Infill Modelling Influence on Dynamic Identification and Model Updating of Reinforced Concrete Framed Buildings

Bovo, Marco; Tondi, Michele; Savoia, Marco

doi:https://doi.org/10.1155/2020/9384080

Advances in Civil Engineering

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Abstract Introduction Results and Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9384080 | https://doi.org/10.1155/2020/9384080

Infill Modelling Influence on Dynamic Identification and Model Updating of Reinforced Concrete Framed Buildings

Marco Bovo,¹Michele Tondi,²and Marco Savoia²

Academic Editor: Flavio Stochino

Received03 Dec 2019

Revised27 Apr 2020

Accepted29 May 2020

Published19 Jun 2020

Abstract

In order to correctly capture the dynamic behavior of infilled framed buildings, the importance to take into account in seismic design the infill panels’ contribution is nowadays well recognized since they could modify in a significant way the global and local response of the whole building. Despite about sixty years of continuous research in the field, the modelling of the frame-infill interaction still represents a serious issue for the daily practical design since there is no reference model proven to be suitable to cover a wide record of possible cases. Moreover, few works are available in the literature, comparing the results of different modelling proposals with outcomes of dynamic tests on a full-scale building. To this regard, starting from the results of induced vibration dynamic tests performed on a 7-story building with reinforced concrete frames with masonry infill, in the present paper, the effects of the infill presence have been evaluated by comparing experimental outcomes, achieved using a MDOF Circle-Fit identification procedure, with the results obtained by means of numerical analyses performed on finite element models. Using a model updating procedure, the optimal width to assign to the masonry equivalent struts modelling the infill panels was defined. Furthermore, several literature proposals for the definition of the equivalent strut width have been analysed. Thirteen different proposals have been selected and implemented in thirteen different finite element models. The reliability of each proposal has been investigated and quantified by comparing the dynamic properties of the models with the building dynamic response obtained by the experimental tests. The main outcomes of the analyses highlight that different proposals provide a great variability for the strut width. This brings to a large variability of the mechanical properties of the equivalent struts, and as a consequence, the modelling choice also influences the dynamic behaviour of the numerical models. Currently, this represents a serious issue for the daily designers’ activity. The outcomes provided in the paper, although established for a specific case study, can be extended to a wide range of buildings and should drive the future research studies in order to provide more robust criteria for the modelling of this worldwide building class.

1. Introduction

The study of frame-infill panel interaction started since the early sixties, and it still represents an open issue. Only with a proper interpretation of the interaction between infill panels and surrounding frames, it is possible to achieve the correct comprehension of the behavior of a building under horizontal dynamic loading [1]. As far as reinforced concrete (RC) frames infilled by masonry panels are concerned, the interaction between infill and frames plays an important role for the design of new buildings and for the seismic assessment of the existing ones. Actually, several experimental campaigns showed that frame-infill interaction can produce an important modification to the whole system stiffness [2–7], and also the force distribution in the frame elements (i.e., beams and columns) can vary significantly if compared to the case of the bare frame [8].

In the last sixty years, a lot of researchers worked on the definition of modelling criteria in order to take into account the infill contribution. Following the consolidated literature [8, 9], the different numerical models could be classified into micro-, meso-, and macromodels. Macromodels are the most used models in applications requiring the modelling of an entire building [10]. The macromodels proposed by researchers in order to reproduce the frame-infill interaction are, almost all, based on the concept of equivalent struts [8–10], i.e., two diagonal finite elements having only axial stiffness and simulating the presence of the infill panels.

Polyakov [11], among the first, modelled the infilled masonry panels as a diagonal strut with axial stiffness only. Then, Stafford Smith [12] refined the models by introducing a dimensionless parameter in order to take into account the relative stiffness between the panel and the surrounding frame (see also [13–15]). An alternative expression in order to consider the frame-infill panel relative stiffness was introduced by Bazan and Meli [16]. Hendry [17] proposed to evaluate and consider, in the definition of the equivalent strut, both the infill column and infill beam contact length. Paulay and Priestley [18] proposed a constant value set equal to 0.25, for the strut width and diagonal panel length (d) ratio. Durrani and Luo [19] refined furtherly the proposal with the introduction of the two parameters γ and m. Recently, the influence of the vertical load on the panel-frame contact length was experimentally tested and numerically analysed by Cavaleri et al. [1], Amato et al. [20], and Campione et al. [21], suggesting expressions for the ratio that explicitly consider the vertical load value.

