20.16
偏心軸荷重の一般的な場合
非対称な曲げは、構造部材に加えられる曲げモーメントがその主軸と一致しない場合に発生します。この位置ずれにより、対称的な曲げとは異なる複雑な応力分布とたわみパターンが生じます。これらは、さまざまな荷重条件に耐えられる構造を設計するために不可欠です。
部材の中立軸と一致しない線に沿って加えられる、等しい…
等しい偏心力と反対の力を受けるメンバーを考えてみましょう。これらの力は、部材の主重心軸から a の水平距離と b の垂直距離で加えられます。
各偏心力は、中心力と 2 つのカップルモーメントに等しく、モーメントアームはメンバーの主重心軸から a と b の距離です。
サン・ヴナンの原理を使用して、等価荷重を使用して、部材の断面における応力の分布を決定できます。解析は、断面が部材の両端に近くない場合に保持されます。
重ね合わせの原理は、中心力と曲げ電対による応力を決定し、応力が断面に沿って直線的に変化することを示しています。
メンバーの形状と偏心荷重の作用線に応じて、全応力はセクション全体で同じまたは反対の符号を持つ場合があります。応力の大きさがゼロの断面に沿った線は、断面の中立軸と呼ばれます。
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Q1: How do eccentric forces differ from centric forces in axial loading?
Eccentric forces are applied at a horizontal distance a and vertical distance b from the principal centroidal axis, creating couple moments in addition to axial stress. Centric forces pass through the centroid and produce only uniform axial stress. Using the superposition principle, eccentric loading combines axial stress from the centric component with bending stresses from the couple moments, resulting in a more complex stress distribution across the section.
Q2: What is the neutral axis in eccentric axial loading?
The neutral axis is the line along the cross-section where stress magnitude equals zero. In eccentric loading, the neutral axis does not necessarily align with the geometric axes of the section. Its position is determined by ensuring the sum of normal stresses across the section equals zero, and it depends on the relationship between applied load eccentricity and the geometric properties of the cross-section.
Q3: How does the Saint-Venant principle apply to eccentric axial loading analysis?
The Saint-Venant principle allows equivalent loadings to replace eccentric forces with a centric force and couple moments. This decomposition enables stress analysis at sections away from the load application point. The principle holds when the section is not close to either end of the member, allowing engineers to use superposition to determine total stress from axial and bending components separately.
Q4: Why does stress vary linearly across a section under eccentric loading?
Stress varies linearly because the superposition principle combines linear stress distributions from the centric force and bending couples. The centric component produces uniform stress, while bending moments create linearly varying stress. The total stress at any point is the algebraic sum of these components, maintaining linear variation across the section.
Q5: Can total stress have the same sign throughout an eccentric loading section?
Yes, depending on the member geometry and eccentricity magnitude, total stress may be entirely tensile, entirely compressive, or mixed throughout the section. When the neutral axis lies outside the cross-section, all stresses share the same sign. When the neutral axis passes through the section, stresses change sign, with tension on one side and compression on the other.
Q6: What role does the product of inertia play in unsymmetrical bending stress analysis?
The product of inertia measures the covariance of area element coordinates relative to the axes and is essential for determining stress distribution in unsymmetrical bending. When axes align with the centroidal axes of the section, the product of inertia simplifies calculations and the neutral axis coincides with these principal axes, making stress analysis more straightforward.
Q7: How do couple moments from eccentricity affect bending in multiple directions?
Eccentric forces create couple moments with moment arms equal to distances a and b from the principal centroidal axis, causing bending about multiple axes simultaneously. These moments counteract the eccentricity of the forces and result in complex three-dimensional stress distributions. The combined effect of moments about different axes determines the final stress state and neutral axis orientation.