第 27 卷
岩石力学与工程学报
Chinese Journal of Rock Mechanics and Engineering
第8 期
2008 年 8 月
Vol.27 No.8
Aug.，2008
INFLUENCES OF HYDRAULIC
UPLIFT PRESSURES ON STABILITY OF
GRAVITY DAM
UTILI Stefano1 2，YIN Zhenyu 2，JIANG Mingjing3
，
(1. Department of Civil Engineering，Politecnico di Milano，Milan 20133，Italy；
2. Department of Civil Engineering，University of Strathclyde，Glasgow G4ONG，United Kingdom；
3. Department of Geotechnical Engineering，Tongji University，Shanghai 200092，China)
Abstract：A study of the influences of the hydraulic uplift pressures underneath the base of a typical concrete gravity dam
on its stability is presented. The dam is located at Cumbidanovu(Sardegna，Italy). The foundation of the dam is made of
heavily fractured rock. Firstly，analytical calculations about the equilibrium of the dam as a free body have been employed
to evaluate the maximum hydraulic pressure before collapsing and to assess the impact of an effective drainage system on
the stability of the dam in a simple way. Secondly，numerical analyses by the distinct element method(DEM) using the
code UDEC have been carried out to evaluate the hydraulic flow taking place within the fractured rock foundation，the
uplift pressure distribution generated by the calculated flow，and its influence on the stability of the dam. For design
purposes，it emerges that availability of reliable data on the hydraulic permeability of rock foundations and a
computationally advanced distinct element modeling might lead to the acceptance of loads significantly higher than the
more conservative estimations obtained from equilibrium analyses.
Key words：hydraulic engineering；distinct element method(DEM)；hydraulic uplift；dam stability；gravity dam
CLC number：TV 642
Document code：A
Article ID：1000–6915(2008)08–1554–15
坝底水浮力对重力坝稳定性的影响分析
UTILI Stefano1 2，尹振宇 2，蒋明镜 3
，
(1. 米兰理工大学 土木工程系，意大利 米兰
20133；2. 斯特莱斯克莱德大学 土木工程系，英国
3. 同济大学 地下建筑与工程系，上海
格拉斯堡
G4ONG；
200092)
摘要：着重研究一个典型的混凝土重力坝的坝底水浮力对大坝稳定性的影响，此大坝位于意大利的 Cumbidanovu
岛。大坝的基础由含有高度开裂的岩石所构成。首先，通过把大坝视为自由体的平衡分析法来评价大坝破坏前的
最大水压力和有效排水系统对大坝稳定性的影响。然后，使用离散元方法来进一步评价开裂基岩中的水流状态，
得到该水流产生的浮托力的分布，最终得到此水浮力对大坝稳定性的影响。对设计而言，上述分析考虑了岩基渗
透，运用离散元方法进行模拟。研究结果表面，相比保守的平衡分析法，此模型可以得到更大的水浮力荷载。
关键词：水利工程；离散元法；水浮力；大坝稳定性；重力坝
Received date：2008–02–22；Revised date：2008–05–20
Foundation item：National Science Foundation Project(50679057)；Pujiang Talents Project，Shanghai，China(06PJ14088)
Corresponding author：UTILI Stefano(1976–)，male，Ph. D.，graduated from Politecnico di Milano in 2000. He is now at Strathclyde University，and his main
research interests are mainly covered in DEM，slope stability，and limit analysis. E-mail：mjjiang@excite.com
第 27 卷 第 8 期
UTILI Stefano，
et al. Influences of Hydraulic Uplift Pressures on Stability of Gravity Dam
• 1555 •
INTRODUCTION
stability since it may give rise to a significant vertical
force making the dam overturning about its toe and
The gravity dam investigated is located at
decreasing its frictional resistance against sliding[3 4].
The following two methods will be used to evaluate
Cumbidanovu，Sardegna，Italy；and it rests on a
the influence of the uplift thrust on dam stability：
heavily fractured rock foundation. The data related to
the foundation have been taken from the results
equilibrium analysis of the dam as a rigid free
body(single wedge analysis) and the distinct element
where a detailed collection of the geological and
method(DEM). The former has been used to achieve
geotechnical
a first rough estimation of the maximum reservoir
level and some insight on the influence of the
1
，
[1]
data
for
the
site
is
available.
Concerning with the dam body instead(see Fig.1)，
the geometry has been taken from the proposal A2 of
the 1999 ICOLD(International Committee of Large
Dams) held at Denver [2]，USA，which is therefore
slightly different from the real one. This assumption，
however ， has no influence on the final results
obtained since the global size of the dam is the same
and the uplift pressure distribution underneath the
dam base is not affected by the shape of the dam
body. The objective of the ICOLD proposal was to
benchmark numerical and analytical methods for the
evaluation of the maximum sustainable reservoir
level before dam collapsing and the evaluation of the
uplift pressure distributions acting along the dam
base.
effectiveness of drainage system on stability of
the dam. The usefulness of this method for
，
stability calculations[4 5] for gravity dams is well
known. From a theoretical viewpoint，an alternative
formulation by the upper bound limit analysis
method could be adopted. According to this method，
instead of the cardinal equations of statics ， the
balance between external work and dissipated energy
is imposed[6]；but in this paper，the single wedge
mechanism formulation has been preferred for sake
of simplicity. The DEM has been used to perform a
more refined analysis calculating the uplift thrust
acting along the base of the dam；as a result of the
actual hydraulic flow taking place into the fractured
rock foundation for both cases of effective and
ineffective drainage system. The use of the DEM for
dam stability analyses[7] is not new ， but its
application to the study of seepage within the rock
foundation as a coupled hydro-mechanical problem，
is quite recent in the literature since seepage in the
foundation is often calculated by finite element
analyses[8] ， which can not directly model the
hydraulic flows within the joints in the foundation.
For gravity dams，the most common drainage
system is made by an inspection gallery running
longitudinally above the dam base and located as
close as technologically achievable to the foundation
and several drainage holes drilled from the gallery
floor into the rock foundation(for details of the
，
drainage system see USACE regulations[9 10]).
