大跨径悬索桥主缆精细化计算研究
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摘要
随着悬索桥跨径的不断增大,主缆的安全系数已从最初的4.0降到目前的2.0左右,安全系数的降低,必然要求对悬索桥进行更加精确的计算。结合悬索桥结构的特点,开发新单元,建立更加符合实际情况的计算模型,从而达到精细化计算的目的。本文基于空间分段悬链线理论和几何非线性有限元理论,开展了悬索桥的几何非线性、索与鞍座的接触非线性、空间主缆线形等方面的研究。
首先,将平面缆索的分段悬链线理论拓展到空间缆索体系,建立了空间主缆与空间鞍座的切点位置的计算方法及空间主缆线形的迭代算法,采用数值解析法编制了相应的空间缆索线形计算程序,可全面考虑空间鞍座、散索鞍及锚跨分散索股等因素的影响,能准确计算主缆及锚跨索股的无应力长度、主缆或索股与鞍座的切点坐标及锚跨索股的张力,可用于空间缆索系统的成桥及空缆状态的线形计算。
然后,在分析索与鞍座的约束关系的基础上,提出并建立了可用于悬索桥有限元计算的两种新单元：鞍座单元和锚碇—锚跨单元,其中鞍座单元用于主缆和塔顶鞍座的接触非线性计算,锚碇—锚跨单元用于锚跨分散索股及边跨主缆与散索鞍的接触非线性计算。新单元自动满足索与鞍座是相切的。首先,以两节点间索的无应力长度保持不变为条件,对单元进行状态求解,确定切点位置,然后,根据空间悬链线理论及索与鞍座的几何关系,推导出精确的单元节点力,同时,以增量代替微分,根据刚度矩阵的定义,推导出单元的切线刚度矩阵,计算表明,新单元具有很高的计算精度。这两种新单元的引入,很好地解决了悬索桥的索与鞍座接触非线性模拟困难的问题,使得计算模型更加符合实际情况。
最后,建立了以杆端内力全量计算方法为基础的增量迭代法作为非线性有限元基本方程的求解方法,这种全量方法计算杆端抗力的精度不依赖切线刚度的精度且不存在误差累积的缺点,很适合开发的两种新单元；阐述了各类单元根据节点总位移求杆端抗力和切线刚度矩阵的方法；介绍了矢量发生空间大转动时的旋转变换关系、空间梁单元梁端总转角的确定方法及几何非线性中带刚臂单元、节点自由度放松及主从约束的处理方法；采用不平衡力向量分级施加的算法,编制了包括悬链线索单元、梁单元、鞍座单元及锚碇—锚跨单元在内的悬索桥几何非线性空间有限元计算程序。
计算分析表明,悬索桥的结构计算中需准确考虑鞍座的影响,若采用传统杆系有限元计算方法而不考虑切点位置相对鞍座的变化,计算的成桥线形误差将大于架设精度要求,同时施工过程中出现的误差可能更大。两种新单元的引入提高了悬索桥结构的计算精度,为施工控制提供了准确的计算依据。
With the increasing span length of suspension bridge, the safety factor of main cable was decreased from the original4.0to the current2.0. Therefore it is necessary to perform more accurate calculation of suspension bridge. In order to achieve the purposes of precise calculation, two new elements were developed combined with the structure characteristics of suspension bridge, and the calculation model is more consistent with the actual situation. On the basis of the segmental catenary theory and nonlinear finite element method, a series of research work was carried on around the geometric nonlinear analysis of suspension bridges, the contact nonlinear between cable and saddle, the alignment of the spatial main cable.
Firstly, the plane cable segmental catenary theory was extended to the spatial cable system. The calculation methods of tangential point position between spatial cable and space saddle and the iterative algorithm of spatial main cable configuration were established. A corresponding spatial cable program, which can comprehensively consider of the space saddle, splay saddle, anchor span strand, has been developed using the numerical analytical method. The unstressed length of main cable and anchor span strand, tangential point and jacking force of strand can also be figured out accurately by using the program. The program can be used to analyse the spatial cable shape in both the finished bridge stage and the cable finished stage.
