Name: plane stress changes with rotation
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 18 s
Channel: JoVE

Consider a point on a cube experiencing plane stress, defined by its stress components.
If the element rotates by an angle, its stress components change. This change is determined by considering a prismatic element with faces perpendicular to its axes.
By considering the area of the oblique face, the areas of the vertical and horizontal faces can be calculated using trigonometric functions of the rotation angle.
The forces exerted on these faces are represented, and it is assumed that no forces act on the triangular faces of the element. The equilibrium equations are then written using components along the rotated axes.
Upon solving these equations, the normal and shearing stresses are derived. The normal and shearing stresses are then re-expressed using trigonometric relations.
An expression for the normal stress on the rotated vertical axis is obtained by substituting the rotation angle in one of the previous expressions with a new angle.
It is observed that the sum of the normal stresses exerted on a cubic element remains independent of the element&#39;s orientation.

transformation of plane stress

Studying stress transformation is essential in understanding how stress components within a material, like a cube under plane stress, change with rotation. This change is analyzed by considering a prismatic element within the cube. As the element rotates, the stress components acting on it—both normal and shearing stresses—change in magnitude and orientation. This change is quantified using trigonometric functions of the rotation angle, relating the forces acting on the rotated element's faces to those on its original perpendicular faces.
The equilibrium equations, formulated by considering only the forces on faces perpendicular to the principal axes (excluding any forces on the triangular faces due to rotation), enable the derivation of new stress components. The normal and shearing stresses are re-expressed in terms of the original stresses.
<img alt="Stress transformation equations, symbols for plane stress analysis, showing σx', τx'y' calculations." src="/files/ftp_upload/15652/15652_Equation_1.svg" style="text-align: center; height: 96px;" data-altupdatedat="1754904545000" data-alt="Stress transformation equations; mechanical engineering; formula; stress analysis; trigonometric identity.">
A new expression for normal stress on the rotated vertical axis is obtained by replacing the rotation angle in a previous expression with a new one.
<img alt="Stress transformation equations, σx' and σy', static equilibrium, mechanical analysis." src="/files/ftp_upload/15652/15652_Equation_2.svg" style="text-align: center; height: 122px;" data-altupdatedat="1754904545000" data-alt="Stress transformation equations, static equilibrium, formula, mechanics analysis.">
A notable outcome of this analysis is that the sum of normal stresses does not change regardless of the orientation of the cubic element. This invariance highlights the material's isotropic response to external stresses and is crucial for predicting material behavior under different loading conditions. Understanding how stress components transform with element orientation is vital for predicting material failure modes and designing materials and structures that are more resilient to applied loads.

Transformation of Plane Stress

Studying stress transformation is essential in understanding how stress components within a material, like a cube under plane stress, change with ...

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JoVE Core

Mechanical Engineering

Unit 1

Transformations of Stress and Strain

KEY TERMS AND CONCEPTS

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 on those perpendicular faces, not the forces on the triangular faces created by the rotation.

From this balance, the new stress components can be written in terms of the original stresses. The normal stress on the rotated vertical axis is also found by using a new angle in a previous expression. One important result is that the sum of the normal stresses stays the same, no matter how the cubic element is oriented.

This invariant sum shows the material’s isotropic response to external stress. It helps explain how stress components shift as orientation changes. That understanding is important for predicting failure modes and for designing materials and structures that can better handle applied loads.

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 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

See how normal and shearing stress change as a plane-stressed element rotates, and why the sum of normal stresses stays constant for any orientation.

Plane Stress Changes with Rotation | JoVE K12

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mechanical engineering

Plane Stress Changes with Rotation

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