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The Spencer Method of Slices was developed by Spencer in 1967. It is one of the most theoretically rigorous Methods of Slices since it satisfies both force and moment equilibrium of the failure mass. Thus, it enables more precise calculations of the Factor of Safety (** FoS**). Spencer’s method can be applied to both circular and non-circular failure surfaces.

As mentioned in previous chapters of the Slope Stability educational series, every method of slices requires certain simplifications to be made to address the fact that the number of unknown parameters is greater than that of the available equations. Even though other methods neglect interslice normal and/or shear forces, Spencer’s method takes into account the interslice forces by substituting them with an equivalent resultant force ** Q** which acts at the midpoint of a slice’s base

The assumptions made in Spencer’s method regarding the interslice forces establish a more complex system of equations that needs to be solved in order to derive the * FoS* of a failure surface.

Firstly, the two force equilibriums for each slice (parallel and perpendicular to the failure surface) yield the following equations:

* *θ is denoted without subscript (i) as it is constant for all slices*.

The utilization of the Mohr-Coulomb failure criterion regarding the shear strength of the ground, hence, yields:

where ** T_{ULT}** is the shear strength of the soil mass,

The shear force * T* acting at the base of each slice is:

By using equations [1] to [4], the equivalent resultant interslice force * Q* is derived:

where:

Subsequently, the equilibrium of the overall sliding mass is considered. If the overall moment with respect to the center of rotation is zero, then the moment of the internal interslice forces * Q* must also be zero, hence:

where * R_{i}* is the distance from the center of rotation to the midpoint of each slice. In a circular failure surface, all radii are equal and can be excluded from equation [9], but in a non-circular surface they must be taken into consideration.

The same principal applies for the force equilibrium. The sum of the internal forces of the entire sliding mass in the vertical and the horizontal direction must be zero. Hence, two more equations emerge:

However, due to the fact that the inclination of the equivalent interslice force * Q* is constant, equations [10] and [11] are mathematically identical and both yield:

Solving equations [9] and [12] results in two FoS values, for a given failure surface, assuming a value of interslice force inclination * θ*. However, there is a single value of

Many schemes have been developed in search for an optimized iterative procedure to calculate * θ* and

- Arbitrarily assume several values of
and calculate the corresponding**θ**values for Force and Moment equilibrium (*FoS***F**and_{F}**F**)_{M} - Draw the
**F**,_{F}**F**versus_{M}**θ**curves - Find the intersection point between the curves that corresponds to the actual
**FoS**and**θ**of the assessed surface

An example illustration of the aforementioned curves is shown in **Figure 2**.

Nonetheless, advanced techniques have been developed over the years which use computational means including efficient iterative processes. The Spencer Method is more time-consuming that other methods (Bishop or Janbu) but the rapid evolution of computing capacities has made this issue obsolete.

If the inclination ** θ** is set to zero, then Spencer’s method becomes identical to Bishop’s simplified method. Therefore, Spencer’s method is considered an extended, albeit more sophisticated, version of Bishop’s method.

Some findings that arise based on the results of the Spencer’s method are:

- Force equilibrium
(*FoS***F**) is affected in a more pronounced manner by_{F}compared to Moment equilibrium*θ*(*FoS***F**), a fact that can also be observed in_{M}**Figure 2** - The accuracy of the method depends on the number of slices selected to conduct the analysis. Nevertheless, increasing the number of slices results in greater computational times required. According to Spencer (1967), the gain in accuracy is minor once the number of slices used is greater than 32. In fact, even a smaller number of slices are adequate to provide an accurate FoS, with each case study being unique.
- Spencer method may not always converge to a single pair of
and**FoS**. Convergence problems emerge when the failure surface presents steep sections, complex geometry or jumps. In that case, a slight change in the input data or the utilization of a simpler method is recommended.**θ**

Agam M.W., Hashim M.H.M., Murad M.I., Zabidi H. (2016). Slope Sensitivity Analysis Using Spencer's Method in Comparison with General Limit Equilibrium Method, Procedia Chemistry, Volume 19, Pages 651-658

Carpenter J.R. (1985). STABL5. The Spencer Method of Slices: Final Report. Joint Highway Research Project Engineering Experiment Station, Purdue University in cooperation with the Indiana Department of Highways, West Lafayette, Indiana.

Spencer E (1967). A Method of Analysis of The Stability of Embankments Assuming Parallel Interslice Forces. Géotechnique; 17: 11-26.

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