CHAPTER 6
INTERNAL STABILITY
6.1 
Basic Concept 
6.2 
Factors of Safety 
6.3 
Internal Stability Analysis 
6.1 BASIC CONCEPT
In the internal stability analysis of GRSRW, geosynthetic of adequate strength are used to resist the earth pressure exerted by the dead weight of backfill soil as well as the external loads acting on the facing.
1. Force Equilibrium Analysis Using TwoWedge Method
In the twowedge (twopart wedge) method, the failure surface behind the reinforced soil zone is varied while determining for the earth pressure acting on the facing.
The force polygon used in the twowedge method is shown in Figure 6.1.
Notations:
W_{s} 
: force acting on the crest of GRSRW 
W_{sf} 
: force acting on the crest of Wedge F 
W_{sb} 
: force acting on the crest of Wedge B 
W_{f} 
: dead weight of Wedge F 
W_{b} 
: dead weight of Wedge B 
W_{gv} 
: deadweight of facing 
H_{f} 
: earthquake force acting on Wedge F 
H_{b} 
: earthquake force acting on Wedge B 
W_{gh} 
: earthquake force acting on facing 
F_{f} 
: factor of safety against pullout 
L_{i} 
: anchorage length of geosynthetic reinforcement 
T_{aj} 
: design strength of geosynthetic reinforcement 
T_{i} 
: geosynthetic reinforcement force (see Equation 6.6) 
P_{f} 
: soilfacing interaction force acting on Wedge F 
P_{b} 
: interwedge force between Wedges B and F 
R_{f} 
: reaction force at the base of Wedge F 
R_{b} 
: reaction force at the base of Wedge B 
f 
: internal friction angle of backfill soil 
f _{f} 
: friction angle between facing and backfill soil 
f _{b} 
: interwedge friction angle 
f _{d} 
: friction angle along the base of facing 
2. Internal Stability Analysis
Based on the earth pressure determined from twowedge mechanism, the factor of safety of the facing to resist direct sliding and overturning are calculated for different loading conditions (dead load, live load, earthquake).
3. Earth Pressure Acting on the Facing
The earth pressure acting on the facing, P_{f}, is determined based on the force polygons of the soil wedges (Figure 6.2).
4. Point of Action of Resultant Earth Pressure
In addition to the magnitude of earth pressure, its point of action is required in evaluating for overturning stability. The distributions of earth pressure, due to external loads, are shown in Figure 6.3. The point of action of the resultant earth pressure is determined from Equation 6.1.
(6.1)
where
h_{f} 
: elevation of the point of action of resultant earth pressure 
P_{f} 
: resultant earth pressure 
P_{w} 
: earth pressure exerted by backfill soil 
h_{w} 
: elevation of point of action of earth pressure exerted by backfill soil 
P_{L} 
: earth pressure due to surcharge load 
h_{L} 
: elevation of point of action of earth pressure by surcharge load 
P_{p} 
: earth pressure due to effect of earthquake on the surcharge load 
h_{p} 
: elevation of point of action of earth pressure due to effect of earthquake on the surcharge load 
Based on the earth pressure determined from twowedge mechanism, the factor of safety of the facing to resist direct sliding and overturning are calculated for different loading conditions (dead load, live load, earthquake).
6.2 FACTORS OF SAFETY
In the internal stability analysis of GRSRW, the calculated factors of safety to resist direct sliding and overturning have to be greater than specified values.
1. Definition of Factor of Safety
The factors of safety to resist direct sliding, F_{s}, and overturning, F_{o}, are defined in Equations (6.2) and (6.3), respectively:
(6.2)
where D H_{r} is the force to resist direct sliding, mobilized along the geosynthetic layer, and D H_{d} is the driving force, which is the horizontal component of the resultant earth pressure acting on the facing.
(6.3)
where D M_{r} is the resisting moment offered by the geosynthetic layer, and D M_{d} is the driving moment, which is the product of horizontal component of the resultant earth pressure acting on the facing and its elevation.
2. Required Value of Factors of Safety in Internal Stability Analysis
1) During service, the required value of the factor of safety to resist direct sliding and overturning are given in Table 6.1.
