Dynamic behaviour of drystone retaining walls: shaking table scaled-down tests

Abstract In this paper, an experimental study aiming at understanding the seismic behaviour of dry stone retaining walls is presented. Harmonic shaking table tests have been carried out on scaled-down dry-joint retaining walls involving parallelepiped bricks. It is found that a thicker wall is more resistant and that a given retaining wall is less sensitive to higher frequencies. For those higher frequencies, the walls accept larger displacements before collapsing. The displacements start to occur from a given threshold, which depends on the wall geometry but not on the frequency of the base motion. The typical toppling failure is observed for slender wall and/or low frequency inputs. For less slender walls or higher frequency inputs, walls experience local sliding failures until the complete collapse of the system. The acceleration at failure reported during the dynamic tests has been compared to the corresponding pseudo-static resistance, enabling a conservative estimate of the seismic behaviour coefficient for pseudo-static analysis of this class of retaining walls. This novel experimental dataset is aimed to serve as a validating framework for future numerical or analytical tools in the field.


Introduction
Dry stone retaining walls (DSRWs) are present worldwide and constitute an important cultural heritage for numerous territories. Today, this old heritage is generally damaged due to weathering but also due to a lack of maintenance. In France, most of the DSRWs have been erected before World War I and no more walls were built after World War II. Since this damaged heritage can be considered as an economical asset for these territories (tourism, agriculture, transportation), the local authorities have some great concerns in relation to their conservation. The required interventions on this heritage are however difficult to make in a context where the behaviour of DSRWs is not completely known.
A lot of studies have been carried out on DSRWs since the first known experimental work performed by Lieutenant Sir John Burgoyne in 1834 (Burgoyne, 1853) who studied the impact of the section geometry on the resistance of DSRWs. These first experiments have been afterwards modelled using analytical methods (Capper & Cassie, 1969;Clayton et al., 2014;Husband & Harby, 1911) or numerical models using the Discrete Element Method (DEM) (Claxton et al., 2005;Harkness et al., 2000;Powrie et al., 2002). At the time, DEM codes had already been used to successfully model the free standing walls of Great Zimbabwe site (UNESCO) (Dickens & Walker, 1996;Walker & Dickens, 1995). Further experiments on slope DSRWs were carried out much more recently when Villemus et al. (2007) and Colas et al. (2010Colas et al. ( , 2013a aimed at finding the critical height of a backfill that triggers the collapse of a DSRW. They developed analytical methods (Colas et al., 2008(Colas et al., , 2013bVillemus et al., 2007) to create charts included in design guides for the French dry stone masons (CAPEB et al., 2008;ENTPE et al., 2017). These works have been complemented by analytical refinements (Terrade et al., 2018) as well as more sophisticated numerical models (Oetomo et al., 2016). In other countries, the behaviour of slope DSRWs is also a new field of research (Alejano et al., 2012a(Alejano et al., , 2012bPulatsu et al., 2020;Walker et al., 2007). In addition, other experimental (Le et al., 2019;McCombie et al., 2012;Mundell et al., 2009Mundell et al., , 2010, analytical (Le et al., 2016) and numerical (Quezada et al., 2016) studies have been carried out on 3D failure of DSRWs induced by a concentrated load on backfill top.
All these studies focused on the static behaviour of DSRWs. However, the assessment of the seismic stability of these structures is a necessary step to guide the repair of old structures, as well as to design new ones, since they may be located in seismic zones. Fukumoto et al. (2014) carried out an experimental study on the seismic behaviour of DSRWs but focused on the optimal shape of dressed stones for the wall seismic resistance through centrifuge scaled-down tests. A pseudo-static study on slope DSRWs was carried out by Savalle et al. (2018aSavalle et al. ( , 2018b with the aim at developing a practical analytical tool for the design of DSRWs in seismic areas. Although seismic recommendations are given for the design of slope DSRWs in low seismic risk zones, their method failed to give realistic recommendations for higher seismic risk zones, as pseudo-static approaches are known to be over conservative in that case (Bellezza, 2014;Richards & Elms, 1979;Savalle et al., 2018a), thus requiring a displacement design approach (Gaudio et al., 2018;Huang et al., 2009;Newmark, 1965). Therefore, dynamic approaches must be followed to get a better insight into the seismic behaviour of slope DSRWs.
Dynamic studies using shaking table tests are common in geotechnical engineering: numerous studies of slope stability Lo Grasso et al., 2004;Huang et al., 2010;Srilatha et al., 2013;Sugimoto et al., 1994) and of unreinforced (or reinforced) retaining walls (Bathurst, 2002;Bathurst et al., 2002;Fairless, 1989;Huang et al., 2009;Iai, 1989;Ishibashi & Fang, 1987;Koseki et al., 1998;Wilson & Elgamal, 2015) have been carried out in the past thirty years. However, none of them dealt with the specificity of DSRWs, where blocks can move during the base motion and hence can dissipate energy before collapsing.
In order to pave the way to numerical and analytical dynamic studies, a vast and novel experimental campaign of shaking-table tests involving dry stone retaining walls is envisioned in this work. First, shaking table tests on scaled-down dry-joint brick retaining walls are presented. Then, the influence of the slenderness of the wall as well as the frequency of the ground motion on the behaviour of the retaining walls are investigated. Finally, a comparison with pseudo-static calculations is conducted and a conservative value for the seismic behaviour coefficient r is proposed.

Experimental set-up
2.1. Shaking-table tests 2.1.1. Principle of the tests Herein, more than forty tests on a shaking table have been carried out using scaled-down models of dryjoint brick walls retaining a sandy backfill. The walls were erected using small paralelepiped shaped clay bricks. The mock-up was loaded by a horizontal harmonic ground motion whose amplitude was increased from 0 until the collapse of the wall.
One could note that the parallelepiped bricks utilised in this work are quite different in shape from the undressed blocks used in the field. This latter point has already been studied by other authors. First, Fukumoto et al. (2014) found that, using a trapezoidal shape for stones, the results of centrifuge tests were actually improved compared to more simple shapes like parallelepipeds. However, they attributed this improvement to the change in the backfill-wall interface friction angle and not directly to the shapes of stones. In addition, all numerical and analytical static studies modelling the actual irregular blocks by parallelepiped blocks have been proved relevant to found the key parameters driving the behaviour of actual dry stone walls as well as their ultimate capacity (Claxton et al., 2005;Colas et al., 2008;2013b;Dickens & Walker, 1996;Harkness et al., 2000;Mundell et al., 2009;Oetomo et al., 2016;Powrie et al., 2002;Pulatsu et al., 2020;Savalle, 2019;Villemus et al., 2007). It means that, as long as the dry stone walls are constructed following the state of the art of dry stone construction, the shape of the blocks does not seem to play a significant role on the wall stability.

Experimental devices
The erected retaining walls were B ¼ 34 mm wide, L ¼ 400 mm long and H ¼ 50-140 mm high and were made of clay bricks of dimensions: height Â width Â length ¼ 11 Â 17 Â 34 mm. They were placed in the aperture of a container of dimensions height Â width Â length ¼ 250 Â 400 Â 400 mm (Figure 1). To reduce the significance of the two walls ends (potentially the most vulnerable parts), the wall length L has been chosen sufficiently large. Indeed, according to Dewoolkar et al. (2000), above a wall ratio L=H ¼ 2:0, end effects do not alter the plane strain conditions.
The backfill was made of a dry Hostun sand S28 1 , filling the container and having an equivalent volume of height Â width Â length ¼ 50-140 Â 400 Â 400 mm (Figures 1 and 2). Using an analytical pseudostatic approach, the backfill width W (Figure 2) was designed so that the failure surface in the backfill did not interfere with the vertical back wall of the container (Savalle et al., 2018a). This aspect was validated throughout the shaking-table tests.
The whole mock-up (30-60 kg) was placed on an electrodynamic shaking table of dimensions 60 Â 60 cm that can handle accelerations of 8 g for loads of 600 N (Serial number: Sonic S201LS3-340/ SPA203). In order to absorb harmonic waves reaching or originated from the vertical back wall of the container, this latter was covered by an absorbing foam which was intended to reproduce free-field conditions ( Figure 2). Moreover, a gap was introduced between the brick wall ends and the lateral container walls to avoid any disturbing friction at this location. Two plastic sheets were placed in this gap to prevent the seepage of the backfill sand ( Figure 1). The first layer of the wall was fixed in order to model a rigid foundation with an infinitely resistant abutment (Figures 1 and 2). In the following, the wall height (H) refers to the height of the free wall and does not take into account this first fixed layer. Therefore, the base friction of the wall was the one of the brick-brick contact itself. In addition, a Linear Variable Displacement Transducer (LVDT6 in Figure 2) was used to monitor the settlement of the backfill. Then, a laser sensor (serial number: FT 50 RLA-40-F-L4S of SensoPart company) was used to measure the displacement of the second to last row of bricks. It will be used later as an indicator of failure for the wall.
Finally, eight accelerometers (ACC) with a minimum precision of 10 mV/g have been placed at different locations on the container to quantify its 3D dynamic response ( Figure 2). The accelerometer labeled ACC2, located on the floor of the container, was a special low-frequencies sensor that has been used as an input for the shaking-table tests (special precision of 1000 mV/g). The accelerometer labeled ACC10 was a triaxial sensor and all the other sensors (ACC3, ACC4, ACC5, ACC7, ACC8, and ACC9) were mono-directional and oriented in the direction of shaking (X ), perpendicular to the wall face ( Figure 2).

