COMPUTER AIDED STABILITY ANALYSIS OF GRAVITY DAMS

4e Conférence spécialisée en génie des structures
de la Société canadienne de génie civil
4th Structural Specialty Conference
of the Canadian Society for Civil Engineering
Montréal, Québec, Canada
5-8 juin 2002 / June 5-8, 2002
COMPUTER AIDED STABILITY ANALYSIS OF GRAVITY DAMS
M. Leclerc, P. Léger, R. Tinawi
Department of Civil Engineering, École Polytechnique de Montréal, Canada
ABSTRACT: This paper presents the main features and organization of CADAM, a computer program
that has been developed for the static and seismic stability evaluations of concrete gravity dams. CADAM
is based on the gravity method using rigid body equilibrium and beam theory to perform stress analyses,
compute crack lengths, and safety factors. Seismic analyses could be done using either the pseudo-static
or a simplified response spectrum method. CADAM is primarily designed to provide support for learning
the principles of structural stability evaluation of gravity dams. It could also be used for research and
development on stability of gravity dams. In adopting several worldwide published dam safety guidelines,
a large number of modeling options have been implemented regarding (a) crack initiation and
propagation, (b) effects of drainage and cracking under static, seismic, and post-seismic uplift pressure
conditions, and (c) safety evaluation formats (deterministic allowable stresses and limit states,
probabilistic analyses using Monte Carlo simulations). Structural stability evaluation of a 30m dam is
presented to illustrate the use of CADAM that is available free from the web site:
http://www.struc.polymtl.ca/cadam/.
1.
INTRODUCTION
The static and seismic safety of existing concrete gravity dams is a continuous concern owing to the
ageing processes altering their strength and stiffness, as well as revised predictions of the maximum
loads associated to severe floods and earthquakes. It is thus required to perform periodic reassessment of
their static and seismic structural stability. A progressive approach is generally followed starting with the
gravity method based on rigid body equilibrium and beam theory before considering linear or non linear
finite element models, if necessary (Ghrib et al. 1997). FERC (2000, 1991), CDA (1999), USACE (1995),
ANCOLD (1991), and USBR (1987) present guidelines for dam safety assessment based on the gravity
method. There are several differences among these guidelines regarding (a) cracking initiation and
propagation criteria, (b) static and seismic uplift pressures along joints and cracks, and (c) safety
evaluation format (allowable stress, limit state method). There is also a growing interest in performing risk
based safety evaluation where the probability of failure of a dam is evaluated considering explicitly
uncertainties in strength and loading modelling parameters through suitable probability density functions
(Kreuzer 2000).
In most engineering offices, in-house spreadsheets are developed and adapted on a case by case basis
to perform dam stability analysis following particular safety guidelines. This is due to the very lengthy and
tedious computations, particularly when pseudo-dynamic seismic analyses are considered. Moreover,
there are no widely available computational tool (a) for learning the principles of stability analysis in the
1
academic or professional environment, and (b) for performing research and development on the structural
safety of gravity dams. We have thus identified a need to develop, and put in the public domain, a
comprehensive computer program, CADAM, to perform stability evaluation of gravity dams based on the
gravity method.
2.
CADAM – OVERVIEW OF MAIN FEATURES AND ANALYSIS OPTIONS
Figure 1 presents CADAM overall organisation. The dam geometry (Fig. 2a), material properties (Fig. 2b),
the various load conditions, cracking options, and load combinations are first specified as input data for
subsequent structural analyses. Additional input data as lift joints, post-tensioning cables, applied forces,
added masses, floating debris, silts and many more may be included in the model. The following analysis
options are currently available:
•
•
•
•
•
3
Static Analyses: Static analyses are performed for the normal operating reservoir elevation or the
flood elevation including overtopping over the crest and floating debris.
Seismic Analyses: Seismic analyses are performed using the pseudo-static method (seismic
coefficient method) or the pseudo-dynamic method by the simplified response spectrum analysis
described by Chopra (1988) for gravity dams.
Post-Seismic Analyses: In post-seismic safety analysis, the crack length induced by the seismic
event could alter the cohesive shear resistance and uplift pressures. The post-seismic uplift
pressures could either (a) build-up to its full value in seismic cracks or (b) return to its initial value
if the seismic crack is closed after the earthquake.
