薄壁梁变形分析的基本理论
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摘要
薄壁杆件的应用和薄壁梁理论的发展相互促进。近年来,随着截面开阔、力学性能优良的金属以及复合等材料的薄壁杆件日益广泛地应用于航空、航天、桥梁、建筑等各个结构领域中,薄壁杆件的理论研究和工程技术水平不断提高。薄壁杆件的截面愈开阔,其力学性能愈好,从而在横截面积一定的情况下可以更有效的利用材料。然而,开阔的横截面也使得薄壁杆件截面的形状更容易发生改变。确定薄壁杆件承载能力时,稳定性常常是起决定作用的重要方面。当越来越多截面更加开阔的薄壁杆件得以应用,考虑截面的变形就逐渐成为杆件稳定性分析中必不可少的因素。在线弹性分析中考虑薄壁杆件截面的变形是稳定性等精确分析的前提。现有的薄壁杆件截面畸变稳定分析以数值方法为主。就数值方法而言,无论是其方法的形成还是对其结果的分析和总结,都是以解析理论为基础的。截面可变形的薄壁梁理论将截面的变形纳入梁理论的范畴,可以到达解析方法与数值方法的完美结合,有助于从全局的高度把握稳定性分析。广义梁理论(GeneralizedBeam Theory或GBT)是符拉索夫薄壁梁理论的推广,是截面可变形的薄壁梁理论。然而,现有广义梁理论的完备性问题长期未能解决。不完备的表现是广义梁理论有板件最少数量的限制。
本文仅限于线弹性范围内关于广义梁理论一些基本问题的讨论,试图解决现有广义梁理论的完备性问题,是关于薄壁梁变形分析的基本理论。本文继承了现有广义梁理论的合理内核,按照应用力学梁理论对弹性力学问题进行近似的一般方法,从经典理论中发掘并建立与现代数值方法的联系,导出完备广义梁理论(Complete GBT或CGBT)。
弹性力学问题难以求解使应用力学近似理论得以发展。本文关于单平板薄壁梁的分析表明,应用力学中的梁板理论是基于研究对象在几何方面横向几何尺度“远小于”纵向几何尺度的特点,利用康托洛维奇(Kantorovich)方法对弹性力学问题进行分离变量、横向插值的结果。用计算力学观点可以将应用力学简化方法概括为通过其基本假定中的运动学假定,将各点在三个坐标方向的位移分别线性表示为关于纵向坐标的未知函数及其导数的代数多项式,多项式的系数是一组线性无关的关于横向坐标的已知函数即基底函数,从而用分离变量的方法将三维物体的弹性力学问题简化为一维或二维问题,形成梁、板等理论分支。在满足完备性基础上,无论是考虑中面内剪切变形的情形,还是考虑中面外弯曲变形的情形,都可以通过提高横向插值多项式最高次数,建立考虑截面变形的完备的高“阶”次的薄壁杆件理论。
本文指出现有广义梁理论的完备性问题分为两个方面。其一是在GBT基本部分的阐述中,不考虑边缘板件的弯曲,从而将边缘板件基底函数(试函数)的最高次数降低为1次,截面自由度不包括边线上的中面法向位移和扭转角。另一在于该理论扩展部分的阐述中虽然允许将包括边线在内的节线中面法向位移做为独立的自由度,但仅将其做为可有可无的选项,并且将节线上的扭转角做为从属自由度利用三弯矩方程等方法予以消去,减少了中间板件上插值基底函数的数量。GBT不完备具体表现在该理论有板件最少数量的限制,即n个板件单元构成的开口薄壁梁,其基本部分方程的基本未知量为n+1(n≥3)或扩展部分的n+3(n≥3)。本文导出单平板薄壁梁广义梁方程是完备的广义梁方程,初步解除了现有广义梁理论最少板件数量的限制,是完备化的基石。
在几何方面,薄壁梁有两个“远小于”,即横截面的宽度或高度远小于杆件的长度,组成板件的厚度远小于横截面的宽度或高度。由平板组成的薄壁梁在其几何形态上又具有自然离散的特点。本文基于这些认识,继单平板薄壁梁分析之后,应用与现有广义梁理论相同的运动学基本假定,即各板件单元的中面内运动满足简单梁的基本假定、中面外运动满足薄板弯曲问题的基本假定,按照各板件单元上满足完备性的要求,提出完备广义梁理论。对于由n块板件构成的开口截面薄壁梁,利用薄板理论的自然变分原理,分别以完全的1次Legendre多项式和3次Hermit多项式为描述板件平面内、外变形的基底函数,导出完备的具有2n+4(n≥1)个广义位移的广义梁方程,彻底解除现有理论对板件数量的限制,使广义梁理论完备化。
在完备广义梁理论的截面分析中,本文建立了全新的完备的自由度(基本未知量)系统。将周线看做连续梁或平面框架,以位移法基本结构做为研究对象,不但适用于无分支开口截面和有分支开口截面,也适用于有计算节点的截面。去除了自由度缩减的过程,仅利用位移协调条件,将截面上任意一点的位移由截面上的独立自由度表出。不再以全部的棱节点、边节点轴向位移为自由度(基本未知量),换之以周线起始节点和各中线单元终止节点的中面法向位移做为自由度(基本未知量),并将节点的转角和边节点的中面法向位移也做为必不可少的自由度来考虑。在现有广义梁理论中,位移列向量转换矩阵的元素以关于中线单元夹角的正弦、正切做为分母,于是就有中线单元夹角非零的要求。在周线直线段上节点即中间节点如计算节点的位移需单独列出,进行特殊处理。本文建立的自由度系统,其转换矩阵元素将正弦、余弦做为分子,不再要求中线单元夹角非零。因而无须将节点区分为棱节点、边节点或中间节点,适用于由平板组成的任意开口截面薄壁梁的分析,减小了计算的复杂性,方便了分析和计算。
通过实例说明了在本文提出的自由度系统中施加连续的以及离散的截面约束的方法。前者为自由度缩减,后者为梁单元端约束。仍然利用自由度缩减,使具有2n+4个广义自由度的CGBT方程退化为现有GBT基本部分的n+1(n≥3)个以及其扩展部分的n+3(n≥3)个广义自由度的方程。方程退化过程的完成进一步明确了现有GBT不完备的根源:1)边缘板件中面外位移线性化;2)在横向应用三弯矩方程的原因不明确;3)过度依赖节点翘曲做为截面自由度。
以CGBT方程为基础,本文对广义梁理论的简化计算方法进行阐述。仅仅将n+1个节点转角做为从自由度消去,而不引入边缘板件的中面外位移的线性化约束,使得方程CGBT方程自由度数量缩减到n+3(n≥1),得到缩减的完备广义梁理论(Reduced CGBT或RCGBT)方程。适当选取前若干阶模态进行简化计算的依据,可得到截断的完备广义梁理论(Truncated CGBT或TCGBT)。特别地,选取前4阶刚周边模态得到的TCGBT即得到刚周边的符拉索夫薄壁梁理论。这两种简化计算的方法都具有重要的意义,是面向工程应用的后续工作中,为考虑截面变形,而进行类似弯扭耦合的近似解析推导,提供了基本依据。
通过算例,利用CGBT方程的有限元列式将CGBT、RCGBT、TCGBT方程的有限元解与现有GBT以及薄板理论有限元解进行比较,初步显示出完备广义梁理论的优越性:1)较高的数值稳定性。2)横向加密节点不能显著提高CGBT结果的精度。3)总自由度数目较少,计算效率得以提高,计算经济性好。
