Analysing Flexural Response in RC Beams: A Closed-Form Solution Designer Perspective from Detailed to Simplified Modelling

Abstract

This paper presents a detailed analytical approach for the bending analysis of reinforced concrete beams, integrating both structural mechanics principles and Eurocode 2 provisions. The general analytical expressions derived for the curvature were applied for the transverse displacement analysis of a simply supported reinforced concrete beam under four-point loading, focusing on key limit states: the initiation of cracking, the yielding of tensile reinforcement and the compressive failure of concrete. The displacement’s results were validated through experimental testing, showing a high degree of accuracy in the elastic and crack propagation phases. Deviations in the yielding phase were attributed to the conservative material assumptions within the Eurocode 2 framework, though the analytical model remained reliable overall. To streamline the computational process for more complex structures, a simplified model utilising a non-linear rotational spring was further developed. This model effectively captures the influence of cracking with significantly reduced computational effort, making it suitable for serviceability limit state analyses in complex loading scenarios, such as seismic impacts. The results demonstrate that combining detailed analytical methods with this simplified model provides an efficient and practical solution for the analysis of reinforced concrete beams, balancing precision with computational efficiency.

1. Introduction

Reinforced concrete (RC) is one of the fundamental building materials. Consequently, it is utilised regularly throughout a plethora of structures worldwide within modern civil engineering [1]. Its exceptional features like economy, stiffness and load-bearing capacity have made it a primary resource for many infrastructure structures, from viaducts and bridges to high-rise structures. Nevertheless, challenges arise when using RC, with cracking being one of the key issues arising from the complex non-linear behaviour of concrete.

Cracks in concrete indicate that the tensile strength of the material has been exceeded, and they are not just a cosmetic concern. A comprehensive analysis of the behaviour of RC structures over time is necessary to understand this process fully. With the appearance of the first cracks, a structure that initially exhibits linear behaviour transitions to non-linear behaviour due to crack propagation. In this context, the concrete takes on the compressive force, while the rebar bears the tensile force. As cracks propagate, the load-carrying capacity of the structural element undergoes changes. Therefore, a meticulous analysis of the impact of cracks on the structure is essential.

Load capacity analysis is crucial for guaranteeing the safety and durability of complex structures, such as bridges, power stations and tunnels. The creation of mathematical models and computational approaches is pivotal to accurately determining load-bearing capacity and ensuring structural safety [2,3]. The structural engineer is thus frequently challenged with designing an idealised mathematical model of the structure to ensure that the ensuing analysis accurately accounts for all the essential attributes of the actual structure, as far as is feasible.

Experimental verification of RC structures is crucial in enhancing the precision of structural analysis and design within the field of civil engineering. Incorporating a broader scope of material traits and integrating them into analytical models can lead to superior simulation of the structure’s behaviour under realistic circumstances. This, in turn, results in safer and more efficient construction practices. To further emphasise the importance of analysing structural behaviour, it is noteworthy that beams, as integral members of structures, have been studied extensively. This is particularly true for composite beams, where studies have included deflection [4], buckling [5] and dynamic analyses [6]. These investigations highlight important aspects of structural performance and provide valuable insights into the complexities of load-bearing members.

To conduct precise computational analyses of RC structures, it is imperative to establish quality performance (stress–strain) diagrams for concrete and steel. The fundamental material characteristics of concrete and steel are their compressive and tensile strength, which can be determined via straightforward uniaxial tensile and compressive tests. Additionally, these tests enable the assessment of the complete stress–strain behaviour of both materials. Although these two fundamental properties (strengths) hold paramount importance, the precise determination of performance diagrams requires the use of additional material properties such as the modulus of elasticity, elongation at yield of steel, or failure of concrete.

In contrast to steel, it is challenging to determine the mathematical model parameters of concrete in the compressive and tensile zones due to its wider spectrum of values obtained from experiments. Conducting additional experimental verifications, e.g., three-point or four-point bending tests of RC beams, enables a more comprehensive assessment of its mechanical properties, including crack and deformation development during loading, up to ultimate failure [7,8,9,10,11].

RC structures exhibit a highly non-linear behaviour, particularly under extreme tensile loads. Cracks form even at relatively low bending loads, caused by the bending moment surpassing the concrete’s tensile strength, leading to stiffness changes and consequent non-linear behaviour. In civil engineering, 1D finite element (FE) models are generally utilised for global analyses of non-linear responses in RC structures comprising primarily beams and columns. The non-linear region of 1D structural elements can be most simply and efficiently modelled using plastic hinges, which are simulated by non-linear rotational spring in the bending analyses [12]. Essentially, plastic hinges represent a concentrated area of the structure where the non-linear behaviour of materials is assumed. The rest of the structure is treated as linearly elastic.

The use of a non-linear rotational spring in 1D models allows for simulating the bending behaviour of a structure at loads beyond its elastic limits with sufficient accuracy. This is especially crucial in extreme conditions like seismic loads, where deformation and cracking may be limited to certain areas [13]. The precision of these analyses within the inelastic area of the structure relies predominantly on defining the moment–rotation diagram of the rotational spring stiffness.

Research on reinforced concrete beams’ response has increasingly focused on enhancing their structural efficiency and durability through innovative materials and methodologies, with particular emphasis on steel bars. Recent studies have developed analytical displacement solutions for statically determinate beams, employing bi-linear stress–strain behaviour with a horizontal top branch for steel and concrete to define a trilinear moment–curvature model for predicting deflection under various loading conditions [14]. Hybrid reinforcement strategies, combining basalt fibre-reinforced polymer (BFRP) and steel bars, have demonstrated significant improvements in flexural behaviour, particularly for under-reinforced configurations [15].

The use of high-strength and high-toughness (HSHT) steel bars has further enhanced energy absorption and crack control in reinforced concrete beams [16]. Due to the broad applicability of high-strength steel bars, not only their standard bending capacity but also their behaviour at elevated temperatures is being investigated [17]. Additionally, steel–basalt fibre composite bars (SBFCBs) have shown promise in addressing corrosion issues while maintaining ductility, indicating the potential for hybrid reinforcement [18].

Constant resistance energy (CRE) steel reinforcement has outperformed traditional high-strength bars in flexural performance, highlighting its advantages in design applications [19]. Research on two-layer fibre-reinforced concrete beams has confirmed that an optimised fibre distribution enhances load-carrying capacity and ductility [20]. Furthermore, hybrid GFRP–steel reinforced concrete beams exhibit superior impact resistance compared to conventional configurations [21]. For very harsh and extreme environmental conditions, such as marine ones, the application of Negative Poisson’s Ratio (NPR) steel bars was also studied [22].

