1 Introduction

For decades, the construction industry has devoted significant efforts and attention to improving the safety and durability of existing civil engineering structures. However, the materials utilized in building such structures, including steel and concrete, deteriorate gradually and cannot retain their original qualities permanently. For this reason, innovations aimed at retrofitting and strengthening reinforced concrete (RC) elements have been developed. Notably, much focus is placed on approaches suggested to improve beam shear strength, as failures in this area are a source of significant risk of catastrophic collapses.

Diverse retrofitting systems have been implemented, employing a variety of techniques and materials. Conventional methods involve the use of fiber-reinforced polymer bars [1], high-strength concrete with steel fibers [2], fiber-reinforced concrete [3], steel plates [4], and steel ribbons [5]. Recent investigations have explored alternatives, such as shape memory alloy fibers [6], prestressed near-surface mounted fiber-reinforced polymer reinforcement [7], engineered cementitious composite [8], and retrofitting against out-of-plane loadings [9]. Additionally, hybrid strengthening materials and methods are developed, such as corrective post-tensioning forces with shape memory alloys [10], ultra high-performance fiber reinforced concrete [11], external prestressed hybrid fiber-reinforced polymer sheet [12], and antibacterial cement composites [13], or conjunction with nano silica and binary oxide systems [14], and carbon-modified TiO\(_{\text {2}}\) [15]. It is noteworthy that in most contemporary technologies, the activation of strengthening materials takes place only after external loads have intensified [16]. This implies that the reinforced structure must undergo deformation up to a certain level of damage before the strengthening material begins to contribute, underscoring the requirement for a predetermined threshold of structural damage as a precursor to the reinforcement mechanism.

On the other hand, shape memory alloys (SMAs) offer the unique advantage of applying prestressing forces [17] on elements right from their installation [18], as well as providing self-centering [19], and their activation does not require the use of intricate [20] or costly auxiliary devices [21]. Classified as a subtype of metallic and intelligent materials, SMAs possess the capability to recover their initial, undeformed shape [22] following deformation induced by external stimuli, such as temperature, stress, or a magnetic field [23]. This characteristic, known as the shape memory effect [24], results from a reversible martensitic phase transformation occurring within the alloy at specific threshold temperatures [25]. In the case of SMAs, this transformation occurs when the alloy is cooled below a certain transition temperature, causing it to change from an austenite phase to a martensite phase [26]. The austenite phase is a high-temperature phase characterized by a cubic crystal structure, while the martensite phase is a low-temperature phase with a more complex crystal structure [27]. Additionally, SMAs exhibit superelasticity, allowing them to undergo substantial reversible strains (up to 8% [28]) during loading and unloading at temperatures exceeding the phase-transformation range, beyond what is called the austenite finish temperature (\(A_{f}\)) [29]. Despite considerable efforts in recent years to explore the application of SMAs in concrete structures, these investigations remain ongoing, and further research is deemed essential [30].

The primary objective of the present study was to replicate the findings of a previous experimental study by Traver et al. [31] with the use of ABAQUS finite element software [32]. Additionally, a parametric analysis was performed by systematically varying features of iron-based SMA (Fe-SMA) strips. To the best knowledge of the authors, the computational modeling of beam strengthening using Fe-SMA material in the context of damage is a pioneering line of research. This unique approach involves the initial loading of the beam to a specified damage level, subsequent relaxation, activation of the Fe-SMA material, and a repeat of the loading process. This modeling methodology is potentially applicable to other numerical analysis studies focused on post-damage strengthening. Furthermore, a parametric analysis was carried out to establish the influence of parameters related to the Fe-SMA strips, including thickness, width, and quantity, particularly on shear load-bearing capacity. The results of such parametric analysis can be used for further predictions of the shear strength using soft computing-based methods [33]. This analytical investigation provides valuable insights into identifying optimal methods for strengthening. Additionally, a mesh sensitivity analysis was carried out to assess the model’s dependency on meshing, considering both the accuracy of the model and the computational efficiency of the analysis.

In this study, the behavior of Fe-SMA is regulated by a predefined variable that delineates the various stages of the analysis, both before and after the installation of the Fe-SMA strips. From the initial stage of the modeling and analysis process, the Fe-SMA strips are explicitly defined. However, their mechanical behavior is controlled, such that during stages preceding their installation, they exhibit a very low Young’s modulus, while in the post-installation stages, they accurately reflect their true Young’s modulus. This approach entails the definition of a single computational model, encompassing the entire analysis process. This eliminates the necessity of conducting analyses in two separate models, for the structure with and without strengthening elements. Further details about this method are presented in Sect. 3.1.3. To the best of the authors’ knowledge, the utilization of such a technique has not been explored previously. It can be considered a novel method employed to simulate the retrofitting of concrete structures within a unified platform and simulation model. This method offers a promising approach for future computational simulations using ABAQUS for the strengthening and retrofitting of structures. It enables a more accurate representation of the Fe-SMA material’s behavior at different stages of the process, leading to more reliable and consistent results.

