ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 1 Estimation of the ground shaking from the response of rigid bodies FILOMENA DE SILVA1, STEFANIA SICA2, FRANCESCO SILVESTRI3, STEFANO AVERSA4 1 Department of Civil, Architectural and Environmental Engineering, Uni- versity of Napoli Federico II, Napoli, Italy; filomena.desilva@unina.it; 2 Department of Engineering, University of Sannio, Benevento, Italy 3 Department of Civil, Architectural and Environmental Engineering, Uni- versity of Napoli Federico II, Napoli, Italy 4 Department of Engineering, University of Napoli Parthenope, Napoli, Italy Abstract The paper illustrates and compares simplified approaches to interpret the mechanisms of damage observed on rigid bodies in the cemetery of Amatrice, after the first strong-motion event (August 24, 2016, MW=6.0) of the seismic sequence occurred in Central Italy. The final goal of the work is to link the ob- served movements of the fallen objects to specific characteristics of the ground motion recorded on site. I. INTRODUCTION fter a strong earthquake, information on ground motion characteristics can be in- directly derived from the observation of the damage suffered by objects with quite simple geometry and known degrees of free- dom, for which the collapse mechanism is straightforward and simple to back-figure. This methodology, borrowed from the archeo- seismology, takes into account not only the in- ventory and amount of the occurred damage, but also the failure mechanisms and the pat- tern of displacement of the fallen objects. The back-analysis of these latter aspects may pro- vide useful information on the amplitude of ground motion parameters (if no seismic sta- tions are available nearby) or on other aspects of the seismic shaking (directivity, impulsive motion, etc.). Previous works on such an issue may be found in Yegian et al. (1994), Athana- sopoulos (1995) and Lanzo et al. (2010). Con- versely to inhabited areas, most of the injured cemeteries are not closed soon after a strong earthquake or - if seriously damaged - they may be inspected in a second stage, before any reconstruction or restoration. The above men- tioned literature showed that their inspection soon after the main shock, or even later, may be very useful to have additional insight into the ground motion occurred on site. During the GEER reconnaissance operations (Stewart et al., 2016) after the MW 6.0 first strong motion event (August 24, 2016, local time 3.32 a.m.) of the seismic sequence in Cen- tral Italy, many interesting cases of fallen gravestones, funerary monuments and statues were observed in the cemetery of Amatrice A ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 2 (Figures 1a). This is located about 400 m South- East and at approximately the same elevation of the main gate of the historical center (Figure 1a), which suffered a local IMCS=10.5 (Galli et al., 2016). All the objects apparently fell down during the first event, and most of them along the NS direction (Figures 1b-e). Among the rigid objects showing regular ge- ometry and clear damage pattern, an obelisk was selected in order to develop a simple analysis to relate the observed failure mecha- nism to the ground motion occurred at the site. (a) (b) (c) (d) (e) Figure 1. Location of the cemetery and of the seismic station at Amatrice (a), damages to statues, monuments and graves in the Amatrice cemetery (b-e). ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 3 II. THE CASE STUDY OF THE MARBLE OBELISK The selected case study concerns the failure of the marble obelisk shown in Figure 2. The obe- lisk has a shape of a truncated pyramid and it is approximately 1.10m high. The width of the square cross section tapers with the height from 0.27m at the base up to 0.13m at the top. The base section was attached through a thin layer of mortar to a stone basement, about 0.80m high. The detected geometry and the height of the gravity center are reported in Ta- ble 1. During the reconnaissance, the obelisk was found lying on the ground, apparently over- turned along the NS direction (Figure 2a), while the weak mortar connection failed (Fig- ure 2b). Table 1. Geometrical parameters of the obelisk. Lower width Upper width Height Gravity center height Slenderness ratio B b h hG  (m) (m) (m) (m) (/) 0.27 0.13 1.10 0.49 1.78 (a) (b) Figure 2. Damaged obelisk in the Amatrice cemetery (a), particular of the base section (b). III. FAILURE