and internal forces are treated as two types of independent field quantities. Two numerical examples are conducted to investigate the validity and accuracy of the proposed theory and beam element. Numerical results indicate that the proposed mixed beam element can accurately predict the stress distributions over the cross-section and produce accurate displacement solutions. Meanwhile, it is noted that the effect of distributed loads related to the transverse stretching deformation should be correctly considered in the analysis of functionally graded beams based on quasi-3D theory.

Quasi-3D flexural analysis of laminated composite plates resting on elastic foundations using higher-order displacement model

This work examines the impact of transverse strains on the static, vibration, and buckling analysis of laminated composite plates that are symmetric and anti-symmetric and rest on elastic foundations. This laminated plate theory, which is called a quasi-3D plate theory since it considers the effects of both transverse shear and normal strains, is provided here. The significance of nonclassical influences on plate deformation, such as shear deformation and thickness stretching, is sufficiently highlighted by the current theory. The theory demonstrates realistic transverse shear stress distributions across the laminated plate's thickness and complies with the traction-free boundary conditions at the upper and bottom surfaces of the plate. Hamilton's principle provides the current theory's equations of motion. For simply supported edges, the laminated plates supported by elastic foundations are investigated. For the aim of verification, the numerical results of the displacements, stresses, frequencies, and buckling load derived using the present theory are, if possible, compared with established literature. The current results and those derived from the other hypotheses found in the literature show a good degree of consistency. Additionally, the distributions of the interlaminar stresses in thickness direction of laminated composite plates sitting on elastic foundations are demonstrated.

Multi-Directional Functionally Graded Sandwich Plates: Buckling and Free Vibration Analysis with Refined Plate Models under Various Boundary Conditions

This study conducts buckling and free vibration analyses of multi-directional functionally graded sandwich plates subjected to various boundary conditions. Two scenarios are considered: a functionally graded (FG) skin with a homogeneous hard core, and an FG skin with a homogeneous soft core. Utilizing refined plate models, which incorporate a parabolic distribution of transverse shear stresses while ensuring zero shear stresses on both the upper and lower surfaces, equations of motion are derived using Hamilton’s principle. Analytical solutions for the buckling and free vibration analyses of multi-directional FG sandwich plates under diverse boundary conditions are developed and presented. The obtained results are validated against the existing literature for both the buckling and free vibration analyses. The composition of metal–ceramic-based FG materials varies longitudinally and transversely, following a power law. Various types of sandwich plates are considered, accounting for plate symmetry and layer thicknesses. This investigation explores the influence of several parameters on buckling and free vibration behaviors.

A hyperbolic theory for the analysis of laminated shallow shells with double curvature

In this paper, higher-order closed-form analytical solutions to the static bending and free vibration problems of laminated composite shells with double curvature are obtained using a hyperbolic shear deformation theory. The current theory is a modification of the shape function provided by Soldatos [30] in his well-known hyperbolic theory. The distributions of transverse shear stresses through the thickness of the shell are precisely predicted by the current theory satisfying traction free boundary conditions at the top and the bottom surfaces of the shell. Hamilton's principle serves as the foundation for the development of equations of motion. Navier's method is used for the analysis of simply-supported laminated shells under static and free vibration conditions. Displacements, stresses, and natural frequencies are presented for different shells with double curvature. The results from past investigations are compared to verify the accuracy and efficacy of the present hyperbolic shell theory.

A new trigonometric shear deformation plate theory for free vibration analysis of FGM plates with two-directional variable thickness

The present research aims to propose a new trigonometric shear deformation plate theory (TSDPT) for free vibration analysis of functionally graded plates with two-directional variable thickness. Trigonometric functions are utilized to determine the distribution of transverse shear stress in the new TSDPT. The thickness variation follows three distinct patterns: linear, concave, and convex. The governing equations of the composite plates are derived via the new theory and the Hamilton's principle. Then, via the Galerkin's method, the current study establishes a solution for determining natural frequencies of variable-thickness FGM plates. The obtained outcomes of FGM plates are calculated by using the MATLAB program. To confirm the accuracy of the current computational model, we conducted comparisons between the obtained results and data previously reported in the open literature. The results obtained from the new theory are compared not only for the case of a plate made of isotropic material but also in comparison to a plate made of functionally graded material. In both scenarios, the proposed theory showcases superior performance when juxtaposed against classical plate theory (CPT), first-order shear deformation plate theory (FSDPT), and other shear deformation plate theories, displaying significantly reduced error rates. Furthermore, graphical representations of the influences of three-parameter stiffness of Kerr foundation, boundary conditions, geometrical parameters, two different FGMs, FGM index, and mode numbers on the vibrational behaviors of FGM plates are provided.