Preliminary results
It has been observed that the container had not a simple response with neither fully rigid nor fully deformable pattern which would not make future numerical simulations easy. Then, an analysis of the response of the container is presented in this part.
Using a white noise input signal given to the base of the model (ACC2), the Transfer Functions 2 of the responses of all other accelerometers have been analysed (Figure 3). Though only the response of the empty container is shown, the response of the container filled with different backfill heights (5,8,10,15 and 20 cm) has been found similar. It is observed that in the range of frequencies between 10 and 30 Hz, the container can be likened to a rigid container. Indeed, all accelerometers in the X-direction (ACC3, ACC4, ACC5, ACC7, ACC8, ACC9 and ACC10X) had a frequency response equal to the one of the input signal, while there were no junk vibrations in the other directions (ACC10Y and ACC10Z). Analysing the data from sensors ACC4 and ACC9 (not placed at the top of the container), and considering that the top surface of the backfill was never above the sensor ACC9, the mock-up can be considered as rigid up to the investigated frequency of 70 Hz.
Then, the response of the backfill itself has been studied. The first resonant frequency of the backfill layer has been evaluated using the classical free-field shear layer model of soils which gives the frequency of the first mode: with H the height of the backfill and C s the speed of shearing waves in the backfill, given by with G the shear modulus of the sand, E its Young's modulus, , its Poisson's ratio and q its unit mass. The elastic modulus E ¼ 18MPa was evaluated using the formula from Biarez and Hicher (1994), e ¼ 0.72 being the sand's void ratio and p 0 ¼ 0:7kPa the confining pressure: Using a classical Poisson's ratio of ¼ 0:3, a backfill height H ranging between 5 and 15 cm and the experimental unit weight of q ¼ 1540kgm À3 , the first eigen-frequency of the backfill is expected to range between 100 and 300 Hz. In addition, one should note that the validity of Eq. (3) for low confined soil is not established. Indeed, close to the soil surface, this equation predicts an excessively deformable material which does not hold any physical meaning. Therefore the actual value might be larger than the predicted one. Therefore, even at 70 Hz, the resonance of the sand layer is not likely to be triggered throughout the experiments. However, this aspect could not have been checked, since the precision of the laser sensor (%0:1 mm) was too poor compared to the amplitude of the signal (%0:01 mm at 70 Hz).

Description of the protocol
First, the wall was erected in the aperture of the container. Then, the sand was carefully deposited from a zero drop-height in order to obtain an homogeneous loose backfill. As the backfill was expected to settle due to the subsequent shakings, a preliminary step for all shaking-table tests was applied to stabilise it to a final state, turning the loose backfill into a dense backfill. 3 It consisted in applying a 5-min duration white noise to the wall-backfill system (constant spectral density of 0.001 g 2 /Hz in the range of [3-300 Hz] leading to a mean acceleration of 0.6 g). This white noise input has been found sufficient to achieve a dense and stable state for the backfill that would not evolve afterwards. The wall was maintained in place by a wood panel fixed to the container, so that no failure of the wall could happen throughout this preliminary stage.
An actual seismic signal is a combination of multiple harmonic waves, whose amplitude is varying in time but also in frequency. In a research context, this complex motion needs to be first simplified in order to understand the basic dynamic features, instead of using actual earthquake recordings where the influence of all parameters is difficult to extract. Therefore, in a first step, the authors chose harmonic waves as input motions in the shaking-table tests, before more complex signals that will be addressed in a future work. Four frequencies compatible with the response of the container have been used ( Figure  3): 10, 30, 50 and 70 Hz, which is in accordance with the classical frequency scaling law (Fairless, 1989;Huang et al., 2010;Ishibashi & Fang, 1987;Krawinkler, 1988): where f scaledÀdown is the frequency of shaking in the scaled model, f fullÀscale is the frequency of shaking at full scale and K is the scale ratio (approximately equal to 1=20 ¼ 0:05 in these experiments). Given that The equation of the harmonic ground motion S(t) is given by with A a constant (see Appendix A for details about its values), t the time throughout the experiment and x the angular frequency defined as x ¼ 2pf , f being the frequency of the motion (Figure 4). The duration of the tests was 30 s, which is a typical, though long, duration for an actual quake.

