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GATE 2023 CE (CE1) – Question 53

Solid Mechanics · Transformation of stress, Mohr circle · 2 marks · Numerical answer

The infinitesimal element shown in the figure (not to scale) represents the state of stress at a point in a body. What is the magnitude of the maximum principal stress (in N/mm$^2$, in integer) at the point?

a parallelogram element with horizontal top and bottom faces carrying a normal stress of 6 N/mm² and a shear stress of 3 N/mm², and with inclined faces at 45° carrying a normal stress of 5 N/mm² and a shear stress of 4 N/mm².

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Correct answer: 7

Explanation

Take the horizontal faces as the faces normal to the y-axis: $\sigma_y = 6$ and $\tau_{xy} = 3$. The unknown $\sigma_x$ follows from the 45° plane, whose normal stress is 5: $\frac{\sigma_x + 6}{2} + \frac{\sigma_x - 6}{2}\cos 90° + 3\sin 90° = 5$, so $\frac{\sigma_x + 6}{2} = 2$ and $\sigma_x = -2$. (The shear on that plane is then $\frac{6 + 2}{2} = 4$, which matches the figure.) The principal stresses are $\frac{-2 + 6}{2} \pm \sqrt{\left(\frac{-2 - 6}{2}\right)^2 + 3^2} = 2 \pm 5$, that is 7 and −3 N/mm². The maximum principal stress is 7 N/mm².