GLOBAL RESEARCH PUBLICATION
Vol.1, No. 1, 2007
SEISMIC ANALYSIS OF EARTH WALL GRAVITY DAMS USING
DECOUPLED MODAL APPROACH
Adedeji, A. A.
Department of Civil Engineering, University of Ilorin, PMB 1515, Ilorin, Nigeria.
email: amadeji@yahoo.com
ABSTRACT
This study aimed at the assessment of the effects of seismic loads on the earth dam for design purposes.
The Unilorin concrete gravity dam is used in comparison with the earth wall gravity dam in seismic zone
1 with a peak ground acceleration of 0.05g. Finite element (FE) method of analysis was used by
employing Lagrangian-Eulerian formulation of 4-node plain quadrilateral elements, with modal analysis
used to decouple system dynamic equations, thereby saving computational time. The loadings were
determined based on EM 1110-2-22000(1995), while the FE model is being implemented using the
MATLAB programming tool. Though, the collapse and response of the earth gravity dam is higher than
that of concrete gravity dam, the dam satisfies the stability and stress requirements in design.
KEY WORDS
Sismic analysis, earth wall gravity dam, Finite element method
1. INTRODUCTION
The structural response of a material to different loads determines how it will be economically
utilized in the design process. The loads that structures are subjected to are either static or dynamic, or a
combination of both. Static loads are time-independent while dynamic loads vary with time. Dynamic
loads can be classified as cyclic, impact or moving but can occur simultaneously.
Earthquake is a natural disaster that has claimed so many lives and destroyed lots of property.
Earthquake hazards had caused the collapse and damage to continual functioning of essential services
such as communication and transportation facilities, buildings, dams, electric installations, ports,
pipelines, water and waste water systems, electric and nuclear power plants with severe economic losses.
Earthquake is a major source of seismic forces that impinge on structures others are Tsunami, seethe etc.
Earth wall is chosen as a material for the dam since its major constituent- earth is abundantly
available and provides a sustainable solution. This necessitates the seismic analysis of concrete gravity
dam. Finite element has been widely used in seismic analysis of concrete gravity dams (Waltz 1997, Lotfi
2003) with a defined approach as presented in this programme, using the most natural method based on
the Lagrangian–Eulerian formulation.
Smith (1985) claimed that dams are critical structures that should be made earthquake-resistant, since
earthquakes cause severe damages, consequent huge economic and life losses. Hence, the need to prevent
the occurrence of these earthquake hazards by carrying out seismic analysis of dams (Polyakov, 1985).
Over the years, a lot of work has been done in making earth-fill and concrete dams earthquakeresistant, with advances in structural vibration and finite element method which have aid in the seismic
Accepted for publication by:
GLOBAL RESEARCH PUBLICATION, No 1 Ramaprasad Roy Lane, Calcutta -70000, India
analysis of theses dams. However, not much attention has been paid to the seismic analysis of an earth
wall dam. More importantly, most recent methods in seismic analysis of concrete gravity dams have not
been employed for the seismic analysis of the earth wall dams. Therefore, the purpose of this is to apply
the decoupled modal approach in the seismic analysis of Earth wall dams.
Earthquakes had caused severe damages and consequently huge economic losses including losses of
lives. On 27th December, 2004, The Punch newspaper reported that Tsunami, a seismic sea wave claimed
a lot of lives, rendered many homeless and destitute, and destroyed essential services with Sri Lanka,,
Indonesia, Somalia, India, etc been the major casualty countries. Nigeria is not left out from the
occurrence of ground motions caused by earth movement as earth tremors had been recorded to occur
some recent years back in Abeokuta and Ibadan. Most structures and infrastructure in the country are not
earth-resistant. Also, earth wall, as a sustainable material is quite economical in construction of dams
since there is abundant availability of good earth or soil, especially in localities closed to road
construction and excavation sites.
This study involved the use of finite element method for the dynamic analysis of both the dam and
reservoir bodies, using a modal approach as the Lagrangian-Eulerian formulation forming the basis. Also,
static loads (weights, hydrostatic pressures) are each visualized as being applied in one separate increment
of time.
