earthquake analysis of arch dams anil k. chopra dsi: devlet su isleri ankara, turkey october 13,...

1

Earthquake Analysis of Arch Dams

Anil K. Chopra

DSI: Devlet Su IsleriAnkara, Turkey

October 13, 2011

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

3

Complex System Geometry

Three-dimensional system

Reservoir: unbounded in the upstream direction

Foundation: semi-unbounded domain

4

Dynamic Analysis Should Consider:

Dam-water interaction

Reservoir boundary absorption

Water compressibility

Dam-foundation rock interaction

Spatial variations in ground motion

5

Early Research at Berkeley

Six Ph.D. theses at U.C. Berkeley (1972-96)

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

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)

7

3D ANALYSIS OF DAM-WATER-FOUNDATION ROCK SYSTEM

8

Arch Dam-Water Foundation Rock System

9

EACD-3D-2008 Model

(a) Finite element model: Dam (b) Finite element model: Fluid Domain

SM10

SM06 SM08 SM07

SM01 SM05

SM02 SM04

SM03 54 60

Recorders Nodal points

Infinite

channel

(c) Boundary element mesh: dam-foundation rock interface

10

Foundation Dynamic Stiffness Matrix, Sf ()

Foundation idealization

Canyon cut in a viscoelastic half-space

Infinitely long canyon

Arbitrary but uniform cross-section of canyon

11

Infinitely Long CanyonArbitrary but Uniform Cross Section

12

Computation of Foundation Dynamic Stiffness Matrix, Sf ()

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

13

Foundation Dynamic Stiffness Matrix, Sf ()

Defined for DOFs in finite element idealization of dam at dam-foundation interface, I

ˆˆSf r R

excitation frequency

R interaction forcest

r interaction displacementst

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

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

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)

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

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)

20

Hoover Dam221-meter high, curved gravity dam

21

Evaluation of Hoover Dam

Stresses computed by state-of-the-art finite element analysis

Dam will crack through the thickness

Did not seem credible to Reclamation engineers

2204 lb/in2 (15196 kPa)

22


23

Hoover Dam: Cross Section

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

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

26

Deadwood Dam50-meter high, single curvature dam

27

Monticello Dam93-meter high, single curvature dam

28

Morrow Point Dam142-meter high, double curvature dam

29


30

Hoover Dam

Dam-foundation interactionMassless foundation rock

(flexibility only)

758 lb/in2 (5226 kPa) 2204 lb/in2 (15196 kPa)

31

Deadwood Dam

Massless foundation rock (flexibility only)

Dam-foundation interaction

476 lb/in2 (3282 kPa)

844 lb/in2 (5819 kPa)

32

Monticello Dam

Massless foundation rock (flexibility only)

Dam-foundation interaction 730 lb/in2 (5033 kPa)

1410 lb/in2 (9722 kPa)

33

Morrow Point Dam

Dam-foundation interactionMassless foundation rock

(flexibility only)

665 lb/in2 (4585 kPa) 1336 lb/in2 (9211 kPa)

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

35

Monticello Dam

Water compressibility considered

Water compressibility neglected

1565 lb/in2 (10790 kPa)

1309 lb/in2 (9025 kPa)

36

Morrow Point Dam

Water compressibility considered Water compressibility neglected

1513 lb/in2 (10431 kPa) 2215 lb/in2 (15272 kPa)

