Numerical analysis of block stone displacements in ancient masonry

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2008; 32:1321–1340Published online 15 November 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.671

Numerical analysis of block stone displacements in ancientmasonry structures: A new method to estimate historic

ground motions

Ronnie Kamai1,2 and Yossef H. Hatzor1,∗,†

1Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel2Department of Civil and Environmental Engineering, University of California, Davis, CA, U.S.A.

SUMMARY

An innovative approach is presented, in which the discontinuous deformation analysis (DDA) method isused to estimate historic ground motions by back analysis of unique structural failures in archaeologicalsites. Two archaeological sites in Israel are investigated using this new approach and results are presentedin terms of displacement evolution of selected structural elements in the studied masonry structure. Theresponse of the structure is studied up to the point of incipient failure, in a mechanism similar to theone observed in the field. Structural response is found to be very sensitive to dynamic parameters of theloading function such as amplitude and frequency.

Prior to back analysis of case studies, two validations are presented. Both compare the performance ofDDA with analytical solutions and present strong agreement between the two.

Using comprehensive sensitivity analyses, the most likely peak ground acceleration (PGA) and frequencythat must have driven the observed block displacements are found for the two case studies—the Nabateancity of Mamshit and the medieval fortress of Nimrod in southern and northern Israel, respectively. Itis found that horizontal peak ground accelerations (HPGA) of 0.5g and 1g were required to generatethe observed deformations in Mamshit and Nimrod, respectively. Although these might seem too high,considering structural and topographic amplifications it is concluded that the analyses suggest groundmotions of 0.2g at a frequency of 1.5Hz for Mamshit and up to 0.4g at a frequency of 1Hz for Nimrod.These values provide constraints on the seismic risk associated with these regions as appears in the localbuilding code using a completely independent approach. Copyright q 2007 John Wiley & Sons, Ltd.

Received 31 August 2006; Revised 15 August 2007; Accepted 15 October 2007

KEY WORDS: ground motions; seismic risk; PGA; discontinuous deformation analysis; masonry struc-tures; validation

∗Correspondence to: Yossef H. Hatzor, Department of Geological and Environmental Sciences, Ben-Gurion Universityof the Negev, Beer-Sheva 84105, Israel.

†E-mail: [email protected]

Copyright q 2007 John Wiley & Sons, Ltd.

1322 R. KAMAI AND Y. H. HATZOR

1. INTRODUCTION

The expected ground motions at a site, a critical component in seismic risk assessments, are gener-ally evaluated using empirical relationships between magnitude, distance, and site effects [1, 2].Here, we present a quantitative approach to estimate historic ground motions using back analysisof block displacements in archaeological masonry structures. Reliable estimates of paleoseismicground motions at a given site can provide constraints on ground motion predictions which aretypically obtained using empirical methods.

The analysis focuses on man-made masonry structures such as arches, where the hewn stonesthat form the building provide a well-defined initial geometrical reference. Where failure is confinedto displaced blocks within an otherwise intact structure, block displacement is measurable and amechanical analysis is possible; this cannot be achieved in completely collapsed structures. Addi-tionally, sound bedrock foundation and high-quality masonry workmanship enhance the reliabilityof the results by eliminating other failure modes associated with poor construction methods orweak foundation material [3].

The suggested method is tested using two archeological sites in Israel: The Nabatean cityof Mamshit in the Negev desert and the Ayyubid-Crusader Fortress of Nimrod, at the footof Mt. Hermon (Figure 1). For each site, a sensitivity analysis is performed to determine themost likely amplitude and frequency of motion that must have generated the observed failure inthe field.

The numerical analysis is performed using the discontinuous deformation analysis (DDA)method, a member of the family of implicit DEM methods [5] developed by Shi during thelate 1980s [6]. Here, we use a research version of DDA software modified by DDA authorDr Gen-hua Shi for the specific targets of this research program. The DDA method solves afinite element type of mesh, where all elements are real isolated blocks and the unknowns ofthe equations are the displacements and deformations of the blocks. The blocks are ‘simplydeformable’, namely stresses and strains are constant throughout the block. When the blocks arein contact, Coulomb’s friction law applies to the contact interface, and the simultaneous equilib-rium equations are formulated and solved for each loading or time increment. The formulation isbased on the minimization of the system potential energy, following the second law of thermo-dynamics.