Incremental Load Analysis: Sensitivity analyses are automatically performed by computing and
plotting the evolution of typical performance indicators (ex: sliding safety factor) as a function of a
progressive application in the applied loading (ex: reservoir elevation, peak ground acceleration).
Probabilistic Safety Analysis: Probabilistic safety analyses are performed to compute the
probability of failure of a dam-foundation-reservoir system as a function of the uncertainties in
loading and strength parameters that are considered as random variables with specified
probability density functions. A Monte-Carlo simulation computational procedure is used. Static,
seismic, as well as post-seismic analyses may be considered.
STATIC LOADING CONDITIONS
The load conditions supported by CADAM are shown in Figure 3. Some particular features are described
in the following. Various dam safety guidelines equations presented to compute the uplift pressures
according to the position of the drain from the upstream (u/s) face, the drain effectiveness and the
elevation of the drainage gallery have been implemented (Fig. 4). It is interesting to note that Federal
agencies in the US (FERC, USACE and USBR) are currently evaluating the need for unified Federal
criteria for the calculation of uplift pressures as well as crack initiation and propagation criteria in the
stability of concrete gravity dams (USACE 2000). It is believed that a computational tool like CADAM
could be of great assistance to conduct extensive parametric analyses for various dam geometry and
drainage conditions to study the effects of modelling assumptions on computed performance indicators.
4
SEISMIC AND POST-SEISMIC SAFETY ANALYSIS
Some original features that have been included for seismic and post-seismic safety analyses are
presented below. CADAM allows cracking to initiate either from the u/s face or the d/s face depending
upon the orientation of the base acceleration and corresponding inertia forces. Existing cracks computed
from the initial static conditions may close depending on the intensity and orientation of the earthquake
forces. Separate analyses could be performed successively with the base acceleration pointing u/s and
d/s to estimate the cumulative damage by reducing the cohesion that could be mobilised along the joint
2
considered. The cohesion is considered null along the seismically induced crack length to compute the
sliding safety factors in seismic and post-seismic conditions.
Since the pseudo-static method does not recognise the oscillatory nature of earthquake loads, CADAM
performs the safety evaluation in two phases: (a) a stress analysis using peak ground acceleration (or
spectral acceleration) values to compute the crack length, and (b) a stability analysis using sustained
acceleration values to compute sliding safety factors. A single acceleration peak might be sufficient to
induce a crack but it may not be of sufficient duration to induce significant sliding displacement.
4.1
Pseudo-static analysis
In a pseudo-static seismic analysis, the inertia forces induced by the earthquake are computed from the
product of the mass and the acceleration. The dynamic amplification of inertia forces along the height of
the dam due to its flexibility is neglected. In a pseudo-static analysis, it is required to specify the peak
ground horizontal and vertical accelerations as well as the sustained accelerations. Westergaard added
mass (USACE 1995) is used to represent the hydrodynamic effects of the reservoir on the dam. Options
are provided to account for (a) water compressibility effects, (b) inclination of the u/s and d/s faces.
4.2
Pseudo-dynamic analysis
A pseudo-dynamic seismic analysis is based on the response spectrum method. A pseudo-dynamic
analysis is conceptually similar to a pseudo-static analysis except that it recognises the dynamic
amplification of the inertia forces along the height of the dam. However, the oscillatory nature of the
amplified inertia forces is not considered. That is the stress and stability analyses are performed with the
inertia forces continuously applied in the same direction. The basic input data required to perform a
pseudo-dynamic analysis using the simplified response spectrum method proposed by Fenves and
Chopra (1987) are: (a) peak ground and spectral accelerations (peak and effective values), (b) dam and
foundation stiffness and damping properties, (c) reservoir bottom damping properties and velocity of an
impulsive pressure wave in water, (d) a modal summation rule.
