This paper introduces a novel approach for reliability assessment with dependent variables. In this work, the boundary of the failure domain, for a computational problem with expensive function evaluations, is approximated using a Support Vector Machine and an adaptive sampling scheme. The approximation is sequentially refined using a new adaptive sampling scheme referred to as generalized "max-min". This technique efficiently targets high probability density regions of the random space. This is achieved by modifying an adaptive sampling scheme originally tailored for deterministic spaces (Explicit Space Design Decomposition). In particular, the approach can handle any joint probability density function, even if the variables are dependent. In the latter case, the joint distribution might be obtained from copula. In addition, uncertainty on the probability of failure estimate are estimated using bootstrapping. A bootstrapped coefficient of variation of the probability of failure is used as an estimate of the true error to determine convergence. The proposed method is then applied to analytical examples and a beam bending reliability assessment using copulas.