The effect of shear deformation on the vibration of circular arches by the finite element method
TL;DR: In this paper, the free-vibration natural frequencies of circular arches subtending a considerable range of central angles and having a wide range of ratios of radius of curvature to radius of gyration (R r ) are obtained.
Abstract: The free-vibration natural frequencies of circular arches subtending a considerable range of central angles and having a wide range of ratios of radius of curvature to radius of gyration ( R r ) are obtained. Both symmetrical and antisymmetrical modes of vibration are investigated and the crossover values of frequencies from one of the modes to the other are obtained. Arches that are fixed at both ends, hinged at both ends and one end fixed whilst the other is pinned are analysed. The finite element method is employed to obtain these results. A strain-based curved beam element, using Timoshenko's deep-beam formulation in a system of curvilinear coordinates, is obtained and employed in the analysis. It is shown that this element does not have any spurious constraints and locking characteristics. Convergence curves show that accurate and stable results are obtained by the use of few elements, enabling the employment of a simple method of obtaining the eigenvalues and eigenvectors representing the frequencies and the corresponding modes of vibrations, respectively.
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TL;DR: In this article, the effect of shear deformation on deflection and shear deformations together with rotatory inertia on natural and cross over frequencies of curved beams are obtained using a simple cubic linear beam element having 4 degrees of freedom per node viz u, w, θ and γ.
80 citations
TL;DR: In this paper, a set of fundamental dynamic equations of a one-dimensional higher-order arch theory for in-plane vibration problems of shallow circular arches is derived through Hamilton's principle.
42 citations
Journal Article•
TL;DR: In this paper, a bibliography lists references to papers, conference proceedings, and theses/dissertations dealing with finite element vibration and dynamic response analysis of engineering structures.
Abstract: This bibliography lists references to papers, conference proceedings, and theses/dissertations dealing with finite element vibration and dynamic response analysis of engineering structures that wer
29 citations
TL;DR: Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material.
Abstract: Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results Out of plane buckling is investigated for circular beams due to radial load
25 citations
TL;DR: In this article, the stiffness and the mass matrices for the in-plane motion of a thin circular beam element are derived respectively from the strain energy and the kinetic energy by using the natural shape functions of the exact inplane displacements.
Abstract: In this paper, the stiffness and the mass matrices for the in-plane motion of a thin circular beam element are derived respectively from the strain energy and the kinetic energy by using the natural shape functions of the exact in-plane displacements which are obtained from an integration of the differential equations of a thin circular beam element in static equilibrium. The matrices are formulated in the local polar coordinate system and in the global Cartesian coordinate system with the effects of shear deformation and rotary inertia. Some numerical examples are performed to verify the element formulation and its analysis capability. The comparison of the FEM results with the theoretical ones shows that the element can describe quite efficiently and accurately the in-plane motion of thin circular beams. The stiffness and the mass matrices with respect to the coefficient vector of shape functions are presented in appendix to be utilized directly in applications without any numerical integration for their formulation.
25 citations
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1,602 citations
TL;DR: In this article, the method of forming curved finite element shape functions from simple independent generalized strain functions is applied to a rectangular cylindrical shell element, which has 20 degrees of freedom and satisfies the conditions for rigid body displacements and constant strain (in so far as this is allowed by compatibility equations).
139 citations
TL;DR: In this article, the authors extended the work of Ashwell and Sabir, assessing the power of curved finite elements by applying them to circular arches, and presented an element I based on the cylindrical shell element of Bogner et al.
129 citations
01 Jun 1953
TL;DR: In this article, a formula for the shear coefficient was obtained by calculating the frequency (from the free-vibration equations of Timoshenko) at which an infinite beam vibrates without transverse deflections and by equating this with the frequency calculated from the 3D equations of small vibrations of an elastic body.
Abstract: : Timoshenko's shear coefficient for flexural vibrations of beams was computed for circular, elliptical, parabolic, and oval cross sections. A formula for the shear coefficient was obtained by calculating the frequency (from the free-vibration equations of Timoshenko) at which an infinite beam vibrates without transverse deflections and by equating this with the frequency calculated from the 3-dimensional equations of small vibrations of an elastic body. Approximate frequencies obtained by Hidaka (Mem. Imp. Marine Observ., Kobe, Japan 4:99-219, 1931) were used for the ovals, but the other frequencies, as calculated from the equations for small vibrations, were exact. The calculated shear coefficients were tabulated, and all were within about 10% of the coefficient for the rectangular section.
108 citations
TL;DR: Several segmental arch problems were solved by finite elements using three shape functions, namely, I, cubic expression for radial, linear for circumferential displacements; II, a form of Cantin and Clough's shape function for cylindrical shells; III, a shallow element form of II as mentioned in this paper.
98 citations