Effect Of Transverse Normal Strain Research Articles

A refined shell theory considering the effects of transverse normal strain for the analysis of functionally graded porous shallow shells with double curvature

In this paper, a static and free vibration analysis of functionally graded porous shallow shells with double curvature containing an even distribution of porosity is investigated using a trigonometric shear and normal deformation theory. Both the effects of transverse shear and normal strains are included in the present theory to investigate their effects on the responses of FGM porous shells. The theory satisfies zero transverse shear stress conditions at the top and bottom surfaces of the shell. Hamilton’s principle is used for obtaining governing equations and boundary conditions of the present theory for functionally graded porous shallow shells. The FGM porous shell is assumed isotropic at any point within the shell domain, with its elastic properties varying across its thickness by a power law in terms of the volume fractions of the shell constituents. The simply-supported FG porous shell is analyzed in the present study using the Navier method. The numerical results of FGM porous plates are obtained and compared with other higher-order theories available in the literature to verify the present theory. Further, the numerical results are presented for the FGM porous shells with double curvature. The effects of a/h ratios, porosity distribution factor, and radii of curvature on deflections, stresses, and fundamental frequencies are presented. The solution of hyperbolic and elliptical FGM porous shells is presented for the first time in this paper.

Improved Finite Element Thermomechanical Analysis of Laminated Composite and Sandwich Plates Using the New Enhanced First-Order Shear Deformation Theory

This paper proposes a simple yet accurate finite element (FE) formulation for the thermomechanical analysis of laminated composites and sandwich plates. To this end, an enhanced first-order shear deformation theory including the transverse normal effect based on the mixed variational theorem (EFSDTM_TN) was employed in the FE implementation. The primary objective of the FE formulation was to systematically interconnect the displacement and transverse stress fields using the mixed variational theorem (MVT). In the MVT, the transverse stress field is derived from the efficient higher-order plate theory including the transverse normal effect (EHOPT_TN), to enhance the solution accuracy, whereas the displacement field is defined by the first-order shear deformation theory including the transverse normal effect (FSDT_TN), to amplify the numerical efficiency. Furthermore, the transverse displacement field is modified by incorporating the components of the external temperature loading, enabling the consideration of the transverse normal strain effect without introducing additional unknown variables. Based on the predefined relationships, the proposed FE formulation can extract the C0-based computational benefits of FSDT_TN, while improving the solution accuracy for thermomechanical analysis. The numerical performance of the proposed FE formulation was demonstrated by comparing the obtained solutions with those available in the literature, including 3-D exact solutions.

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Static and free vibration analysis of sandwich shells with double curvature considering the effects of transverse normal strain

Higher-order closed-form solutions for the static and free vibration problems of sandwich shells with double curvature are presented in the current study based on a new hyperbolic shell theory considering the effects of transverse normal strain. A theory involves six unknowns and satisfies traction-free boundary conditions at the top and the bottom surfaces of the shell. The theory does not require a shear correction factor to account for the strain energy due to the shear deformation effect. A shell consists of three layers, wherein the top and the bottom layers, i.e. face sheets are made up of hard material and the middle layer, i.e. core is made up of soft material. The governing equations and associated boundary conditions of the theory are produced by employing Hamilton’s principle. Semi-analytical closed-form solutions for the static and free vibration problems are produced by the Navier technique for simply supported boundary conditions of the shell. The present results are compared with results that have already been published to confirm the accuracy and efficacy of the current higher-order hyperbolic shell theory.

Effects of transverse normal strain on the deformation of laminated and sandwich arches under the action of concentrated force

It is well-known that the concentrated force develops high-stress concentration at the point of application which is one of the important parameters to be considered while designing the laminated composite structures under the action of concentrated force. The literature available shows that the study on static deformation of laminated sandwich shallow arches under the action of the concentrated force is limited. In the present study, static deformation of laminated sandwich arches under the action of concentrated loading is investigated using higher-order arch theory considering the effects of transverse normal strain. An exponential type higher-order arch theory is developed in this study which satisfies the traction-free boundary conditions at the top and the bottom surfaces of the arch using constitutive relations. Governing equations are derived within the framework of the principle of virtual work. An analytical solution for the static deformation of simply supported laminated and sandwich shallow arches are obtained using Navier’s technique. The effects of the lamination scheme, radius of curvature and aspect ratio on the deflection, and stresses of shallow arches are evaluated. The present results are compared with previously published results wherever possible for the verification of the present theory.