Section 2 provides a brief overview of the conducted experiments and the derived outcomes and results. Section 3 introduces a detailed explanation of the process involved in formulating the computational model. In Sect. 4, a comparative analysis is presented, contrasting the results derived from both the computational simulation and the experimental test. Section 5 contains a comprehensive account of the parametric analysis and explains the findings. Section 6 discusses the implications of the results for subsequent studies and research. Concluding remarks appear in Sect. 7.

2 Description of the experiments

Traver et al. [31] investigated three concrete beams, each with a cross-section of 250 mm by 450 mm and a length of 7 m (see Fig. 1). Each beam’s longitudinal reinforcement consisted of six top and six bottom steel bars, each having a diameter of 20 mm. Beam R-B12 lacked shear reinforcement inside the studied zone shown but beams R-B15C and R-B15S had 0.20% web reinforcement in the form of two-legged closed stirrups (diameter 8 mm) placed at 200 mm intervals. Two displacement-controlled load configurations were used to create shear failures in both statically determinate and indeterminate structures.

These beams were subjected to three shear tests: two cantilever tests (B12C and B15C) and one continuous test (B15S). Steel plates measuring 250 \(\times\) 250 \(\times\) 40 mm\(^3\) were used to apply the load. Both the support and load systems allowed in-plane horizontal displacements and rotations, although during the testing, one support point restricted horizontal displacement. Following failures, both beams were strengthened using 1.5 mm-thick and 30 mm-wide Fe-SMA full wrap strips. Three screws, fastened with nuts and lock washers on both sides of the strips in the lap length at the top of the beam, entirely encircling the beam sections and re-attaching to themselves (see Fig. 2), were used to secure the strips.

Using a heat gun, a temperature of 160\(^{\circ }\)C was applied to the Fe-SMA strips to trigger a recovery stress of around 349 MPa. The beams were loaded again to determine the effect of the strips on the strength of the beam. The principal emphasis of the study was on the shear behavior of the reinforced beams, with particular attention paid to the studied zones assigned for examination [34].

The current investigation concentrated on the beam designated B15-C (with stirrups in the examined zone). This model was chosen to investigate the influence of shear strengthening utilizing Fe-SMA strips on beams already equipped with the existing shear reinforcement, in comparison with model B12C without any stirrups in the analyzed zone. This option is similar to the situations of existing concrete buildings with shear reinforcement that require shear retrofitting owing to the existence of fractures caused by seismic loadings or other severe applied forces.

Fig. 1
figure 1

Geometry, reinforcement, strengthening, and load configuration for each test (dimensions in millimeters), adapted from [35]. The initial damage patterns shown are obtained by DIC: a strengthened RC beam without stirrups; b strengthened RC beam with stirrups loaded on the left side; c strengthened RC beam with stirrups loaded on the right side

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Fig. 2
figure 2

Comprehensive strengthening system used by Traver et al. [31]: a cross-sectional view of the RC beam (dimensions in millimeters); b close-up of the anchoring system; c photograph of the anchoring system

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3 Computational modeling

Subsequent parts elaborate on the approaches and procedures used for constructing the computational model, with Sect. 4 giving a comparison of the results obtained from computational analyses and experiments after establishing the reliability of the computational model. Following the testing of the model’s reliability, a parametric analysis was performed to evaluate the effect of altering the geometries and characteristics of the Fe-SMA strips (see Sect. 5).

3.1 Constitutive models of materials

For ABAQUS simulations, three material components (concrete matrix, reinforcing steel, and Fe-SMA) were provided. These materials’ basics properties and mathematical formulae are presented in this section.

3.1.1 Concrete constitutive model

To define the mechanical characteristics of the concrete material, it was assumed that the material is isotropic and homogeneous. The Concrete Damage Plasticity (CDP) model developed by Lubliner [36], Lee, and Fenves [37] was used to study the behavior of semi-brittle materials. The uniaxial stress–strain relationships of concrete under compressive and tensile loads are depicted in Fig. 3.

Fig. 3
figure 3

The CDP modeling approach for the uniaxial stress–strain correlation of semi-brittle materials: a in compression and b in tension

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The following stress–strain relationships describe the CDP model’s uniaxial tensile and compressive responses:

$$\begin{aligned}{} & {} {\sigma }_{t}=\left( 1-d_{t}\right) E_{0}\left( {\varepsilon }_{t}-\tilde{{\varepsilon }}_t^{pl}\right) \end{aligned}$$
(1)
$$\begin{aligned}{} & {} {\sigma }_{c}=\left( 1-d_{c}\right) E_{0}\left( {\varepsilon }_{c}-\tilde{{\varepsilon }}_c^{pl}\right) . \end{aligned}$$
(2)