Material properties of the mock-up
Retaining brick walls had a width of B ¼ 34 mm and a variable height H (50-140 mm), enabling the slenderness of the studied walls to range between 1.3 and 4. Only one wall has been erected using a bigger width: B ¼ 51 mm and a height of H ¼ 104 mm. A typical assemblage is presented in Figure 5. The detailed assemblages used to construct the walls can be found in Savalle et al. (2020). They were based on dry stone state-of-art rules and enabled stable masonries: headers were placed frequently and all the bricks were installed with staggered joints. In this study, the mechanical parameters were derived from previous experimental works that used the same materials (Amirouche, 2008;Quezada et al., 2016;Savalle et al., 2018a). Table 1 gives all the data in relation to the materials. The mean unit weight of the wall as well as the unit weight of the backfill were measured during the experiments and were found to be equal to 14.4 and 15.4 ± 0.1 kN/m À3 , respectively.
The internal friction angle u 0 sand of the backfill has been evaluated using several indirect approaches since typical geotechnical tests can hardly be used due to the very low pressures involved in these experiments (approximately p 0 ¼ 0:7 kPa at mid-height). Moreover, in the literature there are only few studies focused on low confined sand, and no specific ones on the S28 Hostun sand at dense state. Yet, using literature's analytical expressions and the experimental confining pressure p 0 ¼ 0:7 kPa, a value of u 0 sand ¼ 55 can be found (Appendix B). Given that this value is out of the range of typical values for this material, two further techniques were envisioned to give an estimate of the internal friction angle u 0 sand : The first one consisted in adding a raised floor ( Figure 6) and a vertical wood panel instead of the brick wall. After filling the container with the loose sand, the container was shaken by the same white noise as herebefore. Then, while removing the wood panel, a failure surface developed in the backfill and dense sand flowed out of the container. Along this failure surface, it has been observed that the sand experienced strong shearing strains leading to the generation of dilative volumetric strains. Therefore, the obtained sandy slope corresponded to the angle at the critical state which is close to the angle of repose for loose S28 sand ( Figure 6). Then, a layer (approximately 4 cm) of loose sand laying on the dense sand slope (Figure 7) was carefully removed with a brush. Out of three different tests, nine measurements of the slope angle were performed leading to an average value of u 0 sand ¼ 45 62 : The second experimental approach consisted in using an inverse method with the help of results coming from a previous study (Savalle et al., 2018a). In that study, an analytical tool was developed to predict the tilting angle at failure of an identical mock-up made of a brick wall retaining a sandy backfill. The only difference arises from the density of the backfill which was in a loose state. To evaluate u 0 sand , the container was first filled with the brick wall (H ¼ 8 cm and B ¼ 3.4 cm) and with the loose Hostun S28 sand. Then, the preliminary process that leads to a dense backfill was applied. At the end of the shakings, the mock-up was carefully placed on a tilting platform. After the removal of the vertical wood panel that maintained the brick wall stable, the mock-up was progressively inclined and the tilting angle at failure for the wall was measured. Two tests were carried out for repeatability. Using a trial and error method for the analytical method, the unknown being the internal friction angle of the backfill, values equal to u 0 sand ¼ 41:5 and u 0 sand ¼ 43 were found. These values are close to the internal friction angle evaluated with the previous experimental approach (45 62 ) and are both very different from the one estimated from the literature analysis. Therefore, in the following, an internal friction angle of u 0 sand ¼ 45 was considered. The interface (wall-backfill) friction angle d sand has not been experimentally evaluated but was deduced from past results. Indeed, in a previous study, Savalle et al. (2018a) showed that the interface friction angle can be related to the backfill internal friction angle u 0 sand through the relationship: which is close to the common value of two third typically considered for this kind of interfaces. Thus, the interface friction angle d sand was such that:

