Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.3 Deflections due to Bending 443

Fig.15.16

Deflection of a cantilever beam carrying a concentrated load at its free end (Example 15.5).

The load W causes the cantilever to deflect such that its neutral plane takes up the curved
shapeshownFig.15.16(b);thedeflectionatanysectionZisthenv,whilethatatitsfreeendisvtip.The
axissystemischosensothattheorigincoincideswiththebuilt-inendwherethedeflectionisclearly
zero.
Thebendingmoment,M,atthesectionZis,fromFig.15.16(a),

M=W(L−z) (i)

SubstitutingforMinthesecondofEq.(15.32)

v′′=−

W

EI

(L−z)

orinmoreconvenientform

EIv′′=−W(L−z) (ii)

IntegratingEq.(ii)withrespecttozgives

EIv′′=−W

(

Lz−

z^2

2

)

+C 1