Spencer

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Introduction
Theoretical_Background
   Method of Slices
   Bishop
   Spencer
   Morgenstern & Price
Use of Limit Equilibrium
   Choice of Geometry
   Strengths & Pressures
   Long Term Problems
   Short Term Problems
   Summary
References
Input Instructions
   Geometry
   Boundary Forces
   Pore Pressures
   Strength Parameters
   Earthquake Loading
   Iteration Control
   M-P Method
   Convergence
Examples
   Example 1
   Example 2
   Example 3i
   Example 3ii
   Example 4

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Spencer's Method Spencer's Method assumes that the inter-slice forces are parallel (Spencer, 1967). A typical slice and the corresponding force polygon are shown in Figure 4. The forces on the slice are:

the slice weight, W
the pseudo-static seismic force, kW, in which k is the seismic coefficient
the pore pressure force, U ( = u . l )
the effective normal force on base, P'
the mobilized shear force,
the resultant of boundary forces perpendicular and parallel to the top of the slice, N and M respectively
the resultant of the parallel side forces, Q

Figure 4 Force polygon for Spencer's Method.

The pseudo-static seismic force and the resultants of boundary pressures will not be included in the following discussion; these forces are of known magnitude and direction and therefore do not contribute to the understanding of the theory but do tend to lengthen the equations.

Summation of forces, normal and tangent to the base of each slice, provides two equations of force equilibrium and two unknowns, P' and Q:

(8)

(9)

Solving Equation 8 for P' and substituting the expression into Equation 9 and then solving Equation 9 for Q yields:

(10)

If the external forces on the slope are in equilibrium, the vectorial sum of the inter-slice forces must be zero to assure overall force equilibrium. Since the inter-slice forces are all parallel, this requirement reduces to:

(11)

Furthermore, the normal force and the weight of each slice are assumed to be coincidental at a point on the slip surface with the same X-coordinate as the slice's center of gravity. For each slice to be in moment equilibrium the resultant Q of the inter-slice forces must be concurrent with the remaining forces acting on the slice. In other words, Q must act through the point on the base of each slice where the normal and weight forces act, with sufficient modifications made to account for top-of-slice boundary forces or pseudo-static seismic forces.

If the sum of the moments of the external forces about an arbitrary point, say the origin, is zero, then the sum of the moments of the inter-slice forces about this point must also be zero:

(12)

where x and y are the coordinates of the point on the base of the slice where the forces are acting. Satisfaction of Equations 11 and 12 assures that equilibrium is fully satisfied for each slice. Once a solution is found to these two equations, the line of thrust can be calculated for each slice.

A solution to Equations 11 and 12 is obtained by simultaneously varying F and until the two equations are satisfied. For the initial assumed values of F and , the equations may be in error by the amounts R1 and R2 respectively, that is:

(13)

(14)

where Q is based on the assumed values of F and

, and R1 and R2 are the horizontal force and moment imbalances respectively. Note that Equation 11 has been modified slightly to give the horizontal component of the inter-slice forces.

By the Newton-Raphson method for convergence, F and are varied until R1 and R2 are within acceptable limits. This convergence process is discussed in detail in Wright's dissertation (1969).

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