1D Finite Element Modeling of Bond-Slip Behavior and Deflection in Reinforced Concrete Flexural Members

Abstract

The serviceability limit state (SLS) is a crucial aspect of structural design, ensuring that reinforced concrete structures perform satisfactorily under everyday loading conditions without excessive deflections, vibrations, or cracking that could compromise their functionality or aesthetics. This study investigates the bond-slip relationship in flexural reinforced concrete members. The focus is on the influence of concrete fracture properties on the stress and strain distribution in the cracked zone. A 1D Finite Element Method (FEM) model was developed to better predict the distribution of stress and slip along the length of the reinforcement as well as the deflection. The proposed method uses material models and their interactions to provide a reliable analysis of the nonlinear behavior of RC beams, including crack width and crack spacing. A database built with numerous experimental results available in the bibliographic references allowed for the validation of the model. The results of some phenomenological models were discussed. A comprehensive analysis of the Eurocode 2 (EC2) method for calculating the deflection and cracking control of RC members was also performed. The results indicate a clear enhancement in the precision of deflection prediction in comparison to the perfect bond assumptions outlined in Eurocode 2. Additionally, the research successfully quantifies a 4–17% increase in deflection attributable to bond-slip effects.

1. Introduction

Deflection control in reinforced concrete (RC) beams is a critical serviceability requirement to ensure structural functionality, occupant comfort, and aesthetic integrity under working loads. Excessive deflections can lead to cracking of partitions and finishes and may impair the use of a structure even if safety is not at risk. As a result, design codes impose deflection limits (often as a fraction of span) to safeguard serviceability, and deflection considerations frequently govern the sizing of RC members [1,2]. Indeed, it is widely recognized that the span-to-depth ratio of beams is often limited by deflection rather than by the requirement for ultimate strength [3,4]. Adequate deflection control is essential not only for normal service conditions but also for long-term performance, as time-dependent deflections due to creep and shrinkage can accumulate and cause serviceability problems [5,6]. These considerations underscore the importance of developing reliable deflection prediction methods for RC beams.

In modern structural engineering, the increasing use of high-strength concrete, advanced reinforcement systems, and nonstandard geometries has exposed the limitations of conventional design codes such as Eurocode 2 and ACI 318. These codes often rely on simplified assumptions, including perfect bond and empirical stiffness modifiers, which may not adequately capture the complex interaction between steel and concrete under service loads. In particular, bond-slip behavior becomes critical in long-span members, lightly reinforced sections, and structures subjected to cyclic or time-dependent loading. This study addresses these challenges by developing a physically informed finite element model that explicitly incorporates bond-slip effects, offering improved deflection predictions and crack control strategies for both standard and advanced RC applications.

Despite their widespread use, current code-based deflection predictions, such as those in ACI 318-19 and Eurocode 2, exhibit notable limitations due to their empirical and simplified nature [5,7] Traditional methods like Branson’s effective moment of inertia approach rely on assumptions that overlook complex behaviors such as partial bond-slip and the gradual reduction in tension stiffening. These effects are often generalized through empirical factors, which can lead to inaccuracies, particularly in members with high-strength materials, nonstandard loading or support conditions, and long spans [7,8]. Although ACI 318-19 [9] has adopted a modified inertia expression using Bischoff’s tension stiffening model to enhance accuracy, especially for lightly reinforced members [4], discrepancies between predicted and actual deflections remain significant [5,7]. Furthermore, the lack of consideration for factors like size effects and specific load histories, combined with generalized safety margins, can result in either overly conservative or insufficiently safe designs [1,10] These challenges highlight the need for more advanced and context-sensitive deflection prediction methods.

Accurate deflection prediction in RC beams depends on the proper representation of tension stiffening. This effect enhances post-cracking stiffness and can significantly reduce service load deflections [11]. However, tension stiffening degrades with increased cracking and bond deterioration under sustained or cyclic loading [8,12]. Current provisions address tension stiffening indirectly, e.g., through the effective moment of inertia or stiffness reduction factors, which often fail to reflect its variability [6]. Over-simplification or neglect of this mechanism can lead to deflection and overestimation, particularly in lightly reinforced or fiber-reinforced members [8,13]. Additionally, size effects significantly influence deflection behavior; as beam dimensions increase, differences in crack behavior and reduced nominal tensile strength due to statistical effects become more pronounced [5,7,10]. These variations often result in earlier cracking and larger deflections than predicted by models calibrated on smaller specimens [6]. Moreover, long-term behaviors such as creep and shrinkage exhibit depth-dependent variations that are amplified in larger members [12]. Collectively, these findings emphasize the limitations of simplified code approaches and the need for more advanced, physically informed deflection models.

