the investigation of a concrete gravity dam in a narrow

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21st Century Dam Design

Advances and Adaptations

31st Annual USSD Conference

San Diego, California, April 11-15, 2011

On the CoverArtist's rendition of San Vicente Dam after completion of the dam raise project to increase local storage and provide

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3-D Nonlinear Analysis 189

THE INVESTIGATION OF A CONCRETE GRAVITY DAM IN A NARROW CANYON USING 3-D NONLINEAR ANALYSIS

Mike Knarr, P.E., S.E.1 Matthew Muto, Ph.D.2 Nicolas von Gersdorff 3 John Dong, Ph.D., P.E.4 Ziyad Duron, Ph.D.5 John Yen, P.E.6

ABSTRACT

A series of numerical and field analyses were performed on a gravity dam located in a steep, narrow canyon for the purpose of evaluating performance under extreme flood and earthquake loading. The dam exhibits both stream and cross-stream behavior which is not captured in traditional 2-D stability analyses. To accommodate this behavior, a complete 3-D finite element model of the dam was built and analyzed. The model of the dam, which included the foundation and the reservoir, was validated through the use of low-level field testing. This paper discusses the significant aspects and findings of the 3-D modeling and analysis. This study is part of a larger effort to develop risk-based performance criteria and fragility analyses for dams to address potential failure modes that lead to risk-based decisions for resource allocation and remedial action.

INTRODUCTION

Big Creek Dam No. 7 (Dam 7) is located on the San Joaquin River, about 50 miles northeast of Fresno, California. The dam is owned and operated by the Southern California Edison Company. The concrete gravity dam was completed in 1951 and is composed of 19 blocks. It has a maximum height of 250 feet and is 875 feet long. The dam is located in a steep and narrow canyon as shown in Figure 1. An important aspect of the dam is that the center and higher blocks of the dam are tied together with keyed and grouted joints. This interconnection is not continuous over the full height of the blocks, but extends up from the base of the blocks a part of the block height. This varies from about a quarter to a third of the block height. This connectivity is sufficient to have the tied blocks act in unison and provide load transfer capability between the higher blocks of the dam. The shorter blocks at the ends of the dam are not keyed and grouted, and, thus, will act more independently. However, another important aspect of the dam is that a large portion of the toe (downstream bottom edge) of the dam has been reinforced with toe

1 Principal Structural Engineer, Dam Safety, Southern California Edison, San Dimas, CA 917773, [email protected] 2 Technical Specialist, Civil Engineering, Southern California Edison, San Dimas, CA 91773, [email protected] 3 Structural Engineer, Dam Safety, Southern California Edison, San Dimas, CA 91773, [email protected] 4 Structural Engineer, Dam Safety, Southern California Edison, San Dimas, CA 91773, [email protected] 5 Professor, Department of Engineering, Harvey Mudd College, Claremont, CA 91711, [email protected] 6 Chief Engineer, Dam Safety, Southern California Edison, San Dimas, CA 91773, [email protected]

190 21st Century Dam Design Advances and Adaptations

blocks that extend horizontally into the walls of the rock canyon. These concrete toe blocks were actually pored integral with the dam blocks providing significant restraint to sliding of the dam. These toe blocks can be seen in Figure 1(b).

(a)

(b)

Figure 1. Big Creek Dam No.7, shown in (a) upstream photo

and (b) plan and upstream elevation drawings

Toe Blocks


The performance of the dam was previously evaluated using 2-D analysis techniques that indicated the dam would meet stability criteria under normal loading and during and following a maximum credible earthquake (MCE). However, the evaluation also indicated that some blocks at each end of the dam would not meet stability criteria under probable maximum flood (PMF) conditions. Distinguishing features of these blocks that contributed to this finding included the absence of grouted or key interfaces between blocks and the location over low sloping topographic areas in the canyon. These factors allow these blocks to move independently and predominately in the downstream direction which is why the stability criteria are not met. This is in contrast to the taller blocks near the center of the dam, where keyed interfaces and steeper sloping topographic areas contribute to coupled cross-stream and downstream movements, providing satisfactory stability margins against sliding. Further analyses using 3-D analysis techniques have recently been performed for Dam 7. Features in these analyses that were not previously considered include:

friction between blocks and along the dam-foundation interface, detailed modeling of existing foundation geometry along the dam-foundation

interface, detailed modeling of the steep and narrow canyon configuration, and improved reservoir modeling that captures observed dam-reservoir interactions.

