STRESS FIELD EVALUATION IN CONCRETE GRAVITY DAMS USING THE
PSEUDO

STATIC AND PSEUDO

DYNAMIC APPROACHES
Paulo Marcelo VIEIRA RIBEIRO
Doctorate Student, Universidade de Brasília
–
UnB
Lineu José PEDROSO
Professor, Universidade de Brasília
–
UnB
Silvio C
ALDAS
Civil
Engineer
, Centrais Elétricas do Norte do Brasil

ELETRONORTE
Brazil
1.
INTRODUCTION
Traditionally, concrete gravity dams have been d
esigned and analyzed by a
simplified
procedure, which does not consider the structural elasticity and
the
fluid
compressibility
(Pseudo

Static Method). Some authors have already found
out
that
the consideration of these aspects can produce signific
ant stresses
in the dam,
indicating
that the simple procedure might not be appropriate for the design of
this
type
of structure
[1,2].
A study conducte
d by
[1] indicates an alternative
procedure for evaluation of
the
seismic action
s
, incorporating
the effects of
the
elasticity o
f the structure and the
compressibility of the water. It
is a simplified analytical proced
ure, based on the
spectral response of the structure. Its application consists in the calculation of
the
seismic loading
, whi
ch should be applied to the dam in a equivalent static
analysis.
In design conditions where the dam can be treated as a rigid b
ody (period of
vibration is approximately equal to zero), the proposed model by the Pseudo

Static
method is appropriate, and a constant distribution of seismic accelerations along
the dam’s height can be adopted, neglecting relative motion contribution. Th
is
method considers that the dam is provided with the same acceleration of an
infinitely rigid foundation, and the peak ground acceleration produces the largest
seismic effects on the structure.
However, there are conditions where the effects of the flex
ibility of the dam
should be considered, and analysis of
the
dynamic response of the structure is
needed. There are reports of accidents with dams that were designed by the
traditional procedure, which were submitted to seismic actions much higher than
tho
se laid down by the usual rigid body procedure [1]. The Koyna dam, for
example, was designed with a seismic coefficient of 0.05g. However, the 1967
earthquake, occurred in India, was able to produce actions that exceeded the
design considerations. As a res
ult, many cracks were formed along the dam.
The structural flexibility is of fundamental importance for understanding the
actions produced by the earthquake. A large amplification of the ground
acceleration can occur with the consi
deration of this effect.
Fig.
1 and 2 illustrate
the total acceleration responses produced in a system of 1 degree of freedom
(obtained by numerical integration, using a fourth order Runge

Kutta procedure),
when subjected to a short range of the north

south component of
the
El Ce
ntro
earthquake

1940 [3]. The first system has a period of vibration equal to 0.02s,
while the second system presents a period of vibration equal to 1s. The damping
for the two situations is equal to 2%. It can be observed that
in the first system
(Fig.
1) the response approaches the ground acceleration. Thus, this indicates a
rigid body motion.
T
he second system
presents
a
n amplification of the
acceleration response
,
with respect to the
earthquake
produced
component
. At a
certain instant the total accele
ration achieved by the system is twice the value of
the ground acceleration.
The behavior shown in Fig.
1 reveals an important characteristic of the
seismic design spectra. On this diagram the spectral acceleration tends to move
closer to the peak ground
acceleration as the period of vibration of the structure
decreases. This indicates that the system ac
quires properties similar to those of a
rigid body motion
, with the total
acceleration response approximately equal to the
ground acceleration. In these c
ases the use of Pseudo

Static method is
acceptable. According to
[4] the use of this procedure is valid for dams with
fundamental
periods
of vibration
less than 0.03s. Thus
,
the dam can be treated as
a rigid body, with a constant coefficient equal to the g
round produced acceleration,
distributed throughout its height (sometimes called seismic coefficient method).
Structural relative motion is neglected on this type of analysis. The beh
avior
shown in Fig.
2 illustrates
a dynamic amplification of the ground m
otion. In some
cases the gain reaches 75% of the ground acceleration. A dam with the specific
characteristics of this dynamic system, designed by the traditional procedure,
would have a peak ground acceleration of approximately 0.04g, while the flexible
st
ructure reaches a maximum acceleration of 0.07g.
2.
PSEUDO

