Four-Point Bending of Basic Rails: Theory and Experimental Verification

Abstract

Mathematical models of prediction provide theoretical support for basic rail automation. The three-point bending method for basic rails is characterized by its simplicity and flexibility, and, as such, it is widely used in bending processes. However, due to the significant curvature changes that occur after bending, it is not suitable for scenarios requiring large arc bending, and its range of achievable deflections is limited. This study focuses on four-point bending, dividing the bending process into three stages and using a power-law material hardening model to establish different bending moment expressions for each stage. We derived the relationships between curvature, elastic zone ratio, load, and deflection, ultimately creating a load–deflection model. Based on the simple springback law, we developed the final bending prediction model. Finite element simulations were conducted to simulate the bending process under various conditions, using top punch distances ranging from 200 mm to 400 mm and die distances ranging from 600 mm to 1000 mm. These simulations validated the advantages and accuracy of the four-point bending prediction model in large arc bending. Additionally, a four-point bending experimental setup was established under specified conditions. The experimental results were compared with the theoretical model calculations, showing errors within 0.2 mm and thus verifying the accuracy of the four-point bending prediction model. The mathematical model developed in this study provides theoretical support for the automation of basic rail bending.

1. Introduction

The rapid development of railway technology has led to the development of trains that can tolerate higher speeds and heavier loads, imposing higher demands on the precision of railway track equipment. Basic tracks are the main type of track encountered during train operation, and, as no train journey is entirely straight, this necessitates that basic tracks possess a certain degree of curvature to ensure high smoothness, precision, and reliability in curved sections of railway [1,2,3]. Therefore, top bending is indispensable in railway track machining.

The top bending process of basic tracks involves the plastic deformation of metal. In the initial stage of top bending, the basic track undergoes elastic deformation under external loads. When the external load exceeds the elastic limit load, the basic track undergoes elastic–plastic deformation, resulting in irreversible plastic deformation. Upon removal of the external load, the basic track enters the rebound stage, partially recovering from its deformation [4]. Due to the varying elastic deformations present in different top bending processes, predicting residual deflection in top bending is complex.

In the field of metal elastic–plastic deformation, S.K. Panthi [5] conducted a study on the bending springback of sheet metal bending processes using finite element analysis. Boris Štok [6] analyzed the deflection of beams with rectangular cross-sections under specific loading conditions. A.E. Shelest [7] developed an elastic–plastic alternating bending model for metal strips processed on roller straightening machines, calculating geometric and deformation parameters during the straightening process. H. Jrad [8] established constitutive equations for material plasticity using a linear hardening model. O. Nabochenko [9] proposed the criterion of equivalent bending stiffness for tracks. Additional scholars have studied the bending deflection of sandwich beams under concentrated loads [10,11].

In terms of research into using load–deflection models for bending problems, Jinping Ou [12] investigated reinforced concrete beams and obtained the load–deflection relationship under three-point bending. Tsai and Kan [13] created a load–deflection relationship model for cantilever beams using both rectangular cross-sections and uniformly distributed loads. S. Fallah Nafari [14] established a mathematical correlation between rail deflection, rail stress, and the applied load on the rail. Fu Cui [15] provided relevant concepts for beam straightening and derived the relationship between load and deflection during three-point top bending processes. Jun Li et al. [16] simplified the three-point pressure straightening problem for metals and obtained a predictive model. Youshuo Song [17] analyzed the elastic–plastic mechanics of metals, derived a load–deflection model for T-shaped rails, and conducted finite element simulations and validation experiments.

In studies of metal top bending, the focus has mainly been on three-point pressure bending. However, three-point pressure bending has certain disadvantages when processing large-radius arcs. To better meet the requirements of such scenarios and enhance precision in production processes, a predictive mathematical model for four-point pressure bending was constructed. The correctness of the model was validated through finite element simulation, and a comparative analysis was conducted using results from traditional three-point bending models to verify the superiority of the proposed model [18,19].

2. The Four-Point Bending Mechanical Model

According to the characteristics of the four-point pressure top bending process of a basic track, the following assumptions are made during theoretical modeling:

(a) During the elastic–plastic deformation process of the basic track, it is assumed that the neutral layer is located at the center of the rail thickness.

(b) A cross-section of the plane is available.

(c) Stress is unidirectional.

