Dynamic numerical simulations of dry-stone retaining walls: identification of the seismic behaviour factor

Citation: Savalle, N.; Vincens, E.; Hans, S.; Lourenço, P.B. Dynamic Numerical Simulations of Dry-Stone Retaining Walls: Identification of the Seismic Behaviour Factor. Geosciences 2022,12, 252. https://doi.org/ 10.3390/geosciences12060252 Academic Editors: Mohamed Shahin and Jesus Martinez-Frias Received: 8 May 2022 Accepted: 13 June 2022 Published: 17 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). geosciences Article Dynamic Numerical Simulations of Dry-Stone Retaining Walls: Identification of the Seismic Behaviour Factor Nathanaël Savalle 1,* , Eric Vincens 2,*, Stéphane Hans 3and Paulo B. Lourenço 1 1University of Minho, ISISE, Department of Civil Engineering, 4800-058 Guimarães, Portugal; [email protected] 2Universitéde Lyon, École Centrale de Lyon, LTDS, UMR CNRS 5513, 36 Avenue Guy de Collongue, CEDEX, 69134 Écully, France 3Universitéde Lyon, LTDS, UMR CNRS 5513, ENTPE, 3 rue Maurice Audin, 69518 Vaulx-en-Velin, France; [email protected] *Correspondence: [email protected] (N.S.); [email protected] (E.V.) Abstract: Dry-stone retaining walls can be found worldwide and constitute critical assets of the built heritage for many sloped territories, holding cultural and economic value. Their design currently follows empirical rules, though the first steps towards a static safety assessment have recently been proposed in the scientific and engineering literature. However, the seismic design of these structures still lacks research studies. Therefore, this work conducts discrete element simulations to assess their dynamic behaviour. First, the approach is validated through existing scaled-down shaking table experiments, and it is found that the numerical simulations are conservative (i.e., on the safe side). Next, full-scale dry-stone retaining walls are subjected to harmonic excitations as an idealisation of earthquakes. Finally, based on a simplified limit-equilibrium analytical tool, their seismic behaviour factor is estimated for the first time in the literature, which falls within the proposed values of the European standards (Eurocode 8). This will allow engineers to adopt a validated behaviour factor in practice to assess and design dry-stone retaining walls with a pseudo-static approach. Keywords: discrete element method (DEM); masonry; earthquake engineering; seismic design; standards; geotechnical engineering; slope engineering 1. Introduction Masonry is one of the oldest building materials worldwide and is currently the material adopted in a significant part of the built heritage. Many different building techniques can be found within this general typology. Among them, dry-stone masonry consists of a clever assemblage of rubble (undressed) stones without mortar. This vernacular technique can be found in many countries, including several UNESCO sites (Lavaux’s Terraces, Machu Picchu, Douro’s Valley, the Great Wall of Zimbabwe). More recently, the dry-stone technique and associated know-how have been listed as a World Intangible Heritage by UNESCO, recognising the specificities of this still living technique and its contribution to high-quality landscapes and built heritage. Dry-stone retaining walls (DSRWs), which constitute a high portion of this built heritage, are still used for agricultural or transportation purposes. One can also highlight their advantages, including their low embodied energy [ 1 ], permeability to water flow [ 2 , 3 ] and adaptability to different environmental sites. However, appropriate maintenance actions for the dry-stone heritage are missing due to insufficient knowledge concerning this building technique and, until recently, a lack of design standards [4,5]. This state of the art led many researchers to study the static behaviour of DSRWs, with different goals. While some studies aimed to reproduce the particular shapes, also called bulging phenomena, observed in the field [ 6 – 9 ], others carried out parametric analyses via the Discrete Element Method (DEM) [ 10 , 11 ]. Finally, some research groups produced Geosciences 2022,12, 252. https://doi.org/10.3390/geosciences12060252 https://www.mdpi.com/journal/geosciences Geosciences 2022,12, 252 2 of 20 useful analytical tools to assess the stability of existing structures [ 12 – 14 ]. The literature encompasses experimental [ 9 , 14 – 17 ], analytical [ 12 , 13 , 18 ] and numerical DEM [ 19 – 23 ] works, conducted