PSI - Issue 78

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ScienceDirect

Procedia Structural Integrity 78 (2026) 1759–1766

© 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of XX ANIDIS Conference organizers Keywords: Structural health monitoring; 3D printing; smart-materials; strain sensing; resistor; self-monitoring; carbon-based fillers; carbon microfiber Abstract The automation of concrete constructions through 3D printing has garnered considerable attention in civil engineering due to significant advantages over conventional methods. Nevertheless, the widespread adoption of this technology faces substantial chal lenges stemming from inherent uncertainties associated with the additive manufacturing process. A solution is to functionalize the 3D printed components with self-sensing capabilities to monitor performance during construction and operation and thus assess quality in real-time. Here, we study the local functionalization of 3D printed components through a hybrid 3D printing process. To do so, we build on prior work in self-sensing cementitious composites by integrating graphite powder and carbon microfibers as conductive fillers into cement-based mixtures to generate substantial piezoresistive capabilities. The technology is demonstrated on a 3D printed reinforced concrete beam. The smart beam is fabricated using a self-sensing composite at the bottom, followed by a continuous transition to a traditional cementitious mix. The printed self-sensing layers serve as strain-responsive interfaces capable of mapping strain field evolution by continuously monitoring changes in electrical resistance. A series of quasi-static and dynamic tests were performed to characterize the strain-sensing performance of the developed composite specimens. Results demonstrate the successful integration of self-sensing cementitious materials into the 3DP fabrication process, highlighting their potential for real-time monitoring of construction quality, detection of load-path alterations, and early identification of structural defects. XX ANIDIS Conference 3D-Printed Smart Reinforced Beam for Strain Monitoring HanLiu a , Israel Nilton Lopes Sousa a , Simon Laflamme a,b , Shelby E. Doyle c , Antonella D’Alessandro d , Filippo Ubertini d a Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA, 50010, USA b Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, 50010, USA c Department of Architecture, Iowa State University, Ames, 50010, IA, USA d Department of Civil and Environmental Engineering, University of Perugia, via G. Duranti, 93, 06125, Perugia, Italy

1. INTRODUCTION

Additive manufacturing, particularly three-dimensional printing (3DP) of cementitious materials, has introduced transformative possibilities to the field of civil engineering due to its advantages over traditional construction methods.

∗ Han Liu. Tel.: + 1-515-294-2140 E-mail address: liuhan@iastate.edu

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of XX ANIDIS Conference organizers 10.1016/j.prostr.2025.12.224

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These include reduced labor demands, improved cost e ffi ciency, accelerated fabrication, enhanced quality control, and increased design flexibility enabled by complex geometries and sustainable mix designs Buswell et al. (2018); Zhang et al. (2019); Khan et al. (2021); Xiao et al. (2021). Despite these advantages, challenges such as inconsistency qual ity throughout each print, monitoring of layer adhesion, material variabilit, reinforcement embedment, and ensuring overall structural performance continue to impede its widespread adoption Siddika et al. (2019); Raphael et al. (2023). To address these limitations, the use of structural health monitoring (SHM) techniques has been explored to pro vide in-situ assessment of the performance and integrity of 3D-printed structures Lynch et al. (2016); Mishra et al. (2022). While conventional sensors, such as strain gauges Yoon et al. (2022), acoustic emission transducers Van Steen et al. (2024), piezoelectric elements Gomasa et al. (2023), and fiber-optic systems Alwis et al. (2021) can o ff er valu able data, they often present integration challenges in printed structures due to their fragility, complex wiring, signal interpretation di ffi culties, and the intrusive nature of installation. As a result, recent research has shifted toward the development of self-sensing materials, particularly cementitious composites engineered to respond to mechanical stimuli by intrinsically altering their electrical properties, enabling embedded sensing without the need for discrete transducers Laflamme et al. (2023). Our prior work demonstrated that incorporating small amounts of conductive fillers, such as graphite (G) and car bon microfibers (CMF), into a cementitious matrix can impart a significant piezoresistive response to strain Birgin et al. (2021). These functionalized composites exhibit changes in electrical resistance that correlate with strain lev els under mechanical loading, enabling their use for real-time structural monitoring. This sensing principle has been demonstrated in various applications, including smart pavements Gupta et al. (2021); Gulisano et al. (2024), conduc tive cementitious sensors Han et al. (2020); Bekzhanova et al. (2021), and smart masonry units Wi et al. (2021); Meoni et al. (2022). More recently, we have found that such functionalized cementitious composites can be 3D printed, al lowing the creation of self-sensing nodes that can be embedded within structural components to monitor strain in real time during and after fabrication Laflamme and Ubertini (2020); Liu et al. (2024). The aim of this work is to investigate the integration of self-sensing nodes into structural cementitious elements using additive manufacturing, and demonstrate the technology on a 3D printed reinforced concrete beam. Specifically, we proposed self-sensing nodes made of cementitious mixes doped with G and CMF, using Sikacrete powder as the primiary binder. The beam is printed with a functionally graded design, wherein the bottom layers are made from the conductive Sikacrete composite to form the self-sensing layers, transitioning continuously to a normal Sikacrete composite at the top. These printed self-sensing layers act as strain-sensitive interfaces capable of capturing strain fields through the monitoring of electrical resistance. Commercial carbon steel drop-in anchors are embedded as electrodes to create stable and reliable electrical connections to external data acquisition systems. Percolation behavior was characterized to determine the optimal filler content in Sikacrete, and a series of dynamic tests were conducted to evaluate the strain-sensing performance of the fabricated smart beam. The rest of the paper is organized as follows. Section 2 provides the background on 3D printed self-sensing cemen titious specimens including the material properties, the fabrication process, and the derivation of the electromechan imal model. Section 3 describes the experimental methodology. Section 4 presents and discusses results from the experimental investigation. Section 5 concludes the paper.

