Appraisal of the overburden mass and boundary conditions on the rocking behaviour of the vertical spanning strip wall

Vol.: (0123456789) Meccanica https://doi.org/10.1007/s11012-024-01910-2 RESEARCH Appraisal oftheoverburden mass andboundary conditions ontherocking behaviour ofthevertical spanning strip wall GeorgiosVlachakis · CarlaColombo · DarioVecchio · AnastasiosI.Giouvanidis · PauloB.Lourenço Received: 8 September 2024 / Accepted: 5 November 2024 © The Author(s) 2024 Abstract Unreinforced masonry structures are particularly vulnerable to seismic events. Specifically, local out-of-plane mechanisms form recurrently, posing a serious threat of collapse. Among them, a Vertical Spanning Strip Wall (VSSW) mechanism occurs when a portion of a wall is solely constrained at its top and base, experiencing out-of-plane bending across its height. During the formation of the mechanism and up to collapse, the VSSW presents a highly nonlinear dynamic behaviour governed by the presence of an overlying wall or diaphragm and a diversity of diaphragm-to-wall connections. The present work simulates the dynamics of a VSSW, accounting for the influence of the overburden mass and a variety of boundary conditions through a two-rigidbody single-degree-of-freedom model. In addition, this study examines the energy losses of the VSSW and provides closed-form expressions to estimate the angular coefficient of restitution either analytically for predictive purposes or experimentally for characterisation campaigns. Finally, a series of illustrative comparative freeand forced-rocking analyses highlight the importance of the overburden mass and boundary conditions on the dynamic stability of the VSSW. Keywords Out-of-plane collapse mechanism· Vertically spanning strip wall· Unreinforced masonry structures· Coefficient of restitution· Rocking dynamics· Boundary conditions 1 Introduction Unreinforced masonry structures constitute a significant portion of our built environment, especially when considering heritage constructions [1]. Despite the societal demand for their safeguard, unreinforced masonry structures have shown to be particularly vulnerable to seismic actions [2–4]. Their vulnerability stems mainly from their high mass, weak connections among structural elements and the brittle behaviour of masonry [5]. As a result, cracks develop rather easily during earthquakes, forming Out-OfPlane (OOP) and In-Plane (IP) collapse mechanisms [6]. The former usually occurs when the connections among the structural elements are insufficient, whilst the latter requires a so-called “box-like” or integral structural behaviour. Most importantly, even though OOP mechanisms are more common and destructive than IP ones, their complex dynamic response is less understood [7, 8]. Among the OOP mechanisms, the simple overturning of a rocking wall is undoubtedly the weakest and therefore has attracted the highest G.Vlachakis(*)· C.Colombo· D.Vecchio· P.B.Lourenço ISISE, ARISE, Department ofCivil Engineering, University ofMinho, Guimarães, Portugal e-mail: giorgovlac[email protected] A.I.Giouvanidis Department ofCivil andEnvironmental Engineering, The University ofAuckland, Auckland, NewZealand Meccanica Vol:. (1234567890) attention in the literature [9–11]. Nonetheless, the second most vulnerable OOP mechanism is the Vertical Spanning Strip Wall (VSSW). Such a mechanism appears when a wall is restrained at its top and base. In fact, the Boundary Conditions (BCs) on top, together with the formation of three hinges along the wall’s height and their associated impact interfaces, make the VSSW mechanism a challenging topic of structural dynamics [12–16]. There is a scarcity of experimental studies dedicated to the VSSW [17–23], while several numerical and analytical models have been proposed to replicate its dynamic behaviour. More specifically, some formulations resort to rocking dynamics [10, 12, 13, 24, 25], while others adopt a series of assumptions on the dynamics of the system to simplify the problem and propose practical methodologies [26–29], such as the linearisation of the equation of motion and the kinematics [26, 28] or the adoption of multi-linear moment-rotation diagrams [26–29]. Recent works have extended the investigation to flexible interfaces [24] or flexible BCs [12, 27]. The quantification of the energy losses of the VSSW mechanism is another open issue, which has been addressed by analytical mechanics-based considerations or by direct experimental observations. Sorrentino et al. [13], Mehrotra and DeJong [24] and Prajapati etal. [12] estimated the angular