4.0 Design Criteria - Stability Analysis
4.1 Requirements for Stability
The following are the basic requirements of stability for a gravity dam:
i) Dam shall be safe against sliding at any section in the dam/dam foundation interface/within the foundation.
ii) Safe unit stresses in concrete/masonry shall not be exceeded.
iii) Dam shall be safe against overturning
4.2 Basic Assumptions
For stability analysis the following assumptions are made:
i) That the dam is composed of individual transverse vertical elements each of which carries its load to the foundation without transfer of load from or to adjacent elements.
ii) That the vertical stress varies linearly from upstream face to downstream face on any horizontal section.
4.3 Load Combinations
The following loading combinations have been prescribed by IS:6512 for stability analysis:
Combination A (Construction condition) Dam completed but no water in reservoir and no tail water
Combination B (Normal operating condition) Reservoir at full reservoir elevation, normal dry weather tail water, normal uplift and silt.
Combination C (Flood Discharge condition) Reservoir at maximum flood pool elevation, all gates open, tail water at flood elevation, normal uplift and silt
Combination D Combination A with earthquake
Combination E Combination B with earthquake
Combination F Combination C with extreme uplift (Drains inoperative)
Combination G Combination E with extreme uplift (Drains inoperative)
4.4 Forces acting on a gravity dam
The various external forces considered to be acting on a gravity dam are:
1. Dead loads
2. Reservoir and Tail water loads
3. Uplift pressures
4. Earthquake forces
5. Silt pressures
6. Ice pressure
7. Wave pressure
8. Thermal loads, if applicable
Dead Loads
Self weight of dam and weight of appurtenant works such as, spillway piers, gates, hoists, spillway bridge etc. are considered for computing the dead loads. The unit weights adopted in preliminary designs are 2.3 t/m3 for masonry and 2.4 t/m3 for concrete.
Reservoir & Tail Water Loads
The load due to reservoir water is calculated using hydrostatic triangular pressure distribution and taking unit weight of water as 1 t/m3. The weight of flowing water over spillway is neglected. The load due to tail water is calculated by taking tail water pressure corresponding to tail water elevation in case of NOF sections and for a reduced value in case of OF sections depending on the E.D.A.
Silt Pressures
The deposited silt may be taken as equivalent to a fluid exerting a force with a unit weight 0.36 t/m3 in horizontal direction and 0.925 t/m3 in vertical direction. Thus the horizontal silt and water pressure is determined as if silt and water have a horizontal unit weight of 1.36 t/m3 and vertical silt and water pressure is determined as if silt and water have a vertical unit weight of 1.925 t/m3.
Uplift Pressures
Water seeping through the pores, cracks and fissures of the foundation material, and water seeping through the body of the dam exert an uplift pressure on the base of the dam. It is assumed to act over 100% of the area of base and assumed to vary linearly from upstream to downstream corresponding to water heads. However, in case drainage galleries are provided, there is a relief of uplift pressure at the line of drain equal to two-thirds the difference of the hydrostatic heads at upstream and downstream. It is assumed that uplift pressures are not affected by earthquakes.
Earthquake forces (Ref: IS 1893 – 1984)
Earthquake forces are determined as per IS:1893. Design seismic coefficients in horizontal and vertical direction are worked out as per the above code based on the location of the project on the seismic map of India.
Design horizontal seismic coefficient ( h)
1. By Seismic Coefficient Method (for dams upto 100 m high)
h = I o
where,
h = Design Horizontal Seismic Coefficient
o = Basic Horizontal Seismic Coefficient (from Table 2, IS:1893)
I = Importance factor of the structure (3.0 for dams)
= Coefficient depending upon soil foundation system
(1.0 for dams)
2. By Response Spectrum Method (For dams higher than 100 m)
h = I FoSa/g
where,
Fo = Seismic Zone factor for average acceleration spectra (from
Table 2, IS:1893)
Sa/g = Average Acceleration coefficient read from Fig. 2, IS:1893 for
appropriate natural period and damping.
Design vertical seismic coefficient
The design vertical seismic coefficient is taken as half the design horizontal seismic coefficient.
Inertia forces on the dam
A triangular distribution of acceleration is prescribed for determining inertia forces on the dam. For horizontal inertia forces 1.5 times the design horizontal seismic coefficient is assumed at the top of the dam varying to zero at the base. For vertical inertia forces also 1.5 times the design vertical seismic coefficient is assumed at the top of the dam varying to zero at the base.
Hydrodynamic Pressure on the dam
The basic work in this regard has been done by Westergaard. Subsequently Zanger in 1952 presented formulas for computing hydrodynamic pressures exerted on vertical and sloping faces by horizontal earthquake effects. Based on Zanger’s work, IS:1893 gives the procedure for calculating hydrodynamic pressure on the dam.