Fig.1
Dam geometry
2
In the literature，it is known that hydraulic uplift
thrust due to water percolating underneath the dam
base is an important factor to be assessed for dam
EQUILIBRIUM ANALYSES OF
DAM
In this section，Some calculations based on the
• 1556 •
2008 年
岩石力学与工程学报
performed to determine the maximum sustainable
But the assessment of the effectiveness of the drains
after dam construction and reservoir impoundment is
reservoir level. According to the method adopted，the
a difficult task. Therefore，it is common practice to
dam assumed to be rigid and static equilibrium has
make assumptions on their effectiveness taking
been imposed taking into account all the forces
values recommended by regulations issued by
acting on it：the pull of gravity upon the dam mass，
designated engineering national bodies although
the lateral thrust due to the water impounded in the
rules are not，in general，strictly prescriptive，in
reservoir and the pore water pressure distributed
Europe and the US at least.
equilibrium of the dam as a free rigid body will be
along all or a portion of the boundaries of the dam body.
According to the related Italian and USACE
Regulations[11
，12]
The foundation has been considered impermeable
whereas the joint between dam and foundation has
the uplift pressure along the dam base may be assumed
been assumed fully permeable. To work out the
to vary linearly from the value of hydrostatic pressure
following calculations，the upper right triangular part
generated by the reservoir underneath the dam heel，
of the dam crest(see Fig.2) has been neglected since
which is µγw in our case，to the hydrostatic pressure
it represents only 0.52% of the total area given by
underneath the dam toe due to the water level
in case of absent drainage system，
where B and H are the width of dam base and dam
downstream of the dam that is zero in our case. If a
part of the dam base，adjacent to the upstream
height respectively.
face(marked as a in Fig.2)，is not compressed against
Atot = BH / 2
(1)
the foundation because of a high overturning
moment acting on the dam，the uplift pressure p
under the non-compressed part must be assumed
equal to the hydrostatic pressure corresponding to
the reservoir water level(see Fig.2(a))：
p = µγw
(2)
where µ and γw are the reservoir water level and the
water unit weight respectively.
In case of a properly working drainage system，
the distribution recommended by the regulations is
(a) Absent or ineffective drainage system
bilinear：constant under the non-compressed part of
the dam ， a ， and linearly decreasing in the
compressed part. The uplift pressure variation along
the compressed part of the dam is characterized by a
steeper gradient upstream of the drains line and a
flatter one downstream of it(see Fig.2(b)). This type
of uplift distribution is also assumed in related
results[13]. Underneath the drains line，the regulations
of many countries recommend to assume a value of
pressure ranging from 0.25 to 0.6 times the
difference between the pressure underneath the
(b) Effective drainage system
Fig.2
Uplift water pressure distribution under the dam
upstream face，p = µγw，in our case，and the pressure
underneath the toe，p = 0 in the case. According to
the Italian Regulations[11]，this value must be 0.35
Drainage is the single most effective mean of
times the aforementioned difference. Up taking this
significantly reducing the amount of uplift thrust.
recommendation in our calculations ， the uplift
第 27 卷 第 8 期
UTILI Stefano，
et al. Influences of Hydraulic Uplift Pressures on Stability of Gravity Dam
• 1557 •
pressure acting under the drains line results to be p =
0.35µγw. If the drains are not suitable to guarantee
such a pressure reduction，stability analyses must be
performed assuming no drainage system. When the
non-compressed zone extends downstream of the
drain line(a＞d)，thus，the recommended pressure
distribution becomes constant along the noncompressed zone and linear in the compressed one，
hence falling again into the case shown in Fig.2(a).
In the following section，the stability of the dam
will be examined with regard to two possible
collapse mechanisms：overturning about the toe and
sliding along the dam-foundation interface joint.
Punching of the dam into the foundation or the
development of a failure surface throughout the
foundation has been disregarded since the rock
constituting the foundation is highly resistant to
crushing and shear failure. The two mechanisms，
overturning and sliding，therefore will be considered
separately.
2.1 Overturning mechanism
Overturning of the dam about its toe occurs
when the overturning moment due to the lateral
thrust of the water impounded in the reservoir and to
the uplift forces acting on its base overcomes the
resistance of the stabilizing moment due to its
dead-weight. Considering first a low reservoir level，
the dam rests on the foundation lying with its entire
base in contact with it. If the reservoir level is then
continuously increased，the dam will be progressively
uplifted with an increasingly smaller part of its base
in contact with the foundation until the contact reduces
to one point，its toe，and overturning effectively occurs.
In this case，the vertical net reaction force exchanged
between dam and foundation， N ′ ，passes through
the only point of contact O(see Fig.3). In this case，
the uplift pressure along the dam base is，according
to the regulations，uniform and equal to µγw since a =
B and therefore a＞d. In order to determine the
reservoir level responsible of overturning，µoverturning，
it is enough to impose the equilibrium of moments
around the point O：
H
H
B 2
+ S2
+ U − WB = 0
(3)
3
2
2 3
where S1 and S 2 are the triangular and rectangular
S1
Fig.3
Forces acting on the dam at incipient overturning
(sliding is not considered)
components of the lateral thrust respectively，U is the
uplift thrust，and W is the dam dead-weight.
After calculation of the forces and some passages
(see Appendix)，the reservoir level at incipient overturning is achieved：
µ overturning
⎡ γ c ⎛ B ⎞2 ⎤
2 H ⎢ ⎜ ⎟ + 1⎥
⎥⎦
⎢⎣ γ w ⎝ H ⎠
=
= 80.21 m
2
⎡⎛ B ⎞
⎤
3⎢⎜ ⎟ + 1⎥
⎢⎣⎝ H ⎠
⎥⎦
(4)
where γ c is concrete unit weight， γ c = 24 kN/m3.
Considering now a reservoir level µ＜µoverturning
but high enough to uplift a part of the dam base from
the foundation(a＞0)，the position of N ′ is unknown
and therefore the problem becomes statically
indeterminate. To solve the problem，it is necessary
to introduce a hypothesis on the distribution of the
effective normal stresses along the compressed part
of the dam against the foundation. First，if the dam is
considered as a rigid body，and the reaction offered
by the foundation by means of Winkler′s elastic
springs with constant stiffness，the distribution of the
normal stresses is triangular(see Fig.4(a)). But in
reality，the more the dam is subjected to a higher
overturning moment under an increasing reservoir
level，the more its toe will be compressed against the
foundation，and therefore the zone of the dam near
the toe will be subjected to high compressive stresses
• 1558 •
(a) Triangular
Fig.4
2008 年
岩石力学与工程学报
(b) Real
(c) Uniform
Stress distributions along the compressed zone of the
dam base with x/3＜c＜x/2
leading to plastic stress redistribution. In fact，it is
(a) Absent drainage system
well known that concrete undergoing high compressive
stresses exhibits a stress-strain relationship progressively
drifting away from linearity until crushing occurs.
