Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

19.4 Deflection of Open and Closed Section Beams 555

whereagainthesuffixesrefertotheactualandunitloadingsystems.Inthecasesofbothopenandclosed
sectionbeams,thegeneralexpressionsforshearflowarelongandarebestevaluatedbeforesubstituting
inEq.(19.19).Foranopensectionbeamcomprisingboomsandwallsofdirectstress-carryingthickness
tD,wehave,fromEq.(19.6),

q 0 =−

(

Sx,0Ixx−Sy,0Ixy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDxds+

∑n

r= 1

Brxr

⎞

⎠

−

(

Sy,0Iyy−Sx,0Ixy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDyds+

∑n

r= 1

Bryr

⎞

⎠

(19.20)

and

q 1 =−

(

Sx,1Ixx−Sy,1Ixy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDxds+

∑n

r= 1

Brxr

⎞

⎠

−

(

Sy,1Iyy−Sx,1Ixy

IxxIyy−Ixy^2

)⎛

⎝

∫s

0

tDyds+

∑n

r= 1

Bryr

⎞

⎠

(19.21)

SimilarexpressionsareobtainedforaclosedsectionbeamfromEq.(19.11).

Example 19.5
Calculatethedeflectionofthefreeendofacantilever2000mmlonghavingachannelsectionidentical
tothatinExample19.3andsupportingavertical,upwardloadof4.8kNactingthroughtheshearcenter
ofthesection.Theeffectivedirectstress-carryingthicknessoftheskiniszero,whileitsactualthickness
is1mm.Young’smodulusEandtheshearmodulusGare70000and30000N/mm^2 ,respectively.

Thesectionisdoublysymmetrical(i.e.,thedirectstress-carryingarea)andsupportsaverticalload
producing a vertical deflection. Thus, we apply a unit load through the shear center of the section at
thetipofthecantileverandinthesamedirectionastheappliedload.Sincetheloadisappliedthrough
theshearcenter,thereisnotwistingofthesection,andthetotaldeflectionisgiven,fromEqs.(19.17),
(19.19),(19.20),and(19.21),by

=

∫L

0

Mx,0Mx,1

EIxx

dz+

∫L

0

⎛

⎝

∫

sect

q 0 q 1

Gt

ds

⎞

⎠dz (i)

whereMx,0=−4.8× 103 ( 2000 −z),Mx,1=− 1 ( 2000 −z)

q 0 =−

4.8× 103

Ixx

∑n

r= 1

Bryr q 1 =−

1

Ixx

∑n

r= 1

Bryr