基于共旋坐标法的结构非线性计算理论研究
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摘要
现代桥梁结构不断向大跨、结构复杂与轻型方向发展,迫切需要提高桥梁结构非线性分析的精度和效率,在学习国内外相关文献的基础上,本文开展了如下研究工作：
1、基于共旋坐标法和静力平衡条件,利用微分方法导出了平面杆元在结构坐标系下考虑几何非线性的单元切线刚度矩阵及杆端力的全量算法,列式简单但精度很高。
2、通过采用即时构形几何参数来计算单元状态变量对已有文献提出的几何非线性平面梁单元共旋坐标法列式进行了改进,在此基础上,为考虑实际结构模型中存在的梁端带铰和刚臂等特殊情况,根据带铰处梁端弯矩为零的受力特征及刚臂在受力后只有刚体运动而无变形特点,利用微分方法导出带铰梁元和带刚臂梁元的非线性切线刚度矩阵表达式,该表达式与普通梁元切线刚度矩阵表达式在形式上完全一致,通过多个实例进行了考证。
3、以建立预应力钢筋混凝土几何与材料双非线性梁单元分析模型为例,将上述平面梁单元只考虑几何非线性的算法拓展到同时考虑几何与材料双非线性,先利用虚功原理计算共旋坐标系下完全粘结任意截面预应力钢筋混凝土梁考虑材料非线性的切线刚度矩阵,再通过结构坐标系与共旋坐标系下节点力之间及节点位移之间的总量关系及微分导出的增量关系,获得预应力钢筋混凝土梁在结构坐标系中考虑几何与材料双重非线性的切线刚度矩阵,对多个钢筋混凝土及预应力钢筋混凝土结构进行了考证。
4、基于共旋坐标法导出能模拟预应力钢筋的空间杆元在结构坐标系下考虑几何与材料双非线性的单元切线刚度矩阵,联立已有的非线性实体退化壳单元,根据实体退化壳单元内部的位移场模式,形成了由实体退化壳单元与预应力钢筋空间杆元组成的非线性组合单元,对钢筋混凝土及预应力钢筋混凝土结构进行了考证。
5、以四边形八节点平面应力单元为例,基于共旋坐标法导出平面应力单元在结构坐标系下考虑几何非线性的单元切线刚度矩阵表达式,再将考虑几何非线性的方法推广到同时考虑几何与材料双非线性分析,将平面应力单元视为第三章实体退化壳单元分层模型中主平面13、23上剪应力τ13、τ23均为0的特殊情况,因此可利用实体退化壳单元所采用的混凝土本构关系、屈服准则和破坏准则等,基于钢筋及预应力钢筋与周围混凝土完全粘结的特点,导出共旋坐标系下考虑材料非线性的带预应力钢筋平面应力单元切线刚度矩阵,再利用结构坐标系与共旋坐标系下节点力的总量及增量关系导出结构坐标系下同时考虑几何与材料非线性的单元切线刚度矩阵,对钢筋混凝土及预应力钢筋混凝土结构进行了考证。
6、为提高钢管混凝土拱空间非线性分析的精度和效率,首先基于共旋坐标法,利用Euler转角公式及微分法,导出空间梁在结构坐标系下考虑几何非线性的单元切线刚度矩阵表达式；参照平面梁单元双非线性分析思路,将上述考虑几何非线性的方法推广到同时考虑几何与材料双非线性分析,即基于共旋坐标系下小变形的假定利用线性的应变-位移关系,采取与第二章类似的方法先导出共旋坐标系下考虑材料非线性的单元切线刚度矩阵,再导出结构坐标系下同时考虑几何与材料双非线性的单元切线刚度矩阵,通过平面与空间钢管混凝土拱试验模型进行了考证。
7、为进行大跨柔性混凝土结构考虑几何非线性与徐变效应耦合算法研究,基于能描述空间矢量有限转动的EULAR-Rodrigues公式,采用与第六章相类似的方法导出结构坐标系下考虑几何非线性的空间梁单元切线刚度矩阵表达式,与第六章方法比较而言,该算法更能有效将刚体位移和变形成分完全分离,因而减少在变形较大时计算结果的误差,借鉴平面梁中考虑梁端带铰和刚臂的分析思路,基于带铰处梁端弯矩为零的受力特征和将刚臂视为空间矢量,利用空间矢量有限转动公式及微分方法导出结构坐标系下梁端带铰和刚臂的空间梁元切线刚度矩阵显式表达式,从而使该问题得到彻底解决；在此基础上,参照已有文献建立的考虑混凝土收缩徐变的UL列式虚功增量方程算法,本文建立空间梁中考虑混凝土收缩徐变效应的共旋坐标法列式算法,对一模拟大跨径混合梁斜拉桥混凝土桥塔进行了考虑混凝土徐变效应的几何非线性分析。
With the development of modern bridge structure aiming for long span, complicated structure and light type, the need to improve calculation accuracy and efficiency of spatial nonlinearity analysis for bridge structure is urgent. On the basis of researching the related literature home and abroad, the following work has been done as:
1, Based on co-rotational procedure and static balance condition, a geometrical non-linear tangent stiffness matrix of plane truss element in global coordinate system has been successfully derived by means of differential method, and total algorithm of nodal force has also been acquired. The numerical results demonstrate that these formulations are simple but highly accurate.
