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On the role played by the openings on the first frequency of historic masonry towers

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Abstract

The manuscript provides a wide database for historic masonry towers collecting their modal, mechanical and geometrical features. This large amount of data was collected according to a literature review, then summing up as many as 56 different case studies. The collected data was then used to identify the main physical parameters influencing the modal behaviour of slender masonry towers, including the openings on the façades. After a critical discussion of the data reported in the database, the existing empirical or semi-empirical formulations available for the estimation of the first natural frequency of such structural typology are first evaluated. Subsequently the effects of the openings, as revealed through the analysis of the experimental results reported in the database, are discussed by comparison with a simple but effective numerical model. Despite the difficulties to quantify the effect of the openings along the height of the tower on the main frequency due to a cross-contribution of mass and stiffness, a simple but effective formulation is proposed which is able to account for this parameter.

Introduction

The recent seismic events in Italy (Augenti and Parisi 2010; Borri et al. 2017; Fumagalli et al. 2017) have again underlined the high level of seismic vulnerability of the historic masonry towers. The collapse of several towers drew the attention of the scientific community to the necessity of both improving their level of knowledge and developing proper numerical modelling techniques for the conservation of these structures.

The low level of knowledge usually achieved for heritage structures is mainly connected with the difficulties arising when performing exhaustive experimental static tests for the mechanical characterization of the materials of existing structures. Nevertheless, even if an experimental investigation is possible, the heterogeneity of the historic masonries can lead to several problems in the sampling operations, and, in addition, the experimental results are usually able to represent only the properties of a limited portion of the structure. In this respect, dynamic experimental tests have gained in last decades a growing interest; indeed, as well as providing a knowledge of the modal properties of the structure, the results can be used to calibrate numerical models, and long-time data can be employed for Structural Health Monitoring (Clementi et al. 2017, 2018; Ribilotta et al. 2018; Saisi et al. 2018).

Among the modal parameters, the main frequencies of a tower play a relevant role in the assessment of its seismic vulnerability. Indeed, although the structural response of the construction during the seismic loading depends on the evolution of its dynamic characteristics, the knowledge of the initial main frequencies along the two main directions may represent a relevant datum since both capacity and demand are strongly dependent on this dynamic characteristics of the structure (Milani et al. 2012; Casolo et al. 2013; Bartoli et al. 2016). In some codes and provisions, such as the Italian Guidelines for the assessment and mitigation of the seismic risk of the Cultural Heritage (DPCM 2011), simplified seismic risk assessment approaches for masonry towers are proposed on the base of the main natural frequency of the structure. Moreover, a wide review of the scientific literature reveals, in last years, the growing interest in simplified formulations capable to offer effective and expeditious estimations of the main frequency for these slender structures. Several empirical or semi-analytical formulations have been in fact proposed to improve the prediction of those present in the codes. These formulations range from simple empirical relations, which depend on a single geometric parameter, to more sophisticated expressions depending on both geometric and mechanical parameters of the masonry (Shakya et al. 2016; Bartoli et al. 2017). Among all the parameters affecting the main frequency of these slender structures, the total height certainly plays a relevant role.

Indeed, the largest part of the existing formulations proposes to evaluate the main frequency as a function of the tower height, according to a frequency (period)–height relationship approach developed since the eighties for the simplified design of reinforced concrete framed structures (Goel and Chopra 1997). An example is given by the (ASCE/SEI 7-10) or by the Italian Code (NTC Norme Tecniche per le Costruzioni 2008), even though the expression reported within the Italian Code is suggested for ordinary buildings with a height up to 40 m. As a consequence, its application to slender masonry structures can lead large scattering. The recent update of the Italian Code (NTC Norme Tecniche per le Costruzioni 2018) introduces a new approximate formulation according to which the main frequency is evaluated as a function of the lateral elastic displacement of the top level of the building due to a seismic load combination. This new formulation implements an expression already included in the EC8 (Eurocode 8 1998). A comprehensive discussion of the approximate formulations suggested in recent seismic codes to estimate the fundamental periods of buildings was provided by Crowley and Pinho (2010), to which the interested reader is referred. It is however possible to observe that the approximate expressions included in seismic codes do not explicitly account for historic masonry towers but mainly for ordinary buildings. A formulation specifically intended for masonry tower was instead proposed by Faccio et al. (2010) in the DPCM (2011), where the main period is expressed as a linear function of the total height of a tower, but no references about its calibration are provided.

