Stability Analysis and the Stabilisation of Flexural Toppling Failure

Abstract

Flexural toppling is a mode of failure that may occur in a wide range of layered rock strata in both rock slopes and large underground excavations. Whenever rock mass is composed of a set of parallel discontinuities dipping steeply against the excavated face plane, the rock mass will have the potential of flexural toppling failure as well. In such cases, the rock mass behaves like inclined superimposed cantilever beams that bend under their own weight while transferring the load to the underlying strata. If the bending stress exceeds the rock column’s tensile strength, flexural toppling failure will be initiated. Since the rock columns are “statically indeterminate,” thus, their factors of safety may not be determined solely by equations of equilibrium. The paper describes an analytical model with a sequence of inclined superimposed cantilever rock columns with a potential of flexural topping failure. The model is based on the principle of compatibility equations and leads to a new method by which the magnitudes and points of application of intercolumn forces are determined. On the basis of the proposed model, a safety factor for each rock column can be computed independently. Hence, every rock column will have a unique factor of safety. The least factor of safety that exists in any rock column is selected as the rock mass representative safety factor based on which simple equations are proposed for a conservative rock mass stability analysis and design. As a result, some new relations are established in order to design the length, cross-sectional area and pattern of fully grouted rock bolts for the stabilisation of such rock mass. Finally, the newly proposed equations are compared with the results of existing experimental flexural toppling failure models (base friction and centrifuge tests) for further verification.

Appendix (with reference to Fig. 11)

$$ \overline{{BO}} = \frac{H} {{\sin \theta }} $$

(A1)

$$ \overline{{BM}} = \overline{{BO}} \cdot \cos {\left( {\theta - \delta + \varphi } \right)}\xrightarrow{{{\text{replace}}\;\overline{{BO}} \;{\text{from}}\;{\text{A1}}}}\overline{{BM}} = \frac{{\cos {\left( {\theta - \delta + \varphi } \right)}}} {{\sin \theta }} \cdot H $$

(A2)

$$ \overline{{AN}} = \overline{{BN}} \cdot \tan {\left( {\delta - \varphi } \right)}\xrightarrow{{\overline{{BN}} = \Phi }}\overline{{AN}} = \Phi \cdot \tan {\left( {\delta - \varphi } \right)} $$

(A3)

$$ \overline{{NC}} = \overline{{BN}} \cdot \tan {\left( {\theta - \delta + \varphi } \right)}\xrightarrow{{\overline{{BN}} = \Phi }}\overline{{NC}} = \Phi \cdot \tan {\left( {\theta - \delta + \varphi } \right)} $$

(A4)

$$ \xi = \overline{{AN}} + \overline{{NC}} = \Phi \cdot {\left[ {\tan {\left( {\delta - \varphi } \right)} + \tan {\left( {\theta - \delta + \varphi } \right)}} \right]} $$

(A5)

$$ \xi = \eta \cdot \tan {\left( {\theta - \delta + \varphi } \right)} $$

(A6)

$$ \eta = \Uppsi\cdot \cos \varphi $$

(A7)

$$ S_{{ABC}} = S_{{CDO}} \Rightarrow 0.5\xi \cdot \Phi = 0.5\varsigma \cdot \eta $$

(A8)

Substituting ξ, ζ and Ψ from Eqs. A5, A6 and A7, respectively, in Eq. A8 yields:

$$ \frac{{\tan {\left( {\delta - \varphi } \right)}}} {{\tan {\left( {\delta - \varphi } \right)} + \tan {\left( {\theta - \delta + \varphi } \right)}}} \cdot \eta ^{2} + \frac{{2\cos {\left( {\theta - \delta + \varphi } \right)}}} {{\sin \theta }} \cdot \eta + {\left[ {\frac{{\cos {\left( {\theta - \delta + \varphi } \right)}}} {{\sin \theta }} \cdot H} \right]}^{2} = 0 $$

(A9)

Substituting Eq. A7 in A9 results in:

$$ \frac{{\tan {\left( {\delta - \varphi } \right)} \cdot \cos ^{2} \gamma }} {{\tan {\left( {\delta - \varphi } \right)} + \tan {\left( {\theta - \delta + \varphi } \right)}}} \cdot \Uppsi^{2} + \frac{{2\cos {\left( {\theta - \delta + \varphi } \right)} \cdot \cos \varphi }} {{\sin \theta }} \cdot \Uppsi+ {\left[ {\frac{{\cos {\left( {\theta - \delta + \varphi } \right)}}} {{\sin \theta }} \cdot H} \right]}^{2} = 0 $$

(A10)

$$ A = \frac{{\tan {\left( {\delta - \varphi } \right)} \cdot \cos ^{2} \gamma }} {{\tan {\left( {\delta - \varphi } \right)} + \tan {\left( {\theta - \delta + \varphi } \right)}}}, \quad B = \frac{{2\cos {\left( {\theta - \delta + \varphi } \right)} \cdot \cos \varphi }} {{\sin \theta }} \cdot H, \quad C = {\left[ {\frac{{\cos {\left( {\theta - \delta + \varphi } \right)}}} {{\sin \theta }} \cdot H} \right]}^{2} $$

(A11)

$$ \Uppsi_{1} = \frac{{B - {\left( {B^{2} - 4AC} \right)}^{{0.5}} }} {{2A}}, \quad \Uppsi_{2} = \frac{{B + {\left( {B^{2} - 4AC} \right)}^{{0.5}} }} {{2A}} $$

(A12)

$$ \Uppsi _{2} \succ H, \quad \Uppsi = \Uppsi_{1} = \frac{{B - {\left( {B^{2} - 4AC} \right)}^{{0.5}} }} {{2A}} $$

(A13)