In the present paper, starting from the experimental results collected during tests on a multistory building selected as a case study, a numerical finite element (FE) model was calibrated with the aim of identifying the contribution of the infill panels to the dynamic behavior of a real building. The selected structure, located in Reggiolo (Reggio Emilia, North-East of Italy), is depicted in Figure 1. The building has been designed according to the Italian building code D.M. 27/07/1985 [22] and has been built in 1990 with the bearing system constituted by RC frames infilled by stiff panels of hollow clay blocks. After some inspections and a detailed geometrical survey of the building, preliminary characterization tests (i.e., compression tests on concrete cored samples, tensile tests on steel reinforcement bars, and double flat-jack tests on masonry panels) were performed in order to obtain the main mechanical properties of the materials. Then, by means of dynamic excitation tests with a vibration generator, the natural frequencies and the mode shape components have been obtained.

In the second part of the paper, a model updating procedure has been adopted using a single equivalent diagonal strut replacing the infill panels, in order to capture the global behavior of the structure. In this way, the numerical calibration phase requires the updating of only one parameter (i.e., the width of the equivalent strut). By means of the FE models, a parametric analysis has been carried out, varying the width of the equivalent strut from zero (i.e., absence of infill) to the value 0.40 d, where d represents the diagonal panel length. By comparing the obtained frequencies and mode shape components, respectively, from the numerical model and experimental tests, it was possible to define a reliable value. Then, the effects of considering or excluding the building basement (i.e., the underground story) was studied.

Lastly, in order to evaluate the effectiveness of different literature proposals for the calibration of the equivalent strut dimensions, a comparison between the dynamic response of the building adopting these proposals and the optimal solution obtained through the parametric analysis was set. The main outcomes of the analyses highlight that different existing proposals provide a range of values affected by great variability. All of this brings to a large variability of the mechanical properties that equivalent struts will assume by adopting a proposal rather than another one, but as a consequence, the choice will also influence the dynamic behaviour of the FE model implementing those struts. Currently, this represents a serious issue for the daily designers’ activity. The outcomes provided in the paper, although established for a specific case study, can be extended to a wide range of buildings and should drive the future research studies to provide more robust criteria for the modelling of this worldwide building class.

2. Description of the Case Study

The case study is a 6-story building aboveground with a further basement story. The building has commercial units at the ground floor and residential dwellings at the upper floors. Garages are located in the basement. The building has a floor surface of about 440 m² and a total height of about 20 m above the ground level. The building has RC bearing frames and stairwell and lift at the center of the plan, in a vertical tube structure with a 20 cm thick RC walls. The columns have 30 × 45 cm cross section at the underground and ground floor and 25 × 30 cm at the other floors. The beams have a rectangular section at all stories except the second one having perimeter L-shaped beams. The stories have a stiff RC slab. The foundation elements are reverse T-shaped RC beams. The infill panels are constituted by brickwork with hollow clay blocks and concrete mortar. The perimeter walls are 30 cm thick, while the partitions are 25 cm thick. From the building inspections, it was emerged that there are no connectors between infill walls and RC frames.

The vertical loads, deduced by the original design documentation, were modified based on the outcomes of the detailed survey. The uniformly distributed vertical loads considered in numerical analysis are 4.34 kN/m², 4.48 kN/m², and 3.85 kN/m², respectively, for the first story, stories from two to six, and the roof story. Only dead loads have been considered because when dynamic excitations have been induced in the structure, the building was not used and no live loads were present. In order to consider in a properly way the masses and stiffness of the infill walls in the FE models, they were modelled in their position, considering a specific weight for the masonry equal to 12 kN/m³.

3. Experimental Campaign

3.1. Mechanical Characterization of the Material Properties

With the aim of characterizing the material properties of RC columns and walls, preliminary experimental investigations were conducted on vertical elements. 21 RC drilled core samples were collected in situ. The outcomes emerged from compressive tests provide an R_ck (i.e., characteristic cubic strength) value of 40 MPa for the underground and ground story and 30 MPa for structural elements of other stories. Moreover, by means of four double flat-jack tests on masonry panels, an average Young modulus E = 2995 MPa and an average compressive strength f_m = 3.85 MPa have been obtained for the infill panels.

