MecMovies 4.0 : M13.6: Principal Stresses from Rosette Data

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To determine the principal stresses, use the following equation:

σ_{p 1}, σ_{p 2} = \frac{σ_{x} + σ_{y}}{2} \pm \sqrt{{(\frac{σ_{x} - σ_{y}}{2})}^{2} + {τ_{xy}}^{2}}

= \frac{51.41 + (- 7.53)}{2} \pm \sqrt{{(\frac{51.41 - (- 7.53)}{2})}^{2} + {(51.05)}^{2}}

= 80.89 MPa, - 37.01 MPa

The orientation of the principal stresses is determined from:

\tan 2 θ_{p} = \frac{2 τ_{xy}}{σ_{x} - σ_{y}} = \frac{2 (51.05)}{51.41 - (- 7.53)} = 1.732

∴ θ_{p} = 30.000 °

Note: Since the denominator (σ_x - σ_y)>0, the computed angle is measured from the x axis to the σ_P1 direction. A positive angle is measured counterclockwise from the x axis.

Principal stresses from rosette data