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OCR for page 36
~2
Analysis of Linear Response
PRELIMINARY COMMENTS
The ability to evaluate the effects of earthquake ground motion on concrete
dams is essential to assessing the safety of existing dams, to determining
the adequacy of modifications planned to improve old dams, and to evaluating
designs for proposed new dams. However, the prediction of the performance
of concrete dams during earthquakes is one of the most challenging and
complex problems found in the field of structural dynamics because of the
following factors:
1. Dams and the retained reservoirs are of complicated shapes, as dictated
by the topography of the sites.
2. The response of a dam is influenced to a significant degree by the
interaction of the motions of the dam with the impounded water and the
foundation rock; thus, the deformability of the foundation and the earthquake-
induced response of the reservoir water must be considered.
3. A dam's response may be affected by variations in the intensity and
frequency characteristics of the earthquake motions over the width and height
of the canyon; however, this factor cannot be fully considered at present
because of the lack of instrumental data to define the spatial variation of
ground motion, as discussed in Chapter 2.
When evaluating the earthquake behavior of concrete dams, it is reasonable
in most cases to assume that the response to low- or moderate-intensity
earthquake motions is linear. That is, it is expected that the resulting deformations
of the dam will be directly proportional to the amplitude of the applied
ground shaking. Such an assumption of response linearity greatly simplifies
36
OCR for page 37
37
both the formulation of the mathematical model used to represent the dam,
reservoir water, foundation rock system and also the procedures used to
calculate the response. The results of linear analysis serve to demonstrate
the general character of the dynamic response, and the amplitudes of the
calculated strains and displacements indicate whether the assumption of
linearity is valid. In the case of a major earthquake it is probable that the
calculated strains would exceed the elastic capacity of the dam's concrete,
indicating that damage would occur; in this case a much more complicated
nonlinear analysis would be required to determine the expected extent of
damage. However, a linear analysis still can be very valuable in helping to
understand the nature of the dynamic performance and in deciding whether
a nonlinear analysis will be necessary. In many cases a reasonable estimate
of the expected degree of damage can be made from the linear analysis,
even though the results suggest that a slight to moderate degree of cracking
or other form of nonlinearity is to be expected.
In this chapter both the earliest, rather primitive, techniques of estimating
earthquake performance are described and also the refined, modern, linear
computer analysis procedures that are presently recommended for seismic
safes', evaluations of concrete dams.
STATIC ANALYSIS
Traditional Analysis and Design
Because most dams in the United States were built prior to the development
of modern computer analysis procedures, earthquake effects were accounted
for in the designs by using methods that are now considered oversimplified.
In particular, the dynamic behavior of the dam, reservoir water, foundation
rock system was not recognized in defining the earthquake forces used in
traditional design methods (3-1, 3-2~. Thus, the forces associated with the
inertia of the dam were expressed simply as the product of a seismic coefficient-
taken to be constant over the surface of the dam, with typical values of 0.05
to O.l~and the weight of the dam per unit surface area expressed as a
function of location. Seismic water pressure in addition to hydrostatic
pressure was specified in terms of the seismic coefficient and an additional
pressure coefficient; the latter was evaluated based on the assumptions that
the dam was rigid and had a plane vertical upstream face and that water
compressibility effects were negligible (3-3, 3-4~. Generally, interaction
between the dam and the foundation rock was not considered in evaluating
the aforementioned earthquake forces, but in the seismic stress analysis of
arch dams the flexibility of the foundation rock sometimes was recognized
through the use of Vogt coefficients (3-1~. Stresses in gravity dams with
ungrouted construction joints were usually determined by treating the concrete
OCR for page 38
38
blocks as vertical cantilever beams; for the analysis of arch dams, and
sometimes for gravity dams in narrow canyons or with keyed joints, the
trial load analysis procedure was usually used.
Traditional design criteria (3-1, 3-2) require that the compressive stresses
not exceed either one-fourth of the specified compressive strength or 1,000
psi. Tensile stresses were usually not permitted in gravity dams or, if they
were, were limited to such a small value that cracking was not considered
possible; in arch dams the tensile stresses were required to remain below
150 psi. In the design of gravity dams it was generally believed that stress
levels were not a controlling factor, so the designer was concerned mostly
with satisfying criteria for overturning and sliding stability.