提出自然完全模态谱系的概念。自然完全模态是指CGBT截面分析中不考虑计算节点,仅由构造节点进行截面划分,不进行任何的截面自由度缩减而得到的全部截面模态。由自然完全模态下广义梁方程D矩阵非对角线非0元素相互关联着的模态构成的模态族和模态子族,在更广泛的意义上揭示出尤其是具有对称轴截面的各模态之间的相关性。模态谱系概念刻画了薄壁梁的不同类型的平衡状态,是关于截面变形和位移的规律性认识。借助模态谱系概念可以系统地认识薄壁梁的平衡状态,便于在分析中区分、识别和设定初始平衡状态和失稳平衡状态,使现有薄壁梁稳定理论得到深化并对薄壁梁稳定研究提供指导。
Application of thin-walled members and development of thin-walled beam theory boost reciprocally the progress of each other. In recent years, with the increasing application of thin-walled members of metals and composite materials with slender cross sections and good mechanical behaviors in such structural fields as aeronautics, cosmonautics, bridgework and architecture, etc, theoretical research and engineering level of thin-walled members have been improved constantly. The slender the section of the thin-walled bar is, the higher its mechanical property is. So the material can be used more efficiently for certain cross sectional area. However, a slender cross-section is prone to result in the cross-section deformation. Stability usually plays a decisive role in determining the load carrying capacity of the thin-walled beam. When more and more thin-walled members with slender sections are applied, the sectional deformation gradually becomes an essential factor to be considered in analysis of member stability. To consider the sectional deformation of the thin-walled beam in linear elastic analysis is the prerequisite of an accurate analysis of its stability, especially the distortional buckling. The current analysis of distortional buckling of thin-walled members is primarily numerical-method oriented. The formation of numerical method as well as the analysis and conclusion of its results is grounded on analytic theory. The sectional deformable thin-walled beam theory incorporates the sectional deformation into the category of beam theory, to achieve the ideal combination of analytic method and numerical method, which is conducive to the stability analysis from an overall perspective. Generalised Beam Theory (GBT), being an extension of Vlasov's thin-walled beam theory, is a sectionally deformable thin-walled beam theory. But the incompletion problem of the existing GBT has remained unsettled for a long period. The restriction of least number of plates is the manifestation of the incompleteness of GBT.
This dissertation discusses several fundamental issues of GBT merely in the scope of linear elasticity, trying to settle the incompleteness problem of the present GBT, and is the basic theory on deformation of thin-walled beam. Our work inherits the reasonable core of the existing GBT, and seek to establish the relation between classical theory and modern numerical method according to the general method of approximating elastic mechanics problems with applied mechanics beam theory, so that to bring forward a Complete Generalised Beam Theory (CGBT).
The difficulty in finding the solutions to the elasticity problems leads to the development of the approximation theory, namely beam/plate theories in applied mechanics. The analysis of single-plate thin-walled beams in this work reveals that the beam/plate theories are the outcome of variable separation and lateral interpolation of the elasticity problems using Kantorovich method, based on the characteristic that the transverse geometric dimensions of the research subject are "far less" than the longitudinal ones. By computation mechanics, the applied mechanics simplified method can be generalized as that, through kinematics assumptions, the displacement of each point along the three coordinate axes is linearly expressed as the algebraic polynomial of the unknown function with respect to the longitudinal coordinate and its derivatives. The coefficients of the algebraic polynomial are a group of linearly independent known functions with respect to the transverse coordinate, namely the basis function. Thus, by variable separation, the three-dimensional elasticity problem can be simplified to one-dimensional or two-dimensional problem, giving rise to beam/plate theory branches. If the completeness is satisfied, for the cases of both the shear deformation in the midplane and the bending deformation outside the midplane, the complete high-order thin-walled beam theory considering sectional deformation can be established by enhancing the highest order of the transverse interpolation polynomial.
This work points out that the completeness issue of the existing GBT comprises two aspects. First, in the discussion of the fundamental part of GBT, the bending of edge plates will not be taken into account so that to reduce the highest order of the basis function (trial function) of the edge plates to first order, and the sectional degrees of freedom exclude the normal displacement and torsion of the midplane on the border line. Secondly, in the discussion of the extending part of the theory, though the normal displacements of midplanes on the node lines including border lines are taken as independent degrees of freedom, it is only a dispensable option, and the torsions of the node lines are eliminated with three-moment equation as dependent degrees of freedom, decreasing the number of interpolation basis functions on the middle plates. The incompleteness of GBT is embodied in its "least number of plates" restriction that, for the open section thin-walled beam comprising n plates, the basic unknowns of the equation of the fundamental part of GBT are n+1(n≥3) or those of the extending part are n+3(n≥3). The GBT equation of single-plate thin-walled beams derived in this work is a complete generalized beam equation, which preliminarily relieves the GBT from the restriction of "least-number of plates" and is the footstone of completion.
