Analytical Model and FE Implementation for FRCM-Retrofitted Flat Masonry Under Direct Shear Tests

Abstract

This study presents an analytical and numerical framework to describe the debonding behavior of fiber-reinforced cementitious matrix (FRCM)-reinforced flat masonry elements under direct shear tests. A sawtooth shear stress–slip law, initially proposed for Steel Reinforced Grout (SRG) systems by two of the authors, is calibrated for a PBO-FRCM system based on the experimental results available in the literature. These recent experimental outcomes on flat masonry pillars serve to validate the model by capturing essential interface behaviors, including residual strength and pseudo-linear hardening. Furthermore, a finite element (FE) model of the specimens has been developed to simulate the interface response, allowing for a comparison between numerical predictions and experimental results. The sawtooth law is implemented directly in commercial FE software without the need for custom coding. Additionally, a mesh sensitivity analysis was performed to verify numerical stability and identify the optimal discretization parameters for consistent model response. Results show good agreement among experimental observations, the sawtooth analytical model, and FE simulations. The analytical model slightly underestimates the experimental peak load by about 4–6%, while the FE predictions differ from the experimental results by less than 10%, confirming the reliability of the proposed modeling framework.

1. Introduction

The seismic retrofitting of existing historical masonry structures represents a crucial research area, as these structures embody significant cultural and historical value. Due to their low tensile strength and vulnerability under seismic and environmental actions, adequate strengthening interventions are often necessary to ensure their structural safety and durability. Over the past decades, several retrofitting techniques have been proposed, among which externally bonded fiber-reinforced composites have emerged as efficient and relatively non-invasive solutions [1,2,3,4,5,6,7,8]. Initially, Fiber Reinforced Polymer (FRP) systems were widely adopted owing to their excellent mechanical performance; however, their application to heritage masonry revealed critical limitations, including poor compatibility with inorganic substrates, lack of removability [9], and sensitivity to high temperatures. To overcome these drawbacks, Fiber-Reinforced Cementitious Matrix (FRCM) systems have been developed as a more compatible and sustainable alternative. FRCM composites consist of an open fiber mesh—typically made of glass, basalt, carbon, or polyparaphenylene benzobisoxazole (PBO)—embedded within an inorganic cementitious matrix applied in thin layers on the surface of the masonry. These systems offer improved vapor permeability, better substrate compatibility, and enhanced removability, making them particularly suitable for the conservation and strengthening of historical masonry structures [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16].

Several experimental studies investigated the bond behavior of FRCM systems through direct shear tests, focusing on flat masonry surfaces. These works explored the effects of material properties and testing conditions on load–slip response and failure modes, providing valuable insight into matrix-to-substrate adhesion [17,18,19,20,21,22,23,24,25]. However, their results mostly concern planar interfaces, with limited consideration of more complex surface geometries such as curved masonry. Other studies examined the local stress-transfer mechanisms between FRCM and masonry, highlighting the influence of material properties, grid configuration, and matrix cracking on bond performance [25,26,27,28,29,30,31,32,33,34,35,36,37]. FRCM techniques exhibit more complex bond behavior than FRP systems. In fact, while the failure mechanisms of FRP reinforcements bonded to masonry are generally related to cohesive detachment at the substrate, the failure modes of FRCM reinforcements can involve cohesive debonding in the substrate, detachment at the matrix-to-substrate or at the textile-to-matrix interfaces, the sliding of the textile within the matrix, or the tensile failure of the matrix. These distinct mechanisms result in a stress-transfer process that differs from the brittle delamination observed in FRP-strengthened elements [27].

In the design of structural elements strengthened with FRCM systems, reference is commonly made to international guidelines such as the Italian CNR-DT 215 (2018) [38] and the American ACI 549.6R (2020) [39]. These documents provide procedures for estimating the effective strain of FRCM systems based on experimental characterization of the composite and its interface with the substrate. However, approaches relying on a single effective strain parameter may not fully capture the complex bond–slip mechanisms governing stress transfer between the reinforcement and the masonry substrate. Several analytical and numerical models have been proposed in the literature to characterize the interface behavior of strengthened flat masonry structures [40,41,42,43]. Nevertheless, their application often necessitates the development of customized Finite Element (FE) subroutines [44,45,46].