The bond between steel reinforcement and concrete is crucial to deflection in reinforced concrete (RC) beams. This mechanism governs how effectively cracks transfer tensile forces and affect stiffness. Under loading, micro-slips at the steel-concrete interface cause cracks to widen. Increased slip weakens the tension in the concrete between cracks, reducing stiffness and causing greater deflections, particularly under higher or cyclic loading conditions [2,12]. Conventional models assume perfect compatibility up to cracking, which oversimplifies the behavior [2,7]. In reality, partial interaction causes the reinforcement to elongate independently of the concrete, lowering the effective flexural stiffness. Current design codes do not explicitly account for deformations arising from crack widening due to bond-slip [2]. To improve deflection prediction, researchers have proposed partial-interaction models that incorporate bond–slip relationships, thereby allowing for differential strains and capturing the progressive stiffness degradation as cracks evolve [12]. Such models better reflect the time-dependent nature of tension stiffening loss. Recent investigations have shown that bond-slip is a main cause of increased deflections over time, emphasizing the need for its inclusion in realistic deflection analyses [5]. Unlike comparative studies on bond-slip models, this work focuses on developing a 1D FE model that integrates bond-slip mechanics (Fib-MC2010) and validates its accuracy against experimental data.

Researchers have developed advanced analytical and numerical approaches for improved prediction accuracy in reinforced concrete beams, including the inclusion of tension stiffening and an effective tensile concrete area for stiffness computations [11,14]. These models are often calibrated using empirical data or inverse analysis of experimental results to more accurately represent the transfer of post-cracking tension from steel to concrete. Genetic algorithms are also used to enhance the effective moment of inertia formulas by fitting them to extensive datasets, improving their sensitivity to key variables [1,14].

Nonlinear finite element analysis (FEA) is now a powerful tool for deflection prediction. These simulations can capture complex behaviors such as material nonlinearity, crack propagation, tension softening, and bond–slip interaction between steel and concrete. FEA incorporating bond elements and concrete damage models has shown strong agreement with experimental load–deflection responses, often predicting deflections within 5–15% of measured values [2,12]. Discrete cracking and embedded tension stiffening laws are proving effective. However, the computational demands and need for material parameter calibration are spurring efforts to develop reliable alternatives to full-scale simulations. Machine learning (ML) is a powerful tool for predicting deflection in reinforced concrete (RC) beams. ML algorithms can estimate deflections with high accuracy. Early studies have shown that ML approaches can predict midspan deflections across various reinforcement configurations [15]. Furthermore, ML has been effectively applied to forecast long-term deflections—including creep-related effects—by leveraging early-age data and material properties, offering a faster and reasonably accurate alternative to traditional iterative calculations [16]. These emerging ML techniques, along with advanced analytical models and high-fidelity simulations, aim to overcome the limitations of conventional methods. By accounting for complex factors, these tools provide more reliable and efficient deflection predictions. Their integration into engineering practice promises improved structural safety and more economical designs, mitigating the risks associated with overly conservative or inadequate estimations.

2. Material Models

The choice of Eurocode 2 (EC2) material models for concrete and steel was guided by their widespread use in structural design and their compatibility with nonlinear fracture-based modeling. EC2 provides well-established stress–strain relationships that are suitable for short-term loading conditions and can be extended to incorporate post-cracking behavior. Although EC2 does not explicitly model bond-slip, its formulations serve as a robust base for coupling with bond-slip laws such as those defined in the fib Model Code 2010. This integration allows the model to capture the degradation of stiffness due to interfacial slip while maintaining consistency with design standards.

For a nonlinear analysis, EC2 [17] proposes the following stress-strain relationship under a short-term compressive load (Figure 1a).

$${\displaystyle \frac{{\sigma }_c}{f_{cm}}}={\displaystyle \frac{k\eta -{\eta }^2}{1+\left(k-2\right)\eta }}$$

(1)

where $\eta ={\displaystyle \frac{{\epsilon }_c}{{\epsilon }_{c1}}}$, $k={\displaystyle \frac{1.05E_c{\epsilon }_{c1}}{f_{cm}}}$ and ${\epsilon }_{c1}=0.7f_{cm}^{0.31}<2.8\times {10}^{-3}$. The failure strain ε_cu1 can be taken equal to 3.5 × 10⁻³.

While tensile strength is generally neglected in structural design per EC2 [17], its inclusion is critical for accurately predicting the behavior of reinforced concrete members [18]. Experimental results demonstrate that, under controlled displacement, concrete exhibits linear behavior up to the tensile strength limit, followed by a progressive softening response [3]. This reduction in tensile resistance is associated with crack initiation and propagation. Beyond a critical crack opening, the tensile resistance effectively reduces to zero [19]. The behavior of concrete in direct tension can be described by the model proposed by Wang and Hsu [20]. This model is expressed by Equation (2) and is shown in Figure 1b.

$$\begin{array}{cc}{\sigma }_t=E_c{\epsilon }_t& {\epsilon }_t\le {\epsilon }_{cr}\\ {\sigma }_t=f_{ctm}{\left({\displaystyle \frac{{\epsilon }_{cr}}{\epsilon }}\right)}^{0.4}& {\epsilon }_t>{\epsilon }_{cr}\end{array}$$

(2)

with ${\epsilon }_{cr}={\displaystyle \frac{f_{ctm}}{E_c}}$ the cracking strain.

The adopted model (Equation (2)) incorporates tension stiffening via a power-law softening branch, essential for simulating post-cracking stress redistribution. This approach is validated against direct tensile tests and avoids the unrealistic assumption of zero tensile strength post-cracking in EC2.

For steel reinforcement, the bilinear stress–strain relationship defined in EC2 [17] is adopted (Figure 1c). The initial linear segment extends up to the yield stress f_y, followed by a plastic plateau or a hardening branch extending to the ultimate strain ε_uk, with a maximum stress value $f_{yk}=1.25f_y$. The bilinear stress-strain curve balances simplicity and accuracy, with a defined yield plateau and hardening branch. This is sufficient for serviceability analysis.

