Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

430 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams

Example 15.2
NowdeterminethedistributionofdirectstressinthebeamofExample15.1ifthebendingmomentis
appliedinahorizontalplaneandinaclockwisesenseaboutCywhenviewedinthedirectionyC.

Inthiscase,thebeamwillbendabouttheverticalyaxiswhichthereforebecomestheneutralaxis
ofthesection.Thus,Eq.(15.9)becomes

σz=

M

Iyy

x,(i)

whereIyyisthesecondmomentofareaofthebeamsectionabouttheyaxis.AgainfromSection15.4,

Iyy= 2 ×

20 × 2003

12

+

260 × 253

12

=27.0× 106 mm^4

Hence,substitutingforMandIyyinEq.(i)

σz=

100 × 106

27.0× 106

x=3.7x

We have not specified a sign convention for bending moments applied in a horizontal plane.
However, a physical appreciation of the problem shows that the left-hand edges of the beam are
in compression, while the right-hand edges are in tension. Again the distribution is linear and
varies from 3.7×(−100)=−370N/mm^2 (compression) at the left-hand edges of each flange to
3.7×(+100)=+370N/mm^2 (tension)attheright-handedges.
WenotethatthemaximumstressesinthisexampleareverymuchgreaterthanthoseinExample15.1.
Thisisduetothefactthatthebulkofthematerialinthebeamsectionisconcentratedintheregionof
theneutralaxiswherethestressesarelow.TheuseofanI-sectioninthismannerwouldthereforebe
structurallyinefficient.

Example 15.3
ThebeamsectionofExample15.1issubjectedtoabendingmomentof100kNmappliedinaplane
paralleltothelongitudinalaxisofthebeambutinclinedat30◦totheleftofvertical.Thesenseofthe
bendingmomentisclockwisewhenviewedfromtheleft-handedgeofthebeamsection.Determinethe
distributionofdirectstress.

The bending moment is first resolved into two components,Mxin a vertical plane andMyin a
horizontalplane.Equation(15.9)maythenbewrittenintwoforms

σz=

Mx

Ixx

y σz=

My

Iyy

x (i)

Theseparatedistributionscanthenbedeterminedandsuperimposed.Amoredirectmethodistocombine
thetwoequations(i)togivethetotaldirectstressatanypoint(x,y)inthesection.Thus,

σz=

Mx

Ixx

y+

My

Iyy

x (ii)