First-order Transverse Shear Deformation Research Articles

Nonlinear forced vibration analysis of laminated composite doubly-curved shells enriched by nanocomposites incorporating foundation and thermal effects

Assessment of the geometrically nonlinear forced vibration of the doubly-curved laminated composite shallow shell panels enriched by nanocomposites have been carried out in this paper. The proposed shells incorporate some layers composed of fibres and nanocomposite-reinforced resin, resting on the two types Winkler and Pasternak elastic foundations. Since the thin and moderately thick shells are investigated, the first-order transverse shear deformation theory (FSDT) is employed for extending the governing equations. The achieved system of differential equations incorporating the geometrical nonlinear terms is solved by applying the principles of the Galerkin method. This approach yields a closed-form semi-analytical solution for motion differential equation of the shell which is solved by fourth-order Runge-Kutta method. The consequence of the present approach is a strong and robust tool without any time-consuming computational process. The applicability of the proposed method is verified by considering some benchmark problems from the literature and comparing the relevant results with those are achieved in this study. Finally, by solving several examples, the effects of various factors on the forced-vibration of proposed shells are evaluated.

Geometrically nonlinear electro-thermo-static analysis of piezoelectric/CNT/GPL/fibre/polymer sandwich panels with double curvature resting on elastic foundation

Evaluation of the electro-thermo-static behaviour of the shell panels of double curvature using an analytical method is carried out in this paper. The proposed panels are made of a hybrid composite, and considered to be rested on the Winkler-Pasternak elastic foundation. The hybrid composite shell contains two thin piezoelectric sheets at its bottom and top surfaces, and some plies made of carbon fibres and polymeric resin which is reinforced by the randomly oriented nanocomposites. The employed nanocomposites for some layers are carbon nanotubes (CNTs) and for some other are graphene nanoplatelets (GPLs). The governing nonlinear system of differential equations of orthotropic doubly-curved shell panels are developed based on the first-order transverse shear deformation theory (FSDT) which incorporates moderately thick and also thin shells. The Gauss Quadrature method is applied to obtain the forces and moments due to the thermo-electrical loads, and the Galerkin method is employed to solve the developed system of equations which leads to a closed-form load-deformation relationship (equilibrium path of the shell). The novelty of the present study can be expressed in the solution and employed hybrid composite materials. First, by employing the proposed closed-form solution no time consuming computations are required. On the other hand, the effects of the piezoelectric and two different nanocomposite materials on the thermo-electro-mechanical nonlinear static behaviour of the hybrid composite shells can be evaluated. The accuracy of the present method is validated by comparing the achieved results for the some benchmark problems with those are existent in the literature. The impacts of various electro-thermo-mechanical and geometrical characteristics of the proposed shells on their equilibrium paths are also considered by carrying out an extended parametric study.

Free vibration and stability of hybrid nanocomposite-reinforced shallow toroidal shells using an extended closed-form formula based on the Galerkin method

This paper presents an analytical solution for free-vibration and stability analyses of hybrid-composite toroidal shells. The considered hybrid material incorporates the composition of fibers and nanocomposite-reinforced matrix (NCRM) such as carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs). The governing differential equations are extended based on the first-order transverse shear deformation theory (FSDT) and are solved by employing the approximation of Fourier series and Galerkin method. The considered hybrid composite material in the parametric studies are consisted of three layers in which the polymeric matrix are reinforced by CNTs in some layers and reinforced by GPLs in the other layers.

Flutter characteristics and free vibrations of rectangular functionally graded porous plates

This paper investigates the flutter characteristics and free vibrations of porous functionally graded plates using the analytical and Rayleigh-Ritz methods. The analytical methods can be employed for the solution of six of 21 arbitrary boundary conditions (the combinations of clamped, simply supported and free). The influence of five various models of porosity is studied, including different variations of stiffness/density along the thickness and/or length of a plate. The analysis is carried out for the classical (CPT) and first order transverse shear deformation (FSDT) theory. For CPT analytical solutions the effects of porosity are characterized by two simple correction factors that shows a very good agreement with the numerical Rayleigh-Ritz method. Parametric studies illustrate the possibility of increasing natural frequencies as well as flutter critical aerodynamic pressures and the necessity of implementing the optimization techniques to find the best solutions from the engineering point of view. The solutions are valid for thin and moderately thick porous FGM plates although the proposed methodology of computations can be easily adopted to the application of different approximations of 2D displacement distributions (see Appendix).

Dynamic Analysis of Sandwich Plate Structure by Finite Element Method

An 8-noded quadratic isoparametric plate bending finite element that incorporates first-order transverse shear deformation and rotary inertia is used to predict the free vibration response of sandwich plate structures. A programme has been developed using MATLAB. The finite element results presented here show good agreement with the available semi-analytical solutions and finite element results. Parametric studies have been conducted by incorporating variation in support conditions, fibre angles of the skins and overall thickness and detail interpretations are provided.