Secondly, based on the analysis of cable and saddle constraint relation, two new elements, saddle element and anchorage-anchor span element, have been put forward for suspension bridge finite element calculation. The saddle element is used for contact nonlinear analysis between main cable and saddle on tower, and the anchorage-anchor span element is used for contact nonlinear analysis between anchor span strands, side-span main cable and splay saddle. The new elements automatically satisfy the condition that the main cable should be always tangent to saddle. First of all, on the condition that the total unstressed length of cable between two nodes remains constant, the current state of the element is determined and the location of the tangent points are obtained. After that, according to the spatial catenary theory and the geometric relation between cable and saddle, the accurate element nodal force can be derived. At the same time, with increment instead of differential, the element tangent stiffness matrix can be deduced depending on the definition of the stiffness matrix. Calculation shows that the new elements have high calculation precision. The computational simulation problem of contact nonlinear between cable and saddle has been well solved through the introduction of the two new elements.
Finally, a combinational method of increment and iteration based on the total method for the element rod end resisting force was chosen to solve the equilibrium equation. The precision of calculation doesn't rely on tangent stiffness matrix and no accumulative error would occur. The calculation method of rod end resisting force and tangent stiffness matrix according to total node displacement was described. The rotation transform relationship when the vector occurring spatial limited rotation was derived from the introduction of Euler-Rodrigues formulation. The method to determine the co-rotational coordinate system and total beam end rotation angle of space beam element was introduced. An improved method, which divides the unbalanced force vector into prescribed parts and makes them act gradually in one load step, was proposed. The processing methods for the element with rigid arms, relaxation of degree of freedom, master-slave constraint, were introduced. A calculation program including catenary cable element, space beam element, saddle element and anchorage-anchor span element which can satisfy the geometric nonlinear analysis of suspension bridges was developed.
Calculation analysis demonstrates that the effect of saddle should be accurately considered in the calculation of suspension bridge. If using the traditional bar system FEM without considering the change of tangent point location, the calculation error of the final bridge shape will be greater than the erection accuracy requirements, and moreover, the calculation error occurring in the construction process may be greater than that of finished bridge state. The calculation accuracy of suspension bridge has been improved through the introduction of the two new elements. It provides more accurate calculation basis for construction control.
首先,将平面缆索的分段悬链线理论拓展到空间缆索体系,建立了空间主缆与空间鞍座的切点位置的计算方法及空间主缆线形的迭代算法,采用数值解析法编制了相应的空间缆索线形计算程序,可全面考虑空间鞍座、散索鞍及锚跨分散索股等因素的影响,能准确计算主缆及锚跨索股的无应力长度、主缆或索股与鞍座的切点坐标及锚跨索股的张力,可用于空间缆索系统的成桥及空缆状态的线形计算。
然后,在分析索与鞍座的约束关系的基础上,提出并建立了可用于悬索桥有限元计算的两种新单元：鞍座单元和锚碇—锚跨单元,其中鞍座单元用于主缆和塔顶鞍座的接触非线性计算,锚碇—锚跨单元用于锚跨分散索股及边跨主缆与散索鞍的接触非线性计算。新单元自动满足索与鞍座是相切的。首先,以两节点间索的无应力长度保持不变为条件,对单元进行状态求解,确定切点位置,然后,根据空间悬链线理论及索与鞍座的几何关系,推导出精确的单元节点力,同时,以增量代替微分,根据刚度矩阵的定义,推导出单元的切线刚度矩阵,计算表明,新单元具有很高的计算精度。这两种新单元的引入,很好地解决了悬索桥的索与鞍座接触非线性模拟困难的问题,使得计算模型更加符合实际情况。
最后,建立了以杆端内力全量计算方法为基础的增量迭代法作为非线性有限元基本方程的求解方法,这种全量方法计算杆端抗力的精度不依赖切线刚度的精度且不存在误差累积的缺点,很适合开发的两种新单元；阐述了各类单元根据节点总位移求杆端抗力和切线刚度矩阵的方法；介绍了矢量发生空间大转动时的旋转变换关系、空间梁单元梁端总转角的确定方法及几何非线性中带刚臂单元、节点自由度放松及主从约束的处理方法；采用不平衡力向量分级施加的算法,编制了包括悬链线索单元、梁单元、鞍座单元及锚碇—锚跨单元在内的悬索桥几何非线性空间有限元计算程序。
计算分析表明,悬索桥的结构计算中需准确考虑鞍座的影响,若采用传统杆系有限元计算方法而不考虑切点位置相对鞍座的变化,计算的成桥线形误差将大于架设精度要求,同时施工过程中出现的误差可能更大。两种新单元的引入提高了悬索桥结构的计算精度,为施工控制提供了准确的计算依据。