The compressive and tensile plastic strains are denoted by \(\tilde{{\varepsilon }}_c^{pl}\) and \(\tilde{{\varepsilon }}_t^{pl}\), respectively. The compressive plastic strain \(\tilde{{\varepsilon }}_c^{pl}\) is defined as the total strain \({\varepsilon }_{c}\) minus the elastic strain corresponding to the undamaged material \({\varepsilon }_{0c}^{el}={{\sigma }_{c}}/{E_{0}}\) and then by subtracting the product of the compressive stress \({\sigma }_{c}\) and a term that depends on the compressive softening coefficient \(d_{c}\) and the initial elastic modulus \(E_{0}\) of the undamaged material. Similarly, \(\tilde{{\varepsilon }}_t^{pl}\) is obtained by subtracting from the total tensile strain \({\sigma }_{t}\) the elastic strain corresponding to the undamaged material \({\varepsilon }_{0t}^{el}={{\sigma }_{t}}/{E_{0}}\) and then subtracting the product of the tensile stress \({\sigma }_{t}\) and a term that depends on the tensile softening coefficient \(d_{t}\) and the initial elastic modulus \(E_{0}\). The compressive and tensile scalar softening coefficients may take values from the interval \(0 \le d_{c} \le 1, \, 0 \le d_{t} \le 1\) and are defined as \(d_{c}=1-{\sigma _{c}}/{\sigma _{cu}}\) and \(d_{t}=1-{\sigma _{t}}/{\sigma _{t0}}\) for stress–strain states on the softening branch, respectively. The CDP model assumes that the reduction of the elastic stiffness of concrete can be described by using a scalar damage variable d as \(E=(1-d) E_0\) with \(0 \le d \le 1\). Furthermore, to account for the uniaxial cyclic conditions the overall damage variable d is defined as

$$\begin{aligned} d =1 - \left( 1-s_t d_c\right) \left( 1-s_{c}d_t\right) , \end{aligned}$$
(3)

where \(s_t\) and \(s_{c}\) are switch functions of the stress state, which in the uniaxial case are

$$\begin{aligned} \begin{array}{rcll} s_t &{}=&{} 1 - w_t r^{\star }(\sigma _{11}); &{} \quad 0 \le w_t \le 1 \\ s_c &{}=&{} 1 - w_c (1- r^{\star }(\sigma _{11})); &{} \quad 0 \le w_c \le 1, \end{array} \end{aligned}$$
(4)

wherein

$$\begin{aligned} r^{\star }(\sigma _{11}) = H(\sigma _{11})= \left\{ \begin{array}{lcl} 1, &{} \text {if} &{} \sigma _{11} > 0 \\ 0, &{} \text {if} &{} \sigma _{11} < 0. \end{array} \right. \end{aligned}$$
(5)

In general three-dimensional stress states, the stress–strain relationships are also expressed by the scalar damage elasticity equation

$$\begin{aligned} \varvec{\sigma } = (1-d)\, \varvec{E}_0 \varvec{:} \left( \varvec{\varepsilon } - \varvec{\varepsilon }^{pl} \right) , \end{aligned}$$
(6)

where \(\varvec{E}_0\) is the elasticity stiffness tensor of the undamaged material, and \(\varvec{\sigma }\), \(\varvec{\varepsilon }\), and \(\varvec{\varepsilon }^{pl}\) are the stress tensor, the total strain tensor, and the plastic strain tensor, respectively.

In the 3D stress states, Eq. (6), the scalar overall damage variable d is calculated also according to eqn. (3), but the values of the switch functions \(s_t\) and \(s_{c}\) are now evaluated with replacing the unit step function \(r^{\star }(\sigma _{11})\) by a multiaxial stress weight factor \(r(\varvec{\hat{\sigma }})\)

$$\begin{aligned} r(\varvec{\hat{\sigma }}) = \frac{\sum _{i=1}^3\left< \hat{\sigma }_i\right>}{\sum _{i=1}^3 | \hat{\sigma }_i|}; \quad 0 \le r(\varvec{\hat{\sigma }}) \le 1, \end{aligned}$$
(7)

where \(\hat{\sigma }_i\) denote the principal stresses and \(\left< \varvec{\cdot }\right>\) the Macauley bracket, \(< x > = \frac{1}{2}(|x|+x)\).

In the three-dimensional stress state, the CDP model makes use of the concepts of the yield function F [36], the plastic flow vector \(\varvec{g}=\partial G/\partial \varvec{\sigma }\) normal to the plastic potential function G defining the flow rule \(\varvec{\dot{\varepsilon }}^{pl}= \dot{\lambda }\varvec{g}\) with \(\dot{\lambda } \ge 0\) being the plastic loading multiplier [37], and of the Kuhn–Tucker complementarity conditions (loading/unloading conditions), which govern the evolution of the deformation process [32]. The Kuhn–Tucker complementarity conditions can be expressed as a variational inequality [38], and both the problem of plastic flow and the problem of response with hysteresis loops in shape memory alloys can be solved incrementally as a sequence of linear complementarity problems [39].

Traver et al. [31] gave the values for the characteristic value of compressive strength (\(f_\textrm{ck}\)), modulus of elasticity (\(E_\textrm{c}\)), and axial tensile strength of concrete (\(f_\textrm{ct}\)) as 26.0 MPa, 26.6 GPa, and 2.6 MPa, respectively.

These values, in conjunction with the data presented in Model Code 2010 (MC2010) [40], were employed to define the material inputs for the CDP model utilized in this study to simulate the behavior of the concrete matrix. The International Federation for Structural Concrete (fib) developed the MC2010 as a comprehensive guide for the life cycle of concrete structures. It incorporates safety, serviceability, durability, resilience, and sustainability while highlighting a life-cycle approach.

