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

Karami, G.; Malekzadeh, P.

doi:10.1016/s0045-7825(02)00289-x

Cited by 124 publications

(59 citation statements)

References 26 publications

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“…The governing differential equations of motion can be derived using Hamilton's principle: (26) where is variation of the energy functional and t denotes the time. The energy functional of a stiffened rotating conical shell can thus be written as: (27) Substituting Eqs.…”

Section: Governing Equations Of Motionmentioning

confidence: 99%

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Dynamic Analysis and critical speed of rotating laminated conical shells with orthogonal stiffeners using generalized differential quadrature method

Daneshjou

Talebitooti

et al. 2013

Lat. Am. j. solids struct.

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This paper presents effects of boundary conditions and axial loading on frequency characteristics of rotating laminated conical shells with meridional and circumferential stiffeners, i.e., stringers and rings, using Generalized Differential Quadrature Method (GDQM). Hamilton's principle is applied when the stiffeners are treated as discrete elements. The conical shells are stiffened at uniform intervals and it is assumed that the stiffeners have similar material and geometric properties. Equations of motion as well as equations of the boundary condition are transformed into a set of algebraic equations by applying the GDQM. Obtained results discuss the effects of parameters such as rotating velocities, depth to width ratios of the stiffeners, number of stiffeners, cone angles, and boundary conditions on natural frequency of the shell. The results will then be compared with those of other published works particularly with a non-stiffened conical shell and a special case where angle of the stiffened conical shell approaches zero, i.e. a stiffened cylindrical shell. In addition, another comparison is made with present FE method for a non-rotating stiffened conical shell. These comparisons confirm reliability of the present work as a measure to approximate solutions to the problem of rotating stiffened conical shells.

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Section: Governing Equations Of Motionmentioning

confidence: 99%

“…The mathematical fundamentals and recent developments of the GDQ method as well as its major applications in engineering are discussed in detail by Shu [23]. It is worthwhile to note that the increasing interest of researches [24][25][26][27][28][29][30][31] in this procedure is mainly due to its great simplicity and versatility.…”

mentioning

confidence: 99%

Dynamic Analysis and critical speed of rotating laminated conical shells with orthogonal stiffeners using generalized differential quadrature method

Daneshjou

Talebitooti

et al. 2013

Lat. Am. j. solids struct.

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“…Eqs. (18) may be rearranged and partitioned in order to separate the boundary, domain, and adjacent displacement and rotation components as (Karami & Malekzadeh, 2002) (20)…”

Section: Governing Equationsmentioning

confidence: 99%

Free vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses using DQEM

Torabi¹,

Afshari²,

Heidari-Rarani³

2013

10.5267/j.esm

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In this paper, a differential quadrature element method (DQEM) is developed for free transverse vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses. Governing equations, compatibility and boundary conditions are formulated according to the differential quadrature rules. The compatibility conditions at the position of each concentrated mass are assumed as the continuity in the vertical displacement, rotation and bending moment and discontinuity in the transverse force due to acceleration of the concentrated mass. The effects of number, magnitude and position of the masses on the value of the natural frequencies are investigated. The accuracy, convergence and efficiency of the proposed method are confirmed by comparing the obtained numerical results with the analytical solutions of other researchers. The two main advantages of the proposed method in comparison with the exact solutions available in the literature are: 1) it is less time-consuming and subsequently moreefficient; 2) it is able to analyze the free vibration of the beams whose section varies as an arbitrary function which is difficult or sometimes impossible to solve with analytical methods. }}

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“…In the past decade, the differential quadrature element method (DQEM) has been used to study the behavior of different structural elements [Wang et al 1996;Wang and Gu 1997;Karami and Malekzadeh 2002;Chen 2003]. However, apart from applications of the differential quadrature (DQ) method in fluid mechanics, all the studies in the literature have been restricted to various 2D elasticity problems of isotropic and laminated plates and shells; the method has not been used in micromechanics of heterogeneous materials.…”

Section: Introductionmentioning

confidence: 99%

“…Wu and Liu [2000] proposed a generalized differential quadrature rule, introducing multiple degrees of freedom at boundary points. Recently, Karami and Malekzadeh [2002] proposed a method of applying the multiboundary conditions. In formulations of the weighting coefficients of third and fourth-order derivatives, the second derivatives at the boundary points are viewed as additional independent variables.…”

Section: Introductionmentioning

confidence: 99%

Micromechanical analysis of unidirectional composites using a least-squares-based differential quadrature element method

Bayat

Aghdam

2012

J. Mech. Mater. Struct.

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A generalized plane strain micromechanical model is developed to predict the stress and strain fields and overall elastic properties of a unidirectional fiber-reinforced composite subjected to various axial and transverse normal loading conditions using a least-squares-based differential quadrature element method (DQEM). The representative volume element (RVE) of the composite consists of a quarter of the fiber surrounded by matrix to represent the real composite with a repeating square array of fibers. The cubic serendipity shape functions are used to convert the solution domain to a proper rectangular domain and the new versions of the governing equations and boundary conditions are also derived. The fully bonded fiber-matrix interface condition is considered and the displacement continuity and traction reciprocity are imposed on the fiber-matrix interface. Application of DQEM to the problem leads to an overdetermined system of linear equations mainly due to the particular periodic boundary conditions of the RVE. A least-squares differential quadrature element method is used to obtain solutions for the governing partial differential equations of the problem. The numerical results are in excellent agreement with the available analytical and finite element studies. Moreover, the results of this study reveal that the presented model can provide highly accurate results with a very small number of elements and grid points within each element. In addition, the model shows advantages over conventional analytical models for fewer simplifying assumptions related to the geometry of the RVE.

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

Cited by 124 publications

References 26 publications

Dynamic Analysis and critical speed of rotating laminated conical shells with orthogonal stiffeners using generalized differential quadrature method

Dynamic Analysis and critical speed of rotating laminated conical shells with orthogonal stiffeners using generalized differential quadrature method

Free vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses using DQEM

Micromechanical analysis of unidirectional composites using a least-squares-based differential quadrature element method

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