Gravity Dam Fundamentals

A gravity dam is a massive concrete or masonry structure designed to withstand external forces primarily by its own weight. The stability of a gravity dam is crucial, and engineers must ensure it remains stable under various loading conditions, especially when the reservoir is full. The distribution of forces at the base of the dam determines the stresses, specifically compressive and tensile stresses, within the dam's foundation.

Forces on a Gravity Dam

When a reservoir behind a gravity dam is full, several forces act on the dam structure. Understanding these forces is key to analyzing the stress distribution at its base:

• Water Pressure (Hydrostatic Force): This is the most significant horizontal force acting on the upstream face of the dam. It acts perpendicularly to the wetted surface and increases with depth, being maximum at the base of the dam. This force tends to push the dam downstream.
• Self-weight of the Dam: This is a vertical force acting downwards through the center of gravity of the dam. It provides the primary resisting moment against overturning and sliding.
• Uplift Pressure: Water seeping through the pores and cracks in the dam's foundation or body creates an upward pressure, reducing the effective weight of the dam and its resistance to sliding.
• Silt Pressure: Accumulation of silt against the upstream face can exert additional horizontal pressure.
• Wave Pressure: Waves generated on the reservoir surface can exert pressure on the dam's top portion.
• Ice Pressure: In cold climates, ice formation on the reservoir surface can exert horizontal pressure.

The primary forces influencing the maximum compressive stress are the horizontal water pressure and the vertical self-weight of the dam.

Resultant Force and Stress Distribution

The combined effect of all these forces results in a single resultant force acting on the base of the dam. The location where this resultant force intersects the base determines the stress distribution across the base width. The base of the dam has two critical points: the 'heel' (the upstream edge) and the 'toe' (the downstream edge).

When the reservoir is full, the large horizontal water pressure (acting downstream) tends to shift the point of application of the resultant force towards the downstream side, i.e., towards the toe of the dam. The general formula for normal stress at any point on the base is given by:

$$ \sigma = \frac{N}{B} \left( 1 \pm \frac{6e}{B} \right) $$

Where:

• $\sigma$ is the normal stress at a point on the base.
• $N$ is the total normal (vertical) force acting on the base.
• $B$ is the width of the dam's base.
• $e$ is the eccentricity, which is the distance from the center of the base to the point where the resultant force intersects the base.

When the resultant force moves towards the toe, the eccentricity 'e' is considered positive (or negative, depending on the chosen convention, but its effect is to increase stress on one side and decrease it on the other). This shift causes a non-uniform distribution of stress across the base.

Maximum Compressive Force at the Toe

The maximum compressive force or stress occurs at the edge of the base that is closer to the point of application of the resultant force. When the reservoir is full, the significant horizontal water pressure pushes the resultant force towards the toe of the dam. This concentration of the resultant force near the toe causes the pressure on the foundation at the toe to be much higher than at the heel.

Conversely, the stress at the heel decreases, and if the resultant force falls outside the 'middle third' of the base towards the toe, the heel might even experience tensile stresses, which is undesirable for concrete or masonry dams that are weak in tension. Therefore, to ensure overall stability and safety, the resultant force should ideally remain within the middle third of the base.

Because the large horizontal water load pushes the resultant force close to the toe, the foundation material at the toe bears a much greater share of the vertical load, leading to the maximum compressive force being produced at the toe of the gravity dam when the reservoir is full.