Figure 4 depicts the relationship between compressive stress \(\sigma _{c}\) and shortening strain \(\varepsilon _{c}\) as absolute values for short-term uniaxial loading, which is expressed as

$$\begin{aligned} \frac{\sigma _{c}}{f_{cm}}=\frac{k\eta -\eta ^2}{1+\left( k-2\right) \eta }, \end{aligned}$$
(8)

where \(\eta =\frac{\varepsilon _{c}}{\varepsilon _{c1}}\), \(\varepsilon _{c1}\) is the strain at the peak stress (based on MC2010 is assumed equal to 0.225%), and \(k=1.05E_{cm}\cdot \frac{\mid \varepsilon _{c1} \mid }{f_{cm}}\). Equation 8 corresponds to Fig. 4 for \(0<\mid \varepsilon _c \mid < \mid \varepsilon _{cu1} \mid\), where \(\varepsilon _{cu1}\) represents the nominal ultimate strain. The characteristic (\(f_\textrm{ck}\)) and mean (\(f_\textrm{cm}\)) compressive strengths of concrete are related by \(f_\textrm{ck} = f_\textrm{cm} - 8\).

Fig. 4
figure 4

Schematic representation of the stress–strain relation for structural concrete [40]

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After defining all the parameters using the initial data from Traver et al. and adhering to the standards of MC2010, the non-linear behavior of concrete is characterized using the CDP constitutive model.

Based on the details in MC2010, a Poisson’s ratio of 0.2 is assigned at the beginning of the analysis to control the concrete’s initial elastic behavior. Although MC2010 recommended applying a Poisson’s ratio of 0 for cracked concrete, it is important to note that Poisson’s ratio is one of the variables that determines the material’s initial stiffness. CDP model parameters, such as dilation angle and flow potential eccentricity, become increasingly important once damage begins [41]. In brief, while the Poisson’s ratio is an important parameter, its impact on the findings may become less significant as the material enters the inelastic area and damage occurs (6). That is why, in the current study, the Poisson’s ratio of 0.2 was employed throughout the entire study, whereas the CDP model variables regulate the material’s inelastic behavior.

Additionally, for the concrete matrix, the geometric parameters defining the plasticity surface were assumed to be as follows: \(\in\) (eccentricity) = 0.1; \(\psi\) (dilation angle) = 40\(^\circ\); \(\sigma _{b0}/\sigma _{c0}\) = 1.16; \(K_{c}\) = 0.667; and viscosity parameter \(1\cdot 10^{-10}\).

3.1.2 Steel reinforcement

Based on the information presented by Traver et al. [31], two types of steel materials were employed in the experimental test, for longitudinal (diameter 20 mm) and transversal (diameters 8 and 12 mm) steel reinforcement. The longitudinal steel reinforcement exhibited Young’s modulus, yield stress, ultimate stress, and ultimate strain values of 206 GPa, 531 MPa, 639 MPa, and 18.3%, respectively. For the transversal steel reinforcement, the corresponding values were 189 GPa, 541 MPa, 661 MPa, and 10.9%. The stress–strain behavior of the steel reinforcement materials utilized in this study is illustrated in Fig. 5. Similar mechanical behavior was simulated by specifying an isotropic hardening plasticity model based on the given information.

Fig. 5
figure 5

Stress–strain behavior of the steel materials

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3.1.3 Iron-based shape memory alloy (Fe-SMA) strips

The Fe-SMA material has a unique characteristic known as shape memory recovery, which allows it to partially return to its original shape after deformation when subjected to heat. This property results from a martensitic phase change. In a previous study [31], an experiment was carried out using prestrained Fe-SMA strips (prestrained at 2% of their original length) mounted to the surface of a broken concrete beam. These strips were heated to trigger the shape memory recovery behavior, causing prestressing stresses. To define this behavior in an ABAQUS simulation, one of three methods can be applied: use of a user material subroutine (UMAT) developed based on a reliable constitutive model describing the material’s behavior; direct specification of the prestressing stresses, or definition of the temperature effect on the Fe-SMA material as a form of shrinkage.

To the best of the authors’ knowledge, there are no UMATs that describe the complex behavior of Fe-SMA materials. Furthermore, because the experimental test required the strips to be installed and activated after the concrete beam had already sustained partial damage, imposing prestressing forces on the strips using the option given in the first stage would be unfeasible in the current investigation.

The use of a temperature-dependent behavior to model the characteristics of Fe-SMA has already been described in the works by Abouali et al. [42] and Dolatabadi et al. [43]. In addition, a model of a 300 mm free-length Fe-SMA strip was simulated, using temperature activation, and the determined recovery stress and mechanical behavior were compared between simulations and practical testing (see Figs. 6 and 7). This technique was considered sufficient for further simulations, due to the good agreement of the simulation results with experimental test data. Based on the information provided by [26], the material behavior of the Fe-SMA strips is expected to show isotropic hardening plasticity, with the stress–strain curve depicted in Fig. 7.

