Displacement incompatibility shape functions between masonry infill wall panels and reinforced concrete frames

During an earthquake, the detachment and local interaction between infill wall panels and surrounding frame can occur, potentially leading to significant local damage to both structural and non-structural elements, if not global collapse. Yet, a procedure to assess the relative deformation mechanism in terms of detachment shape and values, rather than, and in addition to, the diagonal compression strut mechanism and associated internal panel strain and stress path, is still missing in the literature. Therefore, in this paper the concept of shape functions is proposed and adopted to assess the seismic displacement incompatibility between infill walls and the surrounding frame structure. A parametric study on different typologies of infilled frames is developed to investigate the key parameters affecting the infill-frame detachment. The proposed concept of shape functions can support the design/retrofit of improved construction details, such as shear keys and/or steel dowels, in view of either decoupling or strengthening retrofit/repair strategies. Moreover, as infill-frame detachment can lead to damage to energy enhancement rehabilitation solutions, such as external thermal insulation systems, which are becoming more common nowadays in view of the international target towards a significant reduction of energy consumption and CO2 emission, it is suggested to implement the proposed displacement-compatible design check to assess and detail for adequate displacement capacity.


Introduction and motivation
Reinforced Concrete (RC) frame structures with masonry infill walls represent a large part of the European building stock. Infill wall panels are usually considered as "non-structural" elements as they are designed to provide facade multi-purpose functions (e.g., thermal and acoustic insulation), thus they are often neglected in the structural design. However, it is well known that infills and surrounding frames have a strong interaction during seismic shakings. Although infilled frame structures exhibit, in a first stage, a significant increase in the lateral strength and stiffness when compared to the bare frame solution, the infill-toframe seismic interaction can lead to local shear failures of structural elements (Fig. 1a) or global failure mechanism (e.g., soft-story mechanism, Fig. 1b) (Magenes and Pampanin 2004). Moreover, past studies (Taghavi and Miranda 2003;Cardone and Perrone 2017) pointed out that approximately 80% of building loss is associated with damage to nonstructural components (e.g., masonry infills, partitions) and contents.
On the other hand, the recent sustainability requirements in European regions (Directive 2018/844 2018) have led to growing attention to the overall performance of facade elements. In Europe, buildings are responsible for 40% of EU energy consumption and 36% of CO 2 emissions (Calvi et al. 2016); therefore, improving structural/seismic safety and energy efficiency of existing buildings has become an important socio-economic and environmental need (Bournas 2018). To achieve this goal, it is becoming increasingly evident that energy and seismic retrofitting should be studied through an integrated multiperformance design approach (Calvi et al. 2016;Marini et al. 2017;Di Vece and Pampanin 2019). Consequently, the seismic performance of facades plays a crucial role since damage to these components can lead to loss of performance to energy refurbishment solutions even for low-intensity earthquakes (Fig. 1c).
Seismic response of infilled RC structures has been widely studied in the past years and effective retrofit solutions capable to reduce the infill-frame seismic interaction have been proposed (e.g., Tsantilis and Triantafillou 2018; Morandi et al. 2018b;Tasligedik and Pampanin 2017;Butenweg 2019, 2022). In the case of seismic load, partial detachment between infill panels and surrounding frame occurs, with a concentration of the compression contact region at the diagonally opposite corners, leading to the formation of a diagonal compression strut load path mechanism (Zamic and Tomazevic 1984). In the past, particular focus has been given to the study of the diagonal compression strut load path, with comprehensive experimental and investigation aiming at characterizing its properties (width, depth, contact regions, strength and hysteresis loop, local mechanisms).
Yet, looking at the same interaction mechanism from an alternative and complementary perspective, infill-frame detachment can be harmful to External Thermal Insulation Composite Systems (ETICS) as insulation materials can exhibit a brittle failure when subjected to tensile stresses (Tang et al. 2019). Moreover, seismic retrofit solutions for facade  (Izmit, 1999, NISEE image collection); c damage to masonry infill walls and external thermal insulation due to an earthquake (available online: http:// www. reluis. it/ images/ stori es/ webin ar-23-11-20/ Prota-daPor to. pdf) components based on strengthening or decoupling techniques would require appropriate structural details, such as shear anchors or steel dowels, to create a composite action or to decouple the two systems while preventing the out-of-plane collapse of the infill panels. These structural details may damage the facade components or result themselves damaged due to the infill-frame detachment process or displacement incompatibility. Therefore, a better understanding of the displacement incompatibility and detachment mechanism between frame and infill panel, supported by specific investigations, is needed.
In this context, this paper aims to assess the detachment shape and values due to the seismic displacement incompatibility between frame and infill wall by adopting the concept of shape functions. Firstly, a brief review of the modelling approach for infilled frames is provided, and initial parametric analyses are performed to assess the seismic response of different typologies of masonry infills through alternative macro-modelling approaches. Secondly, displacement incompatibility is investigated by developing shape functions for different infilled frame configurations. A simplified modelling approach to assess the value of detachment along the frame structure is adopted and preliminary shape functions are presented. Finally, a conceptual framework to support the assessment and design of specific construction details for decoupling/strengthening' retrofit strategies and techniques, as well as for mechanical fixing systems for external thermal insulation composite systems, is presented.