Consequently, the present study builds upon the analytical model developed in [3], which is based on a sawtooth interface law. Originally applied to simulate the behavior of Steel Reinforced Grout (SRG) systems bonded to concrete under direct shear tests, the model is here extended to PBO-FRCM systems externally bonded to flat masonry pillars subjected to single-lap shear loading. This approach captures key interface features, including residual strength, pseudo-linear hardening, and fiber slippage. A preliminary formulation of this analytical approach was previously introduced by the authors [47]. The present work extends that initial study through a comprehensive calibration procedure, sensitivity analysis, and full FE implementation. The analytical model developed in [3] is suitable for direct implementation in commercial FE software without the need for advanced user-defined routines. The sawtooth model is calibrated using experimental data from [1,2], and an accompanying FE model is developed to simulate the interface response and compare numerical predictions with experimental observations. By integrating analytical and numerical approaches, the study provides a reliable and efficient framework for assessing the performance of FRCM–masonry interfaces. This framework facilitates the design and optimization of strengthening interventions for historical masonry structures, supporting conservation efforts through sustainable, compatible, and minimally invasive retrofitting solutions.

2. Materials and Methods

2.1. Reference Experimental Program

The results of the experimental study carried out by [2] at the Laboratory of the University of Florence (Firenze, Italy), concerning the bond behavior of a PBO-FRCM reinforcement externally bonded to both flat and curved masonry pillars, are employed in the current study to validate both the sawtooth cohesive material law and the FE model implemented in Straus7 (release 3.1.6). Since the present research concerns the modeling of FRCM reinforcements externally bonded to plane substrates, only the experimental results in [2] concerning flat specimens will be summarized in the following for completeness; the reader can refer to [2] for further details. The specimens consisted of masonry prisms made of solid pressed clay bricks (supplied by Terreal Italia S.r.l.—SanMarco, Noale, Italy) and ready-mixed mortar joints approximately 10 mm thick. The bricks (250 × 120 × 65 mm) had an average compressive strength of 20.10 MPa (coefficient of variation referring to 18 specimens: CoV = 10.79%), an average elastic modulus of 8712 MPa (n. spec. = 6; CoV = 6.92%), an average direct tensile strength of 2.49 MPa (n. spec. = 6; CoV = 16.90%), and an average flexural tensile strength of 3.36 MPa (n. spec. = 6; CoV = 33.77%). The mortar had an average compressive strength of 5.18 MPa (n. spec. = 12; CoV = 8.21%) and an average flexural tensile strength of 1.85 MPa (n. spec. = 6; CoV = 9.42%).

The FRCM reinforcement consisted of a cement-based mortar (M20) and a bidirectional PBO mesh (70 g/m² in the warp direction and 18 g/m² in the weft direction), both supplied by Ruregold (Laterlite S.p.A., Solignano, Italy), externally bonded to the masonry prisms using a wet lay-up procedure, with two mortar layers (thickness 4 to 5 mm for each layer) sandwiching one mesh layer at mid-thickness. The applied PBO-FRCM reinforcement had a width of 63 mm, a bonded length of 315 mm, and an overall thickness of about 8 mm. The M20 mortar had an average compressive strength of 32.54 MPa (n. spec. = 12; CoV = 7.21%) and an average flexural tensile strength of 7.57 MPa (n. spec. = 6; CoV = 10.61%). As per the product datasheet, the PBO fiber had a Young’s modulus of 241 GPa and toughness of 5.80 GPa.

Six single-lap shear (SLS) tests were conducted to evaluate the bond–slip response between the FRCM and the masonry substrate. The experimental setup and the geometry of the specimens are illustrated in Figure 1. All the load–slip curves exhibited an initial quasi-linear branch up to the formation of the first cracks in the reinforcement in the loaded-end region. Subsequently, the equilibrium paths showed a more scattered ascending branch with decreasing slope up to the maximum load, during which the crack pattern evolved, and detachment phenomena of the reinforcement occurred. Finally, the diagrams exhibited a descending branch up to failure. Of the six flat specimens considered in the experimental program [2], four were selected to validate the analytical and numerical models presented in this paper. The results related to the other two specimens were discarded because their behavior deviated significantly from the average, due to issues that occurred during testing possibly because of imperfect initial positioning of the specimens.

Figure 2 shows the experimental load (P)–global slip (g) curves obtained from the SLS tests on flat specimens. The global slip (g) was calculated as the average of the values recorded by the two LVDTs (see Figure 1a), measuring the relative displacement between the textile mesh and the masonry substrate in the region close to the upper end of the reinforcement. These specimens exhibited statistically consistent behavior, with limited scattering of the obtained mechanical parameters, as is apparent from the parameters reported in

Table 1. Main experimental results from the SLS tests [2].

