Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.3 Deflections due to Bending 445

Thebendingmoment,M,atanysectionZisgivenby

M=

w

2

(L−z)^2 (i)

SubstitutingforMinthesecondofEq.(15.32)andrearranging,wehave

EIv′′=−

w

2

(L−z)^2 =−

w

2

(L^2 − 2 Lz+z^2 ) (ii)

IntegrationofEq.(ii)yields

EIv′=−

w

2

(

L^2 z−Lz^2 +

z^3

3

)

+C 1

Whenz=0atthebuilt-inend,v′=0,sothatC 1 =0and

EIv′=−

w

2

(

L^2 z−Lz^2 +

z^3

3

)

(iii)

IntegratingEq.(iii),wehave

EIv=−

w

2

(

L^2

z^2

2

−

Lz^3

3

+

z^4

12

)

+C 2

andsincev=0whenx=0,C 2 =0.Thedeflectioncurveofthebeam,therefore,hastheequation

v=−

w

24 EI

( 6 L^2 z^2 − 4 Lz^3 +z^4 ) (iv)

andthedeflectionatthefreeendwherex=Lis

vtip=−

wL^4

8 EI

(v)

whichisagainnegativeanddownward.

Example 15.7
Determine the deflection curve and the midspan deflection of the simply supported beam shown in
Fig.15.18(a);thebeamhasadoublysymmetricalcrosssection.

ThesupportreactionsareeachwL/2andthebendingmoment,M,atanysectionZ,adistancezfrom
theleft-handsupportis

M=−

wL

2

z+

wz^2

2

(i)