钢结构非线性分析方法研究及其在软件中的实现
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摘要
钢结构具有诸多优点,目前在国内外的工程建设中得到了越来越广泛的应用,尤其是在大跨度、大空间结构以及超高层结构的设计中更是必然的选择。传统的钢结构设计采用的是两阶段设计法,第一阶段是按线弹性理论计算结构内力,第二阶段是进行构件设计,通过计算长度系数考虑构件之间的影响,但该方法也有一定局限性,随着现代计算机技术和结构计算理论的发展,钢结构高等分析法被提出并逐渐完善。高等分析法是考虑了结构的非线性响应、各种缺陷以及其他影响结构承载能力的因素,通过对结构进行一次全过程的整体分析研究结构的响应。高等分析法的关键是对结构进行非线性分析,本文将对钢结构几何非线性以及材料非线性的理论进行研究和推导,并以通用钢结构设计软件USSCAD为平台进行非线性程序设计,以完善其非线性分析功能。
空间杆单元受力比较简单,本文采用平衡方程法和UL法两种方法推导其切线刚度矩阵,并与TL法进行比较。在对杆单元进行几何非线性分析时,无论采用TL法或UL法推导的切线刚度矩阵,还是直接基于杆件平衡方程推导出的切线刚度矩阵,计算结果都基本一致。
空间梁单元的几何非线性分析要比杆单元复杂很多,影响计算结果的因素也很多。本文分别采用梁柱法和基于UL列式法的非线性有限元法推导了空间梁单元的切线刚度矩阵。论证了空间大转动的不可易性和不可线性叠加的特点,并采用旋转矩阵和四元数的方法对大转动进行叠加；应用共转法的思想更新单元的局部坐标系,并计算出杆端的自然变形。详细介绍了自然变形法和外部刚度法两种单元内力的计算方法。
目前在轻钢厂房中楔形单元使用非常广泛。本文提出了等效惯性矩的方法建立了楔形单元的力学模型,并对节点与截面形心不重合的偏心现象提出了力学解决方法。采用基于UL列式的非线性有限元法对楔形单元的几何非线性问题进行了分析。
对空间梁单元进行材料非线性分析的关键是确定杆件的塑性模型,一般采用分布塑性模型或集中塑性模型。本文针对理想弹塑性材料,分别采用集中塑性模型中的弹塑性铰法和精细塑性铰法,推导了空间梁单元的弹塑性切线刚度矩阵,采用LRFD相关方程判断杆端的弹塑性状态,并计算了弹塑性单元内力,对超出屈服面的内力进行修正。
根据本文的非线性理论进行程序设计,完善USSCAD钢结构设计软件的非线性功能,非线性方程求解采用Newton-Raphson法、修正的Newton-Raphson法和弧长法,收敛准则采用不平衡力的无穷范数。程序采用面向对象技术,采用C++编程语言,借助Autodesk公司提供的ObjectARX开发工具实现了对AutoCAD2007的二次开发。借助Office办公软件提供的程序接口,实现了word计算书的自动生成。应用USSCAD软件对算例进行的非线性分析表明,本文编制的程序的精度满足结构设计的要求,并具备对大型空间结构进行非线性分析的能力。
本文最后对国内外有关结构抵抗连续性倒塌研究情况进行了介绍,对国外结构规范中有关结构连续性倒塌的内容进行了概述。介绍了拉结力设计法和变换荷载路径法两种常用的分析方法。并对某一满足我国钢结构设计规范要求的钢框架采用变换荷载路径法和美国GSA规范提出的荷载修改方法进行了连续性倒塌的初步分析。
Steel structures have many advantages, including high strength, low weight, fast construction, etc., that they have been widely used in engineering, especially for constructing large span, large space structures and high-rise buildings. Traditionally, steel structures are designed by the two-step method, which includes two steps that first calculating the internal force of members using linear-elastic method, and then designing the members separately under the guides of codes (In those guides, the interaction among members is considered by using effective length factor). As been widely recognized, the two-step method has many shortcomings. Alternatively, with the development of modern computer technology and calculation theory of structure, the advanced analysis method has been developed. The advanced analysis method can consider the nonlinear response of structures, the effects of deficiencies and other factors on the loading capacity of structures, etc.. By the advanced analysis method, only a full-phase global analysis is needed to design the structure. The nonlinear analysis is the key component for an advanced analysis. In this dissertation, the theory of geometric nonlinearity and material nonlinearity of steel structures will be presented. A non-linear programming is developed and added to the common design software of steel structure, USSCAD, to realize the nonlinear analyzing ability of the software.
The mechanism of space bars is very simple that the bars can only have axial displacements. The dissertation has derived the tangent stiffness matrix of space bars by equilibrium equations and UL formulations respectively. In nonlinear analysis of space bars, the results by using the derived matrix from equilibrium equations and UL formulations are the same as the results by using the tangent stiffness matrix from TL formulations.
