Timoshenko beam theory

About: Timoshenko beam theory is a research topic. Over the lifetime, 9426 publications have been published within this topic receiving 200570 citations.

Finite deformation beam models and triality theory in dynamical post-buckling analysis☆

Abstract: Two new finitely deformed dynamical beam models are established for serious study on non-linear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola–Kirchhoff’s type stress only) in finite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that if the gap function introduced by Gao and Strang in 1989 in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Duffing equations are obtained.

Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory

Abstract: In this article, the nonlinear free vibration behavior of Timoshenko nanobeams subject to different types of end conditions is investigated. The Gurtin–Murdoch continuum elasticity is incorporated into the Timoshenko beam theory in order to capture surface stress effects. The nonlinear governing equations and corresponding boundary conditions are derived using Hamilton's principle. A numerical approach is used to solve the problem in which the generalized differential quadrature method is applied to discretize the governing equations and boundary conditions. Then, a Galerkin-based method is numerically employed with the aim of reducing the set of partial differential governing equations into a set of time-dependent ordinary differential equations. Discretization on time domain is also done via periodic time differential operators that are defined on the basis of the derivatives of a periodic base function. The resulting nonlinear algebraic parameterized equations are finally solved by means of the pseudo arc-length continuation algorithm through treating the time period as a parameter. Numerical results are given to study the geometrical and surface properties on the nonlinear free vibration of nanobeams.

Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid

Abstract: The flexural vibration of viscoelastic carbon nanotubes (CNTs) conveying fluid and embedded in viscous fluid is investigated by the nonlocal Timoshenko beam model. The governing equations are developed by Hamilton's principle, including the effects of structural damping of the CNT, internal moving fluid, external viscous fluid, temperature change and nonlocal parameter. Applying Galerkin’s approach, the resulting equations are transformed into a set of eigenvalue equations. The validity of the present analysis is confirmed by comparing the results with those obtained in literature. The effects of the main parameters on the vibration characteristics of the CNT are also elucidated. Most results presented in the present investigation have been absent from the literature for the vibration and instability of the CNT conveying fluid.