Slope Stability: The Janbu Method

Description of Method

Janbu’s Method was developed by the Norwegian Professor N. Janbu. The method has similar features with the Bishop Method of Slices regarding the assumptions made on the inter-slice forces. However, a major difference between the two is that Janbu’s Method satisfies force equilibrium as opposed to Bishop’s Method that satisfies moment equilibrium. Moreover, Janbu’s Method can be used for both circular and non-circular failure surfaces. Non-circular failure surfaces are more common in nature (due to the existence of soil layers with different properties or due to geometrical restrictions). Examples of such surfaces are illustrated in Figure 1a and 1b.

The assumptions made by the Janbu’s method are that the interslice forces acting at each slice are equal and can be neglected (Figure 2).

Janbu’s method satisfies horizontal force equilibrium for the entire failure mass, as well as vertical and horizontal equilibrium for each slice. In addition, a correction factor f₀ is incorporated in the analysis and will be further discussed in the Calculations section below.

Moreover, the Mohr-Coulomb failure criterion is utilized to derive the shear strength of the ground, hence, the ultimate shear force that can be developed in a single slice i is:

Where c and φ refer to the cohesion and the friction angle of the ground, respectively, L is the length of the slice’s base, and N’ is the effective normal force acting at the base of the slice.

Calculations

Firstly, the horizontal force equilibrium on each slice is established. The horizontal components of the shear resistance and the reaction of the ground below the failure surface are taken into consideration as:

Subsequently, the correspondent vertical force equilibrium for each slice yields:

Combining equations [2] and [3], we get:

Finally, the overall horizontal equilibrium yields:

Given that:

and considering equations [4] and [5] it is deduced that:

Where B is the width of each slice, B_i=L_i*cos(α_i).

Using the formula of equation [6], the FoS cannot be calculated as the vertical interslice forces V are unknown. To address this issue, Janbu incorporated the aforementioned correction factor f₀ which substitutes the difference (V_i-1 - V_i+1). The final form of equation [6] is:

The correction factor f₀ depends on the strength parameters of the ground and on the geometry of the slope. In particular, the segment that connects the toe and the crown of a potential failure surface is derived and its length is measured (L). Subsequently, the maximum vertical distance between this segment and the failure surface is also determined (d). Finally, the ratio d/L is calculated and the correction factor f₀ is derived via Figure 3. As shown in Figure 3, there are 3 curves that can be used depending on the shear strength parameters of the soil. The “c-only” curve is utilized when total stress analyses are performed hence, the shear strength of the ground is fully represented by the undrained shear strength component (c=c_u, φ=0). The “φ and c soils” and “φ-only soil” curves are utilized once effective stress analyses are conducted and hence, the shear strength of the ground is given via the Mohr-Coulomb criterion:

The correction factor can also be derived via the following equation:

Where b₁ depends on the shear strength assumptions made for the soil mass:

• C-only soils: b₁=0.69
• Φ and C soils: b₁=0.50
• Φ-only soils: b₁=0.31

The correction factor f₀ is greater that one, hence, it may increase the FoS by up to 5% and 12% for granular soils without cohesion and for clayey soils in which a total stress analysis is conducted, respectively. The f₀ was later added by Janbu to compensate for the fact that the assumed interslice shear forces are negligible (an assumption made in a simplified method). Hence, the current method is also known as the Janbu’s Corrected Method.

Unlike Bishop’s, Janbu’s Method does not need an iterative procedure to derive the FoS and hence, it can be conducted via hand calculations without requiring a computational solution. It is a method sufficiently accurate for engineering design projects which prevails over Bishop method when it comes to non-circular surfaces.

References

Bouckovalas (2006). Computational method notes in geotechnical engineering. National Technical University of Athens. School of Civil Engineering. Geotechnical Department.

Rabie M. (2014).  Comparison study between traditional and finite element methods for slopes under heavy rainfall, HBRC Journal, Volume 10, Issue 2, 2014, Pages 160-168, ISSN 1687-4048, doi.org/10.1016/j.hbrcj.2013.10.002.

Samtani, N.C, Nowatzki, E.A. (2006). Soils and Foundations Reference Manual Volume 1. U.S. Department of Transportation, Federal Highway Administration, Washington, D.C. 20590.