The elasticity and resistance of the materials of engineering . shall be covered. It is very important to observe that the equations of con-dition for the determination of ti, and consequently the generalvalues of T^^ and T^^, are wholly independent of any consider-ations regarding the position of the axis of torsion, or the axisof X. It follows from this, that the resistance of pure torsion isprecisely the same wherever may be the axis about which thepiece is twisted. It is to be borne in mind, however, that, ifthe axis of the twisting is not the axis of the cylindrical piece,the latter will
The elasticity and resistance of the materials of engineering . shall be covered. It is very important to observe that the equations of con-dition for the determination of ti, and consequently the generalvalues of T^^ and T^^, are wholly independent of any consider-ations regarding the position of the axis of torsion, or the axisof X. It follows from this, that the resistance of pure torsion isprecisely the same wherever may be the axis about which thepiece is twisted. It is to be borne in mind, however, that, ifthe axis of the twisting is not the axis of the cylindrical piece,the latter will be subjected to combined bending and torsion ;the bending being produced by a force sufficient to cause thepiece to take the helical position ne-cessitated by the torsion. The cylin-drical axis is the straight line locus ofthe centres of gravity of all the normalsections. If, as in Fig. 2, there are n cylinderswhose centres c are all at the samedistance Cc — I from the centre C oftwisting, or motion ; and if M is thetotal moment of torsion of the system, 4. 50 TORSION IN EQUILIBRIUM. [Art. lO. while ffi is the moment of torsion of each cyHnder about itsown axis or centre r, then will M = n7n ; and each cylinderwill be subject to a bending moment whose amount can bedetermined from the condition that the diameter of each piecelying along Cc before torsion, must pass through C after, andduring, torsion, also. Since T^^ and 7^^ act at right angles to each other, the re-sultant intensity of shear at any point in an originally normalsection of the twisted piece will be: T=VTj-{- T^^^ (24) According to the ordinary methods of the calculus, the co-ordinates of the point at which T has its greatest value mustsatisfy the equations: dT dT f . d^T/ , d^ / / d^T \^ _dT d^T /dcp^^ dr \° \dcpdr) dqj dr ~^ After the solution of Eqs. (25), it will usually be necessaryonly to inspect the resulting value of T, in order to determinewhether it is a maximum or minimum, without applying thet
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