One of the strengths of the DDA method, which is extremely important for this research, isthat the mode of failure of the block system is a result of the analysis and not an assumption.This feature enables us to study the response of the structure to different loading functions and toisolate the parameters that produce the unique failure that is observed in the field.

Over the last decade, researchers in the DDA community have dedicated a great deal of effort toprove the accuracy of the method by performing theoretical validation studies. MacLaughlin andDoolin [7] review more than 100 validation studies with respect to analytical solutions, laboratoryand field data, and other numerical techniques. DDA performance was found to be more thanadequate for engineering applications.

Up to date, DDA has been used for numerous rock engineering projects. In Israel alone, DDAhas been applied to solve dynamic stability issues in the highly discontinuous Masada rock slopes[8] to stabilize a discontinuous overhanging cliff [9] and to determine the stability of historicunderground excavations in discontinuous rock [10, 11]. Unlike these applications, which involvepotentially hazardous rock masses, this work uses the DDA method to examine the stability ofman-made masonry structures.

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1321–1340DOI: 10.1002/nag

NUMERICAL ANALYSIS OF BLOCK STONE DISPLACEMENTS 1323

Figure 1. Location map of studied sites (after [4]) shown on two sides of the seismicallyactive Dead Sea transform (DST).



2. NUMERICAL CODE VALIDATION

Validation studies in this section were performed for calibration purposes only, using both existingand newly developed analytical solutions. The validations are fully dynamic, meaning that thevelocity of each block at the end of a time step is fully transferred to the next time step with noenergy dissipation (in DDA notation k01=1).

2.1. Dynamic sliding of a block on an incline

The case of a sliding block on an incline subjected to gravitational loads only has been successfullyanalyzed and discussed in detail in previous papers [12], and hence will not be repeated here. Thecase of a sliding block subjected to dynamic loading, however, is addressed here in detail. Theanalysis presents higher accuracy than previously described [13] due to a correction that has beenmade here to the analytical solution.

A displacement-based sliding block model was first proposed by Newmark [14] as well asGoodman and Seed [15], now largely referred to as ‘Newmark’-type analysis. Determinationof the amount of displacement during an earthquake necessarily involves two steps [15]: (1)determination of horizontal acceleration required to initiate down slope motion, also known as‘yield acceleration’ (ay), which can be found by pseudo-static analysis, and (2) Evaluation ofthe displacement developed during time intervals when yield acceleration is exceeded, by doubleintegration of the acceleration time history, with the yield acceleration used as reference datum.

Figure 2 displays a single block resting on a plane inclined at an angle � with friction along theinterface �, and subjected to gravitational acceleration g and a horizontal, time-based, sinusoidalacceleration input a as driving forces.

Goodman and Seed [15] showed that for this case the yield acceleration is given by

ay= tan(�−�)g (1)

α

a=kgsin( t)ω

gsin +kgsin( t)cosα ω α

g

g(cos -ksin( t )sin )*tgα ω α φ

Figure 2. Schematic presentation of the forces acting on a single block on an incline. The driving forcesare gravity and a horizontal, time-based, sinusoidal acceleration. The resisting frictional force is constant

and depends only on the friction angle �(�= tan �).



For an acceleration record of the form a=kg sin(�t), where � corresponds to the frequency ofthe function and k calibrates the proportion between a and g, the corresponding time interval �until yield acceleration is attained is given by

�= sin−1(ay/kg)

�(2)

The down slope acceleration of the sliding block can be determined by subtracting the resistingforces from the driving forces, as described in Figure 2

at =[kg sin(�t) cos �+g sin �]−[g cos �−kg sin(�t) sin �] tan � (3)

Similarly, the displacement of the block at any time is determined by double integration on theacceleration, with � as reference datum, in a conditional manner:

if a>ay or v>0

dt =∫ t

�v=

∫ ∫�a=g[(sin�−cos� tan�)(t2/2−�t)] (4)

+ ag

�2[(cos�+sin� tan�)(�·cos(��)(t−�)−sin(�t)+sin(��))]

otherwise dt =dt−1

Equation (4) provides the analytical solution for the dynamic displacement of a block on aninclined plane with inclination � and friction angle �, starting from rest and subjected to a sinusoidalloading function with frequency �.

