Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

150 CHAPTER 5 Energy Methods

Fig.5.25

Approximate determination of beam deflection using total potential energy.

inwhichvBisthedisplacementatthemidspanpoint.FromEq.(i),weseethatv=0whenz=0andz=L
and thatv=vBwhenz=L/2. Also dv/dz=0whenz=L/2 so that the displacement function satisfies
theboundaryconditionsofthebeam.
Thestrainenergy,U,duetobendingofthebeamisgiveninStructuralandStressAnalysis[Ref.3]

U=

∫

L

M^2

2 EI

dz (ii)

Also,

M=−EI

d^2 v

dz^2

(seeChapter15) (iii)

SubstitutinginEq.(iii)forvfromEq.(i)andforMinEq.(ii)from(iii)

U=

EI

2

∫L

0

v^2 Bπ^4

L^4

sin^2

πz

L

dz

whichgives

U=

π^4 EIvB^2

4 L^3

TheTPEofthebeamisthengivenby

TPE=U+V=

π^4 EIv^2 B

4 L^3

−WvB

Then,fromtheprincipleofthestationaryvalueoftheTPE,

∂(U+V)

∂vB

=

π^4 EIvB

2 L^3

−W= 0