The analytical computation of the modal approach procedure has been carried out and implemented
using MATLAB programming tool. The pseudo-static seismic coefficient method was adopted in
computing the seismic loads on the dam. The dam used as a case study was assumed to be in seismic
zone 1 with seismic coefficient ranging between 0.0 and 0.05. The dam was analysed seismically using
the decoupled modal approach and the results were compared with that of the concrete gravity dam.
2. SEISMIC RESPONSE OF EARTH AND CONCRETE GRAVITY DAMS
In earth dams, seismic forces or shaking can induce destabilising deformation or outright failure if not
made earthquake resistant. A permanent simplified procedure can be adopted to estimate permanent
horizontal displacements of the dams using finite element method that account for non-linear material
behaviour and strength reduction due to liquefaction or strain softening. It has been shown ((Hatami,
2001) that the seismic performance of earth dams has been related to the nature and state of compaction
of the fill material. Concrete dam’s structural safety and stability are jeopardized due to the hydrodynamic
load of the reservoir that is subjected to ground motion
2.1 Fluid –Structure System
During earthquake occurrence, the dam and reservoir body respond differently, as a result of
hydrodynamic forces impinging on the fluid body and solid structure. As a result of this, interaction will
occur between the fluid–solid structure interfaces as particles move relatively to the mesh points whereas,
the meshes moves with the material particles (Bathe, 1996, Qixiang et al. 2000). Much research work has
been carried out for the dynamic response of the fluid-solid structure systems. Several methods of
analysis for the fluid-structure systems use finite element idealization in the non-linear dynamic response
of the system (Fenves and Vargas- Loli, 1988).
FP
FP
SF
ΩF
ΩF
N
FU
SU
Fig. 1.(a) Domains and boundaries of the fluid-structure system
F
S
F
S
F
S
F
S
Fig. 1(b): Degrees of freedom of the fluid-structure system
In Fig.1(a, b) the following terms are defined as:
F=
Fluid domain
FP'
= Fluid boundary at reservoir end
FP
= Free surface of the fluid
SF
= Fluid-structure common interface
N = Normal to the fluid boundary
S
= Structure domain
SU
= Portion of structure boundary along an earthquake
ground motion path
FU
= Portion of structure boundary along an earthquake
ground motion path
3. LOADINGS
3.1 Static loads.
The static loads are due to (i) The weight of the dam: the unit weight is assumed to be 19.62kN/m3
until an exact unit weight is determined from materials investigation., (ii) Hydrostatic pressure of the
water in the reservoir and (iii) The uplift forces caused by hydrostatic pressure on the foundation at the
interface of the dam and the foundation. Uplift forces are usually considered in stability and stress
analysis to ensure structural adequacy and are assumed to be unchanged by earthquake forces.
3.2 Dynamic loads
Earthquake or seismic loads are the major dynamic loads (Major 1980, Schoeber 1981, Polyakov
1985, Wyatt (1989). being considered in the analysis and design of dams especially in earthquake prone
areas. They are of kinematic origin and owe their existence to vibration caused in the structure by the
movement of the earth’s surface during an earthquake. They have random characteristics and are regarded
as deterministic in practical calculations to simplify the design model. According to EM1110- 22200(1995), the earthquake loading used in the design of concrete gravity dams are based on design
earthquakes and site specific motions determined from seismological evaluation. The seismic coefficient
method is used in determining the resultant location and sliding stability of dams. Seismic analysis of
dams is performed for the most unfavourable direction, despite the fact that earthquake acceleration might
take place in any direction. Seismic coefficient method of analysis is commonly known, as pseudostatic
analysis, and is the ratio of the earthquake acceleration to the acceleration due to gravity. Fig.2 shows
the dynamic loads on a gravity dam.