Analysis of the experimental data
To get back the acceleration corresponding to the triggering of the wall failure, both the data of the laser sensor ( Figure 2) and the data of the harmonic waves are required. The laser sensor registered the displacement of the second row of bricks from top wall (an example is shown in Figure 8). The time t fail associated to the wall failure has been defined as the time required to reach a given displacement threshold, herein defined as 5% of the wall height H (dashed line in Figure 8). This definition is consistent Symbol Ã refers to Amirouche (2008); symbol ᭛ to Quezada et al. (2016); symbol † to Savalle et al. (2018a) and symbol ᭝ to the present study. with the definition of Wu and Prakash (1996) who considered that an acceptable displacement for a rigid retaining wall is generally below 2% of its height, while a displacement characterising its failure is about 10% of it. In this study, the wall being discrete in nature, a smaller value of 5% was chosen as a limit value corresponding to failure. Then, the amplitude A of the sinusoidal acceleration at time t fail was found as a root mean square value of the imposed time-acceleration S(t), recorded by ACC2, in a time window of duration T ¼ 1=f centred on t fail (f being the frequency of the input signal used for the particular test: 10, 30, 50 Hz or 70 Hz). 4

Repeatability of the tests
To have more insight into the repeatability of the experiments, two different walls (base width of 3.4 cm and height equal to 7.9 cm or 11.3 cm) have been studied. Three identical tests were carried out for each of the signal frequencies of 10, 30, 50 and 70 Hz for the wall of 7.9 cm height following the protocol described in Sect. 2.2.2. This repeatability test was restricted to the frequency 10 Hz for the wall with 11.3 cm height. Table 2 gives the acceleration at failure for all the repeatability tests. One can note that the tests are quite repeatable for all frequencies with a coefficient of variation of about 10%. The higher value of 11.2% was found for the imposed frequency signal of 70 Hz. In the following, the study focuses on frequency 10 Hz, as seismic frequencies are usually low: 10 Hz in the model corresponds to actual seismic frequency of 2.2 Hz (Eq. (4)).

Influence of H/B
The influence of the wall slenderness (i.e. its aspect ratio) on its dynamic resistance was first studied. Ten wall geometries with slenderness ranging from H=B ¼ 1:3 to H=B ¼ 4:0 have been tested. Nine of them had a base width of B ¼ 34 mm and various heights, ranging from H ¼ 44-135 mm and one of them had a width of B ¼ 51 mm and a height of H ¼ 104 mm. Figure 9 gives the results in terms of acceleration at failure time t fail : A significant decrease of the wall dynamic resistance with respect to its slenderness can be observed, which was expected. The test with a thicker wall enabled to have arrangements of bricks with no full headers, which is more representative of actual arrangements in DSRWs. Its height was chosen in order to obtain a typical slenderness of H=B ¼ 2:0: This first test (though needed to be confirmed by other tests) leads to the conclusion that slenderness is the main geometrical factor responsible for the dynamic resistance of dry-stack retaining walls, which has also been noticed for pseudo-static tests (Savalle et al., 2018a(Savalle et al., , 2018b. Moreover, the two typical failure modes of DSRWs were experimentally retrieved: sliding (crosses in Figure 9) occurred for the less slender walls whereas toppling (circles in Figure 9) occurred for the more slender ones. A mixed mode of failure was obtained for walls with slenderness ranging between 2 and 2.7 which is the typical range of actual DSRWs. Figures 10 and 11 give snapshots of two walls through the development of toppling and sliding modes of failure. Generally, toppling was easier to identify compared to sliding since for the former type of failure, the wall kept a monolithic behavior until collapse. It was not the case for the sliding mode where local relative displacements occurred in different locations. Walls were considered to experience a sliding mode of failure when the top bricks slid and fell one by one until the total collapse of the retaining wall.

