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

Zwei rechtwinklig angeordnete Ebenen tragen neben Zug- und Druckspannungen auch die Schubspannung. Bestimmen Sie bei bekannten Spannungswerten die Hauptebenen, die maximale Schubspannung und die Hauptspannungen.
Die Hauptebenen werden anhand des Verhältnisses bestimmt, bei dem die Hauptebene dem Verhältnis zwischen der doppelten Schubspannung und der Differenz zwischen Zug- und Druckspannungen entspricht.
Setzen Sie die bekannten Werte der Schubspannung, der Normalspannung und der Druckspannung in die Gleichung ein, um die Werte der Hauptebene zu berechnen.
Die mittlere Normalspannung wird dann durch Berechnung des Mittelwerts der Zug- und Drucknormalspannungen ermittelt.
Berechnen Sie als Nächstes die maximale Schubspannung, indem Sie die Werte der Normal- und Schubspannungen in die Gleichung einfügen.
Berechnen Sie die Haupthauptspannung, indem Sie den durchschnittlichen Normalspannungswert zur maximalen Schubspannung addieren.
Die Differenz zwischen dem durchschnittlichen Normalspannungswert und der maximalen Schubspannung ist die minimale Hauptspannung.

hauptbelastungen: problemlösung

Bei der Analyse zweier Ebenen, die sich unter dem Einfluss von Scher-, Zug- und Druckspannungen im rechten Winkel schneiden, ist es wichtig, die Hauptebenen, die maximale Scherspannung und die Hauptspannungen zu identifizieren. Um die Hauptebenen zu ermitteln, wenden Sie eine Formel an, die sie dem Doppelten der Scherspannung geteilt durch die Differenz zwischen Zug- und Druckspannung gleichsetzt.

<img alt="Equation 1" src="/files/ftp_upload/15654/15654_Equation_1.svg" style="text-align: center; height: 42px;">

Durch Einsetzen der angegebenen Scher, Zug, und Druckspannungswerte in diese Formel kann man die Orientierungen der Hauptebenen berechnen. Betrachten Sie dann den Durchschnitt der Zug- und Druckspannungen, um die durchschnittliche Normalspannung zu bestimmen. Dieser Schritt ist wichtig für das Verständnis der gesamten Spannungsverteilung im Material. Die maximale Scherspannung wird anhand der abgeleiteten Normal- und Scherspannungswerte berechnet.

<img alt="Equation 2" src="/files/ftp_upload/15654/15654_Equation_2.svg" style="text-align: center; height: 52px;">

Diese Berechnung verdeutlicht die maximale Scherbeanspruchung des Materials, die für die Beurteilung des Ausfallrisikos von entscheidender Bedeutung ist. Die größte Hauptspannung, die die maximale Spannung angibt, der das Material standhalten kann, ohne nachzugeben, wird berechnet, indem die maximale Scherspannung zur durchschnittlichen Normalspannung addiert wird. In ähnlicher Weise wird die minimale Hauptspannung durch Subtrahieren der maximalen Scherspannung von der durchschnittlichen Normalspannung ermittelt. Diese Berechnungen sind für technische Anwendungen von grundlegender Bedeutung und liefern Einblicke in das Materialverhalten unter komplexen Belastungsbedingungen.

Hauptbelastungen: Problemlösung

Zwei rechtwinklig angeordnete Ebenen tragen neben Zug- und Druckspannungen auch die Schubspannung. Ermitteln Sie bei bekannten Spannungswerten die Haupt- und ...

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

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 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

Description

Transcript

Tags