Geometrically, there are two "far less" concerning thin-walled beams, that is, the breadth or height of the cross section is far less than the beam length, and secondly the thickness of the constituent plates is far less than the breadth or height of the cross section. The thin-walled beam composed of flat plates is characterized by its naturally discrete geometrical shape. Based on these knowledge, following the analysis of single-plate thin-walled beams, adopting the same basic kinematic hypotheses as applied in the existing GBT (namely, the motion of each plate elements in the midplane satisfies the basic assumption of simple beam theory and the motion outside the midplane satisfies the basic assumption of the thin plate bending theory), and according to the completeness requirement in each plate elements the Complete GBT (CGBT) is proposed. For a thin-walled beam with open section consisting of n plates, applying the natural variation principle of thin plate theory, and using the complete first-order Legendre polynomial and third-order Hermit polynomial as basis functions to describe the deformation in or outside the plane respectively, we deduce the complete generalized beam equation with 2n+4(n≥1) generalized displacements, so that to thoroughly remove the limitation of the current theory for the plate number and complete the generalized beam theory.
In the sectional analysis of CGBT, a brand-new complete system of degrees of freedom (basic unknowns) is set up. To take the cross-section midline as continuous beam or plane frame and the basic structure of the displacement method as the research subject, the system of degrees of freedom not only applies to the open sections with or without branches but also applies to sections with computational nodes. Abandonment of the process of degree of freedom reduction and merely using the displacement compatibility condition, the displacement of an arbitrary point in the section can be expressed by the independent degrees of freedom on the section. The axial displacements of all ridge nodes and border nodes will no longer be the degrees of freedom (the basic unknown), but the normal displacements of midplane of the start nodes of first plate element midline and the ending nodes of plates midline will be the degrees of freedom (the basic unknown). And the node rotations and the midplane normal displacements of border nodes are also considered as indispensable degrees of freedom. In the existing GBT, the elements in the transformation matrix of the displacement column vectors take the sine and tangent of the included angle of plate midline elements as denominators. Thus it is required that the included angles of plate midline elements should be non-zero. The displacements of nodes in the straight-line segments of cross-section midlines, namely the intermediate nodes such as the computational nodes, should be listed and treated specially. In the degree of freedom system in this work, the elements in the transformation matrix take sine and cosine as numerators, and the included angles of plate midline elements are not required to be non-zero. Accordingly, the nodes needn't be classified into ridge nodes, border nodes or intermediate nodes. This works for the analysis of thin-walled beams with arbitrary open sections consisting of flat plates, reducing the computation complexity and facilitating the analysis and calculation.
The method of imposing continuous and discrete sectional constraints on the degree of freedom system is demonstrated by examples. The continuous one is degree of freedom reduction, and the discrete one is beam end restraint. By reducing the degrees of freedom, the CGBT equation with 2n+4 generalized degrees of freedom is degenerated to the existing GBT with n+1(n≥3) generalized degrees of freedom in the basic part and n+3(n≥3) in the extending part. The accomplishment of equation degeneration further confirms the sources of the incompleteness of the existing GBT: (1) the linearization of displacements of the edge plates outside the midplane; (2) the obscureness of the reason for transverse application of three-moment equation; (3) the excessive dependence on nodal warping displacement as the sectional degrees of freedom.
On basis of the CGBT equation, the simplified calculation method of GBT is presented in this dissertation. Removing the rotations of n+1 nodes from the degrees of freedom without introducing the linearization constraint on the displacements outside the midplane of the edge plates, the number of degrees of freedom of CGBT equation is reduced to n+3(n≥1) and the Reduced Complete Generalized Beam theory (Reduced CGBT or RCGBT) is obtained. The Truncated Complete GBT (Truncated CGBT or TCGBT) also can be obtained by selecting a few lowest order modes for the simplified calculation. In particular, the TCGBT obtained by selecting the first 4 lowest order modes, namely the rigid-body modes (extension, major and minor axis bending and torsion), is exactly Vlasov's theory. These two simplified calculation methods are of great significance. They provide foundation for the approximate analytical derivation which considering sectional deformation similar to bending-torsion coupling in succeeding work of engineering application.
With the objective of illustrating the performance of CGBT, an example of a simply supported single angle beam is presented, in which the results of CGBT beam element, GBT beam element and general shell finite element are compared: (1) higher numerical stability; (2) the accuracy of CGBT results can not be significantly improved by increasing transverse node; (3) the total number of degrees of freedom is smaller, leading to more efficient and economic calculation.