The exploration of stainless steel (SS) as a reinforcement has revealed improved moment capacity, especially at lower reinforcement ratios, presenting an alternative to conventional steel [23]. An analytical approach for ultra-high-performance concrete (UHPC) has also been proposed, offering a flexible design methodology for achieving the desired structural performance [24]. Finally, studies on hybrid FRP–steel reinforced beams have provided new insights into predictive modelling for cracking moments and deflections [25]. This paper focuses on analyses of the bending behaviour of RC structures using an analytical model in accordance with the Eurocode 2 standard, assimilating the principles of structural mechanics. The new general analytical expressions derived for the curvature as a function of the bending moment and considering the actual state of stresses and strains in the cross-section were first derived. Afterwards, the derived expressions were implemented for the bending analysis of a beam under four-point loading. The choice of analysing this kind of structure was motivated by the availability of experimental data for this specific loading condition, allowing for a comprehensive investigation and validation of the proposed analytical model. Key aspects such as the first crack initiation, the loss of load-carrying capacity of the steel reinforcement and the ultimate failure of the concrete were examined in detail. This study contributes to the theoretical understanding and mathematical modelling of the RC structure’s behaviour. In the end, it also discusses a new practical computational model with a discrete crack modelled by a rotational spring for predicting its response.

2. Determination of the Closed-Form Solution for the Three-Linear Moment—Curvature Diagram of the Rectangular Cross-Section

In this paper, a new analytical model is developed to calculate the structure’s transverse displacement, taking into account the principles of structural mechanics and assimilating the principles and provisions of the Eurocode 2 standard. The elementary hypothesis is thus the Euler–Bernoulli hypothesis, which is in accordance with the Eurocode 2 standard. The most thorough constitutive law for concrete under compression from the code was used for the uniaxial stress–strain state, which does not explicitly demonstrate any fracture theory. This is standard practice in the flexural design of reinforced concrete members to several standards, not just Eurocode 2 [26,27,28,29].

This study can be divided in the following two main steps:

• Determination of the closed-form solutions for the three-linear moment–curvature (flexural strength) diagram of the beam cross-section;
• Determination of the transverse displacement in the considered point of the structure using the elastic Euler–Bernoulli bending theory based on the moment–curvature diagram of the cross-section provided in point 1.

Further, the simplified beam finite element analysis is presented as a promising potential expansion of the computational process, implementing non-linear rotational spring stiffness. The determination of the closed-form solution for this spring, which implements the three-linear moment–rotation diagram, is based on the first two derivation steps. The complete flexural strength response of the RC rectangular section shows a highly non-linear response to bending due to the both very different constitutive materials. This response has been divided into three primary behavioural phases. The first phase is the elastic non-cracked limit phase (I), followed by the crack propagation limit phase (II) and the bottom reinforcement yielding limit phase (III). These phases are defined by the identification of three limit points. For each of these phases, the moment–curvature relationship (M-ĸ) is assumed to be linear. Further, these three phases together constitute a simplified three-linear diagram of the cross-section (see Figure 1). The key points in this regard are the occurrences of the initial crack in the concrete (M = M_cr, ĸ = ĸ_cr), the initiation of reinforcement yielding (M = M_y, ĸ = ĸ_y) and the failure of the concrete in the compression zone (M = M_u, ĸ = ĸ_u).

The fundamental requirement for determining the moment–curvature diagram of the RC cross-section is to know the behaviour of the two constituent materials. Numerous empirical formulae have been proposed for both concrete [30,31,32] and steel to determine their uniaxial stress–strain behaviour. In this study, in accordance with Eurocode 2 (EC2 [33]), the non-linear constitutive parabola–rectangle stress–strain behaviour of concrete in the compression zone is assumed to be as

$${\sigma }_c({\epsilon }_c)=\left\{\begin{array}{c}{f}_c\left(1-{\left(1-{\displaystyle \frac{{\epsilon }_c}{{\epsilon }_{c2}}}\right)}^2\right)\hspace{1em} for\hspace{1em} 0\le {\epsilon }_c\le {\epsilon }_{c2}\\ \hspace{1em}\hspace{1em} f_c\hspace{1em}\hspace{1em}\hspace{1em} for\hspace{1em} {\epsilon }_{c2}\le {\epsilon }_c\le {\epsilon }_{c3}\end{array}\right.,$$

(1)

where f_c is the concrete compressive strength, ε_c2 is the strain at reaching the maximum strength and ε_c3 is the ultimate strain. It should be noted that the compressive strains and strength in Equation 1 are considered as positive.

An elasto-plastic approximation, as per the EC2 and given by Equation (2), is assumed to idealise the behaviour of the steel in tension and compression as

$${\sigma }_s({\epsilon }_s)=\left\{\begin{array}{c} E_s{\epsilon }_s\hspace{1em} for\hspace{1em} -{\epsilon }_y\le {\epsilon }_s\le {\epsilon }_y\\ f_y\hspace{1em} for\hspace{1em} {\epsilon }_s> {\epsilon }_y\\ -f_y\hspace{1em} for\hspace{1em} {\epsilon }_s< -{\epsilon }_y\end{array}\right.$$

(2)

where E_s is the elastic modulus of steel, f_y is the yield strength and ε_y = f_y/E_s is the yield strain.

The other following basic assumptions were made to determine the moment–curvature diagram of an RC cross-section [34]:

• A straight cross-section before bending remains straight after bending;
• A perfect bond exists between the reinforcement and the concrete;
• Since the tensile strength of concrete is relatively small compared to its compressive strength, a linear distribution of tensile stresses was considered for the non-cracked tensioned part of the cross-section. After a crack occurs, the tensile strength of concrete in the cracked section is neglected;
• The ultimate cross-section flexural capacity is achieved once the ultimate failure deformation ε_c3 is reached in the top concrete compression fibre at the ultimate stage.

2.1. Determination of the Limit Characteristic Point for M = M_cr and ĸ = ĸ_cr

The cross-section characteristics include the cross-section width (b), the cross-section height (h), the distance of the bottom rebar from the top edge of the section (d), the distance of the top rebar from the top edge of the section (d′), the area of the bottom rebar (A_s) and the area of the top rebar (A_s′).

