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

Рассмотрим длинный стержень, подвергающийся равномерно распределенным нагрузкам по бокам.
В таких стержнях преобразование плоской деформации при вращении осей координат включает в себя состояния плоской деформации, при которых деформации происходят в параллельных плоскостях.
Плоское состояние деформации в точке O имеет компоненты деформации, связанные с осями x и y. Затем компоненты деформации выражаются в терминах угла θ.
Выражение получается для нормальной деформации вдоль линии, образующей произвольный угол с осью x.
Затем определяется нормальная деформация в направлении биссектрисы угла, образованного осями x и y. Деформация сдвига затем выражается в терминах этих нормальных деформаций.
Рассматривая тригонометрические соотношения, уравнения для преобразования плоской деформации при вращении оси получаются путем вычисления нормальной деформации вдоль биссектрисы осей x' и y' и выражения деформации сдвига в терминах нормальной деформации.

Преобразование плоской деформации

При анализе удлиненных конструкций, таких как стержни, подвергающиеся равномерно распределенным нагрузкам, важно понимать трансформацию плоской деформации при вращении осей координат. Это преобразование помогает оценить, как характеристики деформации материала изменяются в зависимости от ориентации, что имеет решающее значение в материаловедении и проектировании конструкций. 

В условиях плоской деформации, характерной для элементов, у которых один размер значительно превышает другие, деформации и результирующие напряжения заметны только в одной плоскости. В любой точке, скажем, O, важными компонентами деформации являются компоненты деформации вдоль осей x и y, а компоненты деформации вдоль оси-z незначительны. Эти компоненты всесторонне описывают состояние деформации в плоскости-xy. 

Значимым случаем является расчет нормальной деформации по биссектрисам углов между осями x и y, который составляет 45 градусов. В этом случае учет этого угла упрощает выражение, отражая, как комбинация нормальных деформаций и сдвиговых деформаций влияет на деформацию вдоль этой линии. 

<img alt="Equation 1" src="/files/ftp_upload/15658/15658_Equation_1.svg" style="text-align: center; height: 32px;"> 

Когда система координат поворачивается для совмещения с биссектрисой осей x и y, компоненты деформации в новых координатах могут быть пересчитаны с использованием тригонометрических соотношений, полученных из теории упругости. В частности, нормальные деформации в направлениях, образующих произвольный угол с исходными осями, и соответствующие деформации сдвига могут быть определены через компоненты исходной деформации. 

<img alt="Equation 2" src="/files/ftp_upload/15658/15658_Equation_2.svg" style="text-align: center; height: 188px;">

Рассмотрим длинный стержень, подвергающийся равномерно распределенным нагрузкам по бокам.
В таких стержнях происходит преобразование плоской деформации при вращении координаты...

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

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

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

Plane Strain Under Rotated Axes

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