挫屈梁在側向負荷下的幾何非線性分析

Abstract

挫屈梁在本文中指的是一細長直梁受到超過其臨界端點軸向壓縮位移形成，本研究的第一個目的是探討挫屈梁的初始變形與自然振動，第二個研究目的是探討雙端拱起固定角度的挫屈梁受到側向負荷時的幾何非線性行為，本文中使用方法的是共旋轉有限元素法與數值程序。 本文利用非線性梁理論的一致線性化、d’Alembert原理和虛功原理在當前的元素座標上推導梁元素的節點變形力和節點慣性力，為了考慮軸向與側向變形間的耦合，在推導內力時保留至應變的二次項，將元素節點內力對元素節點變形參數與節點加速度微分，即可得到元素切線剛度矩陣與質量矩陣。 將挫屈梁的系統運動方程式用泰勒級數在穩態變形的位置展開，取到一次項，即為線性振動的運動方程式。 本文使用基於弧長法和牛頓-拉福森法的增量迭代法解非線性平衡方式，探討不同細長比之挫屈梁在不同端點壓縮位移的初始變形，同時以子空間迭代法求得挫屈梁的自然振動頻率與其對應的振動模態，本文使用在平衡方程式中增加臨界條件的延伸系統來追蹤在不同軸向壓縮與不同雙端拱起角度下的挫屈梁受到側向負荷的臨界點的摺線，並以二分法求摺線的轉折點，來分析不同邊界條件下各種臨界點發生的門檻。

The buckled beam considered here is a slender and originally straight beam compressed axially beyond the critical buckling axial displacement. The first objective of this paper is to investigate the deformed configuration and free vibration of the buckled beam. The second objective of this paper is to investigate the geometrically nonlinear behavior of clamped buckled beams with both ends fixed at an adjustable angle subjected to lateral load. The co-rotational finite element formulation is used here. A numerical procedure is proposed here. The element deformation nodal forces and inertia nodal forces are systematically derived by consistent linearization of the fully geometrically non-linear beam theory using the d'Alembert principle and the virtual work principle in the current rotating element coordinates. In this paper the terms up to the second order of deformation parameters and their spatial derivatives are retained in element nodal forces. The element tangent stiffness matrix and mass matrix may be obtained by differentiating the element nodal force vector with respect to the element nodal parameters and their second time derivative, respectively. The governing equations for linear vibration are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position of the buckled beam. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations to investigate the post-buckling configuration of clamped beams with different slenderness ratios under different axial compression. The subspace iterative method is used here to find the natural frequencies and its corresponding vibration mod of buckling beam. An extended system generated by augmenting equilibrium equations with a criticality condition is employed to trace the fold lines of the critical points for the buckling beams with different axial compression and adjustable angle subjected to a lateral load. The threshold of the initial lateral deflection for the existence of the critical point is determined by the bisection method.