Generalized Differential Quadrature Method Research Articles

A generalized differential quadrature approach to the modelling of heat transfer in non-similar flow with nonlinear convection

This study aims to investigate heat transfer and entropy production in the non-similar flow of an incompressible fluid, with nonlinear convection and viscous dissipation. To obtain precise solutions, the Sparrow-Quack-Boerner local non-similarity method is implemented. The non-similarity arises from the nonlinear convection term present in the momentum equation. The non-similarity arises from the nonlinear convection term present in the momentum equation. Consequently, the inclusion of non-similarity terms into the energy equation is achieved through the interconnection of momentum and energy equations. Entropy generation is explored by employing the second law of thermodynamics. Numerical results from several truncation levels are presented in tabular form, and equations for both first and second-level truncations are generated. The one- and two-equation models are solved by implementing the Generalized Differential Quadrature Method (GDQM). Comparison with a second level of truncation reveals significantly greater inaccuracies in numerical results obtained from the first level. This discrepancy arose due to the omission of non-similar terms in the primary governing equations at the first truncation stage. The validity and accuracy of the derived numerical solutions are further demonstrated by the application of the GDQM and the midpoint method with Richardson extrapolation. Additionally, a plot of the numerical data produced from the second level of truncations is presented together with a discussion of the various physical factors. Increasing the mixed convection parameter accelerates fluid velocity, also when the viscous dissipation parameter increases, temperature and the Bejan number increase. The Prandtl number and the thickness of the thermal boundary layer are observed to be inversely related. Moreover, the quantity of entropy generated at the stretching surface and inside the boundary layer rises in proportion to the increases in the Prandtl and Eckert numbers.

Frictional heating effects on entropy generation with nonlinear Rosseland radiation: Implementation of generalized differential quadrature method

This article examines heat transfer and entropy generation analysis for boundary layer flow, considering the impacts of frictional heating and nonlinear Rosseland thermal radiation. The equations governing momentum and energy in a two-dimensional boundary layer are transformed into self-similar equations via similarity operations. The generalized differential quadrature method (GDQM) is then used to numerically solve the no-dimensional governing equations. Entropy generation and Bejan numbers are computed using the acquired solutions and are then plotted against the emerging flow parameters. An in-depth analysis of temperature distribution, entropy generation, velocity profile, and Bejan number. A high level of agreement was seen when the results from the literature were replicated using GDQM to verify the accuracy of the numerical code. It is found that a self-similar solution exists in the presence of viscous dissipation. The primary contributor to entropy generation at the interface is fluid friction, in contrast to heat transfer. Further, the findings indicate that entropy generation positively correlates with the Prandtl number, heating parameter, radiation parameter, and Eckert number. Notably, the results show that a decrease in operational temperature causes a decrease in entropy production.

Symmetric/Asymmetric Vibrational Characteristics of Bi-Directional Functionally Graded Annular Microplates

In this study, a free vibration model of bi-directional functionally graded (BD-FG) annular microplates is established based on the modified couple stress theory (MCST) and the first-order shear deformation theory (FSDT), and the corresponding governing equations are derived from Hamilton’s principle. The annular microplates are assumed to be composed of metal and ceramic materials, and the equivalent material properties are a combination of power-law function and exponential ones. The generalized differential quadrature method (GDQM) is used to obtain the vibration frequencies and mode displacements for the C–C, S–C, C–S, and S–S (S: simply supported; C: clamped) annular microplates. The convergence and validity of the proposed model are verified through selective numerical examples. Further, the effects of gradient index, material length scale parameter, boundary conditions, and geometrical dimensions on the symmetric/asymmetric vibration frequencies of the annular microplate are explored. The modal assurance criterion is applied to estimate the influence of the radial gradient index and material length scale parameter on different-order mode displacements of the annular microplate. The results show that: (1) the vibration frequencies of the annular microplate are sensitive to changes in the geometry and boundary conditions; (2) compared with the axial gradient index, the radial gradient index not only affects the vibration frequencies of the annular microplate, but also impacts the distribution of the highest and lowest points of the vibration mode displacements as well as the vibration mode nodal lines; (3) as the material length scale parameter increases, the vibration frequencies are significantly increased, and the higher-order vibration modes of the annular microplate also change.

