444 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams
whereC 1 isaconstantofintegrationwhichisobtainedfromtheboundaryconditionthatv′=0atthe
built-inendwherez=0.Hence,C 1 =0and
EIv′=−W(
Lz−z^2
2)
(iii)IntegratingEq.(iii),weobtain
EIv=−W(
Lz^2
2−
z^3
6)
+C 2
inwhichC 2 isagainaconstantofintegration.Atthebuilt-inendv=0whenz=0sothatC 2 =0.Hence,
theequationofthedeflectioncurveofthecantileveris
v=−W
6 EI
( 3 Lz^2 −z^3 ) (iv)Thedeflection,vtip,atthefreeendisobtainedbysettingz=LinEq.(iv).Then
vtip=−WL^3
3 EI
(v)andisclearlynegativeanddownward.
Example 15.6
DeterminethedeflectioncurveandthedeflectionofthefreeendofthecantilevershowninFig.15.17(a).
Thecantileverhasadoublysymmetricalcrosssection.
Fig.15.17
Deflection of a cantilever beam carrying a uniformly distributed load.