Modelling approaches for masonry infilled frames: literature review
Modelling methods to assess the seismic performance of infilled RC frames can be grouped into: (1) micro-modelling, (2) macro-modelling approaches and (3) meso-modelling approaches. The micro-modelling approach aims to provide an accurate description of the structural behaviour of the infill wall by modelling masonry units and mortar joints in detail. However, this method requires a great amount of data, resulting to be complex and time-consuming for its high computation effort. Otherwise, in the macro-modelling approach there is no distinction between mortar joints and bricks and an equivalent diagonal strut is typically adopted to model the infill panel. Following the meso-modelling approach, the infill panel is idealized as a continuous bi-dimensional element with no distinction between brick-mortar joints. Therefore, the micro-modelling and meso-modelling approaches are generally recommended to investigate local effects due to the infill-frame interaction, while the macro-modelling approach is recommended to assess the global behaviour of the structure, taking advantage of its low computational effort. A state-of-art of different macro-modelling techniques is presented below, since as a first step the equivalent strut model is adopted to achieve the goals of this paper. Polyakov (1960) was possibly the first author to suggest that the behaviour of infill panels could be described by diagonal struts connected to the two opposite corners of the surrounding frame, while Holmes (1961) firstly applied this concept. The geometrical properties of the equivalent strut are fully described by the thickness of the infill panel t w , the width b w and the length of the diagonal strut d w . The main issue of adopting such an approach is related to the calculation of the width, since the thickness and the length are automatically defined by the panel geometry. Thus, Holmes (1961) proposed an empirical formula to calculate b w . After a few years, Stafford Smith (1967) developed a mathematical expression to calculate the relative stiffness between infill and frame ( ) . The same author also proposed a formulation, widely adopted nowadays, to evaluate the contact 1 3 length between panel and frame based on the analogy of a beam on an elastic foundation. Later, other researchers proposed different equations to calculate the equivalent strut width depending on , including Mainstone (1974) (adopted by FEMA 306, 1998), Liauw andKwan (1984), andDecanini andFantin (1987). It is also acknowledged the work developed by Paulay and Priestley (1992), who proposed a simplified formulation to calculate b w , as well as the research by Papia et al. (2003), who defined a mathematical expression depending on an alternative relative panel-to-frame-stiffness parameter * . In the model proposed by Bertoldi et al. (1993), adopted in this paper, the strut proprieties are evaluated considering four different failure mechanisms: (1) compression failure at the centre of the infill, (2) compression failure at the corners of the infill, (3) sliding shear failure, and (4) diagonal tension failure. The ultimate compression stress is defined as the minimum value obtained for the four failure mechanisms. The whole formulations for this model are reported in Table 1.
Furthermore, two main different types of equivalent diagonal strut modelling can be identified from a literature review: single diagonal strut models and multiple diagonal strut models. The multiple strut models mainly differ in number, type, inclination, and connection position along the frame of the struts (e.g., Chrysostomou et al. 2002;El-Dakhakhni et al. 2003;Crisafulli and Carr 2007;Di Trapani et al. 2018).
By comparing different diagonal strut models (single, double, and triple strut) with a more refined FEM model, Crisafulli et al. (2000) demonstrated that the single diagonal strut model generally allows to assess the seismic global performance of the structure, however, it is less accurate in capturing the actual distribution of the internal actions expected from the infill-frame interaction. Alternatively, the double and triple diagonal strut models can provide a better estimation of the frame bending moment diagrams.
Although out of the scope of this research work, when dealing with the seismic performance of infilled frame structures, the out-of-plane (OOP) response of infills is also deemed a critical issue, as well as the interaction between the in-plane (IP) and OOP response. Clearly, during seismic shakings, the structure is simultaneously subjected to both IP and OOP loading, and damage to infills due to the IP seismic loads could lead to Table 1 Formulations of the equivalent strut model proposed by Bertoldi et al. (1993) E w , E wh , E wv , Elastic modulus of the infill in the diagonal, horizontal and vertical direction, respectively; , Inclination of the strut: G , Shear modulus of the infill; , Poisson modulus of the infill; , Relative stiffness between infill and frame (Stafford Smith 1967); E c , Elastic modulus of concrete; I c , Column moment of inertia; h w , Height of the infill panel; t w , b w , d w , Thickness, width and length of the diagonal strut, respectively; f wh , Compression strength for the horizontal direction; f wv , Compression strength for the vertical direction; f wu , Sliding shear resistance of the mortar joints; f ws , Shear resistance under diagonal compression; v , Vertical compression stress due to gravity loads; w1,2,3,4 , Compression at the centre of the infill, compression at the corners of the infill, sliding shear failure and diagonal tension failure.
a reduction in terms of OOP capacity and vice versa, potentially leading to the collapse of the infill. This topic has been widely studied in the past through both experimental (e.g., Calvi and Bolognini 2001;Furtado et al. 2016;Ricci et al. 2018;Morandi et al. 2022) and numerical (e.g., Cavaleri et al. 2020) investigations. Among others, the results presented by Ricci et al. (2018) can be considered to better understand the IP-OOP interaction in masonry infill panels. Specifically, three infilled RC frames (2:3 scaled), designed according to modern seismic-code provisions in Italy, were tested by applying OOP loads after IP cyclic tests. The three specimens were subjected to an initial IP interstorey drift ratio equal to 0.16%, 0.37%, and 0.58%, respectively; the results were compared to previous experimental investigations carried out by the same authors on a nominally identical specimen subjected to "pure" (i.e., without IP initial damage) OOP loading (Di Domenico et al. 2019). Concerning the OOP strength capacity, a significant reduction was observed when considering initial IP damage, especially for higher initial interstorey drift values: reductions up to -52% and -73% were observed considering an initial interstorey drift equal to 0.37%, and 0.58%, respectively. Recently, some macro-modelling approaches have been proposed for the simulation of the IP and OOP response of infilled frame structures subjected to earthquakes (e.g., Di Trapani et al. 2018;Mazza 2019;Donà et al. 2022). Finally, the hysteretic behaviour of the infill panel needs to be defined in order to describe its strength and stiffness degradation. Crisafulli (1997) proposed a refined analytical formulation (adopted in this paper) of the backbone curve and hysteresis rule based on the observation of experimental tests on masonry panels.