Free vibration and static analysis of sandwich composite plate with auxetic core and GPLRC facing sheets in hygrothermal environment

The main objective of this study is to investigate the vibration and static analysis of a sandwich composite plate in the hygrothermal environment. The plate consists of three layers such as an Aluminum auxetic core and Graphene platelet-reinforced composite (GPLRC) facing sheets. Based on the Halpin-Tsai theory and Gibson's model, the macro mechanical properties of facing sheets and auxetic core are derived. Via Reddy's third-order shear deformation theory (TSDT), the equations of motion of the sandwich plate are obtained and solved by both Navier and the generalized differential quadrature method (GDQM). The current formulation is validated by comparing the numerical results with those that are reported in the literature. Finally, the influence of different parameters such as the inclined angle of auxetic cells, thickness to the length, and core to total thickness on the natural frequencies, deflection and stresses of the sandwich plate, is examined. Even though some studies have been conducted to investigate the bending or vibration behavior of sandwich structures with negative Poisson's ratio (NPR) cores, the bending and vibration behavior of such structures in the hygrothermal environment is remained unexplored. Additionally, the analysis of stress component variations along the thickness of the plate was a novel feature that has not been addressed in prior studies on the similar structure.

Nonlinear Dynamic Analysis of a Curved Sandwich Beam with a Time-Dependent Viscoelastic Core Using the Generalized Differential Quadrature Method (GDQM)

This paper focuses on the geometrically nonlinear dynamic analyses of a three-layered curved sandwich beam with isotropic face layers and a time-dependent viscoelastic core. The boundary conditions and equations of motion governing the forced vibration are derived by using Hamilton’s principle. The first-order shear deformation theory is used to obtain kinematic relations. The spatial discretization of the equations is achieved using the generalized differential quadrature method (GDQM), and the Newmark-Beta algorithm is used to solve the time variation of the equations. The Newton–Raphson method is used to transform nonlinear equations into linear equations. The validation of the proposed model and the GDQM solution’s reliability are provided via comparison with the results that already exist in the literature and finite element method (FEM) analyses using ANSYS. Then, a series of parametric studies are carried out for a curved sandwich beam with aluminum face layers and a time-dependent viscoelastic core. The resonance and cancellation phenomena for the nonlinear moving-load problem of curved sandwich beams with a time-dependent viscoelastic core are performed using the GDQM for the first time, to the best of the authors’ knowledge.

Modified couple stress and artificial intelligence examination of nonlinear buckling in porous variable thickness cylinder micro sport structures

This investigation focuses on the nonlinear behavior of porosity-dependent functionally graded (FG) truncated conical small-scale structures. The modified coupled stress theory, as well as the energy method, are applied to generate the nonlinear partial differential equations (PDEs) related to buckling analysis of simply supported nonuniform micro-cylindrical structures. The material dispersion is gradually changed along the length of the structures between the Nickel and concrete, while the porosity voids are scattered in the radial direction, and the external radius of the structure decreases along the length direction via nonlinear mathematic equations applicable in sports structures. The PDEs are numerically solved via the generalized differential quadrature method (GDQM) coupled with the numerical iterative technique. In this particular context, the aim is to predict nonlinear results using a newly developed methodology that employs artificial neural networks (ANNs). The predictions generated by this approach will be compared against previously obtained data and validated to ensure their accuracy and reliability. The ANN methodology is expected to provide a more robust and comprehensive framework for predicting nonlinear results, which would be helpful in a variety of settings, from scientific research to engineering applications.

Vibration characteristics of FG saturated porous annular plates integrated by piezoelectric patches on visco‐Pasternak foundation

AbstractThe vibrational behavior of a three‐layered annular plate is considered in the present study. The plate is composed of a functionally graded (FG) porous core which is saturated by fluid and two piezoelectric patches that are bonded to the core and are subjected to the electric field. The whole of the structure is also rested on visco‐Pasternak elastic foundation. The material properties of the FG porous core vary through the thickness direction based on different patterns which are called porosity distributions. Love‐Kirchhoff's hypothesis and Hamilton's principle is employed to extract the governing motion equations and boundary conditions and following it, they are solved by generalized differential quadrature method (GDQM) for various boundary conditions. After ensuring the validity of the results by comparing them with known data in the literature, the effect of the most important parameters on the results is considered. It is seen that the effect of the porosity coefficient on the natural frequencies is completely dependent on the pores’ distribution patterns. Also, increasing in externally applied voltage to the piezoelectric facesheets, leads the results to enhance. The outcomes of this work can help to design and create smart structures and systems such as sensors and actuators.