This implies that along the dam zone in compression，
x = B − a ，the normal stresses will vary nonlinearly
as shown in the distribution in Fig.4(b). To find out
Triangular stress distribution with
effective drainage
Uniform stress distribution with
effective drainage
the real stress distribution is a complex problem that
would require nonlinear 3D finite element analyses.
To bracket the real stress distribution，two distributions
(b) Effective drainage system
were assumed：a triangular and a uniform one. In the
former case， N ′ is applied at x/3 from the dam toe，
Fig.5 Reservoir level vs. non-compressed zone length
whereas in the latter one at x/2 with the true line of
action of N ′ lying somewhere in between x/3 and
obtained in case of triangular stress distribution are
x/2.
distribution，since the destabilizing moment due to
N ′ is smaller because of a shorter lever arm(x/3
To determine the extension of the uplifted part
of the dam base，a，given a known reservoir level µ，
it is necessary to impose the equilibrium of moments
for the forces acting on the dam assuming point O，
as the pole：
∑M
overturning
= ∑ M stabilising
(5)
In Appendix，the analytical calculations are
higher than those obtained in case of uniform
instead of x/2).
Looking now at Fig.5(a)，the gap between the
two curves reduces with increasing a，until it becomes
nil at a = B = 60 m，when overturning occurs. Since
the dam is in contact with the foundation only in one
point，its toe，and the location of N ′ becomes
shown. Depending on the two distributions adopted，
independent of the assumed stress distribution.
two different polynomial expressions were found：
In case of effective drainage system，the analytical
expressions µ = µ (a) obtained for a＜d give rise to
µ triang = µ ( a) ⎫⎪
(6)
higher µ values than the case of absent drains，since
Since the real normal stress distribution has
effective drainage allows a significant reduction of
uplift pressures making the dam much more stable
⎬
µ uniform = µ1 ( a) ⎪⎭
been bracketed by the two distributions assumed，
also the real relationship µreal = µ(a) will be bracketed
by the two functions in Eq.(6). In Fig.5，the real
function will lie in the region of space enclosed by
the two lines，the gray one，corresponding to µtriang，
and the black one corresponding to µuniform. In Fig.5(a) it
is shown the case of ineffective or absent drainage
system whereas in Fig.5(b) the case of effective
drainage. In the graphs，the reservoir levels µtriang
(compare Fig.5(a) with Fig.5(b) for a＜ d ) . But
when the non-compressed zone goes beyond the
drains line，the drains must be considered ineffective
according to the regulations[11
，12]
and therefore uplift
water pressures suddenly increase to the value they
assume in case of absent drains(see Fig.6). This is
due to the fact that when the dam is uplifted beyond
the drains line an effective water interception by the
drainage system is no longer possible. In Fig.5(b)，
第 27 卷 第 8 期
Region 1
UTILI Stefano，
et al. Influences of Hydraulic Uplift Pressures on Stability of Gravity Dam
• 1559 •
Region 2
Fig.6 Uplift pore pressure distributions in case of a1＜d
(region 1) and a2＞d(regions 1 + 2)
the dashed curves for a＞d are the same curves as
the solid ones plotted in Fig.5(a) for the case of absent
drainage system. They are dashed to point out that
they are only theoretical ones. In fact，considering
the dam subjected to an increasing reservoir level，as
soon as the uplifted zone goes beyond the drains
line ， a sudden increase of a occurs at constant
reservoir level. In case of uniform distribution，the
reservoir level may continue to be increased even
after the uplifted zone has gone beyond the drains
line until the dam overturns around its toe(a = B)；
whereas in case of triangular distribution ， the
reservoir level cannot be further increased since the
reservoir level reached at a = d = 10 m is larger than
µoverturning. Hence，in case of uniform distribution，the
presence of drains increases the load bearing capacity
of the dam only for values of a＜17 m，whereas in
case of triangular distribution，for any value of a.
2.2 Sliding mechanism
Resistance against horizontal sliding is given by
a frictional component proportional to the net vertical
force exchanged between dam and foundation， N ′ ，
and a cohesive one. This force opposes the lateral
thrust due to the water impounded in the
reservoir(see Fig.7). If Tres is the resisting force，it is
given by：
Tres = N ′ tan ϕ + c′x
calculation for the sliding mechanism
From Eq.(8)，two polynomial functions µ = µ(a)
are obtained ： one in case of absent drainage
system(see Fig.8(a)) and the other in case of
effective drains(see Fig.8(b)). In both cases ， the
reservoir level µ decreases with increasing a. This is
because the higher the value of a ， the less the
cohesive part of the resisting force，and therefore the
lower the value of µ .
＞
＞
(a) Ineffective drains
＞
＞
(7)
where ϕ and c′ are the friction angle and the
cohesion along the dam base with N′ calculated by
imposing the vertical equilibrium of forces.
The relationship between the reservoir level µ，
and the extension of the non-compressed zone a，is
found imposing the horizontal equilibrium(details of
the calculations are given in Appendix)：
Tres = Tact
Fig.7 Forces acting on the dam taken into account in the
(8)
(b) Effective drainage system
Fig.8 Reservoir level vs. uplifted zone in case of ineffective
drains and effective drainage system
• 1560 •
2008 年
岩石力学与工程学报
2.3 Determination of the dam bearing capacity
this condition does not lead to collapse since the
To determine the bearing capacity of the dam，
reservoir level can still be increased prior to the onset
the two investigated mechanisms must be analyzed
of sliding. In reality，the normal stress distribution
together. In Fig.8，a graph is shown in terms of
reservoir level versus length of the uplifted zone
will be neither linear nor uniform but something in
along the dam base，a. The curves related to the
analyses performed if the attainment of the condition
overturning mechanism are given by solid lines
whereas the curves related to sliding are given by
a = d leads to the collapse of the dam or there is still
cross lines. The real collapse load is given by the
system is presented，higher reservoir loads can be
between， so it is not clear from the equilibrium
a margin of safety. However，if an effective drainage
minimum collapsing load determined for each
mechanism taken separately. Therefore，considering
sustained by the dam in so far as the drainage system
remains efficient ， i.e. the uplifted zone remains
the dam under an initially low and then increasing
upstream of the drains line，a＜d.
