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

考虑一个侧面承受均布载荷的长杆。
在这种情况下，坐标轴旋转时平面应变的转换涉及变形发生在平行平面内的平面应变状态。
点 O 处的平面应变状态具有与 x 轴和 y 轴相关的应变分量。这些应变分量随后用角度 θ 表示。.
推导出沿与x轴成任意角度的直线方向的正应变表达式。
然后，确定沿 x 轴和 y 轴所形成夹角的角平分线方向的正应变。剪应变则通过这些正应变来表示。
根据三角关系，通过计算沿 x' 和 y' 轴角平分线方向的正应变，并将剪应变用正应变表示，可推导出轴旋转条件下平面应变转换的方程。

平面应变的变换

Transformation of Plane Strain

<div>
 在分析承受均匀分布载荷作用的杆等细长结构时，必须了解在坐标轴旋转时其平面应变所发生的转变。这种转变有助于评估材料变形的特性是如何随方向发生变化的，这在材料科学和结构工程中是至关重要的。
 
 在平面应变的条件下，对于一个尺寸明显超过其他尺寸的构件来说，这是特殊的，其中的变形和由此产生的应变仅在单个平面内是显著的。在任意一点处（例如 O 点），重要的应变分量是沿 x 轴和 y 轴的应变分量，而沿 z 轴的应变分量则可以忽略不计。这些分量综合描述了在 xy 平面内的变形状态。
 
 一个重要的例子是沿 x 轴和 y 轴之间的角平分线（45 度）来计算其中的法向应变。在这种情况下，这个角度可以简化其表达式，以此来反映法向应变和剪切应变的组合是如何对沿着这条线的变形产生影响。
 
 <img alt="Equation 1" src="/files/ftp_upload/15658/15658_Equation_1.svg" style="text-align: center; height: 32px;">
 当坐标系旋转到与 x 轴和 y 轴对齐的角平分线时，可以使用从弹性理论推导出的三角关系来对新坐标中的应变分量进行重新计算。具体来说，可以根据原始应变分量来确定与原始轴线在任意角度方向上形成的法向应变和相应的剪切应变。
 
 <img alt="Equation 2" src="/files/ftp_upload/15658/15658_Equation_2.svg" style="text-align: center; height: 188px;">
</div>

考虑一根侧边承受均布载荷的长杆。
在此类杆件中，平面应变在坐标系旋转下的变换……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Transformations of Stress and Strain

应力与应变的变换

KEY TERMS AND CONCEPTS

关键术语与概念

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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