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.
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.
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Research question – what ground motions caused
these specific failure mechanism ?
Avdat
Mamshit
Kamai et al. Dynamic back analysis of structural failures
Nimrod Fortress
3
Physical and Mechanical properties of the building stones
are obtained in the Rock Mechanics Laboratory of the
Negev, Ben-Gurion University
Direct shear
Kamai et al. Dynamic back analysis of structural failures
Ultrasonic waves
4
Results for a direct shear test on a sample from Avdat
site, under three different normal stresses.
120
 = 200 psi
100
t (psi)
80
60
 = 100 psi
40
 = 75 psi
20
0
0
5
10
15
20
25
30
35
40
u (mil)
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Mechanical Parameters
Site
Mamshit
Avdat
Nimrod
Density (Kg/m3)
1890
2555
2604
Porosity (%)
30-38
5
3.5
Dynamic Young’s modulus (GPa)
16.9
54.2
-
Dynamic Poisson’s ratio
0.37
0.33
-
Dynamic Shear modulus (GPa)
6.17
20.3
-
Point load index (MPa)
2.64
-
3.6
35
34.2
-
Mechanical Property
Peak Interface friction angle (deg)
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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
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DDA Validations
1. Block on an incline - gravitation only
Equations of Motion:
gcosa*tgf
gsina
ma  mgsin a  m( g cosa tanf )
1
1
st  at 2   g sin a  g cos a tan f t 2
2
2
g
a
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Accumulating displacement of block:
Analytical vs. DDA
2.5
displacement (m)
2
f5
DDA
f10
DDA
f15
DDA
f 20
DDA
f25
DDA
1.5
1
0.5
0
0
0.2
0.4
Kamai et al. Dynamic back analysis of structural failures
0.6
time (s)
0.8
1
10
2. Block on an incline - Dynamic validation
sin shape acceleration input motion
g(cosa-ksin(wt)sina)*tgf
a= kgsin(wt)
gsina+kgsin(wt)cosa
g
a
Equations of Motion:
t
t


U   U  g (sin a  cosa tanf )  dt  g (cosa  sin a tanf ) k sin(wt )dt
cos(w ) cos(wt )
U   U  g (sin a  cosa tanf )  (t   )dt  kg (cosa  sin a tanf ) 

dt
t
t


w
w

t2
 kg
U  g sin a  cosa tanf     t   2 cosa  sin a tanf w cos(w )(t   )  sin(wt )  sin(w ) 
2
 w

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Accumulating displacement of block :
Analytical vs. DDA
analytical
14
DDA
12
displacement (m)
a  f  20
10
8
6
4
at = 0.5sin(2t)
2
0
0
1
2
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time (sec)
4
5
12
Accumulating displacement of block:
Analytical vs. DDA
8
displacement (m)
a20
7
f 22
6
DDA
a<f
f 30
5
DDA
4
3
2
1
at = 0.5sin(2t)
0
0
0.5
1
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1.5
time (sec)
2
2.5
13
3. Input motion mechanism –
displacement to basement block
y
2
x
1
Responding block
Ground- input motion
Basement block - fixed
Equations of Motion:
m2 a2  F friction
*
Conditions for direction of force ( v1  v1  v2 ):
if
v1  0

m2 a2    m2 g

a2    g
and a1  g
if v1  0
Kamai et al. Dynamic back analysis of structural failures
a 2  a1
and a1 < g
and a1  0
a2  g
and a1 < 0
a2  g
and v1  0
a2  g
and v1 < 0
a2  g
14
Input motion into Block 1: Analytical vs. DDA
Analytical
1
DDA
displacement of lower block (m)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
dt = 0.5 (1- cos (2pt))
0
0
0.5
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1
time (sec)
1.5
2
15
Dynamic response of Block 2: Analytical vs. DDA
Influence of  (f = 1Hz; A = 0.5m)
1.2
 0.1
 0.6
 1
Analytical
Analytical
Analytical
DDA
DDA
DDA
displacement of upper block(m)
input motion
1
0.8
0.6
0.4
0.2
dt = 0.5 (1- cos (2pt))
0
0
0.5
1
1.5
2
Kamai et al. Dynamic back analysis of structural failures
2.5
time (s)
3
3.5
4
4.5
5
16
Dynamic response of Block 2: Analytical vs. DDA
Influence of Amplitude (f = 1Hz;  = 0.6)
1.8
displacement of upper block (m)
1.6
A=1m
Analytical
DDA
A=0.5m
Analytical
DDA
A=0.3m
Analytical
DDA
1.4
1.2
1
0.8
0.6
0.4
0.2
dt = A (1- cos (2pt))
0
0
0.2
0.4
0.6
0.8
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1
time (sec)
1.2
1.4
1.6
1.8
2
17
Dynamic response of Block 2: Analytical Vs. DDA
Influence of Frequency (A = 0.02 m ;  = 0.6)
f=5Hz
f=3Hz
f=2Hz
displacement of upper block (m)
0.05
Analytical
Analytical
Analytical
DDA
DDA
DDA
0.04
0.03
0.02
0.01
dt = 0.02 (1- cos (2pwt))
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
time (sec)
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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).
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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
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1.
Mamshit
h
g
f
E
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
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Sensitivity analysis was performed for:
Overburden (h)
Amplitude of earthquake
Frequency of earthquake
stiffness of embedding wall
The vertical displacement of the Key stone
over time is the measured parameter.
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overburden (h)
f = 1.5Hz, A = 0.5g
0.5
time (sec)
0
Vkey block (cm)
0
1
2
3
4
5
6
7
8
9
10
0m
-0.5
0.725 m
-1
1.225 m
-1.5
-2
-2.5
-3
-3.5
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Amplitude (A)
f = 1Hz
time (sec)
0
0
1
2
3
4
-0.5
5
6
7
8
9
10
0.1 g
0.32 g
Vkey block (cm)
0.5 g
-1
0.8 g
-1.5
-2
-2.5
-3
-3.5
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Frequency (f)
A = 0.5g
time (sec)
0
0
1
2
3
4
-0.5
5
6
7
8
9
10
0.5 Hz
1 Hz
1.5 Hz
Vkey block (cm)
-1
5 Hz
15*nueiba
-1.5
-2
-2.5
-3
-3.5
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Stiffness of wall blocks
f = 1.5Hz, A = 0.5g, E1=17GPa
1.5
time (sec)
E2=1 MPa
1
E2=5 MPa
0.5
E2=100 MPa
Vkey block (cm)
0
0
1
2
3
4
5
6
7
8
9
10
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
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Best fit to field evidence after 10sec obtained with:
f =1.5 Hz, A = 0.5g and h = 0
Vkey block= -3 cm
h=0
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2. Avdat
Five blocks have
slid westerly out of
the western wall
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.
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Preliminary results
Sensitivity 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.
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Conclusions
• 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.
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Thank you for your time.
Ronnie Kamai (levyronn@bgu.ac.il)
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