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4
Nonlinear Analysis and
Response Behavior
INTRODUCTION
As stated in Chapter 3, useful information about the expected earthquake
response of concrete dams can be obtained from linear dynamic analysis
based on assumed elastic behavior. However, a severe earthquake may
cause significant damage to concrete dams, as was observed at Koyna Dam,
and such damage usually is associated with important changes in structural
stiffness. Consequently, if significant damage occurs, the actual performance
of the dam can be predicted only by a nonlinear analysis that takes account
of these stiffness changes. For this reason a rigorous evaluation of concrete
dam seismic performance should consider the behavior of the materials and
components in the system, which is characterized by nonlinear force-displacement
relationships, in conjunction with nonlinear response analysis procedures.
Additionally, realistic consideration of the nonlinear behavior of dams may
indicate mechanisms that limit the earthquake response and provide an added
margin of safety against failure.
After definition of the seismic input, a comprehensive safety evaluation
of a concrete dam requires (1) identification of the behavior of materials
and components under dynamic loads, (2) mathematical models that can
represent the nonlinear behavior, (3) efficient numerical procedures for computing
the nonlinear earthquake response of the dam system, and (4) criteria to
assess acceptable performance and evaluate failure modes. In this chapter
the nonlinear behavior important in the earthquake response of concrete
dams and the requirements for nonlinear analysis are identified. In addition,
recent developments in modeling and analysis of such dams are summarized,
and further research needed to improve the seismic safety evaluation is
outlined.
61

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62
CHARACTERISTICS OF NONLINEAR BEHAVIOR
Concrete exhibits a complicated nonlinear relationship between stress
and strain that is dependent on loading rate and history. The nonlinear
behavior of mass concrete becomes significant as stresses approach the
compressive strength or the comparatively small tensile strength. Because
concrete dams are designed to resist their primary loads—gravity and
hydrostatic through compressive stress fields, the tensile stresses induced
by such static loads are minimal (or nonexistent). Moreover, when current
design criteria for static loads are followed, the computed compressive stresses
are found to be much less than the compressive strength of the concrete.
However, strong earthquake ground motion can produce large dynamic
stresses in dams, both compressive and tensile, and the combination of
static and dynamic stresses may exceed the linear response range of the
concrete, particularly with regard to the tensile stresses. Experiments demonstrate
that concrete behavior is essentially linear under cyclic compressive loads
up to approximately 50 to 60 percent of the compressive strength (4-1~. For
example, assuming a compressive strength of 4,000 psi, nonlinear behavior
is important for compressive stresses greater than about 2,000 to 2,400 psi.
But linear elastic dynamic analyses of gravity and arch dams show that
compressive stresses rarely exceed this range during typical earthquakes;
consequently, the nonlinear behavior of concrete in compression, including
hysteretic energy dissipation, can generally be neglected in the earthquake
response analysis of dams.
Tensile stresses in a dam produced by earthquake ground motion can be
resisted only by the tensile strength of concrete, because reinforcement
generally is not provided in the body of a dam. The tensile strength of
concrete is an order of magnitude less that its compressive strength, and
linear analyses demonstrate that the tensile limit may be exceeded at widespread
different locations in a dam during an earthquake (3-8, 3-11~. For example,
in a recent study (4-2) a gravity dam monolith subjected to a wide range of
earthquake ground motions was analyzed assuming linear behavior. A moderate
earthquake with a peak ground acceleration of 0.25 g induced tensile stresses
greater than the tensile strength of typical concrete in both the top part of
the dam and the heel, indicating the potential for cracking.
As the stress in a dam approaches the tensile strength of the concrete,
microcracks (which are always present in concrete) coalesce to form a crack
surface. However, it is important to note that the tensile cracking caused by
moderate earthquakes may not be deleterious to the performance of a dam,
because the dynamic forces open and then recluse the cracks during a cycle
of vibration. After the earthquake, static loads will generally return the
stresses to compression, leaving the cracks in a closed condition and maintaining
the dam's stability. Because tensile cracking in concrete is fundamentally a