Figure 3 displays a comparison between the analytical and DDA solutions for three friction angles(�=22, 30, and 35◦) with �=0.035,0.18, and 0.28 s, respectively. The accumulated displacementsare computed for a plane inclination of 20◦. The numerical error EN shown in the lower panel ofFigure 3 is defined as

EN=∣∣∣∣‖d‖−‖dN‖

‖d‖∣∣∣∣100% (5)

where d and dN are the analytical and the numeric displacement vectors, respectively. ‖·‖ is thenorm operator, which for a 2D displacement vector is

d=√u2+v2 (6)

Other than for the first second, in which the relative error reaches 50%, due to a very smallabsolute error of only 2.6E10−5 m, the relative error is less than 1% for all three frictionangles for all accumulated displacements. Time-step size is kept constant in all DDA runs andis 0.002 s.



Figure 3. Validation of the dynamic case of a block on an incline for three different interface frictionangles: top—input acceleration function; center—comparison between analytical (solid line) and DDA

(symbols) solutions; and bottom—relative error for each simulation.

2.2. Block response to induced displacements in the foundation

In the previous section, a validation was performed for a block on an incline subjected to dynamicinput applied to its center of mass. We now present a validation for the case of a block restingon a horizontal interface and responding to dynamic input applied to the underlying block. Suchan application would be suitable for dynamic analysis of structures which respond to earthquakemotions at their foundations. This approach was not adopted in this current paper because appli-cation of this loading mechanism for cases that involve multiple blocks seems to require furtherresearch. Nevertheless, we feel it is instructive to present this original validation here because thisis the first time such a validation is performed and the obtained results are promising.

The studied block system consists of three blocks: a fixed foundation block (no. 0), the inducedblock (no. 1), and the responding block (no. 2) (Figure 4). Block 1 is subjected to a horizontaldisplacement input function in the form of a cosine, starting from 0 (shown in the upper panel ofFigure 5)

d(t)=D(1−cos(2��t)) (6′)



Block 0

Block 1

Block 2 x

d(t

y

)

Figure 4. The DDA block system and sign convention used for the case of a single block resting on anunderlying block which is subjected to time-dependant displacement.

Inp

ut

Dis

plac

em

ent

(m

)

0

0.5

1

1.5

Dis

plac

em

ent

of

uppe

r blo

ck (

m)

210

Time (sec)

Analytic

Analytic

Analytic

DDA

DDA

DDA

Input Motion

0.1

1

10

rela

tive

err

or (

%)

D=0.3 m

D=0.5 m

D=1 m

2D

Figure 5. The responding block motion (Figure 4) for three different input displacement amplitudes,all at f =1Hz: top—input displacement function of the form d(t)=D(1−cos(2��t)) applied toBlock 1; center—comparison between analytical (solid line) and DDA (symbols) solutions; and

bottom—relative error for each simulation.

The dynamic response of Block 2 is investigated and DDA results are compared with theanalytical solution, the essentials of which are provided below.



The only force acting on Block 2 other than gravity is the frictional force, which immediatelydetermines the acceleration of Block 2:

m2a2=Ffriction (7)⏐⏐⏐�m2a2=�·m2g (8)⏐⏐⏐�a2=�·g (9)

where � is the friction coefficient.The direction of the driving force is determined by the direction of the relative velocity between

Block 1 and 2 (v∗1). When Block 1 moves to the right relative to Block 2, the frictional force pulls

Block 2 in the same direction, and determines the sign of a2.When Block 2 is at rest in relation to the Block 1, the frictional force is determined by the

acceleration of the bottom block (a1). The threshold acceleration, under which the two blocksmove in harmony, is equal to the friction coefficient multiplied by the gravitation acceleration(�g). When the acceleration of Block 1 passes the threshold value, the frictional forces act in thesame direction as a1.