(d) The Bauschinger effect and reverse yield phenomenon are neglected during the unloading of the material.

For more precise calculations, we simplified the forces applied by the upper two anvils on the basic track according to the characteristics of the four-point pressure top bending process, as depicted in Figure 1. During the downward pressing of the anvils, the outer edges of the anvils make linear contact with the basic track. The concentrated force is located at the outer edges of the upper two anvils, with its distance defined as the outer anvil distance $l_m$. The distances between the inner edges of the two anvils are defined as the inner anvil distance $l_n$. The width of each anvil is $l_a$, and the width of the support anvils is $l_b$. The distance from the center position to the support anvils is $l_w$.

The mechanical model of the four-point bending process is shown in Figure 2.

Using the simplified mechanical model, the forces $F$ and moments $M$ at different positions can be calculated.

In the AC segment:

$$F_x=F_A=\frac{1}{2}F$$

(1)

$$M_x=F_{A}x=\frac{1}{2}Fx$$

(2)

In the CD segment:

$$F_x=F_A-\frac{1}{2}F=0$$

(3)

$$M_x=F_{A}x-\frac{1}{2}F\left(x-l_w^{\prime }\right)=\frac{1}{2}F{l}_w^{\prime }$$

(4)

In the BD segment:

$$F_x=-F_B=-\frac{1}{2}F$$

(5)

$$M_x=F_B\left(2l_w-x\right)=\frac{1}{2}F\left(2l_w-x\right)$$

(6)

In elastoplastic mechanics, to simplify the complexity of problems, it is common to simplify the stress–strain relationship using certain mathematical expressions, as shown in Figure 3. Various simplification methods are commonly used. Depending on the material characteristics, an appropriate elastoplastic model is selected for stress–strain calculations.

For the U71Mn material examined in this study, which exhibits distinct strengthening stages, a more accurate material model was required to calculate the bending moment. Therefore, we adopted a power-law strengthening material model to calculate the bending moment of the steel rail, where $K$ is the coefficient related to the material and $n$ is the power-law strengthening coefficient. Figure 4 contrasts the tensile test data with the power-law strengthening model, showing a high degree of overlap.

$$\left\{\begin{array}{l}\sigma =E\epsilon \\ {\sigma }^{\prime }=K{\epsilon }^n\end{array}\right.$$

(7)

Due to the complex cross-sectional shape of a basic rail, calculating its bending moment is very complex and requires simplification. After simplifying the cross-section, as shown in Figure 5, the basic rail can be divided into three parts: the rail head, rail waist, and rail bottom. During the bending process, plastic deformation can be divided into three stages based on the location of the plastic bending. When plastic deformation penetrates into the mutated section, the formula for calculating the bending moment changes accordingly. Reference [17] provides bending moment calculation formulas for different stages; of these, the elastic limit bending moment is denoted as $M_t$, the bending moment of the first stage is denoted as $M_1$, the bending moment of the second stage is denoted as $M_2$, and the bending moment of the third stage is denoted as $M_3$.

(1) Elastic limit bending moment

When the basic track undergoes bending due to external loads, the initial deformation is elastic. The elastic limit bending moment of the basic track during the top bending process is as follows:

$$M_t=2{\displaystyle {\int }_0^{\frac{B_1}{2}}\sigma H_{1}xdx}+2{\displaystyle {\int }_0^{\frac{B_2}{2}}\sigma H_{2}xdx}+2{\displaystyle {\int }_0^{\frac{B_3}{2}}\sigma H_{3}xdx}$$

(8)

By substituting $\sigma =\frac{2x}{B_1}{\sigma }_s$ into the above equation, we obtain the following:

$$M_t=\frac{H_1{B}_1^3+H_2{B}_2^3+H_3{B}_3^3}{6B_1}{\sigma }_s$$

(9)

According to the calculation formula for $W=\frac{M}{\sigma }$, the bending resistance section modulus of the basic track can be determined as follows:

$$W=\frac{H_1{B}_1^3+H_2{B}_2^3+H_3{B}_3^3}{6B_1}$$

(10)

(2) Elastoplastic bending moment

We defined the elastic zone ratio $\xi$ as the ratio of the length of the elastic zone to the total length, denoted as $\xi =H_t/H$. The bending moments experienced by the basic track at different bending stages were calculated using the following three phases.

Phase 1: Plastic deformation occurs only at the rail base.

The expression of the bending moment is established as follows:

$$M_1=2{\displaystyle {\int }_0^{\frac{H_t}{2}}\sigma H_{1}xdx}+2{\displaystyle {\int }_{\frac{H_t}{2}}^{\frac{B_1}{2}}{\sigma }^{\prime }{H}_{1}xdx}+2{\displaystyle {\int }_0^{\frac{B_2}{2}}\sigma H_{2}xdx}+2{\displaystyle {\int }_0^{\frac{B_3}{2}}\sigma H_{3}xdx}$$

(11)

By substituting the expression into the equation and rearranging it, we obtain the following:

$$M_1=(\frac{H_1{B}_1^2{\sigma }_s}{6}-\frac{H_1{B}_1^2{\sigma }_s}{2(n+2)}){\xi }^2+\frac{\left(H_2{B}_2^3+H_3{B}_3^3\right){\sigma }_s}{6B_1}\frac{1}{\xi }+\frac{H_1{B}_1^2{\sigma }_s}{2(n+2)}\frac{1}{{\xi }^n}$$

(12)

Phase 2: Plastic deformation occurs at both the rail base and rail head.

The expression for the bending moment is established as:

$$M_2=2{\displaystyle {\int }_0^{\frac{B_2}{2}}\sigma H_{2}xdx}+2{\displaystyle {\int }_0^{\frac{H_t}{2}}\sigma H_{1}xdx}+2{\displaystyle {\int }_{\frac{H_t}{2}}^{\frac{B_1}{2}}{\sigma }^{\prime }{H}_{1}xdx}+2{\displaystyle {\int }_0^{\frac{B_3}{2}}\sigma H_{3}xdx}+2{\displaystyle {\int }_{\frac{H_t}{2}}^{\frac{B_3}{2}}{\sigma }^{\prime }{H}_{3}xdx}$$

(13)

By substituting the expression into the equation and rearranging it, we obtain the following:

$$M_2=(\frac{(H_1+H_3)B_1^2{\sigma }_s}{6}-\frac{(H_1+H_3)B_1^2{\sigma }_s}{2(n+2)}){\xi }^2+\frac{H_2{B}_2^3{\sigma }_s}{6B_1}\frac{1}{\xi }+(\frac{(H_1{B}_1^{n+2}+H_3{B}_3^{n+2}){\sigma }_s}{2(n+2)B_1^n})\frac{1}{{\xi }^n}$$

(14)

Phase 3: Plastic deformation occurs at the rail base, rail head, and rail waist.

The expression for the bending moment is established as follows:

$$M_3=2{\displaystyle {\int }_0^{\frac{H_t}{2}}\sigma H_{1}xdx}+2{\displaystyle {\int }_{\frac{H_t}{2}}^{\frac{B_1}{2}}{\sigma }^{\prime }{H}_{1}xdx}+2{\displaystyle {\int }_0^{\frac{H_t}{2}}\sigma H_{2}xdx}+2{\displaystyle {\int }_{\frac{H_t}{2}}^{\frac{B_2}{2}}{\sigma }^{\prime }{H}_{2}xdx}+2{\displaystyle {\int }_0^{\frac{H_t}{2}}\sigma H_{3}xdx}+2{\displaystyle {\int }_{\frac{H_t}{2}}^{\frac{B_3}{2}}{\sigma }^{\prime }{H}_{3}xdx}$$

(15)

By substituting the expression into the equation and rearranging it, we obtain the following:

$$M_3=(\frac{(H_1+H_2+H_3)B_1^2{\sigma }_s}{6}-\frac{(H_1+H_2+H_3)B_1^2{\sigma }_s}{2(n+2)}){\xi }^2+(\frac{(H_1{B}_1^{n+2}+H_2{B}_2^{n+2}+H_3{B}_3^{n+2}){\sigma }_s}{2(n+2)B_1^n})\frac{1}{{\xi }^n}$$

(16)

By rearranging the above equations, we can obtain the bending moment expressions for each stage as follows:

$$M_{\lambda }=\left(a{\xi }^2+\frac{b}{\xi }+\frac{c}{{\xi }^n}\right){\sigma }_s=\left(a{\xi }^2+\frac{b}{\xi }+\frac{c}{{\xi }^n}\right)\frac{M_t}{W}$$

(17)

where the values of a, b, and c depend on the stage of plastic deformation during the top bending process.

