A study of the bending of a short beam

Shevlyakov, Yu. A.; Ivanov, Yu. B.

doi:10.1007/bf02362663

Cited by 2 publications

(6 citation statements)

References 2 publications

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“…For the first category of bending, the K x and K y coefficients are taken to be zero, given the neglect of the strain energy due to the effect of the shear force. 1,2,[5][6][7][8][9][10]…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…When φ y = 0, which is the case for curve 7 in dark purple color, the variation of T(z) and M(z) represent the results of bending according to the classical model (first category) without the effect of the shearing force. 1,2,[5][6][7][8][9] Although taking into account the effect of the section geometry, the effect of the shear force becomes part of the results according to our bending model for the second category.…”

Section: N Sectionmentioning

confidence: 99%

“…If the longitudinal dimension is quite large compared to the transverse dimension, this category is based on the consideration of bending moment only and it neglects the effect of shear force in the calculation of strength and stiffness. 5,6 All the theories of inflection consider this assumption to allow formulating the mathematical model. [7][8][9][10] This first theory is called in our work by the classical theory.…”

Section: Introductionmentioning

confidence: 99%

“…For this reason, builders use beams long enough to solve this problem so that the effect of the shear force is negligible. [5][6][7][11][12][13] The third category is obtained when the longitudinal dimension of the beam is very small as the transverse dimensions of the section. In this case, the construct case is obtained, that is to say the bending moment effect becomes negligible and the calculation is based solely on the consideration of the shear force.…”

Section: Introductionmentioning

confidence: 99%

“…The calculation based on the strain energy 1,2,6,7 can be cited, and using one of the theorems known in the literature such as Castiglione's theorem, fictitious forces, unit forces, the method de Mohr, Maxwell Betti's reciprocal theorem, and Vereshchagin's method. 1,2,6,7 The famous FEM method can be found, which will be used in this work as an application for calculating a bending beam. [11][12][13] The bending study problem data are divided into two parts.…”

Section: Introductionmentioning

confidence: 99%

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Strain energy form coefficients for bending of short beams having full and thin-walled arbitrary cross section with application for FEM and Airfoils

Yahiaoui

Zebbiche

2022

Proceedings of the Institution of Mechanical Engineers, Part G:

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The aim of this work is to develop a new generalized formula and a numerical computation program for evaluating the energy form coefficient of a complex and arbitrary cross section for full and thin-walled cross section with respect to any central axis, for the bending of beams of small lengths in comparison with the transverse dimension of the section. This coefficient plays a very important role in the calculation of the deformation energy of beams subjected to bending under the effect of a shearing force for short beams. It also enters in the formulation of FEM bending model, in order to calculate the stresses and the strains due to the external forces. The application is made for complex sections used in various fields of construction and in particular for airfoils designed for aerospace construction. A method is developed to calculate this coefficient as a function of the rotation of the central axes. The calculation of the area, the moments, and the product of inertias with respect to the central axes is necessary. The formula for calculating this coefficient is presented as a definite integral of a non-analytical function determined point by point along the direction of the application of the shear force. This function is based on the calculation of the partial static moments. The calculation of the latter is based on the development of a technique by subdividing the upper part of the section into adjacent common triangles at one point for the full solid section or by segments on the boundary for the thin-walled section. To speed up the process of numerically calculating this integral with high precision and reduced time, Gauss Legendre quadrature of order 40 is used. The calculation of the distribution of the tangential stress as well as its maximum value is determined. A shear shape coefficient is therefore determined. In the second part of this work, an application is made for the static calculation by the FEM of a hyper static beam with a view to determining the influence of this coefficient on all the parameters of resistance and bending stiffness as a correction of the classical model of bending by the FEM. A study of the error made by the classical bending model on our shear effect model is presented. A coefficient of efficiency of a section is presented.

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“…For the first category of bending, the K x and K y coefficients are taken to be zero, given the neglect of the strain energy due to the effect of the shear force. 1,2,[5][6][7][8][9][10]…”

Section: Mathematical Formulationmentioning

confidence: 99%

Section: N Sectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Strain energy form coefficients for bending of short beams having full and thin-walled arbitrary cross section with application for FEM and Airfoils

Yahiaoui

Zebbiche

2022

Proceedings of the Institution of Mechanical Engineers, Part G:

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The Use of Trigonometric Series for the Study of Isotropic Beam Deflection

Botean

2023

Mathematics

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The average deformed fiber is a continuous and smooth function of the fourth order. The deflection and rotation of beams can be determined by various methods available in the literature. Thus, in this paper, the expression of the average deformed fiber is defined in advance, then it is considered an infinite sum of sinusoidal functions that are successively evaluated using sinusoidal trigonometric series if it is considered a periodic function, or using Fourier series if it is considered a non-periodic function. The method is examined by solving several beam problems. The results indicate that the method can be used with confidence for solving any bending beam problem.

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A study of the bending of a short beam

Cited by 2 publications

References 2 publications

Strain energy form coefficients for bending of short beams having full and thin-walled arbitrary cross section with application for FEM and Airfoils

Strain energy form coefficients for bending of short beams having full and thin-walled arbitrary cross section with application for FEM and Airfoils

The Use of Trigonometric Series for the Study of Isotropic Beam Deflection

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