RESEARCH ARTICLES
Recent developments in safety assessment of
concrete gravity dams
J. M. Chandra Kishen
Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India
Most of the dams in India were built in the sixties, when
only simple methods could be used to properly design
them. Forty years later, despite the progress made in
our capability to conduct sophisticated finite element
studies, current analyses procedures for safety assessment remain essentially the same as those used earlier for
design. The main objective of this article is to highlight the work done in the area of fracture mechanics
for safety assessment of gravity dams. It is shown that the
conventional strength of materials-based design leads
to large factors of safety when compared to the more
recent and rational concepts of fracture mechanics.
Keywords: Fracture energy, fracture toughness, gravity
dam, shear friction factor of safety, sliding failure.
FOR over five thousand years, as evidenced in the cradles of
civilization, dams have played a vital role in regulating
water flow to cater to the needs of irrigation. They endow
important quantities of water that are absolutely crucial for
human consumption and industries, in addition to irrigation. Dams form key structures for harnessing hydroelectric power, with all its advantages over other sources of large
energy generation, as well as for control of floods. They
have also played a vital role in taming some of the largest
rivers on earth, in many cases at points upstream from cities
and populous and developed valleys. Their number, height,
economic and social importance have increased exponentially
over the last decades, including their cost. The huge capital
outlay required for dam construction coupled with the
devastating effect it can have on life and property in case of
failure, is a cause of great concern. Thus, it is essential to
ensure the safety of these structures in the design, pre- and
post-construction stages.
Concrete dams, like all concrete structures, suffer from
cracking which is caused by various factors such as construction, alkali-aggregate reaction, load application, etc. Of
greater concern are cracks that develop as a result of hydrostatic load application. Because of the joints generated by
a step-wise successive construction and because of thermal
operating gradients, cracks can have relevant dimensions
from the beginning1. As such, it should not come as a surprise
that numerous dams all over the world have shown disturbing
signs of cracking2. Due to this growing concern, concrete
e-mail: chandrak@civil.iisc.ernet.in
650
dams are increasingly coming under the scrutiny of regulatory
agencies and other groups that are responsible for dam safety3.
From recent researches4, it is well understood that the
concepts of fracture mechanics could be usefully applied for
failure analysis of concrete dams. In a concrete dam, the interface between concrete superstructure and rock foundation is
one of the potential sites of crack formation and subsequent
failure. Not only does it contribute in weakening the mechanical strength, but also constitutes conduits for water to seep
through and exert uplift pressure. Nevertheless, its response to
seismic excitation is not well understood and it is commonly
accepted that it constitutes the weakest link in the safety of
a dam during and after an earthquake. Hence, it is important
that proper mechanical behaviour of this interface is understood in the light of realistic loading conditions.
Fracture mechanics concepts and theories have been
successfully applied to study the cracking phenomena in
dams1,5–13. In all these works, crack propagation was studied
by considering the cracks within the body of the dam. In the
present work, crack propagation at the interface between
concrete superstructure and rock foundation is considered
using the concepts of interfacial fracture mechanics. Crack
lengths and shear friction factor of safety are computed
and compared using different criteria for crack propagation.
Reasons for fracture mechanics approach
Despite the tremendous progress made in our capability to
conduct sophisticated finite element analysis, the design and
rehabilitation of dams are based on the simple Bernoulli
equation (P/A + Mc/I = 0). Such an equation, valid for ‘shallow beams’, cannot be blindly applied to a dam without certain shear corrections. Furthermore, it fails to recognize the
singular nature of the stress at the tip of a crack. Alternatively,
the fracture mechanics approach identifies the singular nature
of stresses at the crack tip. According to linear elastic fracture
mechanics (LEFM), the stresses increase according to r–1/2
in a linear elastic material, where r is the distance to the crack
tip. It is evident that no material can withstand infinite
stresses and the hypothesis of LEFM breakdown near the
crack tip. However, if the length of the zone where the
stresses separate from the ideal LEFM results is small
compared to the characteristic dimensions of the structure,
the fracture criterion provided by LEFM is still valid14. The
length of this zone is controlled by the material characteristic length lch given by15
CURRENT SCIENCE, VOL. 89, NO. 4, 25 AUGUST 2005
RESEARCH ARTICLES
lch =
GF E ′
σ t2
,
(1)
where E′ is the generalized Young’s modulus (E′ = E/(1 – ν )),
ν the Poissons ratio, GF the specific fracture energy and
σt, the concrete tensile strength. According to Kaneko et
al.16, without losing much accuracy, the LEFM approach
could be an appropriate approximation for nonlinear fracture mechanics problems under the following conditions:
2
a0
≥ 0.01,
lch
(2)
where a0 is the physical crack size. For dam concrete, lch
values range17 between 1100 mm and 1700 m. Thus, the above
condition is fulfilled for a minimum initial crack size of about
17 mm, which is possible in the case of large and cracked
structures like dams.
