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

Stellen Sie sich eine lange Stange vor, die an ihren Seiten gleichmäßig verteilten Lasten ausgesetzt ist.
In solchen Stäben beinhaltet die Transformation der ebenen Dehnung unter einer Drehung der Koordinatenachsen Zustände der ebenen Dehnung, bei denen Verformungen innerhalb paralleler Ebenen auftreten.
Ein ebener Dehnungszustand an Punkt O weist Dehnungskomponenten auf, die der x- und y-Achse zugeordnet sind. Die Dehnungskomponenten werden dann in Form des Winkels θ ausgedrückt.
Es wird ein Ausdruck für die Normaldehnung entlang einer Linie abgeleitet, die einen beliebigen Winkel mit der x-Achse bildet.
Dann wird die Normaldehnung in Richtung der Winkelhalbierenden des Winkels bestimmt, der durch die x- und y-Achse gebildet wird. Die Schubdehnung wird dann in Form dieser Normaldehnungen ausgedrückt.
Unter Berücksichtigung trigonometrischer Beziehungen werden die Gleichungen für die Transformation der ebenen Dehnung unter Achsendrehung abgeleitet, indem die Normaldehnung entlang der Halbierenden der x'- und y'-Achse berechnet und die Schubdehnung in Form der Normaldehnung ausgedrückt wird.

transformation der ebenen dehnung

Bei der Analyse langgestreckter Strukturen wie Stäbe, die gleichmäßig verteilten Lasten ausgesetzt sind, ist es wichtig, die Transformation der ebenen Dehnung beim Drehen der Koordinatenachsen zu verstehen. Diese Transformation hilft bei der Beurteilung, wie sich die Materialverformungseigenschaften mit der Ausrichtung ändern, was in der Materialwissenschaft und im Bauingenieurwesen von entscheidender Bedeutung ist.

Unter ebenen Dehnungsbedingungen, die typisch für Bauteile sind, bei denen eine Abmessung die anderen deutlich übersteigt, sind Verformungen und resultierende Dehnungen nur in einer einzigen Ebene spürbar. An jedem Punkt, sagen wir O, sind die wichtigen Dehnungskomponenten diejenigen entlang der x- und y-Achse, und die Dehnungskomponenten entlang der z-Achse sind vernachlässigbar. Diese Komponenten beschreiben umfassend den Verformungszustand innerhalb der xy-Ebene.

Ein wichtiger Fall ist die Berechnung der Normaldehnung entlang der Winkelhalbierenden zwischen der x- und der y-Achse, die 45 Grad beträgt. In diesem Fall vereinfacht die Berücksichtigung dieses Winkels den Ausdruck und spiegelt wider, wie die Kombination aus Normaldehnungen und Scherdehnung die Verformung entlang dieser Linie beeinflusst.

<img alt="Equation 1" src="/files/ftp_upload/15658/15658_Equation_1.svg" style="text-align: center; height: 32px;">

Wenn das Koordinatensystem gedreht wird, um es an der Winkelhalbierenden der x- und y-Achse auszurichten, können die Dehnungskomponenten in den neuen Koordinaten mithilfe trigonometrischer Beziehungen aus der Elastizitätstheorie neu berechnet werden. Insbesondere können die Normaldehnungen in den Richtungen, die einen beliebigen Winkel mit den ursprünglichen Achsen bilden, und die entsprechenden Scherdehnungen anhand der ursprünglichen Dehnungskomponenten bestimmt werden.

<img alt="Equation 2" src="/files/ftp_upload/15658/15658_Equation_2.svg" style="text-align: center; height: 188px;">

Transformation der ebenen Dehnung

Stellen Sie sich eine lange Stange vor, die an ihren Seiten gleichmäßig verteilten Lasten ausgesetzt ist.
In solchen Stäben ist die Transformation der ebenen Dehnung unter einer Drehung der Koordinate ...

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Transformations of Stress and Strain

Spannungs- und Dehnungstransformationen

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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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