Lecture - 06

Comprehensive Stability Analysis of Gravity Dams

Stability analysis

The stability analysis of gravity dams may be easily carried out by

·        Two-dimensional gravity method

·        Three dimensional methods (Slab analogy, trial and twist method)

·        Experimental studies on models

Two-dimensional gravity method

Assumptions

1.     The dam is considered to be composed of a number of cantilevers, each of which is 1 m thick and acts independent of the other

2.     No loads are transferred to the abutments by beam action

3.     The foundation and the dam behave as a single unit

4.     The materials in the dam body and foundation are isotropic and homogeneous

5.     The stresses developed in the dam and foundation are within elastic limits

6.     No movement of the foundation is caused due to transfer of loads.

7.     Small openings made in the body of the da do not affect the general distribution of stresses and they only produce local effects as per St. Venant’s principle

Procedure

·        Analytically

·        Graphically

Analytical Method

1.     Consider a unit length of the dam.

2.     Work out the magnitude and directions of all the vertical forces and the algebraic sum of all vertical forces acting on the dam, ∑ 𝑉

3.     Work out the magnitude and directions of all the horizontal forces and the algebraic sum of all horizontal forces acting on the dam, ∑ 𝐻

4.     Determine the lever arm of all these forces about the toe.

5.     Determine the moments of all these forces about toe and find out the algebraic sum of all those moments ∑ M

6.     Determine the position of the resultants force by determining its distance from the toe as:

𝑿 = ∑ 𝑴/ ∑ 𝑽

7.     Determine the eccentricity, e, of the result R from the toe as 𝒆 = B/𝟐 – 𝑿. It must be less than B/6 in order to ensure that no tension is developed anywhere in the dam.

8.     Determine the vertical stresses at the toe and heel using,

Pv = ∑V / B * [1 ± 6 e / B]

9.     Determine the principal stresses at the toe and heal points as:

σ_{at Toe} = Pv. Sec²α – (P^’ – P_e^’) tan²α

σ_{at Heel} = Pv. Sec²Ø – (P + P_e) tan²Ø,

Ø = is the angle which the u/s face makes with vertical

𝝉 _toe = (Pv – P’) tanα

𝝉 _Heel = [Pv - (P+ P_e) tanØ

They should not exceed the maximum allowable values. The crushing strength of concrete varies depending upon concrete grade.

10.  Find out the factor of safety against overturning

F.O.S = ∑ Stabilizing Moment (+) / ∑ Disturbing Moment (-) >2 to 3

(+) sign for anti-clockwise and

(-) sign for clockwise moments

11. Find out the factor of safety against sliding

Sliding Factor = ɥ∑V/∑H >1

Shear friction factor (S.F.F) = [ɥ∑V+bq] / ∑H > 3 to 5

High and Low Gravity Dams

H <  f / [γ_w (S_c + 1)], Low gravity dam

H  >  f / [γ_w (S_c + 1)], High gravity dam