STATIC METHOD SEISMIC ACTIONS
The Pseudo

Static method seismic actions are derived from the hypoth
esis
of a rigid body moving
towards an incompressible fluid. Thus
,
the structural mass
is treated
with a uniform distri
bution of acce
leration, equal to the
infinitely rigid
foundation
acceleration
. The seismic loading (distributed per unit of height) to be
applied in an equivalent static analysis is composed of two parts: the inertia force
and the hyd
rodynamic pressure
s
. The first one is
given by
the product of the
corresponding mass distribution in the analyze
d section with the design
established
rigid body acceleration. The hydrodynamic pressure distribution is
based on
studies developed by
[5], whic
h were
recently reviewed by
[6], and
represents the inertial effects of the reservoir along the fluid

structure interface.
0.03
0.02
0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
time (s)
acceleration (g)
total acceleration (Runge Kutta)
ground acceleration (El Centro)
Fig. 1
Single degree of freedom total acceleration response (0.02s period)
0.08
0.06
0.04
0.02
0.00
0.02
0.04
0.06
0.08
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
time (s)
acceleration (g)
total acceleration (Runge Kutta)
ground acceleration (El Centro)
Fig. 2
Single degree of freedom total acceleration r
esponse (1s period)
This procedure is also known as the Seismic Coefficient Method and was
widely used for the seismic resistant design of dams. The simplified character and
the routine procedure is one of the major advantages of this method, despite its
drawbacks, which include not considering: the structural elasticity, damping forces,
variation of the foundation acceleration over time, and the alternation and short
duration characteristics of the seismic loading [7].
The Pseudo

Static M
ethod seismic act
ions are given by
Eq.[1]
:
..
g
S
v
C S y w y g p y
g
[
1
]
where
g
v
stands for the horizontal ground acceleration,
g
indicates the
gravity
acceleration,
S
w
represents the distribu
ted dam weight per unit of height and
gp
is related to the hydrodynamic pressure distribution along the fluid

structure
interface (relative to a rigid body motion towards an incompressible fluid [2]). Thus,
the resulting seismic fo
rces and moments in an analyzed section are given by
application of this seismic loading in an equivalent static analysis.
2.1.
E
QUIVALENT STATIC ANALYSIS
Stress analysis procedures can be applied to the structure once the seismic
loading (distributed per
unit of height) has been established. The procedure
involved in the Pseudo

Static method is simple, and consis
ts in applying this
loading to
an equivalent static analysis. The stress analysis calculation may be
driven by analytical or numerical methods. On
e the highlights of the analytical
tools is the Gravity Method [8] with a formulation that provides the internal
stress
distribution
in the dam
geometry
. These are divided into three
in plane
componen
ts
: two normal stress
es
(
to
the vertical and horizontal
planes,
respectively) and a
shear
stress
. These are given
, respectively,
by Polynomial
Equations [2], [3] and [4]
. A great attraction of this method is the ease of use, in
addition to the excellent results obtained
when compared to more refined
numerical s
olutions [9]
.
Fig.
3 and 4 illustrate, respec
tively,
results from
the stress
field
distribution
obtained with this methodology and
with
the application of
the
Finite Element Method. A clear similarity between the
se
two stress fields can be
observed. Genera
lly, this comparison tends to lose quality as the selected section
approaches
the foundation.
In sections where the
hypothesis of
trapezoidal
distribution of vert
ical normal stresses
is not verified
,
the method collapses
, since
this a basic assumption of t
his procedure.
However this method
remains as one of
the most widely used
approaches for preliminary dam design.
( )
Z
y a b y
[
2
]
2
1 1 1
( )
ZY
y a b y c y
[
3]
2 3
2 2 2 2
Y
y a b y c y d y
[
4]
where a, b, a
1
, b
1,
c
1,
a
2,
b
2,
c
2
and d
2
are constants related to the analyzed section,
which depend on parameters such as geometry, forces and moments on the
design elevation.
A more detailed explanation can be found on [2].
Fig. 3
Gravity Method results
(units: kPa)
Fig. 4
Finite Eleme
nt Method results
(units: kPa)
Another alternative for the analytical stress evaluation is the application of
the
procedure developed by [
10].
This author proposed
the
stress
cal
culation
only
at the dam’s upstream and downstream slopes
, which usually
pr
ovide the critical
design
regions of this type of structure when facing
seismic actions. This
reduces
the design equations to a procedure
simpler than the
Gravity Method
,
based on
the infinitesimal prisms equilibrium at
upstream and downstream
slopes
.
On
t
he
se
formulations the
involved
terms
are reduced to:
normal stresses on horizontal
planes
, hydrostatic pressure
s
, hydrodynamic pressure
s
and slope angle
s
.
All
these components are immediately provided
, except for the norma
l stress
es
,
which can be obtained
by application of the classical beam theory
. The latter is a
function of the moment of inertia,
and
the result
ant
of forces
and moments in the
analyzed section. These actions
can be obtained in practice with the interpretation
of seismic l
oading by means o
f
line
segments
[10], forming
trapezoidal loading
areas
. Thus, the calculat
ion of the seismic action is re
duced
to
the solution of a
load
ing
formed by a combination of straight segments, largely simplifying the
procedure for ca
lculating the seismic result
ant forces and moments and providing
excellent
results
if an appropriate number of design divisions
is chosen.
3.
SEISMIC ACTIONS IN THE PSEUDO

