Seismic Demand on a Unreinforced Masonry Wall Restrained by Elasto-Plastic Tie-Rods Under Earthquake Sequences

3.1. Walls characteristics

Four walls are considered, the same as in AlShawa, Liberatore, and Sorrentino (Citation2019), of different aspect ratio and size. A three-leaf uncut-stone masonry was assumed as reference material, but additional analyses were performed with cut-stone masonry with good bond. Material properties are reported in Table 1 and are in accordance with the Commentary to the Italian Building Code (CMIT Citation2009).

As customary in Italian technical literature (Cangi, Caraboni, and De Maria Citation2010; Giuffrè Citation1993; Munari et al. Citation2010), the tie is designed according to a force-based procedure based on the Commentary to the Italian Building Code (CMIT Citation2009), in order to get the following horizontal collapse load multiplier:

(11) ${\alpha }_0=\frac{a_{g}SCF{e}^{\ast }}{q}$(11)

where a_g = ground acceleration corresponding to the life safety limit state in a horizontal rock site, S = site response coefficient (related to topographic and stratigraphic settings), CF = confidence factor, e* = participating mass factor, q = behaviour factor. The following values were assumed as in AlShawa, Liberatore, and Sorrentino (Citation2019): a_g = 0.26 g, S = 1.33, CF = 1.00, e* = 1.00 (as for a monolithic wall according to the Italian standard (Sorrentino et al. Citation2017)), q = 2.00 (according to the Commentary to the Italian Building Code (CMIT Citation2009)).

The tie force, F_y, granting the load multiplier α₀ is equal to:

(12) $F_y=\frac{mg\left({\alpha }_{0}H-B+2u\right)}{2H_t}$(12)

where the hinge indentation u is usually assumed equal to:

(13) $u=\frac{mg}{2\cdot 0.85f_d{L}_h}$(13)

with f_d being the masonry design compressive strength:

(14) $f_d=\frac{f_m}{CF{\gamma }_m}$(14)

and γ_m = material partial safety factor. In the following, it is assumed γ_m = 2.0, according to the recommendation of Italian Building Code (DMI Citation2008) in the case of seismic design. Eq. (13) is the same as the third line of Eq. (4) but for the factor values and the 0.85 coefficient, applied to the masonry compressive strength.

The tie cross-section area, A_t, related to Eq. (12) is equal to:

(15) $A_t=\frac{F_y}{f_y}$(15)

where f_y = steel yield strength. In the following f_y = 223.8 MPa and ε_u = 0.20 were assumed. Tie-rod area is different for walls of the same geometry but with different tie-rod heights. Ties features are kept constant for all seismic sequences, despite such seismic events occurring in different regions and having different return periods, hence being related to a seismicity different from the one assumed in the tie design.

3.2. Ground motion sequences

It is well known that the issue of record selection is a complex one. Many features must be taken into account, as source type, site conditions and different intensity parameters. This complexity increases when dealing with sequences rather than a single shock. In this regard, no internationally established criteria exist for the selection of ground motion sequences. Since the structural assemblies studied herein are representative of situations typical of Italy areas, in terms of both masonry characteristics and tie design assumptions, Italian earthquakes were considered. Starting from this premise, the choice was led by the availability of records belonging to sequences in which the strongest event had a magnitude M_w of at least 6.0. In the last half century, the following earthquakes were found to possess these characteristics: Friuli 1976 (Cheloni et al. Citation2012), Irpinia 1980 (Bernard and Zollo Citation1989), Umbria–Marche 1997 (Decanini, Gavarini, and Mollaioli Citation2000), L’Aquila 2009 (Decanini, Liberatore, and Mollaioli Citation2012), Emilia 2012 (Iervolino, De Luca, and Chioccarelli Citation2012), Central Italy 2016 (Mollaioli et al. Citation2018). These earthquakes were all produced by normal or reverse fault mechanisms.