Methodology
Displacement incompatibility for RC frame structures is a well-known issue in seismic engineering and it has been investigated in past research works. However, these works mainly focused on the seismic displacement incompatibility between structural components (e.g., the "beam elongation", Fenwick and Megget 1993;Peng et al. 2011; vertical displacement incompatibility between beams and precast flooring unit, Matthews et al. 2003;Vides and Pampanin 2015). On the other hand, very few studies have focused on the assessment of the infill-frame interaction in terms of contact and, in turn, the complementary detachment zones. Crisafulli (1997) discussed through numerical investigations the relative deformation mechanism between the infill and surrounding frame; particular attention has been given to the modification of the bending moment and shear diagram in the structural members due to the presence of the infill, both in the case of intact panels and panel with horizontal shear sliding failure. Brodsky et al. (2018) investigated the interaction behaviour in terms of the infill-frame contact regions and interfacial tractions in case of loss of a supporting column. Milanesi et al. (2018a) numerically investigated the local seismic interaction between Autoclaved Aerated Concrete (AAC) masonry infills and surrounding RC frames. The authors performed nonlinear static analyses adopting a 'mesomodelling' approach. A comparison between the bare and the infilled frame in terms of moment and shear demand in the columns was provided, highlighting that substantial differences in the distribution of the internal forces were obtained, even for low levels of interstorey drift. Moreover, considering higher interstorey drift values, a higher contact length between infill and frames was observed. The authors also concluded that analytical expressions available in the literature tend to underestimate the contact length of about 50% with respect to the obtained numerical results. Wararuksajja et al. (2020) studied the local seismic interaction between RC frames and concrete block infill walls through both experimental and numerical investigations in order to provide a design methodology to avoid local brittle failures in the structural members. The increment of shear force in the columns due to the infill-frame interaction has been experimentally evaluated by measuring the strain of the stirrups through uniaxial strain gauges. Results showed that, for interstorey drift values higher than 0.75%, the strain values in the transversal reinforcements increased significantly, especially for strain gauges located at approximately 3/4 of the effective depth of the column (measured from the joint interface). The authors concluded that the corner crushing in the infills can lead to a "short column" mechanism, especially in the case of strong infills, and, consequently, increasing shear demand can be observed, potentially leading to brittle failure mechanisms in the columns. Pashaie and Mohammadi (2021) presented an extended multiple-strut model for infilled steel frames considering different typologies of beam-to-column connection (i.e., rigid, semi-rigid, and pinned). The authors carried out an extensive parametric investigation on refined finite element models characterized by different aspect ratios, friction coefficients, and relative stiffness between infill and frames. Particular focus was given to the ability of the proposed macro-model in estimating the modification of the bending moment diagram in beams and columns due to the infill panel and the three different considered connections. Recently, Facconi and Minelli (2020) studied a strengthening technique for RC weak infilled frames by using thin glass fiber mesh reinforced mortar overlays. The authors paid particular attention to the steel dowel connection between the masonry panel and the RC frame and investigated the different responses of the infill-to-frame interface at different loading levels.
In this work, in order to assess the infill-frame detachment, the concept of displacement compatibility shape functions, introduced by Taylor (2004) and further developed and applied by Vides and Pampanin (2015) to study the vertical displacement incompatibility profiles between the seismic resisting frame and the precast flooring unit, is adopted. Shape functions are defined as the envelope of the maximum (both horizontal and vertical) displacement incompatibility recorded along the interface with the surrounding structural frame.
In order to evaluate the detachment values between the infill wall and surrounding frame structure, the best approach would arguably be to perform a refined model according to micro-modelling techniques. However, as mentioned above, this method requires the identification of several parameters, consequently leading to an excessive effort. It is worth noting that this paper aims to investigate which parameters strongly affect displacement incompatibility as well as to provide a preliminary range of values of detachment expected for different infilled frame configurations. Hence, to achieve this scope, a more simplified modelling approach is adopted. Specifically, a comparative procedure between two different models for the same structural system is carried out, as described in Cavaleri and Di Trapani (2015) where the same method was adopted to predict additional shear actions on frame members due to infills. For each analyzed configuration, two different models are implemented: an equivalent strut model and a continuous bi-dimensional shell model. These models can be considered as equivalent when they exhibit the same lateral secant stiffness under monotonic loading. Hence, operatively the non-linear behaviour of the shell elements is introduced by iteratively reducing their thickness. This procedure aims to obtain for the continuous model the same lateral secant stiffness of the equivalent strut model for a fixed interstorey drift level, as shown in Fig. 2.
It is highlighted that, by a comparison between the multistage linear equivalent bidimensional model and a fully nonlinear bi-dimensional model, Cavaleri and Di Trapani (2015) proved that this modelling strategy can provide good accuracy in assessing both the global response of the structure and the local shear demands on frames. Hence, the structural responses of the two models at specific interstorey drifts are compared. Finally, horizontal and vertical displacements of both structural frame and infill edges are evaluated and shape functions of the displacement incompatibility for a fixed interstorey drift are developed (Fig. 3).
Referring to the adopted modelling strategies, a single strut model seems to be the best choice to assess the global seismic performance of infilled frames. Crisafulli et al. (2000) proved this consideration by performing a comparative analysis of different strut modelling techniques and assuming linear elastic behaviour. However, the more accurate estimation of the frame bending moment diagram derived by double or triple diagonal strut models can modify the nonlinear behaviour of the structure. Therefore, a preliminary parametric Flowchart of the adopted methodology to evaluate the displacement-incompatibility shape functions analysis of the seismic behaviour of infilled frames is also performed to support the choice of the modelling technique.