DYNAMIC METHOD
This analytical pr
ocedure was developed by
[1] as an alternative to more
general procedures, wh
ich required the use of a
computer
in order to evaluate the
structural seismic response. It consists on a simplified analysis of the spectral
response, which determines the structure’s response in the fundamental vibration
mode due to a horiz
ontal ground m
otio
n [11]
. This author
observed that the
response of structures with short periods of vibration, such as concrete gravity
dams
,
when s
ubjected to seismic actions
, was largely influenced by the
fu
ndamental vibration mode. It was
also concluded that
the
ver
tical components
of
the ground acceleration
exerted
little
i
nfluence on the structural response
.
Based
on these conclusions this author s
uggest
ed
a simplified methodology for
preliminary
analysis
of concrete gravity dams
.
The dam, which in the Pseudo

Sta
tic Method was supposed rigid, is now
treated
as flexible (see Fig.
5), and the water in the reservoir considered as a
compressible fluid. The seismic loading now depends on the fundamental mode
seismic response of the structure, associated with the corres
ponding horizontal
ground motion.
Fig. 5
Pseudo

Dynamic Method
seismic response [1]
The seismic coefficient of the Pseudo

Dynamic Method takes into account
the particular characteristics of each ground m
otion
.
The spectral acceleration is
obtained by
a acceleration response spectrum
corresponding to the
design
earthquake
,
and depends on the fundamental vibration period
and
the
structural
damping
. The seismic forces calculated using the spectral acceleration are used in
an equivalent static analysis. Th
e main disadvantages of this
procedure
involve
the alternation and short duration characteristics of the seismic loading, which are
not considered
[7]. It is, in fact, an estimate of the maximum response produced by
the fundamental mode.
A more refined an
alysis would consist on
a study of the
dynamic response of the structure, where the entire history of displacement
s and
other response values
would be studied over time. The seismic loading
of
this
level of analysis, including the effects of the reservoir
,
is given by
Eq.[5]
. A more
detailed explanation of the or
igin and application of this
load
ing
can be obtained in
[
1,11
].
1
..4
a
S
S
CS y w y y g p y
g
[5]
where
a
S
indicates the spectral acceleration corresponding to the structure
’s
fundamental vibration period (considering the reservoir effects),
g
is the gravity
acceleration,
S
w
represents the distributed dam weight per unit of height,
stands for the f
undamental mode shape function and
1
g p
is related to the
hydrodynamic pressure distribution along the fluid

structure int
erface (relative to a
flexible boundary motion towards a
c
ompressible fluid [1
]).
3.1.
P
ROPOSED MODIFICATIONS TO T
HE EQUIVALENT STATIC ANALYSIS
PROCEDURE
The Gravity Method [8] is an excellent
analytical tool for evaluation of
the
stress field distribution. However, its original formulation was conceived for the
Pseudo

Static actions, and does include those provided
by the Pseudo

Dynamic
Method. Through a more detailed interpretation
of this procedure
,
[2] introduced in
a simplified manner
,
the seismic actions pr
oduced by the procedure developed
by
[1].
The modifications proposed by [2] include the adoption
of
a sec
ond degree
polynomial
fundamental
mode shape
function
(simplifying
largely the analytical
proce
dure for evaluation of
the equivalent forces and moments produced by the
inertia force) and the use of hydrodynamic pressure
functions
similar to those
proposed
by
[5], with the
inclusion of
correction
coefficient
s
. Fig.
6 and 7 illustrate
graphics
with comparisons of these
simplifications
with the
consideratio
ns
originally proposed by [1]
.
0.00
0.20
0.40
0.60
0.80
1.00
0.00
0.20
0.40
0.60
0.80
1.00
Mode Shape
y/H
S
CHOPRA
PARABOLIC
=
0.8 (y/H
s
)
2
+ 0.2 (y/H
s
)
Fig. 6
Proposed fundamental mode shape function [2]
0.00
0.20
0.40
0.60
0.80
1.00
0.00
0.20
0.40
0.60
0.80
1.00
non dimensional pressure
y/H
R2=0.99
R2=0.98
R2=0.95
PROPOSED (R2=0.99)
PROPOSED (R2=0.98)
PROPOSED (R2=0.95)
Fig. 7
Proposed
hydrodynamic pressure functions [2]
It can be observed that the proposed fundamental mode shape function
leads to conservative results in evaluation of the seismic loading, producing values
always higher than those obtained with Chopra’s proposed fundamen
tal mode
shape. It should be
also noticed
that the proposed solution to the hydrodynamic
pressure is non

conservative, producing
results low
er than those achieved by [1]
.
In general, it is expected a compensation between the overestimated inertia force
and
the underestimated hydrodynamic pressure
s
. Results of tests conducted by
[2] confirm the latter.
4.
A COMPARATIVE STUDY BETWEEN THE TWO METHODOLOGIES
N THE PSEUDO

DYNAMIC MET
The Pine Flat dam
non

overflow section (see Fig.
8) was analyzed by both
the P
seudo

Static and the Pseudo

Dynamic Methods. This study will present the
main differences obtained on the stress field distribution for this profile. In the
equivalent static analysis the Gravity Method was applied, with the modifications
proposed by [2],
for the introduction of
the
pseudo

dynamic seismic actions in this
procedure.
In the pseudo

static analysis the dam will be treated as rigid, accelerated
with the peak ground acceleration, which will be adopted as equal to 0.2g. The
pseudo

dynamic seismic
loadings are evaluated using the simplifications
proposed
by [2]. Fig.
9 illustrates the seismic response spectrum applied on these analyses.
Due to the difficulties of obtaining a typical seismic response spectrum for
the Brazilian territory, a curve wa
s adopted for a specific earthquake in the North
American region. This spectrum is suitable for seismic design

in regions of firm
ground in California
–
to ground motions with a similar intensity of the earthquakes
recorded in Taft, during the Kern Count
ry earthquake in July 1952 [1]. This is a
characteristic response spectrum of an earthquake with PGA (peak ground
acceleration ground) equal to 0.2g.
Fig. 8
Pine Flat Dam non

overflow section
Fig. 9
Seismic design spectrum [1]
4.1.
S
TATIC ANALYSIS (N
ORMAL OPERATION ACTIONS)
Fig.
10 and 11 illustrate, respectively, the analyses of maximum and
minimum principal stresses (were positive sign indicates compression), obtained
from the normal operation actions, including the concrete self

weight and the
hy
drostatic pressures. Uplift pressures were neglected on both analyses.
4.2.
E
QUIVALENT STATIC ANALYSIS (PSEUDO

STATIC METHOD)
In this type of analysis it is assumed that the structure is rigid moving toward
the incompressible fluid reservoir. For the seism
ic coefficient, values ranging
between 0.05g and 0.10g are usually adopted (or a fraction of the peak ground
acceleration). The response sp
ectrum illustrated on Fig.
9, for a vibration period
equal to zero, results in a seismic coefficient equal to 0.2g, w
hich corresponds to a
value much higher than usually adopted. Still, this is the value that should be used,
because it corresponds to the peak ground acceleration (PGA) in this example.
In this analysis the following loadings are considered: concrete sel
f

weight,
hydrostatic pressures, inertia forces and hydrodynamic pressures. The dam will be
examined with horizontal seismic acceleration towards the upstream direction.
This means that seismic forces will act in the downstream direction.
Fig.
12 and 13
illustrate, respectively, the analyses of maximum and
minimum principal stresses, obtained with the applica
tion this procedure.
Fig. 10
Maximum principal stresses results
(units: kPa)
Fig. 11
Minimum principal stresses results
(units: kPa)
Fig. 12
Fig. 13
Maximum principal stresses results
(units: kPa)
Minimum principal stresses results
(units: kPa)
4.3.
E
QUIVALENT STATIC ANALYSIS (MODIFIED PSEUDO

DYNAMIC METHOD)
In this analysis
the applied seismic loading will be simila
r to the one
propose
d by
[1], with the
simplifications made by
[2]. This is not a pseudo

dynamic
analysis itself, because the seismic loading is not implemented exactly as defined
by [1]. However, the results of previous studies performed by [2] demonstrate an
excellent agree
ment
of
the simplified procedure (
defined as Modified Pseudo