either in 2D [ 6 , 10 , 14 , 24 , 25 ] or 3D [ 9 , 17 , 21 , 26 ]. Therefore, the design of DSRWs under static loading is now relatively easy to process, given that dedicated safety rules and criteria are provided [ 4 , 5 , 27 ]. If prejudice prevents broader use of this building technique, there is, however, still a lack of knowledge regarding the seismic assessment of DSRWs. Contrarily to static designs, the seismic design of such assets has been investigated only recently, with experimental, analytical and numerical pseudo-static studies [ 18 , 22 , 27 ]. Still, most of the building standards worldwide take seismic hazards into account in the design procedures of structures [ 28 ]. The most complete (but also complex) approach used to assess the seismic capacity of structures is the performance-based method. Originally introduced by Newmark [ 29 ], it is still in use nowadays [ 30 , 31 ]. It computes the displacement of a retaining wall during an earthquake motion. It then compares the response (e.g., top displacement) with the maximum allowable displacement, from a structural point of view, to draw conclusions on the acceptability of the design. This method often requires several simulations with different seismic inputs on the same structure [ 30 ] to provide statistically significant results. Thus, a simplified (and more engineering-oriented) approach is also proposed by the Eurocode standards (EC8) for the seismic design of retaining walls [ 32 ], where the seismic assessment of DSRWs is conducted using a pseudo-static method. A seismic behaviour factor (also known as the seismic response modification factor), denoted as r, reduces the reference design acceleration to account for a larger dynamic capacity, related to a reserve of ductility given by the non-linear behaviour of the system. However, such a factor has not been explicitly evaluated for DSRWs. As the structural behaviour of DSRWs (the presence of weak block interfaces) is different from that of classical concrete retaining walls, a dedicated seismic behaviour factor rfor DSRWs should be identified. Therefore, as its contribution to the current state of the art, the present work proposes a research path to characterise a relevant seismic behaviour factor rfor DSRWs. It involved scaled-down dynamic experiments on DSRW mock-ups [ 33 ] to validate a numerical DEM model before simulating full-scale DSRWs. Herein, Discrete Element Modelling (DEM) is preferred to Finite Element Modelling, as large displacements are assumed to occur at interfaces between blocks [ 34 ]. Additionally, DEM has already shown high potential to reproduce the brittle failure of such structures [10,11,19,20]. The study first describes the numerical DEM modelling strategy (Section 2). Next, the DEM model is validated against scaled-down laboratory experiments (Section 3). Section 4 presents numerical simulations of the full-scale DSRWs. Their comparison with pseudostatic predictions gives an initial estimate of the behaviour factor r. The conclusions are drawn in the final section. 2. Numerical DEM Strategy for Dynamic Simulations The numerical strategy uses the plane strain software UDEC (version 7.0) developed by the Itasca consulting group [ 35 ]. The numerical model consists of an assembly of masonry units retaining a homogeneous backfill. Each masonry unit is represented by a deformable homogeneous medium characterised by linear elastic isotropic parameters (Young’s modulus Eand Poisson’s ratio ν ). The backfill is modelled as an elastoplastic medium: its mechanical behaviour is driven by elastic parameters (Young’s modulus Eand Poisson’s ratio ν ) and a Mohr–Coulomb plasticity criterion (friction angle ϕ , cohesion C, tensile strength Tand dilation angle ψ ). Both the units and backfill are therefore meshed using finite elements of size ∆ z, which should respect the following inequality to ensure an appropriate propagation of elastic waves through the media: ∆z < 0.1 ×λmin = 0.1 ×VS×Tmin = 0.1 ×[G/ρ]0.5 ×Tmin (1) where λmin ,V S and T min are the propagated waves’ smallest wavelength, shear velocity and minimum period, respectively. Gand ρ are the medium’s shear modulus and unit Geosciences 2022,12, 252 3 of 20 weight, respectively. Equation (1) does not constrain ∆ zmuch (see Sections 3and 4for a formal check). Therefore, herein, ∆ zis defined based on the masonry unit height so that at least six subcontact points are generated between each unit and the backfill, as