(Solved): The torsion rod with variable cross-section shown in Fig. \( 2.1 \) is clamped at \( A \) and carr ...

The torsion rod with variable cross-section shown in Fig. \( 2.1 \) is clamped at \( A \) and carries point torques at \( B(4 \mathrm{kN} . \mathrm{m}), C(8 \) \( \mathrm{kN} . \mathrm{m} \) ) and \( D \) (unknown value \( T \) ) with the senses indicated in the figure. It is known that the diameter of \( A B \) is \( 25 \mathrm{~mm} \), the diameter of \( B C \) is \( 50 \mathrm{~mm} \) and the diameter of \( C D \) is \( 20 \mathrm{~mm} \). Furthermore, \( 6 L_{A B}=6 \mathrm{~L}_{\mathrm{BC}} \) \( =5 L_{C D}=1200 \mathrm{~mm} \). If the shear modulus of the material is \( 80 \mathrm{GPa} \), find: a) The value of \( \mathrm{T} \) that would make the angle of twist at point \( \mathrm{C} \) with respect to \( \mathrm{A} \) equal to zero. b) The maximum value of \( T \) that can applied with the sense shown such that failure does not occur for an allowable shear stress of \( 1.1 \mathrm{GPa} \). Can the condition from part a be achieved without failure?