基于曲率积分法的板材矫直理论研究
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摘要
板材广泛应用于车辆、船舶、桥梁、航空航天等众多国民经济领域。随着市场经济的深化和科学技术的进步，用户对板材质量要求越来越高。尽管采用先进的轧制设备及工艺可以一定程度上提高板材质量，但是在轧制生产过程中，由于板厚不均、温度变化和冷却不均等原因可能会使板材产生弯曲、中波和边波等板形缺陷；另外，即使是平直的板材，如果残余应力过大，在机械加工、热处理、运输、使用等过程中，由于残余应力释放，还可能出现失稳而产生新的板形缺陷，严重时会产生微观裂纹甚至开裂。因此，板材还需要通过后续的矫直工序改善其平直度和控制残余应力，最终满足各行业的高质量工程标准要求。
板材矫直过程的实质，就是通过矫直机上、下两排交错排列的矫直辊实现连续反复弹塑性弯曲变形，从而达到矫直和减少板形缺陷的目的。板材矫直质量很大程度上取决于矫直机的先进性和矫直工艺的合理性，这两者都依赖于先进的矫直理论。目前国内矫直理论落后于实践的现象比较明显，例如，虽然个别厂家推出了高刚度、全液压和全自动的新一代强力矫直机，但设计机理并不十分清晰，基本上还处于类比设计阶段，而工艺模块仍然要依赖于国外大公司。因此，开展板材矫直理论分析及相关工艺优化的系统深入研究，对于提高我国矫直机设计水平和矫直工艺应用水平、增强国内企业在该领域的核心竞争力具有重要的理论意义和广阔的应用前景。
本文基于曲率积分法建立了反映压下量与反弯曲率和横向残余应力关系的解析模型，采用高效的算法进行了数值求解和试验验证；探讨了板材矫直过程中的曲率、应力、塑性变形率、矫直轨迹、矫直力和扭矩等主要矫直参数的变化规律；结合倾斜压下矫直方案并基于曲率积分法针对典型来料规格研究了辊系结构参数的确定方法；分析了影响矫直质量的主要因素，总结了这些因素与合理压下量的关系，创建了第三代强力液压矫直机弯辊辊形曲线数据库；基于曲率积分法准确绘制了与现场数据吻合的矫直能力边界曲线；以辊系压下量为设计变量，以总矫直力或者出口平直度最小为目标建立了多约束的非线性隐式压下工艺优化模型，并进行数值优化求解，优化结果和现场数据具有较好的一致性。
本文取得的主要创新研究成果如下：
①推导了板材与相邻两辊之间接触角、压下量与曲率分布关系的曲率积分模型，首次采用优化方法对该模型推导的非线性方程组直接进行了数值求解，克服了以曲率积分模型为目标的迭代算法收敛性差和精度不够的缺点，实现了求解方法的创新。以该模型为基础，对横向残余应力进行了解析，探讨了压下量对板材矫后残余应力的影响规律。
②考虑倾斜压下矫直方案并基于曲率积分法准确计算了单辊最大矫直力和扭矩，根据强度理论确定了辊距下限，考虑矫直辊有限的压下行程确定了辊距上限，在保证矫直质量的条件下确定了最少辊数，从计算的所有品种的最小辊距中挑选出最大值作为最优辊距，改变了传统方法按照塑性极限状态确定单辊最大矫直力、矫直扭矩，进而确定辊距的思路，实现了结构参数确定方法的创新。
③从期望塑性变形率和设备最大力能参数限制角度建立了快速确定矫直能力边界的模型，绘制了粗略的矫直能力曲线，分析了其构成原理，以此为基础，结合倾斜压下方案并基于曲率积分法对每个厚度进行屈服极限极大化的准确搜索，获得了更接近实际的矫直能力边界曲线，为快速判定来料是否可矫、一定厚度板材可矫的最大屈服极限或一定屈服极限板材可矫的最大厚度提供了理论依据。
④探讨了来料板材厚度、材料屈服极限、弹性模量、压下量和弯辊凸度对矫直质量的影响，研究了在满足矫直质量条件下，这些因素与合理压下量的关系，所总结的理想弹塑性材料板材的弹性极限曲率与合理压下量之间的规律与创建的弯辊辊形曲线数据库可以为后续矫直工艺制定和在线合理设定弯辊力提供依据。
⑤以总矫直力最小或者出口平直度最优为目标，以辊系压下量为设计变量，以设备安全运行为部分不等式边界条件，以矫直能力边界所对应的实际压下量为另外一部分不等式边界条件，以压下量和辊系反弯能力的关系为等式约束条件建立了大型非线性隐式矫直压下工艺优化模型，并进行了高效的数值优化求解，优化的结果与现场几百个品种数据具有较好的一致性，验证了所建立模型的可靠性，为现场合理制定压下工艺提供了理论依据。
本文在借鉴前人研究成果的基础上，系统和全面地深化了板材矫直基本理论，所取得的一些创新性的研究成果为合作企业提供了有力技术支撑。
Plates are widely used in the national economic areas including automobiles,shipbuilding, bridges construction, aerospace, et al. The required quality of plates hasbeen increasing with the development of the market economy and the progress ofscientific technologies. Although advanced rolling equipments and rolling technologiescan improve the quality of plates in some degree, it could be possible to appear someplate defects such as bending, middle wave, and edge wave due to the of the differenceof the temperature, nonuniformity of the thickness of plates. In addition, flat platesmight be still instable and form a new defect during the secondary manufacturing, heattreatments and transport process if there are great enough residual stresses in plates. Themore serious situation is the appearance of some microscopic crack on the surface ofplates. Therefore, bending plates should be leveled and the residual stress should befurther removed by the leveling process in the actual production for the high qualityproduction purpose of plates.
