Numerical and Experimental Investigation on Cyclic Behavior of Masonry Infill Walls Retrofitted with CFRP

Abstract

This paper presents verification of the numerical model of masonry infill walls against the experimental results. Three cases are investigated: an undamaged model, a damaged model, and a carbon fiber-reinforced polymer (CFRP) strip. ABAQUS commercial finite element model (FEM) software was used in the modeling. Nonlinear behavior as well as cracking and crushing of masonry bricks were simulated using the Concrete Damaged Plasticity (CDP) model. To solve this, a three-dimensional simplified micro-model was used. Experimental and simulation of the hysteresis curve, skeleton curve, damage patterns, maximum and minimum stresses, and plane strain distribution were compared. The changes in natural frequencies, and mode shapes before and after CFRP strengthening masonry wall are evaluated. A sensitivity analysis was done to study the effect of damage and strengthening on the nonlinear behavior of steel frames with masonry infill. This investigation demonstrated that the numerical model was able to effectively simulate and predict the strength of these models. Then a look at the effect on seismic performance is reported and commented on.

1 Introduction

The seismic behavior of masonry-infilled buildings has been a subject of investigation for the last half-century, with numerical and experimental studies concentrating on the interaction of infill walls with reinforced concrete (RC) and/or steel frame structures. Masonry walls are often applied as inner or outer walls in steel and reinforced concrete buildings, creating composite buildings known as infilled frames. These buildings indicate increased lateral stiffness and strength compared to those without walls, generally referred to as bare frames. The existence of walls made of stone considerably affects the performance of the bounding frames, particularly under seismic loading. Nevertheless, many building design engineers typically overestimate the importance of these infills, identifying them as nonstructural components, which can lead to incorrect outcomes. Over the last sixty years, a range of experimental and computational investigations [1,2,3,4,5] have been undertaken to develop rational approaches for accounting for the infill’s contribution to the overall stiffness and strength of structural systems. The progress of computing technology over the past few decades has greatly improved the usage of numerical modeling, incorporated into specialized programs for modeling the behavior of masonry-infilled frames. This has been notably obvious in the carrying out of FEM [6,7,8]. The modeling of infilled frames has been greatly facilitated by discrete element methods [9, 10].

Significant numerical investigations have concentrated on studying the properties of steel frames made from material [4, 11, 12]. At the same time, there has been progress in laboratory studies on infilled frame structures, particularly steel frames. This study has mainly focused on monotonic and cyclic quasi-static testing [13, 14] and, more recently, pseudo-dynamic testing [15]. Mehrabi et al. [16] develop interface models that describe the shear cracking in concrete and mortar joints, as well as the bond-slip behavior of steel reinforcement in concrete. Canbay [17], and Anıl and Altın [18] examined the cyclic behavior of RC infill walls through tests and static pushover analysis for verification. Darwish [19] and Sivri [20] conducted studies on the numerical modeling of RC infill walls subjected to cyclic loading, adopting parametric analysis. Lotfi and Shing [21] proposed a smeared crack model to simulate the nonlinear properties of masonry and concrete in infilled RC frames. Al-Chaar et al. [22] employed smeared crack quadrilateral elements to represent masonry and a cohesive interface model to simulate the behavior of mortar and shear failure in concrete. Stavridis and Shing [23] created a 2D micro-model for masonry-infilled RC frame analysis, adding cohesive crack interface features in order to simulate mortar impacts. Mohyeddin et al. [24] created a 3D micro-model, separating the mortar at joints and establishing an elastic interaction model between mortar layers. Minaie et al. [25] applied the Concrete Damaged Plasticity (CDP) model in ABAQUS to evaluate the one-directional loading behavior of whole and partially coated masonry shear walls. Kim and Yu [26] investigated the effects of masonry strength in infill walls on RC and steel frames subjected to lateral loads. The results revealed that infill RC frames exhibited higher residual strength, initial stiffness, and ultimate strength compared to frames without masonry infill walls. FE modeling of masonry infills divides into micro- and macro-modeling approaches. Micro-models enable a full description of masonry infills, producing exact behaviors. However, this approach needs massive processing resources, making it insufficient for larger structures. On the other hand, macro-models employ a more generalized representation of infill wall panels. This is often achieved by demonstrating the masonry infills as homogeneous isotropic walls or by replacing each infill panel with one or several diagonal struts. The selected method of analysis inside the FE frame considerably affects the cyclic behavior evaluation of masonry-infilled structures: macro-modeling [27,28,29,30,31,32] and micro-modeling [33,34,35]. The implicit method can be applied to a wide range of linear and nonlinear issues, requiring iterative strategies for each load increment to identify solutions. On the contrary, the explicit method is particularly suitable for short transient situations like impact loadings and is commonly applied in research. However, in analysis, the time dependence might create noise in seismic response assessments of structures, potentially leading to incorrect evaluations of the hysteresis behavior in steel plate walls, as reported by Azandarian [36]. Moreover, the addition of infill masonry walls provides additional failure mechanisms, such as short column effects, soft story phenomena, and torsional failures, which are commonly neglected in structural design [37]. Furthermore, the failure mechanisms of infilled frames have shown nonductile characteristics, as observed in studies like [38]. In order to improve our comprehension of how existing infill frames respond to seismic activity and develop efficient strategies for strengthening them, it is essential to employ dependable techniques for predicting the many failure modes that may occur during earthquakes. Numerous seismic retrofitting methodologies, such as steel bracing, column jacketing, and the application of FRP layers, are applied to improve the seismic resistance of infill structures. Recently, FRP has become popular as a modern option for building repair, given its multiple advantages, including a high strength-to-weight ratio, flexibility, and onsite customization. Batikha [39] have been studied about to improve both the load-carrying capacity and stiffness of FRP-RC masonry. For example, Yuksel et al. [40] and Ozkaynak et al. [41] validated that a retrofitting strategy involving cross-diamond bracing efficiently increases shear resistance, exceeding other methods. This method significantly reduces wall corner failures, where cracking commonly occurs under lateral loads. Additionally, Altin et al. [42] have seen a 50% increase in stiffness with the increase in CFRP strip thickness. Butenweg et al. [43] investigated complex nonlinear behavior in concrete infill walls and wall–frame interfaces, finding agreement useful for parametric studies and design recommendations. Finite element models using micro-modeling demonstrate good agreement. To address the weakness of masonry infills, Akın et al. [44] investigated FRP-strengthened infill walls using a numerical model. They validated simulation results by comparing results from OpenSees software with experimental response curves. A parametric study was conducted to further investigate the effect of the aspect ratio on hollow clay tile infill walls. The findings revealed that the use of diagonal CFRP strips significantly enhanced both the lateral strength and stiffness of the frames.