Specimen	${\mathit{K}}_{\mathit{e}}$ [kN/mm]	${\mathit{F}}_{\mathit{b}}$ [kN]	$\mathit{\eta }$
1	7.089	4.559	0.646
2	6.121	4.439	0.629
3	8.475	4.481	0.634
4	9.773	4.309	0.610
Mean	7.865	4.447	0.630
CoV (%)	17.59	2.04	2.06

. The parameter $K_e$ is the average slope of the first linear branch of the load–slip diagrams; $F_b$ is the maximum load; and $\eta$ is the exploitation ratio, calculated as the ratio between the bonding capacity of each specimen and the experimental average tensile strength of textile coupons. The failure mode of these specimens was cohesive and associated with detachment at the textile-to-matrix interface (see Figure 3). More details can be found in the original experimental studies [1,2].

2.2. Analytical Method

This section presents the development of an analytical model for FRCM-strengthened flat masonry. The sawtooth approach, originally proposed in [3] for SRG-strengthened concrete under direct shear, is adopted and tailored for PBO-FRCM systems bonded to masonry. The model is calibrated against the widely used trilinear law to accurately capture the interface bond–slip behavior. First, the trilinear model is described, followed by the sawtooth approach.

2.2.1. Trilinear Model

The trilinear model is a widely adopted approach to describe the bond–slip behavior at the interface between the fiber and the surrounding mortar in FRCM systems [48]. In this model, all non-linearities are concentrated at the fiber–matrix interface, while the reinforcing layer (fiber embedded in mortar) is assumed to remain linearly elastic and the substrate rigid. The experimentally observed failure is governed by the interface debonding mechanism, with negligible damage to the masonry substrate. Therefore, modeling the substrate as rigid does not significantly influence the predicted global response, while local nonlinear phenomena such as mortar cracking and fiber–matrix slip are implicitly captured through the nonlinear bond–slip law assigned to the interface. The interfacial law is expressed as the relationship between the shear stress ($\tau$), which is transferred across the fiber–matrix surface, and the relative slip (s), which represents the displacement difference between the fiber and the matrix mortar. The trilinear law typically consists of three branches (Figure 4):

• Elastic phase: The initial linear region, where the tangential stress (τ) increases proportionally with the slip (s) according to the interface stiffness, up to the peak shear stress ${\tau }_{max}={\tau }_1$ at slip $s_1$. No damage occurs at this stage.
• Softening phase: After reaching the peak shear stress, the interface enters a descending branch where the stiffness progressively decreases. This stage models the degradation of the bond due to micro-cracking and partial debonding.
• Residual plateau: The bond behavior is characterized by a residual plateau, where the shear stress reaches a constant level representing the residual adhesion between the FRCM system and the substrate. This stage is primarily governed by the frictional shear stress (${\tau }_f={\tau }_3$), which remains approximately constant until a critical slip value ($s_f=s_3$) is attained, beyond which the load transfer capacity does not significantly increase.

The area under the curve corresponds to the interface fracture energy $G_{f1}$, which can be calibrated from direct shear test data. This model provides a quite accurate description of the interface response and allows for closed-form analytical solutions under simplified assumptions. However, its practical application is not straightforward since such a law is not directly available in standard commercial FE software [3]. Implementing it typically requires custom subroutines and specialized programming, making it less feasible for routine engineering practice. Therefore, the trilinear model is mainly used as a benchmark for validating simpler and more practical approaches, such as the sawtooth model developed in [3].

2.2.2. Sawtooth Model

To overcome the limitations of the trilinear model in FE analysis, the sawtooth interface model was proposed in [3]. In this approach, all nonlinear phenomena are concentrated at the interface between the reinforcement and the surrounding matrix. The reinforcement is modeled as a linear elastic layer, while the substrate is considered rigid enough to avoid any damage. Within this framework, failure occurs through the progressive slip of the reinforcement relative to the matrix. The purpose of the model is not to reproduce the detailed failure mechanism but to accurately capture the global response of the strengthened system [3].

The mechanical representation can thus be described as an elastic fiber layer bonded to a rigid support through a nonlinear interface, which governs the transfer of stresses. The reinforcement is subjected to uniaxial stress along its length, and the only kinematic parameter of interest is the longitudinal displacement of the fiber layer. At any position along the bond, this displacement corresponds to the slip between fiber and substrate, which induces a shear stress transmitted through the interface. Consequently, the bond is modeled as a pure shear transfer mechanism (Mode II) [3].

