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Abstract - In this article , thermal buckling analysis of functionally graded annular sector plate is studied. The mechanical and thermal properties of the functionally graded sector plate are assumed to be graded in the thickness direction . The equilibrium and stability equations are derived based on the first order shear deformation plate theory (FSDT) in conjunction with nonlinear von-karman assumptions. Differential quadrature method is used to discretize the equilibrium and stability equations. In this method a non-uniform mesh point distribution (Chebyshev-Gauss-Lobatto) is used for provide accuracy of solutions and convergence rate . By using this method, there is no restriction on implementation of boundary conditions and various boundary conditions can be implemented along any edges . Finally, The results compared with other researches and the effects of plate thickness, sector angle, annularity, power law index and various boundary conditions on the critical buckling temperature are discussed in details .

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In this paper a two dimensional elasticity for free vibrations and the effect of elastic foundantion on a two-direction functionally graded beams with integrated surface piezoelectric layers with combination of differential quadrature method and space-state method is presented here. Differential quadrature method in axial direction and space-state method in transverse direction is used. It’s considered that two parameters model or winkler-pasternak for elastic foundation which has been considered two kinds of boundary conditions include simply support and clamped-clamped. Also, It is assumed that beam properties in thickness and axial direction varying exponentially and poison factor is constant which has been considered the effects of materials properties gradient index and number waves on free vibrations beams. The obtained results show that this method has good accuracy and high speed of convergence.

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In this article, nonlinear bending analysis of single-layered circular graphene sheet is studied. The equilibrium equations are derived based on the nonlocal continuum mechanics and principle of virtual work and first order shear deformation plate theory (FSDT). Differential quadrature method is used to discretize the equilibrium equations. In this method a non-uniform mesh point distribution (Chebyshev- Gauss- Lobatto) is used for provide accuracy of solutions and convergence rate. The effect of nonlocal parameter, thickness, number of grid points and lateral loading are investigated on deflection of graphene sheet. The results are compared with valid results reported in the literature.

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In this paper, differential quadrature element method (DQEM) is used to analyze the free transverse vibration of multi-stepped rotors resting on multiple bearings. Timoshenko beam theory is used to show the gyroscopic effects; Also each bearing is replaced with four springs; two translational and two rotational acting on two perpendicular directions. Governing equations, compatibility conditions at the each step and each bearing and external boundary conditions are derived and formulated by the differential quadrature rules. First, convergence and versatility of the proposed method are tested by the presented exact solutions. Then, the Campbell diagram is derived for a desired case study and variation of natural frequencies is investigated versus angular velocity of spin. The most advantage of the proposed method is being less time-consuming in comparison with the other methods, especially for cases with high number of steps and bearings. Accuracy of the proposed method is confirmed by the presented exact solutions and effect of angular velocity of spin on natural frequencies (Campbell diagram) is investigated. Comparison of the proposed method with the exact solutions revealed the convergence and accuracy of the proposed method.

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Axially moving beams are extensively involved in various industries and have significant importance in many mechanical engineering problems. In this paper, the nonlinear forced vibrations of axially moving beam under harmonic force and thermal environment have been studied. In order to considering the effects of transverse shear deformation and rotary inertia, the Timoshenko beam theory has been used to model the axially moving beam. The nonlinear governing equations are derived with the help of Hamilton’s principle. Then the equations and boundary conditions are discretized through generalized differential quadrature method (GDQ) and its differential matrix operators, and accordingly the partial differential equations are converted into the ordinary differential equations. To study the frequency response of the system, the harmonic balance method is used. Also the time responses of the axially moving beam are obtained by the Runge-Kutta method. In a case study, the effects of various parameters such as the axial speed, transverse force acting on the beam, damping coefficient and temperature change on the frequency responses of the axially moving beam with both end simply supported boundary conditions are discussed. The results show that the dynamic behavior of system is significantly affected by any of the mentioned factors.

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In this paper, free vibration of two-dimensional functionally graded (2D-FG) annular sectorial plate surrounded by Winkler-Pasternak elastic foundation has been investigated. It is assumed that the plate properties vary continuously through its both circumference and thickness according to power law distribution of the volume fraction. Primarily, we calculate the forces and resultant moments and then the total potential energy of system. Then, by applying the Hamilton’s principal any by regarding the first order shear deformation plate theory (FSDT) the governing differential equations have been derived. The numerical differential quadrature method, (DQM), has been employed for solving the motion equations. Two different boundary conditions such as simply supported and clamp-simply supported are considered. Initially, the obtained results were verified against those given in the literature and by ANSYS software and we confident from the obtain results. The effects of geometrical and elastic foundation parameters along with FG power indices effects on the natural frequencies have been studied. The study of results shows that, elastic foundation and FG parameters have significant effects on natural frequencies. By doing this research for 2D-FG materials the characteristic vibration of structure can be controlled by more parameters than 1D-FG materials.