The positive direction is determined by the sign convention in Figure 4, and the relative velocityof Block 1 is given by

v∗1 =v1−v2 (10)

The direction of the acceleration of Block 2 is set by the following boundary conditions:

if v∗1 =0 and |a1|<�g, a2=a1

and |a1|>�g and a1>0, a2=�g

and a1<0, a2=−�g

if v∗1 �=0 and v∗

1>0, a2=�g

and v∗1<0, a2=−�g

(11)

In the solution of Equation (11) we employed Matlab 7.0 [16] software package because theanalytical solution must be computed iteratively as the relative velocity and the direction of theforce are dependent on each other.

In order to compare DDA and the analytical solution, the mode of failure of the analyzed blockin DDA has to be constrained to horizontal sliding only, so that it would simulate a single degreeof freedom system as does the analytical solution. Therefore, Block 2 of the DDA block systemwas made very flat, to eliminate possible rotational motions (Figure 4).

Figures 5 and 6 present the accumulated displacement of Block 2 and the relative error betweenanalytical (Matlab) and DDA solutions, using a set of sensitivity analyses for amplitude and frictioncoefficient. Figure 5 presents the response of Block 2 to changing displacement amplitudes (D),



Figure 6. The responding block motion (Figure 4) for three different friction coefficients, all at D=0.5m,f =1Hz: top—the input displacement function applied to Block 1; center—comparison between analytical

(solid line) and DDA (symbols) solutions; and bottom—relative error for each simulation.

with a constant input frequency of 1Hz and a friction coefficient of 0.6. The obtained magnitudeof accumulated displacement is directly proportional to the input amplitude, as expected. Notethat the three displacement curves follow the periodic behavior of the input displacement function(T =1s), and that divergence between curves starts after 0.25s, where the input displacementfunction has an inflection point. The relative error, as shown in the upper panel of Figure 5, ismostly between 1 and 2%, reaching a maximum of 3.5% for D=0.3m, which actually correspondsto a difference of 1 cm only in the displacements.

Figure 6 presents the response of Block 2 to changing friction coefficients across the inter-face, with a constant displacement function of D=0.5m, f =1Hz. Note that the accumulateddisplacement is directly proportional to the friction coefficient up to 0.5 s, after which the inputdisplacement function changes direction. After that point, the accumulated displacement for �=0.6is larger than for �=1, since the higher friction works in both forward and backward directions.

It is interesting to note that the relative errors (lower panel of Figure 6) show large sensitivity tothe friction coefficient. For a reasonable friction coefficient of say 0.6, the relative error remainswithin 2%. However, when the friction coefficient is very low (�=0.1) or relatively high (�=1),the errors increase up to 12%. Note that the errors for the high friction follow the displacement



pattern, with the maximum errors closely corresponding to change in the direction of motion,where the system experiences high accelerations and inertia forces.

In general, it is shown that DDA follows the analytical results in both cases with a remarkableagreement, both for changing friction coefficients and for changing motion amplitudes.

It is important to note that while application of this loading mechanism yielded good agreementfor a single responding block as shown above, when we experimented with problems involvingmultiple blocks we obtained a complex structural response not supported by field observations.We feel that while this loading mechanism is more appropriate for the problem at hand, furtherresearch of its numerical implementation is required.

3. RESULTS

We will present our results in the form of sensitivity analyses for amplitude and frequency of theinput loading function. As mentioned previously, the input acceleration function is applied simul-taneously at the centroid of each block in the mesh, simulating therefore structural accelerations.The relevant ground motion corresponding to this excitation mode is discussed in the next section.

Since the final objective of the analysis is to successfully mimic an existing failure in the field,the sensitivity analyses must be performed in trial and error: an artificial, sinusoidal accelerationrecord is applied and the responses of the masonry structures are studied up to the point of incipientfailure, in a mode similar to the one observed in the field. Note that the magnitude of the measureddisplacement in the field could be a result of structural response to several, consecutive, largeearthquakes over the life span of the studied structure, which in our case is in the order of 1–2×103

years. The analysis herein assumes that the observed failure is a result of a single shaking event, inwhich all shaking characteristics have come together to create the unique pattern of deformation.Therefore, in addition to mimicking the mode of failure, the results are also discussed in terms ofthe magnitude of displacement, where applicable.

