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
Calculate the force - pulley: A string is passing over a smooth pulley. The string supports the mass m on one side which is to be pulled up with acceleration a. calculate the

Calculating the number of kanban card sets: A hospital desires to set up a Kanban system to manage its supply of blood with the regional blood bank. The regional blood bank tr

Answers all questions i. Illustrate Zeroth law of thermodynamics. ii. A gas contained in a cylinder is compressed, the work neede for compression being 5000 KJ. During the pr

Objectives After studying basics troubleshoot of motorcycle, you should be able to check, perform inspection and identify the reason(s) of various problems reported by

Dynamically Equivalent Link: this is link that has point masses at two points such that (a) overall mass is same, (b) centre of gravity is at the similar position, and (c) mass m

Q. Show Hazard assessment of plant operation? Using the hazard assessment of each plant operation will help establish the layout or orientation of process units within the plan

Solve the support reactions at A and D by using the force method. A is a fixed support; B and C are rigid joints; and D is a pin support. EI is constant for all members.

Q. Show Bolt Material? Bolt material must be selected with adequate yield strength at the operating temperature to provide the required strength to properly seat the gasket. It