It is assumed that in this phase, there is a linear relationship between the normal stresses and strains in concrete. This is due to the short first non-cracked limit phase leading to the onset of the first crack in the cross-section and the relatively low compressive stresses. Before the formation of the first crack, the complete cross-section is considered to be in an elastic state, as shown in Figure 2. In the absence of axial force, the neutral axis y_n = y_cr (given with respect to the top edge) and is given as

$$y_{cr}={\displaystyle \frac{\frac{b{h}^2}{2}+(n-1)\left(A_s^{\prime }{d}^{\prime }+A_{s}d\right)}{bh+(n-1)\left(A_s^{\prime }+A_s\right)}},$$

(3)

where n = E_s/E_c represents the ratio between the elastic modulus of steel (E_s) and concrete (E_c).

The moment of inertia of the adjusted (with A₂ = (n − 1) A_s′ and A₃ = (n − 1) A_s, representing adjusted cross-sections of the rebars) cross-sectional centroid $I_n={\displaystyle \sum _{i=1}^3{I}_i+A_i{(y_i-y_{cr})}^2}$, accounting for the small and entirely negligible centroid moment of inertia of the rebar (I_s = I₂ = I₃ ≈ 0), is calculated as

$$I_n={\displaystyle \frac{b{h}^3}{12}}+bh{({\displaystyle \frac{h}{2}}-y_{cr})}^2+(n-1)\left(A_s^{\prime }{(d^{\prime }-y_{cr})}^2+A_s{(d-y_{cr})}^2\right).$$

(4)

The limit moment M_cr and the corresponding curvature ${\kappa }_{cr}={\displaystyle \frac{d{\epsilon }_c\left(y\right)}{dy}}$ at the point when the first crack appears are as follows:

$$M_{cr}={\displaystyle \frac{f_{ct}{I}_n}{h-y_{cr}}},$$

(5)

and

$${\kappa }_{cr}={\displaystyle \frac{f_{ct}}{E_c(h-y_{cr})}}={\displaystyle \frac{{\epsilon }_{ct}}{h-y_{cr}}},$$

(6)

where ε_ct represents the maximum tensile strain of the concrete.

2.2. Determination of the Limit Characteristic Point for M = M_y and ĸ = ĸ_y

In determining the limit moment M_y at which the tensile reinforcement starts to yield, the strain (ε_s) and the resultant force (F_s) in the bottom (tensile) reinforcement have been taken into account as

$${\epsilon }_s={\epsilon }_y,$$

(7)

and consequently,

$$F_s=f_y{A}_s.$$

(8)

The normal stresses in concrete, as given by Equation (1), occur only in the compression zone and are expected to be only parabolically distributed along the height of the non-cracked part of the cross-section, within the limits 0 ≥ ε_c ≥ ε_c2 shown in Figure 3, where parabolic distribution is presented. The figure illustrates the additional required parameters, including the maximum compressive strain in the top fibre of the concrete (ε_cc), strain in the top reinforcement (ε_s′), the resultant compressive force in the concrete (F_c), the resultant force in the top reinforcement (F_s′), and the linear function of the strains over the section height ε_c(y) when the bottom reinforcement reaches yielding.

At the beginning of the yielding of the bottom reinforcement in the cross-section (assuming that the stresses in the compression zone of the concrete are exclusively distributed by the parabola), the compressive forces in concrete F_c and the top reinforcement F_s′, depending on the position of the neutral axis y_n = y_y, must be determined as follows:

$$F_c=b{\displaystyle {\int }_0^{y_y}{\sigma }_c(y)dy}=b{\displaystyle {\int }_0^{y_y}{f}_c\left(1-{\left(1-\frac{{\epsilon }_c(y)}{{\epsilon }_{c2}}\right)}^2\right)dy},$$

(9)

and

$$F_s^{\prime }={\epsilon }_s^{\prime }{A}_{\mathrm{s}}^{\prime }\left(E_{\mathrm{s}}-E_{\mathrm{c}}\right)={\epsilon }_{\mathrm{c}}\left(y=d^{\prime }\right)A_{\mathrm{s}}^{\prime }\left(E_{\mathrm{s}}-E_{\mathrm{c}}\right),$$

(10)

assuming that the top rebar is still behaving elastically.

The strain ε_c(y) can be expressed as a linear function by considering boundary conditions ε_c(y = d) = ε_y and ε_c(y = y_y) = 0. Therefore, the strain function is

$${\epsilon }_c\left(y\right)={\displaystyle \frac{{\epsilon }_y(y-y_y)}{d-y_y}}\hspace{1em} for\hspace{1em} 0\le y\le h.$$

(11)

Considering Equation (11) in Equation (9), the resultant force in the concrete (F_c) is now

$$F_c=-{\displaystyle \frac{b{f}_{\mathrm{c}}{y_y}^2{\epsilon }_y\left(3d{\epsilon }_{\mathrm{c}2}+y_y\left({\epsilon }_y-3{\epsilon }_{\mathrm{c}2}\right)\right)}{3{{\epsilon }_{\mathrm{c}2}}^2{\left(d-y_y\right)}^2}},$$

(12)

where the correct orientation of F_c is already presented in Figure 3.

The neutral axis position y_y is established by balancing all the forces acting on the cross-section F_c + F_s′ + F_s = 0 (considering that F_c and F_s′ are obtained with negative signs). As a result, the cubic characteristic polynomial is formed:

$${\displaystyle \sum _{i=1}^4{\beta }_{i,y}}{y_y}^{i-1}=0,$$

(13)

where the constants β_i,y (i = 1, 2, 3, 4) are defined as

$$\begin{array}{l}{\beta }_{1,y}=3d{{\epsilon }_{c2}}^2\left(A_{s}d{f}_y+A_s\prime {\epsilon }_y{d}^{\prime }\left(E_s-E_c\right)\right)\\ {\beta }_{2,y}=-3{{\epsilon }_{c2}}^2\left(2A_{s}d{f}_y-A_s^{\prime }{\epsilon }_y\left(d+d^{\prime }\right)\left(E_s-E_c\right)\right)\\ {\beta }_{3,y}=3{\epsilon }_{c2}\left(A_s{f}_y{\epsilon }_{c2}-{\epsilon }_y\left(A_s^{\prime }{\epsilon }_{c2}\left(E_s-E_c\right)+bd{f}_c\right)\right)\\ {\beta }_{4,y}=b{f}_c{\epsilon }_y\left(3{\epsilon }_{c2}-{\epsilon }_y\right)\end{array}$$