An analytical model for the static behaviour of honeycomb sandwich plates with auxetic cores using higher-order shear deformation theories

AbstractThis paper presents an analytical model to investigate the static behaviour of sandwich plates comprised of two isotropic face sheets and a honeycomb core. Through-thickness transverse shear stresses were considered using a unified displacement field with which various plate theories were implemented, i.e., exponential, third-order, hyperbolic, sinusoidal, fifth-order, Mindlin, and the classic plate theory. The equilibrium equations of a simply-supported sandwich panel were derived using the principle of virtual work and Navier solution was obtained under static transverse loading. After validating of the model, various mechanical and geometrical parameters were varied to characterise the behaviour of the structure under regular and auxetic response. It was found that the auxeticity of the core strongly affects the mechanical response, e.g., in controlling deflection, in-plane anisotropy, and Poisson’s ratio. Cell wall angle was found to be most critical parameter that can be used to adjust anisotropy, out-of-plane shear modulus, transverse shear stress distribution, and deflection of the panel. Also the cell aspect ratio controls the sensitivity of the core response to other geometrical variations. In terms of the higher-order theories, the deflection-dependent parameter of the unified formulation seems to have more control of maximum deflection compared to independent rotations. Auxeticity of the core showed some benefits in controlling anisotropy, deflection and providing additional out-of-plane shear rigidity. Overall, since there is not one-to-one relationship between specific values of Poisson’s ratio, anisotropy, and shear rigidity, careful design considerations must be invested to obtain a correct mechanical response.

Static analysis of an exponentially varying functionally graded beam using trigonometric shear deformation theory

The present study investigates the bending behavior of a functionally graded beam under transverse loads (sinusoidal and uniform) using trigonometric shear deformation theory (TSDT). Boundary value problem are derive using principle of virtual work. The Navier-type closed-form solution is obtained under simply supported boundary conditions. The material properties across the thickness directions vary according to the exponential law of distribution. The present theory does not need a problem-dependent shear correction factor since it accurately accounts for the distribution of transverse shear stress across the beam thickness. Results from higher-order shear and normal deformation theory (HOSNT), exact elasticity theory, and other theories are compared with the current approach to ensure its accuracy and efficacy. The results of present theory is in excellent agreement with the exact elasticity theory.

Elastic buckling of simply supported bimetallic steel plates

Bimetallic steel plate is a laminated composite one consisting of metallurgically bonded cladding and substrate layers, such as the stainless-clad (SC) or titanium-clad (TC) ones. Due to significant corrosion resistance and cost efficiency, it has gained growing recognition as an alternative to stainless steel plate and is particularly suitable for offshore and marine structures. In this paper, the nominal elastic buckling stress of a bimetallic steel plate is derived by adapting the first-order shear deformation plate theory (FSDT) and compared with the solutions of classical plate theory (CPT). The plates are simply supported and subjected to uniaxial compression; two different types of cladding configurations are considered. The transverse shear stress distributions and shear correction factors are obtained by theoretical analyses. Finite element models of the bimetallic steel plates are developed by MATLAB and ABAQUS to validate the derivation. Parametric studies are then conducted to elucidate the effects of the cladding configuration, clad ratio, elastic modulus ratio and width-to-thickness ratio on the buckling stress. Based on the theoretical analyses, a series of simplified design formulae for the critical buckling stress and design requirements for interface shear strength requirements are proposed for further engineering applications. The research works may provide important bases for understanding and investigating the buckling behaviour of the bimetallic steel structures.

A higher-order morphoelastic beam model for tubes and filaments subjected to biological growth

Slender structures, in which width and thickness are much smaller than the length, are common in the natural world, such as wheat straws, bean sprouts, and biofilaments. Due to biological growth, these structures exhibit diverse morphological profiles with curling hairs, curling leaves, and twining plants as some examples. Accurate prediction of growth induced instabilities and corresponding shear deformation require a higher-order morphoelastic beam theory for growing tubes and filaments, for which we present a theory here. In the theory, the deformation gradient is decomposed into elastic deformation and growth using multiplicative decomposition. Appropriate assumptions for slender structures with circular cross-section are introduced and the displacement field and constitutive relations are derived from three-dimensional morphoelasticity. Corresponding variational principle is established and equilibrium equations are obtained. Our higher-order beam with growth theory can model growth-induced instability and shear deformation of growing tubes and filaments with circular cross-sections. Importantly, it can predict shear stress distribution and locations of maximum transverse shear stress in the circular cross-section. The shear stress, critical growth and post-buckling predictions of our proposed model and analytical solutions were validated by comparing its predictions against results from three-dimensional finite element simulations. Growth-induced instability and corresponding shear stress distributions were analyzed and the results are discussed. Our model can provide improved prediction for critical growth induced instabilities which are overestimated if the shear deformation is not considered.

Vibration analysis of functionally graded beams using a higher-order shear deformable beam model with rational shear stress distribution

This paper extends the higher-order shear deformable mixed beam element model with rational shear stress distribution to vibration analysis of functionally graded (FG) beams. In the element model proposed, the stress equilibrium equation is employed to derive the rational distribution of transverse shear stress. The beam element formulation oriented with vibration analysis is derived on the basis of the variational principle with mixed energy. To visualise the effectiveness and applicability of the proposed mixed beam element model in vibration analysis, numerical examples have been studied. According to the numerical results, in comparison with the classical beam element model and the beam element models based on the conventional first-order or higher-order shear deformation theory, the vibration frequencies obtained using the mixed beam element model are significantly more satisfactory. Moreover, the comparative study demonstrates that, for vibration analysis of FG sandwich beams, the rational distribution of transverse shear stress is of decisive significance to achieve high-precision solutions. With the mixed beam element used, the load–frequency relationship of FG sandwich beams is discussed. As indicated by the results, in addition to the axial force, the bending moment exerts a significant difference upon the vibration frequencies of the FG beams.

Geometrically exact beam element with rational shear stress distribution for nonlinear analysis of FG curved beams

Based on the geometrically exact beam theory, a first-order shear deformable curved beam element is developed for geometrically nonlinear analysis of functionally graded (FG) curved beams. In order to accurately predict the distribution of transverse shear stress, the stress equilibrium condition is introduced into the element formulation by using the mixed finite element method with displacements and internal forces as independent fields. The element nodal force vector and consistent tangent stiffness matrix for nonlinear iterative solution are obtained by going through a consistent linearization procedure. Numerical examples are presented to validate the present formulation. As indicated by the numerical results, the proposed element demonstrates a high level of solution accuracy and good applicability. Furthermore, using the proposed element, an investigation is conducted into the nonlinear stability of FG structures with different material distribution parameters, and the accuracy loss caused by unreasonable distribution of shear stress is discussed.

Finite element bending analysis of symmetric and non-symmetric functionally graded sandwich beams using a novel parabolic shear deformation theory

In this paper, the bending behavior of functionally graded single-layered, symmetric and non-symmetric sandwich beams is investigated according to a new higher order shear deformation theory. Based on this theory, a novel parabolic shear deformation function is developed and applied to investigate the bending response of sandwich beams with homogeneous hardcore and softcore. The present theory provides an accurate parabolic distribution of transverse shear stress across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the functionally graded sandwich beam without using any shear correction factors. The governing equations derived herein are solved by employing the finite element method using a two-node beam element, developed for this purpose. The material properties of functionally graded sandwich beams are graded through the thickness according to the power-law distribution. The predictive capability of the proposed finite element model is demonstrated through illustrative examples. Four types of beam support, i.e. simply-simply, clamped-free, clamped–clamped, and clamped-simply, are used to study how the beam deflection and both axial and transverse shear stresses are affected by the variation of volume fraction index and beam length-to-height ratio. Results of the numerical analysis have been reported and compared with those available in the open literature to evaluate the accuracy and robustness of the proposed finite element model. The comparisons with other higher order shear deformation theories verify that the proposed beam element is accurate, presents fast rate of convergence to the reference results and it is also valid for both thin and thick functionally graded sandwich beams. Further, some new results are reported in the current study, which will serve as a benchmark for future research.

Nonlocal finite element model for the bending and buckling analysis of functionally graded nanobeams using a novel shear deformation theory

In this study, an efficient nonlocal finite element model is developed to investigate the bending and buckling behavior of functionally graded (FG) nanobeams. New two-node beam element with eight degrees of freedom is formulated based on the recently refined higher order shear deformation theory proposed by the authors. The present theory can provide an accurate parabolic distribution of transverse shear stress through the thickness direction satisfying the traction free boundary conditions needless of any shear correction factor. In order to capture the small size effect, Eringen’s nonlocal elasticity theory is incorporated. The material properties of the FG nanobeams are assumed to vary continuously through the thickness direction according to the power-law form. The performance and reliability of the proposed model is demonstrated by comparing the author’s results with those available in the literature. The numerical results show that the present element model is free of shear locking and has a high accuracy and fast rate of convergence. Moreover, a detailed numerical study is carried out to examine the effects of several parameters such as boundary conditions, power-law index, nonlocal parameter and length-to-height ratio on the deflection and critical buckling load of the FG nanobeams. Many new results are also reported in the current study, which will serve as a benchmark for future research.