Hygro-thermo-mechanical analysis of sandwich shallow shells considering the effects of transverse normal strain

The static response of softcore doubly-curved sandwich shallow shells under hygro-thermo-mechanical loading is presented herein using a fifth-order shell theory which considers the effects of transverse normal strains and stresses at the same time. The present theory accounts for non-linear variations of displacements. The strain-displacement relations take into account only small curvatures following a shallow shell description. The present theory ensured the traction-free boundary conditions at the top and the bottom faces of the shell. The governing equations along with the natural and essential boundary conditions are derived using the principle of virtual work. The governing equations are solved by using Navier’s method. Three layered sandwich shallow shells of double curvature subjected to hygro-thermo-mechanical loading are analyzed. The accuracy and efficacy of the present theory are proved by applying it to the benchmark solutions available in the literature and then the present theory is extended for the analysis of softcore sandwich shells of double curvature. Since an exact 3D elasticity solution is not available for many such problems; this paper can be serving as a benchmark reference solution for future researchers working in this area.

Bending analysis of FGM plates using sinusoidal shear and normal deformation theory

This paper presents the bending analysis of functionally graded material (FGM) plates using sinusoidal shear and normal deformation theory. The in-plane displacements include sinusoidal functions in the thickness coordinate to consider the effect of transverse shear deformation, and transverse displacement includes the effect of transverse normal strain using the cosine function in thickness coordinate. The displacement field of the theory enforces to satisfy shear stress-free boundary conditions on the top and bottom surfaces of the plate with realistic variations across the thickness. Plate material properties vary across thickness directions according to a power law. The boundary value problem of the theory is derived using the principle of virtual work. Simply supported plate bending problems are solved using the Navier solution technique. Response of the plate is obtained with respect to the type of load, type of plate, aspect ratio, and power law index. The results of present theory are compared with those of quasi-3D discrete layer theory and semi-analytical solutions based on the theory of elasticity to ensure the accuracy of theory. The current theory showed excellent agreement with more exact theories in bending response.

Higher order computational model considering the effects of transverse normal strain and 2-parameter elastic foundation for the bending of laminated panels

In this study, the authors analyze laminated composite panels supported on an elastic foundation considering the effects of transverse normal strain. A 2-parameter, i.e., Winkler and Pasternak foundation model is assumed to represent the interaction between the panels and the foundation. The theory presented here takes into account the effects of transverse shear and normal strains. The theory plots realistic distributions of the transverse shear stress through the plate thickness and satisfies the shear-free conditions at the extreme surfaces of the panel. The differential equations of the present model are obtained from the principle of virtual work. The laminated composite panel resting on the elastic foundation is analyzed for simply supported boundary conditions. For the verification purpose, the presented problems are also solved using the Reddy's model, Mindlin's model, and the classical model. Good agreement is observed between the numerical results obtained using the present model and the other models.

A Quasi-3D Higher-Order Theory for Bending of FG Nanoplates Embedded in an Elastic Medium in a Thermal Environment

This paper presents the effects of temperature and the nonlocal coefficient on the bending response of functionally graded (FG) nanoplates embedded in an elastic foundation in a thermal environment. The effects of transverse normal strain, as well as transverse shear strains, are considered where the variation of the material properties of the FG nanoplate are considered only in thickness direction. Unlike other shear and deformations theories in which the number of unknown functions is five and more, the present work uses shear and deformations theory with only four unknown functions. The four-unknown normal and shear deformations model, associated with Eringen nonlocal elasticity theory, is used to derive the equations of equilibrium utilizing the principle of virtual displacements. The effects due to nonlocal coefficient, side-to-thickness ratio, aspect ratio, normal and shear deformations, thermal load and elastic foundation parameters, as well as the gradation in FG nanoplate bending, are investigated. In addition, for validation, the results obtained from the present work are compared to ones available in the literature.