The following discusses the details of the 3-D analyses, which includes a description of the numerical model, load cases considered, and significant findings.

3-D NONLINEAR FINITE ELEMENT MODEL To address the steep and narrow canyon effects and the interconnectivity of the blocks with their keyed and grouted joints and to develop a more realistic understanding of the response of Dam 7 to severe loading, a 3-D finite-element model of the dam-foundation-reservoir system was developed using a commercially available computer program (ABAQUS 2010). The model was developed from a combination of construction drawings, and model parameters were obtained from field investigations at the dam. Model Description The numerical (finite element) model for Dam 7 is shown in Figure 2. It is composed of 84,000 elements and has approximately 217,000 degrees of freedom. Contours of the dams foundation were recorded on construction drawings, as shown in Figure 3. These were used to create a detailed model of the interface between the dam and the foundation, as shown in Figure 4.


Figure 2. 3-D Finite-element model for Big Creek Dam 7.

Figure 3. Contours of the dam foundation assembled from construction drawings. View

is looking down at the top of the rock at the rock-concrete interface.

Dam Blocks Reservoir Non-Reflecting

Boundary

Non-Reflecting Boundary

Foundation Gate

Downstream toe

Upstream heel


Figure 4. Finite-element mesh for the section of the foundation immediately underlying

the dam. Dam Blocks: The blocks of the dam were modeled with eight-node brick elements (upper portion of each block) and with four-node tetrahedral elements (lower portion of each block) to capture the complex geometry of the dam-foundation interface. Contact surfaces were used to represent the interfaces at the vertical joints between blocks, and between the dam and the underlying foundation. Depending upon the parameters selected for these contact surfaces, fully open or fully closed joints could be represented between blocks, and where keyed and grouted joints were present in the dam, model nodes at these interfaces tied the adjacent surfaces together. Foundation: The portion of the foundation model immediately underlying the dam was constructed using four-node tetrahedral elements to capture the complex geometry of the dam-rock interface. Farther from this interface, eight-node brick elements were used, and the mesh size was increased to reduce the numerical problem size. The foundation model extends to a distance H (equal to the maximum height of the dam) below, downstream, and left and right of the dam, and extends upstream 2H from the face of the dam. The bottom and sides of the foundation were modeled as infinite non-reflecting boundaries to account for radiation damping present in the dam system. This provided a mechanism for energy absorption and wave propagation inside the foundation model. Reservoir: Dam-reservoir interaction effects have traditionally been modeled using lumped added masses to represent incompressible fluid against the upstream dam face. 3-D modeling of the reservoir allows geometry and compressibility effects to be accounted for in the model. For the analyses conducted at Dam No. 7, acoustic finite elements were used to represent the reservoir water, and parameters for the acoustic element representation were selected based on comparisons against observed hydrodynamic pressure response behavior.

Downstream toe

Upstream heel


Spillway Gates: Flow over the spillway of Dam 7 is controlled by four radial gates. For analysis of the dam, the gates have been modeled as planar, and the configuration of the supporting steel beams are simplified, as shown in Figure 5. For normal and earthquake loading, the gates are resting on the spillway sill. The gates are assumed to be open in the flood loading analyses. The simplified model for the gates is used to generate loads on the gate concrete piers at the trunnion locations.

Figure 5. Radial gate model for simplifying loading on the full dam model.