Coefficient of Restitution (CoR) by resorting to impulsive dynamics and the conservation of momentum. More recent numerical studies either adopted the previous or utilised equivalent viscous damping models [10, 16, 26–30]. Nonetheless, comparisons of the dynamic response of numerical predictions with experiments have been successful only after careful calibration of the damping models [16, 27–29]. At the same time, a few campaigns have quantified the experimental energy losses of the VSSW [17, 20]; however, comparisons with the analytical predictions showed a mismatch [20]. Overall, previous studies have highlighted the influence of several factors on the response of the VSSW mechanism, namely the height of the intermediate hinge, the significance of the overburden load, the influence of the energy losses at impacts, and the effect of flexible BCs and interfaces. The present work complements these studies and investigates in depth the influence of the BCs and the overburden mass on the seismic response of the VSSW. To this end, a refined model is developed to incorporate a parametrised position of the hinge on top of the VSSW and the presence of an overburden mass. The former generalisation allows the replication of a variety of symmetric and asymmetric BCs found in real-world VSSW structures [31], whereas currently available models restrain the BCs at predefined positions [10, 12, 13, 24–29]. Furthermore, the incorporation of an overburden mass essentially replicates an overlaying structure, the weight of the diaphragm, among others. Previous models usually employed an overburden force instead of a mass to simulate this aspect [10, 13, 24, 26, 28, 29]. However, a mass has greater implications on the dynamics of the VSSW, as it contributes to both the inertia and momentum. Importantly, this work demonstrates through extensive comparative analyses how the aforementioned features influence the Equation of Motion (EoM), the hinge height, the instability angle, the energy losses, and the overall dynamic response of the VSSW. The paper is organised as follows: Sect.2 presents the formulation of the model, the description of the system’s kinematics and the derivation of the EoM. Section 3 discusses the energy losses of the system and proposes: (i) an analytical formula to estimate the angular CoR based on impulsive dynamics, and (ii) a formula to extract the angular CoR from experimental response-histories. Section4 illustrates the influence of the BCs and overburden mass on the dynamic response of the VSSW, through freeand forced-rocking analyses. Finally, Sect.5 summarises the conclusions of the study. 2 Model formulation 2.1 Geometry and kinematics Consider the VSSW depicted in Fig. 1a with width 2b, height 2h. Upon ground excitation, assuming no sliding at the contact interfaces and given the horizontal restraint on top of the wall, three hinges are formed for the activation of the VSSW mechanism: (i) at the base of the wall, (ii) at an intermediate height 2h1 (relative to the base), and (iii) at the top of the wall. As a result, the wall is divided into two distinct rigid bodies (Fig.1a): the lower body “1” and the upper body “2”, with heights 2h1 and 2h2 , slenderness angles 𝛼1 and 𝛼2 , and diagonal distances R1 and Meccanica Vol.: (0123456789) R2 , respectively. The aforementioned geometry can be fully described using a set of three parameters, e.g. the half width b and the two half-heights h1 and h2 , or the two slenderness angles 𝛼1 and 𝛼2 , and the diagonal distance R1 . Similar to Sorrentino etal. [13], this study adopts the latter set of variables. The position of the hinge at the top of the wall is usually assumed to be either at the upper left (Fig. 1b) or right corner of the body “2” (Fig. 1c) depending on the sign of rotation [10, 13, 28], or at the centre of the wall [29]. Nevertheless, the position of the hinge is directly related to the Boundary Conditions (BCs) and the connection of the wall with the horizontal diaphragm/constraint. As a matter of fact, a wide diversity of BCs can be found in real structures [31, 32], with Fig. 2a–b showing two cases where the diaphragm rests on a limited portion of the wall. Considering this diversity, the current model parametrises the position 𝜆BC ∈[0, 2] of the top hinge (Fig.1b–c). This allows replicating a variety of BCs using different values of 𝜆BC , with Fig. 2b–f illustrating three indicative examples: (i) “clamped” with 𝜆+ BC =𝜆 − BC = 2 , where pivoting occurs at the upper corners (Fig.2d), (ii) “pinned” with 𝜆+ BC = 𝜆 − BC =1 , where pivoting occurs at the centre (Fig. 2e), and (iii) asymmetric with 𝜆+ BC =0.5 and 𝜆− BC =2 , where pivoting occurs at the upper corner of the VSSW for counter-clockwise rotation and at the corner of the diaphragm for clockwise rotation of the upper body “2”, respectively (Fig. 2f). The superscripts indicate the sign of rotation of the lower body “1” of the VSSW, i.e. “ + ” for positive counter-clockwise rotation and “–” for negative clockwise rotation. The present model considers also the influence of an overburden mass ( mT ) and an overload ( N ) acting at the top of the wall (Fig.1b–c). These replicate the weight of any overlaying structure resting on top of the wall, such as an upper storey, an overlaying parapet or a diaphragm. Previous models in the literature have focused solely on the overload [10, 13]. Nevertheless, in practice both scenarios may occur. Their influence on the VSSW appears comparable, although the mass has additional inertia and momentum directly affecting the system’s dynamics. Note that the overburden mass mT is assumed to act at the hinge point at the top and it is controlled by the parameter 𝜆BC . Similarly, the overburden load N is parametrised by the distance 𝜆N∈[0, 2] shown in Fig.1b–c. Assuming that the two bodies are rigid, can uplift and rock but not slide or rebound, the pure rocking motion of the VSSW can be captured by a single degree of freedom. This is assumed to be the angular rotation 𝜃1 of the lower body “1”, with the angular rotation of the upper body “2” 𝜃2 being related to the former through the constraint condition introduced by the intermediate and top hinges: with: The term C1 is given in the Appendix 1. Similarly, the angular velocities  𝜃1 and  𝜃2 of the two bodies can also be related as: where dC 1 d𝜃 1 is provided in the Appendix1. 2.2 Equation of motion (EoM) The EoM of the system of Fig.1 is derived using the Lagrange’s equation: (1) 𝜃2=arcsin C1−𝜑 (2) 𝜑 =sgn𝜃1arctan ( 𝜆BC 2tan 𝛼2 ) (3)  𝜃 2= dC 1 d𝜃1 1 √ 1−C2 1  𝜃 1 Fig. 1 Scheme of the vertical spanning strip wall: a geometry and displaced configuration with boundary conditions for b positive (counter-clockwise) and c negative (clockwise) rotation of the lower body “1” Meccanica Vol:. (1234567890) Fig. 2 Boundary conditions on top of the vertical spanning strip wall: a structural detail during the construction stage (reproduced from [33] with permission from Springer Nature), b schematic view of the diaphragm-wall connection (reproduced from [31] with permission from Elsevier), c schematic view of the displaced wall, and d–f view of illustrative boundary conditions that the present model replicates Meccanica Vol.: (0123456789) where V and Z represent the potential and kinetic energies of the system, while ΓN and Γg are the work done by the overload force N and the inertia forces due to the horizontal and vertical components of the ground acceleration xg and yg , respectively. The potential energy V is: where m1 and m2 are the mass of the bodies “1” and “2”, respectively, and g is the acceleration of gravity. The kinetic energy Z reads: with IG1 and IG2 being the moment of inertia of the bodies “1” and “2” with respect to their centre of gravity, while the terms C2 , C3 and C4 are provided in the Appendix1. The virtual work of the superimposed load ΓN is: and the virtual work of the inertia forces Γg reads: After expanding Eq.(4), the following non-linear second-order differential equation is obtained: where the terms CA , CS , CH , CV and CN are: (4) d dt ( 𝜕(Z−V) 𝜕 𝜃 1) −𝜕(Z−V) 𝜕𝜃 1 =Γ N+Γ g (5) V =gR1 [( m1+2m2+2mT ) cos ( 𝛼1− || 𝜃1 ||) +m2 sin 𝛼1 sin 𝛼 2 cos ( 𝛼2− || 𝜃2 ||) +2mT sin 𝛼1 tan 𝛼 2 cos 𝜃2 cos 𝜑 ] (6) Z= 1 2 R 2 1  𝜃 2 1 [ m1+m2 ( C 2 2+C 2 3 ) +mTC 2 4 ] + 1 2  𝜃2 1 [ IG1+IG2 ( dC1 d 𝜃1) 21 1− C2 1] (7) 𝛿 ΓN=−NR1 ⎡ ⎢ ⎢ ⎢ ⎣ 2sgn𝜃1sin � 𝛼1− �� 𝜃1 ��� −sin 𝛼1 sin 𝛼2 dC1 d𝜃1 1 � 1−C2 1 � sgn𝜃1𝜆Nsin 𝛼2cos 𝜃2+2 cos 𝛼2sin 𝜃2 �⎤ ⎥ ⎥ ⎥ ⎦ 𝛿𝜃 1 (8) 𝛿 Γ g = { −x g R 1[ m 1 cos ( 𝛼 1 − | | 𝜃 1| |) −m 2 C 2] +y g R 1[ m 1 sgn𝜃 1 sin ( 𝛼 1 − | | 𝜃 1| |) +m 2 C 3 +m T C 4]} 𝛿𝜃 1 (9) CA 𝜃 1 +C S 𝜃 2 1 =−R 1 C H x g +R 1 C V( y g −g ) −R 1 C N N (10a) C A=R2 1 [ m1+m2 ( C2 2+C2 3 ) +mTC2 4 ] +IG1+IG2 (dC 1 d 𝜃1)21 1 −C2 1 (10b) CS=R 2 1 [ m2 ( C2 dC2 d𝜃1 +C3 dC3 d𝜃1 ) +mTC4 dC4 d𝜃1 ] + IG2 (1 −C2 1) 2[d2C1 d𝜃2 1 dC1 d𝜃1 ( 1−C2 1 ) + ( dC1 d𝜃1 ) 3 C1 ] (10c) CH =m 1 cos ( 𝛼 1 − | | 𝜃 1| |) −m 2 C 2 Note that the overburden mass mT is included both in the inertia term CA and the centrifugal and Coriolis term CS . 2.3 Rocking initiation, intermediate hinge height and instability angle The minimum horizontal ground acceleration | | | xg | | |min capable of initiating rocking motion can be computed by setting 𝜃1= 𝜃 1= 𝜃 1=0 in Eq.