Earthquake Analysis of Arch Dams
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1 Earthquake Analysis of Arch Dams Anil K. Chopra Electricity Generating Authority of Thailand Bangkok, Thailand December 7-8, 2010
2 Earthquake Analysis of Arch Dams: Factors To Be Considered Focus on linear analysis Before embarking on nonlinear analysis for any project, the best possible linear analysis should be implemented Comment on nonlinear analysis EGAT EGAT, Thailand Thailand ANIDIS
3 Complex System Geometry Three-dimensional system Reservoir: unbounded in the upstream direction Foundation: semiunbounded domain EGAT EGAT, Thailand Thailand ANIDIS
4 Dynamic Analysis Should Consider: Dam-water interaction Reservoir boundary absorption Water compressibility Dam-foundation rock interaction Spatial variations in ground motion EGAT EGAT, Thailand Thailand ANIDIS
5 Early Research at Berkeley Six Ph.D. theses at U.C. Berkeley ( ) Substructure method for linear systems Frequency domain method Implemented in computer programs distributed by NISEE EAGD-84: Gravity Dams, 1984 EACD-3D-96: Arch Dams, 1996 EGAT EGAT, Thailand Thailand ANIDIS
6 EACD-3D-96 Computer Program Considers 3D semi-unbounded geometry Dam-water interaction Reservoir-boundary absorption Water compressibility Dam-foundation rock interaction Foundation flexibility, inertia, and damping (material and radiation) EGAT EGAT, Thailand Thailand ANIDIS
7 3D ANALYSIS OF DAM-WATER-FOUNDATION ROCK SYSTEM 7
8 Arch Dam-Water Foundation Rock System 8
9 EACD-3D-2008 Model (a) Finite element model: Dam 54 SM03 60 SM02 SM04 SM01 SM05 (b) Finite element model: Fluid Domain Infinite channel SM08 SM07 SM06 Recorders Nodal points SM10 (c) Boundary element mesh: dam-foundation rock interface 9
10 Foundation Dynamic Stiffness Matrix, S f ( ) Foundation idealization Canyon cut in a viscoelastic half-space Infinitely long canyon Arbitrary but uniform cross-section of canyon EGAT EGAT, Thailand Thailand ANIDIS
11 Infinitely Long Canyon Arbitrary but Uniform Cross Section 11
12 Computation of Foundation Dynamic Stiffness Matrix, S f ( ) Direct boundary element procedure Full-space Green s function 3D boundary integral equation Analytical integration along canyon axis Infinite series of 2D problems Each 2D problem for one wave number Superpose solution of 2D problems EGAT EGAT, Thailand Thailand ANIDIS
13 Foundation Dynamic Stiffness Matrix, S f ( ) Defined for DOFs in finite element idealization of dam at dam-foundation interface, I S ˆ f rˆ R excitation frequency R r t interaction forces t interaction displacements EGAT EGAT, Thailand Thailand ANIDIS
14 Earthquake Analysis of Dams: Computer Programs EAGD-84 and EACD-3D-96 include all factors Developed before desktop computers Developed by graduate students Primarily research programs Applied to several actual projects EGAT EGAT, Thailand Thailand ANIDIS
15 Practical Applications of EACD-3D-96 Seismic safety evaluation of Englebright Dam, California, USA Valdecanas Dam, Spain Pardee Dam, California, USA Deadwood Dam, Idaho, USA Morrow Point Dam, Colorado, USA Monticello Dam, California, USA Hoover Dam, Nevada/Arizona, USA EGAT EGAT, Thailand Thailand ANIDIS
16 EACD-3D-96 Computer Program Considers 3D semi-unbounded geometry Dam-water interaction Reservoir-boundary absorption Water compressibility Dam-foundation rock interaction Foundation flexibility, inertia, and damping (material and radiation) 16
17 Popular Finite Element Techniques for Dams Ignore dam-water interaction and water compressibility Ignore wave absorption by sediments at reservoir boundary Assume foundation rock to be massless, i.e., consider only foundation rock flexibility EGAT EGAT, Thailand Thailand ANIDIS
18 BUREAU OF RECLAMATION PROGRAM TO EVALUATE EXISTING DAMS 18
19 Bureau of Reclamation Program to Evaluate Existing Dams Major program, started in 1996 Twelve dams were investigated, including: Hoover dam (221 meter-high curved gravity dam) EGAT EGAT, Thailand Thailand ANIDIS