Effects of shear deformation on Mechanical and thermo-mechanical nonlinear stability of FGM shallow spherical shells subjected to uniform external pressure

In the present paper, the asymmetrical nonlinear response of a clamped functionally graded shallow spherical shell is subjected to uniform external pressure. It considers the effects of thermal stresses by both of the theories, Classical Laminate Theory, CLT and First-Order Shear Deformation Theory FSDT. Material properties are graded in the thickness direction according to the power-law distribution in terms of the volume fraction of the constituents. Mechanical and thermo-mechanical properties are assumed to be temperature-independent and linear elastic. All of the governing equations are derived by aid of first-order transverse shear deformation theory considering geometrical nonlinearity. The nonlinear differential equation system is solved employing Galerkin method. Buckling and post-buckling analysis have been done according to one-term deformation mode by the closed form relation of load-deflection that shows the equilibrium path. Parametric studies are conducted to bring out the effects of shear deformation on the equilibrium path in different geometries and boundary conditions. Numerical results are presented in graphical arrangement, showing that the geometrical nonlinear equilibrium paths. The effects of shear deformation on the equilibrium path are considered by comparing the results from FSDT and CLT and was verified by nonlinear finite-element method.

Developing an element free method for higher order shear deformation analysis of plates

A procedure is presented to eliminate transverse shear locking in analysis of laminated composite plates using Element Free Galerkin (EFG) method based on higher-order transverse shear deformation theory (HSDT). In the procedure, derivatives of the transverse displacement are introduced as independent variables. Thus, a formulation requiring C0 continuity shape functions for approximation is proposed for higher-order transverse shear deformation analysis of plates. Shear locking is avoided considering reduced integration for shear stiffness matrix; a method which is implemented in FEM. Moving Least Squares (MLS) method is utilized for shape functions and the penalty method is implemented to impose approximate boundary conditions. The present solutions are verified with other higher-order shear deformable studies. Moreover, a comparison between the present solutions with those obtained by EFG procedure based on First-order transverse Shear Deformation Theory (FSDT) is performed.

Varied effects of shear correction on thermal vibration of functionally graded material shells

AbstractThe effects of varied shear correction coefficient on the first-order transverse shear deformation result of functionally graded material (FGM) thick circular cylindrical shells under thermal vibration are investigated and computed by using the generalized differential quadrature method. The computed and varied values of shear correction coefficient are usually functions of FGM power law index and environment temperature. In the thermoelastic stress–strain relations, the simpler form stiffness of FGM shells under linear temperature rise is considered. The equation of shear correction coefficient is derived and obtained by using the total strain energy principle. Two parametric effects: environment temperature and FGM power law index on the thermal stress and center deflection of FGM thick circular cylindrical shells are obtained and investigated.

On the simple and mixed first-order theories for functionally graded plates resting on elastic foundations

In this article, the bending response of a functionally graded plate resting on elastic foundations and subjected to a transverse mechanical load is investigated. An accurate solution for the functionally graded plate with simply supported edges resting on elastic foundations is presented. The interaction between the plate and the elastic foundations is considered and included in the equilibrium equations. Pasternak’s model is used to describe the two-parameter elastic foundations, and get a special case of Winkler’s model by considering one-parameter of elastic foundation. A relationship between the simple and mixed first-order transverse shear deformation theories is presented. Numerical results for deflections and stresses of functionally graded plates are investigated. Comparisons between the results of the simple and mixed first-order theories are made, and appropriate conclusion is formulated. Additional boundary conditions at the edges of the present plates are investigated.

Magneto-thermo-elasticity of an electroconductive circular cylindrical shell featuring nonlinear deformations

The magneto-thermo-elastic interactions of a circular cylindrical shell immersed in applied magnetic and thermal fields are modeled and the shell’ nonlinear response and stability are investigated. The shell is made of a transversely isotropic, perfectly electroconductive non-ferromagnetic material. The shell’s deformation is characterized by small strains and moderately small rotations. A geometrically nonlinear, first-order transverse shear deformation shell model is developed and Galerkin’s method is used for spatial semi-discretization. The close connection between the static nonlinear response and the temperature-frequency interaction is explored to interpret the multi-solutions of the nonlinear response. Influence of prebuckling deformations on the buckling results is addressed and the significance of the applied magnetic fields on the response and stability is evaluated.

