Since the step-by-step integration method is the most powerful technique for a nonlinear dynamic analysis and it involves much more computational efforts when compared to a nonlinear static analysis, it is important to improve the computational efficiency of the nonlinear dynamic analysis. A novel family of structure-dependent integration methods is proposed herein. It involves no nonlinear iterations, which is the most important advantage for explicit methods, and the unconditional stability, which is the most important advantage for implicit methods. In addition, it also has a second-order accuracy. Consequently, it has very high computational efficiency. In fact, numerical experiments reveal that for a 2000-DOF system, the CPU time involved by the proposed family method is only about 0.6% of that required by the constant average acceleration method. The computational efficiency can become more evident if the number of the degree of the freedom for the analyzed system increases.