Fig. 6
figure 6

Simulation of the activation of the Fe-SMA strip: a distribution of the recovery stresses generated in the Fe-SMA strip in response to the applied temperature, and b the relationship between the generated recovery stress and temperature changes in the Fe-SMA strip

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Fig. 7
figure 7

Stress–strain behavior of the Fe-SMA material

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In the experimental test, the RC beam underwent an initial loading phase, experiencing partial damage. Subsequently, Fe-SMA strips were installed, and the loading process was repeated. Modeling such an analysis in ABAQUS involves two potential approaches. The first method entails conducting the analysis on an undamaged and unstrengthened beam, saving the residual stress values. Then, a new model is defined with the installed Fe-SMA strips and the residual stress values from the previous analysis are applied as the initial stress values in the current analysis. However, this method may lack accuracy, because some amount of irreparable damage occurs in the unstrengthened beam, which cannot be accurately applied in the second analysis using only stress values as the initial state. The second method involves defining the Fe-SMA strips from the beginning in a manner that eliminates any potential effect they might have on the results of the initial analysis. In subsequent steps, where the strips are installed, they mechanically become part of the analysis. To achieve this, a switch was defined while specifying the Fe-SMA material to constrain the Young’s modulus of the Fe-SMA material. In steps where no strips are installed on the beam, the material exhibits a very low Young’s modulus. In subsequent steps, the material assumes its actual Young’s modulus. This process is illustrated in Fig. 8.

This approach enables the analysis to be conducted within a single model, accurately calculating all parameters without the need to transfer data to another model, especially concerning the damaged elements of the beam. Using this method, Fe-SMA strips sustain very small stress values (about 10\(^{-5}\)) in the initial loading step (before installation), as depicted in Fig. 9a. In the second loading step, with their real Young’s modulus (144 GPa), they endure higher stress values (see Fig. 9b).

Fig. 8
figure 8

The procedure for applying the Fe-SMA material in computational analysis

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Fig. 9
figure 9

Stress distribution in Fe-SMA strips in a the first loading (loading-1 step, \(\alpha =0\)), and b the second loading (loading-2, \(\alpha =1\))

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3.2 Boundary conditions and loading

Figure 10 shows a schematic representation of the boundary conditions, and mechanical and thermal loadings employed in the simulation. A rigid element measuring 250 \(\times\) 250 \(\times\) 40 mm\(^3\) was defined. This element served both as a support and as the interface for applied mechanical loads. These elements were affixed to the top and bottom surfaces of the concrete beam at identical locations, replicating the experimental arrangement. A central reference point was designated on each of these rigid surfaces, utilized for applying boundary conditions, and loads, and measuring reaction forces and displacements. Displacement-controlled loading was applied at points \(\text {P}_1\) and \(\text {P}_2\), akin to the experimental setup, with the measured vertical forces (\(\text {RF}_2\)) recorded at these points.

The temperatures required to activate the Fe-SMA strips were also imposed in this step, utilizing the predefined temperature field module of ABAQUS. The reference temperature was assumed to be 20\(^{\circ }\)C, and the activation temperature was set at 160\(^{\circ }\)C. In the second loading step, following the installation of Fe-SMA strips, the temperature was reset to the reference temperature. Table 1 summarizes the steps and associated boundary conditions used in every step of the simulation process.

Fig. 10
figure 10

Boundary conditions and applied load

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Table 1 Summary of actions and steps in the simulation procedure
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3.3 Interactions and constraint conditions

Figure 11 depicts the various elements of this computational modeling and their interactions with one another. Perfect contact with embedded restrictions is used to establish the connection between longitudinal and transversal reinforcements and the concrete matrix, with reinforcements as embedded and the concrete matrix as host domains. The Fe-SMA strips are firmly fastened to the concrete beam’s exterior surface, allowing prestressing forces to be absorbed by the concrete beam. The displacements of the provided rigid components are restricted to the reference points located at their centers, allowing displacements, support boundary conditions, and load measurement. These rigid components make perfect contact with the concrete beam’s outer surface, establishing the tie contact characteristic.

Fig. 11
figure 11

Interactions and contact properties

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3.4 Mesh sensitivity analysis

In preliminary studies, it became evident that the results of computational modeling in such complex and non-linear analyses could be highly mesh-dependent. Observations revealed that larger mesh sizes led to non-convergence in the analysis while reducing mesh sizes decreased the error factor. Consequently, a mesh sensitivity analysis was conducted to identify the optimal mesh size for this type of analysis. Various models with different mesh sizes were employed, including 125 mm, 62.5 mm, 31.25 mm, and 15.625 mm, as well as one model with a mesh size of 31.25 mm in the studied zone and 125 mm in other locations (i.e., hybrid meshing due to varying mesh sizes). Figure 12 depicts the meshed beam using these different mesh sizes.

The results of these analyses were compared with the experimental test in terms of force–displacement behavior, by determining the error percentage. The results are presented in Fig. 13. The accuracy of the analysis was significantly increased from approximately 88% to almost 98.5% by reducing the mesh sizes from 250 mm to 15.625 mm. However, the analysis time also increased significantly (by about 3500%). Nevertheless, by adopting a mesh size of 31.25 mm in the studied zone and 125 mm in other parts, an acceptable accuracy was achieved, and the analysis time was significantly reduced compared with the analysis with a mesh size of 15.625 mm. Therefore, this method was employed for subsequent computational simulations.