Influence of the motion frequency
In a second study, the influence of the frequencies of shakings has been investigated. For three different walls with three different slendernesses which are typical range of slendernesses of actual DSRWs     (H=B ¼ 2:3, H=B ¼ 2:7 and H=B ¼ 3:3), shaking table tests have been carried out for four signal frequencies 10, 30, 50 and 70 Hz. The results are shown in Figure 12, while full details can be found in the Appendix A. One can note that the walls resistance increases dramatically with the frequency of the base motion. Moreover, for the slenderness of 3.3, the mode of failure was found independent of the signal frequency. It is not the case for the two other studied slendernesses which were smaller and where sliding was generally triggered for higher frequencies. As previously mentioned, sliding generally occurs with local failures of bricks. However for slenderness H=B ¼ 2:7 and frequency 70 Hz, a monolithic slide started to occur at the bottom right corner wall and then a toppling mode of failure propagated through the wall (Figure 13). One could note that this failure did not develop following plane strain assumptions, on the contrary to the vast majority of toppling failed walls. This can easily be explained by local material and geometrical heterogeneities, and especially at the two ends, leading to more vulnerable parts.
The increase of resistance with the frequency may be linked to the following points: The duration of each cycle of loading is shorter (though more frequent), The velocity amplitude is smaller; indeed the Peak Ground Velocity (PGV) is known to be a relevant parameter to characterised the destructive power of a seismic loading (Richards & Elms, 1979), The maximum dynamic earth pressure is lower due to the decrease of the wave length of the input signal with the increase of the frequency (Baziar et al., 2013;Bellezza, 2014;Pain et al., 2015). Indeed, though at f ¼ 10Hz, the wave length is approximately equal to k ¼ Cs f %6m which is much larger than the wall height (H%10cm), at f ¼ 70Hz, the wave length decreases to k%0:9m: For a wall of height H%10cm, the local dynamic earth pressure may not be any more constant along its height. The smaller displacements (at each cycle, as well as total displacements) are more endurable by the structure.
To illustrate this latter point, the author proposes hereafter an analysis of a single brick based on Newmark's theory (Newmark, 1965). Newmark's analysis is here intended as an explaining tool for the increase of failure acceleration with the increase of the frequency of the base motion. Basically, Newmark's theory consists in integrating twice the acceleration record. Therefore, it leads to a displacement amplitude divided by x 2 (x being the angular frequency in rad=s), compared to the acceleration amplitude. In order to obtain the total displacement, we sum the displacements occurring for each cycle, thus multiplying one cycle's displacement by the number of cycles, which is proportional to the angular frequency x. In the end, the total displacement is inversely proportional to the frequency f of the base motion. The following developments give complementary details.
For the sake of simplicity and in a context where just qualitative results are expected, a very basic modelling is chosen, neglecting for instance the earth pressure. The analysis is focused on the stability of one brick of the last layer, since it has been observed that in case of sliding failures, bricks of the topmost layer fell the first. The pseudo-static critical acceleration a max is taken as the tangent of the brickbrick friction angle. Then, a Newmark's scheme is adopted, where at every moment the input acceleration S(t) is bigger than the threshold a max ¼ g Ã tan ðu 0 brick Þ, the total displacement of the studied brick d is computed with a Matlab home code integrating twice the unbalanced acceleration SðtÞ À a max : Since the brick can only move if the acceleration is directed outward, the double integration is carried out only for SðtÞ>a max and not for SðtÞ< À a max : Using the parameters of the experiments (Eq. (5) with a max ¼ 0:62g and A ¼ 1:0g s À1 ), one can obtain the following fitted relationship: with f the frequency of the signal and t the reference time of the test. The numeric values (0.18, 33 and 2.75) enable to fit the original curve, which basically contains steps associated to stable phases. These numerical values depend on the parameters a max and A, though no simple relationships have been found. One could notice however that using constant amplitude harmonic signal, the value 33 is replaced by 0 and the 2.75 one is replaced by 1.
During the experiments, the failure has been detected through a displacement criteria d crit ¼ 5% Ã H: Then, inverting Eq. (8) for a given wall height H, one can find that the failure time t fail is a root function of the frequency f of the input. Therefore, it partially explains why sliding failures were triggered later for higher frequency inputs.
In Figure 14, the displacements measured throughout the tests for the three walls subjected to the four frequency input signals have been plotted. Firstly, it is noted that irrespective of the frequency, noticeable wall displacement (above 1 mm) occurs at the same moment, i.e. at the same acceleration level. Therefore, until a certain acceleration level, the input frequency does not play a significant role: the dynamic behaviour of the wall is not yet mobilised. This acceleration threshold only depends on the wall's static resistance and is slightly higher than the pseudo-static resistance of the wall. Secondly, the rate of increase of the displacement through time is slower for higher frequencies. In addition, one can note that though the wall shaken by 10 Hz harmonic waves can indeed be considered as near collapsing when reaching the 5% displacement threshold, this is not the case when it is shaken with higher frequency signals. In fact, the behaviour of the wall shaken by 10 Hz waves is a rigid one with no displacement in a first phase until a threshold after which huge displacements and brittle failure occur. Conversely, the larger the frequency is, the more ductile its response seems to be: walls can handle huge displacements without collapsing and these displacements are noticeable well before the wall collapse. This capacity to experience stable irreversible displacements is clearly linked to the capacity of the wall to dissipate seismic energy, and thus to resist to an earthquake. This is particularly interesting for actual DSRWs, where their actual resistance could be much higher than the one evaluated through classical pseudo-static approaches.
To conclude, the retaining wall seems to be less vulnerable to higher frequencies. This extra resistance is permitted by small and stable relative movements of the bricks and/or the whole wall. These ones are irreversible, enabling to dissipate the energy of the input signal. Analysing again the experimental results of Huang et al. (2010), sand slopes seem also to be more resistant to higher frequency motions, though this had not been highlighted by the authors. Huang et al. (2010) carried out shaking-table tests on three identical sand slopes using tapered input waves of growing amplitude. The three sand slopes were subjected to three different frequency waves: 3, 6 and 15 Hz. From Figure 6 of their article, it appears that for higher frequency tapered waves, larger amplitudes were required to trigger the slope failure.

