IJRRAS 7 (1) ● April 2011 www.arpapress.com/Volumes/Vol7Issue1/IJRRAS_7_1_11.pdf 70 EXPLICIT EQUATION FOR SAFETY FACTOR OF SIMPLE SLOPES Said M. Easa 1 & Ali R. Vatankhah 2 1 Professor, Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada M5B 2K3. 2 Assistant Professor, Department of Irrigation and Reclamation Engineering, University College of Agriculture and Natural Resources, University of Tehran, P. O. Box 4111, Karaj, Iran. 1 E-mail: seasa@ryerson.ca ABSTRACT The existing methods for determining the safety factor of simple homogeneous slopes are graphical in nature and may require iterations. This technical note presents a simple explicit equation for determining the safety factor for such conditions. A polynomial surface of the stability number was established based on the Taylor’s chart. A dimensionless parameter and a trigonometric series approximation were then used along with the stability number surface to establish the explicit equation. The proposed equation is applicable to the case of homogeneous slopes without seepage as well as the special cases involving complete submergence, complete sudden drawdown, steady seepage, and zero boundary neutral force. Validation of the proposed equation was performed by comparing its results with those of the existing graphical and analytical methods. The results showed that the proposed equation was very accurate. As such, the proposed equation should be useful in many geotechnical applications, especially those that implement safety factor as part of a larger modeling system. Keywords: Explicit equation, Safety factor, Simple homogeneous slopes, Regression, Validation, Seepage. 1. INTRODUCTION The analysis of slope stability is frequently encountered in practical applications, see for example Smith [1], Keskin [2], Greenwoo [3], and Renaud et al. [4]. The computation of the safety factor for analyzing the stability of simple, homogeneous slopes with no tension cracks can be easily performed using stability charts developed by Taylor [5]. Taylor’s charts have a theoretical basis and represent a classical method that is still being used in practical applications. The method, which is based on the friction-circle method, has been described in most geotechnical engineering textbooks, such as Murthy [6], Shroff and Shah [7], and Smith [1]. The basic method assumes a finite slope without seepage, but it may also be used to provide rough determinations and preliminary solutions for more complex cases of Taylor [8]. Taylor presented two charts, one for  = 0 and limited depth that can be used to determine the safety factor directly. For the case where the internal friction angle  > 0, the chart requires a trial procedure to determine the safety factor for a given slope. The reason for this is that the chart is developed in terms of the mobilized friction angle, which is a function of the safety factor. The iterative procedure was later eliminated by Baker [9] who developed a similar chart using a dimensionless parameter. Both procedures, however, relies on charts. This technical note eliminates this graphical procedure by developing an explicit equation for directly calculating the safety factor. Before presenting the development of this equation, it is useful to describe the existing Taylor-based graphical methods. 2. EXISTING GRAPHICAL METHODS Existing graphical methods (trial or direct) are based on Taylor’s stability chart. The chart presents the stability number SN as a function of the slope inclination angle ȕ and the mobilized friction angle  m. . The chart for the case  > 0, which requires a trial procedure, is shown in Fig. 1. As noted from the figure, as the slope inclination angle increases the value of SN increases. On the other hand for a given ȕ when the mobilized friction angle increases the stability number decreases. Taylor’s chart is a general solution of slope stability, based on the friction-circle method which is solved using a numerical trial procedure.