Similarly, other researchers have developed numerous strengthening techniques using different approaches and materials. These include the use of fiber-reinforced polymer (FRP) sheets [45], strengthening with FRP sheets and anchors [46], and strengthening with plaster mortar [47]. Although the aforementioned experimental investigation has contributed significantly to understanding the performance of steel frame interactions, the challenges and high expenses associated with conducting these tests, particularly for extensive parametric studies, have made FE modeling more common.

As a result, there is a pressing need for greater precision in numerical models to permit complete parametric analysis. The experimental study conducted by Padalu et al. [48] demonstrated satisfactory performance in terms of strength, stiffness, ductility, and energy dissipation. The researchers compared the testing findings of RC frame samples with various types of infill strengthening materials, including FRP, fiber mesh, and wire mesh. Similarly, other researchers such as Leeanansaksiri et al. [49], Coccia et al. [50], Tanjung et al. [51], and Furtado et al. [52] focused their investigations on the strengthening of masonry infills. Various researchers have developed numerical models to evaluate the performance of masonry infill walls under both cyclic and monotonic loads, employing macro- and micro-modeling techniques. However, when observing the condition of masonry infill walls after earthquakes, it becomes evident that many problems and questions remain unsolved, necessitating further studies and development. The main challenge lies in representing the nonlinear behavior of elements and modeling the interaction between different materials, such as concrete and masonry. While previous numerical studies have demonstrated the capability of finite element (FE) models for simulating masonry infills or shear walls, some limitations have been identified. The use of 2D macro-models, although simple, proved inadequate in capturing various aspects of infilled frames, including nontypical geometric properties, stress concentration, local reinforcement effects, and out-of-plane behavior. In the case of existing 3D simplified micro-model studies, there is often a lack of information regarding input material parameters, making it challenging for others to replicate the model and reproduce the associated results.

Against this background, the purpose of this study was to conduct numerical analyses using Abaqus software and compare them with the findings of a previous experimental study on typical half-scale steel frames with brick masonry infill. For the current numerical analysis, a 3D finite element model was created to investigate the in-plane behavior of steel frames filled with. An investigation was conducted to examine the effects of different cases, including undamaged, damaged, and strengthened models. The results are discussed in the form of dynamic characteristics such as frequency values and mode shapes, load–displacement hysteretic curves, damage patterns, maximum and minimum principal stresses, and plane strain distribution. The complete validation of the model in contrast to the outcomes of the experiments was discussed. Also, the seismic performance of the damaged and strengthened models was determined.

2 Description of the Experimental Campaign

2.1 Test Setup

The structural behavior of steel frames with masonry infills was analyzed in this study using experimental tests conducted by Samberou [53]. A 1/2-scaled, single-story rectangular infilled steel frame was cast for this purpose. The model comprises two W250 × 58 columns and one W200 × 46 beam, with a height of 1500 mm and a span of 1200 mm. The height/length ratio is approximately 1.25:1. The brick infill has dimensions of 185 mm (length), 285 mm (height), and 100 mm (thickness). The dimensions of the model. The experimental test and the hollow brick units are depicted in Fig. 1. The construction of the model involved single-wythe masonry walls as infill, and the experimental tests subjected the model to reversible and repeated horizontal loads. Experimental measurements were carried out using ambient vibration tests [54, 55].