Numerical Model Considerations The analysis is performed using an explicit integration solution technique. Explicit solution techniques allow computation of system behavior subject to sudden, short duration loading events (as may exist early in the seismic event), where large deformations are suspected, and for models that incorporate complex contact conditions requiring small time steps for accurate integration procedures. Another advantage of the explicit solution approach is the ability to sequentially apply loading to the dam system. For the analysis at Dam No. 7, the following load sequences were followed:

Flood-type loading Earthquake-type loading Gravity Gravity Hydrostatic Hydrostatic Hydrostatic - PMF Dynamic MCE Time Histories Post-seismic

Viscous damping is not used in the model. Energy is dissipated through a combination of radiation damping from the non-reflecting boundaries of the foundation and frictional damping from the contact surfaces. Non-reflecting boundaries have been included at the bottom and sides of the foundation and contact surfaces have been included along the entire base of the dam at the concrete-rock interface and at all of the block vertical joints. These modeling aspects can not be represented by a single damping coefficient.

Trunnion location


Model Calibration Using Data from Field Testing Vibration tests were conducted on Dam 7 to capture the dynamic characteristics of the dam. In June of 2008, both ambient and forced vibration tests were performed on the dam (Scheulen 2008). During these tests, day-time temperatures at the site averaged between 100 F and 115 F. To investigate the role of temperature effects in the dynamic response, additional forced vibration testing was performed in March of 2010 (Jacques), with average day-time temperatures between 55 F and 65 F. Data obtained for the dam, the foundation, and the spillway gates during these tests was used to calibrate the finite element model. The elastic moduli of the concrete and foundation rock were adjusted so that the first three modes of a linear model provided reasonable agreement with the natural frequencies identified through field testing. The final model properties are listed in Table 1.

Table 1. Concrete and Foundation Material Properties Material Density

(pcf) Elastic Modulus

(psi) Poissons

Ratio Concrete 150 2.5106 0.2 Granite Foundation 165 5106 0.3

Reservoir: Reservoir modeling considerations were based on comparisons of predicted hydrodynamic pressure response from reservoir models that used lumped added masses, fluid elements, and acoustic elements against measured hydrodynamic pressures acquired during field tests at the site (Scheulen 2010). The results from these studies suggested the best match with observed behavior at the dam was achieved by modeling the reservoir with acoustic elements, and a sample comparison is shown in Figure 6 (Scheulen 2010). The comparison includes model results obtained using direct integration and modal superposition solution techniques against observed (experimental) behavior.


0 5 10 15 200

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Resp

onse

Mag

nitu

de (g

/Mlb

f)

Numerical and Experimental FRF ComparisonMonolith 7

DirectModalExperimental

0 5 10 15 200

50

100

150

200

Frequency (Hz)

Resp

onse

Pha

se (d

egre

es)

Numerical and Experimental FRF ComparisonMonolith 7

DirectModalExperimental

Figure 6. Magnitude (top) and phase (bottom) of the acceleration frequency response at Block 7.

Enhanced confidence in the ability of acoustic elements to capture dam-reservoir interaction effects was obtained from a closed-form solution of an accelerating wall into a reservoir. Developed by Housner (1954), the solution was compared against a numerical model representation (see Figure 7) of the same configuration in which acoustic elements were used to model the reservoir. The comparison of hydrodynamic pressure on the face of the accelerating wall is shown in Figure 8 and includes predicted pressure response from both an acoustic element and a fluid element representation of the reservoir. Figure 7 shows contours of dynamic pressure for the acoustic finite elements on a section cut through the middle of the model. The acoustic element pressure distribution is within 10% of the Housner closed-form solution and within 20% for the fluid solid element. Figure 9 shows the agreement between the Housner closedform solution and the response of the acoustic elements for a simple sinusoidal input motion to the dam-reservoir system.


The material properties used for the acoustic elements are the density of water and the bulk modulus, which were 62.4 pcf and 3.12105 psi, respectively, which are standard values for water.

Figure 7. Acoustic element pressure from simplified dam-reservoir system.

Figure 8. Comparison of Housner closed-form solution

with acoustic and solid elements.

Rigid boundary at sides & bottom

Rigid boundary at upstream face of dam

Non-reflecting boundary


Figure 9. Comparison of the Housner (1954) closed-form solution for pressures on the wall of a tank subjected to sinusoidal motion to the

solution using acoustic elements in the ABAQUS program.

