Calculating a beam of variable section lying on an elastic foundation

Akhmediev, Serik; Михайлов, В. О.; Tazhenova, Gulzada; Bakirov, Madi; Filippova, T. S.; TOKANOV, Daniyar

doi:10.5937/jaes0-38800

Cited by 3 publications

(2 citation statements)

References 10 publications

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“…Other contributions on beam on elastic foundations are in Akhazhanov et al [17], [18], [19]; Huang et al [20]. Akhmediev et al [21] and Zhang et al [22].…”

Section: Introductionmentioning

confidence: 99%

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

Ike

2024

J., eng., them. sci.

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The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

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“…Other contributions on beam on elastic foundations are in Akhazhanov et al [17], [18], [19]; Huang et al [20]. Akhmediev et al [21] and Zhang et al [22].…”

Section: Introductionmentioning

confidence: 99%

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

Ike

2024

J., eng., them. sci.

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“…Researchers who worked on continuously distributed foundation models include: Vlasov andLeontiev (1966), Jones andXenophontos (1977), Das (1988, 1991), Akhazhanov et al (2020Akhazhanov et al ( , 2022Akhazhanov et al ( , 2023aAkhazhanov et al ( , 2023b, Huang et al ( 2019), Akhmediev et al (2023), and Zhang et al (2020).…”

Section: Introductionmentioning

confidence: 99%

Ritz Variational Method for Buckling Analysis of Euler-Bernoulli Beams Resting on Two-Parameter Foundations

Ike

2024

IJCE

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Critical buckling load analysis of Euler-Bernoulli beam on two-parameter foundations using Galerkin method

Ike

2024

J. mechatron., artif. intell., eng.

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The critical buckling load determination of Euler-Bernoulli beams on two-parameter elastic foundations (EBBo2PFs) is important to avert buckling failures. The governing equation for buckling of thin beam on two-parameter elastic foundation is a homogeneous ordinary differential equation (HODE) of fourth order and constant parameters when the beam is prismatic and homogeneous. The HODE has been solved in this work by Galerkin method for simply supported, clamped and clamped-simply supported ends. One-parameter algebraic shape function formulation was used to reduce the problem to an algebraic eigenvalue problem, which is solved to find the critical buckling load for each studied case. The critical buckling load for EBBo2PF for simply supported boundary conditions was found to be closely identical to the exact solutions. The solutions for clamped-clamped edges and clamped-simple supports were found to be accurate. The merit of the Galerkin method is the simplicity and the accuracy even when one-parameter shape function has been used.

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Calculating a beam of variable section lying on an elastic foundation

Cited by 3 publications

References 10 publications

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

Ritz Variational Method for Buckling Analysis of Euler-Bernoulli Beams Resting on Two-Parameter Foundations

Critical buckling load analysis of Euler-Bernoulli beam on two-parameter foundations using Galerkin method

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