Free vibration of functionally graded beams with arbitrary number of surface cracks

• The material gradient greatly affects the natural frequencies of FGM beams.
• Location, size and number of cracks notably influence the frequencies of FGM beams.
• The type of support also influences the eigenfrequencies of the FGM beams.
• The reduced order of determinant significantly reduces the computational time required.
• Nodal points are determined for each of FGM beam types.

Free vibration of beams made of functionally graded materials (FGMs) containing any arbitrary number of open edge cracks is studied. The study is based on Euler–Bernoulli beam and massless rotational springs connecting two intact segments of the beam. It is assumed that the material gradients follow exponential distribution through beam thickness direction. Frequency equations are obtained for flawed FGM beams with fixed–fixed, fixed–hinged, fixed–free, hinged–hinged, and spring–spring end boundaries. Detailed parametric investigation is carried out to examine the influences of crack depth, crack location, total number of cracks, material property distribution, and boundary conditions on the natural frequencies of the damaged FGM beams. The frequency equation for a damaged FGM beam with any kind of two end supports and any arbitrary number of cracks are established through a third order determinant. Compared to previous studies, this decrease in the determinant order can lead to significant advantages in the computational time.