Differential Quadrature Element Method Research Articles

DQEM analysis of in-plane vibration of curved beam structures

The development of differential quadrature element method (DQEM) in-plane vibration analysis model of arbitrarily curved beam structures was carried out. The DQEM uses the extended differential quadrature to discretize the differential eigenvalue equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results obtained by the developed numerical algorithm show that the convergence of the developed DQEM analysis model is efficient.

Differential Quadrature Element Method Analysis of In‐Plane Deflection of Vertically Curved Beam Structures

Abstract: The development of the differential quadrature element method (DQEM) in‐plane deflection analysis model of arbitrarily curved beam structures was carried out. The DQEM uses the extended differential quadrature to discretize the differential equation defined on each element, the transition conditions defined on the interelement boundary of two adjacent elements, and the boundary conditions of the beam. Numerical results obtained from solving two structural models using a numerical algorithm are presented. The convergence of the developed DQEM analysis model is efficient.

Elasto-Plastic Analysis of Frames by Differential Quadrature Element Method

Elasto-Plastic Analysis of Frames by Differential Quadrature Element Method

A semi-analytical DQEM for free vibration analysis of thick plates with two opposite edges simply supported

This paper is introducing a semi-analytical differential quadrature element method (DQEM) based on first-order shear deformation theory for free vibration analysis of thick plates. By decomposing the plate into a series of sub-domains or elements, any discontinuity in geometry, material properties, and elastic foundations in one direction can be handled conveniently. Classical, as well as non-classical boundary conditions can be imposed. One benefits of this semi-analytical DQEM is reducing the storage requirements and the computational efforts needed to assemble the global equations. The method would also increase the computational efficiency in comparison with the finite element method (FEM) or differential quadrature method (DQM) analysis of such problems. The accuracy, versatility, stability, and convergence of the algorithm are certified.

A modified quadrature element method to perform static analysis of structures

In this paper, an improved version of the quadrature element method (QEM) is proposed. By means of a careful distribution of sampling points, the proposed method overcomes the drawback of the distance δ of the QEM without introducing further degrees of freedom at the ends of the element as in the differential quadrature element method (DQEM). The aim of the paper is to show the potentialities of this modified quadrature element method (MQEM) for several structural examples.

Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method

AbstractA new version of the differential quadrature method is presented in this paper to overcome the difficulty existing in the ordinary differential quadrature method for applying multi‐boundary conditions in two‐dimensional problems. Since the weighting coefficients of the first derivative are the same as for the ordinary differential quadrature method even with the introduction of multi‐degree‐of‐freedom at the boundary points, the method is easier to extend to two‐ or three‐dimensional problems. A new version of the differential quadrature plate element has been established for demonstration. The essential difference from the existing old version of the differential quadrature plate element is the way the weighting coefficients are determined. The methodology is worked out in detail and some numerical examples are given to show the efficiency of the present method. Copyright © 2004 John Wiley & Sons, Ltd.

DQEM and DQFDM irregular elements for analyses of 2-D heat conduction in orthotropic media

The differential quadrature element method and differential quadrature finite difference method have been proposed by the author. The DQ, GDQ and mapping technique have been used to develop DQEM and DQFDM solution algorithms for solving scientific and engineering problems having irregular domain configurations. The two novel numerical methods are use to solve the problems of 2-D heat conduction in orthotropic media having irregular domain configurations. Numerical results are presented. They demonstrate the developed algorithms.

Differential cubature method for buckling analysis of arbitrary quadrilateral thick plates

In this paper, a novel numerical solution technique, the differential cubature method is employed to study the buckling problems of thick plates with arbitrary quadrilateral planforms and non-uniform boundary constraints based on the first order shear deformation theory. By using this method, the governing differential equations at each discrete point are transformed into sets of linear homogeneous algebraic equations. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Detailed formulation and implementation of this method are presented. The buckling parameters are calculated through solving a standard eigenvalue problem by subspace iterative method. Convergence and comparison studies are carried out to verify the reliability and accuracy of the numerical solutions. The applicability, efficiency, and simplicity of the present method are demonstrated through solving several sample plate buckling problems with various mixed boundary constraints. It is shown that the differential cubature method yields comparable numerical solutions with 2.77-times less degrees of freedom than the differential quadrature element method and 2-times less degrees of freedom than the energy method. Due to the lack of published solutions for buckling of thick rectangular plates with mixed edge conditions, the present solutions may serve as benchmark values for further studies in the future.

