Gravity Dam Stability and Tension Prevention

A gravity dam is designed to resist the external forces primarily by its own weight. The stability of the dam is crucial, and one important aspect is ensuring that no tension develops at the base of the dam. Tension can lead to cracks, weakening the structure and potentially causing failure.

Understanding Tension at the Base

When the resultant of all forces acting on the dam (including its weight, water pressure, uplift pressure, etc.) falls at a certain location on the base, it determines the stress distribution across the base. If the stress becomes tensile (negative pressure), it indicates that the base is trying to pull apart, which concrete and masonry dams are generally weak against.

The Middle Third Rule Explained

To avoid tension at the base of a gravity dam, the resultant force must fall within a specific region of the base. This region is known as the "middle third".

• The base of the dam is considered.
• This base is divided into three equal parts.
• The central part is the middle third portion.
• If the resultant force acts anywhere within this middle third portion, the stress distribution across the base will be compressive throughout, meaning there will be no tension.
• If the resultant falls exactly at the edge of the middle third, the pressure at the upstream or downstream toe will be maximum, and at the opposite toe, it will be zero, but still no tension.
• If the resultant falls outside the middle third, tension will develop at one of the toes (upstream if the resultant is towards the downstream third, downstream if the resultant is towards the upstream third).

Condition for No Tension

For no tension to develop in the gravity dam's base, the criterion is that the resultant of all external forces must always lie within the middle third portion of the base. This ensures that the entire base remains under compression.

Mathematically, for a rectangular base of width B, if the resultant force acts at a distance x from the center, the condition for no tension is:

\( |x| \le \frac{B}{6} \)

This means the resultant must fall within a distance of \( \frac{B}{6} \) from the center, which corresponds exactly to the middle third region (from \( \frac{B}{6} \) on one side of the center to \( \frac{B}{6} \) on the other side, totaling \( \frac{B}{3} \) length in the center).

Conclusion

Based on the principles of gravity dam design and stability, the condition for preventing tension at the base is that the resultant force must fall within the middle third of the base. This is a fundamental rule in gravity dam engineering.