薄殼結構在位移負荷作用下之幾何非線性分析

Abstract

本文以共旋轉(co-rotational formulation)有限元素推導法及增量迭代法來探討薄殼結構在位移負荷作用下的幾何非線性行為。 本文採用文獻[24]提出的平面三角殼元素，此元素是由CST(constant strain triangle) 平面元素與DKT(discrete Kirchhoff theory)三角板元素疊加而成。此元素有三個節點，每一節點具有六個自由度。結構的節點座標、增量位移與增量旋轉、以及結構的平衡方程式都是定義在一組固定的總體座標系統上；而殼元素的應變、節點內力以及元素剛度矩陣，則是在當前元素位置上建立的元素座標上定義。 本文採用牛頓-拉福森(Newton-Raphson)法和弧長控制(are length control)法的增量迭代法來解受位移負荷之結構的非線性平衡方程式。 最後，本研究以四個數值例題探討殼結構受到各種位移負荷之幾何非線性行為。例題一為鉸接球殼受到不同的側向位移負荷，例題二為圓柱殼受到不同的側向位移負荷，例題三為含缺口之簡支圓柱殼之邊界受到均勻位移及均勻力負荷，例題四為懸臂板受到不同的側向位移負荷。由例題結果可以知道結構受單一位移負荷及力負荷時，其平衡路徑是一樣的，但受多個位移負荷作用與受多個力負荷作用時，結構的行為有很大的差異。

The geometrically nonlinear behavior of thin shell structure under displacement loading are investigated using the co-rotational finite element formulation and a incremental-iterative method. The shell element employed here is the flat three-node triangular shell element with six degrees-of-freedom per node proposed by Bathe and Ho’s [24] .The shell element is obtained by superimposing CST(constant strain triangle)element and DKT(discrete Kirchhoff theory) triangular plate element. The nodal coordinates, displacements, rotations, and the equilibrium equations of the structure are defined in a fixed global set of coordinates. The strains of shell element, the element internal nodal forces, the element stiffness matrix are defined in terms of element coordinates, which are constructed at the current configuration of the shell element. A incremental-iterative method based on the Newton-Raphson method and constant arc length method is used for solving nonlinear equilibrium equations with displacement loading. Four numerical examples are studied to investigate the geometrically nonlinear behavior of thin shell structures under different proportional displacement loadings. Example 1 is a hinged spherical shell under different lateral displacement loadings, Example 2 is a cylindrical shell under different lateral displacement loadings, Example 3 is a simply supported cylindrical shell with cutout under uniform in plane displacement loadings and force loadings, Example 4 is a cantilever plate under different lateral displacement loadings. It is found that a single concentrated displacement loading is equivalent to a single concentrated force loading as expected. However, the difference between the structure behaviors correspond to multiple displacement loading and force loading is remarked.