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In this study, the vibrations of carbon nanotube are investigated in which the inner fluid flow with constant velocity and the widespread external harmonic force is applied to it. Also the nanotube is embedded in an elastic visco-Pasternak medium and the boundary conditions at two ends of nanotube are simply supported. In order to analyze the system and considering the small scale effects, the couple stress theory is employed and the Timoshenko beam theory is used for modeling the nanotube. The Hamilton's principle is written with taking into account all energies and external works of system and consequently the nonlinear motion equations of the system are achieved. Then with help of generalized differential quadrature method, the obtained partial differential equations are converted to ordinary differential equations and the domain of the beam is discretized. From the MatCont package in MATLAB software, the frequency responses of nanotube are examined. To this aim, the second order differential equations are turned to first order ones with appropriate transformations. So the small scale effect or equivalently the differences between present approach and classical Timoshenko beam theory are presented. Furthermore the effects of the size of nanotube, fluid velocity, applied transverse force and the elastic foundation parameters are studied. It is observed that the dependency of frequency response on each of these parameters is different and it significantly changes with these factors.

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In this paper, pull-in instability of a cantilever beam type nanoactuator made of the functionally graded material (FGM) based on higher order modified strain gradient theory investigated. It is assumed that the functionally graded beam, made of germanium and silicon, follows the volume fraction definition and law of mixtures, and its properties change as a power function through its thickness. By changing the germanium constituent volume fraction percent of the nano-beam, five different types of the nano-beams are investigated. The influences of the volume fraction index, length scale parameter and the intermolecular forces, on the pull-in instability are examined. Principle of minimum total potential energy used to derive the nonlinear governing differential equation and consistent boundary conditions which is then solved using the differential quadrature method (DQM). The present analysis is validated through direct comparisons with published other research methods and experimental results and after comparison excellent agreement has been achieved between new solution method and other experimental and numerical solution results. Besides, the results demonstrate that size effect and amount of volume fraction have a substantial impact on the pull-in instability behavior of beam-type nanoactuator.

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In the present study, thermal buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shells is presented. The effective material properties of FG-CNTRCs are determined using the extended rule of mixture. By employing the Hamilton’s principle and based on first-order shear deformation theory and Donnell strain-displacement relations, the governing equations are obtaind. The membrane solution of linear equilibrium equations is considered to obtain the pre-buckling force resultants. Using the generalized differential quadrature method in axial direction and periodic differential operators in circumferential direction, the stability equations are discretized and the critical buckling temperature difference of shell is obtained. The accuracy of the present work are first validated by the results given in the literature and then the impacts of involved parameters such as volume fractions and types of distributions of carbon nanotubes, boundary conditions and geometrical parameters on thermal buckling of functionally graded nanocomposite conical shell are investigated. The results indicate that the values of volume fractions and types of distributions of carbon nanotubes along the thickness direction play an important role on thermal instability of FG-CNTRC conical shells.

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In this paper, the effect of nonhomogeneous parameter in orthotropic Functionally Graded Material(FGM) in a cracked layer is investigated. It is assumed that the mechanical and thermal properties of material are dependent on x-coordinate (collinear with crack surfaces) in exponential form. The problem is solved for internal and edge crack in two way, integral equations and generalized differential quadrature method. Thermal loading is in a way that temperature distribution in the layer is uniform. Because of variation in mechanical and thermal properties, stress distribution due to this loading is not uniform. In the solution of problem with integral equations method, first, thermo-elasticity problem with no cracks and then isothermal crack problem are separately solved. Afterward with these solutions, the main problem will be solved. In order to solve isothermal crack problem, after conversion and simplifying the equations in orthotropic material, Navier's equations will be solved with the Fourier. Numerical solution of the problem is the generalized differential quadrature element method that is being presented for verification of the results of the integral equations for a specific state in the diagram format. Also the effect of temperature on intensity factor with various values of nonhomogeneous parameter is investigated.

