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)