reservoir level，the more critical failure mechanism
is overturning since it is associated with the lower
3
curve. This means that the dam will be gradually
uplifted because of increasingly higher overturning
moments(see curves with pointed arrows in the
DISTINCT ELEMENT ANALYSES
The specifications of the A2 proposal of the
ICOLD Benchmark 99 do not provide data related to
graphs). The dam uplift，in turn，will progressively
the dam foundation. These have been taken from the
decrease the resistance against sliding along the dam
foundation interface(see the descending curves in
foundation of Cumbidanovu′s dam[1]. Instead，the
geometry of the dam and its mechanical properties
Fig.8 related to the sliding mechanism). When the
have been assumed according to the Benchmark
curve related to the overturning mechanism and the
curve related to sliding meet in point A for the case
specifications that are consistent with the equilibrium
analyses. These are given in Table 1.
of triangular distribution and point B for the case of
Concerning the joints，a Mohr-Coulomb shear-
uniform distribution，the collapse of the dam occurs.
In correspondence of those points，it happens that
strength relation has been adopted. The dam-foundation
Tres becomes smaller than Tact and the dam starts
joint is characterized by a softening behavior as
sliding downstream.
The collapse load calculated by the performed
prescribed by the ICOLD specifications(see Fig.9). The
equilibrium varies within a precise range：from 76.5
mechanical parameters，also reported in Table 1，are
given by：c′ = 0.7 MPa，ϕ ′ = 30°，ψ = 10°(dilation
to 78 m in case of absent/ineffective drainage system
(line AB in Fig.8(a))；from 76.5 to 81.8 m in case of
angle) ， f t = 0(tensile resistance) ， c′res = 0(residual
′ = 30°(residual friction angle). The joints
cohesion)，ϕ res
effective drainage system(line AB in Fig.8(b)).
In case of linear stress distribution，as soon as
in the foundations are characterized by the following
parameters[1]：c′ = 0 MPa，ϕ ′ = 30°，ψ = 0°， f t = 0
the uplifted zone goes beyond the drains line，sliding
MPa. Water viscosity K j has been assumed equal to
occurs whereas in case of uniform stress distribution，
K j = 300 Pa·s 1.
Table 1
Methods
－
Summary of the physico-mechanical parameters used in calculation
Dam
Foundation
Dam-foundation joint
γc/(kN·m 3)
Ec/GPa
νc
γrock/(kN·m 3)
Erock/GPa
νrock
ϕ/(°
)
c′/MPa
ψ/(°
)
ft
ϕres/(°
)
c′res
Equilibrium analysis
24
24.5
0.17
–
–
–
30
0.7
–
0
–
–
DEM
24
24.5
0.17
26
20
0.3
30
0.7
10
0
30
0
Methods
－
－
Joints in foundation
Kj/(Pa·s 1)
ha0 /mm
–
–
–
–
–
–
0
300
1
0.5
5×104
5×102
ϕ/(°
)
c′/MPa
ψ/(°
)
ft/MPa
Equilibrium analysis
–
–
–
DEM
30
0
0
－
hares /mm
KN/(kN·m 1)
－
KS/(kN·m 1)
－
第 27 卷 第 8 期
UTILI Stefano，
et al. Influences of Hydraulic Uplift Pressures on Stability of Gravity Dam
• 1561 •
In order to perform a meaningful analysis the
choice of the hydraulic conductivities along the
joints is critical. To verify that the chosen values，
ha0 = 1 mm and hares = ha0 / 2 ，are adequate for the
problem；the simpler case of non-fractured foundation
has been analyzed in order to check the agreement
Tension σt
Fig.9
with the equilibrium analyses performed in Section 2.
o
Shear strength of the dam-foundation joint according
[2]
to the Mohr-Coulomb criterion
In Fig.11，it is shown the pressure distribution at
incipient collapse with a，the length of the uplifted
part of the dam base beyond the drains line. This
tangential mechanical joint stiffnesses，K N and K S
distribution is well in agreement with that taken from
the referenced regulations and assumed in the
respectively，and the joint hydraulic apertures， ha0
equilibrium analyses(see Fig.2(a)，for the case a＞
Other parameters ， such as the normal and
and hares ，have been selected by the authors.
The regulations taken as a reference for the
d ). This match corroborates the goodness of the
hydraulic apertures adopted in the UDEC analyses.
equilibrium analyses only specify the reduction of
uplift pressure operated by the drains without any
indication in regard to the hydraulic flow along the
drains. Therefore ， the same hydraulic condition
assumed in Section 2 has been implemented into the
distinct element analyses by assigning a fixed pore
pressure value ， p = 0.35µγw ， at the point of
intersection between the drains line and the
dam-foundation joint.
Fig.11 Pore pressure distribution underneath the dam
In Fig.10，it is shown the relationship between
hydraulic aperture and normal stress adopted in the
code UDEC[14
assuming intact foundation(reservoir level，
µ = 80 m)
，15]
：the maximum hydraulic aperture
along a joint is reached when the compressive stress
goes to zero， ha0 in the figure，then the hydraulic
aperture varies linearly to a residual value，hares ，with
m given by：
K N and K S ，the mechanical stiffnesses of the
joints，rule the amount of mechanical overlapping
allowed between two adjacent blocks but they also
greatly influence the time needed to run a simulation
since the stability of the integration scheme used to
m = −1 / K N
(9)
solve the equations of motion is explicit and the time
step adopted by the code is inversely proportional
to the square root of K. Here ， there are two
conflicting objectives which can be labeled as“fast
computations” and“realistically deformable joints”.
Tension σt
After some preliminary analyses，K N = 5×104 kN/m
1
and K S =
K N have been proven suitable for the
100
problem.
3.1 Analyses
o
Fig.10 Hydraulic aperture of joint，ha，vs. stress normal to
joint，σn，within the rock foundation
[14]
The effect of the inclination and spacing of the
joints has been studied assuming four different
• 1562 •
岩石力学与工程学报
geometries. The total of the presented models is
eight for each set of joints both cases of absent and
effective drainage system have been considered(see
Table 2). In the first configuration，the joints form a
+45° angle with the horizon，the spacing between
joints being：in model a，6 m for the first set and 12 m
for the second set；in model b，12 m for the first set
and 6 m for the second set. In model c，the first set of
joint is +15° inclined over the horizon with 6 m
spacing and the second set －69°with 12 m spacing；
whereas in model d，the first set is +15°inclined over
the horizon with 12 m spacing and the second is
－69°
with 6 m spacing.
