Answer to Question #298102 in Mechanical Engineering for Koo

Question

A triaxial stress element as shown in Figure has σx = 40 MPa, σy = –20 MPa, σz = –10 MPa, τxy = 5 MPa, τyx = –1.5 MPa, τzx = 2.5 MPa. Answer following questions:

Find the principal stresses using a numerical method (i.e., using Eigen values method) and draw the resulting Mohr’s circles. (You can use Scientific Calculator for calculating eigen values of the stress matrix)

Draw the resulting Mohr's circles.

Answer 1

Mohr's circle is a graphical representation of the transformation equations for plane stress problems. It is useful in visualizing the relationships between normal and shear stresses acting on a stress element at any desired orientation.

principal stresses

${{\rm{σ }}_1},\;{{\rm{σ }}_2} = \frac{{{{\rm{σ }}_x} + \;{{\rm{σ }}_y}}}{2} \pm \sqrt {{{\left( {\frac{{{{\rm{σ }}_x} - \;{{\rm{σ }}_y}}}{2}} \right)}^2} + {{\left( {{τ _{xy}}} \right)}^2}}$