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fracture process that dissipates energy as the cracks propagate, the amplitude
of dynamic response may be decreased by the cracking if the dam remains
stable during the earthquake. During severe ground motion, however, it is
possible for tensile cracks to propagate completely through a dam, potentially
leading to dynamic instability and to uncontrolled release of water. Both
linear analyses and model studies of arch dams have indicated related failure
modes: extensive tensile cracking that results in the formation of a semicircular
or rectangular notch near the crest, leading to dynamic instability of the
notched portion. Although it is not certain that such a failure can actually
develop in an arch dam due to an earthquake, or that crack propagation
completely through a dam section necessarily results in instability, the major
goal of a safety evaluation is to determine whether unstable response could
develop due to credible levels of earthquake ground motion.
Realistic analytical modeling of tensile cracking in mass concrete should
recognize the variation in properties of the materials in a dam, the incremental
construction procedure, and the effect of pore water on crack development.
The practice adopted in some designs, as mentioned in Chapter 3, of varying
the concrete strength over the dam height may affect the locations where
tensile cracks develop. Also, contraction joints between the monoliths, as
well as horizontal planes of weakness that may exist at lift joints, can be a
major influence on crack location. The presence of pore water in saturated
mass concrete affects the stress state and the initiation and propagation of
cracks, particularly in the lower portions of a dam and at the interface with
the foundation rock. Generally only the static pore water pressure is considered
in an earthquake analysis, as a constituent of the combined (static plus
dynamic) state of stress. The earthquake response is associated with dynamic
total stresses that include both intergranular and pore pressure components,
but typically it is assumed that the pore water does not migrate during the
cyclic pressure changes because of their short duration. Consequently, it is
usually assumed that the earthquake does not significantly influence the
static pore pressure effects. It is recognized that some sort of hydraulic
fracture mechanism might result at the wetted face of the dam due to dynamic
pore pressures during the earthquake, but there is no evidence that such
hydraulic fracturing has actually occurred. Thus, present practice assumes
that pore pressures have no direct effect on dynamic cracking and that the
cracking behavior may be characterized simply in terms of the dynamic
total stresses.
The preceding comments on concrete cracking pertain to both gravity
and arch dams; however, the stress state in many gravity dams tends to be
essentially uniaxial (cantilever stresses), whereas the stresses in arch dams
are considered to be biaxial, involving components in both the cantilever
and arch directions. On the other hand, the development of arch-direction
tensile stresses is inhibited by vertical contraction joints that are provided

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between monoliths, because the joints are unable to resist any significant
net tension. The joints tend to open and close during a severe earthquake
(3-17), and this nonlinear response mechanism has two consequences in the
earthquake response of arch dams. First, the intermittent opening temporarily
reduces the arch-direction resistance, causing transfer of load to both cantilever
action and inertial resistance. The load picked up by cantilever bending
may then lead to flexural overstress and failure of the monoliths. The
second potential consequence of the joint opening-closing is compressive
failure of the joint itself. Model tests have shown that loss of joint integrity
is possible in arch dams (4-3), because as a joint opens the compressive
stresses in the portion of the joint remaining in contact may increase dramatically,
possibly crushing the concrete. On the other hand, the nonlinear behavior
of the joints may limit the earthquake response of large (long-period) dams
by further lengthening the vibration period, thereby reducing the dynamic
amplification that is a function of the vibration properties of the dam and
characteristics of the ground motion (3-17~.
The behavior of the foundation rock supporting a dam is typically nonlinear,
because the rock is often fractured and discontinuous. The nonlinear behavior
of foundation rock will affect the static and dynamic response of a dam.
For example, an elastic modulus based on small strain overestimates the
stiffness of fractured foundation rock because of the rock's inability to
resist large tensile stresses. During a severe earthquake the forces acting at
the abutments of an arch dam also can increase significantly. In some cases
it may be possible for the abutments to fail because of shear failure along
planes of weakness in the foundation rock. Questions about foundation
stability are a safety concern for gravity dams as well. Thorough consideration
of the potential for foundation rock failure is often difficult, because information
on subsurface rock conditions is limited, particularly in the case of older
dams. However, foundation stability must always be studied intensively,
because experience shows that actual failures of concrete dams generally
are initiated in the foundation rock.
The response predicted by a linear model of a concrete dam depends on
both the spatial variation of earthquake ground motion (as described in
Chapter 2) and the temporal variation. Analytically, the response of a
linear dam system can be considered as the summation of responses to
harmonic components of the ground motion specified for one or more support
degrees of freedom. The nonlinear response of concrete dams may depend
on other characteristics of the earthquake ground motion. For example, it
may be affected by the duration of the ground motion and the amplitude of
incremental ground velocity, particularly in the propagation of tensile cracks
and the integrity of joints. Crack propagation under dynamic loads is affected
by the number of loading cycles, which is related to the duration of ground
motion. Similarly, the integrity of a joint may depend on the number of

