Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

18.2 Shear 531

ThesectionissymmetricalaboutCysothatIxy=0,andsinceSx=0,theshearflowdistributioninthe
closedsection3456is,fromEq.(16.15),

qs=−

Sy

Ixx

∫s

0

tyds+qs,0 (i)

Also,theshearloadisappliedthroughtheshearcenterofthecompletesection—thatis,alongtheaxis
ofsymmetry—sothatintheopenportions123and678theshearflowdistributionis,fromEq.(16.14),

qs=−

Sy

Ixx

∫s

0

tyds (ii)

Wenotethattheshearflowiszeroatthepoints1and8,andthereforetheanalysismayconveniently,
thoughnotnecessarily,beginateitherofthesepoints.Thus,referringtoFig.18.2,

q 12 =−

100 × 103

14.5× 106

∫s^1

0

2 (− 25 +s 1 )ds 1

thatis,

q 12 =−69.0× 10 −^4 (− 50 s 1 +s^21 ) (iii)

fromwhichq 2 =−34.5N/mm.
Examination of Eq. (iii) shows thatq 12 is initially positive and changes sign whens 1 =50mm.
Further,q 12 hasaturningvalue(dq 12 /ds 1 =0)ats 1 =25mmof4.3N/mm.Inthewall23,

q 23 =−69.0× 10 −^4

∫s^2

0

2 ×75ds 2 −34.5

thatis,

q 23 =−1.04s 2 −34.5 (iv)

Hence,q 23 varieslinearlyfromavalueof−34.5N/mmat2to−138.5N/mmat3inthewall23.
Theanalysisoftheopenpartofthebeamsectionisnowcomplete,sincetheshearflowdistribution
inthewalls67and78followsfromsymmetry.Todeterminetheshearflowdistributionintheclosed
partofthesection,wemustusethemethoddescribedinSection16.3,inwhichthelineofactionofthe
shearloadisknown.Thus,we“cut”theclosedpartofthesectionatsomeconvenientpoint,obtainthe
qbor“opensection”shearflowsforthecompletesection,andthentakemomentsasinEqs.(16.17)or
(16.18).However,inthiscase,wemayusethesymmetryofthesectionandloadingtodeducethatthe
finalvalueofshearflowmustbezeroatthemidpointsofthewalls36and45—thatis,qs=qs,0=0at
thesepoints.Hence,

q 03 =−69.0× 10 −^4

∫s^3

0

2 ×75ds 3