3. Establishing the Top Bending Deflection Model

According to the equilibrium of the internal and external moments, we can derive the equilibrium relationship for the elastoplastic stage as follows:

In the AC segment: $\frac{1}{2}Fx=\left(a{\xi }^2+\frac{b}{\xi }+\frac{c}{{\xi }^n}\right)\frac{M_t}{W}$

In the CO segment: $\frac{1}{2}F{l}_w^{\prime }=\left(a{\xi }^2+\frac{b}{\xi }+\frac{c}{{\xi }^n}\right)\frac{M_t}{W}$

Figure 6 illustrates the variations in deflection during the four-point bending process of the basic track. Here, $l_t$ represents the length of the elastic region on one side of the basic track, and $l_s$ represents the length of the elastoplastic region on one side.

If we consider the small deformation $d_x$ to be the small arc length $d_s$ in the deflection curve, let us designate the curvature at point x as $C_x$ and the corresponding radius of curvature as ${\rho }_x$. Then, we can calculate the change in angle as $d_{\theta }=d_x/{\rho }_x=C_x{d}_x$ and simplify it to find the change in deflection at point x using $d_y=x{d}_{\theta }=x{C}_x{d}_x$. Hence, the formula for calculating the deflection at the midpoint is as follows:

$${\delta }_{\Sigma }={\displaystyle {\int }_0^{\delta }{d}_y}={\displaystyle {\int }_0^{l_w^{\prime }}x{C}_x}{d}_x$$

(18)

We integrated the above equation piecewise for $0\sim l_t$ and $l_t\sim l_w^{\prime }$. The elastic deformation regions are located at the two ends of the basic track, where the length $l_t$ of the elastic region is constant when the force $\frac{F}{2}$ is constant. Therefore, within the elastic region, the function $C_x$ is a linear function. According to reference [20], the equation can be obtained as follows:

$$C_x=\frac{M_x}{EI}=\frac{Fx}{2EI}$$

(19)

By substituting the expression and rearranging it, we obtained the calculation formula for the first part as follows:

$${\delta }_{{\Sigma }_1}={\displaystyle {\int }_0^{l_t}\frac{F{x}^2}{2EI}}{d}_x+{\displaystyle {\int }_{l_t}^{l_w^{\prime }}x{C}_{x}dx}=\frac{F{l}_t^3}{6EI}+{\displaystyle {\int }_{l_t}^{l_w^{\prime }}x{C}_{x}dx}$$

(20)

The calculation formula for the second part is as follows:

$${\delta }_{{\Sigma }_2}={\displaystyle {\int }_{l_w^{\prime }}^{l_w^{\prime }+\frac{l_m}{2}}x{C}_{x}dx}$$

(21)

4. Establishing the Four-Point Bending Deflection Model

Based on the curvature formula $C_x=\frac{C_t}{\xi }$, according to the previous analysis, the following equation can be derived. Based on the total deflection formula ${\delta }_{\Sigma }={\delta }_{{\Sigma }_1}+{\delta }_{{\Sigma }_2}$ of the bending process, where ${\delta }_{{\Sigma }_1}$ represents the deflection of the first part and ${\delta }_{{\Sigma }_2}$ represents the deflection of the second part, rearranging the formula yields the correspondence between $F$ and ${\delta }_{\Sigma }$. By inputting different parameters, the corresponding mathematical model for predicting four-point pressure bending can be obtained under different conditions. Reference [15] adopts the minimum elastic area ratio ${\xi }_{\min }=0.2$.

$$\left\{\begin{array}{l}{F}_1=\alpha \frac{2M_t}{l_w^{\prime }}\left(\alpha =1\to \frac{a{{\xi }_{\min }}^2+\frac{b}{{\xi }_{\min }}+\frac{c}{{\xi }_{\min }^n}}{W}\right)\\ {\delta }_{{\Sigma }_1}=\frac{F{l}_t^3}{6EI}+{\displaystyle {\int }_{l_t}^{l_w^{\prime }}x{C}_{x}dx}\\ C_x=\frac{C_t}{\xi }\\ x_1=\frac{l_w^{\prime }}{\alpha W}\left(a{\xi }^2+\frac{b}{\xi }+\frac{c}{{\xi }^n}\right)\end{array}\right.$$