(9): (10d) CV =m 1 sgn𝜃 1 sin ( 𝛼 1 − | | 𝜃 1| |) +m 2 C 3 +m T C 4 (10e) CN= 2sgn 𝜃1 sin ( 𝛼1−||𝜃1|| ) − sin 𝛼1 sin 𝛼2 dC1 d𝜃1 1 √1 −C2 1 ( sgn𝜃1𝜆Nsin 𝛼2cos 𝜃2+2 cos 𝛼2sin 𝜃2 ) Meccanica Vol:. (1234567890) Notice that the overburden weight mTg and the overload force N have exactly the same influence on | | | xg | | |min when 𝜆BC =𝜆N and yg =0 . For walls without tensile strength (such as dryjoint masonry), the height of the intermediate hinge 2h1 can be computed as the height that minimises Eq.(11) [13]. Figure3a plots the intermediate hinge height 2h1 normalised by the height of the wall 2h as a function of the overburden mass mT normalised by the total mass of the wall mtot (i.e. mtot =m1+m2 ), which is equivalent to the overload N normalised by the weight of the wall W (i.e. W=mtotg ). Furthermore, Fig.3a depicts the influence of the overburden’s mass position 𝜆BC (and equivalently the overload’s position (11) ||| xg ||| min =tan 𝛼1 ( g−yg )[ m1+m2 ( 2+tan 𝛼2 tan 𝛼1 ) +mT ( 2+𝜆BC tan 𝛼2 tan 𝛼1 )] +N ( 2+𝜆N tan 𝛼2 tan 𝛼1 ) m1+m2 𝜆N , assuming 𝜆BC =𝜆N ) on the hinge height. Overall, Fig.3a shows that the intermediate hinge height changes drastically for small mT/ m tot (or N∕W ) values while it asymptotically stabilises for higher mT/ m tot (or N∕W ) values. Moreover, the position of the overburden mass (or overload) has an inverse effect on the hinge height, with the lower values of 𝜆BC (or 𝜆N ) causing an increase in the hinge height. As a result, the intermediate hinge height drops to almost half of the wall’s height for very high overburden mass (or overload) and large values of 𝜆BC (or 𝜆N ). On the contrary, for very low overburden mass (or overload) and regardless of the value of 𝜆BC (or 𝜆N ), the intermediate hinge height asymptotically tends to one, which resembles the single rocking block configuration. Finally, it is worth noting that, based on experimental evidence, the intermediate hinge height ranges between 0.5 and 0.75 the height of the wall [18, 20–23]. Figure 3a indicates a comparable range of values, while discrepancies between the experimental evidence and the proposed model may arise due to the presence of joints at discrete heights, the presence of mortar with tensile strength, or any other sources of imperfection. Furthermore, the static instability angle ( 𝜃1,inst ) of a VSSW can be computed by setting  𝜃1 =  𝜃 1 =x g =y g = 0 in Eq. (9) and solving for 𝜃1 . In this context, Fig.3b illustrates the instability angle 𝜃1,inst normalised by the slenderness angle of body “1” 𝛼1 , as a function of the normalised overburden mass mT/ m tot (and equivalently the overload N∕W ), for various positions of the overburden mass 𝜆BC (and equivalently positions of the overload 𝜆N , assuming 𝜆BC =𝜆N ) for a fixed intermediate hinge height h1∕h=0.7 . Firstly, Fig.3b shows that 𝜃1,inst is equal to 𝛼1 only when 𝜆BC =𝜆N=2 or for zero overburden mass (or overload), whereas in any other case, 𝜃1,inst is less than 𝛼1 . In general, this counterintuitive behaviour stems from the maximum stabilising lever arm of the overburden mass (or overload) when 𝜆BC =2 (or 𝜆N=2 ), relative to the intermediate hinge. This effect diminishes as 𝜆BC (or 𝜆N ) decreases. More specifically, 𝜃1,inst decreases rapidly for low mT/ m tot (or N∕W ) values, and it plateaus Fig. 3 a Intermediate hinge height as a function of the overburden mass (or load) for various boundary (overload) conditions, and b instability angle as a function of the overburden mass (or load) for various boundary (overload) conditions, assuming a fixed hinge height h1∕h=0.7 Meccanica Vol.: (0123456789) for high mT/ m tot (or N∕W ) values. Furthermore, the position of the overburden mass (overload) influences strongly 𝜃1,inst , with the lower values of 𝜆BC (or 𝜆N ) resulting in smaller 𝜃1,inst . 3 Energy dissipation—Coefficient ofrestitution (CoR) During rocking motion, when 𝜃1 changes sign, simultaneous impacts occur at the interface between the two bodies and their BCs, resulting in sudden energy losses. This complex damping phenomenon is commonly accounted for using the angular CoR e𝜃 , which relates the pre-impact with the post-impact angular velocities e𝜃 =  𝜃 + 1/ 𝜃 − 1 . It is understood that e𝜃 is merely a phenomenological representation that considers implicitly the complex impact phenomena taking place at the contact interfaces and the radiation damping occurring within the bodies. Nonetheless, the simplicity of the angular CoR makes it particularly convenient from a structural perspective [34]. Therefore, this Section presents two ways to compute the angular CoR: (i) using analytical impulsive dynamics, and (ii) from experimental measurements. The former is useful for predictive purposes, while the latter is beneficial for characterisation, model assessment, and calibration after experimental campaigns. 