For the concrete matrix, the element type C3D8R (an 8-node linear brick with reduced integration) was used. The Fe-SMA strips were defined as shell elements with a mesh size of 15 mm and element type S4R (a 4-node doubly curved thin or thick shell with reduced integration). The longitudinal and transverse reinforcement rebars were defined with truss elements having mesh sizes of 15 mm and element type T3D2 (a 2-node linear 3-D truss). A computer system powered by an AMD \(\circledR\) Ryzen\(^{TM}\) 7-4800 CPU@ 2.90 GHz and 64 GB RAM was used for the analysis.

Fig. 12
figure 12

Models with different mesh sizes used for mesh sensitivity analysis

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Fig. 13
figure 13

Comparison between experimental and simulation results

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4 Comparison and interpretation of the results

Figure 14b compares the load-deflection behavior of the cantilever test resulting from the simulations with the experimental test findings for \(\text {P}_1\) and \(\delta _1\). Figure 14a depicts the geometry of the cantilever test, where the force and displacement were measured. It is important to point out that load \(\text {P}_1\) was applied using displacement control until shear failure at a low velocity (0.01 mm/s). To achieve no reaction in support \(\text {B}\) (\(\text {R}_{\text {B}}\approx 0\)), \(\text {P}_2\) was introduced using load control in response to increasing load \(\text {P}_1\).

First, the computational simulation findings are closely consistent with the experimental test results, proving the simulation’s reliability, although there are minor discrepancies. Second, after achieving peak loading in non-strengthened RC beams, both the experimental test and simulation show a brittle shear failure with a precipitous decrease (depicted by red lines). The impact of strengthening with Fe-SMA strips, on the other hand, is shown with blue lines, revealing an improvement in ductility and shear strength for the damaged beam. This enhancement enables the beam to deform to a large degree even after it has been damaged initially.

Fig. 14
figure 14

a Loading-displacement scheme for cantilever test; b comparison between experimental and simulation results for \(\text {P}_1\) and \(\delta _1\)

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The assumption is made that a damaged zone demarcates the reinforced concrete cross-section. A small portion of this zone extends above the support due to the reaction taking place there, while the primary line of damage emerges between the point of force application and the support. This damage is a result of shearing, made evident by an inclination angle of approximately 45\(^{\circ }\), as depicted in Fig. 15a and b for the beam before the second loading stage. Following the second loading stage, as seen in Fig. 15c and d, the crack area significantly expands. The crack patterns are comparable in both the numerical and experimental approaches.

Fig. 15
figure 15

a Comparison of damage distribution in the experimental test and simulation; b the unstrengthened beam after the first loading; c and d the Fe-SMA-strengthened beam after the second loading

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Moreover, during the experimental tests, Fe-SMA strip number 3 (second from the left) experienced failure due to high tensile stresses after the second loading process. In the simulation, while the observation of such failure typically requires the definition of a ductile metal behavior, a closer look at Fig. 16, illustrating the plastic strain distribution in the Fe-SMA strips, suggests that strip number 3, with its specific plastic strain values, would indeed fail in tension.

Fig. 16
figure 16

The distribution of plastic strain in the FE-SMA strips

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Specimen B15 (unstrengthened) sustained a load of 276.8 kN (at point \(\text {P}_{1}\)) in the experimental test; following strengthening and retesting, the load climbed to 354.1 kN (an increase of 28%). Thus, the numerical findings revealed that the beam reached 286.65 kN for the unstrengthened case and 372.97 kN for the strengthened case (an increase of 30%). Furthermore, the cracking pattern seen during the analysis corresponds to the experimental results. The following section applies the same model after a comprehensive comparison of the data obtained from experimental testing and computer simulation, and after identifying the reliability of the computational model. A parametric study is performed by altering several characteristics associated with the Fe-SMA strips, and the results are compared.

5 Parametric analysis

It is of interest to explore whether the properties of Fe-SMA strips, such as their thickness and width, impact the outcomes of the analysis. Hence, in this section, some of these features are modified, and the analysis is reported to compare the results. Each analysis conducted for the parametric study, in addition to the experimental test and the reference model, is labeled and explained in Table 2. For the reference simulation model (\(T_0\)), the geometry, loading, and heating conditions were considered to be similar to the experimental test (\(T_E\)). Four Fe-SMA strips measuring 1.5 mm thick and 30 mm wide were installed on the beam, as illustrated in Fig. 1b. The room temperature of 20\(^{\circ }\)C and the applied heating temperature of 160\(^{\circ }\)C were also determined for this model. Table 2 shows that all models defined for parametric analysis have the same scenario as the reference model, with the exception of one variable indicated.

Figure 17 illustrates the schematic geometry used for the analysis of tests \(\text {T}_5\) and \(\text {T}_6\), involving 2 and 6 Fe-SMA strips, respectively, in comparison with the reference model employing 4 Fe-SMA strips. The results of this parametric study regarding the applied force at point \(\text {P}_{1}\) are presented in Fig. 18. Additionally, Table 3 provides the ratio of damaged elements to the total number of elements in the studied zone, defining three distinct damage levels: strongly damaged (above 75%), partially damaged (between 25% and 75%), and slightly damaged (below 25%).

