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

長いバーが側面に均一に分散された荷重を受けるとします。
このようなバーでは、座標軸の回転下での平面ひずみの変換には、平行平面内で変形が発生する平面ひずみの状態が含まれます。
点Oの平面ひずみ状態には、x軸とy軸に関連付けられたひずみ成分があります。次に、ひずみ成分は角度θで表されます。
式は、x軸に対して任意の角度を形成する線に沿った垂直ひずみに対して導出されます。
次に、x軸とy軸によって形成される角度の二等分線の方向の垂直ひずみが決定されます。次に、せん断ひずみは、これらの垂直ひずみで表されます。
三角関数の関係を考慮すると、軸回転下での平面ひずみ変換の方程式は、x'軸とy'軸の二等分線に沿った垂直ひずみを計算し、せん断ひずみを垂直ひずみで表すことによって導き出されます。

平面ひずみの変換

均一に分布した荷重を受ける棒のような細長い構造物を解析する場合、座標軸を回転させたときの平面ひずみの変換を理解することが不可欠です。この変換は、材料の変形特性が方向によってどのように変化するかを評価するのに役立ちます。これは、材料科学および構造工学において重要です。

平面ひずみ条件下では、一方の寸法が他方の寸法を大幅に超える部材に特有であり、変形とその結果生じるひずみは単一平面内でのみ顕著になります。たとえば、任意の点Oにおいて、重要なひずみ成分は x 軸と y 軸に沿ったものであり、z 軸に沿ったひずみ成分は無視できます。これらの成分は、xy 平面内の変形状態を包括的に記述します。

重要なケースは、x 軸と y 軸の間の角度の二等分線 (45 度) に沿った垂直ひずみの計算です。この場合、この角度を考慮すると式が簡略化され、垂直ひずみとせん断ひずみの組み合わせがこの線に沿った変形にどのような影響を与えるかを反映します。

<img alt="Equation 1" src="/files/ftp_upload/15658/15658_Equation_1.svg" style="height: 32px;">

座標系を回転して x 軸と y 軸の二等分線に一致させると、弾性理論から導かれた三角関係を使用して、新しい座標のひずみ成分を再計算できます。具体的には、元のひずみ成分から、元の軸と任意の角度をなす方向の法線ひずみとそれに対応するせん断ひずみを求めることができます。

<img alt="Equation 2" src="/files/ftp_upload/15658/15658_Equation_2.svg" style="height: 188px;">

長いバーが側面に均一に分散された荷重を受けるとします。
このようなバーでは、座標の回転下での平面ひずみの変換...

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 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

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mechanical engineering

Plane Strain Under Rotated Axes

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