The concept of natural complete mode spectrum is brought forward. The natural complete mode denotes all the cross-sectional modes obtained without any reduction of sectional degrees of freedom, and not considering the computational nodes in the CGBT sectional analysis in which section discretion is performed merely by structural nodes. Complete nature modes are gathered into mode groups and/or sub-groups by off-diagonal components of matrix D in CGBT equation. These groups and subgroups reveal the correlation between various sectional modes, especially between those of sections with symmetry axes, in a more extensive sense. The mode spectrum concept depicts the different types of equilibrium states of thin-walled beams and is a regularity understanding of sectional deformation and displacement. This concept helps us to grasp a systematic knowledge of the equilibrium states of thin-walled beams, facilitates us to distinguish, identify and set the initial equilibrium state and unstable equilibrium state in analysis, intensifying the present stability theory of thin-walled beams and piloting the future research of their stability.
本文仅限于线弹性范围内关于广义梁理论一些基本问题的讨论,试图解决现有广义梁理论的完备性问题,是关于薄壁梁变形分析的基本理论。本文继承了现有广义梁理论的合理内核,按照应用力学梁理论对弹性力学问题进行近似的一般方法,从经典理论中发掘并建立与现代数值方法的联系,导出完备广义梁理论(Complete GBT或CGBT)。
弹性力学问题难以求解使应用力学近似理论得以发展。本文关于单平板薄壁梁的分析表明,应用力学中的梁板理论是基于研究对象在几何方面横向几何尺度“远小于”纵向几何尺度的特点,利用康托洛维奇(Kantorovich)方法对弹性力学问题进行分离变量、横向插值的结果。用计算力学观点可以将应用力学简化方法概括为通过其基本假定中的运动学假定,将各点在三个坐标方向的位移分别线性表示为关于纵向坐标的未知函数及其导数的代数多项式,多项式的系数是一组线性无关的关于横向坐标的已知函数即基底函数,从而用分离变量的方法将三维物体的弹性力学问题简化为一维或二维问题,形成梁、板等理论分支。在满足完备性基础上,无论是考虑中面内剪切变形的情形,还是考虑中面外弯曲变形的情形,都可以通过提高横向插值多项式最高次数,建立考虑截面变形的完备的高“阶”次的薄壁杆件理论。
本文指出现有广义梁理论的完备性问题分为两个方面。其一是在GBT基本部分的阐述中,不考虑边缘板件的弯曲,从而将边缘板件基底函数(试函数)的最高次数降低为1次,截面自由度不包括边线上的中面法向位移和扭转角。另一在于该理论扩展部分的阐述中虽然允许将包括边线在内的节线中面法向位移做为独立的自由度,但仅将其做为可有可无的选项,并且将节线上的扭转角做为从属自由度利用三弯矩方程等方法予以消去,减少了中间板件上插值基底函数的数量。GBT不完备具体表现在该理论有板件最少数量的限制,即n个板件单元构成的开口薄壁梁,其基本部分方程的基本未知量为n+1(n≥3)或扩展部分的n+3(n≥3)。本文导出单平板薄壁梁广义梁方程是完备的广义梁方程,初步解除了现有广义梁理论最少板件数量的限制,是完备化的基石。
在几何方面,薄壁梁有两个“远小于”,即横截面的宽度或高度远小于杆件的长度,组成板件的厚度远小于横截面的宽度或高度。由平板组成的薄壁梁在其几何形态上又具有自然离散的特点。本文基于这些认识,继单平板薄壁梁分析之后,应用与现有广义梁理论相同的运动学基本假定,即各板件单元的中面内运动满足简单梁的基本假定、中面外运动满足薄板弯曲问题的基本假定,按照各板件单元上满足完备性的要求,提出完备广义梁理论。对于由n块板件构成的开口截面薄壁梁,利用薄板理论的自然变分原理,分别以完全的1次Legendre多项式和3次Hermit多项式为描述板件平面内、外变形的基底函数,导出完备的具有2n+4(n≥1)个广义位移的广义梁方程,彻底解除现有理论对板件数量的限制,使广义梁理论完备化。
在完备广义梁理论的截面分析中,本文建立了全新的完备的自由度(基本未知量)系统。将周线看做连续梁或平面框架,以位移法基本结构做为研究对象,不但适用于无分支开口截面和有分支开口截面,也适用于有计算节点的截面。去除了自由度缩减的过程,仅利用位移协调条件,将截面上任意一点的位移由截面上的独立自由度表出。不再以全部的棱节点、边节点轴向位移为自由度(基本未知量),换之以周线起始节点和各中线单元终止节点的中面法向位移做为自由度(基本未知量),并将节点的转角和边节点的中面法向位移也做为必不可少的自由度来考虑。在现有广义梁理论中,位移列向量转换矩阵的元素以关于中线单元夹角的正弦、正切做为分母,于是就有中线单元夹角非零的要求。在周线直线段上节点即中间节点如计算节点的位移需单独列出,进行特殊处理。本文建立的自由度系统,其转换矩阵元素将正弦、余弦做为分子,不再要求中线单元夹角非零。因而无须将节点区分为棱节点、边节点或中间节点,适用于由平板组成的任意开口截面薄壁梁的分析,减小了计算的复杂性,方便了分析和计算。
