Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

246 CHAPTER 7 Bending of Thin Plates

Fig.7.16

Calculation of shear strain corresponding to bending deflection.

reduction in the right angle C 2 AB 1 to the angle C 1 AB 1 of the element in Fig. 7.16 or, rotating C 2 A
with respect to AB 1 to AD in the plane C 1 AB 1 , the angle DAC 1. The displacement C 2 Disequalto
(∂w/∂y)δy,andtheangleDC 2 C 1 is∂w/∂x.Thus,C 1 Disequalto

∂w

∂x

∂w

∂y

δy

andtheangleDAC 1 representingtheshearstraincorrespondingtothebendingdisplacementwis

∂w

∂x

∂w

∂y

sothattheworkdoneontheelementbytheshearforceNxyδxis

1

2

Nxyδx

∂w

∂x

∂w

∂y

Similarly,theworkdonebytheshearforceNxyδyis

1

2

Nxyδy

∂w

∂x

∂w

∂y

andthetotalworkdonetakenoverthecompleteplateis

1

2

∫a

0

∫b

0

2 Nxy

∂w

∂x

∂w

∂y

dxdy