This approach approximates the classic trilinear law while offering significant computational advantages. The model divides the interface behavior into four phases, as in Figure 5a, each characterized by a distinct stiffness, thus providing a piecewise-linear representation of the softening response [3]:

• Phase 1—Initial Elastic Stage: Similar to the first branch of the trilinear model, where the interface behaves elastically with stiffness $k_1$, transferring shear stress to the substrate without damage.
• Phases 2 and 3—Progressive Softening: These two phases approximate the descending branch of the trilinear law by introducing two additional elastic phases with progressively reduced stiffness ($k_2$ and $k_3$). This ensures that global behavior mimics the gradual loss of bond capacity.
• Phase 4—Residual Stage: The last phase corresponds to a horizontal plateau with zero stiffness ($k_4=0$), representing the residual stress after complete degradation of the bond.

Figure 5b illustrates the four phases of the sawtooth analytical interface model and the corresponding shear stress distribution ${\tau }_{\mathrm{x}}$ along the bonded length of the FRCM–substrate system. The first phase represents the initial elastic response of the interface with maximum stiffness $k_1$ and peak stress ${\tau }_1$. As loading increases, softening begins near the loaded end $(x=0)$, leading to the activation of the second phase with reduced stiffness $k_2$ and stress ${\tau }_2$. Further loading causes the onset of the third phase $k_3,{\tau }_3$, representing additional degradation of the interface stiffness and the propagation of the debonding front toward the free end. The final fourth phase corresponds to the residual plateau, where the interface maintains a constant shear strength ${\tau }_3$ due to frictional effects. The characteristic lengths ($l^{*},\tilde{l},\overline{l}$) identify the positions along the bonded length where the transitions between phases occur—specifically, $l^{*}$ marks the activation of Phase 3, $\tilde{l}$ the initiation of Phase 4, and $\overline{l}$ a representative average position of stress redistribution during the progressive debonding process. Overall, Figure 5b summarises the gradual evolution of the interface response, from the elastic regime to complete debonding, under increasing applied load P. This piecewise formulation allows the continuous trilinear cohesive law to be replaced by a sequence of elastic segments, simplifying the problem while accurately reproducing the experimental response. Consequently, closed-form analytical solutions can be derived to describe the evolution of the shear stress distribution τ(x) and the overall load–slip behavior of the FRCM–substrate system. A detailed description of the computation of ($l^{*},\tilde{l},\overline{l}$), as well as further analytical details, is provided in [3].

The main advantage of the sawtooth model is that it maintains linear elasticity within each phase, allowing for analytical solutions of the governing differential equation without the need for numerical solvers such as Runge–Kutta or finite difference methods. Furthermore, the model is easily implemented in standard FE software using cutoff bars (CoBs)—truss elements with limited tensile strength—combined with rigid links to simulate the interaction between the FRCM layer and the substrate. This direct implementation eliminates the need for user-defined subroutines [3].

The model is defined by six key parameters ${(\tau }_1{, \tau }_2, {\tau }_3, s_1, s_2, s_3)$ (see Figure 5a), where ${\tau }_i$ is the interface shear stresses and $s_i$ is the corresponding slip at the phase transition points. Notably, four of these parameters (${\tau }_1, {\tau }_3, s_1, s_3$) are common with the trilinear law, meaning only two additional parameters (${\tau }_2, s_2$) need to be calibrated to reproduce the same macroscopic response. To ensure consistency in the total fracture energy between the trilinear and sawtooth interface models, the fracture energy of the sawtooth model is equated to that of the trilinear model. Since (elastic) phase 1 and phase 4 in Figure 5a are identical in both the trilinear and sawtooth models, such energetic equivalence can only be referred to the portion of the diagrams between $s_1$ and $s_3$. These values can be derived by referring to the trilinear model ($G_{f1}$) and the sawtooth model ($G_{f2}$) as follows:

$$G_{f1}={\displaystyle \frac{{\tau }_1+{\tau }_3}{2}} \left(s_3-s_1\right)$$

(1)

$$G_{f2}={\displaystyle \frac{{\tau }_2}{2}}\left(1+{\displaystyle \frac{s_1}{s_2}}\right)\left(s_2-s_1\right)+{\displaystyle \frac{{\tau }_3}{2}}\left(1+{\displaystyle \frac{s_2}{s_3}}\right)\left(s_3-s_2\right)$$

(2)

