Residual stress control of hot-rolled strips during run-out table cooling

The strips with good flatness after finish rolling will have flatness defects such as edge wave after run-out table cooling. These defects are caused by the residual stress due to inhomogeneous thermal and microstructural behaviors during the cooling stage. The traditional residual stress control methods rely too much on experience, resulting in poor control effect and poor universality. Therefore, this study proposed two residual stress control methods based on model calculation: water flow central crown and center wave compensation control. The former is mainly based on controlling the uniformity of strip transverse temperature and microstructure. The latter is mainly controlled by mutual compensation of internal stresses. In this paper, the mathematical models of water flow central crown cooling and critical center wave compensation stress were established, respectively. Meanwhile, a post-buckling model of hot-rolled strip was established to calculate the aim flatness value at exit of finishing mill under critical compensation stress. Furthermore, a fully coupled finite element model based on temperature, phase transitions, and stress was developed and it was verified using experimental data. Finally, the above two residual stress control methods were realized in the finite element model. The results showed that the two methods can reduce the residual stress and improve the flatness, and the application of center wave compensation control is more effective.


Introduction
With the continuous optimization of hot rolling technology [1,2], coupled with an advanced control approach [3,4], rolled strips can be created with a good strip shape. However, the flatness of finished products is not consistent with the eventual rolled product. These flatness defects may have various origins, particularly run-out table cooling [5,6] or/and cooling after coiling [7]. The root cause of flatness defects is the formation of internal residual stresses [8]. If the residual stresses are not controlled, even strips without manifest defects will deform during subsequent machining or cutting [9]. Thus, this paper focuses on the methods of controlling residual stress during cooling on the run-out table.
There are significant differences in the metallurgical, mechanical, and flatness properties of the final strips following different cooling techniques [10][11][12]. In particular, uneven transverse cooling will generate thermal stress. Furthermore, it will cause non-uniform phase transformation and lead to the formation of phase transformation stress inside the strips. Thermal stress and phase transition stress are both sources of residual stress. Therefore, the reduce residual stress by controlling cooling technology is attracting more and more attention. It is hard to detect the temperature of strips during the actual cooling process because of the relatively rapid movement of the strips. Therefore, many scholars established numerical models to study the cooling process of hot-rolled strip. Zhou et al. [13] used the finite element ABAQUS software to simulate strip residual stresses during different cooling strategies (uniform, early, later and split cooling). They found that the cooling mode changing the length direction cannot reduce the transverse temperature difference and had little effect on the distribution of residual stress. Witek and Milenin [14] studied the residual stress level in the final coil by using early and late laminar cooling mode. The appropriate selection of laminar cooling conditions allowed to reduce the phase transformation in the coil and reduce the level of residual stresses. Moreover, Qiu et al. [15] used the finite element method to show that the residual stress level of strip after sparse cooling was lower than that after posterior cooling. Wu et al. [16] developed a two-dimensional finite element model to simulate the strip temperature and stress behavior during the run-out table cooling. Front section water cooling and back section water cooling were adopted. The results showed that the residual stress of strip in front section water cooling was less than that in back section cooling. These reports focused on cooling process along the length of run-out table (temperature evolution path). Although the temperature evolution path can effectively control the microstructure of steel strip, the effect of regulating residual stress is limited.
Based on the formation mechanism of residual stress, the residual stress can be improved by controlling the transverse temperature gradient of the steel strip. Weisz-Patrault and Koedinger [17] developed a classic thermalmetallurgical-mechanical coupling numerical model to study the development of residual stresses in thin strips during the transverse uniform cooling process. The numerical results showed that uniform cooling along the width of the strip could significantly reduce residual stresses. In general, the methods of temperature control in strip width are edge induction heating and edge masking, as shown in Fig. 1. Edge induction heating is generally in the stage of transfer table between roughing mill and finishing mill [18,19]. After edge heating, the transverse temperature difference of the strip leaving the last finishing mill was small. The researches of Zhou et al. [13] and Sun et al. [20] showed that improving the initial transverse temperature difference when the strip left the finishing mill can effectively reduce the residual stress. However, edge induction heating requires heating equipment, which also increases energy consumption.