3.2. Dynamic Excitation Tests

With the aim of identifying the main dynamic characteristics of the structure, experimental dynamic tests with induced vibrations were performed. The vibration generator was anchored to the roof floor (story #7) close to a symmetry axis (in the direction NW-SE in the global reference system as depicted in Figure 2). The vibration generator is a machine that, firmly anchored to the structure, allows the application of forces that vary over time with the sinusoidal law. It basically consists of two counter-rotating discs around two parallel rotation axes; two masses are constrained to each disk whose relative position is identified by the angle α formed by the joints joining their center of gravity with the axis of rotation. The maximum intensity of the applied force, depending on the mechanical parameters indicated by the manufacturer of the vibration inducer (ISMES, Bergamo, Italy), was 22 kN. Two test phases, associated with two different directions of the vibration generator (Position #1 and Position #2), have been considered. With the vibration generator in Position #1, horizontal forces in the direction NW-SE were applied. With the vibration generator in Position #2 (i.e., 90° rotated with respect to the former), horizontal forces in the direction NE-SW were induced. For convenience reasons, a local reference system with axes x and y in directions NE-SW and NW-SE, respectively, was defined (see Figure 2). In the various positions where the vibration generator is placed, the frequency range deemed of interest for assessing the dynamic behavior of the structure was explored with different values of the force intensity, corresponding to different values of the relative angle between the masses α.

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The instrumentation used to record the acceleration during the tests was constituted by 12 piezoelectric accelerometers PCB/393B12 (labelled in the following as A1–A12) with a sensitivity equal to 10 volt/g. The position of the accelerometers is depicted in Figure 2 in both plan and elevation views. Furthermore, Figure 3 depicts some details relative to the positioning of the vibration generator on the roof and some connection details of the accelerometers on the walls of the structure.

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3.3. Dynamic Identification Procedure

The natural frequencies of the building were identified as the peak values of the inertance function. The mode shape components were obtained by the peak values of the modulus and the phase of the inertance function of each accelerometer in correspondence with the peaks itself using a standard MDOF Circle-Fit procedure [23–25]. The mode shape components recorded for each identified mode shape are reported for Position #1 and Position #2 in Tables 1 and 2, respectively. The identified frequencies range from 2.55 Hz to 9.62 Hz for Position #1 and from 2.47 Hz to 9.57 Hz for Position #2.

Figure 4 shows, for the sake of brevity, the dominant natural vibration modes identified for the structure. The identified modes were the first flexural modes in the two directions (associated to the frequencies 2.47 Hz and 2.55 Hz), two torsional modes (with a frequency of about 2.75 Hz and 8.98 Hz), and the second prevailing flexural modes in the two directions (having frequencies of 18.55 Hz and 21.71 Hz).

4. Literature Proposals for Infill Modelling

In the literature studies focused on infill-frame interaction, the attention of researchers was mainly devoted to the definition of numerical models able to reliably simulate the global behavior of the whole system. In the present study, only macromodels have been considered because micro- and mesomodelling, typically used for analysis of single infill panels or small building portions [10], are computationally very expensive and not suitable for the study of a whole building. The macromodels proposed by researchers in order to reproduce the frame-infill interaction are, almost all, based on the concept of equivalent struts [8–10], but a further subclassification can be based on the number of equivalent struts adopted for the infill modelling. Crisafulli et al. [8] showed that equivalent single-strut models are not suitable to represent the stress distribution in the RC frame, and if these stresses are required, multistrut models must be considered. A detailed description and applications of multistrut models can be found in [26–29]. On the other hand, Asteris et al. [10] showed that single-strut models provide reliable approximations of the global response of the structures, as, for example, the definition of the mode shape components and natural frequencies of a building (see also [7, 30, 31]). Hence, in the present paper, only single-strut models were considered, being the work focused on the global behavior of the whole building.

Following the approach of Polyakov [11], several researchers studied the issue at hand, by modelling the infilled masonry panels as a diagonal strut with axial stiffness only. The literature evaluated in the present paper provides different methodologies and expressions in order to define the equivalent strut width to be multiplied for the panel thickness (t) in order to obtain the gross cross-sectional area of the diagonal strut, whereas the elastic modulus E adopted for the equivalent strut is that of the masonry itself. Table 3 collects the different proposals, from various authors, considered in the present work for the definition of the ratio. Some authors have proposed specific parameters to take into account the particular aspects in their proposals. The most widespread parameters are reported in Table 4. For example, the dimensionless λ_h parameter, introduced by Stafford Smith [12] and then adopted in several other works [13–15], is defined in the following way:where and are, respectively, the thickness and height of the panel; and are, respectively, the Young modulus of masonry and Young modulus of concrete; θ is the slope of the diagonal of the panel with respect to the horizontal plane; is the moment of inertia of the columns; h is the height of the columns.