3. Database for Validation of Proposed Models

This study compiles a curated database of reinforced concrete beams tested under four-point bending (Figure 2), focusing exclusively on conventional concrete to ensure consistency. This dataset serves as a valuable resource for validating numerical models. By focusing on conventional concrete, this work ensures broad applicability while minimizing variability from advanced material compositions (e.g., recycled aggregate, fiber-reinforced or high-performance).

The selection criteria required each study to report complete material properties and steel reinforcement yield strength. Four-point bending was chosen for its ability to produce a pure flexural region between loading points. The database includes key geometric and mechanical parameters to support reliable comparisons. All data is presented in Appendix A. The results retained from each paper are the load-deflection curve, with particular attention to the yielding load and the corresponding displacement.

4. Deflection of Reinforced Concrete Beams

A reinforced concrete beam subjected to monotonic bending undergoes three distinct phases of behavior. Initially, both concrete and steel respond elastically, with linear stress–strain relationships in both tension and compression. During this uncracked elastic phase, the neutral axis is located at the centroid of the uncracked cross-section. As loading increases, flexural cracks initiate in the tension zone near the region of maximum moment when the tensile stress exceeds the tensile strength of concrete. Following crack initiation, these cracks propagate upward toward the neutral axis, which gradually shifts upward due to the loss of tensile stiffness in the cracked concrete. This phase concludes when the tensile reinforcement yields. In the final stage, the failure phase yielding of the steel reinforcement leads to increased crack widths and significant deflection. Continued loading results in the crushing of the concrete in the compression zone, ultimately leading to structural failure [21].

The Serviceability Limit State (SLS) defines the condition under which a structure remains functional, comfortable, and durable under service (working) loads, even if structural collapse is not imminent. One of the primary SLS criteria is deflection control, as excessive deflections can damage non-structural components, exacerbate crack widths, and permit the ingress of moisture and aggressive agents, accelerating the corrosion of embedded reinforcement.

4.1. Short-Term Deflection Calculation According to EC2 [17]

The method adopted by Eurocode 2 (EC2) for calculating short-term deflection is based on evaluating the curvature of the cross-section under a specified bending moment (Figure 3). The deflection is determined by integrating the effective curvature (${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r_{eff}$}\right.}$) along the length of the member, considering both cracked and uncracked sections [17,21]. This curvature can be expressed by the following equation:

$${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r_{eff}$}\right.}=\xi {\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.}\right)}_{cr}+\left(1-\xi \right){\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.}\right)}_{uc}$$

(3)

Here, ξ is a distribution $\xi =1-\beta {\left({\displaystyle \frac{M_{cr}}{M}}\right)}^2$ and β is a stand taken equal to 1.0 for short-term loads.

Figure 3. Stress and strain diagrams in the cracked phase before steel yielding.

Figure 3. Stress and strain diagrams in the cracked phase before steel yielding.

For a given moment, the curvature of the uncracked section based on linear elastic theory is given by

$${\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.}\right)}_{uc}={\displaystyle \frac{M}{E_{cm}{I}_{uc}}}$$

(4)

where I_uc is the second moment of area of the uncracked section, and E_cm is the modulus of elasticity of concrete.

Similarly, the cracked curvature is expressed as

$${\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.}\right)}_{cr}={\displaystyle \frac{M}{E_{cm}{I}_{cr}}}$$

(5)

with I_cr the second moment of area of the cracked transformed section.

The moment of inertia for uncracked and cracked sections can be expressed by Equation (6).

$$\begin{array}{c}{I}_{un}={\displaystyle \frac{b{h}^3}{12}}+bh{\left(0.5h-x_{un}\right)}^2+n_{\alpha }{A}_s{\left(h-x_{un}-d_0\right)}^2\\ I_{cr}={\displaystyle \frac{b{x}^3}{3}}+n_{\alpha }{A}_s{\left(d-x\right)}^2\end{array}$$

(6)

A key challenge in determining I_cr lies in evaluating the depth of the neutral axis, x. Figure 3c assumes a linear strain distribution based on Bernoulli’s hypothesis, implying that cross-sections remain plane after deformation. Under service load conditions, the stress distribution in both concrete and steel can also be considered linear (Figure 3d). Assuming a perfect bond between the steel and concrete and neglecting the tensile strength of the concrete, the force equilibrium between the compressive zone of concrete and the tensile force in the steel reinforcement yields the following expression for the neutral axis depth:

$$x={\displaystyle \frac{-n_{\alpha }{A}_s+\sqrt{n_{\alpha }^2{A}_s^2+2b{n}_{\alpha }{A}_{s}d}}{b}}$$

(7)

These assumptions are valid for under-reinforced sections, where the steel yields in tension prior to the failure of concrete in compression. According to classical beam theory, the midspan deflection under four-point bending (as illustrated in Figure 2) can be calculated using the following equation:

$${\Delta }_{cr}={\displaystyle \frac{M}{24E_c{I}_{eff}}}\left(3L^2-4a^2\right)={\displaystyle \frac{1}{24r_{eff}}}\left(3L^2-4a^2\right)$$

(8)

4.2. Fracture Mechanics-Based Sectional Model

A sectional fracture mechanics-based mechanical model for RC beams is necessary because traditional approaches proposed in design codes are based on empirical assumptions and idealized material behavior [17,21]. Simplified assumptions ignore real material behavior, in particular post-crack resistance in tension, where it is assumed that concrete loses all tensile strength after cracking. Moreover, the assumption of no slip between rebar and concrete overestimates the stiffness of RC members and underestimates the deflection [18].