CONDITIONS Neglecting the vertical component of the seis- mic acceleration, the forces acting on the obe- lisk during the earthquake (Figure 3) are its weight, W, and a horizontal inertia force H=ma, corresponding to the seismic action on the mass, m. Due to the high stiffness of the marble, both the basement and the obelisk be- have as rigid bodies: thus, the horizontal accel- eration of the obelisk can be considered equal to the ground acceleration, a(t), at the site of the cemetery, from the beginning of the shak- ing until when it detached from the basement N ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 4 and failed. Since the likely failure mechanisms are brittle, it is possible to estimate a lower bound for the maximum ground acceleration from the horizontal force, Hf, that produces the failure of the obelisk. Being difficult to quanti- fy the degree of the constraint exerted by the mortar layer, two limit conditions were con- sidered: 1. the horizontal force is determined by the limit equilibrium in terms of sliding and toppling of the obelisk, assuming a purely fric- tional marble-mortar interface (Figure 3a); 2. the horizontal force corresponds to the achievement of the strength, in terms of com- pression and shear stress, in the reacting zone of the base (Figure 3b-c). According to the first approach, the accelera- tion amplitudes (expressed in g) required to produce the horizontal translation and the ro- tation around the y axis (Figure 3a) can be re- spectively computed as follows: Sa   (1) M 1 a 2   (2) where: -  is the friction coefficient of the marble- mortar interface, -  is the slenderness ratio of the obelisk, given by: Gh B   (3) (a) (b) (c) W H hG D B b b h W H hG B b b H h u W H hG D' B b b h 0.80u W B H H W W W H hG D B b b h W H hG B b b H h u W H hG D' B b b h 0.80u W B H H W W W H hG D B b b h W H hG B b b H h u W H hG D' B b b h 0.80u W B H H W W Figure 3. Approach n°1 (a) and n° 2, corresponding to the onset of the yielding (b) and to the collapse (c) of the obelisk. The first translational mechanism does not necessarily lead to a global failure: in fact, the obelisk can accumulate sliding displace- ments along the base recovering a final glob- al equilibrium, according to the Newmark (1965) rigid block model. In the case at hand, the width of the basement is similar to that of the base of the obelisk. For this reason, the accumulation of horizontal displacements can induce a sort of "second order" effect, leading to the toppling of the obelisk when the cumulated displacement overcomes half of its base width. In this way, the accelera- tion producing the sliding mechanism can be ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 5 considered as a lower bound of the maxi- mum acceleration of the basement which is required to produce global instability. Following Maiorano et al. (2015), the onset of rotation with the second type of failure mechanism does not necessarily induce a toppling failure, since the block can recover a stable position after some bouncing on the base. Also in this case, the acceleration am- plitude that triggers the first rotation can be considered as a lower bound of that required for producing failure. The second approach is based on the hy- pothesis that the limit compression stress, σu, is achieved at the edge of the resisting zone of the base section, this latter assumed with a width equal to D. This condition corre- sponds to the onset of the first yielding in the mortar section. From Figure 4a, the equilib- rium of the obelisk is determined by: - vertical translation: uW 0.5 DB  (4) - rotation around the gravity center of the base section: M G u B D ma h 0.5 DB 2 3         (5) - horizontal translation: Sma cDB W  (6) In this latter equation, it is assumed that the shear strength at the interface obeys to the Mohr-Coulomb criterion. From the equilib- rium under the vertical action W (eq. 4), the value of D, corresponding to the achieve- ment of the compression limit stress at the edge of the base section, is calculated. From eqs. (5)-(6), the acceleration amplitudes cor- responding to the ultimate moment, aM, and to the shear strength, aS, can be respectively computed. (a) (b) D  u W maMhG B/2-D/3  c 0.5uDB D' 0.80u W maMhG (B-D')/2  0.80uD'B maSmaS Figure 4. Equilibrium of the base section for the second approach, corresponding to the onset of the yielding (a) and to the collapse (b). With the propagation