(14)

The neutral axis position y_y at reinforcement yielding initiation in the tensile zone is determined using the relevant cubic polynomial solution, as follows:

$$y_y={\displaystyle \sum _{j=1}^2\left(\left(-\frac{1}{2}+{(-1)}^{j+1}\frac{\mathrm{i}\sqrt{3}}{2}\right)\sqrt[3]{-q+{(-1)}^j\sqrt{q^2+p^3}}\right)}-{\displaystyle \frac{{\beta }_{3,y}}{3{\beta }_{4,y}}},$$

(15)

where the coefficients p and q are defined as

$$\begin{array}{l}p={\displaystyle \frac{3{\beta }_{4,y}{\beta }_{2,y}-{{\beta }_{3,y}}^2}{9{{\beta }_{4,y}}^2}}\\ q={\displaystyle \frac{2{{\beta }_{3,y}}^3-9{\beta }_{2,y}{\beta }_{3,y}{\beta }_{4,y}+27{\beta }_{1,y}{{\beta }_{4,y}}^2}{54{{\beta }_{4,y}}^3}}\end{array}.$$

(16)

The two remaining obtained solutions of the cubic polynomial for this problem are irrelevant as they lie outside the boundaries of the cross-section area.

The characteristic limit moment M_y is determined by utilising the moment equilibrium of all the resulting forces in the cross-section:

$$M_y=M_c+F_s^{\prime }{d}^{\prime }+F_{s}d,$$

(17)

where $M_c=b{\displaystyle {\int }_0^{y_y}{\sigma }_c(y)ydy}$ represents the moment of resulting force (F_c) of the non-cracked concrete section, which is denoted as follows:

$$M_c=-{\displaystyle \frac{b{f}_{\mathrm{c}}{y_y}^3{\epsilon }_y\left(4d{\epsilon }_{\mathrm{c}2}+y_y\left({\epsilon }_y-4{\epsilon }_{\mathrm{c}2}\right)\right)}{12{{\epsilon }_{\mathrm{c}2}}^2{\left(d-y_y\right)}^2}}.$$

(18)

The corresponding curvature, obtained as ${\kappa }_y={\displaystyle \frac{d{\epsilon }_c\left(y\right)}{dy}}$, follows as

$${\kappa }_y={\displaystyle \frac{{\epsilon }_y}{d-y_y}}.$$

(19)

The closed-form solution for M_y and ĸ_y is restricted to cases where the bottom reinforcement yields prior to the strain in the concrete reaching ε_c2 from Equation (1). A necessary condition for the validity of the application of the derived expressions is therefore ε_cc = ε_c(y = 0) ≥ ε_c2.

2.3. Determination of the Limit Characteristic Point for M = M_u and ĸ = ĸ_u

The value of the characteristic ultimate moment M_u is determined by considering that, at the initiation of compression failure of the top concrete fibre, the maximum compressive edge strain in the concrete is ε_cc = ε_c3. The linear strain function ε_c(y) is therefore acquired under the following two conditions, ε_c(y = 0) = ε_cc = ε_c3 and ε_c(y = y_u) = 0. Thus, the strain function is

$${\epsilon }_c\left(y\right)={\epsilon }_{c3}\left(1-{\displaystyle \frac{y}{y_u}}\right).$$

(20)

At the initiation of concrete failure at the top compressive edge of the cross-section, the total resulting force (F_c) in the concrete non-cracked area is calculated by combining the resulting forces of two sub-areas. These areas include the region where the concrete strain is within the limits of ε_c2 ≥ ε_c ≥ ε_c3 and the region where the concrete strain is at limit 0 ≥ ε_c ≥ ε_c2, as illustrated in Figure 4.

The location of the fibre separating the two stress functions across the cross-section, as derived from Equation (1), is designated as y₁, originating from the top of the cross-section, and by applying the condition ε_c(y = y₁) = ε_c2, y₁, it is defined as follows:

$$y_1={\displaystyle \frac{y_u\left({\epsilon }_{c3}-{\epsilon }_{c2}\right)}{{\epsilon }_{c3}}}.$$

(21)

Considering Equation (21), the resulting force in concrete $F_c=b\left({\displaystyle {\int }_0^{y_1}{f}_{c}dy}+{\displaystyle {\int }_{y_1}^{y_u}{f}_c\left({\left(1+\frac{{\epsilon }_c\left(y\right)}{{\epsilon }_{c2}}\right)}^2-1\right)dy}\right)$ can be expressed as follows:

$$F_c={\displaystyle \frac{1}{3}}b{f}_c{y}_u\left(3-{\displaystyle \frac{{\epsilon }_{c2}}{{\epsilon }_{c3}}}\right).$$

(22)

The forces acting on the top reinforcement (presumed in an elastic zone) and on the bottom reinforcement are

$$F_s^{\prime }={\epsilon }_s^{\prime }{A}_{\mathrm{s}}^{\prime }{E}_{\mathrm{s}}={\epsilon }_{\mathrm{c}}\left(y=d^{\prime }\right)A_{\mathrm{s}}^{\prime }{E}_{\mathrm{s}},$$

(23)

and

$$F_s=f_y{A}_s.$$

(24)

It should be noted that when calculating the neutral axis y_n = y_u at the initiation of the compressive failure of the concrete, it is necessary to check the following condition |ε_s′| = |ε_c(y = d′)| ≤ ε_y, as it is assumed in Equation (23) that the upper reinforcement remains within the elastic region.

The position of the neutral axis y_u is determined by achieving equilibrium among the resulting forces of F_c + F_s′ + F_s = 0 in the cross-section. As a result, the quadratic polynomial is obtained:

$${\displaystyle \sum _{i=1}^3{\beta }_{i,u}}{y_u}^{i-1}=0.$$

(25)

where the constants β_i,u (i = 1, 2, 3) are defined as

$$\begin{array}{l}{\beta }_{1,u}=-3A_s^{\prime }{d}^{\prime }{E}_s{{\epsilon }_{c3}}^2\\ {\beta }_{2,u}=3{\epsilon }_{c3}\left(A_s{f}_y+A_s{E}_s{\epsilon }_{c3}\right)\\ {\beta }_{3,u}=b{f}_c\left(3{\epsilon }_{c3}-{\epsilon }_{c2}\right)\end{array}.$$

(26)

The position of the neutral axis y_u at the initiation of concrete collapse is defined by the solution of Equation (25), which is thus

$$y_u={\displaystyle \frac{\sqrt{{{\beta }_{2,u}}^2-4{\beta }_{1,u}{\beta }_{3,u}}-{\beta }_{2,u}}{2{\beta }_{3,u}}}.$$

(27)

The other solution of Equation (25) is irrelevant as it lies outside the boundaries of the cross-sectional area.

