Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

490 CHAPTER 16 Shear of Beams

Fig.16.11

(a) Determination ofqs,0; (b) equivalent loading on “open” section beam.

We obtainqbby supposing that the closed beam section is “cut” at some convenient point, thereby
producingan“open”section(seeFig.16.11(b)).Theshearflowdistribution(qb)aroundthis“open”
sectionisgivenby

qb=−

(

SxIxx−SyIxy

IxxIyy−I^2 xy

)∫s

0

txds−

(

SyIyy−SxIxy

IxxIxy−Ixy^2

)∫s

0

tyds

as in Section 16.2. The value of shear flow at the cut (s=0) is then found by equating applied and
internalmomentstakenaboutsomeconvenientmomentcenter.Then,fromFig.16.11(a),

Sxη 0 −Syξ 0 =

∮

pqds=

∮

pqbds+qs,0

∮

pds,

where

∮

denotesintegrationcompletelyaroundthecrosssection.InFig.16.11(a),

δA=

1

2

δsp

sothat
∮
dA=

1

2

∮

pds

Hence,
∮
pds= 2 A

whereAistheareaenclosedbythemidlineofthebeamsectionwall.Hence,

Sxη 0 −Syξ 0 =

∮

pqbds+ 2 Aqs,0 (16.17)