Torsional Deformation of a circular shaft Torsion Formula

Torsional Deformation of a circular shaft, Torsion Formula , Power Transmission 1

Torsional Deformation of a circular shaft Length BD when 2

The Torsion Formula Angular strain is proptional to shear stress: Mean: • highest shear stress: will be at farthest away from center • At the center point, there will be no angular strain and therefore no shear stress is developed. 3

The Torsion Formula J: Polar moment of area Maximum shear stress due to torsion Shear stress due to torsion at radius r 4

Polar Moment of Inertia (J) Solid shaft Tubular shaft 5

Solve it! The solid circular shaft is subjected to an internal torque of T= 5 k. Nm. Determine the shear stress developed at point A and B. Represent each state of stress on a volume of element.

Solve it! The solid circular shaft is subjected to an internal torque of T= 5 k. Nm. Determine the shear stress developed at point A and B. Represent each state of stress on a volume of element.

Example 1 The shaft shown in figure is supported by two bearings and is subjected to three torques. Determine the shear stress developed at points A and B, located at section a-a of the shaft.

Point A R = 0. 075 m J = Jtotal T = internal torque Point B R = 0. 015 m J = Jtotal T = internal torque T=4. 25 – 3. 0 = 1. 25 k. Nm

Solve it! Determine the maximum shear stress developed in the shaft at section a-a. 10

Maximum Torsional Shear T: 2100 Nm = 2100(10)3 Nmm r: 40 mm J: tubular Polar Moment of Area Max torsional shear when r = 40 mm 11

Solve it! The solid shaft has a diameter of 40 mm. Determine the absolute maximum shear stress in the shaft. 12

Maximum Torsional Shear T: 70 Nm = 70(10)3 Nmm r: 20 mm J: solid r =20 mm 13

Power Transmission Power P: power (1 Watt = 1 Nm/s) T: torque (Nm) w: radian/s Input n(rpm) Input frequency of shaft rotation 14

Solve it ! The gear motor can develop 1. 6 k. W when it turns at 450 rev/min. If the shaft has a diameter of 25 mm, determine the maximum shear stress developed in the shaft.

Calculate the T Maximum shear stress

Solve it ! The gear motor can develop 2. 4 k. W when it turns at 150 rev/min. If the allowable shear stress for the shaft is tallow = 84 Mpa, determine the smallest of the shaft to nearest multiples of 5 mm that can be used.

Solve it ! Calculate the T Maximum shear stress Calculate d

Angle Twist Lecture 1 19

Sign Convention Right hand rule +ve: direction of the thumb is away from the part Lecture 1 20

Solve it! The splined ends and gears attached to the A-36 steel shaft are subjected to the torques shown. Determine the angle of twist of end B with respect to end A. The shaft has a diameter of 40 mm.

300 Nm 500 Nm 200 Nm Tac= 300 Nm 500 Nm Tcd=200 Nm 300 Nm 400 Nm Tdb =400 Nm Total angle of twist: 400 Nm

Solve it ! The two shafts are made of A-36 steel. Each has a diameter of 25 mm, and they are supported by bearings at A, B and C, which allow free rotation. If the support at D is fixed, determine of the angle of twist of end B and A when the toques are applied to the assembly as shown.

Equilibrium of shaft AGFB Direction of FF FF Equilibrium at gear F and E Direction of FE FE

Direction of FF Equilibrium of shaft DCE 3 ONm 9 ONm 12 ONm 3 ONm 9 ONm FE

Length of arc 12 = length of 13 3 2 1 Angle of twist at end B = angle of twist at F FB : there is no torque Angle of twist at end A