Shearing Stress in Multiple Shafts Static equilibrium must be satisfied. The vector of all applied torques = 0 T A +T B +T C +T D =0 The shearing stress at any point in the shaft is a function of the Internal torque on a plane containing the point. The maximum Stress is computed from Radius of the shaft The polar moment of inertia Internal torque
The shearing stress at a point on shaft through which section 1 passes can only be defined once the internal torque is determined. From the free-body diagram, the internal torque at the section The shearing stress is given by
The shearing stress at a point on shaft through which section 2 passes can only be defined once the internal torque is determined. From the free-body diagram, the internal torque at the section The shearing stress is given by
Angle of Twist in Multiple Shafts For this shaft sections AB, BC, and CE will each have different - Internal torque (FBD) - Polar moment of inertia - Shear Modulus - Length The angle of twist of the end of a shaft consisting of N sections is expressed as
Relative Rotation In some situations both ends of a shaft rotate. The angle of twist is the angle through which one end rotates w.r.t the other. fixed Both rotates Shaft AB
Shearing Stress on inclined planes
Statically Indeterminate Shafts •Both ends of the shaft are built in, leading to two reaction torques but one has only one moment equilibrium equation. •The compatibility equation is the relative rotation! •See example
Assuming a counterclockwise torque is positive, summing moments about the axis of the shaft results in The total angle of twist of the shaft must be 0 since both ends are fixed.