Page 159 - Aircraft Stuctures for Engineering Student

5.6 Energy method for the bending of thin plates 143

rotation, of the ends of the element is negative as the slope decreases with increasing x.
The bending strain energy due to M, is then

1
-Mx6Y( -ssx)
a2W
2
Similarly, in the yz plane the contribution of My to the bending strain energy is

1
-M,,SX ( $ )
--sy
2
The strain energy due to the twisting moment per unit length, Mxy, applied to the by
edges of the element, is obtained from Fig. 5.14(b). The relative rotation of the by
edges is (#w/axay)sx so that the corresponding strain energy is

Finally, the contribution of the twisting moment Mxy on the Sx edges is, in a similar
fashion
1 @W
- MXYSX -
6y
2 axay
The total strain energy of the element from bending and twisting is thus
a2w a2W
1 ( - M, - My@ + 2Mxy- axay ) sxsy
2

Substitution for M,, My and Mxy from Eqs (5.7), (5.8) and (5.14) gives the total strain
energy of the element as

which on rearranging becomes

Hence the total strain energy U of the rectangular plate a x b is

Note that if the plate is subject to pure bending only, then M,, = 0 and from Eq.
(5.14) @w/axay = 0, so that Eq. (5.37) simplifies to