# 3DEC建模.pdf

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3DEC建模.pdf

**简介：**本文档为《3DEC建模pdf》，可适用于学术研究领域，主题内容包含NONLINEARSTATICANDDYNAMICANALYSISOFTHECUSHMANARCHDAMSUSINGDISTINCTELEMENTS符等。

NON-LINEAR STATIC AND DYNAMIC ANALYSIS OF THE
CUSHMAN ARCH DAMS USING DISTINCT ELEMENTS
D. D. Curtis1 J. P. Aglawe2
E. B. Kollgaard3 D. E. Bowes4
S. H. Fischer5
ABSTRACT
The paper presents the detailed non-linear static and dynamic analysis of the Cushman
No. 1 arch dam. The Cushman No. 1 arch dam is owned and operated by the City of
Tacoma, Department of Public Utilities. The analyses were undertaken as part of the
F.E.R.C. Part 12 investigations. The static and dynamic analyses are unique in that
opening, closing, and sliding along joints is modeled in considerable detail. The distinct
element program 3DEC is used in the non-linear analysis of the dam and foundation. We
believe that the sophisticated non-linear analyses that have been carried out on these
dams’ advances the state-of-the-art of dam assessment under seismic loading.
In the 3DEC analysis, all the dam contraction joints and the dam-foundation interface
joints are allowed to open, close, and slide under static and dynamic loading. In addition,
joints in the foundation rock are modeled such that stability analysis of the dam and
foundation are made within one 3DEC model. The static analysis simulates dam
construction, grouting of contraction joints, and reservoir impoundment. The static
stability of the dam is checked by gradually reducing the frictional strength of the joints
until displacements become excessive. A detailed non-linear static analysis was
undertaken to investigate slip on the dam-foundation interface, particularly at the right
abutment where the contact geometry is adversely sloped downstream.
In the non-linear seismic analysis, the dam joints and dam-foundation interface open and
close during the earthquake. At several time instances during the earthquake, the shear
strength along various joint surfaces was exceeded and this caused relative shear slip
displacements. The post-earthquake stability of the dam was assessed by increasing
uplift and gradually reducing the strength of the joints. By this means the dam was found
to be safe.
_______________
1 Senior Civil Engineer, Acres International, 4342 Queen St., P.O. Box 1001, Niagara Falls, Ontario
Canada, L2E 6W1, Tel: 905-374-5200, Fax: 905-374-1157, dcurtis@acres.com.
2 Senior Geotechnical Engineer, Acres International, 4342 Queen St., P.O. Box 1001, Niagara Falls,
Ontario Canada, L2E 6W1, Tel: 905-374-5200, Fax: 905-374-1157, jaglawe@acres.com.
3 Consulting Engineer, 4820 Eagle Way, Concord, CA, USA, 94521, Tel: 925-798-9475, Fax: 925-689-
3456, ebkollgaard@ca.astound.net.
4 Consulting Engineer, 2922 78th Ave. Ct. N.W., Gig Harbor, WA, USA, 98335, Tel: 253-265-0811, Fax:
253-265-0812, bowespe@halcyon.com.
5 Senior Principal Engineer, Tacoma Power, Generation, 3628 South 35th Street, Tacoma, WA, USA,
98409, Tel: 253-502-8316, Fax: 253-502-8136, sfischer@ci.tacoma.wa.us.
INTRODUCTION
The Cushman Dams are on the lower stretch of the North Fork of the Skokomish River
on the southeastern side of the Olympic peninsula near the southern end of the Hood
Canal. The City of Tacoma owns both the dams.
Cushman Dam 1 is a single curvature concrete arch dam with an overall height of 260 ft
above the streambed. The 400 ft-long crest of the dam is at El. 741.5. The reservoir
normal maximum storage level is El. 738.0. The dam was completed and placed into
operation in 1926.
The Cushman 1 dam/foundation contact is poorly shaped especially at the right abutment
where it is sloped adversely in the downstream direction. This fact led to the question of
ability of the dam to withstand the strong ground motions associated with the seismic
loading. It was recommended to perform a sophisticated non-linear numerical analysis to
ascertain the seriousness of the seismic response and evaluation of remedial
modifications.
The non-linear static and dynamic analyses were performed with 3DEC, a three
dimensional distinct element analysis program. The 3DEC program was used to perform
a non-linear static analysis of dam followed by a non-linear dynamic time history
analysis. The 3DEC program was used to analyze both Cushman 1 and 2 dams, but due
to space limitations, the main results from the Cushman 1 analysis are presented herein.