It is well known that the primary mode of failure in a gravity dam is sliding (rather than overturning or bearing) along
the uncracked ligament. In the classical analysis, the criterion
for horizontal equilibrium requires that the horizontal water
pressure be resisted by frictional forces acting along the
uncracked ligament, which is obtained from the Mohr–
Coulomb theory. Most of the agencies involved in safety
assessment of dams require the evaluation of the shear friction factor of safety, which is defined as
SFF =
cL + ∑ V tan φ
,
∑H
(3)
where c is cohesion, L the uncracked ligament length, V the
vertical force, H the horizontal force and φ the angle of
internal friction. It is noted that cohesion is allowed for shear
strength, but no tensile stresses are permitted. Hence,
there is significant inconsistency between the assumptions
made for the crack length determination and sliding factor
calculations.
All analyses done for design and safety evaluations assume
that the crack propagates along the weak interface between
the dam and its foundation. This is incorrect, because experimental resuts3 have shown that the presence of large shear
stresses would guide an interface crack into the rock
foundation. It will be shown in this study that fracture
mechanics-based analysis can well predict the crack kinking
into the rock.
It has been shown18 that the stresses (both normal and shear)
obtained near the crack tip using the finite element method
is not objective. In other words, the stresses are sensitive to
the mesh size. The finer the mesh, the higher is the normal
stress near a crack tip. This is clearly caused by the linear
elastic stress singularity at the tip of a crack, which is better
captured by a fine mesh. This type of scatter seen in the stress
results is not present when fracture parameter such as the
stress intensity parameter is computed for different mesh
sizes. Hence, fracture mechanics results are objective.
CURRENT SCIENCE, VOL. 89, NO. 4, 25 AUGUST 2005
Fracture mechanics-based analysis of concrete
gravity dam
The dam used for analysis was constructed in the 1930s. It
is 176 ft (53.7 m) high and has 53 monoliths, with a crest
elevation of 1525 ft (464.8 m). Based on hydrological studies
and using classical equilibrium analysis, it has been determined that the probable maximum flood (PMF) elevation
is at 1555.8 ft (474.2 m) and the imminent failure flood (IFF),
which is the pool elevation that would trigger failure, is at
1533 ft (467.3 m) or 22 ft (6.71 m) below the PMF. Hence,
this dam has been targetted for major rehabilitation19. The
cross-section of the dam along with its dimensions are
shown in Figure 2. Table 1 gives the material properties of
the concrete dam, rock foundation and the interface.
LEFM analysis
In LEFM analysis, the crack length along the interface
between the concrete dam and rock foundation is determined
using the concepts of bimaterial fracture mechanics, which
considers the oscillatory nature of the stress singularity (as
shown in Appendix (A)) at the crack tip.
A series of linear elastic fracture mechanics analyses were
performed with different crack lengths. In all the analyses,
the IFF elevation of 1533 ft (467.3 m) was used, since the
classical analysis19 had yielded this water elevation to trigger
failure. The stress intensity factors K1 and K2 in mode I and
mode II respectively, were computed using the Stern contour
integral for bimaterial interfaces. The energy release rate
G was computed using eq. A8 (see Appendix). The following loads were used in the analysis:
(i) Hydrostatic load with water elevation at 1533 ft
(467.3 m).
(ii) Body forces due to the self-weight of the dam.
(iii) Full uplift pressure inside the crack.
