site response analysis with 2d-dda

Site Response Analysis with 2D-DDA

Yossef H. HatzorSam and Edna Lemkin Professor of Rock Mechanics

Dept. of Geological and Environmental Sciences

Ben-Gurion University of the Negev, Beer-Sheva, Israel

Talk Outline

• Dynamic displacement of discrete elements: review of some published DDA verifications and validations

– Dynamic sliding on a single plane

– Dynamic sliding of a wedge on two planes

– Block response to dynamic shaking of foundations

– Dynamic block rocking

• Accuracy of wave propagation modeling with DDA: recent results

– P wave propagation

– S wave propagation

– Site response analysis with DDA

• Case Study: The Western Wall Tunnels in Jerusalem – the significance of local site response

2Y. Hatzor: Site Response Analysis with DDA. 11th ICADD-11, August 27-29, 2013, Fukuoka, Japan

Dynamic Displacement of Discrete Elements with DDA: Verifications and Validations


Single Face Sliding

-5

0

5

10

Inp

ut

mot

ion

(m/s

2)

0

4

8

12D

ispl

acem

ent

of

uppe

r blo

ck (

m)

0 1 2 3 4 5 6

Time (sec)

Analytic

Analytic

Analytic

DDA

DDA

DDA

Input Motion

0.01

0.1

1

10

100

rela

tive

err

or (

%)

=22 0

=30 0

=35 0

Dynamic sliding under

gravitational load only was

studied originally by Mary

McLaughlin in her PhD thesis

(1996) (Berkeley) and

consequent publications with

Sitar and Doolin 2004 - 2006.

Sinusoidal input first studied by

Hatzor and Feintuch (2001),

IJRMMS. Improved 2D solution

presented by Kamai and Hatzor

(2008), NAG. Ning and Zhao

(2012), NAG (From NTU)

recently published a very

detailed study of this problem.


Double Face Sliding

Wedge parameters:

P1=52/063, P2=52/296

1=2=30o

x

y

z

DDA validation originally investigated

by Yeung M. R., Jiang Q. H., Sun N.,

(2003) IJRMMS using physical tests.

Analytical solution proposed and 3D

DDA validation performed by Bakun-

Mazor, Hatzor, and Glaser (2012),

NAG.

0

0.5

1

1.5

2

2.5

Dis

pla

cem

ent

(m)

0.1

1

10

100

Rel

ativ

e E

rro

r (%

)

0 0.4 0.8 1.2 1.6 2

Time (sec)

-6

-4

-2

0

2

4

6

Ho

rizo

nta

l In

put

mo

tio

n (

m/s

2)

0.1

1

10

100

Analytical

3D-DDA

Input Motion (y)

A


Shaking Table Experiments

0

20

40

60

Acc

um

ula

ted

Dis

pla

cem

ent,

mm

0 10 20 30 40Time, sec

10

100

1000

10000

Rel

ativ

e E

rror,

%

-0.2

0

0.2

Acc

eler

atio

n o

fS

hak

ing T

able

, g

Erel; Loading Mode

Erel; Displacement Mode

Shaking Table

3D DDA ; Loading mode

3D DDA ; Displacement mode

C

B

A

ti


Test_CY.MOV

Test_CY.MOV

Test_CY.MOV

Test_CY.MOV

Rate Dependent Friction

0 1 2 3 4 5Shear Displacement, mm

0

1

2

3

4

Shea

r S

tres

s, M

Pa

0 2 4 6Normal Stress, MPa

0

1

2

3

4

Shea

r S

tres

s, M

Pa

v = 0.002 mm/sec

v = 0.020 mm/sec

v = 0.100 mm/sec

=

=2

=22n = 0.98 MPa

n = 1.97 MPa

n = 3.00 MPa

n = 4.03 MPa

n = 5.02 MPa

A B

Upper Shear Box

Shear Cylinder

Lower Shear Box

Roller BearingConcrete Samples

20 cm

Shear System

Normal Cylinder


Observed Block “Run-out”

0

20

40

60

Up.