2. Background

2.1. Materials

The conductive cementitious composite was formulated by incorporating G powder (Fisher Chemical APS 7-11 micron, 99%), milled carbon microfiber (MCMF) (SGL Carbon C M150-4.0 / 240-UN), and chopped carbon microfiber (CCMF) (SGL Carbon C M150-4.0 / 240-G100) into commercially available Sikacrete ® -752 3D powder, which is a 1-part micro-concrete specifically for use with 3D robot or gantry printers and served as the primary binder for the 3D printing process in this study. These materials were selected based on prior research findings, which demonstrated that combining these two forms of carbon microfibers in a hybrid configuration e ff ectively imparts piezoresistive behavior to cement-based composites Birgin et al. (2021); Liu et al. (2024).

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2.2. Fabrication Process

A standardized mixing protocol was used to prepare specimens and maintain consistent rheological properties. As illustrated in Figure 1, two types of Sikacrete composites including conductive Sikacrete composite and normal Sikacrete composite were produced. To fabricate the conductive Sikacrete composites, Sikacrete powder, G, MCMF, and CCMF were first dry-mixed using a spatula for two minutes to break up particle agglomerates. Subsequently, 50 wt% of the designed total mixing water was added, and the mixture was blended with a handheld electric mixer at 200 rpm for 2 minutes. The remaining 50 wt% of the water was then introduced to achieve a water-to-binder ratio (w / b) of 0.26, followed by an additional 3-minute mixing cycle at the same speed to produce a homogeneous and printable cementitious composite. The fabrication of the normal Sikacrete followed the same procedure, excluding the addition of any conductive fillers, and was prepared with a lower w / b ratio of 0.23 to account for the absence of conductive additives.

Fig. 1: Fabrication process of the conductive cementitious composites: (a) combination of cement, G, and CMF; (b) dry mechanical mixing of material; (c) adding water and following with standard mixing procedure; (d) feed-in composites mixture; and (e) mounting on pump and printing paste.

To investigate the percolation behavior of the conductive Sikacrete composites, a series of cube specimens were prepared using various mixture proportions, with three replicates fabricated for each formulation. Specifically, graphite (G) powder was introduced at an initial dosage of 1 wt% and incrementally doubled up to 16 wt%. Similarly, MCMF was incorporated starting at 0.03125 wt% and doubled in each subsequent mixture until reaching 0.25 wt%, with the same stepwise approach applied to CCMF. The water-to-binder (w / b) ratio was held constant at 0.40 across all mixtures. Specimen labels begin with a numeric prefix indicating the weight percentage (wt%) of conductive filler relative to the weight of Sikacrete powder. A hybrid composition consisting of 2 wt% G, 0.25 wt% MCMF, and 0.062 wt% CCMF, referred to as 2G250M62CCMF, was ultimately selected as the mix design for fabricating the conductive Sikacrete composite in this study. This selection was informed by the percolation behavior observed in the tests, as detailed in Section 4, as well as by practical considerations to ensure su ffi cient workability for layer-by-layer extrusion and adequate buildability to support the 3D printing process. An extrusion-based commercial 3D clay printer (3D Potter 7) was used for the 3DP process, and the overall setup is shown in Figure 1. The freshly mixed Sikacrete composite was manually loaded into the extruder tube in three consecutive pours, in which the first pour consisted of conductive Sikacrete composite, while the second and third pours contained the normal Sikacrete mix. A circular nozzle with a diameter of 8 mm was attached to the extruder, and the stand-o ff distance between the nozzle and the printing platform was set at approximately 4 mm. A hollow beam specimen featuring a diagonal reinforcement pattern and overall dimensions of l × w × h = 320 × 120 × 55mm 3 was designed and sliced in Cura into 15 layers, each with a layer height of 3.66 mm. The specimen was printed at an extrusion rate of 320 mm 3 / s and a printing speed (the movement speed of the printer platform) of 120mm / s. Six steel drop-in anchors were embedded as electrodes at equal intervals of 46 mm along the left side of the third layer. Additionally, three #2 reinforcement bars (diameter of 6.35 mm) were placed longitudinally in the