Case-study infilled frame structures
Two single-story one-span infilled frame structures, consisting of a bay length of 3 m and 5 m, respectively, and an interstorey height of 3 m, are considered to implement the study. Construction details are typical of a pre-1970s existing building (i.e., designed for gravity loads only) according to available data of an existing school building in Lucera (FG), Italy. Specifically, no "capacity design" principles are provided, and the structure presents the typical structural weaknesses of existing buildings in Italy designed in that period according to old code provisions (i.e., "Reggio Decreto" RD 2229, 1939) and available technical guidelines (e.g., Santarella 1968), e.g., inadequate transverse reinforcement for shear and confinement, strong beam/weak column, inadequate anchorage details, lower quality of materials. Three different masonry infill walls are selected, to represent weak, medium and strong infill panels, according to Hak et al. (2012). The weak infill panel is a single-leaf masonry wall with horizontally hollowed brick; the medium infill panel is a double-leaf masonry wall with horizontally hollowed brick divided by an internal cavity; the strong infill panel is a single-leaf wall with vertically hollowed brick units. Geometrical properties of the selected infilled frames are reported in Fig. 4, while mechanical proprieties of the three typologies of infill walls are reported in Table 2.
Structural details of RC beams and columns are assumed the same for both structural configurations. The beam-column joint presents no stirrups and plain round beam bars with end-hooks. The mean concrete cylindrical strength is equal to 16 MPa and the mean steel yield stress is equal to 400 MPa. A Young's modulus of 22,850 MPa and 200,000 MPa are considered for concrete and reinforcement steel, respectively. An axial load N = 250kN acting on the columns is applied to each configuration.

Macro-modelling approaches
Parametric analysis of the seismic behaviour of infilled frames is performed considering three different macro-modelling techniques: (1) a single strut joint on the opposite corner node, (2) a single strut with rigid arms at the end sections (conceptually proposed by Fig. 4 Geometrical properties of the infilled frame configurations Pampanin, personal communication, 2002, and adopted in subsequent research works, e.g., Gentile et al. 2019c) and (3) a triple strut model. A qualitative illustration of the three selected techniques is reported in Fig. 5, where A w is the cross-section area of the strut and z is the contact zone length, derived according to Stafford Smith (1967).
Firstly, an analytical prediction for each configuration is performed by using the Simple Lateral Mechanism Analysis (SLaMA) method (NZSEE 2017). SLaMA is an analytical mechanics-based procedure to assess the seismic performance of Reinforced Concrete (RC) and UnReinforced Masonry (URM) existing buildings. The method has been introduced and further developed as part in the New Zealand Society for Earthquake Engineering (NZSEE) "Seismic Assessment Guidelines" 2006(NZSEE 2006Pampanin 2017) and widely adopted and validated in past studies in the literature (e.g., Del Vecchio et al. 2018;Gentile et al. 2019a, b, c, d;Bianchi et al. 2019;Pedone et al. 2022;Sansoni et al. 2022). The main step of the SLaMA procedure can be summarized as ( Fig. 6): (i) assess the flexural and shear capacity (in terms of both moments/forces and rotations/  Recently, a refinement of the SLaMA method including the influence of the infill walls in the global building response (capacity curve) has been carried out and validated by Gentile et al. (2019c, d). The SLaMA method for infilled frames (SLaMA-infill) is based on the calculation of the overturning moment and base shear contributions related to the infills and the frame, independently. Specifically, the infill panels are idealized as equivalent struts and, in order to evaluate their contribution to the global base shear, the strut forces are decomposed into their horizontal and vertical components; then, as pointed out by Gentile et al. (2019c), the overturning moment contribution provided to the infill (OTM infills ) can be calculated, similarly to the coupling contribution of the beams within the frame, considering the vertical components of the strut forces, which create tension-compression couples. Thus, the OTM infills can be converted into base shear V b,infills considering the effective height of the structure H eff , according to the original SLaMA method, V b,infills = OTM infills / H eff . Finally, the sum of the bare-frame and infills contributions defines the global response of the structure (Fig. 6). More details on the SLaMA-infill methodology can be found in Gentile et al. (2019c).
In this preliminary study, a refinement of the SLaMA method is proposed and adopted to compute the elastic stiffness of the bare frame capacity curve. In fact, the original SLaMA method approximates the force-displacement capacity curve of the structure through a bilinear elastic-perfectly-plastic curve. In other words, the global capacity curve is obtained by assessing the displacement related to the first attainment of the ultimate limit state (ULS) rotation or moment/shear (in case of brittle failure mechanisms) in any beamcolumn-joint subassembly and the base shear associated to the ultimate global inelastic mechanism, while the base shear at the yielding point of the structure is assumed to be equal to the ultimate strength capacity. Therefore, a possible refinement to the method consists of evaluating the base shear at intermediate stages (e.g., considering different limit states), as recently suggested and implemented in Sansoni et al. (2022) for the application of the SLaMA method to URM structures (i.e., referred to as SLaMA-URM). In particular, in this work, the base shear corresponding to the achievement of the first yielding rotation of the structural element is evaluated by using an equilibrium approach at an intermediate (yielding) point similar to the one adopted for the ultimate point. Therefore, once knowing the yielding displacement of the capacity curve, the elastic stiffness of the capacity curve can be better identified. This refinement of the procedure can lead to a better evaluation of the capacity curve for infilled frames, since the typical overestimations of the base shear at yielding, due to the use of a bilinear capacity curve, are reduced.
Then, numerical nonlinear static analyses are performed for the case-study structures using the software SAP2000 (CSI 2019). Details of the adopted modelling approach are reported in Fig. 7.