The reliability of several existing formulations was initially checked by Rainieri and Fabbrocino (2012) and Shakya et al. (2016). The authors compared the results collected through extensive experimental investigations with those provided by simplified expressions (the first on the base of output-only modal identification of 30 masonry towers in southern Italy, the second with respect to a database of 58 slender structures, composed by 32 masonry towers, 16 minarets, 7 chimneys and 3 Pagoda temples). More recently Diaferio et al. (2018) have developed novel formulations by means of a regression analysis on the base of a collected database, distinguishing between isolated and confined towers.

The cited formulations express the main frequency as a function of the height of the tower only, but the introduction of additional geometrical and mechanical parameters can improve their predictive capability. In this respect, the Spanish Standard (NCSE 2002) proposes the main frequency estimation as a function of the total height and the minimum length of the outer side of the base cross-section of the tower. This formulation is not specifically intended for masonry towers but the same structure was maintained by Shakya et al. (2016), suggesting new coefficients based on the experimental data of the 58 slender structures included in their database. In addition to these empirical formulations, the theoretical formulation of a cantilever beam with a hollow square section can be taken into account. In this respect, Shakya et al. (2016) introduces an empirical coefficient to improve the prediction capability of the theoretical formulation. In a recent paper of the authors, (Bartoli et al. 2017), a systematic study was performed for the identification of the main structural features affecting the main natural frequency of masonry towers, based on experimental results collected on a wide and homogeneous database. This research revealed that, in case of not isolated towers, the predictions of the existing formulations improves if the effective height of the structure (i.e. the length of the portion of the tower that is free from the restraint offered by adjacent buildings) is used rather than the total height. Additionally, by operating on the theoretical expression for the main frequency of cantilever beams, a novel semi-empirical formulation was introduced with a clear physical soundness. It leads to considerable improvements if compared with the prediction capability of other formulations.

Based on the previous results, this paper aims at investigating the role of the openings along the façade of the towers in the estimation of the main frequency. To the authors’ knowledge, this aspect has not been deeply studied. To illustrate these aspects, the paper is organized as follows. In the first part, the database of geometric information and experimental results is reorganized and updated with additional cases. The configuration of the openings along the façade of the towers is reported and its effect on the main frequency are discussed. Subsequently, a review of the existing empirical formulations is performed, in which the available expressions are analysed and discussed. The effect of different configurations of the openings on the main frequency estimation is first evaluated through the analysis of the results of an extensive number of numerical simulations. Subsequently, a new simple but effective formulation for isolated tower is proposed which is able to account for the effect of the openings along the height of the tower. Despite the difficulties to quantify the effect of the openings on the main frequency of the tower due to a cross-contribution of mass and stiffness, the proposed formulation (easy-to-use) is able to reproduce the experimental results with great confidence.

Masonry towers database

The database collecting experimental data on masonry towers reported in Bartoli et al. (2017) was here expanded by including 8 additional case studies. The 56 masonry towers there contained are all located in the European territory (most of them in Italy) and, despite some common aspects, the geometric and mechanic characteristics show a significant variability mainly due to: (i) different intended use (e.g. bell-tower, civic tower, tower house, etc.), (ii) different constructive techniques and (iii) different period of construction. From a geometric point of view, the main differences among the towers involve two aspects: (i) the openings (i.e. windows) along the height and (ii) the lateral restraint conditions (i.e. isolated and/or confined towers).

In order to evaluate these effects on the modal frequencies of the towers, the original database has been reorganized by differentiating isolated towers (IT) from confined towers (CT), and including the opening ratio ϕ of the tower façades for each case study. This latter parameter, evaluated as the ratio between the area of the openings on the façade and the whole area of the façade, is expressed as a percentage.

With this classification, the database is thus made up of 11 isolated towers (Table 1) and 45 confined towers (Table 2), and includes both towers with a limited opening ratio, e.g. the Capocci tower in Rome (Fig. 1a) or the Cugnanesi tower in San Gimignano (Fig. 1b), and towers with large opening ratio, e.g. the Giotto bell-tower in Florence (Fig. 1c) or the bell-tower of Trani Cathedral (Fig. 1d).

