A steel beam may be strengthened in flexure by bonding a carbon fibre-reinforced polymer (CFRP) plate to the tension face. Such a beam may fail by debonding of the CFRP plate that initiates at one of the plate ends (i.e. plate end debonding) or by debonding that initiates at a local damage (e.g. a crack or concentrated yielding) away from the plate ends (intermediate debonding). This paper presents the first finite element (FE) approach that is capable of accurate predictions of such debonding failures, with particular attention to plate-end debonding. In the proposed FE approach, a mixed-mode cohesive law is employed to depict interfacial behaviour under a combination of normal stresses (i.e. mode-I loading) and shear stresses (i.e. mode-II loading); the interfacial behaviour under pure mode-I loading or pure mode-II loading is represented by bi-linear traction-separation models. Damage initiation is defined using a quadratic strength criterion, and damage evolution is defined using a linear fracture energy-based criterion. Detailed FE models of steel beams tested by previous researchers are presented, and their predictions are shown to be in close agreement with the test results. Using the proposed FE approach, the behaviour of CFRP-strengthened steel beams is examined, indicating that: (1) if the failure is governed by plate end debonding, the use of a CFRP plate with a higher elastic modulus and/or a larger thickness may lead to a lower ultimate load because plate end debonding may then occur earlier; (2) plate end debonding is more likely to occur when a short CFRP plate is used, as is commonly expected; and (3) the failure mode may change to intermediate debonding or other failure modes such as compression flange buckling if a longer plate is used.