Structural members are modelled by elastic components with lumped plasticity at the end sections. The plastic hinges are described using proper moment-curvature relationships and the plastic hinge length is calculated according to Priestley et al. (2007). The shear failure mechanism is also evaluated. Panel zones are modelled by rigid arms with an additional rotational spring to take into account the possible failure of beam-column joints (Pampanin et al. 2003). Specifically, the moment-rotation springs are located between beams and columns and their behaviour is described by equivalent column moment vs. drift curves, following the provisions reported in the New Zealand guidelines for seismic assessment of existing buildings (NZSEE 2017). Joint ultimate drift capacity is set equal to 1% and 1.5% for exterior and interior joints, respectively, consistently with past experimental observations and according to Pampanin et al. (2003). Equivalent strut properties are obtained according to the procedure proposed by Bertoldi et al. (1993), discussed above. Diagonal struts are modelled by a non-linear axial spring element described by a simplified tri-linear stress-strain relationship, derived from the Crisafulli hysteresis. Hence, only two parameters are needed, the strain at the peak stress ′ w and the ultimate strain u , while the ultimate stress f ′ m is obtained according to Bertoldi et al. (1993). In this preliminary study, a value of � w = 0.0013 and u = 0.0045 are adopted, as reported in Hak et al. (2012). For the single strut approach, an additional model is also performed by using the software OpenSees (python library, Zhu et al. 2018), in order to consider an alternative description Fig. 7 Schematic illustration of the adopted numerical modelling strategy of the infill stress-strain hysteresis. In this model, a fiber section is defined for the equivalent strut and the stress-strain law of the fibers is described by a Kent-Scott-Park concrete material (Kent and Park 1971), setting the parameters in order to best approximate the Crisafulli hysteresis, as for the previous numerical models implemented in SAP2000 (both SAP2000 and OpenSees do not include the Crisafulli hysteresis in their libraries).

Nonlinear static analyses
The obtained numerical results are presented in Fig. 8. An ultimate interstorey drift of 1.0% is assumed for all the analyses since the panel joint shows a critical damage level at this drift level (Pampanin et al. 2002).
The different macro-modelling approaches provide a very similar nonlinear global behaviour for all the considered configurations. Furthermore, the SLaMA-based approach allows predicting the seismic response of the infilled frame with good accuracy. An increase in global strength is observed when considering a stronger infill panel. However, it is worth noting that this increased capacity is related to an increased level of local shear demand in the structural elements, which can result in a brittle failure mechanism.
Looking at the results of this preliminary study, it can be concluded that all the considered macro-modelling approaches (as well as the analytical SLaMA-based approach) are valuable to achieve the objectives of this paper. In the following section, the fiber section strut model (i.e., OpenSees model) will be adopted.

Numerical versus experimental results
In the previous section, preliminary parametric analyses with different macro-modelling approaches were reported, based on the adoption of the Crisafulli stress-strain relationship and fixed values for the strain parameters, according to Hak et al. (2012). However, these values were calibrated by referring to available experimental results on RC frames with weak masonry infills. Moreover, Hak et al. (2012) highlighted the need to extend the procedure to different masonry infill typologies, once relevant experimental data would be available. Hence, this section aims to calibrate strain parameters for medium and strong infills, according to experimental tests on RC infilled frame structures. To achieve this scope, the experimental tests performed by Mehrabi et al. (1996), Cavaleri and Di Trapani (2014) and Morandi et al. (2018a) are selected. Geometrical details and material properties of the infill panels are reported in Tables 3 and  4. More information about the specimens can be found in the previously cited papers.
The experimental results are compared with the numerical results developed in OpenSees. It is worth noting that, for the experimental testing performed by Mehrabi et al. (1996), the elastic modulus in the diagonal direction is assumed to be equal to the one in the vertical direction, according to Di Trapani et al. (2017), considering that the elastic modulus in the horizontal direction is not provided. Results of the implemented calibration are presented in Fig. 9.
Although out of the scope of this study, the numerical vs. experimental comparison in terms of cyclic response of the infilled frame tested by Morandi et al. (2018a) is shown in Fig. 9a. The results highlight a good agreement when considering a strain at maximum stress � m = 0.0013 and an ultimate strain u = 20 � w . The force-displacement behaviour is well captured, while the hysteretic behaviour of the numerical model slightly overestimates the energy dissipation of the infilled structure. Regarding the experimental tests performed by Cavaleri and Di Trapani (2014) and Mehrabi et al. (1996), the comparison between the backbone curve of cyclic tests and the numerical results shows a good accuracy when adopting an ultimate strain of the strut u = 10 � w . Table 5 summarizes the main parameters of the equivalent strut models.
The implemented study highlights the great influence of the thickness, instead of the infill strength or stiffness, in the choice of the strain parameters during the calibration process. According to these results and considering that this section aimed to evaluate a range of strain parameters for medium and strong masonry infill types, in the next section the numerical investigation is carried out referring to values of ultimate strains u equal to 10 and 20 times the peak strain ′ w for these two categories, respectively, while a peak strain value of � w = 0.0013 is adopted for both cases, according to the results.