It should be noted that in many parts of Italy complex fault systems exist and, consequently, earthquakes usually occur as a cluster. This feature is mainly due to the initial rupture, which causes the first earthquake, not usually relieving all accumulated strain but producing high stresses at different locations of the fault system, which in turn may cause sequential ruptures leading to multiple earthquakes. In other words, the energy released after a mainshock event in a certain fault could trigger another mainshock in a nearby fault that could affect the same region. In this case, a sequence of earthquakes may be observed from proximate fault segments, different from the classic earthquake sequence defined as fore-, main-, and after-shocks. The selected dataset includes events belonging to both types of sequences.

All the records in the sequences (Tables 2, 3), which were retrieved from the ITACA database (Luzi, Pacor, and Puglia Citation2017), were selected to comply with the following conditions: (1) there is sufficient geological and geotechnical information at the recording station; (2) the magnitude of each event of the sequences is not less than 3.7; (3) the ground motion was recorded possibly at a free field station, or at the ground level of a building so as to avoid possible effects due to soil-structure-foundation interaction; (4) all sequences have four events; (5) the selection of four events for each earthquake is based on Peak Ground Velocity (PGV) values, since this parameter is considered to be adequately correlated with damage; the four events with the largest PGV are, therefore, included in each sequence (Table 2). In order to meet the last criterion, the events considered for a given station are not the same for all stations affected by the earthquakes. For instance, with reference to the Friuli earthquake, accelerograms considered for the FRC station are those recorded during events d, f, g and h, whereas events a, b, c and d are those considered for the TLM1 station (see Table 2).

It is worth noting that no other Italian earthquakes with the above characteristics were found other than those used in this study. The suite of records covers a range of magnitudes between 3.7 and 6.9. The aftershock records usually do not have significant long-period spectral content, being characterised by lower magnitudes. This is not the case when the records belong to a sequence with more than one mainshock.

In all cases, a time buffer of 20 s is used between the earthquake records that are applied in series. The reason for using this time buffer between successive records is to ensure that the structural behaviour under the subsequent earthquake is not influenced by any remaining dynamic action due to the previous earthquake.

4.1. Parametric analyses

A parametric analysis was performed, considering: two types of masonry (Table 1), four wall geometries obtained varying the H/B ratio and the wall thickness (H/B = 8 and 12, B = 0.6 m and 0.9 m), six tie-rod normalised heights H_t/H (Figure 1a) varying from 0.5 to 1, and 30 accelerograms (15 stations, 2 components, Table 2). For each wall and for each component recorded at a station, two time-history analyses were performed: the first one considering only the main event, which is not necessarily the first one in the sequence, and the second analysis considering the sequence of four events. In the latter case, to study the effect of the sequence, the results were examined after the first event, then considering the cumulated effect of the first and the second events and so on up to the complete sequence of four events.

The results for the East component of the GMN record (1976 Friuli Earthquake) are plotted in Figure 4, where the role of the seismic sequence is shown for each of the four walls considered and for different tie-rod normalised heights, H_t/H. The response is expressed in terms of normalised maximum rotation, θ_max/α, and the elastic (E), plastic (P) or failure (F) tie-rod response is reported by means of a letter on top of each bar. The bars represent the sequence progression: just event 1 for the first bar, a partial sequence of events 1 and 2 for the second bar, then a partial sequence of events from 1 to 3, or the complete sequence of events from 1 to 4. In the case of East-West component of the accelerograms recorded during the 1976 Gemona sequence in Friuli, we can observe that during the first two events the tie-rod remained elastic, while it became plastic for the last two with the walls experiencing rather large rotations. However, this behaviour is not systematic because, in other sequences, such as that in Figure 5 recorded in Valle Aterno during the 2009 L’Aquila Earthquake, the first event is the one inducing the maximum rotation, with later shaking only marginally increasing or not increasing at all the response. It is worth emphasising that each bar in Figure 4 and in Figure 5 shows the largest rotation occurred in the (partial or complete) sequence (Figure 6), not just the largest rotation in the last event of the sequence, considered by itself.

Figure 4. Normalised maximum rotation, θ_max/α, of a façade restrained by a tie-rod varying its normalised height, H_t/H (Figure 1a), and wall geometry (a-d). Elastic, E, or plastic, P, response of the tie emphasised. Three-leaf uncut-stone masonry.