Efficient analysis on buckling of FG-CNT reinforced composite joined conical–cylindrical laminated shells based on GDQ method under multiple loading conditions

This work presents an efficient analysis of buckling for functionally graded carbon nanotube reinforced composite (FG-CNTRC) joined conical–cylindrical laminated shells with lateral pressure and axial compressive load considering thermal effect. The governing equilibrium equation of the joined conical–cylindrical shell (CO-CYL) is determined using the first-order shear deformation theory (FSDT), Donnell kinematic assumptions, von-Kármán geometrical nonlinearity, and considering thermal effect. A semi-analytical solution, generalized differential quadrature (GDQ) method, and artificial springs are applied during the solution procedure. The comparison studies have demonstrated a remarkable 97% reduction in running time compared to the finite element method (FEM).

Artificial neural networks coupled with numerical approach for the stability prediction of non-uniform functionally graded microscale cylindrical structures

Functionally graded materials (FGMs) have garnered significant attention in research due to their potential for creating innovative materials with diverse applications. This study focuses on investigating the behavior of steel-concrete functionally graded structures under axial loading, specifically exploring nonlinear buckling. The research considers cylindrical structures with simply-supported boundary conditions and analyzes four different configurations with varying thickness. The governing equations are derived using the modified couple stress theory, incorporating assumptions of nonlinear von-Kármán theory coupled with classical beam theory. Numerical solutions are obtained using the general differential quadrature (GDQ) method. A notable aspect of this work is the use of Artificial Neural Networks (ANN) to predict nonlinear buckling forces, greatly enhancing the accuracy and efficiency of the predictions. This approach offers a novel method for analyzing concrete structures. The results are validated by comparing them with data from other published works, using dimensionless parameters for consistent comparisons. The findings provide a comprehensive analysis of various parameters, including wall thickness function and porosity, on the buckling stability of the cylindrical structure. Both numerical and ANN approaches are employed to report the effects. The study concludes that the uniform thickness cylinder exhibits the highest resistance to axial loading, resulting in buckling.

Dynamic stability and frequency responses of the tilted curved nanopipes in a supersonic airflow via 2D hybrid nonlocal strain gradient theory

In the presented work, vibrational behavior related to a curved nanopipe which is under supersonic airflow and conveying fluid flow, is examined. To model the size-dependent nanopipe, the Quasi-2D hybrid type of nonlocal strain gradient theory (QHNSGT) is presented. Formulations are obtained by means of Hamilton’s principle for bi-directional functionally graded (Bi-FG) nanopipe. Also, formulations are solved by means of the generalized differential quadrature method (GDQM). By taking into account that the fluid flow is infinite, incompressible, uniform flow, Newtonian, laminar, as well, as viscous, and with the help of the Navier-Stokes equation, the fluid-structure interaction is obtained. A quasi-2D hybrid type of higher-order shear deformation theory is employed to introduce the displacement fields. The verification section shows that the results of this paper are very near to the results of the published article in the literature. One of the important findings of the current research is critical values of the Mach number could have increased with the aid of increasing the rigidity of edges and decreasing fluid flow velocity. Another marvelous output is that opening angle and airflow stability have an indirect relation, and increasing the opening angle provides an expansion in the unstable area.

A variational quantum algorithm-based numerical method for solving potential and Stokes flows

This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems. In this method, the governing equations for potential and Stokes flows can be respectively written in the form of Laplace's equation and Stokes equations using velocity potential, stream function and vorticity formulations. Then the finite difference method and the generalized differential quadrature (GDQ) method are applied to discretize the governing equations. For the prescribed boundary conditions, the corresponding linear systems of equations can be obtained. These linear systems are solved by using the variational quantum linear solver (VQLS), which resolves the potential and Stokes flow problems equivalently. To the best of authors' knowledge, this is the first study that incorporates the GDQ method which is inherently a high-order discretization method with the VQLS algorithm. Since the GDQ method can utilize much fewer grid points than the finite difference method to approximate derivatives with a higher order of accuracy, the size of the input matrix for the VQLS algorithm can be smaller. In this way, the computational cost may be saved. The performance of the present method is comprehensively assessed by two representative examples, namely, the potential flow around a circular cylinder and Stokes flow in a lid-driven cavity. Numerical results validate the applicability and accuracy of the present VQLS-based method. Furthermore, its time complexity is evaluated by the heuristic scaling, which demonstrates that the present method scales efficiently in the number of qubits n and the precision ε. This work brings quantum computing to the field of computational fluid dynamics. By virtue of quantum advantage over classical methods, promising advances in solving large-scale fluid mechanics problems of engineering interest may be prompted.