The essence of the leveling process for plates is a continuous elastic-plasticitybending and reverse bending deformation process by staggered roller gaps so that thebending shape and residual stress can be removed finally. The quality of plates afterleveling is determined by the advantage of levelers and reasonability of the levelingtechnology which depend largely on the advanced leveling theory. Current levelingtheory is too simple to describe the leveling process which is not enough to satisfy thepractical application. Although some steel industries at home have proposed their newgeneration levelers with high stiffness characteristic and hydraulic system, they still donot understand the basic design principle fully. It means they are still at analogy-baseddesign level and rely on the major company aboard about the leveling process module.Therefore, the study of the leveling theory and research of the leveling technologyoptimization have important theoretical meaning and wide application foreground forimproving the equipment design and leveling technology level and enhancing the corecompetitiveness for steel companies at home.
This paper built an analytical model related the roller gap, bending curvature andresidual stress horizontally based on the curvature integration method. This model wassolved numerically by an effective algorithm and the result was verified byexperimental data. The curvature, stress, plastic ratio, trajectory and mechanical parameters distribution during leveling process for the plate were discussed. A methodof designing the structure parameters of the roller system was proposed based on thecurvature integration method and linearly decreasing roller gap schedule. The mainfactors which influenced the leveling quality were analyzed. The relationship betweenthe factors and the reasonable roller gap was summarized. The database of roll bendingcurves was built for the third generation hydraulic leveler. The curves of levelingcapacity border matched well with field data were drawn based on the curvatureintegration method. A nonlinear and implicit technology optimization model withequality and inequality constraints was built. The roller gaps were selected as designvariables and the target was to minimize the total leveling force or the final flatness inthis model. It was solved by a numerical optimization algorithm. There was aconsistency between the optimization results and field data.
The achievement of this paper could be listed as follow:
①The curvature integration model was proposed. This model could describe therelationship of contact angles between the plate and adjacent rollers, the roller gap andthe curvature distribution as well. An optimization method was used for the first time tosolve the nonlinear system of equations deduced by the curvature integration model.This method could avoid the disadvantage of poor convergence and precision of theiterative algorithm. The residual stress horizontally was analyzed based on this method.The influence of the roller gap on the residual stress after leveling was discussed.
