Curved composite laminates such as L-beams are frequently used in wind turbine blade structures such as spars and ribs. It is widely assumed that delamination initiates at the curved region of the L-shaped laminate leading to loss of loading carrying capacity. However, as shown in this paper, under certain conditions a second failure mode in thick L-shaped laminates is observed in which a secondary crack initiates at the arm region. Delamination in L-shaped laminates is modeled using a sequential analysis with implicit analysis followed by explicit dynamic (explicit) finite element analysis in conjunction with cohesive zone methods. The 2-D model consists of 24 plies of unidirectional CFRP laminate with an initial crack at the center of curved region. Loading is applied parallel to one arm quasi-statically and the observed delamination occurs dynamically. For thin laminates and larger precracks, delamination starts from the initial crack and propagates towards the arms. For thicker L-shaped laminates and smaller precracks at the center of curved region, formation of secondary crack in the arm region is observed. Therefore the size of the initial crack as a function of thickness at the center of the curved region mainly decides the failure mode.