MODEL RESPONSE TO LOADING CONDITIONS The response of the finite element model was studied under three different loading conditions: normal, flood, and seismic. These load conditions are described below, and this is followed by a discussion of the analysis results for each of the load conditions. Normal Loading The normal load case represents the typical operating condition of the dam, with reservoir level approximately 10.5 feet below the crest. Applied loads include gravity and hydrostatic forces from the impounded water and the tail water, which are applied as pressure on the submerged upstream and downstream surfaces. Uplift forces are also applied as pressures acting on the bottom of each dam block. Dam No. 7 has an extensive network of pressure relief drains, and the influence of these drains is considered in the application of these loads. The base of each block in the model is divided into four to five strips and the uplift is varied linearly along the stream direction from full head (reduced by the drain effectiveness at the upstream heel) to full tail water pressure at the downstream toe as shown in Figure 10.


Figure 10. Uplift pressure distribution used in 3-D model.

Drain effectiveness for the normal load case is assumed to be 50%, which is conservative compared to drain effectiveness values estimated from pressure measurements taken over a ten year period (1998-2008). The measured and assumed drainage effectiveness is shown in Figure 11.

Measured and Assumed Drainage Effectiveness of Dam 7

0%

20%

40%

60%

80%

100%

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Block Number

Dra

inag

e Ef

fect

iven

ess

Measured Drainage Effectiveness

Assumed Drainage Effectiveness

Block 19/No Drain at B19

Figure 11. Comparison of measured and assumed drain effectiveness under normal loading.

Flood Loading Flood loading conditions were based on a probable maximum flood height of 11 ft above the crest of the dam. Hydrostatic forces associated from impounded water, tail water, water overtopping the dam, nappe pressures, and uplift were considered. Uplift is applied with pressure varying linearly in the stream direction from full reservoir head to tail water pressure, modified by the drain effectiveness as shown in Figure 10 above. Two levels of drain effectiveness were considered - 0% effectiveness corresponding to maximum uplift conditions under the dam, and 20% effectiveness that allows some reduction in uplift conditions during PMF loading. Contours of uplift pressure for these two levels are shown in Figure 8.


Figure 8. Contours of uplift pressures on the base of the dam under PMF loading for 0%

and 20% drain effectiveness. Friction effects at the interfaces between blocks were evaluated by considering friction angles of zero and 35 degrees, and set at 35 degrees along the dam-foundation interface. Four load cases were considered based on the various combinations of drain effectiveness and contact surface friction parameters and are listed in Table 2.

Table 2. Four load cases for analysis Drain Effectiveness (%) Friction Angle (Deg) Case

Normal PMF Dam to Rock Block to Block 1 50 0 35 35 2 50 0 35 0 3 50 20 35 35 4 50 20 35 0

Seismic Loading Seismic loading for the analysis was based on the maximum credible earthquake (MCE). The MCE hazard spectrum for Dam 7 is shown in Figure 13. The seismic loading case includes hydrostatic forces from water impounded to the normal pool, which is at the top of the radial gates, uplift from normal loading, and dynamic loading represented by seismic time history records appropriate for the site. The corresponding 3-D time histories are applied along a section in the foundation above the infinite or radiation boundary to minimize artificially induced and unwanted boundary effects.

Uplift under PMF condition, 0% drain effectiveness

Uplift under PMF condition, 20% drain effectiveness

Units = psf


Figure 13. Response spectrum for the Dam 7 MCE event.

Drain effectiveness was assumed the same as under normal loading at 50% to provide added conservatism during seismic loading. Similar to flood loading, two conditions for friction between the block to block contact surfaces were analyzed - the first assumed a zero friction angle and the second a 35 degree friction angle. Contact surfaces between the dam and the foundation incorporated a friction angle of 35 degrees. Analysis Results Predicted Dam Performance under Normal Loading: The application of normal loading not only causes the blocks to slide and twist in the downstream direction, but also in the cross-canyon direction. This movement was calculated to be very slight, on the order of a couple hundredths of an inch. As expected, there is very little movement under normal loading, which is consistent with observations. Predicted Dam Performance under Flood Loading: The results of the analyses for all of the blocks in the dam for the four cases are shown in Figures 9 and 10. Block 14 is most affected by the flooding.


