Failure modes and stability criteria of gravity dams

A gravity dam resists the water thrust and other forces mainly by its own weight. It can fail in four ways; each one gives a stability criterion that the design must satisfy for all load combinations (empty reservoir, full reservoir, earthquake, flood).

     |\           W = self weight
 P-> | \          P = water thrust
     |  \         U = uplift
 ~~~~|   \        R = resultant
     | W  \
     |  \  \
heel |___R__\ toe
     ^ U ^ ^ ^

1. Overturning about the toe

• Mode: the horizontal forces (water pressure, silt, wave, earthquake) and uplift produce a moment that tends to rotate the dam about its toe.
• Criterion: factor of safety against overturning

$$FS_o = \frac{\sum M_R}{\sum M_O} \geq 1.5\ (\text{usually } 2 \text{ to } 3)$$

In practice, a dam that satisfies the no-tension and crushing criteria will not overturn.

2. Sliding (shear failure)

• Mode: the dam slides along the base or along a weak joint/seam in the foundation when the horizontal force exceeds the frictional and shear resistance.
• Criteria:
  • Factor of safety against sliding (friction only): $FS_s = \dfrac{\mu \sum V}{\sum H} \geq 1$ (usually 1 to 1.5).
  • Shear friction factor (friction plus shear strength $q$ of the joint over base width $B$): $SFF = \dfrac{\mu \sum V + B q}{\sum H} \geq 3$ to 5.

3. Compression or crushing

• Mode: the vertical stress at the toe (full reservoir) or heel (empty reservoir) exceeds the allowable compressive strength of concrete or rock.
• Criterion:

$$p_{max} = \frac{\sum V}{B}\left(1 + \frac{6e}{B}\right) \leq \text{allowable stress}$$

The principal stress at the toe, $\sigma_1 = p_v\sec^2\alpha - p'\tan^2\alpha$, must also be below the allowable value.

4. Tension cracking

• Mode: if the resultant falls outside the middle third of the base ($e > B/6$), tension develops at the heel. Concrete is weak in tension, so a crack opens, uplift enters the crack, the effective base width reduces and stresses at the toe rise, which can lead to failure.
• Criterion (middle-third rule):

$$e \leq \frac{B}{6} \quad\Rightarrow\quad p_{min} = \frac{\sum V}{B}\left(1 - \frac{6e}{B}\right) \geq 0$$

A small tension (e.g. up to about 500 kN/m²) is sometimes allowed for extreme load cases such as earthquakes.

Summary of criteria