Shear Strength Applications in Plastic and Limit Equilibrium - Unsaturated Soil Mechanics in Engineering Practice - page 646

state points located in every stage is eventually found by
tracing back from the final stage to the initial stage. This
optimal path defines the critical slip surface.
The value of the overall factor of safety F s in Eq. 12.94 has
not been defined in advance, and therefore an initial valuemust
be assumed. The trial value of F s is updated using the value
of F s evaluated after each trial of the search. The optimization
process will stop when a predefined convergence is reached.

stage. Trial segments of the slip surface are formed by
connecting all state points of the initial stage to all state
points located in the second stage. Values of the optimal
function are obtained at all state points of the second stage
using Eqs. 12.101 and 12.102 and the assumed factor of
safety F s . The number of optimal functions to be calculated
at one state point of the second stage is equal to the number
of state points located in the initial stage.
The minimum value of the optimal function is determined
at each state point in the second stage. The correspond-
ing state point in the previous stage (i.e., the initial stage
for the first segment) is identified. The process proceeds to
the next stage with the same routine until the final stage is
reached.
A comparison is made of the values of the optimal func-
tions obtained at all state points of the final stage and the
determination of the state point at which the correspond-
ing value of the optimal function was a minimum. The
determined state point will be the first optimal point of the
optimal path. There is then a trace back to the previous stage
to find the corresponding state point with the first optimal
point. This corresponding state point will be the second opti-
mal point of the optimal path. The trace back to the initial
stage is continued to determine the entire optimal path. The
actual factor of safety corresponds to the optimal path previ-
ously obtained. A new value for the factor of safety is then
calculated.
The process is repeated until the difference between the
assumed and the actual factor of safety is within the selected
convergence criterion δ . The critical slip surface is defined
using the entry and exit points which correspond to the opti-
mal path.

12.6.1.2 Finite Element Analysis for Stress State
A general partial differential equation solver can be used to
solve the partial differential equations for the stress states
throughout the soil mass. (The computed strain values are
of no interest to this analysis.) For a two-dimensional, plane
strain analysis (i.e., ε z = 0), a soil element subjected to its

body forces has partial differential equations representing
the stress balance in the x - and y -directions:

∂τ yx
∂y

∂σ x
∂x

+ + F x = 0 (12.104)

∂τ xy
∂x

∂σ y
∂y

+ + F y = 0 (12.105)

where:

σ x ,σ y = normal stress in the Cartesian x - and

y -coordinate directions,

τ yx ,τ xy = shear stress in x-y planes, and

F x ,F y = body forces in the x - and y -coordinate direc-

tions.

Partial differential equations 12.104 and 12.105 can be
solved in conjunction with appropriate specified boundary
conditions. The stresses are computed and stored at the
Gaussian points over the domain of the problem. The stresses
can then be interpolated from the Gaussian nodes to any
location on the arbitrary grid that is superimposed for the
slope stability analysis. The stresses at the center point of each
grid element are determined using the interpolation shape
functions. The stress interpolation process is undertaken prior
to the performance of the dynamic programming search.

12.6.1.4 Advantages of Using Dynamic Programming
Procedure
The dynamic programming method combined with a finite
element stress analysis forms a valuable means of calculat-
ing the factor of safety of a slope. The dynamic program-
ming procedure provides a solution with increased flexibility
over limit equilibrium methods of slices (Pham and Fred-
lund, 2003).
The critical slip surface can be irregular in shape and can
be determined as part of the slope stability solution when
using the dynamic programming method. The shape of the
failure surface must be assumed by the modeler when using
methods of slices. No assumption is required regarding the
shape or the location of the critical slip surface when using
the dynamic programming procedure.
More complex stress-strain models for the soil can be
used in the assessment of the stress state in the soil mass.
The effect of weather-related environmental conditions such
as infiltration (and associated matric suction decreases) can
readily be taken into account when using the dynamic pro-
gramming procedure (Gitirana et al., 2006).

12.6.1.3 Dynamic Programming Solution Procedure
The analytical scheme of the dynamic programming method
for performing a slope stability analysis is illustrated in
Fig. 12.82 (Pham and Fredlund 2003). A search boundary
is defined such that it extends outside the geometry of the
problem at the top and bottom of the slope. There needs to
be one point near midheight where the search boundary dips
below the soil surface of the geometry. This point will force
all possible slip surfaces to lie within the soil mass. The
stresses at each grid intersection point are computed from
the nodal stresses.
The initial factor of safety is assumed to be 1.0 and a
search is launched through state points located in the initial