With the increasing span length of suspension bridge, the safety factor of main cable was decreased from the original4.0to the current2.0. Therefore it is necessary to perform more accurate calculation of suspension bridge. In order to achieve the purposes of precise calculation, two new elements were developed combined with the structure characteristics of suspension bridge, and the calculation model is more consistent with the actual situation. On the basis of the segmental catenary theory and nonlinear finite element method, a series of research work was carried on around the geometric nonlinear analysis of suspension bridges, the contact nonlinear between cable and saddle, the alignment of the spatial main cable.
Firstly, the plane cable segmental catenary theory was extended to the spatial cable system. The calculation methods of tangential point position between spatial cable and space saddle and the iterative algorithm of spatial main cable configuration were established. A corresponding spatial cable program, which can comprehensively consider of the space saddle, splay saddle, anchor span strand, has been developed using the numerical analytical method. The unstressed length of main cable and anchor span strand, tangential point and jacking force of strand can also be figured out accurately by using the program. The program can be used to analyse the spatial cable shape in both the finished bridge stage and the cable finished stage.
Secondly, based on the analysis of cable and saddle constraint relation, two new elements, saddle element and anchorage-anchor span element, have been put forward for suspension bridge finite element calculation. The saddle element is used for contact nonlinear analysis between main cable and saddle on tower, and the anchorage-anchor span element is used for contact nonlinear analysis between anchor span strands, side-span main cable and splay saddle. The new elements automatically satisfy the condition that the main cable should be always tangent to saddle. First of all, on the condition that the total unstressed length of cable between two nodes remains constant, the current state of the element is determined and the location of the tangent points are obtained. After that, according to the spatial catenary theory and the geometric relation between cable and saddle, the accurate element nodal force can be derived. At the same time, with increment instead of differential, the element tangent stiffness matrix can be deduced depending on the definition of the stiffness matrix. Calculation shows that the new elements have high calculation precision. The computational simulation problem of contact nonlinear between cable and saddle has been well solved through the introduction of the two new elements.
Finally, a combinational method of increment and iteration based on the total method for the element rod end resisting force was chosen to solve the equilibrium equation. The precision of calculation doesn't rely on tangent stiffness matrix and no accumulative error would occur. The calculation method of rod end resisting force and tangent stiffness matrix according to total node displacement was described. The rotation transform relationship when the vector occurring spatial limited rotation was derived from the introduction of Euler-Rodrigues formulation. The method to determine the co-rotational coordinate system and total beam end rotation angle of space beam element was introduced. An improved method, which divides the unbalanced force vector into prescribed parts and makes them act gradually in one load step, was proposed. The processing methods for the element with rigid arms, relaxation of degree of freedom, master-slave constraint, were introduced. A calculation program including catenary cable element, space beam element, saddle element and anchorage-anchor span element which can satisfy the geometric nonlinear analysis of suspension bridges was developed.
Calculation analysis demonstrates that the effect of saddle should be accurately considered in the calculation of suspension bridge. If using the traditional bar system FEM without considering the change of tangent point location, the calculation error of the final bridge shape will be greater than the erection accuracy requirements, and moreover, the calculation error occurring in the construction process may be greater than that of finished bridge state. The calculation accuracy of suspension bridge has been improved through the introduction of the two new elements. It provides more accurate calculation basis for construction control.
引文
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