2, The co-rotational formulations of plane beam element for geometrical nonlinear in existing literatures has been improved because that geometric parameters of instant configuration rather than initial configuration is adopted to calculate displacement and strain in this paper. Moreover, according to the mechanical characteristics of hinged beam that bending moment is equal to zero and the characteristics of no deformation but rigid motion of the forced rigid arm, the explicit expressions of nonlinear tangent stiffness matrix of plane beam element with rigid arms or hinges in both ends are obtained by employing differential method. The Numerical results demonstrate that the proposed plane beam element can solve the structural analysis of plane beam with rigid link or hinge. Moreover, the developed beam element is valid and practicable due to the same format as the general beam element without rigid arms or hinge.
3, The above-mentioned has been extended from only geometrical nonlinearity to geometrical and material bi-nonlinearity algorithm by taking a prestressed concrete as an example. A numerical model for a given section considering material and geometrical nonlinear analysis of prestressed concrete beam element is developed. Firstly, by means of virtual work, a tangent stiffness matrix for material nonlinearity of perfectly-bonded prestressed concrete beam element is derived in co-rotational coordinate system. Then, through building total and incremental relationships derived from differential equations of nodal displacements and forces between global coordinate system and co-rotational coordinate system, respectively, tangent stiffness in global coordinate system prestressed concrete beam element is developed considering geometric and material nonlinearity. Nonlinear analysis of several reinforced concrete and prestressed concrete structure have been performed to demonstrate the usefulness of the developed method.
4, Based on co-rotational procedure, the geometrical and material bi-nonlinearity tangent stiffness matrix of spatial truss element in global coordinate system has been developed because that this element can simulate prestressing steels. Combined with the existing nonlinear degenerated shell element, the tangent stiffness matrix of nonlinear composite elements has been derived by use of internal displacement field model of degenerated shell element. The nonlinear analysis of several reinforced concrete and prestressed concrete structures have been employed to verify the effectiveness of the developed method.
5, Taking quadrilateral8-node plane element as an example, geometrical nonlinear element tangent stiffness matrix for plane stress element under a large rotation with small strain is presented based on co-rotational procedure. The above-mentioned algorithm has been extended from only geometrical nonlinearity to geometrical and material bi-nonlinearity when shear stress τ13and τ23of principal plane are0in layered model of degenerated shell element mentioned in the third chapter, it is special for plane stress element, Thus the constitutive relation yield criterion and failure criteria of concrete for degenerated shell element can been used for bi-nonlinearity plane stress element, Firstly, by means of virtual work, a tangent stiffness matrix for material nonlinearity of perfectly-bonded reinforced concrete beam element is derived in co-rotational coordinate system. Then, through building total and incremental relationships derived from differential equations of nodal displacements and forces between global coordinate system and co-rotational coordinate system, respectively, tangent stiffness in global coordinate system reinforced concrete beam element is developed considering geometric and material nonlinearity. Nonlinear analysis of several reinforced concrete and prestressed concrete structure have been performed.
