scholarly journals Differential Quadrature and Differential Transformation Methods in Buckling Analysis of Nanobeams

2019 ◽  
Vol 6 (1) ◽  
pp. 68-76 ◽  
Author(s):  
Subrat Kumar Jena ◽  
S. Chakraverty
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Transformation Method ◽  
Nonlocal Theory ◽  
Buckling Analysis ◽  
Differential Quadrature ◽  
Load Parameter ◽  
Computationally Efficient ◽  
Differential Transformation

AbstractIn this paper, two computationally efficient techniques viz. Differential Quadrature Method (DQM) and Differential Transformation Method (DTM) have been used for buckling analysis of Euler-Bernoulli nanobeam incorporation with the nonlocal theory of Eringen. Complete procedures of both the methods along with their mathematical formulations are discussed, and MATLAB codes have been developed for both the methods to handle the boundary conditions. Various classical boundary conditions such as SS, CS, and CC have been considered for investigation. A comparative study for the convergence of DQM and DTM approaches are carried out, and the obtained results are also illustrated to demonstrate the effects of the nonlocal parameter, aspect ratio (L/h) and the boundary condition on the critical buckling load parameter.

Structural Analysis by the Differential Quadrature Method Using Modified Weighting Matrices

1998 ◽  
Author(s):  
Siu-Tong Choi ◽  
Yu-Tuan Chou
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Sampling Point ◽  
Order Partial Differential Equation ◽  
Quadrature Method ◽  
The Finite Element Method ◽  
Computationally Efficient ◽  
Base Points ◽  
Sampling Points

Abstract The differential quadrature method has lately been more and more often used for analysis of engineering problems as an alternative for the finite element method or finite difference method. In this paper, static, dynamic and buckling analyses of structural components are performed by the differential quadrature method. To improve the accuracy of this method, an approach is proposed for selecting the sampling points which include base points and conditional points. The base points are taken as the roots of the Legendre polynomials. Accuracy of the problems analyzed will be increased by using the base points. The conditional points are determined according to boundary conditions and specified conditions of external load. A modified algorithm is proposed for applying two or more boundary conditions in a sampling point at boundary of domain, such that the higher-order partial differential equation can be solved without adding new sampling points. By applying this approach to variety problems, such as deflections of beam under nonuniformly distributed loading, vibration and buckling analyses of beam and plate, it is found that numerical results of the present approach are more accurate than those obtained by the equally-spaced differential quadrature method and is computationally efficient.

scholarly journals Vibration Analysis of Magneto-Electro-Thermo NanoBeam Resting on Nonlinear Elastic Foundation Using Sinc and Discrete Singular Convolution Differential Quadrature Method

2019 ◽  
Vol 13 (7) ◽  
pp. 49 ◽  
Author(s):  
Ola Ragb ◽  
Mokhtar Mohamed ◽  
M.S. Matbuly
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Differential Quadrature ◽  
Mode Shapes ◽  
Electric Voltage ◽  
Nonlinear Elastic Foundation ◽  
Axial Forces ◽  
Nonlinear Elastic

Magneto-Electro-Thermo nanobeam resting on a nonlinear elastic foundation is presented. This beam is subjected to the external electric voltage and magnetic potential, mechanical potential and temperature change. Also, we added the new material PTZ-5H-COFe2O4. The governing equations and boundary conditions are derived using Hamilton principle. These equations are discretized by using three differential quadrature methods and iterative quadrature technique to determine the natural frequencies and mode shapes. Numerical analysis is introduced to explain the influence of computational characteristics of the proposed schemes on convergence, accuracy and efficiency of the obtained results. The obtained results agreed with the previous analytical and numerical ones. A detailed parametric study is conducted to investigate the influences of different boundary conditions, various composite materials, nonlinear elastic foundation, nonlocal parameter, the length-to-thickness ratio, external electric and magnetic potentials, axial forces, temperature and their effects on the vibration characteristics of Magneto-Electro-Thermo-Elastic nanobeam.

scholarly journals A Mixed Layerwise-Differential Quadrature Method for 3-D Vibration and Buckling Analyses of Orthogonally Stiffened Composite Conical Shell