Fig. 1

Front view geometry of the steel frame and hollow brick dimensions [39]

2.2 Lateral Loading Cycles

Figure 2 depicts the application of a displacement control loading approach in the current study. The loading methodology was established in conformity with FEMA 461 [56]. A low-cycle cyclic load was applied to the centroid of the top frame beam.

Fig. 2

Applied horizontal displacement protocol (FEMA 461) [40]

This load is frequently used for seismic performance assessment of nonstructural components, enabling the measurement of all levels of damage to develop appropriate fragility models [57].

Three different cases are taken into account to simulate the states that include intact, impaired, and reinforced. The undamaged infill wall is designated as Model-W0, the damaged as Model-W1, and the strengthened with CFRP strip as Model-W2. The experimental testing consisted of applying cyclic loadings to the masonry infill wall, with a range of −100 to 100 kN (Fig. 3).

Fig. 3

With and without CFRP strengthening experimental models [53]

3 Description of the Numerical Simulation Approach

This study has concentrated on the numerical verification of three different model cases: the undamaged infill model (Model-W0), the damaged infill model (Model-W1), and the infill frame strengthened with CFRP strips. The FE modeling in this study adopts ABAQUS [58]. An advanced nonlinear software package known for numerous material constitutive laws and strong contact features is perfect for simulating brick infill walls. This commonly utilized commercial FE software simulates modal analyses and cyclic nonlinear quasi-static analyses. The subsequent parts detail the structure of geometry, the meshing procedure, constitutive material models, and the particulars of the numerical studies.

3.1 Definition of Materials

The material properties utilized in the analysis were selected based on findings from literature studies investigating similar materials. For brick units, the elasticity modulus and density values were considered within the ranges of 3.5–34 GPa and 1200–2000 kg/m², respectively [59, 60]. The mechanical properties of the steel profiles were obtained from ASTM-A36 [61]. Both linear elastic and nonlinear material properties employed in the analyses are detailed in Table 1.

Table 1 Linear elastic material properties used in FE analyses [62]

The literature includes a number of constitutive models based on the fundamental concepts of elasticity, plasticity, and continuum damage mechanisms. In this study, the Concrete Damaged Plasticity (CDP) model is applied to evaluate the mechanical properties of bricks, based upon the micro-modeling approach initially proposed by Lourenco [63] and further improved by Lourenço et al. [64]. This modeling approach is particularly suitable for simulating isotropic brittle materials, such as masonry units, as demonstrated by Moradabadi et al. [64] and Page [65]. CDP was chosen to represent the tensile behavior of the masonry infill wall [66]. The damage parameters and variables in the CDP model are based on the assumption that damage occurs only in the softening range [67]. The specific values of the parameters in the CDP model may vary based on the type of concrete being modeled and the loading conditions. Therefore, it is advisable to calibrate the CDP model parameters using experimental data before employing them in numerical simulations [68]. The CDP model can be implemented in ABAQUS to analyze concrete and other quasi-brittle materials, and it stands as one of the most widely used models for simulating concrete’s nonlinear behavior [69]. Modal analysis and nonlinear time history analysis were conducted to assess the structural performance of each minaret. In the nonlinear analyses, the CDP material model was employed. This constitutive model combines plasticity theory and damage mechanics, assuming that the main failure mechanisms of the material are tensile cracking and compressive crushing. The evolution of the yield (or failure) surface is controlled by two hardening variables, $\varepsilon_t^{pl} \;{\text{and}}\;\varepsilon_c^{pl} .\;{\text{Where,}}\;\varepsilon_t^{pl} \;{\text{and}}\;\varepsilon_c^{pl}$ linked to failure mechanisms under tension and compression loading and refer to as tensile and compressive equivalent plastic strains, respectively. These variables are referred to as tensile and compressive equivalent plastic strains, respectively. The stress–strain diagrams of the CDP model, adaptable to masonry structures to define nonlinear material behavior, are presented in Fig. 4.

Fig. 4

Response of material to uniaxial a tension and b compression [63]

The stress–strain response exhibits a linear elastic relationship under uniaxial tension until reaching the peak stress, $\sigma_{t0}$. The initiation of micro-cracking in the material corresponds to the peak tension stress. Subsequently, the stress–strain response is characterized by material softening. Under uniaxial compression, the stress–strain response follows a linear elastic relationship until reaching the initial yield value, $\sigma_{c0}$. The plastic regime is marked by stress hardening, followed by strain softening beyond the ultimate stress, $\sigma_{cu}$. The stress–strain relations under uniaxial tension and compression loading are defined in Eqs. 1 and 2.

$$\sigma_t = (1 - d_t )E_0 (\varepsilon_t - \varepsilon_t^{pl} ),$$

(1)

$$\sigma_c = (1 - d_c )E_0 (\varepsilon_c - \varepsilon_c^{pl} ),$$

(2)

In these equations, ε_c and ε_t represent the total strain in compressive and tension conditions, E₀ is the initial modulus of elasticity, σ_c and σ_t are compressive and tension stresses, and d_c and d_t are compressive and tension damage parameters, respectively.

