A simplified mathematical approach for the evaluation of the stabilizing forces applied by a passive cemented bolt to a sliding rock block

Passive bolting is used to stabilise unstable rock blocks in surface and underground structures due to the various advantages it offers. Despite its use, the design phase still presents aspects of considerable complexity because the fact that the load of the bolt and therefore, its static action depends on its interaction with the block and the stable rock. In the present work, a mathematical model was developed which is capable of directly calculating the stabilisation forces as a function of the characteristic parameters of the bolt and of its interaction with the rock. This discussion is based on a simplified hypothesis of bolt behaviour, which provides negligible errors, and on the observation that the critical point is positioned at the intersection of the bolt with one of the lateral surfaces that separate it from the portion of stable rock. The formulation of the stabilisation forces obtained made it possible to evaluate the static contribution of each single bolt to the stability of the rock block, by varying the diameter of the steel bar and then designing the bolting operation to achieve acceptable stability conditions for the rock block. The application of stabilising equations to a real case, for which the results of load tests on bolt tests were available, allowed us to outline steps to be taken in the bolt design process.