(22)

$$\left\{\begin{array}{l}{F}_2=F_1\\ {\delta }_{{\Sigma }_2}={\displaystyle {\int }_{l_w^{\prime }}^{l_w^{\prime }+\frac{l_m}{2}}x{C}_{x}dx}\\ C_x=\frac{C_t}{\xi }\\ \alpha W=a{\xi }^2+\frac{b}{\xi }+\frac{c}{{\xi }^n}\end{array}\right.$$

(23)

5. Finite Element Simulation Analysis

This paper focuses on the four-point bending of the base rail. Based on the actual field conditions, the bending process was modeled as shown in Figure 7, which depicts the simplified model used in ANSYS finite element simulation.

Applying pressure to the basic track using two anvils causes elastoplastic deformation. To facilitate the simulation, the corresponding boundary conditions and suitable meshes were set according to the parameters shown in

Table 1. Finite element simulation parameters for four-point top bending.

Rail Material	Distance between Top Picks (mm)	Outer Distance of the Top Pick (mm)	Distance between Support Picks (mm)
U71Mn	8	100, 200, 300	600, 700, 800, 900, 1000

, and simulations were conducted for different models.

The corresponding boundary conditions were set as shown in Figure 8. The contact between the top die and the base rail was set to have a friction coefficient of 0.15, similar to the setting between the support die and the top die. As depicted in Figure 9, the contact between the base rail and the supporting plate was set to have a friction coefficient of 0.2.

By inputting the parameters set for the simulation into the four-point top bending predictive model and setting the material parameters to those of the fitted power-law strengthening model, with a downward stroke of 8 mm for the two anvils, finite element simulation data were obtained. Due to space constraints, only partial data are displayed. The theoretical calculation results and corresponding simulation results for a top anvil outer distance of 200 mm are shown in

Table 2. Theoretical and simulated strokes for top anvil outer distance of 200 mm.

Support Distance (mm)	Theoretical Calculated Stroke (mm)	Finite Element Simulation Stroke (mm)	Error (mm)
600	3.83	3.65	0.18
700	2.41	2.55	−0.14
800	1.66	1.60	0.06
900	0.75	0.88	−0.13
1000	0.20	0.36	−0.16

.

To observe the patterns of variation more clearly, the obtained data were plotted as load–deflection graphs, using the same top anvil outer distance but different support distances in each graph. The results are shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

Using Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, we can observe that under the same top anvil outer distance conditions the load–deflection model forms for different support distances are the same, but numerical differences exist. When the top anvil outer distance remains the same, a smaller support distance requires a greater load for top bending, resulting in larger deflection changes in the rail and larger residual deflection after top bending. This indicates that in situations where significant deflection is required during top bending reducing the support distance appropriately can achieve larger deflection while meeting the load requirements.

As shown in Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19, under the same support distance conditions, the load–deflection model forms for different top anvil outer distances are also the same. When the support distance remains the same, a larger top anvil outer distance requires a greater load for top bending, resulting in larger deflection changes in the rail and larger residual deflection after top bending. This indicates that in situations where significant deflection is required during top bending increasing the top anvil outer distance appropriately can achieve larger deflection while meeting the load requirements.

Based on these two properties, we selected the most suitable top anvil outer distance and support distance according to the deflection requirements of the top bending process, thus achieving the desired deflection for top bending.

The deflection curves of the working edge after top bending under different conditions are plotted in Figure 20, Figure 21, Figure 22, Figure 23 and Figure 24.

As shown in Figure 20, Figure 21, Figure 22, Figure 23 and Figure 24, it is clear that under the same support distance and the same downward stroke a larger top anvil outer distance results in a greater residual deflection and a larger area of plastic deformation in the rail.

By comparing the simulation model of four-point top bending with the traditional three-point top bending model in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23 and Figure 24, it can be seen that under the same support distance four-point top bending has a wider range of top bending with the same top bending stroke. By increasing the top anvil outer distance, the residual deflection after top bending can be increased, thereby enlarging the top bending range. In terms of bending curvature, the deflection curve of the working edge of the rail after four-point top bending is smoother and therefore more suitable for the requirement of bending into a large arc. The curvature changes more rapidly in three-point top bending, which is more suitable for folding requirements.