Dam Bottom Relative Displacement(U)with Friction between Blocks

0.00

0.05

0.10

0.15

0.20

0.25

0.30

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Block Number

Def

orm

atio

n (In

ches

)DE=0%/ Case 1

DE=20%/ Case 3

DE=drainage effectiveness

Figure 9. Sliding displacements under PMF loading at the upstream heel of each block for cases considering friction in the unbonded vertical joints.

Dam Bottom Relative Displacement(U)

No Friction between Blocks

0.00

0.05

0.10

0.15

0.20

0.25

0.30

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Block Number

Def

orm

atio

n (In

ches

) DE=0%/ Case 2

DE=20%/ Case 4

DE=drainage effectiveness

Figure 10. Sliding displacements under PMF loading at the upstream heel of each block for cases with frictionless behavior in the unbonded vertical joints.


Table 3 summarizes the displacements at Block 14 for the four flood loading cases analyzed.

Table 3. PMF Analysis Results Drain Effectiveness (%) Friction Angle (Deg) Case

Normal PMF Dam to Rock Block to Block Max U

(in) Block

1 50 0 35 35 0.26 14 2 50 0 35 0 0.22 14 3 50 20 35 35 0.23 14 4 50 20 35 0 0. 14

Block 14 exhibits the largest sliding displacements in all cases. It is estimated to slide about -inch downstream under the PMF loading. This particular block is just outside the keyed and grouted section of the dam and the underlying foundation has an adverse (downstream) slope. The block also does not have a toe stability block as there are for many of the other blocks. Under Case 1, where there is friction in the unbonded vertical joints in the dam, Block 14 is restrained by a combination of shear forces resulting from frictional contact with the foundation and neighboring blocks and normal forces caused by contact with the foundation at the downstream toe of the block, as shown in Figures 11(a) and (b). In Case 2, contacts between unbonded vertical joints are modeled as frictionless. Figure 12(a) shows that there are no longer frictional forces along the sides of the block. However, Block 14 twists slightly around its vertical axis, bringing the edge of the upstream face into contact with Blocks 13 and 15, which exerts normal contact forces that restrain sliding, as shown in Figure 12(b). This shows that the presence of the adjacent blocks, which are interacting through friction and twisting, is one of the keys to limiting the movement of the dam blocks during flood loading. The other major contributor to limiting movement is the horizontal dilatation or movement of the blocks down the canyon side slopes toward the river bottom. The steep, narrow canyon allows the blocks to creep in the cross-canyon direction and down slope toward the river bottom of the canyon if the loading is high enough to cause the blocks to slide and twist. This movement will tend to compress the vertical joints, which will increase friction and impact between the joints, and, thus, transfer transverse forces to the larger and interconnected blocks in the canyon bottom. The vertical joints between the nine central blocks, Blocks 5 to 13 inclusive, are keyed and grouted over their lower third, making it possible for these nine blocks to respond together.


Figure 11. Under PMF loading Case 1, Block 14 (other blocks hidden from view) is restrained by a combination of (a) frictional shear forces from sliding along the dam-rock interface and along the unbonded vertical joints with neighboring blocks and (b) normal forces at the downstream toe. Arrows represent resultant force vectors from contact at

element nodes.

Reaction forces

(a) Shear Forces

(b) Normal Forces


Figure 12. Under PMF loading Case 2, Block 14 (other blocks hidden from view) is no longer restrained by frictional contact with other blocks, as shown by (a) the resultant

shear forces, but is restrained by (b) normal contact forces caused by Block 14 rotating to contact Block 13. Arrows represent resultant force vectors from contact at element

nodes.

(b) Normal Forces

(a) Shear Forces

Reaction forces


Predicted Dam Performance under Seismic Loading: The results of the analyses for all of the blocks in the dam for Case 2 in Table 2 are shown in Figure 13.