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

38

COMPUTED VERSUS RECORDED RESPONSES

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

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

41

Mauvoisin Dam, Switzerland250 meters high

42

Mauvoisin Dam, SwitzerlandLocation of Recorders

43

Recorded Motions at Mauvoisin Dam Stream Direction

1996 Valpelline earthquake: Magnitude 4.6, 12 km away

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09

SM0FLocated 600 mdownstream

11.113.0

9.8 2.0

2.7

3.2

2.6

5.93.1

2.2

4.1

2.8

8 10 12 14 16 18 20-20

-10

0

10

20

Time, sec

Acc

eler

atio

n, c

m/s

2

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09


SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09


11.113.0

9.8 2.0

2.7

3.2

2.6

5.93.1

2.2

4.1

2.8

8 10 12 14 16 18 20-20

-10

0

10

20

Time, sec

Acc

eler

atio

n, c

m/s

2

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09


SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09


11.113.0

9.8 2.0

2.7

3.2

2.6

5.93.1

2.2

4.1

2.8

8 10 12 14 16 18 20-20

-10

0

10

20

Time, sec

Acc

eler

atio

n, c

m/s

2

11.113.0

9.8 2.0

2.7

3.2

2.6

5.93.1

2.2

4.1

2.8

8 10 12 14 16 18 20-20

-10

0

10

20

Time, sec

Acc

eler

atio

n, c

m/s

2

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09


SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09


11.113.0

9.8 2.0

2.7

3.2

2.6

5.93.1

2.2

4.1

2.8

8 10 12 14 16 18 20-20

-10

0

10

20

Time, sec

Acc

eler

atio

n, c

m/s

2

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09

SM05

SM08

SM04 SM03 SM02 SM01

SM07 SM06

SM11 SM10 SM09


11.113.0

9.8 2.0

2.7

3.2

2.6

5.93.1

2.2

4.1

2.8

8 10 12 14 16 18 20-20

-10

0

10

20

Time, sec

Acc

eler

atio

n, c

m/s

2

11.113.0

9.8 2.0

2.7

3.2

2.6

5.93.1

2.2

4.1

2.8

8 10 12 14 16 18 20-20

-10

0

10

20

Time, sec

Acc

eler

atio

n, c

m/s

2

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

45


Using measured 3% damping, response was overestimated

46


8% damping provided better match

How to justify 8% damping in model when measured value is 2-3%?

47


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

48

EACD-3D 2008 Model

(a) Finite element model: Dam (b) Finite element model: Fluid Domain

SM10

SM06 SM08 SM07

SM01 SM05

SM02 SM04

SM03 54 60

Recorders Nodal points


Infinite

channel

49

Selection of DampingBased on Frequency Response Functions

Damping: Dam 1%; Rock 3% 2% in overall system

Frequency, Hz

0

25

50

(a)

Acc

eler

atio

n A

mpl

itude

0 1 2 3 4 5 60

25

50

(b)

Freqency, Hz

Acc

eler

atio

n A

mpl

itude

0

25

50

(a)

Acc

eler

atio

n A

mpl

itud

e

0 1 2 3 4 5 60

25

50

(b)

Freqency, Hz

Acc

eler

atio

n A

mpl

itud

e

Acc

eler

atio

n A

mp

litu

de

Cross-Stream ResponseStream Response

50

Improved Agreement betweenComputed and Recorded ResponseWhen Foundation Inertia and Damping Included


-20

-10

0

10

20Computed: node 54

Acc

eler

atio

n, c

m/s

2

5 10 15 20 25-20

-10

0

10

20

Time, sec

Recorded: SM03

51



0 1 2 3 4 5 60

10

20 Computed: dam 1%, rock 3% Recorded: SM03

Freqency, Hz

Fo

uri

er A

mp

litu

de,

cm

/s2

52


8 10 12 14 16 18 20-0.4

-0.2

0.0

0.2

0.4 Computed RecordedVertical

Time, sec

-0.4

-0.2

0.0

0.2

0.4

Dis

pla

cem

ent,

mm

Cross-stream

-0.4

-0.2

0.0

0.2

0.4Stream

53

Pacoima Dam, California, USA113 meters high

54

Instrumentation at Pacoima Dam

CDMG Sensor Locations

55

Recorded Motions at Pacoima Dam2001 Earthquake, Stream Direction

1 2 3

9

1 2 7 6 8 1 5

160

115

1 29

34

4 5

43

24

4 3

13

4 5 6 7 8 9 10-10 0

-5 0

0

5 0

10 0

Time , sec

Acc

eler

atio

n,

cm/s

2

56

EACD-3D-2008 Model

211 84 21

47 88 25

Recorders

Nodal points

Infinite channel ∞

(a) Finite element model: dam (b) Finite element model: reservoir


57

Selection of DampingBased on Frequency Response Functions

Damping: Dam 2%; Rock 4% 6.2-6.6% in overall system

Frequency, Hz

Acc

eler

atio

n A

mp

litu

de

Second ModeFirst Mode0

5

10

(a)

0

5

10

(b)

Acceleration Amplitude

0 2 4 6 8 10 120

5

10(c)

Freqency, Hz

0

5

10

(a)

0

5

10

(b)

Acceleration Amplitude

0 2 4 6 8 10 120

5

10(c)

Freqency, Hz

58

Comparison of Computed and Recorded DisplacementsPacoima Dam, 2001 Earthquake

Dis

plac

emen

t, m

m

4 5 6 7 8 9 10-4

-2

0

2

4

crest left quarter pointChannel 5: Radial component at

-4

-2

0

2

4

crest centerChannel 4: Tangential component at

-4

-2

0

2

4

crest centerChannel 2: Radial component at

-4

-2

0

2

4

crest right third pointChannel 1: Radial component at Computed Recorded

Time, sec

59

SPATIAL VARIATIONS IN GROUND MOTION

60

Extended Analysis Procedure 2007-2008






EACD-3D-2008 computer program

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

62

Mauvoisin Dam: Spatial Variations in Interface Motions Are Small

SM05

SM08

SM01

SM06

SM10

2.0

4.1

2.6

3.1

2.8

8 10 12 14 16 18 20-5

0

5

Time, sec

Accele

rati

on

, cm

/s2

63

Quasi-Static Component Is Only a Small Part of Mauvoisin Dam Response

5 10 15 20 25-0.4

-0.2

0.0

0.2

0.4

Time, sec

Vertical Quasi-static Total response-0.4

-0.2

0.0

0.2

0.4

Dis

pla

cem

ent,

mm

-0.4

-0.2

0.0

0.2

0.4

Cross-stream

Stream

64

Spatial Variations in Ground MotionSmall Influence on Stresses in Mauvoisin Dam

Spatially-Uniform Excitation Spatially-Varying Excitation

Arch stressses on upstream face in kPa

65

Pacoima Dam: Spatial Variations in Interface Motions Are LargeNorthridge Earthquake, 1994

