Mohr's circle for plane stress shows the relationship between the state of stress, principal stresses, and maximum shear stress. This article will cover the basics of Mohr's circle in plane stress, and how it is easily created using MEboost.
MEboost also handles bodies under plane strain. To learn about this scenario, refer to the article on plane strain.
Let's start with some definitions.
Principal Planes In an element under plane stress, there are two planes where shear stress is zero. These are referred to as the principal planes.
Principal Stress A tensile or compressive stress that acts perpendicular to a principal plane.
Consider the element below that is under plane stress.
We need to define a convention for shear. Shear yx that results in clockwise rotation is positive, and shear yx that results in counterclockwise rotation is negative. In the element above, we see that shear yx is positive. Also note that shear yx = -shear xy. When Mohr's circle is plotted, we will see how this affects the principal plane angles.
To illustrate, let's consider an example. Stress values are:
- Stress in x direction: 5,000 psi
- Stress in y direction: -4,000 psi
- Shear: 3,000 psi
Positive stress is tensile, and negative stress is compressive.
To create Mohr's circle, we plot two points: (stress x, -shear yx) and (stress y, shear yx). Shear is the vertical axis and stress is the horizontal axis. A line is drawn between these points, and a circle is drawn that goes through both points. A Mohr's circle is shown below with labels to illustrate the properties.
The principal stresses are ordered according to σ1 > σ2 > σ3. For plane stress, one of the principal stresses will be zero.
In the previous example, principal stress 1 was positive and principal stress 3 was negative. When the maximum principal stress is positive and the minimum principal stress is negative, the in-plane shear stress is the greatest shear stress so only one circle is drawn.
When the in-plane principal stresses are both positive, or both negative the out of plane shear stress is the maximum. To show this, three circles are drawn to show in-plane and out of plane shear stresses. Out of plane shear stress is either shear xz or shear yz.
Let's do another example where both in-plane principal stresses are positive. Stress values are:
- Stress in x direction: 10,000 psi
- Stress in y direction: 5,000 psi
- Shear: 2,000 psi
The inner blue circle intersects principal stress 1 and principal stress 2 on the x-axis. The inner green circle intersects principal stress 2 and principal stress 3 on the x-axis. Principal stress 3 is zero. The outer circle intersects principal stress 1 and principal stress 3 on the x-axis.
In this situation, and when in-plane principal stresses are < 0, the out of plane shear stress is greatest.
Max. in-plane shear stress: 3201.56 psi
Max. out of plane shear stress: 5350.78
Properties of Mohr's Circle for Plane Stress
Mohr space refers to properties of Mohr's circle. Real space refers to properties of the actual element under stress.
- The principal stresses, σ1, σ2, and σ3 are located on the horizontal axis where shear is zero.
- Mohr space: the principal planes are 180 degrees apart. Real space: the principal planes are 90 degrees apart.
- Mohr space: the principal planes are located at an angle of 2Θ from the red line. Real space: the principal planes are at an angle of θ from the x-direction.
- Maximum shear is at the top and bottom of the circle.
- Mohr space: maximum shear occurs 90 degrees from the horizontal axis. Real space: maximum shear occurs at 45 degrees from the principal planes.
- The radius of the circle is maximum shear. Alternatively, it is 0.5(σ1 - σ2) , 0.5(σ2 - σ3) or 0.5(σ1 - σ3) depending on the circle.
The principal planes of the element are shown below.
Drawing the Circle with MEboost
In the MEboost ribbon, clicking on the Mohr's Circle button will show the Mohr's Circle form. Using the example above, we enter stress values in the form. The value and sign for shear stress is for shear yx.
After clicking the Create button, a report will be created in a new worksheet.
In addition to the chart, the report also shows the values of principal stresses, maximum in-plane shear stress, maximum out of plane shear stress, and the principal plane angle.
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