Arbitrary Combination Of Boundary Conditions Research Articles

Dynamic response of moderately thick graphene reinforced composite cylindrical panels under the action of moving load

An attempt is made is this research to study the response of cylindrical panels subjected to a moving load. Cylindrical panel is made from a composite laminated material where each layer of the media is reinforced with graphene platelets. Also the amount of graphene platelets in the layers may be different which results in a piecewise functionally graded media. With the aid of the first order shear deformation theory of shells, Donnell type of kinematics and Hamilton principle, the basic components of energies in the panel are established. With the aid of general idea of Ritz method whose shape functions are constructed using the Chebyshev polynomials, the matrix representation of the governing equations is established. While the developed formulation is general and may be used for moving load with arbitrary path and velocity, here it is assumed that velocity of the moving load is constant and load is passing on a straight line. The obtained equations are traced in time domain via the Newmark time marching scheme. Results of this study may be used for cylindrical panels with arbitrary combinations of boundary conditions. At first convergence and comparison studies are given to assure the validity of the proposed solution method and formulation. After that, novel numerical results are given to explore the effects of different parameters. It is shown that with introduction of even a low amount of reinforcements, dynamic deflections in the panel may be alleviated significantly.

On the response of graphene platelet reinforced composite laminated plates subjected to instantaneous thermal shock

Current investigation deals with the dynamics of a novel class of composites, known as functionally graded graphene reinforced composite (FG-GPLRC) laminated plates subjected to thermal shock. It is assumed that each layer of the composite laminated plate is reinforced with different amount of GPLs which results in a piecewise functionally graded media. Plate is subjected to different types of thermal boundary conditions such as rapid heat flux, rapid temperature elevation or thermally insulated on the top and bottom surfaces. The one dimensional heat conduction equation is established and solved considering the boundary conditions on the top and bottom surfaces and also continuity of temperature and heat flux among the layers. Thermal force and thermal moment resultants are evaluated at each time step after evaluation of temperature profile. The dynamic equations of the plate are established by means of the first order shear deformation theory and the Ritz method where shape functions are estimated via the Chebyshev polynomials. The adopted method is general and may be used for arbitrary combinations of boundary conditions. Thermally induced vibrations of FG-GPLRC plates are studied for various weight fractions of GPLs, different patterns of GPLs, various mechanical and thermal boundary conditions. It is shown that thermally induced vibrations indeed exist.

Study of the effects of shear piezoelectric actuators on the performance of laminated composite shells by an isogeometric approach

This paper presents an isogeometric approach based on the Non-Uniform Rational B-Splines (NURBS) to investigate static and free vibration responses of smart composite shells integrated with shear piezoelectric actuators. The degenerated shell formulation according to the Mindlin-Reissner shell theory is combined with the isogeometric approach. To model the laminated smart shells, the Equivalent Single Layer (ESL) theory is used. To consider the electric potential in the shear piezoelectric actuator layers, a sub-layer approach is adopted that assumes linear variation in the thickness direction of the sub-layer. The effect of different mechanical and electrical boundary conditions on transverse deformation and natural frequencies of laminated smart shells by applying the electric field have been investigated. In the case studies, two parallel edges of the considered shell structures are assumed simply supported and the other two with an arbitrary combination of boundary conditions including clamped, free or simply supports. Also, open-circuit and closed-circuit conditions are used as electric boundary conditions. Investigation of the effects of the shear piezoelectric actuator layers on various factors, including the simultaneous mechanical and electrical loadings as well as the radius of curvature of the shell are amongst the objects of this paper. Also, several numerical examples are presented to demonstrate the efficiency and accuracy of the isogeometric approach in the study of shear effects of the piezoelectric actuator layers. The obtained results indicate the reliability and desirability of the proposed approach.

NURBS-based thermal buckling analysis of graphene platelet reinforced composite laminated skew plates

A nonuniform rational B-spline isogeometric finite element formulation is presented in this research to analyze the thermal buckling behavior of composite laminated skew plates reinforced by graphene platelets. Formulation is based on the first-order shear deformation plate theory. It is assumed that each layer of the composite laminated plate may have different volume fraction of graphene platelets leading to a through-the-thickness piecewise functionally graded medium. The equivalent properties of the plate are obtained by means of the Halpin–Tsai rule. The developed solution method may be used for arbitrary combinations of boundary conditions. The accuracy of the developed formulation is depicted via comparison studies with respect to the available data in the open literature. Novel numerical results are also given to show the effects of volume fraction of graphene platelets, distributed patterns of graphene platelets, and geometric characteristics of the skew plate.