this parameter appears to be critical to reasonably reproduce the actual force distribution at the interfaces [20,22,36,37]. Zero-thickness contact interfaces are assumed between the units and between the unit and the backfill. They follow an elastoplastic Mohr–Coulomb criterion. In UDEC, equations of motion are solved explicitly via a central finite difference scheme. For each computational node of the finite element mesh (both units and backfill), and at each time t, the dynamic equations of motions are solved as follows, thus giving new positions and velocities at time t + ∆t: miai(t) = ΣFi(t) vi(t + ∆t/2) = vi(t −∆t/2) + ai(t) ×∆t ui(t + ∆t) = ui(t) + vi(t + ∆t/2) ×∆t (2) where ∆ tcorresponds to the timestep of the simulations; F i corresponds to all the applied loads, namely the external loads (gravity, dynamic shaking at the bottom of the model), contact interactions between the different deformable bodies, and internal loads given by classical continuum mechanics calculations [ 35 ]; m i ,a i ,v i and u i correspond to the associated mass, acceleration, velocity and displacement of the computational node, respectively. At the new simulation time t+ ∆ t, the positions of each computational node give the strain of each finite element, which, in turn, updates the stress as a result of the constitutive law of the material. Contact forces are also recomputed at this stage according to the following equations: ∆Fn= Ac×kn×∆un= Ac×kn×[un(t + ∆t) −un(t)] ∆Fs= Ac×ks×∆us= Ac×ks ×[us(t + ∆t) −us(t)] (3) where ∆ F n and ∆ F s are the normal and shear contact force increments at a given subcontact, A c corresponds to the subcontact area, k n (respectively, k s ) is the normal (respectively shear) elastic contact stiffness, and u n (respectively u s ) represents the normal (respectively shear) relative displacement between the two contact points. The constitutive law of the interface is then applied (the Mohr–Coulomb model in this case), which reads: Fn(t + ∆t) = Fn(t) + ∆Fn, Fn< T ×Ac Fs(t + ∆t) = Fs(t) + ∆Fs, |Fs|<Fs, max = C ×Ac−Fn×tan(ϕ)(4) Additional expressions exist to include the dilation effect but are not presented because hereafter, the dilation angle is always taken to be equal to zero [ 35 ]. Updating all the forces in the system allows the resolution of the equations of motion (Equation (2)) for the next simulation time t+∆t. Finally, numerical damping is used to model the actual energy dissipation. In dynamics, Rayleigh damping is generally preferred [ 37 , 38 ], including massand stiffnessproportional contributions, because it gives constant damping across a relatively wide frequency range. However, only mass-proportional damping is considered here, leading to a reasonable timestep ∆ t[ 38 ]. Additionally, the input signals are all harmonic. Therefore, the under-evaluation of damping at higher frequencies when disregarding stiffness-proportional damping is insignificant for the current simulations. In Sections 3and 4, the numerical models are characterised by three different boundaries: free, fixed-velocity and free-field boundaries (Figure 1). In particular, the velocity of the bottom computational nodes (the backfill and bottom wall units) are fixed to match the dynamic solicitation. Next, a free-field boundary is applied to the right part of the model (vertical boundary). An elastic 1D numerical model is first run parallel to the primary model. It has the same dynamic input signal, elastic properties and computational nodes Geosciences 2022,12, 252 4 of 20 as the backfill boundary. At each time t, the velocities produced from this 1D model are directly applied to the vertical boundary of the main simulation. In other words, it assumes that the free-field boundary is far enough from the masonry wall (i.e., no impact of reflected waves on the wall in the media close to the boundary). Wall Cohesive layer Backfill Applied velocity Free-field boundary Free boundaries Figure 1. UDEC model of the experiments (see Section 3). The different boundary conditions are detailed. Spurious large deformations are obtained for non-cohesive backfill free layers because there is zero confining pressure, and hence zero shear resistance, of the elements involved. Therefore, an artificial thin cohesive layer is added to the backfill to stabilise the behaviour of the shallow elements (Figure 1). Cohesion of only 2 kPa with a tiny depth avoids unrealistic deformation without significantly modifying the global response of the system. The following two sections focus on the numerical modelling of the scaled-down experiments (Section 3) and full-scale walls (Section 4). The specifics of each model and their numerical parameters are described in the corresponding sections. 