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

큐브의 한 점이 응력 성분에 의해 정의되는 평면 응력을 받고 있다고 가정해 보겠습니다.
요소가 각도만큼 회전하면 응력 요소가 변경됩니다. 이 변경은 축에 수직인 면이 있는 프리즘 요소를 고려하여 결정됩니다.
비스듬한 면의 면적을 고려하여 수직 및 수평 면의 면적은 회전 각도의 삼각 함수를 사용하여 계산할 수 있습니다.
이러한 면에 가해지는 힘이 표시되며, 요소의 삼각형 면에는 어떤 힘도 작용하지 않는 것으로 가정합니다. 그런 다음 평형 방정식은 회전된 축을 따라 구성 요소를 사용하여 작성됩니다.
이 방정식을 풀면 법선 응력과 전단 응력이 도출됩니다. 그런 다음 수직 응력과 전단 응력을 삼각 관계 를 사용하여 다시 표현합니다.
회전된 수직 축의 수직 응력에 대한 표현식은 이전 표현식 중 하나의 회전 각도를 새 각도로 대체하여 얻습니다.
입방 요소에 가해지는 수직 응력의 합은 요소의 방향과 독립적으로 유지되는 것으로 관찰됩니다.

평면 응력의 변환

응력 변환을 연구하는 것은 평면 응력을 받는 큐브와 같은 재료 내의 응력 구성 요소가 회전에 따라 어떻게 변하는지 이해하는 데 필수적입니다. 이러한 변화는 큐브 내의 프리즘 요소를 고려하여 분석됩니다. 요소가 회전함에 따라 요소에 작용하는 응력 구성 요소(수직 응력과 전단 응력 모두)의 크기와 방향이 변경됩니다. 이러한 변화는 회전된 요소의 면에 작용하는 힘을 원래 수직 면의 힘과 연관시키는 회전 각도의 삼각 함수를 사용하여 정량화됩니다.

주축에 수직인 면에 작용하는 힘(회전으로 인해 삼각형 면에 작용하는 힘은 제외)만을 고려하여 공식화된 평형 방정식을 통해 새로운 응력 성분을 도출할 수 있습니다. 수직 응력과 전단 응력은 원래 응력으로 다시 표현됩니다.

<img alt="Equation 1" src="/files/ftp_upload/15652/15652_Equation_1.svg" style="height: 96px;">

회전된 수직축의 수직 응력에 대한 새로운 표현식은 이전 표현식의 회전 각도를 새 표현식으로 대체하여 얻습니다.

<img alt="Equation 2" src="/files/ftp_upload/15652/15652_Equation_2.svg" style="height: 122px;">

이 분석의 주목할만한 결과는 입방 요소의 방향에 관계없이 법선 응력의 합이 변하지 않는다는 것입니다. 이러한 불변성은 외부 응력에 대한 재료의 등방성 반응을 강조하며 다양한 하중 조건에서 재료 거동을 예측하는 데 중요합니다. 응력 구성요소가 요소 방향에 따라 어떻게 변환되는지 이해하는 것은 재료 파손 모드를 예측하고 적용된 하중에 더 탄력적인 재료 및 구조를 설계하는 데 필수적입니다.

큐브의 한 점이 응력 성분에 의해 정의되는 평면 응력을 받고 있다고 가정해 보겠습니다.
요소가 각도만큼 회전하면 응력 요소가 변경됩니다. ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Transformations of Stress and Strain

응력과 변형률의 변환

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Plane Stress Changes with Rotation

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