Name: reading stress components with mohr's circle
Uploaded: 2024-05-23T12:59:54.000Z
Duration: 1 min 23 s
Channel: JoVE

Mohr의 원은 요소에 대한 응력 변환을 시각적으로 표현한 것입니다. 이 요소의 응력 요소는 평면 응력을 받는 정사각형 요소에 대한 Mohr의 원을 만들기 위해 그래프에 표시됩니다.
전단 응력이 양수이면 점 A는 수평 축 아래에 플롯되고 점 B는 위에 플롯됩니다. 음수이면 위치가 반전됩니다.
점 A와 B를 연결하는 선의 중간점 O는 가로축에 있으며 원의 중심 역할을 합니다. O를 중심으로, 선 AB를 지름으로 하여 그려진 원을 모어원이라고 합니다.
원 이 수평 축과 교차하는 점 X와 Y의 가로 좌표는 각각 최대 응력과 최소 주응력을 나타냅니다.
각도 AOX는 각도 θp의 두 배와 같습니다. 점 X에 해당하는 주평면의 방향 θp는 Mohr의 원에서 측정된 각도 AOX를 절반으로 줄임으로써 얻을 수 있습니다.
수직 방향에서 Mohr의 원 반지름은 최대 전단 응력의 크기에 해당합니다.

평면 응력에 대한 모어의 원

모어의 원은 재료의 한 지점에서 응력 상태를 식별하는 그래픽 방법으로, 평면 응력 조건에서 응력 변환을 더 쉽게 분석할 수 있도록 해줍니다. 이 2차원 기술은 요소의 수직 응력과 전단 응력을 모두 시각화합니다.

데카르트 좌표 집합을 고려해보세요. 수평 및 수직 축은 각각 수직 응력(σ) 및 전단 응력(τ)에 해당합니다. 두 점 A와 B는 요소의 수직 응력과 전단 응력으로 정의됩니다. 점 A의 좌표는 요소의 전단응력과 수직응력을 기준으로 평면에 위치합니다. A점의 좌표는 (σ_x, -τ_xy)이고, B점의 좌표는 (σ_x, τ_xy)입니다. 모어의 원은 A와 B 사이에 선을 그려서 만들어집니다. 가로축과 교차하는 점 O가 모어의 원의 중심입니다. O는 A점과 B점의 정확히 중간에 있습니다.

원이 수평(수직 응력) 축과 교차하는 점 X와 Y는 최대 및 최소 주 응력을 나타냅니다. θ_p로 표시되는 이러한 기본 평면의 방향은 O에서 점 X(최대 주 응력)까지의 선과 점 A와 B를 연결하는 선 사이의 각도의 절반입니다. θ_p는 기본 평면과 원래 좌표 사이의 각도입니다. 체계. O부터 원의 가장 높은 지점까지의 반경은 최대 전단 응력을 나타냅니다.

모어의 원은 구조 설계 및 재료 파손 분석에 필수적인 주응력 및 전단 응력의 크기와 방향을 강조하여 재료의 거동에 대한 중요한 통찰력을 제공합니다.

Mohr의 원은 요소에 대한 응력 변환을 시각적으로 표현한 것입니다. 이 요소의 응력 요소는 그래프에 ...

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

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, τxy).

Mohr's circle is built by drawing a line between points A and B. The center of the circle is point O, and it lies on the horizontal axis. O is exactly halfway between A and B.

The points where the circle crosses the horizontal axis are X and Y. These points show the maximum and minimum principal stresses. The principal plane orientation, written as θp, is half the angle between the line from O to X and the line joining A and B. This angle tells you how the principal plane sits relative to the original coordinate system.

The radius from O to the highest point on the circle gives the maximum shearing stress. Mohr's circle helps show both the size and direction of principal stresses and shearing stresses. That information is important for structural design and for studying material failure.

Reading Stress Components with Mohr's Circle

reading-stress-components-with-mohrs-circle

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

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

Learn how Mohr's circle maps normal and shearing stress on a graph, showing principal stresses, plane angle, and maximum shear for plane stress.

Reading Stress Components with Mohr's Circle | JoVE K12

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mechanical engineering

Reading Stress Components with Mohr's Circle

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