衍生自彈性力學含多種變形機制之梁理論

Abstract

本文發展包含六種變形機制的均質等剖面直梁理論，分別是軸向變形機制、扭轉變形機制、剪切變形機制、撓曲變形機制、翹曲變形機制與剖面縮脹變形機制。前四種變形機制在傳統的桿、軸、梁、柱理論中常見，特別的是翹曲變形機制，共分為三項，與三個調和函數相乘後，使縱向產生翹曲。此三個調和函數分別是扭轉翹曲函數與剖面兩向的撓曲函數。而剖面形狀可以是實心、空心、薄壁、厚壁、開剖面、閉剖面、單連通、多連通等任意形狀。 另外引進力學層次的概念，使得本梁理論建立在滿足彈性力學的機動協調、組成律以及力平衡的基礎上。再利用能量守恆原理，令虛功相等，並透過對位移的假設，將六種變形機制疊加，使得梁理論得以由彈性力學中的三維微元，推廣衍生至材料力學中的一維微元以及結構學的離散構件與結構中。 文中並介紹矩形梁受到純彎矩作用時，利用彎矩與曲率的曲線(材料力學層次的組成律)能夠反算求得應力與應變的曲線(彈性力學層次的組成律)方法，並且與實驗結果進行對照。由於梁受彎矩後在剖面上以中性軸為分界，一端受拉，一端受壓，因此受拉側與受壓側的組成律是否相同會影響計算的結果，因而理論中將區分為拉壓組成律相同的材料以及拉壓組成律不相同的材料。實驗則採用鋁合金6061，拉壓組成律相同，中性軸不移動的材料，經實驗結果與理論對照後，發現在彎矩加載、卸載以及反向加載時都能有不錯的結果。另以鑄鐵模擬拉壓組成律不相同，中性軸會移動的材料，但受限於材料性質不穩定，因而導致實驗結果與理論對照結果不甚理想。

In this thesis we consider a beam theory with six deformation modes: axial deformation, torsional deformation, shear deformation, bending deformation, warping deformation and section homothetic deformations. The first four modes are often seen in rods, shafts, beams and columns. The special one is warping deformation. We use three harmonic functions to describe warping. The cross section can be arbitrary, for example solid, hollow, thin-walled, thick-walled, open section, close section. We also introduce the concept of levels of mechanics, and then our beam theory can satisfy level one, elasticity. Based on principle of virtual work, the beam theory can expand to level two, mechanics of materials, level three and four, structural theory. Finally, we introduce a method that stress-strain curves can be calculated by using moment-curvature curves, when the beam is subjected to bending moment. Two experimental data from bending tests and axial tests are also presented with aluminum 6061-O specimens. By using this method to calculate stress-strain curves from bending test and then comparing with axial tests, we find that the results are compatible.