The stress–strain behavior of brick masonry is indicated in Fig. 5. The stress–strain response follows a linear elastic relationship under uniaxial tension until the peak stress σ_t0 is reached, indicating the initiation of micro-cracking. After reaching the peak tension stress, the material exhibits softening. To address the issue of convergence difficulties encountered when elements exhibit softening behavior, the factor μ_v is introduced. This parameter plays a critical role in stabilizing the numerical model and ensuring reliable convergence during the simulation of softening responses in materials [70]. In implicit analysis, material models that display softening behavior and stiffness degradation can frequently result in significant convergence challenges. This approach modifies the consistent tangent stiffness of the softening material to maintain a positive value, particularly when sufficiently small time increments are used. This method is crucial for ensuring stability and accuracy in simulations involving materials with softening characteristics (Tables 2 and 3).

Fig. 5

Stress–strain properties of brick unit: a compression and b tension [63]

Table 2 Mechanical parameters of CDP model used in brick wall [63]

Table 3 Compression and tensile stress–strain values of brick wall [63]

In this study, the CFRP material is modeled as a linear elastic isotropic material through the moment of failure, showing negligible yielding when compared to steel reinforcement. Figure 6 depicts the selected model, in conjunction with its parameters: E_f represents the elastic modulus of the fiber; σ_fu is the ultimate stress of the fiber; and ε_fu is the tensile failure strain of the fiber. The above method of modeling CFRP material represents its particular mechanical characteristics, especially its linear elastic response until failure.

Fig. 6

Constitutive model for CFRP strip [71]

3.2 Contact Model

The cohesive interaction in materials has been defined as a function of the displacement separation between the edges of potential cracks [72]. Permeability studies have also been conducted to investigate the parameters that affect the performance of cohesive interaction in loose materials. The studies suggest that the mechanical behavior of cohesive elements can be described in three modes: (1) based on uniaxial stress, (2) based on continuity, and (3) based on a tension-separation structural model. The traction-splitting pattern represents the corresponding initial separation caused by the normal stress (Fig. 7).

Fig. 7

The view of three traction-splitting patterns

The mortar was not explicitly modeled in the numerical simulation. Contact elements are used at the interfaces of the bricks to allow for the presence of the mortar. The contact elements employ a cohesive surface-based behavior that models the mortar joints using traction separation of the interfaces [5]. This approach allows for a linear elastic transmission of compressive stresses, tensile stresses, and shear stresses until the maximum strength of the mortar is reached. In addition to maximum strength, friction that contributes to the shear stress is activated. For the transfer of compressive stresses, a hard contact method is used whenever the surfaces are in contact.

3.2.1 Definition of Interaction

The constitutive behavior of masonry bricks and interfaces was defined using material models. The present study employed nonthickness cohesive elements. The linear elastic traction separation law prior to damage is written in terms of the tractions, t, and separations, δ, constitutive matrix, as shown in Eq. 3.

$$t = \left\{ {\begin{array}{*{20}c} {t_n } \\ {t_s } \\ {t_t } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {K_{nn} } & {K_{ns} } & {K_{nt} } \\ {K_{ns} } & {K_{ss} } & {K_{st} } \\ {K_{nt} } & {K_{st} } & {K_{tt} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\delta_n } \\ {\delta_s } \\ {\delta_t } \\ \end{array} } \right\}.$$

(3)

In the context provided, the nominal traction stress vector t comprises three components: t_nt_s, and t_t, which represent the normal and two shear tractions, respectively. Corresponding to these tractions are the associated separations, δ_n, δ_s, and δ_t. The stiffness of the interface, denoted as K, is expressed in Eqs. 4 and 5. This definition of K is crucial in modeling the behavior of the interface, particularly in relation to the tractions and separations.

$$K_{nn} = \frac{E_u E_m }{{h_m \left( {E_u - E_m } \right)}},$$

(4)

$$K_{ss} = K_{tt} = \frac{G_u G_m }{{h_m \left( {G_u - G_m } \right)}}.$$

(5)

In the given context, $E_u$ and $E_m$ represent the Young’s modulus, while $G_u$ and $G_m$ are the shear modulus for the block and mortar, respectively. Additionally, h denotes the actual thickness of the mortar layer. The validity of this methodology has been corroborated in [73] study, through detailed discontinuum finite element analyses. In this study, the tensile strength ($t_n^{\max }$) and shear strengths ($t_s^{\max }$) and ($t_t^{\max }$) were calibrated based on the diagonal compressive testing of masonry triplets, as reported in [15], employing the formulas presented in Eq. 6. This calibration process ensures that the model accurately reflects the mechanical properties of the materials used in masonry construction.

$$\left( {\frac{t_n }{{t_n^{\max } }}} \right)^2 + \left( {\frac{t_s }{{t_s^{\max } }}} \right)^2 + \left( {\frac{t_t }{{t_t^{\max } }}} \right)^2 = 1.$$

(6)

The initial behavior of the element is assumed to be linearly elastic. This is followed by the initiation and progression of the damage. The criteria for damage initiation and propagation are defined by the mixed modes of tensile behavior (Mode I) and shear behavior (Mode II). The quadratic stress criterion is adopted for the initiation of the damage.