Blo

ckV

eloci

ty,

mm

/sec

0

20

40

60

80

100

Up. B

lock

Acc

um

. D

ispla

cem

ent,

mm

0 20 40 60Time, sec

-0.2

0

0.2

Acc

eler

atio

nof

Shak

ing

Tab

le,

g

Measured

Calculated

= 29.5o

= 29.0o

= 27.0o

A8Y. Hatzor: Site Response Analysis with DDA. 11th ICADD-11, August 27-29, 2013, Fukuoka, Japan

Friction Angle Degradation

0.0001 0.001 0.01 0.1 1 10 100Velocity, mm/sec

0.3

0.4

0.5

0.6

0.7

Fri

ctio

n C

oef

fici

ent

0.0001 0.001 0.01 0.1 1 10 100Velocity, mm/sec

0.3

0.4

0.5

0.6

0.7

Fri

ctio

n C

oeff

icie

nt

Shaking Table

Coulomb-Mohr

= -0.0079 * ln(V) + 0.6071

R2 = 0.909

= -0.0174 * ln(V) + 0.5668

R2 = 0.948

Direct shear test results

Shaking table experiments

Conclusion: frictional resistance of geological sliding interfaces may exhibit both velocity dependence as well as degradation as a function of velocity and/or displacement. This is particularly relevant for dynamic analysis of landslides, where sliding is assumed to have taken place under high velocities. Therefore, a modification of DDA to account for friction angle degradation is called for. This has already been suggested by Sitar et al. (2005), JGGE –ASCE; a new approach has recently been proposed by LZ Wang et al. (in press), COGE (from Zhejiang University).


Dynamic interaction of discrete blocks


The dynamic interaction between discrete blocks subjected to dynamic loads such as earthquake vibrations is of high importance in seismic risk studies both for preservation of historic monuments as well as geotechnical earthquake engineering design.

Kamai and Hatzor (2008), NAG, have suggested to use this approach to constrain the paleoseismic PGA in seismic regions by back analysis of stone displacements in historic masonry structures.

Several DDA research groups have began to explore this issue. Notably, Professor YuzoOhnishi’s DDA research group has recently made some important contributions to this field, e.g. Miki et al. (2010), IJCM; Sasaki et al. (2011), IJCM.

Direct accelerationinput simultaneously to all blocks: we call it

“QUAKE” mode

Y. Hatzor: Site Response Analysis with DDA. 11th ICADD-11, August 27-29, 2013, Fukuoka, Japan 11


Direct displacement input

to foundation block: we call it

“DISP” mode

Response of overriding block to cyclic motion of foundation block: DISP mode

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

time (sec)

D2

(m)

D=0.3m (1.2g) analytical DDA

D=0.5m (2g) analytical DDA

D=1m (4g) analytical DDA

Input motion: dt = D(1- cos (2pt)); f = 1Hz; = 0.6


For complete analytical solution see:

Kamai and Hatzor (2008), NAG.

Dynamic Block Rocking: QUAKE mode 2b

2h

R

c.m.

h

b

0’ 0

ü(t)

2b

2h

R

c.m.

h

b

0’ 0

ü(t)

Analytical solution proposed by:

Makris and Roussos (2000), Geotechnique.

DDA validation with applications:

Yagoda-Biran and Hatzor (2010), EESD.


apeak slightly lower than PGA required for toppling


apeak slightly higher than PGA required for toppling


Accuracy of Wave Propagation Modeling with DDA: Benchmark Tests and Field Investigations


DDA accuracy in simulations of P wave propagation: 1D elastic bar


Work in progress with Huirong Bao , Xin Huang, and Ravit Zelig

Measurement pointLoading point

1m

2m

0.5m

5m

50m 50m

1m

Input function for P wave and model properties


0.000 0.002 0.004 0.006 0.008 0.010

-1000

-500

0

500

1000

Lo

ad

(K

N)

Time (s)

Time history of input load Input parameters for block and joint materials

𝑭 𝒕 = 𝟏𝟎𝟎𝟎𝒔𝒊𝒏 𝟐𝟎𝟎𝝅𝒕 𝑲𝑵

Block material

Unit mass

(kg/m3)2650

Young’s modulus

(GPa)50

Poisson ratio 0.25

Joint material

Friction angle 35˚

Cohesion (MPa) 24

Tensile strength

(MPa)18

0.0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

Str

ess r

ela

tive

err

or

(%)

Time interval (ms)

5m

2m

1m

0.5m

Time interval effect on the accuracy of P wave stress

Y. Hatzor: Site Response Analysis with DDA. 11th ICADD-11, August 27-29, 2013, Fukuoka, Japan20

Note very significant effect of time interval on P wave stress accuracy, block size is much less important.

where A1 is the measured wave amplitude or calculated wave velocity at a reference measurement point in the model, and Ao is the incident wave amplitude or analytical wave velocity at a given point.

𝒆 =𝑨𝟏 − 𝑨𝟎𝑨𝟎

× 𝟏𝟎𝟎%

0.0 0.1 0.2 0.3 0.4 0.5-1

0

1

2

3

4

5

P-w

ave

ve

locity r

ela

tive

err

or

(%)

Time interval (ms)

5m

2m

1m

0.5m

Time interval effect on the accuracy of P wave velocity


Note complicated block size effect on P wave velocity; time interval is much less important. There seems to be an optimal block size below which the error increases!