Furthermore, edge masking is also a common lateral temperature control method. Zhang et al. [21] established a novel 1.5D finite element model to simulate the transverse temperature of strip under edge masking process. The model improved the temperature uniformity in plate width by changing the edge masking distance. Tian and Guo [22] calculated the residual stress of the strip after cooling when the edge masking distance was 200 and 300 mm, respectively. The results showed that the masking amount was too large, it would increase the edge residual stress of steel strips. For the edge masking distance, it is mainly dependent on experience or test method. Besides, edge masking also needs to install new masking equipment on the production line.
In addition to the above methods, more efficient control methods with less energy consumption are urgently needed in actual production. In response to these industrial requirements, this study proposes two residual stress control methods during run-out table cooling process. One is water flow central crown, and the other is center wave compensation control. The former main idea is that cooling water flow at the edge of the strip is reduced so that the distribution of cooling water along the width of the strip presents a central crown shape, as shown in Fig. 2. Similar studies such as Kerdabadi et al. [23] studied the influence of changing the water spray mode and layout on the transverse temperature control of plates. However, there is a lack of systematic research on water flow arrangement in water flow center crown cooling.
Up to now the phenomenon of edge wave in hot-rolled strip during the run-out table cooling has been studied extensively [24][25][26]. Hence, center wave compensation control is that strip is rolled with appropriate center wave at the exit of finishing mill to compensate the edge wave defect during the cooling stage. Wang et al. [27] studied the central wave compensation control method with different initial strip flatness values by numerical experiments. Their subsequent study showed that setting the target strip flatness values at the exit of the finishing mill to 10 ~ 20 I Unit was effective in reducing residual stresses and suppressing flatness defects [28]. However, the target flatness of 10 ~ 20 I Unit is suitable for strips with widths and thicknesses around 1250 mm and 2.75 mm, which is difficult to be generally applicable. Obviously, the insufficiency of the above is that setting the appropriate initial flatness value also depends on experience or test methods, and there is no accurate calculation model.
The above researchers have studied in detail the control methods of residual stress in the run-out table cooling process through numerical simulation. However, it is mainly based on the means of experience or trial and error, and no control theory with universal applicability has been proposed. Therefore, to get rid of the residual stress control methods that rely on experience and lack of model calculations, we conducted a series of studies on the control methods proposed in this paper. First, we constructed various functions for the heat transfer coefficient distribution of water flow central crown cooling. The water distribution form of water flow central crown cooling was inversely calculated. Next, the mathematical model of critical center wave compensation stress was established, and the flatness (at the exit of finishing mill) of strip under critical compensation stress was calculated by post-buckling model. Moreover, a high precision nonlinear finite element model based on temperature, phase transitions, and stress was developed. Finally, the above two methods were realized in the finite element model, and the results showed that they could effectively reduce the residual stress level.

Water flow central crown
Generally, the edge temperature of the steel strip was lower than that in the middle area [13]. In order to achieve the purpose of horizontal cooling uniformity, it is necessary to reduce the cooling water volume in the edge area of the strip, where is essentially to reduce the heat transfer coefficient. As shown in Fig. 2, it is assumed that the strip width is 2b and the distance of edge temperature reduction is b E . It means that the heat transfer coefficient in the range of shall be less than that in the range of b E , 2b − b E .There are linear, parabolic, sinusoidal, exponential and other function types of decline modes of strip edge heat transfer coefficient. It can be assumed that the heat transfer coefficient is h c in the range of b E , 2b − b E and h e at the edge. Then the crown ratio (ξ is ratio factor) of water flow central crown cooling can be expressed as: where ω is the water flow density (m 3 /(min•m 2 )); T and T w are strip surface and cooling water temperature (°C), respectively; D is the nozzle diameter (m), P l and P c are the nozzle spacing in the rolling direction and the nozzle pitch in the width direction (m), respectively. According to the study of Miyake et al. [29], formula (6) is very accurate within the range of Reynolds number (Re = vD/υ) being 10 4 ≤ Re ≤ 10 5 during the laminar cooling process. Where, v is the velocity of water flow at the nozzle mouth (m/s) and coefficient of kinematic viscosity υ (m 2 /s) is approximately expressed by υ = 2.5 − 1.15lgT w . Therefore, the distribution of cooling water can be inversely calculated according to formula (6).