In a pseudo-dynamic analysis, the moment and axial force acting on the lift joint considered are computed
from the selected modal combination rule. The resulting moment and axial force are then used to compute
the related stresses and crack length. This approach is generally conservative. In linear (uncracked)
analysis, it is more appropriate to compute stresses separately for the first mode and the higher modes
and then apply the modal combination rule to stresses. However, this approach, adopted in linear
analysis, is not suitable to estimate crack length, especially if uplift pressures are to be varied within the
seismic crack (ex. with no uplift pressure in an opened crack).
5
CRACKING OPTIONS AND EVOLUTION OF UPLIFT PRESSURES IN CRACKS
CADAM provides various options for specification of (a) tensile strengths for crack initiation and
propagation, (b) dynamic amplification factor for the tensile strength, (c) the incidence of cracking on static
uplift pressure distributions (drain effectiveness), (d) the effect of cracking on the transient evolution of
uplift pressures during earthquakes, and (e) the evolution of uplift pressures in the post-seismic
conditions. When cracking is allowed, a distinction is made between the criteria for crack initiation and
crack propagation. After crack initiation, say at the u/s end of a joint where stress concentration is minimal,
it is likely that stress concentration will occur near the tip of the propagating crack (ANCOLD 1991). For
example the crack initiation criterion could be set to a tensile strength of 1000 kPa but once the crack is
initiated, it should be propagated to a length sufficient to develop compression at the crack tip (no-tension
condition for crack propagation). The allowable tensile strengths for crack initiation and propagation are
specified for different usual, flood, seismic and post-seismic load combinations.
Upon cracking when drainage is considered, four options are offered (Fig. 5): (1) no drain effectiveness
under any cracking condition, (2) no drain effectiveness when the crack reaches the drain line; (3) full
drain effectiveness, but with full uplift pressures applied between the reservoir and the drain line; (4) full
3
drain effectiveness with a linear decrement in uplift pressure starting from full reservoir pressure at the
reservoir level to the drainage pressure at the drain line.
Due to the lack of historical and experimental evidences, there is still a poor knowledge of the transient
evolution of uplift pressures in a crack due to its cyclic movements during earthquakes (Fig. 6). ICOLD
(1986) states: “The assumption that pore pressure equal to the reservoir head is instantly attained in
cracks is probably adequate and safe“. USACE (1995) and FERC (2000, 1991) assume that uplift
pressures are unchanged by earthquake load (i.e. at the pre-earthquake intensity during the earthquake).
USBR (1987) mentions: “When a crack develops during an earthquake event, uplift pressure within the
crack is assumed to be zero”. This is based on the assumption that a rapid crack opening reduces the
uplift pressures and that the cyclic crack motions are too fast to allow reservoir water to penetrate and
build-up the pressure. CDSA (1997) mentions: “In areas of low seismicity, the uplift pressure prior to the
seismic event is normally assumed to be maintained during the earthquake even if cracking occurs. In
areas of high seismicity, the assumption is frequently made that the uplift pressure on the crack surface is
zero during the earthquake when the seismic force are tending to open the crack”.
6
LOAD COMBINATIONS AND SAFETY EVALUATION FORMAT
Five load combinations are supported by CADAM: (a) normal operating, (b) flood, (c) seismic 1, (d)
seismic 2, and (e) post-seismic. For each load combination, multiplication factors could be specified for
each basic load conditions. This option is very useful when an applied load is increased until a safety
factor equal to 1 is reached and thus determines the ultimate strength of the dam. For each load
combination, the required safety factors to ensure an adequate safety margin for structural stability are
specified. These values are not used in the computational algorithm of the program. They are reported in
the output results to facilitate the interpretation of the computed safety factors in comparison with the
corresponding allowable values.
The Australian National Committee on Large Dams (1991) presented a dam safety evaluation format
based on a limit state approach. Various magnification and reduction factors are applied to basic load
conditions and material strength parameters to reflect related uncertainties. By adjusting the input material
parameters, and applying the specified load multiplication factors, CADAM could be used to perform limit
analysis of gravity dams as described by ANCOLD (1991).
7
PROBABILISTIC AND RISK ANALYSES
The objectives of CADAM probabilistic analysis module is to compute the probability of failure of a gravity
dam as a function of the uncertainties in loading and strength parameters that are considered as random
variables (Fig. 7). A probabilistic analysis requires more information than a deterministic analysis. For
example, probability density functions (PDF) (uniform, normal, log-normal or user defined) are to be
selected for the friction coefficient and cohesion; the mean values, and the standard deviation must then
be specified.