Table 2
Summary of the models analyzed
Joint
inclination
Model
Drains
to the
)
horizon/(°
a
b
c
d
UDEC
LE analysis
Block
analysis bearing
bearing
size
/(m×m) capacity µ/m capacity µ/m
a1
+45
No
12×6
87
76.5–78.5
a2
－45
Yes
12×6
91
76.5–81.8
b1
+45
No
6×12
85
76.5–78.5
b2
－45
Yes
6×12
87
76.5–81.8
c1
+15
No
12×6
77
76.5–78.5
c2
－69
Yes
12×6
80
76.5–81.8
d1
+15
No
6×12
75
76.5–78.5
d2
－69
Yes
6×12
80
76.5–81.8
2008 年
The loads(dam weight and reservoir level) were
applied progressively in successive steps in order to
achieve equilibrium conditions in a quasi-static way
avoiding to introduce any inertial effects. First，the
load of the dam was applied to the foundation
increasing the dam dead-weight by steps；and then
the reservoir load was assigned by steps until the
dam collapsed. After each step，a number of cycles
large enough for the system to reach static
equilibrium was run. Some preliminary analyses
were performed to determine the minimum number
of steps for which the influence of the loading
procedure on the final results is negligible.
The reservoir level was increased by 5 m steps
up to the collapse of the dam. Then，the analysis was
resumed from the last step of loading before collapse
and smaller increments of reservoir level were assigned
to identify with better accuracy the reservoir level
causing collapse. Collapse was detected looking at
the displacement undergone by the dam. After the
total displacement of the crest exceeded 0.5 m，the
analysis was stopped.
In all the simulations run，a small numerical
damping was used. This damping is given by decreasing
block accelerations and increasing block decelerations
Each block of rock and the dam have been
assumed linearly elastically deformable. To impose
the elastic law，each rock block in the foundation has
been subdivided into four regular triangles whereas
the dam has been subdivided into non-regular triangular
elements(see gray lines in Fig.12). Then，stresses and
strains were calculated by UDEC in the nodes of
each mesh every time-step.
by a factor proportional to the unbalanced forces
acting on each block. It was used only to shorten the
computational time needed to run a simulation and
therefore it has no particular meaning but its use is
justified as long as it does not affect the global
response of the assembly of blocks to the mechanical
and hydraulic actions imposed. Some preliminary
tests with zero damping were run to verify that the
introduced damping had a negligible effect in terms
of the analyzed global variables.
3.2 Results
For all the joint configurations analyzed，the
maximum hydraulic flow takes place along the
dam-foundation joint and it has been noted that the
greater the distance of a joint from the dam base is，
the smaller the flow is. This is due to the fact that
the hydraulic conductivity of the joints depends on
the compressive stresses. For increasing depths，
Fig.12
Domain of analysis(model c)：black lines indicate
compressive stresses increase and the hydraulic
edges of blocks；gray lines indicate the mesh contour
conductivity decreases as a result.
第 27 卷 第 8 期
UTILI Stefano，
et al. Influences of Hydraulic Uplift Pressures on Stability of Gravity Dam
• 1563 •
In Fig.13 ， it is shown the hydraulic flow
monitored along the dam-foundation joint with the
the dam，when a = 0 ，the distribution is quite in
dam at incipient failure：each point in the graph
In case of absent/ineffective drainages，it is linear
indicates the flow along one hydraulic domain
extending between two consecutive rock joints
(see Fig.15(a))；whereas in case of effective drains in
operation，the distribution changes：it is still linear
intersecting the dam-foundation joint. It can be noted
upstream of the drains but becomes parabolic-like
that in case of absent drainage system，the flow is
almost constant. In case of effective drainage system
instead，the flow along the upstream part of the joint
is more than twice as high as the flow in the previous
case，whereas the flow along the downstream part
of the joint，from the drains line to the dam toe，
remains lower than the previous case with a jump in
correspondence of the drainage point. Moreover，the
agreement with what prescribed by the regulations.
downstream of them(see Fig.15(b)) slightly different
from what prescribed by the regulations. After the
dam starts uplifting，a＞0，the obtained pressure
distributions strongly depend on the geometry of the
rock joints，drifting substantially from the prescriptions
of the regulations.
Dam base/m
the hydraulic flow in a large area of the foundation.
Depending on the inclination of the rock joints，it
may also happen that the flow along the damfoundation interface goes upstream for a short
length(see Fig.14，just before the drains line).
Pore pressure/kPa
presence of the drainage system greatly influences
0
100
200
300
400
500
600
700
800
0
10
20
30
40
50
60
Model b
Model d
Regulations
h = 85 mm(no drainage system)
h = 87 mm(drainage system)
20
Dam base/m
15
10
5
0
0
10
20
30
40
50
60
Dam-foundation joint/m
Fig.13
Hydraulic flow along the dam-foundation interface
at incipient collapse：black line in case of effective
drainage system and gray one in case of absent
drainage system
Pore pressure/kPa
－
Hydraulic flow /(L·s 1)
(a) No drainage
25
0
100
200
300
400
500
600
700
800
0
10
20
30
40
50
60
Model b
Model d
Regulations
(b) Active drainage
Fig.15 Uplift pressures with the dam base all in contact
against the foundation，a = 0，in case of models
b and d with reservoir level µ = 70 m
The load bearing capacity of the dam varies a
lot，depending on the geometry of the joints in the
foundation(see Table 2). Making a comparison with
the equilibrium analyses of Section 2，it is possible
to say that in case of joints inclined ± 45°on the
horizontal line，the resulting bearing capacity is，on
average，7 m higher than the equilibrium analysis
Fig.14
Hydraulic flow directions underneath the dam
with drainage system in operation
With regard to the uplift pressure underneath
predictions；whereas，in case of joints inclined +15°/
－ 69 °on the horizontal line(cases c and d) ， the
bearing capacity is within the range predicted by
equilibrium analyses. The physical explanation is as
• 1564 •
2008 年
岩石力学与工程学报
follows：in models a and b，given a known reservoir
level µ，a smaller length a and lower water pressures
Fig.17 provides the horizontal displacements of
the left edge of the dam crest during the reservoir
underneath the dam were generated leading to a
filling until the last step of loading before failure.
larger resistance against sliding. On the contrary，in
models c and d，given the same reservoir level µ，a
The trend shown in Fig.17 is reasonable ： at the
beginning， the crest moves upstream，being the dam
larger length a and higher uplift pressures were
rotating counterclockwise，and then moves downstream
generated leading to a smaller resistance against
sliding. In fact，resistance against sliding depends，
with the dam rotating clockwise. The horizontal
asymptote in the figure indicates the onset of the
for its cohesive term，on the extension of the zone in
sliding of the dam along the foundation.
contact with the foundation ， B-a ， and ， for its
frictional term，on the amount of N ′ (see Eq.(7))
that in turn depends on the uplift pressures.
