## 4 Conclusions  
The stand-up time of the rock mass around an unsupported tunnel plays a crucial role in underground construction in general, and tunnelling in particular. The estimation of the values of stand-up time significantly affects the decision-making procedures of choosing effective excavation methods as well as the supporting systems. In the present paper, the authors conducted experimental and theoretical studies on the effect of stand-up time for rock masses around the tunnel on an unsupported span, based on mechanical parameters of rock such as the strain modulus, and the geological strength index (GSI). The analyses took into account values of allowable displacement, as well as hereditary creep properties. Using the two approaches of analysing the results for scrutinising the overall time-dependent vertical displacements of circular tunnel walls under the influence of hydrostatic in situ stress using empirical and FEM methods, it was revealed that the effect of the GSI on stand-up time calculation is reliable and allows optimisation of supporting structure design in tunnelling, to avoid potential risks and hazards during tunnel excavation works. The research on how the permissible displacement influences the stand-up time of argillite stone, utilising a specified set of input data validates the influence of permissible displacement of tunnel walls. Consequently, an increase in the allowable displacements of tunnel walls leads to an enhancement in the stand-up time of an unsupported tunnel span. Clearly, the permissible displacement values are significantly influenced by the types of rock materials surrounding the tunnel. As a result, the authors highlight the matching relationships established using [PERSON]'s methods for rock mass; [PERSON]'s modulus; GSI; tunnel depth; and values of allowable displacement to stand-up time of an unsupported tunnel. The results also confirm the findings obtained using [PERSON]'s method, as well as enhance the study results obtained by [3], by taking into consideration time factors in the calculation of rock mass stand-up time. A noteworthy advancement in the present study is the incorporation of hereditary creep properties into the calculations. This temporal consideration adds a dynamic dimension to the estimation of stand-up time, addressing a crucial aspect that has usually been  
Figure 11: Relationship of unsupported tunnel span with stand-up time corresponding to various values of allowable displacement of tunnel. neglected in previous research. By acknowledging the time factor, the authors bring a more realistic perspective to the assessment of rock mass stability, thereby contributing to the refinement of existing models. In conclusion, the authors' work not only reaffirms established correlations, but also pioneers the integration of time-dependent factors, providing a more accurate understanding of the stand-up time for rock masses around unsupported tunnels. The present research contributes significantly to the optimisation of underground construction practices, offering a valuable resource for engineers and practitioners in the scientific field of underground construction. Conceptualisation, P.D.N., H.H.N. and M.V.N. ; methodology, P.D.N. and H.H.N. ; software, P.D.N., H.H.N. and H.H.D. ; validation, M.V.N. and H.H.D. ; formal analysis, E.K., P.D.N., P.O. and H.H.N. ; investigation, H.H.D. and H.H.N. ; resources, P.D.N. ; data curation, H.H.D. and H.H.N. ; writing--original draft preparation, P.D.N., P.O. and E.K. ; writing--PO. and E.K. ; visualisation, H.H.N., P.O. and E.K. ; supervision, M.V.N., P.D.N. and E.K. ; project administration, P.D.N. and H.H.N. ; funding acquisition, P.D.N., E.K. and P.O. All authors have read and agreed to the published version of the manuscript. This research was funded by the Vietnamese Ministry of Education and Training under Grant No. B2022-MDA-07. Not applicable. Data Availability StatementData are contained within the article.