Figure 13. Sliding displacements under MCE loading for each block.

Block 14 is affected significantly more than the other blocks by the seismic loading. It exhibits the largest sliding displacements in all cases. Table 4 summarizes the displacements at Block 14 for the four seismic loading cases analyzed.

Table 4. MCE Analysis Results Drain Effectiveness (%) Friction Angle (Deg) Case

Normal Dam to Rock Block to Block Max U

(in) Block

1 50 35 35 0.46 14 2 50 35 0 0.61 14 3 Measured 35 35 0.39 14 4 Measured 35 0 0.49 14

Block 14 is estimated to slide about 0.6 of an inch downstream under the MCE loading. As noted for flood loading, this particular block is just outside the keyed and grouted section of the dam and the underlying foundation has an adverse (downstream) slope, and the block also does not have a toe stability block as there are for many of the other blocks. Under seismic loading the blocks experience motion similar to that for flood loading, i.e. they interact in frictional contact and twisting and they tend to squeeze together from movement down slope toward the river bottom. More specifically, the torsional motion of the blocks provides restraint to neighboring blocks even without friction in the vertical joints. In addition to movement upstream and downstream during seismic shaking, the blocks also tend to vibrate and move into and away from one another in the cross-canyon direction. This motion can have a tendency to allow the blocks to act independently, depending on the direction of motion for each block. Blocks with adverse slopes would tend to be the worst affected by this motion. This is the case at Dam 7, where Block 14,


being on an adverse slope, tends to ratchet its way down slope with a movement of about 0.6 of an inch. Other blocks of the dam are generally on favorable slopes, i.e. sloped downward toward the reservoir from the downstream toe to the upstream heel. There is also a large part of the dam where the blocks are tied together (keyed and grouted) and move together. Blocks 3 and 4 and 15 through 19 are keyed in and also dont move. The motion of the blocks was tracked in the seismic dynamic analysis. Plots of the time histories of displacement of Block 9, which is a spillway block near the center and highest part of the dam, and Block 17, which is three blocks from the right side of the dam in a flatter base area, are shown in Figures 15 and 16, respectively. Time histories are plotted for the center and each end of the block at the heel of the dam. Block 9 exhibits very little torsion, i.e. the time histories track each other as shown in Figure 15, and Block 17 exhibits considerable torsion as shown in Figure 16, where there is separation between the time histories.

Figure 15. Time histories of displacement at the right (blue), center (red) and

left (green) of the upstream heel of blocks 9.

Figure 16. Time histories of displacement at the right (blue), center (red) and

left (green) of the upstream heel of blocks 17.

The analyses performed for this study for seismic loading indicate that the dam may experience some minor movements. These movements are considered minor and not


capable of causing instability or an uncontrolled rapid release of water in a post-earthquake condition.

SUMMARY The analysis of Dam 7 is an example of the use of advanced computational modeling and analysis techniques informed by field investigations and engineering judgment to present a realistic picture of the anticipated behavior of a large concrete dam under severe loading conditions.

REFERENCES

ABAQUS 2010. ABAQUS manual. Version 6.10. Dassault Systemes Simulia Corp. Scheulen, F., Ellis, E., Duron, Z., 2008, Experimental and finite element studies of the forced vibration response of Big Creek Dam No. 7, Report to Southern California Edison, Harvey Mudd College, Claremont, CA. Jacques, C., Lownsbery, K., McAfee, K., Powers, E., Smith, E., Duron, Z., 2010. Big Creek Dam No.7 supplemental field testing. Report to Southern California Edison, Harvey Mudd College, Claremont, CA. Scheulen, F., von Gersdorff, N., Duron, Z., Knarr, M., 2010. Numerical model validation for large concrete gravity dams, 2010 United States Society on Dams Conference. Housner, G.W., 1954. Earthquake pressures on fluid containers. Report EERL-1954-3, Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, CA. http://resolver.caltech.edu/CaltechEERL:1954.EERL.1954.003

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