The characteristic moment M_u is determined by applying the moment equilibrium of all the resulting forces in the cross-section:

$$M_u=M_c+F_s^{\prime }{d}^{\prime }+F_{s}d,$$

(28)

where $M_c=b\left({\displaystyle {\int }_0^{y_1}{\sigma }_{c}ydy+{\displaystyle {\int }_{-y_1}^{y_u}{\sigma }_c(y)ydy}}\right)$ denotes the moment of the stresses in the non-cracked concrete section, which is expressed as

$$M_c={\displaystyle \frac{b{f}_c{y_u}^2\left({{\epsilon }_{c2}}^2-4{\epsilon }_{c2}{\epsilon }_{c3}+6{{\epsilon }_{c3}}^2\right)}{12{{\epsilon }_{c3}}^2}},$$

(29)

and the corresponding curvature ${\kappa }_u={\displaystyle \frac{d{\epsilon }_c\left(y\right)}{dy}}$ is as follows:

$${\kappa }_u=-{\displaystyle \frac{{\epsilon }_{c3}}{y_u}}.$$

(30)

It should be noted that the presented expressions (Equations (3)–(30)), which were derived for doubly reinforced cross-sections, are applicable also to the single rebar cross-section, simply taking into account the cross-section of the top rebar as A_s′ = 0.

3. Practical Application and Experimental Testing

The expressions derived in the previous section can be implemented for a pure bending analysis of an arbitrary rectangular cross-section with single or double reinforcement. Although they are derived in complete accordance with the laws of structural mechanics, they also include some details related to the EC2 standard requirements. To gain insight into matching the assumptions of structural mechanics and the standard’s requirements, an analysis of the bending of a simply supported beam, for which the experimentally obtained values of transverse displacement at the centre were known [10], was performed using the presented expressions. Since it is a simple statically determined structure, a preliminary analysis was first performed, which led to analytical expressions for the magnitude of the transverse displacement for different phases related to the degree of cracking of different parts of the concrete beam depending on the size of the load. These derived specific expressions for the magnitude of the transverse displacement of the structure in question enabled a faster analysis of the values for comparison, but they are not necessary since all the integrals that appear in the analysis can equally be qualitatively evaluated purely numerically.

3.1. Determination of a Closed-Form Solution for the Transverse Displacement at the Centre of a Simply Supported Beam

In this section, the limit value of the Euler–Bernoulli transverse displacement at the centre of the four-point loaded RC beam from Figure 5 is calculated. The end spans and the middle span areas had different longitudinally reinforced rectangular cross-sections, with single reinforcement (subscript 1) in the end spans and double reinforcement (subscript 2) in the middle span. The beam characteristics include the total span of the beam between the two supports (l), the end span from the support to the concentrated force (l₁) and the mid-span between the concentrated forces (l₂). The beam is loaded with two vertical concentrated loads, acting at the borders of the middle span area. This load causes a constant bending moment value in the beam between both forces in the middle span area, and linear ones in both side areas.

Transverse displacement is calculated for all three limit moments ($M_{cr}^1$ at the initiation of the first crack, $M_y^1$ at the occurrence of the first yielding of the tensile reinforcement and $M_u^1$ at the compressive failure of the concrete, all appearing at the middle span area of the beam). Due to the symmetry of the beam and the loading, an alternative simplified model for the left half of the beam is considered for calculating the transverse displacements. The change in the bending stiffness, which occurs at the first crack, does not affect the linear relationship between the load and the bending moments due to the static determinacy of the structure. Therefore, the three limit moments $M_i^1$ (i = cr, y, u) that occur at the middle span area of the beam between the two forces, as shown in Figure 6, determine the two bending moment functions for the two areas:

$$M_i\left(x\right)={\displaystyle \frac{M_i^1}{l_1}}x\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0\le x\le l_1,$$

(31)

and

$$M_i\left(x\right)=M_i^1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{l}_1\le x\le {\displaystyle \frac{l}{2}},$$

(32)

where $M_i^1=P_i{l}_1/2$ (i = cr, y, u) represents the maximum bending moment at the middle span area of the beam as a function of the applied load P_i.

The transverse displacement of the beam at the centre (Δ) caused by bending is determined analytically for all three characteristic moments using the virtual work principle $\Delta ={\displaystyle {\int }_{l}M\left(x\right)/EI\delta M\left(x\right)} dx$, where δM(x) is the moment that occurs due to a virtual vertical unit force at the centre of the beam. Considering the relation between the bending moment and the curvature, the transverse displacement Δ is expressed by the following equation:

$$\Delta ={\displaystyle {\int }_0^{\frac{l}{2}}\kappa (x)x}dx.$$

(33)

3.1.1. Determination of the Limit Characteristic Transverse Displacement Δ_cr

The transverse displacement Δ_cr that characterises the occurrence of the first crack in area II is determined utilising Equation (33) for the two elastic zones as

$${\Delta }_{cr}={\displaystyle {\int }_0^{l_1}{\kappa }_{cr,M_{cr}^1}^2(x)x}dx+{\displaystyle {\int }_{l_1}^{\frac{l}{2}}{\kappa }_{cr,M_{cr}^1}^{1}x}dx,$$

(34)

where the corresponding curvatures are defined as

$$\begin{array}{l}{\kappa }_{cr,M_{cr}^1}^2(x)={\displaystyle \frac{{\kappa }_{cr}^2}{M_{cr}^2}}{\displaystyle \frac{M_{cr}^1}{l_1}}x\hspace{1em}0\le x\le l_1,\\ {\kappa }_{cr,M_{cr}^1}^1={\kappa }_{cr}^1\hspace{1em}{l}_1\le x\le {\displaystyle \frac{l}{2}}.\end{array}$$

(35)