Using UDEC，it is also possible to investigate
the displacements undergone by the dam，otherwise
not achievable by the equilibrium analyses performed
in Section 2. During its process of construction and
until the reservoir level becomes half full(about 35 m)，
the dam tends to tilt towards the reservoir. Then，
further increases of the reservoir level lead to a
Fig.17
Horizontal displacements recorded at the dam crest
vs. reservoir level
change of direction ， the dam tilting towards the
opposite side(see Fig.16).
4
CONCLUSIONS
By means of static equilibrium analyses，it has
been possible to achieve analytical expressions for
the bearing capacity of a typical Italian gravity dam
resting on a rock foundation. To this end，the uplift
thrust due to the hydraulic pressures underneath the
dam and the beneficial effect of an effective drainage
system were taken into account by assuming uplift
(a) µ = 35 m
pressure distributions recommended by the Italian
and USACE regulations.
Secondly，a more refined analysis was carried
out by the DEM. The hydraulic flow taking place
within the fractured foundation and the influence of
different geometries of rock joints on the uplift thrust
and，as a result，on the dam bearing capacity was
investigated. With this numerical method，it was also
possible to monitor the displacements undergone by
the dam.
Both methods are shown to be viable ways of
(b) µ = 85 m
Fig.16
investigating the influence of the hydraulic uplift
Dam and rock foundation for model b with reservoir
pressures on the stability of dams. Comparing the
level(deformations magnified by 100 times)
obtained results，equilibrium analyses give rise to
第 27 卷 第 8 期
UTILI Stefano，
et al. Influences of Hydraulic Uplift Pressures on Stability of Gravity Dam
more conservative predictions in terms of dam
[6]
徐千军，李
• 1565 •
旭，陈祖煜 . 百色水利枢纽主坝坝基三维抗滑稳
bearing capacity ， whereas the DEM is a more
定分析 [J]. 岩石力学与工程学报，2006 ，25(3)：533 –538.(XU
powerful numerical tool which can be relied on to
Qianjun ， LI Xu ， CHEN Zuyu. Three-dimensional stability
accept less conservative predictions，since it can also
analysis of dam foundation of Baise hydro-junction[J]. Chinese
model the hydraulic flow within fractured rock
Journal of Rock Mechanics and Engineering，2006，25(3)：533–
foundations.
538.(in Chinese))
For design of new dams or evaluation of existing
[7]
张
冲，侯艳丽，金
峰，等. 拱坝–坝肩三维可变形离散元
ones，in light of the analyses performed，it emerges
整体稳定分析 [J]. 岩石力学与工程学报，2006，25(6)：1 226–
the importance of having both reliable data about the
1 232.(ZHANG Chong，HOU Yanli，JIN Feng，et al. Analysis of
hydraulic permeability of rock foundations and a
arch dam-abutment stability by 3D deformable distinct elements[J].
computationally advanced modeling，e.g. DEM，so
Chinese Journal of Rock Mechanics and Engineering，2006，25(6)：
that loads significantly higher than the estimations
1 226–1 232.(in Chinese))
obtained from equilibrium analyses might be accepted.
[8]
岩石力学与工程学报，2007，26(增 1)：3 017–3 024.(XIANG
ACKNOWLEDGEMENTS The authors would like
Yan ， WU Zhongru. Analysis theory and method of crossfeed
to thank the staff at Politecnico di Milano(Milan，
between concrete dam body and foundation[J]. Chinese Journal of
Italy)，in particular Professor R. Nova and Professor F.
Rock Mechanics and Engineering，2007，26(Supp.1)：3 017–
Calvetti for their support. The support of Itasca C.G.
is also gratefully acknowledged.
3 024.(in Chinese))
[9]
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[10]
BONINI M. Studio delle condizioni di flusso idraulico nell′ammasso
Department of the Army，1986.
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[12]
LIU J， FENG X T， DING X L. Stability assessment of the Three
modeling，part II：numerical modeling[J]. International Journal of
[5]
U.S. Army Corps of Engineers. Gravity dam design— engineer
manual 1110–2–2200[S]. Washington，D. C.：Department of the
Gorges dam foundation， China ， using physical and numerical
[4]
MINISTERO L P. Norme tecniche per la progettazione e la
costruzione delle dighe di sbarramento(DM No.44)[S]. Roma：[s.
ICOLD. A2 benchmark proposal for numerical and analytical
methods for gravity dams[R]. Denver，USA：[s. n.]，1999.
[3]
U.S. Army Corps of Engineers. Seepage analysis and control for
dams—engineer manual 1110–2–1901[S]. Washington， D. C.：
roccioso di fondazione di una diga a gravita′ mediante il metodo
degli elementi distinti[Ph. D. Thesis][D]. Politecnico Torino：[s. n.]，
U. S. Army Corps of Engineers. Grouting technology—engineer
manual 1110–2–3506[S]. Washington：Department of the Army，
References(参考文献)：
[1]
向 衍，吴中如. 混凝土坝坝体与坝基互馈的分析理论和方法[J].
Army，1995.
[13]
常晓林，蒋春艳，周
伟，等. 岩质坝基稳定分析的等安全系
Rock Mechanics and Mining Sciences，2003，40(5)：633–652.
数法及可靠度研究 [J]. 岩石力学与工程学报， 2007 ， 26(8) ：
YU X，ZHOU Y F，PENG S Z. Stability analyses of dam abutments
1 594– 1 602.(CHANG Xiaolin， JIANG Chunyan， ZHOU Wei，
by 3D elastoplastic finite-element method：a case study of Houhe
et al. Equal safety factor method and its reliability analysis for rock
gravity-arch dam in China[J]. International Journal of Rock
foundation of dam[J]. Chinese Journal of Rock Mechanics and
Mechanics and Mining Sciences，2005，42(3)：415–430.