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65
times it opens and closes. A large increment of ground velocity, typical of
ground motion near an earthquake source, will impart an impulse to a dam
that can initiate and propagate tensile cracks or affect the dynamic stability
after extensive cracking. For these reasons a realistic safety evaluation
based on nonlinear analysis would require a detailed seismotectonic study
of a dam site and the development of appropriate site-specific ground motion
records.
The nonlinear behavior mechanisms mentioned above (tensile cracking
of concrete, loss of joint integrity, and foundation failure) result from dam
vibration in response to earthquake ground motion. Overall sliding and
overturning of a dam due to ground motion, the traditional criteria for static
stability, are generally unrealistic failure modes during earthquakes. Indirect
earthquake failures of dams due to fault displacement or overtopping also
are important considerations, but they are not directly related to dynamic
response.
MATERIAL MODELS AND RESPONSE ANALYSIS
As discussed in Chapter 3, linear dynamic analysis of a concrete dam
system is valuable in understanding the general characteristics of earthquake
response. However, if the linear analysis shows repeated tensile stresses
significantly greater than the tensile strength, the concrete can be expected
to crack, and the linear response results would no longer be valid. The
ultimate earthquake behavior cannot be predicted from such linear models,
because the strength of the materials is not represented, nor is the redistribution
of forces due to tensile cracking, joint opening-closing, or foundation instability.
A realistic nonlinear earthquake response evaluation of a concrete dam
requires analysis of mathematical models that include tensile crack propagation
in concrete, cyclic displacement behavior of joints and abutments, and
deformability of the foundation rock. Because these phenomena induce nonlinear
relationships between resisting forces and displacements, nonlinear methods
of response analysis are required. These are considerably more complicated
than linear analysis. A rigorous nonlinear response analysis should include
the important darn-water and dam-foundation rock interaction effects mentioned
in Chapter 3, together with nonlinear models of the dam concrete, joints,
and foundation rock. The finite element method is presently recognized as
the best approach for spatial discretization of the equations of motion for
concrete dams because of its ability to represent arbitrary geometry and to
incorporate arbitrary variations of material behavior. Finite elements can
also be used to discretize the reservoir water and foundation rock, although
alternatives such as boundary integral elements may be more efficient in
modeling large reservoir and foundation regions.
When concrete dam systems exhibit nonlinear response to severe earthquake

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66
ground motion, the principle of superposition is no longer valid, and frequency-
domain analysis may not be applied directly. Usually, the equations of
motion are solved in the time domain using a time-stepping procedure.
Because of the large amount of computation required to obtain the nonlinear
dynamic response of a concrete dam system, it is mandatory to truncate the
size of the water and foundation rock domains; however, the boundaries
should be modeled in a manner that allows radiation of energy from the
system. Methods for reducing the number of generalized coordinates while
still representing the nonlinear behavior of a dam system may be explored
(4-4~. The use of a substructure formulation also may provide an efficient
approach for nonlinear earthquake analysis of dams. Reference 4-5 presents
a numerical method in which a structure is divided into linear and nonlinear
substructures. The method was used to analyze a concrete arch dam, including
opening of vertical joints, which were modeled as local nonlinearities using
gap elements. Other dynamic analysis procedures for concrete structures
are described in reference 4-6. Finally, evaluation of the dynamic stability
of portions of a dam after extensive cracking requires numerical procedures
that properly represent the impact and sliding of the sections and that conserve
momentum.
An important decision in the analysis of the earthquake response of concrete
dams is whether to use a two-dimensional or a three-dimensional model of
the dam system. The complicated geometry of arch dams and associated
valleys or canyons necessitates use of a three-dimensional model to represent
their complex resistance mechanisms. Two-dimensional models are often
employed for analysis of gravity dam monoliths with the resmctions discussed
in Chapter 3, and nonlinear analysis of two-dimensional models is considerably
less complicated than that for three-dimensional models.
The complete nonlinear response analysis of concrete dams introduces
considerations not normally required in a linear analysis. As mathematical
models become more representative of true nonlinear material behavior, the
numerical results may be sensitive to additional parameters of the model.
Further research should be able to evolve mathematical models that can be
expressed in terms of measured material properties with confidence, but the
cracking of concrete is likely to be very sensitive to its failure-strain limit
as well as to the static state of stress and strain that exists at the time the
earthquake occurs. This initial state is primarily due to the gravity load and
hydrostatic pressure, but it is also affected by shrinkage and temperature
strains that accumulate during the incremental construction process and after
completion up to the time of the earthquake. Diurnal and seasonal temperature
variations control the temperature gradient through the dam, and this may
have a major effect on the initial stress and strain distribution, particularly
for arch dams, as does creep of the concrete over the service life of the dam.
The importance of thermal and shrinkage strains (and possibly creep) is

