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.

Kamai et al. Dynamic back analysis of structural failures 2 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.

Kamai et al. Dynamic back analysis of structural failures 3 Research question – what ground motions caused these specific failure mechanism ? Avdat Mamshit Nimrod Fortress

Kamai et al. Dynamic back analysis of structural failures 4 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

Kamai et al. Dynamic back analysis of structural failures 5 Results for a direct shear test on a sample from Avdat site, under three different normal stresses.

Kamai et al. Dynamic back analysis of structural failures 6 Mechanical Parameters Mechanical Property MamshitAvdatNimrod Density (Kg/m3) Porosity (%) Dynamic Young’s modulus (GPa) Dynamic Poisson’s ratio Dynamic Shear modulus (GPa) Point load index (MPa) Peak Interface friction angle (deg) Site

Kamai et al. Dynamic back analysis of structural failures 7 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

Kamai et al. Dynamic back analysis of structural failures 9 gsin  g gcos  *tg   1. Block on an incline - gravitation only Equations of Motion:

Kamai et al. Dynamic back analysis of structural failures 10 Accumulating displacement of block: Analytical vs. DDA

Kamai et al. Dynamic back analysis of structural failures 11 gsin  +kgsin(  t)cos  g g(cos  -ksin(  t)sin  )*tg  a= kgsin(  t)  2. Block on an incline - Dynamic validation sin shape acceleration input motion Equations of Motion:

Kamai et al. Dynamic back analysis of structural failures 12 Accumulating displacement of block : Analytical vs. DDA  a t = 0.5sin(2t)

Kamai et al. Dynamic back analysis of structural failures 13 Accumulating displacement of block: Analytical vs. DDA  a t = 0.5sin(2t)

Kamai et al. Dynamic back analysis of structural failures 14 Basement block - fixed Ground- input motion Responding block Input motion mechanism – displacement to basement block Equations of Motion: Conditions for direction of force ( ): y x

Kamai et al. Dynamic back analysis of structural failures 15 Input motion into Block 1: Analytical vs. DDA d t = 0.5 (1- cos (2  t))

Kamai et al. Dynamic back analysis of structural failures 16 Dynamic response of Block 2: Analytical vs. DDA Influence of  (f = 1Hz; A = 0.5m) d t = 0.5 (1- cos (2  t))

Kamai et al. Dynamic back analysis of structural failures 17 Dynamic response of Block 2: Analytical vs. DDA Influence of Amplitude (f = 1Hz;  = 0.6) d t = A (1- cos (2  t))

Kamai et al. Dynamic back analysis of structural failures 18 Dynamic response of Block 2: Analytical Vs. DDA Influence of Frequency (A = 0.02 m ;  = 0.6) d t = 0.02 (1- cos (2  t))

Kamai et al. Dynamic back analysis of structural failures 19 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).

Kamai et al. Dynamic back analysis of structural failures 20 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

Kamai et al. Dynamic back analysis of structural failures 21 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   

Kamai et al. Dynamic back analysis of structural failures 22 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.

Kamai et al. Dynamic back analysis of structural failures 23 overburden (h) f = 1.5Hz, A = 0.5g

Kamai et al. Dynamic back analysis of structural failures 24 Amplitude (A) f = 1Hz

Kamai et al. Dynamic back analysis of structural failures 25 Frequency (f) A = 0.5g

Kamai et al. Dynamic back analysis of structural failures 26 Stiffness of wall blocks f = 1.5Hz, A = 0.5g, E 1 =17GPa

Kamai et al. Dynamic back analysis of structural failures 27 Best fit to field evidence after 10sec obtained with: f =1.5 Hz, A = 0.5g and h = 0 V key block = -3 cm h=0

Kamai et al. Dynamic back analysis of structural failures 28 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. Avdat Five blocks have slid westerly out of the western wall

Kamai et al. Dynamic back analysis of structural failures 29 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.

Kamai et al. Dynamic back analysis of structural failures 30 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

Kamai et al. Dynamic back analysis of structural failures 31 Thank you for your time. Ronnie Kamai