Fig. 1 Schematic of the curved beam. Right side is clamped. R 1 = 40 mm, R 2 = 45 mm, out-of-plane thickness t = 1 mm. More

Fig. 2 Variations of the normalized elastic modulus and density along the beam as θ increases from 0 to 180 deg for ( a ) decreasing material properties with m = 0.9 and various n values, ( b ) decreasing material properties with n = 2 and various m values, ( c ) increasing material prop... More

Fig. 3 A differential element of the curved beam showing both degrees-of-freedom for deformation of a cross section of the beam: ( a ) tangential displacement; u and ( b ) radial displacement; w More

Fig. 4 Free body diagram of a differential element of the curved beam showing forces and moments (left) and equivalent inertial forces (right) More

Fig. 5 Schematic of the crack in the curved beam at θ C with a crack size of d . t C = 0.35 mm for all cracked curved beams studied. More

Fig. 6 Convergence of the normalized amplitude of ( a ) u ( x ) and ( b ) w ( x ) as the approximation terms increase. Results are obtained from the analytical model. E / E 0 = ρ / ρ 0 = (1 − m ( θ / π ) n ), m = n = 0.9. More

Fig. 7 Normalized first natural frequency values as a function of modulation parameter n from the semi-analytical and FE models. E / E 0 = ρ / ρ 0 = (1 − m ( θ / π ) n ), m = 0.9. u and w are tangential and radial displacements of the centriod of the cross section and approximated... More

Fig. 8 Normalized first natural frequency of the FG curved beam with ( a,b ) decreasing and ( c,d ) increasing material properties along the curved beam for 0 < n < 30 and 0 < m < 1, and material properties (mass density and elastic modulus) are modulated in the tangential directio... More