1 Introduction

The conventional design of buildings in seismic zones entails energy dissipation in the structural elements. The capacity design, adopted in the main national and international design standards for seismic actions, as the Eurocode 8-1 (CEN 2005a), ensures that plastic hinges form in specific points of the structure to foster a ductile collapse mechanism. For instance, in moment resisting frames (MRF), attention is given to forming plastic hinges at the beam ends to protect columns and avoid soft storey mechanisms. In addition, different strategies were proposed to ensure plasticisation of the MRF beams, e.g. weakening the beams at their ends (Plumier 1990; Iwankiw and Carter 1996). As an alternative, studies have been carried out to concentrate the dissipation in partial-strength joints (Castro et al. 2005; Chen et al. 1996; Pucinotti et al. 2015; Tondini et al. 2018). Although these strategies allow for designing structures capable of dissipating energy under seismic loading, they do not guarantee ease of repair after an earthquake. This results in economic losses due to the long-term downtime of the structure. Moreover, buildings designed according to these approaches may undergo significant damage, whose repair work is often not feasible or too expensive. Therefore, reducing damage to structural and non-structural elements after a disaster is fundamental for costs and functionality. Thus, partial-strength joint solutions with low damage were proposed (Francavilla et al. 2020; Latour et al. 2011). Slit dampers applied to MRF (Oh et al. 2009) and eccentrically braced frames (EBF) showed beneficial hysteretic behaviour (Chan and Albermani 2008). At the same time, focus has been given to solutions that can limit damage and guarantee replaceability after seismic events to enhance structural resilience (Wang et al. 2020).

In this context, this work extensively describes the experimental campaign carried out at the University of Trento on a steel frame, whose seismic lateral-resisting system is composed by a “Dissipative Replaceable Link Frame” component, which consists of two rigid columns connected by locally weakened beams. The objective is to provide experimental evidence of large energy dissipation capability and ease of repair of steel structures equipped with such dissipative components. The DRLF is a structural solution that can be referred to as a Linked Column Frame (LCF) and it was called as such in the Research Fund for Coal and Steel DISSIPABLE project (Kanyilmaz et al. 2022). Several authors have extensively studied LCF systems over the last few decades and they involve connecting steel columns with replaceable links that act as fuses (Dusicka and Iwai 2007; Fortney et al. 2007; Lopes et al. 2012). These studies demonstrated the effectiveness of such solutions in dissipating energy and protecting the main structure from damage. In analogy with EBF, LCF systems dissipate energy in the beam links by shear. Conversely, in the present study, energy dissipation occurs in reduced beam sections (RBS) by bending moment, for which geometric rules that avoid shear-bending interaction are employed (Vayas 2017). In addition, Montuori et al. (2023) proposed an effective design procedure based on the Theory of Plastic Mechanism Control (Mazzolani and Piluso 1996). Malakoutian et al. (2013) conducted a comprehensive study on the seismic behaviour of the LCF system coupled with a moment-resisting frame. The analyses proved that the devices have excellent energy dissipation capabilities. However, the study also showed that the structural frame cannot be rapidly repaired for high seismic hazard levels. Therefore, while the devices effectively prevent structural collapse, they do not guarantee efficient repairability of the structure. In this work, gravity beams were hinged to the columns to enhance the structural resilience and post-earthquake recovery, preventing their participation in the dissipative behaviour and possible yielding. Although theoretical models and simulations provide valuable insights, only a limited number of experimental tests have been conducted on full-scale frames (Lopes et al. 2014). Additionally, the actual replaceability of beam links within these structures has not yet been investigated. On these premises, this paper reports the results of an experimental campaign on a full-scale frame, which also demonstrates the actual repairability of the structure. Another significant aspect of the project is the use of the Hybrid Simulation testing technique, which allows the test of large-scale structures by physically reproducing only a meaningful portion in the laboratory whilst numerically simulating the remainder. Hybrid Simulation (HS) was proposed in the early '70 (Nakashima 2020) and has been successfully applied and validated since then in the seismic engineering field (Abbiati et al. 2015; Del Carpio Ramos et al. 2016; Di Benedetto et al. 2020). Nevertheless, its use has been extended in recent developments to other fields, e.g., fire engineering (Sauca et al. 2018; Abbiati et al. 2020). As a result, only the structure's ground floor was physically tested in the laboratory, whilst the remainder was numerically simulated. Bidimensional frames, representative of a 3D case study, were tested under different seismic intensity levels: Damage Limitation (DL), Significant Damage (SD) and Near Collapse (NC). The aim was to demonstrate the elastic behaviour of the structure expected for the DL limit state, as well as the stable hysteretic behaviour and the replaceability of the dissipative components at SD limit state, at which they should protect the irreplaceable parts from yielding.