3. Validation of the Model on Scaled-Down Experiments 3.1. Experimental Results The numerical strategy is validated against an existing experimental campaign [ 33 ]. The reader is referred to the original study for a detailed description, while the primary outcomes are gathered herein. Several scaled-down dry-joint retaining walls with different slenderness H/B values (constant width Band different heights H) were subjected to harmonic horizontal shaking (Figure 2). The walls were 400 mm long, 34 mm wide and between 50 mm and 140 mm high. The bottom course of the masonry units was fixed to the container. The walls were retaining a sandy backfill of the same height and length, with a depth of 400 mm. An acceleration input signal (S(t)) of increasing amplitude and constant frequency f(equal to 10 Hz, 30 Hz, 50 Hz or 70 Hz) was applied through a shaking table to the bottom of the mock-up, as follows: S(t) = A ×t×sin(ω×t) (5) where Ais a constant that describes the amplitude rate, and ω is the angular frequency, defined as: ω = 2 ×π× f. The primary campaign used a frequency fof 10 Hz. The horizontal displacement d top of the second-to-last (top) course of bricks was monitored through a laser sensor (Figure 2). Section 3.3 uses d top to compare the experimental and numerical outcomes. In reference [ 33 ], wall failure was declared when d top exceeded 5% of the wall height H. Note that Hrefers to the wall height free to move throughout the shaking, disregarding the first fixed masonry course (see Figures 2and 3). Geosciences 2022,12, 252 5 of 20 H B Figure 2. Retaining wall mock-up [33], for a wall of slenderness H/B = 3.3. Fixed course Section 1 Section 2 Section 3 Section 4 p₁ = 7/23 p₂ = 7/23 p₃ = 7/23 p₄ = 2/23 H B Figure 3. Four different cross-sections in the wall length illustrated for the walls with a slenderness H/B = 2.7. Recall that Hrefers to the wall height free to move throughout the shaking, disregarding the first fixed masonry course. 3.2. Numerical Parameters of the UDEC Simulations The numerical models’ geometric data (external shape and unit dimensions) is drawn from the experiments (Figure 1). Table 1gathers all the parameters considered in the simulations. The material properties considered in the numerical model are either collected from the original study [ 33 ] or from numerical studies that modelled the same material [21,22] when no experimental data were available. Apart from the small layer on the top of the backfill, the cohesion C (continuum and interface) is always equal to zero (dry sand backfill and dry contact wall assembly). Note that the timestep ∆ tis automatically computed by UDEC once all the numerical parameters are defined [ 35 ]. This led to 12 h of computation (on an HP computer with a speed of 2.3 GHz) for 1 s of simulation, while the experimental signals range from 15 to 30 s. According to Equation (1), the maximum allowable mesh size is 70 mm in the backfill and 800 mm in the wall, both being much larger than the actual numerical mesh size ∆z, which is equal to 2 mm. Geosciences 2022,12, 252 6 of 20 Table 1. Properties used for the numerical simulations of the scaled-down experimental tests. Units Backfill Wall–Backfill Interface Volumetric properties Unit weight γ(kN/m3) [33]14.4 15.4 - Elastic modulus E(MPa) 1000 10 - Poisson ratio ν0.2 0.3 - Friction angle ϕ(◦) [33] - 45 - Interface properties Friction angle ϕ(◦) [33] 32 - 32 Normal stiffness kn(Pa/m3) 2.6 ×1012 -3.0 ×1010 Tangential stiffness ks(Pa/m3) 2.6 ×1012 -3.0 ×1010 Numerical parameters Timestep ∆t(s) ≈1×10−6 Rayleigh viscous damping ξ10% at 10Hz The viscous damping coefficient ξ used in the numerical simulations is based on the energy dissipation of the backfill. Preliminary simulations have allowed an evaluation of the cyclic strain of the backfill. By using computational nodes intentionally placed far from the wall (and the associated failure zones), a cyclic strain of 0.025% is found, which is the typical threshold where damping greater than 5% can be observed [ 39 ]. As two preliminary simulations using a damping ξ of 5% and 15% gave identical results (a relative difference less than 2%), a damping value ξ of 5% is considered thereafter. Since only the