Name: plane stress changes with rotation
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 18 s
Channel: JoVE

立方体上の点が平面応力を受けており、応力成分によって定義されるとします。
要素が角度付きで回転すると、その応力成分が変化します。この変更は、軸に対して垂直な面を持つプリズム要素を考慮することによって決定されます。
斜め面の面積を考慮することにより、回転角度の三角関数を使用して垂直面と水平面の面積を計算できます。
これらの面に加えられる力が表され、要素の三角形の面には力が作用しないと仮定されます。次に、平衡方程式は、回転した軸に沿ったコンポーネントを使用して記述されます。
これらの方程式を解くと、垂直応力とせん断応力が導出されます。次に、垂直応力とせん断応力は、三角関数の関係を使用して再表されます。
回転した垂直軸の垂直応力の式は、前の式の 1 つの回転角度を新しい角度に置き換えることによって得られます。
立方体要素に加えられる垂直応力の合計は、要素の向きとは無関係であることが観察されます。

平面応力の変換

応力変形の研究は、平面応力を受けた立方体など、材料内の応力成分が回転に伴ってどのように変化するかを理解するために不可欠です。この変化を立方体の中の角柱要素を考慮して解析します。要素が回転すると、要素に作用する応力成分 (法線応力とせん断応力の両方) の大きさと方向が変化します。この変化は、回転角度の三角関数を使用して定量化され、回転した要素の面に作用する力を元の垂直面に作用する力に関連付けます。

主軸に垂直な面にかかる力のみを考慮して定式化された平衡方程式 (回転による三角形面にかかる力は除く) により、新しい応力成分を導出することができます。垂直応力とせん断応力は、元の応力を基準にして再表現されます。

<img alt="Equation 1" src="/files/ftp_upload/15652/15652_Equation_1.svg" style="height: 96px;">

回転した垂直軸上の垂直応力の新しい式は、前の式の回転角度を新しい式に置き換えることによって得られます。

<img alt="Equation 2" src="/files/ftp_upload/15652/15652_Equation_2.svg" style="height: 122px;">

この解析の注目すべき結果は、立方体要素の方向に関係なく垂直応力の合計が変化しないことです。この不変性は、外部応力に対する材料の等方性応答を強調し、さまざまな荷重条件下での材料の挙動を予測するために重要です。要素の方向に応じて応力成分がどのように変化するかを理解することは、材料の破壊モードを予測し、加えられた荷重に対してより弾力性のある材料や構造を設計するために不可欠です。

立方体上の点が平面応力を受けており、応力成分によって定義されるとします。
要素が角度付きで回転すると、その応力成分が変化します。...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Transformations of Stress and Strain

応力とひずみの変換

KEY TERMS AND CONCEPTS

主要用語および概念

Transformation of Plane Stress

plane stress changes with rotation

Plane stress changes with rotation in a material element. The analysis looks at a prismatic element inside a cube under plane stress. As the element turns, the normal stress and shearing stress on its faces change in both size and direction.

These changes are described with trigonometric functions of the rotation angle. The forces on the rotated faces are compared with the forces on the original faces that are perpendicular to each principal axis. The equilibrium equations use only the forces on those perpendicular faces, not the forces on the triangular faces created by the rotation.

From this balance, the new stress components can be written in terms of the original stresses. The normal stress on the rotated vertical axis is also found by using a new angle in a previous expression. One important result is that the sum of the normal stresses stays the same, no matter how the cubic element is oriented.

This invariant sum shows the material’s isotropic response to external stress. It helps explain how stress components shift as orientation changes. That understanding is important for predicting failure modes and for designing materials and structures that can better handle applied loads.

