GATE Mechanical Engineering (ME) 2019 Shift 1 Solved Paper

Question : 57 of 65

Marks: +1, -0

A plane-strain compression (forging) of a block is shown in the figure. The strain in the z direction is zero. The yield strength (Sy) in uniaxial tension\/compression of the material of the block is 300 MPa and it follows the Tresca (maximum shear stress) criterion. Assume that the entire block has started yielding. At a point where

σ_{x}=40

MPa (compressive) and

τ_{x y}=0

, the stress component

σ_{ {y}}

$340 {MPa}$ (tensile)
$260 {MPa}$ (compressive).
260 MPa (tensile)
$340 {MPa}$ (compressive)

Validate

Solution:

For plane strain,

E_{ {z}}=\;{σ_{z}}/{E}-µ(\;{σ_{x}}/{E}+\;{σ_{y}}/{E})=0

So,

σ_{z}=µ(σ_{x}+σ_{y})=µ(-40+σ_{v})

Maximum shear stress

=

larger of

|[\;{σ_{x}-σ_{y}}/{2}],[\;{σ_{y}-σ_{z}}/{2}],[\;{σ_{x}-σ_{x}}/{2}]|

Maximum shear stress = larger of

|[\;{-40-σ_{y}}/{2}],[\;{σ_{y}-µ(σ_{y}-40)}/{2}],[\;{µ(-40+σ_{y})+40}/{2}]|

From the above equation,

[\;{-40-σ_{y}}/{2}]

will be maximum
So,

|\;{-40-σ_{y}}/{2}| ≤ \;{s_{j t}}/{2 F O S}

-40-σ_{y}=300

⇒ σ_{y}=-340 {MPa}=340(

compression

)

-40-σ_{y}=-300

⇒ σ_{y}=260 {MPa}=260

(compression)
But in forging, force cannot be tensile, hence the correct answer must be 340MPa (compressive)

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