# STABILITY ANALYSIS METHODS

The following are the various methods which can be adopted for carrying out the stability analysis of gravity dams.

- Gravity method.
- Trial load twist method.
- Slab analogy method.
- Lattice analogy method.
- Experimental methods. Brief description of each method is given here.

**1. Gravity Method. **

- This method of stability analysis, is also sometimes know two dimensional method.
- In this method, the dam is consider to be compos of parallel side.
- Each cantilever is consider free to act without any attachment with the adjoining cantilevers.
- The loads acting on the dam are resist entirely by the weight of the individual cantilever.
- All the loads are ultimately transferr to the foundation by cantilever action.
- This method of checking the stability of the dam may be further divid into two parts.

** (a) Graphical method.**

- In this case the dam is divid into different sections according to height by drawing horizontal section lines 1–1, 2–2, 3–3 and 4–4.
- Section lines are generally drawn at places where the slope of the dam face changes.
- Starting from the top each part of the dam is analys separately.
- For each section. the sum of all horizontal forces ΣH, and sum of all the vertical forces, ΣV, acting above that section are calculated and their line of action are graphically drawn.
- For each part of the dam resultant (R) of all the forces acting on any part is calculate and its line of action on section line lying immediately below that section is locat.
- This line should evidently lie within the middle third, so that tension may not develop in the dam.
- Such resultant lies are draw for both the conditions, namely, a reservoir full as well as reservoir empty.
- Both the lines of resultant pressure so draw should lie in the middle third portion of the dam. See Fig. 13.6.

**(b) Analytical method.**

- Stability analysis by following steps:

(i) Consider unit length of the dam.

(ii) Find out the algebraic sum of all the vertical forces, acting on the dam.

(iii) Determine the algebraic sum of all the horizontal forces acting on the dam

(iv) Determine the overturn. Find out the algebraic sum of moments ΣM as follows

ΣM = ΣMr – ΣMo

(v) Determine the position of resultant force R, from the. toe (x ) as follows

\[\overline{x}= \frac{\Sigma M}{\Sigma V}\]

(vi) Determine the eccentricity (e) of resultant R from the centre of the base (b)

\[e= \frac{b}{2}-\overline{x}\]

where b is the base width of the dam.

(vii) Find out the normal stress at toe and heel, by the following equation

\[p_{n}= \frac{\Sigma }{b}\left ( 1+\frac{6e}{b} \right )\]

(viii) Determine the principal and shear stresses at toe and heel as explained in Sec. 13.5.

(ix) Find out factor of safety against over turning as follows:

\[F.S.= \frac{\Sigma M_{r}}{\Sigma M_{0}}\]

(x) Find out the factor of safety against sliding by following expressions

\[(a)-sliding-factor= \frac{\Sigma V}{\Sigma H}\]

\[(b)-shear-friction-factor= \frac{\Sigma V+bq}{\Sigma H}\]

It should be remembered that ΣV is the net vertical load, inclusive of the uplift.

**2. Trial Load Twist Method. **

- In this method of analysis, the entire dam is divided into a number of vertical cantilevers and horizontal beams.
- The water pressure is shared between vertical cantilevers and horizontal beams in such a way that the deflections at the common points on the dam face are equal.
- In this method, twisting effect is also developed, as all the cantilevers are of different heights.
- The vertical cantilever is of maximum height at the centre of the dam length.
- The length of vertical cantilevers goes on reducing as we proceed towards the ends of the dam, as valley banks rise quite steeply.
- This develops a twisting effect on each cantilever.

**3. Slab Analogy Method.**

- In this method the analysis.
- Horizontal and vertical beams.
- This is a very laborious method and hence not used much.

**4. Lattice Analogy Method. **

- This method is, though, simpler than slab analogy method, but still quite cumbersome.
- In it, the dam is consider framework of interconnecting square frames, each square being diagonally connected at the corners.

**5. Experimental Methods.**

- Experimental methods may the direct method and indirect method.
- The direct method is also know.

**(i) Three dimensional model analysis. **

- This method of analysis has been found very useful in the case of high dams.
- In it, three dimensional models of dams.
- The size of the models.
- The models are locat in similar manner as the prototype and also subject to loading of the prototype.
- The structural actions, stress conditions, and deformations at various points of the model.
- From this correlation, the stress.