Middle third rule for gravity dam

Statement

For a gravity dam, the resultant of all forces on any horizontal section should cut the base within its middle third, i.e. the eccentricity from the centre must satisfy $e\le B/6$. Then there is no tension (at the heel when the reservoir is full, at the toe when it is empty).

Derivation

Consider a horizontal section of width $B$ and unit length. Let $\Sigma V$ be the total vertical load and $R$ the resultant, cutting the base at a distance $e$ from the centre of the base.

The resultant is equivalent to a vertical force $\Sigma V$ at the centroid of the base plus a moment $M=\Sigma V\cdot e$. The stress at the base is (axial plus bending):

$$\sigma = \frac{\Sigma V}{A} \pm \frac{M\,y}{I}$$

with $A = B\times1$, $I = \dfrac{B^3}{12}$, and $y=\dfrac{B}{2}$ at the extreme edge:

$$\sigma = \frac{\Sigma V}{B} \pm \frac{\Sigma V\,e\,(B/2)}{B^3/12} = \frac{\Sigma V}{B}\left(1 \pm \frac{6e}{B}\right)$$

So

$$\sigma_{max} = \frac{\Sigma V}{B}\left(1+\frac{6e}{B}\right), \qquad \sigma_{min} = \frac{\Sigma V}{B}\left(1-\frac{6e}{B}\right)$$

The minimum stress is at the heel when the resultant lies towards the toe (reservoir full), and it becomes tensile if

$$1-\frac{6e}{B} < 0 \;\Rightarrow\; e > \frac{B}{6}$$

To avoid tension, $\sigma_{min}\ge0$, hence

$$e \le \frac{B}{6}$$

An eccentricity of $e=B/6$ means the resultant passes through the limit of the middle third, a distance $B/3$ from the toe. Thus the resultant must lie within the middle third of the base ($B/3$ to $2B/3$ from the toe). At the limit $e=B/6$, the stress at the heel is zero and at the toe $\sigma_{max}=2\Sigma V/B$.

Where $e$ comes from

$$\bar x = \frac{\Sigma M_R - \Sigma M_O}{\Sigma V}\ (\text{from the toe}),\qquad e = \frac{B}{2}-\bar x$$

Why it is used

• It keeps the entire base in compression, so no crack opens at the heel to admit water (which would raise the uplift).
• The dam is economical because the compressive strength of concrete is used well.
• It also ensures an adequate margin of safety against overturning.