Damage evolution is linear, and the shrinkage and shear fracture of the mortar joint depend on the energy release rate G_I and G_II. These fracture shrinkage energy rates were estimated based on the [21] study. According to the Benzeggagh and Kenane criterion [74], when the shear failure energy velocity is reached, the fracture of the mortar joint has occurred. In the simulation investigation, the mortar interfacing the bricks is considered a cohesive surface interaction, providing a variety of interface elements, notably cohesive elements that are frequently used to define bonded or sliding phenomena, as noted by Diehl [75].

These cohesive features are divided into two distinct categories: element-based and surface-based cohesive elements, both capable of performing the behavior of mortar joints. In the study conducted by Aref et al. [76], element-based cohesive elements were used, combining a set of users’ constitutive models to precisely represent the cyclic behavior of mortar.

3.2.2 Type of Contact

This study includes the interaction between steel and masonry walls, applying friction as an important component. The model combines tangential behavior and the concept of “Hard” contact, allowing no softening and no penetration between surfaces. The friction coefficients are chosen to properly illustrate the materials involved, with 0.75 for brick masonry and 0.57 for brick-to-steel contact, providing the most accurate representation of the physical interactions in the investigation. The properties of interfaces between bricks and infill to frame joints are listed in Table 4.

Table 4 Properties of interfaces between brick units and infill to frame [66]

On the other hand, the mortar joints between adjacent masonry units are modeled using contact pairs, an approach referred to as “simplified micro-modelling” in the literature. This method strikes an optimal balance between simulation accuracy and computational efficiency, contrasting with the more intricate “simplified micro-modelling” technique where mortar joints are explicitly modeled [2]. The interactions between the bricks themselves, however, are represented by zero-thickness interface interactions. This methodological choice is pivotal in accurately depicting the mechanical behavior of masonry structures while maintaining computational manageability.

The “General Contact” interaction technique in ABAQUS was applied to simulate the interactions between steel frames and masonry infill walls, as well as among masonry units following mortar joint failure. This technique successfully captures the linear relationship between shear and normal forces on contacting surfaces, therefore generating frictional forces at the boundaries of these surfaces. For this purpose, the friction coefficient μ was chosen at 0.4. The interactions between the bricks themselves, however, are represented by zero-thickness interface interactions, employing contact components as shown in Fig. 8. This methodical approach is crucial for accurately capturing the mechanical behavior of masonry structures while maintaining computational efficiency.

Fig. 8

Simplified micro-modeling of masonry brick [77]

3.3 FE Modeling

3.3.1 Interaction of CFRP–Wall

The CFRP in the FE model is represented using shell elements. In the experimental test, the FRP was seen to attach effectively to the damaged masonry wall, with no signs of surface deboning from either the masonry wall or the steel frame. Consequently, in the numerical study, the surface interactions between the CFRP and the masonry wall are established using a tie constraint. In this configuration, the wall surface operates as the master surface, while the CFRP is designated as the slave surface. This approach ensures a robust and accurate depiction of the strengthening case in the model. The CFRP-strengthened model is represented in Fig. 9.

Fig. 9

ABAQUS model of the CFRP-strengthened case

3.3.2 Type of Boundary Conditions and Loading

In the numerical models, horizontal loads were applied to the beams as per the experimental tests. FE model’s schematic is illustrated in Fig. 10a, where horizontal excitation is applied monotonically to the upper flange of the steel beam. Fixed boundary conditions, as shown in Fig. 10b, are obtained by restricting both translational and rotational movements at all nodes on the bottom of the model. To replicate real-world loading situations and avoid the effects of abrupt peak loading, the load was applied monotonically with a smooth amplitude. Furthermore, the model includes geometric nonlinearity to describe the real in-plane behavior of the masonry infill wall models under load. All interactions between beams and columns are assumed to be rigid, with any damages or fractures at these connections not considered in the analysis. The interactions between the steel columns and beams are established using tie constraints, guaranteeing that the load transmission mechanism is appropriately represented.

Fig. 10

ABAQUS model a model geometry and b boundary conditions

3.3.3 Type of Element and Meshes

In the modeling process, brick units were represented using eight-node three-dimensional components with reduced integration and a Gaussian integration point (C3D8R). The steel frame and CFRP strips were modeled using four-node shell elements (S4R). To establish the appropriate mesh size and type, mesh sensitivity analyses were done. To achieve an equilibrium between computational effectiveness and precision, multiple mesh sizes were chosen for various structural components. Specifically, a coarser mesh of 100 mm was chosen for the T-shaped steel beam. In contrast, more precise meshes were utilized for the infill and columns, with sizes of 40 and 60 mm, respectively. This mesh size was demonstrated to be sufficient for accurate findings and appropriate analysis time through mesh convergence research. The same mesh size was extended to the CFRP. The chosen mesh arrangements for both the masonry-infilled RC frame and the brick units are presented in Fig. 11.

