This paper focuses mainly on the development of composite sub-step explicit algorithms for solving nonlinear dynamic problems. The proposed explicit algorithms are required to achieve the truly self-starting property, so avoiding computing the initial acceleration vector, and the controllable numerical dissipation at the bifurcation point, so eliminating spurious high-frequency components. With these two requirements, the single and two sub-step explicit algorithms with truly self-starting property and dissipation control are developed and analyzed. The present single sub-step algorithm shares the same spectral accuracy as the known Tchamwa–Wielgosz scheme, but the former possesses some advantages for solving wave propagation problems. The present two sub-step algorithm provides a larger stability limit, twice than those of single step schemes, due to explicit solutions of linear systems twice within each time increment. Numerical examples are also simulated to show numerical performance and superiority of two novel explicit methods over other explicit schemes.