Bending Analysis of Sandwich Plate Under Concentrated Force Using Quasi-Three-Dimensional Theory

This study presents the quasi-three-dimensional, higher-order shear-deformation theory for the stress and displacement analysis of sandwich plate subjected to concentrated force. Using expansion of thickness coordinate in the displacement field, the effects of both transverse shear and normal deformations are included. The theory satisfies the zero shear stress conditions on the top and bottom surfaces of sandwich plate. Hence, the shear correction factor is not required, generally associated with the First order shear deformation theory (FSDT). The boundary-value problem of the theory is derived using the principle of virtual work. Transverse shear and normal stresses are recovered from the stress-equilibrium equations of the theory of elasticity, and the continuity condition is satisfied at the interface between layers and stress boundary condition at the external surfaces. In addition, this study provides the effects of transverse normal strain on flexural response of sandwich plate. Closed-form solutions for simply supported sandwich plate subjected to concentrated force are obtained for various side-to-thickness ratios. The results of the present theory are compared with third-order shear and normal deformation theory, first-order shear-deformation theory, and the classical plate theory. The inclusion of the effect of transverse normal strain in the theory showed significant deviations in displacements and stresses.

Aeroelastic analysis of foam-filled composite corrugated sandwich plates using a higher-order layerwise model

Due to the versatility of load bearing and heat insulation, foam-filled composite corrugated sandwich plates have a broad application in aerospace industry as thermal insulation structures for hypersonic vehicles. In this paper, a C0 higher-order layerwise finite element model is presented for aeroelastic analysis of foam-filled composite corrugated sandwich plates in supersonic flow with equivalent properties. The homogenization method for the foam-filled corrugated core is formulated by force and energy equivalence under different assumptions of force or moment distributions. In the proposed layerwise model, the composite sandwich plate is divided into three sections through thickness, while a higher-order displacement field is assumed for the single-layer orthotropic core section and a first-order displacement field for the two face-sheet sections made of multi-layer laminates. The effect of transverse normal strain is also taken into consideration. The unsteady aerodynamic pressure is evaluated by the first-order Piston theory. An eight-noded isoparametric element with 16 degrees of freedom per node is used for finite element method. The accuracy of the present model is verified with existing results in the literature. The influence of geometric parameters and material properties on critical dynamic pressure are discussed.

Isogeometric static analysis of laminated plates with curvilinear fibers based on Refined Zigzag Theory

In this study, we propose a new IsoGeometric formulation based on Refined Zigzag Theory (RZT), abbreviated as IG-RZT, for static analysis of laminated plates and sandwich panels with curvilinear fiber paths for the first time in literature. The original RZT formulation defines constant, linear, and zigzag deformation contributions of thickness coordinate to the in-plane displacements. This kinematic relation can accurately predict in-plane displacement of composite with straight fibers. However, estimating a realistic variation of in-plane displacements in a variable angle tow (VAT) composite is a more challenging problem as compared to the straight fiber composites. This difficulty has been addressed herein by including a quartic (fourth-order) polynomial thickness expansion that includes the transverse normal strain effects to the kinematic displacement fields of RZT. Moreover, the modelling of VAT composites results in the RZT zigzag functions to depend not only on the thickness coordinate but also on the in-plane positions. The present IG-RZT methodology is free of shear-locking and shear correction factors due to the integration of RZT with Non-Uniform Rational B-Splines (NURBS) functions of isogeometric analysis. The use of NURBS functions also enable the exact geometry data to be taken directly from a computer aided design (CAD) software, e.g., Rhinoceros, into an in-house Mathematica code. The accuracy and efficiency of the present IG-RZT formulation is assessed and validated by solving various examples of curvilinear fiber laminated plates with different aspect and span-to-thickness ratios. Comparison of IG-RZT results with reference solutions available in literature and generated by a commercial software (ANSYS) using 3D finite elements have demonstrated the remarkable benefits of the proposed IG-RZT method for predicting highly accurate both displacement and stress distributions of VAT composite structures.

