REALISTIC DYNAMIC ANALYSIS OF REALISTIC DYNAMIC ANALYSIS OF JOINTED ROCK SLOPES USING DDAJOINTED ROCK SLOPES USING DDA

Yossef H. HatzorDept. of Geological and Environmental Sciences, BGU

Funded by National Park and Nature Authority, Israel

Geological and Seismological Setting…

Seismicity along the Dead Sea Rift System

4.0 < ML < 4.95.0 < ML < 5.96.0 < ML

Topographic Site Effect in Masada

Spectral amplification of ground motions (Zaslavsky and Shapira, 2000)

Station 1Station 2

..

.Station 3

Typical Failure Modes

King Herod Palace: combined sliding (East face) and toppling (West Face ) leading to deterioration of entire slope segments

Snake Path Cliff: sliding, and “block slumping”, of large, removable key blocks + time dependentpropagation of tensile cracks

Rock Mechanics Considerations…

Mechanical Properties of Intact Rock

0

60

120

180

240

300

0 0.1 0.2 0.3 0.4 0.5Axial Strain (%)

Axi

al S

tress

(MP

a)

0

60

120

180

240

300

0 0.1 0.2 0.3 0.4 0.5Axial Strain (%)

Axi

al S

tress

(MP

a)

•Uniaxial Compressive Strength of intact Dolomite σc > 300 MPa;

•Elastic Modulus E = 43 GPa

•Poisson’s Ratio ν = 0.18

Shear Strength of Bedding Planes

An example of a natural, filled, bedding planeIn Masada dolomites

φ saw-cut = 28o

φ residual = 23o

Determination of residual friction angle using tilt testson “saw cut”, and “ground”surfaces

Triaxial Tests of Inclined, Filled, Saw-Cut

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6

Axial strain (%)

Stre

ss D

iffer

ence

(MPa

)

)2.25.1

6.3

11.7

13

σ3 = 16.2 MPa

8.9

c = 0 MPaφ = 22.7o

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6

Axial strain (%)

Stre

ss D

iffer

ence

(MPa

)

)2.25.1

6.3

11.7

13

σ3 = 16.2 MPa

8.9

0

5

10

15

20

25

0 0.1 0.2 0.3 0.40

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6

Axial strain (%)

Stre

ss D

iffer

ence

(MPa

)

)2.25.1

6.3

11.7

13

σ3 = 16.2 MPa

8.9

c = 0 MPaφ = 22.7o

BB

Direct Shear Tests of Natural Bedding Planes

Direct Shear Tests for Rough Bedding Planes

0 0.4 0.8 1.2 1.6Shear Displacement (mm)

0

0.4

0.8

1.2

She

ar S

tress

(MP

a)

Normal Stress 1.38 MPa

1.2 MPa

1.03 MPa0.86 MPa0.7 MPa0.52 MPa

0.35 MPa

0.17 MPa

φ peak=41o

φ residual= 23o

0 5 10 15 20 25Normal Stress (MPa)

0

4

8

12

Shea

r Stre

ss (M

Pa)

Direct Shear of Natural Bedding PlanesTriaxial Shear of Filled Saw-cut

Summary: Mechanical Properties of Rock Mass

••Strength of intact rock > Strength of intact rock > 300 MPa300 MPa

••Elastic modulus = Elastic modulus = 43 43 GPaGPa

••PoissonPoisson’’s ratio = s ratio = 0.180.18

••Peak friction angle of bedding planes = Peak friction angle of bedding planes = 4141oo

••Residual friction angle of bedding planes = Residual friction angle of bedding planes = 2323oo

Measured keyblock Displacements…

Block 2Block 1

LVDT based “Joint Meter”

