A Gao beam subjected to a moving inertial point load

Bajer, Czesław I.; Dyniewicz, Bartłomiej; Shillor, Meir

doi:10.1177/1081286517718229

Cited by 6 publications

(5 citation statements)

References 17 publications

(29 reference statements)

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“…There exists an extensive literature about the Timoshenko beam, see e.g., [2][3][4] and the references therein. Currently, there is a rapidly growing scientific literature about the Gao beam, see, e.g., [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. These publications deal with the modeling, mathematical analysis and computer simulations of the various aspects of the Gao beam model.…”

Section: Introductionmentioning

confidence: 99%

“…The total potentials of these beam models were non-convex, with a double-well structure, which was used in post-buckling analysis and also frictional contact problems. The analytical and numerical solutions of the model were applied to a structure subjected to a moving inertial load [14,15], representing a wheel moving on a rail. It has been shown that an inertial load significantly altered the dynamic response of the system and that omitting the nonlinear term in the differential equation led to incorrect qualitative and quantitative solutions.…”

Section: Introductionmentioning

confidence: 99%

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An extended 2D Gao beam model

Dyniewicz,

Shillor,

Bajer

2023

Meccanica

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This work derives and simulates a two-dimensional extension of the nonlinear Gao beam, by adding the cross-sectional shear variable, similarly to the extension of the usual Bernoulli–Euler beam into the Timoshenko beam. The model allows for oscillatory motion about a buckled state, as well as adds vertical shear of the cross sections, thus reflecting better nonlinear thick beams. The static model is derived from the principle of virtual elastic energy, and is in the form of a highly nonlinear coupled system for the beams transverse vibrations and the motion of the cross sections. The model allows for general distributive transversal and longitudinal loads and a compressive horizontal load acting on its edges. The model is simulated numerically, using the dynamic version for better insight into the steady solutions. The terms that distinguish our numerical solutions from the solutions of the Gao beam, described in the literature, are highlighted. The numerical scheme and its characteristic finite element matrices allow us to obtain simulation results that demonstrate the type of vibrations of our extended and modified beam, and also the differences between these solutions and those of the Gao beam model.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An extended 2D Gao beam model

Dyniewicz,

Shillor,

Bajer

2023

Meccanica

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“…This paper continues the study of the vibrations of a nonlinear Gao beam that is subjected to a moving mass [11,12] by adding a oscillating axial force. Such problems arise naturally in transportation systems such as trains or trams.…”

Section: Introductionmentioning

confidence: 86%

The Gao beam under a moving inertial load and harmonic compression

Dyniewicz

Bajer

2019

MATEC Web Conf.

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In the present work the dynamics of the system of a mass moving on the beam is investigated in detail numerically in the case of vibrations about a buckled state. The differential equation that describes the motion is strongly nonlinear. Simulations are based on the space-time finite element method. It enabled us easily determine the influence of the moving inertial particle. At the computational stage it becomes a real problem when the mass particle traverses joints of neighbouring elements. The results of representative and interesting computer simulations are enclosed.

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“…By equation (2) we know that the axial deformation could be relatively large, while the nonconvexity of the total potential shows that this nonlinear beam model can be used for studying both pre and post-buckling problems [4,30]. Recently, the Gao beam model has been generalized for many real-world applications in engineering and sciences [1,2,3,22,23,25,26,28].…”

Section: Nonconvex Problem and Canonical Duality Theorymentioning

confidence: 99%

On SDP method for solving canonical dual problem in post buckling of large deformed elastic beam

Ali

Gao

2018

Communications in Mathematical Sciences

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This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre-and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

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A Gao beam subjected to a moving inertial point load

Cited by 6 publications

References 17 publications

An extended 2D Gao beam model

An extended 2D Gao beam model

The Gao beam under a moving inertial load and harmonic compression

On SDP method for solving canonical dual problem in post buckling of large deformed elastic beam

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