摘要
网格结构有限元模型的选择会对计算精度产生很大的影响。如果将网格结构中的每一根杆件和每一个节点都用细小的实体/壳体单元来离散,该有限元模型的计算精度是很高的。但这种方法会造成巨大的计算量,这是现有的计算机水平所无法接受的。通常将网格结构中的每一根杆件采用一个梁/杆单元来离散,并忽略节点的影响。然而,数值分析和模型试验表明,这种有限元模型往往高估网格结构的承载力和延性,精度较差。因此,本文对网格结构的精细化有限元分析方法进行了深入的研究,该方法可以根据网格结构的受力情况选择单元的类型、尺寸、以及离散区域,在保证计算精度的前提下尽可能降低计算量。
当网格结构的载荷较小,杆件和节点都没有发生塑性屈服时,网格结构可以采用梁单元模型来进行分析。为了判断网格结构有限元模型中梁单元的长度和插值函数是否合理并对此进行相应的调整,首先推导了梁在受拉、受压、纯弯三种情况下的挠曲微分方程,以有限元试算得到的梁单元两端节点力为边界条件,求出了梁单元广义力分布场的解析解,然后根据Zienkiewicz-Zhu后验误差估计理论,以该解析解为广义力相对精确解,推导了广义力有限元解和广义力相对精确解的能量范数以确定梁单元的相对误差。在试算过程中,如果网格结构中每个梁单元的有限元解相对误差满足都精度要求,则终止试算过程,否则调整梁单元的插值函数或长度后再进行试算。
在节点发生塑性屈服前,节点的刚度是可测的。为了分析节点刚度和体积对结构分析精度的影响,采用与节点半径等长的刚臂来模拟节点的体积,在刚臂与杆件之间设置弹簧单元来模拟节点的刚度,根据经典梁理论,推导了考虑刚臂和节点刚度的梁的单元刚度矩阵；确定了影响节点刚度的主要因素；然后,以焊接球节点为例,根据节点刚度的回归公式,计算出符合施工构造要求的焊接球节点对杆件轴向刚度和弯曲刚度的影响范围。在此基础上,推导了节点体积和刚度对位移解影响的误差估计公式。
随着载荷的增大,结构的节点和杆件可能会发生塑性屈服和弯曲破坏,无法继续采用全梁单元有限元模型来分析结构的受力,‘需要采用多种不同类型的单元来离散网格结构。
为了考虑杆件弯曲破坏对网壳性能的影响,将杆件中可能发生弯曲破坏的部分作为微观尺度模型采用壳单元离散,杆件的其余部分作为宏观尺度模型采用梁单元离散,给出了梁-壳单元的耦合方法。在进行梁单元模型自适应分析的基础上,根据梁单元的内力分布来确定弯曲破坏可能发生的位置,并以塑性功的增量为指标来调整微观尺度模型的长度。
为了分析节点塑性屈服以后网格结构的受力,将可能发生塑性屈服的焊接球节点以及它所连接的杆件长为S的一小段采用作为微观尺度模型采用实体单元离散,结构的其余部分作为宏观尺度模型采用梁单元离散。根据经典梁理论的平截面假定推导了两种尺度模型界面上的位移增量约束方程,给出了梁-实体单元的耦合方法。同时,给出了S的估计值,并采用Zienkiewicz-Zhu后验误差估计理论来检验它是否合理。
理论推导和数值算例表明,精细化有限元分析方法可以根据结构的载荷状况,来调整有限元模型,它的计算精度和全实体/壳体单元有限元模型基本相似,但可以大幅度降低计算量,具有一定的工程实用价值。最后,论文还给出网格结构精细化有限元分析方法进一步研究工作的建议。
Different finite element models of the grid strctures can affect the analysis accuracy of analysis greatly. If the pipes and joints in the grid structure are all discreted by fine solid elements, this finite element model will achieve the best precision. However, this model has to involve too much computational task to be accecpted by current computer capability. Therefore, each pipe in the structure is always discreted by one beam element, and the joints are igored. Model test and numerical analysis shows this kind of finte element model always overestimate the ductility and loading capacity of the structrure. Therefore, refined finite element method for grid structure is studied in this paper, this method can achieve the same precision as the model using solid element, and reduce the computational task greatly, by selecting the type, size, and zone discrretized automatically.
If the load on the grid structure is small, the pipes in the structure can be discreted by beam element. To judge if the length and shape functions of a beam element are suitable, firstly the deflection function is derived when the beam are tensiled,pressed, and purely bended according to the boundary conditions from finite element trial analysis, and then the generalised force field are obtained. Secondly, based on the Zienkiewicz-Zhu error post-processing technique, this generalised force field can be used as a relativly accurate solution, the energy norm of the finite element solution and the relatively accurate solution is given to decide the raltive error of the beam element. The computation trials proceeds after the shape function and the length of beam element is adjusted, until the relative error of every beam element in the structure is satisfied.
Before the joint has plastic strain, its rigidity can be measured. To consider the volume and rigidity of the joint, a rigid panel element with the same length as the radius of the welded hollow spherical joint is set to simulate the volume of the joint, and a spring element is set between the rigid panel and the pipe to simulate the rigidity of the joint. The element matrix of the beam considering both the rigid panel and the joint rigidity is given. With the empirical stiffness equation of hollow sphere joint, the influencing range of the joint to the axial stiffness and the rotation stiffness are calculated. The energy norm is used to analyze the axial rigidness and the rotation rigidness error to the displacement solution.
The joints and the pipes will yield or collapse as the load increases, and the finite element model with only beam element can not simulate this phenomenon. Multi-scale finite element model is employed to solve this problem.
To consider the bending collapse of the pipes in the reticulated frames, based on the multi-scale simulation, the part of member that having bending collapse is divided by the shell elements or solid element as a micro-model, and the other part of the member was simulated by beam elements as a macro-model. The method to couple the beam elements and the shell element is given. The location of the micro-model is predicted by the stress field of the beam element, after the beam element is adjusted by adptive finite element method. The length of the micro model is adjusted by the plastic work increment in the microl model.
To analyze the structure after the joints yielded, based on the multi-scale simulation, the joint and a shot part of each member connecting to it is divided by the solid element as a micro-model, and the other part of structure is simulated by beam elements as a macro-model. The incremental displacement constraint equations for the nodes on the section between the two models are derived based on the plane section premise of classical beam theory. A method to couple the beam element and the solid element is given. The pipe length connecting to the joint is estimated and the Zienkiewicz-Zhu error post-processing technique is used to check its validity.
Theroy analysis and numerical test show that the refined finite element method can achieve almost the same accuracy with the finite element model using only solid element, while reduce computianal task greatly. Finally, the conclusions and the problems of refined finite element method that can be studied further are summarized about reifined finite element method at the end of this thesis.
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