20 Hoover Dam 221-meter high, curved gravity dam 20
21 Evaluation of Hoover Dam Stresses computed by state-of-the-art finite element analysis 2204 lb/in 2 (15196 kpa) Dam will crack through the thickness Did not seem credible to Reclamation engineers EGAT EGAT, Thailand Thailand ANIDIS
22 Hoover Dam 221-meter high, curved gravity dam 22
23 Hoover Dam: Cross Section 23
24 Bureau of Reclamation Program (1996- ) Found it necessary to consider: Dam-foundation rock interaction Dam-water interaction Water compressibility Reservoir boundary absorption Started using EACD-3D-96 computer program for linear analysis LS-DYNA for nonlinear analysis Realistic models based on field tests EGAT EGAT, Thailand Thailand ANIDIS
25 Reclamation Program To Evaluate Existing Dams Deadwood Dam, 50-meters high, single curvature Monticello Dam, 93-meters high, single cuvature Morrow Point Dam, 142-meters high, double curvature Hoover Dam, 221-meters high, thick arch Other dams EGAT EGAT, Thailand Thailand ANIDIS
26 Deadwood Dam 50-meter high, single curvature dam 26
27 Monticello Dam 93-meter high, single curvature dam 27
28 Morrow Point Dam 142-meter high, double curvature dam 28
29 Hoover Dam 221-meter high, curved gravity dam 29
30 Hoover Dam Dam-foundation interaction Massless foundation rock (flexibility only) 758 lb/in 2 (5226 kpa) 2204 lb/in 2 (15196 kpa) 30
31 Deadwood Dam Dam-foundation interaction 476 lb/in 2 (3282 kpa) Massless foundation rock (flexibility only) 844 lb/in 2 (5819 kpa) 31
32 Monticello Dam Dam-foundation interaction 730 lb/in 2 (5033 kpa) Massless foundation rock (flexibility only) 1410 lb/in 2 (9722 kpa) 32
33 Morrow Point Dam Dam-foundation interaction Massless foundation rock (flexibility only) 665 lb/in 2 (4585 kpa) 1336 lb/in 2 (9211 kpa) 33
34 Neglecting Foundation Rock Inertia and Damping Stresses are overestimated by a factor of 2 to 3 Such overestimation may lead to Overconservative designs of new dams Erroneous conclusion that an existing dam requires remediation. Analysis must include dam-foundation rock interaction Ignored in most practical analyses only rock flexibility is considered EGAT EGAT, Thailand Thailand ANIDIS
35 Monticello Dam Water compressibility considered 1565 lb/in 2 (10790 kpa) Water compressibility neglected 1309 lb/in 2 (9025 kpa) 35
36 Morrow Point Dam Water compressibility considered Water compressibility neglected 1513 lb/in 2 (10431 kpa) 2215 lb/in 2 (15272 kpa) 36
37 Neglecting Water Compressibility Stresses may be significantly Underestimated (e.g., Monticello Dam) Overestimated (e.g., Morrow Point Dam) Must include water compressibility Ignored in most practical analyses hydrodynamic effects approximated by added mass of water EGAT EGAT, Thailand Thailand ANIDIS
38 COMPUTED VERSUS RECORDED RESPONSES 38
39 Comparison of Computed and Recorded Responses Large disparity in results depending on numerical model used Important to calibrate numerical models against motions of dams recorded during: Forced vibration tests Earthquakes EGAT EGAT, Thailand Thailand ANIDIS
40 Forced Vibration Tests: Morrow Point Dam Bureau of Reclamation concluded: Massless foundation rock model far from matching measured response Including dam-foundation rock interaction (EACD-3D-96 model) reasonably matched measured response EGAT EGAT, Thailand Thailand ANIDIS
41 Mauvoisin Dam, Switzerland 250 meters high 41
42 Mauvoisin Dam, Switzerland Location of Recorders 42
43 Acceleration, cm/s 2 Recorded Motions at Mauvoisin Dam Stream Direction SM05 SM04 SM03 SM02 SM SM0F Located 600 m downstream SM08 SM SM Time, sec SM11 SM10 SM Valpelline earthquake: Magnitude 4.6, 12 km away 43
44 Analysis of Mauvoisin Dam: Massless Foundation (Proulx, Darbre, and Kamileris, 2004) Finite element model properties calibrated against ambient vibration test data Using measured 2-3% damping, response was overestimated 8% damping provided better match 15% damping required in model for Emosson Dam EGAT EGAT, Thailand Thailand ANIDIS