Application of a new differential quadrature element method to free vibrational analysis of beams and frame structures

Application of a new differential quadrature element method to free vibrational analysis of beams and frame structures

DQEM for free vibration analysis of Timoshenko beams on elastic foundations

A differential quadrature element method (DQEM) based on first order shear deformation theory is developed for free vibration analysis of non-uniform beams on elastic foundations. By decomposing the system into a series of sub-domains or elements, any discontinuity in loading, geometry, material properties, and even elastic foundations can be considered conveniently. Using this method, the vibration analysis of general beam-like structures is to be studied. The governing equations of each element, natural compatibility conditions at the interface of two adjacent elements and the external boundary conditions are developed in a systematic manner, using Hamilton's principle. The present DQEM is to be implemented to Timoshenko beams resting on partially supported elastic foundations with various types of boundary conditions under the action of axial loading. The general versality, accuracy, and efficiency of the presented DQEM are demonstrated having solved different examples and compared to the exact or other numerical procedure solutions.

DQEM for Buckling Analysis of Nonprismatic Columns with and Without Elastic Foundation

The differential quadrature element method is used to solve the buckling problems of nonprismatic column structures with and without elastic foundation. The extended differential quadrature is used to discretize the governing differential eigenvalue equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions. Numerical results obtained by DQEM are presented. They demonstrate the developed numerical solution procedure.

A DQEM for vibration of shear deformable nonuniform beams with general boundary conditions

A differential quadrature element method (DQEM) for free vibration analysis of arbitrary non-uniform Timoshenko beams with attachments, i.e. concentrated mass and rotary inertia and resting on elastic supports is proposed. Using the Hamilton’s principle the governing equation and the natural compatibility conditions at the interface of adjacent elements are devised in a systematic manner. The differential quadrature (DQ) analogs to the governing equation, the compatibility conditions and the external boundary conditions implementations are explicitly formulated. The versatility, accuracy and efficiency of the presented DQEM for free vibration analysis of Timoshenko beams are tested against other solution procedures. The examples include; stepped non-uniform Timoshenko beams, non-uniform beams with attached heavy masses and supported elastically in transverse directions, and also non-uniform beams on elastic foundations. Accurate solutions are archived via few grid points for all the cases considered.

Out-of-plane deflection of nonprismatic curved beam structures solved by DQEM

The differential quadrature element method (DQEM) is used to solve the out-of-plane deflections of nonprismatic curved beam structures. The extended differential quadrature is used to discretize the governing differential equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions. Numerical results obtained by DQEM are presented. They demonstrate the developed numerical solution procedure.

Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method

A new version differential quadrature method (DQM) has been proposed to obtain buckling loads of thin anisotropic rectangular and isotropic skew plates. The essential difference from the old version DQM is the introduction of two degrees of freedom for boundary points and from the existing differential quadrature element method (DQEM) is the determination of the weighting coefficients. The methodology is worked out in detail and a variety of buckling problems shown slow convergence earlier by Rayleigh-Ritz method with beam functions, including isotropic skew plates with various skew angles and anisotropic rectangular plates with simply supported or clamped boundary conditions, are solved by the proposed DQM. Numerical results indicate that fast convergence is achieved and excellent results are obtained by the proposed DQM.

