This question was previously asked in

HPPSC AE Civil 2015 (MPP-PPCL) Official Paper

Option 1 : \(\tau _{xy}^2 = {\sigma _x}{\sigma _{y}}\)

__Concept:__

The principal stresses σ1 and σ2 on a plane are given by

\({\sigma _1} = \;\frac{{{\sigma _{xx}}\;+ \;{\sigma _{yy}}}}{2} + \frac{1}{2}\sqrt {{{\left( {{\sigma _{xx}} - \;{\sigma _{yy}}} \right)}^2} + \;4\tau _{xy}^2} \)

\({\sigma _2} = \;\frac{{{\sigma _{xx}}\;+ \;{\sigma _{yy}}}}{2} - \;\frac{1}{2}\sqrt {{{\left( {{\sigma _{xx}} - \;{\sigma _{yy}}} \right)}^2} + \;4\tau _{xy}^2} \)

__Calculation:__

__Given:__

Let σ2 = 0

\(\frac{{{\sigma _{xx}} + \;{\sigma _{yy}}}}{2} - \frac{1}{2}\sqrt {{{\left( {{\sigma _{xx}} + \;{\sigma _{yy}}} \right)}^2} + \;4\tau _{xy}^2} \; = 0\)

\({\sigma _{xx}} + \;{\sigma _{yy}} = \;\sqrt {{{\left( {{\sigma _{xx}} + \;{\sigma _{yy}}} \right)}^2} + \;4\tau _{xy}^2} \)

Squaring on both sides,

\({\left( {{\sigma _{xx}} + \;{\sigma _{yy}}} \right)^2} = \left\{ {{{\left( {{\sigma _{xx}} - \;{\sigma _{yy}}} \right)}^2} + \;4\tau _{xy}^2} \right\}\)

\(4\sigma_{xx}\sigma_{yy}=4\tau_{xy}^2\)

\(\tau _{xy}^2 = {\sigma _{xx}}{\sigma _{yy}}\)