By equating $G_{f1}$ and $G_{f2}$, ${\tau }_2$ can be calculated as a function of $s_2$ as follows:

$${\tau }_2={\displaystyle \frac{\left({\tau }_1+{\tau }_3\right)\left(s_3-s_1\right)-{\tau }_3\left(1+\frac{s_2}{s_3}\right)\left(s_3-s_2\right)}{\left(1+\frac{s_1}{s_2}\right)\left(s_2-s_1\right)}}$$

(3)

Finally, the slope of each linear ascending branch in Figure 5a can be easily obtained as:

$$k_i={\displaystyle \frac{{\tau }_i}{s_i}}$$

(4)

This procedure allows generating multiple sawtooth interface laws that preserve the same overall fracture energy of the trilinear model while varying ${\tau }_2$ according to the chosen $s_2$. Because each phase is elastic (except the residual plateau), the slip, normal stress in the fibers, and interfacial shear stress can be expressed in exponential form along the bonded length, similar to the trilinear model. This makes the sawtooth approach both accurate and practical, suitable for large-scale FE simulations of FRCM-strengthened members without requiring complex programming. Figure 5c,d illustrate how the sawtooth model derives the global force–slip response of the joint. The left schematic shows the progressive activation of phases along the bonded length, while the right graph presents the corresponding global P-g curve, with key points marking each phase transition. Figure 6 illustrates the flowchart of the main steps of the sawtooth method. For further details, see the reference paper [3].

2.3. Finite Element Method

One of the main advantages of describing the bond–slip law with a sawtooth formulation is that it can be directly reproduced in commercial FE software. This is possible because most FE codes already include simple truss or spring elements with predefined constitutive laws, such as linear elastic with brittle cutoff (perfectly brittle) or linear elastic with plastic cutoff (perfectly ductile). By combining these basic elements, the nonlinear response of the bond interface can be simulated without writing dedicated subroutines [3].

This approach is particularly attractive for practical applications:

• It relies on the simplest type of elements available in FE libraries;
• It ensures fast and stable nonlinear simulations;
• It requires only a minimal theoretical background from the user.

The implementation strategy is schematized in Figure 7. The reinforcement is discretized into finite-length “batteries” composed of three cutoff bars (springs) arranged in parallel, which are connected to the support and to the elastic FRCM layer by rigid beams and appropriate node constraints. This setup reproduces the bond mechanism at the interface. The bars have a finite length $L_{bar}$, chosen by the user (Figure 8) [3].

Each spring is characterized by three parameters:

• $A_{fi}$: Cross-sectional area;
• $E_{fi}$: Elastic modulus;
• $F_{fi}$: Ultimate tensile force of the i-th spring.

For convenience, the area ($A_{fi}$) of each spring can be set equal to unity, which simplifies the calculation of the equivalent stiffness. The top and middle springs are assigned an elastic–brittle behavior, while the bottom spring follows an elastic–ductile law (Figure 7). Together, the three springs reproduce the three phases of the sawtooth law. The equivalent elastic moduli of the springs are computed so that the FE model produces the same slip at each node as the analytical sawtooth interface model. This condition leads to the following triangular system of equations where $p_f$ is the perimeter of a single yarn [3]:

$$\left\{\begin{array}{c}{E_{f1}A}_{f1}+{E_{f2}A}_{f2}+{E_{f3}A}_{f3}=k_1{p}_f{L}_{bar}^2\\ {E_{f2}A}_{f2}+{E_{f3}A}_{f3}=k_2{p}_f{L}_{bar}^2\\ {E_{f3}A}_{f3}=k_3{p}_f{L}_{bar}^2\end{array}\right.$$

(5)

The ultimate resistances $F_{fi}$ of the three springs are computed by formulating and solving a system of three equilibrium equations in three unknowns, corresponding to the completion of Phase 1, Phase 2, and Phase 3. The complete system of equations is reported in the following expression. More information on this procedure is provided in [3].

$$\left\{\begin{array}{c}{F_{f1}+F}_{f2}{\displaystyle \frac{s_1}{s_2}}+F_{f3}{\displaystyle \frac{s_1}{s_3}}={\tau }_1{p}_f{L}_{bar}\\ {F_{f2}+F}_{f3}{\displaystyle \frac{s_2}{s_3}}={\tau }_2{p}_f{L}_{bar}\\ F_{f3}={\tau }_3{p}_f{L}_{bar}\end{array}\right.$$

(6)

3. Results

The results of the study are presented in this section. Section 3.1 focuses on the calibration of the trilinear and sawtooth interface models using the experimental measurements from the SLS tests. Section 3.2 describes the implementation of the sawtooth model in a commercial FE code and reports the corresponding model parameters. Finally, Section 3.3 presents the load–displacement response obtained from both the analytical sawtooth model and the FE simulations and compares them with the experimental results to assess the accuracy of the proposed models.