There are different ways of computing earthquake loads on dams. The deterministic approach will be
employed where the ground acceleration in terms of g (acceleration due to gravity) is specified for the
region where the dam will be constructed. Hence, the exciting force on the structure is,
P(t) =
Max
(1)
and
ax = αg
(2)
where ax, α, g are the ground acceleration, seismic coefficient and acceleration due to gravity
respectively.
y
h
Pew
Pew
0.4h
Fig. 2: Seismically loaded gravity dam
From Fig. 2 and equation ( ) therefore, the equilibrium system is expressed as:
In which
Pex = Max = Wαg /g = Wα
(3a)
ax = αg and
(3b)
W = Mg
along vertical direction
Pew =(2 * Ce * α * y * √ (h *y)) / 3
(4a)
and
Ce = 51 / √ (1 – 0.72 * (h / (1000te))2)
(4b)
where Pex, M, ax, W, α, g are the horizontal earthquake force on the dam, mass horizontal earthquake
acceleration, weight, acceleration due to gravity and seismic coefficient respectively. Also Pe w, h, te are
the additional total water load down to depth y, total height of reservoir, and period of vibration
respectively.
3.3 Forced – damped Vibration.
Forced vibration is the vibration caused by a time- dependent disturbing forced. The governing
equation for forced-damped vibration is
MÜ + CŮ + KU = P(t)
(5)
where M, C, K, Ü, Ů, U, and P(t) are the mass, damping , stiffness, acceleration, velocity, displacement
and exciting force on the body.
4. FINITE ELEMENT METHOD (FEM).
The finite element method is the best method used in analysing the internal forces of element(s) by
solving differential equation obtained through their discretisation in space dimensions (Adedeji, 2004).
Lotfi (2003) and Iroko (2001) stated that finite element method has been the most widely used method for
seismic analysis of concrete gravity dams. Finite element (FE) models are used for linear static and
dynamic analyses and for non-linear analyses that account for interaction of the dam and foundation. The
FEM provides the capability of modeling complex geometries and wide vibrations in material properties.
Two-dimensional finite element analysis is generally appropriate for concrete gravity dams.
Lotfi (2003) argued that the Lagrangian-Eulerian formulation is the most natural method, and
employs nodal displacements and pressure degrees of freedom for the dam and reservoir region
respectively. Bathe (1996) had previously put down this idea stating that in this type of formulation, the
mesh points move but not necessarily with the material particles, due to fluid-solid coupling and
interaction (Fenves and Vargas, 1988) that allows the approach to be used in modelling the fluid flowstructures interaction. The direct integration methods are used to solve the linear dynamic response of a
system of finite elements
There are different methods used in the direct integration process and are the Central difference, the
Houbolt, the Wilson, the coupling of different integration operators and Newmark methods (Bathe,
1996). Lotfi (2003) claimed, that the direct integration process is usually carried out by applying
Newmark’s algorithm or method, with the following assumptions used by (Bathe, 1996), are
t +Δt
t +Δt
Ů = tŮ + [(1-δ)tÜ + δt +ΔtÜ] Δt
U = tU + tŮΔt + [(0.5-α)tÜ + αt +ΔtÜ] Δt2
(6)
(7)
where t, Δt, tÜ, tŮ, tU, δ, α are time, time step of integration, acceleration at time t, velocity at time t,
displacement at time t, and integration parameters respectively.
However, Lotfi (2001) reported that a non- symmetric system of equations to be solved at each time
step will be encountered and not direct method by Newmark’s algorithm will not be computationally
efficient.
4.1 Decoupled Modal Technique
Bathe (1996) stated that if the integration will be carried out for many time steps, it will be more
effective to first transform the equilibrium equations of equation (8) into a form in which the step-by-step
solution is less costly. Based on this, Lotfi (2003) applied the modal approach to obtain the natural
frequencies and mode shape of the fluid–structure system by rewriting the equilibrium equation (5) as:
KΦ = ω2MΦ
(8)
where ω, Φ, K, M are the eigenvalues, eigenvectors, stiffness and mass of the system. The above
equation is the generalised eigenproblem from which Φ (mode shapes or eigenvectors) and ω
(frequencies or eigenvalues) are determined (Bathe, 1996). Thereafter, the solution at different time steps
can be estimated based on the combination of the modes at the different time steps (Lotfi, 2003).