Fig. 9
Comparison between experimental and numerical results: a cyclic and b monotonic response of the infilled frame tested by Morandi et al. (2018a); monotonic response of the infilled frame tested by c Cavaleri and Di Trapani (2014) and d Mehrabi et al. (1996)

Case-study infilled frames and modelling approach
A parametric analysis is implemented to determine the shape functions describing the displacement incompatibility between the infill wall and RC frame. The selected case-study infilled frames are the same as the preliminary analysis (Fig. 4) N = 120kN, 270kN and 420kN) to represent portal frames located at three different story levels, considering a low-rise building. According to the modelling strategies adopted, firstly, the global seismic response of each configuration is obtained by using the equivalent strut model approach. Secondly, two interstorey drift values ( = 0.4% and = 0.9% ) are selected to represent Damage Limit State (DLS) and Ultimate Limit State (ULS) of the infilled frame, according to the numerical and experimental tests available in the literature (Magenes and Pampanin 2004;Hak et al. 2012;Morandi et al. 2018a). Then, a continuous bi-dimensional model is developed for each configuration and the equal secant stiffness to the equivalent strut model at each fixed interstorey drift is obtained by iteratively scaling the thickness of shell elements. Concerning the bi-dimensional model, the infill wall is modelled as a continuous element by using elastic orthotropic shell elements. Gap elements, able to transfer compression stresses only, are adopted to model the contact interface between the frame and infill. A similar modelling technique can be found in Cavaleri and Di Trapani (2015) and Doudoumis (2007). Friction phenomena are neglected in the numerical investigation due to the complexity of defining a reliable value of the friction coefficient (according to the Coulomb law, commonly used). Moreover, considering that friction progressively varies in the case of cyclic loading (Cavaleri and Di Trapani 2015) not influencing the overall system behaviour (Fiore et al. 2012), it is reasonable to neglect this contribution in a preliminary assessment of the detachment mechanism. A qualitative illustration of the bi-dimensional model is presented in Fig. 10. Hence, detachment values are evaluated for each case study and the shape functions of the infill-frame displacement incompatibility at the fixed interstorey drifts are finally obtained. Operatively, the first step consists in assessing the horizontal and vertical displacement of the structural frame and infill panel edges. Displacement incompatibility is computed as the difference between the displacement of the frame and the displacement of the infill panel along the vertical and horizontal directions (i.e., Horizontal Displacement Incompatibilities, HDI; Vertical Displacement Incompatibilities, VDI). Thus, shape

Results and discussion
In Table 6 the details of the specimens taken into account for the parametric analysis as well as the mechanical properties of the equivalent strut models are reported, while Fig. 11 shows the results for all the infilled frames in terms of seismic performance (force-displacement capacity curves).
Results show that the axial load on columns strongly influences the strength and stiffness of the structure under seismic loads. Furthermore, the strength and stiffness of the structures also increase in the case of strong infill panels, as well as considering a longer beam span length.
The shape functions of the displacement incompatibility between the infill wall and frame are reported in Fig. 12 for the beam span length l = 3 m and Fig. 13 for the beam span length l = 5 m . Maximum detachment values in the horizontal and vertical directions are provided in Table 7, together with the observed contact length values, expressed in terms of beam/column length percentage.
As expected, displacement incompatibility increases considering higher drift values for all configurations. It can be noticed that when the beam span length increases, the seismic horizontal displacement incompatibility increases while the vertical displacement incompatibility decreases. The main differences can be found for the strong infill when compared to weak and medium infills. It is worth highlighting that, as shown in previous Table 2, strong infill walls are characterized by a higher ratio between stiffness in vertical and horizontal directions when compared to weak and medium infills. Considering l = 3 m , the strong infill leads to higher values of detachment in the horizontal direction (columninfill wall) and smaller values in the vertical direction (beam-infill wall) than weak and medium infills. The maximum displacement incompatibility values are recorded for the top corner of the frame. The detachment increases when considering higher values of axial load and stronger infill: strong infill with N = 420 kN leads to the maximum detachments Δ s = 5.2 mm for interstorey drift = 0.4% and Δ s = 11.3 mm for = 0.9% . Generally, weak and medium infills show similar behaviour, while strong infill leads to higher values of displacement incompatibility. Moreover, the contact length has clearly an opposite trend than the detachment (i.e., it decreases when considering a stronger infill or lower axial load on columns). This behaviour is in agreement with the relationship proposed by Stafford Smith (1967). On the other hand, no significative differences in terms of contact length have been observed considering higher values of interstorey drift. Generally, the higher the interstorey drift ratio, the higher the contact length values, especially considering the beam and left column. The higher contact length values are observed for the right column-infill in the case of weak/medium infills and high axial load (N = 420kN), %L contact = 36%. For the beam-infill interface, the contact length range values are 15-31%.
Considering L = 5 m , similar considerations can be made. Displacement incompatibility increases considering a higher value of beam span length. The maximum value of detachment is recorded at the top corner in the horizontal direction Δ s = 6.9 mm and Δ s = 14.9 mm for interstorey drift = 0.4% and = 0.9% , respectively. Displacement incompatibility increases when considering a stronger infill panel even in this configuration. However, when considering higher axial load values, detachment decreases in most of the configurations. Main differences are noticed for the vertical detachment, especially Table 6 Characteristics of the infilled frame specimens for parametric analysis Capacity curves of the infilled frame structure for each configuration Fig. 12 Shape functions of Horizontal and Vertical Displacement Incompatibilities (HDI, VDI) between masonry infill wall and frame structure for 3 m beam span length for = 0.9% since the plastic hinges sequence of the structural frame significantly influences the displacement incompatibility between beam and infill wall. It is highlighted that in this study a pre-seismic-code frame structure is considered and that the failure mode is a mixed sidesway mechanism (i.e., lack of capacity design principles). Increasing the axial load on columns leads to a higher capacity of the panel zone, so when N = 420 kN plastic hinge occurs on the beam for a positive bending moment, while when N = 120 kN shear failure of the panel zone occurs. Furthermore, the residual strength of the infill wall may modify the bending moments in the beam leading to different shape behaviour. For these reasons, it is not easy to identify a common behaviour for VDI. However, it is observed that vertical displacement incompatibility values are always smaller than horizontal ones and thus of lesser interest. It is interesting to notice that the configuration with strong infill and N = 120 kN shows shape functions significantly different from the others, since both   the joint panels achieve the failure and the residual strength of infill modifies the bending moment in the beam. Concerning the contact length, similarly to the L = 3 m configuration, it decreases in the case of a stronger/stiffer infill or lower axial load on columns. Generally, the L = 5 m configurations lead to lower values of contact length with respect to the L = 3 m configurations. Even in this case, for the infill-column contact length, no significative differences are observed considering higher values of interstory drift ratio (e.g., %L contact = 18-27% for right column-infill). On the other hand, the interstory drift value strongly affects the contact/detachment between infill and beam, leading to lower contact length values if higher drift values are considered. For the beam-infill interface, lower contact length values are obtained when considering higher drift values (ϑ = 0.9%), lower axial load, and strong infill, %L contact = 4%; once again, this result is associated to shear failure of the joint panels, as previously discussed.