On aeroelastic flutter assessment of thick nanoshell convening two-phase fluid flow using mathematical simulation and deep-neural networks

It is well known that flow-induced vibration occurs often in many different sectors. Dynamic fluid forces are produced by the flow, and these forces cause the system to become excited. One of the most prominent flows in nature and industrial domains, such as industrial facilities, is the multiphase flow which may contribute to creating noise and vibration. As one specific example of a multiphase flow, a two-phase flow relies heavily on flow arrangement for its characteristics. Furthermore, despite the most recent developments and advancements, flow-induced vibration is still considered an open topic. So, in this work, for the first time, frequency analysis of a thick nanoshell convening two-phase fluid flow subjected to aerodynamic pressure is presented. Via nonlocal strain gradient theory (NSGT), Hamilton's principle, trigonometric shear deformation theory, the reappraisal of the fluid-structure interaction relation, and non-conservative aerodynamic pressure formulation, the governing equations, and various boundary conditions of the system are presented. After that, using the generalized differential quadrature method (GDQM) the obtained governing equations are solved, numerically. For more verification of the results, the results are compared with the outcomes of the artificial neural network (ANN) model. For this purpose, the obtained results using mathematical modeling are presented to generate the training dataset and test dataset of the neural network in batches. Finally, some suggestions for improving the stability of the thick nanoshell convening two-phase fluid flow subjected to aerodynamic pressure are presented for future handbooks.

Higher order theories for the modal analysis of anisotropic doubly-curved shells with a three-dimensional variation of the material properties

The manuscript investigates the dynamic properties of doubly-curved shell structures laminated with innovative materials employing the Generalized Differential Quadrature (GDQ) method. A higher order generalized expression of the displacement field variable is assumed following the Equivalent Single Layer (ESL) approach. The geometrical description of the structures follows the differential geometry basics, whereas arbitrarily-shaped domains are distorted by means of generalized isogeometric blending functions. A three-dimensional generally anisotropic elastic constitutive behaviour is assumed within each layer of laminates, and a smooth variation of the material properties is implemented. In particular, a two-dimensional distribution alongside the physical domain is considered, whereas a through-the-thickness dispersion of the material properties is associated to each layer, taking into account both polynomial and non-polynomial analytical expressions. The fundamental equations are derived from the Hamiltonian principle, and they are solved directly in strong form via the GDQ method taking into account a non-uniform discrete computational grid. After some preliminary validating examples, some parametric investigations are performed systematically, showing the influence of the variation of the material properties on the modal response of the structures characterized by different lamination schemes, curvatures and boundary conditions. The proposed model allows for the dynamic analysis of the structures of different curvatures characterized by very complicated dispersions of the material properties following a simple and accurate methodology.

Dynamic stability examination of perovskite solar cells: Application of numerical analysis, GAN and African vulture optimization algorithms

The most plentiful energy source, solar energy, can be used even while it's cloudy outside. Around 10,000 times more solar energy is being absorbed by the Earth than is being used by people. For several functions, solar systems may provide fuel, power, heat, cooling, and natural lighting. Solar technology may use photovoltaic panels or solar radiation-concentrating mirrors to turn sunlight into electrical energy. Multiple contexts may benefit from the utilization of solar energy. Hence, enhancing the natural frequency of such a system and comprehending its dynamics are crucial in the engineering sector. The objective of the present study is to reinforce the metal layer of solar cells with multi-phase hybrid nanocomposite materials in order to improve its natural frequency. Carbon fiber (CF) and carbon nanotube (CNT), which are reinforcements with macro- and nanoscale dimensions, respectively, are used to strengthen an aluminum matrix that makes up the final layer of the structure. The introduction of infinite and finite displacement fields simulates the compression applied to a solar panel that is either fully clamped or merely supported. The Kronecker tensor product utilizes the generalized differential quadrature method (GDQM) to resolve the dynamic response of the system. As well as numerical analysis, we highlight advanced methods such as Generative Adversarial Networks (GAN) and African Vulture Optimization (AVO) algorithms, used to address dynamic stability challenges in perovskite solar cells. After that, by coupling the mentioned algorithms, an innovative algorithm for solving the engineering problems is presented. By generating synthetic data and optimizing system performance, scientists aim to analyze dynamic stability, promoting broader adoption of perovskite solar cells in the renewable energy industry. Finally, the dynamics of the perovskite solar cells are then suggested to be enhanced by a variety of physical and geometrical factors utilizing the numerical analysis and methods previously discussed.