Results of LEFM analysis: Figure 2 illustrates the variation of mode I stress intensity factor (K1), mode II stress
intensity factor (K2) and energy release rate (G) with crack
Table 1.
Material properties used in the analyses
Concrete
Weight density
Elastic modulus
Poisson ratio
150 lbs/ft3 (23.6 kN/m3)
4.867 × 106 psi (33.54 GPa)
0.255
Rock
Weight density
Elastic modulus
Poisson ratio
0
3.952 × 106 psi (27.23 GPa)
0.165
Interface
Cohesion
Angle of friction
42 psi (0.289 MPa)
51°
651
RESEARCH ARTICLES
Figure 1.
Simplified geometry of dam used in the analyses.
critical fracture toughness for limestone/concrete interface
was experimentally determined3 to be 248 psi in (0.28 MPa
m ). With this value, a crack length of 56 ft (17 m) is
obtained. This indicates that predicting a crack length
based on zero fracture toughness is conservative.
Nonlinear fracture mechanics analysis
Figure 2.
length.
Stress intensity factors and fracture energy versus crack
length. It is seen from Figure 2, that the energy release rate
is constant until a crack length of 33 ft (10 m), after which
it starts increasing. This implies that the crack remains
within the interface up to a length of 33 ft (10 m) and departs from it thereafter. Using the traditional criteria of
K1c = 0, the crack length obtained is 65 ft (19.8 m). The
652
Contrarily to the LEFM analysis, wherein all the energy is
transferred at the crack tip, in a nonlinear fracture mechanics
(NLFM) analysis, energy is also transferred along the crack.
The Interface Crack Model20 is used in NLFM analysis.
The criterion for crack propagation is a strength-based one,
in which the major principal stress cannot exceed the tensile
strength. Interface elements are provided along the rock–
concrete joint. These elements are capable of modelling
shear failure along the interface and may fail either due to
crack opening or excessive shear stresses. The analysis is
carried out by incrementally increasing the water elevations,
with the first increment being the gravity dead load. Other
load increments are water elevations starting from a low
value of 50 ft (15.3 m), and increasing in increments of
50 ft (15.3 m) until the imminent failure flood level, as determined from the classical equilibrium analysis. Thereafter, each increment is increased by 1 ft (0.31 m), until the
CURRENT SCIENCE, VOL. 89, NO. 4, 25 AUGUST 2005
RESEARCH ARTICLES
probable maximum flood level is reached. The incremental load size is reduced for accuracy in predicting the
crack profile and to achieve faster convergence of the nonlinear solution. Crack kinking was considered in the direction
perpendicular to that of maximum principal stress.
Results of NLFM analysis: Figure 3 shows the final crack
profile with respect to changing water elevations. It is seen
that the crack which remains within the interface for 8.3 ft
(2.5 m) (point A), will start to kink into the adjoining
rock when the water elevation is at 175 ft (53.4 m). This
elevation is 1 ft (0.31 m) above the IFF level determined
from classical equilibrium method. The kinked crack propagates alone until the water elevation reaches 184 ft (56.1 m).
At this point the crack branches out, i.e. the interface crack
starts to propagate again. When the water reached 190.8 ft
(58.2 m) (which is 8 ft (2.4 m) below the PMF), the nonlinear solution failed to converge due to the presence of
high shear stresses. This indicated final failure due to sliding.
Furthermore, at final failure, the lower face of the interface
crack was pushed upwards by the uplift forces within the
kinked crack. This helped in partially closing the interface
crack.
Strength of shear key in a gravity dam
The wedge crack model proposed by Kaneko et al.16 and described in the Appendix (B) is used to estimate the strength of
shear key in a gravity dam. The dam used for analysis is a
concrete gravity-type dam of 47 m height, with water at full
Figure 3.
tions.
reservoir level. The cross-section of a dam along with the
simplified dimensions are shown in Figure 4. The concrete
is assumed to be linearly elastic with modulus of elasticity
Ec, 27,340 MPa, mass density 2400 kg/m3 and Poisson ratio
0.2. The dam is founded on a linearly elastic massless
foundation block with Poisson ratio of 0.25. The unit
weight of the rock foundations is neglected so that the stresses
computed in the foundation are those caused by the dam and
the reservoir (i.e. in situ stresses are neglected). The modulus
of elasticity of the foundation has been assumed to be 1.5
times the modulus elasticity of concrete21.