In order to expand the deflection range of top bending and meet the bending requirements for basic track processing, research on four-point top bending is vital. As mentioned above, different deflection curves of the working edge can be obtained by changing the support distance and the top anvil outer distance to meet the deflection and curvature requirements of top bending.

6. Result Validation

When setting up the four-point pressure bending test platform to validate the bending prediction model, the parameter information of the basic rail was as shown in

Table 3. Information for the basic rail parameters.

Parameters	Numerical Values
E	214 GPa
${\sigma }_s$	650 GPa
$H_1$	12 mm
$H_2$	98 mm
$H_3$	30 mm
$B_1$	114 mm
$B_2$	14.5 mm
$B_3$	70 mm

.

${\sigma }_s$$H_1$$H_2$$H_3$$B_1$$B_2$$B_3$ Experiments were conducted using a 1500 mm base rail for bending. The experiments involved varying the distance between the top dies and the support dies. Specifically, the top die distance was set to either 200 mm, 250 mm, 300 mm, 350 mm, or 400 mm, while the support die distance was set to either 600 mm, 700 mm, 800 mm, 900 mm, or 1000 mm. Multiple bending operations were performed under different conditions. Figure 25 depicts an actual on-site photo of the top bending process where the top dies were symmetrically placed. The rail was subjected to loading by using two top dies to induce bending. The deflection of the rail was detected using displacement sensors, while the load on the rail was measured using pressure sensors, thus validating the correctness of the calculated load–deflection model and the predictive model.

Using the experimental data, a specific set of parameters was selected: the top die outer distance was set to 200 mm, the support distance was set to 800 mm, and the top die downward stroke was set to 8 mm. Under these parameters, calculations were performed using the four-point top bending predictive model established in this study, resulting in the comparison graph shown in Figure 26. It can be observed that during the experimental process a pronounced elastic stage occurred, with the slope closely matching that calculated by the model. The rebound stage in the experiment exhibited an approximately linear distribution, with a slope similar to that of the elastic stage.

Due to space constraints, the data presented in

Table 4. Experimental data.

Bending Deflection/mm	Simulation Error/mm	Experimental Error/mm
1	0.10	0.09
2	0.15	0.11
3	0.13	0.11
4	0.14	0.13
5	0.16	0.15
6	0.17	0.14
7	0.15	0.16
8	0.18	0.15

correspond to a top die distance of 200 mm and a support die distance of 800 mm. After obtaining a set of data using the four-point bending prediction model, finite element simulations were performed on the bending process. The deviation between the rail bending deflection and the target deflection was less than 0.2 mm/m. Similarly, using the same parameters in our experiments resulted in a deviation from the target deflection of less than 0.2 mm/m. Therefore, the four-point bending prediction mathematical model established in this study can provide theoretical support for the automation of rail bending.

7. Conclusions

This study investigates the four-point bending of basic rails and analyzes parameters such as bending moment and curvature during the bending process. A power-law strengthening model for the rail material is fitted, and a predictive mathematical model for four-point bending is derived.

Our analysis of existing data reveals that under the same distance between the upper anvils a smaller distance between the support anvils leads to greater load requirements and deflection changes in the rail during bending, resulting in larger residual deflections after bending. Similarly, under the same support anvil distance, a greater distance between the upper anvils leads to higher load requirements and larger deflection changes, resulting in larger residual deflections after bending. Within the safety load limits, the desired deflections can be obtained by adjusting the support anvil and upper anvil distances.

A comparison with the traditional three-point bending model described in reference [21] shows that this model applies bilinear material strengthening. When the mechanical model is simplified, the bearing is simplified into a concentrated force, yielding a three-point bending model. While this ensures that the basic rail does not form plastic hinges, the bending range is limited, and processing requirements can only be met by increasing the number of bending cycles; constant deflection means that the post-bending curvature is large and unchangeable and is more suitable for bending scenarios.

The four-point pressure bending model established in this study under the same support spacing has a wider bending range. By increasing the distance between bending points, the residual deflection after bending can be increased, thereby enlarging the bending range. In terms of bending curvature, the working edge deflection curve of the rail after four-point bending is smoother and more in line with the requirement of bending into a large arc, whereas three-point bending exhibits rapid curvature changes which are more suitable for folding requirements.

The four-point bending predictive mathematical model established in this study provides new insights for studying the bending of other complex cross-section beams and theoretical support for automated bending equipment for basic rails.