Joint Displacement Monitoring Devices

Flat Jack

Block 3 – the “control” block

Joint Meter Output - Block 3

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

14 / 1

/ 98

18 / 1

/ 98

23 / 1

/ 98

27 / 1

/ 98

31 / 1

/ 98

4 / 2

/ 98

8 / 2

/ 98

12 / 2

/ 98

16 / 2

/ 98

21 / 2

/ 98

25 / 2

/ 98

1 / 3

/ 98

5 / 3

/ 98

9 / 3

/ 98

13 / 3

/ 98

18 / 3

/ 98

22 / 3

/ 98

28 / 3

/ 98

02/04

/9814

/ 4 / 9

827

/ 4 / 9

8 9

/ 5 / 9

8 21

/ 5 / 9

8 3

/ 6 / 9

8 15

/ 6 / 9

8 28

/ 6 / 9

8

Date

Rel

ativ

e D

ispl

acem

ent

0

50

100

150

200

250

Flat

jack

Pre

ssur

e

FJ 3

JM 10

JM 11

JM = Joint Meter(LVDT) FJ = Flat Jack

JM10

JM11

FJ3

Joint meter output - Blocks 1,2,3

Gulf of Eilat Seismicity???April 7, 98: M=4.4

April 10, 98: M = 4.3

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

14 / 1

/ 98

18 / 1

/ 98

23 / 1

/ 98

27 / 1

/ 98

31 / 1

/ 98

4 / 2

/ 98

8 / 2

/ 98

12 / 2

/ 98

16 / 2

/ 98

21 / 2

/ 98

25 / 2

/ 98

1 / 3

/ 98

5 / 3

/ 98

9 / 3

/ 98

13 / 3

/ 98

18 / 3

/ 98

22 / 3

/ 98

28 / 3

/ 98

02/04

/9814

/ 4 / 9

827

/ 4 / 9

8 9

/ 5 / 9

8 21

/ 5 / 9

8 3

/ 6 / 9

8 15

/ 6 / 9

8 28

/ 6 / 9

8 12

/ 7 / 9

8

Date

Rel

ativ

e D

ispl

acem

ent

Bedrock

Block 1

Block 3

Block 2

Influence of climatic fluctuations– Block 3

Summary: Joint Monitoring Program

•• The monitored The monitored keyblockskeyblocks exhibit dynamic exhibit dynamic behaviorbehavior

•• Periodic Periodic keyblockkeyblock displacements with amplitude displacements with amplitude of up to 1mm and period of 6 months of up to 1mm and period of 6 months –– most likely most likely climatically controlledclimatically controlled

••High High keyblockkeyblock sensitivity sensitivity –– most likely due to most likely due to limiting state of static equilibriumlimiting state of static equilibrium

Limit Equilibrium Analysis…

Representative Keyblock in the East Face (Block 1)

ψ

H = 15mV

W

U

bA A’

J1J3

J2

A

A’

Plan view of Block 1

N

5 m

α=2oo

DDATUM, Zw = 0

Influence of Water Head in “Tension Crack” on Factor of Safety Against Sliding and Rotation – Block 1

φ φ availableavailable = 22.7= 22.7oo

True Geometry of Block 1

Elements of 3D Graphical Solution

Static Solution Pseudo - Static Solution

Total Required Support Force (Ton) for Block 1

0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7PEAK HORIZONTAL GROUND ACCELERATION (g)

SU

PPO

RT

FOR

CE

Dry Case

Saturated Case

FACTOR OF SAFETY = 1 . 5

Summary:Limit Equilibrium Analysis

•• Fully three dimensional analysis is required when Fully three dimensional analysis is required when stability of individual, prismatic, stability of individual, prismatic, keyblockskeyblocks is of is of concern.concern.

•• Dominant failure mode in east slope of mountain Dominant failure mode in east slope of mountain is block slidingis block sliding

•• Relatively small water head will induce sliding of Relatively small water head will induce sliding of critical critical keyblockskeyblocks

Support Installation…

Support installation in progress Preparations for support installation

Horizontal Drilling

The Final “Product” in the Snake Path Cliff

Fully Dynamic DDA…

Upper Terrace of King Herod’s Palace

Joint Trace Map Generation…

Joint Spacing and Orientation – North Palace

0102030405060

0.15

2.78

5.42

8.05

10.69 More

Joint Length (m)

Freq

uenc

y

N = 100Mean = 2.7m

05

10152025

0.15 0.61 1.08 1.54

Bed Spacing (m)

Freq

uenc

y N = 59Mean = 60cm

0

10

20

30

2.82 11.11 19.41 27.70 MoreJ2 Spacing (cm)

Freq

uenc

y

N = 80Mean = 14 cm

0

5

10

15

3.00 10.93 18.87 26.80 More

J3 Spacing (cm)

Freq

uenc

y N = 69Mean = 16.8 cm

J1 (bedding)

J2

J3

A Synthetic Joint Trace Map Model

A Photo-geological Discontinuity Trace Map

- E - - W -

DC Realization of Digitized Photo-geological Map

- W -- E -

Discontinuous Deformation Analysis (Shi, 1993)

• A Discrete Element Method (Completely Discontinuous)

• A Rigorous Contact Detection Algorithm

• Normal and Shear Springs at Contact Points Between Blocks

• No Tension / No Penetration Between Blocks

• Sliding Deformation Modeled Using Coulomb’s Law

• Solution for Displacement Obtained by Minimization of Total Potential Energy (as in FEM).

