Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

4.3 Applications of the Principle of Virtual Work 103

tookplace.Theexternalvirtualworkdonebytheunitloadis,fromFig.4.10(b),1υB.Thedeflection,υB,
isassumedtobecausedbybendingonly;inotherwords,weareignoringanydeflectionsduetoshear.
TheinternalvirtualworkisgivenbyEq.(4.21),which,sinceonlyonememberisinvolved,becomes

Wi,M=

∫L

0

MAMv

EI

dx (i)

Thevirtualmoments,Mv,areproducedbyaunitloadsothatweshallreplaceMvbyM 1 .Then

Wi,M=

∫L

0

MAM 1

EI

dx (ii)

Atanysectionofthebeamadistancexfromthebuilt-inend

MA=−

w

2

(L−x)^2 M 1 =− 1 (L−x)

SubstitutingforMAandM 1 inEq.(ii)andequatingtheexternalvirtualworkdonebytheunitloadto
theinternalvirtualwork,wehave

1 υB=

∫L

0

w

2 EI

(L−x)^3 dx

whichgives

υB=−

w

2 EI

[

1

4

(L−x)^4

]L

0

sothat

υB=

wL^4

8 EI

NotethatυBisinfactnegative,butthepositivesignhereindicatesthatitisinthesamedirectionasthe
unitload.

Example 4.5
Determinetherotation—thatis,theslope—ofthebeamABCshowninFig.4.11(a)atA.

The actual rotation of the beam at A produced by the actual concentrated load,W,isθA. Let us
supposethatavirtualunitmomentisappliedatAbeforetheactualrotationtakesplace,asshownin
Fig.4.11(b).ThevirtualunitmomentinducesvirtualsupportreactionsofRv,A(=1/L)actingdownward
andRv,C(=1/L)actingupward.Theactualinternalbendingmomentsare

MA=+

W

2

x 0 ≤x≤L/ 2

MA=+

W

2

(L−x) L/ 2 ≤x≤L