通过实例说明了在本文提出的自由度系统中施加连续的以及离散的截面约束的方法。前者为自由度缩减,后者为梁单元端约束。仍然利用自由度缩减,使具有2n+4个广义自由度的CGBT方程退化为现有GBT基本部分的n+1(n≥3)个以及其扩展部分的n+3(n≥3)个广义自由度的方程。方程退化过程的完成进一步明确了现有GBT不完备的根源:1)边缘板件中面外位移线性化;2)在横向应用三弯矩方程的原因不明确;3)过度依赖节点翘曲做为截面自由度。
以CGBT方程为基础,本文对广义梁理论的简化计算方法进行阐述。仅仅将n+1个节点转角做为从自由度消去,而不引入边缘板件的中面外位移的线性化约束,使得方程CGBT方程自由度数量缩减到n+3(n≥1),得到缩减的完备广义梁理论(Reduced CGBT或RCGBT)方程。适当选取前若干阶模态进行简化计算的依据,可得到截断的完备广义梁理论(Truncated CGBT或TCGBT)。特别地,选取前4阶刚周边模态得到的TCGBT即得到刚周边的符拉索夫薄壁梁理论。这两种简化计算的方法都具有重要的意义,是面向工程应用的后续工作中,为考虑截面变形,而进行类似弯扭耦合的近似解析推导,提供了基本依据。
通过算例,利用CGBT方程的有限元列式将CGBT、RCGBT、TCGBT方程的有限元解与现有GBT以及薄板理论有限元解进行比较,初步显示出完备广义梁理论的优越性:1)较高的数值稳定性。2)横向加密节点不能显著提高CGBT结果的精度。3)总自由度数目较少,计算效率得以提高,计算经济性好。
提出自然完全模态谱系的概念。自然完全模态是指CGBT截面分析中不考虑计算节点,仅由构造节点进行截面划分,不进行任何的截面自由度缩减而得到的全部截面模态。由自然完全模态下广义梁方程D矩阵非对角线非0元素相互关联着的模态构成的模态族和模态子族,在更广泛的意义上揭示出尤其是具有对称轴截面的各模态之间的相关性。模态谱系概念刻画了薄壁梁的不同类型的平衡状态,是关于截面变形和位移的规律性认识。借助模态谱系概念可以系统地认识薄壁梁的平衡状态,便于在分析中区分、识别和设定初始平衡状态和失稳平衡状态,使现有薄壁梁稳定理论得到深化并对薄壁梁稳定研究提供指导。
Application of thin-walled members and development of thin-walled beam theory boost reciprocally the progress of each other. In recent years, with the increasing application of thin-walled members of metals and composite materials with slender cross sections and good mechanical behaviors in such structural fields as aeronautics, cosmonautics, bridgework and architecture, etc, theoretical research and engineering level of thin-walled members have been improved constantly. The slender the section of the thin-walled bar is, the higher its mechanical property is. So the material can be used more efficiently for certain cross sectional area. However, a slender cross-section is prone to result in the cross-section deformation. Stability usually plays a decisive role in determining the load carrying capacity of the thin-walled beam. When more and more thin-walled members with slender sections are applied, the sectional deformation gradually becomes an essential factor to be considered in analysis of member stability. To consider the sectional deformation of the thin-walled beam in linear elastic analysis is the prerequisite of an accurate analysis of its stability, especially the distortional buckling. The current analysis of distortional buckling of thin-walled members is primarily numerical-method oriented. The formation of numerical method as well as the analysis and conclusion of its results is grounded on analytic theory. The sectional deformable thin-walled beam theory incorporates the sectional deformation into the category of beam theory, to achieve the ideal combination of analytic method and numerical method, which is conducive to the stability analysis from an overall perspective. Generalised Beam Theory (GBT), being an extension of Vlasov's thin-walled beam theory, is a sectionally deformable thin-walled beam theory. But the incompletion problem of the existing GBT has remained unsettled for a long period. The restriction of least number of plates is the manifestation of the incompleteness of GBT.
