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XI ICOLD BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Valencia, October 20‐21, 2011
OPEN TECHNICAL SESSION
SEISMIC BEHAVIOR OF CONCRETE FACE ROCKFILL DAMS
Dakoulas, Panos1
Summary The seismic behavior of concrete faced rockfill dams built in narrow canyons, subjected to upstream‐downstream
and vertical excitation, is
investigated. Emphasis is placed on
the evaluation of the structural
integrity and waterproofing performance of
the upstream face concrete
slab. The
static analysis uses a nonlinear hyperbolic model, whereas the dynamic analysis uses a nonlinear hysteretic model which accounts
for the dynamic properties of rockfill. A damage plasticity model
is used
for the reinforced concrete slab. The numerical study simulates the phased construction process, creep settlements,
reservoir
impoundment and seismic shaking. The dynamic analysis
takes
into account the wave energy radiated back into the canyon, the hydrodynamic pressures and the development of dynamic
settlements. The compressive and
tensile stresses in the concrete
slab panels and
the opening of the vertical joints are investigated, considering the effect of dynamic settlements.
1. Introduction
Figure 1: Messochora Dam (a) Cross‐section and material zones of the studied dam and (b) numerical
discretization of the embankment and concrete face slab consisting of 23 panels
This work
is motivated by the need to explain the seismic damage of the concrete face slab of the 156m‐high Zipingpu Dam (China) during the May 12, 2008, Wenchuan earthquake (M=8). Subjected to a peak ground acceleration
larger than 0.5g, Zipingpu Dam
suffered compressive failure of
the concrete slab panels along the vertical joints.
The structural integrity and water
tightness of the concrete slab
is of crucial importance for
the safety and good performance of the dam [3]. In the case of dams built in narrow canyons, a realistic modeling of (a) the 3D geometry of the slab panels, (b) the nonlinear material behaviour and (c) the
1 Associate Professor, University of Thessaly, Department of Civil Engineering, Volos, Greece
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contact conditions at the concrete
base and the vertical joints
between adjacent slab panels
is essential.
Ignoring the 3D nature of the problem or the presence of the vertical
joints would result into predictions
that do not capture the
real behaviour of slab. Also, a
realistic assessment of the initial
static stresses
is necessary, as evidence shows
that in some CFRDs,
tensile cracking or even failure of the concrete slab occurred during impoundment.
The objective of this paper is to investigate the seismic behavior of a concrete face rockfill dam built in
a narrow canyon, placing special
emphasis on the proper modeling
of the slab. To this end,
a refined numerical model
is utilized, which accounts for
the 3D geometry of
the embankment and concrete slab panels,
the phased construction process, the
loading due to reservoir
impoundment, the rockfill creep
settlement, the flexibility of the
canyon rock,
the hydrodynamic effects and
the potential dynamic settlements of the rockfill.
2. Constitutive models The phased construction and reservoir impoundment is based on the Duncan and Chang constitutive model
for the rockfill material [2].
The seismic analysis uses a
nonlinear hysteretic model
that accounts for
the dynamic properties of
rockfill. A damage plasticity model
[5]
for both monotonic and cyclic loading is used for the reinforced concrete.
Figure 2: Dynamic properties of gravel from experiments by Rollins et al. [1] and hysteretic model predictions: (a) secant shear modulus ratio,
0/sG G
and (b) damping ratio, ξ, vs cyclic shear strain
2.1 Hyperbolic and hysteretic models for rockfill
The hyperbolic model by Duncan and Chang [2] accounts for the dependency of the elastic moduli on the current stress state and the
loading/unloading stress path. The main advantage of the model
is the considerable accumulated
experience regarding the model
parameters for various types
of rockfill materials for which laboratory testing is difficult.
For the dynamic analysis, the tangent shear modulus
tG for monotonic loading is given by
0 2
exp( (log ) / )/1 exp( (log ) / ) (1 exp( (log ) / )) ln10
et
e e
a c baG Gc b b c b
(1)
-
where , ,a b c = material constants
and e = the equivalent shear
strain. For unloading
and reloading, eq. (1) is combined with the Masing criterion to form hysteresis loops. Fig. 2 presents the mean
value and one
standard deviation of the
shear modulus ratio 0/sG G
and damping ratio
versus the cyclic shear strain for gravelly soils from a comprehensive study based on results from 15 independent
experimental investigations [6]. The
combined static‐plus‐dynamic model has
been implemented into the FE code ABAQUS [1]. Fig. 3 illustrates an imposed simple shear deformation on a soil element and the simulated stress‐strain relationship derived from the hysteretic model.
Figure 3: Hysteretic model: simulation of a simple shear test (a) shear strain (b) stress‐strain relationship
Figure 4: Damage plasticity model: cyclic uniaxial loading test (a) tension and (b) compression
Table 1: Properties and model parameters for rockfill and gravel
TYPE Zone 3B Zone 3C
Zone 2B Density, ρ, kN/m3 2150
2150 2150 Poisson’s ratio, v
0.35 0.35 0.35 K
600 450 1200 Kur
1500 1125 3000 Kb
150 112
300 n 0.45 0.45
0.45 m 0.22
0.22 0.22 Rf 0.59
0.59 0.59 φο
51° 51°
51° Δφο
9°
9°
9°
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2.2 Damage plasticity model for concrete
The damage plasticity constitutive model for cyclic
loading of concrete [5]
is used for modeling the behavior
of the slab. The model takes
into account the effects of
strain softening,
distinguishing between the damage variables for tension and compression. Here, the concrete slab
is assumed to have a density
=2350, kg/m3, a compressive strength
of 25 MPa, a tensile strength
of 3 MPa, Young’s modulus E
= 29 GPa and Poisson’s ratio
= 0.2. Figure 4 plots
simulations of the
stress‐strain behavior of a concrete cubic specimen in uniaxial loading using the damage plasticity model.
3. Messochora Dam model and impoundment
The dam has a height of 150 m and
crest length of 330 m
(Fig. 1a). The upstream slope
is 1:1.4, whereas the downstream slope is 1:1.4 at the lower 110 m and 1:1.55 at the upper 40 m. Zones 3B and 3C consist of rockfill from healthy or slightly weathered limestone. The concrete slab was placed upon a 4 m thick layer of well graded gravel (zone 2B), which is quite stiffer than zone 3B. A zoned fill was
placed upon the slab up to
a height of 55 m from the
dam base. Fig. 1b illustrates
the discetization of the 3‐D
geometry of the embankment and
the 23 concrete panels. The
phased construction process is
simulated with 40 height increments.
The complex geometry of
the embankment and abutment boundaries is discretized using modified quadratic tetrahedral (C3D10M) that are suitable for contact problems. The width of each panel
is 15 m, except of the first and
last ones, where it is about half of this value. The thickness of each panel is 0.30 m at crest and 0.74 m at the toe of the maximum section. The slab panels are discretized using two layers of hexahedral solid elements. Reinforcement
is placed at the mid‐thickness of
the
slab, consisting of 25 mm diameter steel bars at 15 cm distance
in the
two directions of the concrete panels. Each panel has
interface properties between its base
and the underlying gravel,
as well as between the
vertical walls of adjacent slabs.
The concrete‐to‐gravel friction coefficient
is =0.7 and the
concrete‐to‐concrete friction coefficient is c
=0.5. Table 1 summarizes the values of all model parameters for the rockfill and gravel materials. After placement of the concrete panels, a fill is placed on the lower part of the slab to the height of 55 m (Fig. 1a). Creep settlements are simulated approximately by increasing and decreasing to the previous value the gravity load on the dam body, yielding a net settlement that is equal to the measured creep (≈0.15m). Fig. 5a plots the distribution of settlement after creep versus depth, whereas Fig. 5b plots contours of settlements at mid‐section, reaching a maximum of 2.37 m. Finally, the reservoir water is raised to the maximum level of 148 m.
