Torsion of a cylinder, Mechanical Engineering

Torsion of a Cylinder

Let a cylinder of length l and radius R be clamped at one end and twisted at the other end by applying a torque τ about the axis of the cylinder. Each cross-section of the cylinder is rotated about the axis by an amount proportional to its distance from fixed end; hence the twist is maximum at the free end.


Consider the fig. let the cylinder be divided into a large number of thin co-axial cylindrical shells. Consider a shell of radius rand thickness dr. When the cylinder is twisted, a generating line PQ of this shell is displaced to position PQ'. The angle θrepresents angle of shear, while angle Ø (Q O' Q') is called angle of twist. From geometry, we have

QQ' = l θ = r Ø

Or, θ = rθ/l

The angle of shear θ is same on the entire surface of the shell. It increases linearly with r and hence the shearing strain is maximum at the outermost surface of the cylinder.

Let F be the tangential force acting on the lower face of the given shell; then the shearing stress is given by

Stress = F/(2 π r dr)

where, 2 π r dr is the face area of the shell. Now modulus of rigidity η, by definition is

η = (F/(2 π r dr))/θ

or, F = 2 π η θ rdr

= (2 π η Ø )/l r2 dr

The torque applied by this force about the axis OO' of the cylinder is,

d τ = Fr = (2 π η Ø )/l r3 dr

The above is the torque required to twist a thin cylindrical shell of radius r by an angle Ø. The torque required to twist the entire cylinder is therefore,

149_download (6).png 

The quantity τ/Ø is called torsional rigidity of the cylinder. It is equal to torque required to twist the cylinder by one radian (Ø = 1).

Posted Date: 9/18/2012 1:20:00 AM | Location : United States







Related Discussions:- Torsion of a cylinder, Assignment Help, Ask Question on Torsion of a cylinder, Get Answer, Expert's Help, Torsion of a cylinder Discussions

Write discussion on Torsion of a cylinder
Your posts are moderated
Related Questions
Inlet Manifold: The pipe which connects the intake system to the inlet valve of the engine through which air or air-fuel mixture is drawn into the cylinder is called the inlet ma

Prove that energy is a property of system

Component s of Steam Power Plant: There are four components of steam power plant: 1. The boiler: Hot-source reservoir in which the combustion gases raise steam. 2. Engi


Types in trusses

direct shear and torsional shear of the coil

Determine the minimum thickness required: A thin spherical shell of 2 m diameter is subjected to an internal pressure of 1.5 N/mm 2 . If the allowable tensile stress of the ma

Resultant of Concurrent Force System: The resolution of forces helps in finding out the resultant of a number of forces working on a body as it decrease vectorial addition to

Heat Treatment of Stainless Steels: Stainless steels, as other steels, react to heat treatments over broad range. They are subjected to one or more of the given heat treatmen

Special Tool: Use special tools wherever applicable. Special tools are designed to dismantle or assemble a specific part or assemblies without causing damage to them. Cleaning