45 Analysis of Mauvoisin Dam: Massless Foundation (Proulx, Darbre, and Kamileris, 2004) Using measured 3% damping, response was overestimated EGAT EGAT, Thailand Thailand ANIDIS
46 Analysis of Mauvoisin Dam: Massless Foundation (Proulx, Darbre, and Kamileris, 2004) 8% damping provided better match How to justify 8% damping in model when measured value is 2-3%? 46
47 Analysis of Mauvoisin Dam: Massless Foundation (Proulx, Darbre, and Kamileris, 2004) Finite element model properties calibrated against ambient vibration test data Using measured 2-3% damping, response was overestimated 8% damping provided better match 15% damping required in model for Emosson Dam EGAT EGAT, Thailand Thailand ANIDIS
48 EACD-3D 2008 Model (a) Finite element model: Dam 54 SM03 60 SM02 SM04 SM01 SM05 (b) Finite element model: Fluid Domain Infinite channel SM08 SM07 SM06 Recorders Nodal points SM10 (c) Boundary element mesh: dam-foundation rock interface 48
49 mplitude Acceleration Amplitude Acceleration Amplitude Acceleration Amplitude Acceleration Amplitude Selection of Damping 50 Based on Frequency Response Functions (a) 25 Damping: Dam 1%; Rock 3% 2% in overall system Stream Response 0 Cross-Stream Response 50 (a) 50 (b) Frequency, (b) Hz Freqency, Hz 49
50 Improved Agreement between Computed and Recorded Response When Foundation Inertia and Damping Included Damping: Dam 1%; Rock 3% 2% in overall system Computed: node 54 Acceleration, cm/s Recorded: SM Time, sec 50
51 Improved Agreement between Computed and Recorded Response When Foundation Inertia and Damping Included Damping: Dam 1%; Rock 3% 2% in overall system 20 Computed: dam 1%, rock 3% Recorded: SM03 Fourier Amplitude, cm/s Freqency, Hz 51
52 Displacement, mm Improved Agreement between Computed and Recorded Response When Foundation Inertia and Damping Included Stream Cross-stream Vertical Computed Recorded Time, sec 52
53 Pacoima Dam, California, USA 113 meters high 53
54 Instrumentation at Pacoima Dam CDMG Sensor Locations 54
55 Recorded Motions at Pacoima Dam 2001 Earthquake, Stream Direction 55
56 EACD-3D-2008 Model 21 (a) Finite element model: dam (b) Finite element model: reservoir Infinite channel Recorders Nodal points (c) Boundary element mesh: dam-foundation rock interface 56
57 Acceleration Amplitude Selection of Damping Based on Frequency Response Functions Damping: Dam 2%; Rock 4% % in overall system 10 First Mode (b) 10 Second Mode (c) Frequency, Hz 57
58 Displacement, mm Comparison of Computed and Recorded Displacements Pacoima Dam, 2001 Earthquake Channel 1: Radial component at Computed Recorded crest right third point Channel 2: Radial component at crest center Channel 4: Tangential component at crest center Channel 5: Radial component at crest left quarter point Time, sec 58
59 SPATIAL VARIATIONS IN GROUND MOTION 59
60 Extended Analysis Procedure Spatial variations in ground motion Dam-water interaction Reservoir boundary absorption Water compressibility Dam-foundation rock interaction EACD-3D-2008 computer program EGAT EGAT, Thailand Thailand ANIDIS
61 Significance of Spatial Variations in Ground Motion Structural response split in two parts: Quasi-static component: due to static application of interface displacements at each time instant Dynamic component Key factor is significance of quasi-static component Depends on degree to which ground motion varies spatially EGAT EGAT, Thailand Thailand ANIDIS
62 Mauvoisin Dam: Spatial Variations in Interface Motions Are Small SM SM Acceleration, cm/s 2 5 SM SM Time, sec SM
63 Displacement, mm Quasi-Static Component Is Only a Small Part of Mauvoisin Dam Response Stream Cross-stream Vertical Quasi-static Total response Time, sec 63
64 Spatial Variations in Ground Motion Small Influence on Stresses in Mauvoisin Dam Arch stressses on upstream face in kpa Spatially-Uniform Excitation Spatially-Varying Excitation EGAT EGAT, Thailand Thailand ANIDIS
65 Pacoima Dam: Spatial Variations in Interface Motions Are Large Northridge Earthquake, 1994 Missing segments estimated by Alves & Hall (2004) 65
66 Displacement, cm Quasi-Static Component Dominates Pacoima Dam Response 8 4 Radial Tangential Vertical Quasi-static Total response Time, sec 66
67 Spatial Variations in Ground Motion Major Influence on Stresses in Pacoima Dam during 1994 Earthquake Arch stressses on upstream face in MPa Spatially-Uniform: Base Spatially-Varying Excitation EGAT EGAT, Thailand Thailand ANIDIS
68 Pacoima Dam, California, USA 113 meters high 68
69 Pacoima Dam, Cracking Visible 69
70 Applications to Evaluation and Remediation of Existing Dams 70
71 Seismic Evaluation of Existing Dams Geological and seismological investigations Probabilistic seismic hazard analysis Uniform Hazard Spectrum Ground motion selection and scaling Dynamic analysis Concrete testing: tensile strength Performance evaluation Remediation strategies EGAT EGAT, Thailand Thailand ANIDIS
72 Deadwood Dam 50-meter high, single curvature dam EGAT EGAT, Thailand Thailand ANIDIS
73 Seismic Upgrading of Deadwood Dam EACD-3D-96 analysis including dam-waterfoundation interaction (2001) Compute forces transmitted to foundation Stabilize 3 unstable foundation blocks 60 rock bolts Cost: US $1.0 M Higher cost if analyses assumed massless foundation rock EGAT EGAT, Thailand Thailand ANIDIS
74 Stewart Mountain Dam Concrete arch dam Built 1928 to 1930 Height : 207 ft 63 m Crest width: Base width: 8 feet 2.4 m 33 feet 10 m Arizona EGAT EGAT, Thailand Thailand ANIDIS
75 Problems: Concrete placed very wet - Segregated concrete No lift line cleanup - Unbonded lift lines (16 of 23 unbonded) Alkali-aggregate reaction -Crest expanded 6-inches (15 cm) upstream Earthquake shaking - Generates 2.6 g at dam crest - Concrete blocks move upstream EGAT EGAT, Thailand Thailand ANIDIS
76 Seismic Upgrading of Stewart Mountain Dam 62 post-tensioned anchors 10-ft spacing EGAT EGAT, Thailand Thailand ANIDIS 2009 Dam passes flood 76
77 Special Drilling and Surveying Required Because of Thin Arch Dam Crest 8-feet (2.4 m) thick Maximum height: 212 ft (65 m) Cables as close as possible to neutral axis EGAT EGAT, Thailand Thailand ANIDIS 2009 Base 34-feet (10.4 m) thick 77
78 Seismic Upgrading of Stewart Mountain Dam Earthquake analyses assumed massless foundation rock (1994) 62 post-tensioned ft Cost: US $6.8 M Lower cost if analyses included damwater-foundation rock interaction EGAT EGAT, Thailand Thailand ANIDIS
79 Pardee Dam, California 345 ft high 79
80 Englebright Dam, California 280 ft high 80
81 East Canyon Dam, Utah 260 ft high 81
82 Valdecanas Dam, Spain 332 ft high 82
83 CLOSURE 83
84 Dynamic Analysis Should Consider: Dam-water interaction Reservoir boundary absorption Water compressibility Dam-foundation rock interaction Spatial variations in ground motion EGAT EGAT, Thailand Thailand ANIDIS
85 Slow Adoption in Engineering Practice Most analytical advances to include dam-waterfoundation rock interaction were reported Over 20 years ago for gravity dams Over 10 years ago for arch dams : Extended to include spatial variations in ground motion EACD-3D-2008 computer program for linear analysis User-friendly software is needed EGAT EGAT, Thailand Thailand ANIDIS
86 Dynamic Analysis Should Consider: Dam-water interaction Reservoir boundary absorption Water compressibility Dam-foundation rock interaction Spatial variations in ground motion 86
87 Nonlinear Analysis of Dams If radiation boundary is simple, large FE model is necessary to simulate semiunbounded domains and dam-waterfoundation rock interaction feet 1905 feet 1817 feet 5466 feet feet feet 1744 feet Bureau of Reclamation 87
88 LS-DYNA Finite Element Model Finite Elements: Dam = 12,000; Foundation = 92,000; and Water = 38,000 Bureau of Reclamation EGAT EGAT, Thailand Thailand ANIDIS
89 Nonlinear Analysis of Dams Recently developed PML boundary drastically reduces size of model, now implemented in LS-DYNA 2391 feet 1905 feet 1817 feet 5466 feet feet feet 1744 feet 89
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