This thesis presents a discussion of design tools for analysing block stability around a tunnel. First, it was determined that joint length and field stress have a significant influence on estimating block stability. The results of calculations using methods based on kinematic limit equilibrium (KLE) were compared with the results of filtered DFN-DEM, which are closer to reality. The comparison shows that none of the KLE approaches– conventional, limited joint length, limited joint length with stress and probabilistic KLE – could provide results similar to DFN-DEM. This is due to KLE’s unrealistic assumptions in estimating either volume or clamping forces. A simple mechanism for estimating clamping forces such as continuum mechanics or the solution proposed by Crawford-Bray leads to an overestimation of clamping forces, and thus unsafe design. The results of such approaches were compared to those of DEM, and it was determined that these simple mechanisms ignore a key stage of relaxation of clamping forces due to joint existence. The amount of relaxation is a function of many parameters, such as stiffness of the joint and surrounding rock, the joint friction angle and the block half-apical angle. Based on a conceptual model, the key stage was considered in a new analytical solution for symmetric blocks, and the amount of joint relaxation was quantified. The results of the new analytical solution compared to those of DEM and the model uncertainty of the new solution were quantified. Further numerical investigations based on local and regional stress models were performed to study initial clamping forces. Numerical analyses reveal that local stresses, which are a product of regional stress and joint stiffness, govern block stability. Models with a block assembly show that the clamping forces in a block assembly are equal to the clamping forces in a regional stress model. Therefore, considering a single block in massive rock results in lower clamping forces and thus safer design compared to a block assembly in the same condition of in-situ stress and properties. Furthermore, a sensitivity analysis was conducted to determine which is  the most important parameter by assessing sensitivity factors and studying the applicability of the partial coefficient method for designing block stability. It was determined that the governing parameter is the dispersion of the half-apical angle. For a dip angle with a high dispersion, partial factors become very large and the design value for clamping forces is close to zero. This suggests that in cases with a high dispersion of the half-apical angle, the clamping forces could be ignored in a stability analysis, unlike in cases with a lower dispersion. The costs of gathering more information about the joint dip angle could be compared to the costs of overdesign. The use of partial factors is uncertain, at least without dividing the problem into sub-classes. The application of partial factors is possible in some circumstances but not always, and a FORM analysis is preferable. / QC 20111201

Block stability is common and has to be studied in detail for designing tunnels. Stability of block depends upon the shape and size of the blocks, stresses around the block and factors such as clamping forces and the ratio between joint stiffness. These factors are studied in detail and are the main objective of this thesis. In this thesis influence of loading and unloading of blocks on joint stiffness and thus on ultimate pullout force are analyzed. Normal stress on the joint plane is linked with shear stiffness of the joint and relaxation of forces. Changes of forces were considered to estimate joint stiffness and ultimate pullout force using new methods in the present thesis. First method takes into account changing clamping forces considering stiffness ratio constant (Crawford and Bray). The second method was developed in which the ratio between normal and shear stiffness was taken as a function of normal stress (Bagheri and Stille). In third method, gradually pullout force is increased which changes the normal stress and joint stiffness. The lower limit of joint stiffness gives a very conservative design. So a stiffness value based on the average of lower and upper limit of normal force has also been considered. A comparison between the new methods and the previous method proposed by Crawford and Bray which considers a constant ratio of normal and shear stiffness and constant clamping forces shows that Crawford and Bray’s solution overestimates the pullout forces hence the design is unsafe. It was observed that stiffness ratio is an important factor for estimating required rock support and safety.