LAUSR

Abstract Curved beams such as arches find ubiquitous applications in civil, mechanical and aerospace engineering, e.g., stiffened floors, fuselage, railway compartments, and wind turbine blades. The analysis of free vibrations of curved structures plays a critical role in their design to avoid transient loads with dominant frequencies close to their natural frequencies. One way to increase their applications and possibly make them lighter without sacrificing strength is to comprise them of Functionally Graded Materials (FGMs) that are composites with continuously varying material properties in one or more directions. Here, we study free vibrations of FGM circular beams by using a shear deformation theory that incorporates through-the-thickness logarithmic variation of the circumferential displacement, does not require a shear correction factor, and has a parabolic through-the-thickness distribution of the shear strain. The radial displacement of a point is assumed to depend only upon its angular position. Thus the beam theory generalizes the Timoshenko beam theory. Equations governing transient deformations of the beam are derived by using Hamilton’s principle. Assuming a time harmonic variation of displacements, and by utilizing a generalized differential quadrature method (GDQM), the free vibration problem is reduced to solving an algebraic eigenvalue problem whose solution provides frequencies and corresponding mode shapes. Results are presented for different spatial variations of the material properties, boundary conditions, and the beam aspect ratio. It is found that frequencies of the FGM beam are bounded by those of two geometrically identical homogeneous beams composed of the two constituents of the beam. Keeping other variables fixed, the change in the beam opening angle results in very close frequencies of the first two modes of vibration at a critical value of the opening angle, a phenomenon usually called mode transition. The critical opening angle is essentially the same for radially graded, bidirectionally graded and monolithic beams. It equals about 80 o (60 o ) for clamped-clamped (hinged-hinged) beams.