Thermal buckling analysis of cylindrical shells of carbon nanotubes reinforced composites

The purpose of this study is to investigate the thermal buckling analysis of cylindrical shells of carbon nanotubes reinforced composites (CNTRC). In this study, a hybrid laminate composite plate consisting of N layers which are of thickness t is designed. N is the number of layers and the thickness of each functionally graded carbon nanotube reinforced composite (CNTR-FG) layer is h, and three types of the distribution of carbon nanotubes (CNT) are considered for each layer. CNTR-FG are multi-layered plates which have the length "a", width "b", thickness "t", and an arbitrary combination of boundary conditions along the four edges of the layer. The distribution of carbon nanotubes along the thickness of each layer of CNTR-FG is considered to be given. Implementation is done by using Abaqus software. In the investigation of the thermal buckling of cylindrical shells of CNTRC, in three different types of distribution of carbon nanotubes, the behavior of buckling of a cylindrical shell is investigated. In each case, the amount of buckling generated in the shell is calculated and analyzed. The results show that the behaviors of these three types of distributions are close, and the nanotubes with uniform distribution against thermal buckling loads perform better than the other two distributions.

Enhancement of non-linear thermal stability of temperature dependent laminated beams with graphene reinforcements

Present investigation deals with the thermal postbuckling behaviour of a composite laminated beam where graphene is used as reinforcement of each lamina. The composite laminated beam may be piecewise functionally graded by changing the volume fraction of graphene in each lamina. Mechanical properties in each layer are obtained according to the modified Halpin-Tsai approach which contains efficiency parameters to capture the size dependency of the nanocomposite properties. Thermomechanical properties of the matrix and reinforcements are assumed to be temperature dependent. Beam is resting on an elastic foundation which acts in compression as well as in tension. The first order shear deformation beam theory and von Kármán type of geometrical nonlinearity are used as the basic assumptions to establish the total strain energy of the system. Beam is subjected to uniform temperature elevation. With the aid of conventional Ritz method and the simple polynomials as the basic functions, the matrix representation of the governing equations is derived. The adopted solution method may be used for arbitrary combinations of boundary conditions. The obtained system of equations is nonlinear in terms of both temperature and displacements. An iterative displacement control strategy is proposed to extract the thermal postbuckling curves of the beam resting on a conventional elastic foundation. Numerical results are given to discuss the effects of graphene distribution, stacking sequence, boundary conditions, side to thickness ratio and foundation stiffness on critical buckling temperature and thermal postbuckling equilibrium path of the beam. It is shown that, through a piecewise functionally graded distribution of graphene in matrix, critical buckling temperature may be enhanced significantly and thermal postbuckling deflection may be alleviated.

Thermal post-buckling of FG-CNT reinforced composite plates

Present research deals with the postbuckling problem of carbon nanotube reinforced composite plates subjected to uniform temperature rise loading. Distribution of carbon nanotubes as reinforcements may be uniform or functionally graded. To account for the large deformations of the plate, von-Kármán type of geometrical nonlinearity is included into the formulation. The virtual displacements principle associated with the conventional Ritz formulation whose shape functions are selected as the Chebyshev polynomials is used to obtain the matrix representation of the nonlinear equilibrium equations. The solution method is general and may be used for arbitrary combination of boundary conditions. The postbuckling equilibrium path which is governed by a nonlinear eigenvalue problem is traced using a displacement control strategy. Results of this study are compared with the available data in the open literature for the cases of isotropic homogeneous plates and cross-ply laminated plates. Afterwards numerical results are given for FG-CNTRC plates. It is shown that, FG-X pattern results in higher buckling temperature and also decreases the postbuckling deflection of the plate. Furthermore, this type of composites are eager to exhibit the secondary instability which is designated with a snap-through phenomenon in the post-buckling equilibrium path.

Analysis of free vibration of functionally graded material cylinders by Hermitian collocation meshless method

In the present work, a new axisymmetric Hermitian collocation meshless method is presented for analysis of free vibration of functionally graded material (FGM) cylinders. This method is based on strong form of equilibrium equation and Hermitian moving least squares approximation. Collocation method is purely meshless and numerical integration is not needed in this method. But this method has difficulty of instability in analysis of P.D.E.s with natural boundary conditions. In the present work, Hermitian shape functions were used to overcome this difficulty. Material is assumed to be functionally graded in the radial direction. Variations in the material properties such as Young’s modulus and Poisson’s ratio may be arbitrary functions of the radial coordinate. The FGM cylinder material varies continuously from silicon carbide on the inner surface to Stainless steel (SUS304) on the outer surface. Free vibration analysis of FGM cylinders with any arbitrary combination of boundary conditions is possible by the proposed model. Natural frequencies obtained from the presented model are in a good agreement with analytical solutions. Effect of various types of boundary conditions, geometrical parameters and mechanical properties on the natural frequencies are studied.