A simple higher-order beam model that is represented by two kinematic variables and three section constants

This article presents a new higher-order beam model. The present beam model is governed by differential equations that are similar to those present in some existing higher-order beam models; however, the present beam model makes use of a novel method of calculating the transverse shear stiffness, which facilitates the calculation of a shear-warping stiffness without the need for an assumed warping displacement field, and without introducing any additional kinematic variables. The present beam model also facilitates the recovery of the distributions of longitudinal normal stresses and transverse shear stresses. The authors postulate that the bending and shear terms in first-order shear deformation theory represent the first two terms in an infinite series that would constitute an ideal one-dimensional beam model, and it is suggested that the present beam model constitutes the first four terms in this hypothetical infinite series. The present beam model is solved for several example beams, and the results are compared with those of existing classical and higher-order beam models, as well as computational results from finite element analyses. It is shown that the present beam model is able to accurately represent deformed shapes and stress distributions pertaining to beams that exhibit non-trivial shear compliance as well as non-trivial shear-warping stiffness. In the case of laminated composite beams comprising a large number of laminae, the present beam model offers a level of analytical fidelity that is comparable to that of existing zigzag beam models; however, unlike zigzag beam models, the present beam model is equally well suited for the analyses of beams comprising any number of laminae.

Dynamic Response of Angle Ply Laminates with Uncertainties Using MARS, ANN-PSO, GPR and ANFIS.

In the present work, for the first time, free vibration response of angle ply laminates with uncertainties is attempted using Multivariate Adaptive Regression Spline (MARS), Artificial Neural Network-Particle Swarm Optimization (ANN-PSO), Gaussian Process Regression (GPR), and Adaptive Network Fuzzy Inference System (ANFIS). The present approach employed 2D C0 stochastic finite element (FE) model based on the Third Order Shear Deformation Theory (TSDT) in conjunction with MARS, ANN-PSO, GPR, and ANFIS. The TSDT model used eliminates the requirement of shear correction factor owing to the consideration of the actual parabolic distribution of transverse shear stress. Zero transverse shear stress at the top and bottom of the plate is enforced to compute higher-order unknowns. C0 FE model makes it commercially viable. Stochastic FE analysis done with Monte Carlo Simulation (MCS) FORTRAN inhouse code, selection of design points using a random variable framework, and soft computing with MARS, ANN-PSO, GPR, and ANFIS is implemented using MATLAB in-house code. Following the random variable frame, design points were selected from the input data generated through Monte Carlo Simulation. A total of four-mode shapes are analyzed in the present study. The comparison study was done to compare present work with results in the literature and they were found in good agreement. The stochastic parameters are Young’s elastic modulus, shear modulus, and the Poisson ratio. Lognormal distribution of properties is assumed in the present work. The current soft computation models shrink the number of trials and were found computationally efficient as the MCS-based FE modelling. The paper presents a comparison of MARS, ANN-PSO, GPR, and ANFIS algorithm performance with the stochastic FE model based on TSDT.

Mechanical and thermal buckling analysis of laminated composite plates

The mechanical and thermal buckling analysis of laminated composite plates is presented in this document. Different theories of thick plates taking into account the parabolic distribution of transverse shear stresses and satisfying the condition of zero shear stresses on the top and bottom surfaces without using shear correction factor are presented and a comparison between the results obtained by these theories is also illustrated. The high order nonlinear stress-displacement relation of the plates was taken into consideration. The principle of potential energy is used to obtain the equations of equilibrium. The closed-form solutions of symmetric and antisymmetric cross-ply are obtained using Navier solution. Using math software Maple, the temperatures and the critical loads of buckling are determined. Finally, a parametric study of the influence of the various parameters such as: mode of buckling, the geometrical ratios a / b and a / h, Young's modulus, Coefficient of thermal expansion, loading type, the orientation of the fibers and the number of layers on the critical buckling temperature and the critical buckling charge is shown and discussed. Numerical results indicate that deformation due to transverse shear has a significant effect on both mechanical and thermal behavior of buckling of laminated simply supported plates.

