Page 160 - Aircraft Stuctures for Engineering Student

144 Bending of thin plates

5.6.2 Potential energy of a transverse load

An element Sx x Sy of the transversely loaded plate of Fig. 5.8 supports a load qSxSy.
If the displacement of the element normal to the plate is w then the potential energy
SV of the load on the element referred to the undeflected plate position is
SV = -wqSxSy (See Section 4.2)
Therefore, the potential energy V of the total load on the plate is given by
ab
V= -1 1 wqdxdy (5.39)
0 0

5.6.3 Potential energy of in-plane loads

We may consider each load N,, Ny and Nxy in turn, then use the principle of super-
position to determine the potential energy of the loading system when they act
simultaneously. Consider an elemental strip of width Sy along the length a of the
plate in Fig. 5.15(a). The compressive load on this strip is N,Sy and due to the bending
of the plate the horizontal length of the strip decreases by an amount A, as shown in
Fig. 5.15(b). The potential energy SV, of the load NJy, referred to the undeflected
position of the plate as the datum, is then

SV, = -N,ASy (5.40)
From Fig. 5.15(b) the length of a small element Sa of the strip is

Sa = (Sx2 + SW2)t
)’I
and since awlax is small then
Sa M sx [1+; (g

Hence

giving
dl
a=d+JoT(ax> dx
dw
and

A=a-d=j afl -(-) d~
dw
0 2 ax
Since

jr (g ) ’ dx only differs from j:i (g)*dx