Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.3 Deflections due to Bending 441

or,whensecond-ordertermsareneglected

Sy=

∂Mx

∂z

Wemaycombinetheseresultsintoasingleexpression

−wy=

∂Sy

∂z

=

∂^2 Mx

∂z^2

(15.23)

Similarlyforloadsinthexzplane,

−wx=

∂Sx

∂z

=

∂^2 My

∂z^2

(15.24)

15.3 DeflectionsduetoBending.........................................................................

Wehavenotedthatabeambendsaboutitsneutralaxiswhoseinclinationrelativetoarbitrarycentroidal
axes is determined from Eq. (15.22). Suppose that at some section of an unsymmetrical beam the
deflectionnormaltotheneutralaxis(andthereforeanabsolutedeflection)isζ,asshowninFig.15.15.
Inotherwords,thecentroidCisdisplacedfromitsinitialpositionCIthroughanamountζtoitsfinal
positionCF.SupposealsothatthecenterofcurvatureRofthebeamatthisparticularsectionisonthe
oppositesideoftheneutralaxistothedirectionofthedisplacementζandthattheradiusofcurvature
isρ. For this position of the center of curvature and from the usual approximate expression for cur-
vature,wehave

1

ρ

=

d^2 ζ

dz^2

(15.25)

Fig.15.15

Determination of beam deflection due to bending.