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67
evidenced by the cracks that developed in several monoliths in the Dworshak
and Richard B.- Russell dams under static loading conditions (4-7~. An
aerial photograph of the Richard B. Russell Dam is shown in Figure 4-1.
The realistic analytical evaluation of earthquake response based on nonlinear
behavior must recognize the distribution of stress and strain prior to an
earthquake. This implies that for a given earthquake several analyses of a
dam may be necessary, corresponding to the initial conditions at different
times during the expected service life.
The challenge of nonlinear dynamic analysis of concrete dams lies in
developing mathematical models that represent the true behavior of concrete
and joints and in incorporating these models in efficient numerical procedures.
Although researchers and practitioners have investigated limited aspects of
the nonlinear behavior of concrete dams, a great deal of innovative analytical
research must yet be done to develop practical nonlinear response analysis
procedures; in addition, the results of such analyses must be verified by
careful experimentation before the procedures can be fully accepted for
earthquake safety evaluation of concrete dams.
CRITERIA FOR SAFETY EVALUATION
Although the criteria for evaluating the seismic performance of concrete
dams are discussed in detail in Chapter 6, it is pertinent to note here that
performance criteria based on nonlinear response evaluations are especially
important. Traditionally, a no-tension stress criterion has been used in the
design of concrete dams. However, microcracking is always present in
concrete, and the acceptance of moderate tensile cracking that does not
impair the function of a dam is a realistic point of view for earthquake
loads. The complete nonlinear earthquake analysis of new and existing
dams is not likely to be undertaken in the near future, and linear analysis
will remain the normal practice for some time to come. The U.S. Army
Corps of Engineers (4-~) has proposed an evaluation procedure that uses
several linear dynamic analyses to estimate the extent of cracking by varying
the level of acceptable tensile stresses and modifying the viscous damping
ratio to represent energy losses associated with cracking. If cracking is
clearly indicated by large tensile stresses, a separate stability analysis must
be performed based on the estimated elevation of the cracks. An evaluation
(4-9) of the proposed criteria shows that, although they are conservative in
predicting the presence of tensile cracking, incorrect elevations of the potential
cracks are indicated. Further work is definitely required to improve the
criteria for acceptable tensile cracking.

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69
SUMMARY OF RECENT RESEARCH
To understand the observed damage to Hsinfengkiang, Koyna, and Pacoima
dams, experimental and analytical studies have been undertaken to determine
the importance of nonlinear behavior in the earthquake response of concrete
dams. Much of the work in concrete dams is related to or based on research
in other applications of plain and reinforced concrete, for which excellent
descriptions are available in references 4-10 and 4-11.
Experimental Research
Experimental investigations of the nonlinear behavior and failure mechanisms
for concrete dams are discussed in Chapter 5 of this report, but the testing
of the materials used in such dams is discussed briefly in the following
paragraphs.
Most mathematical models for tensile cracking are based on data from
direct uniaxial, split-cylinder, and flexural tension tests. The measurement
of tensile behavior is affected by the type of test and by the size, configuration,
and curing conditions of the specimen; this fact clouds the selection of
parameters for material models (4-12~. Most tests use plain concrete specimens
of small aggregate that may not properly represent the effect of the large
aggregate used in typical mass concrete. Tests of core samples properly
taken from actual dams are more representative of the mix and in situ
properties of the concrete. Because of the sensitivity of tensile tests to the
loading and measurement system, optical interferometry techniques have
been used to provide a continuous measurement of deformation in test specimens
(4-13~. The results show the nonuniform distribution of strain longitudinally
and transversely in the specimens and strain discontinuities in the microcracked
and fractured regions. Analyses of the test results indicate that the energy
dissipated in the fracture process is due to separation of the two sides of the
crack, while the extension of the crack front produces relatively little energy
dissipation.
There have been few experimental investigations of the dynamic tensile
behavior of concrete. Many mathematical models are based on early uniaxial
tests (4-14) that showed increased tensile strength and stiffness with increasing
monotonic strain rate. Recent biaxial tests of hollow concrete cylinders
have been performed (4-15) using an impulsive loading system. The biaxial
tensile strength in tension-compression loading also increased with decreasing
rise time of the impulsive load, and the strain at failure was essentially
independent of the rise time, which confirms the conclusions from earlier
studies. These results suggest that a cracking criterion for concrete should
be based on a maximum tensile strain instead of tensile strength. The
dependence of the behavior on the loading rate, which is characteristic of a
viscoelastic or viscoplastic material, should be accounted for in the mathematical

