All the design tools used to design against failure have some simplifications and assumptions. Model uncertainty, which is associated with imperfect assumption in the design tools, defines the deviation from the true failure surface. Quantification of model uncertainty is one of the critical issues in the design process which is easily neglected and it has great effect on the safety of project. One of the most common failures in underground openings is the block failure, and analytical solution based on joint relaxation is one design tool used to analyze block stability. The purpose of this paper is to show the reliability of the analytical solution. This has been done by comparing the result of it against the result of DEM in different stress conditions. The results show that the analytical solution is an accurate solution when the vertical stress is negligible, while for the cases with higher value of vertical in situ stress or in the cases where the tip of wedge is located out of distressed zone, the model uncertainty increases. Therefore the vertical stress is one important factor in the safety prediction of roof rock blocks which neglecting it makes model uncertainty and unsafe design.


One of the most common failure modes of underground openings is block failure. Variety of different design tools such as analytical solution, kinematics analysis and numerical analysis are available for estimating block stability. Among them ‘Kinematic’ refers to the motion of bodies without reference to the forces that cause to move [1]. The problem with Kinematic analysis of wedge is the method ignores forces acting on joints [2] therefore design based on full wedge weight and frictional force would be conservative. The conservatism is more pronounced in cases where the block has low apical angle [3,4]; therefore the analytical solution based on joint relaxation [5] was developed in order to consider the influences of horizontal stress field and joint stiffness on the stability of finite and removable blocks (according to Goodman and Shi definition, [6]). Crawford and Bray [5] proposed a two dimensional plain strain analytical solution for stability analysis of blocks in the roof for both symmetric and asymmetric wedges. The solution is based on joint relaxation to calculate joint deformation based on horizontal clamping forces and joint stiffness. However the solution is simplified based on some assumptions. The designer needs to know how good is the analytical design tool with the simplified assumptions and how much is the model uncertainty. The model uncertainty is associated with the imperfect representation of reality, and this is given by the simplification and idealization of joint relaxation. Model uncertainty is classified as epistemic uncertainty. The epistemic uncertainty is a knowledge based uncertainty and it could be reduced. In this paper the model uncertainty of Crawford-Bray solution is quantified.

Model Uncertainty

Generally all types of engineering works are subjected to the uncertainty. The word uncertain means feeling doubt about something (Longman dictionary). Baecher and Christian [7] dividedthe uncertainty in to three group.

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