Center wave compensation control
The transverse temperature and microstructure uniformity of strip can be improved by water flow central crown cooling, and finally reduce the residual stress. However, the different thermophysical parameters and microstructure transition temperatures of different grades of steel, which also causes the difference of cooling process. Center wave compensation control is not affected by steel grade. Strip was rolled with appropriate center wave at the exit of the last stand of finishing mill to compensate the flatness change trend of edge wave occurring at cooling stage. Essentially, it is the internal stresses of strip cancel each other, as shown in Fig. 4. Where x, y are transverse and longitudinal coordinates, respectively.
As shown in Fig. 4, the internal stress distribution of strip at the end of the run-out table cooling is assumed to σ y (x) EW , and the center wave compensation stress distribution at the finish rolling exit is σ y (x) CW . The determination of σ y (x) CW is shown in Fig. 5 (The stress distribution is based on assumptions).
As shown in Fig. 5a, the area under σ y (x) CW stress was exactly the same as that under σ y (x) EW stress, which meant that at the initial σ y (x) CW stress, the internal stress of the strip was cancelled after the run-out table cooling. When coiling tension was applied to the compensating stress in Fig. 5a, the compensating stress distribution becomes as shown in Fig. 5b. It can be found that the compressive stress at the edge would be offset into tensile stress  after cooling, and the tensile stress state in the middle remained unchanged. This can also eliminate strip flatness defects. On the contrary, when the stress distribution of the strip after finishing rolling is shown in Fig. 5c, the edge compressive stress cannot be completely offset during subsequent cooling on the run-out table. Furthermore, when the stress distribution of the strip after finishing rolling is shown in Fig. 5d, excessive compensation will occur, resulting in central wave defects of the strip after cooling. However, in some cases, the compensation stress distribution shown in Fig. 5c and d could also make the cooling residual stress significantly lower than the critical buckling stress. This also restrained residual stress so that the strip did not deform during subsequent cutting. Therefore, the critical center wave compensation stress distribution is defined as: where b CW and b EW are the distances from the center wave buckling position and edge wave buckling position to the center of the strip, respectively. b is half the width of the strip. ΔF is the difference between residual stress and compensated stress after cooling.
Polynomial can be used to describe the internal stress function.
where σ 0 is stress amplitude, e i (i = 0, 2, 4, 6, 8) is the stress distribution coefficient; e c is a constant determined by stress self-equilibrium.
Considering the stress inside the strip need to reach self-equilibrium, the following equation should be meet: where h(x) is the thickness, and h(x) = h 0 .
By combining Eq. (9) and Eq. (10): For the flatness of the strip at the finish rolling outlet, the buckling wave shape function is defined as follows: where r is the half of wave height, r i (i = 0, 2, 4, 6, 8) is a constant to control the mode of the wave, a is the half of wavelength.
And the wave shape function needs to meet the following constraints: The Von-Karman's large deflection equation for center buckling of hot-rolled steel strip was established. The equation was described in detail in the Appendix. The buckling degree of wave in strip was obtained by using energy method to solve the equation [31][32][33]. In the buckling phase, according to the principle of large deflection energy of thin plate, the of bending strain energy U 1 and mid-plane strain energy U 2 were calculated.
where E is the elastic modulus, μ is Poisson's ratio.
When the strip head exits the run-out table and enters the coiler, coiling occurs. Therefore, the post buckling solution under small tension was carried out. Assuming that the compression strain is positive, the longitudinal total strain distributed along the width direction can be expressed as: where CW is the center wave compensation strain; T is tension strain; δ is the longitudinal displacement.
The condition of longitudinal uniform distribution can be obtained by differentiating Eq. (16).
Integrating δ in Eq. (17) twice in combination with the buckling deflection function, the following expression is obtained: The boundary conditions of longitudinal displacement are expressed as follows: The final total longitudinal strain is expressed as follows: The principle of minimum potential energy is invoked to solve for the unknown coefficients in the assumed outof-plane deflections and wave degree.