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The objective of this investigation is to present a semi-analytical method for studying the buckling of the moderately thick composite conical shells under axial compressive load. In order to derive the equilibrium equations of the conical shell, first order shear deformation shell theory is used. The equilibrium equations are derived by applying the principle of minimum potential energy to the energy function that they are in the type of partial differential equations. In the following, the partial differential equations are transformed to algebraic type by using Galerkin and differential quadrature methods and then the standard eigenvalue equation is formed and critical buckling load is calculated. Also, to validate the results obtained in this study, comparisons are made with outcomes of previous literatures and the results of Abaqus finite element software. Analyzing the results, shows the convergence speed and good accuracy of differential quadrature method and desired precision of Galerkin method in calculating the critical buckling load. Finally, the effect of cone angle, fiber orientation, boundary conditions, ratios of thickness to radius and length to radius of the critical buckling load are studied.

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Noncircular lobed journal bearing performance, in comparison with circular types, depends on various design parameters such as tilt and mount angles. Mounting orientation of this kind of bearings with respect to machine frame (mount angle) and also the way of setting their lobes with respect to each other (tilt angle), can change the bearings configuration and as the result their performances. In present study the thermo-hydrodynamic performance of noncircular two, three and four lobed journal bearings for different values of tilt and mount angles, using generalized differential quadrature (GDQ) method, are investigated. The results show that the thermal effects on these bearings performance are considerable and that the thermal consideration makes the results closer to real performance situations. The results of bearings performances due to rise in temperature in rotor, lubricant fluid and bearing shell, when compared to their isothermal conditions, show that viscosity of lubricant as well as load carrying capacity of bearings are decreased, depending on tilt and mount angles especially in case of two lobed bearings. The results also show that the effects of tilt and mount angles on bearing performance are periodic and so it is possible to select these angles suitably for bearings to be optimum.

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Conical shells are widely used in various engineering applications such as mechanical, civil and aerospace engineering. In the present paper, based on the first order shear deformation theory (FSDT) of shells, the nonlinear vibration behavior of truncated conical shells with different boundary conditions is investigated using a numerical approach. To this end, the governing equations of motion and corresponding boundary conditions are derived by the use of Hamilton's principle. After catching the dimensionless form of equations, the generalized differential quadrature (GDQ) method is employed to obtain a discretized set of nonlinear governing equations. Thereafter, a Galerkin-based scheme is applied to achieve a time-varying set of ordinary differential equations and a method called periodic grid discretization is used to discretize the equations on the time domain. The pseudo arc-length continuation method is finally applied to obtain the frequency-amplitude response of conical shells. Selected numerical results are presented to examine the effects of different parameters such as thickness-to-radius ratio, small-to-large edge radius ratio, semi-vertex angle of cone, circumferential wave number and boundary conditions. It is concluded that the changes of the vibrational mode shapes and circumferential wave number have significant effects on the nonlinear vibration characteristics and hardening effects.

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In this paper, the elastoplastic buckling of rectangular plates over the Pasternak foundation has been analyzed with the fixed and simply supported boundary conditions. Associated with the uniform loading conditions on the plate by the in- plane compression and tension, the influence of the elastic foundation is investigated in terms of two stiffness parameters; including the Winkler spring and the Pasternak shear coefficients. In order to extract governing equations, two theories are used from the plasticity: deformation theory (DT) with the Hencky constitutive relations and the incremental theory (IT) based on the Prandtl-Reuss constitutive relations. By implementing the generalized differential quadrature method to discrete the differential equations, influences of loading ratio, length to width ratio, plate thickness, and the elastic foundation characters are studied. By comparing the obtained results with the data reported in references, the accuracy of the model is verified. Consideration of results shows that applying the elastic foundation causes to increase critical buckling load. In addition, enhancing the elastic foundation parameters leads to amplifying the difference between buckling loads obtained from two theories, especially in the larger thicknesses. Moreover, according to increasing the plate thickness in the tensile state of the loading, application of the elastic foundation causes to reach plate stress to a value more than the ultimate stress of the specimen.

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In this study, based on the third-order shear deformation theory the equations of motion are obtained to analyses the deformation of a long and slender composite beam. The beam has initial geometric imperfection and subjected to impact load. The impact procedures are applied by rigid body with a specific speed, off-center and at a certain distance from the beam's surface. Hamilton’s principle and the von-Karman nonlinear strain-displacement relationship are used to obtain the equations of motion that they are based on displacement and in a set of coupled nonlinear partial differential equations in dynamic mode. The generalized differential quadrature Method (GDQM) is used to discretize the obtained equations and convert them into a set of ordinary differential equations. Newton-Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Then, by solving the equations of the system, the effects of initial geometric imperfection on the beam’s deflection have been studied. Also the effects of mass and the initial velocity of the impactor on the beam’s deformation are investigated. The results of this research show that an increase in the amount of the initial velocity and mass of the impactor entail an increase in the beam deformation.

