stresses, gravity method, analytical method, Practical profile of gravity dam, elementary profile, design considerations,

1. Gravity Dams
Asst Prof: Mitali Shelke
St. John College of Engineering and Management, Palghar
Department of Civil Engineering

2. Principal and Shear Stresses:
• The vertical stress intensity Pmax or Pmin determined from the equation,
is not the maximum direct stress produced anywhere in the dam.
• When the the reservoir is full the vertical direct stress is maximum at the toe as the the resultant is
nearer to the toe.
• To study the principal stresses that will develop near the toe let us consider a small element ABC
near the toe of the dam.

3. • Let the downstream face of the dam be inclined at an angle ᾳ to to the vertical. This face of
the dam will act as a principal plane because the water pressure p’ acts at right angle to the
face, and also there is no shear stress acting on this plane. Since the principal planes are at
right angles to each other the plane BC drawn right angles to the face AB will be the second
principal plane. Let the stress acting on this plane be σ.

4. • Let ds, dr and db be the lengths of of AB, BC and CA respectively.
• p’ is the intensity of water pressure on the face AB and pv is the intensity of vertical pressure on
face AC and σ is the intensity of normal stress on face BC.
• Considering unit length of dam forces acting on faces AB, BC and CA are p’ds, σdr and pv.db
respectively.

5. • For σ to be maximum, p’ should be zero i.e. when there is no tail water then in such
case:
• If hydrodynamic pressure (pe’) exerted by tail water during an earthquake moving
towards the reservoir is also considered, then net pressure on face AB will be (p’-pe’)
because the effect of this earthquake will be to reduce the tail water pressure.
• The principal stress σ can be given by:

6. Shear stress on horizontal plane near the toe:
• A shear stress ԏ will act on face CA on which vertical stress is acting.
• Resolving all forces in horizontal direction we get:

7. • Neglecting tail water, shear stress is given by,
• If effect of hydrodynamic pressure produced by earthquake moving towards reservoir
is also considered then equation for shear stress near the toe becomes:

8. Gravity Method for stability analysis:
• The preliminary analysis of all gravity dams can be made easily by isolating a typical cross
section of dam of unit width.
• The dam is considered to be made up of a number of cantilevers of unit width each which act
independently of each other.
• Assumptions:
• The dam is considered to be composed of a number of cantilever each of which is 1m thick and
each of which acts independent of each other.
• No loads are transferred to the abutments by beam action.
• The foundation of the dam behave as a single unit
• The materials in the foundation and body of the dam are isotropic and homogeneous.
• The stresses developed in the foundation and body of the dam are within elastic limits.
• No movements of the foundations are caused due to transference of loads.
• Small openings made in the body of the dam do not affect the general distribution of stresses and
they only produce local effects as per St. Venant’s principle.

9. Procedure by analytical method:
• Consider a unit length of the dam.
• Work out the magnitude and directions of all vertical and horizontal forces acting on the dam and
their algebraic sum.
• Determine the lever arm of all these forces about the toe.
• Determine the moments of all these forces about the twoe and find out algebraic sum of all these
moments.
• Find out the location of resultant force by determining its distance from toe x= ∑M/ ∑V
• Find out the eccentricity of the resultant using e= (B/2)-x.
• Determine the vertical stresses at the toe and heel using the equation

10. • Determine the maximum normal stresses that is principal stresses at the toe and heel using the
equation: ԏ= (pv – p’)tan σ and
• Determine the factor of safety against overturning as equal
∑stabilising moment(+)
∑disturbing moment(−)
• Determine the factor of safety against sliding using sliding factor = µ (∑V+Bq)/ ∑H
• Shear friction factor
• Sliding factor must be greater than unity and SFF must be greater than 3 to 5. The analysis should
be carried out for reserve full as well as empty case.

11. Practical profile of a gravity dam:
• The elementary profile of gravity dam is only theoretical. Certain changes will have to be
made in this profile in order to cater to practical needs. These needs are:
a) Providing a straight top width for road construction over the top of the dam.
b) Providing a free board above the top water surface so that water may not spill over the top of
dam due to wave action.

12. This addition will cause resultant force to shift towards the heel. Hence tension will be developed at
the toe.
In order to avoid this tension some masonry or concrete will have to be added to the upstream side as
shown in the figure.

13. Elementary profile of gravity dam:
The elementary profile of a dam subjected only to external water pressure on upstream side will be
right angle triangle having zero width at the water level and base width B at bottom.

14. When the reservoir is empty the only single force acting on it is self weight W of the dam and it acts
at a distance B/3 from the heel.
The vertical stress distribution at the base when the reservoir is empty is given as
The maximum vertical stress equal to 2W/B will act at heel and the vertical stress at toe will be 0.

15. When the reservoir is full the base width his governed by:
1. The resultant of all forces that is P, W and U passes through the outermost middle third point.
2. The dam is safe in sliding.
For the first condition to be satisfied the equation should be given by
Where Sc= specific gravity of concrete i.e. material of the dam.
C = constant called as seepage coefficient
According to USBR recommendation value of C is equal to 1 in calculation and 0 when no uplift is
considered.

16. If B is taken equal to or greater than H/√(Sc-C) no tension will be developed at the heel with full
reservoir,
when C = 1
If uplift pressure is not considered,

17. For second condition to be satisfied (the dam is safe in sliding) the frictional resistance should be
equal to or more than the horizontal forces
Equation can be given as:
From the above two equations of B the greater value should be chosen for design purpose.

18. In vertical stress distribution maximum stress will occur at the toe because the resultant is near the
toe. Hence the equation is given as:
And Pmin at the heel = 0.
The principal stress σ near the toe which is maximum normal stress is given by equation:
The shear stress ԏ at horizontal plane near the toe is given by equation:

19. Design considerations and fixing the section of dam:
1. Freeboard:
The margin between the maximum reservoir level and top of the dam is known as freeboard.
This must be provided in order to to avoid the possibility of water spilling over the dam top due to
wave action. This can also help as a safety for unforeseen floods higher than the designed flood.
The freeboard is generally provided equal to 3hw / 2.
Where,
These days freeboard equal to 4% to 5% of the the dam height is provided.

20. 2. Top width:
The effects produced by the addition of of top width at the apex of elementary dam profile and their
remedies are explaind below:
Let AEF with the triangular profile of dam of height H1.
Let element ABQA be added at the apex for providing top width ‘a’ for road construction.
Let M1 and M2 be the inner third and outer third points on base. Thus AM1 and AM2 are the inner
third and outer third lines.
The weight of element W1 will act through the CG of
this triangle i.e. along CM.
Let CM and AM1 cross at H, and CM and AM2 cross at
K.

21. 1. Reservoir empty case: We know that in the elementary profile the resultant of the force passes
through the inner third point when reservoir is empty.
The height H1’ below which the upstream batter is required can be worked out as:
Thus for the height greater than H1’ upstream batter is necessary.

22. 2. Reservoir full case: When the reservoir is full the resultant of all the forces acting on elementary
profile passes through the outer third point.
When W1 is added to this initial resultant at any plane below the plane PKQ the resultant will shift
towards upstream side of dam.
For economic point of view the resultant should lie near the the downstream face of dam and hence
the slope of the downstream face may be flattened from QE to QE’.

23. Thus an increase in top width will increase the masonry or concrete in the added element and increase
it on upstream face, but shall reduce it on downstream face.
The most economical top width without considering earthquake forces has been found by Creager to
be equal to 14% of the dam height.
It's useful value varies between 6m to 10m and is generally taken approximately equal to √H, where
H is the height of maximum water level above the bed.

24. Thank You.