薄壁開口梁之自由振動分析及幾何非線性動態反應研究

Abstract

本研究的主要目的是以一致性共旋轉法推導一個薄壁開口梁元素，並將其應用在梁結構的自由振動分析及幾何非線性動態分析。 本文中推導的梁元素有兩個節點，每個節點有七個自由度，本研究用傳統的力、力矩及雙力矩為廣義的節點力。本文中將元素節點定在元素兩端斷面的剪心，並取剪心軸當作描述元素變形的參考軸。本研究在一移動元素座標上描述元素的變形，本研究用三個旋轉參數描述元素斷面在移動元素座標上的方位，但用對固定座標的旋轉向量描述元素間共同節點的旋轉。本研究在梁元素當前的變形位置上，利用元素節點的座標及斷面方位建立一個移動元素座標並決定元素節點的旋轉參數，對應於元素節點旋轉參數擾動量的廣義節點力為一廣義力矩，為推導傳統力和力矩與該廣義力矩的關係，本研究在一個與當前的移動元素座標重合的固定元素座標上，推導出元素節點在當前固定元素座標的擾動位移和擾動旋轉與元素節點旋轉參數的擾動量的關係。本研究利用元素節點在當前固定元素座標的位移和旋轉及其擾動量、速度、加速度、角速度、角加速度，推導出移動元素座標的角速度及角加速度及元素節點的變形參數對時間的一次及二次微分。本研究利用虛功原理和D’Alembert原理，以及完整的幾何非線性梁理論的一致性二次線性化在當前的固定元素座標推導元素節點變形力及慣性力，本研究中保留了變形力中撓曲、扭曲及軸向變形間之耦合項、軸向扭轉率的三階項、慣性力中速度間的耦合項。為了推導上的方便，本研究用虛功原理推導梁元素節點變形力時，先推導出廣義節點力矩，再用controgradient law求得傳統節點力和力矩。本研究在推導元素節點在當前固定元素座標的擾動位移和擾動旋轉與元素節點旋轉參數的擾動量的關係時，保留了值為零的節點位移及旋轉向量，故可由元素節點變形力對節點參數微分求得元素切線剛度矩陣。本研究推導元素的節點慣性力時，先將元素擾動位移表示成當前固定元素座標的擾動位移和旋轉之函數，故可直接求得元素的節點慣性力，元素的一致性質量矩陣 (consistent mass matrix) 可由元素節點的慣性力對元素節點的加速度微分求得。 本研究採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程式，本研究採用次空間法(Subspace Iteration Method)解梁結構的自然頻率及振動模態。本文應用Newmark直接積分法和牛頓-拉福森法的增量迭代法解非線性運動方程式。本研究以數值例題探討不同斷面、邊界條件及負載對開口薄壁梁之自然頻率、振動模態及幾何非線性之動態反應之影響，以說明本研究提出之非線性開口薄壁梁元素的正確性及有效性，並驗證文獻上梁結構之自然頻率及幾何非線性之動態反應之正確性。

A consistent co-rotational finite element formulation for the free vibration analysis and geometric nonlinear dynamic analysis of thin-walled beams with generic open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in a current moving element coordinate system constructed at the current configuration of the beam element. Three rotation parameters are used to describe the orientation of the beam cross section in the moving element coordinate system. However, the rotation vector is used to describe the element nodal rotations in fixed coordinates. The values of the nodal rotation vectors are reset to zero at current configuration. The element equations are derived in a fixed current element coordinates which are coincident with the current moving element coordinates. The perturbed moving element coordinates and the variation of the element nodal rotation parameters corresponding to the perturbation of element nodal displacements and rotations referred to the current fixed element coordinates is consistently determined using the first order linearization of the way used to determine the current element coordinates and element nodal rotation parameters corresponding to the incremental element nodal displacements and rotations referred to the global coordinates. The angular velocity and acceleration of the current moving element coordinates and the first and the second time derivative of the element nodal rotation parameters are consistently determined using the current element nodal displacements and rotations, nodal velocities and accelerations, and nodal angular velocities and accelerations. The element deformation and inertia nodal forces are derived using the virtual work principle, the d’Alembert principle, and the consistent second order linearization of the fully geometrically nonlinear beam theory. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered. For convenience, in the derivation of the element deformation nodal force, the generalized nodal moments corresponding to the variation of the nodal rotation parameters are derived first, and then transformed to the conventional moments and forces using controgradient law. Because the element nodal displacements and rotations with value of zero are retained in the relationship between the variation of the element nodal rotation parameters and the variation of element nodal displacements and rotations, the element tangent stiffness matrix may be obtained by differentiating the element deformation nodal force with respect the element nodal parameters. 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. The subspace iterative method is used for the solution of natural frequencies and vibration modes for the free vibration of beam structures. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed for the solution of nonlinear equations of motion. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. The effect of different cross sections, boundary conditions and different loads on the natural frequencies, vibration modes, and nonlinear dynamic behavior of three dimensional thin-wall beam structures are also investigated through numerical examples.