②The maximum leveling force and torque of the single roller were calculatedaccurately considering the linearly decreasing roller gap schedule based on theintegration method. The lower roller spacing was determined by the strength theory. Theupper roller spacing was calculated according to the limited stroke of rollers. Theminimum roller number was selected in the condition of leveling quality of plates. Themaximum value among all the lower roller spacing of the plates was chosen as the finalroller spacing. This method changed the traditional idea on how to design the rollerspacing because the traditional idea calculated the maximum roller force and torque byan assumption of the plastic limit state. It means the proposed method is innovative fordesigning the structure parameters.
③A model constrained by expected plastic ratio and maximum mechanicalparameters was built to disseminate the leveling border quickly. The rough levelingcapacity curves were drawn approximately and the constitutive principle of final curvewas discussed. The maximum yield stress value was searched for each thickness considering the linearly decreasing roller gap schedule and the curvature integrationmethod. The precision leveling capacity curves closer enough to the actual curves werefound. The curves of the leveling capacity border could be used to judge whether theincoming plate could be leveled, the maximum yield stress of a specific thickness andthe maximum thickness of a specific yield stress.
④The influence of the thickness, yield stress, elasticity modulus of the incomingplate, roller gap and bending value of rollers on the leveling quality were discussed. Therelationship between these single variables and the reasonable roller gap wassummarized in the condition of good leveling quality for plates. This relation and thedatabase of the bending roller curves could provide evidences on how to set roller gapsand reasonable bending forces.
⑤A large scale nonlinear and implicit technology optimization model was built.The total leveling force and final flatness was selected as the target. The roller gap waschosen as the design variables. The final quality and the condition of safety operationfor the leveler were viewed as some inequality boundary conditions. The otherinequality boundary conditions were the maximum value of the roller gap related withthe leveling capacity border. The equality boundary condition was the relationshipbetween the roller gap and the bending curvature based on the curvature integrationmethod. This model was solved by an effective algorithm. The optimum results couldmatch hundreds of field data well which verify the reliability of the proposed model.This model could provide an theoretical foundation about setting reasonable roller gaps.
This paper studied the basic leveling theory for plates systematically based on theachievements of former researchers. The innovative achievements could provide somestrong technical support for the cooperative enterprise.
板材矫直过程的实质，就是通过矫直机上、下两排交错排列的矫直辊实现连续反复弹塑性弯曲变形，从而达到矫直和减少板形缺陷的目的。板材矫直质量很大程度上取决于矫直机的先进性和矫直工艺的合理性，这两者都依赖于先进的矫直理论。目前国内矫直理论落后于实践的现象比较明显，例如，虽然个别厂家推出了高刚度、全液压和全自动的新一代强力矫直机，但设计机理并不十分清晰，基本上还处于类比设计阶段，而工艺模块仍然要依赖于国外大公司。因此，开展板材矫直理论分析及相关工艺优化的系统深入研究，对于提高我国矫直机设计水平和矫直工艺应用水平、增强国内企业在该领域的核心竞争力具有重要的理论意义和广阔的应用前景。