In order for the analysis to be as accurate as possible, genuine mechanical properties of theanalyzed structures are used as input for DDA. Original building stones were taken from thearchaeological sites, under supervision of the Israel Nature and Parks Authority, to the RockMechanics Laboratory of the Negev at Ben-Gurion University. Lab tests were performed in orderto obtain physical and mechanical properties of intact rock samples and results are summarized inTable I. The input stiffness of the embedding wall is reduced by four orders of magnitude in order tosimulate the soft filling material between stones in the wall. Higher stiffness values for the confiningwall were investigated, and it was found that the arch rocking behavior is extremely limited at highconfining wall stiffness, restricting the initiation of the unique failure mechanisms observed in thefield. Choosing a low value, in the order of soil stiffness, allows for large deformations under lowstresses. Consequently, the ground motion that is claimed to have caused the deformations is alower bound on the expected ground motions at that site. A friction angle of 35◦ for the stoneinterfaces was chosen as a representative value for all analyses. It was found that the responseis not very sensitive to the friction angle, and that friction angles between 25 and 40◦ yieldedkeystone displacements of 2.2–3.2 cm, respectively, with the same unique failure mechanism.

All numerical analyses are performed with kinetic damping of 1% (k01=0.99), namely, theinitial velocity in each time step is 1% lower than the terminal velocity of the previous step. Theintroduction of kinetic damping in the analysis stems from previous work that suggests that withan increasing number of blocks in the mesh, a certain degree of energy dissipation is required for



Table I. Mechanical properties obtained in the lab from the original buildingblocks taken from the studied sites.

Mechanical property Mamshit Nimrod FortressLithology and formation Hazera limestone Hermon limestone

Density (kg/m3) 1890 2604Porosity (%) 30–38 3.5Dynamic 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 3.6Uniaxial compressive strength (MPa) 66 90Peak interface friction angle (◦) 35 35

∗Owing to sampling difficulties at NF, these parameters are derived from tests performed on blockssampled at the Avdat site [17] with similar rock properties.

Figure 7. The deformed arch at Mamshit: (a) general view of the arch which is embedded in a veryheterogenic wall and (b) the keystone has slid 4 cm downwards while the rest of the arch remained intact.

realistic modeling [8, 18], because DDA does not consider inelastic energy dissipation mechanismssuch as crushing at block ends, out-of-plane rotations, heat generation, etc.

3.1. The case studies

Two archaeological sites in Israel, where a confined structural failure is identified, measured andmapped are selected for analysis (Figure 1).

Mamshit is the last Nabatean city built in the Negev on the trade route between Petra, Hebron,and Jerusalem [19, 20]. A unique structural failure is noticed in a tower at the corner of theEastern Church, where a key stone has slid approximately 4 cm downwards out of a still standingsemi-circular arch (Figure 7).

The numerical analysis of the arch at Mamshit is performed on a block system (Figure 8)that contains the arch, with its accurate measurements from the field, confined by a uniformmasonry wall. The arch is intersected by ‘material lines’ which enable us to assign separate sets ofmechanical parameters for the wall and the arch. These simulate the great difference between the



Figure 8. The DDA block system for the embedded arch at Mamshit. The modeled masonry wall restson two fixed blocks. The lines intersecting the arch blocks represent material lines and a measurement

point (circle) is assigned for the keystone.

hewn stones of the arch itself and the heterogenic confining wall material (Figure 7) by assigningintact rock stiffness values to the arch stones (Earch=17GPa) and soil-like stiffness values to thewall stones (Ewall=1MPa). A measurement point is assigned at the keystone, and its verticaldisplacement versus time is plotted in the relevant figures.

