Question

Consider a solid round elastic bar with constant shear modulus, G, and cross-sectional area, A. The bar is built-in at both ends and subject to a spatially varying distributed torsional load t(x) = p sin( 2π L x) , where p is a constant with units of torque per unit length. Determine the location and magnitude of the maximum internal torque in the bar.

Answer 1

`\t(x)_(max) =(p* L)/(2* \pi)`

Step-by-step explanation:

Given that

Shear modulus= G

Sectional area = A

Torsional load,

`t(x) = p sin( (2\pi)/( L) x)`

For the maximum value of internal torque

`(dt(x))/(dx)=0`

Therefore

`(dt(x))/(dx) = p cos( (2\pi)/( L) x)* (2\pi)/(L)\\ p cos( (2\pi)/( L) x)* (2\pi)/(L)=0\\cos( (2\pi)/( L) x)=0\\ (2\pi)/( L) x=(\pi)/(2)\\\\x=(L)/(4)`

Thus the maximum internal torque will be at x= 0.25 L

`t(x)_(max) = \int_(0)^(0.25L)p sin( (2\pi)/( L) x)dx\\t(x)_(max) =\left [p* (-cos( (2\pi)/( L) x))/((2\pi)/( L)) \right ]_0^(0.25L)\\t(x)_(max) =(p* L)/(2* \pi)`