Predicting Load–Deflection Response of Enclosed Composite Concrete Beams Using the Mansur Nonlinear Constitutive Model

The load-deflection relationship is a pivotal measure of performance of moment-resisting reinforced and compacted fiber-reinforced cement composites (CFRCCs) beams that takes into account the influence of stiffness degradation, cracking performance, and ultimate strength. It provides valuable information on the influences of flexural rigidity, ductility, and failure modes and is of great significance with regard to the design of serviceability and ultimate limit states. Correct modeling of this relationship is not only crucial for the sake of safety in calculations, but also enables optimization of material usage and long-term service under different service loads [1, 2]. In practical design codes such as American Concrete Institute (ACI) 318-19 and American Institute of Steel Construction (AISC) 360-16, the stress-strain relationship of the concrete is often represented by simplified parabolic or bilinear form, which might ignore the nonlinear behaviors that exist in the vicinity of peak and post-peak concrete zones. More advanced constitutive models that can effectively reproduce the stress-strain behavior of both the ascending and descending branches of the curve are necessary in refined analysis, especially in modern composite systems [3, 4].

There are many concepts of nonlinear constitutive models that were created to capture how concrete behaves under flexural loads. Each of these models does relatively well when they are applied to the behavior of concrete. However, the amount of confinement on concrete after it has peaked will make the models behave differently from one another. The most common nonlinear constitutive models include those developed by Hognestad, Saenz, and Sargin [5-7]. On the other hand, Carreira and Chu in 1985 [8] developed a model, which is commonly used, consisting of a general rational function well known for its broad range of applicability to the data in experiments. However, there is still a slight lack of confidence in the confinement in enclosed composite structure, where steel and concrete are working together in a confined shape using the above-mentioned model, which gives the reason that essential calibration and verification based on full-scale applications are vital for concrete columns subjected to uniaxial and biaxial flexure [9-11].

Providing a crucial refinement in modelling the nonlinear behavior of concrete under high amplitude monotonic load, Mansur in 1999 introduced the Mansur nonlinear constitutive model, which exhibits a) smooth transition between the elastic and plastic ranges (unlike the aforementioned models) and b) a smooth descending branch reflecting experimental failure patterns observed in reinforced concrete elements [12]. Although the Mansur model has been used successfully in reinforced concrete frame analyses and prestressed beam evaluations, there has been limited use in assessing load–deflection response of composite enclosed beams. While some studies, e.g., Nicolaides 's study in 2015 [13], have shown the Mansur model’s ability in simulating the propagation of cracks and energy absorption characteristics in beams, there is a lack of comprehensive studies within the realm of steel–concrete composite beams [14, 15].

As the usage of composite constructions increases, and in need of more realistic simulation tools, evaluating the behaviour of the Mansur model within enclosed composite concrete beams becomes an interesting investigation. In confined composite sections, the interaction between the steel flanges and the surrounding concrete leads to adopting the proper stress redistribution behaviour, which needs an appropriate constitutive model that could adjust to these complications [16, 17].

The purpose of the present study is to assess the capability of the Mansur Nonlinear Constitutive Model to predict the overall load-deflection behaviour of encased composite concrete beams under monotonic flexure. To do this, the predicted behaviour of encased composite beams (full-scale testing), as reported by Ibrahim et al. [18] and Li et al. [19], is used as a benchmark. Additionally, these predicted behaviours will be compared with typical design approaches as per (ACI 318-19, AISC 360-16) [20, 21]. While ACI and AISC essentially provide design bases that assume that concrete exhibits a predictable pattern of behaviour in relation to the strength of concrete and its corresponding stress–strain relationship that is based upon a simplistically idealised Hognestad model ascending branch shape with a defined ultimate strain limit, the Mansur model provides an overall constitutive representation of encased composite concrete beams, including nonlinear behaviour pre-peak and softening behaviour post-peak, including residual strength effects. This allows for the prediction of the response of encased composite concrete beams through the cracking stage and into the ultimate condition where the degradation of stiffness and softening behaviour determines the form of the load-deflection curve. Finally, the present work demonstrates that the Mannur-based sectional analysis most effectively predicts the nonlinear response of encased composite concrete beams, while still allowing the support of typical design code databases. Further, the results of this investigation serve as a basis for extending the Mansur model to additional structural configurations and for supporting more accurate nonlinear finite-element simulations in advanced engineering applications.

