Semi-Analytical Solution of In-Plane Vibrations of Circular Arches Carrying Added Point Masses

摘要

In spite of the current use in modern civil engineering of straight structural components made of reinforced concrete, such as beams and slabs, arches still remain quite often adopted in architectural design, due to their interesting mechanical and esthetical properties. Therefore, practical analytical and numerical tools should be available in order to allow designers to ensure the resistance and stability of this type of structures, especially in the dynamic regime and when point masses are added. The sixth degree differential equation of motion of inextensible arches is known to be quite difficult to deal with and is not applicable to the case examined here of arches with added point masses. The procedure adopted in this article consisted on finding relatively easy particular solutions of this equation satisfying the end conditions and using them as trial functions in the Rayleigh-Ritz formulations of the practical cases of interest. Circular arches of different opening angles are examined here under the classical hypotheses: (1) the effect of shear deformation and rotary inertia are neglected (2) the arch axis is inextensible (3) the dimensions of the cross-section are small in comparisons with the radius;(4) the cross-section is uniform. Three different kinds of end conditions are considered: simply supported-simply supported (SS), clamped-simply supported (CS) and clamped-clamped (CC). The Rayleigh-Ritz method allowed in each case to determine, via an algebraic solution of the associated eigenvalue problem, the first seven mode shapes and natural frequencies of arches with different values of the opening angle. The comparison with the results available in the bibliography was satisfactory and the mode shapes were plotted. The second part of this work was devoted to arches with added concentrated masses: likewise, the numerical values found for the natural frequencies compare quite well with the few references available and the mode shapes are plotted showing the effects of the opening angle, the mass and the positions of the added point masses.