In Nimrod fortress, the largest medieval fortress in Israel [21, 22], evidence of destruction causedby a severe earthquake can be seen throughout the fortress. The damage is dated to one of twolarge earthquakes of the year 1759 (Ms=6.6 and 7.4) [23]. The most impressive evidence from theearthquake that has damaged the fortress can be observed in the gate tower. There, three parallelarches as well as an arched passageway display the same mode of failure, in which a single block,not necessarily the ‘keystone’, has slid downwards while the rest of the arch remained intact(Figure 9).

In the modeled block system (Figure 10), material lines intersect the arch blocks and allow us toassign separate sets of mechanical parameters to the arch and the wall, as before. The wall in whichthe arch is embedded is confined by a boundary block on its right-hand side and its base, addedin order to simulate the geographical constraints at the site: the fortress is built on an elongatedrange, cut off by deep canyons from three sides. As a result, the gate tower is supported by a 30mhigh retaining wall on its western side, while the eastern side rests directly on bedrock. This settingmay certainly cause asymmetric structural response and possibly seismic wave amplifications inpreferred orientations.

Five measurement points are assigned to five blocks in the arch, the label and position of whichare shown in Figure 11.

During shaking, the entire modeled structure is gradually displaced to the left, most probablydue to the confining wall on the right-hand side which represents a fixed boundary. In order torepresent correctly the computed displacement of block AR (Figure 9(a)), the block that exhibitsthe greatest displacement in the field in relation to the computed displacements of the entire blocksystem, the selected parameter for sensitivity analyses was the relative displacement of AR withrespect to the average displacement of the other four keystones. The following equations describethe derivation of that parameter:

uavg =∑

ui4

, vavg=∑

vi

4(12)

u = u(AR)−uavg, v=v(AR)−vavg (13)



Figure 9. Evidence for a destructive earthquake that hit the Nimrod Fortress most probably at 1759 (seetext for explanation): (a) the same stone has slid downwards out of three parallel arches at the main gatearch; (b) a similar deformation is noticed at a passage-way parallel to the gate arch; (c) an adjacent arch

facing east was not deformed; and (d) location of the passage-way, 10m west of main gate.

where u versus v is the relative spatial displacement of block AR, plotted in the relevantfigures.

3.2. Sensitivity analyses

A very interesting and unique behavior of the two structures is revealed through the sensitivityanalyses of structural response to changing amplitude of motion. Instead of an intuitive proportionbetween induced motion amplitude and degree of damage, measured here by the magnitude of thedisplacement vector, there is apparently a structural ‘preference’ to a certain range of amplitudes:



Figure 10. The DDA block system for the studied arch at Nimrod Fortress. Four fixed points (squares)are assigned to the confining block and five measurement points (circles) are assigned to the top arch

blocks. The lines intersecting the arch blocks represent material lines.

Figure 11. Sign convention for the Nimrod arch blocks: K =keystone, A and B for the first and secondblock from the keystone respectively, and R or L to indicate right or left.

in both Mamshit and Nimrod, the studied block exhibits the greatest displacement under specificamplitudes, not necessarily the largest. Figure 12 shows that the keystone of the arch at Mamshitexhibits the greatest downward displacement under an amplitude of 0.5g , when everything else iskept equal. Downward displacement increases with acceleration amplitude up to an amplitude of0.5g. When the acceleration amplitude is greater than 0.5g, the keystone response exhibits strongfluctuations and even a shift in displacement direction.

A similar phenomenon is observed in Nimrod. Figure 13 shows the spatial displacement ofblock AR in response to changing amplitudes. Under the smallest amplitude presented (A=0.8g),a very minor response is detected where the block displaces inwards and later outwards, resultingin almost no accumulated displacement. The largest amplitude (A=1.5g), however, has a strongaffect on the entire structure leading to partial destruction, but consequently block AR is locked



6420 8Time (sec)

10

-3

-2

-1

0

Key B

lock

Displace

ment

(cm

)

0.1 g

1 g

0.5 g0.32 g

0.8 g

Figure 12. Influence of input motion amplitude on vertical keystone displacement at Mamshit ( f =1Hz).

