Free vibration analysis of arches using curved beam elements

Abstract

SUMMARYThe natural frequencies and mode shapes for the radial (in-plane) bending vibrations of the uniform circular arches were investigated by means of the ÿnite arch (curved beam) elements. Instead of the complicated explicit shape functions of the arch element given by the existing literature, the simple implicit shape functions associated with the tangential, radial (or normal) and rotational displacements of the arch element were derived and presented in matrix form. Based on the relationship between the nodal forces and the nodal displacements of a two-node six-degree-of-freedom arch element, the elemental sti ness matrix was derived, and based on the equation of kinetic energy and the implicit shape functions of an arch element the elemental consistent mass matrix with rotary inertia e ect considered was obtained. Assembly of the foregoing elemental property matrices yields the overall sti ness and mass matrices of the complete curved beam. The standard techniques were used to determine the natural frequencies and mode shapes for the curved beam with various boundary conditions and subtended angles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed by using an arch segment connected with a straight beam segment at each of its two ends was also studied. For simplicity, a lumped mass model for the arch element was also presented. All numerical results were compared with the existing literature or those obtained from the ÿnite element method based on the conventional straight beam element and good agreements were achieved.

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

“…It is noted that the dynamic characteristics of a vibrating system obtained from ''consistent-mass matrix'' and those from ''lumpedmass matrix'' are very close to each other as one may see Ref. [17]. For convenience, the FEM using ''consistentmass matrix'' of a rod element is called ''FEMC'' and that using ''lumped-mass matrix'' of a rod element is called ''FEML'' in this paper.…”

“…In the conventional FEM [15,16], there exist two types of mass matrices for a rod element (or the other structural elements): consistent-mass matrix and lumped-mass matrix. Since the theories of the presented Methods 1 and 2 are similar to those of consistent-mass matrix and lumped-mass matrix, respectively, and the natural frequencies of a structural system based on the consistent-mass matrix are very close to the corresponding ones based on the lumped-mass matrix [17], it is under expectation that the lowest five natural frequencies and mode shapes of a beam carrying multiple spring-mass systems with mass of each helical spring considered obtained from Method 1 are very close to those obtained from Method 2.…”

“…(16a)-(16d), that for the final bean segment, (Ā bn ;B bn ;C bn andD bn ), is given by Eqs. (16) and (17) with i ¼ n À 1, while that for the ith rod, (Ā ri andB ri Þ, is given by Eqs. (18) and (16d), one can obtain the equations for the relationships between integration constants associated with all beam segments and attached rods for a simply supported beam carrying any number of rod-mass systems from Eqs.…”

Geometric nonlinear dynamic analysis of curved beams using curved beam element

“…Friedman et al [12] utilized the trigonometric functions as the interpolation function. Wu et al [13] presented the curved element by a set of simple implicit interpolation function on the basis of Ref. [12], and established transformation relations between the curved beam and the straight beam elements to investigate the in plane vibration of hybrid beam.…”

On the Validity of Flat-Plate Finite Strip Approximation of Circular Cylindrical Shells

“…To cater for such couplings, curved elements for modeling practical structures with curvature in one or two dimensions have been developed since the seventies; some examples of these developments can be found in [1][2][3][4]. Despite such extensive research works, curved beam and shell-type structures are still very often in practice modeled as an assemblage of straight beam elements or flat plate elements, due to their conceptual simplicity, ease of modeling, and availability of such elements in most commercial softwares.…”