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Error propagation in the response calculation of non-viscously damped systems by the decrease of time-step-size may lead to significant numerical difficulties in the analysis. In this paper, the authors investigate the time-step-size sensitivity of the proposed direct integration method to study the propagation of errors in the response calculation of dynamic systems. The state-space matrix of the asymmetric system by utilizing the anelastic displacement field model (ADF) is formulated. Then, an asymmetric formulation based direct integration method involving a piecewise linear interpolation is proposed. The effect of time-step-size sensitivity on the proposed method performance is studied in the benchmark example of a continuum system. It is shown that the linear displacement-velocity based integration method (DIMLDV) is highly sensitive as compared to the proposed method. The propagation of errors in the DIM-LDV method increases rapidly by the reduction of time-step-size, preventing in practice the use of time refinement as compared to the proposed method. The outcomes obtained from computational analysis reveal that the proposed method performance in terms of computational accuracy at two different step-sizes is better than the DIM-LDV method, making it suitable for analysis of complex linear dynamic systems.