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

Il cerchio di Mohr è una rappresentazione visiva della trasformazione dello stress su un elemento. I componenti di sollecitazione in questo elemento vengono tracciati su un grafico per creare il cerchio di Mohr per un elemento quadrato sottoposto a sollecitazione piana.
Se la sollecitazione di taglio è positiva, il punto A viene tracciato sotto l'asse orizzontale e il punto B viene tracciato sopra. Se è negativo, le loro posizioni vengono invertite.
Il punto medio, O, della linea che collega i punti A e B si trova sull'asse orizzontale e funge da centro del cerchio. Un cerchio, disegnato con O come centro e la linea AB come diametro, è chiamato cerchio di Mohr.
Le ascisse dei punti X e Y, dove il cerchio interseca l'asse orizzontale, rappresentano rispettivamente le sollecitazioni principali massime e minime.
L'angolo AOX è uguale al doppio dell'angolo θp. L'orientamento, θp, del piano principale corrispondente al punto X, può essere ottenuto dimezzando l'angolo AOX misurato sul cerchio di Mohr.
Il raggio del cerchio di Mohr in direzione verticale corrisponde all'entità della massima sollecitazione di taglio.

cerchio di mohr per lo stress piano

Il cerchio di Mohr è un metodo grafico per identificare lo stato di stress in un punto di un materiale, rendendo più semplice l'analisi delle trasformazioni di stress in condizioni di stress piana. Questa tecnica bi-dimensionale visualizza sia lo stress normale che quello di taglio su un elemento.

Consideriamo un insieme di coordinate cartesiane. Gli assi orizzontale e verticale corrispondono rispettivamente allo stress normale (σ) e allo stress di taglio (τ) Due punti, A e B, sono definiti dallo stress normale e di taglio sull'elemento. Le coordinate del punto A si trovano sul piano basato sullo stress di taglio e normale sull'elemento. Le coordinate del punto A sono (σ_x, -τ_xy), e le coordinate del punto B sono (σ_x, τ_xy). Il cerchio di Mohr viene creato tracciando una linea tra A e B. Il punto O, che attraversa l'asse orizzontale, è il centro del cerchio di Mohr. O è esattamente a metà strada tra i punti A e B.

I punti X e Y, dove il cerchio interseca l'asse orizzontale (tensione normale), indicano le tensioni principali massime e minime. L'orientamento di questi piani principali, indicato con θ_p, è la metà dell'angolo tra una linea da O al punto X (il massimo stress principale) e la linea che collega i punti A e B. θ_p è l'angolo tra il piano principale e la coordinata originale sistema. Il raggio da O al punto più alto del cerchio rappresenta il massimo stress di taglio.

Il circolo di Mohr offre spunti vitali sul comportamento dei materiali, evidenziando l'entità e l'orientamento dello stress principale e di taglio, che è essenziale per la progettazione strutturale e l'analisi del cedimento dei materiali.

Cerchio di Mohr per lo stress piano

Il cerchio di Mohr è una rappresentazione visiva della trasformazione dello stress su un elemento. Le componenti di sollecitazione in questo elemento sono tracciate su un grafico per ...

Education

Didattica

JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Transformations of Stress and Strain

Trasformazioni di tensione e deformazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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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