Vibration analysis of axially moving line supported functionally graded plates with temperature-dependent properties

Vibration analysis of axially moving functionally graded plates with internal line supports and temperature-dependent properties is investigated using harmonic differential quadrature method. The plate is subjected to static in-plane forces while out-of-plane loading is dynamic. Stability of an axially moving plate, traveling at a constant velocity between different supports and experiencing small transverse vibrations are considered. The series of internal rigid line supports parallel to the plate edges are considered together with various arbitrary combinations of boundary conditions. Material properties of the plate are assumed temperature-dependent which is a non-linear function of temperature and differ continuously through thickness according to a power-law distribution of the volume fractions of the plate constituents. Two types of micromechanical models, namely, the Voigt and Mori–Tanaka models are considered. Based on the classical plate theory, the governing equations are obtained for functionally graded plate using the Hamilton’s principle. In the frame of a general dynamic analysis, it is shown that the onset of instability takes place in the form of divergence. The plate may experience divergence or flutter instability at a super critical velocity. Results for dynamic analysis of isotropic and laminated plates are validated with available data in the existing literature, which show excellent agreement. Furthermore, some new results are presented for vibration analysis of functionally graded material plates to study effects of the location of line supports, material properties, volume fraction, temperature, loading, aspect ratio and speed.

Discussion: “Closed Form Expression for the Vibration Problem of a Transversely Isotropic Magneto-Electro-Elastic Plate” (Liu, M. F., and Chang, T. P., 2010, ASME J. Appl. Mech., 77, 024502)

In Ref. [1] the authors developed a plate theory for bending and vibration of magnetoelectroelastic (MEE) thin plates by employing the Kirchhoff hypothesis, and numerical results are presented and compared with certain existing solutions. The work should be interesting and valuable since MEE plate structures have a wide application prospect in various fields. However, in the derivation of the constitutive relations for thin plates, i.e., Eqs. (15)–(17) in Ref. [1], the authors improperly substituted the zero normal strain (ɛz=0) in the three-dimensional constitutive equations. It can be easily shown that, the presented moment-deflection relations, i.e., Eqs. (33)–(35) in Ref. [1], cannot be reduced to the classical results for the degenerated case of an isotropic elastic material.The correct way to derive the proper constitutive relations for thin plate actually starts from the relation σz=0, since the plate is very thin and a plane-stress state dominates [2–4]. From this relation, one can solve for ɛz in terms of other strain components. Substituting ɛz back into the 3D constitutive equations of MEE materials, the ones for MEE thin plates can be given as follows(1)σx=c¯11(-z∂2w∂x2)+c¯12(-z∂2w∂y2)+e¯31∂ϕ∂z+q¯31∂ψ∂zσy=c¯12(-z∂2w∂x2)+c¯11(-z∂2w∂y2)+e¯31∂ϕ∂z+q¯31∂ψ∂zτxy=-2c66z∂2w∂x∂yDz=e¯31∂u∂x+e¯31∂v∂y-ɛ¯33∂ϕ∂z-d¯33∂ψ∂zBz=q¯31∂u∂x+q¯31∂v∂y-d¯33∂ϕ∂z-μ¯33∂ψ∂zwhere(2)c¯11=c11-c13c13/c33, c¯12=c12-c13c13/c33e¯31=e31-c13e33/c33, q¯31=q31-c13q33/c33ɛ¯33=ɛ33+e33e33/c33, μ¯33=μ33+q33q33/c33d¯33=d33+e33q33/c33Obviously, Eq. (1) is different from Eqs. (15)–(17) presented in Ref. [1], and can be reduced to the classical results for an isotropic elastic plate. A systematic deduction of the two-dimensional theories for MEE beams, plates and shells as well as their applications may be found in Refs. [5–], to name a few.Furthermore, to the writer's knowledge, it is impossible to write the natural frequency of a rectangular plate with an arbitrary combination of boundary conditions in the simple form of Eq. (42) in Ref. [1]. On the other hand, comprehensive discussions on vibration of elastic rectangular plates, as documented in Ref. [9], can be similarly applied to MEE thin plates.