3.1 Analytical angular coefficient of restitution This Section provides an analytical expression of the angular CoR e𝜃 of the VSSW of Fig.1. To derive it, this work adopts the common assumptions of impulsive rocking dynamics, which can be summarised as follows [13, 35–37]: (i) instantaneous duration of impact, (ii) change of angular velocity with no variation of rotation, (iii) sticking impact (i.e. rocking motion is sustained, without sliding, bouncing or free-flight), and (iv) non-impulsive forces (e.g. body weights, overload force, reaction forces etc.) are negligible to the outcome of impact. Within this framework, the angular CoR is computed assuming the conservation of angular momentum over the pivot point of the post-impact configuration. Figure 4 illustrates the examined impact scheme, depicting the VSSW: (i) before impact where bodies “1” and “2” have linear ( 𝐯 − G1 , 𝐯 − G2 ) and angular (  𝜃− 1 ,  𝜃− 2 ) pre-impact velocities with the top mass having only linear ( 𝐯 − m ) pre-impact velocity (Fig.4a), (ii) during impact where contact impulses ( ∫ F Odt , ∫ F Hdt , ∫ F Tdt ) at the impact points “O”, “H” and “T” are introduced (Fig.4b), and iii) after impact where bodies “1” and “2” have linear ( 𝐯+ G1 , 𝐯+ G2 ) and angular (  𝜃+ 1 ,  𝜃+ 2 ) post-impact velocities with the top mass having only linear ( 𝐯+ m ) post-impact velocity (Fig. 4c). The conservation of angular momentum over point “O” reads: where 𝐫G1,O , 𝐫G2,O and 𝐫m,O , are the relative position vectors of the Centre of Gravity (CG) of body “1”, the CG of body “2” and the CG of the top mass with respect to the impact point O, while the superscript sign symbols “–” and “ + ” refer to the time-instants before and after the impact, respectively. Here, two assumptions related to the mass on top are adopted: (i) the position of its CGm at the time-instant of impact is assumed to be above the CG1, CG2 of the VSSW, and (ii) it has negligible rotational momentum. These (12) H− O =H+ O⇒ I G1  𝜃 − 1 +m 1 (𝐫 G1,O ×𝐯− G1 ) +IG2 𝜃− 2+m2(𝐫G2,O×𝐯− G2)+mT(𝐫m,O×𝐯− m) = =IG1 𝜃+ 1+m1 ( 𝐫G1,O×𝐯+ G1 ) +IG2 𝜃+ 2 +m 2( 𝐫 G2,O ×𝐯+ G2) +m T( 𝐫 m,O ×𝐯+ m) Fig. 4 Impact instances scheme and associated impulses from a a counter-clockwise rotation to b impact and c clockwise rotation of the lower body Meccanica Vol:. (1234567890) assumptions are valid when the overburden mass stems from a wall resting on top of the VSSW (e.g. a parapet, an upper storey or a gable) but entail an approximation when the overburden mass is due to a diaphragm with distributed mass across its span. In the latter case, one may follow the same procedure and include the additional term of the rotational inertia of the top mass together with its angular velocity (related also to its boundary condition). Nevertheless, such analysis merits further investigation which is beyond the scope of the present work. After expanding Eq. (12), the angular CoR e𝜃 becomes: where, again, the superscript sign symbols “–” and “ + ” refer to the time-instant prior and after impact, respectively. In particular, if the height of the intermediate hinge remains the same before and after impact, the superscript sign concerns solely the term related to 𝜆BC (i.e. m TR2 1 ( cos2𝛼1𝜆BC tan 𝛼2 tan 𝛼1 +2 sin2𝛼1 ) in Eq.(13)). Moreover, in the case of symmetric BCs (e.g. Figure2d, e) and the same intermediate hinge height before and after impact, the superscript sign is redundant. Contrary to the mass on top mT , the overload N does not influence the angular CoR. Instead, the different BCs play a role only if there is a mass on top, since the term 𝜆BC affects solely the term related to mT . In addition, Eq.(13) confirms the equation of Sorrentino etal. [13] when mT=0 . Considering that the total energy of the system ( Z+V ) cannot increase and that the potential energy remains constant at impact implies that before and after impact (13) e 𝜃= ⟨ IG1−IG2 tan 𝛼 2 tan 𝛼1 +m1R2 1 ( cos2𝛼1−sin2𝛼1 ) +m2R2 1[cos2𝛼1(2+ tan 𝛼1 tan 𝛼2)−sin2𝛼1(2+ tan 𝛼2 tan 𝛼1)] −mTR2 1(cos2𝛼1𝜆BC tan 𝛼2 tan 𝛼1 +2 sin2𝛼1)⟩ − ⟨ IG1−IG2 tan 𝛼2 tan 𝛼1 +m1R2 1 +m2R2 1(cos2𝛼1 tan 𝛼1 tan 𝛼2 +sin2𝛼1 tan 𝛼2 tan 𝛼1 +2) +mTR2 1 ( cos2𝛼1𝜆BC tan 𝛼2 tan 𝛼 1 +2 sin2𝛼1 ) ⟩ + with the term C5 given in the Appendix1. This condition is always satisfied for symmetric configurations (e.g. Figure 2d,e), but not necessarily for asymmetric ones (e.g. Figure2f). Nonetheless, for any VSSW system, one can compute the angular CoR e𝜃 using Eq.