2.6 Bending of an End-Loaded Cantilever 59Thedeflectioncurvefortheneutralplaneis
(v)y= 0 =Px^3
6 EI−
Pl^2 x
2 EI+
Pl^3
3 EI(xiii)from which the tip deflection (x=0)isPl^3 / 3 EI. This value is that predicted by simple beam theory
(Chapter 15) and does not include the contribution to deflection of the shear strain. This was elim-
inated when we assumed that the slope of the neutral plane at the built-in end was zero. A more
detailed examination of this effect is instructive. The shear strain at any point in the beam is given
byEq.(vi)
γxy=−P
8 IG
(
b^2 − 4 y^2)
andisobviouslyindependentofx.Therefore,atallpointsontheneutralplanetheshearstrainisconstant
andequalto
γxy=−Pb^2
8 IG,
whichamountstoarotationoftheneutralplaneasshowninFig.2.7.Thedeflectionoftheneutralplane
duetothisshearstrainatanysectionofthebeamisthereforeequalto
Pb^2
8 IG(l−x)andEq.(xiii)mayberewrittentoincludetheeffectofshearas
(v)y= 0 =Px^3
6 EI−
Pl^2 x
2 EI+
Pl^3
3 EI+
Pb^2
8 IG(l−x) (xiv)Letusnowexaminethedistortedshapeofthebeamsection,whichtheanalysisassumesisfreeto
takeplace.Atthebuilt-inendwhenx=lthedisplacementofanypointis,fromEq.(xi)
u=νPy^3
6 EI+
Py^3
6 IG−
Pb^2 y
8 IG(xv)Fig.2.7
Rotation of neutral plane due to shear in end-loaded cantilever.