Design of structural and energy retrofitting construction details
Improving the overall performance of the building envelope (facade) represents a crucial aspect to enhance seismic safety and energy efficiency of existing buildings. However, generally, local displacement incompatibility issues are not directly considered because of the lack of a standardized and easy-to-apply procedure to assess local infill-frame interaction. The proposed concept of shape functions, as well as the modelling approach adopted in this paper, may be used to design construction details for both seismic and energy retrofit solutions in view of an integrated intervention. A possible operative flowchart for the design of construction details of both seismic and energy retrofit solutions for an integrated requalification of RC buildings is reported in Fig. 14.
The methodology presented in this paper can be used for any infilled frame configuration in order to derive shape functions of displacement incompatibility without performing Fig. 14 Conceptually flow-chart for the design of integrated seismic-energy retrofit interventions using the proposed concept of displacement incompatibility shape functions a more complex nonlinear FEM (micro-model) analysis. Alternatively, as a preliminary step, the most similar configuration among those reported in this work can be adopted and built on. By assessing the expected values and shape of detachment at the infill-frame interface, appropriate structural details can be designed for either strengthening or decoupling approaches and techniques. When considering decupling techniques, shape functions can be used to identify the best location for shear anchors (i.e., contact zones) and to design their length to guarantee their effectiveness. Clearly, in the case of decoupling techniques, shear anchors should be designed and detailed to only prevent out-of-plane failure mechanisms of the panel, thus in-plane load transfer between the infill and frame should be avoided. Milanesi et al. (2018b) pointed out that the use of steel ribbed rebars as infill-frame connections to increase the out-of-plane capacity of the panel could not allow for in-plane differential movements between the infill and the surrounding frames, consequently leading to damage to the blocks in which the rebars are located. Therefore, it is suggested to use anchors/rebars not fully embedded in the masonry units to avoid possible local interaction. In that direction, some possible solutions to avoid stress concentrations in the blocks may include the use of shear anchors placed in the columns and in a side opening of the block filled with elastomer to allow for in-plane movements, as experimentally investigated by Marinković and Butenweg (2022) (Fig. 15a). Moreover, other decoupling solutions for infilled frame structures in which the OOP capacity is provided by specific infill-frame connection, based on ad-hoc shear keys, have been presented (e.g., Preti et al. 2015;Morandi et al. 2018b). As an example, in the decoupling solution investigated by Morandi et al. (2018b), steel shear keys connected to the columns are provided and the masonry units at the infill-column interface are C-shaped to accommodate the shear key (Fig. 15b). In view of the implementation of this type of construction detail, the proposed concept of shape functions of displacement incompatibility may provide useful information to adequately design the geometrical dimension of the shear keys and the C-shaped blocks.
Otherwise, considering strengthening solutions, steel dowels are typically adopted to improve the shear resistance at the interface and avoid detachments. In this case, shape functions allow to identify the location and value of the maximum detachment at the interface and therefore the expected most stressed dowel. A step-by-step design/worked Fig. 15 Examples of construction details to avoid out-of-plane failure in the case of decoupling retrofit techniques: a shear anchors (modified after Marinković and Butenweg 2022); b shear keys with C-shaped masonry blocks (modified after Morandi et al. 2018b) example of shear dowels in strengthening retrofit techniques according to the herein proposed displacement incompatibility shape function methodology is provided in the "Appendix". Finally, it is recommended to adequately design the connection between bare structure and energy efficiency rehabilitation solutions, as in the case of external thermal insulation systems, to allow for an adequate relative displacement movement and avoid possible local stress concentration and failure of the new high-performance envelope panel. In such a case, the required displacement capacity can be evaluated by using the proposed concept of shape functions. Specifically, a practical and efficient solution could rely upon mechanically fixed connection systems (rather than glued-on solutions), with a specifically designed displacement capacity for the critical insulation panels (i.e., those panel units located at the infill-frame interface).

Conclusions
In this paper, a methodology to assess the seismic displacement incompatibility between RC frame and masonry infill panel based on the concept of shape functions has been presented. Parametric analyses of different infilled frame configurations have been performed to investigate the key parameters affecting displacement incompatibility and provide a preliminary range of detachment values as well as identify the maximum detachment location along the surrounding frame.
Firstly, a comparison between the seismic response of infilled frames obtained by adopting different macro-modelling techniques has been carried out. This study highlights that all the considered models provide very similar global non-linear force-displacement behaviour and are in good agreement with the experimental response available in the literature. Then, a more refined model where the infill wall is idealized using shell elements has been prepared and displacement-incompatibilities shape functions between the infill wall and frame at fixed interstorey drift have been developed. Nonlinear behaviour of shell elements has been obtained by iteratively scaling their thickness, hence equating the secant stiffness of the continuous bi-dimensional shell-element model and that of the equivalent strut model. This simplified approach allows to capture and assess the local displacement incompatibility issues without performing a more complex nonlinear micro-model. Results demonstrate that the aspect ratio of infilled frame structures strongly affects the infill-frame detachment. Moreover, it is highlighted that horizontal displacement incompatibility (i.e., infill-to-columns) is generally more severe than vertical displacement incompatibility (i.e., infill-to-beam).
