Solution: Determination of the cross sectional profile of concrete gravity dam for the following data and plot the profile by the single step method of design. (maximum water depth to be retained)= 45m Tail water depth, h=0 Top (or crest) width, L=4m Unit weight of concrete,=25KN/m3 Unit weight of water,
w=10KN/m3
Uplift intensity factor,=1.0 Allowable coefficient of friction within concrete and between concrete and foundation, (f=0.65) Ultimate shear resistance of dam and foundation, Sa=4MPa Minimum permissible shear friction safety factor ,S.F.F=5 Wind velocity over water, Fetch Maximum allowable inclined stress in dam or foundation,=3MPa
Step one: Let to derive the base width of base of the elementary profile B=
=
= 36.9m ==37m
Now it is possible to calculate the base width of the triangular profile b=37-4 =33m Let’s consider the downstream inclination (Φd)
Using this value we can calculate the height of the triangular profile:
Before checking whether this dam is stable or not first lets determine the free board of the dam as follow Computing the significant wave height:Hs = significant wave height for this we use the equation :-
Hs = 0.0032
+0.76-0.24
= 0.0032
+0.76-0.24
= 1.95m
Wave rise above still water level (Freeboard),
then
Take the free board 3.0 m Therefore, height of the dam
4.80
Fig 1: the cross section of the dam Step 2: Now let’s check the stability of the dam for empty reservoir condition Computation of forces:Item
W1 W2
Description and dimension
Force KN Vertical force
Weight of dam 4*48*25 0.5*33*40.2*25 Sum
4800 16582.5 21382.5
Horizontal force
Moment arm from toe
Moment about the toe
35 22
168000 364815 532815
Step 3: Cheek whether the resultant is in the middle third (kern) i.e.
(The negative sign indicates the resultant passes towards the heel)
The resultant force was outside of the middle third at empty reservoir condition. Hence it needs to provide an upstream batter. Step 4: With the upstream slope provided, take moment of the upstream middle third point, for the now base width. Dam with upstream flare (batter)
Figure: free body diagram force analysis for the providing upstream batter By Applying Moment Area Method, lets calculate the centroid of the crest mass ( ) Total area (
AeF)* BE = 4(3) (2) + 1/2*4.9*4(2.67)
(12 + 9.8)BE = 24 + 26.17 BE = 2.3 m From triangle similarity theorem ACD
ABE
Y= 45- 8.4=36.6m
Now consider the dam for empty reservoir condition For reservoir empty condition (Taking summation of moment about the new upstream middle third
item
Description and dimension
Force in KN
Lever arm from point Horiz ‘m’ ontal force
Moment about point m
Vertical force
Antic lock wise (+ve)
Clockwise (-ve)
self weight W1
0.5*25*Bf*36.6
457.5Bf
12.33 - Bf
5640.98Bf-305
W2
4*48*25
4800
10.33
49584
W3
0.5*25*40.2*33
16582.5
2/3*33(12.33-4)
226682.8
=5640.98BF-305BF2+49584-226682.8 Using quadratic equation we can solve the values of ????? Take the positive value of
Check the stability of the dam for this section about the heel for reservoir empty condition.
Table1: Stability analysis for empty reservoir including upstream batter item Description and dimension
W1 W2 W3
self weight 0.5*25*Bf*36.6 4*48*25 0.5*25*40.2*33
Force in KN Vertical Horizont force al force
Lever arm Moment about point m from the toe Antic lock wise Clockwise (+ve) (-ve)
3202.5 4800 16582.5
39.33 35 24.67
125954.33 168000 409090.28
=703044.61
Therefore the resultant force lies in the middle third. Now let’s check the stability of the dam for reservoir empty condition A. Stability against overturning Not necessary since it is stable in empty condition! B. overstress
C. Vertical stress
D. Horizontal Normal Stress Upstream:.36(
) =38.32
Downstream: = 55.8 =37.6 E. Shear stress
= -45.76 KN/m2, .........Safe
F. Principal Stress Upstream
= 1099.96
-------------------------safe
Down stream
92.4048
--- -----------------------safe
for full reservoirs condition
Figure: Free body diagram for full reservoirs condition The detail force analysis in the below table Table: Force analysis for full reservoir condition Item
W1 W2 W3
Ph
Description and dimension I/self weight 0.5*7*36.6*25 0.5*25*37*45 25*3.6*3+(0.5*3.6*4.2 5*25) Sum II/water pressure 0.5*10*(45)2
Force KN Vertical force
Lever arm Horizont From the al force toe
3202.5 4800 16582.5
39.33 35 24.67
24585
Moment about from the toe Anticlockwise Clockwise (+ve) (-ve) 125954.33 168000 409090.28 703044.61
10125
15
151875
Pv1 Pv2
7*8.4*10 .5*10*7*36.6 lll. Wave Pressure Pwave 2*10* lV.uplift pressure Pv 10*44*.5*45
588 1281
40.5 41.67 76.13
-(9900) 16544
Including uplift pressure,
23814 53379.27
48
3654.24
29.33
290400
=16544KN
445929.24(Including Up lift) 334308.4 Excluding uplift pressure, Check the resultant force in kern
Check the stability I)
stability against overturning
II)
Stability against sliding Sliding factor , Shear friction factor
where S - shear resistance ,
III) stability against stressing (here All computation of stress is Excluding Uplift)
Vertical normal stress ,σz -------safe
Shear stress Upstream face
= Downstream face,
Horizontal normal stress , σy Upstream face
Downstream face,
Principal Stresses Up stream -450(0.0361) =
503.89
-------------safe
Downstream --here P’= 0 (no tail water) =1159.913
-----------------safe
Figure: Final profile of the dam Remark: Silt load, tail water and earthquake acceleration are neglected
