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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


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