3.1. Calibration of Trilinear and Sawtooth Models

This section presents the procedure and parameter settings for both the trilinear and sawtooth models applied to a PBO-FRCM composite on flat masonry subjected to direct shear tests, as reported in [2].

Table 2. Mechanical and geometric properties of the PBO mesh [1,2].

${\mathit{E}}_{\mathit{f}}$ [MPa]	${\mathit{A}}_{\mathit{f}}$ [mm2]	${\mathit{p}}_{\mathit{f}}$ [mm]	${\mathit{L}}_{\mathit{b}}$ [mm]
241,000	1.260	28	315

lists the elastic modulus ($E_f$) of the mesh along with its geometric characteristics, including the cross-sectional area ($A_f$), the perimeter ($p_f$) of a single yarn, and the total bonding length of the reinforcement ($L_b$), which are used for calibrating the sawtooth model. $p_f$ is estimated here based on the observed experimental failure mode of the specimens (see Figure 3). It is assumed that only one face of the yarn is effectively bonded to the mortar and contributes to load transfer.

Table 3. Parameters defining the tri-linear cohesive material law (CML) [49,50].

${\mathit{\tau }}_1$ [MPa]	${\mathit{\tau }}_3$ [MPa]	${\mathit{s}}_1$ [mm]	${\mathit{s}}_3$ [mm]
1.750	0.250	0.065	0.420

summarizes the parameters of the trilinear model based on [49,50].

In the present study, the sensitivity of the procedure was analyzed with respect to ($s_2$, ${\tau }_2$). Specifically, $s_2$ is considered at three different values, as reported in

Table 4. Parameters define the sawtooth model.

${\mathit{\tau }}_2$ [MPa]	${\mathit{s}}_2$ [mm]	${\mathit{l}}^{\mathit{*}}$ [mm]	$\tilde{\mathit{l}}$ [mm]
3.90	0.18	22.22	50.22
2.88	0.24	33.75	54.61
2.29	0.30	45.24	59.06

; corresponding to one-third, one-half, and two-thirds of the distance between $s_1$ and $s_3$. Consequently, three sawtooth curves with identical fracture energy are obtained, as illustrated in Figure 9. Consistent with these values, three corresponding results for both $l^{*}$ and $l_2$ are also provided in Table 4, following the analytical equations described in [3].

The sawtooth model can be integrated using the procedure described in [3], to which the reader can refer for further details. In Figure 10, for example, and for completeness, the predictions of this model are reported for ${\tau }_2=2.88$ MPa.

The tangential stress distribution along the bonding length is shown at four instants of the loading process, once Phase 4$\le$ is fully developed, corresponding to a tangential stress of ${\tau }_2=2.88$ MPa (Figure 11).

3.2. Finite Element Model: Parameters and Implementation

Here, the implementation of the sawtooth model in a commercial FE code is described. The section reports the definition of the FE model, the selection of element types, and the calibration of parameters such as stiffness and strength. A schematic representation of the FE setup is also provided. The parameters of the cutoff bars, which were described in Section 2.3, were implemented in the commercial FE software Straus7 (release 3.1.6) for the numerical analysis.

Table 5. Parameters of cutoff bars for ${\tau }_2=2.88$ MPa.

${\mathit{E}}_{\mathit{f}\mathit{i}}$ [MPa]	${\mathit{F}}_{\mathit{f}\mathit{i}}$ [N]	${\mathit{k}}_{\mathit{i}}$ [N/mm]
37,233.56	290.42	26.92
27,269.65	793.55	11.73
1458.33	73.50	0.60

presents the input parameters in Strand7 for ${\tau }_2=2.88$ MPa.