Modal characteristics of cross-ply laminated smart beams using various beam theories

Analytical models and solutions are obtained for the free vibration of cross-ply laminated beams with extension piezoelectric actuators. The piezoelectric actuators are bonded along the beam surface. The models are based on the higher-order shear deformation beam theory (HOBT), first-order shear deformation beam theory (FOBT) and the classical Euler-Bernoulli beam theory (CBT). The equations of motion and associated boundary conditions are derived for the beam models using Hamilton's principle. The state-space approach is used to find accurate natural frequencies and mode shapes for arbitrary combinations of boundary conditions. The exact analytical solutions obtained are illustrated numerically in a number of figures and tables revealing the effects of the number of layers, the actuators' thickness, the length to thickness ratio and the character of boundary conditions on the non-dimensional frequencies. Finally comparison of frequencies obtained by CBT, FOBT and HOBT is made to assess the importance of shear deformation.

Analysis of free vibration of functionally graded material (FGM) cylinders by a meshless method

In the present work, a new axisymmetric weak form meshless method is presented for analysis of free vibration of functionally graded material (FGM) cylinders. This method is based on weak form of equilibrium equation and moving least squares (MLS) approximation. Essential boundary conditions are imposed by transformation method. In this method, shape functions that do not satisfy the Kronecker delta condition are corrected, then essential boundary conditions are imposed easily as in the finite element method (FEM). In the present work, the material is assumed to be functionally graded in the radial direction. Variations in the material properties such as Young’s modulus and Poisson’s ratio may be arbitrary functions of the radial coordinate. The FGM cylinder material varies continuously from silicon carbide (SiC) on the inner surface to stainless steel (SUS304) on the outer surface. Free vibration analysis of FGM cylinders with any arbitrary combination of boundary conditions is possible by the proposed model. Natural frequencies obtained from the presented model are in good agreement with results of finite element simulation. Effects of various types of boundary conditions, geometrical parameters, and mechanical properties on the natural frequencies are studied.

FREE VIBRATION OF SHEAR-DEFORMABLE GENERAL TRIANGULAR PLATES

An analysis of free vibration of shear-deformable general triangular plates with arbitrary combinations of boundary conditions is presented. The Reissner-Mindlin plate theory is used to incorporate shear deformation effects in the analysis. The triangular plate is first mapped onto a basic square plate. The Rayleigh-Ritz method with an admissible displacement function expressed as a product of a two-dimensional simple polynomial and a basic function is then used to obtain the governing eigenvalue equation. The basic function is chosen as the product of boundary expressions of the basic square plate, each raised to an appropriate power to satisfy the various geometric boundary conditions of the actual triangular plate Gaussian quadrature is used for numerical evaluation of stiffness and mass matrices. The natural frequencies of general triangular Mindlin plates with different combinations of free, simply supported and clamped conditions are determined. Wherever possible, the results are verified by comparison with existing published solutions. A comprehensive parametric study of natural frequencies of general triangular plates with all three edges clamped is presented graphically. No previous results are known to exist for general triangular Mindlin plates having arbitrary combinations of boundary conditions.

Analytical method for vibration of angle-ply cylindrical shells having arbitrary edges

A theoretical method is presented for solving the free vibration of angle-ply laminated cylindrical shells. The angle-ply laminated shell is macroscopically modeled as a thin shell of general anisotropy by using the classical lamination theory. The functional derived from the Flugge-type shell theory is minimized by following the Ritz procedure, and arbitrary combinations of boundary conditions at both ends are accommodated by introducing newly developed admissible functions. A computer code is made to verify the validity of the present method, and the natural frequencies and mode shapes of the laminated cylindrical shells are given for some typical edge conditions numerically.

境界条件の自由な組合せを考慮したＦＲＰ積層長方形板の振動解析法

境界条件の自由な組合せを考慮したＦＲＰ積層長方形板の振動解析法

Bending of Ring-Sector Plates of Variable Stiffness

For the bending of ring plates of varying thickness having rotational symmetry in rigidity, loading, and boundary conditions, a wide range of solutions have been reported in the literature. On the other hand, know solutions for ring-sector plates without such symmetry, are restricted to cases having rigidity proportional to some power of the radius and simply supported along the radial edges. No work on bending the ring-sector plates having clamped radial edges appears to have been reported so far. The work reported herein presents a solution of the problem of bending of ring-sector plates of variable rigidity, having clamped radial edges and the circumferential edges having arbitrary combinations of boundary conditions.