• Solution Progresses by a Time-Step Marching Scheme

• First Order Displacement Approximation

• Simply Deformable Blocks

DDA Validation…

Block on an Incline:Gravitational Loading - Constant Acceleration

(MacLaughlin and Sitar, 1997)

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5TIME (s)

DIS

PLA

CEM

ENT

(m)

φ=5

DDA, g1=0.002φ=10

DDA, g1=0.005φ=15

DDA, g1=0.05φ=20

DDA, g1=0.1

Block on an Incline – Dynamic Loading(Hatzor and Feintuch, 2001)

0 1 2 3 4 5

Time (sec)-5

0

5

10

15

20

a (m

/s2 )

, v (m

/s),

s (m

)

AccelerationVelocityDisplacementDDA displacement

a(t) = 2sint + 3sin2t

Shaking Table Experiments (Wartman, 1999)

Shaking Table

11.37o

45 in

48 in

Inclined Plane

Rigid Block

1

2

34

5

No. Instrument Direction of Measurement 1 accelerometer parallel to plane 2 accelerometer parallel to plane 3 accelerometer horizontal 4 displacement transducer horizontal 5 displacement transducer parallel to plane

Comparison Between Shaking Table and DDA (Tsesarsky, Hatzor, and Sitar 2002)

-0.3

-0.2

-0.1

0

0.1

0.2

Acc

eler

atio

n (g

)

0 1 2 3 4 5Time (sec)

0

0.02

0.04

0.06

0.08

0.1

Dis

plac

emen

t (m

m)

ay(φ = 16o) = 0.08g

ty(φ = 16o) 0.64sec

Measuredk01 = 1k01 = 0.985k01 = 0.98k01 = 0.975K01 = 0.95

Summary:Validation Study•• Good agreement between numeric and analytical Good agreement between numeric and analytical solutions solutions -- provided that the control parameters provided that the control parameters (g01, g02) are optimized in constant acceleration (g01, g02) are optimized in constant acceleration cases.cases.

•• Similar results are found for time dependent Similar results are found for time dependent acceleration.acceleration.

•• When comparing time dependent acceleration When comparing time dependent acceleration analysis with physical models (e.g. shaking table) analysis with physical models (e.g. shaking table) some velocity damping must be introduced in order some velocity damping must be introduced in order to get good agreement. For a single block problem to get good agreement. For a single block problem the optimal value of velocity damping seems to be the optimal value of velocity damping seems to be not greater than 2%.not greater than 2%.

The Real Case…

The Nueiba 7.1 Earthquake of Nov. 1995Fill Response (Eilat)

0 10 20 30 40 50 60Time (sec)

-0.08

-0.04

0

0.04

0.08

0.12

Accl

. (g)

-0.08

-0.04

0

0.04

0.08

0.12

Accl

. (g)

Vertical

E - W

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

Acc

l. (g

) N-S

De-convolution: Bed-Rock Response (Zaslavski, Shapira, and Arzi, 2000)

0 5 10 15 20 25 30 35 40 45 50 55 60Time (sec)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Acc

l. (g

)

E-W

-0.08

-0.04

0

0.04

0.08

Acc

l. (g

)

N-S

Convolution: Topographic Site Effect

0 5 10 15 20 25 30 35 40 45 50 55 60Time (sec)

-0.12

-0.08

-0.04

0

0.04

0.08

Acc

l. (g

)

-0.08

-0.04

0

0.04

0.08A

ccl.

(g)

N-S

E-W

Predicted Damage by DDA

No Energy Dissipation2.5% Kinetic Damping5% Kinetic Damping

Rock Bolt Reinforcement:Modified Record Normalized to a 0.6g PGA

Dense Bolting Pattern: L = 6m, s = 2mSparse Bolting Pattern: L = 6m, s = 4m

Conclusions•• The DDA method can be used to perform a fully The DDA method can be used to perform a fully dynamic analysis of a jointed rock slope.dynamic analysis of a jointed rock slope.

•• Historical evidence indicate that the modeled slope has Historical evidence indicate that the modeled slope has sustained at least 4 episodes of PGA = 0.2g earthquakes sustained at least 4 episodes of PGA = 0.2g earthquakes in the past.in the past.

•• This historical evidence is confirmed by DDA.This historical evidence is confirmed by DDA.

•• DDA with no energy dissipation clearly over predicts DDA with no energy dissipation clearly over predicts damage due to shaking.damage due to shaking.

•• On the basis of historical performance we believe that On the basis of historical performance we believe that a reasonable damping value for a multi block problem in a reasonable damping value for a multi block problem in DDA should be in the order of 2% .DDA should be in the order of 2% .