Name: stress circle for principal and shear stress
Uploaded: 2024-05-23T12:59:54.000Z
Duration: 1 min 24 s
Channel: JoVE

The normal and shearing stress equations of the transformed plane, when graphed, form a circle that demonstrates their relationship for any given angular parameter.
The circle&#39;s center, relative to the vertical axis, represents the average normal stress, with its radius indicating the spread of these stress values.
The circle intersects the horizontal axis at two points, signifying the maximum and minimum normal stresses, occurring with zero shearing stress. These points define principal planes of stress where only normal stress, known as principal stress, exists.
The maximum and minimum normal stresses can be identified by adding or subtracting the average stress from the radius.
The principal plane that experiences maximum or minimum normal stress is identified by substituting the angular parameter into the normal stress equation.
The largest shearing stress is represented by points on the circle&#39;s vertical diameter, obtained when the normal stress equals the average stress, resulting in two 90&#176; orientations that predict maximum shearing stress.
The planes of maximum shearing stress and the principal planes are 45&#176; apart.

principal stresses

Principal Stresses

The graphical depiction of normal and shearing stress equations is represented by a circle, demonstrating the interplay between these stresses under different angular conditions. The center of this circle C, located on the vertical axis, represents the average normal stress, while its radius shows the range of stress variations. At points A and B, where the circle intersects the horizontal axis, the maximum and minimum normal stresses are observed, occurring without shearing stress. These pivotal points define the principal stress planes, where only normal stress, known as principal stress, exists.
To determine the values of the maximum and minimum normal stresses, one must adjust the mean stress by the circle&#39;s radius. Identifying the principal plane that carries the maximum or minimum normal stress requires inputting the angular parameter into the normal stress equation.
The points along the circle&#39;s vertical diameter indicate areas of maximum shearing stress, which arise when the normal stress equals the average stress. This condition leads to two orientations, each 90&#176; apart, identifying the peak shearing stress locations.
An important observation is the 45&#176; angular difference between the planes experiencing maximum shearing stress, and the principal stress planes. This geometrical relation highlights the essential connection between normal and shearing stresses, providing fundamental insight into stress distribution and interaction within materials under various loading scenarios.

The normal and shearing stress equations of the transformed plane, when graphed, form a circle that demonstrates their relationship for any given angular ...

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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 stress, acts.

The maximum and minimum normal stresses are found by adding and subtracting the circle radius from the mean stress. To locate the principal plane that carries either the maximum or minimum normal stress, the angular value is placed into the normal stress equation. The points on the vertical diameter show the maximum shearing stress, and at those locations the normal stress equals the average stress.

This condition gives two orientations that are 90° apart and identify the peak shearing stress locations. There is also a 45° angular difference between the principal stress planes and the planes of maximum shearing stress. This geometric link helps explain how normal and shearing stresses interact in a material under different loading conditions.

Stress Circle for Principal and Shear Stress

stress-circle-for-principal-and-shear-stress

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...

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 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

Explore the stress circle for principal and shear stress, including average normal stress, maximum and minimum normal stress, and the 45° link to peak shear.

Stress Circle for Principal and Shear Stress | JoVE K12

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Stress Circle for Principal and Shear Stress

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