Table 1 Mechanical and geometrical parameters, and experimental frequencies of Isolated Towers (IT)
Table 2 Mechanical and geometrical parameters, and experimental frequencies of confined towers (CT)
Fig. 1
figure1

a Capocci tower in Rome, b Cugnanesi tower in San Gimignano, c Giotto bell-tower in Florence and d Trani Cathedral bell-tower

The database, summarized in Tables 1 and 2 for the isolated (IT) and confined (CT) case studies respectively, reports the mechanical and geometric characteristics of the considered masonry towers together with the available experimental frequencies. The experimental frequencies were measured by different authors using ambient vibration tests and employing OMA (Operational Modal Analysis) or EMA (Experimental Modal Analysis) techniques depending on the specific case study. The analysis of the experimental results shows that the first two modes of the towers are always bending modes along the two main directions while, for the most part of the case studies, the third mode is a torsional one.

The data included in the database are: the elastic modulus (E); the estimated velocity of propagation of the elastic compression P-waves (vp); the total height of the tower (H) and the effective one (Heff); the outer side lengths of the base cross-section (a(x) and b(y)); the façade opening ratio (ϕ), and the first two frequencies in the main directions of the tower (f1,x and f1,y, corresponding to the direction of the sides a(x) and b(y)). As far as the value of vp is concerned, the following relationship has been considered:

$$ v_{p} = \sqrt {\frac{E}{\rho } \cdot \frac{{\left( {1 - \nu } \right)}}{{\left( {1 + \nu } \right) \cdot \left( {1 - 2 \nu } \right)}}} \cong \sqrt {\frac{E}{\rho }} $$
(1)

where ρ denotes the mass density, and \( \nu \) the Poisson coefficient. The second term under the first square root in Eq. (1) can be neglected; indeed, it assumes values around 1.0 when \( \nu \) ranges from 0 to 0.25 (typical for masonry). The benefit of the introduction of the P-waves propagation velocity is twofold: firstly, it allows to define a soundly physical relationship between E and ρ and secondly, this parameter can be obtained through sonic tests, which represents a non-destructive investigation technique widely adopted for masonry structures (even if the notable heterogeneity of the historic masonries makes still complicated using these results to gain the mechanical properties of the masonry).

The elastic modulus and the velocity of propagation of the elastic compression P-waves included in the database were directly derived by the corresponding papers, only in a few cases (marked with a star), these properties were assumed according to available values on similar masonries (Bartoli et al. 2013; Vignoli et al. 2016; Boschi et al. 2019). Finally, it is worth noting that for the CT cases, the opening ratio was calculated on the façade portion of the tower above the restraint level offered by the adjacent buildings.

The database: from a geometrical point of view

The analysis of the database from a geometrical point of view shows that the total height of the towers ranges from 18.50 m [#12, bell-tower of San Giorgio church in Trignano, (Bongiovanni et al. 2000)] to 94.0 m [#37, tower of Arnolfo in Florence, (Pieraccini et al. 2009)]. The width of the walls at the lower level varies from 0.5 m [#12, bell-tower of San Giorgio church in Trignano, (Bongiovanni et al. 2000)] to 3.8 m [#6, bell-tower of Aversa, (Ferraioli et al. 2011)]. The cross-section is almost square, except for a few cases: Diavolo tower in San Gimignano (#53), Aquila tower [#14, (Ceriotti et al. 2009)], and bell-tower of SS. Annunziata church [#13, (Bonato et al. 2000)], which have a unambiguously rectangular plan. The length of the outer side of the base cross-section ranges from 3 m [#12, bell-tower of San Giorgio church in Trignano, (Bongiovanni et al. 2000)] to 14 m [#6, bell-tower of Aversa, (Ferraioli et al. 2011)]. Evaluating the slenderness as the ratio between the total height H and the minimum length of the outer side of the base cross-section, its value ranges from 3.1 for the Propositura tower in San Gimignano [#49, (Bartoli et al. 2017)] to 10.3 for the Mangia tower in Siena [#28, (Pieraccini et al. 2014)]. These two extreme cases well highlight the great variety of typology of masonry towers included in the database, ranging from squat medieval towers, such as the Propositura tower, to very slender bell-towers, such as the Mangia tower. These two towers are also characterized by the maximum and minimum value of the experimental main frequency equal to 4.20 Hz and 0.35 Hz, respectively.

