Question

Example (5.1): A steel wire of (5mm) diameter is bent into a circular shape of (5m) radius. Determine the maximum stress induced in the wire. Take E=200 GPa. o max 5 mm 104

Answer 1

Data

Diameter of wire, d = 5 mm

Radius of circle, R = 5 m = 5000 mm

Young's modulus, E = 200 GPa = 200*10⁹ N/m² = 200*10³ N/mm²

Solution

We know that the bending equation is given by,

$\frac{E}{R}=\frac{M}{I}=\frac{\sigma}{y}$

where,

R= Radius of curvature

$\sigma=Bending \: stress$

Clearly, maximum stress occurs at the outermost element of the circular cross-section of the wire,

$y=\frac{d}{2}=\frac{5}{2}=2.5 \: mm$

Thus, using the first and third term of the above equation we get,

$\frac{E}{R}=\frac{\sigma}{y}$

$\Rightarrow {\sigma}=\frac{Ey}{R}=\frac{200*10^3*2.5}{5000}=100 \: N/mm^2=100*10^6 \: N/m^2 = 100\, MPa$

Thus, the maximum stress induced in the wire is 100 MPa