Effect of porosity distribution on natural frequencies of thin and thick cylinders based on Mirsky–Hermann’s shear deformation theory

In this paper, the effect of the porosity coefficient and its distribution on the natural frequencies and mode shapes of a thin and thick cylindrical shell made of asymmetric porous materials is investigated analytically. The mechanical properties of the porous shell are non-uniform in the thickness and isotropic in the circumferential directions. The displacement field is assumed based on the first-order Mirsky–Hermann theory by considering the effects of transverse normal strain and rotary inertia. For more accurate stress resultants, the trapezoidal shape factor is considered in the formulation. The constitutive relations follow Biot’s theory for the porous materials and the kinematic of the problem is linear. The governing equations include four coupled partial differential equations that are solved using an analytical method and they are compared with the finite element analysis. By the sensitivity analysis, the effects of porosity distribution and geometry of the cylinder on the natural frequencies and mode shapes are studied. The results show that the largest frequency corresponds to the material which has the maximum density inside of the cylinder. Abbreviations: FSDT, first-order shear-deformation theory; 2D, two-dimensional; 3D, three-dimensional; FG, functionally graded; FGM, functionally graded materials; DQ, differential quadrature; MDEW, maximum density in outer (external) wall; MDIW, maximum density in inner wall; C.C, complex conjugate (of proceeding terms)

Dynamic response determination of viscoelastic annular plates using FSDT – perturbation approach

In this paper, the transient response of a viscoelastic annular plate which has time-dependent properties is determined mathematically under dynamic transverse load. The axisymmetric conditions are considered in the problem. The viscoelastic properties obey the standard linear solid model in shear and the bulk behavior in elastic. The equations of motion are extracted using Hamilton’s principle by small deformation assumption for the elastic condition and they are extended to the viscoelastic form by defining viscoelastic operators based on the separating the bulk and shear behaviors. The displacement field is defined with the first order shear deformation theory by considering the transverse normal strain effect. These equations which contain four coupled partial differential equations with variable coefficients are solved using the perturbation technique. The results are compared with those obtained from the classical plate theory and the finite element method. The presented formulation is useful for parametric study because it does not need to generate mesh and selecting time step for each model; also the running time is short with respect to the finite elements method. For sensitivity analysis, the effects of geometrical and mechanical parameters on the response are investigated by parametric study.

Effect of different viscoelastic models on free vibrations of thick cylindrical shells through FSDT under various boundary conditions

This paper investigates the free vibrations of cylindrical shells made of time-dependent materials for different viscoelastic models under various boundary conditions. During the extraction of equations, the displacement field is estimated through the first-order shear deformation theory taking into account the transverse normal strain effect. The constitutive equations follow Hooke\'s Law, and the kinematic relations are linear. The assumption of axisymmetric is included in the problem. The governing equations of thick viscoelastic cylindrical shell are determined for Maxwell, Kelvin-Voigt and the first and second types of Zener\'s models based on Hamilton\'s principle. The motion equations involve four coupled partial differential equations and an analytical method based on the elementary theory of differential equations is used for its solution. Relying on the results, the natural frequencies and mode shapes of viscoelastic shells are identified. Conducting a parametric study, we examine the effects of geometric and mechanical properties and boundary conditions, as well as the effect of transverse normal strain on natural frequencies. The results in this paper are compared against the results obtained from the finite elements analysis. The results suggest that solutions achieved from the two methods are ideally consistent in a special range.

