Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

52 CHAPTER 2 Two-Dimensional Problems in Elasticity

Further,

∂^4 φ

∂x^4

= 0

∂^4 φ

∂y^4

=− 120 Dy

∂^4 φ

∂x^2 ∂y^2

= 60 Dy

SubstitutinginEq.(2.9)gives

∂^4 φ

∂x^4

+ 2

∂^4 φ

∂x^2 ∂y^2

+

∂^4 φ

∂y^4

= 2 × 60 Dy− 120 Dy= 0

Therefore,thebiharmonicequationissatisfied,andthestressfunctionisvalid.
FromFig.2.3,σy=0aty=hsothat,fromEq.(i)

2 A+ 2 BH+ 10 Dh^3 =0(iv)

Also,fromFig.2.3,σy=−qaty=−hsothat,fromEq.(i)

2 A− 2 BH− 10 Dh^3 =−q (v)

Again,fromFig.2.3,τxy=0aty=±hgiving,fromEq.(iii)

2 Bx+ 30 Dxh^2 = 0

sothat

2 B+ 30 Dh^2 = 0 (vi)

Atx=0,thereisnoresultantmomentappliedtothebeam;thatis,

Mx= 0 =

∫h

−h

σxydy=

∫h

−h

( 6 Cy^2 − 20 Dy^4 )dy= 0

thatis,

Mx= 0 =[2Cy^3 − 4 Dy^5 ]h−h= 0

or

C− 2 Dh^2 = 0 (vii)

SubtractingEq.(v)from(iv)

4 Bh+ 20 Dh^3 =q

or

B+ 5 Dh^2 =

q

4 h

(viii)