(13), check the energy condition using Eq.(14), and if the condition is not satisfied, reduce the angular CoR e𝜃 to: (14) Z + ≤ Z−⇒e2 𝜃 ≤⟨ C5 ⟩− ⟨ C 5⟩ + To illustrate the behaviour of the analytical angular CoR, Fig. 5 plots e𝜃 for a range of aspect ratios (slenderness) of VSSW masonry structures. The plot shows four representative values of the top mass mT corresponding to 0, 0.2, 0.5 and 1.0 of the total mass of the wall, including all three BCs shown in Fig.2d–f, while fixing the hinge height at h1∕h=0.7 for all cases. It is worth highlighting that mT/ m tot = 0.2 corresponds approximately to the weight of an overlaying diaphragm in the case of a single-storey masonry building [16], mT/ m tot = 0.5 corresponds to a parapet resting on the top of the wall together with an overlaying diaphragm, and mT/ m tot = 1 corresponds to an overlaying storey above the VSSW with the same height. In general, Fig. 5 shows that stockier VSSW walls dissipate more energy, in accordance with the case of a single (15) e 𝜃= �⟨ C5 ⟩ − ⟨ C 5⟩ + Meccanica Vol.: (0123456789) rocking block [35]. Importantly, both the mass on top mT and the BCs have an influence on the energy dissipation of the VSSW. More specifically, the presence of a higher mass on top results in a reduction of the angular CoR and, thus, an increase in the energy dissipation for all examined BCs. Moreover, Fig.5 indicates that the “pinned” BC (Fig. 2e) dissipates less energy than the “clamped” configuration (Fig. 2d), especially for larger values of mT . Furthermore, the angular CoR of the asymmetric BCs (Fig.2f) is characterised by different values of e𝜃 , depending on the direction of impact, especially for higher values of mT . Finally, Fig. 5 implicitly illustrates the importance of the top mass mT compared to the overload N on the angular CoR e𝜃 , since N does not affect e𝜃 and thus corresponds to the case of zero top mass in Fig.5 (solid blue line). Overall, Fig.5 illustrates that both the mass on top and the BCs have a noteworthy influence on the energy dissipation of the VSSW, even in the case where the hinge height remains the same before and after impact. 3.2 Extraction of the angular coefficient of restitution from the response-history The quantification of the angular CoR after an experimental campaign may be done following the original definition of e𝜃 , i.e. by measuring the angular velocities of the VSSW right before and after each impact. Despite the seemingly straightforward procedure of this approach, two main reasons make the experimental acquisition of the angular velocities a challenging task [20, 34, 38–41]. Firstly, the time-instants that an impact “starts” and “ends” cannot be defined objectively, as the impact phenomenon is physically continuous and not instantaneous, as assumed by the phenomenological definition of the angular CoR [35]. In addition, during impacts, the velocity of the structure increases at the time-instants of impact. Thus, accurate acquisition of the velocities requires a high sampling rate, which is not always experimentally feasible. Therefore, the need for a simpler approach to extract the angular CoR after an experimental campaign is evident. To this end, one may assume that energy is preserved during the pivoting phase (energy is lost only during impacts), employ the maximum rotations of the system ( 𝜃1,max ) for the half cycles before and after the impact, and compute the associated energy loss and the angular CoR e𝜃 . The conservation of energy for each half-cycle reads: (16) V 𝜃1,max −V𝜃1=0+ 𝜃 1,max ∫ 𝜃 1=0 ΓNd𝜃1=Z𝜃1= 0 Fig. 5 Angular coefficient of restitution e𝜃 for different wall aspect ratios and overburden mass, assuming a fixed hinge height h1∕h=0.7 and alternative boundary conditions: a “clamped”, b “pinned”, and c asymmetric Meccanica Vol:. (1234567890) the “clamped” case (solid black lines) outperforming the “pinned” (densely dashdotted green lines) and asymmetric (dotted red lines) cases. Interestingly, the last two cases show marginal differences for stockier walls (Fig.11a–c). In sum, Fig.11 provides a holistic overview of the response of the VSSW model subjected to recorded ground motions and highlights the importance of both the overburden mass and BCs on its seismic vulnerability. 