Name: principal strain directions in 3d
Uploaded: 2024-05-23T12:59:55.000Z
Duration: 1 min 29 s
Channel: JoVE

3차원 변형률 해석은 주 응력 축을 사용하여 탄성이 있고 균일한 재료의 변형을 평가합니다.
작은 입방체 요소는 이러한 축 주위로 확장될 때 직사각형 평행 육면체로 변형되어 변형을 보여줍니다.
변형률 해석에는 n축과 같은 주축을 중심으로 요소를 회전하고 이 축에 수직인 면의 변형률 성분을 평가하는 작업이 포함됩니다. 이 방법은 평면 변형 변환을 기반으로 합니다.
Mohr의 원을 사용하여 큐브가 주축을 중심으로 회전할 때 변형률 변환을 검사합니다. 이 접근법은 특정 지점에서 최대 전단 변형률을 식별하며, 이는 세 원 중 가장 큰 원의 지름과 동일합니다.
평면 변형에서 n축은 Mohr의 원 다이어그램의 원점에서 변형률이 0인 주축으로 변환됩니다. 이 원점의 반대쪽에 있는 주 변형률은 최대 및 최소 수직 변형률을 나타냅니다.
평면 응력을 받는 구조의 얇은 플레이트에서 n축은 주 응력 축으로 바뀝니다.
n축을 따른 주 변형률은 평면 내 변형률과 연관되어 있습니다. m축을 기준으로 한 회전은 최대 전단 변형률을 정의합니다.

변형률의 3차원 분석

Three-Dimensional Analysis of Strain

3차원 변형률 분석은 특히 탄성이 있는 균질한 재료에서 응력이 가해지면 재료가 어떻게 변형되는지 이해하는 데 중요합니다. 이 방법은 주요 응력 축을 사용하여 복잡한 응력 상태를 보다 이해하기 쉬운 형태로 단순화합니다. 응력을 받으면 재료 내의 작은 입방체 요소가 이 축을 따라 팽창하거나 수축하여 직육면체로 변형됩니다. 이 변환은 재료의 변형을 효과적으로 보여줍니다. 주 응력 축은 직교하며 응력이 재료 내에서 전단을 유발하지 않는 방향을 나타냅니다.

모어의 원은 변형률 분석에 필수적인 도구입니다. 이는 한 지점의 응력 상태에 대한 그래픽 표현을 제공하고 요소가 n축과 같은 주축을 중심으로 회전할 때 변형 구성 요소를 평가합니다. 이 분석은 n축 원점의 변형률이 0인 평면 변형률 변환에 중점을 두어 모어의 원의 반대편에 표시된 최대 및 최소 변형률 결정을 단순화합니다.

평면 응력을 받는 얇은 판과 같은 시나리오에서는 n축이 주요 응력 축이 됩니다. 여기서 n축을 따른 주요 변형률은 재료의 평면 내 변형률과 직접적인 상관관계가 있습니다. m축과 같은 다른 주축을 중심으로 한 회전은 최대 전단 변형률의 위치와 크기를 정확히 찾아내는 데 도움이 되며, 이는 하중이 가해진 상태에서 재료 거동을 예측하고 구조 설계의 안전성과 신뢰성을 보장하는 데 중요합니다.

3차원 변형률 해석은 주 응력 축을 사용하여 탄성이 있고 균일한 재료의 변형을 평가합니다.
작은 입방 요소, 때 ...

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

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 material. This simplifies a complex stress state into a form that is easier to analyze and compare.

Mohr's circle is a key graph used in strain analysis. It shows the stress state at a point and helps evaluate strain components as the element rotates about a principal axis, such as the n-axis. In plane strain transformations, the strain at the origin of the n-axis is zero. This makes it easier to find the maximum and minimum strains, which appear on opposite sides of Mohr's circle.

For thin plates under plane stress, the n-axis becomes a principal stress axis. In that case, the principal strain along the n-axis is directly related to the in-plane strains of the material. Rotation about another principal axis, such as the m-axis, helps locate the maximum shearing strain and its magnitude. These results are important for predicting how materials behave under load and for supporting safe, reliable structural design.

Principal Strain Directions in 3D

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 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

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 principal stress axes, Mohr's circle, and plane stress help show 3D strain, including maximum, minimum, and shearing strain in materials.

Principal Strain Directions in 3D | 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

Principal Strain Directions in 3D

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