
MECHANICS  THEORY



Maximum and Minimum Normal Stress

Rotating Stresses from xy 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 σ_{x′} with
respective to the rotation angle θ is equated to zero.
This gives,
dσ_{x′} /
dθ = 0  (σ_{x}  σ_{x})
sin2θ_{p} + 2τ_{xy} cos2θ_{p} =
0
where subscript p represents the principal angle that produces the maximum or
minimum. Rearranging gives,




Principal Stresses, σ_{1} and σ_{2},
at Principal Angle, θ_{p} 

The angle θ_{p} 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 σ_{1} (maximum)
and σ_{2} (minimum),
For certain stress configurations, the absolute value of σ_{2} (minimum)
may actually be be larger than σ_{1} (maximum).
For convenience, the principal stresses, σ_{1} and σ_{2},
are generally written as,
where the +/ is the only difference between the two stress equations.
It is interesting to note that the shear stress, τ_{x′y′} will go to zero when the stress element is rotated θ_{p}. 





Maximum Shear Stress

Maximum Shear Stresses, τ_{max},
at Angle, θ_{τmax} 

Like the normal stress, the shear stress will also have a maximum at a given
angle, θ_{τmax}. 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, σ_{1} and σ_{2},
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 θ_{p} and θ_{τmax} angles
because of the periodical nature of the equations. However, they will give
the same absolute values.
At the principal stress angle, θ_{p}, 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, σ_{1} and σ_{2},
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, σ_{1}, σ_{2}
and σ_{3}.
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 (σ_{x}, σ_{y} and σ_{z}) and the three shear stresses (τ_{xy}, τ_{yz} and τ_{zx}).