Figure 12 shows the numerical representation of the specimen built in Straus7. The model is composed of 30 identical “batteries”: in each one, three stacked cutoff bar segments (one per layer) of length $L_{bar}=10.5$ mm are placed below a single FRCM 2-node linear truss element. At the right side of each “battery,” a rigid beam (beam type with high stiffness) is introduced to connect the three cutoff bars elements with the FRCM truss, ensuring kinematic compatibility of the unit. Consecutive FRCM trusses are then linked by rigid-truss (truss type with high stiffness) connectors of 0.5 mm, which function as stiff ties between adjacent cells. Repeating these units—each consisting of a 10.5 mm cutoff bar segment and a 0.5 mm rigid truss—for 30 segments resulted in a total specimen length of 330 mm in the FE model. These rigid elements are numerical artifacts introduced to couple the truss segments and are excluded from the bonding length $L_b=315$ mm, where the analytical calibration was consistently carried out with $L_{bar}=10.5 \mathrm{m}\mathrm{m}$. The loading was applied under displacement control (prescribed displacement towards the right) in agreement with the experimental setup. At the left end of each cutoff bar, a restraint was applied, fixing all translational and rotational degrees of freedom (DX, DY, DZ, RX, RY, RZ), thus fully constraining the bar end. Meanwhile, at the left end of the FRCM truss element, a restraint fixing only the vertical displacement component (DY) was applied. Finally, at the right end of the last cutoff bar—corresponding to the connection between the bottom bar and the rigid beam—a restraint fixing the vertical displacement and rotation about the global Z-axis (DY, RZ) was assigned. The distribution of axial forces in the cutoff bars for ${\tau }_2=2.29$ MPa, with a maximum value of 4459.37 N in the FE model, is illustrated in Figure 13.

3.3. Load–Displacement Response: Comparison with Experimental Results

This section presents the load–displacement curves obtained from both the analytical sawtooth model and the FE simulations. The results are compared with the experimental data to assess the accuracy of the models, as in Figure 14. Both models show a high capability in capturing the global response of the PBO-FRCM masonry reinforcement. Specifically, the initial elastic phase and the peak load are well-predicted, falling within the experimental scatter. It is worth noting that while the experimental curves show some variability due to the heterogeneous nature of masonry, both the analytical sawtooth model and the FE model provide a consistent representation of the bond–slip mechanism.

The slight oscillations observed in the FE response are attributed to the discretization of the interface into finite truss segments and would be reduced with a finer mesh, as also noted in the reference study [3]. To investigate this effect, a sensitivity analysis was conducted in the present study on the cutoff bar segment length $L_{bar}$—a step not previously addressed in the reference study [3]. In this context, while the value of $L_{bar}=10.5$ mm was adopted for all simulations presented in Section 3.3, three additional values $L_{bar}=21, 5.25, \mathrm{a}\mathrm{n}\mathrm{d} 2.625$ mm were considered here for comparison. Figure 15 illustrates the load–slip response for these four different cases. The results demonstrate that a coarser mesh $L_{bar}=21$ mm results in high-amplitude oscillations and a slight overestimation of the peak load. Conversely, reducing the segment length leads to a smoother response, confirming that the oscillations are a numerical artifact of the discretization level. The results justify that $L_{bar}=10.5$ mm (approximately 3.3% of the total bonding length $L_b=315$ mm) provides an optimal trade-off between capturing the characteristic sawtooth peaks and maintaining computational efficiency. This analysis provides practical guidance for the numerical implementation of the sawtooth model, ensuring stable results across different mesh densities.

4. Discussion

In this section, the numerical and analytical results are compared and discussed, emphasizing the influence of the constitutive parameters $s_2$ and ${\tau }_2$ on the overall bond response. To better illustrate their effect, all load–slip curves obtained from both the sawtooth analytical model and the FE simulations were combined in Figure 16, allowing direct comparison among the different cases. The results in Figure 16 show a particularly good agreement between the sawtooth and FE predictions, confirming the reliability of the analytical law within the numerical framework. Both models successfully capture the major features of the experimental behavior, including the initial stiffness, the peak load, and the post-peak softening trend. The variation in $s_2$ provides a clear means of tuning the balance between bond strength and ductility. The consistency between the analytical and FE models confirms that the numerical implementation is both robust and accurate.

The sensitivity analysis highlights that the parameter $s_2$—and its corresponding stress level ${\tau }_2$—has an appreciable impact on the ductility of the model and a moderate effect on the bonding capacity (which, as is known, depends mainly on the fracture energy, which is kept constant in the three models considered). As the parameter $s_2$ increases, so that $s_2$ decreases, the analytical curves exhibit a gradual reduction in peak load but a significant improvement in ductility. When $s_2=0.18$ mm (${\tau }_2=3.90$ MPa), the model predicts the highest bond strength but with a brittle post-peak response. Increasing $s_2$ to $0.24$ mm (${\tau }_2=2.88$ MPa) slightly reduces the peak load while leading to a smoother softening branch. For the largest value, $s_2=0.30$ mm (${\tau }_2=2.29$ MPa), the peak load decreases further, but the response becomes more ductile. These results demonstrate that the sawtooth model effectively reproduces the bond–slip behavior with a small number of intuitive parameters.