Material model
The strip grade E235B was determined to be studied, and its chemical composition is shown in Table 1. The (20) physical parameters of the strip changes with temperature were provided by JMatPro thermal dynamics calculation software, whose calculations are widely considered to be reliable. The thermophysical parameters are illustrated in in Fig. 6.
During the run-out table cooling process of hot-rolled strip, ̇q is the internal heat source due to the transformation of austenite and this term can be expressed as [34].
where F and P denote ferrite and pearlite, respectively; ΔX i is the transformed volume fraction within a time increment of Δt.
The kinetics of the diffusional transformation of austenite to ferrite and pearlite have been described for the run-out table cooling condition by Esaka model [28,35] as follows: where X max is the maximum volume fraction of a phase; B is the phase transition parameter, where the new phase will be ferrite or pearlite, expressed by 4 or 100, respectively; d γ is the austenite grain size of 30 μm; t and τ are cooling time and incubation period of phase transition, respectively; The function Δε is defined by: where f D and f N are the volume fraction of dynamic recrystallization and non-dynamic recrystallization of the steel strip, respectively; ε C is the critical deformation at the beginning of dynamic recrystallization; H and h denote the thickness values of the strip before and after the rolling process, respectively. The parameters τ, k and n can also be expressed by: where [%C] and[%Mn] are the contents of carbon and manganese in steel strip. The physical parameters of the steel strip with temperature changes significantly affected the calculated results. Figure 6 shows the thermal physical parameters of the steel strip; however, the constitutive model required more detailed physical parameters. The curves of Young's modulus, Poisson's ratio, and yield strength varied with temperature, and the true stress-strain curves are shown in Fig. 7.
According to the research conducted by Wang et al. [28], thermal expansion coefficient 10 −6 ⋅ • C −1 and phase transformation expansion coefficient can be expressed by: As the strip ran continuously on the actual run-out table, a length of 6 m was selected for study, and the width and thickness were 1200 mm and 2 mm, respectively. Then, the finite element model was established by the commercial finite element software ANSYS on the basis of above material model.

Model evaluation
To verify the accuracy of the finite element model, the real-time temperature of the strip during the actual cooling process was measured using an infrared thermal imager in a hot-rolled strip (E235B) production line. Large amounts of measurements were conducted and recorded, and then processed. The actual initial lateral temperature was measured when the strip left the last stand of the finishing mill, as shown in Fig. 8a. It was input into the finite element model, and the calculated values at any given time were compared with the corresponding actual measured values. The results are illustrated in Fig. 8b-d. It can be seen from Fig. 8, the transverse temperature of the steel strip, as calculated by the finite element model was essentially the same as the actual transverse temperature. The temperature error of each lateral position was mainly concentrated within − 1 to 14 °C, as shown in Fig. 9. In addition, the subsequent phase transformation, stress, and flatness evolution in the strip during cooling were all determined by the temperature. Therefore, it is reasonable to discuss the residual stress control using the established finite element model.

Initial temperature conditions
In order to study the regulation of subsequent residual stress and improve the calculation efficiency, the new initial transverse temperatures are adopted. According to the work by Zhou et al. [13], the temperature in the middle of the strip was uniformly 880 °C, and the temperature drops at the edges were approximately linear, with a minimum temperature of 820 °C. Two initial temperature conditions were set to approximate the actual distribution, as shown in Fig. 10.