Error decreases with decreasing block size

Error increases with decreasing block size

𝑽𝒑 =𝑬

𝝆𝟎

For the special case of a one dimensional bar (Kolsky, 1964):

Time interval effect on waveform accuracy


Block length = 1 mStress measured at mid section

0.01 0.02 0.03 0.04 0.05

-1.0

-0.5

0.0

0.5

1.0

Str

ess (

Mp

a)

Time (s)

Analytical

0.5 ms

0.1 ms

0.05 ms

0.01 ms

Wave form accuracy greatly improves with decreasing time step!

Contact stiffness (k) effect on P wave stress accuracy


10E 20E 40E 80E 160E 320E 640E

2

3

4

5

6

7

8

9

10

11S

tre

ss R

ela

tive

Err

or

(%)

Contact Stiffness

Error of Min Stress

Error of Max Stress

Error of Max-Min

Not much effect of kvalue on P wave stress

Block length = 1 mStress measured at mid section

Contact stiffness (k) effect on P wave velocity accuracy


Block length = 1m

10E 20E 40E 80E 160E 320E 640E

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5P

-wa

ve

ve

locity r

ela

tive

err

or

(%)

Contact stiffness

P-wave velocity relative error

There clearly exists an optimal k value beyond which the error begins to increase

Contact stiffness (k) effect on waveform accuracy


0.01 0.02 0.03 0.04 0.05

-1.0

-0.5

0.0

0.5

1.0

Str

ess (

MP

a)

Time (s)

Analytical

1E

5E

10E

40E

100E

Block length = 1m

Wave from accuracy greatly improves with increasing k value!

Relationship between wavelength and block size

Y. Hatzor: Site Response Analysis with DDA. 11th ICADD-11, August 27-29, 2013, Fukuoka, Japan26

x

=

The relationship between element (block) side length (x) and wave length (), has a strong influence on numerical accuracy. In FEM the optimal ratio η should be smaller than 1/12 (0.083) where:

We have performed a series of tests for various values of η and obtained the following results:

Δt (ms) Block length (m) 0.5 1 2 5 10

η (Δx/λ) 0.012 0.023 0.046 0.115 0.230

0.01Velocity (m/s) 4219.16 4310.34 4359.20 4436.36 4432.43

Error 2.87% 0.77% 0.36% 2.13% 2.04%

0.1Velocity (m/s) 4884.24 4699.91 4553.73 4493.57 4490.35

Error 12% 8% 5% 3% 3%

In our simulations:

T (s) 0.01v (m/s) 4343λ (m) 43.43

Influence of η on velocity accuracy


η < 1/12

We see again that the error decreases with decreasing block length, but when η is smaller than 1/22 the numeric error in fact increases! This corresponds to the result we obtained regarding the effect of block size.

Clearly accuracy improves with decreasing time interval!

η = 1/22

Influence of η on stress accuracy


Δt (ms) block length (m) 0.5 1 2 5 10

η (Δx/λ) 0.012 0.023 0.046 0.115 0.230

0.01Amplitude (KPa) 983.6 985.4 999.8 981.6 927.5

error 1.6% 1.5% 0.0% 1.8% 7.2%

0.1Amplitude (KPa) 884.6 887.3 888.2 870.3 835.3

error 11.5% 11.3% 11.2% 13.0% 16.5%

Great improvement in stress accuracy with decreasing block length down to a minimum at η = 1/22 below which the error increases for both time intervals studied.

Very significant accuracy improvement with decreasing time interval.

η = 1/22

The problem with adding joints artificially


Analytical model:

In the analytical model there is no additional stiffness from contact springs as in the DDA model, and the total stiffness of a block system of length L is:

b

bL

n

KK =

where nb is the number of blocks in specified length L.

In the DDA model, the total stiffness of the block system in length L is:

KL

=1

KL

+n

c

Kc

æ

ææ

æ

æ÷

-1

< KL

where nc is the number of contact springs in specified length L.

DDA model:

Artificially decreasing the size of blocks down to a certain value may increase stress accuracy, but below a value of η ≈ 1/22 errors both in stress and velocity will increase because of the inaccurate representation of the real stiffness of the system due the large number of contacts.