6, In order to improve calculation accuracy and efficiency of spatial nonlinearity analysis for concrete filled steel tube arch. In this paper, based on co-rotational procedure, a numerical model considering material and geometrical nonlinear analysis for concrete filled steel tube beam element is developed. Firstly, using Euler formulas and variable method, a tangent stiffness matrix for material nonlinearity of perfectly-bonded concrete filled steel tube beam element is derived in co-rotational coordinate system by means of virtual work. Then, based the method as nonlinear analysis of plane beam element, the method has been extended from geometric nonlinearity to Bi-nonlinearity, that is to say, the small strain is assumed as linear strain-stress relation in co-rotational coordinate system. The tangent stiffness for material nonlinearity in co-rotational coordinate system is developed based on method mentioned in chapter two. Thus, a tangent stiffness in global coordinate system concrete filled steel tube beam element is developed considering geometric and material nonlinearity. Plane and spatial comparisons between the results in this paper and those from model test of concrete-filled steel tubular arch rib shows the accuracy of the developed method is very high.
7, In order to analyze the coupled effect of geometric nonlinearity and concrete creep for long-span concrete structures, based on Euler-Rodrigues formula for spatial rotation, a tangent stiffness matrix of spatial beam considering geometric nonlinearity using the method mentioned in Chapter6in co-rotational coordinate. Compared with method in Chapter6, the developed method in this chapter can separate the displacement and deformation of rigid body, and then decrease calculation error when the deformation is large. Using the method of plane beam considering hinge and rigid arm of beam end, based on the mechanical characteristics of hinged beam that bending moment is equal to zero and taking rigid arm as spatial vector, the explicit expressions of nonlinear tangent stiffness matrix of plane beam element with rigid arms or hinges in both ends are obtained by employing limit rotation formulas of spatial vector differential method. Based on this, refer to the algorithm of concrete shrinkage and creep, the explicit expressions of spatial beam element considering concrete shrinkage and creep is developed in co-rotational coordinate system. Finally, a geometric nonlinearity analysis of concrete tower of hybrid beam cable-stayed bridge has been performed considering concrete shrinkage and creep.
1、基于共旋坐标法和静力平衡条件,利用微分方法导出了平面杆元在结构坐标系下考虑几何非线性的单元切线刚度矩阵及杆端力的全量算法,列式简单但精度很高。
2、通过采用即时构形几何参数来计算单元状态变量对已有文献提出的几何非线性平面梁单元共旋坐标法列式进行了改进,在此基础上,为考虑实际结构模型中存在的梁端带铰和刚臂等特殊情况,根据带铰处梁端弯矩为零的受力特征及刚臂在受力后只有刚体运动而无变形特点,利用微分方法导出带铰梁元和带刚臂梁元的非线性切线刚度矩阵表达式,该表达式与普通梁元切线刚度矩阵表达式在形式上完全一致,通过多个实例进行了考证。