MODELING APPROACH
Brief Description of 3DEC
The 3DEC (3-Dimensional Distinct Element Code) program is the three-dimensional
extension of Itasca's two-dimensional code UDEC. It is specifically designed for
simulating either the quasi-static or dynamic response to loading of rock media
containing multiple, intersecting joint structures.
The 3DEC model is an assemblage of discrete polyhedras representing discontinuous
medium. Discontinuities are treated as boundary conditions between blocks. Large
displacement on the discontinuities such as slip and opening is simulated in a
discontinuous medium. Relative motion along discontinuities is governed by linear and
non-linear force-displacement relations for movement in both the normal and shear
directions. The program uses an explicit solution scheme, which gives a stable solution
to unstable physical processes.
3DEC is particularly well suited to simulate blocky structures, such as stone masonry
arches. Assessment of the safety conditions of old masonry bridges (Lemos, 1997) and
the seismic behavior of stone masonry arches (Lemos, 1995) has been done using 3DEC.
It has been successfully employed to simulate the behavior of a concrete arch dam
constructed on a jointed rock foundation (Lemos, 1996) and also to perform stability
analysis of underground powerhouse station (Dasgupta and Lorig, 1995, Dasgupta, et al.,
1995).
Geological Setting
The brief review of the engineering geology of Cushman Dam No 1 is given by Coombs
(1972). Recent geological compilation and review was done by Hamilton (2001) for the
right abutment of Cushman Dam 1. The bedrock in the area of Cushman projects
consists of thick sequence of basalt and andesite flows with local interflow layers of tuff
and agglomerate, of the Eocene age Crescent Formation. At the dam site the Crescent
Formation layering strikes about NE-SW, crossing the canyon at a high angle and dips 45
to 60 degrees SE downstream. Various joints in the foundation rock are present.
Strike and dip angles for the various joints are given in Table 1. The joint plane A1
intersects near the contraction joint at station 3+64.79 of the dam. Figure 1 shows the
joint planes D, A1, A2 and B. The right abutment wedge is formed by an assumed
vertical plane D on the upstream, ramp fracture plane B below the dam-foundation
contact, and the joint planes A1 and A2. These joint planes form a right abutment wedge
with a total weight of about 30,000 tons.
Table 1. Orientations of the Discontinuities
Joint Plane Strike Dip
D N25W Vertical
B N56E 53 NW
A1 N67E 55SE
A2 N67E 55SE
Model Development
3DEC model was developed as an assemblage of discrete blocks using commercially
available program DISPLAY (EMRC, 1997). Foundation rock, concrete dam, and
reservoir water elements were discretized. An exploded view of foundation rock,
concrete dam, and the reservoir is shown in Figure 2. In Figure 2, the reservoir is shown
in the upper part of the figure, the dam in the middle and the reservoir in the lower part of
the figure. The reservoir was extended more than three times the dam height in the
upstream direction. It is noted that the bulk of the model was created using a 3D
AutoCad model, which was supplied by Tacoma Power. In the foundation rock, four
joint sets were used. In the concrete dam, the seven vertical contraction joints and the
dam/foundation contact joints were modeled. Various rock joints forming a right
abutment wedge are shown in Figure 1. Figure 3 provides an illustration of right
abutment wedge along with dam. Right abutment gravity sections and the start of the
arch section of the dam are also shown to obtain spatial location and orientation of the
wedge with respect to the dam.
Figure 1. Dam-Foundation Contact, Contraction Joints and
Foundation Rock Mass Discontinuities
Figure 2. Exploded View of Foundation, Dam and Reservoir
Figure 1 3DEC model of Cushman Arch Dam No. 1 Figure 9a.
Figure 3. Right Abutment Wedge Geometry and the Dam.
The Rock Wedge Alone is shown in the Inset.
Various FISH functions were written to simulate the effect of grouting of the independent
cantilever monoliths at the contraction joints, the uplift pressure distribution at the dam-
foundation contact and water pressure in the joints on the right abutment wedge surfaces.
FISH functions provide a programming capability in 3DEC that allows the user to
program such features as grouting joints, i.e., closing gaps at the dam joints.
Modeling Sequence
Figure 4 presents various stages during the modeling sequence to establish initial state of
stress. Initial rock stresses were computed using gravity loading. The dam monolithic
blocks were then constructed. Grouting of the independent cantilevers was simulated by
specifying a closed gap between the contraction joints after gravity loads were
equilibrated. The reservoir elements were turned on to load the dam hydrostatically.
Parametric studies were performed to examine the sensitivity of the 3DEC to reduced
frictional strength at the dam-foundation contact and on the joints forming the right
abutment wedge. Finally, the dynamic analysis was performed using Juan de Fuca and
Cascadia seismic records for MCE loading.