Nonlinear two-dimensional transient thermoelastic analysis of functionally graded composite plates subjected to localised cooling loads

In this study, for the first time, localized thermal cooling is considered for the transient thermoelastic response of an axisymmetric plate, addressing the underexplored area of localized low-temperature thermoelasticity. Functionally graded composite plates are analyzed which consist of a mixture of stainless steel (SUS304) and low-carbon steel (AISI1020). A two-dimensional transient heat conduction equation is employed to capture localized cooling effects, featuring various adaptive time-dependent thermal boundary conditions which mirror both loading and unloading scenarios, effectively simulating real-world phenomena. Novel parameters are introduced to facilitate a comprehensive understanding of various aspects of thermal load handling and their impact on thermoelastic responses. The heat conduction equations are solved using the Generalized Differential Quadrature Method (GDQM) and the Crank-Nicolson scheme. The nonlinear governing equations, incorporating geometrical nonlinearity through the von Kármán assumption within the First Shear Deformation Theory (FSDT) framework, are solved using the GDQM and the Picard's technique. The model is validated using Finite Element Method (FEM) through Abaqus. A comprehensive analysis is provided that considers influence of the ratio of thermally affected and unaffected plate surface, thermal load magnitude, rapidity of thermal loading and unloading, duration of the cooling load, geometrical nonlinearity, and temperature dependence of material. The study shows that the maximum von Mises stress within the structure remains consistent regardless of the duration of the cooling load, as long as the rate at which the cooling load is applied stays the same. Additionally, the findings reveal that even minor localized thermal loads can produce substantial stress, especially at the intersection of thermally affected and non-affected zones on the exposed surface in some situations.

Nonlinear vibration of porous truncated conical shell under unified boundary condition and mechanical load

The aim of this article is to research nonlinear vibration responses of a metal porous truncated conical shell undergoing mechanical harmonic loads in thickness direction. The artificial spring technique is employed to unify boundary condition which can arrive at arbitrary classical boundary condition in real-word by changing the stiffness of springs. Utilizing Hamilton's principle, theoretical formulations of the conical shell are developed in view of FSDT and von-Kármán geometric relation. Afterwards, by employing generalized differential quadrature method (GDQM), the governing equation of nonlinear vibration in the form of partial differential equation is truncated and discretized into an ordinary differential equation which is a high-dimensional nonlinear dynamic system described in physical space rather than the mode space. On this basis, the analysis of effect of exciting frequency on the amplitude is carried out for three different distributing types of porous truncated conical shells by numerical integration. The frequency-amplitude responses of the system are depicted utilizing the sweeping frequency. The bifurcation diagrams, time series and phase diagrams are utilized to analyze the nonlinear dynamics of the porous truncated conical shell under unified boundary condition. It is found that the multiple solutions high-incidence area is near the 1st primary resonance, and vibration amplitude is dominated by the 1st order primary resonance and the 3rd one. The hardening-spring nature decreases for Type3, Type2 and Type1 successively.

A comprehensive study on the free vibration of adhesive lap joint laminated conical–conical shells adherends

In this research, the free vibration analysis of lap joint laminated conical shell connected to laminated conical shell with adhesive layer for various boundary conditions are investigated. Based on the first-order shear deformation theory (FSDT), the main field equations of the analytical model are derived and the governing differential equations of laminated conical shell to laminated conical shell adherends lap joint are obtained for homogeneous and elasticity adhesive layers by using Hamilton principle method. Then, based on the generalized differential quadrature method (GDQM), the equilibrium equations of the structure adherends are investigated. The influence of the boundary conditions, the type of the composite materials and adhesive layer, the circumferential wave number, the length to radius ratio of the shell, and the length of overlap to length of shell ratio are studied. Furthermore, the thickness to radius ratio and the thickness of adhesive to thickness of shell ratio, as well as the various cone angles of conical shell on the natural frequency of laminated conical adherends shells connected to the conical shells with adhesive layer are investigated. For validation of the numerical results, the results are compared with the previous research and the comparisons show very good agreement. The numerical results show that with the increase of the internal shell and external shell of semi-angle, the non-dimensional frequency of lap joint connection of two laminated conical shells with adhesive layer decreased and increased, respectively. Furthermore, with the increase of the adhesive thickness to conical shell thickness ratio and lap joint length to length of conical shell ratio, the dimensionless natural frequency of the structure increased.