In addition, to take into account the infill-frame contact length (z) in the evaluation, Stafford Smith [12] proposed the following expression:where h represents the height of the columns.

An alternative formulation to evaluate the frame-infill relative stiffness has been proposed by Bazan and Meli [16], through the definition of the dimensionless parameter β:where is the gross cross-section area of the columns, is the area of the horizontal section of the infill panel (panel length × panel thickness), and is the shear modulus of the panel. Hendry [17] proposed to evaluate separately the panel-beam relative stiffness () from the panel-column relative stiffness () with the following expressions:

Analogously, the panel-beam contact length () and the panel-column contact length () are computed with the following expressions:

Durrani and Luo [19] proposed to evaluate the relative stiffness panel-frame as reported in the following equation:by means of the m coefficient reported in the following equation:

Recently, a panel-frame relative stiffness parameter () was proposed by Papia and Cavaleri [40] and Papia et al. [41]:where is the beam gross cross-section area, is the length of the beam measured between the centrelines of the columns, and is the height of the columns measured between the centrelines of the beams.

Furthermore, recently, the influence of the vertical load value on the panel-frame contact length was experimentally tested and numerically analysed by Cavaleri et al. [1], Amato et al. [20], and Campione et al. [21], suggesting expressions for the w/d ratio that explicitly consider the vertical load value.

5. Model Updating

5.1. Finite Element Modelling

For the investigated building, an FE model has been achieved by means of the software SAP2000 [42]. The frame elements were modelled through monodimensional linear elastic Timoshenko beam/column elements. The RC walls of the underground story and the lift tube were modelled with classic bidimensional linear elastic 4-node Reissner–Mindlin shell elements. The base nodes have been fully restrained. The stories were considered as rigid diaphragms in their plane because of the presence of the RC slabs.

The geometric properties of elements have been deduced by a detailed survey of the structure. The material properties have been obtained from the experimental characterization tests. An elastic modulus equal to 35030 MPa has been assigned to RC elements in the underground and the ground story, while a modulus equal to 29994 MPa has been used for the elements in the upper stories.

The infill panels have been modelled through an equivalent single strut for each diagonal with only axial stiffness. The Young modulus of masonry has been assumed to be equal to E_m = 2995 MPa. The elastic material properties adopted in the FE models are summarized in Table 4. The infill panels inserted in the model have been the perimeter walls (30 cm thick) and the partitions (25 cm thick). The stiffness has been reduced by means of proper reduction factors to take into account the opening presence [43]. With the aim to not introduce in the model a double strut stiffness, due to the presence of a strut for each diagonal, a value equal to one-half of the real thickness has been introduced in the FE model. The structural model obtained is depicted in Figure 5. In order to define the more suitable width for the equivalent struts, the model updating process described in the following section has been adopted.

Finally, in order to obtain the numerical natural frequencies and mode shape components, to compare with the experimental outcomes, modal decomposition analyses have been performed on the FE models of the building.

5.2. Definition of Target Function H

In order to investigate the effects of the infill panels’ presence on the dynamic behavior of the building, a parametric study was performed by varying the ratio in the range 0.0–0.40.

For each considered value, natural frequencies and mode shape components of the building have been recorded. Then, matching between numerical and experimental outcomes was checked through a target function H defined as follows:where N represents the number of eigenmodes, and are the two weight functions, and (normalized modal difference) is defined as follows [25]:

The H function represents the relative error between frequencies and mode shape components obtained via the FE model , for a fixed set of parameters, and the same quantities obtained experimentally . The modal assurance criterion (MAC) represents a parameter ranging between 0 and 1. It is used in order to provide indications on the matching between numerical and experimental mode shapes (e.g., MAC close to 1 indicates that two modes are very similar). The following expression for the MAC has been assumed [23, 44]:where represents the dimension of the analysed vectors.

The MAC parameters, collected in a histogram matrix form, result in a useful tool to identify the fitting between numerical and experimental modes.

By means of the dynamic excitation tests on the building, frequencies and mode shape components reported in Table 5 were identified. In the model updating process, N (number of modes) and (number of components) parameters were set to 5 and 12, respectively. The weight functions is defined as follows [25]:where is the participating mass of the i-th mode with the introduction of the further assumption:

For the weight function , two different values, associated with two different hypothesis, have been considered:or in alternative

With the first criterion, only the errors on frequencies (i.e., without any contribution of errors connected to mode shape components) have been considered in the evaluation of the target function H. Instead, by adopting the second hypothesis, even the error contribution of the mode shape components have been introduced in the evaluation of H. Therefore, H function was computed for two specific pair of weight functions: and .

