Since the study of stress singularities became central to failure mechanics, mixed-mode failure initiation at reentrant corners has been recognized as a critical phenomenon for understanding the strength of ceramics and other elastic brittle materials. Failure initiation in the vicinity of a sharp V-notch tip---particularly under complex states of stress---requires a robust framework to be accurately interpreted for practical use. In this work, the phase-field fracture theory of Kumar et al., which has been established as a complete theory of fracture capable of accurately describing the nucleation and propagation of cracks in elastic brittle materials under arbitrary quasistatic loading conditions, is employed as a robust variational framework for investigating crack initiation. The study focuses on V-notched beam specimens under four-point bending, using experimentally validated configurations that vary mixed-mode loading through asymmetric load placement. The primary objective is to conduct a quantitative analysis of failure loads and crack initiation angles in V-notched structures.