3.1.2. Determination of the Limit Characteristic Transverse Displacement Δ_y

The characteristic transverse displacement Δ_y at yielding initiation is determined by considering $M_y^1$. At the occurrence of the initial reinforcement yielding, the part of the beam from the left support to location x₁ still remains in the elastic zone (i.e., non-cracked state), while the rest of the beam from location x₁ to the centre of the beam is in the state of the crack propagation limit phase. The location x₁ is found from the moment function when considering $M\left(x=x_1\right)=M_y^1{x}_1/l_1=M_{cr}^2$. Thus, the location x₁ is

$$x_1={\displaystyle \frac{M_{cr}^2}{M_y^1}}{l}_1.$$

(36)

The characteristic displacement Δ_y at yielding initiation is therefore

$${\Delta }_y={\displaystyle {\int }_0^{x_1}{\kappa }_{cr,M_y^1}^2(x)x}dx+{\displaystyle {\int }_{x_1}^{l_1}{\kappa }_{y,M_y^1}^2(x)x}dx+{\displaystyle {\int }_{l_1}^{\frac{l}{2}}{\kappa }_{y,M_y^1}^{1}x}dx,$$

(37)

where the corresponding curvatures are defined as

$$\begin{array}{l}{\kappa }_{cr,M_y^1}^2(x)={\displaystyle \frac{{\kappa }_{cr}^2}{M_{cr}^2}}{\displaystyle \frac{M_y^1}{l_1}}\cdot x\hspace{1em}0\le x\le x_1,\\ {\kappa }_{y,M_y^1}^2(x)={\displaystyle \frac{{\kappa }_{cr}^2-{\kappa }_y^2}{M_{cr}^2-M_y^2}}{\displaystyle \frac{M_y^1}{l_1}}x+{\displaystyle \frac{M_{cr}^2{\kappa }_y^2-M_y^2{\kappa }_{cr}^2}{M_{cr}^2-M_y^2}}\hspace{1em}{x}_1\le x\le l_1,\\ {\kappa }_{y,M_y^1}^1={\kappa }_y^1\hspace{1em}{l}_1\le x\le {\displaystyle \frac{l}{2}}.\end{array}$$

(38)

3.1.3. Determination of the Limit Characteristic Transverse Displacement Δ_u

The characteristic displacement Δ_u is computed under the assumption of the bending moment reaching the value of $M_u^1$ in the middle span area. At the initiation of the compressive failure of the concrete, the section of the beam between the left support and location x₂ is in the elastic zone, while the section between locations x₂ and x₃ is in the cracked zone, and the area between x₃ and the centre of the beam is in the yielding zone. Locations x₂ and x₃ are determined by utilising the moment functions with respect to $M\left(x=x_2\right)=M_u^1{x}_2/l_1=M_{cr}^2$ and $M\left(x=x_3\right)=M_u^1{x}_3/l_1=M_y^2$. Thus, locations x₂ and x₃ are

$$x_2={\displaystyle \frac{M_{cr}^2}{M_u^1}}{l}_1,$$

(39)

and

$$x_3={\displaystyle \frac{M_y^2}{M_u^1}}{l}_1.$$

(40)

The characteristic displacement Δ_u at the compressive failure of the concrete is therefore

$${\Delta }_u={\displaystyle {\int }_0^{x_2}{\kappa }_{cr,M_u^1}^2(x)x}dx+{\displaystyle {\int }_{x_2}^{x_3}{\kappa }_{y,M_u^1}^2(x)x}dx+{\displaystyle {\int }_{x_3}^{l_1}{\kappa }_{u,M_u^1}^2(x)x}dx+{\displaystyle {\int }_{l_1}^{\frac{l}{2}}{\kappa }_{u,M_u^1}^{1}x}dx,$$

(41)

where the corresponding curvatures are defined as

$$\begin{array}{l}{\kappa }_{cr,M_u^1}^2(x)={\displaystyle \frac{{\kappa }_{cr}^2}{M_{cr}^2}}{\displaystyle \frac{M_u^1}{l_1}}x\hspace{1em}0\le x\le x_2,\\ {\kappa }_{y,M_u^1}^2(x)={\displaystyle \frac{{\kappa }_{cr}^2-{\kappa }_y^2}{M_{cr}^2-M_y^2}}{\displaystyle \frac{M_u^1}{l_1}}\cdot x+{\displaystyle \frac{M_{cr}^2{\kappa }_y^2-M_y^2{\kappa }_{cr}^2}{M_{cr}^2-M_y^2}}\hspace{1em}{x}_2\le x\le x_3,\\ {\kappa }_{u,M_u^1}^2(x)={\displaystyle \frac{{\kappa }_u^2-{\kappa }_y^2}{M_u^2-M_y^2}}{\displaystyle \frac{M_u^1}{l_1}}x+{\displaystyle \frac{M_u^2{\kappa }_y^2-M_y^2{\kappa }_u^2}{M_u^2-M_y^2}}\hspace{1em}{x}_3\le x\le l_1,\\ {\kappa }_{u,M_u^1}^1={\kappa }_u^1\hspace{1em}{l}_1\le x\le {\displaystyle \frac{l}{2}}.\end{array}$$

(42)

3.2. Experimental Verification and Discussion

Prior to the bending experiment, two preliminary tests were performed to obtain the mechanical characteristics of the steel and the concrete. The yield strength of the rebar is f_y = 605 MPa and represents the average value of two tensile tests performed separately of the rebar being installed. The measured strain at the yielding of the rebar is ε_y = 3‰. The elastic modulus of the steel (calculated from the measured stresses and strains) is E_s = f_y/ε_y = 201.7 GPa. The compressive mean value of the concrete strength is f_cm,cube = −44.8 MPa and represents the average of the uniaxial compressive strength of the embedded concrete on six standardised cubes.

The beam geometric characteristics from the bending experiment, which have also been implemented in the analytical model, are summarised in

Table 1. Default beam characteristics from the bending experiment.

Parameter	Value
Beam span between supports (l)	1.45 m
Span from support to concentrated force (l1)	0.475 m
Span between concentrated forces (l2)	0.50 m
Cross-section width (b)	0.10 m
Cross-section height (h)	0.15 m
Distance from top edge to centroid of bottom rebar (d)	12.1 cm
Distance from top edge to centroid of top rebar (d′)	2.9 cm
Area of bottom rebar (As)	1.571 cm2
Area of top rebar (As′)	1.571 cm2

.

All the remaining material properties of the concrete that were not determined experimentally during the concrete testing phase were determined using the empirical expressions defined by EC2 and are given in

Table 2. Default material characteristics of concrete in analytical model in accordance with EC2.

