Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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

Fig.15.18

Deflection of a simply supported beam carrying a uniformly distributed load.

SubstitutingforMinthesecondofEq.(15.32),weobtain

EIv′′=

w

2

(Lz−z^2 ) (ii)

Integrating,wehave

EIv′=

w

2

(

Lz^2

2

−

z^3

3

)

+C 1

Fromsymmetryitisclearthatatthemidspansectionthegradientv′=0.
Hence,

0 =

w

2

(

L^3

8

−

L^3

24

)

+C 1

whichgives

C 1 =−

wL^3

24

Therefore,

EIv′=

w

24

( 6 Lz^2 − 4 z^3 −L^3 ) (iii)

Integratingagaingives

EIv=

w

24

( 2 Lz^3 −z^4 −L^3 z)+C 2