Figure 5: (a) Measured and computed settlements at the end of creep versus distance from the crest
(b) Distribution of settlements at the end of creep
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Figure 6: Imperial Valley 1940 earthquake, El Centro record (a) comp. 180 (b) vertical component
Figure 7: Concrete slab: (a) Peak major principal stress and (b) Peak minor principal stress
(Dynamic settlement = 0 m)
Figure 8: Concrete slab: (a) Peak minor principal stress and (b) Displacement in the y direction
(Dynamic settlement = 0.5 m)
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4 Dynamic analysis and results
Fig. 6 plots the excitation
imposed in the form of traction
vectors. Energy radiation is
accounted through viscous dashpots placed at the dam‐canyon
interface. The canyon rock has S‐wave velocity
0cV =2800 m/s, density c =2400
kg/m3 and Poisson’s ratio cv =0.2.
The hydrodynamic pressures
acting on the upstream face are approximated using the added‐mass formulation by Zangar [7].
The peak upstream‐downstream and vertical acceleration histories at mid‐crest are equal
to 1.37g and 0.89g, respectively. The high accelerations are caused by “wave focusing effects” associated with the narrow canyon geometry and rockfill
inhomogeneity [3]. Fig. 7a plots the minor principal stress
1
in the slab, showing that tension develops from the perimeter to the central upper region, as the dam moves downstream. Although tensile cracks are expected in this region during shaking, they will close again after shaking due to static compressive stresses. Fig. 7b plots the minor principal stress
3
when the slab moves upstream. The compressive stresses reach a maximum value of ‐17 MPa.
As the rockfill settles, the high friction is dragging the panels towards the central section of the dam, thereby
increasing the compressive stresses y
, especially in
the central region; also, it
is pushing the upper part of the slabs towards the toe plinth, thereby increasing the compressive stresses
x
. Figs. 8a plots the distribution of
3
when the dam is experiencing a maximum upstream movement for dynamic settlement
dynS
= 0.5 m. The maximum compressive stress increases to ‐19 MPa. Figs. 8b plots
the distribution of
the horizontal displacement yU at
the moment of maximum downstream displacement. As
the dynamic settlement increases,
the concrete panels tend to move
towards to mid‐section causing widening
of the joint gaps near the
left and right sides of the
slab. During shaking, the maximum
horizontal displacement of the slab
panels is ≈0.12 m, but the
joint
gap openings are ≈ 0.065 m (less than the 0.1 m waterstop limit).
5. Conclusions 1. During shaking,
tensile stresses may reach the
strength of concrete and cause
some cracks
which, most likely, will close again when the prevailing compressive stresses return after shaking. 2.
The compressive stresses developing
in the central part of the
slab, as the dam body moves
upstream, are significantly higher than their static values. 3.
Dynamic settlements tend to increase the compressive stresses, decrease any tensile stresses in
the central and upper part of the concrete slab, and cause horizontal movements of the panels towards the dam middle section.
6. References [1]
ABAQUS (2008), Users’ Manual, Simulia, Pawtucket, Rhode Island. [2]
Duncan J. M. & Chang C. Y.
(1970), Nonlinear analysis of
stress and strain in soils,
J. of Soil
Mech. and Found. Engineering, ASCE, 96(5): 1629‐1653. [3]
Gazetas, G., & Dakoulas, P.
(1992), Seismic Analysis and Design of Rockfill Dams: State of the
Art, J. of Soil Dynamics and Earthquake Engineering; 11(1): 27‐61. [4]
Guan, Z. (2009), Investigation of
the 5.12 Wenchuan Earthquake damages
to Zipingpu Water
Control Project and an assessment of its safety state, Sc. in China, E, Tech. Sc.; 52(4):820‐834. [5]
Lee J., & Fenves, G.L
(1998), “A plastic‐damage
concrete model for earthquake analysis
of
dams”, Journal of Earthq. Eng. & Struct. Dynamics; 27: 937‐596. [6]
Rollins, K., Evans, M., Diehl, N.B., & Daily, W.D. (1998), Shear modulus and damping relations
for gravel, J. Geotechn. and Geoenvironm. Engineering, ASCE, 124(5): 396‐405 [7]
USBR, State‐of‐Practice for the
nonlinear analysis of concrete dams
at the Bureau of
Reclamation, US Department of Interior, Bureau of Reclamation, January, 2006.
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XI ICOLD BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Valencia, October 20‐21, 2011
OPEN TECHNICAL SESSION
EFFECT OF LONGITUDINAL VIBRATIONS ON THE BEHAVIOR OF CONCRETE FACED ROCKFILL DAMS
Dakoulas, Panos1
Summary The effect of longitudinal seismic excitation on the behavior of concrete face rockfill dams (CFRDs) in narrow canyons
is investigated. Emphasis
is placed on the evaluation of the structural
integrity and waterproofing performance of the upstream
face concrete slab. The numerical study simulates the phased
construction process, creep settlements,
reservoir impoundment and seismic
shaking. The dynamic analysis takes
into account the wave energy
radiated back into the canyon
and
the development of dynamic settlements. The study uses an existing 150 m–high CFRD built in a narrow canyon,
to investigate some key
issues on the seismic behavior of
such dams to excitation in
the longitudinal direction. Emphasis
is placed on the evaluation of compressive stresses along the slab‐to‐slab vertical
interfaces and the opening of
the vertical joints. Moreover,
the effect of potential dynamic settlements on both the slab stresses and joint openings is investigated.
1.
Introduction Longitudinal vibrations of concrete
faced rockfill dams may increase
the compressive
stresses and cause concrete failure
in the vertical
joints of the slab and opening of the
joints, resulting to water flow
through the rockfill. If significant
leakage occurs through the dam
rockfill, it may
cause additional settlement and therefore further damage of the concrete slab.
Figure 1. Zipingpu Dam damage by the May 12, 2008, earthquake: (a) compressive concrete failure
along the vertical joints of the slab [10] and (b) settlement along the crest [6]
1 Associate Professor, University of Thessaly, Department of Civil Engineering, Volos, Greece
-
A recent example of a concrete face damage has been observed in Zipingpu Dam (China) during the May 12, 2008, Wenchuan earthquake (M=8). The dam has a height of 156 m and crest length of 635 m.
Located at 17.17 km from the
earthquake epicentre, it was
subjected to a peak
ground acceleration larger
than 0.5g, as reported by Guan
[6]. Designed
for a peak ground acceleration of only 0.26g, Zipingpu Dam suffered compressive failure of the concrete slab panels along the vertical joint, as shown in Fig. 1a. It is noted that the direction of wave propagation forms a small angle with the direction of
the longitudinal axis of
the dam. Crest accelerations
recorded by seismigraphs
at Zipingpu dam have shown very high values
( 2g). Figure 1b plots
the observed settlement due
to seismic shaking along the crest of the dam, having a maximum value of 0.76 m [6].
The objective of this paper is to investigate the seismic behavior of a concrete face rockfill dam in a narrow canyon subjected to longitudinal and vertical excitation. Emphasis is placed on the evaluation of
the structural integrity
and waterproofing performance of the
upstream face concrete
slab. A refined numerical model
of Messochora Dam (Greece) is
utilized, which accounts for the
3D geometry of the embankment
and concrete slab panels, the
phased construction process,
the reservoir impoundment, the rockfill creep settlement, the flexibility of the canyon rock and potential dynamic settlements. Fig. 2 shows Messochora Dam during construction of the slab. Fig. 3a shows a cross‐section of the dam, which has a height of 150 m and crest length of 330 m. The upstream slope is 1:1.4, whereas the downstream slope
is 1:1.4 at the
lower 110 m and 1:1.55 at the upper 40 m. Zones 3B and 3C
consist of limestone rockfill. The
slab was placed upon a 4 m
thick
layer of well graded gravel (zone 2B). A zoned fill was placed upon the slab up to a height of 55 m from the dam base. Fig. 3b illustrates the numerical discretization of the embankment and the 23 concrete panels.