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Figure 5. Normalised maximum rotation, θ_max/α, of a façade restrained by a tie-rod varying its normalised height, H_t/H (Figure 1a), and wall geometry (a-d). Elastic, E, or plastic, P, response of the tie emphasised. Three-leaf uncut-stone masonry.

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Figure 6. AQV station, East-West component. Event 1: a) Time history of normalised rotation, b) time history of ground motion accelerations. Event 1 through 3: c) Time history of normalised rotation, d) time history of ground motion accelerations. Wall features: H/B = 8, B = 0.6 m, H_t/H = 0.5, three-leaf uncut-stone masonry. Time buffer between events reduced to improve readability.

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4.2. Influence of the interface normal stiffness

As mentioned above, the model proposed in this study adopts a deformable interface at the wall base, following recent research by Costa et al. (Citation2013) and Mehrotra and DeJong (Citation2018). In the analyses, interface normal stiffness k_n was set equal to 6 MPa/mm, which is the largest value investigated by Costa et al. (Citation2013). This value is still rather low because, assuming a large joint thickness of 30 mm, Young’s modulus as low as 180 MPa is necessary to get k_n = 6 MPa/mm. It is worth mentioning that after analysing rather poor historical mortars, the lowest value of Young’s modulus reported by Mirabile Gattia et al. (Citation2019) is 232 MPa. Researches on tuff masonry reviewed by Marotta, Liberatore, and Sorrentino (Citation2016) reported mortar elastic moduli in excess of 1500 MPa. Therefore, to investigate the influence of parameter k_n, the analyses were repeated also assuming k_n equal to 0.6, 2.0, 4.0, 75 and 500 MPa/mm.

In Figure 7(a) the moment rotation law of an unrestrained wall is shown, varying the base flexible interface stiffness k_n. It is evident that for the lower figures among those investigated, there is an evident deviation from the bilinear curve of a rigid wall, with a smoothening effect increasing with decreasing k_n. If a tie-rod is introduced (Figure 7b), the role of interface stiffness is similar but, related to the largest moment, less relevant. This behaviour is shown in Figure 8, which highlights how k_n has no systematic effect on wall maximum rotation, thus downplaying the relevance of this uncertain parameter on tie-rod design or on restrained-wall assessment.

Figure 7. Normalised restoring moment–rotation law of a wall that is: a) unrestrained, b) restrained by a tie-rod, varying the base flexible interface stiffness k_n. Wall features: H/B = 8, B = 0.6 m, H_t/H = 0.7, three-leaf uncut-stone masonry.

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Figure 8. Normalised maximum rotation, θ_max/α, of a façade with tie-rod varying the base flexible interface stiffness kn: a) B = 0.6 m, b) B = 0.9 m. Wall features: H/B = 8, H_t/H = 0.7, three-leaf uncut-stone masonry.

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4.3. Low-cycle fatigue effects

It is well known that when steel experiences large inelastic cyclic deformations, like those which can be caused by earthquake-induced forces, cyclic degradation phenomena can occur. According to Panthaki (Citation1992), an earthquake load can result in 2 to 10 full cycles for common structures, and up to 30 cycles for structures with high natural frequencies. Therefore, the degradation effect may be ascribed to the so-called ultra-low-cycle fatigue phenomenon. In this case, the number of cycles is less important than the extension of the plastic deformations range.

To investigate possible effects due to cyclic degradation in the steel tie, the time-history analyses were repeated for the three-leaf uncut-stone masonry walls, introducing in the steel constitutive model a low-cycle fatigue law. Several fatigue-life models were proposed to predict the effect of low-cycle fatigue on steel elements. These models are usually defined in terms of total or plastic strain as a function of the number of cycles leading to failure. One of the main shortcomings of most existing formulations is that they require regular cycles to predict material failure (Martinez et al. Citation2015). However, the response of structures to earthquakes is not regular, with frequencies and amplitudes varying at each cycle. To overcome this drawback, approaches were proposed based on accumulated plastic strain experienced by the material throughout the loading history (e.g. Mendes and Castro Citation2014) or on the energy dissipated during the cyclic process (Martinez et al. Citation2015).

