A body is subjected to a direct stress 'σ' in one plane accompani
A. <span class="math-tex">\(\frac{\sigma }{2} + \frac{1}{2}\sqrt {{\sigma ^2} + 4{\tau ^2}} \)</span>
B. <span class="math-tex">\(\frac{\sigma }{2} - \frac{1}{2}\sqrt {{\sigma ^2} + 4{\tau ^2}} \)</span>
C. <span class="math-tex">\(\frac{\sigma }{2} + \frac{1}{2}\sqrt {{\sigma ^2} - 4{\tau ^2}} \)</span>
D. <span class="math-tex">\(\frac{\sigma }{2} - \frac{1}{2}\sqrt {{\sigma ^2} - 4{\tau ^2}} \)</span>
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Right Answer is: A
SOLUTION
Concept:
The maximum and minimum normal stress is given by
\({\sigma _{max,\;\;min}} = \frac{{{\sigma _x} + {\sigma _y}}}{2}\; \pm \sqrt {{{\left( {\frac{{{\sigma _x}\; - \;{\sigma _y}}}{2}} \right)}^2} + τ _{xy}^2} \)
where σx and σy are direct stress and τxy is the shear stress
Calculation:
Given:
From the given condition we have,
σx = σ, σy = 0, τxy = τ
Now maximum normal stress is:
\({\sigma _{max}} = \frac{{{\sigma _x} + {\sigma _y}}}{2} + \sqrt {{{\left( {\frac{{{\sigma _x}\; - \;{\sigma _y}}}{2}} \right)}^2} + τ _{xy}^2} \)
\(\sigma_{max}= \frac{\sigma }{2} + \frac{1}{2}\sqrt {{\sigma ^2} + 4{τ ^2}} \)