Shear Strength Applications in Plastic and Limit Equilibrium - Unsaturated Soil Mechanics in Engineering Practice - page 644

Y

Stage

A

State point

B

n + 1 X

1 i

Figure 12.80 Slip surface AB selected through use of an optimization technique such as dynamic
programming.

state points located in two successive stages. The stage-state
system forms a grid consisting of rectangular elements called
the search grid. The rectangular elements formed by the
search grid are called grid elements. The overall factor of
safety for the slip surface AB is defined as follows for the
discretized soil mass:

n = total number of discrete segments comprising the

slip surface.

The minimum value of the auxiliary function, G min , is
defined as

n

G min = min (R i − F s S i ) (12.95)

n

i = 1

τ f i L i

The shear strength for a saturated-unsaturated soil can be
calculated and applied along the i th segment (Fredlund and
Rahardjo, 1993a):

i = 1

F s = (12.93)

n

τ i L i

i = 1

c + u a ) tan φ + u w ) tan φ b τ f i = (σ n − (u a − (12.96)

where:

where:

n = number of discrete segments,

τ i = mobilized shear stress,

c ,φ ,φ b = shear strength parameters of a saturated-

unsaturated soil,

τ f i = shear strength, and

L i = length of i th segment.

σ n − u a = net normal stress acting on the i th segment,

and

12.6.1.1 Theory of Dynamic Programming Method
A minimization technique is necessary to obtain the fac-
tor of safety F s (Eq. 12.93). Baker (1980) showed that the
minimum of F s in Eq. 12.93 can be found by using an auxil-
iary function G . The auxiliary function is also known as the
return function and can be defined as follows (Fig. 12.81):

u a − u w =

matric suction.

The normal and shear stresses acting on the i th segment
can be computed from a stress analysis as follows:

σ x sin 2 θ σ y cos 2 θ σ n = + − τ xy sin 2 θ (12.97)

σ y − σ x

τ xy ( sin 2 θ cos 2 θ)

τ n = − − sin 2 θ (12.98)

n

2

G = (R i − F s S i ) (12.94)

where:

i = 1

σ n = normal stresses acting on the i th segment,

where:

τ n = shear stresses acting on the i th segment,

S i = actuating forces acting on the i th segment of the

slip surface,

θ = inclined angle of the i th segment with the

horizontal direction, and

R i = resisting forces acting on the i th segment of the slip

surface, and

σ x ,σ y ,τ xy = normal and shear stresses acting in the x -

and y -coordinate directions.