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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.


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Journal ArticleDOI
Sunil K. Sinha1
TL;DR: In this article, the dynamic response of a rotating cantilever twisted and inclined airfoil blade subjected to contact loads at the free end is considered. And the Rayleigh-Ritz method is used to convert the set of coupled partial differential equations into equivalent classical mass, stiffness, damping, and gyroscopic matrices.
Abstract: In this paper, consideration is given to the dynamic response of a rotating cantilever twisted and inclined airfoil blade subjected to contact loads at the free end. Starting with the basic geometrical relations and energy formulation for a rotating Timoshenko beam constrained at the hub in a centrifugal force field, a system of coupled partial differential equations are derived for the combined axial, lateral and twisting motions which includes the transverse shear, rotary inertia, and Coriolis effects, as well. In the mathematical formulation, the torsion of the thin airfoil also considers a very general case of shear center not being coincident with the CG (center of gravity) of the cross section, which allows the equations to be used also for analyzing eccentric tip-rub loading of the blade. Equations are presented in terms of axial load along the longitudinal direction of the beam which enables us to solve the dynamic pulse buckling due to the tip being loaded in the longitudinal as well as transverse directions of the beam column. The Rayleigh-Ritz method is used to convert the set of four coupled-partial differential equations into equivalent classical mass, stiffness, damping, and gyroscopic matrices. Natural frequencies are computed for beams with varying "slenderness ratio" and "aspect ratio" as well as "twist angles." " Dynamical equations account for the full coupling effect of the transverse flexural motion of the beam with the torsional and axial motions due to pretwist in the airfoil. Some transient dynamic responses of a rotating beam repeatedly rubbing against the outer casing is shown for a typical airfoil with and without a pretwist.

78 citations

Journal ArticleDOI
TL;DR: In this paper, the surface Cauchy-Born model was used to quantify the coupled effects of surface stresses and boundary conditions on the resonant properties of silicon nanowires.
Abstract: The purpose of the present work is to quantify the coupled effects of surface stresses and boundary conditions on the resonant properties of silicon nanowires. We accomplish this by using the surface Cauchy–Born model, which is a nonlinear, finite deformation continuum mechanics model that enables the determination of the nanowire resonant frequencies including surface stress effects through solution of a standard finite element eigenvalue problem. By calculating the resonant frequencies of both fixed/fixed and fixed/free ⟨100⟩ silicon nanowires with unreconstructed {100} surfaces using two formulations, one that accounts for surface stresses and one that does not, it is quantified how surface stresses cause variations in nanowire resonant frequencies from those expected from continuum beam theory. We find that surface stresses significantly reduce the resonant frequencies of fixed/fixed nanowires as compared to continuum beam theory predictions, while small increases in resonant frequency with respect to continuum beam theory are found for fixed/free nanowires. It is also found that the nanowire aspect ratio, and not the surface area to volume ratio, is the key parameter that correlates deviations in nanowire resonant frequencies due to surface stresses from continuum beam theory.

78 citations

Journal ArticleDOI
TL;DR: In this paper, a method for finding exact out-of-plane natural frequencies of plane structures composed of curved Timoshenko beams is presented. Butler et al. used a modification to a well-established algorithm, which ensures that no natural frequencies can be missed and avoids the usual approximations associated with traditional finite elements.
Abstract: A powerful and efficient method is presented for finding exact out-of-plane natural frequencies of plane structures composed of curved Timoshenko beams. Initially, exact dynamic stiffnesses are derived from the governing differential equations of motion in a form that can be used directly in the stiffness method of analysis. This enables any appropriate structure to be modeled according to standard techniques, which, in this case, yield a transcendental eigenvalue problem. Then it is shown how any desired natural frequency may be obtained with certainty by employing a modification to a well-established algorithm, which ensures that no natural frequencies can be missed and avoids the usual approximations associated with traditional finite elements. Finally, comparisons are made with published results and an example shows how the natural frequencies of a continuous curved beam are altered when the effects of shear deflection and rotary inertia are considered.

78 citations

Journal ArticleDOI
TL;DR: In this article, a theoretical treatment of Timoshenko [S. Timoshenko, Philos. Mag. 41, 744 (1921)] beams is presented, in which the influences of shear deformation, rotary inertia, and scale coefficient are taken into account.
Abstract: This letter presents a theoretical treatment of Timoshenko [S. Timoshenko, Philos. Mag. 41, 744 (1921)] beams, in which the influences of shear deformation, rotary inertia, and scale coefficient are taken into account. Based on the nonlocal elasticity theory, coupled equations for transverse deflection and rotation of cross section are derived. Free vibration of several typical beams is analyzed. Explicit expressions for modal shapes of vibration are presented. Natural frequencies are evaluated for free vibration of simply supported beams, clamped beams, cantilever beams, and clamped-hinged beams. The effects of the nonlocal parameter on natural frequencies and modal shapes are discussed in detail.

78 citations

Journal ArticleDOI
TL;DR: In this article, a higher-order shear-deformable beam model is proposed, which is based on a higher order displacement model and incorporates linear and quadratic variation of transverse normal strain and transverse shearing strain through the beam thickness.

78 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023194
2022437
2021509
2020487
2019540
2018508