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International Journal of Rock Mechanics and Mining SciencesAnalysis of geo-structural defects in flexural toppling failureAbbas Majdi and 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.Keywords Geo-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 stresses, and the effect of maximum shearing force is also negligible compared with the effect of tensile stress. Thus, for the purpose of the ultimate stability, structural defects reduce the cross-sectional area of load bearing capacity of the rock columns and, consequently, increase the stress concentration in neighboring solid areas. Thus, the in-situ tensile strength of the rock columns, the shearing effect might be neglected and only the tensile stress caused due to maximum bending stress could be used.2.2. Analysis of geo-structural defectsDetermination of the quantitative effect of geo-structural defects in rock masses can be investigated on the basis of the following two approaches.2.2.1. Solid mechanics approachIn this method, which is, indeed, an old approach, the loads from the weak areas are removed and likewise will be transferred to the neighboring solid areas. Therefore, the solid areas of the rock columns, due to overloading and high stress concentration, will eventually encounter with the premature failure. In this paper, for analysis of the geo-structural defects in flexural toppling failure, a set of cracks in critical cross-sectional area has been modeled as shown in Fig. 5. By employing Eq. (9) and assuming that the loads from weak areas are transferred to the solid areas with higher load bearing capacity (Fig. 6), the maximum stresses could be computed by the following equation (see Appendix A for more details): （11）Hence, with regard to Eq. (11), for determination of the factor of safety against flexural toppling failure in open excavations and underground openings including geo-structural defects the following equation is suggested: （12）From Eq. (12) it can be inferred that the factor of safety against flexural toppling failure obtained on the basis of principles of solid mechanics is irrelevant to the length of geo-structural defects or the crack length, directly. However, it is related to the dimensionless parameter “joint persistence”, k, as it was defined earlier in this paper. Fig. 2 represents the effect of parameter k on the critical height of the rock slope. This figure also shows the limiting equilibrium of the rock mass (Fs=1) with a potential of flexural toppling failure.Fig. 2. Determination of the critical height of rock slopes with a potential of flexural toppling failure on the basis of principles of solid mechanics.View Within Article2.2.2. Fracture mechanics approachGriffith in 1924 13, by performing comprehensive laboratory tests on the glasses, concluded that fracture of brittle materials is due to high stress concentrations produced on the crack tips which causes the cracks to extend (Fig. 3). Williams in 1952 and 1957 and Irwin in 1957 had proposed some relations by which the stress around the single ended crack tips subjected to tensile loading at infinite is determined 14, 15 and 16. They introduced a new factor in their equations called the “stress intensity factor” which indicates the stress condition at the crack tips. Therefore if this factor could be determined quantitatively in laboratorial, then, the factor of safety corresponding to the failure criterion based on principles of fracture mechanics might be computed.Fig. 3. Stress concentration at the tip of a single ended crack under tensile loadingSimilarly, the geo-structural 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 geo-structural 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 “length 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. 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 geo-structural defects on flexural toppling failure on the basis of principles of solid mechanics and fracture mechanics accordingly. For the sake of comparison, these equations are applied to one kind of rock mass (limestone) with the following physical and mechanical properties 16: , , =20kN/m3, k=0.75.In any case studies, a safe and stable slope height can be determined by using Eqs. (12) and (13), independently. The two equations yield two different slope heights out of which the minimum height must be taken as the most acceptable one. By equating Eqs. (12) and (13), the following relation has been derived by which a crack length, in this paper called critical length of crack, can be computed: （14a）where ac is the half of the average critical length of the cracks. Since ac appears on both sides of Eq. (14a), the critical length of the crack could be computed by trial and error method. If the length of the crack is too small with respect to rock column thickness, then the ratio t/(t2ac) is slightly greater than one. Therefore one may ignore the length of crack in denominator, and then this