Variational derivation of equilibrium equations of arbitrarily loaded pre‐stressed shear deformable non‐prismatic composite beams and solution by the DQEM buckling analysis

AbstractThe generic equilibrium equations of arbitrarily loaded pre‐stressed non‐prismatic shear deformable composite beams can be derived. The buckling equilibrium equations can be obtained by simplifying the generic equilibrium equations. These equations can be used to develop the differential quadrature element method (DQEM) analysis model of the buckling of shear deformable beams. The DQEM uses the DQ to discretize the buckling equilibrium equation defined on each element, the transition conditions defined on the inter‐element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results of the buckling of shear deformable bar structures solved by the developed numerical algorithm are presented. They prove that the DQEM buckling analysis model is efficient in comparison of convergence rate and numerical stability with the existing numerical techniques such as traditional FDM and some FEM models. Copyright © 2003 John Wiley & Sons, Ltd.

Buckling equilibrium equations of arbitrarily loaded nonprismatic composite beams and the DQEM buckling analysis using EDQ

The generic buckling equilibrium equations of arbitrarily loaded nonprismatic composite beams are derived. These equations can be used to develop the differential quadrature element method (DQEM) analysis model of buckling of generic nonprismatic beam structures. In developing the DQEM analysis model, the extended differential quadrature can be used to discretize the buckling equilibrium equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results of the buckling of sample bar structures solved by the DQEM are presented. They prove that the DQEM buckling analysis model is efficient.

DQEM and DQFDM for the analysis of composite two-dimensional elasticity problems

The differential quadrature element method (DQEM) and differential quadrature finite difference method (DQFDM) have been proposed by the author. The differential quadrature, generic differential quadrature and mapping technique are used in developing DQEM and DQFDM. The two novel numerical methods are effective for solving generic continuum mechanics problems. In this paper, these two numerical methods are used to solve the composite two-dimensional elasticity problems. Numerical procedures are summarized and presented. Numerical results are also presented. They demonstrate the DQEM and DQFDM.

A derivation and solution of dynamic equilibrium equations of shear undeformable composite anisotropic beams using the DQEM

In this paper, Hamilton's principle is used to derive the dynamic equilibrium equations of composite nonprismatic beams made of anisotropic materials. The effects of transverse shear deformations are neglected. The displacements are defined on an arbitrarily selected coordinate system. For Hamilton's principle, the dynamic behavior of nonprismatic beams is characterized by two energy functions: a kinetic energy and a potential energy. The formulation uses the procedure of variational operations. The obtained dynamic equilibrium equations and natural boundary conditions are strongly coupled. They can be solved by the differential quadrature element method (DQEM).

A new differential quadrature methodology for beam analysis and the associated differential quadrature element method

This paper presents a computationally efficient and an accurate new methodology in differential quadrature analysis of beam elements. The methodology would overcome the difficulties in boundary conditions implementations of fourth-order differential equations encountered in such problems. The methodology benefited from defining the second-order derivatives along the boundaries as independent degrees of freedom would enable the differential quadrature method to exactly satisfy some types of boundary conditions; where by most other conventional algorithms have to be satisfied approximately. The weighting coefficients employed are not exclusive, and any accurate and efficient method such as the generalized differential quadrature method may be used to produce the methods weighting coefficients. By solving some typical stability, deflection and frequency analysis beam problems and by comparing the results with those of exact solutions and/or those of other methodologies, accuracy, convergency and efficiency of the methodology is asserted. In order to generalize its application to large-scale beam structures and to the cases with the discontinuity in loading conditions and geometry, a new one-dimensional differential quadrature element method formulation is presented and implemented.

DQEM VIBRATION ANALYSES OF NONPRISMATIC BEAMS RESTING ON ELASTIC FOUNDATIONS

The development of differential quadrature element method (DQEM) free vibration analysis model of nonprismatic Bernoulli–Euler beams resting on Winkler elastic foundations was carried out. The DQEM uses the extended differential quadrature (EDQ) to discretize the differential eigenvalue equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithm are presented. They prove that the DQEM is efficient. The developed numerical algorithm can be used to analyze the related pressure vessel and piping structures.