Empirical Expressions (According to EC2)	Material Characteristics
$f_{cm}=0.8 f_{cm,cube}$	fc = fcm = −35.84 MPa
$f_{ck}=f_{cm}-8$(MPa)	fck = −27.84 MPa
$f_{ctm}=0.30\cdot {\left(f_{ck}\right)}^{{\displaystyle \frac{2}{3}}}$	fct= fctm = 2.75 MPa
$E_{cm}=22\cdot {\left(f_c/10\right)}^{0.3}$	Ec = Ecm = 32.3 GPa
fck < 50 MPa	εc2 = −2.0‰
fck < 50 MPa	εc3= εcu2 = −3.5‰

. As can be seen from the table, the mean values of the compressive strength (f_c = f_cm), tensile strength (f_ct = f_ctm) and modulus of elasticity (E_c = E_cm) of the cylinder were implemented in the analytical model for the concrete.

The testing device consisted of two parts, the lower support structure and the upper hydraulic piston, through which the load was introduced (Figure 7). The used hydraulic piston had a nominal force of Fv = 100 kN, and the working span of the piston was 250 mm. The piston was connected to a control unit with an analogue load display in pressure units, and the loading rate was around 32 N/s. The displacements of the concrete beam under the applied force Fv were simultaneously measured with three MarCator 1086 digital indicators located beneath the applied forces and at the mid-span of the beam. All the applied devices were calibrated by the corresponding actual testing standards.

The analytical model has produced results which include the limit points for the moment–curvature diagrams ($M^1-{\kappa }^1$ and $M^2-{\kappa }^2$) of the middle span and end span areas, and the force–displacement ($P-\Delta$) diagrams. All of these results have been determined based on the data given for this case, and are conveniently summarised in

Table 3. Results of the characteristic limit points of the analytical model.

i	cr	y	u
$M_i^1$ [kNm]	1.12787	9.98333	10.2068
${\kappa }_i^1$ [10−3 rad/m]	1.17264	38.7849	106.822
$M_i^2$ [kNm]	1.15910	9.93093	10.1910
${\kappa }_i^2$ [10−3 rad/m]	1.13519	37.4166	114.095
$P_i$ [kN]	4.74891	42.0351	42.9761
${\Delta }_i$ [mm]	0.25897	8.51234	19.0298

.

The transverse displacements from the experiment (presented in Figure 8) clearly confirm three phases for the structure in question, as was also foreseen when preparing the analytical expressions, and in the first two phases, the match between the calculated and measured values is almost perfect from the engineering point of view.

The experimental elastic phase (I) shows a slightly shorter linear range of measured transverse displacements at the centre of the beam compared to the analytical model with implemented expressions from EC2. This is due to the use of an apparent overestimation of the concrete tensile strength value (f_ct), obtained through the code. Therefore, the actual tensile strength is apparently much lower than assumed. On the other hand, the remaining material properties of the concrete defined according to EC2 provide a satisfactory accuracy.

In the crack propagation limit phase (II), the transverse displacements match well with the experimental results, providing an accurate representation of the real behaviour of the materials used in the analytical model.

During the reinforcement’s yielding limit phase (III), when the bending moment due to the applied load approaches the ultimate limit state of bearing capacity, the analytical model exhibits a somewhat diminished agreement with the experimentally obtained values. In this phase, the stress resultant in the concrete area is an increasing function of stresses and the simultaneously decreasing function of the concrete area in compression. Such an inverse relationship between the two parameters certainly depends on the quality of both material’s constitutive laws, which are, as we know, approximations. Therefore, the observed differences are attributed to the use of a simplified and conservative non-linear behaviour (in accordance to standard EC2) for both materials in the analytical model. In the preliminary experimental testing of the steel bars, the elastic limit was conservatively set at the limit of proportionality, without consideration of their non-linear elastic (i.e., with the inclined top branch of the steel design stress–strain diagram) behaviour. Nevertheless, the analytically derived definitions not only yield more than satisfactory results considering the uncertainties inherent in the experimental results in the ultimate plastic range, but the experimentally obtained results are even more favourable from the design practice point of view.

4. Expanding Utilisation of Detailed Solutions to Simplified Substitutive Computational Model

The analysis of the case of a simple structure in the previous study not only showed that the matching of values in the more significant engineering phase of the structural element’s cracking is very good, but at the same time, it also exposed that such a thorough non-linear incremental analysis would be very time- and computational effort-consuming for more demanding structures.

When considering the serviceability limit states, where we usually focus just on the displacements, or in situations with a not accurate enough knowledge of the load (unpredictable both in magnitude as well as in direction), such as, e.g., in the analysis of the seismic load, it becomes clear that the analysis of the practically infinite possible combinations of orientations and the magnitude of the seismic load in this way is at least meaningless, if not even impossible. At the same time, it is clear that with such extreme loads, it is impossible to avoid concrete cracking, which will appear quite unpredictably (both in terms of location and size), because this is also the nature of an earthquake impact. Therefore, various standards (Eurocode 8, FEMA) enable a compromise between the thorough analysis and the complete ignoring of cracking. The impact of the cracks is thus introduced as a simple uniform reduction of bending stiffness over the entire length of each individual structural element. Such an elementary approach of including cracking is computationally straightforward, but at the same time, the quality of its results is questionable.

4.1. Determination of the Simplified Computational Model Mechanical Parameters

A much better engineering alternative, also allowed by the structural codes and also already implemented in existing software, is where the displacement response of a cracked element or beam is being modelled as a combination of two rigid parts, connected by a rotational spring (Figure 9). The position of the spring coincides with the location of the analysed displacement. The stiffness K_r of this spring must now summarise the effects from both distributions: the pure elastic deformation as well as the cracked state within the complete structural element (in contrast to the rotational spring stiffness genuine definition, which is an exclusive function of the local state of the crack, i.e., at the discrete location of the crack).

The total characteristic rotations φ_tot,_i (i = cr, y, u) in spring with rotational stiffness K_r situated at the midpoint of the beam, shown in Figure 9, which encompass both the elastic as well as non-elastic deformations in the beam, have been determined using three analytically computed characteristic transverse displacements Δ_i (i = cr, y, u) and the kinematic relationship that is valid for the selected substitutive model. Each total characteristic rotation φ_tot,i value was thus calculated as

$${\phi }_{tot,i}={\displaystyle \frac{4{\Delta }_i}{l}}.$$

(43)

and the obtained values are presented in the first row of Table 3.