Properties
The joints between the dam-reservoir and reservoir-foundation are assumed to be elastic.
Contraction joints within the dam and dam-foundation contact are assumed to have zero
cohesion and 55 degrees friction angle. The dam foundation was quite rough, therefore,
the assumed friction angle is considered conservative. In the right abutment wedge
analysis, the cohesion on the joint planes was set to zero. The assumed total combined
(cohesive and frictional) shear strength on joints B, D, A1 and A2 is taken as 55 degrees.
R k
It was assessed that the total shear strength of the joint planes is at least equivalent to that
with a combined friction angle of 55 degrees.
Figure 4 Modeling Methodology Adopted for Dynamic Analysis
of Cushman-1 Arch Dam
Create Dam + Rock Foundation + Reservoir
blocks from SADSAP model
Extend far-field boundaries for rock foundation
and reservoir
Discretize model
Apply boundary conditions
Rock only analysis
Equilibriate rock foundation under gravity
Construct arch dam
+ right and left wing wall dams as monoliths with
7 contraction joints
Contraction joint : coh = 0 psi, fric 5 deg
Rock dam contact : coh = 300 psi, fric 55 deg
Equilibriate the model
Grouting
Close the gaps on the contraction joints
between the concrete monoliths
Contraction joint : coh = 0 psi, fric 55 deg
Activate reservoir elements
water el. 741' (3 ft above Normal)
Elastic contact between
dam & reservoir
reservoir & foundation
Reduce strength parameters
on Rock-dam contact
coh = 0 psi, fric 55 deg
Dynamic Analysis
Reduce stiffness of joints to 5.588e4 psi/in
Apply viscous boudaries to the sides
Perform Time History Analysis
Include quiet period at end of time history
The bulk modulus of the elements in the reservoir region was set to 2.90 x 105 psi. The
shear modulus was calculated using Poisson’s ratio of 0.495, i.e., a compressible fluid.
Damping Parameters and Discretization: The hydrodynamic interaction between the dam
and the reservoir is modeled using solid elements for the dam concrete and reservoir
water. The highly uneven surface topography in the upstream region resulted in complex
geometrical shape for reservoir region.
The dominant frequency range for the earthquake and dam response is between 0 and
5 Hz. The mass-proportional component of Rayleigh damping is employed. The fraction
of critical damping of 4.35% was obtained as operating at the center frequency of
3.43 Hz. The 3DEC zone (element) size was adjusted to ensure this frequency was
captured in the analysis.
STATIC ANALYSES
Initial Setup
Two reservoir elevations corresponding to usual loading case (738 ft) and the PMF
unusual loading case (745 ft) were considered. The loading due to reservoir
impoundment was simulated by activating the reservoir elements. A comparison of
results was made using the reservoir modeled with solid element and merely applying the
hydraulic loads as nodal forces. Similar response of the dam was observed when the
reservoir was simulated by applied hydraulic loads on the upstream face of the dam. For
the usual load condition with a joint friction angle of 55 degrees, the maximum computed
slip displacement at the right abutment contact was 0.06 in. When the friction angle was
reduced to 35 degrees, the maximum computed joint shear displacement of 0.129 in. is
observed at the dam foundation contact as shown in Figure 5. The shear displacement is
concentrated on the right side of the mid-cantilever, i.e., where the contact is adversely
sloped downstream. It is noted that Figure 5 does not show the dam elements but rather
the geometry as input to 3DEC. The model contains more than 200,000 elements.
Wedge Analysis
The static numerical analyses carried out showed that the right abutment wedge at the
Cushman Dam 1 is stable with friction angles of reduced to 25 degrees on the
discontinuities within the rock mass. The global displacement vector pattern within dam
did not alter significantly until the friction angle was reduced to 25 degrees. However,
even for a friction angle of 25 degrees, the maximum displacements after the reservoir
impounding are less than 0.3 in. within the dam and less within the foundation.
Therefore, for both usual and unusual loading the factor of safety against right abutment
wedge failure is greater than 3 [i.e., tan (55)/tan (25) is greater than 3]. A similar
conclusion on the wedge stability was reached using a manual stability analysis.
Figure 5. Shear Displacement at the Dam-Foundation Contact Surface.
DYNAMIC ANALYSES
The dynamic loading of the dam represents the extreme loading condition for the dam
stability assessment. Two controlling earthquakes ground motions for the Cushman dam
sites are the Cascadia (inter-plate) and the Juan de Fuca (intra-plate) (Abrahamson,
2001). The analyses were performed with both the records. The results for the Juna-de-
Fuca are discussed in this paper, although the Cascadia record gave similar results.
