Evaluating paleoseismic ground motions using dynamic back analysis of structural failures in...

Evaluating paleoseismic ground motions using dynamic back

analysis of structural failures in archaeological sites

Ronnie Kamai (1), Yossef Hatzor (1), Shmulik Marco (2)

(1) Department of Geological and Environmental Sciences, Ben Gurion University of the Negev, Beer – Sheva.

(2) Department of Geophysics and Planetary Sciences, Tel Aviv University.

2Kamai et al. Dynamic back analysis of structural failures

Research objective

To develop an alternative method for obtaining strong ground-motion data:

by back analysis of structural failures in archaeological sites.

Results will provide constraints on PGA estimates, generated by the existing seismological strong motion

catalogue.


Research question – what ground motions caused these specific failure mechanism ?

Avdat

Mamshit Nimrod Fortress


Physical and Mechanical properties of the building stones are obtained in the Rock Mechanics Laboratory of the

Negev, Ben-Gurion University

Direct shear Ultrasonic waves


Results for a direct shear test on a sample from Avdat site, under three different normal stresses.

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35 40

u (mil)

t (p

si)

= 75 psi

= 100 psi

= 200 psi


Mechanical Parameters

Mechanical Property

MamshitAvdatNimrod

Density (Kg/m3) 189025552604

Porosity (%) 30-3853.5

Dynamic Young’s modulus (GPa) 16.954.2-

Dynamic Poisson’s ratio 0.370.33-

Dynamic Shear modulus (GPa) 6.1720.3-

Point load index (MPa)2.64-3.6

Peak Interface friction angle (deg)3534.2-

Site


Dynamic analysis was performed with the Discontinuous Deformation Analysis method

(DDA): • A numerical method of the distinct element family• Based on the second law of thermodynamics - minimization of

energy in every time step• The numerical elements are real isolated blocks, having 6

degrees of freedom• No tension or penetration is allowed between blocks• The interfacial friction obeys the Coulomb-Mohr criterion

• Limitations:– This research was performed with the 2-D model– Stresses and strains are constant through the blocks

DDA Validations


gsin

g

gcos*tg

1. Block on an incline - gravitation only

)tancos(sin gmmgma

22 tancossin2

1

2

1tggatst

Equations of Motion:


Accumulating displacement of block: Analytical vs. DDA

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

time (s)

dis

pla

cem

ent

(m)

5 DDA

10 DDA

15 DDA

20 DDA

25 DDA


gsin+kgsin(t)cos

g

g(cos-ksin(t)sin)*tga= kgsin(t)

tt

dttkgdtgUU

)sin()tansin(cos)tancos(sin

dtt

kgdttgUUt t

)cos()cos()tansin(cos)()tancos(sin

)sin()sin())(cos(tansincos2

tancossin2

2

tt

kgt

tgU

2. Block on an incline - Dynamic validation sin shape acceleration input motion



Accumulating displacement of block : Analytical vs. DDA

20

0

2

4

6

8

10

12

14

0 1 2 3 4 5time (sec)

dis

pla

cem

ent

(m)

analytical

DDA

at = 0.5sin(2t)


Accumulating displacement of block: Analytical vs. DDA

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5

time (sec)

dis

plac

emen

t (m

)

20

22

DDA 30

DDA

at = 0.5sin(2t)


Basement block - fixed

Ground- input motion

Responding block

1

2

3. Input motion mechanism – displacement to basement block


ga

gmam

Fam friction

2

222

22

01 vif 01 vand ga 2

01 vand ga 2

gaand 101 aand ga 2

01 aand ga 2

01 vif gaand 1 12 aa

Conditions for direction of force ( ):21*1 vvv

y

x


Input motion into Block 1: Analytical vs. DDA

dt = 0.5 (1- cos (2t))0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

time (sec)

dis

pla

cem

ent

of lo

wer

blo

ck (

m)

Analytical

DDA


0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5time (s)

dis

pla

cem

ent

of u

pp

er b

lock

(m)

0.1 Analytical DDA 0.6 Analytical DDA 1 Analytical DDA

input motion

Dynamic response of Block 2: Analytical vs. DDAInfluence of (f = 1Hz; A = 0.5m)

dt = 0.5 (1- cos (2t))


