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Maximum and Minimum Normal Stress Rotating Stresses from x-y Coordinate
System to new x'-y' Coordinate System

Rotating the stress state of a stress element can give stresses for any angle. But usually, the maximum normal or shear stresses are the most important. Thus, this section will find the angle which will give the maximum (or minimum) normal stress.

Start with the basic stress transformation equation for the x or y direction. To maximize (or minimize) the stress, the derivative of σx′ with respective to the rotation angle θ is equated to zero. This gives,

dσx′ / dθ = 0 - (σx - σx) sin2θp + 2τxy cos2θp = 0

where subscript p represents the principal angle that produces the maximum or minimum. Rearranging gives,  Principal Stresses, σ1 and σ2,
at Principal Angle, θp

The angle θp can be substituted back into the rotation stress equation to give the actual maximum and minimum stress values. These stresses are commonly referred to as σ1 (maximum) and σ2 (minimum), For certain stress configurations, the absolute value of σ2 (minimum) may actually be be larger than σ1 (maximum).

For convenience, the principal stresses, σ1 and σ2, are generally written as, where the +/- is the only difference between the two stress equations.

It is interesting to note that the shear stress, τx′y′ will go to zero when the stress element is rotated θp.

Maximum Shear Stress Maximum Shear Stresses, τmax,
at Angle, θτ-max

Like the normal stress, the shear stress will also have a maximum at a given angle, θτ-max. This angle can be determined by taking a derivative of the shear stress rotation equation with respect to the angle and set equate to zero.  When the angle is substituted back into the shear stress transformation equation, the shear stress maximum is The minimum shear stress will be the same absolute value as the maximum, but in the opposite direction. The maximum shear stress can also be found from the principal stresses, σ1 and σ2, as Plotting Stresses vs Angle Stresses as a Function of Angle

The relationships between principal normal stresses and maximum shear stress can be better understood by examining a plot of the stresses as a function of the rotation angle.

Notice that there are multiple θp and θτ-max angles because of the periodical nature of the equations. However, they will give the same absolute values.

At the principal stress angle, θp, the shear stress will always be zero, as shown in the diagram. And the maximum shear stress will occur when the two principal normal stresses, σ1 and σ2, are equal.

Principal Stresses in 3D

In some situations, stresses (both normal and shear) are known in all three directions. This would give three normal stresses and three shear stresses (some may be zero, of course). It is possible to rotate a 3D plane so that there are no shear stresses on that plane. Then the three normal stresses at that orientation would be the three principal normal stresses, σ1, σ2 and σ3.

These three principal stress can be found by solving the following cubic equation, This equation will give three roots, which will be the three principal stresses for the given three normal stresses (σx, σy and σz) and the three shear stresses (τxy, τyz and τzx).

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