$8.2 Torsion 179 This force will produce a moment about the centre axis of the shaft,providing a contribution to the torque =(t'X2πrdr)Xr =2πx'r2dr The total torque on the section T will then be the sum of all such contributions across the section, i.e. T= 2nt'r2 dr 0 Now the shear stress t'will vary with the radius r and must therefore be replaced in terms of r before the integral is evaluated. From eqn.(8.3) GO t'= L T= 2 L R 2nr3dr L 0 The integral 2dr is called the polar second moment ofarea J,and may be evaluated asa standard form for solid and hollow shafts as shown in $8.2 below. T GO or 万= (8.4) Combining egns.(8.3)and(8.4)produces the so-called simple theory of torsion: Tt GO 了=R=L (8.5) 8.2.Polar second moment of area As stated above the polar second moment of area J is defined as 2nr3dr$8.2 Torsion 179 This force will produce a moment about the centre axis of the shaft, providing a contribution to the torque = (7' x 2nrdr) x r = 2n7'r2 dr The total torque on the section T will then be the sum of all such contributions across the section, i.e. T = 2nz'r2dr J i 0 Now the shear stress z' will vary with the radius rand must therefore be replaced in terms of r before the integral is evaluated. From eqn. (8.3) R .. 0 = E L jkr3 dr 0 The integral 5 0" 2nr3 dr is called the polar second moment of area J, and may be evaluated as a standard form for solid and hollow shafts as shown in $8.2 below. GO L .. T=-J or T GO JL - =- Combining eqns. (8.3) and (8.4) produces the so-called simple theory of torsion: T z G8 -- - J-R-L 8.2. Polar second moment of area As stated above the polar second moment of area J is defined as J = 2nr3dr 0 i