mass-proportional part of the Rayleigh damping is used, a multiplicative factor of 2 is applied to the chosen value to ensure appropriate effective damping (5%) at the critical frequency (Table 1) [35]. In the experiments, an absorbing foam placed at the back of the backfill was intended to reproduce a “free-field boundary condition”. Preliminary simulations directly modelling the foam with its properties (taken from [ 40 ]) revealed that modelling the right vertical boundary with either an absorbing foam or with a free-field boundary led to almost identical behaviour of the mock-up. The latter modelling strategy, being faster, has been considered in the rest of this section. UDEC assumes a unique cross-section along the wall while the actual walls are 3D assemblages with four different cross-sections (Figure 3); the first three are equally distributed, while the fourth corresponds to the two ends of the wall. As the position of the headers (and, in general, the assemblage) in the height of the wall significantly influences the behaviour of dry-jointed structures [ 22 , 41 , 42 ], the four different cross-sections are expected to have different responses (Figure 3). Figure 4shows the horizontal displacement d top of the second-to-last course obtained from the numerical simulations of the wall with a slenderness H/B = 2.7 using the first three cross-sections; the fourth one has the same response as the second one (Figure 3). Next, the global curve is computed, weighting the individual responses according to the proportions in Figure 3[ 22 ]. For each d top ,one finds the corresponding times in each response and then weights them to obtain the global response time. This is illustrated in Table 2for a single displacement d top equal to 5 mm. As can be noticed in Figure 4, the weighted response is very close to the response of Cross-section 3 (a difference of less than 0.25 s for all points in the signal). Therefore, Cross-section 3 is assumed to represent the global response and is used alone in the following numerical simulations to save computational time. Geosciences 2022,12, 252 7 of 20 Figure 4. Comparison of the displacement at the top of the wall (slenderness H/B = 2.7) for the different cross-sections (Figure 3) and the global weighted response. Table 2. Computation of the weighted response illustrated with the threshold dtop equal to 5 mm. Section No. Proportion Time for dtop = 5 mm (s) 1 7/23 18.4 2 7/23 21.5 3 7/23 19.9 4 2/23 21.5 Weighted average 20.1 3.3. Validation of the Modelling Strategy As the first step, Figure 5compares the numerical response d top (only for Cross-section 3) with the experimental curves from five repeated tests on a wall with a slenderness H/B = 2.3. The obtained trends are globally similar (also found for all the studied configurations; not shown for brevity). In addition, the numerical response is encompassed by the experimental ones, highlighting the validity of the numerical strategy. Wall (numerical or experimental) failure was declared when d top exceeded 5% of the wall height H[ 33 ]. The acceleration at the instant of failure, taken from the dynamic signal, is denoted A fail .Figure 6and Table 3compare the outcomes, both in terms of acceleration at failure Afail and the observed failure mode, between the experiments and simulations. Geosciences 2022,12, 252 8 of 20 Figure 5. Comparison of the numerical response with the envelop of the experimental results [ 33 ] for walls with a slenderness H/B = 2.3. Figure 6. Comparison of the acceleration at failure (A fail ) between the numerical simulations and experimental tests for different wall slenderness (H/B). Geosciences 2022,12, 252 9 of 20 Table 3. Experimental and numerical acceleration at failure (A fail ) for different wall slenderness: the mean value is given when several experimental points are available. The relative difference is computed by taking the experimental value as a reference. Finally, the detected failure modes are also indicated. Wall Slenderness (H/B)Exp. Afail (m/s2)Exp. Failure Mode Num. Afail (m/s2)Num. Failure Mode Error (%) H/B = 1.3 4.2 S 3.7 S −11% H/B = 1.7 4.6 S 3.4 S −27% H/B = 2.3 3.3 S/T 3.2 S −5% H/B = 2.7 3.2 T 2.6 T −18% H/B = 3 2.7 T 1.7 T −35% H/B = 3.3 2.0 T 1.4 T −32% Mean −21% Firstly, it can be observed that the numerical simulations accurately reproduce the observed failure mechanisms. Secondly, the numerical simulations underestimate, on average, the actual experimental responses