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In this paper, numerical and analytical solution of composite metal cylindrical vessel are investigated under dynamic load using first-order shear deformation theory and differential quadrature method. For this purpose, the shell equilibrium equations are derived based on the first order shear deformation theory. The load applied to the shell is achieved from the experimental test of a double-base propellant and then, is applied to the model in numerical and theoretical analysis. The aim of this paper is study and investigate the behavior of the composite metal cylindrical vessel under dynamic load with first-order shear deformation theory and comparing its results with the numerical solution. Therefore, after extracting the shell equilibrium equations are used from differential quadrature method for solve the equations. Then, the governing equations are extracted in a composite metal cylindrical vessel to form the matrix equations to solve with differential quadrature method. To apply boundary conditions from free and support clamping conditions are used and the results of these two modes are compared together. The MATLAB programming code is used to solve differential quadrature equations. To validate theoretical results, modeling and numerical analysis done by Abaqus finite element software and then, results are compared with the analytical solution using the differential quadrature method.

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In this paper, axial buckling of a composite cylindrical shell with and without a rectangular cutout is studied based on the first-order shear deformation theory. The equations are derived in a general form and can be converted to Donnell`s, Love`s, and Sanders` theories. To investigate the perforated shell, a physical domain is decomposed into several elements with uniform boundary and loading conditions in each element edges. In each element, the governing equations are discretized in both longitudinal and circumferential directions by the use of generalized differential quadrature method (GDQM). By assembling these discretized relations, a system of algebraic equations is generated. The boundary conditions at the shell and cutout edges, and the compatibility conditions at the interface boundaries of adjacent elements are also discretized by GDQM. Finally, the buckling load is calculated by an eigenvalue solution. To validate the presented method, the results of GDQM are compared with the available ones in the literature and also with ABAQUS finite element model. Then a parametric analysis is performed to investigate the effects of different parameters on the buckling behavior of the shells with and without cutouts. This study illustrates that the shell layup has a great effect on the buckling load of a shell. In addition, the influence of increasing the cutout size is not identical for different layups. However, the buckling behavior is independent of the shell material. Moreover, it was concluded that the shell with a square cutout has higher critical load than the one with a rectangular opening.

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In this paper, numerical analysis and experimental study on Friction Stir Welding (FSW) is considered. Generalized Differential Quadrature (GDQ) method was used to solve the equations of the material flow during the process. This method which is known as the highest-order finite difference scheme is one of the meshless method and has a very high convergence speed respect to ordinary finite difference and finite element methods. After validating the application of this procedure with the results of experiments on aluminium alloy, friction stir welding of mild steel considered and the results compared with the published results of other researchers. Numerical analyses show that at high rotational speed of the welding tool the analysis of the process should be done in 3-dimentional framework. The results of FSW on aluminum features along with the welding results on steel ones considered in order to better understanding of the process nature of dissimilar alloys. Results of this study show that the macroscopic behavior of both materials during friction stir welding is the same. Furthermore, viscosity spectrum shows high fluidity of steel in the range of solidity to melting temperatures, so the ratio of rotational to welding speeds (ω/v) in friction stir welding of steel work pieces could be higher which it should be mentioned whenever joining of aluminium to mild steel work pieces is planned.

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Biologic tissues modeling play an important role in understanding the tissue behavior and development of synthetic materials for medical applications. It is also a vital action to develop the predictive models for a wide range of uses including medical and tissue engineering. Various strain energy functions have been introduced to model arteries to date. The newest introduced strain energy function is the Nolan strain energy function. Two-layer arterial modeling using this strain energy function has not been performed so far. In this paper, modeling the arteries was carried out in the form of double layers including media and adventitia and hyperelastic material assumption. At first, governing equations were driven based on continuum mechanics. Boundary conditions including inner pressure of artery, axial load and torque as well as static equilibrium were applied. Moreover, Cauchy stress components were gotten by using the continuum mechanics relations. Then, the equilibrium equations in cylindrical coordinate were obtained by using the Cauchy stress. Stress distribution through the artery wall was specified by solving the resulting nonlinear partial differential equations based on generalized differential quadrature method. In the beginning, the artery modeling was conducted in the form of monolayer including the media layer and the results were compared with experimental ones, comparison between stresses in the artery wall and experimental data showed that the volcanic energy function of Nolan is suitable for modeling. After that, the stress distribution was obtained by artery modeling in the form of double layers including the media and adventitia layers.