Page 154 - Aircraft Stuctures for Engineering Student

138 Bending of thin plates

aNx
.- 6X
ax
SX

Y

Fig. 5.12 In-plane forces on plate element.

the element in these directions. For equilibrium parallel to Ox

aw a2w
( 2 ) (ax ax2 ) -N,~YCOS-
aW
SYCOS -+-SX
N,+-SX
ax
SX - NYxSx = 0

For small deflections awlax and (awlax) + (#w/a?)Sx are small and the cosines of
these angles are therefore approximately equal to one. The equilibrium equation thus
simplifies to

(5.31)

Similarly for equilibrium in they direction we have

(5.32)

Note that the components of the in-plane shear loads per unit length are, to a first
order of approximation, the value of the shear load multiplied by the projection of
the element on the relevant axis.
The determination of the contribution of the shear loads to the equilibrium of the
element in the z direction is complicated by the fact that the element possesses
curvature in both xz and yz planes. Therefore, from Fig. 5.13 the component in the