2.6 Bending of an End-Loaded Cantilever 57(2) the distribution of shear and direct stresses at the built-in end is the same as those given by
Eqs.(iii)
(3) allsectionsofthebeam,includingthebuilt-inend,arefreetodistort
Inpracticalcasesnoneoftheseconditionsissatisfied,butbyvirtueofSt.Venant’sprinciplewemay
assume that the solution is exact for regions of the beam away from the built-in end and the applied
load. For many solid sections, the inaccuracies in these regions are small. However, for thin-walled
structures,withwhichweareprimarilyconcerned,significantchangesoccur.
We now proceed to determine the displacements corresponding to the stress system of Eqs. (iii).
Applying the strain–displacement and stress–strain relationships, Eqs. (1.27), (1.28), and (1.47), we
have
εx=∂u
∂x=
σx
E=−
Pxy
EI(iv)εy=∂v
∂y=−
νσx
E=
νPxy
EI(v)γxy=∂u
∂y+
∂v
∂x=
τxy
G=−
P
8 IG
(
b^2 − 4 y^2)
(vi)IntegratingEqs.(iv)and(v)andnotingthatεxandεyarepartialderivativesofthedisplacements,we
find
u=−Px^2 y
2 EI+f 1 (y) v=νPxy^2
2 EI+f 2 x (vii)wheref 1 (y)andf 2 (x)areunknownfunctionsofxandy.SubstitutingthesevaluesofuandvinEq.(vi)
−
Px^2
2 EI+
∂f 1 (y)
∂y+
νPy^2
2 EI+
∂f 2 (x)
∂x=−
P
8 IG
(
b^2 − 4 y^2)
Separatingthetermscontainingxandyinthisequationandwriting
F 1 (x)=−Px^2
2 EI+
∂f 2 (x)
∂xF 2 (y)=νPy^2
2 EI−
Py^2
2 IG+
∂f 1 (y)
∂ywehave
F 1 (x)+F 2 (y)=−Pb^2
8 IGThetermontheright-handsideofthisequationisaconstant,whichmeansthatF 1 (x)andF 2 (y)
mustbeconstants,otherwiseavariationofeitherxorywoulddestroytheequality.DenotingF 1 (x)by
CandF 2 (y)byDgives
C+D=−
Pb^2
8 IG(viii)