ratio becomes 1. In this case Eq. (14a) reduces to the following equation, by which the critical length of the crack can be computed directly: (14b)It must be born in mind that Eq. (14b) leads to underestimate the critical length of the crack compared with Eq. (14a). Therefore, for an appropriate determination of the quantitative effect of geo-structural defects in rock mass against flexural toppling failure, the following 3 conditions must be considered: (1) a=0; (2) aac.In case 1, there are no geo-structural defects in rock columns and so Eq. (3) will be used for flexural toppling analysis. In case 2, the lengths of geo-structural 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 geo-structural 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 comparative studies the results of Eq. (3), which represents the rock columns with no geo-structural 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 in-depth comparison, 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 slopes with a potential of flexural toppling failureIn rock slopes and trenches, except for the soil and rock fills, the heights are dictated by the natural topography. Hence, the desired slopes must be designed safely. In rock masses with the potential of flexural toppling failure, with regard to the length of the cracks extant in rock columns the slopes can be computed by Eqs. (3), (12), and (13) proposed in this paper. These equations can easily be converted into a series of design curves for selection of the slopes to replace the lengthy manual computations as well. Fig. 12, Fig. 13, Fig. 14 and Fig. 15 show several such design curves with the potential of flexural topping failures. If the lengths of existing cracks in the rock columns are smaller than the critical length of the crack, one can use the design curves, obtained on the basis of principles of solid mechanics, shown in Fig. 12 and Fig. 13, for the rock slope design purpose. If the lengths of the cracks existing in rock columns are greater than the critical length of the crack, then the design curves derived based on principles of fracture mechanics and shown in Fig. 14 and Fig. 15 must be used for the slope design intention. In all, these design curves, with knowing the height of the rock slopes and the thickness of the rock columns, parameter (H2/t) is computed, and then from the design curves the stable slope is calculated. It must be born in mind that all the aforementioned design curves are valid for the equilibrium condition only, that is, when FS=1. Hence, the calculated slopes from the above design curves, for the final safe design purpose must be reduced based on the desired factor of safety. For example, if the information regarding to one particular rock slope are given 17: k=0.25, =10, t=10MPa, =20kN/m3, =45, H=100m, t=1m, aca=0.1m, and then according to Fig. 12 the design slope will be 63, which represents the condition of equilibrium only. Hence, the final and safe slope can be taken any values less than the above mentioned one, which is solely dependent on the desired factor of safety.Fig. 5. Selection of critical slopes for rock columns with the potential of flexural toppling failure on the basis of principles of solid mechanics when k=0.25.Fig. 6. Selection of critical slopes for rock columns with the potential of flexural toppling failure based on principles of solid mechanics when k=0.75.Fig. 7. Selection of critical slopes for rock columns with the potential of flexural toppling failure based on principles of fracture mechanics when k=0.25.Fig. 8. Selection of critical slopes for rock columns with the potential of flexural toppling failure based on principles of fracture mechanics when k=0.75.5. Stabilization of the rock mass with the potential of flexural toppling failureIn flexural toppling failure, rock columns slide over each other so that the tensile loading induced due to their self-weighting grounds causes the existing cracks to grow and thus failure occurs. Hence, if these slides, somehow, are prevented then the expected instability will be reduced significantly. Therefore, employing fully grouted rock bolts, as a useful tool, is great assistance in increasing the degree of stability of the rock columns as shown in Fig. 16 5 and 6. However, care must be taken into account that employing fully grouted rock bolts is not the only approach to stabilize the rock mass with potential of flexural toppling failure. Therefore, depending up on the case, combined methods such as decreasing the slope inclination, grouting, anchoring, retaining walls, etc., may even have more effective application than fully grouted rock bolts alone. In this paper a method has been presented to determine the specification of fully grouted rock bolts to stabilize such a rock mass. It is important to mention that Eqs. (15), (16), (17), (18), (19) and (20) proposed in this paper may also be used as guidelines to assist practitioners and engineers to define the specifications of the d