Comparison with a pseudo-static study
The results obtained throughout the experimental dynamic tests have been compared to results obtained throughout a pseudo-static approach of the problem in order to evaluate how much this latter is conservative. The pseudo-static method considered herein is based on the limit equilibrium of a Coulomb's wedge of soil, which includes the pseudo-static force modelling the seismic action. Then, the stability of the wall is checked according to two possible modes of failure, sliding and toppling. This method has been validated on pseudo-static experiments (Savalle et al., 2018a). Table 3 gives the relative difference, denoted gap, between the acceleration at failure for the dynamic approach (for 10 Hz input signals) and the one derived analytically from the pseudo-static approach according to the wall slenderness.
gap ¼ acc dynamic ð10HzÞ À acc pseudoÀstatic acc pseudoÀstatic First, the results confirm that the pseudo-static approach (which is equivalent to a 0 Hz frequency input motion) is generally conservative since almost only positive values were obtained. The special case where slenderness is equal to 1.3 should require more tests before confirmation. When the walls tended to fail by toppling (high slendernesses), the relative difference lies around 45%, which is very significant. On the other hand, when the walls tended to fail by sliding (low slendernesses), the relative difference falls down to 15%. Typical slendernesses for DSRWs ranges between H=B ¼ 2 and H=B ¼ 2:7 where the relative difference decreased from 43% to 12%. Then, depending on the properties of the studied DSRW, a pseudo-static approach could be just conservative or very conservative.
In fact, these experimental results have to be analysed in the light of the conclusions made previously: Increasing the frequency of the motion increases the resistance of a dry-joint retaining wall, and since natural seismic motions usually have a frequency content that extends beyond the frequency of 2.2 Hz (equivalent to the 10 Hz frequency input in the shaking-table tests), the relative difference gap may be most of the time higher than those noted in Table 3 (Table 4). From some other tests not shown here, it has been noted that the larger the duration of the base motion was, the lower the wall's resistance was. Earthquakes often last less than 30 s, which is the average duration of the harmonic input used herein. Therefore, the actual seismic resistance of slope DSRWs may probably be higher than that identified in this work.
Finally, a seismic behaviour coefficient r ¼ 1.2 can be proposed from the present experimental campaign. This can already be used in conjunction with a pseudo-static approach. As explained above, this value can still be considered as too conservative in many cases. Therefore, further numerical simulations involving true seismic motions would be helpful to support a higher value for this coefficient. On the other hand, one could conduct Newmark's analysis on the displacement of the retaining wall, as it is commonly performed for geosynthetic reinforced earth retaining walls (Gaudio et al., 2018;Masini et al., 2015). For these particular systems, since it has been found that the failure mode predicted by pseudo-static analysis was similar to the one developed throughout dynamic simulations (Masini et al., 2015), pseudo-static methods can be very efficient for an in-field application, since less time-consuming than dynamic simulations. However, back to DSRWs, the validity of Newmark's analysis for this class of retaining walls still needs to be demonstrated, and above all regarding the ability of Newmark's method to handle toppling failure modes.