Fig. 11

Generated mesh of the model

For the representation of the masonry infill wall’s behavior, a homogenized, simplified micro-modeling approach was used [78, 79]. This micro-model is especially important for our investigation, which focuses on the accurate behavior of infilled frames, demanding a comprehensive nonlinear analysis.

3.3.4 Type of Analysis and Convergence

In this verification study, two types of analysis were conducted: modal analysis and nonlinear cyclic analysis. The nonlinear cyclic analysis was done in two phases using numerical model. In the first step, the structural self-weight and additional dead loads were applied. The second phase involved a quasi-static analysis via explicit solver, material nonlinear analysis. The ability of ABAQUS for such analyses has been verified in numerous analytical and numerical studies [81,82,83,84,84]. However, it is noteworthy that highly nonlinear problems, like those modeling masonry buildings, typically have challenges in FEM solutions, including convergence difficulties and premature termination of the solution in the early phases of the nonlinear regime. The current study focuses on the application of explicit analysis utilizing ABAQUS software. This method is crucial for ensuring stability and accuracy in simulations involving materials with softening characteristics.

4 Results and Discussion

4.1 Modal Analyses

In this section, three different cases (undamaged, damaged, and strengthened) are considered for the masonry-infilled steel frame model to show the changes in dynamic characteristics. This is one of the most important analyses to demonstrate the accuracy of the FE model. However, the FE model includes many uncertain parameters, such as material property, boundary condition, mesh size, element type, and modeling assumptions. For this reason, it is necessary to verify the numerical results with experimental measurement and reduce the possible differences to acceptable limits by using a model updating approach. Table 4 summarizes the numerical and experimentally identified first three frequency values for undamaged, damaged, and strengthened masonry infill walls. It can be seen that there is a good agreement between numerical and experimental results; the average differences were calculated as 0.75% for Model-W0, 3.87% for Model-W1, and 0.99% for Model-W2. Significant decreases in frequencies occurred with the damage. The maximum reduction was calculated as approximately 21% in the second mode. After the strengthening, there was a significant increase in the frequencies. Compared to the damaged state, the maximum increase percentage is set at about 36% in the first mode. It is seen that the strengthened model becomes more rigid than the undamaged condition. The first three frequency values for each case obtained by numerical and experimental methods are summarized in Table 5.

Table 5 The first three frequency values for each case obtained by numerical and experimental methods

The numerical and experimental first three mode shapes are depicted in Table 6. The first mode is transverse mode in-plane, followed by torsion mode in-plane, and finally out-of-plane mode. It can be seen that the numerical and experimental mode shapes obtained for the undamaged, damaged, and strengthened cases are in harmony with each other. It has been observed that the mod shapes obtained for the damaged and undamaged scenarios are the same, and there is no change in the mod shapes after the damage. However, after the strengthening, the 1st and 3rd mode shapes have changed places. Table 6 shows the first three mode shapes for each case obtained by numerical and experimental [53].

Table 6 The first three mode shapes for each case

4.2 Nonlinear Analyses

The numerical simulations carried out for this investigation contained two important steps: modal analysis and cycle analysis. Figures 12 and 13 present comparative findings between the outcomes of previous experimental testing and the simulated damage patterns for the damaged and strengthened cases, respectively. As indicated in Fig. 12, a comparison between the experimental test results and the FEM of Model-W1. The masonry infill wall was characterized by diagonal and transverse cracking at the horizontal mortar joints. Particularly, the diagonal cracks were mainly situated at the upper right and lower left corners of the brick masonry infill wall, as depicted in Fig. 12b. After to the application of CFRP strips, damage took place mainly in the bottom left corner of Model-W2, as shown in Fig. 13b. It is particularly important to cracking that estimating the extent of damage in a CFRP-strengthened infill wall through visual examination alone can be challenging, especially in cases of small failures. In the simulation results, areas highlighted in red indicate where the stress surpassed the cracking or tensile strength, while green areas indicated imminent cracking or failure. The present pattern corresponds well with the quasi-static test results. The agreement between the actual numerical simulation results and experimental findings data demonstrates the efficacy of the simplified micro-modeling approach applied in the FE study. This approach successfully replicated the practical in-plane behavior observed in the damaged and strengthened models. In all models, corner crushing appeared as the ultimate failure mechanism, typically following the development and propagation of diagonal cracking. This knowledge is crucial for understanding the failure mechanisms of masonry infill wall under different load scenarios.

