## Sliding stability analysis

Basic formula for a horizontal sliding plane (static loads)

The basic formula for the calculation of the sliding safety factor (SSF) is given by:

= Sum of vertical forces excluding uplift pressure

U = Resulting force of uplift pressure

ϕ = Friction angle (peak or residual value)

c = cohesion (apparent or real, for the apparent cohesion a minimum compression value, σn determines the area in compression over which cohesion can be mobilised can be specified

AC = Area in compression

= Sum of horizontal forces

Basic formula for a horizontal sliding plane (seismic loads, vertical upstream wall)

In a seismic analysis, the sliding safety factor (SSF) is calculated using:

= Sum of vertical forces excluding uplift pressure

QV = Concrete vertical inertia force

U = Resultant uplift pressure force

= Concrete horizontal inertia force

Qh = Horizontal hydrodynamic force

ϕ = Friction angle (peak or residual value)

c = cohesion (apparent ou real)

AC = Area in compression

= Sum of horizontal static forces

CADAM3D calculates sliding safety factors by considering the peak and residual shear strength.

Influence of post-tension forces (e.g. static loading, horizontal analysis plane)

Post-tension anchors are often used to increase the compression along an analysis plane for control of tensile cracking and to increase sliding resistance.

Post-tension forces considered as active load: In most cases, the post-tension forces were considered as active forces, i.e. the horizontal component of the post-tension force Pdh is in the denominator of the formula for the sliding safety factor. In this case, Pdh is added algebraically to the other external horizontal forces applied to the structure (e.g. hydrostatic thrust).

= Sum of vertical forces excluding uplift pressure

U = Resultant of uplift pressures

ϕ = Friction angle (peak or residual value)

c = Friction angle (peak or residual value)

Ac = Area in compression

= Sum of horizontal static forces

Pv = Vertical component of post-tension force (Pc, PdV)

Pdh = Horizontal component of post-tension force

Post-tension forces considered as passive load: In this case, Pdh is placed in the numerator of the formula for the sliding safety factor. In this approach, Pdh is added directly to the sliding resistance provided by the vertical force of the anchor. This approach is more conservative than the one where Pdh is considered as an active load. See Corns et al. 1988b (p.593) for a more elaborate discussion:

Inclined joints (e.g. static loads) - Evaluation of the sliding safety factor using the limit equilibrium method:

When the joint is inclined, the resultant forces shall be calculated in the directions normal and tangential to the joint to evaluate the sliding safety factor.

= Sum of forces normal to sliding plane

= Sum of forces tangential to sliding plane

U = Resultant of uplift pressures normal to inclined sliding plane

α = Friction angle with respect to the horizontal sliding plane

Cracked length and compressive stresses: Stability against overturning can be checked by limiting the cracked length such that the admissible compressive stresses are not exceeded.

Resultant force position:

The position of the resultant force along the joint is another performance indicator that is used to assess the overturning stability of the cracked section above the considered plane. The position of the resultant force relative to the upstream end of the joint is calculated using:

ΣMU/S = Sum of all moments relative to the U/S position of the joint

ΣV = Sum of normal forces including uplift pressure

In the results of CADAM3D, LFR is expressed as a percentage of the total length of the joint calculated with respect to the upstream end of the joint. When the resultant forces are within the middle third of the section, there are no tensile stresses, provided the joint does not vary in thickness longitudinally. For well-proportioned structures, overturning from upstream to downstream is unlikely. Rather, a sliding mechanism will tend to occur after an upstream toe lift.

Flotation Safety Factor

In the case of significant uplift pressures, the structure must resist the lifting forces resulting from the water pressure that tends to initiate flotation. The factor of safety against flotation is calculated as follows:

= Sum of vertical forces excluding uplift pressure (but including the weight of water above the submerged component),

U = Force resulting from uplift pressure

Uplifting Safety Factor

In the case of a significant overflow, the structure must resist the lifting forces resulting from the water pressure that tends to lift it. The factor of safety against uplifting is calculated as follows:

= Sum of normal negative forces (stabilizing),

= Sum of normal positive forces (destabilizing).