Missing segments estimated by Alves & Hall (2004)

8 8 2

7 3 5

4 2 9

0 1 2 3 4 5 6 7 8 9 1 0 1 1-1 000

-500

0

500

1 000

T im e , s ec

Acc

eler

atio

n,

cm/s

2

9

12 15

F IG U R E 4 .6 A C C E L E R A T IO N S (C M /S E C 2 ) G E N E R A T E D B Y A L V E S [ 2 0 0 4 ] IN S T R E A M D IR E C T IO N A T C H A N N E L S 9 , 1 2 A N D 1 5 T O R E P R E S E N T M O T IO N S

D U R IN G N O R T H R ID G E E A R T H Q U A K E .

66

Quasi-Static Component Dominates Pacoima Dam Response

0 1 2 3 4 5 6 7 8 9 10 11-8

-4

0

4

8

Time, sec

Quasi-static Total responseVertical-8

-4

0

4

8

Dis

pla

cem

ent,

cm -8

-4

0

4

8

Tangential

Radial

67

Spatial Variations in Ground MotionMajor Influence on Stresses in Pacoima Dam during 1994 Earthquake

Spatially-Uniform: Base Spatially-Varying Excitation

Arch stressses on upstream face in MPa

68

Pacoima Dam, California, USA113 meters high

69

Pacoima Dam, Cracking Visible

70

Applications to Evaluation and Remediation of Existing Dams

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

72

Deadwood Dam

50-meter high, single curvature dam

73

Seismic Upgrading of Deadwood Dam

EACD-3D-96 analysis including dam-water-foundation 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

74

Stewart Mountain Dam

Concrete arch damBuilt 1928 to 1930Height : 207 ft

63 m

Crest width: 8 feet2.4 m

Base width: 33 feet10 m

Arizona

75

Problems:Concrete placed very wet

- Segregated concreteNo lift line cleanup- Unbonded lift lines (16 of 23 unbonded)Alkali-aggregate reaction

- Crest expanded 6-inches (15 cm) upstreamEarthquake shaking

- Generates 2.6 g at dam crest- Concrete blocks move upstream

76

Seismic Upgrading of Stewart Mountain Dam

62 post-tensioned anchors10-ft spacing Dam passes flood

77

Special Drilling and Surveying Required Because of Thin Arch Dam

Crest 8-feet (2.4 m) thick

Base 34-feet (10.4 m) thick

Cables as close as possible to neutral axis

Maximum height:212 ft(65 m)

78

Seismic Upgrading of Stewart Mountain Dam

Earthquake analyses assumed massless foundation rock (1994)

62 post-tensioned anchors @10 ft Cost: US $6.8 M Lower cost if analyses included

dam-water-foundation rock interaction

79

Pardee Dam, California345 ft high

80

Englebright Dam, California280 ft high

81

East Canyon Dam, Utah260 ft high

82

Valdecanas Dam, Spain

332 ft high

83

CLOSURE

84







85

Slow Adoption in Engineering Practice

Most analytical advances to include dam-water-foundation rock interaction were reported Over 20 years ago for gravity dams Over 10 years ago for arch dams

2007-2008: Extended to include spatial variations in ground motion

EACD-3D-2008 computer program for linear analysis

User-friendly software is needed

86







87

Bureau of Reclamation

Nonlinear Analysis of Dams

5466

feet

1744

feet1817

feet1905

feet

2391

feet

13

30

fee

t

92

5

fee

t

5466

feet

1744

feet1817

feet1905

feet

2391

feet

13

30

fee

t

92

5

fee

t

If radiation boundary is simple, large FE model is necessary to simulate semi-unbounded domains and dam-water-foundation rock interaction.

88

LS-DYNA Finite Element Model

Bureau of Reclamation

Finite Elements: Dam = 12,000; Foundation = 92,000; and Water = 38,000

89

Nonlinear Analysis of Dams

Recently developed PML boundary drastically reduces size of model, now implemented inLS-DYNA

5466

feet

1744

feet1817

feet1905

feet

2391

feet

13

30

fee

t

92

5

fee

t

5466

feet

1744

feet1817

feet1905

feet

2391

feet

13

30

fee

t

92

5

fee

t

earthquake analysis of arch dams anil k. chopra dsi: devlet su isleri ankara, turkey october 13,...

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