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

Круг Мора — это визуальное представление преобразования напряжения на элементе. Компоненты напряжения в этом элементе наносятся на график для создания круга Мора для квадратного элемента, испытывающего плоское напряжение.
Если напряжение сдвига положительное, точка A наносится ниже горизонтальной оси, а точка B — выше. Если она отрицательная, их позиции меняются местами.
Средняя точка О линии, соединяющей точки А и В, лежит на горизонтальной оси и служит центром окружности. Круг, нарисованный с буквой O в центре и линией AB в качестве диаметра, называется кругом Мора.
Абсциссы точек X и Y, где окружность пересекает горизонтальную ось, представляют собой соответственно максимальное и минимальное главные напряжения.
Угол AOX в два раза больше угла θp. Ориентация θp главной плоскости, соответствующей точке X, может быть получена путем уменьшения вдвое угла AOX, измеренного на окружности Мора.
Радиус окружности Мора в вертикальном направлении соответствует величине максимального напряжения сдвига.

Круг Мора для определения плоского напряжения

Круг Мора — это графический метод определения состояния напряжения в определенной точке материала, упрощающий анализ трансформации напряжений в условиях плоского напряжения. Этот двумерный метод визуализирует как нормальные, так и касательные напряжения в элементе.

Рассмотрим набор декартовых координат. Горизонтальная и вертикальная оси соответствуют нормальному напряжению (σ) и касательному напряжению (τ) соответственно. Две точки, точки A и B, определяются нормальными и касательными напряжениями на элементе. Координаты точки А расположены на плоскости с учетом сдвиговых и нормальных напряжений на элементе. Координаты точки А — (σ_x, -τ_xy), а координаты точки B — (σ_x, τ_xy). Круг Мора создается путем рисования линии между A и B. Точка O, пересекающая горизонтальную ось, является центром круга Мора. О находится ровно посередине между точками А и В.

Точки X и Y, в которых окружность пересекает горизонтальную ось (нормальное напряжение), обозначают максимальное и минимальное главные напряжения. Ориентация этих главных плоскостей, обозначаемая θ_p, равна половине угла между линией, ведущей от точки O до точки X (максимальное главное напряжение), и линией, соединяющей точки A и B. θ_p— это угол между главной плоскостью и исходной системой координат. Радиус от О до самой высокой точки круга представляет собой максимальное касательное напряжение.

Круг Мора позволяет понять информацию о поведении материалов, подчеркивая величины и направления главных и касательных напряжений, которые необходимы для проектирования конструкций и анализа разрушения материалов.

Круг Мора — это визуальное представление преобразования напряжения на элементе. Компоненты напряжения в этом элементе нанесены на график для ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Transformations of Stress and Strain

Преобразования напряжений и деформаций

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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