4.2 Structural vibration
It is well known that, the linear dynamic equation for the solid structure (dam), is given as in equal
and are reproduced here.
MÜ + CŮ + KU = P(t)
(9)
Therefore, the linear dynamic equation for the fluid region (reservoir) is written as,
GÜ + LŮ + HU = R(t)
(10)
where M, C, K, Ü, Ů, U, and P(t) are the mass, damping , stiffness, acceleration, velocity, displacement
and exciting force on the dam. G, L, H, Ü, Ů, U, and P(t) are the mass, damping , stiffness, acceleration,
velocity, displacement and exciting force on the reservoir.
4.3 Lagrangian-Eulerian formulation
For the Lagrangian-Eulerian formulation for the fluid-structure system, both the Lagrangian and
Eulerian formulations are coupled together. The Lagrangian formulation as stated before is used for
formulating the solid domain as given in equation (11)
∫oV tSij δ tεij d0V = tR
(11)
and the Eulerian formulation is employed for the fluid domain,
τij = -tp* δij + 2μ* teij
(12a)
eij = (tνi,j + tνj,i)/2
(12b)
t
t
where tSij, teij, 0V, tR, tτij, tp, δij, μ are the Piola-Kirchoff stress tensor at time t, Green-Lagrange strain
tensor at time t, volume at time t = 0, Cauchy stress tensor at time t, pressure displacement at time t,
Kronecker delta and dynamic viscosity respectively.
4.4 Finite Element Model of a Dam-Reservoir System
Discretisation of the dam-reservoir system is shown in Fig. 3.1 while in Fig. 3.2 shows twodimensional plane solid and fluid quadrilateral continuum elements used in the discretisation of the
coupled dam-reservoir system.
Linear Dynamic Analysis for the Coupled Dam-Reservoir System is shown in details by Lotfi (2003),
while the damping matrix is totally symmetric and the following relation holds.
Ku = -MuT
(13)
It will be computationally inefficient to use the direct integration process (usually carried out by
Newmark’s algorithm) since a non-symmetric system of equations to be solved at each time step will be
encountered. On the other hand, the modal approach depends on obtaining the natural frequencies and
mode shapes of the system.
4.5 Decoupled Modal Approach
The eigenvalue problem corresponding to equation 3.7 is
K Xj = λjM Xj
(14)
where λj = eigenvalue of the system and eigenvector matrix
It is clear that the eigenvalues of this system are real and free vibration modes exist. However, it is
noted from the form of matrices K and M that the system is not symmetric and standard eigenvalue
computation method are not directly applicable. It would be computationally costly since additional
variables are required in using the available techniques to arrive at a symmetric form and reduce the
problem to a standard eigenvalue one. On the contrary, eigenvalues and vectors extracted from the
following eigenproblem will be used as:
KsXj = λjMsXj
(15)
The eigenvectors of the (15) are not the true mode shapes of the coupled system but can be presumed
as Ritz’s vectors and be combined to estimate the true solution. Standard methods are used to obtain the
solutions of this substitute eigenproblem. Using the orthogonal condition and normalizing the modal
matrix with respect to mass matrix, one could have:
XTMsX = I
(16a)
XTKsX = Λ
(16b)
where I is the identity matrix and Λ is a diagonal matrix containing the eigenvalues of the symmetric
substitute system. The following relations are derived from relations in equations (15) and (16).
XTMX = I +XTMuX
(17a)
XT KX = Λ – XTMuTX
(17b)
As in modal techniques, the solution is written as a combination of different modes.
U = XY
(18)
The vector Y contains participation factors of the modes. Applying the Newmark’s technique for each
new time step,
KYn+1 = Fn+1
(19)
K and Fn+1 are denoted as the generalized effective stiffness matrix and the generalized effective
force vector of the system at Time step n+1 respectively. They are defined as:
K = a0(I + XTMuX) + a1C* + (Λ - XTMuTX)
(20)
Fn+1 + F*n+1 + ( I + XTMuX)(a0Yn + a2Yn + a3Yn)
+ C*(a1Yn + a4Yn + a5Yn)
And,
(21)
In general, the vector of participation factors can be solved through relation in equation (19).