Calculation of [formula omitted] for a multidirectional composite double cantilever beam on two-parametric elastic foundation

In this research, a Timoshenko beam (TB) resting on two-parametric (or Pasternak) elastic foundation (PEF) is developed for determination of strain energy release rate (SERR) of mode-I delamination in multidirectional double cantilever beam (DCB) specimens. Based upon the compliance approach, a closed-form solution is obtained for SERR versus delamination length, applied load, effective mechanical properties, and geometrical dimensions of DCB specimen. The required effective flexural modulus in this model is determined with three different methodologies, plane stress resultant method, plies arrangement method, and using longitudinal tensile and compressive moduli method. The proposed model is assessed by experimental and numerical results available in the literature for various lay-ups. A comprehensive evaluation is also carried out among various theories which are used to model mode-I interlaminar fracture toughness. Results show that the developed model predicts G I at the onset of delamination growth very well for unidirectional and multidirectional DCB specimens. However, the main advantages of this model are: 1) It gives a simple and accurate explicit closed-form solution for SERR of symmetric unidirectional and multidirectional lay-ups in comparison with the high-order shear deformation beam theories with a very lengthy and tedious procedure. 2) It models both first-order transverse shear deformation and local effects at the crack tip (root rotation) to improve the split beam solution. 3) By simplifying the Timoshenko beam on two-parametric foundation, as a generalized case, special cases like Euler–Bernoulli and Timoshenko beams on the Winkler elastic foundation (WEF) are obtainable.

Free-Vibration Analysis of Ring-Stiffened Branched Composite Shells of Revolution

Application of the multisegment numerical integration technique is extended to the free-vibration analysis of macroscopically anisotropic filament-wound branched shells of revolution with ring stiffeners, considering the variation of the thickness and winding angle. The solution procedure is based on a modified-frequency trial method, which processes on the numerically integrated transformed fundamental shell equations that are obtained in terms of finite exponential Fourier transform of the fundamental shell variables. The full macroscopically anisotropic form of the constitutive relations, including first-order transverse shear deformation and all components of translatory and rotary inertia, are included in the analysis. To handle branched shells of revolution, modifications that are necessary to incorporate junctions are added to the solution procedure. Inclusion of asymmetric circumferential stiffeners, with respect to the middle surface of the shell, into the semi-analytical solution method is demonstrated by presenting two alternative methods of analysis. The present solution methodology also incorporates the variation of the thickness and winding angle along the meridian of filament-wound shells of revolution, with general meridional curvature, by assuming placement of filaments along the geodesic fiber path on the surface of the shell of revolution.

English

This paper presents a finite element simulation and buckling analysis of layered composite structures. The shell element is based on the Reissner-Mindlin first-order shear deformation theory and the finite element method account for full geometric nonlinearity imposing large deformations and rotations. Finite rotations are treated by Rodrigues parameterization. The combinations of enhanced assumed strain (EAS) in the membrane strains and assumed natural strains in the shear strains are implemented to improve the shell element behavior. Stability analysis of anisotropic short open cylinder, layered cylindrical shells with different lamina sequenceand a hinged roof structure are presented including snap-through and snap-back problems. The present simulations are compared with those obtained by finite element analyses based on first-order transverse shear deformation moderate or large rotation or refined von Karmantype theories in earlier literature.

Semi-analytical study of free vibration characteristics of shear deformable filament wound anisotropic shells of revolution

Free vibration characteristics of filament wound anisotropic shells of revolution are investigated by using multisegment numerical integration technique in combination with a modified frequency trial method. The applicability of multisegment numerical integration technique is extended to the solution of free vibration problem of anisotropic composite shells of revolution through the use of finite exponential Fourier transform of the fundamental shell equations. The governing shell equations comprise the full anisotropic form of the constitutive relations, including first-order transverse shear deformation, and all components of translatory and rotary inertia. The variation of the stiffness coefficients along the axis of the shell is also incorporated into the solution method. Filaments are assumed to be placed along the geodesic fiber path on the shell of revolution resulting in the variation of the stiffness coefficients along the axis of the composite shell of revolution with general meridional curvature. Sample solutions have been performed on the effect of the variation of the stiffness coefficients on the free vibration behavior of filament wound truncated conical and spherical shells of revolution.

Large rotations in first-order shear deformation FE analysis of laminated shells

The paper deals with the geometrically non-linear analysis of laminated composite beams, plates and shells in the framework of the first-order transverse shear deformation (FOSD) theory. A central point of the present paper is the discussion of the relevance of five- and six-parameter variants, respectively, of the FOSD hypothesis for large rotation plate and shell problems. In particular, it is shown that the assumption of constant through-thickness distribution of the transverse normal displacements is acceptable only for small and moderate rotation problems. Implications inherent in this assumption that are incompatible with large rotations are discussed from the point of view of the transverse normal strain–displacement relations as well as in the light of an enhanced, accurate large rotation formulation based on the use of Euler angles. The latter one is implemented as an updating process within a Total Lagrangian formulation of the six-parameter FOSD large rotation plate and shell theory. Numerical solutions are obtained by using isoparametric eight-node Serendipity-type shell finite elements with reduced integration. The Riks–Wempner–Ramm arc-length control method is used to trace primary and secondary equilibrium paths in the pre- and post-buckling range of deformation. A number of sample problems of non-linear, large rotation response of composite laminated plate and shell structures are presented including symmetric and asymmetric snap-through and snap-back problems.

