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Strength of Materials: Mohr's Circle

2d Mohr's Circle

The 2d Mohr's circle is a graphical representation of the equations seen above. To view a 2d graphical Mohr's circle refer to the figure below.

2 Dimensional Mohr's Circle

It can be seen from this graph that the center of the circle represents the average stress, while the two ends on the x axis represent the principle stresses, and the peaks of the circle on the y axis represents the maximum shear stress.

3d Mohr's Circle

The 3d Mohr's circle is developed by the same methods as the 2d Mohr's circle except there are 2 additional shear stresses to consider, and one more principle stress to consider. To solve for the principle stresses for a 3d problem the roots of equation 1 would have to be determined. Also, equations 2 thru 4 would have to be used to solve equation 1.

Principle Stress Equation for 3d Mohr's Circle (1)

I1 for 3d Principle Stress Equation (2)

I2 for 3d Principle Stress Equation (3)

I3 for 3d Principle Stress Equation (4)

σ = Normal Stress

τ = Shear Stress

Equation 4 is a 3X3 determinant. If you're not sure how to solve a determinant follow this link.

To solve for the maximum shear stresses equation 5 would be used.

Calculate Max Shear Stresses for 3d Mohr's Circle (5)

Refer to the figure below to derive a 3d Mohr's Circle from equations 1-5.

3 Dimensional Mohr's Circle

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