S wave propagation: DDA vs. SHAKE

15m

1m

x15=

15m

M1

M2

ug

block 1

block 15

1m

x15

=15m

ug

layer 1

layer 2

bedrock

layer 15

DDA Model SHAKE model

Bao, Yagoda-Biran, and Hatzor, Earthquake Engineering and Structural Dynamics (in press)


Input Ground Motions

0 10 20 30 40 50 60

-0.10

-0.05

0.00

0.05

0.10 displacement

dis

pla

ce

me

nt

(m)

time (s)

0 10 20 30 40 50 60-2

-1

0

1

2 acceleration

acce

lera

tio

n (

m/s

2)

time (s)

CHI-CHI 09/20/99, ALS, E (CWB): acceleration for SHAKE and displacement for DDA


Modeling Procedure

Time step size

Damping ratio

DDA

SHAKE

Damping ratio transfer from DDA into SHAKE utilizing DDA algorithmic damping (for details on the algorithmic in DDA See Doolin and Sitar 2004)

Modeled material parameters in layered model

Layer/BlockUnit

mass (kg/m3)

Young’s Modulus

(GPa)

Shear Modulus

(GPa)

Poisson ratio

1-3 2403 4.5 1.8 0.25

4-6 2162 4.1 1.64 0.25

7-9 2243 4.2 1.68 0.25

10-12 2483 4.0 1.6 0.25

13-15 2643 4.8 1.92 0.25


Spectral Amplification Ratio

time step size (s): 0.001

Total steps: 60000

Spring stiffness (N/m): 1.50E+12

Calculated damping ratio: 2.3%

0 5 10 15 20 250

5

10

15

20

25

30

Am

plif

ica

tio

n r

atio

Frequency (Hz)

DDA

SHAKE

DDA SHAKE

Natural frequency (Hz) 14.23 14.50

Max amplification 28.76 27.79

DDA numerical control parameters


2D Site Response: DDA vs. Field Test


Bao, Yagoda-Biran, and Hatzor, Earthquake Engineering and Structural Dynamics (in press)

“Static” Push and Release at top column

“Dynamic” blow with sledgehammer at column base

Results of Geophysical Field Measurements

aTypical vibrations of the Column top and base in the X and Y directions due to force excitation in the Y direction by horizontal stroke of sledgehammer at the base of the Column

The corresponding Fourier amplitude spectra for the top and base of the Column.


The DDA model of a multi-drum column

Parameter Value

Young's modulus 17 GPa

Poisson's ratio 0.22

Interface Friction 30o

Density 2250 kg/m3

Time step 0.01-0.001 sec.

Displacement ratio

0.01 – 0.001

k (penalty value) 2x107 - 2x107 N/m


DDA response to dynamic pulse of 10,000 N

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0 5 10 15 20 25 30

time (sec)

dis

pla

cem

ent

(cm

)


load

time

0

F

t1-t t1t1+t

load

time

0

F

t1-t t1t1+t

Dynamic response of uppermost block in time domain

Top column response to static push: DDA vs. Geophysical survey


1 2 3 4 5 6 7 8 9 100

0.01

frequency (Hz)

dis

pla

cem

en

t am

pli

tud

e (

cm

)

1 2 3 4 5 6 7 8 9 100

0.01

frequency (Hz)

dis

pla

cem

en

t am

pli

tud

e (

cm

)

0

500

1000

1500

velo

cit

y a

mp

litu

de (

m/s

)

DDA results

experimental

results

1 2 3 4 5 6 7 8 9 100

0.01

frequency (Hz)

dis

pla

cem

en

t am

pli

tud

e (

cm

)

1 2 3 4 5 6 7 8 9 100

0.01

frequency (Hz)

dis

pla

cem

en

t am

pli

tud

e (

cm

)

0

500

1000

1500

velo

cit

y a

mp

litu

de (

m/s

)

DDA results

experimental

results

load

time

0

F

t2t2+tt1

load

time

0

F

t2t2+tt1

k = 4x108 N/m

1 2 3 4 5 6 7 8 9 100

0.01

dis

pla

cem

en

t am

pli

tud

e (

cm

)

frequency (Hz)

0

100

200

300

velo

cit

y a

mp

litu

de (

m/s

)

DDA results

experimental

results

1 2 3 4 5 6 7 8 9 100

0.01

dis

pla

cem

en

t am

pli

tud

e (

cm

)

frequency (Hz)

0

100

200

300

velo

cit

y a

mp

litu

de (

m/s

)

DDA results

experimental

results

Top column response to dynamic blow: DDA vs. Geophysical survey


load

time

0

F

t1-t t1t1+t

load

time

0

F

t1-t t1t1+t

k = 4x108 N/m

DDA sensitivity of resonance frequency to penalty value k

1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Frequency (Hz)

Am

pli

tud

e (c

m)

k = 1x108 N/m

k = 2x108 N/m

k = 4x108 N/m

k = 7x108 N/m

k = 1x109 N/m


FFT of uppermost block displacement

Best agreement with field test

Decreasing resonance frequency and motion amplitude with increasing penalty value k

site response analysis with 2d-dda

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