A novel application of the FPM to the buckling differential equation of non-uniform beams

Totry, Essam; Altus, Eli; Proskura, Aleksey

doi:10.1016/j.probengmech.2007.12.009

Cited by 4 publications

(6 citation statements)

References 16 publications

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“…By using symbolic computing software, such as MAPLE, one can generate the explicit expressions of F m and D m through their recursive formulas Equations ( 14) and (15), respectively. Substituting F m and D m into Equation (11), the closed-form solutions of the random buckling eigenvalue F(𝛏) and corresponding buckling eigenvector D(𝛏) can be obtained as…”

Section: Homotopy Series Expansions For Isolated Buckling Eigenvalues...mentioning

confidence: 99%

“…The calculation steps to determine the explicit forms of Z • , including the coefficients in the homotopy series for the buckling eigenvalue or buckling eigenvector in Equation ( 16), are provided in the following. Substituting the recursive formulas of F m and D m in Equations ( 14) and ( 15) into Equation (11) and arranging the expression according to the power of the variables, then the homotopy series in Equation ( 16) can be obtained. The homotopy series for the buckling eigenvalue and buckling eigenvector are written, respectively, as…”

Section: Data Availability Statementmentioning

confidence: 99%

“…Raham and Siragy 9 conducted a stability analysis of a stiffened panel with small variation in structural parameters. Altus and Totry 10,11 proposed a functional perturbation method to analytically determine the solution of the buckling load for simple structures with random material properties. Recently, by combining the stochastic element-free Galerkin method and second-order perturbation technique, Gupta and Arun 12 studied the elastic buckling of columns with small fluctuation of uncertain parameters.…”

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confidence: 99%

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A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

Zhang

Huang

Wang

et al. 2022

Numerical Meth Engineering

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The large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed‐form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, a pth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing the pth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross‐section column, a 7‐story frame, and a Kiewitt single‐layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.

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Section: Homotopy Series Expansions For Isolated Buckling Eigenvalues...mentioning

confidence: 99%

Section: Data Availability Statementmentioning

confidence: 99%

mentioning

confidence: 99%

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A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

Zhang

Huang

Wang

et al. 2022

Numerical Meth Engineering

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“…As the last example, we consider a nonlinear differential equation 19) with u(0) = 1 and u ,x (0) = 1. By rewriting Eq.…”

Section: Examplementioning

confidence: 99%

“…Also in the same year the buckling load of heterogeneous columns has been found by applying the FPM directly to the buckling differential equation in [18]. The FPM is generalized in [19] for solving eigenvalue functional differential equations in 2008. In the current study, we use the FPM to approximate the solution of linear and nonlinear ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

A new approximation method to solve boundary value problems by using functional perturbation concepts

Pourghanbar

Ranjbar

2018

B Soc Paran Mat

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Functional perturbation method (FPM) is presented for the solution of differential equations with boundary conditions. Some properties of FPM are utilized to reduce the differential equation with variable coefficients to the equations with constant coefficients. The FPM can be applied directly for many types of differential equations. The exact solution is obtained by only the first term of the Frechet series for polynomial cases. Four examples are included to demonstrate the method.

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Buckling loads for steel tube with flattened ends

2017

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A novel application of the FPM to the buckling differential equation of non-uniform beams

Cited by 4 publications

References 16 publications

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

A new approximation method to solve boundary value problems by using functional perturbation concepts

Buckling loads for steel tube with flattened ends

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