Janbu

Janbu is a general method of slices developed on the basis of limit equilibrium. It requires satisfying equilibrium of forces and moments acting on individual blocks (only moment equilibrium at last uppermost block is not satisfied). The blocks are created by dividing the soil above the slip surface by dividing planes. Forces acting on individual blocks are displayed in the following figure:

Static scheme – Janbu method

Each block is assumed to contribute due to the following forces:

Wi	-	block weight, including material surcharge having the character of weight including the influence of the coefficient of vertical earthquake Kv
Kh.Wi	-	horizontal inertia force representing the effect of earthquake, Kh is the factor of horizontal acceleration during earthquake
Ni	-	normal force on the slip surface
Ti	-	shear force on the slip surface
Ei ,Ei+1	-	forces exerted by neighboring blocks, they are inclined from horizontal plane by angle δi resp. δi+1 and lie at the height zi resp. zi+1 above slip surface
Fxi Fyi	-	other horizontal and vertical forces acting on block
M1i	-	moment from forces Fxi Fyi rotating about point M, which is the center of the i–th segment of slip surface
Ui	-	pore pressure resultant on the i–th segment of slip surface

The following assumptions are introduced in the Janbu method to calculate the limit equilibrium of forces and moment on individual blocks:

-	dividing planes between blocks are always vertical
-	the line of action of weight of block Wi passes through the center of the i–th segment of slip surface represented by point M
-	the normal force Ni is acting in the center of the i–th segment of slip surface, at point M
-	position zi of forces Ei acting between blocks is assumed, at slip surface end points is z=0

Choice of position z_i can have significant infuence on convergency of method. If we make a bad assumption of position z_i for a given slope, it can become impossible to satisfy equilibrium conditions (algorithm does not converge). Heights z_i above slip surface are set approximately to one third of height of interface between the blocks. In case of unsatisfying equilibrium conditions algorithm changes heights to a different position, e.g. slightly higher within passive zone, near the toe, and lower within active zone, near the crest of slope.   

The solution adopts the following expressions:

	(1)
	(2)
	(3)
	(4)
	(5)

Equation (1) represents the relationship between effective and total value of the normal force acting on the slip surface. Equation (2) corresponds to the Mohr-Coulomb condition representing the relation between the normal and shear forces on a given segment of the slip surface. Equation (3) represents the force equation of equilibrium in the direction normal to the i–th segment of the slip surface, whereas Equation (4) represents equilibrium along the i–th segment of the slip surface. FS is the factor of safety, which is used to reduce the soil parameters. Equation (5) corresponds to the moment equation of equilibrium about point M, where y_gi is the vertical coordinate of the point of application of the weight of block and y_M is the vertical coordinate of point M.

Modifying equations (3) and (4) provides the following recursive formula (6):

(6)

This formula allows calculating all forces E_i acting between blocks for given values of δ and FS. This solution assumes that at the slip surface origin the value of E is known and equal to E₁=0.

Formula for calculating angles δ_i (7) follows from the moment equation of equilibrium (5) as:

(7)

This formula allows calculating angles δ_i for a given values of z_i, knowing the values at slip surface end points δ=0. 

The factor of safety FS is determined by employing the following iteration process:

1.	The initial value of angles are set to zero δi = 0 and positions zi to approximately one third of interface height.
2.	The factor of safety FS for a given value of δi follows from equation (6), while assuming the value of En+1 = 0 at the end of the slip surface.
3.	The value of δi is provided by equation (7) using the values of Ei determined in the previous step.
4.	Steps 2 and 3 are then repeated until the value of FS does not change.

It is necessary to avoid unstable solutions for successful iteration process. Such instabilities occur at points where division by zero in expression (6) takes place, i.e.: 

Another check preventing numerical unstability is verification of parameter m_α  - following condition must be satisfied:

Therefore before iteration run it is required to find the highest of critical values (FS_min) satisfying above mentioned conditions. Values below this critical value FS_min are in area of unstable solution, therefore iteration begins by setting FS to a value "just" above FS_min and all result values of FS from iteration runs are higher than FS_min.

Generally rigorous methods converge worse than the simpler methods (Bishop, Fellenius). Examples with convergence problems include too steep sections of slip surface, complex geometry, a significant jump in surcharge etc. If no result is obtained, we recommend slight change of inputted data, e.g. less steep slip surface, input more points into the slip surface etc. or using of some of the simpler methods.

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Literature:

Janbu, N. 1954. Application of Composite Slip Surface for Stability Analysis. European Conference on Stability Analysis, Stockholm, Sweden.

Janbu, N. 1973. Slope Stability Computations. Embankment Dam Engineering - Casagrande Volume, R.C. Hirschfeld and S.J. Poulos, eds., John Wiley and Sons, New York, pp 47-86.