CADAM probabilistic analysis module could be used:
•
•
•
For educational purpose to develop a basic understanding of the concepts and procedure
required to perform a risk analysis, where risk is evaluated as the product of the probability of
failure (pf) and the related consequences.
To actually perform probabilistic (risk) analysis for a particular dam. It is then possible to construct
a fragility curve, that defines the probability of failure as a function of an applied load level and
compute reliability indices (as a function of (1 – pf)).
To perform R&D in risk based dam safety assessment (ex. calibration of nominal strength
(resistance R) reduction factor, φ, and load (L) factor,γ, to develop limit state based safety
evaluation format; φR ≥ γ L).
4
•
To study different safety approaches (ex: strength requirements to ensure uniform risk during the
service life of a dam).
Due to concrete cracking, and related modifications in uplift pressures, the stress and stability analysis of
a dam is in general a non-linear process. Monte Carlo simulation is used as the computational procedure
to perform the probabilistic “non-linear” analysis in CADAM. Monte Carlo simulation technique “involve
sampling at random to simulate artificially a large number of experiments and to observe the results”
(Melchers 1999):
(1) a large number (up to 250,000) of loading and strength parameters are “sampled” at random
within bounds of user specified PDF to perform a large number of possible strength-loading
scenarios;
(2) stress and stability analyses are performed;
(3) statistics are performed on the results (ex. sliding safety factors, SSF) to determine the probability
of failure.
The output results can also be analysed statistically to define the mean, the variance, the PDF and the
cumulative density function (CDF).
8
APPLICATION EXAMPLE
8.1
system Analyzed
The 30.48m (100ft) high gravity dam to illustrate some of CADAM potentials is shown in Figure 2a. This
dam that was used in USACE (2000) to evaluate and compare stability analysis and uplifting criteria for
gravity dams by three US Federal agencies. The analyses are performed considering the material
properties shown in Figure 2b. The usual upstream and downstream reservoir elevations are set to
27.432m (90ft) and 1.524m (5ft), respectively. Lift joints are spaced at every 3.048m (10ft) in elevation
from the base. The drainage system is initially considered according to USACE (1995) guideline, the drain
position, efficiency and the elevation of the drainage gallery are given in Figure 4.
8.2
Probabilistic safety analysis
As an illustrative example, Figure 8 shows the results of probabilistic safety analyses where the peak
shear strength parameters (cohesion and friction angle) are considered as random variables following a
normal PDF with means and variances indicated in Figure 7. To construct the fragility curve shown in
Figure 8, about 20 probabilistic analyses were performed by increasing the flood reservoir elevation. A
probability of failure is obtained for each reservoir elevation, thus defining a point of the curve. By
definition, the dam failure occurs when the peak sliding safety factor is less than one (shear resistance is
exceeded). Annual exceedance frequency of particular reservoir elevations is also plotted in Figure 8 for
illustrative purposes, allowing computing the frequency of dam failure per year due to hydrological events
(McCann et al., 1985).
9
PERSPECTIVES FOR FUTURE DEVELOPMENTS
There are almost endless possibilities for further developments of a computer program like CADAM for
structural safety assessment of gravity dams. Currently, the plan is to add the following features:
•
From a pseudo-static or a pseudo-dynamic seismic analysis, the lift joint most susceptible to
cracking can be easily obtained using CADAM. Calculation of seismic sliding displacements and
rocking response of cracked dam components using transient dynamic analysis of rigid body is
envisaged.
5
•
•
•
•
10
Computation of displacements using beam theory for the dams and Boussinesq coefficients for
the semi-infinite elastic foundation. Thermal analysis will be performed along lift joints using finite
differences to evaluate the thermal field required for thermal displacement and stress
computations. The displacement response of a 2D model could be calibrated against that of a
preliminary 3D finite element model to determine the fraction of the hydrostatic load that is
resisted in a pure cantilever mode. Unit thermal loads could also be used for calibration purposes.