In this study, the approach suggested by Mendes and Castro (Citation2014) is used. In the model, the ultra-low-cyclic fatigue effect is taken into account by means of reduction factor, ${\gamma }_f$, applied to yield strength at the end of each cycle and expressed as:

(16) ${\gamma }_f=1.0-c_f{\left(\frac{{\epsilon }_p^{ac}}{{\epsilon }_f}\right)}^{n_f}$(16)

where ${\epsilon }_p^{ac}$ is the accumulated plastic strain experienced by the tie at the end of the cycle, ${\epsilon }_f$ is the value of the accumulated plastic strain at failure, $c_f$ and $n_f$ are parameters which control the fatigue evolution. The model parameters were set as follows: ${\epsilon }_f={\epsilon }_u$, $c_f=0.25$ and $n_f$ = 0.5, which, compared to those suggested by Mendes and Castro (Citation2014), are somewhat conservative. Clearly, when ${\epsilon }_p^{ac}\ge {\epsilon }_f$, the steel tie breaks and ${\gamma }_f$ must be set equal to zero.

The severity factor, which takes into account that for the same level of accumulated plastic strain, larger plastic cycle amplitudes induce higher fatigue-type damage, is not considered in this study because, as suggested by Mendes and Castro (Citation2014), it should be applied only in the presence of effective load reversals, in which the stress changes from tension to compression. Such reversals do not occur in the present case because, due to wall restraint conditions, the tie is never compressed.

The calculated yield stress reduction factor, ${\gamma }_f$, ranges from 0.84 to 1, being 1 when steel remains in the elastic range. It is worth noting that the ${\gamma }_f$ factor is less than 0.90 only in 3% of the analysed cases, whereas it is greater than 0.95 in 80% of cases. This is due to the fact that the number of cycles performed in the plastic range, and therefore the accumulated plastic strain, is somewhat low in the cases at hand.

The maximum reduction of the yield stress is obtained with the GMN sequence (Figure 9), where ${\gamma }_f$ = 0.84. In this case, the rod elongation (Figure 9c) increases slightly, leading to a marginal increase in the maximum normalised rotation (Figure 9a). However, the maximum effect on wall rotation is related to the AMT station (Figure 10), where ${\gamma }_f$ = 0.88 and θ_max/α = 0.47, which is noticeably greater than the value obtained with no consideration of the fatigue effect (θ_max/α = 0.20). This large difference is due to the fact that yield stress reduction leads to the occurrence of a third cycle, with the consequent increase of the strain in the tie-rod (Figure 10c).

Figure 9. GMN station, East-West component: a) time history of normalised rotation, θ/α, b) time history of normalised axial force in the tie-rod, F_t/F_y, c) normalised axial force, F_t/F_y, versus normalised elongation in the tie-rod, εt/εu. Wall features: H/B = 12, B = 0.6 m, H_t/H = 1.0, three-leaf uncut-stone masonry.

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Figure 10. AMT station, East-West component: a) time history of normalised rotation, θ/α, b) time history of normalised axial force in the tie-rod, F_t/F_y, c) normalised axial force, F_t/F_y, versus dilation in the tie-rod, ε_t/ε_u. Wall features: H/B = 12, B = 0.6 m, H_t/H = 1.0, three-leaf uncut-stone masonry.

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However, in most cases, the effect of tie degradation on wall rotations is limited. In fact, in 93% of cases, the rotation increase, if present, is less than 5%, and in no case did the fatigue phenomenon lead to tie-rod failure or to overturning of the wall.