This dissertation discusses several fundamental issues of GBT merely in the scope of linear elasticity, trying to settle the incompleteness problem of the present GBT, and is the basic theory on deformation of thin-walled beam. Our work inherits the reasonable core of the existing GBT, and seek to establish the relation between classical theory and modern numerical method according to the general method of approximating elastic mechanics problems with applied mechanics beam theory, so that to bring forward a Complete Generalised Beam Theory (CGBT).
The difficulty in finding the solutions to the elasticity problems leads to the development of the approximation theory, namely beam/plate theories in applied mechanics. The analysis of single-plate thin-walled beams in this work reveals that the beam/plate theories are the outcome of variable separation and lateral interpolation of the elasticity problems using Kantorovich method, based on the characteristic that the transverse geometric dimensions of the research subject are "far less" than the longitudinal ones. By computation mechanics, the applied mechanics simplified method can be generalized as that, through kinematics assumptions, the displacement of each point along the three coordinate axes is linearly expressed as the algebraic polynomial of the unknown function with respect to the longitudinal coordinate and its derivatives. The coefficients of the algebraic polynomial are a group of linearly independent known functions with respect to the transverse coordinate, namely the basis function. Thus, by variable separation, the three-dimensional elasticity problem can be simplified to one-dimensional or two-dimensional problem, giving rise to beam/plate theory branches. If the completeness is satisfied, for the cases of both the shear deformation in the midplane and the bending deformation outside the midplane, the complete high-order thin-walled beam theory considering sectional deformation can be established by enhancing the highest order of the transverse interpolation polynomial.
This work points out that the completeness issue of the existing GBT comprises two aspects. First, in the discussion of the fundamental part of GBT, the bending of edge plates will not be taken into account so that to reduce the highest order of the basis function (trial function) of the edge plates to first order, and the sectional degrees of freedom exclude the normal displacement and torsion of the midplane on the border line. Secondly, in the discussion of the extending part of the theory, though the normal displacements of midplanes on the node lines including border lines are taken as independent degrees of freedom, it is only a dispensable option, and the torsions of the node lines are eliminated with three-moment equation as dependent degrees of freedom, decreasing the number of interpolation basis functions on the middle plates. The incompleteness of GBT is embodied in its "least number of plates" restriction that, for the open section thin-walled beam comprising n plates, the basic unknowns of the equation of the fundamental part of GBT are n+1(n≥3) or those of the extending part are n+3(n≥3). The GBT equation of single-plate thin-walled beams derived in this work is a complete generalized beam equation, which preliminarily relieves the GBT from the restriction of "least-number of plates" and is the footstone of completion.
Geometrically, there are two "far less" concerning thin-walled beams, that is, the breadth or height of the cross section is far less than the beam length, and secondly the thickness of the constituent plates is far less than the breadth or height of the cross section. The thin-walled beam composed of flat plates is characterized by its naturally discrete geometrical shape. Based on these knowledge, following the analysis of single-plate thin-walled beams, adopting the same basic kinematic hypotheses as applied in the existing GBT (namely, the motion of each plate elements in the midplane satisfies the basic assumption of simple beam theory and the motion outside the midplane satisfies the basic assumption of the thin plate bending theory), and according to the completeness requirement in each plate elements the Complete GBT (CGBT) is proposed. For a thin-walled beam with open section consisting of n plates, applying the natural variation principle of thin plate theory, and using the complete first-order Legendre polynomial and third-order Hermit polynomial as basis functions to describe the deformation in or outside the plane respectively, we deduce the complete generalized beam equation with 2n+4(n≥1) generalized displacements, so that to thoroughly remove the limitation of the current theory for the plate number and complete the generalized beam theory.