Table 2 Models used for the parametric analysis
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Fig. 17
figure 17

Arrangement of Fe-SMA strips for model \(T_5\) with 2 and model \(T_6\) with 6 strips (dimensions in mm)

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Fig. 18
figure 18

Comparison between experimental and simulation results for the parametric analysis

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Table 3 Percentage of damaged elements in the parametric analysis models
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Based on the results of the parametric studies, the following observations can be made:

  • Effect of thickness: The impact of thickness on the load-bearing capacity of the beam is observed to be insignificant, ranging from 1.9% to 3.1%. With an increase in thickness, there is a decrease in the percentage of strongly damaged elements.

  • Effect of width: Altering the width of SMA does not reveal a direct correlation with the damage in the RC beam, although it does have a slight impact on the shear forces obtained. It is important to note that localized concrete damage may result from the crushing of concrete during heating in the case of extremely narrow strips. Therefore, it can be concluded that increasing the width of the strips is an effective method for improving the load-bearing capacity of the beam.

  • Effect of amount: The results indicate that the number of Fe-SMA strips used has a significant impact on the load-bearing capacity of the beam. However, this is associated with a higher percentage of damaged elements.

  • Effect of temperature: The quantity of damaged elements increases with temperature. It is noteworthy that at a temperature of 140\(^{\circ }\)C, a higher load-bearing capacity was achieved than in the reference model. This suggests that the temperature of the strips should be adjusted based on the concrete’s properties to provide the highest prestressing forces while minimizing damage due to the activation of the Fe-SMAs. However, due to the lack of fully detailed information on the concrete material used in this study, further elaborations are required to determine the exact relation between the applied temperature on Fe-SMA strips and the damage occurring during the activation process.

  • Comparison with steel strips: An analysis was conducted using standard steel strips instead of Fe-SMA strips. As anticipated, the beam’s load-bearing capacity showed the least improvement for this variant. However, the number of damaged elements was lower than with Fe-SMA.

6 Implications of the results for future studies

The findings contribute to a better understanding of the optimum utilization of Fe-SMA strips in the strengthening and retrofitting of damaged RC beams. The authors examined the load-bearing capacity and the number of damaged elements obtained through parametric analyses of temperature, thickness, width, and the number of strips, among other factors.

While all analyses were conducted for beams with shear strength assumed as the dominating factor, it is noteworthy that damaged columns are more prevalent than damaged beams in practice as suggested by González-Góez et al. [44] and Siddika et al. [45]. This is because columns are mainly vertical members that extend from the sub-structure to the superstructure playing a vital role in the transfer of loads from the top of the structure to the foundation [46]. Therefore, any damage to the columns can have a significant impact on the overall stability of the structure [47]. Investigating how strengthening with Fe-SMA strips can specifically address column-related issues could be a valuable perspective for future research. In cases where the column deterioration is significant, unloading the column is usually required, so that the entire cross-section of the repaired column is capable of carrying the reintroduced design load. If it is not possible to remove the load from the column, then a supplemental column system can provide an alternative method of support in combination with the repair of the existing column.

Figure 18 illustrates that the largest variations in load capacity were observed when the beam was prestressed by varying Fe-SMA strip counts. Using more Fe-SMA strips provides the highest load-bearing capacity, although this method is also the most expensive due to the amount of material used [48]. While a uniform distribution seems logical, exploring different strip arrangements in the studied zone could provide valuable insights [49]. For instance, among different CFRP strip arrangements in beam models, diagonal reinforcement at a 45\(^{\circ }\) angle is reported to be the most effective, increasing the shear capacity up to 49%. Vertical U-wrap side arrangements caused the lowest increase in the bearing capacity for the beams [50].

The impact of the temperature applied to the Fe-SMA strips on the shear strength of RC beams is indeed a promising area of research. Fe-SMA strips have been developed in recent years and ongoing research shows that they may become a competitive material for civil engineering applications due to their low cost, mechanical properties, and corrosion resistance [18]. The main highlighted property of the Fe-SMA used is the ability to recover around 1% of its shape when heated up to 160\(^{\circ }\)C after being previously deformed [51]. More importantly, bars and strips made of Fe-SMA can be used to prestress concrete members if the free recovery of the SMA element is restrained, obtaining recovery stresses upon 350 MPa [52]. Optimizing the position of these strips is a potential area for further research. The placement and activation of these strips can significantly influence the behavior of the beams. For instance, beams retrofitted with activated strips showed a clear delay in the appearance of shear cracks compared with reference beams [53]. However, it is important to note that high temperatures and long-term application of these temperatures might negatively influence the concrete’s stiffness. High temperatures can lead to a non-uniform distribution of the products of hydration, which can affect the strength of the concrete [54]. Analyzing the effects of temperature on various concrete classes is another important aspect. High temperatures can affect the strength of concrete in several ways. For instance, abrupt temperature changes can cause cracking and spalling due to thermal shock, and aggregate expansion can also produce distress within the concrete. Research has shown high-strength concrete to be more vulnerable than normal-strength concrete.