Plane Stress Changes with Rotation

transformation of plane stress

Plane stress changes with rotation in a material element. The analysis looks at a prismatic element inside a cube under plane stress. As the element turns, the normal stress and shearing stress on its faces change in both size and direction. These changes are described with trigonometric functions of the rotation angle. The forces on the rotated faces are compared with the forces on the original faces that are perpendicular to each principal axis. The equilibrium equations use only the forces...

principal stresses

Stress Circle for Principal and Shear Stress

The stress circle shows how normal stress and shearing stress change with angle on a transformed plane. Its center, C, lies on the vertical axis and represents the average normal stress. The radius shows the amount of stress variation as the plane rotates. Points A and B, where the circle crosses the horizontal axis, mark the maximum and minimum normal stresses. At these points, shearing stress is zero. These points define the principal stress planes, where only normal stress, called principal...

principal stresses: problem solving

Finding Principal Planes and Stress Limits

Principal planes and principal stresses are found by analyzing two planes that meet at right angles under shearing, tensile, and compressive stresses. The goal is to use the given stress values to describe how the material is loaded and to locate the planes where the stress state changes. To find the principal planes, use the formula that relates the plane angle to twice the shearing stress divided by the difference between tensile and compressive stresses. After substituting the known stress...

mohr&#39;s circle for plane stress

Reading Stress Components with Mohr's Circle

Mohr's circle is a graphical way to read the stress state at a point in a material under plane stress. It turns normal stress and shearing stress into a two-dimensional picture. This makes stress transformation easier to analyze. The method uses Cartesian coordinates. The horizontal axis shows normal stress, written as σ, and the vertical axis shows shearing stress, written as τ. Two points, A and B, are plotted from the stresses on the element. Point A is at (σx, -τxy), and point B is at (σx,...

general state of stress

Stress Tensor and Principal Stresses

The stress tensor describes the internal forces inside a material. It is used when a material is under external force or deformation. At a point such as O, the tensor can show the stress components on any plane through that point. A tetrahedral element helps show how these forces act. One face, XYZ, is perpendicular to the line OA. The other faces line up with the coordinate axes, and point O is the origin. The stress on the tetrahedron is split into normal and shear components on each face. <br...

yield criteria for ductile materials under plane stress

yield-criteria-for-ductile-materials-under-plane-stress

Tresca and Von Mises Yield Criteria

Ductile materials can yield when they are loaded beyond a safe stress level. Engineers use yield criteria to check whether structural elements and machine parts can handle the expected load without permanent deformation. The yield point is first found with a tensile test, which measures how the material responds to uniaxial stress. A tensile test is not enough when a part faces biaxial or plane stress conditions. In those cases, more advanced criteria are needed to predict failure. One common...

transformation of plane strain

Plane Strain Under Rotated Axes

Plane strain changes when the coordinate axes are rotated. This matters for long bars and similar elongated structures that are loaded uniformly on their sides. In these members, one dimension is much larger than the others, so deformation is mainly studied in a single plane. At a point such as O, the key strain components lie along the x and y axes. The strain along the z-axis is negligible in plane strain. Together, the x and y strain components describe the deformation state in the xy-plane.

mohr&#39;s circle for plane strain

Principal and Shear Strains on Mohr’s Circle

Mohr’s circle for plane strain shows how normal strain and shear strain change on a graph. The horizontal axis is normal strain, written as ε, and the vertical axis is shear strain, written as γ. It uses two plotted points, X and Y, to represent the strain state. Point X has coordinates (εx, -γXY), and point Y has coordinates (εY, γXY). The center of the circle, O, gives the average normal strain. The radius comes from the relationship between normal strain and shear strain, so the circle...

three-dimensional analysis of strain

Principal Strain Directions in 3D

Three-dimensional strain analysis shows how a material changes shape when it is loaded. It is especially useful for elastic, homogeneous materials. A small cubic element inside the material can expand or contract along principal stress axes. That change turns the cube into a rectangular parallelepiped, which makes the deformation easier to picture. The principal stress axes are orthogonal, or at right angles to each other. Along these directions, the stress does not create shear inside the...

measurements of strain

How Strain Gauges Measure Deformation

Strain describes how a material deforms when a force is applied. Normal strain compares the change in length with the original length. Engineers use this measure to track stretching or compression in materials. For higher accuracy, they often use electrical strain gauges. A strain gauge has a conductive wire on a paper backing that sticks to the surface of the material. It works through the piezoresistive effect, which means the wire’s electrical resistance changes when the material is bent,...

key terms and concepts

See how normal and shearing stress change as a plane-stressed element rotates, and why the sum of normal stresses stays constant for any orientation.

Plane Stress Changes with Rotation | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Plane Stress Changes with Rotation

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