Saint Venant’s torsion of homogeneous and composite bars by the finite volume method

Chen, Heze; Gómez, J.; Pindera, Marek‐Jerzy

doi:10.1016/j.compstruct.2020.112128

Cited by 14 publications

(7 citation statements)

References 33 publications

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“…Chen [11] successfully implemented the finite volume method for the analysis of unrestrained torsion. The stresses and torsional rigidities obtained in the study were accurate as compared with previous works in thee technical literature.…”

Section: Introductionmentioning

confidence: 99%

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

Ike

2024

J., mech. eng. autom., control. syst.

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The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

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Section: Introductionmentioning

confidence: 99%

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

Ike

2024

J., mech. eng. autom., control. syst.

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“…The additional nodes permitted to apply the governing equations at the boundary nodes and to satisfy the boundary conditions. In a thesis Chen [7] developed a new approach to Saint-Venant's torsion problems based on the finite-volume method. The approach employed the displacement formulation expressed in terms of the warping function subject to Neumann-type boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Saint-Venant Torsion of Beams with Rectangular Cross-Section Using the Fourier Analysis

Fogang

2024

Preprint

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This paper presents an approach to the analysis of beams with rectangular cross-sections subjected to Saint-Venant torsion using the Fourier analysis, body forces being not considered. The Saint-Venant torsion of beams, also called free torsion or unrestrained torsion, is characterized by the absence of axial stresses due to torsion; only shear stresses are developed. The solution to this torsion problem consists of finding the Prandtl stress function that satisfies the governing equation and the boundary conditions, the governing equation being such that the Laplacian of the stress function has a constant value within the cross-section. In this paper, Fourier sine series was utilized to describe the constant value of the above mentioned Laplacian, while double sine series or simple sine series was utilized to describe the Prandtl stress function. Closed-form solutions in form of infinite series were derived for shear stresses, torsional constant and warping function. Those solutions were identical to the exact solutions.

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“…Besides the finite difference and finite element method, many authors have applied other numerical methods such as the boundary element method (BEM) [11][12][13][14][15][16], line element-less method (LEM) [17][18], a null-field integral technique [19], and finite-volume method [20][21] to analyse the torsion problem. Katsikadelis, J. T. et al [11] examined the torsion of general composite bars by BEM while Gaspari, D. et al [12] tackled orthotropic beams with a polygonal cross-section.…”

Section: Introductionmentioning

confidence: 99%

“…Chen, J-T. et al [19] introduced the null-field integral technique to analyse the torsion problem of circular cross-sections with round holes. Chen, H. et al [20][21] developed a new finite-volume based on Bansal and Pindera's work to investigate Saint Venant's torsion problems of homogeneous and composite prismatic bars with multiply connected domain.…”

Section: Introductionmentioning

confidence: 99%

Torsional Shear Stress in Prismatic Beams With Arbitrary Cross-Sections Using Finite Element Method

Tran¹

2021

CEJ

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Determining the shear stress of a structural element caused by torsion is a vital problem. The analytical solution of the Saint-Venant torsion is only suitable for simple cross-sections. The numerical method to evaluate the shear stress of complicated cross-sections is indispensable. Many numerical methods have been studied by scientists. Among these studies, Gruttmann proposed an excellent numerical method, which inherited the Saint-Venant theory. However, the use of isoparametric four-noded quadrilateral elements made the method not to reach the best optimization. The objective of this paper is to improve Gruttmann‘s method by using isoparametric eight-noded quadrilateral elements. MATLAB is the language for programming the numerical method. The validated examples have demonstrated that the author’s numerical method is more effective than Gruttmann‘s method.

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Saint Venant’s torsion of homogeneous and composite bars by the finite volume method

Cited by 14 publications

References 33 publications

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

Saint-Venant Torsion of Beams with Rectangular Cross-Section Using the Fourier Analysis

Torsional Shear Stress in Prismatic Beams With Arbitrary Cross-Sections Using Finite Element Method

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