Closed form solution for free vibrations analysis of FGPM thick cylinders employing FSDT under various boundary conditions

A porous material contains a structure in which its density reduces when the volume increases due to voids. The high strength, low weight and absorption of the sound or impact convert the porous materials desirable with vast applications to different fields of science and technology. In this paper, an analytical method is proposed for investigating the vibrations behavior of thick porous cylinders for various boundary conditions. The porosity variation is function of the thickness as symmetric, asymmetric or uniform. For mathematical modeling, the first-order shear deformation theory is used as displacement field, by considering the transverse normal strain effect. Hamilton’s principle in conjunction with the linear kinematic relations and Biot constitutive equations are employed to extract the motion equations. The governing equations contain four coupled partial differential equations. These equations are solved analytically and the natural frequencies and mode shapes are determined. A parametric study is performed and the effect of the materials and mechanical properties is studied for different boundary conditions. The results are compared with the finite element methods and the available results in the literature.

Dynamic response of functionally graded plates resting on two-parameter-based elastic foundation model using a quasi-3D theory

In this article, an improved quasi 3-D theory, which considers transverse normal strain effect along with inverse tangential variation of in-plane displacements across plate thickness, is employed to encapsulate the response of thick functionally graded plates. The present model satisfies both non-linear variation and zero transverse shear stresses across plate thickness. The governing equations of plate are obtained using Lagrange equation and solved by applying finite element procedure. The adaptability of the proposed theory is handled by considering the impact of various parameters on dynamic response of plate through several illustrations. The present model gives incredible concurrence with accessible literature.

Effects of transverse normal strain on bending of laminated composite beams

Effect of transverse normal strain on bending of laminated composite beams is proposed in this paper. A Quasi-3D beam theory which accounts for a higher-order variation of both axial and transverse displacements is used to consider the effects of both transverse shear and normal strains on bending behaviours of laminated composite beams. Ritz method is used to solve characteristic equations in which trigonometric shape functions are proposed. Numerical results for different boundary conditions are presented to compare with those from earlier works, and to investigate the effects of thickness stretching, fibre angles, span-to-height ratio and material anisotropy on the displacement and stresses of laminated composite beams.

Bending of functionally graded plates via a refined quasi-3D shear and normal deformation theory

Abstract Bending of functionally graded plate with two reverse simply supported edges is studied based upon a refined quasi three-dimensional (quasi-3D) shear and normal deformation theory using a third-order shape function. The present theory accounts for the distribution of transvers shear stresses that satisfies the free transverse shear stresses condition on the upper and lower surfaces of the plate. Therefore, the strain distribution does not include the unwanted influences of transverse shear correction factor. The effect of transverse normal strain is included. Unlike the traditional normal and shear deformation theories, the present theory have four unknowns only. The equilibrium equations are derived by using the principle of virtual work. The influence of material properties, aspect and side-to-thickness ratios, mechanical loads and inhomogeneity parameter are discussed. The efficiency and correctness of the present theory results are established by comparisons with available theories results.

Generalization of the C0-type zigzag theory for accurate thermomechanical analysis of laminated composites

Abstract A new method based on the C0-type efficient higher-order zigzag theory (EHOZT) is proposed for the efficient and accurate thermomechanical analysis of laminated composite plates. In this method, the transverse shear strain energy is modified according to the mixed variational theorem (MVT). The transverse stress field is defined by the arbitrary m-th order zigzag model including the transverse normal strain effect to improve the accuracy, while the displacement field is obtained from the C0-type EHOZT to amplify the benefits of the numerical efficiency. The transverse displacement field is assumed to be a smooth parabolic distribution to consider the transverse normal strain effect, which has a significant role in predicting the thermal deformation. Derivative components of the transverse displacement field are eliminated from the in-plane displacement field of the C0-type EHOZT by the transverse shear stress condition. The resulting strain energy expressions are collectively referred to as the C0-type EHOZT via the MVT. The proposed theory has the computational advantage of utilizing the C0-continuity condition for finite element implementation while allowing local distributions of the transverse stress to be improved via the recovery procedure. The obtained displacements and stress distributions were compared with data available in the literature for validation.