Figure 2: Messochora Dam (Greece) during concrete slab construction
Figure 3: (a) Cross‐section and material zones and (b) numerical discretization of the embankment
and the concrete face slab consisting of 23 panels
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2. Constitutive models for rockfill and concrete For the static analysis, the hyperbolic model by Duncan & Chang [3] accounts for the dependency of the
elastic moduli on the current
stress state and the
loading/unloading stress path.
The main advantage of the model is the considerable accumulated experience regarding the model parameters for various types of rockfill materials for which laboratory testing is difficult.
For the dynamic analysis, the tangent shear modulus
tG for monotonic loading is given by
0 2
exp( (log ) / )/1 exp( (log ) / ) (1 exp( (log ) / )) ln10
et
e e
a c baG Gc b b c b
(1)
where e
= the equivalent shear strain and
, ,a b c
= material constants derived from experimental data
[9]. For unloading and reloading,
eq. (1) is combined with the
Masing criterion to
form hysteresis loops.
The damage plasticity constitutive model for cyclic
loading of concrete [7]
is used for modeling the behavior
of the slab. The model takes
into account the effects of
strain softening,
distinguishing between the damage variables for tension and compression. Here, the concrete slab
is assumed to have a density
=2350, kg/m3, a compressive strength
of 25 MPa, a tensile strength
of 3 MPa, Young’s modulus E
= 29 GPa and Poisson’s ratio
= 0.2. More information about
the constitutive models for rockfill and concrete, as well as values for all model parameters, are given in a companion paper on
the seismic behavior of concrete
faced rockfill dams subjected
to upstream‐downstream vibrations [2].
3. Phased construction and impoundment
The phased construction is simulated with 40 height increments. The geometry of the embankment is discretized using modified quadratic tetrahedral elements that are suitable for contact problems, whereas the slab panels are discretized with brick elements. The width of each panel is 15 m, except of the first and last ones, where it is ≈7.5m. Reinforcement is placed at the mid‐thickness of the slab, consisting of 25 mm diameter
steel bars at 15 cm distance
in the two directions of the
slab. Each panel has
interface properties between
its base and
the underlying gravel, as well as between
the vertical walls of adjacent slabs. The concrete‐to‐gravel friction coefficient is
=0.7 and the concrete‐to‐concrete friction coefficient is
c
=0.5. A fill is placed on the lower part of the slab to the height of 55 m
(Fig. 1a). Creep settlements are simulated approximately by
increasing and decreasing to the previous value the gravity load on the dam body, yielding a net settlement is equal to the measured creep (≈0.15m). Finally, the reservoir water is raised to the maximum level of 148 m.
4 Dynamic analysis and results
Fig. 4 plots the excitation
imposed in the longitudinal and
vertical direction, consisting of the
El Centro record No. 9
(Imperial Valley 1940 earthquake). The
longitudinal acceleration is scaled
to a peak value of 0.35g, whereas
the vertical acceleration
to 0.175g. The excitation is
imposed in the form of
traction vectors. Energy radiation
is accounted for
through viscous dashpots placed at
the dam‐canyon
interface. The canyon rock has S‐wave velocity
0cV =2800 m/s, density c =2400 kg/m
3 and Poisson’s ratio cv
=0.2.
The computed peak longitudinal acceleration at mid‐crest is equal to ≈1.5g. Additional analyses using different earthquake excitations have shown a range of mid‐crest longitudinal acceleration response from
1g to 1.5g. Similarly
high mid‐crest accelerations have been
also predicted for the case
of upstream‐downstream excitation of
the dam [2]. Such high
accelerations are caused by “wave
-
focusing effects” associated with
the narrow canyon geometry and
rockfill inhomogeneity with depth, in
agreement with previous analytical
studies and field observations
[4,5,6,8]. The
peak relative mid‐crest displacement in the longitudinal direction for the El Centro record is 0.08 m.
Due to the high friction developing between concrete and gravel, sliding of the slab during shaking occurs only
in the upper part of the
slab (approximately along the
top 20% of the panel
length). Consequently, large inertial
forces of the embankment exerted
in the longitudinal direction
are transferred through friction to the concrete slab, which is acting as a diaphragm wall. These inertial forces
are causing an increase in
compressive stresses in the slab
panels. For the case of
zero dynamic settlement ( dynS =0 m),
Fig. 5a plots the minor principal
stress 3 in the slab
showing a maximum compressive stresses of
‐17 MPa. It
is noted that the compressive stresses
in the central region of the dam are smaller during
longitudinal vibration, compared to those experienced during upstream‐downstream vibration, shown in the companion paper [2].
Fig. 5b plots the horizontal
displacement yU in the longitudinal
direction at the moment
of maximum movement.
Figure 4: Imperial Valley 1940 earthquake, El Centro record No. 9 (a) comp. 180 (b) vertical comp.
Figure 5: Concrete slab: (a) Peak minor principal stress and (b) Peak relative horizontal movement
yU
in the longitudinal direction (Dynamic settlement = 0 m)
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Figure 6: Concrete slab: (a) Peak major principal stress and (b) Peak minor principal stress
(Dynamic settlement = 0.5 m)
Figure 7: Concrete slab: (a) Peak minor principal stress and (b) Displacement in the y direction
(Dynamic settlement = 0.5 m)
For the case of dynamic settlement having a maximum value at mid‐crest equal to
dynS
= 0.5 m, Fig. 6a shows a significant increase of the compressive stresses in the slab. As the rockfill settles, the high friction
is dragging the panels towards
the central section of the dam,
thereby increasing the compressive
stresses y , especially in
the central region; also, it
is pushing the upper part of
the slabs towards the toe
plinth, thereby increasing the
compressive stresses x . Depending on
the particular moment during shaking, the maximum compressive stress may appear in the central part of the slab or near the side of the slab, where there is an abrupt change of the plinth inclination, as shown in Fig. 6a. Here the maximum compressive stress is ‐19 MPa. Fig. 6b shows that the horizontal relative movement of
the panels also increases
from 0.1 m to 0.16 m, as
the dynamic
settlement increases from 0 m to 0.5 m.
Figs. 5 and 6 correspond to vibrations only in the longitudinal and vertical direction. If simultaneous vibrations in the upstream‐downstream direction are also considered and the magnitude of dynamic settlement
increases further to 1 m, then
for the excitation record considered
the maximum compressive stress reaches
a value of ‐22 MPa. To
control the development of high
compressive stresses and increase
safety against local concrete failure
along the vertical joints, one
possible solution is to allow some
limited movement of the panels. This can be achieved by
introducing a 5‐
-
cm wide gap between slab panels in selected vertical joints. In this case such cuts are considered for joints J6, J12, J13 and J19.
In practice, these cuts should be filled with a compressible material and their watertightness should be secured.
For dynamic settlement dynS
= 0.5 m, Figs. 7a plots
the distribution of 3 showing a
reduction of maximum compressive stress from
‐19 MPa to
‐16 MPa, due to the presence of the four slab cuts. Fig.
7b plots the distribution of
the horizontal displacement yU at
the moment of
maximum downstream displacement, having a peak value of about 0.16 m.
It is noted that despite the
large movement of the panels near the crest due to their rotation and bending, the joint gap openings in Figs. 6b and 7b are less than the 0.1 m waterstop limit.
5. Conclusions The effect of
longitudinal seismic excitation on
the behavior of CFRDs in narrow
canyons was investigated numerically.
Emphasis was placed on the
evaluation of the structural
integrity
and waterproofing performance of the upstream face concrete slab. The main conclusions of this study are the following:
1. Longitudinal vibrations of concrete
faced
rockfill dams, combined with dynamic settlements of rockfill, may cause significant compressive stresses in the concrete panels, and therefore should be taken into account.
2.
Horizontal movements of the slab panels
in the longitudinal direction may
increase significantly with dynamic settlement.
3.
A reduction of compressive stresses may be achieved by
introducing 5‐cm wide cuts along few selected vertical joints of the concrete slab.