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)

dis

pla

cem

ent

of u

pp

er b

lock

(m

)

A=1m Analytical DDA

A=0.5m Analytical DDA

A=0.3m Analytical DDA

Dynamic response of Block 2: Analytical vs. DDAInfluence of Amplitude (f = 1Hz; = 0.6)

dt = A (1- cos (2t))


0

0.01

0.02

0.03

0.04

0.05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

time (sec)

dis

pla

cem

ent

of

up

per

blo

ck (

m)

f=5Hz Analytical DDAf=3Hz Analytical DDA

f=2Hz Analytical DDA

Dynamic response of Block 2: Analytical Vs. DDAInfluence of Frequency (A = 0.02 m ; = 0.6)

dt = 0.02 (1- cos (2t))


Validation Conclusions

• A remarkable agreement between DDA and analytical solutions of various mechanisms is shown

• DDA is sensitive to interface friction and loading function parameters (Amplitude and frequency).


Case studies• Careful and accurate mapping of the structure is performed

• The 2-D DDA model is built in attempt to best represent the structural situation

• An Earthquake record, either a synthetic sinusoidal one, or an amplified record of Nuweiba 1995 is induced into all block centroids

• A sensitivity analysis for varying Amplitudes and frequencies is performed

• The dynamic displacements and stresses at pre-defined measurement points is recorded and analyzed


h

The model:• The embedding wall is very heterogenic, so material lines (red) define different mechanical parameters for arch and wall• Dots are “fixed points”, fixating the basement block

1. Mamshit


Sensitivity analysis was performed for:Overburden (h)

Amplitude of earthquakeFrequency of earthquake

stiffness of embedding wall

The vertical displacement of the Key stone over time is the measured parameter.


overburden (h)f = 1.5Hz, A = 0.5g

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 1 2 3 4 5 6 7 8 9 10

time (sec)

Vke

y bl

ock (

cm)

0 m

0.725 m

1.225 m


-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5 6 7 8 9 10

time (sec)

Vk

ey b

lock

(cm

)

0.1 g

0.32 g

0.5 g

0.8 g

Amplitude (A)f = 1Hz


-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5 6 7 8 9 10

time (sec)

Vk

ey b

lock

(cm

)

0.5 Hz

1 Hz

1.5 Hz

5 Hz

15*nueiba

Frequency (f)A = 0.5g


-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10

time (sec)

Vk

ey b

lock

(cm

)

E2=1 MPa

E2=5 MPa

E2=100 MPa

Stiffness of wall blocks f = 1.5Hz, A = 0.5g, E1=17GPa


Best fit to field evidence after 10sec obtained with: f =1.5 Hz, A = 0.5g and h = 0

Vkey block= -3 cmh=0


The model:• Because of 2-D limitations, the model is of the northern wall, in order to see the westerly sliding• The model is confined on its left side because of a later structure attached to the wall to the left of the door•Red dots are “fixed” points, yellow dots are measurement points.

2. AvdatFive blocks have slid westerly out of the western wall


Preliminary resultsSensitivity analysis is not completed yet, best fit up to this point:

The observed blocks were displaced 4-10cm after 10sec with an earthquake of A=1g, f=3Hz for 10 sec.


• Analysis of a structural failure in archaeological sites was performed successfully using DDA.

• The new procedure can be applied to other sites in the world, provided that the displacement of a distinct element in the structure can be measured.

• We find that frequency, amplitude, and duration of shaking have a strong influence on the structural response.

• Specifically, for the case studies presented we find that:1. For the site of Mamshit:– The downward displacement of the arch-keystone became possible

after the collapse of the overlying layers due to the relaxation of arching stresses.

– The critical frequency and amplitude for the detected failure mode in the analyzed arch is 1Hz and 0.5g, respectively.

2. For the site of Avdat: – The door opening causes an arching of the stresses, and therefore the

displaced blocks are not the ones with the least vertical load.– The critical frequency and amplitude for the detected failure mode in

the analyzed structure is 3Hz and 1g, respectively.

Conclusions


Thank you for your time.

Ronnie Kamai ([email protected])

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