Effect of Structural Defects on the Analysis of Flexural Toppling Failure of Rock Slopes (Based on Fracture Mechanics)

Flexural toppling failure occurs due to tensile stress caused by in-situ rock column moments. Observations and theoretical analyses carried out by researchers show that the total failure plane is perpendicular to the rock mass discontinuity plane. In this paper, to analyze the flexural toppling failure, each rock column is modeled as a cantilever beam. Then using the laws of the strength of materials, along with the limit of equilibrium, the safety factor is found. Although the result of this analysis is comparable with those of laboratory tests, their use in real slopes shows a safety factor more than what it has to be. This is due to stress concentration around and near the tips of structural defects in the rock mass. Calculation of the amount of stress concentration around structural defects, based on the laws of the strength of materials, is cumbersome and has not been observed in the analysis of flexural toppling failure. In this paper, for the first time, structural defects of in-situ rock columns, with a potential of flexural toppling failure, enter the analysis. In nature, structural defects in rock masses appear haphazardly and unevenly in different locations. However, due to brittleness of the rocks, the rock defects generally appear in the form of ended cracks. Keeping this in mind, to analyze rock columns with structural defects, we considered a single ended crack perpendicular to the column length resulting total failure. Hence, in this case, based on the theory of fracture mechanics, each rock column, with respect to the length of the crack, acts like a beam with two infinite ends. With the above presumptions and employing the equations of limit of equilibrium, we can find the forces and moments acting on the section involving the crack are found as follows: (1) (2) (3) Where:: Rock column weight.: The angle between the rock slope and the horizontal plane: Rock inter-column coefficient of friction.: Pore water pressure at the base of the rock column.: Rock column width. : Side length of the rock column.: Water pressure force at each side of the rock column.: Rock inter-column normal forces.: Rock inter-column shear forces (equal to respectively).: Distance between force points and the support.: Distance between force points and the support.Bending moment, compressive and shear forces cause, respectively, tensile, compressive, and shear stresses on the plane involving the crack, and their combined action produces the stress intensity around the structural defect of in-situ rock columns. Using empirical TADA functions, the “Stress Intensity Factor”, related to the above stress field is defined as follows:(4-a) (4-b) (5-a) (5-b) (6-a) (6-b) Where;: Normal stress intensity factor caused by the bending moment.: Normal stress intensity factor caused by the compressive force.: Shear stress intensity factor caused by the shear force. : Length of the crack.Since the factor of safety against failure is:, (7)if equations 1 to 6 are substituted into equation7, then the rock inter-column normal force is computed as follows: (8) To analyze a given slope against flexural toppling failure by using equation 8, first we should draw the total failure plane. Then rock columns are numbered, from the toe of the slope to the last column that has a potential to failure. Next, we substitute and for and (the force points between two columns) respectively, along with a prescribed factor of safety. Other related parameters, including the slope geometry, geomechanical properties of the rock mass and the ground water level are also taken into account. Now we apply equation 8 for the last column (column n) to calculate by letting. Repeating the calculations, knowing for column, we can compute and, finally for the first column, we can determine the value of. Knowing we can decide about the flexural toppling failure with the following conditions:If, safety factor of the slope is equal to the allowable safety factor.If, safety factor of the slope is less than the allowable safety factor.If, safety factor of the slope is more than the allowable safety factor.In cases 2 and 3 is substituted with a new value (in case 2 less than the allowable safety factor and in case 3 more) and the calculations are repeated. This process continues until the difference between two given consecutive safety factors becomes less than the acceptable error. In this case, the last assumed safety factor is considered equal to the factor of safety of the rock slope stability against flexural toppling failure.