Conclusion
The seismic behaviour of slope dry stone retaining walls has been studied through a simplified case of harmonic shaking-table tests where the input acceleration is a sinus associated to a single frequency. Experiments have been carried out on a model of actual dry stone retaining walls composed of parallelepiped bricks. Their resistance to shaking has been investigated for input signals of increasing amplitude, with different harmonics.
As expected, their resistance was found decreasing with slenderness. Classical toppling failures have been observed for slender walls, while less slender ones experienced local failures of bricks that could be akin to a sliding failure. As long as the frequency of the harmonic input wave increases, the wall's resistance increases as well and the toppling mode of failure is progressively replaced by sliding ones. It has been noted that a dry-stack retaining wall can experience small local relative displacements throughout the shaking, particularly when subjected to higher frequency motions. It may be an advantage since relative displacements contribute to dissipate energy. Based on these experimental results and on a pseudo-static approach, the seismic Table 3. Comparison between resistance to 10 Hz harmonic motion and pseudo-static resistance of dry-joint walls depending on the slenderness of the wall. The failure mode, sliding (S) or toppling (T), is also depicted. behaviour coefficient has been derived for the experimental walls. A reasonable value of r ¼ 1.2 can be selected for a pseudo-static analysis of DSRWs. Finally, this experimental campaign paves the way for future studies in the field, either for numerical computations furnishing a complete, detailed and wide experimental data set. It also sets the basis for similar experiments with a consolidated and accurate protocol. To conclude, the contribution of the paper to the field can be summarised as follows: 1. It provides the first experimental data set of shaking-table tests involving dry-stack masonry retaining walls, which will be helpful for numerical validations. It should lead to a better understanding of the seismic behaviour of Dry Stone Retaining Walls (DSRWs). 2. It gives details about the best way to conduct shaking-table tests on retaining structures with high reproducibility, with some insights about the way to handle a densifying backfill or to identify the internal friction angle of a low confined sand. 3. It brings to light the capacity of dry-stack retaining walls to experience small stable relative movements, and especially when the frequency of the input signal is high. It leads to a greater seismic capacity of walls for higher frequencies. 4. It evidences that for a given wall, irrespective of the frequency of the base motion, noticeable displacements starts to occur for the same acceleration threshold. 5. It identifies a first and conservative seismic behaviour coefficient for pseudo-static studies of DSRWs, which is equal to 1.2. This behaviour coefficient is generally sensitive to the wall slenderness, the duration of the motion or the frequency content.

Notes
1. The S28 sand had minimum and maximum void ratios equal to e min ¼ 0.689 and e max ¼ 1.031 (Combe, 1998). 2. The Transfer Function corresponds to the Fourier Transform of the response of an accelerometer divided by the Fourier Transform of the input signal (ACC2), which is herein characterised by a plateau between 3 and 70 Hz. 3. Mean void ratios and relative densities of e loose ¼ 0.97, R D, loose ¼ 19%, e dense ¼ 0.72 and R D, dense ¼ 91% have been found throughout the tests.

A ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 Â <S 2 ðtÞ> q :  Figure A1. Example of a typical acceleration record used during the experiments. First phase (before t ¼ 5s) corresponds to the automatic calibration of the shaking-table. Then, the acceleration is linearly growing during the second phase. At around t ¼ 25s, the wall collapses, generating huge noise in the acceleration record. Finally, the test is manually stopped at t ¼ 29s: resistance. Table A1 sums up the ending accelerations of each test depending on the aspect ratio (H/B) of the wall. The relatively short duration of each test enables further numerical simulations of the experiments. Similarly, full details of the tests carried out using different frequencies are given in Table A2. In this table, the duration refers to the duration between the beginning of the test and the failure of the wall, including the 5 s of automatic calibration of the shaking-table at low-level shaking ( Figure A1).

Appendix B: Evaluation of the friction angle of the sand
In this appendix, the authors sum up a literature analysis on friction angle of low confined sand, with the aim of having some reference values. Sands are known for increasing their friction angle as confining pressure decreases (Lancelot et al., 1996). Some analytical formulas (Bolton, 1986;Maksimovic, 1989) have been calibrated on low-pressure experiments (Lancelot et al., 1996) involving Hostun RF sand, which is very close to Hostun S28 sand used in the present work (Combe, 1998). Some analytical equations have also been calibrated on experimental results for Toyoura sand (Chakraborty & Salgado, 2009). Finally, very low-pressure tests have been conducted out of gravity on Ottawa sand (Alshibli et al., 2003;Sture et al., 1998). All the results from these studies are gathered in Table B1, leading to values well above u 0 sand ¼ 50 : From these results, the value found by the authors of u 0 sand ¼ 45 is plausible. Table B1. Friction angle of dense sands at very low confining pressure according to the literature (Alshibli et al., 2003;Bolton, 1986;Chakraborty & Salgado, 2009;Lancelot et al., 1996;Maksimovic, 1989;Sture et al., 1998).

Authors
Type Sand u 0 ( ) Comments Lancelot et al. (1996) and Maksimovic, (1989) Anal. Hostun RF 55 Validated up to a minimum pressure of 20 kPa Lancelot et al. (1996) and Bolton, (1986) Anal. Hostun RF 55 Validated up to a minimum pressure of 20 kPa Chakraborty and Salgado (2009) Anal. Toyoura 53 Validated up to a minimum pressure of 4 kPa Sture et al. (1998) Exp. Ottawa 54 Mean from two tests at 0.52 kPa. Friction rapidly Alshibli et al. (2003) dropped from 1% strain, reaching 38 at 5% strain For analytical expression, these values were derived using p 0 ¼ 0:7kPa, as it was the mean pressure along the experiments.