The main focus of this article is on the load-bearing capacity of the RC beam (before and after strengthening with Fe-SMA strips). However, a more comprehensive approach could involve leveraging the capabilities of the program for calculating shear capacity and comparing it with widely recognized technical standards such as MC2010 and Eurocode [55]. The classification of damaged elements based on the extent of the destruction is a crucial aspect of structural analysis. This classification can be based on a uniform and comprehensive system, often referred to as taxonomy, which is fundamental for the characterization of building portfolios for natural hazard risk assessment [56]. The damaged elements can be ranked based on the severity of the damage, for example: negligible damage, slight damage, moderate damage, major damage, and collapse. Finding alternative comparisons to experimental results can indeed be challenging. Experimental testing and numerical modeling are common methods used in construction to compare behaviors of materials and structures, and also, various software packages are employed in computational simulations that are mainly based on the finite element method (ABAQUS, ANSYS) [57].

The authors used a tie-in model to describe contact between Fe-SMA strips and concrete, but in practice, there is always some degree of sliding, which may change the findings, particularly in the case of cross-sectional corners. An adhesive behavior should be provided in ABAQUS to precisely model the contact characteristics between the concrete and Fe-SMA surface. This necessitates defining bonding strength and friction variables [58]. It is critical to perform the installation of the reinforcing system correctly when strengthening concrete structures. The considered method guarantees that the highest prestressing forces are employed while minimizing the damage caused during installation [59]. It also helps to save money and time by eliminating the requirement for specialized equipment or skilled staff.

The employment of specific adhesives for adhering the Fe-SMA strips to the concrete surface is a feasible alternative to mechanical installation (i.e., using bolts). Adhesives, such as polyimides, silicones, and polysulfides, have been demonstrated to perform effectively in a variety of circumstances and on a variety of surfaces. On the other hand, the chemical composition of these adhesives can vary significantly due to the addition of mineral fillers [60], and their characteristics can be tuned to specific purposes [61]. Some adhesives, for example, are intended to bind solely to concrete, while others can adhere to a wide range of materials [62]. The glue used would thus be determined by the unique needs of the concrete structure being reinforced.

Moreover, due to the fact that thermal treatment is required to return Fe-SMA to its original shape, it is necessary to consider actions that minimize the negative impact of temperature on concrete. In further research, it is worth considering the use of pro-adhesive polyurethane polymers, which have good insulating properties (a low thermal conductivity coefficient) and high thermal resistance 300\(^{\circ }\)C) [63]. Moreover, when selecting an adhesive, the relationship between shear modulus, elongation, and lap shear strength is important. Polyurethane adhesives can be controlled over the entire range of mechanical properties, from very flexible to very rigid. By selecting molecular building blocks, chain length, and the number of functional groups, the cross-linking density can be controlled, and consequently, a high modulus of elasticity with sufficient elongation at break can be obtained.

Alternatively, shape memory alloys can be replaced by biopolymer-oxide systems. However, this type of composition will be the subject of new research related to sustainable development technologies.

The method employed in this study to control the behavior of the Fe-SMA material according to a predefined variable (here denoted as \(\alpha\)) could prove advantageous in future computational simulations using ABAQUS for the strengthening and retrofitting of structures. This approach enables the specification of a very low Young’s modulus for the Fe-SMA material at specific analysis steps. Consequently, in the stages preceding their installation, the material exhibits an almost negligible Young’s modulus. In the subsequent steps, post-installation, the material accurately reflects its true Young’s modulus. This strategic manipulation ensures that, before installation, the material behaves akin to an elastic substance without exerting any influence on the structure’s behavior. However, after installation, it demonstrates its genuine mechanical properties, actively engaging in stress absorption. This method offers the advantage of employing a single simulation, although it essentially encompasses at least two distinct tests: one on the intact and undamaged structure and another on the strengthened structure. Consequently, there exists consistency between the data and the results obtained at each defined step.

7 Conclusions

The finite element ABAQUS software was used in this study to develop computational models for examining the behavior of reinforced concrete beams strengthened with Fe-SMA strips. When the results were compared with the experimental data, the following key conclusions were drawn:

  • Employing data available in the literature related to experimental tests, a computational model was constructed using ABAQUS software, and the simulation outcomes closely align with the results of the experimental tests. This robust correlation between simulation and tests positions this study as a valuable reference for analyzing beams strengthened with Fe-SMA under various conditions.

  • The layout of damaged finite elements corresponds well with experimental results, revealing primary crack propagation at the support related to bending moment, and diagonal cracks in the span due to shear force. Fe-SMA strips notably enhance load-bearing capacity and ductility, facilitating the identification of potential failures.

  • The quantity of Fe-SMA strips has the most significant influence on load-bearing capacity. A decrease in load capacity was observed with fewer strips, while an increase was noted with a higher quantity. However, the latter represents the most expensive option among those investigated.

  • Alterations in the thickness, width, and applied temperature of Fe-SMA strips exhibit no significant impact on load-bearing capacity. Nevertheless, careful consideration is necessary to prevent cracking or local concrete crushing.

  • While the use of steel-based strips could enhance the load-bearing capacity of the beam, the improvement is relatively marginal when compared to the scenario involving Fe-SMA strips.