Page 147 - Aircraft Stuctures for Engineering Student

5.3 Distributed transverse load 131

Taking moments about the x axis

Simplifying this equation and neglecting small quantities of a higher order than those
retained gives
aiwXy aM, (5.17)
+Q,=O
ax ay
Similarly taking moments about the y axis we have
aMx
ay ax +Q,=O (5.18)
Substituting in Eq. (5.16) for Q, and Qy from Eqs (5.18) and (5.17) we obtain

a2Mx #M,, +---- #My #MXy
--- - -4
ax2 axay ay2 axay
or

(5.19)

Replacing M,, Mxy and My in Eq. (5.19) from Eqs (5.7), (5.14) and (5.8) gives

(5.20)

This equation may also be written

or

The operator (#/ax2 + #/ay2) is the well-known Laplace operator in two dimen-
sions and is sometimes written as V2. Thus

Generally, the transverse distributed load q is a function of x and y so that the
determination of the deflected form of the plate reduces to obtaining a solution of
Eq. (5.20), which satisfies the known boundary conditions of the problem. The
bending and twisting moments follow from Eqs (5.7), (5.8) and (5.14), and the
shear forces per unit length Qx and Q, are found from Eqs (5.17) and (5.18) by