Free in-plane vibration of general curved beams using finite element method

A displacement finite volume formulation for the static and dynamic analysis of shear deformable circular curved beams

“…It is observed that very close agreement exists between the present results and those published in Refs. [11,13,36]. The associated modal shapes are also presented in Fig. 14 where an excellent agreement is observed with the results given in Ref.…”

“…The beam has already been studied in Ref. [13] where R, r = A I , l = R φ, and γ are the radius of curvature, the radius of gyration, length of the curved beam and the material density, respectively.…”

“…The geometrical and material properties are the same as those used in Ref. [13]: R/r = 15.915; l/r= 25; φ= 90; R= 0.6366m; A= 1m 2 ; I = 0.0016m 4 , E= 70GPa; k s = 0.85; k s G/E = 0.3; ρ = 2777 kg/m 3 .…”

“…Also, Ref. [13] applied an element of four nodes with 12 degrees of freedom. However, in the proposed formulation we use a curved cell of two points with three degrees of freedom in which even assuming a linear variation of unknowns between the computational points can provide accurate results.…”

“…Friedman and Kosmatka [12] utilized the Ritz method based on the trigonometric functions and derived the exact static stiffness matrix for the uniform circular curved beams. Yang et al [13] have used a four-node Lagrangian type curved beam element and a polynomial equation of order 5 describing the tangential displacement, for modeling the uniform and non-uniform curved beams with variable curvatures. 3 All of the above-mentioned studies have been successfully applied to achieve the locking free elements with different accuracy levels.…”