The motivation for such tests lies in the DISSIPABLE project that was funded to provide experimental evidence of large energy dissipation capability and ease of repair of steel structures equipped with innovative dissipative components by means of large-scale demonstrative tests. Three different components were examined, namely the “Dissipative Replaceable Braced Connections” (DRBrC), the “Dissipative Replaceable Link Frame” (DRLF) and the “Dissipative Replaceable Beam Splices” (DRBeS), which were named after the INERD, FUSEIS type 1 and FUSEIS type 2 connections (Valente et al. 2016), respectively. They were the result of a design improvement of components developed in previous RFCS projects, i.e., INERD project (Plumier et al. 2006) and FUSEIS project (Castiglioni et al. 2013). The monotonic and the cyclic behaviour of these devices were investigated at the component level; however, experimental evidence of the behaviour of structures equipped with such components under real earthquake conditions still needs to be provided. In Andreotti et al. (2023a, b), the outcomes of the tests on full-scale specimens equipped with DRBeS and DRBrC components are respectively presented, whilst this work focuses on the experimental campaign carried out on a steel frame equipped with the DRLF component.

The paper is organised as follows: a brief description of the dissipative component under investigation is given in Sect. 2; in Sect. 3 the numerical models of both the component and the building prototypes are reported, whilst the description of the ground motion selection is presented in Sect. 4. The description of the hybrid simulation procedure is presented in Sect. 5; whereas in Sect. 6 the experimental test results are described. The calibration of the numerical model is reported in Sect. 7, while Sect. 8 draws some conclusive remarks.

2 Description of the dissipative replaceable link frame (DRLF) system

The dissipative replaceable link frame is a lateral-resisting system conceived to be used in the external frames of a steel or a steel–concrete building. It comprises two columns connected by multiple beams, making the system a Vierendeel beam (Fig. 1). The beam links work mainly in bending or shear, depending on the span-to-height ratio, whilst the columns are subjected to strong axial force components (Vayas 2017). The beam is typically weakened at the ends to force the formation of the plastic hinges at those locations. For this system, replaceability is guaranteed by means of bolted connections between the devices and the columns. Moreover, the beam links are not part of the gravity load-carrying system. In the tests performed at the University of Trento, IPE sections were employed as beam links and weakened by reducing the gross section in accordance with Eurocode 8-3 (CEN 2005b).

Fig. 1
figure 1

DRLF system configuration: a link beams and b mechanical behaviour (Vayas 2017)

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2.1 Numerical modelling of the DRLF component

In the first stages of the work, a preliminary modelling of the DRLF component was performed. Then, after the tests, the numerical model was calibrated, as shown in Sect. 7. In greater detail, the DRLF component was numerically modelled in the finite element (FE) software OpenSees (McKenna et al. 2006). The model is shown in Fig. 2, where the nonlinear part representing the reduced beam sections (RBSs) was modelled with the "TwoNodeLink" element, whilst the remaining parts were modelled through elastic beam elements with the properties of the gross section. To avoid additional flexibility, the model included a rigid link of length equal to half of the column section height at both ends of the beam. The rigid link reproduced the moment-resisting joint between the beam and the column. Moreover, this way of modelling allowed to model the DRLF beam link with the actual physical length (Vayas 2017).

Fig. 2
figure 2

Beam link numerical model

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The nonlinear rotational part of the DRLF component was modelled with the Bouc–Wen (Wen 1976) model. The parameters were determined by means of the tool Multical (Chisari et al. 2017), by minimising the difference between a 3D numerical model developed in ABAQUS (Smith 2009) and the beam model developed in OpenSees in terms of energy dissipation and monotonic envelope, as shown in Fig. 3. The dissipated energy differences are 0.9%, 0.4% and 0.1% for the IPE160, IPE140 and IPE100 profiles, respectively. To guarantee a uniform dissipative behaviour along the structure height, three different sections were employed in the prototype model of the building, namely IPE 160, IPE 140 and IPE 100.