4.4. Correlation to intensity measures

Maximum rotations were compared with most intensity measures considered by Mollaioli et al. (Citation2013), excluding only those related to a specific period of vibration of a linear-viscous-elastic oscillator. A sample plot, assuming a log-log scale, is presented in Figure 11 for PGA and PGV, considering three-leaf uncut-stone masonry. Each marker in the plots is related to a wall geometry, a tie-rod height, a station, and a component, while the intensity measure is computed over the (partial or complete) sequence of events (1 alone, 1–2, 1–3, 1–4). The only exception is the significant duration, t_d, and related intensity measures, with this duration defined as:

(17) $t_d=t\left|0.95AI-t\right|0.05AI$(17)

Figure 11. Normalised maximum rotation, θ_max/α, of a façade with tie-rod varying the intensity measure of the ground motion. Each marker related to a wall geometry, a tie-rod height, a station, a component, a partial or complete sequence of events (1 alone, 1–2, 1–3, 1–4). a) Peak Ground Acceleration (PGA); b) Peak Ground Velocity (PGV). Three-leaf uncut-stone masonry.

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where t| is the time instant of the accelerogram identifying the attainment of 95% and 5% of Arias Intensity, AI, defined as:

(18) $AI=\frac{\pi }{2g}{\int }_0^{t_f}{\ddot{x}}_g^2\mathrm{d}t$(18)

where t_f = record duration.

If t_d is computed for the (partial or complete) sequence its value is sensitive to the time buffer included between two events. Therefore, t_d was computed for each event independently, and then individual values were summed up for the sequence under consideration.

In Table 4, intensity measures were compared in terms of the best coefficient of determination, R², which can be computed using different regression equations to predict maximum rotation, as in Figure 11. It is evident that the best intensity measures are PGA, PGV and ASI (acceleration spectrum intensity), defined as:

(19) $ASI={\int }_{0.1\mathrm{s}}^{0.5\mathrm{s}}{S}_{pa}\mathrm{d}T$(19)

where S_pa = spectral pseudo-acceleration at period of vibration T of a 5% damped linear-elastic oscillator.

Almost coincident results are obtained if the same analyses are repeated assuming cut-stone masonry with good bond. Finally, it is worth mentioning that a coefficient of determination as large as 0.69 is obtained when considering an intensity measure equal to PGA²∙PGV³.

4.5. Code considerations

None of the three above-mentioned parameters (PGA, PGV and ASI) is related to a building up during a time history or a sequence of time histories, but rather to single (PGA, PGV) or integral (ASI) maximum values. Therefore, the wall restrained by a tie-rod seems marginally sensitive to a sequence of events and very sensitive to maximum values, either occurring in a single event or within a sequence of events.

In order to investigate this behaviour, hysteretic energy V_TR,H dissipated by the tie-rod during a (partial or complete) sequence is normalised by the sum of wall potential energy V_W and tie-rod elastic potential energy V_TR,E:

(20) $V_{TR,H}=E_s{A}_t{L}_t\left\{\begin{array}{c}\begin{array}{cc}0& {\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le {\epsilon }_y\end{array}\\ \begin{array}{cc}{\epsilon }_y\left({\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\epsilon }_y\right)& {\epsilon }_y<{\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le {\epsilon }_u\end{array}\\ \begin{array}{cc}{\epsilon }_y\left({\epsilon }_u-{\epsilon }_y\right)& {\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}>{\epsilon }_u\end{array}\end{array}\right.$(20)

(21) $V_W=mgR\left[\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha -{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)-\mathrm{c}\mathrm{o}\mathrm{s}\alpha \right]$(21)

(22) $V_{TR,E}=\frac{1}{2}{E}_s{A}_t{L}_t\left\{\begin{array}{c}\begin{array}{cc}{\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}^2& {\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le {\epsilon }_y\end{array}\\ \begin{array}{cc}{\epsilon }_y^2& {\epsilon }_y<{\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le {\epsilon }_u\end{array}\\ \begin{array}{cc}0& {\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}>{\epsilon }_u\end{array}\end{array}\right.$(22)

where ${\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is now evaluated for the complete time history, E_s = steel Young’s modulus, equal to 210 GPa in all analyses. A plot for a given sequence is shown in Figure 12, which is qualitatively similar to Figure 4. Most hysteretic energy is dissipated in a concentrated fashion when maximum rotation occurs, rather than continuously along the entire time history. This behaviour is almost systematic, as shown in Figure 13, with the energy dissipated in the largest plastic cycle, V_TR,H1, accounting on average for 73% of all hysteretic energy. Additionally, it is possible to observe that in most cases the largest plastic cycle occurs when PGA occurs, and, even more, when PGV occurs, as shown in Figure 14. The most notable exception is given by the 2016 Central Italy Earthquakes, characterised by distinct mainshocks. If this sequence is removed from the regression, the coefficient of determination rises from 0.49 to 0.91, for PGA, and 0.74 to 0.92 for PGV.