of yielding, the full plastic failure occurs with non-linear stress distribution at the stone-mortar interface (see Figure 4b). Just like commonly assumed in concrete structures, it can be hypothesized that plastic failure corresponds to the achievement of a mean 0.80σu in the whole resisting zone of the base section, the width of which reduces to D’. In this condition, the equilibrium equations become: ' uW 0.80 D B  (7) ' ' M G u B D ma h 0.80 D B 2           (8) ' Sma cD B W  (9) ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 6 IV. RESULTS The two approaches illustrated in the previ- ous section were applied to the case study of the obelisk, in order to compute two respec- tive couples of limit accelerations, aS and aM. The geometrical dimensions, including the slenderness ratio are those reported in Table 1, while the physical and mechanical param- eters adopted in the calculations are summa- rized in Table 2. Table 2. Parameters assumed in the calculations. Unit weight Weight Compression limit stress Shear strength App.  W σu c  (kN/m3) (kN) (MPa) (MPa) (/) 1 27.00 1.30 / / 0.40 2 1.00 0.34 0.40 In details: · W was evaluated from the geometry and the marble unit weight, ; · the compression limit stress, σu, was as- sumed on the basis of the lowest class (M1) of the European classification of the mortar (EN 998:2), included in EC6 (EN 1996-1-1; 2006); · the frictional strength coefficient, , is the typical value adopted for the mortar adopted in the masonry (EN 1996-1-1; 2006); · the cohesion, c, was back-calculated from σu and  through the Mohr-Coulomb criterion, adopted to model the shear strength of the marble-mortar interface. Since the only vertical load considered is the weight of the obelisk, the limit value σu is achieved in the section for a very low length of the resisting zone (D=0.01m; D’=0.005m). As a consequence, the solutions of the equi- librium in correspondence of the first yield- ing and of the full plastic failure are coinci- dent. Moreover, they are close to the results of the first approach, as synthesized in Table 3. Table 3. Computed limit accelerations. Approach 1 Approach 2 aM aS aM aS (g) (g) (g) (g) 0.28 0.40 0.27 0.40 The obtained results highlight that: · the relatively high slenderness of the ob- elisk makes it overturn under a seismic action lower than the value correspond- ing to the onset of sliding, as well as the acceleration associated to the ultimate moment at the base section results lower than that related to the ultimate shear force; · according to both approaches, the ground acceleration in the NS direction, i.e. the falling direction of the obelisk, should have been higher than 0.27g. IV. COMPARISON WITH THE RECORDED GROUND MOTION Figure 5a-b show the NS and EW compo- nents of the acceleration time history record- ed at the Amatrice seismic station (AMT), located 75 m downhill and about 1 km away of the cemetery (see Figure 1a). The data downloaded from the ESM database (ESM working group, 2015) are plotted in the most critical time window of the shaking, i.e. from t=6s to t=12s. ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 7 (a) -1 -0.5 0 0.5 1 6 7 8 9 10 11 12 |a N S |- |a E W | (g ) t (s) -1 -0.5 0 0.5 1 6 7 8 9 10 11 12a N S (g ) t (s) max(|NS|-|EW|) NS exceedance EW exceedance -1 -0.5 0 0.5 1 6 7 8 9 10 11 12a E W (g ) t (s) aS aM aM aS aS aM aM aS (b) -1 -0.5 0 0.5 1 6 7 8 9 10 11 12 |a N S |- |a E W | (g ) t (s) -1 -0.5 0 0.5 1 6 7 8 9 10 11 12a N S (g ) t (s) max(|NS|-|EW|) NS exceedance EW exceedance -1 -0.5 0 0.5 1 6 7 8 9 10 11 12a E W (g ) t (s) aS aM aM aS aS aM aM aS (c) -1 -0.5 0 0.5 1 6 7 8 9 10 11 12 |a N S |- |a E W | (g ) t (s) -1 -0.5 0 0.5 1 6 7 8 9 10 11 12a N S (g ) t (s) max(|NS|-|EW|) NS exceedance EW exceedance -1 -0.5 0 0.5 1 6 7 8 9 10 11 12a E W (g ) t (s) aS aM aM aS aS aM aM aS Figure 5. NS (a) and EW (b) acceleration time histo- ries recorded in Amatrice station (AMT) and difference (c) between the absolute values of the two horizontal components. The two limit accelerations of the obelisk, aM and aS, are exceeded by several peak values of the EW component (Figure 5b), with the first of them occurring around t=9s. At al- most the same time, the NS component tres- passes the limit overturning acceleration, i.e. aM = 0.27g, having approached this threshold already two times before (Figure 5a). Since the obelisk fell down along the NS direction, its collapse likely occurred when