Fig. 3
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RBSs numerical modelling

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3 Design of the building prototype structure

The prototype building under investigation comprised two spans in the transversal X-direction, three spans in the longitudinal Y-direction and six stories along the height (Z-direction). Due to laboratory constraints, the span length was 4.275 m, while the inter-storey height was equal to 3.5 m. In the Y-direction, the lateral-resisting system consisted of two external braced frames equipped with Dissipative Replaceable Bracing Connections. In the X-direction, two parallel DRLF systems were employed and coupled at the top floor by bracing elements (see Fig. 4a) to increase the building stiffness and comply with the deformability limits. As depicted in Fig. 4a, fixed column bases were considered in the direction of the DRLF system, whilst the columns were pinned in the direction of the DRBrC system. Table 1 collects the dimensions of the building dissipative components and structural members along the DRBrC system direction. The static design was performed according to Eurocode 3-1-1 (CEN 2005c), whilst for the seismic design Eurocode 8-1 (CEN 2005a) and Eurocode 8-3 (CEN 2005b) prescriptions were adopted, as suggested by Pinkawa et al. (2017). The structural seismic design was performed by means of linear dynamic analysis of the 3D building developed in SAP2000 (CSI 2019). Member profiles and component dimensions are depicted in Fig. 4b. The deck was composed of a 150 mm high concrete slab with a 55 mm high steel sheeting. The steel grade for all the non-dissipative members was S355, whilst for the dissipative components, i.e., IPE profiles that constitute the beam links, S235 steel grade was employed.

Fig. 4
figure 4

Case study building: a SAP2000 model; b 2D model in the direction of the DRLF system. Dimensions in m

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Table 1 DRBrC component dimensions in millimetres and element profiles along longitudinal direction
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Moreover, since a nonlinear model of the numerical substructure was needed for the experimental tests, the building was also modelled in OpenSees, including the nonlinearities of the DRLF system, as described in Sect. 2. In contrast, all other structural members were modelled with elastic beam elements. The comparison between the SAP2000 and the OpenSees models in terms of modal properties is reported in Table 2, where a very good agreement is highlighted.

Table 2 Modal properties
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Pushover and nonlinear time-history analyses were performed to investigate the inelastic behaviour. In this respect, accelerograms at Significant Damage (SD) and Near Collapse (NC) limit states were employed. Figure 5 shows the hysteretic behaviour of RBSs installed in the building subjected to one SD accelerogram. For brevity, only the moment-rotation diagrams at the floor level of the first frame are presented here. The structure experienced large and uniformly distributed hysteretic behaviour, except for the 6th floor, where the components remained almost elastic due to the braces, which provided additional stiffness (see Fig. 5).

Fig. 5
figure 5

Hysteretic behaviour of RBSs at SD limit state

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Since the experimental tests were performed on 2D frames, reducing the 3D numerical model to a 2D frame and, successively, to a 2D substructured frame that comprised only the numerical substructure was necessary. Hence, only one frame comprising the DRLF system was modelled, and, given the building layout, half of the mass was considered. Moreover, the effect of one internal gravity frame was accounted for by means of a lumped column (Flores et al. 2016), whose modulus of inertia was calculated, floor by floor, as the sum of the modulus of inertia of all the columns belonging to the gravity frame. As a result, the analysed frame is depicted in Fig. 4b.

Then, a further substructuring step was performed to consider the actual laboratory configuration. A maximum of two actuators can be simultaneously controlled. Moreover, only the horizontal degree of freedom can be effectively controlled in the laboratory. On these premises, preliminary analyses were conducted to obtain consistent results between the 2D frame and the 2D substructured frame (SF) shown in Fig. 6, simulating the actuators' presence by continuity constraints between the numerical and experimental substructures. In this respect, the optimal position of the second actuator was assessed by changing the location of the hinge along the column height of the second floor. The mid-height position, which represents the contraflexure point with good approximation, was the location that minimised the difference between the 2D monolithic and the 2D substructure frames. Moreover, the P-Δ effects were also investigated and relevant comparisons showed that discrepancies were negligible. In greater detail, Fig. 7 compares the top floor displacement during the strong motion phase at the near collapse limit state using both linear and P-Δ formulations. The results showed that considering second-order effects did not improve the accuracy of the results. Thus, it was decided not to apply the axial loads to the columns and the vertical loads to the beams in the physical substructure.

Fig. 6
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2D Substructured Frame

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Fig. 7
figure 7

Top floor displacement at the near collapse limit state

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The comparison between the modal properties of the 3D building, the 2D frame and the 2D substructured frame in the direction of the DRLF are reported in Tables 3, 4, 5. In Tables 4 and 5, the modal assurance criterion (MAC) (Pastor et al. 2012) was computed to investigate possible discrepancies between the mode shapes. Very good agreement, both in terms of periods and mode shapes, can be observed.

Table 3 Comparison of the modal properties between 3D building, 2D frame and 2D substructure frame
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Table 4 MAC matrix between 3D building and 2D frame
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Table 5 MAC matrix between 2D frame and 2D substructured frame
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As shown in Fig. 8, the pushover analyses highlight that the nonlinear behaviour of the 2D frame and the 2D substructured frame agree well after the yielding of the devices, whilst the initial stiffness turns out to be different due to the discontinuity and the vertical restraint on the numerical substructure.

