17.6
応力: 一般的な荷重条件
複数の荷重が同時に構造物にかかる現実世界の複雑な状況を把握するには、物体内の特定の点を通過する断面を xy 平面に平行に並べて視覚化するとよいでしょう。 このセクションには、元の荷重、垂直力、せん断力などのさまざまな力がかかります。
断面の平面内で潜在的な方向性を持つせん断力は、x 軸と y 軸に平…
複数の負荷がかかる体を考えてみましょう。xy平面に平行な平面を使用して点を解剖すると、せん断力が点の周囲の小さな領域に作用することが観察されます。
前に示した荷重を受ける本体の一部について考えてみて、垂直抗力がz軸に沿って作用します
せん断力は、断面平面上で明確に定義された方向を持たないため、x軸とy軸に平行な2つの成分力に分解されます。
次に、各力の大きさを面積で割って、応力成分を求めます。
応力の最初の添え字は、応力が Z 軸に垂直なサーフェスに作用することを示しています。せん断応力の 2 番目の添え字は、方向を識別します。
垂直面の反対側を解析すると、同じ応力成分が得られますが、方向は反対です。
yz平面とzx平面に平行な点を通る断面を通すことで、追加の応力成分を定義できます。
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Q1: What are stress components and how are they determined at a point in a loaded body?
Stress components are determined by passing a sectional plane through a point in a body and analyzing forces acting on that section. Normal forces and shearing forces are resolved into components parallel to coordinate axes. Each force component is then divided by the area to obtain stress values. This process yields three distinct stress components at any point when sections are passed parallel to different planes.
Q2: What do the subscripts in stress notation indicate?
In stress notation, the first subscript indicates the plane on which the stress acts—specifically, the direction of the surface normal. The second subscript identifies the direction of the stress component itself. For example, τzy indicates shearing stress acting on a surface perpendicular to the z-axis, directed along the y-axis. This notation system clarifies both location and orientation of each stress component.
Q3: How do shearing forces differ from normal forces in stress analysis?
Normal forces act perpendicular to a sectional plane, typically along one axis. Shearing forces, however, lack a single well-defined direction on the sectional plane and must be resolved into two component forces parallel to the coordinate axes. This resolution allows shearing stresses to be calculated by dividing each component by the section area, yielding multiple shearing stress components at a point.
Q4: Why do opposite sides of a sectional plane show stress components in opposite directions?
When analyzing opposite sides of a vertical plane through a point, Newton's third law applies: action and reaction forces are equal and opposite. Therefore, the stress components calculated on one side of the plane are identical in magnitude but opposite in direction on the other side. This symmetry reflects the equilibrium of forces within the body under load.
Q5: How many stress components exist at a point in a three-dimensional loaded body?
At any point in a three-dimensional body, nine stress components exist: three normal stresses (σx, σy, σz) and six shearing stresses (τxy, τyx, τxz, τzx, τyz, τzy). These components are determined by passing sectional planes parallel to the xy, yz, and zx planes through the point. Together, they fully describe the stress state at that location under general loading conditions.
Q6: What do positive and negative signs indicate in stress component values?
The signs of stress components indicate the type of loading: positive values represent tension, where the material is being pulled apart, while negative values represent compression, where the material is being pushed together. This sign convention applies to both normal and shearing stress components, allowing engineers to quickly identify whether a region experiences tensile or compressive loading.
Q7: How does analyzing multiple sectional planes help understand stress under complex loading?
Analyzing sections parallel to different coordinate planes reveals the complete stress state at a point. By examining sections parallel to xy, yz, and zx planes, all nine stress components become defined. This comprehensive approach unravels the complex interplay of forces and stresses under intricate loading conditions, enabling accurate prediction of material behavior and structural failure.