Basic Theory On Deformation Of Thin-walled Beam

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.