Results and discussion
The two residual stress control methods were studied based on conventional cooling finite element model. Various water flow central crown cooling heat transfer coefficient functions were used as boundary conditions of high precision finite element model. The strip flatness under center wave compensation stress is taken as the initial condition of finite element model. The residual stresses under these two control methods were calculated respectively. The results were compared with the residual stress and flatness of strip under conventional cooling. The specific calculation process is shown in Fig. 11.

Cooling water distribution
To be specific, we choose the equipment parameters of laminar cooling process on a hot rolling production line are shown in the following table: According to the parameters in Table 2, the heat transfer coefficient in the middle of the steel strip can be calculated as 420 W/m 2 •℃ from the formula (6). The ratio factor ξ was determined to be 1.3 by numerical experiments. Hence, eight kinds of central coronal cooling heat transfer coefficient distribution functions can be calculated. Due to the heat transfer coefficient in the middle area was uniform, only the edge heat transfer coefficient distribution was studied, as shown in Fig. 12.
In order to realize the cooling effect shown in Fig. 12, the flow density distribution of cooling water in the edge region needs to be calculated by Eq. (7). According to Eq. (7), water flow density is related to strip surface temperature. Therefore, the analytical solution of the transverse temperature field of steel strip, which may be derived from the technique known as the method of separation of variables, is as follows: where 0 (x) = T 0 (x) − T w is the initial excess temperature; n is the eigenvalue of equation Bi = cot ( ) , Bi is the Biot number; α is the thermal diffusivity of the strip; t is the cooling time, in the water cooling region t = L v s , L is the length of the run-out table, and v s is the running speed of strip.
To facilitate the calculation, only the temperature field and cooling water distribution at the edge of the strip with temperature drop were considered. Therefore, Eq. (28) was simplified according to the cooling boundary conditions, and the specific form of strip edge temperature model in the water-cooled zone was obtained as follows: To verify the accuracy of Eq. (29), the calculation results are compared with the finite element model, as shown in Fig. 13. It can be seen from Fig. 13 that the accuracy errors of Eq. (29) are mainly concentrated in 5 ~ 6.5 ℃, so Eq. (29) can be used for the calculation of water flow density.
The distribution of cooling water flow density at strip edge was as follows: where L w is the length of the water zone,100 m; v s = 10 m∕ s.
The cooling water flow distribution shown in Fig. 14a-h corresponds to the heat transfer coefficients of various types in Fig. 12. The water flow density calculated in Fig. 14 was also within the range of Miyake et al. [29] experiment, so the results were also relatively accurate.