In contrast to the reference study [3], the present FE model does not show a minor underestimation of the load-carrying capacity and ultimate ductility. In fact, the FE predictions are slightly higher than those obtained from the sawtooth analytical model as reported in

Table 6. Predicted maximum loads from sawtooth model and FE analysis.

${\mathit{\tau }}_2$ [MPa]	P max Sawtooth [kN]	P max FE [kN]
3.90	4.25	4.85
2.88	4.20	4.61
2.29	4.16	4.46

, due to the specific mesh discretization and loading implementation used in this simulation. Overall, the results demonstrate that the proposed approach is reliable and, due to its simplicity, suitable for practical design applications. Future research will extend this methodology to more complex scenarios, including curved masonry elements, experimental strengthening of full arches with various FRCM systems, and analytical studies on arches reinforced with various FRCM systems, considering unique design codes.

A key contribution of this study is the investigation of mesh sensitivity in the FE model, a technical aspect not investigated in [3]. The analysis justifies that selecting an appropriate segment length $L_{bar}$ is essential to stabilize the numerical response and prevent the overestimation of peak loads as mentioned in Section 3.3. By providing this practical guidance, the current FE framework ensures more robust and reliable predictions for the FRCM-masonry interface. While the initial results are encouraging, the current model validation is based on a specific PBO-FRCM system and a limited set of flat masonry tests. Additionally, the simplified 1D interface inherently lacks the capacity to capture 3D stress effects or local material heterogeneities. Future research could expand the experimental scope to include diverse FRCM systems, complex geometries, and full-scale members to fully establish the model’s robustness for practical applications.

5. Conclusions

An analytical model and a finite element approach developed to simulate and investigate the bond behavior of PBO-FRCM reinforcing systems externally bonded to flat masonry elements subjected to single lap shear tests have been described in this study. The analytical sawtooth interface model, originally developed for SRG systems in [3], was calibrated and extended to represent the bond–slip response of FRCM-to-masonry reinforcements using experimental results available in the literature [1,2]. The calibration procedure allowed the sawtooth law to reproduce the main phases of the load–slip experimental response, namely, the initial linear stage up to the occurrence of the first cracks; the decrease in stiffness due to progressive crack growth; and the residual capacity related to frictional sliding of the textile within the mortar matrix. A finite element framework was developed based on the characteristics of the calibrated analytical model. Specifically, a simplified discretization strategy using truss elements and cutoff bars was adopted, enabling the nonlinear interface response to be reproduced without the need for user-defined subroutines. This simplified approach does not require specialized numerical tools and is therefore also suitable for applications in professional practice. Furthermore, the investigation into mesh sensitivity revealed that the segment length $L_{bar}$ influences the numerical stability and the smoothness of the predicted load–global slip response. A segment length of 10.5 mm was found to be the optimal discretization size, as it effectively captures the local sawtooth oscillations and prevents the overestimation of the peak load, providing a reliable balance between accuracy and computational efficiency. A sensitivity analysis was carried out by varying the intermediate slip parameter $s_2$ while maintaining constant fracture energy. The results highlighted that this parameter affects the ductility of the response and the shape of the post-peak branch, while its influence on the predicted bond capacity remains moderate. Smaller values of $s_2$ lead to higher intermediate ${\tau }_2$ stresses and a more brittle response, whereas larger values produce smoother softening and increased ductility. This demonstrates that the sawtooth formulation provides a flexible and intuitive way to control the post-elastic behavior of the interface. The comparison between analytical predictions, FE simulations, and experimental results showed good overall agreement in terms of global load–slip response and failure evolution. In particular, the models satisfactorily reproduced the initial stiffness, the peak load, and the post-peak performance observed experimentally. The predicted peak loads ranged (with respect to the considered $s_2$ values) between approximately 4.16 and 4.25 kN for the analytical sawtooth model and 4.46 to 4.85 kN for the FE simulations, showing only limited deviations between the two approaches. Such predictions are in good agreement with the experimental average value (4.45 kN). These results confirm the reliability of the proposed modeling strategy for describing the stress transfer mechanism and debonding evolution along the bonded interface for the considered reinforcements.