Page 143 - Aircraft Stuctures for Engineering Student

5.2 Plates subjected to bending and twisting 127

(Compare Eqs (5.10) and (5.1 1) with Eqs (1.8) and (1.9).) We observe from Eq. (5.11)
that there are two values of a, differing by 90” and given by

2Mxy
tan20 = -
Mx - My
for which Mt = 0, leaving normal moments of intensity M, on two mutually
perpendicular planes. These moments are termed principal moments and their
corresponding curvatures principal curvatures. For a plate subjected to pure bending
and twisting in which M,, My and Mxy are invariable throughout the plate, the
principal moments are the algebraically greatest and least moments in the plate. It
follows that there are no shear stresses on these planes and that the corresponding
direct stresses, for a given value of z and moment intensity, are the algebraically
greatest and least values of direct stress in the plate.
Let us now return to the loaded plate of Fig. 5.5(a). We have established, in
Eqs (5.7) and (5.8), the relationships between the bending moment intensities M,
and My and the deflection w of the plate. The next step is to relate the twisting
moment Mxy to w. From the principle of superposition we may consider Mxy
acting separately from M, and My. As stated previously Mxy is resisted by a
system of horizontal complementary shear stresses on the vertical faces of sections
taken throughout the thickness of the plate parallel to the x and y axes. Consider
an element of the plate formed by such sections, as shown in Fig. 5.6. The complemen-
tary shear stresses on a lamina of the element a distance z below the neutral plane are,
in accordance with the sign convention of Section 1.2, ?xy. Therefore, on the face
ABCD

D
Fig. 5.6 Complementary shear stresses due to twisting moments Mv.