The geometric nonlinearity analysis is more complex for space beams than that for space bars. The dissertation derives the tangent stiffness matrix of space beams by beam-column theory and the theory of nonlinear finite element by UL formulations. The characteristics of unchangeable and disable to be directly added for space rotations are proved, and then the methods of rotation matrix and quaternion are present for the superposition of space rotations. The idea of co-rotational is used to update the local coordinate system for space beams in each iteration process and to calculate the natural deformations of the ends of beams. In the process of nonlinear analysis, the unbalance forces are very important, which can affect the accuracy of final result. Here, two methods of calculating the internal force are present in detail, which are the natural deformation method and the external stiffens method.
Currently the tapered members are used widely in the light steel plants, but there has no accurate mechanical model for this kind of element. The dissertation proposes a new approach to solve this problem, by adopting the method of equivalent moment of inertia to establish the mechanical model of tapered members. For the condition where the centriod of the section does not coincide with the corresponding node, two revised matrixes are used. The dissertation derives the theory of geometric nonlinear analysis for tapered members by the UL formulation.
The key of material nonlinear analysis for space beams is that of defining the plastic model of members. Generally there are two models:distributed plasticity model and concentrated plasticity model. For elastic-perfectly plastic material the dissertation takes the elastic-plastic hinge method and the refine plastic hinge method deriving the elastic-plastic tangent stiffness matrix. The elastic-plastic state of members is judged by the limit yield surface equation of LRFD. Then the method of calculating the elastic-plastic internal force of beams is presented. At last in order to follow the yield criterion, it is necessary to revise the force-points which are deviant from the yield surface.
By the nonlinear analysis theory of this dissertation, the programming can be easily designed, to complete the ability of nonlinear analysis for the steel software of USSCAD. The methods for solving nonlinear equations are Newton-Raphson method, modified Newton-Raphson method and arc-length method. The convergence criterion of iterations uses the infinite norm of unbalanced forces. The programming takes the object-oriented technique and C++ is adopted as the programming language. With the help of ObjectARX provided by the Autodesk Company, it achieves the secondary development of AutoCAD2007. Using the programming interface provided by the Microsoft Office, it is conveniently to realize the automatical generation of calculation book in the style of Microsoft Word. The results of several numerical examples designed by USSCAD show that the accuracy of nonlinear analysis theory can fulfill the requirements for engineering usages. Furthermore, the software is able to implement nonlinear analysis for large-sized space structures.
Finally, the dissertation introduces the studies on the progressive collapse, and the provisions in some foreign codes for structure design about progressive collapse. The tie force method and the alternate path method are included for analyzing the capacity of structures to resist the progressive collapse. Following with the alternate path method and the modification method of load specified by USA' GSA code, a case study on progressive collapse analysis of a steel frame (which is designed under the guides of Chinese steel structures code) is given.
空间杆单元受力比较简单,本文采用平衡方程法和UL法两种方法推导其切线刚度矩阵,并与TL法进行比较。在对杆单元进行几何非线性分析时,无论采用TL法或UL法推导的切线刚度矩阵,还是直接基于杆件平衡方程推导出的切线刚度矩阵,计算结果都基本一致。
空间梁单元的几何非线性分析要比杆单元复杂很多,影响计算结果的因素也很多。本文分别采用梁柱法和基于UL列式法的非线性有限元法推导了空间梁单元的切线刚度矩阵。论证了空间大转动的不可易性和不可线性叠加的特点,并采用旋转矩阵和四元数的方法对大转动进行叠加；应用共转法的思想更新单元的局部坐标系,并计算出杆端的自然变形。详细介绍了自然变形法和外部刚度法两种单元内力的计算方法。
目前在轻钢厂房中楔形单元使用非常广泛。本文提出了等效惯性矩的方法建立了楔形单元的力学模型,并对节点与截面形心不重合的偏心现象提出了力学解决方法。采用基于UL列式的非线性有限元法对楔形单元的几何非线性问题进行了分析。
对空间梁单元进行材料非线性分析的关键是确定杆件的塑性模型,一般采用分布塑性模型或集中塑性模型。本文针对理想弹塑性材料,分别采用集中塑性模型中的弹塑性铰法和精细塑性铰法,推导了空间梁单元的弹塑性切线刚度矩阵,采用LRFD相关方程判断杆端的弹塑性状态,并计算了弹塑性单元内力,对超出屈服面的内力进行修正。
根据本文的非线性理论进行程序设计,完善USSCAD钢结构设计软件的非线性功能,非线性方程求解采用Newton-Raphson法、修正的Newton-Raphson法和弧长法,收敛准则采用不平衡力的无穷范数。程序采用面向对象技术,采用C++编程语言,借助Autodesk公司提供的ObjectARX开发工具实现了对AutoCAD2007的二次开发。借助Office办公软件提供的程序接口,实现了word计算书的自动生成。应用USSCAD软件对算例进行的非线性分析表明,本文编制的程序的精度满足结构设计的要求,并具备对大型空间结构进行非线性分析的能力。
本文最后对国内外有关结构抵抗连续性倒塌研究情况进行了介绍,对国外结构规范中有关结构连续性倒塌的内容进行了概述。介绍了拉结力设计法和变换荷载路径法两种常用的分析方法。并对某一满足我国钢结构设计规范要求的钢框架采用变换荷载路径法和美国GSA规范提出的荷载修改方法进行了连续性倒塌的初步分析。