by 20%. However, since most numerical parameters have been taken by default, the agreement is considered to be good. Furthermore, in [ 33 ], the authors noted that the friction angle of the backfill sand could be as high as 55 ◦ (larger than the considered value of 45 ◦ ). Table 4shows the numerical underestimations of A fail for the wall with a slenderness H/B = 2.7 for three different backfill friction angles (45 ◦ , 50 ◦ and 55◦), which significantly affect the capacity of the walls. Table 4. Effect of the backfill friction angle on the numerical acceleration at failure A fail of the wall with a slenderness H/B = 2.7. Backfill Friction Angle ϕ(◦) 45◦50◦55◦ Error compared with experiment (%) −18% −13% −10% In addition, in Figures 4and 5, one can also note that the numerical responses are softer than the experimental ones, indicating that the estimated material properties may need correction (e.g., units of Young’s modulus or joint stiffness). The comparison of the accelerations at failure becomes sensitive to the chosen threshold value. For example, considering a threshold of 10% (instead of the original 5%) to define the experimental and numerical failures (Figure 5) decreases the average error from 21% to 16%. Finally, UDEC does not catch 3D effects that are also partly responsible for the observed discrepancies. In particular, the most resistant cross-section (Figure 3) may play a more significant role than the weaker cross-sections, contrary to the assumptions of Table 2. 3.4. Effect of the Frequency Figure 7compares the numerical and experimental responses obtained for an input signal with f= 30 Hz (instead of 10 Hz). An excellent agreement (a difference of less than 1%) is also noticeable when computing the acceleration at failure A fail (a threshold of 5%), similar to the case with f= 10 Hz (Table 3). To conclude, the precise calibration of the numerical model to fit the experimental results is not sought here, since the number of parameters is significant. Nevertheless, it has been shown that the numerical strategy is accurate enough for modelling the influence of both the wall’s slenderness and the signal frequency on the wall’s response. Therefore, this strategy is used further to simulate larger-scale systems (Section 4). Finally, the modelling approach underestimates the actual capacity, which is conservative and particularly important, as design recommendations are proposed in the following section. Geosciences 2022,12, 252 16 of 20 Figure 15. Comparison of the response of DSRW1 subjected to dynamic input signals with and without vertical acceleration. Table 7. Collapse acceleration A col and time t col for all simulations performed. Here, t col is almost constant for all the simulations (with a range between 7.6 s and 9.5 s, and a coefficient of variation of 7%). DSRW Name DSRW1 DSRW2 DSRW3 DSRW4 Frequency (Hz) 3Hz 5Hz 10Hz 5Hz 5Hz 5Hz 5Hz 5Hz Vertical acceleration - - - in-phase opposition - - - tcol (s) 7.6 8.5 9.5 8.9 8.6 9.4 8.3 8.5 Acol (m/s2)4.83 5.38 6.03 5.34 5.19 1.83 5.06 4.63 4.4. Comparison of Pseudo-Static Predictions and the Estimated Behaviour Factor r Table 8estimates the peak acceleration A peak for each DSRW, considering the ratio of 60% found in Section 4.2. In addition, Table 8also shows the pseudo-static resistance A pseudo of each wall computed using an analytical approach based on the limit-equilibrium of a Coulomb’s wedge of soil [ 18 , 27 ]. The properties used in the analytical model are the same as for the numerical simulations (Tables 5and 6). Furthermore, the stones are assumed to be perfect, which leads to a high interface stiffness in the numerical simulations (Table 6) and no internal rotation of the stones in the analytical approach (see the details in [56]). Finally, Table 8estimates the behaviour factor rfor the investigated DRSWs as the ratio between the peak acceleration A peak , and the pseudo-static resistance A pseudo . On average, the behaviour factor requals 2.08, with a minimum value of 1.83. Therefore, to fit the thresholds (1, 1.5 or 2) recommended by Eurocode 8 for retaining walls [ 32 ], the authors propose r= 1.5, which is conservative, as it is intended to be used for practical design/assessment. Furthermore, the latter can readily be used in conjunction with a pseudo-static approach to give fast practical recommendations on-site. This is of particular Geosciences 2022,12, 252 17 of 20 interest when many DSRWs need to be assessed/designed, requiring