GATE CE 2023 Set 2 — Question 39
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Structural Engineering → Torsion, Stress Transformation & Columns → Principal Stresses & Planes
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Question
In a two-dimensional stress analysis, the state of stress at a point is shown in the figure. The values of length of , , and are 4, 3, and 5 units, respectively. The principal stresses are ________. (round off to one decimal place)
Correct answer
(C) σₓ = 67.5 MPa, σ_y = 213.3 MPa
Solution
Given the lengths of the sides of the triangular wedge :
units (vertical face)
units (horizontal face)
units (inclined face)Let be the angle that the inclined plane makes with the vertical face . From the geometry:
The normal stress and shear stress on an inclined plane at an angle from the vertical plane (where acts) are given by:
From the figure, there is no shear stress on the vertical and horizontal faces, so . This means and are the principal stresses. Substituting the given values MPa and MPa:1)
--- (Eq. 1)2)
MPa
--- (Eq. 2)Substitute Eq. 2 into Eq. 1:
MPaNow find :
MPaRounding to one decimal place, we get MPa and MPa. This matches option (C).
units (vertical face)
units (horizontal face)
units (inclined face)Let be the angle that the inclined plane makes with the vertical face . From the geometry:
The normal stress and shear stress on an inclined plane at an angle from the vertical plane (where acts) are given by:
From the figure, there is no shear stress on the vertical and horizontal faces, so . This means and are the principal stresses. Substituting the given values MPa and MPa:1)
--- (Eq. 1)2)
MPa
--- (Eq. 2)Substitute Eq. 2 into Eq. 1:
MPaNow find :
MPaRounding to one decimal place, we get MPa and MPa. This matches option (C).
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