This article introduces a simple and effective adaptive surrogate model to structural reliability analysis using deep neural network (DNN). In this paradigm, initial design of experiments (DoEs) are randomly selected from a given Monte Carlo Simulation (MCS) population to build the global approximate model of performance function (PF). More important points on the boundary of limit state function (LSF) and their vicinities are subsequently added relied on the surrogate model to enhance its accuracy without any complex techniques. A threshold is proposed to switch from a globally predicting model to a locally one for the approximation of LSF by eradicating previously used unimportant and noise points. Accordingly, the surrogate model becomes more precise for the MCS-based failure probability assessment with only a small number of experiments. Six numerical examples with highly nonlinear properties, various distributions of random variables and multiple failure modes, namely three benchmark ones regarding explicit mathematical PFs and the others relating to finite element method (FEM)-programmed truss structures under free vibration, are examined to validate the present approach. •A DNN-based adaptive surrogate model for structural reliability analysis is proposed.•The performance and limit state functions are evaluated by the surrogate model.•A threshold is suggested to switch from a globally predicting model to a locally one.•The paradigm estimates the failure probability with only a small number of samples.•Six examples are investigated to confirm the reliability of the current methodology.