Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.6 Energy Method for the Bending of Thin Plates 245

andsince∂w/∂xissmall,then

δa≈δx

[

1 +

1

2

(

∂w

∂x

) 2 ]

Hence,

a=

∫a′

0

[

1 +

1

2

(

∂w

∂x

) 2 ]

dx

giving

a=a′+

∫a′

0

1

2

(

∂w

∂x

) 2

dx

and

λ=a−a′=

∫a′

0

1

2

(

∂w

∂x

) 2

dx

Since

∫a′

0

1

2

(

∂w

∂x

) 2

dx onlydiffersfrom

∫a

0

1

2

(

∂w

∂x

) 2

dx

byatermofnegligibleorder,wewrite

λ=

∫a

0

1

2

(

∂w

∂x

) 2

dx (7.41)

ThepotentialenergyVxoftheNxloadingfollowsfromEqs.(7.40)and(7.41);thus,

Vx=−

1

2

∫a

0

∫b

0

Nx

(

∂w

∂x

) 2

dxdy (7.42)

Similarly,

Vy=−

1

2

∫a

0

∫b

0

Ny

(

∂w

∂y

) 2

dxdy (7.43)

Thepotentialenergyofthein-planeshearloadNxymaybefoundbyconsideringtheworkdoneby
Nxyduringthesheardistortioncorrespondingtothedeflectionwofanelement.Thisshearstrainisthe