Spencer

The Spencer method 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. 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 – Spencer 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 δ
Fxi Fyi	-	other horizontal and vertical forces acting on block
M1i	-	moment of 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 Spencer 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
-	inclination of forces Ei acting between blocks is constant for all blocks and equals to δ, only at slip surface end points is δ=0

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:

This formula allows us to calculate for a given value of δ all arms z of forces acting between blocks, knowing the value on the left at the slip surface origin, where z₁=0=0. 

Additional recursive formula follows from the moment equation of equilibrium (5) as:

This formula allows us calculating for a given value of δ all arms z of forces acting between blocks, knowing the value on the left at the slip surface origin, where z₁=0. 

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

1.	The initial value of δ is set to zero δ = 0.
2.	The factor of safety FS for a given value of δ follows from Equation (6), while assuming the value of En+1 = 0 at the end of the slip surface.
3.	The value of δ is provided by equation (7) using the values of E determined in the previous step with the requirement of having the moment on the last block equal to zero. Equation (7) does not provide the value of zn+1 as it is equal to zero. For this value the moment equation of equilibrium (5) must be satisfied.
4.	Steps 2 and 3 are then repeated until the value of δ does not change.

For the process of iteration to be stable it is necessary to avoid unstable solutions. Such instabilities occur at points where division by zero in expressions (6) and (7) takes place. In equation (7), division by zero is encountered for δ = π/2 or δ = -π/2. Therefore, the value of angle δ must be found in the interval (-π/2;π/2).

Division by zero in expression (6) appears when:

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:

Spencer, E. 1967. A method of analysis of the stability of embankments assuming parallel interslice forces. Géotechnique, 17(1): 11–26.