【正文】 sses are equal to that exist in the corresponding longitudinal section [18]. Hence, the total shear force S in the longitudinal section of the beam can be calculated as follows: The inserted shear force in the crosssectional area of the rock bolt is equal to the total force exerted longitudinally as well. Therefore, the shear force exerted to the rock bolt39。 Selection of critical slopes for rock columns with the potential of flexural toppling failure on the basis of principles of solid mechanics when k=.Fig. 6.m, t=1, σt=10 Determination of the critical height of rock slopes with a potential of flexural toppling failure on the basis of principle of fracture mechanics.View Within Article3. Comparison of the results of the two approachesThe curves shown in Fig. represent Eqs. (12) and (13), respectively. The figures reflect the quantitative effect of the geostructural defects on flexural toppling failure on the basis of principles of solid mechanics and fracture mechanics accordingly. For the sake of parison, these equations are applied to one kind of rock mass (limestone) with the following physical and mechanical properties [16]: , , γ=20International Journal of Rock Mechanics and Mining SciencesAnalysis of geostructural defects in flexural toppling failureAbbas Majdi and Mehdi AminiAbstractThe insitu rock structural weaknesses, referred to herein as geostructural defects, such as naturally induced microcracks, are extremely responsive to tensile stresses. Flexural toppling failure occurs by tensile stress caused by the moment due to the weight of the inclined superimposed cantileverlike rock columns. Hence, geostructural defects that may naturally exist in rock columns are modeled by a series of cracks in maximum tensile stress plane. The magnitude and location of the maximum tensile stress in rock columns with potential flexural toppling failure are determined. Then, the minimum factor of safety for rock columns are puted by means of principles of solid and fracture mechanics, independently. Next, a new equation is proposed to determine the length of critical crack in such rock columns. It has been shown that if the length of natural crack is smaller than the length of critical crack, then the result based on solid mechanics approach is more appropriate。 Stress concentration at the tip of a single ended crack under tensile loadingSimilarly, the geostructural defects exist in rock columns with a potential of flexural toppling failure could be modeled. As it was mentioned earlier in this paper, cracks could be modeled in a conservative approach such that the location of maximum tensile stress at presumed failure plane to be considered as the cracks locations (Fig. 3). If the existing geostructural defects in a rock mass, are modeled with a series cracks in the total failure plane, then by means of principles of fracture mechanics, an equation for determination of the factor of safety against flexural toppling failure could be proposed as follows: （13）where KIC is the critical stress intensity factor. Eq. (13) clarifies that the factor of safety against flexural toppling failure derived based on the method of fracture mechanics is directly related to both the “joint persistence” and the “l(fā)ength of cracks”. As such the length of cracks existing in the rock columns plays important roles in stress analysis. Fig. 10 shows the influence of the crack length on the critical height of rock slopes. This figure represents the limiting equilibrium of the rock mass with the potential of flexural toppling failure. As it can be seen, an increase of the crack length causes a decrease in the critical height of the rock slopes. In contrast to the principles of solid mechanics, Eq. (13) or Fig. 4 indicates either the onset of failure of the rock columns or the inception of fracture development.Fig. 4. (3) aac.In case 1, there are no geostructural defects in rock columns and so Eq. (3) will be used for flexural toppling analysis. In case 2, the lengths of geostructural defects are smaller than the critical length of the crack. In this case failure of rock column occurs due to tensile stresses for which Eq. (12), based on the principles of solid mechanics, should be used. In case 3, the lengths of existing geostructural defects are greater than the critical length. In this case failure will occur due to growing cracks for which Eq. (13), based on the principles of fracture mechanics, should be used for the analysis.The results of Eqs. (12) and (13) for the limiting equilibrium both are shown in Fig. 11. For the sake of more accurate parative studies the results of Eq. (3), which represents the rock columns with no geostructural defects are also shown in the same figure. As it was mentioned earlier in this paper, an increase of the crack length has no direct effect on Eq. (12), which was derived based on principles of solid mechanics, whereas according to the principles of fracture mechanics, it causes to reduce the value of factor of safety. Therefore, for more indepth parison, the results of Eq. (13), for different values of the crack length, are also shown in Fig. As can be seen from the figure, if the length of crack is less than the critical length (dotted curve shown in Fig. 11), failure is considered to follow the principles of solid mechanics which results the least slope height. However, if the length of crack increases beyond the critical length, the rock column fails due to high stress concentration at the crack tips according to the principles of fracture mechanics, which provides the least slope height. Hence, calculation of critical length of crack is of paramount importance.4. Estimation of stable rock sl