The following are the various methods which can be adopted for carrying out the stability analysis of gravity dams:
1. Gravity Method:
This method of stability analysis is also sometimes known as two dimensional methods. In this method, the dam is considered to be composed of parallel sided vertical cantilevers. Each cantilever is considered free to act without any attachment with the adjoining cantilevers.
The loads acting on the dam are resisted entirely by the weight of the individual cantilever. All the loads are ultimately transferred to the foundation by cantilever action.
This method of checking the stability of the dam may be further divided into two parts:
(a) Graphical Method.
In this case the dam is divided 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 analysed 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 calculated and its line of action on section line lying immediately below that section is located.
Finally a line is drawn joining the points at which the resultant cut the various section lines. This line should evidently lie within the middle third, so that tension may not develop in the dam. Such resultant lines are drawn for both the conditions, namely, reservoir full as well as reservoir empty. Both the lines of resultant pressure so drawn should lie in the middle third portion of the dam. See Fig. 13.6.
(b) Analytical Method:
Stability analysis by this method is done as per following steps:
(i) Consider unit length of the dam.
(ii) Find out the algebraic sum of all the vertical forces, acting on the dam. The forces that come in this category are weight of the dam, weight of water acting on inclined faces of the dam, uplift pressure etc. Let this algebraic sum be denoted by ΣV.
(iii) Determine the algebraic sum of all the horizontal forces acting on the dam. Let it be denoted by ΣH.
(iv) Determine the overturning moment (ΣM0) and the resisting or stabilizing moment (ΣMr) about the toe of the dam.
Find out the algebraic sum of moments ΣM as follows –
ΣM = ΣMr-ΣM0
(v) Determine the position of resultant force R, from the toe (x̅) as follows
x̅ = ΣM / ΣV
(vi) Determine the eccentricity (e) of resultant R from the centre of the base (b)
e = b/2 – x̅
Where b is the base width of the dam.
(vii) Find out the normal stress at toe and heel, by the following equation –
(viii) Determine the principal and shear stresses at toe and heel.
(ix) Find out factor of safety against over-turning as follows –
F.S. = ΣMr / ΣM0
(x) Find out the factor of safety against sliding by following expressions –
(a) Sliding factor = μΣV / ΣH
(b) Shear friction factor = μΣV + bq / ΣH
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. Each cantilever element is thus dragged on its one face to greater deflections, by deeper adjacent cantilevers and is held on by the shorter cantilever on hill side. This develops a twisting effect on each cantilever.
3. Slab Analogy Method:
In this method the analysis is made by dividing the analogous slab into horizontal and vertical beams. Horizontal and vertical beams are brought into slope and deflection agreement by trial loads. 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 considered as a frame work of interconnected square frames, each square being diagonally connected at the corners.
5. Experimental Methods:
Experimental methods may be direct method and indirect method. Direct method is also known as three dimensional, model analyses, whereas indirect method as photoelastic model analysis.
(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 are made from elastic materials. The size of the models is made proportionate. The models are located in similar manner as the prototype and also subjected to loading of the prototype.
The structural actions, stress conditions, and deformations at various points of the model, are thereafter measured and a correlation is developed between the model and prototype, in regard to stress and deformation. From this correlation, the stresses and deformations for the prototype can be easily obtained.
(ii) Photo-Elastic Models:
In this method, photo-elastic models of dam are prepared from elastic materials. The materials used for photo-elastic models should be elastic, transparent, isotropic and free from initial stresses. Bakelite, celluloid, gelatin, and glass are the usual photo-elastic materials.
The stresses in the photo-elastic models are determined by a special optical instrument known as photo-elastic polariscope. Polarised light is passed through it and stress and strain conditions in the model are studied. This method locates such points or regions in the model, where stresses become concentrated. This method has limited application.