Prepared for : MOHD JAMALUDIN MD NOOR (ASSOC. PROF.DR.)

Prepared by:NOOR ILIFADHILAH SAMSUDIN2008741015

NOORFARHAH BINTI MD DOM2009869506

18th January 2011

During past three decades, numerous methods have been proposed for performing the two-dimensional limit equilibrium method of slices (Wright,1969).

One of the method commonly used is:• The Spencer Method

Developed based on method of slices of Fellenius (1927) and Bishop (1955)

The analysis is in terms of effective stress

It satisfies two equation of equilibrium• With respect to forces• With respect to moments

Assume forces on the sides of slices are parallel

The Spencer method assumes a constant relationship between magnitude of the interslice shear and normal forces (Spencer,1967)

Spencer (1967) summed forces perpendicular to the interslice forces to derive the normal force equation

Figure (a) shows a section through an embankment of height H and slope β with a circular slip surface and a typical slice of mean height h and width b.

Figure (b) and (c) shows an enlarged sketch of the slice with the forces acting upon it and the resolved forces to the base of the slice.

There are five forces taken into consideration :-• Weight (w)• Total reaction (P) normal to the base of slice

which have two components [ P = P’ + ub sec α ]

The force (P’) due to effective or inter-granular stress

The force (ub sec α) due to the pore pressure (u)

• Mobilized shear force• Inter-slice forces (Zn) and (Zn+1); for

equilibrium, the resultant (Q) of these two forces must pass through the point of intersection of the three other forces

The expression of factor of safety obtained:-

Considering inter-slice forces, these forces in a fully rigorous solution would be separated into two components like force P.

Total force is used to derive the components for the sake of simplicity.

Resolving the five forces to the base of slice, the following expression is obtained for the resultant (Q) of the two inter-slice forces

Where:• Weight, • Mean pore pressure on base of slice (assuming homogeneous pore-pressure

distribution as proposed by Bishop and Morgenstern (1960)) with ru as the pore-pressure coefficient.

Therefore, the equation can be re-written in a dimensionless form as follows:-

If external forces on embankment are in equilibrium• Vectorial sum of inter-slice force = 0• Sum of horizontal components = 0• Sum of vertical components = 0

If sum of moments of external forces about centre of rotation = 0• Sum of moments of inter-slice forces about

centre of rotation = 0• Assuming slip surface to be circular, radius of

curvature (R) is constant

Therefore, there are three equations to be solved.

Values of F and θ must satisfy all three equations

Assuming inter-slice forces are parallel (θ is constant), the equation for sum of horizontal and vertical components become identical, Σ Q = 0, thus leaving only two equations to be solved.

Result was then checked in a few trial cases.

Several points emerged from the solution.

• The value of θ1 (which satisfies both force and moment equation) was less than the slope of embankment (β)

• Variation in the value of θ affected the values of Ff (obtained using force equilibrium equation) to a very much greater extent than those of Fm (obtained using moment equilibrium equation); in fact when θ was less than θ1, the variation in Fm was very small indeed; in consequence there was not much difference between the values of Fmo (obtained using moment equilibrium equation and taking θ as zero) and F1 (intersection of two curves plotted showing the relationship between Ff and θ and also Fm and θ)

•The line passing through the point of action of the inter-slice forces passed very close to the lower ‘third point’ on each of the inter-slice boundaries; this implies an approximately triangular pressure distribution on these boundaries which was an acceptable result.

Spencer have developed stability chart that was obtained from curves given by the relationship of mobilized angle of shearing resistance, and the stability number based on different values of ru using graphical method.

The stability charts gives the critical slope (F=1) when the height of an embankment and the soil properties are known.

The factor of safety is expressed as:-

Other factors are expressed as:-•Mobilized angle of shear resistance

•Pore pressure ratio

•Stability factor

Determine c, γ, H, β, Ø, and ru for the given slope

Assume a value of Fs

Calculate c/[Fs (assumed) γH] With the value of c/Fs γH calculated and

slope angle β, enter the proper chart to obtain Øm. Note that the charts are for ru values of 0, 0.25 and 0.5.

Calculate Fs = tan Ø / tan Øm

If the value of Fs assumed in step 2 are not the same as those calculated in step 5, repeat the steps 2,3,4 and 5 until they are the same.

Spencer, E. (1967) “A Method of Analysis of the Stability of Embankments Assuming Parallel Inter-slice Forces” Geotechnique, Vol. 17, No. 1, pp 11-26

D.G Fredlund,J Krahn, D.E Pufahl “The Relationship between Limit Equilibrium Slope Stability Methods” ,Dept of CE,Univ.of Saskatchewan, Canada.