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fifth layer, spaced 30 mm apart, to provide reinforcement. A schematic drawing of the printed specimen and detailed geometric dimensions are shown in Figure 2. Fabricated specimens were then cured in the laboratory for an additional 28 days to promote further hydration and enhance their mechanical properties, as is for standard concrete.

Fig. 2: Schematic drawing showing the (a) top view; (b) side view; (c) front view; and (d) 3D view of the printed Specimen.

2.3. Electromechanical model

The self-sensing behavior of cementitious composites primarily stems from their piezoresistive response under mechanical loading, which allows the electrical resistance to change in response to strain. Assuming only the internal resistance is a ff ected by the mechanical deformation Han et al. (2012); Dong et al. (2019), the conductive cement composite can be simplified as an ideal resistor with nominal resistance R 0 , expressed as:

d A

R 0 = ρ

(1)

where ρ is the bulk resistivity of the composite, d is the spacing between the sensing electrodes, and A represents the cross-sectional area perpendicular to current flow. To evaluate the sensing response, the fractional change in resistance (FCR) is used and defined as:

R − R 0 R 0

∆ R R 0

(2)

FCR =

=

where ∆ R is the resistance change from its initial value R 0 . In the case of a beam subjected to three-point bending, the maximum bending strain ε max , appearing at the outermost fibers (top and bottom surfaces of the beam) of the section, is a function of mid-span deflection δ and computed as:

6 · δ · h L 2

(3)

ε max =

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where δ represents the midpoint deflection, taken as the maximum vertical displacement recorded at mid-span; h is the height of the beam; and L is the span length between the two supports. Assuming a linear strain profile across the beam depth, with maximum strain occurring at the extreme fibers and zero strain at the neutral axis, the average bending strain ε avg at the middle of the sensing layer, the third printed layer located 11 mm from the base of the 55 mm specimen’s overall height, can be approximated using geometric scaling as:

24 · δ · h 5 L 2

44 · ε max 55

(4)

ε avg =

=

By applying logarithmic di ff erentiation to equation 1, the FCR reflecting geometric strain e ff ects that account for strain-dependent changes in resistivity, can be expressed as a function of the average bending strain ε avg :

∆ e

∆ A A

∆ ρ ρ

∆ ρ ρ

+ (1 + 2 ν ) ε avg

(5)

FCR =

e −

+

=

where ν is the Poisson’s ratio of the material, and ∆ ρ/ρ is the piezoresistive e ff ect that often dominates when at electrical percolation. Therefore, the gauge factor λ bend of the self-sensing beam under bending can be written:

1 ε avg

FCR ε avg

∆ ρ ρ

= (1 + 2 ν ) +

(6)

λ bend =

3. Experiments

The strain sensing performance of the self-sensing Sikacrete beam was characterized and evaluated through a series of three-point bending test. Figure 3(a) shows the overall experimental setup. Flexural testing was carried out using an Instron 5944 universal testing machine equipped with a 1 kN load cell (2580 series). Each beam specimen was horizontally supported on a three-point bending fixture with a span length of 200 mm along its longitudinal axis. A loading nose was positioned at the midpoint of the span to apply a vertically concentrated force, and fiberglass plates were placed between the specimen and both the loading nose and support rollers to provide electrical insulation, as illustrated in Figure 3(b). The loading protocol consisted of five triangular waveform cycles. The first two cycles were applied with a peak load of 150 N at a rate of 30 N / s, followed by two additional cycles with a peak load of 300 N at 60N / s. The final cycle reached a maximum load of 600 N with a loading rate of 120 N / s. A preload of 20 N was applied prior to testing to ensure full contact between the specimen and supports and to eliminate minor mechanical slack or seating e ff ects in the test setup. Load and displacement data were captured at a sampling frequency of 100 Hz, while electrical resistance was measured independently between each set of adjacent electrodes, AB, BC, CD, DE, and EF, at 10 Hz using a resistance-reactance (R–X) model of an LCR meter implemented through LabVIEW.