Fig. 8
figure 8

Pushover comparison

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Given the large number of degrees of freedom (DoFs), to reduce the computational burden during the hybrid simulations, the DRLF model was reduced to a simplified one. As shown in Fig. 9, condensation of the DoFs was performed under the assumption of the structure's shear-type deformation, leading to considering only seven horizontal degrees of freedom. These DoFs represented the displacements at each floor level plus the one at the substructuring level. Lumped masses, connected by means of nonlinear shear springs, were located on each DoF. To calibrate the nonlinear spring parameters, a displacement control analysis was performed by imposing a cyclic displacement at the top floor of the reference model. The calibration results are reported in Fig. 10, which highlights good agreement between the reference model and the model used in the experimental tests.

Fig. 9
figure 9

Model reduction. Dimensions in m

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Fig. 10
figure 10

Comparison between the reference and reduced experimental models regarding inelastic behaviour in the DRLF system

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4 Ground motion selection

The tests were conducted at three limit states, namely Damage Limitation (DL), Significant Damage (SD) and Near Collapse (NC) limit states. For each of them, an accelerogram was selected within a set of seven natural ground motions. The selection criteria were (i) the spectral compatibility in accordance with Eurocode 8-1 (CEN 2005a); (ii) the minimisation of the errors between the 2D monolithic and the 2D substructured frame; (iii) the uniformity of inelastic behaviour in the frame at the SD and NC limit states. As a result, Fig. 11 and Table 6 report the selected accelerograms and the related spectra, where the scaling factor was adopted to ensure spectral compatibility.

Fig. 11
figure 11

Selected accelerograms and spectral compatibility

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Table 6 Accelerograms
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As shown in Fig. 11c, the selected ground motion for NC limit state was a pulse-like record and did not respect the imposed limits within the range of periods indicated by the draft of the new Eurocode 8-1C8 (CEN 2018). However, Eurocode 8-1C8 allows for the use of such accelerograms, hence it was employed to obtain structural damage substantially different from the one at the SD limit state.

Furthermore, for each selected accelerogram, the difference between the response of the monolithic 2D frame and the 2D substructured frame was assessed by computing the following error measures:

  • Percentage error on the total hysteretic energy dissipated by the structure.

  • Statistical indicators on bending moment history at the RBS.

The aforementioned error indicators are computed as indicated in Eqs. (1), (2) and (3).

$${\text{Energy}}\,{\text{Error}} = \frac{{\left| {E_{i} - E_{j} } \right|}}{{E_{j} }}$$
(1)
$${\text{NRMSE}} = \frac{{{{\left\| {{\text{x}}_{{\text{i}}} - {\text{x}}_{{{\text{j}}2}} } \right\|} \mathord{\left/ {\vphantom {{\left\| {{\text{x}}_{{\text{i}}} - {\text{x}}_{{{\text{j}}2}} } \right\|} {\sqrt {\text{N}} }}} \right. \kern-\nulldelimiterspace} {\sqrt {\text{N}} }}}}{{{\text{x}}_{{{\text{j}},\max }} - {\text{x}}_{{{\text{j}},\min }} }}$$
(2)
$${\text{NENERR}} = \left| {\frac{{\left\| {{\text{x}}_{{\text{i}}} } \right\|_{2} - \left\| {{\text{x}}_{{\text{j}}} } \right\|_{2} }}{{\left\| {{\text{x}}_{{\text{j}}} } \right\|_{2} }}} \right|$$
(3)

The energy error in Eq. (1) is the difference between two scalar quantities, i.e., the hysteretic energy dissipated by the DRLF components for the two models, i.e. the substructured 2D frame i and monolithic 2D frame j, in which the j dataset is taken as reference. The parameters reported in Eqs. (2) and (3), compare the bending moment evolution in the DRLF components for the two models. NRMSE is sensitive to frequency while NENERR to amplitude (Abbiati 2014). The mean error among all the components for the selected accelerograms are listed in Table 7, whose magnitude is reasonably low.

Table 7 Error between monolithic and substructured frame responses. Values in %
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The structural performance of the frame was evaluated by checking the rotations of RBSs. At the DL limit state all the RBSs attained a maximum rotation lower than the yield rotation, as shown in Fig. 12a. Specifically, the yield rotation was computed using the mean value of the yield strength of an S235 fy,m = 281.10 MPa, with a coefficient of variation equal to 0.1 (Hess et al. 2002), according to D.M. 9/1/1996 (Ministero dei Lavori Pubblici 1996). The following values were obtained: 2.15 mrad, 3.19 mrad, 4.92 mrad, respectively for IPE160, IPE 140 and IPE100. Moreover, as illustrated in Fig. 12b and c, a uniform dissipative behaviour was achieved for both the SD and the NC limit states. It should also be highlighted that the higher stiffness given by the bracing system caused smaller rotations of the RBSs located at the top floor.