The Mansur model leads to several hypotheses as regards the mechanical behavior of HSFC under compression. First, it is hypothesized that incorporating steel fibers into the matrix increases ductility and peak strain, primarily when fibers are aligned in a favorable configuration within the matrix, as in the case of vertically cast specimens, since fibers encounter primary crack paths. Second, it is thought that the initial tangent modulus (E_it) with fiber addition drops slightly in vertically cast specimens due to an increase in heterogeneity and localized deformation zones [12]. A third hypothesis states that the combination of fiber shape and specimen casting configuration affects the stress–strain behavior. For example, it is anticipated that post-peak toughness may be higher for vertically cast prisms when compared to either cylinders or horizontally cast prisms, especially because fiber bridging is more efficient in these prisms. Thirdly, without fibers, the post-peak portion of the stress–strain curve of a plain concrete specimen starts to soften rapidly, while it is postulated that with the addition of fibers, the Stress–strain curve softens much more gradually. Correction factors k₁ and k₂ account for the residual stress contribution of activated fibers. The confirmation of these hypotheses is backed up by an experimental validation and regression-based equations, enabling the Mansur model to predict full-range behavior using measurable input parameters, such as compressive strength and fiber geometry.

The Constitutive Framework of Mansur lends itself to the formulation of Response Level Hypotheses with potential application to Structural Prediction. The use of a complete Concrete Stress Strain Relationship that includes both the ascending branch up to peak stress and the descending or Softening Branch will cover more accurately the factors contributing to Load Deflection Curves of Encased Composite Beams subjected to constant bending loads after initial cracking and nearing failure than do any of the Code Based ACI318 Idealizations using the Hognestad type ascending curves plus the prescribed ultimate Strain Limits for analysis, as well as that developed within AISC360 for Steel Members only (when compared to continuously obtained Nonlinear Beam responses).

A validation study was performed to validate the predictive capability of the proposed model incorporating the Mansur constitutive formulation using steel-concrete composite beams, which were previously tested. Two sets of experimental data by the studies of Ibrahim et al. [18] and Li et al. [19] on load-deflection under three-point bending and simply-supported conditions, respectively, were employed as a benchmark. These tests effectively captured the full behavioral range from elastic response through cracking to ultimate failure. Geometrical and material inputs for each specimen were directly adopted from the original studies to maintain consistency and ensure methodological transparency. Utilizing the Mansur Model allowed for the account of both fibre-induced ductility and strain softening effects on concrete in numerical simulations and has shown that these numerical simulations compare favourably with the experimental data, especially in the post-cracking region.

Ibrahim et al. [18] and Li et al. [19]’s studies provide two experimental data sets as shown in Table 1, for evaluating the performance of the Mansur model [12]. The two sets consist of a composite beam tested under a three-point bending test (Figure 1). The test configurations used in these experiments represent a typical flexural behavior of composite materials. In order to assess how well the Mansur model represents the nonlinear load-deflection relationship for composite systems, these two sets of experimental data were compared to the Mansur model prediction.

Table 1. Geometrical and material properties of the composite beam

Figure 1. Details of the loading on the composite beam

This investigation utilises a statistical approach to determine the ability of the Mansur model to predict flexural performance of reinforced concrete (RC) beams, comparing its predictions to those obtained from standard design codes (ACI 318-19; AISC 360-16) using experimental results as a basis for evaluation. Statistical performance indicators calculated include: RMSE; NRMSE; MAPE; R²; and Pearson Correlation Coefficient (R), as outlined in Eqs. (1)-(5). These statistical performance indicators provide a valid statistical method for determining if the Mansur model can accurately predict the load-deflection response of RC beams subjected to moment loading conditions.

$R M S E=\sqrt{\frac{1}{N} \sum_{n=1}^N\left(A_n-P_n\right)^2}$                  (1)

$N R M S E=\frac{R M S E}{S}$                  (2)

$R^2=1-\frac{\sum\left(A_n-P_n\right)^2}{\sum\left(A_n-S_n\right)^2}$                    (3)

$MAPE =\frac{\left(\sum \frac{\mid A-E\mid}{A}\right) * 100}{N}$                  (4)

$A A \%=100 \%-M A P E$                     (5)

where, (A_n) actual and predicted (P_n) values and the estimated (N) number of points within the dataset (S) are normalized against the mean of the actual values.

