Research Papers

Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

[+] Author and Article Information
J. Awrejcewicz

Department of Automation, Biomechanics
and Mechatronics,
Lodz University of Technology,
1/15 Stefanowskiego Street,
Lodz 90-924, Poland;
Department of Vehicles,
Warsaw University of Technology,
84 Narbutta Street,
Warsaw 02-524, Poland
e-mail: awrejcew@p.lodz.pl

A. V. Krysko

Department of Applied Mathematics and
Systems Analysis,
Saratov State Technical University,
77 Politehnicheskaya,
Saratov 410054, Russia;
Cybernetic Institute,
National Research Tomsk Polytechnic University,
30 Lenin Avenue,
Tomsk 634050, Russian Federation
e-mail: anton.krysko@gmail.com

S. P. Pavlov, M. V. Zhigalov

Department of Mathematics and Modeling,
Saratov State Technical University,
77 Politehnicheskaya,
Saratov 410054, Russian Federation

V. A. Krysko

Department of Mathematics and Modeling,
Saratov State Technical University,
77 Politehnicheskaya,
Saratov 410054, Russian Federation
e-mail: tak@san.ru

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 23, 2016; final manuscript received December 23, 2016; published online February 9, 2017. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(4), 041018 (Feb 09, 2017) (8 pages) Paper No: CND-16-1451; doi: 10.1115/1.4035668 History: Received September 23, 2016; Revised December 23, 2016

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

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Grahic Jump Location
Fig. 1

Geometry of the Timoshenko beam and the acting load

Grahic Jump Location
Fig. 2

Load-deflection curve w(0.5;q) (a) and time histories of the deflection w(0.5;t) (b)

Grahic Jump Location
Fig. 4

Deflection-load functions for kx=24: (a) variants 1–3, γ2=0.3, (b) variants 4–6, γ2=0, and (c) homogeneous beam

Grahic Jump Location
Fig. 3

Solutions to static problems of the Timoshenko beam: (a) w(0.5;q) for variants 1–3, γ2=0.3, (b) w(0.5;q) for variants 4–6, γ2=0, (c) w(0.5;q) for homogeneous beams, and (d) solution for w(n) and q=150




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