## LESSON 30. Stability Analysis of Gravity Dams: Stability

30.1 Introduction: In this lesion we will learn two important safety requirements viz (i) stability against overturning and (ii) stability against sliding. Safety against induced stresses will be discussed in the next lesson.

Cross-section of a typical gravity dam with all relevant forces is shown in Figure 30.1.

Fig. 30.1.

where,

W = Self weight of the gravity dam. It acts at a distance x1  from the  vertical line passing through the toe of the dam.

FV = Weight of water on the inclined part of the upstream face. It acts at a distance x2  from the  vertical line passing through the toe of the dam.

FH = ${1 \over 2}{\gamma _w}h_u^2$ = horizontal water pressure on upstream face. It acts at a distance y from the base of the dam.

U =  ${1 \over 2}{\gamma _w}\left( {{h_d} + {h_u}} \right)b$  Uplift force. It acts at a distance x3  from the  vertical line passing through the toe of the dam.

R = Resultant of W, FV, FH  and U. Rx and Ry are the components of R.

Force due to horizontal water pressure (FH) and uplift pressure (U) will cause overturning moment about the toe of the dam. This overturning moment will be stabilized mostly by the self weight of the dam W. Weight of water on the inclined part of the upstream face FV will also produce some stabilizing moment. A gravity dam is considered to be safe against overturning if the stabilizing moment is higher than the overturning moment. The factor of safety against overturning is defined as,

$FOS{\rm{ against overturning }}={\rm{ }}{{{\rm{Total stabilizing moment}}} \over {{\rm{Total overturning moment}}}}$

The horizontal force acting on the dam is balanced either by friction alone or by friction and shear strength of the joint. A dam will fail in sliding at its base, or at any other level, if the net horizontal force causing sliding is more than the resistance available at that level. The factor of safety against sliding is defined as,

Where,

μ = Coefficient of friction

tc = Permissible cohesion/ shear stress

A = cross-sectional area

Ff = partial safety factor in friction (Table 1 in IS:6512 – 1984)

Fc = partial safety factor in cohesion/shear (Table 1 in IS:6512 – 1984)