Thereafter, the unknown vector is obtained by equation (15) as usual in the modal procedure, it must be
also mentioned that the generalized effective stiffness matrix employed in equation (19) is inherently
unsymmetrical. However, it may be easily transformed to a symmetric matrix by multiplying certain rows
of the matrix in equation (19) by an appropriate factor.
The symmetric matrices were utilized in the substitute eigenproblem that corresponds to the
decoupled dam-reservoir system. Therefore, the natural frequencies and eigenvectors are actually related
to this decoupled system. In actual programming, one can modify the usual subspace iteration routines to
converge to the desired lowest modes of the dam first, and similarly for the finite reservoir region
afterwards by an appropriate selection of initial vectors. Meanwhile, they can also be obtained as two
separate problems. Assuming for simplicity, without loss of generality that the mode shapes for the dam
are ordered first and the ones for the finite reservoir considered subsequently in the modal matrix. All
similar arrangements for the eigenvalues in the diagonal matrix Λ and the relative matrix formations are
expressed by Lotfi (2003).
It is assumed that the damping matrix of the dam is of viscous type for the analysis carried out by
modal approach in time domain. Therefore, this leads to following relationship, so that the structural
damping matrix for the stress-strain relationship.
C1* = 2βdΛ1/2
(22)
where βd is the equivalent damping factor, which is assumed constant for all modes. Substituting equation
(21) into equation (20) and (21), the expanded form of equation (19) is now concluded as ,
a0I1 + a1C1* + Λ1
-XTBTX2
Y1
F1
=
T
a0X2 BX1
a0I2 + a1X2 LX2 + Λ2
T
Y2
n+1
(23)
F2
n+1
The vector of the participation factors is also assumed to be partitioned into two parts in this relation,
and as before the indices 1, 2 correspond to dam and reservoir modes respectively. Equation (23) is
equivalent to (19), which is initially an unsymmetrical system of equation. This matrix relation could
become symmetric by multiplying the lower matrix equation by a factor of –a0-1, which yields the
following equation.
a0I1+ a1C1*+ Λ1
-XTBTX2
Y1
F1
=
- X2TBX1
–a0-1 (a0I2+a1X2TLX2+Λ2)
Y2
n+1
(24)
–a0-1F2
n+1
The above equation (24) can now be readily solved for the vector of participation factors at each time
step, and the original unknown vector is obtained from equation (15).
5. ANALYSIS AND DISCUSSION OF RESULTS
5.1 Case Study: A Typical Analytical Example
To verify the capability of this numerical procedure, the on-going construction of University of Ilorin
(Unilorin) dam is considered. And since the country is generally less prone to earthquakes or tremors, the
seismic zone of this area is taken as Zone 1 whose seismic coefficient ranges from 0 to 0.05g. The dam
was previously designed as a concrete gravity dam whose data is utilized in this study. Fig. 3 shows the
dam geometry which is to retain 1.8 million cubic meters of water with a reservoir length and height of
200m and 8m respectively.
2.5m
2.5m
8m
7.6m
200m
9.2m
Fig. 3: The Unilorin concrete gravity dam geometry.
5. 2 Material Properties and Basic Data
The concrete and earth wall is assumed to be homogenous and isotropic with the following basic
properties in Table 1.
Table 1. Properties of materials
Property of material
Concrete
Earth
Elastic modulus, Ec (GPa)
22.75
21.2
Poisson ratio, νc
0.20
0.15
Unit weight, γc (kN/m3)
24.80
19.62
Dry density γd
18.20
Damping coefficient βd
0.05
0.05
5.3 Finite Element Analysis and Formulation for the System
In Fig. 4 a typical numbering of the dam system indicates the fluid- and structural wall domain, while
an isoparameter of an element showing fluid-structural members domain is shown in Figs. 5 to 6.