Dynamic Method) with the results obtained
by [1].
Fig.
14 and 15 illustrate the maximum and minimum
principal stress
distribution
, obtained with the application of the Modified Pseudo

Dynamic
Me
thod, for a horizontal seismic acceleration oriented towards the upstream
direction. It can be observed an increase in the magnitude of the seismic actions
when compared to the previous item (resulting in
smaller stresses at the upstream
slope and in an in
crease of these values at the down
stream
slope
).
Fig. 14
Maximum principal stresses results
(units: kPa)
Fig. 15
Minimum principal stresses results
(units: kPa)
5.
CONCLUSIONS
Two application exa
mples of procedures for seismic analysis of
typica
l
profile
s
of concrete gravity dam
s
, subjected to a design spectrum, were presented.
The pseudo

static procedure treats the dam as a rigid body, ignoring the
amplification effects of the dynamic response. However, there are
some case were
this effect can
no
t be neglected. Only structures with very small periods of vibration
present dynamic responses that appr
oach the ground motion (see Fig.
1),
featuring a
rigid body
movement
. In other si
tuations, the dynamic characteris
tics of
the dam should be taken
into a
ccount
,
in order to include the
structural response
amplification effects
(see Fig.
2).
The pseudo

static procedure, which
was traditionally
used in the design of
many dams (also known as the Seismic Coefficient Method), produces results
that
underestimat
e the seismic actions when the structure can
not be treated as a rigid
body. Application of this procedure is recommended only to structures with
a
fundamental
vibration
period
less than 0.03s [4]. The
achieved
results provide that
the actions produced by t
he Pseudo

Static Method are far lower than those
produced by the Pseudo

Dynamic procedure.
The stress distribution in the pseudo

static analysis exerts little influence in
the design of a dam, because the tensions ar
e almost nonexistent (see Fig.
13,
with
maximum tension equal to 50 kPa), and the added compression will hardly
exceed the concrete strength usually employed in this type of struc
ture. However,
in the modified
pseudo

dynamic procedure, tension
s can be significant (see Fig.
15, with values up to
1500 kPa), directly influencing the choice of the concrete
design resistance, due to the low tensile strength resisted by this material. In some
cases
,
special mixtures are needed to ensure the necessary strength to regions
were very high tensions are exp
ected.
ACKNOWLEDGEMENTS
The authors are grateful for the financial support provided by CAPES and
CNPq agencies.
REFERENCES
[1]
CHOPRA A .K.
"Earthquake resistant design of concrete gravity da
m
s"
ASCE Journal of Structural Division
, vol. 104, n. ST6,
pp. 953

971, Jun. 1978.
[2]
RIBEIRO P. M. V.
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m
po
de tensões em barragens de concreto durante terremoto
s
"
.
Master’s thesis.
Universidade de Brasília,
2006.
[3]
CHOPRA A. K.
Dynamics of Structures
–
Theory and Applic
ation to
Earthquake Engineering
. Prentice Hall, 2001.
[4]
G
HRIB, F., LÉGER P.,
T
INAWI
R.
,
L
UPIEN
R.
,
V
EILLEUX
M. “Seismic
safety evaluation of gravity dams”.
In:
International Journal on Hydropower &
Dams,
v. 4,
n. 2, p. 126

138, 1997
[5]
WESTERGAARD H. M.
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er pressure on dams during earthquakes"
Transactions ASCE
, v. 98, n. 1835, pp. 418

433, 1933.
[6]
SILVA S. F., PEDROSO L. J.
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ó
rio
Técn
i
co de Pesquisa. R
TP

SF2

05

2005, 2005.
[7]
PRISCU R.
Earthquake Engineering for Large Dams
. Bucaresti
, 1985.
[8]
USBR,
Design of Gravity Dams
. Denver: United States Department of the
Interior
–
Bureau of Reclamation, 1976.
[9]
RIBEIRO P. M. V., PEDROSO L. J
, "Analytical procedure for
stress field
solution in concrete gravity dams," in
International Symposium on Solid
Mechanics.
São Paulo: Universidade de São Paulo, 2007
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[10]
RIBEIRO P. M. V., PEDROSO L. J
.
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método pseudo

dinâmico para a análise sísmica de b
arragens de concr
e
to
gravidade" in
XXVII Seminário Nacional de Grandes Barragen
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Belém, 2007
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[11]
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2002
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