We investigate a new class of nondiffracting optical beams carrying fractional orbital angular momentum (FOAM Bessel beams), which can be constructed by modifying the phase step of the spiral phase plate in a setup originally for the generation of traditional Bessel beams. The FOAM Bessel beams have entirely stable vortex structures, and form an infinite number of orthogonal and complete subsets for monochromatic scalar light in free space. With the advantages of structural stability and orthogonality-completeness, the FOAM Bessel beams are expected to be useful for study and application in various fields of modern optics.