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70
model. These trends are also indicated in a review of uniaxial tension tests
of concrete specimens (4-14), where it is recommended that the tensile
strength be increased by 50 percent to include the effect of strain rates
typical in the earthquake response of dams.
Models for Concrete Cracking
Early work in evaluating the static behavior of reinforced concrete members,
including tensile cracking, used the finite element method with predefined
cracks modeled by separations between elements (4-16~. The modeling of
tensile cracking by discrete gaps in the mesh was extended to allow automatic
locations of cracks (4-17~. When the average stress between two elements
exceeded the tensile strength of the concrete, the common nodal points were
separated, altering the finite element mesh. This discrete-crack approach
involves a number of computational difficulties because the finite element
mesh is redefined at each loading stage, and it is difficult to determine in
which direction a crack will propagate during the next load increment. Moreover,
in this approach the crack propagation is dependent on the size, shape,
orientation, and order of the elements used in the mesh.
To overcome problems with the discrete crack approach, the tensile cracks
can be considered "smeared" over an element (4-18~. In the smeared-crack
approach the discontinuous displacement field caused by cracking is averaged
over the element and represented by the continuous displacement functions
used to derive the element. A crack is assumed to form in any element in
which the principal stress reaches the tensile strength of the concrete in the
direction perpendicular to this stress; then the isotropic model of the concrete
is modified to an orthotropic one with zero stiffness in the tensile direction
but with possible shear transfer across the crack. Such transfer of shear
stresses across cracks is important in mass concrete because of the large
aggregate size and the probability of aggregate interlocking, but the smeared
crack approach can represent shear transfer only approximately, because
there is no direct information on the width and distribution of the smeared
cracks. The tensile strength used to determine crack initiation can be modified
to account for a multiaxial stress state (4-19), and the smeared-crack model
behavior can be easily incorporated into nonlinear finite element analysis
procedures. This requires only modification of the tangent stiffness matrix
for the current state of cracking in an element and release of stress perpendicular
to newly formed cracks.
The smeared-crack approach has been criticized because the numerical
results are not objective with respect to the finite element mesh (4-20~. As
the element size decreases, the zone of fracture decreases, and the force
required to propagate the crack can decrease to a negligible value. To
remedy this lack of objectivity, the theory of fracture mechanics has been

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71
used to modify the smeared-crack approach in the so-called blunt-crack
approach (4-20, 4-21, 4-22~. This approach recognizes that tensile cracking
is a fracture process in which the concrete at the crack front exhibits strain
softening (decreasing stress with increasing strain) as microcracks coalesce
to extend the crack. The crack front has a characteristic width, usually
related to the aggregate size in the concrete mix. The criterion for crack
initiation uses the fracture energy for the material (energy released in formation
of a crack with unit area), and objectivity is achieved by selecting the
strain-softening modulus, and possibly the tensile strength, to give the fracture
energy of the material. An important advantage of the blunt-crack model is
that it includes the effect of aggregate size, which is important when attempting
to compute the response of large mass concrete dams using material properties
obtained from testing small specimens. In a finite element analysis the
crack band-width can be assumed to be the element size (within certain
limits), and the strain-softening modulus then depends on the fracture energy
and tensile strength of the concrete (4-20~. The blunt-crack approach has
been used in a similar form in reference 4-23, and it is related to the line (or
fictitious) crack theory (4-24~. One problem with the smeared-crack approach
is the difficulty in representing impact and sliding of sections in the dam
after extensive crack propagation, because the discontinuous displacements
across the crack are not well defined; in some cases a discrete-crack has
been combined with a smeared-crack model to represent such behavior.
Analytical Research
Numerical computation of the nonlinear earthquake response of concrete
dams has received more attention than physical testing of models and materials.
An early investigation of the nonlinear response of gravity dam monoliths
used a biaxial failure model for concrete (4-25~. The nonlinear compressive
and tensile behavior of concrete was recognized by modifying an equivalent
uniaxial stress-strain relationship (tension and compression) in accordance
with the current state of stress. A smeared-crack approach was used to
represent tensile cracking based on a strength criterion for crack initiation.
Analysis of Koyna Dam, neglecting water and foundation rock interaction
effects, showed that tensile cracks formed near the top of the dam close to
the change in downstream slope, but the cracks did not propagate through
the cross section. Including strain rate effects in the concrete model stiffened
the dam, increasing the participation of the higher-vibration modes and
producing more extensive cracking. An interesting finding was that the
amount of tensile cracking was very sensitive to variations in the assumed
concrete tensile strength; variations such as those observed in a set of typical
tensile cracking experiments produced dramatically different amounts of
cracking. The study also showed that tensile cracking predicted by this