The database: from an openings point of view

It is straightforward to observe that the fundamental frequencies are highly influenced by the height of the towers (Figs. 2, 3) and their slenderness, as already consolidated in the scientific literature.

Fig. 2
figure2

IT cases: first natural frequency as a function of the total height (H)

Fig. 3
figure3

CT cases: first frequency as a function of a the total height (H) and b the effective height (Heff)

However, to have a clear correlation between these geometric characteristics and the main natural frequency, additional parameters should be taken into account. Indeed, if on one hand Figs. 2, 3 reveal an almost regular behaviour of the first experimental natural frequency (\( f_{{1,{ \exp }}} \)) as a function of the height of the towers, on the other hand some comparisons between similar case studies included in the database highlight interesting differences. Just to comment some examples, the bell-tower of S. Vittore church in Arcisate (Varese, Italy) [#16, (Gentile and Saisi 2013)] and the bell-tower of Bonrepos I Mirambell church (Valencia, Spain) [#17, (Ivorra and Cervera 2001)] have significant different values of the first experimental frequency (equal to 1.21 Hz and 0.73 Hz, respectively) but they show very similar characteristics in terms of geometrical configuration and mechanical properties of the masonry. What significantly differs, however, is the opening ratio, which varies from 9.8 to 22.3%. An analogous consideration can be made comparing the bell-tower of S. Justa and Rufina church (Orihuela, Spain) [#15, (Ivorra et al. 2010)] and the bell-tower of Serra S. Quirico (Ancona, Italy) [#24, (Cosenza and Iervolino 2007)]. In these two cases, again, the difference of the experimental frequency (from 2.15 to 1.95 Hz) can be ascribed to opening ratio, which is equal to 15.4% in the first case and 4.3% in the second one.

Therefore, it appears significant to investigate the role that the openings along the height of the towers play in the estimation of the natural frequencies of this structural typology. To point out the variability of the façade openings in the collected database, Figs. 4 and 5 report the opening ratio for all the cases included, distinguish by IT and CT cases, respectively.

Fig. 4
figure4

IT cases: opening ratio

Fig. 5
figure5

CT cases: opening ratio

The analysis of the façade openings, together with their position along the façade, allows to identify some recurring configurations. The first one, which is typical of the medieval tower houses, is characterized by an opening ratio lower than 5%: limited opening are distributed along the façades with small and scattered windows. This configuration is well represented by most of the towers of San Gimignano (Siena, Italy): e.g. the Salvucci towers (#46, #47), the Propositura tower (#49), the Becci tower (#54) and the Cugnanesi tower (#55) (Bartoli et al. 2017). The second configuration includes the bell-towers, that have an opening ratio up to 15%. The openings are typically located at the upper level, in correspondence of the belfry, while the remaining part of the structure is usually without openings. Some examples are: the Collegiata bell-tower of San Gimignano (#48), the S. Nicolas bell-tower (Valencia, Spain) [#45, (Ivorra et al. 2009)], the bell-tower of the Monza Cathedral [#22, (Gentile and Saisi 2007)], the bell-tower of S. Vittore church in Arcisate (Varese, Italy) [#16, (Gentile and Saisi 2013)] and the bell-tower of Santa Maria del Corrobiolo (Monza, Italy) [#34, (Saisi et al. 2016)]. The third configuration is represented by bell-towers with large and regular openings, distributed along the façade, usually in correspondence of different levels of belfries. Typical examples of this configuration are: the Giotto bell-tower in Florence [#11, (Pieraccini et al. 2013)], the Mirandola bell-tower (Modena, Italy) [#43, (Zanotti Fragonara et al. 2017)], the San Federico bell-tower in Lucca (Italy) [#41, (Azzara et al. 2018)] and the Ghirlandina bell-tower (Modena, Italy) [#27, (Lancellotta and Sabia 2015)].

Finally, the database collects four particular cases that show large openings at the base of the tower: the S. Domenico bell-tower [#10, (Gentile and Saisi 2014)] (Mantua, Italy), the medieval Torre Aquila (Trento, Italy) [#14, (Ceriotti et al. 2009)], the bell-tower of Trani Cathedral [#20, (Diaferio et al. 2014)] and the Grossa tower in San Gimignano (Italy) [#50, (Bartoli et al. 2017)].

