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Table 2. Natural frequencies of both the Earth wall and
concrete gravity dams
It was noticed that the natural frequencies of the Earth wall gravity dam is greater than that of the concrete gravity dam in first two modes being used.
The first mode shapes of the Earth wall and concrete gravity dams are shown in Figs 10 and 11 respectively.
Fig. 10 The first mode shape of the Earth wall
Fig. 11. The first mode shape of the concrete
Similarly, the mode shapes show that the Earth wall gravity dam is more affected than the concrete dam by the seismic load. The first mode shapes of both dams depict that the Earth wall gravity is more deflected than the concrete gravity dam at the same seismic load of peak ground acceleration of 0.05g.
6. STABILITY AND STRESS ANALYSES
The following assumptions are made for the Earth wall gravity dam
Freeboard = 30% of the reservoir height. Crest width = 0.23 times dam’s height. This is used to allow the passage of small vehicles, Base width = 0.87 times dam’s height. This is used to avoid tension in the base. Using similar triangles, θ = 48.80 and φ = 41.20 . See Fig. 12.
Vertical force = W1 (=γhl) + W2 (= 0.5γhl) + uplift (U = 0.5γhl) = 583.16kN
Horizontal force Pw
Pw (= 0.5γh2 ) = 313.92kN
F.S. = Net vertical force = 1.86
1.86 > 1.6. Hence, sliding criteria is favourably satisfied.
Sum of Overturning moment = 3051.16kNm
Sum of stabilizing moment = 5883.02kNm
So that F.S.= 1.93 > 1.6. Hence, overturning criteria is favourably satisfied.
Normal Stress at the toe considering the limiting case at e = 9.2/6 = 1.53m, then, Pn = 0.127N/mm2
Principal Stress at the toe, σ1 = 0.072N/mm2
Shear Stress at the toe, τ = 0.111N/mm2
The stresses obtained are less than the allowable values, therefore safe against overstressing.
Earth wall as a material has shown adequate resistance against seismic forces, as there are a lot of referenced evidences in this case. In the same vein, the comparison of the results between that of earth wall and concrete shows that the earth wall is much more affected than the concrete in the same seismic zone. This implies that in higher seismic zones such as zones 2 to 4, the collapse or response of the earth wall gravity dam will be significantly higher compared to that of concrete gravity dam.
However, this work has shown that earth wall gravity dams could be constructed in areas of yet a low seismic activity, especially in rural areas of a country like Nigeria, so as to support and provide more electric and water supply to people without fear of sever structural damages to the dam.
Instead of using 4-node plain quadrilateral elements, as done in this work, for the system discretisation and idealization, 8-node plain quadrilateral elements should be employed in order to refine the solution parameters with convergence criteria put into consideration. In the same vein, the number of modes of the analysis should be increased to further depict the response of the dams.
Moreover, this work shows that as a result of engineering and environmental sustainability, earth wall can be used in building small dams in rural areas in order to provide more hydro-electric power (HEP) and water supply, especially in developing nations of low seismic activity .
I wish to appreciate the thorough work of Mr. .S. T Owolabi for the programming of this research work.
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10. LIST OF SYMBOLS
ax, ag = ground accelerations;
B =interaction matrix, strain displacement matrix;
0BL0, 0BL1 = linear strain displacement transformation matrices;
0BNL = nonlinear strain displacement transformation matrix;
C = structure’s damping matrix; stress-strain material property;
tC = incremental stress-strain material property;
cp = fluid’s specific heat capacity
eij = components of velocity strain tensor;
F*(t) = matrix is defined in equation ;
Fn+1 = generalized effective force vector;
G = fluid mass matrix;
g = acceleration due to gravity;
H = fluid stiffness matrix; coordinates interpolation matrix;
h = interpolation or shape functions;
I = matrix is defined in equation ;
J = jacobian operator;
K = stiffness matrix;
K = generalized effective stiffness matrix;
0KL = linear stiffness matrix at configuration time t = 0;
0KNL = non linear stiffness matrix at configuration time t = 0;
L = fluid damping matrix;
M = structure mass matrix;
Pex = resultant seismic load on the structure;
Pew = resultant seismic load on the fluid;
p = nodal pressures degrees of freedom;
tR = total external load at configuration time t;
0S, tS, t+ΔtS = Piola-Kirchoff stress tensor at configuration time t = 0, t, t+Δt;
tij = thickness of the element;
te =period of integration;
U = nodal displacements;
0V = volume of integation at configuration time t = 0;
W = weight of the structure;
X = eigenvectors matrix;
Y = participation factors matrix;
Α = seismic coefficient; Newmark’s integration parameter;
αij = weighting factors;
βd = damping coefficient;
Γ = boundary of fluid or structure’s domain;
Δt = time step of integration;
δ = Newmark’s integration parameter;
δij = kronecker’s delta;
Λ = matrix is defined in equation ( );
λ = eigenvalues of the system;
μ = fluid dynamic viscosity;
ν = poisson’s ratio;
νi = velocity of fluid flow in direction xi;
ρ = fluid’s density;
τij = Cauchy’s stress tensor;
Φ = eigenvectors;
Ω = fluid or solid domain;
Ω = eigenvalues;
F = fluid quantity;
S = structure quantity;
T = matrix transposition;
t = configuration t;
F = fluid domain;
A1 Loadings on Dams
A1.1 Static loads
The static loading on the dam body comprises its weight and uplift forces. This has been computed during the dam design.
The net load on the dam body = 944.17kN
For the reservoir, The total load = 313.92kN
A1.2 Dynamic loads
In this aspect, the total loads acting on both the dam and reservoir is computed with the static loads put into consideration
The dynamic load at α =0.05, W = 47.2 kN, on the dam is
Pex = 47.21kN
For the reservoir at te = 0.01 , Ce =69.45 ,
Pew = 148.16 kN
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