The dam with the shear key is analysed by using the finite
element program FRANC-2D22. The various loads considered
for the analysis are:
(i) Hydrostatic load with water level up to the crest (47 m).
(ii) Body forces due to self-weight of the dam.
(iii) Full uplift pressure inside the crack.
Results and discussion
The analysis is done with the application of the above loads,
considering complete dam foundation, as shown in Figure 4.
It was observed that the major principal stress (tensile)
exceeded the tensile strength of concrete at point ‘O’
(Figure 5). Hence, a crack is assumed to nucleate at this
point. The upstream side of the shear key got separated
from the foundation due to the presence of high shear stress
near the dam–foundation interface.
A series of analyses were carried out with a small initial
crack of length 1 m at point O (Figure 5), and the mode I
stress intensity factor was computed. Automatic crack
propagation option in the finite element code was used,
wherein the crack propagates when the mode I stress intensity
factor reaches the fracture toughness (KIC) of the material.
At each crack increment, the mesh near the crack tip is
Final crack profile with respect to changing water eleva-
CURRENT SCIENCE, VOL. 89, NO. 4, 25 AUGUST 2005
Figure 4.
Geometry of dam with shear key.
653
RESEARCH ARTICLES
changed by the introduction of a rosette of eight six-noded,
singular, quarter-point elements. Figure 6 shows variation
of mode I stress intensity factor with crack length. It is seen
that the mode I stress intensity factor decreases for larger
crack lengths. This is due to the effect of compressive
body forces which have a tendency to close the crack. At a
total crack length of 5.9 m the analysis was stopped, since
the mode I stress intensity factor at this crack length was less
than the fracture toughness (KIC). For this total crack length
of 5.9 m, the lumped forces F and F′ (Figure 7) are found,
where F′ is the force required to close the crack and F is force
required to open up the crack (Appendix (B)).
Assuming the critical stress intensity factor (KIC) of concrete to be 1040 kN/m3/2 in eq. (B-3) (see Appendix), the
strength of the shear key is computed to be 3358 kN/m.
Conventional analysis based on the strength of materials
approach computes the strength of the shear key as
3850 kN/m. It is observed that the conventional approach
overestimates the strength by 14%, which in turn indicates
a higher factor of safety that is unsafe. Thus, a fracture
mechanics-based approach should be applied in safety assessment of dams.
Conclusions
The following conclusions are made from the fracture
mechanics-based analysis of gravity dams:
Figure 5.
Crack location.
(1) The LEFM analysis, considering the bimaterial nature
of the interface, yielded a crack length of 35 ft (10.7 m).
The classical equilibrium analysis estimated the crack
length to be 65 ft (19.8 m). The NLFM analysis yielded
a total crack length of 25 ft (7.6 m) in the interface
and 33 ft (10 m) into the rock. These indicate that the
classical equilibrium analysis highly overestimates the
crack length, thus making the rehabilitation scheme
(if needed) an un-economical process.
(2) The NLFM analysis indicates a shorter crack length
within the interface. This is due to the closure of interface
crack due to full uplift pressure in the kinked crack.
(3) The strength of a shear key was computed by considering
a fracture mechanics-based approach. It is seen that
the conventional approach based on strength of materials
overestimates the strength of the shear key, which in
turn indicates an higher factor of safety that is unsafe.
Thus, a fracture mechanics-based approach should be
made use of in safety assessment of dams.
Figure 6. Variation of stress intensity factor with crack length for
crack at the junction of shear key and dam.
Figure 7.
654
Wedge crack model16.
Figure 8.
Geometry and conventions of an interface crack.