Figure 12. Hysteretic energy, V_TR,H, dissipated by the tie-rod during a (partial or complete) sequence normalised by the sum of wall, V_W, and tie-rod, V_TR,E, elastic potential energies during the time history of a façade restrained by a tie-rod varying its normalised height, H_t/H (Figure 1a), and wall geometry (a-d). Elastic, E, or plastic, P, response of the tie emphasised. Three-leaf uncut-stone masonry.

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Figure 13. Hysteretic energy, V_TR,H,1, dissipated by the tie-rod during the most dissipative cycle normalised by the total hysteretic energy, V_TR,H, as a function of: a) Acceleration Spectrum Intensity (ASI); b) Velocity spectrum intensity (VSI). Each marker related to a wall geometry, a tie-rod height, a station, a component, a partial or complete sequence of events (1 alone, 1–2, 1–3, 1–4). Cut-stone masonry with good bond.

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Figure 14. Time of occurrence of the most dissipative cycle, t| V_TR,H,1, and time of occurrence of: a) t| Peak Ground Acceleration, b) t| Peak Ground Velocity. Each marker related to a wall geometry, a tie-rod height, a station, a component, a complete sequence of events (1–4). Cut-stone masonry with good bond.

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In order to further investigate the 2016 Central Italy Earthquakes, the maximum rotation associated with the single event having the largest PGA or the largest PGV is compared to the maximum rotation experienced during the complete sequence (Figure 15). It is possible to observe that the consideration of a single event leads to an underestimation of the actual rotation demand by up to 50%, emphasising again how severe the damage accumulation was in 2016 (Sorrentino et al. Citation2018a).

Figure 15. Normalised maximum rotation, θ_max/α, of a façade with tie-rod due to the complete sequence of events (1–4) or the single event having the largest: a) Peak Ground Acceleration (PGA); b) Peak Ground Velocity (PGV). Each marker related to a wall geometry, a tie-rod height, a station, a component, of the 2016 Central Italy Earthquakes. Three-leaf uncut-stone masonry.

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With just this exception, the results presented suggest that the design performed for maximum expected values is sound also with most Italian sequences and does not impose on practitioners the need to perform nonlinear dynamic analyses, with damage accumulation, to account for seismicity involving sequences of events. Of course, this behaviour is related to the features of the model investigated, with masonry accumulating no damage. The investigation of different configurations (Casolo Citation2017) may lead to substantially different results.

On the contrary, if a sequence of mainshocks is predicted by hazard studies, a possible alternative to performing non-linear time-history analyses is the use of a reduced behaviour factor, according to the approach adopted by Rinaldin and Amadio (Citation2018). In Figure 16, the maximum rotation on the vertical axis is related to ties designed assuming behaviour factor q = 1.5 and a full sequence, accounting for a hazard where multiple mainshocks are predicted, whereas, on the horizontal axis, the maximum rotation is related to ties designed assuming behaviour factor q = 2.0 and a single event, accounting for a hazard where a single mainshock is predicted. It is possible to observe that rotation demands are now much closer than in Figure 15.

Figure 16. Normalised maximum rotation, θ_max/α, of a façade with tie-rod designed according to a behaviour factor q = 1.5 under a complete sequence of events (1–4) and tie-rod designed according to a behaviour factor q = 2.0 under a single event having the largest: a) Peak Ground Acceleration (PGA); b) Peak Ground Velocity (PGV). Each marker related to a wall geometry, a tie-rod height, a station, a component, of the 2016 Central Italy Earthquakes. Three-leaf uncut-stone masonry.

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