Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

454 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams

Inthisproblem,anexternalmomentM 0 isappliedtothebeamatB.Thesupportreactionsarefound
inthenormalwayandare

RA=−

M 0

L

(downward) RC=

M 0

L

(upward)

ThebendingmomentatanysectionZbetweenBandCisthengivenby

M=−RAz−M 0 (i)

Equation(i)isvalidonlyfortheregionBCandclearlydoesnotcontainasingularityfunctionwhich
wouldcauseM 0 tovanishforz≤b.Weovercomethisdifficultybywriting

M=−RAz−M 0 [z−b]^0 (Note:[z−b]^0 = 1 ) (ii)

Equation (ii) has the same value as Eq. (i) but is now applicable to all sections of the beam,
since [z−b]^0 disappears whenz≤b. Substituting forMfrom Eq. (ii) in the second of Eq. (15.32),
weobtain

EIv′′=RAz+M 0 [z−b]^0 (iii)

IntegrationofEq.(iii)yields

EIv′=RA

z^2

2

+M 0 [z−b]+C 1 (iv)

and

EIv=RA

z^3

6

+

M 0

2

[z−b]^2 +C 1 z+C 2 ,(v)

whereC 1 andC 2 arearbitraryconstants.Theboundaryconditionsarev=0whenz=0andz=L.From
thefirstofthesewehaveC 2 =0,whilethesecondgives

0 =−

M 0

L

L^3

6

+

M 0

2

[L−b]^2 +C 1 L

fromwhich

C 1 =−

M 0

6 L

( 2 L^2 − 6 Lb+ 3 b^2 )

Theequationofthedeflectioncurveofthebeamisthen

v=

M 0

6 EIL

{z^3 + 3 L[z−b]^2 −( 2 L^2 − 6 Lb+ 3 b^2 )z} (vi)