The first version of the model used in the present work was proposed by Mohamad et al. [18] and was then slightly modified by Bakleh et al. [22], but both versions assumed perfect steel-concrete bonds. The model assumes that plane sections remain plane (Bernoulli’s hypothesis). This means that strains vary linearly over the depth of the section (Figure 4). As a function of the curvature, κ, and the mean normal strain, ε₀, at the beam axis, the deformation of each fiber in the cross-section can be defined. The strain in the strip positioned at y distances from the center of the beam can be calculated using Equation (9).

$$\epsilon (y)={\epsilon }_0+\left({\displaystyle \frac{h}{2}}-y\right)\kappa$$

(9)

The beam’s cross-section is made up of a reinforcement bar and layered strips of concrete. Each strip can only be elongated, so the task is to create a correlation between forces and strains in both compression and tension (Figure 4b). Equation (1) is used to calculate the compressive stress in the compression zone while Equation (2) is used to calculate the stress in the tension zone (Figure 4d). The stress of the reinforcement strip is determined by means of the EC2 relationship represented in Figure 1c.

The equilibrium equations for the internal and external forces are mathematically nonlinear due to the nonlinear behavior of the assumed material and therefore an analytical model is required to solve the system expressed by Equation (10). Solving these nonlinear equations consists of finding the values of curvature, κ, and mean strain, ε₀, that satisfy the equilibrium of the section. The equilibrium equations are solved using the Newton–Raphson method with two variables. The cross-section is divided into n + 1 strips, defining n layers, to eliminate the integral of Equation (10). The force of each layer is computed by multiplying the average stress by the layer’s area which allows sums to be used in place of the integral.

$$\left\{\begin{array}{c}{N}_{internal}={\int }_{-h/2}^{+h/2}\sigma .dA=b.∆h{\sum }_{i=1}^n{\sigma }_i+A_s.{\sigma }_s=0 \\ M_{internal}={\int }_{-h/2}^{+h/2}y.\sigma .dA=b.∆h{\sum }_{i=1}^n{y}_i.{\sigma }_i+A_s.{\sigma }_s.\left({\displaystyle \frac{h}{2}}-d_0\right)=M_{external}\end{array} \right.$$

(10)

Once the curvature has been determined, the height of the compression zone can be given by the expression $x={\displaystyle \frac{{\epsilon }_c}{\kappa }}$.

The crack spacing, s_r, can be expressed by the following equation proposed by EC2 [17].

$$s_r=k_{3}c+k_1{k}_2{k}_3{\displaystyle \frac{\varphi }{{\rho }_{p,eff}}}$$

(11)

with ${\rho }_{p,eff}={\displaystyle \frac{A_s}{A_{c,eff}}}$ and $A_{c,eff}=b\times \min (2.5\left(h-d\right);{\displaystyle \frac{h}{2}};{\displaystyle \frac{h-x}{3}})$. For ribbed bars under bending (k₂ = 1.0), EC2 suggests that $s_{r,max}=3.4c+0.425{\displaystyle \frac{\varphi }{{\rho }_{p,eff}}}$ which represents the maximum expected crack spacing.

The final crack width (w_k) is calculated as

$$w_k=s_{r,max}\left({\epsilon }_{sm}-{\epsilon }_{cm}\right)$$

(12)

where $\left({\epsilon }_{sm}-{\epsilon }_{cm}\right)$ is the difference in mean strains between steel and concrete.

4.3. Stress and Strain Distribution Between Two Cracks

Figure 5a shows the distribution of forces at two adjacent cracks that are subjected to a constant bending moment. Moving towards the center of the block, the stress in the steel decreases while the tensile stress in concrete increases due to bond mechanisms [23]. The bond-slip relationship describes how shear stress, τ_b, develops at the steel-concrete interface due to relative displacement called slip, s (Figure 5b).

The stress–slip relationship used in this work is the Fib-MC2010 [24] relationship, which is described by Equation (13).

$${\mathsf{\tau }}_{\mathrm{b}}=\left\{\begin{array}{cc}{\mathsf{\tau }}_{\mathrm{b},\mathrm{m}\mathrm{a}\mathrm{x}}{\left({\displaystyle \frac{\mathrm{s}}{{\mathrm{s}}_1}}\right)}^{\mathsf{\alpha }}& 0<\mathrm{s}\le {\mathrm{s}}_1\\ {\mathsf{\tau }}_{\mathrm{b},\mathrm{m}\mathrm{a}\mathrm{x}}-({\mathsf{\tau }}_{\mathrm{b},\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathsf{\tau }}_{\mathrm{b}\mathrm{f}})\left({\displaystyle \frac{\mathrm{s}-{\mathrm{s}}_1}{{\mathrm{s}}_2-{\mathrm{s}}_1}}\right)& {\mathrm{s}}_1<\mathrm{s}\le {\mathrm{s}}_2\\ {\mathsf{\tau }}_{\mathrm{b},\mathrm{m}\mathrm{a}\mathrm{x}}={\mathsf{\tau }}_{\mathrm{b}\mathrm{f}}& \mathrm{s}>{\mathrm{s}}_2\end{array}\right.$$

(13)

For ribbed bar and when the bond conditions are qualified as good ${\tau }_{b,max}=2.5\sqrt{f_{cm}}$ and ${\tau }_{bm}=0.4{\tau }_{b,max}$. The slip s₁ is taken equal to 1.0 mm while s₂ is taken equal to 2.0 mm. Finally, the power α equals to 0.4.