Engineering，2007，26(8)：1 594–1 602.(in Chinese))
侯艳丽，张
冲，张楚汉，等. 拱坝沿建基面上滑溃决的可变
[14]
Itasca Consulting Group Inc.. Universal distinct element code，user
形离散元仿真 [J]. 岩土工程学报，2005，27(6)：657–661.(HOU
manual(Version 3.0)[R]. Denver： Itasca Consulting Group Inc.，
Yanli，ZHANG Chong，ZHANG Chuhan，et al. Simulation of
1999.
upward-sliding failure of interface in arch dams by deformable
[15]
CUNDALL P A. UDEC—a generalized distinct element program
distinct elements[J]. Chinese Journal of Geotechnical Engineering，
of modeling jointed rock[R]. [S. l.]：European Research Office，
2005，27(6)：657–661.(in Chinese))
US Army，1980.
• 1566 •
2008 年
岩石力学与工程学报
APPENDIX
A.1 Overturning mechanism
In the following，it is shown the detail of the
derivation of Eq.(4) from Eq.(3). The forces in Eq.(3)
are given by：
⎫
H2
S1 = γ w
⎪
2
⎪
S 2 = γ w ( µ − H ) H ⎪⎪
(A1)
⎬
U = γ w µB
⎪
⎪
BH
⎪
W =γc
⎪⎭
2
Therefore，substituting them in Eq.(3)，it can be
obtained：
H3
H2
B2
H
γw
+ γ w (µ − H )
+γ wµ
= γ c B2
(A2)
6
2
2
3
Then，it can be written：
⎛ B2 H 2 ⎞
H γ wH3 ⎫
⎟⎟ = γ c B 2
+
+
µγ w ⎜⎜
⎪
2 ⎠
3
3 ⎬ (A3)
⎝ 2
⎪
µ 3γ w ( B 2 + H 2 ) = 2γ c B 2 H + 2γ w H 3 ⎭
(a) a＜d
And finally：
⎡ γ ⎛ B ⎞2 ⎤
2 H ⎢ c ⎜ ⎟ + 1⎥
⎢⎣ γ w ⎝ H ⎠
⎥⎦
µ=
2
⎡⎛ B ⎞
⎤
3⎢⎜ ⎟ + 1⎥
⎢⎣⎝ H ⎠
⎥⎦
(A4)
Hereafter，it is shown the derivation of the analytical
functions µ i = µ i (a) giving rise to the graphs plotted
in Fig.5. Let′s start with the case of effective drainage
system and a＜d as shown in Fig.A1(a). In this case µ＜
H，the forces acting on the dam are：
Imposing the equilibrium of moments about point
O(see Fig.A1(a))，leads to：
µ2
⎫
⎪
2
⎪
⎪
f = 0.35
⎪
⎪
⎪
U 1 = γ w µa
⎪
⎪
1
U 2 = γ w µ (1 − f ) ( d − a )⎬
2
⎪
⎪
U 3 = γ w µf ( d − a )
⎪
⎪
1
U 4 = γ w µf ( B − d )
⎪
2
⎪
⎪
BH
⎪
W =γc
2
⎭
S1 = γ w
(b) a＞d
Fig.A1 Forces acting on the dam overturning mechanism
S1
(A5)
µ
a⎞
⎡2
⎤
⎛
+ U 1 ⎜ B − ⎟ + U 2 ⎢ (d − a ) + B − d ⎥ +
3
2⎠
⎣3
⎦
⎝
2
⎡1
⎤
U 3 ⎢ (d − a ) + B − d ⎥ + U 4 ( B − d ) +
3
⎣2
⎦
2
(A6)
N' x − W B = 0
3
Imposing the vertical equilibrium ， N ′ can be
expressed by：
N ′ = W − U1 − U 2 − U 3 − U 4
(A7)
Let us consider first the case of uniform stress
第 27 卷 第 8 期
distribution， x =
UTILI Stefano，
et al. Influences of Hydraulic Uplift Pressures on Stability of Gravity Dam
B−a
. Substituting N ′ and x into
2
Eq.(A6)：
S1
µ
3
a⎞
d
⎛
⎛ 2
⎞
+ U1 ⎜ B − ⎟ + U 2 ⎜ − a − + B ⎟ +
2
3
3
⎝
⎠
⎝
⎠
• 1567 •
The function f (a，µ) = 0 may be rewritten as a
third degree polynomial function：
A0 µ 3 + A1 µ 2 + A2 µ + A3 = 0
(A12)
with the coefficients Ai depending on the parameter a：
A0 = 1.67
⎫
⎪
A1 = 0
⎪⎪
⎬
A2 = 0.541 7a 2 + 164.2a + 2 033⎪
⎪
⎪⎭
A3 = −28 800a − 576 000
2 ⎞
⎛ B a⎞
⎛2
⎞
⎛ a d
U 3 ⎜ − − + B⎟ + U 4 ⎜ B − d ⎟ +W ⎜ − ⎟ −
3 ⎠
⎝ 2 2⎠
⎝3
⎠
⎝ 2 2
⎛ B a⎞
⎛ B a⎞
⎛ B a⎞
U1 ⎜ − ⎟ − U 2 ⎜ − ⎟ − U 3 ⎜ − ⎟ −
⎝ 2 2⎠
⎝ 2 2⎠
⎝ 2 2⎠
2
⎛ B a⎞
U4⎜ − ⎟ −W B = 0
3
⎝ 2 2⎠
(A8)
Then，
(A13)
A third degree function always presents three
solutions，but in this case only one is real. So，using
the available well known algebraic formulae，after
µ
a B a⎞
B a⎞
⎛
⎛ 2 d
S1 +U1⎜ B − − + ⎟ +U 2 ⎜ − a − + B − + ⎟ +
3
2 2 2⎠
2 2⎠
⎝
⎝ 3 3
some more calculations ， the sought expression
µ = µ (a) with B = −2 168.6 × 10 3 a − 43 373 × 10 3 and
B a⎞
B a⎞
2
⎛2
⎛ a d
U3 ⎜ − − + B − + ⎟ + U4 ⎜ B − d − + ⎟ +
2 2⎠
3
2 2⎠
⎝3
⎝ 2 2
C = 2.713 9a 2 + 822.44a + 10 186 can be obtained：
⎛B a 2 ⎞
W ⎜ − − B⎟ = 0
⎝2 2 3 ⎠
(A9)
µ=
µ
3
+U1
B
⎛ a d B⎞
+U 2 ⎜− − + ⎟ +
2
⎝ 6 3 2⎠
case of a＞d the pore pressure distribution changes
radically(see Fig.A1(b)). The forces involved in the
equilibrium of moments are the same as before except
(A10)
for the pore water pressure. The forces due to the
water pressure are(see Fig.5(b))：
Now all the forces can be replaced by their
analytical expressions obtaining：
γw
U 1 = γ w µa
⎫
⎪
⎬
1
U 2 = γ w µ (B − a )⎪
2
⎭
µ3
B
1
+ γ w µa + γ w µ (1 − f ) (d − a ) ⋅