Fig. 12

Comparison of damage patterns for damage case between test and FE results. a Experimental [53]. b Numerical

Fig. 13

Comparison of damage patterns for strengthening case between test and FE results. a Experimental [53]. b Numerical

Figure 14 displays the displacement contours of the infill masonry wall model in two different cases: with and without CFRP strengthening. This component of the study examines the impact of applying CFRP strips on the performance of Model-W2 when subjected to cyclic loading. The results, as represented in Fig. 14b, reveal a significant reduction in wall displacement by about 62% related to the addition of CFRP strips. This study points out the beneficial effects of CFRP strips on increasing the structural strength of masonry walls against loads, indicating their potential as a possible strengthening solution.

Fig. 14

Maximum displacement contour diagram (in m unit). a Model-W1. b Model-W2

Figures 15 and 16 provide a comparison study between the outcomes of previous tests and the simulated hysteretic curves for both damaged and strengthened cases, respectively. Achieving aspect of this comparison is mostly due to the absence of experimental data in its raw form. Despite this, the approach used permits for an accurate and clear comparison, even without the data being displayed on the same graph. The hysteretic and backbone curves are crucial in assessing the nonlinear seismic response of the models, since they offer insights into the strength and deformation properties. Figures 15a and 16a illustrate that before yielding, both FE models displayed a constant increase in load-carrying capacity with displacement, minimum residual deformation, and no significant stiffness loss. However, as the displacement increased, cracking developed. Post-yielding, the hysteretic curve of both models demonstrated an “arc” shape, with pinching effects due to the opening and closing of cracks in the FE models.

Fig. 15

Experimental by Samberou [53] and numerical results of the Model-W1. a Experimental. b Numerical

Fig. 16

Experimental by Samberou [53] and numerical results of the Model-W2. a Experimental. b Numerical

A finding is that the hysteretic curves of the simulated and experimental models closely coincide, verifying the robustness of the FEM. Differences in the shape of these curves could be attributable to the weak restricting during the experimental test. Figure 15 depicts comparisons of representing and predicted the cyclic load–displacement curves with the numerical results for Model-W1. It is notable that the peaks on the graph are at 96 and −96 kN. The initial peak is attained at a displacement close to 9.5 mm, aligning closely with the 9.55 mm displacement observed in the micro-modeling analysis, resulting in a minor error margin of 0.3%. Subsequently, the final tensile peak is observed when the force approximates 60 kN, coinciding with a displacement of 18 mm.

Figure 16 illustrates comparisons of representing and predicting the cyclic load–displacement curves for Model-W2. In the graph on the wall, it is indicated that a displacement of approximately 17 mm corresponds to around 13 kN. However, the initial peaks occur at about 136.62 kN for displacements of 15.45 mm in compression. In tension, the first peak is achieved with less than 18 mm of displacement. As the forces are absorbed, the wall weakens, resulting in the second peak being observed at 130 kN for a displacement of 25 mm. Regarding tension, the last peak occurs at 95 kN with a displacement of approximately 32 mm.

Figure 17 illustrates the backbone curves for Model-W1 and Model-W2, derived from the hysteretic cyclic loading data. The bearing capacity is obtained by connecting the peak points of the load–displacement graphs derived from the numerical simulation. By comparison, the backbone curve of the strengthened wall in Fig. 10b is significantly higher than that of the damaged wall in Fig. 10a. Similar behavior of bearing capacity is described in the literature [85].

Fig. 17

Numerically obtained backbone curves of infill wall models. a Model-W1. b Model-W2

To further aid in understanding and distinctly comparing the load-bearing capacity and maximum displacements before and after damaged and strengthened models. Table 4 shows the ultimate load damaged and strengthened models and the displacement value under the ultimate load obtained by simulation result and experimental finding (Table 7).

Table 7 Numerical and experimental load-carrying capacities and the maximum displacements

The maximum difference ratio between the numerical and experimental load-bearing capacity values for both models is 3.63%. Similarly, the maximum displacement difference ratio is also acceptable, with a maximum of 3.00%. These results indicate that the numerical model is effective in predicting the behavior of masonry infill walls subjected to cyclic loading. Therefore, the validation study confirmed that the developed FE micro-model could effectively predict the lateral capacity and crack pattern for each case. The numerical modeling predicted quite well the results of the in-plane quasi-static test.

A contour plot displaying the maximum principal (tensile) stresses and minimum principal (compressive) stresses at the initial peak point for both the damaged and strengthened cases is illustrated in Figs. 18 and 19, respectively. In the damaged case, the compressive stresses primarily localize along the loaded diagonal, as depicted in Fig. 18a. These minimum principal stress values occur only in a small and few local areas where they exceed the wall compressive strength, as detailed in Table 1. Simultaneously, in the center and upper left of the wall, principal tensile stresses are observed (Fig. 18b). As expected, the highest tensile stress obtained for the model was 0.39 MPa, which is over 0.22 and 0.15 MPa for the brick units and mortar, respectively. This stress value is considerably high, especially in comparison to the limited tensile strength of the wall.

