Name: finding principal planes and stress limits
Uploaded: 2024-05-23T12:59:54.000Z
Duration: 1 min 15 s
Channel: JoVE

Две плоскости, расположенные под прямым углом, несут напряжение сдвига наряду с растягивающими и сжимающими напряжениями. С известными значениями напряжений определите главные плоскости, максимальное напряжение сдвига и главные напряжения.
Главные плоскости определяются с использованием соотношения, в котором главная плоскость равна отношению удвоенного напряжения сдвига к разности между растягивающими и сжимающими напряжениями.
Подставьте известные значения напряжения сдвига, нормального напряжения и сжимающего напряжения в уравнение, чтобы рассчитать значения главной плоскости.
Среднее нормальное напряжение затем определяется путем вычисления среднего значения нормальных напряжений при растяжении и сжатии.
Затем рассчитайте максимальное напряжение сдвига, подставив в уравнение значения нормальных и сдвиговых напряжений.
Рассчитайте основное главное напряжение, сложив среднее значение нормального напряжения с максимальным напряжением сдвига.
Разница между средним значением нормального напряжения и максимальным напряжением сдвига является минимальным главным напряжением.

Главные напряжения: решение проблем

Principal Stresses: Problem Solving

При анализе двух плоскостей, пересекающихся под прямым углом под действием сдвиговых, растягивающих и сжимающих напряжений, важно определить главные плоскости, максимальное касательное напряжение и главные напряжения. Чтобы найти главные плоскости, примените формулу, которая приравнивает их к удвоенному напряжению сдвига, деленному на разницу между напряжениями растяжения и сжатия. 

<img alt="Equation 1" src="/files/ftp_upload/15654/15654_Equation_1.svg" style="text-align: center; height: 42px;">

Подставив в эту формулу заданные значения напряжения сдвига, растяжения и сжатия, можно рассчитать ориентации главных плоскостей. Затем рассмотрим среднее значение растягивающего и сжимающего напряжений, чтобы определить среднее нормальное напряжение. Этот шаг важен для понимания общего распределения напряжений по материалу. Максимальное напряжение сдвига рассчитывается с использованием полученных значений нормального напряжения и напряжения сдвига. 

<img alt="Equation 2" src="/files/ftp_upload/15654/15654_Equation_2.svg" style="text-align: center; height: 52px;">

Этот расчет подчеркивает пиковое напряжение сдвига, которое испытывает материал, что крайне важно для оценки риска его разрушения. Главное напряжение, которое указывает максимальное напряжение, которое материал может выдержать без деформации, рассчитывается путем прибавления максимального напряжения сдвига к среднему нормальному напряжению. Аналогичным образом минимальное главное напряжение находится путем вычитания максимального напряжения сдвига из среднего нормального напряжения. Эти расчеты имеют фундаментальное значение для инженерных приложений, позволяя получить представление о поведении материалов в условиях сложных напряжений.

Две плоскости, расположенные под прямым углом, несут напряжение сдвига наряду с растягивающими и сжимающими напряжениями. С известными значениями напряжений определите главные...

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

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 values, the orientations of the principal planes can be calculated. This step identifies the directions where the normal stress is principal.

Next, find the average normal stress by taking the average of the tensile and compressive stresses. This value shows the overall normal stress level across the material. It helps describe the stress distribution before the shear and principal values are finalized.

The maximum shearing stress is then calculated from the derived normal and shearing stress values. This result shows the peak shear the material experiences, which is important for judging failure risk. From there, the major principal stress is found by adding the maximum shearing stress to the average normal stress. The minimum principal stress is found by subtracting the maximum shearing stress from the average normal stress. These calculations are useful in engineering because they show how a material behaves under combined loading.

Finding Principal Planes and Stress Limits

finding-principal-planes-and-stress-limits

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

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

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

Calculate principal planes, average normal stress, maximum shearing stress, and principal stresses for two perpendicular planes under combined loading.

Finding Principal Planes and Stress Limits | 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

Finding Principal Planes and Stress Limits

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