Earthquake Performance of Koyna Dam
As mentioned in Chapter 1, Koyna Dam in India is one of two concrete
dams that have suffered significant earthquake damage (3-5, 3-6~. A photograph
of this dam is shown in Figure 3-1, and it is useful to discuss its earthquake
performance in some detail in this report because it was designed by the
traditional static analysis procedure using a seismic coefficient of 0.05.
Even though a "no-tension" criterion was satisfied in the design procedures
considering seismic as well as all other forces, the earthquake of 11 December
1967 caused important horizontal cracks on the upstream and/or downstream
faces of a number of nonoverflow monoliths near the elevation at which
there is an abrupt change in slope of the downstream face. Although the
dam survived the earthquake without any sudden release of water, the cracking
appeared serious, and it was decided to strengthen the dam by constructing
buttresses on the downstream face of the nonoverflow monoliths; the overflow
monoliths were not damaged.
To understand why the damage occurred, the dynamic response of the
tallest nonoverflow monolith was calculated, assuming linear behavior. The
results indicated large tensile stresses on both faces, with the greatest values
near the elevation of the downstream-face change of slope. These calculated
stresses (shown in Figure 3-2), which exceeded 600 psi on the upstream
face and 900 psi on the downstream face, were about two to three times the
estimated 350-psi tensile strength of the concrete at that elevation. Hence,
significant cracking consistent with what was observed could have been
expected during an earthquake of this intensity. The maximum compressive
stress in the monolith (not shown in Figure 3-2) was about 1,100 psi, well
within the compressive capacity of the concrete. A similar analysis of the
nonoverflow monoliths indicated that little or no cracking should have occurred
there, which also is consistent with the observed behavior.
OCR for page 39
39
FIGURE 3-1 Koyna Dam, India, was damaged by a magnitude 6.5 earthquake in December
1967 (3-5, 3-6).
Limitations of Traditional Procedures
It is apparent from the preceding discussion that the dynamic stresses
that develop in gravity dams due to earthquake ground motion bear little
resemblance to the results given by standard static design procedures. In
the case of Koyna Dam the earthquake forces based on a seismic coefficient
of 0.05, uniform over the height, were expected to cause no tensile stresses;
however, the earthquake caused significant cracking in the dam. The discrepancy
is the result of not recognizing the dynamic amplification effects that occur
in the dam's response to earthquake motions.
The typical design seismic coefficients, 0.05 to 0.10, used in designing
concrete dams are much smaller in the typical period range for such dams
than are the ordinates of the pseudoacceleration response spectra for intense
earthquake motions, as shown in Figure 3-3. It is of interest to note that the
seismic coefficients used for dams are similar to the base shear coefficients
specified for buildings. However, the Uniform Building Code provisions
(3-7) are based on the premise that the structures should be able to:
OCR for page 40
40
200
300
400
500
600
600
1 1 00 psi
43OoO
600
iN\ N:
11~
100
Envelope Values of Maximum
Principal Stresses for Koyna
Dam due to Koyna Earthquake
2900 290
_
3500 350
L
4100 410
Compressive Tensile
Concrete Strength, PSI
FIGURE 3-2 Maximum stresses in the tallest Koyna Dam monolith calculated by input of the
Koyna earthquake record, compared with the estimated strength of the concrete (3-6).
1. resist minor levels of earthquake ground motion without damage;
2. resist moderate levels of earthquake ground motion without structural
damage, but possibly some nonstructural damage; and
3. resist major levels of earthquake ground motion . . . without collapse,
but possibly with some structural as well as nonstructural damage.
While these may be appropriate design objectives for buildings, major dams
should be designed more conservatively, and this intended conservatism is
reflected in the no-tension or at most small-tension limitation used in traditional
methods for designing dams. What the traditional methods fail to recognize,
however, is that in order for dams to satisfy these criteria during earthquakes,
consideration must be given to their dynamic behavior. For linearly elastic
structures the dynamic aspect of the response is indicated by the response
spectra and by the dynamic displacement patterns, which are conveniently
expressed in terms of free-vibration-mode shapes.