Fig. 18

Maximum and minimum principal stress of Model-W1 numerically obtained, in Pa

Fig. 19

Maximum and minimum principal stress of Model-W2 numerically obtained, in Pa. a Maximum principal stress. b Minimum principal stress

Figure 19 depicts the maximum principal stress from the FE simulation results, indicating that the upper corner areas of the masonry wall are subjected to the highest stress, reaching 3.12 MPa. This value is slightly above the ultimate compressive stress of the brick. Nevertheless, it is slightly greater than the mortar strength as recorded in the testing data (Table 1). It is also important to illustrate that the tensile stress values obtained were below the tensile strength. The enhanced numerical masonry wall gained a higher load capacity due to the high tensile strength of the CFRP strips bonded to the wall surface.

The plastic strain diagram (PE) module in ABAQUS describes the accumulation result of plastic strain during the entire deformation process. According to Fig. 20a the value of plastic strain of Model-W1 is greater than 0, it means that the material has reached the yield capacity, which can approximate the location and state of wall cracks which might cause the failure.

Fig. 20

Maximum principal plastic strains a Model-W1 and b Model-W2

In Fig. 20b, the cracks propagated along the head and bed joints and in the masonry units themselves of the unstrengthened region of Model-W2. These cracks resulted in the redistribution of tensile stresses over the uncracked regions. Plastic strain contours are a useful tool for understanding where plastic deformation is occurring within a material. While they may not directly represent cracks, they can provide insights into regions of high stress and strain where cracks are more likely to initiate or propagate. It is noted here that plastic strains occur in the damaged areas of the wall, primarily due to tension.

5 Conclusions

A numerical study was conducted to evaluate and compare experimental findings on the in-plane behavior of masonry infill walls. The research focused on predicting dynamic characteristics such as frequency values and mode shapes, load–displacement hysteretic curves, damage patterns, and the stress distribution and contribution to its mode of failure. Based on the FE results, three different cases, such as undamaged, damaged, and strengthened models, were modeled and simulated using 3D-simplified micro-modeling through ABAQUS 2019. Based on this investigation, the following conclusions are presented:

• Model-W2, compared to its Model-W1, has enabled an increase in frequency values of about 57% in first mode. When comparing the CFRP strengthened to an undamaged case, there is an observed increase in frequency value of approximately 24%. This highlights the efficacy of CFRP in restoring the wall’s strength capacities. Since the mass remains constant, the observed increase in strength can be attributed to the enhanced rigidity of the brick infill wall due to CFRP strengthening.

• When comparing numerical and experimental results for undamaged and strengthened cases, the outcomes are remarkably similar, with frequency discrepancies less than 1%. However, this similarity further intensifies in the case of damaged models, where the error percentages remain below 4%.

• Test results indicated that CFRP strips remarkably increase the flexural capacity of infill masonry walls. Moreover, the strengthening systems were effective in enhancing the stiffness of the strengthened model.

• The strength capacity of the Model-W2 was improved by 42% in the case of strengthening.

• The hysteresis curve of Model-W1 indicates that the maximum peak values reach approximately ±97 KN. The largest displacements on this curve fluctuate between 10 and 18 mm. However, one of these peaks is attained at a displacement of 9.55 mm.

• The maximum loads at which peaks are obtained range between 130 and 136 KN. This indicates an enhancement in the mechanical properties of the wall due to the application of CFRP.

• The numerical results for dynamic characteristics, such as frequency values and mode shapes, show a promising correlation with experimental data. The average discrepancies in frequency values are calculated to be 0.75% for Model-W0, 3.87% for Model-W1, and 0.99% for Model-W2. This close alignment suggests that the FE model is sufficiently accurate for practical engineering applications.

• The analysis results demonstrate that the constituted FE model software can predict the load-bearing capacity and maximum displacement values of masonry infill walls under cyclic loading with high accuracy, with a maximum difference ratio of less than 4% for Model-W1 and Model-W2.

• The simplified micro-modeling model without and with strengthened masonry infill walls demonstrated computational efficiency while maintaining accurate prediction of nonlinear behavior under the cyclic loading protocol.

• The results obtained from the nonlinear dynamic analysis indicate that compression at the corners, shear sliding, and diagonal tension failure modes that can occur in the infill wall are highly vulnerable to seismic loads.

• In scenarios involving both damaged and strengthened walls, the minimum and maximum principal stress values decrease by factor times of 0.8 and 2, respectively. These stress values do not exceed the strength capacity of the brick masonry, though they slightly exceed the mortar’s strength. This suggests potential for enhancement of the model through increased CFRP strip-to-masonry infill wall. Additionally, it is noted that the distribution of maximum principal stress is predominantly concentrated in the unstrengthened areas of Model-W2.

These findings suggest that the numerical FE model has significant potential for designing and strengthening models for improved seismic performance. Despite the fact that masonry infill walls exhibit complex behavior due to the distinct material properties and geometries of the brick unit, mortar, steel frame, and CFRP layer, the nonlinear finite element analysis outcomes are in agreement with the experimental test results.