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Fig. 3: (a) Overall experimental configuration; and (b) close-up view of the flexural bending setup.

4. Results and Discussion

Figure 4(a) presents 28-day three-specimen averaged resistivity ρ computed using Eq. 1 as a function of CMF inclusion level, ranging from 0 wt% to 0.25 wt%, with error bars indicating the minimum-to-maximum range of resistivity measured across three specimens. The inset presents a similar analysis for varying G from 0 wt% to 16 wt%. It can be observed that the resistivity of the Sikacrete specimens significantly decreased with increasing G content, particularly at lower percentages from 0 wt% to 3 wt%. Percolation behavior for both MCMF and CCMF exhibited a similar trend of rapidly decreasing resistivity at low concentrations, with the most notable reduction occurring both within the range of 0.032 wt% to 0.125 wt%. Following these results, a hybrid composition consisting of 2 wt% G, 0.25 wt% MCMF, and 0.062 wt% CCMF, referred to as 2G250M62CCMF, was ultimately selected as the mix design for fabricating the conductive Sikacrete composite. Figures 4(c) and (d) present time-series plots of the FCR of specimens measured between AB, BC, CD, DE, and EF electrodes, respectively, compared against the strain input during the dynamic test. Data presented here was post processed using a detrend filter to eliminate intrinsic polarization drift caused by the dielectric nature Wen and Chung (2002); Bekzhanova et al. (2021) and / or direct piezoelectric e ff ect Sun et al. (2000); Chung (2002), and the bending strain was computed using Eq. 4. The results show a close match between the electrical signal and strain time histories, indicating that the beam specimen is capable of tracking the strain input. Notably, the FCR amplitudes measured between electrodes AB and EF were smaller than those measured at the central electrode pairs. This observation is consistent with the strain distribution along the beam, as bending strain is reduced near the ends and maximized toward the midspan. Additionally, all resistance measurements exhibited a relatively high level of signal noise, likely due to the contact resistance fluctuations at the embedded electrode interfaces.

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Fig. 4: (a) Electrical percolation curves showing the three-specimen average resistivity ( ρ ) as a function of CMF doping level; and (b)-(f) time-series plots of the FCR measured between AB, BC, CD, DE, and EF electrodes of the beam specimen, respectively.

5. Conclusion

This study demonstrated the integration of self-sensing functionality into 3D-printed cementitious beams by se lectively doping Sikacrete with G and CMF. A functionally graded beam was fabricated, with conductive layers at the bottom enabling strain sensing and normal concrete layers at the top providing structural strength. The optimal mix design (2G250M62CCMF) was identified based on percolation behavior and printability. Embedded steel an chors served as electrodes, allowing real-time electrical resistance measurements. Dynamic flexural testing confirmed that the self-sensing layers e ff ectively tracked bending strain through piezoresistive responses, with spatial resolution reflecting the strain distribution. These results demonstrated the potential of 3D printing for embedding distributed sensing directly into cement-based structures, o ff ering a scalable approach for real-time structural health monitoring in future construction applications.

Acknowledgements

This material is partly supported by the National Science Foundation under Grants No. 2349792 and 2431765, and by the Italian Ministry of University and Research (MUR) via the FIS2021 Advanced Grant “SMS-SAFEST - Smart Masonry enabling SAFEty-assessing STructures after earthquakes” (FIS00001797). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation and the MUR.

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Procedia Structural Integrity 78 (2026) 1529–1536

© 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of XX ANIDIS Conference organizers Keywords: Nonlinear three-dimensional macroelement; strengthening interventions; equivalent-frame modeling; masonry structures; seismic analysis. This paper introduces a novel three-dimensional macroelement developed to effectively capture the coupled in-plane and out-of plane response of unstrengthened and strengthened masonry elements, balancing computational efficiency with constitutive law versatility through a stripe or fiber discretization of the cross-section. Indeed, the proposed formulation facilitates the introduction of additional reinforcement, enabling the explicit modeling of a wide range of strengthening solutions. The effectiveness of the proposed macroelement in reproducing the lateral response of masonry piers are proved through the numerical simulation of experimental quasi-static cyclic shear-compression tests. XX ANIDIS Conference A 3D macroelement formulation for modeling unstrengthened and strengthened masonry piers Christian Salvatori a, *, Gabriele Guerrini a , Alessandro Galasco a , Andrea Penna a a Department of Civil Engineering and Architecture (DICAr), University of Pavia, Via Ferrata 3, 27100 Pavia, Italy Abstract The widespread presence and seismic vulnerability of unreinforced masonry (URM) structures have prompted extensive research into their assessment and retrofit. Even when local vulnerabilities are mitigated and a global three-dimensional behavior is promoted, the seismic performance under horizontal excitations might still be insufficient. To address this limitation, strengthening interventions using composite materials, such as Fabric-Reinforced Cementitious Matrices (FRCM) or Composite-Reinforced Mortars (CRM), as well as retrofitting systems based on steel or timber exoskeletons, are often applied to one or both sides of the masonry walls.