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72
material model did not dissipate a significant amount of energy, suggesting
that nonlinear behavior may not reduce the earthquake forces developed in
the dam. However, this conclusion may not be correct, because the fracture
of concrete may not have been properly accounted for in an objective manner.
Another series of studies (4-26, 4-27) involved analysis of a finite element
model of a complete gravity dam-water-foundation rock system. The concrete
was modeled as a rate- and history-dependent material using an elast~c-
viscoplastic relationship with a Mohr-Coulomb failure surface for compression
and tension. The material model used viscoplastic work to measure accumulated
damage in the dam; the anisotropy due to tensile cracking was not considered.
Analysis of Koyna Dam showed a substantial residual displacement with
large energy dissipation during the strong-motion part of the input record.
It appears, however, that the energy dissipation is from viscoplastic work as
the concrete softens in tension, which would not occur if cracking were
allowed; thus, the significance of these results may be questioned.
Two recent studies illustrate the application of smeared-crack models for
gravity dams. In the first application the crack-band model, described previously,
was incorporated into a nonlinear dynamic analysis procedure for dam-
water systems (4-28~. The analysis of a 400-ft-high gravity dam monolith
with a full reservoir due to a horizontal ground motion with a maximum
acceleration of 0.36 g showed extensive cracking of the concrete, as shown
in Figure 4-2. The cracking is located near the stress concentration caused
by the change in geometry of the downstream face and the dam-foundation
interface. The distributed crack zone indicated in Figure 4-2 is in conflict
with the experimental results, summarized in Chapter 5, which show crack
propagation in a narrow zone.
In another analytical study (4-29, 4-30) the crack-band model was used
with several modifications: (1) a number of features designed to eliminate
crack spreading, including a special formulation of the finite element to
eliminate spurious stiffness, and (2) user control over the elements that are
susceptible to cracking during the response analysis. The latter modification
allows control over the direction of the crack, but it also indicates the lack
of theoretical knowledge of crack propagation in concrete dams. Figure 4-3
shows the crack that forms in the same 400-ft-high gravity dam monolith
with full reservoir when subjected to horizontal and vertical ground motions
with maximum acceleration of 0.50 g and 0.30 g, respectively. The narrow
crack zone extends from the stress concentration at the downstream face,
turning down near and parallel to the upstream face. In addition, a crack
formed near the base of the dam. Although the analysis showed that the
dam remained stable, many numerical difficulties were encountered.
As noted above, the alternate approach for modeling tensile cracking is
to represent the formation, propagation, and closure of discrete cracks in the
concrete. The discrete-crack approach has been used in the finite element

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73
/
/
/
/
/
/
/
/
/
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~ _ _
/~x ~ ~ ~ ~ ~ ~ ~ ·.
FIGURE 4-2 Cracking calculated in 400-ft-high monolith of Pine Flat Dam with full reservoir
due to 0.36-g peak horizontal ground motion; solid lines show cracks open at one instant of
iune, dashed lines show cracks that opened previously and then closed (4-28).
earthquake analysis of gravity dam monoliths, neglecting interaction with
the water and foundation rock (4-31~. Each crack was monitored, and the
topology of the element mesh was redefined to represent the current state of
cracking. Analysis of Koyna Dam showed that the top part of the dam
would become unstable as a tensile crack propagated across the cross section
due to an artificial earthquake with a peak acceleration of 0.50 g. As in the
earlier research mentioned above, the extent of cracking was very sensitive
to the assumed value for concrete tensile strength. The response was also

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74
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FIGURE 4-3 Cracking calculated in system of Figure 4-2 due to combined horizontal and
vertical ground motions with peak values of 0.50 g and 0.30 g, respectively; cracked elements
are shown in black (4-30).
dependent on the finite element mesh size and orientation, although the
effect of aggregate interlock across the cracks was relatively small. Another
approach (4-6) used fracture mechanics techniques to overcome the sensitivity
of the tensile stress field to the finite element discretization near a crack.
Applying the technique to a monolith of Pine Flat Dam, including compressible
water in the reservoir, showed that a tensile crack propagated from the
upstream face at the change in slope, but stopped short of the downstream
face as the compressive stresses arrested the crack growth. The upper
portion of the dam appeared to remain stable; however, the cracking seemed
extensive for the relatively small peak ground acceleration of 0.1 g. Other
applications of fracture mechanics concepts to concrete dams are described
in references 4-32 and 4-33.
There have been two particularly noteworthy earthquake studies of arch
dams that included the nonlinear behavior of construction joints. The first
used an approximate representation of the discrete cracks formed by the
joints (4-34~. The stiffness of the elements in the vicinity of the joints was
modified to represent the current state of tensile stresses near the joints.
This approach has also been used to represent opening of a horizontal joint

