Question 3.3.1: A circular bar AB, of radius r metres, lies on the horizonta...
A circular bar AB, of radius r metres, lies on the horizontal plane (Fig. 3.3-1 (a)) and is rigidly attached to two vertical walls at A and B. A uniformly distributed twisting moment of intensity T newton-metres per metre of bar length acts on the bar.
Determine the reactions at A and B.
Consider an infinitesimal arc length of the bar subtending an angle dθ radians at the centre. The twisting moment acting on this length r dθ is Tr dθ newtonmetres. As explained in Section 1.2, this moment can be represented by the vector shown in Fig. 3.3-1(b). Thus the distributed moment acting on the bar is represented by an infinite number of moment vectors following one another head to tail along the axis of the bar. The moment resultant M_{R} of these moment vectors is given by the line ab.
M_{R} = length ab = T_{r}/\cos 45° Nm
By symmetry the reactions at A and B are each — M_{R}/2, where the negative sign shows that the reactions are equal and opposite to the resultant M_{R}. Specifically,
Bending moment at A =\frac{1}{2} M_{R} \cos 45° =\underline{ \frac{1}{2} T_{r} Nm}
Twisting moment at A = \frac{1}{2} M_{R} \sin 45° =\underline{ \frac{1}{2} T_{r} Nm}
and similarly at B.
