This paper addresses the problem of designing LDPC decoders robust to transient errors introduced by a faulty hardware. We assume that the faulty hardware introduces errors during the message passing updates and we propose a general framework for the definition of the message update faulty functions. Within this framework, we define symmetry conditions for the faulty functions, and derive two simple error models used in the analysis. With this analysis, we propose a new interpretation of the functional Density Evolution threshold previously introduced, and show its limitations in case of highly unreliable hardware. However, we show that under restricted decoder noise conditions, the functional threshold can be used to predict the convergence behavior of FAIDs under faulty hardware. In particular, we reveal the existence of robust and non-robust FAIDs and propose a framework for the design of robust decoders. We finally illustrate robust and non-robust decoders behaviors of finite length codes using Monte Carlo simulations.