CURRENT SCIENCE, VOL. 89, NO. 4, 25 AUGUST 2005
RESEARCH ARTICLES
Appendix
The crack flank displacements for plane strain are given
by23,24:
A. Stress and displacement fields for cracks
between dissimilar materials
δ y + iδ x =
Considering the bimaterial interface crack in Figure 8 for
traction-free plane problems, the near-tip normal and shear
stresses σyy and τxy, may conveniently be expressed in complex stress intensity factors23,24:
σ yy + iτ xy =
( K1 + iK 2 )r iε
,
(2πr )
(A1)
where i = − 1, K1 and K2 are components of the complex
stress intensity factor K = K1 + iK2. The expressions for
these bimaterial stress intensity factors, derived by Rice
and Sih25 are
σ [cos(ε log 2a) + 2ε sin(ε log 2a)] +
τ [sin(ε log 2a) − 2ε cos(ε log 2a)]
K1 =
a,
cosh πε
(A2)
τ [cos(ε log 2a ) + 2ε sin(ε log 2a )] −
K2 =
σ [sin(ε log 2a ) − 2ε cos(ε log 2a )]
cosh πε
a.
(A3)
In the above equations, ε is the oscillation index given by
1 − β 
ε = 1 ln 
,
2π 1 + β 
(A4)
where β is one of Dunders’ elastic mismatch parameters26.
For plane strain β is given by
β=
µ1(1 − 2ν 2 ) − µ2 (1 − 2ν1)
,
2[µ1(1 − ν 2 ) + µ 2 (1 − ν1)]
(A5)
in which µ, ν, and a are the shear modulus, Poisson ratio and
crack length respectively, and subscripts 1 and 2 refer to the
materials above and below the interface respectively.
Dunders defines an additional mismatch parameter α:
α=
E1 − E2
,
E1 + E2
(A6)
where E is the plane strain Young’s modulus, E = E/
(1 – ν2). Note that β, α and ε vanish when the materials
above and below the interface are identical. When ε ≠ 0, eq.
(A1) shows that the stresses oscillate heavily as the crack
tip is approached (r → 0). Furthermore, the relative proportion of interfacial normal and shear stresses varies slowly
with distance from the crack tip because of the factor riε.
Thus K1 and K2 cannot be decoupled to represent the intensities of interfacial normal and shear stresses as in homogeneous fracture.
CURRENT SCIENCE, VOL. 89, NO. 4, 25 AUGUST 2005
4 (1 / E1 + 1 / E2 )( K1 + iK 2 ) rr iε ,
(1 + 2iε ) cosh(πε )
2π
(A7)
where δy and δx are the opening and sliding displacements of
two initially coincident points on the crack surfaces behind
the crack tip, as shown in Figure 8.
The energy release rate G for extension of the crack
along the interface, for plane strain is given by24:
G=
B.
(1/ E1 + 1/ E2 )( K12 + K 22 )
2 cosh 2 (πε )
.
(A8)
Wedge crack model
Based on the experimental and analytical observation of shear
failure of concrete, Kaneko et al.16 have developed a simple
mechanical model for the analysis of shear key joints. The
wedge crack model is employed to describe the formation of a
short crack in analogy to the propagation of a short crack
emanating from the edge of a semi-infinite body under
wedging force, as shown in Figure 7. This short crack is
nucleated at the corner of the key joint as a macroscopic
mode I crack, when the flaws near the corner of the key
are easily mobilized by tensile stresses induced due to high
shear stress concentration. As shown in Figure 7, it is assumed that both the vertical and horizontal stresses acting
on the shear key can be lumped to the edge of the crack
(loads F and F′ respectively), thus neglecting a moment
that tends to increase the stress intensity factors, which is
assumed small.
The stress intensity factors at the tip of the short crack,
produced by the wedging forces F and F′ are expressed as:
K I = 2( F sinθ − F ′ cos θ )
K II = 2( F cos θ + F ′ sin θ )
π
(π − 4)l
2
,
(B1)
,
(B2)
π
(π − 4)l
2
where F′ = σplcosθ, l is the crack length, θ the angle between
the crack and the vertical, F the vertical force required to
open up the crack and σp is the stress acting on shear key, as
shown in Figure 7, KI and KII are the stress intensity factors
in mode I and mode II respectively.
In eq. (B1), setting KI equal to the fracture toughness KIC,
the force F can be obtained as
K IC (π 2 − 4)l
F=
2 π
+ σ pl cos 2 θ
sin θ
.
(B3)
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RESEARCH ARTICLES
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Received 31 December 2004; accepted 18 April 2005
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