The database: some simple comparisons

The structural configuration of the masonry towers allows to assume their modal behaviour very close to the one of an elastic cantilever beam (with a hollow square section). Nevertheless, the masonry towers differ from the theoretical cantilever beam for at least two aspects: (i) the lateral restraint conditions offered by the neighbouring buildings (if present), and (ii) the irregularity of the cross-section along the height of the tower (e.g., due to the openings). The first aspect can be taken into account by identifying the effective position of the fixed restraint. Thus, the first natural frequency of masonry towers (\( f_{1,0} \)) may be obtained through the following expression:

$$ f_{1,0} = \frac{{1.875^{2} }}{{2\pi \cdot H_{\text{eff}}^{2} }}\sqrt {\frac{{{\text{E}} \cdot {\text{J}}}}{{\uprho \cdot {\text{A}}}}} $$
(2)

where Heff denotes the effective height (i.e. the total height minus the height of the confining buildings), J denotes the moment of inertia and A represents the area of the cross section. The ratio between the experimental frequency (\( f_{{1,{\text{e}}}} \)) and the frequency obtained by Eq. (2) represents the discrepancy between the actual behaviour of the tower and the cantilever beam behaviour. This ratio includes the irregularity in the cross-section and should be related to the opening ratio. This ratio is greater than one when Eq. (2) underestimates the first frequency and, vice versa, is less than one when Eq. (2) overestimates the first frequency. With respect to the cases included in the database, Fig. 6a, b show the value of this ratio for IT and CT cases, respectively.

Fig. 6
figure6

Ratio between the experimental (\( f_{{1,{\text{e}}}} \)) and the estimated main frequency through Eq. (2) (\( f_{1,0} \)) for a IT and b CT cases

It is possible to observe that: (i) for CT cases Eq. (2) always overestimates the first experimental frequency, except for one case: the bell-tower of the Trani cathedral (Fig. 1c). Due to the specific geometric configuration of the tower, characterized by large openings at the base and along the height and a strong interaction with the neighbouring church, this result can be considered as an outlier. (ii) The ratio between the actual behaviour of the tower and the theoretical cantilever beam model depends on the opening ratio but other parameters should be taken into account, and (iii) for CT cases greater errors occur compared to IT cases. This latter aspect should be related to the uncertainties in the definition of the effective height of the tower, which depends to the restraint conditions offered by the lateral buildings.

No clear or straightforward connection between the main frequency and the openings is provided by the analysis of the case studies. Indeed, it is evident that on an equal percentage of openings, a key role is also played by their position along the height of the tower. As a matter of fact, the openings located at the base of the tower affect mainly the stiffness of the structure, while those located at the top influence the modal mass. Thus, a numerical model seems useful to understand the role of the openings in the main frequency estimation in order to gather some general conclusions to be also used in the real case studies.

Existing formulations for the first frequency estimation

As reported in the introduction, several simplified formulations have been proposed to estimate the first natural frequency of masonry towers. For the sake of clarity, these formulations are herein resumed in Tables 3, 4 and 5, collecting together those that are functions of the same parameters (geometrical and/or mechanical). To compare the overall prediction capabilities of the formulations, based on the experimental results collected in the database, the following global error is introduced:

$$ {\text{e}} = \frac{{\mathop \sum \nolimits_{{{\text{i}} = 1}}^{\text{N}} \frac{{\left| {f_{\text{i}} - f_{{{\text{i}},{\text{e}}}} } \right|}}{{f_{{{\text{i}},{\text{e}}}} }}}}{\text{N}} $$
(3)

where N denotes the number of samples, fi is the estimated frequency of the tower while \( f_{{{\text{i}},{\text{e}}}} \) denotes the experimental frequency. The errors obtained for each simplified formulation (considering both the isolated and confined tower configuration) are summarized in Tables 3, 4 and 5.