Figure 13. Influence of input motion amplitude on relative spatial displacement of BlockAR at Nimrod Fortress ( f =2Hz).

after 16 s of shaking. A continuous, accumulating inward displacement at a constant rate anddirection is achieved only when A=1g. The inward displacement of block AR decelerates onlyafter 70 s when further displacement is locked, as shown in Figure 14. Note that also in the realevent the structure must have locked at a certain point, and the measured displacement that waspreserved matches the modeled displacement with A=1g.

Structural response to frequency of motion is usually discussed in terms of the natural period ofthe structure, at which the structure will develop a resonance mode and collapse. Since the studiedfailures are local and not complete, each mode of failure will have its own natural period whichcan be different than that of the whole structure. The structural sensitivity to frequency, revealedin our sensitivity analyses, is significant considering that the common terminology for seismic riskevaluation uses mainly PGA (peak ground acceleration) and largely ignores frequency.



Figure 14. Normalized displacement of stone AR in the direction of the displacement vector over time(d=√

u2+v2). After approximately 70 s, the displacement stops and the structural deformation is ‘locked’.

6420 8Time (sec)

10

-3

-2

-1

0

Key B

lock

Displace

ment

(cm

)

0.5 Hz

5 Hz

1 Hz

15*Nuweiba

1.5 Hz

10 Hz

Figure 15. Influence of input motion frequency on vertical displacement of keystone at Mamshit (A=0.5g).

In the case of the keystone at Mamshit (Figure 15), a clear preference is detected for frequenciesin the range of 1–1.5Hz; only under those input frequencies the downward displacement of thekeystone is continuous, and accumulates more than 3 cm of displacement, similar to the amount ofdisplacement measured in the field. Lower or higher frequencies result in other modes of failuresuch as oscillations in the case of low frequencies (e.g. 0.5Hz) and ‘locking’ of the structure inthe case of higher frequencies (e.g. 5–10Hz). A very important observation is made when a realearthquake record, that of Nuweiba 1995 (Figure 16), is used as input motion. The original record,de-convoluted to rock response [8], is amplified by 15 in order for its PGA to reach the sameamplitude of the synthetic motions used for the results that are plotted in Figure 15, namely 0.5g.Although the Nuweiba quake loads the structure with a wide range of frequencies and with twosimultaneous components of motion (horizontal and vertical), the structural response to the natural



-0.1

-0.05

0

0.05

0.1

0.15

0 10 20 30 40 50 6time (sec)

a (g

)

0

g(z)

g(EW)

g(NS)

Figure 16. The Nuweiba 1995 record after de-convolution to rock response. The rectangle marks the 10 sthat were used for the analysis of the Mamshit block system.

Figure 17. Influence of frequency on relative spatial displacement ofBlock AR at Nimrod Fortress (A=1g).

quake is very similar to that of the simple sinusoidal ones, though more moderate. This findingstrongly suggests that the results of the sensitivity analysis, using synthetic records of horizontalmotion only, are valid enough to be further discussed. A similar observation is made in Figure 17,which shows that the mode of failure of the arch at Nimrod Fortress has a clear preference to afrequency of 2Hz over both 1 and 3Hz.

4. DISCUSSION

4.1. Limitations of our approach

The suggested method uses a 2D numerical model with simple deformable blocks, meaning thatstresses and strains are identical everywhere in the block. Although the mode of failure in DDA is aresult of the analysis and not an assumption, the 2D constraint imposes limitations on the mode of



failure, for example, it does not allow for out-of-plane rotations, etc. To minimize this constraint,the analysis was performed on planar structures such as arches, where most of the deformationis concentrated in the plane of analysis. An analysis of a rectangular structure, such as a tower,could have been significantly biased by the 2D constraint. Moreover, the ‘simple deformable block’assumption may cause small deviations in the energy involved in the deformation, since in realitystress concentrations at block corners may result in block cracking or chipping, a mechanicalprocess that cannot be modeled in this version of DDA. We assume, though, that an energydissipation of 1%, which is implemented through the numerical kinetic damping, is enough tocompensate for such losses [18].

