There are several methods for risk management, I am conducting a study in the area and I would like to know which are the main ones used for equipment reliability
I would say FMECA seems the good one for equipment reliability - fault tree analysis may be used in that framework for representing possible failure scenarios
There are several, but it starts with equipment design. For machine safety design criteria I would recommend a review of ISO 13849. This will provide information on how to determine the Performance Level of machine guarding and safety circuitry.
There are several methods already mentioned regarding risk analysis, but I would recommend the Systems-Theoretic Accident Model and Processes (STAMP) model as a newer approach to risk assessment and accident causation that is very robust. Attached is a reference document from MIT that you may find useful.