GATE 2023 CE (CE2) – Question 39
In a two-dimensional stress analysis, the state of stress at a point is shown in the figure. The values of length of PQ, QR, and RP are 4, 3, and 5 units, respectively. The principal stresses are ______. (round off to one decimal place)
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Correct answer: (C) $\sigma_x=67.5$ MPa, $\sigma_y=213.3$ MPa
Explanation
**Setup.** The wedge element has vertices Q (origin), P on the $y$-axis ($QP=4$) and R on the $x$-axis ($QR=3$), with the inclined face $PR=5$. On the inclined face the stress is a normal stress of 120 MPa and a shear stress of 70 MPa. The faces on the axes carry only normal stresses $\sigma_x$ (on the vertical face QP) and $\sigma_y$ (on the horizontal face QR).
**Step 1: direction vectors of the inclined face.**
- Outward normal: $\mathbf n=(4/5,\ 3/5)$ (since the face goes from $(0,4)$ to $(3,0)$).
- Tangent (upward along the face): $\mathbf t=(-3/5,\ 4/5)$.
**Step 2: traction on the inclined face.**
$$t_x=120\left(\tfrac45\right)-70\left(\tfrac35\right)=96-42=54\text{ MPa},$$
$$t_y=120\left(\tfrac35\right)+70\left(\tfrac45\right)=72+56=128\text{ MPa}.$$
**Step 3: equilibrium of the wedge.** The traction on the inclined face times its length 5 must balance the forces on the other two faces:
- $x$-direction: $\sigma_x\times4=54\times5\Rightarrow\sigma_x=\dfrac{270}{4}=67.5$ MPa.
- $y$-direction: $\sigma_y\times3=128\times5\Rightarrow\sigma_y=\dfrac{640}{3}=213.3$ MPa.
Since there is no shear stress on the coordinate faces, these are the principal stresses: $\sigma_x=67.5$ MPa and $\sigma_y=213.3$ MPa (option C).