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Introduction

A bar of uniform section fixed at one end and subject to a torque at the extreme end which is applied normal toits axis will twist to some angle which is proportional to the applied torque. This assumes that the bar is notstressed to a level greater than its elastic limit. This page includes various formulas which allow calculationof the angles of twist and the resulting maximums stresses. The equations are based on the followingassumptions

1) The bar is straight and of uniform section2) The material of the bar is has uniform properties.3) The only loading is the applied torque which is applied normal to the axis of the bar.4) The bar is stressed within its elastic limit.

Nomenclature

T = torque (Nm)l = length of bar (m)

J = Polar moment of inertia.(Circular Sections) ( m4

)J' = Polar moment of inertia.(Non circluar sections) ( m

4)

K = Factor replacing J for non-circular sections.( m4

)r = radial distance of point from center of section (m)ro = radius of section OD (m) = shear stress (N/m 2)G Modulus of rigidity (N/m

2)

angle of twist (radians)

Formulas

Formulas for bars of circular section

Formulas for bars of non - circular section.

Bars of non -circular section tend to behave non-symmetrically when under torque and plane sections to notremain plane. Also the distribution of stress in a section is not necessarily linear.

The general formula of torsional stiffness of bars of non-circular section are as shown below the factor J' isdependent of the dimensions of the section and some typical values are shown below. For the circularsection J' = J.


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Testing the values of J' obtained using the above equations (with = 1) with the values obtained from thetable below the following values result.

1)Channel section 430x100x64 calculated J' = 63,118 cm 4 . Table value 63 = 12) Channel section 100x50x10 calculated J' = 2,39 cm

4. Table value 2,53 = 1,06

3)Tee section 305x457x127 calculated J' = 296 cm4. Table value 312 = 1,06

4)Tee section 133x102x137 calculated J' = 2.69 cm4. Table value 2,97 = 1,10

Torsion in Sections. Structrual design

Important Note : In the notes and tables below J is used throughout for the torsion constant for circular andnon circular sections. .This is the convention in structural design

In structural design the use of sections i.e I sections, channel section, angle sections etc. should be avoidedfor applications designed to withstand torsional loading. Hollow rectangular sections are best suited for


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these applications. Note: Values for J and C for square and hollow rectangular sections are provided onwebpages as indexed on webpag e Sections Index

In the steel Sections tables i.e BS EN 10210-2: 1997"Hot finished Rectangular Hollow Sections" & BSEN 10219-2:"Cold Formed Circular Hollow Sections" The Torsion Constant J and the Torsionmodulus constant C are listed. These are calculated as follows.

The Torsion constant (J) for Hollow Rolled Sections are calculated as follows:

For circular hollow sections.... J = 2 I

For square and rectangular hollow sections ...... J = 1/3 t3h + 2k A h

where:I is the second moment of areat is the thickness of sectionh is the mean perimeter = 2 [(B - t) + (D - t)] - 2 R c (4 - p)

Ah is the area enclosed by mean perimeter = (B - t) (D - t) -Rc2

(4 - p)k =2 A h t / hB is the breadth of section

D is the depth of sectionRc is the average of internal and external corner radii.

Torsion modulus constant C

For circular hollow sections.............C = 2.ZZ = elastic modulus = J /r

For square and rectangular hollow sections.............C = J / ( t + k / t )

Note: Values for J and C for square and hollow rectangular sections are provided on webpagesas indexed on webpag e Sections Index

Torsional /Buckling Properties for Hot rolled Sections

A few tables providing Torsional /Buckling properties for some steel sections are indexed below.

Universal BeamsUniversal beams Torsion/BucklingProperties

BS 4:-1 2005)

Universal ColumnsUniversal Columns Torsion/BucklingProperties

BS 4 Part 1: 2005

Rolled Steel Joists Joists Torsion/Buckling Properties BS 4 Part 1: 2005

ChannelsChannels Torsion/BucklingProperties

BS 4 Part 1: 2005

Channels-Parallel FlangedParallel Flanged ChannelsTorsion/Buckling Properties

BS 4 Part 1: 2005

Tees cut for Universal columnsTees from UC Torsion/BucklingProperties

BS 4 Part 1: 2005

Tees cut for Universal BeamsTees from UB Torsion/BucklingProperties

BS 4 Part 1: 2005
http://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htmhttp://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htmhttp://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htmhttp://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htmhttp://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htmhttp://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htmhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/J_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/J_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/P_Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/P_Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/P_Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/P_Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/P_Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/Ch_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/J_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UC_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/UB_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htmhttp://www.roymech.co.uk/Useful_Tables/Sections/steel_section_index.htm


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Tees cut from Joists Tees from Joists /Buckling Properties BS 4 Part 1: 2005
http://www.roymech.co.uk/Useful_Tables/Torsion/Joists_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/Joists_Tee_Torque.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/Joists_Tee_Torque.html

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