* Corresponding author. Tel.: +39 0382 98 5469 E-mail address: christian.salvatori@unipv.it

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of XX ANIDIS Conference organizers 10.1016/j.prostr.2025.12.195

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1. Introduction Unreinforced masonry (URM) buildings represent a significant portion of the global building stock, particularly in historic centers and rural areas. These structures often exhibit substantial seismic vulnerability due to the inherent material weaknesses and the lack of adequate provisions to resist lateral loads, frequently resulting in out-of-plane failure mechanisms. Even when such vulnerabilities are mitigated and a global three-dimensional behavior is promoted, the seismic performance under horizontal actions may remain insufficient. As a result, extensive efforts have been devoted to developing efficient and effective strategies for retrofitting existing URM buildings and designing new ones. Recently, Composite-Reinforced Mortars (CRM) and Fabric-Reinforced Cementitious Matrices (FRCM) systems allowed to overcome the undesirable increase in panel weight and stiffness of the traditional reinforce plaster and the chemical and physical compatibility issues with masonry substrates related to Fiber-Reinforced Polymer (FRP) stripes or sheets (Papanicolaou et al, 2008; Valluzzi et al., 2014). Furthermore, the need to comply with environmental requirements has led researchers to explore the potential of steel (Albanesi et al., 2023) and timber (Guerrini et al., 2021) exoskeletons as viable strengthening strategies. This paper presents a three-dimensional equivalent-frame macroelement developed to incorporate the biaxial contribution of lumped and distributed reinforcing and strengthening layouts into the static and dynamic response of masonry panels. The proposed formulation builds upon the macroelement proposed by Penna et al. (2014), which proved effective and efficient for the static and dynamic analyses of masonry structures within the equivalent-frame approach (Lagomarsino et al., 2013). However, despite the improvements carried out over the years (Bracchi et al., 2021; Bracchi and Penna, 2021), its formulation has always been restricted to the in-plane response of URM panels. In this work, an additional axial-flexural interface is provided at the center of the macroelement, allowing to reproduce the correct axial and flexural elastic without requiring manual adjustments to the mechanical properties or iterative algorithms (Bracchi et al., 2021). The biaxial axial-flexural response is obtained through either a stripe or fiber discretization of the interfaces, prioritizing computational efficiency on the one hand, as stresses are analytically integrated along each stripe (Penna et al., 2014), or constitutive law versatility on the other hand, as each fiber is assigned a uniaxial stress-strain relationship. Notably, the assembling algorithm responsible for the sectional integration proved suitable for explicitly introducing additional elements, such as lumped or distributed reinforcement, enforcing their collaboration with the macroelement interfaces via kinematic constraints. In previous studies, the proposed macroelement successfully reproduced the cyclic response of stone masonry piers strengthened with CRM applications (Salvatori et al., 2025a,b). In this paper, its capabilities are further investigated through the numerical simulation of quasi-static cyclic shear-compression tests conducted on two calcium silicate masonry piers. The first specimen is unstrengthened and serves as a reference, while the second is retrofitted with a timber exoskeleton and oriented strand boards (OSBs), mechanically connected to the masonry substrate (Guerrini et al., 2021). 2. Three-dimensional macroelement formulation Similarly to the formulation of Penna et al. (2014), the macroelement discussed in this paper consists of an assemblage of axial-flexural interfaces separated by shear-deformable central bodies (Fig. 1a). However, the proposed macroelement introduces a central interface in addition to the two located at the panel extremities (Vanin et al. 2020). This additional interface releases the kinematics of the macroelement and allows to natively capture the axial and flexural elastic stiffness of a masonry element without relying on manual adjustments or iterative corrections (Bracchi et al., 2021). Indeed, the three nonlinear interfaces serve as Gauss-Lobatto integration points, enabling the exact integration of a third-order polynomial expression, which corresponds to a second-order curvature profile. In this context, integration lengths of 1/6 and 2/3 the panel height are assigned to the end and central interfaces, respectively. The nonlinear biaxial axial-flexural response of the interfaces is computed using either a stripe (Fig. 1c) or fiber (Fig. 1d) discretization. In this context, a linear strain profile is enforced via kinematic constraints, as a function of the degrees of freedom associated to the interfaces (Fig. 1b).