The proposed concept of shape functions can be used to design structural details, such as shear keys and/or steel dowels, in view of either decoupling or strengthening retrofit strategies, in order to guarantee their effectiveness and avoid possible local failure. On the other hand, infill-frame detachment can lead to damage to energy retrofit solutions such as external thermal insulation systems, so also these techniques should be designed to provide an appropriate displacement capacity.
In this paper, structural configurations representative of existing RC frame buildings (i.e., designed to resist vertical loads only) have been selected, however, a wider class of infilled frame structures can be used to improve the study, including newly designed RC infilled frame structures. Furthermore, an extensive experimental program as well as more refined numerical investigations, based on micro-modelling approaches, would be useful to gain a better understanding of infill-frame displacement incompatibility. Finally, the proposed study could be extended to different infill typologies. Further advancements would include the development of tables and/or analytical formulations to simply and quickly derive shape functions starting from the geometry and material properties of the infilled frame structure. Finally, it is worth noting that when beam plastic hinges are developed, the "beam elongation" effects may increase the infill-frame horizontal detachment. Future numerical/experimental investigations are needed to achieve a comprehensive understanding of this topic.

Appendix: worked example
This appendix illustrates a worked example of the possible application of the herein proposed displacement incompatibility shape functions for the design of shear dowels in strengthening retrofit techniques. This typology of retrofit techniques has been widely investigated in the past (e.g., Koutromanos et al. 2013;Facconi and Minelli 2020), as well as the use of shear dowels for the connection between infill and surrounding frames, especially for frames with RC infills (Sugano 1996;Altin et al. 2008;Moretti et al. 2014). For the latter, some provisions on the embedment length, minimum spacing of dowel bars, and minimum edge distance have been also included in international codes and guidelines (e.g., ASCE/SEI 41-17 2017; EOTA 2019).
In this application, the strengthening solution reported in Facconi and Minelli (2020) is considered, based on the use of thin glass fiber mesh reinforced mortar overlays for strengthening RC weak infilled frames. In this retrofit technique, the relative shear sliding between the masonry panel and the frame is reduced using steel dowel connections (Fig. 16). According to the authors, this investigated retrofitting solution may represent a feasible alternative for low-rise RC frame buildings located in low-to-moderate seismicity areas.
The dowels consist of a plain round steel bar (diameter d dow = 16 mm ) with nominal yield strength f y = 235 MPa . In order to implement the solution as a retrofit technique suitable for practical applications, each dowel is hammered into a pre-drilled hole located in the RC frame and is located in a "pocket" into the masonry brick; the pockets are thus filled with a thixotropic mortar. More details are available in Facconi and Minelli (2020). Conceptually, this solution could be applied to any typology of infills; however, due to the need to realize pockets into the bricks located at the infillframe interface for the steel dowels, masonry infills with hollowed brick may be deemed as more suitable for its practical implementation. Firstly, the model proposed by Gelfi et al. (2002) is adopted to design steel dowels length. This model allows to prevent failure mechanisms governed by crushing of both concrete and mortar. The whole formulations are reported in Table 8.
A design example is reported for all the infilled frames analyzed in the previous sections, considering a thixotropic mortar with mean cylindrical compressive strength of f c,thix = 60 MPa . The obtained minimum dowel length L dow,min according to Gelfi et al. (2002) is reported in Table 9.
The proposed concept of displacement incompatibility shape function can be thus adopted to prevent dowel failure due to infill-frame detachment. Specifically, detachment values evaluated considering the as-built configuration can be assumed, in a conservative way, as displacement/strain demand for the dowels. The minimum lengths of the dowels placed in beams and columns for all the analyzed configurations are evaluated (Table 10) considering an interstorey drift of 0.4% and an ultimate dowel strain of 6%. It is worth mentioning that the drift value to be considered in the analyses should be a design choice, in line with the philosophy of performance-based design/retrofit. Since strengthening retrofit techniques for infilled frames cannot typically provide an enhancement of the ductility capacity of the structure (unless a selective weakening intervention is applied, in order to develop a low-damage infill system, Tasligedik and Pampanin 2017), these retrofit interventions should rather be designed in order to reduce the displacement for the design earthquake by increasing the strength/stiffness capacity of the structure. Therefore, in view of a retrofit strategy aiming to prevent the infill wall failure by limiting the displacement demand, a drift value equal to 0.4% is herein selected for the design of the shear dowels for illustrative purposes. Clearly, other design choices can be made according to the specific needs of the project.  Gelfi et al. (2002) L dow , Dowel length; L s,c , Embedment length in concrete; L s,m , Embedment length in mortar; d dow , Dowel diameter; f c,con(thix) , Mean cylindrical compressive strength of concrete (thixotropic mortar); f hc,con , Crushing strength of concrete around the dowel Results highlight that the proposed concept of shape functions would suggest increasing the length of the dowels located at the infill-to-column interface for all the 5 m-bay frame configurations. On the other hand, for the dowels located at the infill-to-beam interface, the expected (vertical) detachment value is less severe, and the dowel length designed according to Gelfi et al. (2002) already allows to protect the dowel failure. It is worth noting that in the retrofitted configuration, a less severe displacement incompatibility than in the as-build configuration is expected. Therefore, the proposed displacement-compatible design check should be considered as a conservative hypothesis in a preliminary design. Further research efforts are needed to develop operative design guidelines including suggestions for amended construction details based on the concept of displacement incompatibility shape functions. comments and internal revision of an earlier version of this work. The authors also acknowledge the financial support of the Italian Ministry of University and Research (MUR) for funding the Doctoral Scholarship of Livio Pedone.
Funding Open access funding provided by Università degli Studi di Roma La Sapienza within the CRUI-CARE Agreement. The authors acknowledge the financial support of the Italian Ministry of University and Research (MUR) for funding the Doctoral Scholarship of Livio Pedone.