6. Results and Discussion

6.1. Determination of the Optimal Ratio

The FE model considered in the study has been firstly analysed by considering a strut width equal to (i.e., bare frames without infill). In this case, the experimental frequencies of the structure result are greatly underestimated as shown in Figure 6 and Table 6, respectively, reporting the main frequencies and the MAC matrix for the case neglecting the infill presence.

The H function, considering and excluding the mode shape contribution, assumes the value of 0.354 and 0.310, respectively. Obviously, it is worth noting that the model neglecting the infill presence does not match with the experimental outcomes, providing numerical frequencies very far from the experimental ones. Afterwards, the infill stiffness contribution has been introduced in the model as equivalent strut elements, and 40 different FE models have been analysed by considering 40 values. Figure 7 depicts the evolution of the MAC matrix for some ratios. For low values, the MAC matrix results are sparse, and only three experimental modes are adequately captured by the numerical model. By increasing the ratio, the matrix tends to stabilize, and for values greater than 0.20, it results are almost diagonal. The H function versus the ratio has the trend as depicted in Figure 8. For both criteria, the minimum of H function is achieved for the ratio equal to 0.24. For this specific value, the MAC matrix is reported in Table 7. For the sake of a fast comparison, numerical frequencies obtained for = 0.24 and experimental ones are reported in Table 8.

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When suitable values are introduced in the model, the four main frequencies (i.e., frequencies #1, #3, #4, and #5 with the highest mode participating factors) can be identified with high MAC values. With regard to the second mode, instead, a low MAC value has been obtained. This is probably because the second and the third experimentally identified modes have frequencies very close between them, respectively, 2.55 Hz and 2.79 Hz, and the related mode shapes are both characterized by a torque-bending coupling behavior. So, in order to capture these particular features of the real structure, a deep knowledge of the building details is necessary (e.g., the real distribution and position of masses over the floor, the role of the floor openings, and the role of infill openings). Furthermore, it is very complex to take into account some of these aspects in a numerical model since they can be considered as modelling uncertainties. Anyway, by adopting the value = 0.24, which minimizes the H function, all numerical frequencies, with the exception on the second one, match very well with the experimental outcomes with the largest frequency error resulting lower than 10%.

6.2. FE Model without Underground Story

In order to numerically evaluate the effects of the presence of the underground story, a different FE model has been created excluding the underground story and introducing a full restraint at the base of the columns of the ground level. This model produces results altogether similar to the previous one, for every value. This effect was expected because, in the experimental mode shapes, the deformation contribution of the underground story resulted almost negligible due to the considerable stiffness of the RC perimeter walls of the basement. Figure 9 and Table 9 show, as representative instance, results provided by the model without the underground story for = 0.24. By comparing Tables 8 and 9, it appears clearly that the underground floor, because of its stiffness, does not influence the dynamic behavior of the building.

6.3. Comparison between FE Models Adopting Ratio Suggested by the Literature

Several expressions for the ratio have been provided from the scientific literature, and some of them have been reported in the previous sections. In order to evaluate the agreement and the applicability of the different proposals to full-scale buildings, 13 different proposals have been selected and used for the generation of a specific FE model adopting the ratio. The H values, obtained by considering the ratio provided by the 13 selected proposals, are collected in Table 10 and shown in Figure 10. For the sake of comparison, in the table and in the figure, the H values obtained from the model with = 0.24 have been also reported.

From the analysis of the different models, it emerges that more recent proposals provide results that are generally similar between them and rather close to results obtained for = 0.24. For the case at hand, considering for the evaluation of the target function H by only the natural frequencies, the model which produces the best result is the method proposed by Bazan and Meli [16]. It is worth noting that the study performed by Aniendhita and Data [45], which considers 14 different proposals for the ratio, also indicates that Bazan and Meli is one of the models (i.e., is the second over a total of 14 analysed models) providing the best fitting with experimental tests. On the other hand, following the results by Aniendhita and Data [45], Holmes’ method has the best performance, whereas in the present study, it provides an assessment of far from the optimal value. Conversely, considering also the mode shapes in the H evaluation, the best fitting is obtained by the expression of Paulay and Priestley [18], providing a value of ratio equal to 0.25, very close to the optimal value of 0.24.

For the case study investigated here, the comparison between the different literature proposals provided results for with a similar order of magnitude but generally very different in values. In some cases, these results were also in disagreement with experimental evidence presented here.

