Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.3 Deflections due to Bending 453

fromwhich

C 1 =−

27 wL^3

2048

Equations(iv)and(v)thenbecome

EIv′=

3

64

wLz^2 −

w

6

[

z−

L

2

] 3

+

w

6

[

z−

3 L

4

] 3

−

27 wL^3

2048

(vi)

and

EIv=

wLz^3

64

−

w

24

[

z−

L

2

] 4

+

w

24

[

z−

3 L

4

] 4

−

27 wL^3

2048

z (vii)

Inthisproblem,themaximumdeflectionclearlyoccursintheregionBCofthebeam.Thus,equating
theslopetozeroforBC,wehave

0 =

3

64

wLz^2 −

w

6

[

z−

L

2

] 3

−

27 wL^3

2048

whichsimplifiesto

z^3 −1.78Lz^2 +0.75zL^2 −0.046L^3 = 0 (viii)

SolvingEq.(viii)bytrialanderror,weseethattheslopeiszeroatz 0.6L.HencefromEq.(vii),the
maximumdeflectionis

vmax=−

4.53× 10 −^3 wL^4

EI

Example 15.11
DeterminethedeflectedshapeofthebeamshowninFig.15.23.

Fig.15.23

Deflection of a simply supported beam carrying a point moment.