20.2 Open and Closed Section Beams 565Fig.20.3
Shear flow (N/mm) distribution at Section AA in Example 20.1.
thatis,
q 12 =6.52× 10 −^4 (−s^2 + 300 s+ 60000 ) (ii)Themaximumvalueofq 12 occurswhens=150mmandq 12 (max)=53.8N/mm.Thevaluesofshear
flow at points 1 (s=0)and2(s=300mm) areq 1 =39.1 N/mm andq 2 =39.1 N/mm; the complete
distributionisshowninFig.20.3.
20.2 OpenandClosedSectionBeams...................................................................
Weshallnowconsiderthemoregeneralcaseofabeamtaperedintwodirectionsalongitslengthand
comprisinganarrangementofboomsandskin.Practicalexamplesofsuchabeamarecompletewings
and fuselages. The beam may be of open or closed section; the effects of taper are determined in an
identicalmannerineithercase.
Figure 20.4(a) shows a short lengthδzof a beam carrying shear loadsSxandSyat the sectionz;
SxandSyarepositivewhenactinginthedirectionsshown.Notethatifthebeamwereofopencross
section, the shear loads would be applied through its shear center so that no twisting of the beam
occurred. In addition to shear loads, the beam is subjected to bending momentsMxandMy,which
produce direct stressesσzin the booms and skin. Suppose that in therth boom the direct stress in a
direction parallel to thezaxis isσz,r, which may be found using either Eq. (15.18) or Eq. (15.19).
ThecomponentPz,roftheaxialloadPrintherthboomisthengivenby
Pz,r=σz,rBr (20.8)whereBristhecross-sectionalareaoftherthboom.
FromFig.20.4(b),
Py,r=Pz,rδyr
δz