Name: principal and shear strains on mohr’s circle
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 18 s
Channel: JoVE

Mohr&#39;s circle analyzes plane strain by plotting points with the abscissa equal to the normal strain and the ordinate equal to half the shearing strain.
The center O of Mohr&#39;s circle is defined, with the abscissa of O and the radius equal to the average strain and the radius equation, respectively.
The direction of rotation of the sides associated with the strain components indicates where the corresponding points on Mohr&#39;s circle are plotted.
The intersections of Mohr&#39;s circle with the horizontal axis correspond to the maximum and minimum principal strains, calculated as the sum and difference of the average strain and the radius, respectively.
During elastic deformation in homogeneous, isotropic materials, the principal strain axes coincide with the stress axes following the application of Hooke&#39;s law for shearing stress and strain.
The maximum in-plane shearing strain equals the diameter of Mohr&#39;s circle. The components of strain corresponding to a rotation of the coordinate axes through an angle &#952; are obtained by rotating the diameter of Mohr&#39;s circle through an angle 2&#952;.

mohr&#39;s circle for plane strain

Mohr&#39;s Circle for Plane Strain

Mohr&#39;s circle is a crucial graphical method used to analyze plane strain by plotting strain on a set of cartesian coordinates, where the abscissa is normal strain &#8712; and the ordinate is shear strain &#947;. Similarly to Mohr&#8217;s circle for plane stress, two points X and Y are plotted. Their coordinates are (&#8712;x, -&#947;XY) and (&#8712;Y, &#947;XY), respectively.
Mohr&#39;s circle visually represents the strain states under various conditions, which is essential for understanding material behavior. The center of Mohr&#39;s circle, labeled O, corresponds to the average normal strain, with the circle&#39;s radius derived from the relationship between normal and shearing strains. This helps to visualize how strains transform under different loading conditions by depicting the rotations and shifts in the circle as the coordinate axes rotate.
The points where Mohr&#39;s circle intersects the horizontal axis are particularly significant, representing the maximum and minimum principal strains. These principal strains are calculated from the average strain plus and minus the circle&#39;s radius, respectively, and indicate the strain limits a material can sustain under a given load. In homogeneous, isotropic materials undergoing elastic deformation, the principal strain axes align with the stress axes, a correlation established by Hooke&#39;s law for shearing stress and strain. This alignment aids in predicting material responses under stress.
In addition, the diameter of Mohr&#39;s circle represents the maximum in-plane shearing strain. For analysis involving rotated coordinate axes, rotating the diameter XY of Mohr&#8217;s circle through an angle 2&#952; effectively determines the strain components at that orientation for coordinate axes rotated through the angle &#952;.

Mohr&#39;s circle is a crucial graphical method used to analyze plane strain by plotting strain on a set of cartesian coordinates, where the abscissa is ...

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Mechanical Engineering

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Transformations of Stress and Strain

KEY TERMS AND CONCEPTS

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 summarizes the full strain state in one picture.

As the coordinate axes rotate, the circle shifts and rotates with them. This makes it easier to see how strain changes under different loading conditions. For a new axis orientation, rotating the diameter XY by 2θ gives the strain components for axes rotated by θ.

The points where the circle crosses the horizontal axis are important because they show the maximum and minimum principal strains. These principal strains equal the average strain plus or minus the circle’s radius. They mark the strain limits a material can sustain under a given load.

For homogeneous, isotropic materials in elastic deformation, the principal strain axes line up with the stress axes. This link follows from Hooke’s law for shear stress and shear strain. It helps predict how the material will respond under stress.

The diameter of Mohr’s circle also shows the maximum in-plane shearing strain. Together, the circle’s center, radius, and axis intersections give a clear way to read strain values. This graphical method is especially useful for understanding material behavior in plane strain.

Principal and Shear Strains on Mohr’s Circle

principal-and-shear-strains-on-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

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’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 shows normal strain, shear strain, principal strains, and maximum in-plane shear for plane strain in a clear graphical form.

Principal and Shear Strains on Mohr’s Circle | JoVE K12

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Principal and Shear Strains on Mohr’s Circle

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