Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

58 CHAPTER 2 Two-Dimensional Problems in Elasticity

and

∂f 2 (x)

∂x

=

Px^2

2 EI

+C

∂f 1 (y)

∂y

=

Py^2

2 IG

−

νPy^2

2 EI

+D

sothat

f 2 (x)=

Px^3

6 EI

+Cx+F

and

f 1 (y)=

Py^3

6 IG

−

νPy^3

6 EI

+Dy+H

Therefore,fromEqs.(vii)

u=−

Px^2 y

2 EI

−

νPy^3

6 EI

+

Py^3

6 IG

+Dy+H (ix)

v=

νPxy^2

2 EI

+

Px^3

6 EI

+Cx+F (x)

The constantsC,D,F,andHare now determined from Eq. (viii) and the displacement boundary
conditionsimposedbythesupportsystem.Assumingthatthesupportpreventsmovementofthepoint
Kinthebeamcrosssectionatthebuilt-inend,thenu=v=0atx=l,y=0,andfromEqs.(ix)and(x)

H= 0 F=−

Pl^3

6 EI

−Cl

Ifwenowassumethattheslopeoftheneutralplaneiszeroatthebuilt-inend,then∂v/∂x=0atx=l,
y=0,andfromEq.(x)

C=−

Pl^2

2 EI

Itfollowsimmediatelythat

F=

Pl^3

2 EI

and,fromEq.(viii)

D=

Pl^2

2 EI

−

Pb^2
8 IG
SubstitutionfortheconstantsC,D,F,andHinEqs.(ix)and(x)nowproducestheequationsforthe
componentsofdisplacementatanypointinthebeam.Thus,

u=−

Px^2 y

2 EI

−

νPy^3

6 EI

+

Py^3

6 IG

+

(

Pl^2

2 EI

−

Pb^2

8 IG

)

y (xi)

v=

νPxy^2

2 EI

+

Px^3

6 EI

−

Pl^2 x

2 EI

+

Pl^3

3 EI

(xii)