the NS ac- celeration component was higher than aM, and before the EW acceleration component exceeded the same threshold value. The overturning mechanism might have there- fore occurred in the time interval between 8.5 and 9s, i.e. when the difference between the absolute values of the NS and EW com- ponents (shown in Figure 5c) is mostly posi- tive. Table 4 summarizes the acceleration values corresponding to the instants of the maxi- mum difference between the two horizontal amplitudes (red dots in Figure 5) and when the NS (blue dot) and EW (black dot) com- ponents achieve the aM threshold for the first time. The NS component overcomes significantly the contemporary EW value at t=8.66s and t=8.82s. In addition, at the same instants, the NS acceleration amplitude are 0.22g and 0.24g, respectively, i.e. very close to aM (0.27g). The overturning critical acceleration is at- tained almost simultaneously in both direc- tions, with the NS component reaching the threshold 0.01s before the EW one. Table 4. Acceleration values corresponding to the in- stants of the maximum difference between the two hor- izontal components and to the attainment of the over- turning threshold in both directions. a IA t NS EW | NS |-|EW| NS EW (s) (g) (g) (g) (m/s) (m/s) max (|NS|-|EW|) 8.66 -0.22 -0.02 0.20 0.08 0.06 max (|NS|-|EW|) 8.82 -0.24 0.00 0.24 0.14 0.10 NS exceedance 8.92 0.27 0.21 0.05 0.19 0.11 EW exceedance 8.93 0.33 0.27 0.04 0.20 0.12 ANNALS OF GEOPHYSICS, 59, FAST TRACK 5, 2016; DOI: 10.4401/AG-7296 8 Figure 6 shows the evolution of the Arias In- tensity with time, while Table 4 reports the accumulated values at the four critical in- stants discussed above. Although the final value of IA along EW results more than twice that relevant to the NS direction, the detail in Figure 6 reveals that, before t=9s, more ener- gy is accumulated along NS than in the EW direction. The position of the dots in the same plot and the relevant values of IA in Table 4 confirm that the temporary predom- inance of the NS component on that along EW direction is coherent with the direction of the failure. 0 1 2 6 7 8 9 10 11 12 I A (m /s ) t (s) EW NS 0.00 0.25 0.50 8.6 8.8 9 9.2 max(|NS|-|EW|) NS exceedance EW exceedance Figure 6. Arias Intensity of the NS and EW compo- nents, recorded in Amatrice. A predominance of the NS component with respect to the EW has also been shown by the displacement inferred from the high-rate GPS data by the INGV GPS analysis group (2016), with the NS peak-to-peak value (≈15 cm) resulting twice that corresponding to the EW direction. Moreover, the above interpretation of the ac- celeration recording could not account for site effects due to the different locations of the AMT station, placed on the Laga Flysch formation representing the seismic bedrock, and the cemetery, lying on alluvial deposits constituting the hill of Amatrice (Regione Lazio, 2016), close to the steep slope border- ing the northern ridge. Thus, stratigraphic and topographic amplifi- cations of the recorded ground motion likely occurred at the cemetery during the earth- quake. Considering that the morphology of the Amatrice hill is elongated along the EW direction, the topographic amplification might have affected more significantly the NS component with respect to the EW shak- ing. In such hypothesis, the shaking energy associated with the NS component in the cemetery area might have been even more significant than that relevant to the EW di- rection. It is expected that seismic response analyses, which will be soon carried out for the seismic microzonation of the area, will corroborate such assumption. IV. CONCLUSION AND FUTURE DEVELOPMENTS The paper illustrated two simple analytical approaches to interpret the observed failure of a funerary obelisk in the cemetery of Ama- trice, due to the first strong motion event of the Central Italy seismic sequence. A lower bound for the peak ground acceleration oc- curred at the site was estimated by back- analyzing different possible failure mecha- nisms of the object (sliding and toppling) with reference to global limit equilibrium and yield as well as plastic local failure. The estimated peak acceleration in the NS direc- tion may be representative of the severity of the ground motion occurred in this urban area, located far enough from the available seismic station (AMT) and in quite different site conditions. 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