23.2
主応力
垂直応力方程式とせん断応力方程式のグラフ表示は円で表され、異なる角度条件下でのこれらの応力間の相互作用を示しています。縦軸上にあるこの円 C の中心は平均垂直応力を表し、その半径は応力の変動範囲を示します。円が水平軸と交差する点 A と B では、せん断応力なしで発生する最大垂直応力と最小垂直応力が…
変換された平面の法線応力方程式とせん断応力方程式をグラフ化すると、任意の角度パラメータに対するそれらの関係を示す円が形成されます。
円の中心は、垂直軸を基準にして平均垂直応力を表し、半径はこれらの応力値の広がりを示します。
円は 2 点で水平軸と交差し、せん断応力がゼロの場合に発生する最大垂直応力と最小垂直応力を示します。これらの点は、主応力と呼ばれる垂直応力のみが存在する応力の主平面を定義します。
最大垂直応力と最小垂直応力は、半径から平均応力を加算または減算することで識別できます。
最大または最小の垂直応力が発生する主平面は、角度パラメータを垂直応力方程式に代入することによって識別されます。
最大せん断応力は、垂直応力が平均応力と等しい場合に得られる円の垂直直径上の点で表され、最大せん断応力を予測する 2 つの 90° 方向が得られます。
最大せん断応力の平面と主平面は45°離れています。
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Q1: What does the circle represent when graphing normal and shearing stress equations?
The circle graphically depicts the relationship between normal and shearing stresses across different angular orientations. Its center represents the average normal stress, while its radius indicates the range of stress variations. This geometric representation, known as Mohr circle for plane stress, allows engineers to visualize how stresses transform as the plane orientation changes.
Q2: Where do principal stresses occur on the stress circle?
Principal stresses occur at points A and B where the circle intersects the horizontal axis, representing the maximum and minimum normal stresses with zero shearing stress. These intersection points define the principal planes of stress, where only normal stress exists. The principal plane carrying maximum or minimum normal stress is identified by substituting the angular parameter into the normal stress equation.
Q3: How are maximum and minimum normal stresses calculated from the circle?
Maximum and minimum normal stresses are determined by adding or subtracting the circle's radius to or from the average normal stress at the circle's center. The radius represents the spread of stress values, while the center value is the mean stress. This calculation provides the extreme normal stress values that materials experience under the given loading conditions.
Q4: What conditions produce maximum shearing stress in the stress circle?
Maximum shearing stress occurs at points along the circle's vertical diameter, where the normal stress equals the average stress. This condition produces two orientations, each 90 degrees apart, that predict peak shearing stress locations. The magnitude of maximum shearing stress equals the circle's radius.
Q5: What is the angular relationship between principal planes and maximum shearing stress planes?
The planes experiencing maximum shearing stress and the principal stress planes are oriented 45 degrees apart. This geometric relationship is fundamental to understanding stress distribution, as it reveals how normal and shearing stresses interact. This 45-degree offset is a key characteristic observed in stress transformations.
Q6: How do you identify which principal plane carries maximum versus minimum stress?
The principal plane experiencing maximum or minimum normal stress is identified by substituting the angular parameter into the normal stress equation. This calculation determines the specific orientation angle at which each principal stress acts. The result indicates whether that plane corresponds to the maximum or minimum stress value from the circle's intersection points.
Q7: Why is understanding principal stresses important for material analysis?
Principal stresses represent the extreme normal stresses a material experiences, which are critical for predicting failure and material behavior. Materials fail when stresses exceed their strength limits, making principal stress identification essential for design and safety analysis. This understanding connects directly to yield criteria for ductile materials under plane stress and other failure prediction methods.