Furthermore, it is useful to highlight how a correct prediction of the equivalent strut dimensions has a fundamental importance on the dynamic response of the numerical model. In fact, for the case studied here, it is possible to see how to adopt a model proposal rather than others which lead to evaluations ranging from about 0.11 to 0.52, with the optimal value equal to 0.24. This means that the various existing proposals provide a range of values affected by great variability. For the case studied here, they can vary from 46% (i.e., 0.11/0.24 × 100) to 217% (0.52/0.24 × 100) of the optimal value. This entails large variability of the mechanical properties that equivalent struts will assume by adopting a proposal rather than another one, but, as a consequence, the choice will also influence the dynamic behavior of the FE model implementing those struts. Currently, this represents a serious issue for the daily designers’ activity.

In addition, these models generally provided evaluations lower than the required value for the optimal matching between experimental and numerical dynamic response, at least in the linear range (i.e., for low force values). This is because many of the proposals for evaluation are based on the results of static tests in which the evaluation of the optimal value is made by taking the peak strength of the nonlinear experimental force-displacement curve (see Figure 11).

This aspect, from a practical point of view, results in the clear possibility of underestimating the current elastic stiffness of the infill, and therefore leading to the definition of numerical models characterized by longer vibrating periods (in Figure 11), generally affected by lower seismic actions. Obviously, all of this can be translated into a nonsafety assessment of the seismic state of the building.

Finally, it seems questionable if computation and modelling complications deriving from some very refined parameters as, for example, those connected to relative panel-frame stiffness or vertical loads acting on frame, are necessary for a reliable study of complex systems as real buildings. In fact, even for a more complex model, the introduction of those additional terms did not lead to significant improvements with respect to the simplified choice of adopting, in the FE model, the same equivalent strut width for all the infill panels of the building.

7. Final Remarks

The present paper describes the main results connected to dynamic identification and model updating of a building with RC frames infilled with masonry panels. As a first phase, experimental tests were carried out in order to properly characterize the mechanical properties of the materials. Then, dynamic tests with induced vibrations were performed in order to identify the natural frequencies and mode shape components. By means of numerical analyses, it was observed that only with the introduction of infill panels, it is possible to reproduce, with good approximation, the dynamic behavior of the whole structure.

By means of a parametric study, in which the ratio (i.e., ratio between width and diagonal length of the equivalent axial strut modelling the infill panels) was used as a varying parameter, and an optimal value of the ratio equal to 0.24 was identified for the case at hand. For this specific value, the target function, measuring the error between experimental and numerical quantities, assumes the minimum value and maximizes the matching between experimental and numerical dynamic response. This value has been then compared with values reported in the literature.

Furthermore, it has been observed that modelling or neglecting the basement of the building does not lead to substantial modification of both target function values and building dynamic behavior.

Finally, with the aim of measuring the accuracy of some literature proposals defining the ratio, the original FE model has been modified, introducing diagonal struts as obtained by the application of 13 different expressions considered in the work. Comparing the results achieved by adopting the literature proposals with the experimental outcomes, the H function, computed considering only natural frequencies, resulted in a minimum value by adopting the Bazan and Meli’s model [16], and the numerical results were in good agreement with the ones obtained with the model using a constant ratio (equal to 0.24). This outcome is confirmed by similar works reported in the literature.

Conversely, the target function considering both frequencies and mode shape components presents the minimum value if the Paulay and Priestley’s model [18] is used.

Moreover, from a practical point of view, different proposals provide very different values, ranging from 46% to 217% of the optimal value. This aspect, as a consequence, involves a large variability of the mechanical properties to be adopted in the FE modelling of buildings and could represent a serious issue for the daily practical design since this can lead the designer into nonsafety assessment of the seismic state of the building.

With reference to the case study building investigated here, it can be stated that most refined expressions, defining the ratio, provide results substantially equal, and in some cases worse, with respect to a simpler model considering a constant ratio for all masonry infill panels.

At the present state of the research, it seems that available literature proposals could provide an assessment of the optimal mechanical characteristics of the equivalent strut, but, for more refined evaluations, the building FE model that considers the presence of the infill should be updated based on in situ experimental tests since the complex frame-infill interaction is strongly case dependent. These final remarks should drive the future studies to provide more robust criteria for the modelling of these very spread class of buildings.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The financial support of the Italian Department of Civil Protection (ReLUIS 2019 Grant-Innovative Materials) is gratefully acknowledged.

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