6. References [1]
ABAQUS (2008), Users’ Manual, Simulia, Pawtucket, Rhode Island. [2]
Dakoulas, P. (2011), Seismic behavior of concrete faced rockfill dams, Proceedings of the 11th
ICOLD Benchmark Workshop on Numerical Analysis of Dams, 20‐21 October 2011, Valencia, Spain.
[3] Duncan
J. M. & Chang C. Y.
(1970), Nonlinear analysis of
stress and strain in soils,
J. of
Soil Mech. and Found. Engineering, ASCE, 96(5): 1629‐1653.
[4] Elgamal, A.W.M. & Gunturi
(1993), R. Dynamic behavior and seismic response of El
Infernillo dam, J. of Earthquake Engineering and Structural Dynamics; 22(8):665‐684.
[5] Gazetas, G., & Dakoulas, P.
(1992), Seismic Analysis and Design of Rockfill Dams: State of the Art, J. of Soil Dynamics and Earthquake Engineering; 11(1): 27‐61.
[6] Guan, Z. (2009),
Investigation of
the 5.12 Wenchuan Earthquake damages
to Zipingpu Water Control Project
and an assessment of its safety
state, Science in China, Series
E, Tech. Sc.; 52(4):820‐834.
[7] Lee J., & Fenves, G.L
(1998), “A plastic‐damage
concrete model for earthquake analysis
of dams”, Journal of Earthq. Eng. & Struct. Dynamics; 27: 937‐596.
[8]
Mejia, L.H., & Seed, H.B. (1983), Comparison of 2D and 3D analyses of earth dams, Journal of Geotechnical Engineering, ASCE 1983; 109(11):1383‐1398.
[9]
Rollins, K., Evans, M., Diehl, N.B., & Daily, W.D. (1998), Shear modulus and damping relations for gravel, J. Geotechn. and Geoenvironm. Engineering, ASCE, 124(5): 396‐405
[10] Wieland, M. (2009), “Concrete
face rockfill dams in highly
seismic regions”, 1st
International Symposium on Rockfill Dams, 18‐21 October, Chengdu, China.
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XI ICOLD BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Valencia, October 20‐21, 2011
OPEN TECHNICAL SESSION
NON‐ELASTIC MOVEMENTS ANALYSIS IN ORDER TO EVALUATE ARCH DAM SAFETY
Sánchez Caro, Francisco Javier1
CONTACT Francisco Javier Sánchez Caro GEOTECNIA DE PROYECTOS Y OBRAS (GEOprob) Ctra. Carabanchel a Aravaca, 9 28024 MADRID Phone: (+34) 91 5120437 e‐mail: [email protected]
Summary This paper examines
the process of analysis and
monitoring of arch dam
deformations, establishing procedures, criteria and/or limitations, in order to detect and prevent problematic behavior, basically
in the operating phase. From
this point of view,
this paper assigns great importance
to the analysis of the
evolution of non‐elastic deformations,
as a key tool for evaluating
dam safety. This analysis procedure
has the advantage of fully
objectivity (detachment) because, in fact, it is to see the movements’ data in another way.
The aim of this paper
is to reach practical conclusions based exclusively on research
into real events (El Atazar dam). Nowadays, its application allows to “Canal de Isabel II” (Water Supply Authority of Madrid City and its Province) not only the daily control of the behavior of the dam movements but also the retrospective analysis of the behavior evaluating past evolution.
1. Preliminary aspects. Creep deformations
All materials modify their
properties with time.
These modifications are noticeable in
the effects that
loads have on them. Of the former
is the well known creep or slow strain under the influence of constant stress. The stages of creep are:
• Primary or initial creep, where
the strain rate is relatively
high, but slows
with increasing strain.
•
Secondary or steady‐state creep, where the strain rate eventually reaches a minimum and becomes near constant.
• Tertiary creep, where the strain
rate exponentially increases with
stress till eventual failure. This
stage does not always happen and, depending on
the material and
the stress level, can never be reached (hence never reach failure).
1 Ph. D. Civil and Geotechnical Engineering. GEOprob
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⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=∂∂
cCCe
tet α
σσ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=∂∂
cCC
tt α
σεσε
Figure 1. Strain phases in the creep process.
According to the edometric model (for normal consolidated clay), where Cc and Cα values are the compression index and the secondary compression index (respectively) in the formulae:
The rate Cα/Cc is very similar
in many materials
(Mesri y Castro, 1987; Soriano, 1992) and
it can be assumed that it is a constant value and non dependent on the material. The following expression can be then assumed:
It can be concluded
that movements induced by constant
loads in a period of time
are proportional to the instant movements when a certain load is applied. In fact, this simplified model has been used
for the numerical analysis of long
term movements
in embankments, earth dams, etc. (this type of model has been also developed in other engineering fields, such as
Branson in 1971 for the
estimation of long term deflection
in reinforced
concrete structures). Let’s keep in mind this simple idea, for now.
2. El Atazar dam
2.1. General features
El Atazar dam
(Lozoya river, Spain)
is an arch dam
(double curvature) 134 m high
(maximum over foundation), and it is the most important dam of the water supply system of Madrid. The design engineer was Dr. Laguinha Serafim and
it was built between 1966 and 1972. The dam foundation
consists of metamorphic rock from
the Paleozoic era (Silurian slate
and schist), affected by several
families of discontinuities and
faults. At that time,
the dam project was quite controversial, since, on
the opinion of various technicians,
the conditions of
the valley were not suitable for the construction of a arch dam of this type; the geometry of the valley was a
little too wide and the
foundation was found also to be
too deformable (deformation module
thought to be were around
100.000‐175.000 kp/cm2). The foundation
excavation triggered stability problems
in the upstream side of the
left abutment, which was
solved building a reticular structure
(conformed by large reinforced
concrete beams,
strongly anchored to the rock by 150 t anchors).
-
Although during the construction
almost all the stability problems
happened at the left abutment,
the geotechnical parameters of the
foundation at the right abutment
were considered somehow worse, which
leaded to the construction of a
large foundation
plinth (beam anchored) in most of the right side (downstream).
Figure 2. El Atazar dam (Plan view)
2.2. Beginning of the dam operation. Incident of February 1978.
El Atazar dam started to be
filled in 1971 (after the
partial injection of its
joints), without apparent incidents
(the dam was
quite well monitored), until the
incident of 18th
February 1978, which had not been previously detected. In the first days of 1978 the water level at the dam was about 5 m below
the spill level and the
filtrated water flow through
the dam was about 30 l/s (four times higher than the filtrated flows registered in 1975, when the water level was similar). At the beginning of February that year the water level went rapidly high and new drains started draining.
In the evening of 18th February 1978 there was a sudden
increase of the filtrated flow (a
crack in one of the
galleries was detected) that suddenly
flooded
the galleries. The reservoir was at its historical maximum (elevation 868.72 m), only 1,28 m below the spill
level at the top of
the dam. The infiltrated flow at
that moment was inferred
to be more than 130 l/s.
By opening the dam
low outlets, the water level was
lowered and the filtrated flows reduced some days after. The following investigations discovered a fracture at elevation 770 m, 160 m long and a significantly wide (estimated between 1/3 and 3/4 of the dam total width), reaching one of the galleries. The two main conclusions of the report of D. Nicolas Navalón were:
1. The main origin of the
fracture had a thermal nature,
due to the micro‐breaching developed
in the concrete by the strong thermal gradients existing
in the
inter‐phase cooled concrete – non‐cooled concrete.
2.
The progress of the main fracture was eased by the continuous presence of water in it. The reasons for that situation are difficult to explain.
The fracture was sealed and properly drained, getting to the conclusion that the operation of the dam could continue with no further precautions.
-
2.3. Dam operation after February 1978. “Initial Model” and “Model 1990”
After the former measures were taken (mainly sealing the fracture and the installation of new instrumentation),
the dam entered
a new period of regular operation.
