## 5 Conclusions and Remarks  
In this paper, the essences of structural robustness, the philosophy of robustness quantification as well as some robustness indices, i.e., the robustness quantification methodologies are given. Then a probabilistic framework for robustness quantification of precast concrete frames under seismic loading is developed. According to the findings above, the following conclusions can be drawn. The proposed probabilistic robustness quantification framework is demonstrated to be feasible and effective by employing an example illustration. It is believed that the proposed framework in this work is theoretically rigorous, easy to use and generally applicable in most situations. It could be a good practical tool for those designers who need to evaluate structural safety from the viewpoint of robustness. The proposed framework shows a good balance between simplicity and accuracy and therefore indicates its limitations of failing to fully account for the nonlinear effects in concrete structures and the coupling effect of axial force, bending moment and shear force. The illustration example demonstrates that the framework provides clear and useful information about the procedures to quantify structural robustness under a column removal scenario for precast concrete frame structures. In the example, flexural failure is assumed for beams and the failures due to axial force, bending moment and shear force are under consideration for column components. To quantify structural robustness in a probabilistic manner, the failure probability of the considered precast frame structure in the intact and the damaged states needs to be properly assessed, and the beta-unzipping method at level 1 used in this work proved to be effective. The redundancy (or vulnerability) based robustness indices (\(RI\), \(\beta_{R}\) and \(V\)) can be used to quantify the robustness of precast frames as they have the advantage of quantifying the difference between the intact and the damaged states of the structure. However, the interpretation of the results of these robustness indices should be careful because the judgement system of robustness is still not fully developed and deserves further attention. For simplicity purposes, the strength-based limit state function is considered in the formulation of the framework. This treatment applies to both the precast and cast-in-place conventional concrete structures since the strength performances of the precast concrete components are believed to be comparable to that of the conventional concrete structures. In case the limit state function is based on other indicators, e.g., displacement or energy dissipation, the differences between precast concrete structures and cast-in-place concrete structures should be taken into consideration. The analyses with respect to robustness quantification in this work are related to a specific damage situation (i.e., a central column removal scenario) in a simple frame with specific configurations. The associated results might only hold within the example and should be treated with care. In the given example, only the regular flexural capacity in beam components is taken into account and other favourable effects due to, e.g., the membrane actions in slabs and beams [42], as well as the hardening effect in steel reinforcements are not considered. The influences due to these effects on robustness quantification are of interest and deserve more investigation in the future. Y.Z. : Conceptualization, Methodology, Supervision, Writing--Original Draft, Review and Editing; Y.H. : Data Curation, Writing--Original Draft; Z.S. : Writing--Review and Supervision. All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science Foundation Program of Jiangsu Province (BK20180393) and the National Natural Science Foundation of China (Grant No. 52008089 and 52178120). Not applicable.