3、以建立预应力钢筋混凝土几何与材料双非线性梁单元分析模型为例,将上述平面梁单元只考虑几何非线性的算法拓展到同时考虑几何与材料双非线性,先利用虚功原理计算共旋坐标系下完全粘结任意截面预应力钢筋混凝土梁考虑材料非线性的切线刚度矩阵,再通过结构坐标系与共旋坐标系下节点力之间及节点位移之间的总量关系及微分导出的增量关系,获得预应力钢筋混凝土梁在结构坐标系中考虑几何与材料双重非线性的切线刚度矩阵,对多个钢筋混凝土及预应力钢筋混凝土结构进行了考证。
4、基于共旋坐标法导出能模拟预应力钢筋的空间杆元在结构坐标系下考虑几何与材料双非线性的单元切线刚度矩阵,联立已有的非线性实体退化壳单元,根据实体退化壳单元内部的位移场模式,形成了由实体退化壳单元与预应力钢筋空间杆元组成的非线性组合单元,对钢筋混凝土及预应力钢筋混凝土结构进行了考证。
5、以四边形八节点平面应力单元为例,基于共旋坐标法导出平面应力单元在结构坐标系下考虑几何非线性的单元切线刚度矩阵表达式,再将考虑几何非线性的方法推广到同时考虑几何与材料双非线性分析,将平面应力单元视为第三章实体退化壳单元分层模型中主平面13、23上剪应力τ13、τ23均为0的特殊情况,因此可利用实体退化壳单元所采用的混凝土本构关系、屈服准则和破坏准则等,基于钢筋及预应力钢筋与周围混凝土完全粘结的特点,导出共旋坐标系下考虑材料非线性的带预应力钢筋平面应力单元切线刚度矩阵,再利用结构坐标系与共旋坐标系下节点力的总量及增量关系导出结构坐标系下同时考虑几何与材料非线性的单元切线刚度矩阵,对钢筋混凝土及预应力钢筋混凝土结构进行了考证。
6、为提高钢管混凝土拱空间非线性分析的精度和效率,首先基于共旋坐标法,利用Euler转角公式及微分法,导出空间梁在结构坐标系下考虑几何非线性的单元切线刚度矩阵表达式；参照平面梁单元双非线性分析思路,将上述考虑几何非线性的方法推广到同时考虑几何与材料双非线性分析,即基于共旋坐标系下小变形的假定利用线性的应变-位移关系,采取与第二章类似的方法先导出共旋坐标系下考虑材料非线性的单元切线刚度矩阵,再导出结构坐标系下同时考虑几何与材料双非线性的单元切线刚度矩阵,通过平面与空间钢管混凝土拱试验模型进行了考证。
7、为进行大跨柔性混凝土结构考虑几何非线性与徐变效应耦合算法研究,基于能描述空间矢量有限转动的EULAR-Rodrigues公式,采用与第六章相类似的方法导出结构坐标系下考虑几何非线性的空间梁单元切线刚度矩阵表达式,与第六章方法比较而言,该算法更能有效将刚体位移和变形成分完全分离,因而减少在变形较大时计算结果的误差,借鉴平面梁中考虑梁端带铰和刚臂的分析思路,基于带铰处梁端弯矩为零的受力特征和将刚臂视为空间矢量,利用空间矢量有限转动公式及微分方法导出结构坐标系下梁端带铰和刚臂的空间梁元切线刚度矩阵显式表达式,从而使该问题得到彻底解决；在此基础上,参照已有文献建立的考虑混凝土收缩徐变的UL列式虚功增量方程算法,本文建立空间梁中考虑混凝土收缩徐变效应的共旋坐标法列式算法,对一模拟大跨径混合梁斜拉桥混凝土桥塔进行了考虑混凝土徐变效应的几何非线性分析。
With the development of modern bridge structure aiming for long span, complicated structure and light type, the need to improve calculation accuracy and efficiency of spatial nonlinearity analysis for bridge structure is urgent. On the basis of researching the related literature home and abroad, the following work has been done as:
1, Based on co-rotational procedure and static balance condition, a geometrical non-linear tangent stiffness matrix of plane truss element in global coordinate system has been successfully derived by means of differential method, and total algorithm of nodal force has also been acquired. The numerical results demonstrate that these formulations are simple but highly accurate.
2, The co-rotational formulations of plane beam element for geometrical nonlinear in existing literatures has been improved because that geometric parameters of instant configuration rather than initial configuration is adopted to calculate displacement and strain in this paper. Moreover, according to the mechanical characteristics of hinged beam that bending moment is equal to zero and the characteristics of no deformation but rigid motion of the forced rigid arm, the explicit expressions of nonlinear tangent stiffness matrix of plane beam element with rigid arms or hinges in both ends are obtained by employing differential method. The Numerical results demonstrate that the proposed plane beam element can solve the structural analysis of plane beam with rigid link or hinge. Moreover, the developed beam element is valid and practicable due to the same format as the general beam element without rigid arms or hinge.
