Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.6 Energy Method for the Bending of Thin Plates 243

Thetotalstrainenergyoftheelementfrombendingandtwistingisthus

1

2

(

−Mx

∂^2 w

∂x^2

−My

∂^2 w

∂y^2

+ 2 Mxy

∂^2 w

∂x∂y

)

δxδy

SubstitutionforMx,My,andMxyfromEqs.(7.7),(7.8),and(7.14)givesthetotalstrainenergyofthe
elementas

D

2

[(

∂^2 w

∂x^2

) 2

+

(

∂^2 w

∂y^2

) 2

+ 2 ν

∂^2 w

∂x^2

∂^2 w

∂y^2

+ 2 ( 1 −ν)

(

∂^2 w

∂x∂y

) 2 ]

δxδy

whichonrearrangingbecomes

D

2

{(

∂^2 w

∂x^2

+

∂^2 w

∂y^2

) 2

− 2 ( 1 −ν)

[

∂^2 w

∂x^2

∂^2 w

∂y^2

−

(

∂^2 w

∂x∂y

) 2 ]}

δxδy

Hence,thetotalstrainenergyUoftherectangularplatea×bis

U=

D

2

∫a

0

∫b

0

{(

∂^2 w

∂x^2

+

∂^2 w

∂y^2

) 2

− 2 ( 1 −ν)

[

∂^2 w

∂x^2

∂^2 w

∂y^2

−

(

∂^2 w

∂x∂y

) 2 ]}

dxdy (7.37)

Notethatiftheplateissubjecttopurebendingonly,thenMxy=0,andfromEq.(7.14)∂^2 w/∂x∂y=0,
sothatEq.(7.37)simplifiesto

U=

D

2

∫a

0

∫b

0

[(

∂^2 w

∂x^2

) 2

+

(

∂^2 w

∂y^2

) 2

+ 2 ν

∂^2 w

∂x^2

∂^2 w

∂y^2

]

dxdy (7.38)

7.6.2 Potential Energy of a Transverse Load

Anelementδx×δyofthetransverselyloadedplateofFig.7.8supportsaloadqδxδy.Ifthedisplacement
oftheelementnormaltotheplateisw,thenthepotentialenergyδVoftheloadontheelementreferred
totheundeflectedplatepositionis

δV=−wqδxδy SeeSection5.7

Therefore,thepotentialenergyVofthetotalloadontheplateisgivenby

V=−

∫a

0

∫b

0

wqdxdy (7.39)