fast methods where performance-based approaches are less competitive, as they require more computational time and often need several expensive time-history analysis to give reliable results. Table 8. Comparison of the collapse acceleration A col , the estimated peak acceleration A peak and the pseudo-static resistance A pseudo , along with the estimated behaviour factor r for every single numerical case in this study. When the vertical acceleration is accounted for, the displayed accelerations only show the horizontal components, the vertical ones being always half of the latter. DSRW Geometry DSRW1 DSRW2 DSRW3 DSRW4 Frequency (Hz) 3Hz 5Hz 10Hz 5Hz 5Hz 5Hz 5Hz 5Hz Vertical acceleration - - - in-phase opposition - - - Acol (m/s2)4.83 5.38 6.03 5.34 5.19 1.83 5.06 4.63 Apeak = 0.6 ×Acol (m/s2)2.90 3.23 3.62 3.21 3.11 1.10 3.04 2.78 Apseudo (m/s2)1.58 1.58 1.58 1.50 1.50 0.49 1.36 1.52 r=Apeak/Apseudo 1.83 2.04 2.29 2.14 2.07 2.26 2.00 2.04 5. Conclusions This work presents dynamic numerical simulations involving dry-stone retaining walls (DSRWs). Firstly, a discrete element modelling (DEM) tool is validated against a scaled-down laboratory mock-up subjected to horizontal harmonic shaking. The outcomes show good agreement regarding the collapse mechanism and the shape of the response while predicting a slightly lower failure capacity, thus being conservative. It is also noted that the agreement holds for various slenderness and frequency values. Next, time-history dynamic simulations involving full-scale DSRWs are carried out. The associated data are taken from the literature. The dynamic input signals are harmonic, aiming to model seismic motions in a simplified and fully controlled manner. The signals’ properties are based on the most critical seismicity of French metropolitan areas. However, the outcomes also hold for any country with a similar (or lower) seismicity. The effects of the frequency and the vertical acceleration are investigated. It is found that the frequency of the input signals is the most critical characteristic of the motion, and it noticeably modifies the maximum acceleration supported by the DSRW. Finally, the behaviour factor ris deduced for each simulation, comparing the pseudostatic resistance with that obtained from the time-history dynamic simulations. An average factor r= 2 is found, with a minimum value (for low frequencies) of 1.8. Therefore—and this is the novelty of the present study—the authors advise a conservative value of r= 1.5 for DSRWs, chosen among the recommended values by the Eurocode 8 standard. This value can readily be used in a practical framework to assist engineers and practitioners in designing new (or assessing existing) DSRWs following a classical pseudo-static approach for fast analyses. Future work will involve sets of artificial and recorded seismic motions to confirm the recommended value for the seismic behaviour factor r. Author Contributions: Conceptualisation, N.S., E.V. and S.H.; methodology, E.V. and P.B.L.; software, E.V.; validation, N.S. and S.H.; investigation, N.S.; resources, E.V., P.B.L. and S.H.; writing—original draft preparation, N.S.; writing—review and editing, E.V., P.B.L. and S.H.; supervision, E.V. and S.H.; funding acquisition, E.V. and P.B.L. All authors have read and agreed to the published version of the manuscript. Geosciences 2022,12, 252 18 of 20 Funding: This research was funded by the French Ministry of Education and Research, through the PhD grant attributed to the first author. This work was also partly financed by FCT/MCTES through national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB/04029/2020. Finally, it has been partly funded by the STAND4HERITAGE project (new standards for the seismic assessment of built cultural heritage), which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 833123), as an Advanced Grant. Its support is gratefully acknowledged. The opinions and conclusions presented in this article are those of the authors and do not necessarily reflect the views of the sponsoring organisations. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data related to this study have not been made publicly available. Acknowledgments: The authors would like to thank Fabian Dedecker and Xuan-Phu Nguyen for their high scientific quality technical support when preparing the numerical models. Conflicts of Interest: The authors declare no conflict of interest. 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