To facilitate the use of the Mansur model for benchmarking purposes, deflection values were also determined from existing design codes. In assessing the elastic response of the Mansur model, ACI 318-19 [20] was used for gross section properties of the uncracked (i.e., all portions that are still in elastic behaviour) and modified stiffness properties for the cracked regions. To determine the ultimate capacity of the structure for bending, AISC 360-16 [21] was used; incremental calculations of deflection were performed across the entire loading spectrum.

$y_{\max}=\frac{p \cdot(\text {span})^3}{48 \cdot E_c I_g} for\,uncracked\,section$                    (7)

$y_{\max}=\frac{p \cdot(s p a n)^3}{48 \cdot E_c \cdot\left\{\left[\left(\frac{M_{c r}}{M_a}\right)^3 I_g\right]+\left[1-\left(\frac{M_{c r}}{M_a}\right)^3\right] I_{c r}\right\}}$ for cracked section                     (8)

The applied moment is denoted as M_a, and the cracking moment as M_cr. Uncracked and cracked moment of inertia are represented by I_g and I_cr, respectively.

Figure 2. Load deflection curve for beam Cb.1

As shown in Figure 2, the response predicted by ACI 318-19 and AISC 360-16 [20, 21] closely follows the experimental trend in the elastic range but gradually diverges beyond cracking. The design codes tend to predict ultimate capacity over, as reflected by a failure load of 168.18 kN compared to the experimental value of 159.79 kN. This deviation can be attributed to the built-in safety factors within code-based formulations [20, 21].

As opposed to the previous model, the Mansur Model provides significantly better agreement with actual experimental data at all stages. It predicts a failure load of 162.86 kN at 23.87 mm of deflection, which closely matches what was found during testing. In addition, because it captures both the post-cracking stiffness degradation and strain-softening effects, the Mansur model provides a better approximation of nonlinear structural response behavior, therefore improving the reliability of flexural analysis of composite sections, compared to previous models.

The experimental evidence presented illustrates that following the peak load, there was an overall strength decrease in the specimen as a result of factors such as micro-cracking, stress redistribution, and bond-slip behaviors. Although the Mansur model captured the initial softening behavior displayed by the experimental results, it was unable to predict the subsequent plateau or residual strength achieved at the conclusion of the test.

The application or predictions from the conventional code-based approach such as ACI 318-19 and AISC 360-16 [20, 21] show better correlation with the experimental values compared to the predictions from Mansur nonlinear concrete constitutive model for both initial cracking and failure load cases, with root mean square error (RMSE) of 16.50 kN, mean average percentage errors (MAPEs) of 9.98%, higher coefficients of determinations, R² = 0.7580, and high Pearson’s correlation, R = 0.9719. The code-based approach is conservative of calibration as it over-predicted the capacity reasonably within design tolerances.

Contrarily, the Mansur model produces a MAPE of 11.10% and a 12.80kN RMSE, with lower R² (0.5638) and Pearson R (0.8496) when compared to the Marie model. The quantitative disparity notwithstanding, the Mansur model appreciates important nonlinear problems such as, for example, strain-softening or post-cracking stiffness degradation that linear code models commonly omit. Indeed, this virtuous behavior allows a more realistic simulation of the structural performance after the elastic phase under load all statistical performance indicators for beam Cb.1 shown in Figure 3. Furthermore, Golafshani's study in 2020 supports the idea that the use of constitutive models gives a more accurate and improved prediction of the inelastic mechanisms impacting the performance of the material in the future, even though their statistical accuracy is less than that of other types of models [22].

3.png

Figure 3. Statistical performance evaluation for beam Cb.1

1. The Mansur nonlinear model accurately simulated the post-cracking response of enclosed composite beams, achieving RMSE = 6.41 kN and R² = 0.9812 for Beam Cb.2.
2. Compared to ACI/AISC codes, the Mansur model better captured strain softening and stiffness degradation, particularly beyond peak loads in both beam cases.
3. Statistical indicators confirmed the model’s improved agreement in nonlinear stages, with MAPE reduced to 6.27% and accuracy agreement (AA%) reaching 93.73% in Beam Cb.2.
4. In Beam Cb.1, the model slightly underestimated peak strength (162.86 kN vs. 159.79 kN experimental), yet still outperformed linear code approximations in nonlinear accuracy.
5. Overall, the Mansur model proves more reliable for nonlinear flexural analysis of composite beams, making it a robust alternative to conventional design standards.