Temperature and microstructure
In order to analyze the influence of the distribution of the heat transfer coefficient on the temperature uniformity in Fig. 12, the temperature evolution of the strip at the initial temperature (a) and (b) was calculated using the finite element model. Figures 15 and 16 show the transverse temperature of the strip at the end of run-out table cooling at two initial temperatures, respectively. The temperature difference along the width was greatly reduced after cooling by the water flow central crown cooling. It can be seen by comparing Figs. 10, 15, and 16 that the convexityconcavity of strip edge heat transfer coefficient function was the same as the convexity-concavity of initial edge temperature, and the lateral uniformity of final cooling was better. In Fig. 15, the temperature uniformity of strip cooling was the best under linear and exponential function type heat transfer coefficients (Black and wine line). In The International Journal of Advanced Manufacturing Technology (2023) 125:3205-3227 3034 Fig. 16, the temperature uniformity of strip cooling was the best under convex parabola and sinusoidal function type heat transfer coefficients (magenta and dark yellow line). This also indicated that the type of initial edge temperature was also a factor affecting the uniform transverse cooling of the strip.
When the edge heat transfer coefficient decreased as power function and exponentially, there was a "high temperature" zone in the area 20 ~ 80 mm away from the edge of the strip. On the contrary, the edge heat transfer coefficient decreased as convex parabola and sinusoidally, and there was a "low temperature" region in this region. According to Newton's law of cooling, the heat flux under each heat transfer coefficient can be calculated, as shown in Fig. 17. direction indicated by the arrow in Fig. 17. It can be found that the heat flux under concave function was smaller than that under convex function at the same position. And the concave function was more concave, the heat flux was less, convex function was the opposite. The higher the heat flux, the more heat the strip loses, and the temperature at that location was relatively low. That is why high temperature and low temperature zones appeared at the edge of the strip in Figs. 15 and 16. Based on the above analysis, the actual selection of the distribution mode of water flow central crown cooling requires comprehensive consideration of the length of the water-cooling area and the initial edge transverse temperature of the strip.
Since the temperature and microstructure evolution of strip were similar under the heat transfer coefficient in Fig. 12, the exponential function type heat transfer coefficient was used for specific analysis. As shown in Fig. 18a, the temperature difference between the strip edge and the middle hardly changed in the early aircooling stage. After entering the water-cooling area, the transverse temperature difference began to decrease due to the water flow central crown, and the transverse temperature difference was very small when leaving the water-cooling area. Moreover, when a certain degree of phase transition occurred, the heat released during the decomposition of austenite (latent heat phase transition) could reduce strip cooling rate, as it is shown in Fig. 18b (the blue dotted circle).
As indicated in Fig. 19, the phase transition also experienced a process from early non-uniformity to late synchronization.
It can be seen from Fig. 19a and d that phase transformation first occurred at the edge of the steel strip in the early water-cooling region, and phase transformation was not synchronous between the middle and the edge. In the case of water flow central crown cooling, the temperature at the edge and in the middle of the strip tended to be uniform and the cooling rate tended to be similar, resulting in the gradual synchronization of the subsequent microstructure transformation, as shown in Fig. 19b and c. As indicated in Fig. 19d, until the end of cooling on the entire run-out table, the content of microstructure in the middle and the edge of the strip was almost uniform. This also shows that the water flow central crown cooling can better regulate the transverse temperature and the uniformity of microstructure.

Stress and flatness
Based on the calculation of temperature and microstructure, the internal stress of the strip was calculated, and the final longitudinal stress along the width direction of the strip under various types, heat transfer coefficient was calculated respectively. It can be seen by comparing Figs. 20 and 21 that although the longitudinal stress value of water flow center crown was similar to that of strip under conventional cooling, the position of maximum compressive stress was different. The maximum compressive stress at strip edge under conventional cooling occurred at a certain distance close to the edge. This was because the transverse temperature difference of strip did not tend to be uniform under conventional cooling conditions, which led to large internal stress exceeding yield stress, resulting in plastic deformation at the extreme edge of the strip. Finally, the larger residual stress appeared in the strip. However, when the strip was cooled by water flow center crown, the internal stress was only formed during the early cooling, so this situation did not occur. This greatly reduces the residual stress in the strip at room temperature. Moreover, it can also be seen from Fig. 21 that the distance of compressive stress concentration at the edge of the strip decreased after the water flow central crown cooling.
In addition, it can be seen by comparing Fig. 22a and b-i that the wave degree of strip cooled by water flow center crown was generally smaller than that by conventional cooling. The average wave degree was 0.4% smaller. Based on the comparison of temperature, phase transformation and stress/flatness, it can be determined that the water flow center crown cooling can effectively control the residual stress and flatness of the strip.

