•Finite element formulation for dynamic characteristics of FG shafts with multiple breathing cracks is developed.•Whirl frequencies and critical speeds of a rotor-bearing system with multiple cracked FG shaft depend on power law gradient.•In an FG shaft, power law gradient could be decided to minimize the undesirable effects in the event of multiple breathing cracks appearing during service. Finite element (FE) formulation of a functionally graded (FG) shaft having multiple cracks has been presented to study the transverse vibration of such shafts in a rotor bearing system. Two nodded Timoshenko beam elements with four degrees of freedom (DOFs) per node are used to model cracked FG shaft considering the translational and rotary inertia, transverse shear deformations and gyroscopic moments. Local flexibility coefficients (LFCs) for the cracked FG shaft are derived using Paris's equation and Castigliano's theorem. Breathing effect of cracks has been considered taking into account the stiffness variation during rotation of the FG shaft. Zirconia (ZrO2) and stainless steel (SS) with temperature dependent material properties have been considered as the constituent materials of a radially graded FG shaft. Using the FE code developed, the forward and backward whirl frequencies and critical speeds of the cracked FG shaft are obtained to study the effects of important parameters and power law gradient index. Results show that besides being affected by the crack locations, orientations and size, the extent of percentage reductions in fundamental frequencies and critical speeds are also influenced by the power-law gradient index of the FG shaft. Therefore, from the view point of damage tolerant design of FG shafts, the choice of power-law gradient index has significant importance where multiple transverse cracks appear in the shaft during service. Display omitted