OCR for page 41
41
0.6
of
LL
0.5
LL
LL
O 0.4
LD
C:
O 03
On
LLJ
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~ 0.1
LL
o
Average Aceleration Response Spectrum . g
~Peak Ground Acceleration = 0.339, Damping = 5%
l ~
I \
/ \ Seismic Base Shear Coefficient
\ for Buildings: 1/1 5~/ T
Seismic Coefficient
0.2 for Concrete Gravity
Dams
_ ~
. , , , ~ I ~ ~ ~
I , , I I I
0 1 2 3
VIBRATION PERIOD (in seconds)
FIGURE 3-3 Companson of seismic coefficients used in traditional design with the response
spectrum of a strong earthquake (3-29).
In linear analyses the effective modal earthquake forces may be expressed
as the product of a seismic coefficient (which depends on the earthquake
pseudoacceleration response spectrum and the vibration period of the mode,
and varies according to the shape of the mode) and the unit weight of
concrete. The seismic coefficient associated with forces in the fundamental
mode of a gravity dam varies with height, somewhat as shown in Figure 3-
4. For the first two modes of a symmetric arch dam (fundamental symmetric
and antisymmetric modes), the coefficient may vary over the dam face, as
shown in Figure 3-5. These figures are in sharp contrast with the uniform
distribution of seismic coefficient that has been assumed traditionally and
that has led to erroneous distribution of lateral forces and hence of stresses
in the dam.
One of the results of assuming a heightwise-uniform seismic coefficient
is Hat calculated stresses in gravity dams are found to be greatest at the
base of the dam. This has led to the concept of decreasing the concrete
strength with increases of elevation, as has been done for some dams (e.g.,
Koyna Dam in India and Dworshak Dam in the United States). However,
OCR for page 42
42
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Traditional
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Fundamental
Vibration Mode
FIGURE 3-4 Vanaiion of seismic coefficient along height of gravity dams: traditional constant
value versus dynamic vanai~on (3-29).
the results of dynamic analysis of Koyna Dam (Figure 3-2), as well as the
location of the earthquake-induced cracks, demonstrate that the largest stresses
actually occur at the two faces in the upper part of gravity dams. Therefore,
those are the regions of gravity dams where the highest-strength concrete
should be provided if the designer chooses to vary the concrete strength
within the dam.
Another undesirable consequence of specifying a heightwise constant
seismic coefficient is that the detrimental effects of concrete added near the
dam crest are not apparent, as has been shown by analytical study of Pine
Flat Dam (3-31~. This typical gravity dam, shown in Figure 3-6, was built
by the U.S. Army Corps of Engineers in California. As can be seen in the
photograph, it was widened at the crest to provide for a roadway. The
resulting added crest concrete may appear to have a beneficial effect in
reducing the stresses predicted by traditional static analysis and also may
serve useful functions in providing freeboard above the maximum water
level, in resisting the impact of floating objects, and in affording the roadway.
However, because of the sharp increase of seismic coefficient in the crest
region (Figure 3-4), the crest mass may cause a dramatic increase in the
dynamic stresses approximately doubling them in the earthquake response
of Pine Flat Dam, as shown in Figure 3-7. An interesting consequence of
this type of unfavorable response mechanism was seen in monolith 18 of
Koyna Dam, which suffered the worst damage during the earthquake; it is
believed that this exaggerated damage resulted from an elevator tower that
extended 50 ft above the top of the block and therefore was subjected to
greatly increased inertial forces.
OCR for page 43
43
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7
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FUNDAMENTAL ANTISYMMETRIC
VIBRATION MODE
FIGURE 3-5 Variation of seismic coefficient over face of arch dams (3-12).
OCR for page 44
44
OCR for page 45
45
Structural Section
600
400
200
100 i
100
3Q~
~100 psi ACTUAL
SECTION
p1~
~0
Additional Crest Mass
3001
200
W00
~ \\~200
^00
STRUCTURAL
SECTION
\
100
FIGURE 3-7 Effects of nonstructural crest mass on maximum principal stresses calculated in
the tallest Pine Flat Dam monolith due to Koyna earthquake input (3-31).