Table 3 Empirical formulations as function of a single geometrical parameter (H in m)
Table 4 Empirical formulations as function of two geometrical parameters
Table 5 Semi-empirical formulations as function of geometrical and mechanical parameters

The Eqs. reported in Tables 3 and 4 do not show significant differences in the predictive capability of the empirical formulations. Instead, it is interesting to note that the errors of Eq. T5 reported in Table 5 are significantly different for the isolated and confined towers. More in detail, the formulation of a cantilever elastic beam seems to better represent the dynamic behaviour of the isolated towers compared to the confined cases. These results can be connected to the effect of the neighbouring buildings which, in case of confined towers, modify the free-standing part of the towers. Although the total height (H) of the tower is one of the main parameter affecting its fundamental frequency, Bartoli et al. (2017) first showed the pivotal role played by the height of the tower out of the restraints offered by the adjacent buildings introducing the notion of effective height (Heff). The introduction of the concept of effective height allows to improve the predictive capability of empirical and semi-empirical formulations. In addition, they proposed two formulations, reported in Eqs. (4) and (5) (Table 6). The first was derived by manipulating the theoretical expression of a simple cantilever beam by introducing small approximations and a correction coefficient in order to consider some differences between the hypothesis used in the theoretical expression and the actual tower configuration. The openings along the height of the tower are not directly taken into account and the effective degree of restraint offered by the neighbouring buildings is lower than the clamped one considered in the theoretical formulation. Therefore, in Eq. (4) the effective height Heff is considered instead of the total one. The ratio between the thickness and the side of the cross-section of the tower was also taken into account by introducing the dimensionless parameter n. The mechanical parameters of the masonry are included in the formulation through the P-waves velocity vp according to the following expression:

$$ f_{1} = \frac{0.2 a}{{H_{\text{eff}}^{2} }} \cdot \left( {1 - n} \right) \cdot v_{p} $$
(4)

With the aim to propose a simpler expression, Bartoli et al. (2017) consider the mean values of n and vp by obtaining the following equation only function of the effective height and of the side length of the tower:

Table 6 Errors in the prediction of the first natural frequency of the isolated and confined towers included in the database
$$ f_{1} = 150 \left[ {m/s} \right] \cdot \frac{a}{{H_{\text{eff}}^{2} }} $$
(5)

Similar considerations have been recently developed by Diaferio et al. (2018) through the following two formulations for confined towers:

$$ f_{1} = 12.96 \cdot H_{\text{eff}}^{ - 0.686} $$
(6)
$$ f_{1} = 14.61 \cdot L_{\text{min} }^{ - 0.254} \cdot H_{\text{eff}}^{ - 0.341} \cdot H^{ - 0.216} $$
(7)

where all the dimensions are expressed in meters and Lmin is the minimum side of the base cross section, Heff the effective height and H the total height of the tower. The errors in the prediction of the first natural frequency are compared for the isolated and confined towers and summarized in Table 6. Equations (4) and (5) are valid both for isolated and confined towers, but it is worth noting that for isolated towers the effective height corresponds to the total height.

Effect of the openings on the first frequency

To the authors’ knowledge, despite the openings along the height of the towers affect their fundamental frequencies—as discussed in the previous sections—by reducing both the effective stiffness and mass, this parameter is not explicitly taken into account in the existing formulations. Only a recent research starts to deal with this issue (Najafgholipour et al. 2019).

In order to evaluate this effect, some considerations have been developed starting from the data collected in the database (Tables 1 and 2). The results reported in the previous sections have not highlighted a clear, or straightforward, correlation between the main frequency and the percentage of openings. The openings affect the mass and the stiffness distribution of the tower, but other parameters have the same effects: e.g. a variation of the thickness of the cross-section or different mechanical properties of the masonry along the height of the tower. The mechanical and geometrical characteristics are usually considered homogeneous thus neglecting their variations along the height of the towers, which clearly affects the modal properties of the tower. Due to these considerations, the difficulties in the identification of the effect of the openings in the main frequency estimation for the real case studies could be justified. Moreover, as well as the opening ratio, the modal frequencies of the tower are affected by the position of the openings themselves. It is straightforward to suppose that the openings located at the upper level of the tower reduce the modal mass of the tower rather than the stiffness. Conversely, openings located at the base of the tower could induce a remarkable effect on the tower stiffness with negligible variations in the modal mass. The combined effect of these openings’ configuration on the main frequency estimation is difficult to be identified for real case studies, where other non-homogeneous parameters may have the same effects.

Parametric numerical analyses

In order to draw some general conclusions on the role played by the openings on the first frequency, a simple 3D-numerical finite element model has been employed to reproduce the modal behaviour of masonry towers by using the commercial code ANSYS. The numerical model, which reproduces a masonry cantilever beam with square cross-section and constant walls thickness, was built employing 2D four-node finite elements (SHELL181). The openings were symmetrically placed in each façade of the tower. A schematic representation of the reference geometric model and of one of the numerical models is shown in Fig. 7.

