
MECHANICS  THEORY



Stress Rotation Equation

Rotated Stresses using
Stress Rotation Equations


Previously, the stress
transformation equations were developed to calculate the stress state at different
orientations. These equations were
Plotting these equations show that every 180 degrees rotation, the stress state
repeats. In 1882, Otto Mohr noticed that these relationships could be graphically
represented with a circle. This was a tremendous help in the days of slide rulers when using
complex equations, like the stress transformation equations, was time consuming. 





Mohr's Circle



Mohr's circle is not actually a new derived formula, but just a new way to visualize
the relationships between normal stresses and shear stresses as the rotation
angle changes. To determine the actual equation for Mohr's circle, the stress transformation equations can be rearranged to give,
Each side of these equations can be squared and then added together to
give
Grouping similar terms and canceling other terms gives
Using the trigonometry identity, cos^{2}2θ +
sin^{2}2θ = 1, gives




Basic Mohr's Circle for Stress 

This is basically an equation of a circle. The circle equation can be better
visualized if it is simplified to

where


This circle equation is plotted at the left using r and σ_{ave}.
One advantage of Mohr's circle is that the principal stresses, σ_{1}, σ_{2} ,
and the maximum shear stress, τ_{max}, are easily
identified on the circle without further calculations. 





Rotating Stresses with Mohr's Circle

Stress Rotation with Mohr's Circle 

In addition to identifying principal stress and maximum shear stress,
Mohr's circle can be used to graphically rotate the stress state. This involves
a number of steps.
 On the horizontal axis, plot the circle center at σ_{avg} = (σ_{x} + σ_{y})/2.
 Plot either the point (σ_{x} , τ_{xy})
or (σ_{y} , τ_{xy}).
Note the sign change if plotting σ_{y} and the
vertical axis, τ_{xy}, is positive downward.
 Draw a line from the center to the point plotted in step two (blue line in
the diagram). This line should extend from one side of the circle to the
other. Radius, r, can now be measured from the graph.
 The circle itself can be drawn since the center and one point on the circle
is known (compass works well for this).
 The principal stresses and maximum shear stress can be identified on the
graph.
 The line drawn in step 3 can be rotated twice the rotation angle, 2θ,
in the counterclockwise direction. It is important that the angle is twice the desired rotated angle.
 The new stress state is the intersection of the new line (green in the diagram)
and the circle.






The angle, 2θ_{p}, for the principal stresses
is simply the half the angle from the blue line to the horizontal axis.
Remember, Mohr's circle is just another way to visualize the stress state. It
does not give additional information. Both the
stress transformation equations
and Mohr's circle will give the exactly same values. 



