International Journal of Rock Mechanics and Mining SciencesAnalysis of geo-structural defects in flexural toppling failureAbbas Majdiand Mehdi AminiAbstractThe in-situ rock structural weaknesses, referred to herein as geo-structural defects, such as naturally induced micro-cracks, 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 cantilever-like rock columns. Hence, geo-structural 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 computed 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; otherwise, the result obtained based on the principles of fracture mechanics is more acceptable. Subsequently, for stabilization of the prescribed rock slopes, some new analytical relationships are suggested for determination the length and diameter of the required fully grouted rock bolts. Finally, for quick design of rock slopes against flexural toppling failure, a graphical approach along with some design curves are presented by which an admissible inclination of such rock slopes and or length of all required fully grouted rock bolts are determined. In addition, a case study has been used for practical verification of the proposed approaches.KeywordsGeo-structural defects, In-situ rock structural weaknesses, Critical crack length1. IntroductionRock masses are natural materials formed in the course of millions of years. Since during their formation and afterwards, they have been subjected to high variable pressures both vertically and horizontally, usually, they are not continuous, and contain numerous cracks and fractures. The exerted pressures, sometimes, produce joint sets. Since these pressures sometimes may not be sufficiently high to create separate joint sets in rock masses, they can produce micro joints and micro-cracks. However, the results cannot be considered as independent joint sets. Although the effects of these micro-cracks are not that pronounced compared with large size joint sets, yet they may cause a drastic change of in-situ geomechanical properties of rock masses. Also, in many instances, due to dissolution of in-situ rock masses, minute bubble-like cavities, etc., are produced, which cause a severe reduction of in-situ tensile strength. Therefore, one should not replace this in-situ strength by that obtained in the laboratory. On the other hand, measuring the in-situ rock tensile strength due to the interaction of complex parameters is impractical. Hence, an appropriate approach for estimation of the tensile strength should be sought. In this paper, by means of principles of solid and fracture mechanics, a new approach for determination of the effect of geo-structural defects on flexural toppling failure is proposed. 2.Effect of geo-structural defects on flexural toppling failure2.1. Critical section of the flexural toppling failureAs mentioned earlier, Majdi and Amini 10 and Amini et al. 11 have proved that the accurate factor of safety is equal to that calculated for a series of inclined rock columns, which, by analogy, is equivalent to the superimposed inclined cantilever beams as shown in Fig. 3. According to the equations of limit equilibrium, the moment M and the shearing force V existing in various cross-sectional areas in the beams can be calculated as follows: (5) ( 6)Since the superimposed inclined rock columns are subjected to uniformly distributed loads caused by their own weight, hence, the maximum shearing force and moment exist at the very fixed end, that is, at x=: (7) (8)If the magnitude of from Eq. (1) is substituted into Eqs. (7) and (8), then the magnitudes of shearing force and the maximum moment of equivalent beam for rock slopes are computed as follows: (9) (10)where C is a dimensionless geometrical parameter that is related to the inclinations of the rock slope, the total failure plane and the dip of the rock discontinuities that exist in rock masses, and can be determined by means of curves shown in Fig. Mmax and Vmax will produce the normal (tensile and compressive) and the shear stresses in critical cross-sectional area, respectively. However, the combined effect of them will cause rock columns to fail. It is well understood that the rocks are very susceptible to tensile