Fig. 7
figure7

Scheme of the opening position and one of the numerical models

Through the numerical model, built parametrically, an extensive study was performed by considering a significant variety of openings configurations on the tower façade (both in terms of dimension and location) and tower dimensions. It is in fact noteworthy that in order to analyse the effect of the openings on the tower façade the geometric parameters of the tower which mainly affect its modal behaviour have been varied: the slenderness (by varying the total height H of the tower) and the thickness s of the walls. The reference values assumed for the parametric study have been selected based on the geometric characteristics of the towers included in the database which are summarized in Table 7. The Table shows the range of variability of the geometrical and mechanical properties for the real towers and compares them with the values assumed in this study. For each tower model (i.e. for each of the 9 combinations of H and s), 8 values of opening height (hf) and 20 values of the position of the openings (zi) were considered, summing up to 1440 modal analyses.

Table 7 Geometrical and mechanical characteristics of the reference towers

The results of the parametric analyses are summarized in Fig. 8. The figure shows the results obtained with the numerical models expressed in terms of first frequency ratio (FFR). This number was evaluated as the ratio between the current numerical model result (the first frequency \( f_{1} \) of the tower with the opening along the height) and the theoretical formulation of a cantilever beam with the same hollow square section but without openings along the height (\( f_{1,0} \)). When this ratio is equal to one, there are no differences between the first frequency of the model with openings and the theoretical model without openings (in other terms the cross-contribution of mass and stiffness does not change the value of the first frequency). The figure, with the aim to include and to compare all the parametric results, shows the FFR as a function of the height of the opening and its position, both normalized with respect to the total height H of the towers. The results of the numerical model allow to highlight some interesting aspects, which are useful for the developments of this research:

Fig. 8
figure8

Results of the parametric analyses expressed as FFR

  • the openings located on the upper levels of the tower increase the frequency (the first frequency ratio is greater than one). These openings reduce the mass of the tower but have a light effect of the stiffness of the structure. This result, which may seem obvious, is only found if the experimental frequency is compared with the corresponding value provided by the theoretical formulation of the cantilever beam, as shown in terms of FFR in Table 8. In fact, if the experimental value is compared with the value obtained with the empirical or semi-empirical expressions this increasing is not directly observable. This is probably mainly due to the fact that these formulations were calibrated on the basis of experimental data which already include the contribution of the openings;

    Table 8 Performance of Eq. (13) on the isolated towers (dotted red line represents the neutral height)
  • the variability of the wall thickness influences the FFR when the façade opening ratio (ϕ) increases. This is clearly visible in Fig. 8 when the ratio hf/H is greater than about 0.35-0.40. These values, anyway, are outside the range of variability of the opening ratio of the towers included in the database and the FFR can be considered as not being affected by the wall thickness. Therefore, the whole set of results reported in Fig. 8 can be represented by the mean-surface shown in Fig. 9;

    Fig. 9
    figure9

    Mean-surface of the FFR

  • when the FFR is equal to one, there are no differences between the first frequency of the numerical model with openings and the one of the theoretical model without openings; this suggests the existence of a neutral height (here denoted with zn) which identifies an opening location which not affect the first frequency value. This insight can be observed in Fig. 9 reporting the contour map of the FFR. It is noteworthy to observe that the location of the neutral height, represented by the contour line “1”, is not affected by the height H of the towers.

A new simplified formulation for isolated towers

The analysis of the results of the parametric analyses offers useful hint for the investigation of a new formulation able to include the effect of the openings. To this aim, the results illustrated in Fig. 8 can be expressed by the mean-surface of Fig. 9. This surface can be approximated by the following equation:

$$ \frac{{f_{1} }}{{f_{1,0} }} = 1 + \left( {2 \cdot \frac{{z_{i} }}{H} - 1.3} \right)\frac{{h_{f} }}{H} $$
(8)

Equation (8), which represents the equation of the FFR as a function of opening dimension and location; setting FFR equal to one, allows to obtain the following:

$$ \left( {2 \cdot \frac{{z_{i} }}{H} - 1.3} \right)\frac{{h_{f} }}{H} = 0 $$
(9)

By excluding the trivial solution, Eq. (9) allows to identify the expression of the neutral height (zn):

$$ z_{n} \cong \frac{2}{3}H $$
(10)