This definition of a rotational spring is appropriate for analyses that use non-deformable line elements. However, structural computational models (like finite element models) already account for the bending elastic deformations of elastic segments. Therefore, the elastic (linear) component φ_e,i is obtained through the implementation of linear constitutive moment–rotation law and similar triangles:

$${\phi }_{e,i}={\phi }_{cr}{\displaystyle \frac{M_i^s}{M_{cr}^s}}$$

(44)

Further, this value is subtracted from the total rotation of the rotational spring φ_tot,_i [35], providing the pure inelastic component for all three characteristic points (i = cr, y, u):

$${\phi }_{p,i}={\phi }_{tot,i}-{\phi }_{e,i}.$$

(45)

Figure 10 shows the generalised moment–rotation diagrams for all three definitions of rotational springs.

4.2. Application Demonstration and Discussion

All of the characteristic corresponding rotation values for the previously considered beam have been determined by Equations (34), (37), (41) and (43)–(45) implementing the example data given in Section 3.2, and are conveniently summarised in

Table 4. Results of the characteristic rotation limit points of the rotational spring stiffness.

i	cr	y	u
${\phi }_{tot,i}$ [10−3 rad]	0.71440	23.4823	52.4959
${\phi }_{p,i}$ [10−3 rad]	0.00000	17.1588	46.0308
${\phi }_{e,i}$ [10−3 rad]	0.71440	6.32356	6.46511

. These values, combined with the corresponding moments ($M_i^s$) from Table 3, allow for the actual tri-linear moment–rotation diagram for a rotational spring to be constructed. Further, the corresponding rotational spring stiffness K_r value for phases II and III is obtained as the ratio of the difference of the limiting moments of each phase to the difference of the limiting rotation of the same phase (while for phase I, the stiffness value equals infinity).

To demonstrate the possibilities of the alternative simplified computational model, an FE model was prepared to perform a non-linear static bending pushover analysis of the previously considered four-point-loaded, simply supported RC beam. Due to the different cross-sections of the beam, as well as the applied concentrated loads, several FEs had to be implemented (Figure 11). Since the primary goal of this model was to determine the displacement at the mid-span, i.e., at the point where the experimentally obtained values of the displacement were known, the cracked beam three-noded finite element (CB3NFE) with an embedded non-linear rotational spring (K_r), with an additional degree of freedom at the location of the rotational spring [36,37], was applied for the middle-span area (element 2). By reducing the number of degrees of freedom, the utilisation of this element allows for a slightly smaller computational model than the implementation of four standard elements, simultaneously yielding an identical quality of the results. The applied FE model allows for more convenient modelling than the analytical solving of differential equations, and at the same time significantly simplifies the modelling for the analysis performed.

In the discussed finite element analysis procedure, the bending moment at the loca-tion that governs the state and stresses in the cross-section was first obtained for each con-sidered load P. Afterwards, the bending stiffnesses in the applied finite elements was calculated by multiplying the concrete modulus of elasticity (E_c) with the moment of inertia (I_n) defined in Equation (4). Further, the implemented embedded rotational spring stiffness K_r was taken into account as a function of the applied moment, incorporating the limit points of the moment–rotation diagrams ($M^s-\phi$) for three defined rotations, summarised in Table 4, allowing also the stiffness matrix for the second finite element to be evaluated. The additional process of analysis follows the standard steps for finite element analyses. Therefore, all three stiffness matrices of the finite elements are assembled in the structure’s stiffness matrix.

For the example, for the load P = 42.0351 kN, the following stiffness matrix was obtained:

$$\left[K\right]=\left[\begin{array}{ccccccc}8598.42& -27152.9& 4299.21& 0& 0& 0& 0\\ -27152.9& 324460.& 22265.3& -235595.& 25462.9& -3250.92& 0\\ 4299.21& 22265.3& 20663.5& -52669.1& 3250.92& -523.309& 0\\ 0& -235595.& -52669.1& 471190.& -235595.& 52669.1& 0\\ 0& 25462.9& 3250.92& -235595.& 324460.& -22265.3& 27152.9\\ 0& -3250.92& -523.309& 52669.1& -22265.3& 20663.5& 4299.21\\ 0& 0& 0& 0& 27152.9& 4299.21& 8598.42\end{array}\right]$$

By also applying the load vector, discrete displacements and rotations in the model’s nodes are obtained. Afterwards, this procedure is repeated for all load values of interest. However, it should be noted that the load must be applied incrementally (which is common in non-linear static analysis). Namely, when applying a load, structures redesign themselves accordingly, as the bending moment change can alter the bending stiffness, and the change of the bending stiffness in statically indeterminate structures even consequently causes a redistribution of the bending moment.

These results obtained from the presented FE model for the transverse displacement at the mid-span were identical to the results of the analytical model presented previously.

The relation established in this model between the derived tri-linear moment–curvature and moment–rotation diagrams provides a foundation for future research aimed at refining the definition of plastic hinge lengths. This link between the moment–curvature behaviour of the cross-section and the moment–rotation characteristics of a rotational spring allows for more accurate modelling of the plastic behaviour of RC beams.

5. Conclusions

This paper introduces an analytical model for predicting the transverse displacement of RC beams, using a three-linear moment–curvature diagram of the cross-section. The main contributions of the study are as follows:

• A closed-form solution for each of the three-phase moment–curvature relationships was developed, covering critical limit points: initial cracking, tensile reinforcement yielding, and ultimate concrete failure in compression;
• The model provides accurate predictions of the flexural behaviour of RC beams, validated through comparisons with experimental data, particularly in the elastic and crack propagation phases;
• The model accommodates both single and double reinforcement configurations, broadening its applicability across different beam designs;
• The analytical framework is adaptable and can be integrated into more complex structural systems to predict behaviour under varying loading conditions;
• A simplified computational model was also considered, enhancing the efficiency of the structural analysis and making it more practical for engineering applications.

Despite its strengths (efficiency and proven accuracy in the first two response phases), the model assumes perfect bonding between concrete and reinforcement and uses idealised material behaviours, which may limit its precision in more complex scenarios, such as irregular loading, high strain rates or non-standard geometries. Future research should address these limitations by refining material constitutive models, also incorporating shear effects and exploring the impact of cyclic and dynamic loading conditions.

Future studies might also build on the model with the rotational spring to develop more refined approaches that account for varying material properties, loading conditions and complex cross-sectional geometries, ultimately enhancing the reliability of plastic hinge length predictions in design and analysis.