Steel structures have many advantages, including high strength, low weight, fast construction, etc., that they have been widely used in engineering, especially for constructing large span, large space structures and high-rise buildings. Traditionally, steel structures are designed by the two-step method, which includes two steps that first calculating the internal force of members using linear-elastic method, and then designing the members separately under the guides of codes (In those guides, the interaction among members is considered by using effective length factor). As been widely recognized, the two-step method has many shortcomings. Alternatively, with the development of modern computer technology and calculation theory of structure, the advanced analysis method has been developed. The advanced analysis method can consider the nonlinear response of structures, the effects of deficiencies and other factors on the loading capacity of structures, etc.. By the advanced analysis method, only a full-phase global analysis is needed to design the structure. The nonlinear analysis is the key component for an advanced analysis. In this dissertation, the theory of geometric nonlinearity and material nonlinearity of steel structures will be presented. A non-linear programming is developed and added to the common design software of steel structure, USSCAD, to realize the nonlinear analyzing ability of the software.
The mechanism of space bars is very simple that the bars can only have axial displacements. The dissertation has derived the tangent stiffness matrix of space bars by equilibrium equations and UL formulations respectively. In nonlinear analysis of space bars, the results by using the derived matrix from equilibrium equations and UL formulations are the same as the results by using the tangent stiffness matrix from TL formulations.
The geometric nonlinearity analysis is more complex for space beams than that for space bars. The dissertation derives the tangent stiffness matrix of space beams by beam-column theory and the theory of nonlinear finite element by UL formulations. The characteristics of unchangeable and disable to be directly added for space rotations are proved, and then the methods of rotation matrix and quaternion are present for the superposition of space rotations. The idea of co-rotational is used to update the local coordinate system for space beams in each iteration process and to calculate the natural deformations of the ends of beams. In the process of nonlinear analysis, the unbalance forces are very important, which can affect the accuracy of final result. Here, two methods of calculating the internal force are present in detail, which are the natural deformation method and the external stiffens method.
Currently the tapered members are used widely in the light steel plants, but there has no accurate mechanical model for this kind of element. The dissertation proposes a new approach to solve this problem, by adopting the method of equivalent moment of inertia to establish the mechanical model of tapered members. For the condition where the centriod of the section does not coincide with the corresponding node, two revised matrixes are used. The dissertation derives the theory of geometric nonlinear analysis for tapered members by the UL formulation.
The key of material nonlinear analysis for space beams is that of defining the plastic model of members. Generally there are two models:distributed plasticity model and concentrated plasticity model. For elastic-perfectly plastic material the dissertation takes the elastic-plastic hinge method and the refine plastic hinge method deriving the elastic-plastic tangent stiffness matrix. The elastic-plastic state of members is judged by the limit yield surface equation of LRFD. Then the method of calculating the elastic-plastic internal force of beams is presented. At last in order to follow the yield criterion, it is necessary to revise the force-points which are deviant from the yield surface.
By the nonlinear analysis theory of this dissertation, the programming can be easily designed, to complete the ability of nonlinear analysis for the steel software of USSCAD. The methods for solving nonlinear equations are Newton-Raphson method, modified Newton-Raphson method and arc-length method. The convergence criterion of iterations uses the infinite norm of unbalanced forces. The programming takes the object-oriented technique and C++ is adopted as the programming language. With the help of ObjectARX provided by the Autodesk Company, it achieves the secondary development of AutoCAD2007. Using the programming interface provided by the Microsoft Office, it is conveniently to realize the automatical generation of calculation book in the style of Microsoft Word. The results of several numerical examples designed by USSCAD show that the accuracy of nonlinear analysis theory can fulfill the requirements for engineering usages. Furthermore, the software is able to implement nonlinear analysis for large-sized space structures.
Finally, the dissertation introduces the studies on the progressive collapse, and the provisions in some foreign codes for structure design about progressive collapse. The tie force method and the alternate path method are included for analyzing the capacity of structures to resist the progressive collapse. Following with the alternate path method and the modification method of load specified by USA' GSA code, a case study on progressive collapse analysis of a steel frame (which is designed under the guides of Chinese steel structures code) is given.
引文
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