本文基于曲率积分法建立了反映压下量与反弯曲率和横向残余应力关系的解析模型，采用高效的算法进行了数值求解和试验验证；探讨了板材矫直过程中的曲率、应力、塑性变形率、矫直轨迹、矫直力和扭矩等主要矫直参数的变化规律；结合倾斜压下矫直方案并基于曲率积分法针对典型来料规格研究了辊系结构参数的确定方法；分析了影响矫直质量的主要因素，总结了这些因素与合理压下量的关系，创建了第三代强力液压矫直机弯辊辊形曲线数据库；基于曲率积分法准确绘制了与现场数据吻合的矫直能力边界曲线；以辊系压下量为设计变量，以总矫直力或者出口平直度最小为目标建立了多约束的非线性隐式压下工艺优化模型，并进行数值优化求解，优化结果和现场数据具有较好的一致性。
本文取得的主要创新研究成果如下：
①推导了板材与相邻两辊之间接触角、压下量与曲率分布关系的曲率积分模型，首次采用优化方法对该模型推导的非线性方程组直接进行了数值求解，克服了以曲率积分模型为目标的迭代算法收敛性差和精度不够的缺点，实现了求解方法的创新。以该模型为基础，对横向残余应力进行了解析，探讨了压下量对板材矫后残余应力的影响规律。
②考虑倾斜压下矫直方案并基于曲率积分法准确计算了单辊最大矫直力和扭矩，根据强度理论确定了辊距下限，考虑矫直辊有限的压下行程确定了辊距上限，在保证矫直质量的条件下确定了最少辊数，从计算的所有品种的最小辊距中挑选出最大值作为最优辊距，改变了传统方法按照塑性极限状态确定单辊最大矫直力、矫直扭矩，进而确定辊距的思路，实现了结构参数确定方法的创新。
③从期望塑性变形率和设备最大力能参数限制角度建立了快速确定矫直能力边界的模型，绘制了粗略的矫直能力曲线，分析了其构成原理，以此为基础，结合倾斜压下方案并基于曲率积分法对每个厚度进行屈服极限极大化的准确搜索，获得了更接近实际的矫直能力边界曲线，为快速判定来料是否可矫、一定厚度板材可矫的最大屈服极限或一定屈服极限板材可矫的最大厚度提供了理论依据。
④探讨了来料板材厚度、材料屈服极限、弹性模量、压下量和弯辊凸度对矫直质量的影响，研究了在满足矫直质量条件下，这些因素与合理压下量的关系，所总结的理想弹塑性材料板材的弹性极限曲率与合理压下量之间的规律与创建的弯辊辊形曲线数据库可以为后续矫直工艺制定和在线合理设定弯辊力提供依据。
⑤以总矫直力最小或者出口平直度最优为目标，以辊系压下量为设计变量，以设备安全运行为部分不等式边界条件，以矫直能力边界所对应的实际压下量为另外一部分不等式边界条件，以压下量和辊系反弯能力的关系为等式约束条件建立了大型非线性隐式矫直压下工艺优化模型，并进行了高效的数值优化求解，优化的结果与现场几百个品种数据具有较好的一致性，验证了所建立模型的可靠性，为现场合理制定压下工艺提供了理论依据。
本文在借鉴前人研究成果的基础上，系统和全面地深化了板材矫直基本理论，所取得的一些创新性的研究成果为合作企业提供了有力技术支撑。
Plates are widely used in the national economic areas including automobiles,shipbuilding, bridges construction, aerospace, et al. The required quality of plates hasbeen increasing with the development of the market economy and the progress ofscientific technologies. Although advanced rolling equipments and rolling technologiescan improve the quality of plates in some degree, it could be possible to appear someplate defects such as bending, middle wave, and edge wave due to the of the differenceof the temperature, nonuniformity of the thickness of plates. In addition, flat platesmight be still instable and form a new defect during the secondary manufacturing, heattreatments and transport process if there are great enough residual stresses in plates. Themore serious situation is the appearance of some microscopic crack on the surface ofplates. Therefore, bending plates should be leveled and the residual stress should befurther removed by the leveling process in the actual production for the high qualityproduction purpose of plates.
The essence of the leveling process for plates is a continuous elastic-plasticitybending and reverse bending deformation process by staggered roller gaps so that thebending shape and residual stress can be removed finally. The quality of plates afterleveling is determined by the advantage of levelers and reasonability of the levelingtechnology which depend largely on the advanced leveling theory. Current levelingtheory is too simple to describe the leveling process which is not enough to satisfy thepractical application. Although some steel industries at home have proposed their newgeneration levelers with high stiffness characteristic and hydraulic system, they still donot understand the basic design principle fully. It means they are still at analogy-baseddesign level and rely on the major company aboard about the leveling process module.Therefore, the study of the leveling theory and research of the leveling technologyoptimization have important theoretical meaning and wide application foreground forimproving the equipment design and leveling technology level and enhancing the corecompetitiveness for steel companies at home.