The behavior of
the dam seems acceptable until
the summer of 1985
(Madrid suffered a severe draught
in 1983 and 1984).
It was not until the
summer of 1985 that El Atazar
reservoir reached again high water
levels, close to
its historical maximum (on
the 24th July 1985 the water
level
reached elevation 868,79 m). Although the filtrated flows measured in the drains kept small, the truth is that it was registered a slight flow increase, which evidenced the re‐activation of the crack. At
that moment different prediction /
evolution models of the different
dam
sensors (pendulums, extensometers, concrete thermometers, etc) were under construction.
Anyhow, the operation of
the dam continued until, after a
relative draught period, the
first spill operation, on the
spillway at elevation 870 m,
took place on 25th
July 1988. This event evidenced
clearly the re‐activation of the
fracture (crack), in
terms of measured
infiltration, with appreciable increasing
flows. Besides, it was acknowledged
that
the prediction models then elaborated
(especially the prediction of the radial movements of the pendulum system, which we will call “Initial Model”) were found not suitable to control the behavior of the dam.
From that moment, an
important effort aiming to guarantee
the dam safety
started. These consisted, among others, on developing new prediction models to control the dam behavior, mainly
radial movements on pendulum system,
to compare real movements with
those expected. That was how the “Radial Movement Prediction Model” (“1990 Model” from now) was developed, which was operational
till 2006 and that has been
found to be probably the most
useful tool to track the
behavior of the dam (being the
author of this article
the developer of that model, together with A. Soriano)
After the filling of 1988
it was recommended not to surpass the maximum radial movements that
the dam had previously
experienced, which practically meant
not to exceed in winter (January
to mid March, where
the dam has its maximum movements
towards downstream due to thermal reasons) the water elevation 860 m. This restriction still applies today.
As the events in 1988
re‐activated the fracture, it was
considered convenient to carry
out some additional sealing work
by the end of 1990 and
the beginning of 1991. In
these conditions a second spill
took place on May
the 17th 1991. The Radial Movement Prediction Model for the pendulum system (“1990 Model”, that then was already completed and under trials) behaved satisfactory, so soon later was definitively implemented.
The water level in the
following years varied largely, from
strong draughts to close to
spill situations, but always
respecting the restriction of not
filling the reservoir totally in
the winters. During the last 20
years and every time an
increase in the infiltrated flows
was observed, improvements in the
sealing of the
fracture were carried out using more modern and
flexible epoxy bi‐component resins than
those used after
the event of 1978. Of course, every time the fracture was treated, the drainage system was restored.
3. Need of a new forecasting model (“2005 Model")
In 2005, the possibility of
improving the Radial Movement
Prediction Model (the
so‐called "2005 Model", also developed by
the undersigned) was brought up. This “2005 Model” was
-
designed to replace the previous model, "1990 Model",
in service since November 1990. The performance of
the "1990 Model" in those
fifteen years of operation must be
classified
as "good enough", because it had complied with the goal of monitoring the dam in high pool level scenarios.
Its accuracy was also reasonable, since
maximum errors of a few millimeters (within a
range of nearly 6 cm in
the radial movements) are to be
considered excellent for
this complex structure, which has the following singularities:
• A pretty high ratio of
the width of the valley at
the level of the dam’s crest
to
the height of the dam (“span/height” ratio)
• A hyperannual reservoir •
Large variation in daily radial deformations
•
Existing cracks in the vault – double curvature ‐ arch dam (with periodic treatments)
•
The obvious existence of permanent displacements, of plastic character
•
Other (solar radiation, high temperatures
in summer with a strong effect of reservoir
temperature on the upstream face, etc.)
The need to improve
the Radial Displacement Prediction Model
arose as a consequence of wanting
more control over a future
filling of the reservoir in
winter (in different stages, designed
to assess the suitability of
the existing "pool level restriction").
Thus, the
"2005 Model" could be approached from two different viewpoints:
• Performing
"ex novo" a different prediction model, using,
for example, the
inherent potential found in fuzzy logic formulations, genetic algorithms, ...
• Upgrading the previous model, by
improving its formulation (considering
the
"1990 Model" had been very reliable and had a longer data set).
Obviously, the work presented here is in line with the second approach.
4. Differences between the 1990 and 2005 models
The so‐called "Initial Model" was purely statistical
(and, despite
it was correctly drawn up,
it did not work properly).
There’s no intention to go into
detail about the
mathematical formulation of the "1990 Model", but it’s important to note that it was a non‐statistical model. The
reason for not making a purely
statistical model was the
failure of the "Initial Model".
Therefore,
in the "1990 Model" all parameters were "manually set" considering their
isolated evolution against reservoir
level, temperature, etc. In return,
it was possible for
the model parameters to have a fairly clear physical meaning.
The new prediction model
("2005 Model") is constructed upon
the previous model, but has some differences.
It
is now a pure statistical model
(when developing the model, the author was convinced it would work properly), which makes an automatic adjustment of parameters (minimum root‐mean‐square error and null mean error).
The selected adjustment period runs from January 1st, 1991 to December 31st, 2005.
In addition, both 1990 and
2005 models share an elastic
formulation to predict
radial movements, that is, there
are no permanent displacements
considered. Soon, the
first impressions about
the "2005 Model" performance were obtained.
The accuracy achieved by
-
this model seemed better than the previous one. For example, the standard deviation of the "2005 Model" made up a third of that corresponding to the "1990 Model".
5. Error analysis in the 1990 and 2005 models
Regardless of the overall precision or goodness of a model, it is interesting to carryout an error analysis.
The evolution over time of this
error may indicate the existence
of a
permanent displacement or plastic deformation in the dam (these models lack this analysis, because they have
an "elastic" formulation). This has
been carried out on both 1990
and
2005 models, delivering several graphics for the 46 pendulum bases (coordinometers), that is, points where the coordinates of the steel wires that make up the plumb‐lines are determined:
•
Evolution over time of real and modeled radial displacements
•
Evolution over time of the deviation (difference between real and modeled movements)
• Evolution of this deviation against
reservoir level, setting apart five
different periods
(1991‐93, 1994‐96, 1997‐99, 2000‐02, 2003‐05) •
Evolution of this deviation against temperature T30 (temperature moving average of the
last 30 days), setting apart the same five periods
• Evolution of this deviation against
seasons of the year (according
to a standard year),
distinguishing also those five periods
All these 460 graphics are quite clear and suggest the existence of an evolution over time
in the model error.
Figure 3. Several graphics for Error analysis, applied to radial movements. Real (observed) minus Modeled. Pendulum 3 Base 2.
-
6. Permanent displacements and rate of error evolution over time
For each of the 46 measuring points (pendulum bases), a graphical summary of evolution over time for the annual mean error and the annual mean squared error was developed (this error is the difference between measured and modeled deformations), as well as an adjustment of the linear regression calculations associated to both the "1990 Model" and the "2005 Model".
Obviously, the evolution over time of the annual mean error may be an indication of temporal drift
(creep), that is, the dam and
its foundation develop permanent deformations.
In short, the slope of the
linear regression line could note
the existence of certain
unelastic displacement pattern in the
vault and/or its foundation. If these
charts are analyzed,
the following conclusions can be drawn:
• For both models ("1990" and
"2005") and all 46 reading
points, a fairly
good correlation is achieved
•
For both models and all reading points, the regression line shows a positive slope
Do not think this
is a coincidence. It
is quite evident that the dam
is undergoing, even today, plastic
deformations which, according to this
approach, can be measured quite
accurately. Obviously, once the
regression line is estimated, it’s
straightforward to find the
estimated permanent displacement rate.
It is also another way to look at the data, which is unbiased (not open to question), and reproducible by any engineer. To sum up, the non‐elastic/permanent radial displacement rate of a given point in the dam is a piece of information provided by the monitoring system, by means of the plumb lines (pendulums).
The fact that these rates were the data seen otherwise, and not a conclusion of the dam behavior, led the author to further explore the issue.
Figure 4. Non‐elastic radial movement rate (i.e. Pendulum 2 and 4). Note that the rate of movement is identical in both models. They are data seen in another way
7. Fatigue and creep structural deformations
Fatigue deformations are due to
constant amplitude cyclic/alternating actions
(loads and/or temperature). It is known that these "fatigue" deformations can be likened to a certain "creep" deformations for an equivalent effective loading.
-
Assumed that "equivalence or assimilation" between these processes of "creep and fatigue" in a more general one could be called "global creep", we can conclude (see Section 1) that radial creep movements
in a period of time Δt
(inelastic) are proportional to the
instantaneous movement produced by applying a given load (a certain situation of reservoir level).
But there's more. Assuming this fact, being Δt any value (1 day, 1 month, 1 year, 1 century, ...) must also be satisfied that the creep movement rates in a certain period of time Δt (inelastic) are proportional to the instantaneous movements produced by applying a given load (a certain situation of reservoir level).
The next step is to assume
that these movement rates are
comparable global creep strain rates.
If this is accepted,
the conclusion is clear: how
to distribute the strain rate
should be deferred similar
(proportional) to the deformed shape
(elastic curve) produced by
the application of a given load (a certain situation of reservoir level).
It is necessary to make an
important clarification to
the above statement: how
to distribute the global creep strain rate will be similar (proportional) to the deformed shape (elastic curve), provided that the failure condition is not reached at some point of the structure.
With these ideas it
is possible, from a qualitative point of view, analyzing the behavior of the dam
of El Atazar, assigning the
"existence of any cracks" in
the areas of highest
inelastic deformation rate gradient. The result of this analysis must be described as very consistent.
Figure 5. Qualitative analysis of the non‐elastic radial movement rate
8. Historical evolution of global creep deformations
A general law for the evolution of the creep strain rates could be like:
For most of Civil Engineering processes, it holds that m = 1. The simplest formulation would be like:
),( TFtt
ttK ε
m
R
Rm
σ⎟⎟⎠
⎞⎜⎜⎝
⎛−
=&),( teett
ttA RT
QD
m
R
Rm
σε α Ψ⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎠⎞
⎜⎝⎛−
&
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=tt
ttk ε
R
R&
-
This is not to detail here
the aspects of formulating this
law. This is just to comment
that clearly stable creep processes
(i.e., those in the time
it reaches the failure are
long enough), are often expressions where the product of the strain rate with time is approximately constant (at
least for materials containing
free hydrogen, which is
a process of embrittlement).
This results
in the typical approximate
logarithmic expressions for the evolution of these
inelastic creep deformations.
9. Conclusions
From the analysis so far,
seemed to imply that the recent
creep movements may point
to pronounced plastic deformations in the area of the foundation of the left bank. Is this a recent process? Does this process have something to do with the incident of 1978?
Logically, if we try to
apply "retroactively" Model‐2005 at
the whole history of
radial movements of the dam, we
can try to analyze the
significance of unadjusted
differences (deviations, in short). This exercise was also carried out (graphics of temporal evolution of the average annual deviation and
logarithmic fitting
in the period 1971‐2005) and the analysis of this information resulted in two obvious consequences:
•
The crack appears and develops greatly during the
first filling, prior to 1973 (another thing is that the incident will not occur until 1978, which were found the intersection thereof with the gallery). The typical division to analyze the different behavior of the dam
before and after 1978 has,
therefore, no rationale, from a
structural point of view.
•
The fit degree of the logarithmic law (for temporal evolution of creep movements) can be described as excellent. Thus, the creep process is unique and is fully active since the reservoir
filling began, being the root
cause of cracking observed. The
process continues today its natural course (creep law).
Figure 6. Application of “2005 Model” (logarithmic fit) to the entire history of the dam (Example: Pendulum 2 Base 3)
-
In short, the
rates of creep movements that
took place at the end of 1973
(much higher, of course, than
those which occurred in the
period 1991‐2005), had already
identified
the essential features of the process (sliding of the dam foundation in the area of the left bank).
The described procedure
is basically another way of viewing the data, which
is fully objective (little debatable) and reproducible by any technician.
“Canal de Isabel II” not only has a new model for predicting radial movements ("Model 2005"), but
also a safety simple criterion:
the dam will be safe
enough while the rate of
inelastic movements remains downward.
Figure 7. Non‐elastic radial movement rates in El Atazar dam. Initial (1973) and current (1991‐2005) rates.
References
Bonaldi, P.; Di Monaco, A.; Fanelli, M.; Giuseppetti, G.; Riccioni, R.
(1982): “Concrete Dam Problems: An
Outline of the Role, Potential
and Limitations of
Numerical Analysis”. Numerical Analysis of Dams. Ed. Naylor, Stagg, Zienkiewicz. pp 28‐81.
Lombardi, G. (1998): “Informe final sobre la seguridad de la obra y de su cimentación”. Report Nº 718.5‐R‐8.
Mesri, G. and Castro, A.
(1987): “Cα/Cc Concept and Ko
during
Secondary Compression”. Joumal of Geotechnical Engineering, ASCE, Vol. 113, (3): 230‐247.
Sánchez Caro, F.J. (2007):
“Seguridad de presas: Aportación al
análisis y control de deformaciones
como elemento
de prevención de patologías de origen geotécnico“. Tesis Doctoral. July 2007. U.P. Madrid.
Soriano Peña, A., Sánchez Caro, F. J. and Arcones Torrejón, A. (1992): “Back analysis of arch dam movements”.
International Conference: Geotechnics and Computers. París. September 1992.
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XI ICOLD BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Valencia, October 20‐21, 2011
PROBABILISTIC ANALYSES OF THERMAL INDUCED CRACKING IN A CONCRETE BUTTRESS DAM
Malm, Richard1
Eriksson, Daniel2
Gasch, Tobias 3
Hassanzadeh, Manouchehr4
CONTACT Richard Malm, Vattenfall
Research and Development AB, BU
Engineering, Asset Development,
Jämtlands‐gatan 99, SE‐ 162 16 Stockholm, Sweden, Phone +46 8 739 57 21, [email protected]
Summary Recent assessments and investigations of buttress dams in northern Sweden reveal several types of cracks. The theoretical analysis and field measurements have showed that the most of the cracks are either developed or propagated as a result of the seasonal temperature variations. Cracks influence the
behaviour of the dams in
different ways, such as reducing
the tightness of the dam
and increasing
the hydraulic pressure within
the material/structure. Furthermore,
cracks may have an impact on
the stiffness and stability of
the dam. The ordinary sliding
and overturning
stability analyses are not sufficient when
the supporting structure
is cracked. The cracks may comprise
the integrity and the homogeneity
of the structure. A cracked,
and for that matter even
repaired structure, can’t be regarded
as a homogenous structure and
should be treated
accordingly. Consequently, other types of models instead of the conventional design models should be utilized for the stability analyses of the cracked and repaired dams.
The mode of the failure is one of the decisive elements considering determination of the probability of the failure. The conditions for crack initiation and the trajectory of the crack propagation are the decisive
factors which govern the
failure mode. Ordinary design methods and advanced numerical models which
are based on the elastic
behaviour of the structure can’t
be utilized, since these models
are not able to describe
the non‐linear behaviour and
to predict the failure mode of
the structure.
A finite‐element model based on
non‐linear fracture mechanics is being
utilized to study crack development
in a buttress dam. The aim of the study was to reveal crack trajectories and different probable
failure modes, and moreover to
determine the influences of the
cracks on the overall behaviour
of the structure. In a real
structure the loading (mechanical and
environmental) and boundary conditions
are decisive factors regarding
initiation, propagation and trajectory
of the
1 Vattenfall Research and Development AB, Engineering, Stockholm, Sweden /KTH Royal Institute of Technology 2 Vattenfall Research and Development AB, Engineering, Stockholm, Sweden 3 Vattenfall Research and Development AB, Engineering, Stockholm, Sweden 4 Vattenfall Research and Development AB, Engineering, Stockholm, Sweden /Lund University
-
cracks. Furthermore, the material
properties and their statistical
distribution may influence
the formation of cracks and the mode of failure.
1. Introduction In this paper,
a large reinforced
concrete buttress dam in northern
Sweden, called
Storfinnforsen hydropower dam, has been studied. The dam consists of 100 concrete monoliths, each with a front‐plate facing the water and supported by a buttress, as illustrated in Figure 1. The total length of the dam is about 1200 m, where 800 m consists of concrete monoliths. The tallest monolith is about 40 m high, has an 8 m wide front‐plate; which is about 2.5 m thick at the base and 1.0−1.5 m thick at the dam crest. Each buttress
is 2 m thick and the
largest monoliths are about 30‐35 m wide near
the bedrock. There is an inspection gangway going through each buttress and besides this there are also vertical insulating walls, which together with the front‐plate form spaces where the temperature and humidity can be controlled.
The Storfinnforsen hydropower plant
was completed in 1954 and is
one of the major
Swedish buttress dams. Almost
immediately after completion, horizontal cracks were detected
in the
lower part of the front‐plates, causing
leakage of water from the reservoir, [1].
In addition,
ice formation was found on the upstream side of the front‐plate. To reduce the risk of ice formation on the front‐plate and to reduce the thermal gradient through the front‐plate, an insulating wall was installed in the early 1990’s. This was built of polystyrene‐insulated steel sheets and the area between the front‐plate and
the wall was now heated with a controlled warm air
flow from
the underground power station, [1]. In the inspection performed prior to the installation of the insulating wall, in addition to the horizontal cracks
in the front‐plate,
inclined cracks between the front‐plate and the foundation was also observed. These cracks are defined as
type 1 and 2 respectively
in Figure 1. A
few years after the
installment of the
insulation wall, new cracks were found on some of the monoliths. One type of
crack that had appeared was an
inclined crack from the inspection
gangway towards the front‐plate, called
type 3
in Figure 1. Also, vertical cracks
from the foundation on
the downstream side of the inspection gangway, called type 4 in Figure 1, was found
Insulating wall
Water level
Buttress wall
Inspection gangway
Upstream side
Downstream side
Front-plate
Figure 1: Illustration of three monoliths of the studied buttress dam and the observed cracks.
2. Non‐linear finite element model Finite
element simulations have previously
been performed on Storfinnforsen,
[2]. Non‐linear material properties
was considered where the monolith
was subjected to cyclic variation
in temperature distribution for summer and winter conditions respectively; the first years were cycled without the insulating wall and the following cycles included the insulating wall, [3]. All cracks shown in Figure 1 were obtained in the FE analysis, where the horizontal cracks on the upstream side of the front‐plate,
i.e. type 1, occurred during the summer. The horizontal cracks on the downstream side and the inclined cracks, i.e. type 2, all occurred during the following winter. The inclined crack from
-
the inspection gangway, i.e.
type 3, was first obtained in
the analyses during a summer
load case after the instalment of the insulating wall.
In this project the influence of crack propagation due to a variation in material properties are studied for the inclined crack, type 3 in Figure 1, propagating from the inspection gangway of Storfinnforsen dam. A
3D FE model with shell elements
have been developed in Abaqus
version 6.10, where a continuum
plasticity damage based model called
concrete damaged plasticity has been
used
to simulate the nonlinear effects of concrete.
2.1. Stochastic material properties
To study the effects of
cracking of concrete due to
stochastically assigned material properties,
a Monte Carlo Simulation has been performed on Storfinnforsen.
In the previous analyses, homogenous material properties were assumed for the whole monolith. In this paper, material properties with a stochastic distribution will be assumed for the monolith. The material properties used
in the previous simulations are considered as mean values.
In the present analyses, material
properties for tensile strength,
elastic modulus and fracture energy
have been generated. All have
been assumed to have a
log‐normal distribution with mean and
covariance presented in Table 1.
The distribution and covariance have
been selected according to
the Probabilistic Model Code, [4].
Table 1: Concrete material properties
Mean
Covariance Tensile strength
2.5 MPa
COV = 0.30 Elastic modulus
25 GPa
COV = 0.15 Fracture energy
120 Nm/m2 COV = 0.30
Based on the mean material
properties given in Table 1, a
characteristic length of the
fracture process is equal
to 0.48 m calculated according
to Equation 1. The size of
the characteristic
length determines the brittleness of the material, where a higher characteristic length gives a more ductile material. With the randomly generated log‐normal distributed material properties the characteristic length varies between 0.25 m and 0.78 m.
2
t
fch f
EGl (1)
The compressive strength has not been studied in the analyses, since previous analyses showed that the compressive stresses are below 10 MPa, hence, the non‐linear effects of the compressive curve are not affecting
the results. In the
figures below,
the randomly generated material properties are shown
in histograms together with the
sample size (N), calculated mean
(µ), calculated
standard deviation (σ) and covariance (COV).
0 1 2 3 4 5 6 70
0.05
0.10
0.15
0.20
0.25
Tensile strength (MPa)50 100 150 200 250 300
0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Fracture energy (Nm/m )210 15 20 25 30 35 40 45
Elastic modulus (GPa)
Pro
babi
lity
N = 1000 = 2.53 MPa = 0.78 MPaCOV = 0.31
Statistics of randomgenerated properties
Statistics of randomgenerated properties
2
Statistics of randomgenerated properties
N = 1000 = 120.9 Nm/m = 35.8 Nm/mCOV = 0.30
N = 1000 = 25.2 GPa = 3.8 GPaCOV = 0.15
2
Pro
babi
lity
Pro
babi
lity
0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Figure 2: Log‐normal distribution of the tensile strength, fracture energy and elastic modulus.
-
The material properties have been generated
independently, meaning that the
initial correlation
is almost zero. However, the concrete material properties are dependent on each other; for instance, a high
tensile strength most likely result
in both high
fracture energy and elastic modulus. Normally, equations to calculate the different material properties from another property are derived, usually, based on
the compressive strength other material properties can be calculated. This means
that a correlation equal to one
is usually assumed. In reality,
there is a variation between
the material properties, where for
instance two specimens with the
same tensile strength can have
different fracture energy or elastic modulus.
In order to represent the
correlation between the material
properties, all vectors with
randomly generated material properties have been sorted. This resulted in the mean correlation between the material
data shown in Table 2. With
this approach the lowest correlation
between
the material properties are obtained for the samples with the smallest and highest values respectively, which
is what could be expected in reality.
Table 2: Correlation between material properties
Tensile strength
Elastic modulus
Fracture energy
Tensile strength 1 0.988
0.996 Elastic modulus 0.988 1
0.996 Fracture energy 0.996 0.996
1
2.2. FE sub‐model
As a first step of
the analyses, a partially cracked monolith was
considered, as shown
in Figure 3. Previous analyses
showed that the inclined crack
propagated towards the front‐plate.
The pre‐cracked global model was
therefore used to simulate
further crack propagation and
to make sure that the same
crack propagation was obtained as
in the previous analyses. The
obtained displacements from the pre‐cracked global model were after this transferred as boundary conditions to a sub‐model that had a refined mesh. The sub‐model with a high mesh density made it possible to, in a more detailed manner, describe the crack pattern. The size of the sub‐model has been chosen to be larger than the expected area that may be subjected to cracking. Based on the initial analyses, it was shown that a region of about 2.5 m around the crack tip had tensile stresses that were at least 0.5 MPa. In addition, the initial analyses showed that the crack had a higher likelihood to propagate upwards. Therefore, a
larger area above the pre‐defined crack was selected. The
length of the sub‐model perpendicular to the crack was chosen as 6 m and the length parallel to the crack was defined equal to 9 m.
Sub-model
Figure 3: Illustration of the pre‐cracked global model and the area that constitute the sub‐model.