The computation of displacement using beam theory will allow simple and effective coupled
thermo-mechanical analyses to link the deterministic model of a dam with its statistical model
derived from field measurements of pendulum displacements. This can be viewed as an
intermediate step before undertaking detailed coupled thermo-mechanical finite element analyses,
which requires large resources.
Definition of more complex 2D geometry, spillway and water intake sections, eventually 3D
sections.
Arbitrary user defined uplift pressure distributions.
Link with finite element programs: automatic transfer of model data to finite element programs for
detailed static, thermal, seepage and seismic analyses.
CONCLUSIONS
CADAM provides a very versatile computing environment to learn or investigate modelling assumptions
and computational processes related to the static and seismic structural stability of gravity dams based on
the gravity method. It has been shown in this paper that several assumptions related to load conditions,
cracking criteria, uplift pressures intensities and analysis procedure could be used for static, seismic, and
post-seismic safety assessments. In general, the computations are complex to perform due to the
coupling between the uplift pressure and crack length. In an actual situation, parametric analyses are
most often performed to cover uncertainties in strength and loading parameters to take appropriate
decision concerning a particular structure. The authors have successfully used CADAM as a
computational laboratory in seminars, to engineers from practice, involved in dam safety evaluation.
CADAM is also used for industrial applications and R&D in dam engineering and has been extensively
validated during the past years. The organisation of the program and the particular features that have
been presented herein are useful for those interested in the development and application of computer
aided stability analysis of gravity dams.
Acknowledgements
The development of the computer program CADAM was funded by NSERC (Natural Sciences and
Engineering Research Council of Canada), Hydro-Québec and Alcan. The support of these organisations
is gratefully acknowledged.
References
ANCOLD (1991) Guidelines on design criteria for concrete gravity dams. Australian National Committee
for Large Dams.
Canadian Dam Association (CDA) (1999) Dam safety guidelines. Edmonton, Alberta.
Canadian Dam Safety Association (CDSA) (1997, 1995) Dam safety guidelines and commentaries,
Edmonton, Alberta, Canada.
Fenves, G. and Chopra, A.K. (1987) Simplified earthquake analysis of concrete gravity dams. Journal of
Structural Engineering, ASCE, 113:8:1688-1708.
FERC (Federal Energy Regulatory Commission), 2000. Engineering guidelines for evaluation of
hydropower projects – Draft Chapter III Gravity Dams. Federal Energy Regulatory Commission,
Office of Energy Projects, Division of Dam Safety and Inspections, Washington D.C., USA.
FERC (Federal Energy Regulatory Commission) (1991) Engineering guidelines for evaluation of
hydropower projects - Chapter III Gravity Dams. Federal Energy Regulatory Commission, Office of
Hydropower Licensing, Report No. FERC 0119-2, Washington D.C., USA.
6
Ghrib, F., Léger, P., Tinawi, R., Lupien, R., Veilleux, M. (1997) Seismic safety evaluation of gravity dams.
International Journal of Hydropower and Dams. Vol. 4, No. 2, pp. 126-138.
Hall, J.F. 1988. The dynamic and earthquake behaviour of concrete dams: review of experimental
behaviour and observational evidence. Soil Dynamics and Earthquake Engineering, Vol. 7, No. 2,
pp. 58-117.
International Commission on Large Dams (ICOLD). (1986) Earthquake analysis for dams, ICOLD Bulletin
52, Paris.
Kreuzer, H. (2000) The use of risk analysis to support dam Safety decisions and Management.
Proceedings ICOLD 20th Congress Beijing, China, Gr. Q.76, p.769-834.
McCann, M. W., Franzini, J. B., Kavazanjian, E. and Shah, H. C. (1985) Preliminary safety evaluation of
existang dams, Volume I, The John A. Blume Earthquake Engineering Center, Depertment of Civil
Engineering, Stanford University, Report No. 69.
Melchers, R. E. (1999) Structural reliability analysis and prediction, Second Edition. John Wiley & Sons.
USACE (US Army Corps of Engineers) (2000) Evaluation and comparison of stability analysis and uplift
criteria for concrete gravity dams by three federal agencies. Engineering Research and Development
Center – Information Technology Laboratory. Report ERDC/ITL TR-00-1, Washington, D.C.
