# ELEMENTARY PROFILE OF A GRAVITY DAM

- If a gravity dam is subject to horizontal pressure due to water along with its own weight, the elementary profile of the dam would be a right angled triangle.
- The thickness of the dam at the level of water surface will be zero as there is no hydrostatic pressure to be resist.
- The thickness at base would be maximum as there will be maximum hydrostatic pressure to be resisted by the dam.
- As the variation in hydrostatic pressure from water level to the bottom, is linear; variation in elementary profile from top to the dam to its bottom would also be linear.
- Elementary profile of the low dam is show in Fig. 13.8 (a).
- The triangular elementary profile also provides the maximum possible stabilizing force against overturning without causing tension in the base when reservoir is empty.
- This is because in this case weight of the dam acts at 3 b from the U/S vertical face.
- Should there be any other triangular profile other than rt. angled, the stability of the dam is further increased but tension would develop at the toe when reservoir is empty.
- Elementary profile of the dam is also sometimes know theoretical profile of the dam.
- Following forces are consider acting on the dam while determining the elementary profile.

**(i) Weight of the dam (W) **

(W) = 1/ 2 bHρw. It acts vertically downwards and acts at C.G. of the dam.

**(ii) Water Pressure (P)**

\[P= \frac{1}{2}wH^{2}\]

Force P is horizontal and acts at 3 H from the bottom of the profile.

**(iii)Uplift Pressure (U) **

U = 1 /2 cwbxH

Uplift pressure acts vertically upwards.

Its direct effect is that it reduces the gravity effect or load of the dam.

In all the three expression given above,

- b = Base width of the dam.
- H = Height of the elementary profile.
- ρ = Specific density of the dam material.
- w = Unit weight or density of water.
- C = Uplift pressure intensity coefficient.
- U = Uplift pressure.
- P = Horizontal water pressure.
- W = Weight of the dam as a whole. Base Width of Elementary Profile.

**The base width of elementary profile is to be determined for following conditions. **

- Stress basis.
- Stability of sliding basis.

**1. Stress Basis. **

We know that for no tension to develop the resultant should pass through the inner middle third point, when reservoir is empty and through outer middle third point when reservoir in full of water.

Take moments of all the forces acting on the profile about the outer middle third point, and equate it to zero.

\[Hence\frac{1}{2}wH^{2}\times \frac{H}{2}cwbH\times \frac{b}{3}-\frac{1}{2}bHrw\times \frac{b}{3}= 0.\]

\[On-solving-b= \frac{H}{\sqrt{p-c}}\]

If uplift pressure is not considered c = 0 and

\[hence-b= \frac{H}{\sqrt{p}}\]

**Stability on Sliding Basis.**

- In order to prevent sliding of the dam, the horizontal force causing sliding should be either equal to or less than the frictional resistance opposing the sliding of the dam.
- Critical condition occurs when horizontal force (ρ) is equal to the frictional resistance.

i.e

\[\rho = \mu (W-u)\]

\[\frac{1}{2}wH^{2}= \mu (\frac{1}{2}bH\rho w-\frac{1}{2}cbwH)\]

\[solving-b= \frac{H}{\mu (p-c)}\]

if uplift is neglected

\[b= \frac{H}{\mu \rho }\]

The value of b should be adopted greater than the values obtained on stress basis and sliding basis.