Figure 5c,d show the reinforcement and concrete interfaces for an infinitesimal element of length Δx. The equilibrium equations read:

$$\left\{\begin{array}{c}\Delta N_s\left(x\right)=-\pi \varphi {\tau }_b\left(x\right)\Delta x\\ \Delta N_c\left(x\right)=\pi \varphi {\tau }_b\left(x\right)\Delta x\end{array}\right.\Rightarrow \left\{\begin{array}{c}{\displaystyle \frac{d{\sigma }_s\left(x\right)}{dx}}=-{\displaystyle \frac{4}{\varphi }}{\tau }_b\left(x\right)\\ {\displaystyle \frac{d{\sigma }_c\left(x\right)}{dx}}={\displaystyle \frac{\pi \varphi }{A_{c,eff}}}{\tau }_b\left(x\right)\end{array}\right.$$

(14)

The slip at the interface between the reinforcement and the concrete is defined by

$$s\left(x\right)=u_s\left(x\right)-u_{c,t}\left(x\right)$$

(15)

where u_s(x) and u_c,t(x) are the displacements along the x-axis of the reinforcement and concrete, respectively. The derivative of Equation (15) gives the variation of slip with x:

$${\displaystyle \frac{ds\left(x\right)}{dx}}={\epsilon }_s\left(x\right)-{\epsilon }_{c,t}\left(x\right)$$

(16)

Beyond the Fib-MC2010 bond-slip model adopted here, alternative approaches exist to simulate steel-concrete interaction, including discrete interface layers (e.g., cohesive zone models) for localized slip, spring elements to represent interfacial shear transfer, and damage-plasticity models for micro-mechanical behavior. However, these methods typically require 2D/3D finite element frameworks, which are computationally intensive and incompatible with the 1D Euler-Bernoulli beam theory.

4.4. 1D Finite Element Modeling of the Stress Distribution Between Two Cracks

This paragraph presents a one-dimensional finite element (FE) model for analyzing the distribution of slip and curvature in reinforced concrete (RC) beams under bending, with two flexural cracks. The proposed model adopts the Fib-MC2010 bond-slip relationship (Equation (13)) due to its empirical validation and widespread use in RC analysis. While other bond-slip models exist (e.g., interface layers, spring elements), this study prioritizes computational efficiency and compatibility with 1D Euler-Bernoulli beam theory. The model accounts for the nonlinear interaction between concrete and steel reinforcement through the bond-slip relationship given in Equation (13). The strong form of equilibrium Equation (14) is discretized using linear shape functions, which leads to a nonlinear stiffness matrix that incorporates contributions from concrete (${\mathit{K}}_{\mathit{C}}=A_{ct}{E}_C\left(\epsilon \right)\int {\mathit{B}}_{\mathit{C}}^{\mathit{T}}{\mathit{B}}_{\mathit{C}}dx$), steel (${\mathit{K}}_{\mathit{S}}=A_S{E}_S\left(\epsilon \right)\int {\mathit{B}}_{\mathit{S}}^{\mathit{T}}{\mathit{B}}_{\mathit{S}}dx$), and bond (${\mathit{K}}_{\mathit{b}\mathit{o}\mathit{n}\mathit{d}}=\int {\mathit{N}}^{\mathit{T}}\left({\displaystyle \frac{d{\tau }_b}{dx}}\right)\mathit{N}dx$). These matrices are superimposed into the global stiffness matrix ${\mathit{K}}_{\mathit{g}\mathit{l}\mathit{o}\mathit{b}\mathit{a}\mathit{l}}={\mathit{K}}_{\mathit{C}}+{\mathit{K}}_{\mathit{S}}+{\mathit{K}}_{\mathit{b}\mathit{o}\mathit{n}\mathit{d}}$, which is updated iteratively within a Newton–Raphson scheme to resolve geometric and material nonlinearities, ensuring equilibrium between internal forces and external loads at each load step. This approach enables accurate prediction of slip, curvature, and stress redistribution in the cracked RC beam under progressive loading.

The equilibrium equations are solved using an incremental-iterative Newton–Raphson scheme, which is a type of nonlinear finite element analysis. The load is applied incrementally, with each step initiating an iteration loop to minimize the residual force vector $\mathit{R}={\mathit{F}}_{\mathit{e}\mathit{x}\mathit{t}}-{\mathit{F}}_{\mathit{i}\mathit{n}\mathit{t}}$, where F_ext is the external load and F_int is the internal force vector derived from stresses in concrete, steel, and bond-slip.