In the sectional analysis of CGBT, a brand-new complete system of degrees of freedom (basic unknowns) is set up. To take the cross-section midline as continuous beam or plane frame and the basic structure of the displacement method as the research subject, the system of degrees of freedom not only applies to the open sections with or without branches but also applies to sections with computational nodes. Abandonment of the process of degree of freedom reduction and merely using the displacement compatibility condition, the displacement of an arbitrary point in the section can be expressed by the independent degrees of freedom on the section. The axial displacements of all ridge nodes and border nodes will no longer be the degrees of freedom (the basic unknown), but the normal displacements of midplane of the start nodes of first plate element midline and the ending nodes of plates midline will be the degrees of freedom (the basic unknown). And the node rotations and the midplane normal displacements of border nodes are also considered as indispensable degrees of freedom. In the existing GBT, the elements in the transformation matrix of the displacement column vectors take the sine and tangent of the included angle of plate midline elements as denominators. Thus it is required that the included angles of plate midline elements should be non-zero. The displacements of nodes in the straight-line segments of cross-section midlines, namely the intermediate nodes such as the computational nodes, should be listed and treated specially. In the degree of freedom system in this work, the elements in the transformation matrix take sine and cosine as numerators, and the included angles of plate midline elements are not required to be non-zero. Accordingly, the nodes needn't be classified into ridge nodes, border nodes or intermediate nodes. This works for the analysis of thin-walled beams with arbitrary open sections consisting of flat plates, reducing the computation complexity and facilitating the analysis and calculation.
The method of imposing continuous and discrete sectional constraints on the degree of freedom system is demonstrated by examples. The continuous one is degree of freedom reduction, and the discrete one is beam end restraint. By reducing the degrees of freedom, the CGBT equation with 2n+4 generalized degrees of freedom is degenerated to the existing GBT with n+1(n≥3) generalized degrees of freedom in the basic part and n+3(n≥3) in the extending part. The accomplishment of equation degeneration further confirms the sources of the incompleteness of the existing GBT: (1) the linearization of displacements of the edge plates outside the midplane; (2) the obscureness of the reason for transverse application of three-moment equation; (3) the excessive dependence on nodal warping displacement as the sectional degrees of freedom.
On basis of the CGBT equation, the simplified calculation method of GBT is presented in this dissertation. Removing the rotations of n+1 nodes from the degrees of freedom without introducing the linearization constraint on the displacements outside the midplane of the edge plates, the number of degrees of freedom of CGBT equation is reduced to n+3(n≥1) and the Reduced Complete Generalized Beam theory (Reduced CGBT or RCGBT) is obtained. The Truncated Complete GBT (Truncated CGBT or TCGBT) also can be obtained by selecting a few lowest order modes for the simplified calculation. In particular, the TCGBT obtained by selecting the first 4 lowest order modes, namely the rigid-body modes (extension, major and minor axis bending and torsion), is exactly Vlasov's theory. These two simplified calculation methods are of great significance. They provide foundation for the approximate analytical derivation which considering sectional deformation similar to bending-torsion coupling in succeeding work of engineering application.
With the objective of illustrating the performance of CGBT, an example of a simply supported single angle beam is presented, in which the results of CGBT beam element, GBT beam element and general shell finite element are compared: (1) higher numerical stability; (2) the accuracy of CGBT results can not be significantly improved by increasing transverse node; (3) the total number of degrees of freedom is smaller, leading to more efficient and economic calculation.
The concept of natural complete mode spectrum is brought forward. The natural complete mode denotes all the cross-sectional modes obtained without any reduction of sectional degrees of freedom, and not considering the computational nodes in the CGBT sectional analysis in which section discretion is performed merely by structural nodes. Complete nature modes are gathered into mode groups and/or sub-groups by off-diagonal components of matrix D in CGBT equation. These groups and subgroups reveal the correlation between various sectional modes, especially between those of sections with symmetry axes, in a more extensive sense. The mode spectrum concept depicts the different types of equilibrium states of thin-walled beams and is a regularity understanding of sectional deformation and displacement. This concept helps us to grasp a systematic knowledge of the equilibrium states of thin-walled beams, facilitates us to distinguish, identify and set the initial equilibrium state and unstable equilibrium state in analysis, intensifying the present stability theory of thin-walled beams and piloting the future research of their stability.
引文
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