The failure of a key is due to:

Explanation:

Key

• A Key is a machine element that is used along with coupling in order to transmit power from one shaft to another.
• Key is the weakest member in the assembly of shaft, pulley and key.
• Key acts as a safety device, whenever there is excess load appears on the pulley key fails first and it keeps safer to shaft and pulley.
• A key is the piece of low carbon steel inserted between shaft and hub to connect these together in order to prevent relative motion between them.
• It is always inserted parallel to the axis of the shaft.
• Keys are used as temporary fastenings and are subjected to considerable crushing and shearing stresses.
• A keyway is a slot or recess in a shaft or hub of the pulley to accommodate a key.

Forces acting on a key

Assumption: Exact location of force P on face AC is not exactly known. Therefore, to simplify analysis it is considered that P is tangential to the shaft diameter.

Now P = \(\frac{{{M_t}}}{{\frac{d}{2}}} = \frac{{2{M_t}}}{d}\)    ----(1)

Case (a): Shearing in the key:

τ = \(\frac{P}{{{A_{shear}}}} = \;\frac{P}{{bl}}\)  

\(\tau = \frac{{2{M_t}}}{{dbl}}\)    

\({\left( {{M_t}} \right)_{shear}} = \frac{\tau }{2} \times dbl\)   

where τ = shear stress of the material

Case (b): Crushing in key:

\({\sigma _c} = \frac{P}{{\frac{h}{2} \times l}}\)

\({\sigma _c} = \frac{{4{M_t}}}{{dhl}}\)    

Crushing will occur if \({\sigma _c} = {\sigma _{yt}}\)

\({\left( {{M_t}} \right)_{crushing}} = \frac{{{\sigma _{yt}}}}{4} \times dhl\)

Out of given options, only shearing is present therefore option 1 is correct.