5 Conclusions This paper presents a numerical model for the analysis of a vertical spanning strip wall based on rocking dynamics. This work extends current knowledge by incorporating the positional parametrisation of the boundary conditions at the top of the wall and accounting for the presence of the overburden mass. These extensions enable a more accurate representation of a wide variety of boundary conditions and overburden scenarios found in realistic unreinforced masonry structures. Importantly, this study demonstrates the significance of these parameters on the response of the vertical spanning strip wall through a series of comparative analyses. The nonlinear equation of motion is derived using Lagrangian dynamics, followed by the minimum ground acceleration capable of initiating rocking motion, the estimation of the expected hinge height, and the computation of the instability angle. It is shown that the last two are decidedly influenced by the boundary conditions on top of the wall. The energy losses during the impacts of the vertical spanning strip wall are also examined, and two methods are proposed for quantifying the angular coefficient of restitution. The first method provides an analytical estimation of the coefficient of restitution, useful for predictive applications. To this end, impulsive dynamics are employed and, within postulated assumptions, a formula for the angular coefficient of restitution is derived. This demonstrates the dependence of the angular coefficient of restitution on the overburden mass and the boundary conditions at the top of the vertical spanning strip wall. The second method facilitates the extraction of the angular coefficient of restitution from a given free-rocking time-history response assuming that energy is lost solely during impacts. This method is particularly useful for the characterisation and calibration of the energy losses after an experimental campaign, while its validity and use are demonstrated for controlled numerical tests. Finally, the work highlights various aspects of the proposed model using comparative analyses. These include freeand forced-rocking simulations under sinusoidal pulse excitations and real ground motion records. More specifically, the free-rocking comparisons demonstrate the effect of an overburden mass and boundary conditions on the energy dissipation of a representative vertical spanning strip wall. Furthermore, the analyses of the same wall under sine pulse excitations demonstrate that variations in the overburden mass and boundary conditions alter the system’s dynamics and, therefore, influence the corresponding safe and overturning regions. Importantly, the directivity of the sine pulse appears to be detrimental in the case of asymmetric boundary conditions. Ultimately, the most comprehensive view of the vertical spanning strip wall’s vulnerability is attained through a series of real ground motion forced-rocking analyses. Therein, empirical fragility curves highlight the importance of the overburden mass and boundary conditions on the seismic response of nine different walls. 6 Appendix1 Supplementary terms ofSects.2 and3 This Section presents the supplementary terms used in Sects.2 and 3. Meccanica Vol.: (0123456789) (18) C 1=sgn𝜃1cos 𝜑tan 𝛼2 [ sin ( 𝛼1− || 𝜃1 ||) sin 𝛼1 + 𝜆BC 2−1 ] (19) dC 1 d𝜃 1 =−cos 𝜑tan 𝛼2 cos ( 𝛼1− | | 𝜃1 | |) sin 𝛼 1 (20) d 2C1 d𝜃2 1 =−sgn𝜃1cos 𝜑tan 𝛼2 sin ( 𝛼1− | | 𝜃1 | |) sin 𝛼 1 (21) C 2=−2 cos ( 𝛼1− || 𝜃1 ||) − sin 𝛼 1 sin 𝛼2 dC 1 d𝜃1 1 √ 1−C2 1 cos ( 𝛼2− || 𝜃2 ||) (22) dC2 d𝜃1 =− 2sgn 𝜃1 sin ( 𝛼1−||𝜃1|| ) − sin 𝛼1 sin 𝛼2(1−C2 1) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎡⎢⎢⎢⎣ d2C1 d𝜃2 1√1−C2 1+(dC1 d𝜃1)2C1 √1−C2 1 ⎤⎥⎥⎥⎦ cos (𝛼2−||𝜃2||) −sgn𝜃1 ( dC1 d𝜃 1) 2 sin ( 𝛼2− || 𝜃2 ||) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (23) C3=sgn𝜃1 ⎡ ⎢ ⎢ ⎢ ⎣ 2 sin ( 𝛼1− || 𝜃1 ||) − sin 𝛼1 sin 𝛼2 dC1 d𝜃1 1 √ 1−C2 1 sin ( 𝛼2− || 𝜃2 ||)⎤ ⎥ ⎥ ⎥ ⎦ (24) dC3 d𝜃1 =− 2 cos ( 𝛼1−||𝜃1|| ) − sgn 𝜃1 sin 𝛼1 sin 𝛼2(1−C2 1) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎡⎢⎢⎢⎣ d2C1 d𝜃2 1√1−C2 1+(dC1 d𝜃1)2C1 √1−C2 1 ⎤⎥⎥⎥⎦ sin (𝛼2−||𝜃2||) +sgn𝜃1 ( dC1 d 𝜃1) 2 cos ( 𝛼2− || 𝜃2 ||) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (25) C 4 =2sgn 𝜃1 sin ( 𝛼1 −| |𝜃1 | | ) −2sin 𝛼1 tan 𝛼2 dC1 d𝜃1 1 √ 1−C2 1 sin (𝜑+𝜃2) cos 𝜑 (26) dC4 d𝜃1 =− 2 cos ( 𝛼1−||𝜃1|| ) − 2 sin 𝛼1 tan 𝛼2(1−C2 1) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [d2C1 d𝜃2 1√1−C2 1−(dC1 d𝜃1)2 C1]sin (𝜑+𝜃2) cos 𝜑 + ( dC1 d𝜃1) 2cos ( 𝜑+𝜃2 ) cos 𝜑 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (27) C5=R2 1 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ m1+m2 [ cos2𝛼1+sin2𝛼1 ( 2+ tan 𝛼2 tan 𝛼1 )2] +mTsin2𝛼1(2+𝜆BC tan 𝛼2 tan 𝛼1)2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ +IG1+IG2 ( tan 𝛼2 tan 𝛼1) 2 (28) C6= ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢ ⎢⎢⎢⎣ gR1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ( m1+ 2 m2+ 2 mT )[cos ( 𝛼1−||𝜃1,max|| ) − cos 𝛼1 ] +m2 sin 𝛼1 sin 𝛼2[cos (𝛼2−||𝜃2,max||)−cos 𝛼2] +2mT sin 𝛼1 tan 𝛼2cos 𝜑⎡⎢⎢⎢⎣ √ √ √ √ √1−(cos 𝜑tan 𝛼2[sin (𝛼1−||𝜃1,max||) sin 𝛼1 −1+ 𝜆BC 2])2 −√1−(𝜆BC 2cos 𝜑tan 𝛼2)2⎤⎥⎥⎥⎦ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ +NR1 { 2 [ cos ( 𝛼1− || 𝜃1,max ||) −cos 𝛼1 ] +sin 𝛼1 [ 𝜆Nsin (|| 𝜃2,max ||) +2 cos (|| 𝜃2,max ||) −1 tan 𝛼 2]} ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 7 Appendix2 Details ofground motions used inSect.4.3 This Section tabulates the characteristics of the ground motions adopted in Sect.4.3 see Table1. Meccanica Vol:. (1234567890) Acknowledgements This work is financed by national funds through FCT – Foundation for Science and Technology, under grant agreements 2020.07325.BD, PRT/BD/152830/2021 and 2023.03854.BDANA attributed to the first, second and third authors, respectively. This study has been partly funded by the STAND4HERITAGE project (new STANDards FOR seismic assessment of built cultural HERITAGE) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No. 833123) as an Advanced Grant. This study was also partly funded by FCT / MECI through project 2022.05425.PTDC (RESISTANCE) and project national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB / 04029/2020 (doi.org/https:// doi. org/ 10. 54499/ UIDB/ 04029/ 2020), and under the Associate Laboratory Advanced Production and Intelligent Systems ARISE under reference LA/P/0112/2020. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organisations. Author contribution G.V.: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing—Original Draft, Visualization C.C.: Writing—Review & Editing, Visualization D.V.: Writing—Review & Editing A.I.G.: Writing—Review & Editing, Visualization, Supervision, Project administration, Funding acquisition P.B.L.: Resources, Writing—Review & Editing, Supervision, Project administration, Funding acquisition. Table 1 Details of ground motion records of Sect.4.3, collected from [50] No Event Station Component [°] Soil class Magnitude Mw Rupture distance [km] PGA [g] 1 Loma Prieta, 1989 Agnes State Hospital 90 C, D 6.9 28.2 0.159 2 Northridge, 1994 LA, Baldwin Hills 90 B, B 6.7 31.3 0.239 3 Imperial Valley, 1979 Compuertas 285 C, D 6.5 32.6 0.147 4 Imperial Valley, 1979 Plaster City 135 C, D 6.5 31.7 0.057 5 Loma Prieta, 1989 Hollister Diff. Array 255 –, D 6.9 25.8 0.279 6 San Fernando, 1971 LA, Hollywood Stor. Lot 180 C, D 6.6 21.2 0.174 7 Loma Prieta, 1989 Anderson Dam Downstrm 270 B, D 6.9 21.4 0.244 8 Loma Prieta, 1989 Coyote Lake Dam Downstream 285 B, D 6.9 22.3 0.179 9 Imperial Valley, 1979 El Centro Array #12 140 C, D 6.5 18.2 0.143 10 Imperial Valley, 1979 Cucapah 85 C, D 6.5 23.6 0.309 11 Northridge, 1994 LA, Hollywood Storage FF 360 C, D 6.7 25.5 0.358 12 Loma Prieta, 1989 Sunnyvale Colton Ave 270 C, D 6.9 28.8 0.207 13 Loma Prieta, 1989 Anderson Dam Downstrm 360 B, D 6.9 21.4 0.24 14 Imperial Valley, 1979 Chihuahua 12 C, D 6.5 28.7 0.27 15 Imperial Valley, 1979 El Centro Array #13 140 C, D 6.5 21.9 0.117 16 Imperial Valley, 1979 Westmoreland Fire Station 90 C, D 6.5 15.1 0.074 17 Loma Prieta, 1989 Hollister South & Pine 0 –, D 6.9 28.8 0.371 18 Loma Prieta, 1989 Sunnyvale Colton Ave 360 C, D 6.9 28.8 0.209 19 Superstition Hills, 1987 Wildlife Liquefaction Array 90 C, D 6.7 24.4 0.18 20 Imperial Valley, 1979 Chihuahua 282 C, D 6.5 28.7 0.254 21 Imperial Valley, 1979 El Centro Array #13 230 C, D 6.5 21.9 0.139 22 Imperial Valley, 1979 Westmoreland Fire Station 180 C, D 6.5 15.1 0.11 23 Loma Prieta, 1989 Halls Valley 90 C, C 6.9 31.6 0.103 24 Loma Prieta, 1989 WAHO 0 –, D 6.9 16.9 0.37 25 Superstition Hills, 1987 Wildlife Liquefaction Array 360 C, D 6.7 24.4 0.2 26 Imperial Valley, 1979 Compuertas 15 C, D 6.5 32.6 0.186 27 Imperial Valley, 1979 Plaster City 45 C, D 6.5 31.7 0.042 28 Loma Prieta, 1989 Hollister Diff. Array 165 –, D 6.9 25.8 0.269 29 San Fernando, 1971 LA, Hollywood Stor. Lot 90 C, D 6.6 21.2 0.21 30 Loma Prieta, 1989 WAHO 90 –, D 6.9 16.9 0.638 Meccanica Vol.: (0123456789) Funding Open access funding provided by FCT|FCCN (b-on). Funding was provided by European Research Council, 833123, 833123,833123,833123, Fundação para a Ciência e a Tecnologia, 2020.07325.BD, PRT/ BD/152830/2021,2023.03854.BDANA, 2022.05425.PTDC, 2022.05425.PTDC Data availability Data will be made available on request. Declarations Conflict of interest The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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