What are the three stress invariants (in MPa) respectively in a body under the stress states of, σ= 10, σ= 8, σ= -5, τyz = τzy = 5, τzx = τxz = -4, and τxy = τyx = -8, where all stresses are in MPa ? 

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UPSC ESE (Prelims) Civil Engineering 19 Feb 2023 Official Paper
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  1. 23, -128 and 125
  2. 13, 115 and -138
  3. 115, 13, -128
  4. -138, 23, 125

Answer (Detailed Solution Below)

Option 2 : 13, 115 and -138
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Detailed Solution

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Concept:

Stress invariants in 3D stress plane:

\(J_1=σ_x+σ_y+σ_z\)

\(J_2=σ_xσ_y+σ_yσ_z+σ_zσ_x-\tau_{xy}^2-\tau_{yz}^2-\tau_{xz}^2\)

\(J_3=σ_xσ_yσ_z+2\tau_{xy}\tau_{yz}\tau_{xz}-σ_x\tau_{yz}^2-σ_y\tau_{xz}^2-σ_z\tau_{xy}^2\)

Calculation:

Given

σx = 10 MPa, σy = 8 MPa, σz = -5 MPa, \(\tau_{yz}=\tau_{zy}=5 \, MPa, \tau_{zx}=\tau_{xz}=-4 \, MPa, and \,\tau_{xy}=\tau_{yx}=-8 \, MPa.\)

J1 = 10 + 8 + (-5) = 13 MPa

J2 = 10 × 8 + 8 × -5 + (-5) × 10 - (5)2 - (-4)2 - (-8)2 = - 115 MPa

J3 = 10 × 8 × (-5) + 2 × 5 × (-4) × (-8) - 10 × 52 - 8 × (-4)2 - (-5) × (-8)2 = -138 MPa

J1 = 13 MPa, J2 = -115 MPa, and J3 = -138 MPa.

Hence option (2) is correct.

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