Traditional design earthquake loading for concrete dams includes seismic
water pressures in addition to hydrostatic pressure, as specified by various
formulas (3-2, 3-9~. These formulas differ somewhat in detail and in numerical
values but not in the underlying assumptions, because they are all based on
the classical results of Westergaard (3-3) and Zangar (3-4~. In a typical
formula the seismic water pressure is given as Pe = CSwH, where C is a
coefficient that varies nonlinearly from zero at the water surface to about
0.7 at the reservoir bottom, S is the specified seismic coefficient, w is the
unit weight of water, and H is the total depth of water. For a seismic
coefficient of 0.10 the additional pressure at the base of the dam is 7 percent
of the hydrostatic pressure; values at higher elevations also are small. These
additional pressures have little influence on the computed stresses and hence
on the geometry of the gravity section that satisfies the standard design
criteria. On the other hand, when the true dynamic behavior of the dam is
considered' including dam-water interaction and water compressibility, numerous
analyses have demonstrated that hydrodynamic effects are significant in the
OCR for page 46
46
earthquake response of concrete dams and can lead to important stress increases,
especially in arch dams (3-S, 3-10, 3-11~. It is apparent, therefore, that
hydrodynamic effects are not properly modeled in the traditional design
procedures.
Finally, it is evident that the static overturning and sliding stability criteria
that have been used in traditional gravity dam design procedures have little
meaning in the context of the oscillatory responses produced in dams by
earthquake motions.
DYNAMIC ANALYSIS
Arch Dam Analysis Forerunner
Recognizing the limitations of the static seismic coefficient method, design
and research engineers became interested in dynamic analysis procedures to
reliably predict the earthquake response of dams. In one of the earliest
(1963) dynamic analyses applied to arch dams (3-12), the uniform seismic
coefficient used in static methods was replaced by a spatially varying coefficient
computed for the first two vibration modes of a symmetric dam (first symmetric
and first antisymmetric mode), based on the modal periods and shapes and
on an earthquake response spectrum. For ground motion in the upstream-
downstream direction, the hydrodynamic effects were based on Westergaard's
formula for a rigid gravity dam; for cross-stream motions a formula based
on the work of Zienkiewicz and Nath (3-13) was applied. The stress analysis
for each mode was done by the trial load method, and the modal stresses
were combined appropriately.
Finite Element Modeling
The procedures for earthquake analysis of dams began to change rapidly
in the late 1960s with the adoption of finite element modeling procedures,
with advances in methods of dynamic analysis, and with the increasing
availability of large-capacity, high-speed computers. One of the earliest
applications of this new technology to analysis of the earthquake response
of an arch dam was reported in 1969 (3-14~. In this investigation the dam
was modeled as an assemblage of three-dimensional finite elements on a
rigid base, and the impounded water was modeled as a mesh of incompressible
liquid elements. The dynamic response was calculated by the mode-supe~position
method.
Subsequently, the Bureau of Reclamation funded the development of a
computer program based on similar concepts intended specifically for the
static and dynamic analysis of the Bureau's arch dams (2-5~. This program,
called ADAP (Arch Dam Analysis Program), modeled the dam body by
OCR for page 50
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~ Incompressible water
\~ ~\~\3350~//~//50 ( \ J
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Compressible water, = 0.9
two \/ J
an ~
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Compressible water, = 0.5
FIGURE 3-10 Calculated effects of water compressibility and reservoir boundary absorption
on upstream face maximum envelope stress contours for Monticello Dam subjected to Morgan
Hill earthquake record (3-26).