Fig. 12
figure 12

Maximum RBSs rotation for the selected accelerograms at a DL, b SD and c NC limit state. Dimensions in m

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5 Hybrid simulation

The tests were conducted by means of the HS technique, which allows testing a large structure by physically building only part of the structure in the laboratory, which is the physical substructure (PS), whilst the remaining part is numerically simulated, i.e. the numerical substructure (NS). The procedure employed in the experimental campaign is shown in Fig. 13, and it relies on a partitioned algorithm, the main features of which will be described below. In particular, at each time step, the reaction forces of the structure are measured and sent back to the laboratory PC, where the partitioned algorithm computes the displacements to be imposed on the structure at the next time step by solving the equations of motion which combine the two substructures. Pseudodynamic tests were performed, and the mass contribution of the physical and numerical substructure was only numerically simulated. Moreover, the damping contribution was also numerically accounted for the numerical subdomain, with a damping ratio of 2%, whilst it was not considered in the physical subdomain since is inherent in the nonlinear phenomena. The time-scale factor λ given in Eq. (4) expanded the time scale to avoid considering the inertia forces on the physical substructure. The factor λ is given as the ratio between the time integration step \({\Delta }t_{c}\) used to solve the equations of motion and the wall clock time \({\Delta }t\) corresponding to the time interval between two consecutive commands sent by the controller to the actuators (Bursi and Wagg 2009).

$$\lambda = \Delta t_{c} /\Delta t$$
(4)
Fig. 13
figure 13

Conceptual scheme of hybrid simulation

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The value of λ was equal to 50, for the test at DL limit state, and to 100, for both SD and NC tests. A larger λ was employed for the tests where large inelastic behaviour was expected.

Regarding the partitioned algorithm, the G-α algorithm described by Abbiati et al. (2019) was implemented to solve the equations of motion. Briefly, the algorithm rewrites the system of equations of motion in the state space form, which read

$$M\dot{Y} + R\left( Y \right) = F\left( t \right)$$
(5)

where

$$ Y = \left( {\begin{array}{*{20}c} u \\ v \\ s \\ \end{array} } \right)\! ,\ M = \left( {\begin{array}{*{20}c} I & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I \\ \end{array} } \right)\! ,\ R = \left( {\begin{array}{*{20}c} { - v} \\ {r\left( {u,v,s} \right)} \\ {g\left( {u,v,s} \right)} \\ \end{array} } \right)\! ,\ F = \left( {\begin{array}{*{20}c} 0 \\ {f\left( t \right)} \\ 0 \\ \end{array} } \right) $$
(6)

u, v and r are the displacement, velocity and restoring force vectors, \(s\) is the additional state vector used to model nonlinearities, while I and m are the identity and mass matrices. The term \(g\left( {u,v,s} \right)\) is the nonlinear function that models the evolution of the additional state vector. As described in Sect. 2, the Bouc–Wen (Bouc 1967; Wen 1976) model was implemented to simulate the evolution of the RBSs nonlinear behaviour (Bonelli and Bursi 2005; Li et al. 2012; Abbiati et al. 2015). In the model, the restoring force reads:

$$r\left( t \right) = \alpha ku\left( t \right) + \left( {1 - \alpha } \right)kz\left( t \right)$$
(7)

where \(\alpha\) is the ratio between the post-yielding and initial elastic stiffness, \(k\) is the initial elastic stiffness and z(t) is the hysteretic displacement whose constitutive law is given as the solution of the following nonlinear differential equation.

$$\dot{z} = \frac{{\left[ {A \dot{u} - v\left( {\beta \left| {\dot{u}} \right|\left| z \right|^{n - 1} z + \gamma \dot{u} \left| z \right|^{n} } \right)} \right]}}{\eta }$$
(8)

In Eq. (8), A, β, γ and n are parameters that control the shape of the hysteretic loops whilst η and ν govern the main stiffness and strength degradation phenomena, respectively. In the context of the algorithm implementation, the hysteretic displacement z(t) was selected as the additional state vector whilst the differential equation, i.e. Eq. (8), represents the nonlinear function \(g\left( {\dot{u},z} \right)\) which was implemented in the laboratory PC.

As mentioned, only a relevant part of the structure was physically built in the laboratory, whilst the remainder was numerically modelled. The physical and numerical substructure response, PS and NS, respectively, were coupled with the partitioned algorithm, which employs Lagrange multipliers. From a physical point of view, the latter represents the interface forces exchanged between NS and PS. Moreover, the compatibility is imposed on the velocity of the interface degrees of freedom by means of the following equation,

$$G^{NS} \dot{Y}_{n + 1}^{NS} + G^{PS} \dot{Y}_{n + 1}^{PS} = 0$$
(9)

where \(G^{NS}\) and \(G^{PS}\) are the Boolean matrices that localise the interface degrees of freedom on the velocities, \(\dot{Y}_{n + 1}^{NS}\) and \(\dot{Y}_{n + 1}^{PS}\) and the state vector first derivative with respect to time. The NS and PS apexes indicate the numerical and physical substructure, respectively.

5.1 HS configuration and instrumentation

As mentioned in Sect. 3, the selected substructured configuration involved the use of two actuators in the test set-up. The physical substructure of the frame, depicted in Fig. 14, was composed of five columns, four of which were part of the DRLF systems that carry the horizontal loads. To impose the same displacement at the top of each column, beams with high axial stiffness were placed at the level of the top actuator. Furthermore, the application of a significant axial force on the device would have affected the response of the beam links. Therefore, two rigid axial beams were laterally installed at the first-floor level, which allowed to replicate the rigid diaphragm effect. In addition, a truss system was adopted to brace the frame laterally and prevent out-of-plane instability.

Fig. 14
figure 14

Experimental test set-up

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Figures 15, 16 and 17 depict the instrumentation configuration and the RBS numbering whilst in Table 8 the main characteristics of the sensors are listed. Since it was impossible to instrument all RBSs given the number of available channels, eleven significant sections were selected for their expected dissipative behaviour and instrumented by both strain gauges and displacement transducers. In greater detail, strain gauges were positioned outside the dissipative zone of the DRLF component, in an elastic region near the RBS, to estimate the bending moment according to elastic theory. In this respect, the section's upper and lower edges of the section were instrumented to measure the strain. The curvature could be calculated by assuming plane sections according to Eq. (10)

$$\chi = \frac{{\varepsilon_{top} - \varepsilon_{bot} }}{{H_{sec} }}$$
(10)

where εtop is the strain measured at the top flange, εbot the strain measured at the bottom flange and Hsec is the height of the cross section. An estimation of the bending moment on each instrumented section was then obtained by means of Euler–Bernoulli beam theory. In the following formula, \(I_{beam}\) is the modulus of inertia of the gross section and Es is Young's steel modulus.

$$M = E_{s} I_{beam} \chi$$
(11)
Fig. 15
figure 15

Strain gauges configuration: a instrumentation scheme, b sensor position

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Fig. 16
figure 16

Displacement transducers configuration: a instrumentation scheme, b sensor position

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Fig. 17
figure 17

Inclinometers configuration: a instrumentation scheme, b sensor position

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Table 8 Instrumentation main characteristics
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By assuming the prevalence of the seismic actions with respect to the gravity loads on the devices, the bending moment at the reduced beam section was obtained by linear interpolation. Moreover, the rotation of RBSs was calculated from the measurements of the displacement transducers as:

$$\varphi = \frac{{\Delta_{top} - \Delta_{bot} }}{{H_{sec} }}$$
(12)

where Δtop and Δbot are the top and bottom relative displacements of the RBSs. Two inclinometers were applied on the first column to calculate the base bending moment and to evaluate whether the column remained in the elastic range.

5.2 Material properties

The actual mechanical properties of all the materials are reported in Tables 9 and 10. The beam links were constructed using welded plates and the capacity proprieties of the reduced beam section, based on the laboratory profile dimensions and actual mechanical characteristics, are listed in Table 11.

Table 9 Mechanical properties of S355 steel profiles
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Table 10 Mechanical properties of S235 IPE160 profile
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Table 11 Bending capacity of RBS
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5.3 Test procedure

To fully characterise the seismic response of the frame equipped with coupled DRLF systems, a series of hybrid simulation tests at different limit states were carried out in increasing order of intensity. The test programme was conducted as follows:

  1. 1)

    Damage Limitation (DL) limit state test. After the test, the elastic behaviour of the dissipative components and of the structural members was checked.

  2. 2)

    Significant Damage (SD) limit state test. After the test, the elastic behaviour of the structural members and the inelastic behaviour of the dissipative components were checked.

  3. 3)

    The damaged DRLF components were replaced with new ones. The self-centring capacity of the prototype building was verified and the residual interstorey drift was measured. Moreover, the repairability times were computed.

  4. 4)

    Near Collapse (NC) limit state test.

6 Hybrid simulation results

6.1 Damage limitation limit state

At the DL limit state, the elastic behaviour of the structure was verified for both the primary elements and the dissipative components. The instrumentation located on the RBSs highlighted that the maximum moment attained during the test was 37.5% of the elastic moment, as can be observed from Fig. 18.

Fig. 18
figure 18

DL—Bending moment of the RBSs at the floor level, see Fig. 15a

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Given that the DRLF system behaves as a cantilever Vierendeel beam and the first column of the frame did not belong to the DRLF system, this column was the one that was subjected to the highest bending moment at its base. Moreover, it was also one of the primary elements that could not be replaced after an earthquake. The column base bending moment was computed based on the inclinometer measurement and elastic theory. As depicted in Fig. 19, the maximum bending moment did not exceed the yield limit, confirming that it remained in the elastic range.

Fig. 19
figure 19

DL—Bending moment at the base of the first column

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6.2 Significant damage limit state

For the SD limit state, the structure underwent plastic deformation localised on the RBSs of the DRLF links. The results in terms of the two actuators displacement are reported in Fig. 20. The maximum displacement attained at the floor level was of 19.79 mm, which corresponds to an interstorey drift ratio (IDR) of 0.57% whereas the residual displacement after the test was equal to 1.33 mm. This value corresponds to the 0.04% of IDR which demonstrate the high re-centring capability of the structure and, as an important outcome, the ease of replacement of the beam link. Following the test, all beam links were replaced, with an overall repair time corresponding to six working hours of one person, which represents a beneficial outcome in the context of structural resilience. The bending moment history of the first column, depicted in Fig. 21, shows that the yield limit was not exceeded even for the SD limit state, confirming that the column remained elastic, which is another favourable outcome of the project. Given the fact that the structure was designed for the SD limit state, the latter result confirms the DRLF components' capability to protect the frame's irreplaceable parts, i.e., beams and columns, by localising the damage in the RBSs of the beam links.

Fig. 20
figure 20

SD—Actuator displacements: a Actuator 1, b Actuator 2

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Fig. 21
figure 21

SD—Bending moment at the base of the first column

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The local behaviour of the selected RBSs is reported in terms of bending moment histories. It can be noticed that most of the sections yielded at this intensity level, see Fig. 22a-c; nevertheless, some RBSs remained elastic, Fig. 22d. In Fig. 23, the hysteretic loops of two selected RBSs are reported, and RBS n°4 reached a maximum rotation of 5 mrad. Finally, an out-of-plane rotation of the RBSs was detected, mainly caused by inherent imperfections of the specimen and in the load application point. For this reason, in the Near Collapse limit state test, additional displacement transducers were installed on RBSs 9 and 11 to quantify the extent of such a phenomenon.

Fig. 22
figure 22

SD—Bending moment on selected RBSs

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Fig. 23
figure 23

SD—Cyclic behaviour of selected RBS

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6.3 Near collapse limit state

A strong nonlinearity characterised the structural behaviour at the NC limit state. The moment-rotation diagram reported in Fig. 24 demonstrates that the RBSs underwent significant plastic deformation, with a residual rotation of about 5 mrad. Moreover, strength reduction can be noticed from the experimental cycles. This is due to the interaction between the bending moment along the strong axis My and the bending moment along the weak axis Mz. As shown in Fig. 25, the displacement at the top flange of RBS n°10 is different between the two edges, highlighting the presence of an out-of-plane rotation. The time history of the estimated bending moments My and Mz is reported in Fig. 26, where the resisting plastic bending moments along the two axes were computed by neglecting the mutual interaction. Even in this case, the maximum value of both My and Mz exceeded the plastic resistance. In Fig. 27 the evolution of the (Mz,N,My) state/history for the hybrid simulation test at NC limit state is compared with the interaction domain, as defined in Eurocode 3-1-1 (CEN 2005c). For clarity, the points outside the safe domain are reported in red. It is important to note that, despite the use of the lateral beams to transfer the displacements to the specimen nodes, an axial force was detected in the beam links. Nevertheless, as shown in Fig. 27, this force is small, i.e. less than 20% of the axial plastic resistance. Moreover, it is interesting to note that after the first yielding, the second one occurred for bending along the weak axis (Mz), involving a very low value of My. This suggests that strength degradation observed in the experimental cycles is obtained mainly due to the My–Mz interaction.

Fig. 24
figure 24

NC—Cyclic behaviour of selected RBS

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Fig. 25
figure 25

NC—Out-of-plane displacement recorded

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Fig. 26
figure 26

NC—Bending moment along the two principal axes

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Fig. 27
figure 27

NC—N-Mz-My Interaction domain according to Eurocode 3-1-1

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Concerning the displacement of the two actuators, reported in Fig. 28, the maximum displacement achieved at the floor level was 31.61 mm, which corresponds to an interstorey drift ratio (IDR) of 0.90%. Figure 29 shows the bending moment history at the base of the first column, highlighting that it did not exceed the plastic limit even at the NC limit state test. This emphasises the protection that the devices provide to the structure in case of higher earthquake intensity than the design one. In addition, a residual interstorey drift of 0.03% was recorded, demonstrating the structure's re-centring capability and, as an important outcome, the ease of repair of the innovative frame even after undergoing NC limit state earthquake actions. Figure 30 reports the base shear vs top floor displacement for the three limit states. The three diagrams are superimposed with the pushover curve. Regarding the limit states, it should be noted that the structure remained in the elastic field for the DL limit state, while inelastic behaviour was observed for both the SD and NC limit states.

Fig. 28
figure 28

NC—Actuator displacements: a Actuator 1, b Actuator 2

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Fig. 29
figure 29

NC—Bending moment at the base of the first column

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Fig. 30
figure 30

Base shear vs. Top floor displacement at the three limit states

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7 Calibration of the RBS nonlinear spring

After the tests, the nonlinear model of the DRLF component was calibrated on the experimental results using the Multical tool. The estimation of the stiffness parameters, needed for calibration, was based on three reliable RBSs hysteretic loops at the Near Collapse limit state, as shown in Fig. 31. For each loop, an elastic and hardening stiffness were evaluated for both the positive and negative branches. The stiffness values were determined considering the linear part of the cycles for both elastic and plastic ranges. Despite the experimental behaviour of the RBSs was affected by those of the numerical substructure, this effect was limited given the fact that the elastic and the hardening stiffness are independent of the loading history.

Fig. 31
figure 31

Elastic and hardening stiffness calibration on near collapse cycles

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Figure 31, shows the calibrated stiffness values, defined as the mean value of the elastic and hardening stiffness of each hysteretic loop, as reported in Table 12. The mean values were employed to calibrate the model by fixing these parameters on Multical and using the tool to calibrate the Bouc–Wen model's additional parameters, which are detailed in Table 13. Both strength and stiffness degradation were not considered in the calibration since it was impossible to separate the contribution of out-of-plane effects from that of the material's own degradation.

Table 12 Estimated stiffness parameters
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Table 13 Calibrated Bouc–Wen parameters
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Following the calibration, time-history analyses on the updated model were carried out using the same set of ground motion records employed for the Hybrid Simulation. The results of the calibrated and experimental outcomes are compared in Table 14, where the NRMSE parameters in terms of displacement and forces are listed. The NRMSE parameter did not exceed 20%, revealing that a satisfactory agreement was achieved.

Table 14 NRMSE between the test and the calibrated model results
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8 Conclusions

This article thoroughly described the results of a series of experimental tests performed on a full-scale steel frame equipped with DRLF components. Through the implementation of the partitioned G − α algorithm, Hybrid Simulation technique was successfully employed to characterise the seismic behaviour of the entire frame at three different limit states, namely, DL, SD and NC. Only the ground floor was physically built, whereas the remainder of the structure was numerically simulated. This enabled laboratory costs to be limited while simulating the full-scale structure response. The experimental behaviour satisfied the design assumptions, showing that all structural elements remained at the DL limit state in the elastic field. Only the DRLF links underwent significant plastic deformation at the SD limit state, resulting in large hysteretic loops, whilst the irreplaceable parts remained in the elastic range. Non-negligible out-of-plane behaviour of the DRLF components was observed with consequent bending interaction in the two principal axes, especially regarding the DRLF beam links at the NC limit state. Moreover, the DRLF system allowed for small residual lateral displacement, highlighting an excellent self-centring capability. In terms of repairability, it took 6 man/h to replace the components, which represents a beneficial outcome in the context of structural resilience. At the NC limit state, no damage was detected on the irreplaceable parts, i.e., beams, columns and joints, whilst the RBSs of the DRLF system experienced large plastic deformations, with residual plastic rotation. Since the structure was designed for the SD limit state, the protection given by the components to the irreplaceable elements at the NC limit state is a notable outcome of the experimental campaign. Finally, the numerical nonlinear behaviour of the RBSs was calibrated according to the moment-rotation cycle obtained from the NC limit state. The Finite Element model of the structure was subsequently updated, and the model results were compared with the experimental response, demonstrating a satisfactory agreement. Future perspectives include the employment of the calibrated components for numerical applications, e.g., Incremental Dynamic Analyses and seismic fragility curves.