An anisotropic beam theory based on the extension of Boley’s method

Gahleitner, J.; Schoeftner, J.

doi:10.1016/j.compstruct.2020.112149

Cited by 9 publications

(8 citation statements)

References 8 publications

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“…e books can provide a general overview about the Euler-Bernoulli beam theory, please see [1][2][3]. Some important studies related to beams modeled in the sense of classical beam theory are also summarized as follows, but not limited to [4][5][6][7][8][9][10][11][12][13][14][15]. e beam systems in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] have the integer order derivatives of the state function.…”

Section: Introductionmentioning

confidence: 99%

“…Some important studies related to beams modeled in the sense of classical beam theory are also summarized as follows, but not limited to [4][5][6][7][8][9][10][11][12][13][14][15]. e beam systems in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] have the integer order derivatives of the state function. In the beginning of 1930s, fractional derivative was introduced for describing the constitutive relation of some beam materials [16], and after 1980s, since fractional order equations have good memory and can be used to describe material properties more accurately with fewer parameters, they are considered to be good mathematical models for describing the dynamic mechanical behavior of materials [17].…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam $^{*}$

Yildirim

Alkan

2021

Journal of Mathematics

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In this paper, dynamic response analysis of a forced fractional viscoelastic beam under moving external load is studied. The beauty of this study is that the effect of values of fractional order, the effect of internal damping, and the effect of intensity value of the moving force load on the dynamic response of the beam are analyzed. Constitutive equations for fractional order viscoelastic beam are constructed in the manner of Euler–Bernoulli beam theory. Solution of the fractional beam system is obtained by using Bernoulli collocation method. Obtained results are presented in the tables and graphical forms for two different beam systems, which are polybutadiene beam and butyl B252 beam.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam $^{*}$

Yildirim

Alkan

2021

Journal of Mathematics

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“…The latter constraint for the cross-sectional rotation was suggested by Szabo [27] and applied in the work of Gahleitner and Schoeftner [25] in order to get similar results as for a rigid clamped end with u(0, y) = v(0, y) = 0.…”

Section: Numerical Examplementioning

confidence: 99%

Extension of Boley’s method to functionally graded beams

Gahleitner

Schoeftner

2020

Acta Mech

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The objective of this contribution is the computation of the Airy stress function for functionally graded beam-type structures subjected to transverse and shear loads. For simplification, the material parameters are kept constant in the axial direction and vary only in the thickness direction. The proposed method can be easily extended to material varying in the axial and thickness direction. In the first part an iterative procedure is applied for the determination of the stress function by means of Boley’s method. This method was successfully applied by Boley for two-dimensional (2D) isotropic plates under plane stress conditions in order to compute the stress distribution and the displacement field. In the second part, a shear loaded cantilever made of isotropic, functionally graded material is studied in order to verify our theory with finite element results. It is assumed that the Young’s modulus varies exponentially in the transverse direction and the Poisson ratio is constant. Stresses and displacements are analytically determined by applying our derived theory. Results are compared to a 2D finite element analysis performed with the commercial software ABAQUS. It is found that the analytical and numerical results are in perfect agreement.

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“…If classical boundary conditions are assumed, where either the kinematical or the dynamical boundary conditions vanish at x = 0 and L, Eqs. (14) and (15)…”

Section: Ect For a Beam With Rectangular Cross Sectionmentioning

confidence: 99%

“…Irschik [13] extended this method by finding the stress function of the fourth iteration and thus finds a solution for a statically indeterminate (i.e., clamped-hinged) beam. Gahleitner and Schoeftner [14] modified the Boley-Tolins approximation method for anisotropic materials: the compatibility equation, which is a more complicated form of the bipotential equation in case of anisotropy, is iteratively solved for the Airy stress function. Analytical results perfectly agree with two-dimensional FE results in ABAQUS for a clamped-hinged beam.…”

Section: Introductionmentioning

confidence: 99%

Extension of Castigliano’s method for isotropic beams

Schoeftner

2020

Acta Mech

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In the present contribution Castigliano’s theorem is extended to find more accurate results for the deflection curves of beam-type structures. The notion extension in the context of the second Castigliano’s theorem means that all stress components are included for the computation of the complementary strain energy, and not only the dominant axial stress and the shear stress. The derivation shows that the partial derivative of the complementary strain energy with respect to a scalar dummy parameter is equal to the displacement field multiplied by the normalized traction vector caused by the dummy load distribution. Knowing the Airy stress function of an isotropic beam as a function of the bending moment, the normal force, the shear force and the axial and vertical load distributions, higher-order formulae for the deflection curves and the cross section rotation are obtained. The analytical results for statically determinate and indeterminate beams for various load cases are validated by analytical and finite element results. Furthermore, the results of the extended Castigliano theory (ECT) are compared to Bernoulli–Euler and Timoshenko results, which are special cases of ECT, if only the energies caused by the bending moment and the shear force are considered. It is shown that lower-order terms for the vertical deflection exist that yield more accurate results than the Timoshenko theory. Additionally, it is shown that a distributed load is responsible for shrinking or elongation in the axial direction.

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An anisotropic beam theory based on the extension of Boley’s method

Cited by 9 publications

References 8 publications

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam $^{*}$

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam $^{*}$

Extension of Boley’s method to functionally graded beams

Extension of Castigliano’s method for isotropic beams

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An anisotropic beam theory based on the extension of Boley’s method

Cited by 9 publications

References 8 publications

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam ∗

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam ∗

Extension of Boley’s method to functionally graded beams

Extension of Castigliano’s method for isotropic beams

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Product

Resources

About

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam $^{*}$

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam $^{*}$