Stochastic normal mode frequency analysis of hybrid angle ply laminated composite skew plate with opening using a novel approach

In the present work, for the first time, stochastic normal mode frequency analysis of hybrid angle ply laminated composite skew plate with an opening is attempted by a novel approach. The present novel approach employed 2 D C0 stochastic finite element (FE) model based on higher-order shear deformation theory (HSDT) in conjunction with artificial neural network (ANN) and artificial neural network-particle swarm optimization (ANN-PSO). HSDT model used eliminates the requirement of shear correction factor owing to consideration of the actual parabolic distribution of transverse shear stress. Zero transverse shear stress at the top and bottom of the plate is enforced to compute higher-order unknowns. C0 FE model makes it commercially viable. Stochastic vibration using FE analysis with Monte Carlo simulation, design of experiment (DOE), and soft computing with ANN, ANN-PSO is implemented. With the DOE method, a total of 128 design points are generated. Since there are no results for the present problem in the literature is available. Hence suitable comparisons have been made and found in the good agreement. The stochastic parameters are Young’s elastic modulus, shear modulus, and the Poisson ratio. Lognormal distribution of properties is assumed in the present work. The present soft computation models shrink the trial and are found computationally in agreement with the Monte Carlo simulation-based finite element modeling (MCS-FEM). Current work reveals that ANN-PSO is a fast and straightforward method with result accuracy close to MCS-FEM, and the future researcher in this area may use ANN-PSO.

Dynamic analysis of folded laminated composite plate using nonpolynomial shear deformation theory

In this paper, an efficient C0 finite element modeling based on higher-order nonpolynomial shear deformation theory (NPSDT) is proposed for the dynamic analysis of folded laminated composite plate. The theoretical formulations are based on nine-noded Lagrange isoparametric finite element and inverse hyperbolic shear deformation theory (IHSDT) as NPSDT. The employed theory, IHSDT, assumes the nonlinear and realistic distribution of transverse shear stresses, and also satisfies the traction-free boundary conditions at the top and bottom surfaces of the plate. Hamilton's principle has been adopted to derive the system's governing equation. A penalty approach has been used to take into account the artificial constraints generated due to incorporation of C0 Lagrange element. The free vibration pertaining to eigenvalue problem is solved using subspace iteration method. The transient analysis subjected to pressure load and vertical load is carried out using Newmark's direct integration scheme. The formulation has been validated with the available solution in the literature, and several novel solutions have been proposed to address the various practical aspects of folded plate. Numerical illustrations are presented to investigate the effect of various parameters such as crank angle, fiber angle, lamination scheme, fold location, and boundary conditions on the natural frequency and transient response of laminated composite plate.

A material-based higher-order shear beam model for accurate analyses of FG beams with arbitrary material distribution

A material-based higher-order shear beam model is proposed for accurate analyses of functionally graded (FG) beams with arbitrary material distribution through thickness. Distinct from the existing higher-order shear beam models, the shear function of the present model is dependent on the material distribution. The specific form of this material-based shear function is described by a piecewise linear interpolation field and determined by ensuring the consistency of transverse shear stress distributions between Euler-Bernoulli beam theory and the higher-order shear beam theory. Based on the proposed beam model, a 3-node beam finite element is developed for static and buckling analyses. Numerical examples are studied to visualize the accuracy and effectiveness of the proposed model and finite element. Solutions of static displacements and critical loads obtained by various beam elements including the proposed element, the existing displacement-based beam elements and the mixed higher-order shear beam element are compared. The accuracy loss of the existing higher-order shear beam models is discussed. As demonstrated by the results, the proposed beam model and finite element have excellent solution accuracy and desirable applicability.

Determination of the transverse shear stress in layered composites

An engineering approach to estimation of the transverse shear stresses in layered composites is developed. The technique is based on the well-known D. I. Zhuravsky equation for shear stresses in an isotropic beam upon transverse bending. In general, application of this equation to a composite beam is incorrect due to the heterogeneity of the composite structure. According to the proposed method, at the first stage of its implementation, a transition to the equivalent model of a homogeneous beam is made, for which the Zhuravsky formula is valid. The transition is carried out by changing the shape of the cross section of the beam, provided that the bending stiffness and generalized elastic modulus remain the same. The calculated shear stresses in the equivalent beam are then converted to the stress values in the original composite beam from the equilibrium condition. The main equations and definitions of the method as well as the analytical equation for estimation of the transverse shear stress in a composite beam are presented. The method is verified by comparing the analytical solution and the results of the numerical solution of the problem by finite element method (FEM). It is shown that laminate stacking sequence has a significant impact both on the character and on the value of the transverse shear stress distribution. The limits of the applicability of the developed technique attributed to the conditions of the validity of the hypothesis of straight normal are considered. It is noted that under this hypothesis the shear stresses do not depend on the layer shear modulus, which explains the absence of this parameter in the obtained equation. The classical theory of laminate composites is based on the similar assumptions, which gives ground to use this equation for an approximate estimation of the transverse shear stresses in in a layered composite package.