考慮剪切變形與旋轉慣性效應的三維構架之自由振動分析

Free Vibration Analysis of the 3-D Frameworks with Effects of Shear Deformation and Rotary Inertia Considered

The literature regarding free vibration analysis of frameworks using the Euler-Bernoulli beam elements is plenty particularly fur the two-dimensional (2-D) frameworks, but the report concerning the dynamic analysis of three- dimensional (3-D) ones using the 3-U Timoshenko beans elements is little Thus, the objective of this paper is to provide some information in this aspect. First of all, a consistent approach is used to derive the mass matrix and stiffness matrix of a 3-D Timoshenko beans element using the associated shape functions. Next, the reliability of the presented approach is confirmed by comparing the natural frequencies and mode shapes of a 3-D uniform Timoshenko beam obtained front this paper with the corresponding exact ones of the same Timoshenko beam performing 2-D free vibrations in its two principal planes, respectively. Finally, the free vibration analysis of the 3-D beamlike lattice girders is performed using the derived property matrices of the 3-D Timoshenko beam elements and the influence of flexural rigidity, shear deformation and rotary inertia of the constituent structural members on the free vibration characteristics of the ”triangular-type” and ”rectangular-type” 3-D beamlike (Vierendeel and X-braced) lattice girders is studied. 使用Euler樑元素，來進行構架（特別是二維構架）之自由振動分析的文獻很多，但使用三維的Timoshenko樑元素，來進行三維構架之動態分析的報告卻很少因此、本文之目的即在提供一些這方面的資料。首先，本文使用相關的形狀函數，以一致的方式來推導三維Timoshenko樑元素的質量矩陣與勁度矩陣。接著，吾人利用本文所得的陸質矩陣來計算一根三維Timoshenko樑的最低五個自然頻率及振態，並分別與同一根Timoshenko樑，在其兩個主平面內，進行二維的自由振動之對應自然頻率及振態的正解比較之，以驗證本文理論的可靠性。最後，本文利用上述三維Timoshenko樑元素的性質矩陣，來進行三維構架的自由振動分析，並探討各樑元素的撓曲剛度、剪切變形與旋轉慣性，對三角形及四方形的樑形格狀桁，自由振動特性的影響。