Civil Engineering April 2022 | Vol 30 No 3

Civil Engineering April 2022 19 L imit equilibrium methods for calculating the factor of safety (FoS) require assumptions to make formulations statically determinate. Janbu (1954) assumed interslice forces were horizontal and considered horizontal and vertical force equilibrium. Bishop (1955) similarly assumed horizontal interslice forces but considered both moment and vertical force equilibrium. Spencer (1967) and Morgenstern and Price (1965) assumed different force functions and iterated the ratio of interslice normal to shear forces to match moment and force equilibrium. Consequently, calculated FoS differ. METHODOLOGY While others elaborate on differences (Krahn, 2003; Duncan et al., 2014) a consideration in terms of soil strength is absent. To address this, a parametric study (see Figure 1) in which soil strength was varied (20° ≤ φ ≤ 45° and c = 0) was undertaken using Rocscience’s Slide2 using the Bishop simplified, Janbu simplified and Spencer methods. Janbu FoS in all cases were smallest, and Spencer FoS and Bishop FoS were the same (to two decimal places). Thus, only Janbu FoS and Spencer FoS are considered further. FINDINGS As FoS is a linear function of tanφ, dif- ferences can be visualised in terms of φ, by finding the difference Δφ, required to obtain Janbu FoS equal to Spencer FoS. However, as Δφ increases with φ, the differ- ence was determined at φ = 25° and 45°. Figure 2 shows that as failure becomes more rotational (i.e. steeper slope ge- ometries), differences increased. As the standard deviation of φ for soils is ±3° (Ching and Schweckendiek, 2021), we see that method has a small influence on non- vertical slopes (i.e. an accurate estimate of φ is more important than method). For vertical slopes, while an accurate estimate of φ is equally important, method will have a larger impact on FoS. This makes applying judgement more important for vertical slopes. This conclusion likely applies to all slope failures characterised by significant rotational motion.  REFERENCES Bishop, A.W. 1955. The use of the slip circle in the stability analysis of slopes. Géotechnique, 5, 7-17.doi.org/10.1680/ geot.1955.5.1.7. Ching, J. & Schweckendiek, T. 2021. State-of- the-art review of inherent variability and uncertainty in geotechnical properties and models. International Society of Soil Mechanics and Geotechnical Engineering (ISSMGE) – Technical Committee TC304 ‘Engineering Practice of Risk Assessment and Management. Duncan, J.M., Wright, S.G. & Brandon, T.L. 2014. Soil strength and slope stability, John Wiley & Sons. Janbu, N. Applications of composite slip surfaces for stability analysis. European Conference on the Stability of Earth Slopes, 1954 Stockholm. 39-43. Krahn, J. 2003. The 2001 RM Hardy Lecture: The limits of limit equilibrium analyses. Canadian Geotechnical Journal, 40, 643-660.10.1139/T03-024. Morgenstern, N.R. & Price, V.E. 1965. The analysis of the stability of general slip surfaces. Géotechnique, 15, 79-93.10.1680/ geot.1965.15.1.79. Spencer, E. 1967. A method of analysis of the stability of embankments assuming parallel inter-slice forces. Géotechnique, 17, 11-26.10.1680/geot.1967.17.1.11. Stellenbosch Uni vers i ty geotechni cal research abstracts Limit equilibrium: method or parameters? Billa Uys Junior Engineer Grobler Structural Design billauys2@gmail.com Dr Charles John MacRobert Pr Eng Senior Lecturer | Department of Civil Engineering Stellenbosch University macrobert@sun.ac.za Stellenbosch University has compiled a series of geotechnical research abstracts outlining research undertaken by the Department of Civil Engineering. The third abstract examines limit equilibrium methods for calculating factor of safety. Figure 1 Geometries analysed (H: 5 and 40 m, X: 1,2,3 and 4; Phreatic surface: Absent and present): a) Non-vertical slopes and b) Vertical slopes Figure 2 Differences in friction angle to obtain Janbu FoS equal to Spencer FoS