Free oscillations of porous multi-graded graphene oxide nanocomposites coupled hemispherical-cylindrical shell-configuration structures under various edge constraints

Quantitative investigations of the vibrational characteristics coupled hemispherical-cylindrical shells (CHSCSs) formed of three types, including of porous, nanocomposite, and porous nanocomposite, under various edge conditions are carried out here: (1) a matrix strengthened by graded graphene oxide powder (GOP), (2) a matrix with the effects of the porosity, and (3) a matrix strengthened by multi-graded GOP with the porosity characteristic. To modify the vibrational characteristics of the nanocomposite, nine different functionally graded models are evaluated for GOP distribution within the matrix and CHSCS thickness. The Halpin-Tsai homogenization formula is used to compute the GOP nanocomposite (GOPN) material's mechanical characteristics. Addedly, a functionally graded type of the effect of the porosity is used. The CHSCS equations may be derived from Donnell's theory as well as from the first shear deformation theory (FSDT). The governing equations of the CHSCSs are determined using variation calculation, named Hamilton's principle. After that, this equation system is built using the generalized differential quadrature method (GDQM). The frequencies used as benchmarks are compared to those produced here to test the system. Remarkably, the lowest frequencies of the CHSCS made of pure matrix and multi functionally graded GOPN were decreased for all BCs by extending the value of the porosity factor. In addition, by increasing the value of the porosity factor for all BCs, the porous FGÓ-GOPN CHSCS had the greatest variation related to decreasing the minimum frequencies.

Buckling analysis of sandwich plates reinforced with various loading of multi-walled carbon nanotubes

The current investigation aims to examine the buckling behavior of sandwich structures reinforced with faces containing varying weight percentages (wt.%) of multi-walled carbon nanotubes (MWCNT), through the combined employment of experimental and numerical analysis techniques. In particular, the numerical analysis is executed through the implementation of the generalized differential quadrature method (GDQM). The study first determines the mechanical properties of the pure and reinforced samples via experimental tests. The buckling equations of the sandwich plate are then derived using the Frostig approach and a high-order shear deformation theory (SDT) for the flexible core. The continuity conditions between the faces and the core are considered to obtain the structural equations of the system, and Lagrange coefficients are introduced to enhance the accuracy of the governing equations. Subsequently, the study investigates the impact of various parameters, such as core thickness, aspect ratio, different boundary conditions (BC), and the addition of MWCNT nanoparticles on the critical buckling load of the system, utilizing the mechanical properties obtained for the composite faces. The present investigation reveals that the incorporation of MWCNT into the composite matrix brings about notable enhancements in the mechanical characteristics of sandwich structures. Specifically, the elastic modulus and ultimate strength of the faces are found to increase following the addition of MWCNT. Additionally, the critical buckling load of the sandwich plates is observed to rise with an increase in aspect ratio and length-to-thickness ratio, while a decrease in axial load yields similar results. The clamped boundary condition has higher critical buckling loads than the simply supported boundary condition. The addition of MWCNT nanoparticles decreases the dimensionless critical buckling force as the b/a ratio increases. Finally, the reported results of the current investigation present noteworthy findings about the advancement and improvement of sandwich composite structures to meet various application requirements.

Bending Vibration and Buckling Analysis Of Nonhomogeneous Nanotubes Using Nonlocal Elasticity Theory and Gdq Method

In this paper structural analysis of nonhomogeneous nanotubes has been carried out using nonlocal elasticity theory. Governing differential equations of nonhomogeneous nanotubes are derived. Nonlocal theory of elasticity has been employed to include the scale effect of the nanotubes. Nonlocal parameter, elastic modulus, density and diameter of the cross sections are assumed to be functions of spatial coordinates. General Differential Quadrature (GDQ) method has been employed to solve the governing differential equations of the nanotubes. Various boundary conditions have been applied to the nanotubes. Present results considering nonlocal theory are in good agreement with the results available in the literature. Effect of various geometrical and material parameters on the structural response of the nonhomogeneous nanotubes has been investigated. Present results of the nonhomogeneous nanotubes are useful in the design of the nanotubes.