At each iteration, the tangent stiffness matrix K_global is computed to solve for the displacement correction $\mathsf{\Delta }\mathit{U}={\mathit{K}}_{\mathit{g}\mathit{l}\mathit{o}\mathit{b}\mathit{a}\mathit{l}}^{-1}\times \mathit{R}$. The displacements are updated as ${\mathit{U}}^{\mathit{k}+1}={\mathit{U}}^{\mathit{k}}+\mathsf{\Delta }\mathit{U}$, with a relaxation factor (e.g., 0.8) to enhance convergence. The convergence is checked against a tolerance $‖\mathit{R}‖\le {10}^{-6}$; if unmet after a maximum number of iterations, the step size is reduced. This robust algorithm accurately models the progressive slip, curvature development and load redistribution in the RC beam, thereby ensuring equilibrium is always achieved. Finally, the curvature is calculated using the expression ${\kappa }^i={\displaystyle \frac{{\epsilon }_s^i}{d-x^i}}$. The detailed algorithm is given in Appendix C.

The primary objective of our study was not to compare multiple commercial or academic software packages, but rather to develop and validate a dedicated 1D finite element model that explicitly incorporates bond-slip behavior. The model was implemented in a custom numerical environment to allow full control over the constitutive laws and numerical schemes. The proposed model is more accurate than Manfredi’s initial model [25], which omits the contribution of concrete to traction in the equilibrium equations. Many researchers later adopted the aforementioned model, as did Oliveira et al. [26] and more recently Jakubovskis et al. [27]. Oliveira et al. [26] assumed that the cross-section cracked at nodes, and that, between two cracked nodes, the behavior of concrete in tension is linear up to the tensile strength, after which it remains constant along the tensile zone. The second limitation of their work is that they proposed a linear slip between cracks, which may impact the results. The approach of Jakubovskis et al. [27] omitted the post-cracking behavior of concrete and also simplified the tensile behavior in non-cracked sections between two cracked nodes, assuming a linear stress varied linearly from the neutral axis towards the extremely tensile fiber.

In terms of finite element modeling, previous approaches have involved setting an initial slip value at the level of the first crack. An inner loop then propagates stresses and strains along the length of the beam using fixed spatial increments, while considering the bond stress at each point. The inner loop terminates when the error, defined as the difference between the target steel stress and the computed stress at the final node, falls below the tolerance threshold. If the inner loop does not converge, the initial slip of the outer loop is modified and the process is repeated until convergence is reached. The computational efficiency of this bond-slip analysis method depends on several factors, such as nested iterative loops, fixed spatial discretization and convergence criteria. The outer bisection loop adjusts the initial slip value and requires multiple iterations to converge. Each iteration involves an inner loop that moves along the length of the beam to calculate stresses and strains at specific points. The algorithm proposed in this work is the most efficient in terms of computational cost while correctly accounting for nonlinear slip and post-cracking behaviors, which have been simplified in other approaches [25,26,27].

4.5. 1D Finite Element Matrix System for RC Beam

The finite element formulation for the Euler-Bernoulli beam problem is derived from the principle of virtual work, which enforces equilibrium by balancing work [28,29]. The weak form of the governing equation is expressed as

$${\int }_0^{L}EI(\kappa ){\displaystyle \frac{d^{2}w}{{\mathit{dx}}^2}}{\displaystyle \frac{d^2\left(\delta w\right)}{{\mathit{dx}}^2}}dx={\int }_0^{K}q\left(x\right)\delta wdx+Boundary terms$$

(17)

where $w\left(x\right)$ is the transverse displacement, $EI\left(\kappa \right)$ is the flexural rigidity, $q\left(x\right)$ is the distributed load, and $\delta w$ is a kinematically admissible virtual displacement. The displacement field $w\left(x\right)$ is approximated using cubic Hermitian shape functions N(x) in order to discretize this equation. This yields $w\left(x\right)\cong \mathit{N}\left(\mathit{x}\right){\mathit{u}}_{\mathit{e}}$, where ${\mathit{u}}_{\mathit{e}}={\left[{\mathit{w}}_{\mathbf{1}},{\mathit{\theta }}_{\mathbf{1}},{\mathit{w}}_{\mathbf{2}},{\mathit{\theta }}_{\mathbf{2}}\right]}^{\mathit{T}}$ contains nodal displacements and rotations. The curvature $\kappa \left(x\right)$, defined as the second derivative of displacement, is derived from the curvature-displacement matrix $\mathit{B}\left(\mathit{x}\right)={\displaystyle \frac{{\mathit{d}}^{\mathbf{2}}\mathit{N}}{{\mathit{dx}}^{\mathbf{2}}}}$, resulting in $\kappa \left(x\right)=\mathit{B}\left(\mathit{x}\right){\mathit{u}}_{\mathit{e}}$.