OCR for page 53
53
UPSTREAM COMPONENT
Full Reservoir
Rigid Reservoir Boundary, a= 1
Cantilever Stress Arch Stress
\ 500
\ 400
\ 300
100
1 50~
Full Reservoir
Rigid Reservoir Boundary, a= 1
Cantilever Stress Arch Stress
200~\
Full Reservoir
Rigid Reservoir Boundary, a= 1
Cantilever Stress Arch Stress
~ ~ /~ 20'0
~O)J /0~
300 / ~
WAHOO/
~ /
i7400 ~
Em/ ~
VERTICAL COMPONENT
iOO
\ ~/oo Ado / / /uu [/ 400
:~ ~200
Full Reservoir
Rigid Reservoir Boundary, a = 1
Full Reservoir
Rigid Reservoir Boundary,a= 1
Cantilever Stress Arch Stress
~800 A\
5
100 /
. ~
CROSS-STREAM COMPONENT
Full Reservoir
Rigid Reservoir Boundary,a= 1
Cantilever Stress Arch Stress Cantilever Stress Arch Stress
100
~25
\ \50\
\ 7511
~JI
~0
/
FIGURE 3-11 Calculated effects of reservoir boundary absorption on upstream face maximum
envelope stress contours for Morrow Point Dam due to Taft earthquake record (3-11).
OCR for page 54
54
response of a slender arch dam than for a massive gravity dam.
3. The assumption of water incompressibility that is commonly used in
practical analysis may lead to errors on either the conservative side or the
unconservative side for upstream-downstream earthquake motion, but it is
more likely to be unconservative in predicting response to the vertical and
cross-stream components of motion (3-10, 3-20, 3-21~.
4. An important deficiency of the incompressible water approximation is
that the hydrodynamic-wave absorption effects of the underlying rock or
reservoir boundary sediments cannot be taken into account, and it has been
shown that neglecting boundary wave absorption may lead to unrealistically
large estimates of seismic response.
5. Neglecting the dynamic interaction of gravity dams with deformations
of the foundation rock also will generally lead to overestimation of the
seismic response. Because arch dams resist the reservoir water pressures
and the thermal and earthquake forces, at least in part, by transmitting them
by arch action to the canyon walls, dam-foundation rock interaction also is
likely to be significant in the earthquake response of arch dams, possibly
more so than in the case of gravity dams.
The preceding observations lead to the conclusion that dam-reservoir
water interaction, including water compressibility and pressure wave absorption
at the reservoir boundaries, and dam-foundation rock interaction all should
be considered in the earthquake response analysis of concrete dams.
Dynamic Analysis Procedures and Computer Programs
Two-Dimensional Analysis
Analysis procedures and computer programs that take account of all of
these factors have been developed for evaluating the earthquake performance
of concrete dams idealized as two-dimensional systems (2-4, 3-22), and
these procedures and programs should be used in those cases where two-
dimensional analyses are appropriate because of their computational efficiency.
In gravity dams with plane veridical joints the monoliths tend to vibrate
independently, as evidenced by the spelled concrete and water leakage at
the joints of Koyna Dam during the 1967 earthquake (3-5, 3-6~; hence, a
two-dimensional plane-stress model of the individual monoliths is usually
appropriate for predicting the response of such dams to moderate or intense
earthquake motions. In some cases of gravity dams built in a broad valley,
especially roller-compacted concrete dams built without vertical joints, a
plane-strain idealization may be adopted in place of the plane-stress model.
On the other hand, if the gravity dam has effectively keyed contraction
joints or is located in a narrow canyon, the assumption of independent
response of the blocks may not be appropriate.
OCR for page 55
ss
One computer program that has been developed to perform the earthquake
analysis of two-dimensional dam-water-foundation rock systems, called EAGD-
84, is based on a substructure formulation of the problem (2-4, 3-22~. In
this program a finite element idealization is used to model arbitrary geometry
and variations of material properties of the dam; consequently, both overflow
and nonoverflow sections and also appurtenant structures can be modeled
satisfactorily. The impounded water is treated as a continuum in order to
efficiently represent its large extent as well as the radiation of hydrodynamic
pressure waves upstream. The effects of alluvium and silts that accumulate
at the bottom of the reservoir and of the underlying rock are modeled
approximately by a boundary that may partially absorb the incident hydrodynamic
pressure waves; more rigorous methods for treating this effect without such
a simplifying approximation have recently become available (3-23~. The
foundation rock supporting the dam is idealized as a viscoelastic half-plane
continuum that, as mentioned earlier, accounts for the energy radiation effects
of the dam-foundai~on rock interaction. With this computer program a
complete interaction analysis can be performed of the dynamic response of
a gravity dam to the upstream and vertical components of free-field earthquake
motion, with both components acting simultaneously at the dam-foundation
rock interface.