(document from the web: http://www.wes.army.mil/ITL/itlpubl.html)
USACE (US Army Corps of Engineers) (1995) Engineering and design: Gravity dam design. Report EM
1110-2-2000, Washington, D.C.
USBR (United States Bureau of Reclamation) (1987) Design of small dams. Denver, Colorado.
1
CADAM USER'S INTERFACE
· File management, modelling analysis options;
· Graphical display, output results, link with spreadsheets.
2
DAM MODEL
· Geometry, added masses, material properties, lift joints.
3
STATIC LOADING CONDITIONS
BASIC CONDITIONS
·
·
·
·
Reservoir elevation;
Ice, silt;
Post-tensioning;
User defined forces.
4
UPLIFT PRESSURES
5
· Dam safety guidelines;
· Drainage efficiency.
NO
6
FLOOD
· Floating debris;
· Overtopping.
SEISMIC
LOADS?
YES
7
SEISMIC LOADING CONDITIONS
PSEUDO-STATIC
8
PSEUDO-DYNAMIC
· Analysis input data.
9
· Analysis input data.
CRACKING OPTIONS
10
· Initiation / propagation criteria;
· Effect of cracking on uplift pressures
(static, flood, seismic, post-seismic).
11
LOAD COMBINATIONS
(Static, Flood, Seismic & Post-seismic)
OUTPUTS
LOOP
STRUCTURAL ANALYSIS
(Static, Flood, Seismic & Post-seismic)
INCREMENTAL LOAD ANALYSIS
12
· Printed reports;
· Graphical display;
· ASCII files.
PROBABILISTIC ANALYSIS
(Monte-Carlo simulations)
14
15
· Static, flood, seismic.
· Definition of a probability density function;
· static, flood, seismic.
Figure 1. CADAM overall organization.
7
13
(a)
(b)
Figure 2. Definition of dam model: (a) Dam geometry; (b) Material properties.
Vnc or Vfc
Flood
level
Pc
FD
Normal
level
D:
M:
H:
V:
U:
I:
S:
P:
F:
FD:
Dead load
Masses
Horizontal hydrostatic
Vertical hydrostatic
Uplift
Ice
Silt
Post-tension
Applied force
Floating debris
u:
d:
c:
h:
v:
n:
f:
Upsream
Downstream
Crest
Horizontal
Vertical
Normal level
Flood level
I
Pd
Vfu
Vnu
D
Mv
Silt
level
Hfu
Flood
level
Hnu
Sv
φ
Sh
Fv
Normal
level
Vfd
Vnd
Fh
Hfd
Hnd
l
UPSTREAM
DOWNSTREAM
Un
Uf
X
Figure 3. CADAM basic static loading conditions.
8
Figure 4. Uplift pressures and drainage system options.
Figure 5. Cracking options: Effect of cracking on the drainage system.
Inertia forces
opening
closing
Initial uplift distribution
zero uplift pressure in crack
(USBR 1995, CDSA 1997 (high seismicity))
Pre-earthquake uplift pressures in crack
(USACE 1995, FERC 1991, CDSA 1997 (low seismicity))
Full uplift pressures in crack
(ICOLD 1986)
Figure 6. Transient evolutions of uplift pressures in seismically induced crack.
9
Strength Random Variables:
Loading Random Variables:
Tensile strength;
Peak cohesion;
Residual cohesion;
Peak friction coefficient;
Residual friction coefficient;
Normal upstream reservoir elevation;
Flood upstream reservoir increase;
Silt elevation;
Silt volumetric weight;
Drain efficiency;
Floating debris;
Ice load;
Last applied force;
Horizontal peak ground acceleration.
Figure 7. Probabilistic analysis input data.
1
0.8
1x10-3
Annual frequency
of exceedence
of reservoir elevation
0.6
0.4
-4
1x10
0.2
Probability
of failure
1x10-5
0
30.6
30.8
31
31.2
31.4
31.6
Reservoir elevation (m)
Figure 8. Probabilistic safety analyses.
10
31.8
Probability of failure (sliding mode)
Annual frequency of exceedance
1x10-2

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