Cracking and bond-slip cause an increase in localized curvature, modifying the moment-curvature relationship (M-κ). To capture the effect of cracking on the curvature, the secant stiffness EI(κ) is evaluated at each integration point as

$$EI\left(k\right)=\left\{\begin{array}{cc}{\displaystyle \frac{M}{\kappa }}& cracked regime\\ I_{un}& uncracked regime\end{array}\right.$$

(18)

The stiffness matrix and internal forces are evaluated using Lobatto quadrature, which includes endpoint evaluations to better capture localized cracking effects near supports or high-moment regions:

$$\begin{array}{c}{\mathit{K}}_{\mathit{e}}={\sum }_{i=1}^{n_{gp}}{w}_{i}EI\left(x_i\right){\mathit{B}}^{\mathit{T}}\left(x_i\right)\mathit{B}\left(x_i\right)\mathit{J}\\ {\mathit{F}}_{\mathit{e},\mathit{i}\mathit{n}\mathit{t}}={\sum }_{i=1}^{n_{gp}}{w}_i{\mathit{B}}^{\mathit{T}}\left(x_i\right)\mathit{M}\mathit{J}\end{array}$$

(19)

Here, EI(x_i) and M are updated at each integration point based on the current curvature, κ(x_i), ensuring that the model reflects progressive stiffness degradation due to cracking. As cracks propagate under increasing loads, the global stiffness matrix evolves. This requires the iterative solution technique described in the previous paragraph.

5. Results

5.1. Bond Stress Distribution Between Two Flexural Cracks

The beam selected for validating the models developed in this study is the H50-0 beam, which was tested by Seara-Paz et al. [30]. The geometry of the beam and the properties of the materials used are summarized in

Table A1. Database used in the present study.

Paper	Beam	L (mm)	b (mm)	h (mm)	a (mm)	As (mm2)	fy (MPa)	fcm (MPa)	fctm (MPa)	Ec (MPa)
Ajdukiewicz [31]	BNNm-b2	2400	200	300	800	452.4	410	61.8	4.5	41,900
	BNNh-b2					452.4		103	6.7	51,900
	BNNm-b1					804.2		60.8	4.2	41,500
	BNNh-b1					804.2		100.9	7.2	51,900
Ignjatoviç [32]	NAC-1	3000	200	300	1000	150.8	555	43.7	3.1	26,600
	NAC-2					763.4
Kang [33]	H0-1.8	2700	135	270	900	576.1	410.8	64.5	4.2	37,700
	H0-1.5					402.1	389.1
	H0-1					265.5	407.6
	H0-0.5					157.1	377.0
	N0-1.8					576.1	410.8	38.6	3.3	29,200
	N0-1.5					402.1	389.1
	N0-1					265.5	407.6
	N0-0.5					157.1	377.0
Wardeh [34]	BN-1	1500	150	200	500	402.1	550	38.6	3.6	39,400
	BN-2					307.9
Sato [35]	v-01-10WB	2200	150	200	700	157.1	332	30.6	2.9	30,342
	v-01-13WB					265.5	353
	v-01-16WB					402.1	342
	v-01-10WB-d					157.1	332	32.5	3.0	29,200
	v-01-13WB-d					265.5	353
	v-01-16WB-d					402.1	342
	hv-01-13WB					265.5	331	68.7	3.0	31,854
	v30-03-wb					265.5	331	106.4	6.3	44,080
	v45-03-wb					265.5	331	57.0	3.0	31,854
	v60-03-wb					265.5	331	40.2	3.5	32,113
	vex45-03-wb					265.5	331	55.3	3.6	36,347
Seara-Paz [30]	H50-0	3400	200	300	1275.5	402.1	500	60.7	4.3	36,300
	H65-0							46.9	4.0	35,200
Arezomandi [36]	CC-F-6-1	2700	300	460	900	567.1	568	37.2	3.5	34,500
	CC-F-7-1					760.3	517
	CC-F-6-2					567.1	568	34.2	3.0	33,100
	CC-F-7-2					760.3	517
Dar [37]	OB	5000	300	500	1666	1885	561	19.2	-	21,300
Shen [38]	B0	1600	150	300	550	226.2	548	42.1	-	33,200
Choobbor [39]	US	1690	120	240	563.3	157.1	540.1	38.78	-	-
Wei [40]	Ref	2250	200	300	975	258	400	40	-	39,000
Song [41]	B-1	2800	200	300	1000	1140.4	455	39.1	-	32,100
Zhang [42]	RC	2100	150	250	900	628.4	410	45.2	2.26	34,500
Nguyen [43]	RC	2400	130	300	850	307.9	500	35	2.0	30,000
Wu [44]	R	2000	150	250	700	508.9	420	60.5	-	34,634
Ahmed [45]	BM	4550	190	340	1925	942.5	565	35	-	31,900
Deng [46]	RC1	1600	150	250	550	157.1	435	37.4	3.35	35,000
Ren [47]	NR	1700	150	200	600	226.2	500	32.2	-	-
Wang [48]	L0	1960	230	270	880	480.7	340	34.8	-	-
Yoo [49]	RC-PI	2222	125	250	743	567.06	479	45	-	-
Zhang [50]	BO	1600	150	300	500	265.5	440	30	-	30,000
El-Emam [51]	CB1	2200	200	300	850	157.1	448	30.8	2.63	-
Feng [52]	RCBS	1800	200	300	600	157.1	300	32.7	-	-
Karabulut [16]	SC20-1	1000	150	200	300	157.1	420	20.0	-	-
Shi [53]	RC	1400	150	200	500	265.5	400	37.8	2.66	31,100

.

Analysis of the crack map shows that the average spacing between the primary cracks is 136 mm, which is close to half the height of the section, ${\displaystyle \frac{h}{2}}=150 \mathrm{m}\mathrm{m}$, and the cracking spacing value calculated using Equation (11), $s_r=146 \mathrm{m}\mathrm{m}$.

