Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

244 CHAPTER 7 Bending of Thin Plates

7.6.3 Potential Energy of In-Plane Loads

We may consider each loadNx,Ny,andNxyin turn, and 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 widthδyalong the lengthaof the plate in Fig. 7.15(a). The compressive load on
thisstripisNxδy,andduetothebendingoftheplate,thehorizontallengthofthestripdecreasesbyan
amountλ,asshowninFig.7.15(b).ThepotentialenergyδVxoftheloadNxδy,referredtotheundeflected
positionoftheplateasthedatum,isthen

δVx=−Nxλδy (7.40)

FromFig.7.15(b),thelengthofasmallelementδaofthestripis

δa=(δx^2 +δw^2 )

1

2

Fig.7.15

(a) In-plane loads on plate; (b) shortening of element due to bending.