Introducing the first moment of area (S), and considering the neutral height (zn), it is easy to manage Eq. (8) to finally obtain following simple expression of FFR:

$$ FFR = \frac{{f_{1} }}{{f_{1,0} }} = 1 - \frac{{S_{{f,z_{n} }} }}{{S_{{0,z_{n} }} }} = 1 - \varPhi $$
(11)

where \( S_{{f,z_{n} }} \) denotes the first moment of area of the openings on the tower façade evaluated with respect to the neutral height zn, while \( S_{{0,z_{n} }} \) represents the first moment of area of the entire façade (still evaluated with respect to the neutral height). Equation (11) shows that the behaviour of the FFR can be simple interpreted by analysing the ratio between these two moments and provides a physical interpretation of the cross-contribution of mass and stiffness in the value of the first frequency. Considering a tower with m-openings distributed on the façade it is possible to generalize Eq. (11) and write the FFR as follows (where the notation employed in Fig. 7 has been adopted):

$$ FFR = \frac{{f_{1} }}{{f_{1,0} }} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{m} h_{f,i} \cdot a_{f,i} \cdot \left( {z_{i} - z_{n} } \right)}}{{a \cdot H \cdot \left( {\frac{H}{2} - z_{n} } \right)}} = 1 - \varPhi $$
(12)

As a result, Eq. (12) provides a correction coefficient (\( 1 - \varPhi \)) to be applied to the theoretical formulation of a cantilever beam, in order to estimate the first frequency of an isolated tower.

$$ f_{1} = \left( {1 - \varPhi } \right)f_{1,0} = \left( {1 - \varPhi } \right)\frac{{1.875^{2} }}{{2\pi \cdot H^{2} }}\sqrt {\frac{E \cdot J}{\rho \cdot A}} $$
(13)

In order to evaluate the predictive capability of Eq. (13), the expression has been tested on the IT database (Table 1). In this way Eq. (13) has been applied when: i) more than one opening is along the height of the tower and ii) the towers exhibit significant geometric scattering (Table 8). Table 8 reports the FFR and compares this value with the one provided by Eq. (13). An additional comparison is illustrated in Fig. 10 where the frequency estimation of the proposed formulation is benchmarked with those reported in Table 3, 4, 5 and 6. It is possible to observe that the capacity of prediction of Eq. (13) is generally much better than the others, providing in addition a significant improvement even respect to the previous formulation proposed by the authors and reported in Eq. (4). Equation (13) has been developed for isolated towers and its predictive capability has been tested on the IT database. The achieved agreement between experimental data and formulation results suggest, as future challenges, to look for a similar expression for the case of confined masonry towers.

Fig. 10
figure10

Box plot reporting the scattering of the errors in the existing formulations for the first frequency estimation, based on the IT database. The dotted line refers to the results obtained by using Eq. (13) while the continuous one reports the results obtained by using Eq. (4)

Conclusive remarks

This paper discussed the role played by the openings in the main frequency estimation of historic masonry towers by using the experimental results included in a wide database including 11 Isolated Towers and 45 Confined Towers and by performing an extensive number of parametric analyses employing numerical models. The analysis of the collected data has highlighted that some improvements of the existing empirical or semi-empirical formulations already available for the estimation of the first natural frequency could be achieved when: (i) the effect of the façade openings is considered and (ii) different formulations are considered for isolated and confined cases. The analysis of the results of an extensive number of numerical simulations, where the effect of different configurations of the openings on the main frequency was investigated, allowed to propose a new simple but effective formulation for isolated tower. The proposed formulation is able to reproduce the available experimental results with great confidence (if compared with other expressions) and is able to explain the cross-contribution of mass and stiffness introduced by the opening through the definition of a physical parameter here called neutral height.

Change history

  • 13 June 2019

    The mistake to be corrected is the one named “1” in the attached pdf (i.e. to add the subscript ”,e” to the symbol “f1” at the beginning of the second row of Table 8).

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Bartoli, G., Betti, M., Marra, A.M. et al. On the role played by the openings on the first frequency of historic masonry towers. Bull Earthquake Eng 18, 427–451 (2020). https://doi.org/10.1007/s10518-019-00662-9

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Keywords

  • Masonry towers
  • Database
  • Main frequency
  • Semi-empirical formulations
  • Façade openings
  • Neutral height