When a body is subjected to a direct tensile stress $$\left( {{\sigma _{\text{x}}}} \right)$$ in one plane accompanied by a simple shear stress $$\left( {{\tau _{{\text{xy}}}}} \right),$$ the maximum normal stress is
A. $$\frac{{{\sigma _{\text{x}}}}}{2} + \frac{1}{2} \times \sqrt {\sigma _{\text{x}}^2 + 4\tau _{{\text{xy}}}^2} $$
B. $$\frac{{{\sigma _{\text{x}}}}}{2} - \frac{1}{2} \times \sqrt {\sigma _{\text{x}}^2 + 4\tau _{{\text{xy}}}^2} $$
C. $$\frac{{{\sigma _{\text{x}}}}}{2} + \frac{1}{2} \times \sqrt {\sigma _{\text{x}}^2 - 4\tau _{{\text{xy}}}^2} $$
D. $$\frac{1}{2} \times \sqrt {\sigma _{\text{x}}^2 + 4\tau _{{\text{xy}}}^2} $$
Answer: Option A
This Question Belongs to Mechanical Engineering >> Strength Of Materials In ME
When a body is subjected to a direct tensile stress ((sigma )) and a simple shear stress (( au _{xy})) in one plane, the maximum normal stress (also called the major principal stress) is calculated using the formula:(sigma _{max}= rac{sigma }{2}+sqrt{left( rac{sigma }{2}
ight)^{2}+ au _{xy}^{2}})Why This Happens (The Concept)The Normal Stress ((sigma )): Stretches the material in one direction.The Shear Stress (( au _{xy})): Attempts to slide or twist the material.The Combined Effect: The material's true maximum stress doesn't happen straight up-and-down or side-to-side. Instead, it occurs on a "principal plane" that is tilted at an angle. This tilted angle experiences the combined pulling force of the direct tensile stress and the twisting force of the shear stress, resulting in a higher total stress than either one alone.