# TORSION OF SOLID AND HOLLOW SHAFTS

Torsion of Solid and Hollow Shaft Calculator was developed to calculate shear stress, angle of twist and polar moment of inertia  parameters of a shaft which is under torsion. The calculator is only valid for solid/hollow circular shafts and can be used for sizing of the shafts. The formulas used for torsion of shaft calculations are given in the List of Equations section.

A shaft is a rotation member usually with cylindrical shape which is used to transmit torque, power and motion between various elements such as electric or combustion motors and gear sets, wheels, cams, flywheels, pulleys, or  turbines and electric generators. Shafts can be solid or hollow. During power transmission, shafts twist and stresses and deformations are taking place.

Torsion is twisting of an object due to an applied torque.  When a shaft twists, one end rotates relative to the other and shear stresses are produced on any cross section.

Shear stress is zero on the axis passing through the center of a shaft under torsion and maximum at the outside surface of a shaft. On an element where shear stress is maximum, normal stress is 0. This element where maximum shear stress occurs is oriented in such a way that its faces are either parallel or perpendicular to the axis of the shaft as shown in the figure. To obtain stress in other orientations, plane stress transformation is needed for shear stresses which are found with this calculator. ### Torsion of Shaft Calculator:  Shaft style Solid ShaftHollow Shaft INPUT PARAMETERS Parameter Value Torque [T] N*m lbf*in lbf*ft Rotation speed [w] rpm Shaft outer radius [c2] mm cm m inch ft Shaft inner radius [c1] Shaft length [L] Modulus of rigidity [G] GPa psi*10^6

Note: Use dot "." as decimal separator.

 RESULTS Parameter Value Maximum shear stress [τmax] --- MPa psi Angle of twist [θ] --- Radian Degree Power requirement [P] --- W kW hp Polar moment of inertia [J] --- mm^4 cm^4 inch^4 ft^4

### Definitions:

Angle of Twist: Shaft is rotated from its normal position when a torque is applied.

Dynamometer: Dynamometer is a device to measure torque or power. There are different types of absorption unit in dynamometers such as Eddy current brake, magnetic powder brake, hysteresis brake.

Modulus of rigidity (modulus of elasticity in shear): The rate of change of unit shear stress with respect to unit shear strain for the condition of pure shear within the proportional limit. Typical values Aluminum 6061-T6: 24 GPa, Structural Steel: 79.3 GPa.

Notch Sensitivity: A measure of how sensitive a material is to notches or geometric discontinuities.

Polar Moment of Inertia: A geometric property of cross section. Measure of ability how a beam resists torsion.

Stress Concentration Factor: Dimensional changes and discontinuities of a member in a loaded structure causes variations of stress and high stresses concentrate near these dimensional changes. This situation of high stresses near dimensional changes and discontinuities of a member (holes, sharp corners, cracks etc.) is called stress concentration. Ratio of peak stress near stress riser to average stress over a member is called stress concentration factor.

Torque meter: Torque meter is a device for measuring torque on a rotating system.

### Supplements:

 Link Usage Stress concentration factors Stress concentration factor for different types of stress raisers can be calculated for tension, bending and torsional loading type.

### List of Equations:

 Parameter Symbol Equation Shear stress τ $$\tau =\frac { Tp }{ J }$$ Angle of twist θ $$\theta =\frac { TL }{ GJ }$$ Maximum shear stress τmax $${ \tau }_{ max }=\frac { T{ c }_{ 2 } }{ J }$$ Polar moment of inertia of solid shaft J $$J=\frac { \pi }{ 2 } ({ c }_{ 2 }^{ 4 })$$ Polar moment of inertia of hollow shaft J $$J=\frac { \pi }{ 2 } ({ c }_{ 2 }^{ 4 }-{ c }_{ 1 }^{ 4 })$$ Power P $$P=T\cdot w$$

 Symbol Parameter T Torque to be transmitted J Polar moment of inertia p Radial distance to center of shaft c1 Hollow shaft inner radius c2 Shaft outer radius L Length of the shaft G Modulus of rigidity w Rotation speed P Power

### Examples:

 Link Usage Torsional Stress Calculation of a Stepped Shaft An example about the calculation of torsional stress on stepped shaft. After calculation of torsional stress, principal stresses are calculated and evaluation of yield criteria of material is done with these stresses.

### Reference:

• Budynas.R , Nisbett.K . (2014) . Shigley's Mechanical Engineering Design . 10th edition.  McGraw-Hill
• Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill