Reliability-oriented sensitivity analysis methods have been developed for understanding the influence of model inputs relatively to events characterizing the failure of a system (e.g., a threshold exceedance of the model output). In this field, the target sensitivity analysis focuses primarily on capturing the influence of the inputs on the occurrence of such a critical event. This paper proposes new target sensitivity indices, based on the Shapley values and called "target Shapley effects", allowing for interpretable influence measures of each input in the case of dependence between the inputs. Two algorithms (a Monte Carlo sampling one, and a given-data algorithm) are proposed for the estimation of these target Shapley effects based on the $\ell^2$ norm. Additionally, the behavior of these target Shapley effects are theoretically and empirically studied through various toy-cases. Finally, applications on two realistic use-cases (a river flood model and a COVID-19 epidemiological model) are discussed.