In the first step of
the sub‐model, each element was
randomly given one of
the sets of stochastic material
properties, i.e. tensile strength,
elastic modulus and fracture energy.
The material
-
properties thereby vary over the whole sub‐model and are randomly defined at each element. Due to
the random definition, the mean
values for the whole sub‐model
was always close to
the previously defined mean values
for tensile strength,
fracture energy and elastic modulus. The sub‐model
is defined with 4‐noded elements
S4R with hourglass control and a
characteristic element length equal to
0.12 m, corresponding to a total
of approximately 4000 elements. The
element length of the sub‐model is defined to be less than half the characteristic length for the case with the lowest material properties generated.
In Figure 4, an example of
the material distribution used in
the simulations with the sub‐model
is shown. The left figure in Figure 4 shows the distribution of tensile strength assigned to the elements in
the sub‐model
for one analysis; below the left
figure the average tensile strength
for the whole region
is presented. The figure to the
right in Figure 4 shows
the distribution of the elements
that have a tensile strength
higher and lower than the
expected mean value. Similar figures
were obtained for
the elastic modulus and
fracture energy and every
simulation was performed with a different geometrical distribution in the assigned material properties.
The aim of the study was
to reveal the affect on crack
trajectories that a varying
distribution in material properties
may have. With a Monte Carlo
Simulation, where a sufficient amount
of simulations are performed, it will be possible to determine the area that is likely to be subjected to cracking and also to determine the crack trajectory that is most likely to occur.
Figure 4: Distribution of tensile strength for the elements in the sub‐model.
3. Results The calculated crack patterns obtained
from 500 simulations are presented
in Figure 5. In the
left figure of Figure 5, every element that fully cracked,
i.e. has obtained a macro‐crack, is illustrated as black, while the result from the original analysis with mean values assigned to all elements is shown as magenta. This figure thereby shows the area that
is
likely to be subjected to cracking. However, this doesn’t illustrate the likelihood of cracking to occur within this region. Therefore, one additional figure
is presented, where the number
of times each element was
completely cracked in all
the simulations has been calculated. Based on this, the probability of cracking occurring in each element is calculated based on the total number of simulations performed. The results are presented in form of contour plots of the probability of each element being cracked based on 500 simulations, where the elements that have a probability higher than 20% of cracking are shown in black. It can be seen that the crack trajectory obtained from the original simulation with mean values over the whole sub‐model does not exactly correspond to the area with the highest probability of cracking. Based on the
-
contour plots, the
crack pattern with highest probability has a higher
inclination than
the original model.
Figure 5: Calculated extent of cracking and calculated probability of cracking.
4. Conclusions The simulations presented in this paper shows that the crack propagation and crack trajectories are influenced
by a variation in the material
properties. One conclusion that is
drawn from these simulations
is that the crack trajectory obtained with the original simulation with mean values over the whole
region differs from the crack
trajectories with the highest
probability from the Monte Carlo
simulation. The crack trajectory from
the Monte Carlo simulation is
steeper and
in addition propagates further than the reference model.
One possible overturning failure mode that could occur
is caused by widening of the
inclined crack between the front‐plate and the inspection gangway, thereby splitting the monolith in two halves. If this crack has a steeper inclination, a smaller mass will oppose the overturning of the upper part of the
cracked monolith. In addition, the
steeper crack inclination obtained
from the Monte Carlo simulation
results in that the crack takes
a shorter distance between the
front‐plate and
the inspection gangway. Hence, the total frictional force transmitted through the crack plane decreases, thereby reducing the resistance of an overturning failure further. The results presented in this paper only constitute a first step in using probabilistic analyses to determine possible failure modes and in future research will continue to further develop the methodology.
Acknowledgements The presented work was financially supported by the ELFORSK AB, Swedish Power Companies R&D association. The authors wish to thank Maziar Partovi and Giovanna Lilliu from TNO DIANA.
References [1]
Eriksson, H., Investigation and Rehabilitation of the Storfinnforsen Dam, ICOLD 18th
International Congress on Large Dams, V. 1, Q68, R19, 1994, pp. 247‐259. [2]
Björnström J., Ekström T., Hassanzadeh M. Cracked concrete dams – overview and calculation
methods, Report 06:29, Elforsk AB, Stockholm. 2006 (in Swedish). [3]
Malm, R. and Ansell A. 2011. Cracking of a Concrete Buttress Dam Due to Seasonal Temperature
Variation. ACI Structural Journal 108 (1), pp 13‐22. [4]
JCSS, Probabilistic Model Code. 12th draft. 2001.
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XI ICOLD BENCHMARK WORKSHOP ON
NUMERICAL ANALYSIS OF DAMS
Valencia, October 20-‐21, 2011
OPEN TECHNICAL SESSION
NUMERICAL ANALYSIS AND SAFETY EVALUATION
OF A LARGE ARCH DAM FOUNDED
ON FRACTURED ROCK, USING
ZERO-‐THICKNESS INTERFACE ELEMENTS AND
A C-‐Φ REDUCTION
METHOD
Aliguer, I.1 Carol, I.1 Alonso,
E.E.1
Río, F. 2 Griñó, R. 2
CONTACT Geotechnical Engineering and
Geo-‐Sciences Department (DETCG)
ETSECCPB (School of Civil Engineering)
UPC (UniversitatPolitècnica de
Catalunya), 08034 Barcelona
phone: 0034
934011695email: [email protected],
[email protected], [email protected]
Summary A 140m high arch
dam in the Pyrenees, built
in the 50’s, is founded
on fractured limestone rock. Since
the beginning of the design
process two main families of
discontinuities were identified. The
dam was built very close to
the end of the narrow part
of the valley, which raised
stability concerns early on. In
the late 80’s – early 90’s,
a numerical study of the dam
was developed at the Dept. of
Geotechnical Engineering and Geo-‐Sciences
(School of Civil Engineering)
UPC, using a progressively more
realistic series of models and
approaches, culminating with a 3D
discretization of the dam plus
rock mass, in which
discontinuities were explicitly represented
using zero-‐thickness interface elements
with frictional constitutive laws in
terms of stress tractions and
the corresponding normal and shear
relative displacements. In the
present study, that dam and its
foundation are revisited and
reanalyzed with current, more
advanced numerical tools and including
also a third family of
rock joints which has been
identified more recently. The same
mesh is used as a
departure point, although a much
more detailed description is now
possible. The analysis is also
approached in a different way,
now using the traditional c-‐φ
reduction method developed and
implemented specifically for non-‐linear
zero-‐thickness interfaces.
1. Introduction Canelles dam is
a 151m-‐high arch dam located in
the Pyrenees (Catalunya, Spain),
which was completed in April
1958. Since this date, different
kinds of analyses have been
carried out. Monitoring systems,
reduced-‐scale model tests and
numerical analyses have been combined
to provide engineering evaluations
of the dam safety [3]. One
of the main numerical studies,
concerning the stability of the
dam and both abutments, was
performed during the 90’s. The
analysis procedures and main
conclusions are reported in several
references [1,2]. That first study
led to the conclusion that the
dam was basically safe and the
worst scenario corresponded to a
safety coefficient between 2 and
3.
1GeotechnicalEngineering and
Geo-‐SciencesDepartment (DETCG) UPC,
Barcelona, Spain 2ENDESA, Madrid,
Spain
-
The dam is founded on cretaceous
massive limestone that is fractured
by two sets of discontinuities
(Fig.1). A main set of vertical
joints is oriented parallel to
the valley. The other family is
a set of N-‐S planes which
dip an average of 55°towards
the valley (nearly downstream).
A Laser-‐Scanner field campaign in
2009 led to the identification
of that later set of
discontinuities. In addition, bedding
planes dip 45°upstream.
Figure 1: General view of
the left abutment and anchorage
tunnels
Due to the spatial arrangement
of the three rock discontinuity
families, several rock blocks
have fallen down to the canyon,
which kee