Seismic Record
The peak accelerations in the horizontal plane are 0.49 g. The peak acceleration in the
vertical plane is 0.28 g. In the analysis, the three components of the earthquake are
applied simultaneously. A quiet period of five seconds was used to check that the
relative movement to joints had stopped after the earthquake. The input acceleration
record was integrated to obtain the velocities. The velocities were used as the primary
seismic input for the 3DEC analysis.
Results of Dynamic Analysis
The response of an arch dam to an earthquake loading is influenced to a great extent by
seismic input characteristics and the physico-mechanical properties of the intact concrete,
rock blocks, and the discontinuities within them. The static state is an initial condition
for the dynamic analysis.
The relative displacements of the crown cantilever were found and snapshots of stresses
on the dam at the critical times were studied to examine the stresses within the dam body
and shearing at the dam-foundation contact. The state of stress was also examined at the
time of maximum opening of the contraction joints within the dam. It was found that
these various snapshots, of stress, the stresses remained within acceptable limits.
3D stress vector plot on the dam surface, before and after the seismic shaking on the
downstream face are shown in Figures 6 and 7 respectively. It can be noted that the
principal stress vectors are oriented normal to the foundation surface and are parallel to
the arch direction at the top of the dam. Comparison of stress vectors before and after the
seismic shaking shows that significant stress redistribution within the concrete arch dam
takes place as a result of the seismic shaking. For example, as slip displacements occur at
the right abutment, then stresses are transferred to the left abutment by arch action.
Figure 6. Stress Distribution on Downstream Face Before Earthquake
Figure 7. Stress Distribution on Downstream Face After Earthquake
The non-linear analysis performed with 3DEC offers an interesting insight into various
failure mechanisms that can develop as the blocks slide, slip and open during static and
dynamic loading. The 3DEC results were used in the DISPLAY program to plot
deformed shapes. Figure 8 shows the exaggerated deformed shapes of the dam. It should
be noted that the displacements are greatly exaggerated; otherwise it would be difficult to
see the movements. Figure 9 is a zoomed view of the dam blocks on the right side.
Relative separation and rotation of the dam blocks can be easily seen in these figures.
From these figures the relative movement of various dam blocks can be observed.
Figure 8. Exaggerated Deformed Shape of the Dam
at the End of Earthquake Record
Figure 9. Zoomed of Exaggerated Deformed Shape of the
Dam at the End of Earthquake Record
The maximum computed shear slip displacement was in the range of 0.7 to 1.3 in. A
post-earthquake analysis was undertaken with the dam in its displaced configuration and
the uplift was increased to full reservoir head on the upstream half of the contact. A
sensitivity analysis of frictional strength showed the dam remains stable in its post-
earthquake condition.
It is interesting to note that the dam was less sensitive to reduced frictional strength in its
post-earthquake deformed condition compared to results from its pre-earthquake
sensitivity analysis.
CONCLUSIONS
The following conclusions are drawn
• The dam is acceptably stable for the static, dynamic seismic and post-earthquake
loading conditions
• Under severe seismic loading, the dam will experience permanent movement on the
dam contraction and dam/foundation joints. The computed movements are
considered conservative because the dam shear keys are not modeled and the
cohesive strength of the rough dam/foundation contact is ignored in the analysis. The
magnitude of the movements is found to be acceptable.
• Modeling of movements on joints as shown herein, allows realistic assessment of
dams in static, dynamic and post-earthquake conditions.
REFERENCES
Abrahamson, N., 2001, Time Histories for Cushman Dam, Report to Steve Fischer,
January 14, 2001.
Coombs, H.A., 1972, The Skokomish River Projects, Cushman Dam No. 1 and Cushman
Dam No. 2, Engineering Geology in Washington, Volume 1, Washington Division of
Geology and Earth Resources Bulletin, 78, pp 311-316.
Dasgupta, B. and Lorig, L. J, 1995, Numerical Modeling of Underground Powerhouses in
India, in Proceedings of the International Workshop on Observational Method of
Construction of Large Underground Caverns in Difficult Ground Conditions, (8th ISRM
International Congress on Rock Mechanics, Tokyo, September 1995, pp 65-74,
S.Sakurai, Ed.
EMRC (Engineering Mechanics Research Corporation), DISPLAY III, Pre and Post
Processing Program, Version 7.0, February 1997, Troy, Michigan, USA.
Hamilton, D.H., 2001, Evaluation of the Engineer