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75
in a gravity dam (4-35~. Another study (4-36) used a simple discrete-crack
model of the joint that required only one finite element through the thickness
of the arch. The analytical results for a single arch rib showed the expected
opening-closing behavior and large compressive stresses in the portion of
the joint in contact.
Later work (4-37) extended the crack model to represent vertical construction
joint opening and horizontal joint opening at lift surfaces in arch dams. In
one of the most complete nonlinear analyses to date, the earthquake response
of Pacoima Dam (Figure 24) was computed, including dam-foundation interaction
(with massless foundation rock) and dam-incompressible water interaction
(4-38~. An example of the calculated response of Pacoima is shown in
Figure 4-4. The earthquake ground motion for this study was the Pacoima
Dam record scaled to a maximum horizontal acceleration of 0.50 g and
vertical ground acceleration of 0.30 g. Figure 4-4 shows the joint stresses
and displaced shape of the crest arch and crown cantilever at two instants of
time during the response. The response results, as exemplified by this
figure, showed that the vertical contraction joints in the upper part of the
dam and the joint assumed at the dam-foundation interface opened under
moderate ground motion. The loss of arch stiffness due to vertical joint
opening also caused horizontal joint opening in the upper parts of the cantilevers.
During severe ground motion some cantilever blocks lifted off their supports,
and large compressive cantilever stresses developed, possibly invalidating
the assumption of no joint slip and linear behavior of the concrete in compression.
Nonlinear analysis of concrete dams allows for the inclusion of other
phenomena that may affect the earthquake response. One such possible
effect is cavitation of the impounded water, in which gaseous regions form
if the absolute pressure in the water becomes less than the vapor pressure.
The possibility of cavitation has been shown analytically and observed in
model tests (4-3~. The formation and collapse of gaseous regions in the
water would alter the hydrodynamic pressure acting on the upstream face of
a dam and hence change the dynamic response. One analytical study of
cavitation (4-39) for a gravity dam monolith, assuming incompressible water,
showed that impact of the water resulting from collapse of the cavitation
bubble can increase tensile stresses in the top part of the dam by 20 to 40
percent. In contrast, an evaluation of dam-water interaction including
compressible water concluded that cavitation does not significantly affect
the maximum stresses due to earthquake ground motion (4-40~. Recent
Walk illUluuill~ cumpreSSIDIllly also confirms the latter conclusion, in which
it was shown that cavitation has a very small effect on peak displacements
and stresses in a dam (4-41~. However, cavitation can double the peak
acceleration at the dam crest, which may affect appurtenances and facilities
at the crest. To date, there has been no research into cavitation effects for
arch dams, where dam-water interaction effects may be more important than
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in gravity dams; however, it has been noted that cavitation may have a
beneficial effect by limiting the upstream hydrodynamic force acting to
open the vertical joints (4-39~.
RESEARCH NEEDS
Recognizing the importance of evaluating the earthquake safety of concrete
dams and the limited knowledge that exists concerning the nonlinear behavior
of concrete dams, the following items should be addressed in a future
comprehensive research effort:
1. Material Testing of Mass Concrete
Additional data on the behavior of mass concrete under dynamic loads is
urgently needed and will require an extensive physical testing program.
The testing should emphasize the tensile cracking of the mass concrete
under multiaxial stress states that are representative of the in situ stresses,
including strain rate effects. The test program should recognize the special
properties of the mass concrete used in dam construction, such as its large
aggregate size. Concrete samples should be representative of the actual
curing conditions in dams. To accomplish this, the testing should include
laboratory samples as well as cores from actual dams, and some specimens
should be taken from roller-compacted concrete dams. To perform these
tests it may be necessary to develop or adapt equipment to provide biaxial
cyclic loads on large specimens.
2. Development of Materials Models for Concrete
Using the data obtained from testing of mass concrete specimens, it is
important to develop realistic mathematical models for tensile cracking under
dynamic loads. Identification of a set of parameters that are predictive of
nonlinear dynamic behavior of concrete would be an important advance.
The material model should allow multiaxial stress states, including criteria
for tensile cracking and propagation of cracks, strain rate effects, and shear
stress transfer by aggregate interlock. The models must make reasonable
trade-offs between fitting the experimental data, satisfying conditions such
as objectivity, and providing computational efficiency. Further study of the
smeared-crack approach and use of fracture mechanics principles should be
pursued. With advances in computational techniques and computer hardware,
the discrete-crack approach for dynamic response may provide an alternative.