Figure 6a shows bond stress peaking at 7.03 MPa at crack ends (±0.102 mm slip) and dropping to zero at midspan. The parabolic stress–slip relationship demonstrates load transfer between concrete and steel, critical for crack control and anchorage design. The symmetrical parabolic curve reflects load transfer mechanics, with higher stress where deformation is resisted most. Figure 6b shows that slip follows a near-linear variation along the crack spacing, transitioning from negative to positive values with zero-crossing at mid-spacing, indicating a balanced bond stress distribution between steel and concrete. Although the slip value increases with larger spacing, this symmetry suggests that there is uniform interfacial shear transfer, with the magnitude of the slip scaling proportionally with the distance between two cracks. Figure 6c demonstrates remarkably stable curvature values (~1.23 × 10⁻⁵ mm⁻¹) despite varying slip conditions, confirming that global flexural stiffness dominates local bond-slip effects. The findings show that the beam’s composite action is robust so the spacing of the cracks will be determined by Equation (11).

Figure 7a shows a comparison of the experimental load-deflection curve and the results of the finite element (FE) model, with perfect bond and bond-slip effects assumed. The experimental curve shows typical nonlinear behavior, progressing from elastic response through cracking and yielding. The FE perfect bond model overestimates stiffness and load capacity, particularly post-cracking, resulting in lower deflections than observed experimentally. The FE bond-slip model, accounting for relative movement between reinforcement and concrete, better captures the experimental response. Figure 7b compares the deflections of experimental, FE, and analytical models at the yielding moment, revealing key insights into bond-slip behavior. The experimental deflection (20.83 mm) exceeds all predictions, suggesting complexities like material variability or imperfect boundary conditions.

5.2. Validation of the Proposed FE Model

Beams in the database were analyzed using Equation (11) (perfect bond) and the method in Appendix B (slip effect on deflection) as shown in Figure 8. It can be noticed that, when considering the effect of slip, the correlation between the experimental results and the predicted values improves from R² = 0.51 to R² = 0.64. The bond-slip model is the better choice for deflection predictions, as it aligns more closely with experimental results within the ±10% confidence level. The perfect bond model, while simpler, shows a higher deviation. The bond-slip model outperforms the perfect bond model with lower MAE (2.12 vs. 2.78 mm) and RMSE (3.02 vs. 3.52 mm), and more points within ±10% bounds (79.2% vs. 75.5%). Both models show similar negatively skewed residuals, suggesting slight underprediction of larger deflections, but the bond-slip approach provides better overall accuracy.

Figure 9 shows that the 1D finite element model developed in this study slightly improves the prediction of deflections when a perfect bond is considered. However, the model performs better when the bond-slip effect is considered. The close agreement between analytical and experimental deflections in Figure 9b confirms that bond-slip effects must be explicitly considered in design calculations to ensure structural reliability. The results demonstrate that the Bond-Slip Model outperforms the Perfect Bond Model in all evaluated metrics, exhibiting lower prediction errors (MAE: 1.72 mm vs. 2.51 mm; RMSE: 2.47 mm vs. 3.26 mm), a higher percentage of points within the ±10% confidence band (86.8% vs. 79.2%), and more pronounced negative skewness (−1.30 vs. −0.90) with heavier-tailed error distribution (kurtosis: 1.71 vs. 0.71), indicating the Bond-Slip Model’s superior accuracy and reliability in predicting deflection behavior while maintaining a conservative bias (consistent underprediction trend) that is particularly valuable for structural safety considerations.

While the proposed 1D FE model demonstrates a 19% improvement in deflection prediction accuracy when bond-slip effects are considered, residual discrepancies remain. These can be attributed to several factors. First, boundary conditions in experimental setups may deviate from the idealized supports assumed in the model, introducing localized stiffness or rotation that affects deflection measurements. Second, material variability—particularly in concrete tensile strength and bond quality—can lead to deviations between predicted and observed behavior. Third, experimental measurement uncertainties, especially in crack width and deflection tracking, may contribute to the scatter in the validation data.

Building on these findings, the choice of modeling approach should reflect the specific goals of the analysis. For serviceability-focused assessments where accurate deflection prediction is paramount, we recommend using models that explicitly incorporate bond-slip behavior, such as the one developed in this study. Conversely, for ultimate limit state evaluations or large-scale structural simulations, more comprehensive 2D or 3D models with interface elements or damage-plasticity formulations may be more suitable.

6. Conclusions

The study presented a one-dimensional (1D) finite element model for predicting the deflection of reinforced concrete beams by integrating tension stiffening, bond-slip, and material nonlinearity. The numerical results were validated against a database of 51 tested beams and compared with analytical predictions from the EC2 design standard. The developed 1D FE model outperforms conventional methods by explicitly modeling bond-slip effects, improving deflection prediction accuracy from R² = 0.64 to R² = 0.76 when bond-slip is considered. The developed 1D FE model demonstrates superior performance in comparison with conventional methods through explicit bond-slip modeling. This superior performance results in a 19% improvement in deflection prediction accuracy (R² increasing from 0.64 to 0.76) when interfacial slip effects are accounted for.

Residual discrepancies may stem from testing conditions (e.g., boundary constraints and material variability) that warrant further investigation. Future work should focus on isolating these experimental factors to refine the model’s robustness.