Finite Element Multiple-Mode Approach to Nonlinear Free Vibrations of Shallow Shells

Two finite element (FE) modal formulations for large-amplitude free vibration of isotropic and arbitrary laminated composite shallow shells are presented. The system equation of motion is formulated first in the physical structural node degrees of freedom (DOF). Then the system is transformed into two distinctly different sets of general Duffing-type modal equations based on 1) modal amplitudes of coupled linear bending and in-plane modes, where in-plane inertia is included in the formulation, and 2) modal amplitudes of linear bending modes only, where the in-plane inertia is neglected. Multiple modes and the first-order transverse shear deformation are considered in the formulations. A shallow-shell finite element is developed as an extension from the triangular Mindlin (MIN3) plate element with the improved shear correction factor by Tessler. Time numerical integration is employed to determine the nonlinear periodic frequency characteristics. The inaccuracy in characterizing a shallow-shell response with coupled linear bending and in-plane modes is demonstrated and discussed by comparing with the FE solution in structural node DOF. Study cases include isotropic and composite panels of different shallow-shell geometries.

Buckling of fiber-reinforced viscoelastic composite plates using various plate theories

This paper studies the quasi-static stability analysis of fiber-reinforced viscoelastic composite plates subjected to in-plane edge load systems. The study is based on a unified shear-deformable plate theory. This theory enables the trial and testing of different through-thickness transverse shear-strain distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse shear correction factors. Using the method of effective moduli solves the equations governing the stability of simply supported fiber-reinforced viscoelastic composite plates. The solution concerns the determination of the critical in-plane edge loads associated with the asymptotic instability of plates. In a study of this problem the general quasi-static stability solutions are compared with those based on the classical, first-order and sinusoidal transverse shear-deformation theories. Numerical applications using higher-order shear-deformation theory are presented and comparisons with the results of other theories are formulated.

Application of theorem of minimum potential energy to a complex structure Part I: two-dimensional analysis

A non-circular shell cross-section with flat sides and circular arc corners is analyzed using the theorem of minimum potential energy. Two-dimensional, plane strain assumptions are utilized, and the potential energy (PE) expression for the structure is developed, including first-order transverse shear deformation effects. The unknown displacements are represented by power series, and the PE expression is rewritten in terms of the summation convention for the power series. The variation of the PE expression is taken, leading to a linear system of equations that is solved for the unknown power series coefficients. With the displacements determined, stresses are calculated for a composite sandwich construction. Excellent agreement is found with other analytical methods and with finite element analyses.

Application of theorem of minimum potential energy to a complex structure: Part II: three-dimensional analysis

A cylindrical shell with a non-circular cross-section consisting of flat sides and circular arc corners is analyzed using the theorem of minimum potential energy. The three-dimensional analysis builds on previous two-dimensional work. The potential energy expression for the structure is developed, including first-order transverse shear deformation effects. All unknown displacements are represented by power series, and the potential energy expression is rewritten in terms of the summation convention for the power series. The variation of the potential energy expression is taken, leading to a linear system of equations that is solved for the unknown power series coefficients. With the displacements determined, stresses are calculated for a composite sandwich construction. An examination of both short shells (less than twice the boundary layer length) and long shells (more than twice the boundary layer length) is made. The MPE method with power series is found to predict behavior well for short shells, but not for long shells.

Non-linear thermal effects on the bending response of cross-ply laminated plates using refined first-order theory

Based on the representation of the displacement field proper to the first-order transverse shear deformation theory, and using Reissner’s mixed variational principle modified as to include thermoelastic effects, an equivalent single-layer model of composite laminated plates is presented including the geometric non-linearity. In contrast to the usual approach of such models, the present model has the advantage of eliminating the need of the inclusion of a shear correction factor. Moreover, this model predicts continuous stress distributions through the laminate thickness. The equations of motion, the constitutive equations and the required boundary conditions are obtained. Based on the obtained equations, linear and non-linear numerical results on the static state of stresses and displacements of cross-ply symmetric and antisymmetric laminated plates are supplied. To assess the present model, some of the obtained linear results are compared with their counterparts in the literature. The effect of the geometric non-linearity on the thermal response of the laminate is investigated.