Stress Transformation - Mechanics of Materials

Transforming stress components from one set coordinate system to a coordinate system with a different orientation.



The general state of stress for an element (3-Dimensional) will have 6 stress components. σx, σy, σz, xy, xz, and yz

Plane stress (2-Dimensional) will have 2 normal stress components and 1 shear stress component: σx, σy, and xy

Determining positive and negative stresses: the normal stress(σ) will be positive if pointing out from the element. The shear stress() will be positive if it points up on the right side of the element.


Mohr's Circle

This is a graphical solution for plane stress equations. This graph is a circle with the center on the normal stress(σ) axis (x-axis). The shear stress() axis (or y-axis) will be positive downward(counterclockwise) and negative upward(clockwise). To determine if the shear stress if clockwise(cw) or counterclockwise(ccw) look at the element's state of stress. Locate the shear stress on the right side and the top of the element, now determine if the stress is acting in a cw or ccw motion around the center of the element.


Steps for Drawing Mohr's Circle:

1. Set up the coordinate system (σ, )
2. Plot the reference points that are in the state of stress element.
3. Locate the center: σavg = (σx+σy)/2
4. Draw a line through these 3 points, this is the line that represents where θ is to be measured from.
5. Use trigonometry to determine the radius.
6. Draw the circle.

Principle stresses(σmax and σmin): located on the σ-axis. σmax and σmin will be at opposite ends of the circle. These stresses are transformed an angle of θp1 and θp2 from the reference line to the σ-axis. These two angles are 90 degrees apart. Tan(2θp) = xy / ((σx-σy)/2) to solve for the angle(θp).


Max In-Plane Shear Stress: located at the top and bottom of the circle. The θs1 and θs2 give the orientation of the transformed components. Each θs is 45 degrees apart.
Tan(2θs) = -((σx-σy)/2) / xy to solve for the angle(θs).



Absolute Max Shear Stress (3-D)

In 3-D the same procedure will be followed as listed above. There will be another normal stress added along with 2 more shear stresses.
For σx,σy, and σz there will be a corresponding σmax, σint, and σmin, where σintermediate is between or equal to σmax and/or σmin

Now instead of 1 Mohr's circle there will be 3. There will be one large circle with two smaller circles inside. The large circle will have a diameter from σmin to σmax. The smaller 2 circles will have a diameter from σint to σmax/min.

The absolute maximum shear stress can be found: abs. max = (σmax - σmin) / 2
Max In-Plane stress can be found using the smaller circles.





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