This paper built an analytical model related the roller gap, bending curvature andresidual stress horizontally based on the curvature integration method. This model wassolved numerically by an effective algorithm and the result was verified byexperimental data. The curvature, stress, plastic ratio, trajectory and mechanical parameters distribution during leveling process for the plate were discussed. A methodof designing the structure parameters of the roller system was proposed based on thecurvature integration method and linearly decreasing roller gap schedule. The mainfactors which influenced the leveling quality were analyzed. The relationship betweenthe factors and the reasonable roller gap was summarized. The database of roll bendingcurves was built for the third generation hydraulic leveler. The curves of levelingcapacity border matched well with field data were drawn based on the curvatureintegration method. A nonlinear and implicit technology optimization model withequality and inequality constraints was built. The roller gaps were selected as designvariables and the target was to minimize the total leveling force or the final flatness inthis model. It was solved by a numerical optimization algorithm. There was aconsistency between the optimization results and field data.
The achievement of this paper could be listed as follow:
①The curvature integration model was proposed. This model could describe therelationship of contact angles between the plate and adjacent rollers, the roller gap andthe curvature distribution as well. An optimization method was used for the first time tosolve the nonlinear system of equations deduced by the curvature integration model.This method could avoid the disadvantage of poor convergence and precision of theiterative algorithm. The residual stress horizontally was analyzed based on this method.The influence of the roller gap on the residual stress after leveling was discussed.
②The maximum leveling force and torque of the single roller were calculatedaccurately considering the linearly decreasing roller gap schedule based on theintegration method. The lower roller spacing was determined by the strength theory. Theupper roller spacing was calculated according to the limited stroke of rollers. Theminimum roller number was selected in the condition of leveling quality of plates. Themaximum value among all the lower roller spacing of the plates was chosen as the finalroller spacing. This method changed the traditional idea on how to design the rollerspacing because the traditional idea calculated the maximum roller force and torque byan assumption of the plastic limit state. It means the proposed method is innovative fordesigning the structure parameters.
③A model constrained by expected plastic ratio and maximum mechanicalparameters was built to disseminate the leveling border quickly. The rough levelingcapacity curves were drawn approximately and the constitutive principle of final curvewas discussed. The maximum yield stress value was searched for each thickness considering the linearly decreasing roller gap schedule and the curvature integrationmethod. The precision leveling capacity curves closer enough to the actual curves werefound. The curves of the leveling capacity border could be used to judge whether theincoming plate could be leveled, the maximum yield stress of a specific thickness andthe maximum thickness of a specific yield stress.
④The influence of the thickness, yield stress, elasticity modulus of the incomingplate, roller gap and bending value of rollers on the leveling quality were discussed. Therelationship between these single variables and the reasonable roller gap wassummarized in the condition of good leveling quality for plates. This relation and thedatabase of the bending roller curves could provide evidences on how to set roller gapsand reasonable bending forces.
⑤A large scale nonlinear and implicit technology optimization model was built.The total leveling force and final flatness was selected as the target. The roller gap waschosen as the design variables. The final quality and the condition of safety operationfor the leveler were viewed as some inequality boundary conditions. The otherinequality boundary conditions were the maximum value of the roller gap related withthe leveling capacity border. The equality boundary condition was the relationshipbetween the roller gap and the bending curvature based on the curvature integrationmethod. This model was solved by an effective algorithm. The optimum results couldmatch hundreds of field data well which verify the reliability of the proposed model.This model could provide an theoretical foundation about setting reasonable roller gaps.
This paper studied the basic leveling theory for plates systematically based on theachievements of former researchers. The innovative achievements could provide somestrong technical support for the cooperative enterprise.
引文
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