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Fig. 1. (a) Local degrees of freedom of the three-dimensional macroelement, (b) interface degrees of freedom, (c) stripe and (d) fiber discretization of the cross-section.

In the first case, the interfaces are subdivided into analytical two-dimensional stripes, and the biaxial response is obtained by numerically integrating the contribution of each stripe along the thickness of the macroelement. As the in-plane response of each stripe is computed through the analytical integration of stresses proposed by Penna et al. (2014), it results in a computational efficient formulation. In this context, a bilinear material model with recentering unloading is adopted in compression, while an elasto-fragile relationship is assumed in tension (Fig. 2a). Alternatively, the interfaces are subdivided in a series of fibers, whose nonlinear contribution is numerically integrated along the two principal directions to obtain the three-dimensional response of the cross-section. As each fiber is assigned a uniaxial stress-strain relationship, more detailed material models can be adopted. In this context, the constitutive law proposed by Bracchi et al. (2021) (Fig. 2b) and the multilinear model of Fig. 2c are provided. The shear and torsional responses of the macroelement are concentrated in the central bodies, and are considered independent from other actions and along orthogonal directions. As in the formulation of Penna et al. (2014), the shear behavior is based on the continuum damage model for masonry developed by Gambarotta and Lagomarsino (1997a,b), which has been macroscopically integrated to align with the macroelement formulation (Penna et al., 2014). On the other hand, the torsional response is maintained linear elastic, as typically of minor concern for masonry elements.

Fig. 2. Interface constitutive laws: (a) Penna et al. (2014), (b) Bracchi et al. (2021), and (c) multilinear material models.

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As previously discussed, the proposed macroelement enables the explicit modeling of strengthening or reinforcement solutions. This is accomplished by introducing additional stripe or fiber elements with different mechanical properties or constitutive laws along the macroelement interfaces, while ensuring compatibility by enforcing an a-priori linear strain profile. Strengthening and reinforcement of masonry members typically involves applying or embedding materials with significant tensile strength to overcome one of the main deficiencies of the material. In this context, single- or double sided jacketing interventions can be modeled as additional surface layers discretized according to a stripe or fiber formulation (Salvatori et al., 2025a). On the other hand, internal or external lumped reinforcement can be represented through additional fiber elements. Unlike the axial-flexural response, which is explicitly accounted for, the shear contribution of the additional elements is not directly considered in the current implementation. 3. Validation of the macroelement formulation The proposed macroelement is adopted to simulate the experimental response of two calcium silicate (CS) masonry piers subjected to in-plane quasi-static cyclic shear-compression tests, conducted at the EUCENTRE Foundation and University of Pavia facilities in Italy (Guerrini et al., 2021). 3.1. Specimens and testing protocol The two single-wythe CS masonry piers present identical geometrical dimensions and mechanical properties, measuring 2.70 m in height, 2.00 m in length, and 0.10 m in thickness (Fig. 3). The first specimen is unstrengthened (Fig. 3a), acting as a reference, whereas the second is retrofitted through a timber frame linked to the masonry underneath, to the top reinforced concrete (RC) beam, and to the RC footing through steel connections (Fig. 3b). The timber frame is characterized by vertical posts and horizontal nogging elements. The vertical posts are fastened to the top and bottom sill plates and to the top and bottom RC elements through specific tie-down anchorages, designed to yield before reaching the timber strength (Fig. 3b). Finally, 18-mm-thick oriented strand boards (OSBs) are nailed to the timber frame to enhance the in-plane shear strength and stiffness of the masonry panel (Guerrini et al., 2021). A comprehensive characterization campaign carried out at the DICAr Laboratory of the University of Pavia (Guerrini et al., 2021) allowed obtaining the mechanical properties of the CS masonry. Table 1 summarizes the main findings in terms of Young’s modulus ( E ), tensile and compressive strength of masonry ( f t and f c ) and bricks ( f bt and f bc ), cohesion ( f v0 ) and friction ( μ ) coefficients, and density ( ρ ). The shear modulus ( G ) is conventionally taken as 0.3 E , since no dedicated tests were performed.

Fig. 3. Geometry and details of the tested masonry piers: (a) bare and (b) retrofitted configurations (Guerrini et al., 2021).

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Table 1. Mechanical properties and density of the calcium silicate masonry (Guerrini et al., 2021). E [MPa] G [MPa] f t [MPa] f c [MPa] f v0 [MPa] μ [-] f bt [MPa] f bc [MPa]

ρ [kg/m

3 ]

6593

1978

0.28

10.1

0.62

0.71

2.5

19.8

1837

The specimens were subjected to in-plane quasi-static shear-compression tests with increasing target displacements through a horizontal actuator, while two vertical actuators maintained a constant axial load corresponding to an average stress of about σ 0 =0.5 MPa at the top of the piers and ensured double-fixed boundary conditions. Additionally, a restraining system prevented out-of-plane movements of the piers. The testing protocol provided three cycles per displacement increment to investigate stiffness and strength degradation up to sever damage conditions.

3.2. Numerical modeling

The two masonry piers are modeled with their actual geometric dimensions and subjected to double-fixed boundary conditions. The mechanical properties assigned to the masonry reflect the experimental values obtained from the material characterization campaign (Table 1). However, when using the bilinear constitutive laws (Fig. 2a,b), the compressive strength is reduced to 85% of the experimental value, since these models cannot capture the post-peak softening behavior of the material. In contrast, when the multilinear model shown in Fig. 2c is adopted, no reduction is required. In this case, the full compressive strength is used, with a residual strength of αf c =0.4 f c at a strain of μ α ε c =0.35%. The axial-flexural contribution of the strengthening system is explicitly modeled by introducing lumped elements with the elasto-plastic behavior deduced from the J 2 -plasticity theory within the macroelement interfaces. Specifically, four additional elements are placed at the end interfaces to represent the tie-down anchorages (Fig. 3b). The equivalent Young’s modulus, cross -sectional area, and yielding stress of these elements are computed to match the actual axial stiffness and tensile strength of the steel connectors (126000 kN/m and 12.8 kN, respectively, as reported in Guerrini et al., 2021), based on the integration lengths of the end interfaces. The strengthening is irrelevant in the central interface, as double-fixed boundary conditions prevent relative rotations. The shear strength of the piers is defined as the minimum value between two failure criteria, following the provisions of the Italian building code for masonry with regular texture (MIT, 2019): shear sliding over a cracked length ( V u,s ) and stair-stepped diagonal cracking ( V u,d ). Notably, the parameter f v0,lim , which accounts for the tensile failure of units, is assumed equal to the brick tensile strength f bt . The shear strength improvement provided by the OSB panels is implicitly accounted for by assigning increased cohesion and friction coefficients, according to the approach described in previous works (Guerrini et al., 2024). In this case, only the diagonal cracking criterion is activated, as the strengthening system effectively prevents shear sliding failure. Gc t =5 and β = 0.5 are assigned to the shear model of the macroelement. The numerical simulations are carried out by applying the experimental loading protocol in terms of amplitude and number of cycles. Accordingly, a constant vertical load of 101.45 kN is applied at the top of the specimens. 3.3. Numerical results and comparison Numerical results are discussed in terms of hysteresis cycles, namely the horizontal top displacements against the base shear restoring forces, and failure mechanisms. The numerical failure mode of the unstrengthened pier closely reflects the experimental observations (Fig. 4). The pier initially exhibits a rocking response, followed by a sudden drop in strength due to the activation of a shear-sliding mechanism at approximately 0.20% drift. The hysteresis cycles show good agreement with experimental data in terms of elastic stiffness, lateral strength, and energy dissipation. Differences in peak and residual lateral strength are limited to 5.5% and 3.9%, respectively. It is worth noticing that the numerical peak strength is governed by the f v0.lim limitation, making the results highly sensitive to this parameter. Fig. 4a, 4b, and 4c report the numerical response by adopting the constitutive laws depicted in Fig. 2a, 2b, and 2c, respectively. As the compressive strength is not reached during the loading protocol, the results are essentially equivalent across the three models.

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Fig. 4. Hysteretic cycles of the bare pier using the (a) Penna et al. (2014), (b) Bracchi et al. (2021), and (c) multilinear material models. Experimental and numerical responses in gray and black, respectively.

The numerical simulation of the retrofitted pier also shows good agreement with the experimental results (Fig. 5), with a minor discrepancy of about 2% in the peak lateral strength estimation across the three constitutive models adopted for the axial-flexural response of the interfaces. Experimentally, the timber frame effectively inhibited the shear-sliding mechanism, allowing the pier to reach its flexural capacity. This resulted in pronounced toe-crushing at the base, which progressively reduced the effective dimensions of the pier, leading to lateral strength degradation and the development of a diagonal crack at around 0.80% drift, ultimately causing shear failure at 2% drift. All the adopted material models for the axial-flexural interfaces successfully capture the extensive masonry plasticization. Nevertheless, the cyclic response is significantly influenced by the choice of constitutive law. More specifically, the bilinear model with parallel-elastic unloading (Fig. 4b) provides a better representation of damage accumulation compared to the recentering unloading (Fig. 4a). However, the plastic plateau limits the accuracy of the formulation. Conversely, the multilinear model successfully reproduces the lateral strength degradation associated with toe crushing, up to the onset of the diagonal crack observed during the experimental test, which ultimately led to shear failure of the specimen. This crack is a consequence of the reduction in the effective dimensions of the pier due to severe damage, a phenomenon that cannot be properly captured through this macroelement model. Notably, previous studies proved that stripe and full fiber discretizations with the material model of Fig. 2a yield almost overlapping results (Salvatori et al., 2025a), with minor discrepancies due to the approximations involved in the analytical integration procedure (Penna et al., 2014).

Fig. 5. Hysteretic cycles of the retrofitted pier using the (a) Penna et al. (2014), (b) Bracchi et al. (2021), and (c) multilinear material models. Experimental and numerical responses in gray and black, respectively.

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4. Conclusions This paper presents a novel three-dimensional equivalent-frame macroelement that extends the well-established two-dimensional formulation of Penna et al. (2014) to simulate the biaxial response of unstrengthened and strengthened masonry panels. More specifically, the out-of-plane and torsional degrees of freedom are incorporated, and an additional central interface is introduced to natively reproduce the correct axial-flexural elastic stiffness of the element. The biaxial axial-flexural response of the interfaces is computed according to a stripe or fiber discretization, favoring computational efficiency on the one hand, and constitutive model versatility on the other hand. Notably, the biaxial contribution of strengthening and reinforcement solutions can be explicitly modeled by adding analytical stripes or fibers with different mechanical properties to the macroelement interfaces, enforcing a linear strain profile through kinematic constraints to ensure proper collaboration. The proposed macroelement is validated against in-plane quasi-static shear-compression tests on two calcium silicate (CS) masonry piers with identical dimensions. The first pier is unstrengthened, while the second is retrofitted with a timber frame and OSB panels to enhance its flexural and shear performance, respectively. The numerical model of the bare pier shows good agreement with the experimental results, accurately capturing the initial stiffness, peak and residual strength, energy dissipation, and failure mechanisms. The choice of constitutive law for masonry in compression has irrelevant influence on the results, as the compressive strength is not reached during the applied displacement history. The model of the retrofitted pier shows only minor deviations from the experimental results in terms of lateral strength. However, the choice of constitutive law for the masonry influences the accuracy in predicting cyclic behavior and residual strength. While a parallel-elastic unloading branch improves the simulation of cyclic response, a bilinear model cannot capture the progressive strength degradation observed experimentally. Moreover, the multilinear model provides a closer match to the hysteretic response, accurately reproducing the lateral strength degradation up to the onset of the diagonal crack observed in the test. Overall, the numerical simulations confirm the proposed macroelement as a reliable and versatile tool for modeling unstrengthened and strengthened masonry elements, accommodating a wide range of retrofit solutions while offering different trade-offs between accuracy of results and computational efficiency. Future developments will incorporate the contribution of strengthening and reinforcing materials to shear behavior. Additionally, the macroelement will be further validated against experimental results from URM elements retrofitted with alternative techniques, such as CRM and FRCM systems. The proposed macroelement will subsequently be employed in the global modeling of entire masonry buildings to evaluate the effectiveness of various strengthening strategies in enhancing their seismic performance. Furthermore, thanks to its efficient formulation, the proposed macroelement will be used to investigate the effects of local or global strengthening interventions on URM building aggregates with reasonable computational effort. Acknowledgements This study is conducted within the DPC-ReLUIS (2024- 2026) Work Package 5 “Integrated and sustainable interventions for existing buildings” and Work Package 10 “Code contributions for existing masonry constructions” funded by the Italian Department of Civil Protection (DPC). The authors would like to thank Nederlandse Aardolie Maatschappij BV (NAM) and the EUCENTRE Foundation, Italy, for funding and coordinating the experimental program. Note that the opinions and conclusions presented by the authors do not necessarily reflect those of the funding entities. References Albanesi, L., Manzini, C. F., Morandi, P., 2023. Experimental In-Plane Seismic Performance of an Innovative Steel Modular Strengthening System for URM Walls. Journal of Earthquake Engineering, 28(5), 1331–1357. https://doi.org/10.1080/13632469.2023.2231087 Bracchi, S., Galasco, A., Penna, A., 2021. A novel macroelement model for the nonlinear analysis of masonry buildings. part 1: axial and flexural behavior. Earthquake Engng. Struct. Dyn. 50(8), 2233-2252. https://doi.org/10.1002/eqe.3445 Bracchi, S., Penna, A., 2021. A novel macroelement model for the nonlinear analysis of masonry buildings. part 2: shear behavior. Earthquake Engng. Struct. Dyn., 50(8), 2212-2232. https://doi.org/10.1002/eqe.3444