Degree of center wave at finishing mill exit
Distribution mode of internal stress in hot-rolled strip after run-out table cooling, as shown in Fig. 23. The longitudinal stress distribution along the width of the strip can generally be measured by X-ray diffractometer [36] or predicted by numerical simulation [28]. These stress data are coupled into stress function by polynomial as the controlled target of center wave compensation control. The longitudinal stress distribution of strip steel under conventional cooling was used as a controlled target, as shown in Fig. 20 (orange line). The compensation stress of center wave was determined according to Eq. (8). Therefore, the expression of the center wave stress function: Moreover, the distribution of compensation stress (σ y (x) CW ) is shown in Fig. 24.
The wave shape coefficients in Eq. (12) were determined according to boundary conditions. The wave shape function and the center wave stress function are substituted into the post-buckling model for calculation.  cooling. The initial displacement and stress states of the strip in the finite element model are shown in Fig. 26. Since the displacement function was loaded on the element node, the initial stress shown in the finite element model had a small deviation from the theoretical initial stress.
After conventional cooling of the strip with initial center wave stress state, the distribution of longitudinal stress in the strip along the width at the end of cooling in the runout table is shown in Fig. 27. When the coiling tension was removed, it can be seen that there was a small compressive stress in the central region of the strip, which was mainly concentrated at 4 MPa. Tensile stresses occurred in both sides of the strip. In consequence, the residual stress was much less than the critical buckling stress of steel strip.
Compared with the conventional cooling stress and flatness, the center wave compensation control can reduce the residual stress and eliminate edge wave defects, as shown in Fig. 28.

Comparison of residual stress control methods
At present, the existing control methods of residual stress in run-out table cooling process mainly include improving the cooling strategies (uniform, early, later and split cooling), edge induction heating before finishing rolling and the edge masking of laminar cooling process. Improving cooling strategy can effectively control the microstructure but cannot control the residual stress well. Edge  19 Microstructure evolution during cooling process induction heating can well restrain residual stress, but it is not widely used due to expensive heating equipment and large energy consumption. Edge masking is often based on experience to determine the masking distance, resulting in the residual stress control effect cannot be guaranteed. Furthermore, the above control methods are easy to be effective for a specific steel grade or production process, but difficult to be universally applied.
The residual stress control method based on model calculation proposed in this study has strong universal applicability. Although the residual stress distribution of the strip is uneven after water flow center crown cooling, the concentration range of the compressive stress in the edge area was greatly reduced. Compared with conventional cooling, the final edge wave defects were significantly reduced. Meanwhile, the transverse microstructure of strip became more uniform after water flow central crown cooling. Therefore, water flow center crown cooling can control the residual stress and improve the uniformity of microstructure. Moreover, the residual stress distribution of strip under center wave compensation control was opposite to that under conventional cooling, and small compressive stress appeared in the middle area. Based on the internal stress distribution, center wave compensation control can completely suppressed the edge wave defects. Meanwhile, the implementation of center wave compensation control depends on high-precision rolling force model [37] and bending roll technology [38]. Therefore, the center wave compensation control can be realized based on the current technology. All in all, the two methods proposed in this paper can reduce residual stress and improve flatness. If the flatness quality of hot-rolled strips is further pursued, the effect of center wave compensation control is better.

Conclusions
The present study systematically introduces the hot-rolled strips residual stress control methods of run-out table cooling. The distribution function of heat transfer coefficient of water flow central crown cooling and the mathematical model of center wave compensation control were established. The calculation of mathematical model replaced experience and provided theoretical guidance for the implementation of control methods. The validity of these two methods was evaluated by using high precision finite element model. The results are summarized as follows.
1. The distribution function of heat transfer coefficient at the edge of water flow central crown cooling was defined and the corresponding water flow distribution form was calculated. The comparison results indicated the optimal When the strip has large deflection buckling deformation, the buckling deflection is much greater than the thickness of the strip, and the internal stress in the thickness direction of the strip is considered to be the same. The internal stress relationship is expressed as follows: (43) N x = h 0 x N y = h 0 y N xy = h 0 xy Constitutive relation of mid-plane strain and mid-plane force, is given by The strip deformation compatibility equation is shown as follows: Differential equation of force equilibrium Buckling equilibrium differential equation To simplify the expression, Airy stress function F(x, y) is introduced. The relationship between stress function and stress component is as follows:  The Von-Karman's large deflection equation is obtained by combining the above equations.
For hot-rolled strip, only the longitudinal stress is considered, so the equation can be simplified as follows: