[1]School of Civil Engineering, Central South University, Changsha 410075, China;
[1]School of Civil Engineering, Central South University, Changsha 410075, China;
[1]School of Civil Engineering, Central South University, Changsha 410075, China;
[1]School of Civil Engineering, Central South University, Changsha 410075, China;
karst; functional; catastrophe; theory; Hoek-Brown; criterion; safety; thickness; tunnel; floor; upper; bound;