Three-Dimensional Analysis
As mentioned earlier, computer programs employing finite element
idealizations for the earthquake analysis of arch dam-water-foundation systems
have been in use for as long as two decades. ADAP (2-5), the first widely
available program developed specifically for dynamic analysis of arch dams,
also uses a mesh of finite elements to model the foundation rock. However,
as mentioned earlier, these elements are assumed to be massless, so they
model the foundation flexibility but do not account for wave propagation in
the rock and the consequent radiation damping effect. Also mentioned
earlier is the fact that the liquid finite elements used in recent versions (3-
15) of this program are assumed to be incompressible. Thus, the water
compressibility effects and hydrodynamic wave absorption effects of the
reservoir boundary, which as stated earlier can be significant in the seismic
response, are not considered in ADAP.
These limitations are overcome in a computer program that is based on
the substructure method and was developed recently for the three-dimensional
analysis of concrete dams. The program, named EACD-3D (2-3, 3-24),
accounts for dam-water interaction, including the effects of compressibility
and reservoir boundary pressure wave absorption, using procedures analogous
to those employed in the two-dimensional program EAGD-84. So far, however,
OCR for page 56
56
it has not been possible to take full account of the dam-foundation rock
interaction. All three substructures—dam, reservoir water, and supporting
rock are idealized as finite element systems to represent the complicated
dam geometry and site topography, but special techniques were introduced
to efficiently recognize the great upstream extent of the reservoir. The
massless finite element model of the foundation rock is similar to that used
in ADAP, and thus it also is deficient in representing radiation energy loss.
On the other hand, recent research in Japan (3-27) has been directed toward
modeling of arch dam-foundation rock interaction, but it has not yet advanced
sufficiently to be of use in practical arch dam earthquake response analyses.
Both EACD-3D and ADAP can be used to perform a complete dynamic
analysis of a concrete arch dam subjected to the simultaneous action of
upstream, vertical, and cross-stream components of the free-field motion
specified at the interface between dam and foundation rock. In the model
with massless foundation rock the free-field surface motion at the dam-rock
interface is the same as the motion at the rigid boundary of the foundation
block. In principle, spatial variation of the input earthquake motions could
be specified either for the free-field input used in EACD-3D or for the
foundation block boundary input used in ADAP, and it is evident that such
spatial variation does occur across dam sites, as discussed in Chapter 2.
However, reliable descriptions of the earthquake motions to be expected at
such locations are not available at present, and "multiple-support" excitation
is seldom used for arch dam analysis, even though it is technically feasible
(3-25).
Selection of numerical values for the parameters necessary to describe a
dam-water-foundation rock system for analysis by the aforementioned computer
programs should be based on appropriate experimental tests. Clearly, the
properties of the reservoir water present no problem, and the properties of
the concrete comprising the dam can be defined adequately by standard
procedures. Evaluation of the elastic modulus and damping of the foundation
rock is not so simple, but, as discussed in Chapter 5, numerous field measurement
studies have demonstrated that vibration properties calculated using typical
finite element system models agree well with measured values. However,
techniques are not presently available to determine the reservoir wave reflection
coefficient, which is seen in Figures 3-10 and 3-11 to have a major influence
on the seismic stresses. Until such measured data are available, it is suggested
that values for this coefficient be calculated based on the properties of the
water and the underlying rock, as described in references 3-22 and 3-24.
This approach neglects the unknown cushioning effect of reservoir boundary
sediments and thereby will probably overestimate the earthquake response
in most cases.
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57
The procedures applied in the analysis of arch dams can be used to
evaluate the earthquake response of other types of dams that also must be
modeled as three-dimensional systems; buttress dams (including both flat
slab and multiple arch dams) are a typical example. In the early decades of
this century, buttress dams were often used in preference to gravity dams
because they require about 60 percent less concrete. But as stated in a
Bureau of Reclamation publication (3-28, p. 10), "the increased formwork
and reinforcing steel required usually offset the savings in concrete....