6
2
2
γ cB
as illustrated above，another third degree polynomial
function is achieved which again has only one real root；
1
⎛a B 2 ⎞
( B − d )⎜ + − d ⎟ +
2
⎝2 6 3 ⎠
H ⎛ a B⎞
⎜− − ⎟ = 0
2 ⎝ 2 2⎠
(A15)
Then，following the same calculation procedure
⎛ d B⎞
⎛ a d B⎞
⎜ − − + ⎟ + γ w µf ( d − a)⎜ − + ⎟ +
⎝ 2 2⎠
⎝ 6 3 3⎠
γ w µf
⎤
⎥
− B + B 2 + 4C 3 ⎥⎦
2
black solid curve in the right side(a＜d) of Fig.5(b). In
⎛a B 2 ⎞
⎛ d B⎞
U3⎜− + ⎟ + U4 ⎜ + − d ⎟ +
⎝2 6 3 ⎠
⎝ 2 2⎠
⎛ a B⎞
W⎜− − ⎟ = 0
⎝ 2 6⎠
3
(A14)
The function achieved in Eq.(A14) is plotted as a
And then，
S1
1 ⎡⎢ − B + B 2 + 4C 3
−C
5⎢
2
⎣
then applying again the algebraic formulae with B =
− 2 168.6 × 10 3 a − 43 373 × 10 3 and C = 2.713 9a 2 +
(A11)
Substituting the known parameters with their
numerical values，an expression of the type f(a，µ)=0
can be obtained.
This expression Eq.(A11) can be solved either in
822.44a + 10 186 ：
µ=
3
12 ⎛⎜
6 ⎜⎜
⎝
3
− B + B 2 + 4C 3
−C
2
3
⎞
⎟
2
3 ⎟
− B + B + 4C ⎟⎠
2
(A16)
The function µ = µ (a) achieved in Eq.(A16) is
a or µ. In the following calculations ， µ has been
considered as a variable and a as a parameter so that a
plotted as a black partly solid and partly dashed curve
function of the type µ = µ (a) is obtained consistently
drainage system，the forces acting on the dam are
with Fig.5，where µ is plotted against a.
in the left side(a＞d) of Fig.5(b). In case of absent
those shown in Fig.A1(b). Imposing the equilibrium of
• 1568 •
2008 年
岩石力学与工程学报
moments leads to Eq.(A16) which，in this case，is valid
γ w µa and U 2 = γ w µ ( B − a) / 2 . Therefore：
for any value of a. The graphs obtained are visible in
N ′ = W − U 1 − U 2 = γ c BH / 2 −
Fig.5(a).
In case of triangular stress distribution，the same
calculations are performed. The only difference is the
B−a
value of x which becomes x =
. In case of d＜a，
3
it happens that µ＞H and therefore the lateral thrust
γ w µa − γ w µ ( B − a) / 2
It is now possible to get the two forces Tact and
Tres ：
Tres
γ w H / 2 and S2 = (µ − H )γ w H . The obtained functions
2
have been plotted as gray curves in Fig.5(a) and
Fig.5(b).
Let start with the case of effective drains. Two
different expressions hold true depending on the
extension of a，which can be larger or smaller than d.
For the sake of space，here only one case will be
shown：a＞d，being this the case for which sliding
effectively takes place(see Fig.8). The derivation
shown hereafter is also valid for the case of absent/
ineffective drainages since the uplift forces acting on
the dam are the same(see Fig.2(a)).
Considering the case of µ＞ H ， the horizontal
forces due to the lateral thrust caused by the reservoir
are S1 = γ w H 2 / 2 and S 2 = ( µ − H )γ w H . Defining
the horizontal force causing sliding Tact ，and the one
opposing sliding Tres ，it can be written：
Tact = S1 + S 2
Tres
⎫⎪
⎬
= N ′ tan ϕ + c ′( B − a) ⎪⎭
(A17)
N ′ is achieved by the vertical equilibrium as
shown for the overturning mechanism. The forces due
to the water pressure，as shown in Fig.7，are U 1 =
⎫
⎪
⎪
⎪⎪
BH
= tan ϕγ c
− tan ϕγ w µa − ⎬
2
⎪
⎪
µ ( B − a)
tan ϕγ w
+ c ′( B − a ) ⎪
⎪⎭
2
Tact = γ w µH − γ w
is no longer expressed by one force but two： S 1 =
A.2 Sliding mechanism
(A18)
H2
2
(A19)
Imposing the horizontal equilibrium Tact = Tres
leads to：
γ w µH − γ w
tan ϕγ w
H2
B
⎛
= µ ⎜ − tan ϕγ w a − tan ϕγ w +
2
2
⎝
a⎞
BH
+ c ′( B − a)
⎟ + tan ϕγ c
2⎠
2
(A20)
Rearranging with the objective of getting a
function of the type µ = µ (a) ：
⎛
⎝
µ ⎜ γ w H + tan ϕγ w a + tan ϕγ w
B
a⎞
− tan ϕγ w ⎟ =
2
2⎠
H2
BH
+ tan ϕ
+ c′B − c′a
2
2
and then：
γw
µ=
(A21)
− 2c′a + 2c′B + γ w H 2 + tan ϕγ c BH
=
tan ϕγ w a + 2γ w H + tan ϕγ w B
−1 400 a + 214 470
5.77a + 1 946
(A22)
In case of µ＜H，the same calculations hold true
with the only difference due to Tact that becomes：Tact =
µ2
γw
.
2