3, The above-mentioned has been extended from only geometrical nonlinearity to geometrical and material bi-nonlinearity algorithm by taking a prestressed concrete as an example. A numerical model for a given section considering material and geometrical nonlinear analysis of prestressed concrete beam element is developed. Firstly, by means of virtual work, a tangent stiffness matrix for material nonlinearity of perfectly-bonded prestressed concrete beam element is derived in co-rotational coordinate system. Then, through building total and incremental relationships derived from differential equations of nodal displacements and forces between global coordinate system and co-rotational coordinate system, respectively, tangent stiffness in global coordinate system prestressed concrete beam element is developed considering geometric and material nonlinearity. Nonlinear analysis of several reinforced concrete and prestressed concrete structure have been performed to demonstrate the usefulness of the developed method.
4, Based on co-rotational procedure, the geometrical and material bi-nonlinearity tangent stiffness matrix of spatial truss element in global coordinate system has been developed because that this element can simulate prestressing steels. Combined with the existing nonlinear degenerated shell element, the tangent stiffness matrix of nonlinear composite elements has been derived by use of internal displacement field model of degenerated shell element. The nonlinear analysis of several reinforced concrete and prestressed concrete structures have been employed to verify the effectiveness of the developed method.
5, Taking quadrilateral8-node plane element as an example, geometrical nonlinear element tangent stiffness matrix for plane stress element under a large rotation with small strain is presented based on co-rotational procedure. The above-mentioned algorithm has been extended from only geometrical nonlinearity to geometrical and material bi-nonlinearity when shear stress τ13and τ23of principal plane are0in layered model of degenerated shell element mentioned in the third chapter, it is special for plane stress element, Thus the constitutive relation yield criterion and failure criteria of concrete for degenerated shell element can been used for bi-nonlinearity plane stress element, Firstly, by means of virtual work, a tangent stiffness matrix for material nonlinearity of perfectly-bonded reinforced concrete beam element is derived in co-rotational coordinate system. Then, through building total and incremental relationships derived from differential equations of nodal displacements and forces between global coordinate system and co-rotational coordinate system, respectively, tangent stiffness in global coordinate system reinforced concrete beam element is developed considering geometric and material nonlinearity. Nonlinear analysis of several reinforced concrete and prestressed concrete structure have been performed.
6, In order to improve calculation accuracy and efficiency of spatial nonlinearity analysis for concrete filled steel tube arch. In this paper, based on co-rotational procedure, a numerical model considering material and geometrical nonlinear analysis for concrete filled steel tube beam element is developed. Firstly, using Euler formulas and variable method, a tangent stiffness matrix for material nonlinearity of perfectly-bonded concrete filled steel tube beam element is derived in co-rotational coordinate system by means of virtual work. Then, based the method as nonlinear analysis of plane beam element, the method has been extended from geometric nonlinearity to Bi-nonlinearity, that is to say, the small strain is assumed as linear strain-stress relation in co-rotational coordinate system. The tangent stiffness for material nonlinearity in co-rotational coordinate system is developed based on method mentioned in chapter two. Thus, a tangent stiffness in global coordinate system concrete filled steel tube beam element is developed considering geometric and material nonlinearity. Plane and spatial comparisons between the results in this paper and those from model test of concrete-filled steel tubular arch rib shows the accuracy of the developed method is very high.
7, In order to analyze the coupled effect of geometric nonlinearity and concrete creep for long-span concrete structures, based on Euler-Rodrigues formula for spatial rotation, a tangent stiffness matrix of spatial beam considering geometric nonlinearity using the method mentioned in Chapter6in co-rotational coordinate. Compared with method in Chapter6, the developed method in this chapter can separate the displacement and deformation of rigid body, and then decrease calculation error when the deformation is large. Using the method of plane beam considering hinge and rigid arm of beam end, based on the mechanical characteristics of hinged beam that bending moment is equal to zero and taking rigid arm as spatial vector, the explicit expressions of nonlinear tangent stiffness matrix of plane beam element with rigid arms or hinges in both ends are obtained by employing limit rotation formulas of spatial vector differential method. Based on this, refer to the algorithm of concrete shrinkage and creep, the explicit expressions of spatial beam element considering concrete shrinkage and creep is developed in co-rotational coordinate system. Finally, a geometric nonlinearity analysis of concrete tower of hybrid beam cable-stayed bridge has been performed considering concrete shrinkage and creep.
引文
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