Table 6.1 Required Factor of Safety

DeadLoad Condition 
LiveLoad Condition 
Earthquake Condition 
Factor of Safety 
2.0 
1.5 
1.25 
In the conventional retaining wall design, the factor of safety to resist direct sliding is specified as 1.5 under deadload condition. For GRSRW, the definition of the factor of safety to resist direct sliding is different, thus it is specified as 2.0.
2) During construction, that is during the stage of backfilling, that is before the installation of wall facing, the geosynthetic length L has to be greater than 35% the wall height or 1.5 m long, whichever the greater. Additional analysis is not required.
6.3 INTERNAL STABILITY ANALYSIS
The location and angle of inclination of the potential failure surfaces are varied in the analysis, until both of them satisfy the required factors of safety against direct sliding and overturning.
1. Procedure of Internal Stability Calculation
In the internal stability analysis, the location and angles (q _{f}, q _{b}) of potential failure surfaces are varied accordingly. For an assumed failure surface, the factor of safety against direct sliding F_{s} and overturning F_{o} are calculated from the resultant earth pressure P_{f} and resisting force until the minimum values {F_{s}}_{min} and {F_{o}}_{min} are obtained. If one or both of {F_{s}}_{min} and {F_{o}}_{min} will not meet the required values, the geosynthetic layout and/or strength should be modified so that the calculation is repeated.
2. Driving and Resisting Forces in Direct Sliding Analysis
a) Driving force: D H_{d}
With reference to Figure 6.4, the driving force is obtained as
(6.4)
where P_{fh} is the horizontal component of resultant earth pressure acting on the facing and W_{gh} is the seismic inertia force of the facing.
b) Resisting force: D H_{r}
With reference to Figure 6.5, the resisting force is obtained as
(6.5)
where S T_{i} is the tensile force of the geosynthetic layers that pass through the potential failure surface. W_{gv}tanf _{d} is the frictional force between the base of facing and foundation.
However, in calculation, the frictional force at the base of facing is typically neglected to give conservative results.
The geosynthetic tensile force is obtained from the following equation. However, T_{i} should not exceed the design strength of geosynthetic T_{aj} (see 4.1 on details of T_{aj}).
(6.6)
where
L_{i} 
: anchorage length of geosynthetic, located beyond the failure surface (m) 
s _{v} 
: normal stress acting on the geosynthetic (kN/m) 
f 
: internal friction angle of backfill soil (degrees) 
F_{f} 
: factor of safety against pullout (see Table 6.1) 
In calculating for the pullout strength, the effect of surcharge may be added to the normal stress considering the load distribution shown in Figure 6.6.
Thus, s _{v} is calculated from the following equation:
(6.7)
where
g 
: unit weight of backfill soil (kN/m^{3}) 
p 
: surcharge (kPa) 
b 
: length of surcharge load (m) 
y 
: overburden depth (m) 
q 
: angle of distribution of surcharge load (typically 30^{o}) 
3. Driving and Resisting Moments in Overturning Analysis
a) Driving moment: D M_{d}
With reference to Figure 6.7, the driving moment is obtained as
(6.8)
where P_{fh}h_{f} is the moment of the horizontal component of resultant earth pressure acting on the facing and W_{gh}h_{g} is the moment of the seismic inertia force of the facing.
b) Resisting moment: D M_{r}
With reference to Figure 6.8, the resisting moment is obtained as
(6.9)
where
S (T_{i}X_{i}) 
: moment of tensile force of geosynthetic layers that pass through potential failure surface 
(P_{fv}P_{bv})Y_{f} 
: moment of the vertical component of earth pressure in Wedge F 
P_{bv}Y_{b} 
: moment of the vertical component of earth pressure in Wedge B 
W_{gv}Y_{w} 
: moment of the dead weight of facing 
Notes: The embedment of facing into the foundation is not considered during the stability calculation. The upper limit of P_{fv} should not exceed the allowable bearing capacity of the foundation that supports the facing.
c) Other external forces
The external load exerted by the structures located on the crest of facing, and that acts on the backfill soil, are to be considered in the stability analysis.
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