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3. Modeling of Other Nonlinear Mechanisms
Additional research is required to develop and apply models of construction
joint behavior in the earthquake analysis of concrete dams. Such models
must represent the redistribution of forces as the joints open and close and
allow for degradation of the joint resulting from a large number of loading
cycles. Models for concrete and rock abutments that include potential shear
failure modes are also necessary. Although not directly related to dynamic
response, analytical procedures must be used for determining the state of
stress and strain in a concrete dam at any point in time—due to gravity and
hydrostatic loads, shrinkage, temperature variations, temperature gradients,
and creep for accurate assessment of nonlinear dynamic response of dams
at the time of an earthquake.
4. Numerical Procedures for Computing Nonlinear Response
Once realistic mathematical models for concrete, joints, and foundation
rock are available, they can be incorporated into several classes of numerical
procedures for nonlinear dynamic response analysis of concrete dam systems,
including interaction with the impounded water and flexible foundation rock
and the effects of reservoir-bottom shock-wave absorption. Research should
identify methods for reducing the computational effort, while still representing
the important nonlinear behavior of the system, so that nonlinear analysis
can be used for routine design and evaluation of dams.
Although the finite element method and time integration of the equations
of motion appear to be the most useful techniques for nonlinear dynamic
analysis of concrete dams, other discretization methods and procedures for
solving nonlinear equations of motion should not be precluded. The use of
algorithms that take advantage of the vector and parallel processors in
supercomputers should be pursued.
5. Parametric and Detailed Response Studies
With the development of efficient numerical procedures, response analyses
of typical concrete dams can be used to improve understanding of the nonlinear
behavior during earthquake ground motion. In particular the studies should
identify the significance of tensile cracking and joint opening with respect
to either precipitating failure or limiting dynamic response. It is important
to determine the sensitivity of the response to parameters describing the
nonlinear models. Postcracking stability in gravity and arch dams and
postearthquake joint integrity in arch dams (as well as in some gravity
dams) must be evaluated. In addition, the influence of the following factors
should be determined: (1) ground motion characteristics, particularly amplitude,

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frequency content, duration, and occurrence of velocity pulses, (2) water
compressibility with respect to nonlinear response, and (3) modeling issues,
such as degree of spatial discretization, extent of reservoir and foundation
rock domains, and radiation conditions at the boundaries.
6. Dynamic Testing of Dam Models
In parallel with the analytical studies, further dynamic testing of dam
models is essential for verifying nonlinear behavior and calibrating mathematical
models and numerical results. If at all possible, testing of gravity dam and
arch dam models including water and foundation rock should be undertaken.
New, larger earthquake simulators such as that recently installed at the
research laboratory of the Ministry for Water Conservancy and Hydroelectric
Power in Beijing, China, may be required for realistic testing of concrete
dam systems.
7. Identification of Design Criteria
Realistic design criteria for concrete gravity and arch dams should be
developed using the results of analytical and experimental studies. Although
nonlinear earthquake analysis may not become a standard practice in design
offices, design criteria should recognize the tensile strength of concrete and
postcracking stability of gravity and arch dams and identify acceptable limits
of such behavior. Comparison of the predicted nonlinear response with the
more easily calculated linear response would be useful to determine the
limits of assessing nonlinear behavior based on the results of linear elastic
analysis. Current design criteria should be evaluated in this regard.
8. Investigation of Earthquake-Resistant Design Measures
Nonlinear response analysis capabilities will allow investigation of innovative
measures for increasing the earthquake safety of concrete dams. Based on
the results of nonlinear analyses, it may be possible to alter the geometry of
dams and foundation abutments to improve resistance to earthquakes. Different
jointing schemes, including use of joint materials that dissipate energy,
should be investigated.
It must be emphasized that these research needs are not only important in
improving the seismic safety of concrete dams but would also represent
significant advances in other applications of earthquake engineering. Similarly,
research in the nonlinear dynamic behavior of other structural systems will
have a positive effect on the study of concrete dams.