Effects of Horizontal Earthquake Acceleration
Due to horizontal acceleration of the foundation and dam there is an instantaneous hydrodynamic pressure (or suction) exerted against the dam in addition to hydrostatic forces. The direction of hydrodynamic force is opposite to the direction of earthquake acceleration. Based on the assumption that water is incompressible, the hydrodynamic pressure at depth y below the reservoir surface shall be determined as follows :
p = Cs h wh
where,
p = hydrodynamic pressure in kg/m² at depth y,
Cs = coefficient which varies with shape and depth
h = design horizontal seismic coefficient
w = unit weight of water in kg/m³, and
h = depth of reservoir in m.
The approximate values of Cs for dams with vertical or constant upstream slopes may be obtained as follows :
where,
Cm = maximum value of Cs obtained from Fig.10, IS:1893
y = depth below surface in m, and
h = depth of reservoir in m
Fig. 1 : Maximum Values of Pressure Coefficient (Cm)
for Constant Sloping Faces
The approximate values of total horizontal shear and moment about the center of gravity of a section due to hydrodynamic pressure are given by the relations :
Vh = 0.726 py
Mh = 0.299 py²
where
Vh = hydroldynamic shear in kg/m at any depth, and
Mh = moment in kg.m/m due to hydrodynamic force at any depth y.
Inertia forces on the dam
1. Seismic coefficient method (For dams upto 100 m high)
A triangular distribution of acceleration is prescribed for determining inertia forces on the dam. For horizontal inertia forces 1.5 times the design horizontal seismic coefficient is assumed at the top of the dam varying to zero at the base. For vertical inertia forces also 1.5 times the design vertical seismic coefficient is assumed at the top of the dam varying to zero at the base. The design vertical seismic coefficient is taken as half the design horizontal seismic coefficient.
2. Response Spectrum Method (For dams more than 100 m high)
The fundamental period of vibration is calculated as under :
T = 5.55 H2/B (Wm/g/Es)0.5
where,
T = Fundamental period of vibration of the dam in secs.
H = Height of the dam in meters
B = Base width of the dam in meters
Wm = Unit weight of material of dam in kg/m
g = Acceleration due to gravity in m/sec
Es = Modulus of Elasticity of material in kg/m
Damping used for concrete dams = 5%
The design horizontal seismic coefficient is calculated using the above time period and for a damping of 5% from the average acceleration spectra given in IS:1893.
The basic shear and moment due to the horizontal inertia forces is obtained by the formulae given below:
Base shear = VB = 0.6 W. h
Base Moment = MB = 0.9 W.hCG h
where,
W = Self weight of dam in kg
hCG = Height of C.G. of dam above base in meters
h = Design Horizontal Seismic Coefficient
The vertical inertia forces are calculated using the same distribution as outlined in seismic coefficient method but using the seismic coefficient as calculated above.
5.0 Check for permissible stresses
Check for Compressive Stresses
1. Concrete
• Strength of concrete after 1 year should be 4 times the maximum computed stress in the dam or 14 N/mm whichever is more.
• Allowable working stress in any part of the structure shall not exceed 7 N/mm.
2. Masonry
• Strength of masonry after 1 year should be 5 times the maximum computed strength in the dam or 12.5 N/mm which is more.
• Compressive strength of masonry can be determined by compressing to failure 75 cm cubes (or 45 cm x 90 cm cylinders) cored out of the structures.
Check for Tensile Stresses
Nominal tensile stresses permitted in concrete/masonry gravity dams (as per is: 6512)
Load Combination Permissible Tensile Stress
Concrete dams Masonry dams
A Small Tension Small Tension
B No Tension No Tension
C 0.01 fc 0.005 fc
D Small Tension Small Tension
E 0.02 fc 0.01 fc
F 0.02 fc 0.01 fc
G 0.04 fc 0.02 fc
where, fc = Cube Compressive Strength of Concrete/Masonry
6.0 Check for Sliding
The dam should be safe against sliding across any plane/combination of planes passing through:
- The body of the dam
- Dam foundation interface
- Foundation
The partial factors of safety against sliding as per IS:6512 are given below:
Loading
Condition F Fc
For dams and the Contact plane with Foundation For foundation
Thoroughly investigated Others
A,B,C 1.5 3.6 4.0 4.5
D,E 1.2 2.4 2.7 3.0
F,G 1.0 1.2 1.35 1.5
The factor of safety against sliding shall be computed from the following equation and it shall not be less than 1.0.
(W – U) tan + c.A
F = F Fc
P
Where,
F = factor of safety against sliding
W = total mass of the dam
U = total uplift force
tan = coefficient of internal friction of the material
c = cohesion of the material at the plane considered
A = area under consideration for cohesion
F = partial factor of safety in respect of friction
Fc = partial factor of safety in respect of cohesion, and
P = total horizontal force
No comments:
Post a Comment