[Hence, this] type of construction usually is not competitive with other
types of dams when labor costs are high." However, even though new dams
of this type are not being built in the United States, it still is necessary to
carry out seismic safety evaluations of the existing structures. Littlerock
Dam in California, shown in Figure 3-12, is a typical multiple arch water
supply and flood control dam for which a seismic safety evaluation has
been performed.
Simplified Dynamic Analysis Procedures
While the response history analysis procedures described above are appropriate
for final-stage analyses of the seismic safety of existing dams and proposed
new dams, simplified analysis procedures would be preferable for the preliminary
evaluation or design stages.
In response to this need a simplified procedure was developed in 1978
for the analysis of gravity dams in which the maximum response due to the
fundamental mode of vibration was represented by equivalent lateral forces
computed directly from the earthquake design spectrum (3-29~. Recently,
this simplified two-dimensional analysis of the fundamental mode response
has been extended to include the effects of dam-foundation rock interaction
and wave absorption at the reservoir bottom (3-30), in addition to the effects
of the compressible water-dam interaction considered in the earlier procedure.
Also now included in the simplified procedure is a "static correction" method
to approximate the response contributions of the higher-vibration modes.
The simplified procedure is sufficiently accurate for preliminary design and
safety evaluation of gravity dams.
While many of the basic concepts underlying the procedure may be applicable
to arch dams, the extension of such a method to treat three-dimensional
systems is likely to be very difficult for several reasons: (1) the geometry
of arch dams varies considerably from one site to another, thereby reducing
the possibility of defining "standard" values for vibration properties and
parameters, and (2) their response is generally not dominated by a single
mode of vibration.
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58
it::
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FIGURE 3-12 Littlerock Multiple Arch Dam, Califomia, completed in 1924 for use in irrigation,
has a maximum height of 168 ft and length of 800 ft.
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s9
RESEARCH NEEDS
Although considerable progress has been made in the past 20 years,
much additional research needs to be done to improve the reliability of
methods for the seismic analysis, design, and safety evaluation of concrete
dams. To meet this objective the following tasks should be pursued:
1. Instrumentation
Major dams in seismic areas of the United States should be instrumented
to measure their responses during future earthquakes. The instrumentation
should be designed to provide adequate information on the characteristics
and spatial variation of the ground motion at the site, on the response of the
dam, and on the hydrodynamic pressures exerted on the dam. Because of
the urgent need for such data, dams in highly seismic regions of other
countries also should be considered for instrumentation. This effort should
be coordinated with the plans for the seismic arrays recently installed in
Taiwan, India, and the People's Republic of China under cooperative agreements
with the United States.
2. Field Forced-Vibration Tests
Forced-vibration tests should be conducted on selected dams using more
than one water level where feasible and the resulting hydrodynamic pressures
and motions of the structures and their foundations should be recorded and
analyzed.
3. Evaluation of Analytical Methods for Response Analysis
Existing analytical methods for computing the response of all types of
concrete dams to earthquakes should be evaluated by comparing calculated
results with the responses recorded during forced-vibration tests and, more
importantly, during actual earthquakes when significant ground motions are
recorded at appropriate dam sites. If necessary, the methods should be
refined and the computer programs needed for their implementation prepared
in a form convenient for application in engineering practice.
4. Improvement of Arch Dam Analysis
The methods used for input of the earthquake motions in present methods
of earthquake analysis of arch dams urgently need improvement. Similarly,
improvements are needed in procedures used to account for the interaction
between arch dams and their supporting foundation rock.
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5. Simplified Analysis Procedures
Simplified analysis procedures should be developed that are suitable for
the preliminary phase of design and safety evaluation of arch dams.
6. Evaluation of Dynamic Sliding and Rocking Response of Gravity Dams
Analysis procedures should be developed to determine the dynamic sliding
and rocking response of gravity dam monoliths. Utilizing these procedures,
rational stability criteria should be derived, replacing the traditional sliding
and overturning criteria that do not recognize the oscillatory response of
dams during earthquakes.
Representative terms from entire chapter:
arch dams