Fig. 4. Numbering of the elements in both the fluid and structure domains
s
3
4
b
r
2
1
a
Fig. 5: Isoperimetric element definition
For the reservoir, the two element types are:
s
3
4
s
3
0.765m
4
r
0.35m
2
1m
Element type I
1
r
2
1m
Element type II
Fig. 6: Element types in the fluid domain
1
For the dam body, the element types are as shown in Fig. 7.
L2
L2
s
3
4
0.765m
1
L1
Element type I
r
s
0.765m
r
2
s
3
4
1
L1
Element type III
1
L1
Element type II
L2
2
L2
3
4
0.765m
r
2
s
3
4
0.765m
r
1
L1
Element type IV
2
s
3
4
3
0.50m
r
2
0.625m
1
Element type VI
s
4
0.35m
r
2
0.625m
1
Element type V
Fig. 7: Element types of the dam domain
5.4. Programming the FEM model
The steps carried out above are followed in the programming process. Figs. 8 and 9 below shows the
MATLAB environment in which all the steps were performed.
Fig. 8: Function M-file for implementing the fluid domain model
Fig. 9: Function M-file for implementing the solid domain model
5.5 Seismic Response of the Earth Wall and Concrete Gravity Dams
The analysis is carried out by modal approach. The eigen-problem is solved in the first step based on
the symmetric parts of the coupled equation of the system. The first two mode-numbers were considered
to obtain the first two natural frequencies of both the earth wall and concrete gravity dam as shown in
Table 2
Table 2. Natural frequencies of both the Earth wall and
concrete gravity dams
Mode Number
Natural Frequency (Hz)
Earthwall
Concrete
1
0.9876
0.7421
2
0.9997
0.8126
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
gravity dam
Fig. 11. The first mode shape of the concrete
gravity dam
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.
2.5m
2.5m
W1
Ф
8m
W2
7.6m
Hk
θ
U
Fig. 12: Stability analysis diagram for the Earth wall gravity dam
Vertical force
Vertical force = W1 (=γhl) + W2 (= 0.5γhl) + uplift (U = 0.5γhl) = 583.16kN
Horizontal force Pw
Pw (= 0.5γh2 ) = 313.92kN
Sliding Criteria
F.S. = Net vertical force = 1.86
Horizontal force
1.86 > 1.6. Hence, sliding criteria is favourably satisfied.
Overturning Criteria
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.
Stress Analysis
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.
7. CONCLUSION AND RECOMMENDATION
7.1 Conclusion
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.
7.2 Recommendation
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 .
8. AKNOWLEDGEMENT
I wish to appreciate the thorough work of Mr. .S. T Owolabi for the programming of this research work.
9. REFERENCES
Adedeji, A. A. (2004); Finite Element Method, CVE 567 Lecture Notes, Department of Civil
Engineering, University of Ilorin, Ilorin.
Bathe, K. J. (1996); Finite Element Procedures, Seventh Edition, Prentice-Hall Inc, New Jersey.
CIWAT ENGINEERS (1996); Design Report of the University of Ilorin Dam, CIWAT ENGINEERS,
Ilorin, pp 14 – 25.
EM 1110-2-2200 (1995); Design of Concrete Gravity Dams, Engineer Manual of the US Army Corps of
Engineers, US Army Corps Publications Depot, National Technical Information Service, Springfield,
VA, www.us.armycorps.engineers.com/engineer’s manual.
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Fourth Edition, Butterworth, London, pp 19/14 – 19/ 15.
10. LIST OF SYMBOLS
ax, ag = ground accelerations;
B =interaction matrix, strain displacement matrix;
0
BL0, 0BL1 = linear strain displacement transformation
matrices;
0
BNL = 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;
0
KL = linear stiffness matrix at configuration time t =
0;
0
KNL = 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;
t
R = total external load at configuration time t;
0
S, 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;
0
V = 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;
Superscripts
F = fluid quantity;
S = structure quantity;
T = matrix transposition;
t = configuration t;
Subscripts
F = fluid domain;
APPENDIX I
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