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

モールの円は、要素の応力変換を視覚的に表現したものです。この要素の応力成分は、平面応力を受ける正方形の要素のモール円を作成するためにグラフにプロットされます。
せん断応力が正の場合、点 A は横軸の下にプロットされ、点 B は上にプロットされます。負の場合、それらの位置は逆になります。
点 A と点 B を結ぶ線の中点 O は、水平軸上にあり、円の中心として機能します。Oを中心とし、線ABを直径として描かれた円は、モールの円と呼ばれます。
点 X と Y の横軸は、円が水平軸と交差し、それぞれ最大主応力と最小主応力を表します。
角度AOXは角度θpの2倍に等しくなります。点Xに対応する主平面の向きθpは、モールの円で測定された角度AOXを半分にすることで取得できます。
モールの円の垂直方向の半径は、最大せん断応力の大きさに対応します。

平面応力のモール円

モール円は、材料内の点における応力状態を識別するためのグラフィカルな方法であり、平面応力条件下での応力変形の解析を容易にします。 この 2 次元技術は、要素上の垂直応力とせん断応力の両方を視覚化します。

一連の直交座標系について考えてみましょう。 横軸は垂直応力 (σ)、縦軸はせん断応力 (τ) に対応します。 2 つの点、点 A と点 B は、要素の垂直応力とせん断応力によって定義されます。 点 A の座標は、要素のせん断応力と垂直応力に基づいて平面上に配置されます。 点 A の座標は (σ_x, -τ_xy)、点 B の座標は (σ_x, τ_xy)です。 モール円は、A と B の間に線を引くことによって作成されます。横軸と交差する点 O がモール円の中心です。 O は点 A と点 B のちょうど中間です。

円が水平 (法線応力) 軸と交差する点 X および Y は、最大主応力と最小主応力を示します。 これらの主面の向きは θ_p で示され、O から点 X (最大主応力) までの線と点 A と B を結ぶ線との間の角度の半分です。 θ_p は主面と元の座標系の間の角度です。 O から円の最高点までの半径は、最大せん断応力を表します。

モール円は、材料の挙動に関する重要な洞察を提供し、構造設計と材料破損の解析に不可欠な主応力とせん断応力の大きさと方向を強調します。

モールの円は、要素の応力変換を視覚的に表現したものです。この要素の応力成分は、グラフ上にプロットされて...

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

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

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

Reading Stress Components with Mohr's Circle

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