A Refined Model for Analysis of Beams on Two-Parameter Foundations by Iterative Method

Yue, Feng

doi:10.1155/2021/5562212

Cited by 5 publications

(2 citation statements)

References 28 publications

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“…Hariz et al [44] investigated the buckling analysis of Timoshenko beam resting on two-parameter elastic foundations. Yue [45] used a refined beam model to study the behaviour of beams rested on two-parameter foundations by iterative method. Akgoz et al [46] studied the flexural analysis of beams on elastic foundations using the method of discrete singular convolution, but did not study buckling analysis.…”

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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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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“…A number of scientists [11][12][13] studied beams on two-parameter foundation by the Galerkin method, the power series method and the method of differential operator series. Results of other approaches, such as iterative methods [14], discrete singular convolution method [15], and finite element method [16][17][18], can be found in the research.…”

Section: Introductionmentioning

confidence: 99%

Simplified Method of Calculating a Beam on a Two-Parameter Elastic Foundation

Akhazhanov

2023

GEOMATE

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When calculating beams lying on a solid elastic foundation, the simplest foundation model proposed by Winkler-Zimmerman is often used. This hypothesis has been repeatedly subjected to wellfounded criticism, because the deformation of the base occurs only in the area of the load applied to it. In order to clarify the Winkler-Zimmerman hypothesis, many authors have proposed other models that allow mitigating the disadvantage of this model to varying degrees. The paper deals with the method of obtaining an analytical solution to the problem of beam bending on a two-parameter elastic foundation. A new version of the elastic foundation model is proposed. The account of the elastic foundation is produced by means of parameter of the flexural rigidity. The resolving differential equation of bending of beams resting on a twoparameter elastic foundation is obtained. An analytical solution for this equation is derived, and solutions for various boundary conditions of beams resting on a two-parameter elastic foundation are discussed. The accuracy of the proposed method is verified through examples, and they showed an excellent agreement with the Winkler and Vlasov models.

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Thermally stressed thermoelectric microbeam supported by Winkler foundation via the modified Moore–Gibson–Thompson thermoelasticity theory

2023

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This article offers the first investigation into the thermoelectric vibration of microscale Euler‐Bernoulli beams resting on an elastic Winkler base. We developed the system of equations for thermoelastic microbeams using elastic basis theory and the generalized Moore–Gibson–Thompson (MGT) thermal transport framework with a single‐phase delay. At the front of the microbeam, a graphene strip connected to an electrical power source was employed to generate heat. In addition, the influence of an ultra‐short‐pulsed laser on the vibration of a microbeam is studied, taking into account its heating effect. The microbeam system variables were analyzed after solving the mathematical equations by applying the Laplace transform technique. Graphs showing how different inputs affect different mechanical fields, such as Winkler substrate stiffness, voltage, electrical resistance, and temperature modulus pulses, are also presented. An increase in the Winkler modulus and the shear modulus of the foundation was found to decrease the amount of deflection and axial deformation in the microbeams. The increased beam stiffness has resulted in this decrease.

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A Refined Model for Analysis of Beams on Two-Parameter Foundations by Iterative Method

Cited by 5 publications

References 28 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

Simplified Method of Calculating a Beam on a Two-Parameter Elastic Foundation

Thermally stressed thermoelectric microbeam supported by Winkler foundation via the modified Moore–Gibson–Thompson thermoelasticity theory

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