Name: plane strain under rotated axes
Uploaded: 2024-05-23T12:59:54.000Z
Duration: 1 min 12 s
Channel: JoVE

측면에 균일하게 분포된 하중을 받는 긴 막대를 생각해 보십시오.
이러한 막대에서 좌표 축의 회전 하에서 평면 변형의 변환은 평행 평면 내에서 변형이 발생하는 평면 변형 상태를 포함합니다.
점 O에서의 평면 변형률 상태에는 x축과 y축과 관련된 변형률 성분이 있습니다. 그런 다음 변형률 성분은 각도 θ로 표현됩니다.
x축과 임의의 각도를 형성하는 선을 따라 수직 변형에 대한 표현식이 파생됩니다.
그런 다음, x축과 y축에 의해 형성된 각도의 이등분 방향의 수직 변형률이 결정됩니다. 그런 다음 전단 변형률은 이러한 정상 변형률로 표현됩니다.
삼각법 관계를 고려할 때, 축 회전 하에서 평면 변형률 변환에 대한 방정식은 x' 및 y' 축의 이등분선을 따라 수직 변형률을 계산하고 수직 변형률로 전단 변형률을 표현하여 파생됩니다.

평면 변형의 변환

균일하게 분포된 하중을 받는 막대와 같은 긴 구조를 해석할 때 좌표축이 회전할 때 평면 변형의 변형을 이해하는 것이 필수적입니다. 이러한 변환은 재료 과학 및 구조 공학에서 중요한 방향에 따라 재료 변형 특성이 어떻게 달라지는지 평가하는 데 도움이 됩니다.

한 치수가 다른 치수를 크게 초과하는 부재의 일반적인 평면 변형 조건에서는 변형 및 결과 변형이 단일 평면에서만 두드러집니다. 임의의 지점(예: O)에서 중요한 변형 구성요소는 x 및 y축을 따른 구성요소이고 z축을 따른 변형 구성요소는 무시할 수 있습니다. 이러한 구성 요소는 xy 평면 내의 변형 상태를 포괄적으로 설명합니다.

중요한 사례는 x축과 y축 사이 각도(45도)의 이등분선을 따라 수직 변형률을 계산하는 것입니다. 이 경우, 이 각도를 고려하면 수직 변형률과 전단 변형률의 조합이 이 선을 따라 변형에 어떻게 영향을 미치는지 반영하여 표현이 단순화됩니다.

<img alt="Equation 1" src="/files/ftp_upload/15658/15658_Equation_1.svg" style="height: 32px;">

x축과 y축의 이등분선에 정렬되도록 좌표계를 회전하면 탄성 이론에서 파생된 삼각 관계를 사용하여 새 좌표의 변형률 성분을 다시 계산할 수 있습니다. 구체적으로는, 원래 축과 임의의 각도를 이루는 방향의 수직 변형률과 이에 대응하는 전단 변형률을 원래 변형률 성분으로 결정할 수 있습니다.

<img alt="Equation 2" src="/files/ftp_upload/15658/15658_Equation_2.svg" style="height: 188px;">

측면에 균일하게 분포된 하중을 받는 긴 막대를 생각해 보십시오.
이러한 막대에서, 좌표의 회전에 따른 평면 변형의 변형 ...

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 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.

A useful special case is the normal strain along the bisector of the x and y axes. That bisector is at 45 degrees. Using this angle makes the strain expression simpler and shows how normal strain and shearing strain combine along that direction.

When the coordinate system is rotated to match the bisector, the strain components can be recalculated in the new axes. The needed equations use trigonometric relations from elasticity theory. They give the normal strain in any direction that makes an angle with the original axes, along with the corresponding shearing strain.

Plane Strain Under Rotated Axes

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...

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 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

Learn how plane strain changes when axes are rotated in a long bar under side loads, including 45-degree strain and new normal and shear components.

Plane Strain Under Rotated Axes | 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 Strain Under Rotated Axes

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