Another limitation of this study is the use of a synthetic sinusoidal loading function as directinput, consisting of a single amplitude and frequency value. This is obviously not the case in areal earthquake which consists of a train of frequencies at different amplitudes. Nevertheless, theresults presented in Figure 15 indicate that the structural response in this case does not vary allthat much when a real earthquake record is used as input.

4.2. Applicability of our results

The obtained PGA and frequency values in this research can be compared with Israel BuildingCode # 413 [2], generated largely on empirical grounds by the Geophysical Institute of Israel.IBC 413 predicts PGA values of 0.11g for Mamshit and 0.25g for Nimrod Fortress. Although ourresults are significantly higher, it can be argued that at least two amplification factors can explainthe discrepancies: structural and topographic.

The loading function in the numerical analysis is inserted into all the blocks simultaneously,namely, the analyzed block is subjected to a uniform structural input motion. The building codehowever relates to bedrock PGA, which may be amplified significantly at the top of the structuredue to inherent structural amplifications as well as rock–structure interactions. Common responsespectra predict amplification of up to 2.5 [24], due to wave propagation phenomena that take placewhen the motion is transferred from bedrock to the structure. Therefore, the PGA values found byinversion in our study must be reduced by a factor of 2.5, resulting in bedrock PGA predictionsof 0.2–0.5g and 0.4–1g for the Mamshit and Nimrod sites, respectively. No further modificationsshould be made for the PGA value obtained for Mamshit, because topographic effects are notexpected in this site.

In Kalaat Nimrod, a further modification should be made due to the possible topographic effectat that site, the precise magnitude of which is not known and must be measured in the field. Inthe absence of such data we can only suggest that the actual PGA value for the rock foundationsat Kalaat Nimrod should be lower than 0.4g by up to a factor of 4, if we adopt the results of thetopographic site effect measurements at four different sites in Israel [25].

5. SUMMARY AND CONCLUSIONS

DDA method was validated with respect to analytical solutions for two separate cases of a‘Newmark sliding block’ type of analysis: in one, a block is subjected to time-dependent inputacceleration applied at its center of mass, and in the other, the block responds to a time-dependentinput displacement applied to an underlying block. In both simulations, DDA results matched theanalytical solutions very well.



The structural deformation pattern observed in the two case studies was duplicated very nicelywith DDA, with localized failure of particular blocks in the mesh-matching field observations.Both deformation pattern and displacement magnitude were duplicated accurately in two differentstructural geometries and in two different sites.

For the semi-circular arch in Mamshit, DDA results imply that most of the damage tookplace during the first two seconds of motion. Our best estimate for the structural accelerationparameters for Mamshit is f =1Hz, A=0.5g. Considering structural amplifications these resultscan be downscaled to f =1Hz, A=0.2g. The obtained amplitude is similar to PGA=0.11gpredicted by Israel seismic building code for that region. Interestingly, introduction of verticalmotions using a real earthquake record does not change the obtained deformation picture verysignificantly, justifying our simplified methodology of using synthetic sinusoidal input motionswhich are restricted to the horizontal axis only.

For the asymmetric arch at Nimrod our best estimate for the structural acceleration parametersis f =2Hz, A=1g. Considering both structural and topographic amplifications, the actual rangeof the paleoseismic PGA values can be between 0.1g and 0.4g. Assuming structural amplificationof 2.5 and topographic amplification for this elevated site of 2.0, the expected paleoseismic PGAis 0.2g. This estimation can be compared with Israel seismic building code prediction of 0.25gfor this region.

Further research is required concerning failure modes associated with application of the loadingfunction at the foundation of a multi-block structure. The application of this loading mechanismwill generate true structural amplifications at the top of the structures and will enable more rigorousconclusions regarding paleoseismic PGA values.

ACKNOWLEDGEMENTS

We thank Gen-Hua Shi for modifying his latest version of the DDA code for the purpose of this research,Yuli Zaslavski for the de-convoluted Nuweiba earthquake record, Shmuel Marco and Itai Leviathan fordiscussions, and two anonymous reviewers for critical reading and instructive comments that greatlyimproved this paper.

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