Page 146 - Aircraft Stuctures for Engineering Student

130 Bending of thin plates

Fig. 5.9 Plate element subjected to bending, twisting and transverse loads.

plane the shear strains yxz and yyz must be zero. However, the transverse load
produces transverse shear forces (and therefore stresses) as shown in Fig. 5.9. We
therefore assume that although -yxz = rxz/G and yyz = ryi/G are negligible the
corresponding shear forces are of the same order of magnitude as the applied load
q and the moments M,, My and Mxy. This assumption is analogous to that made
in a slender beam theory in which shear strains are ignored.
The element of plate shown in Fig. 5.9 supports bending and twisting moments as
previously described and, in addition, vertical shear forces Q, and Qy per unit length
on faces perpendicular to the x and y axes respectively. The variation of shear stresses
rxz and ryz along the small edges Sx, Sy of the element is neglected and the resultant
shear forces QxSy and Qy6x are assumed to act through the centroid of the faces of the
element. From the previous sections

In a similar fashion

tl2 t/2
Q, = j-t12 T~ h, Qy = jPtl2 ryZ dz (5.15)

For equilibrium of the element parallel to Oz and assuming that the weight of the
plate is included in q

or, after simplification

de, aQy (5.16)
+-+q=o
ax ay