Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

17.2 Torsion of Open Section Beams 517

Fig.17.11

Warping of an open section beam.

onthesignofAR,whichinturndependsonthesignofpR,theperpendiculardistancefromthecenter
oftwisttothetangentatanypoint.Again,asforclosedsectionbeams,thesignofpRdependsonthe
assumed direction of a positive torque, in this case anticlockwise. Therefore,pR(and thereforeAR)
is positive if movement of the foot ofpRalong the tangent in the assumed direction ofsleads to an
anticlockwise rotation ofpRabout the center of twist. Note that for open section beams the positive
directionofsmaybechosenarbitrarily,since,foragiventorque,thesignofthewarpingdisplacement
dependsonlyonthesignofthesweptareaAR.

Example 17.3
Determine the maximum shear stress and the warping distribution in the channel section shown in
Fig.17.12whenitissubjectedtoananticlockwisetorqueof10Nm.G=25000N/mm^2.

FromthesecondofEqs.(17.13),itcanbeseenthatthemaximumshearstressoccursinthewebof
thesectionwherethethicknessisgreatest.Also,fromthefirstofEqs.(17.11),

J=

1

3

( 2 × 25 ×1.5^3 + 50 ×2.5^3 )=316.7mm^4

sothat

τmax=±

2.5× 10 × 103

316.7

=±78.9N/mm^2

ThewarpingdistributionisobtainedusingEq.(17.20)inwhichtheoriginfors(andhenceAR)istaken
attheintersectionofthewebandtheaxisofsymmetrywherethewarpingiszero.Further,thecenterof
twistRofthesectioncoincideswithitsshearcenterS,withapositionthatisfoundusingthemethod
describedinSection16.2.1,whichgivesξS=8.04mm.InthewallO2

AR=

1

2

×8.04s 1 (pRispositive)