Analysis of Cracked Plates Using Localized Multi¬Domain Differential Quadrature Method

Osman, Tharwat; Matbuly, M. S.; Mohamed, S.A.; Nassar, May Fouad

doi:10.11648/j.acm.20130204.12

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…To ensure the validity of proposed scheme, the obtained results are compared with previous ones for cracked and un-cracked Euler-Bernoulli and Timoshenko problems. A quadrature numerical scheme is designed to solve cracked Euler-Bernoulli beam problems, equations (22)(23)(24)(25)(26)(27)(28)(29). For each sub-beam, N is to be varied from 5-50 to determine N leading to accurate convergent results.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Elkhalek¹,

Osman²,

Matbuly³

2019

ENGMATH

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This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler-Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multisegments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Recently, differential quadrature method is introduced as promising numerical technique. This method leads to very accurate results by using small number of nodal points [23][24][25][26]. Rotational spring model is employed to simulate the crack existence.…”

Section: Introductionmentioning

confidence: 99%

Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Elkhalek¹,

Osman²,

Matbuly³

2019

ENGMATH

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“…Lagrange interpolation polynomials, the cardinal sine function, the delta Lagrange kernel (DLK), and the regularized Shannon kernel (RSK) are some examples of such functions which have led to the development of the polynomial-based differential quadrature method (PDQM) [21], the sinc differential quadrature method (SDQM) [22], and the discrete singular convolution differential quadrature method (DSCDQM), respectively [23]. Thus, the DQ method has emerged as a powerful numerical discretization tool for solving a variety of problems in the engineering and physical sciences [24][25][26][27][28][29][30]. Bellman et al [19] suggested that the nth-order derivative of the function with respect to a grid point can be approximated as a linear summation of the values of the function for all of the sample points in the domain.…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Analysis of Drug Diffusion through a Thin Membrane Using the Differential Quadrature Method

2023

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The primary goal of this work is to solve the problem of drug diffusion through a thin membrane using a differential quadrature approach with drastically different shape functions, such as Lagrange interpolation and discrete singular convolution (the delta Lagrange kernel and the regularized Shannon kernel). A nonlinear partial differential equation with two time- and space-dependent variables governs the system. To reduce the two independent variables by one, the partial differential equation is transformed into an ordinary differential equation using a one-parameter group transformation. With the aid of the iterative technique, the differential quadrature methods change this equation into an algebraic equation. Then, using a MATLAB program, a code is created that solves this equation for each shape function. To ensure the validity, efficiency, and accuracy of the developed techniques, the computed results are compared to previous numerical and analytical solutions. In addition, the L∞ error is applied. As a consequence of the numerical outcomes, the differential quadrature method, which is primarily based on a discrete singular convolution shape function, is an effective numerical method that can be used to solve the problem of drug diffusion through a thin membrane, guaranteeing a higher accuracy, faster convergence, and greater reliability than other techniques.

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Dynamic analysis of double cracked bi-directional functionally graded nanobeam using the differential quadrature method

Attia,

Matbuly,

Osman

et al. 2024

Acta Mech

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This study investigates the free vibration behavior of a double cracked nanobeam composed of bi-directional functionally graded material. The analysis incorporates Eringen’s nonlocal elasticity theory and the Euler–Bernoulli theory. The material properties are considered to vary in both the thickness and length directions. The cracked nanobeam is modeled as a series of interconnected sub-beams, with rotational springs placed at the cracked sections. This modeling approach accounts for the discontinuities in rotational displacement resulting from bending, which is directly related to the bending moment transmitted by the cracked section. The problem is solved using the differential quadrature method, which approximates the derivatives of the field quantities by employing a weighted linear sum of the nodal values. By doing so, the problem is transformed into a linear algebraic system. Various supporting cases are examined, and a parametric study is conducted to analyze the impact of the axial and transverse gradient indices, nonlocal parameter, and crack severity on the obtained results.

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Analysis of Cracked Plates Using Localized Multi¬Domain Differential Quadrature Method

Cited by 5 publications

References 24 publications

Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Semi-Analytical Analysis of Drug Diffusion through a Thin Membrane Using the Differential Quadrature Method

Dynamic analysis of double cracked bi-directional functionally graded nanobeam using the differential quadrature method

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