三維聖維南撓曲扭轉與翹曲之理論及應用

Abstract

大部分聖維南理論只停留在扭轉，而本研究除了扭轉之外，更加入了撓曲，更重要的是本研究還加入翹曲扭矩與翹曲彎矩，並且利用雙彎矩更好的詮釋軸向應力。聖維南撓曲扭轉理論用偏微分方程組的方式解出應力函數與翹曲函數，可以知曉任何位置的應力與變形，更重要的是聖維南撓曲扭轉理論可以適用於各種截面。但是聖維南撓曲扭轉理論在軸向並沒有探討，意即應力函數與翹曲函數並不會隨軸向改變。而本研究目的使聖維南撓曲扭矩理論可以更全面，融入一維的翹曲扭轉問題，翹曲扭轉會因為支承條件而在軸向有所變化，以懸臂梁為例，在固定端翹曲扭矩最大，而自由端則沒有翹曲扭矩。 本研究成功將聖維南撓曲扭轉理論與翹曲扭轉問題結合，使原本軸向固定的應力函數與翹曲函數可以隨軸向改變。並且對軸向應力有更好的詮釋，在經過軸向性質與截面性質的驗證，確認本研究可以同時呈現三維的性質。 剪力中心是重要的截面性質，只要集中橫力施加在剪力中心上，就不會產生扭轉。但剪力中心的解法目前只停留在薄壁截面，而本研究能得到同時適用於薄壁截面與實心截面。本研究可得包含撓曲、扭轉考量翹曲之應力矩陣，該應力矩陣可應用在各種降伏條件與後續更複雜的塑流規則。

Most Saint-Venant theories often focus on torsion, but this thesis goes beyond torsion by incorporating flexure. What's more, it introduces warping torsion and warping moment, and utilizes bimoment to better summarize axial stresses on the cross section. Saint-Venant flexure-torsion theory solves the stress function and warping function governed by Poisson's equation and Laplace's equation, respectively, providing insight into stress and warping at any location. Importantly, Saint-Venant theory can be applied to arbitrary cross section. However, this theory does not consider axial effects, meaning that the stress and warping functions do not account for axial variations. The objective of this thesis is to make Saint-Venant theory analysis more comprehensive by incorporating one more dimension for warping torsion where warping and torsion vary along the axial direction, so that the enhanced Saint-Venant theories become a truely unified three-dimensonal theory. This study successfully unifies Saint-Venant flexure-torsion theory with warping torsion, allowing the stress function and warping function to vary along the axial direction, and also providing formulae between bimoment and axial stress. After verifying the axial and cross-sectional properties, it has been confirmed that this study can well display three-dimensional behavior. The shear center, an important cross-sectional property, ensures that no torsion occurs when transverse (shear) forces externaliy applied at the shear center. However, the existing solutions for the shear center are primarily limited to thin-walled sections. This thesis develops formulae applicable both to thin-walled and to solid sections. Finally, the methodology proposed in this thesis further explores in the field of plasticity, applied to various yield conditions and more complex plastic flow rules.