2019 ◽  
Vol 24 (2) ◽  
pp. 217-227
Author(s):  
Mostafa Talebitooti
Keyword(s):  
Boundary Conditions ◽  
Conical Shell ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Algebraic Equations ◽  
Discrete Elements ◽  
Arbitrary Boundary ◽  
Stiffened Shells

A layerwise-differential quadrature method (LW-DQM) is developed for the vibration analysis of a stiffened laminated conical shell. The circumferential stiffeners (rings) and meridional stiffeners (stringers) are treated as discrete elements. The motion equations are derived by applying the Hamilton’s principle. In order to accurately account for the thickness effects and the displacement field of stiffeners, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations applying the DQM in the meridional direction. The advantage of the proposed model is its applicability to thin and thick unstiffened and stiffened shells with arbitrary boundary conditions. In addition, the axial load and external pressure is applied to the shell as a ratio of the global buckling load and pressure. This study demonstrates the accuracy, stability, and the fast rate of convergence of the present method, for the buckling and vibration analyses of stiffened conical shells. The presented results are compared with those of other shell theories and a special case where the angle of conical shell approaches zero, i.e. a cylindrical shell, and excellent agreements are achieved.

scholarly journals Analysis of beam-column elements on non-homogeneous soil using the differential transformation method

2021 ◽  
Author(s):  
Juan Sebastián Carvajal-Muñoz ◽  
Carlos Alberto Vega-Posada ◽  
Julio César Saldarriaga-Molina
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Mathematical Formulation ◽  
Transformation Method ◽  
Transverse Load ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Pasternak Elastic Foundation ◽  
Wide Range ◽  
Homogeneous Soil

This paper describes an analytical approach to conduct an analysis of beam-column elements with generalized end-boundary conditions on a homogeneous or non-homogeneous Pasternak elastic foundation. The mathematical formulation utilized herein is that presented by the senior author in a recent work. The differential equation (DE) governing the behavior of the beam-column element is solved using the differential transformation method (DTM). The DTM offers practical advantages over other conventional approaches when solving the proposed structural model. The proposed formulation provides the flexibility to account for i) combined lateral and axial load at the ends of the element, ii) homogeneous or non-homogeneous soil, iii) Pasternak elastic foundation, and iv) an external arbitrary transverse load acting on the element. The effects of various slenderness ratios, pile-soil stiffness ratios, and classical and semirigid boundary conditions can be easily studied with the proposed formulation. Examples are presented to validate the accuracy of the model and its applicability over a wide range of analyses.

Analysis of free vibration of an isotropic plate with surface or internal long crack using generalized differential quadrature method

2019 ◽  
Vol 55 (1-2) ◽  
pp. 42-52
Author(s):  
Milad Ranjbaran ◽  
Rahman Seifi
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential

This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

Bi-Directional Functionally Graded Nanotubes: Fluid Conveying Dynamics

2018 ◽  
Vol 10 (04) ◽  
pp. 1850041 ◽  
Author(s):  
Ye Tang ◽  
Tianzhi Yang
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Graded Materials ◽  
Critical Flow Velocity ◽  
Dynamic Behaviors ◽  
Fundamental Frequencies ◽  
Material Gradation

In the paper, a novel model of fluid-conveying nanotubes made of bi-directional functionally graded materials is presented for investigating the dynamic behaviors and stability. For the first time, the material properties of the nanotubes along both radical and axial directions are under consideration. Based on Euler–Bernoulli beam and Eringen’s nonlocal elasticity theories, the governing equation of the nanotubes and associated boundary conditions are developed using Hamilton’s principle. Differential quadrature method (DQM) is applied for discretizing the equation to determine the numerical solutions of the nanotubes with different boundary conditions. Numerical examples are presented to examine the effects of the material gradation, nonlocal parameter, and mode order on the dynamics and stability. It is shown that the two-directional materials distribution can significantly change the critical flow velocity, fundamental frequencies and stability. Comparing with traditional one-directional distribution, such 2D is more flexible to tune overall dynamic behaviors, this may provide new avenues for smart pipes.

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