GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

Question : 53 of 65

Marks: +1, -0

A carpenter glues a pair of cylindrical wooden logs by bonding their end faces at an angle of as shown in the figure.

The glue used at the interface fails if Criterion

1:

the maximum normal stress exceeds

2.5 {MPa}

Criterion 2 : the maximum shear stress exceeds

1.5 {MPa}

Assume that the interface fails before the logs fail. When a uniform tensile stress of 4 MPa is applied, the interface

Solution:

General equation learned for
Normal stress on inclined plane

σ_{n}=\;{σ_{x}+σ_{y}}/{2}+\;{σ_{x}-σ_{y}}/{2} \cos 2 θ+τ_{x y} \sin 2 θ

And shear stress for inclined plane

τ_{n}=-\;{σ_{x}-σ_{y}}/{2} \sin 2 θ+τ_{x y} \cos 2 θ

when

θ

is measured clockwise from left face.
Hence

θ

must be replaced by

(-θ)

in above equations and

σ_{y}=0, τ_{x y}=0

σ_{n}=\;{σ_{x}}/{2}+\;{σ_{x}}/{2} \cos 2(-30)+0

σ_{n}=\;{4}/{2}+\;{4}/{2} \cos (-60)

σ_{n}=3 M P a

τ_{n}=-\;{σ_{x}}/{2} \sin 2(-θ)

τ_{n}=\;{σ_{x}}/{2} \sin (2 θ)

τ_{n}=\;{4}/{2} \sin (60)

τ_{n}=1.73

Since both the stress exceeds the given limits, answer is option (C).

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