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• The new algorithm is second-order accurate, unconditionally stable (L-stable) and self-starting. • The new algorithm shares the identical effective stiffness matrices inside three sub-step. • There is no overshoot for the proposed algorithm when nonzero initial displacement and/or velocity are used. • The second-order accuracy is obtained in its final form, but it is not required in each sub-step. In this paper, a novel three sub-step composite algorithm with desired numerical properties is developed. The proposed method is a self-starting, unconditionally stable and second-order accurate implicit algorithm without overshoot. Particularly, the second-order accuracy in time is achieved in its final form, but it is not required in each sub-step. Its unique algorithmic parameter is analyzed to achieve the unconditional stability and it shares the identical effective stiffness matrix inside three sub-steps to save the computational cost in linear analyses. The same as the Bathe algorithm, the proposed algorithm is always L-stable, meaning that the spurious high-frequency modes can be effectively eliminated. Three numerical examples are simulated to illustrate the superiority of the proposed algorithm over some existing implicit algorithms. The first numerical simulation, solving a linear single-degree-of-freedom system, shows less period elongation errors and the second-order accuracy of the present scheme. The second one, a clamped-free bar excited by the end load, shows the ability of effectively damping out the unexpected high-frequency modes. The last example solves the nonlinear mass-spring system with variable degree-of-freedoms and illustrates that the composite sub-step algorithm can save more computational cost than the traditional implicit algorithm when the integration step size is selected appropriately. [ABSTRACT FROM AUTHOR]