Thin walled beam:
We may exploit the thin-walled nature of aircraft structures to make simplifying assumptions in the determination of stresses and deflections produced by bending. Thus, the thickness t of thin-walled sections is assumed to be small compared with their cross-sectional dimensions so that stresses may be regarded as being constant across the thickness. Furthermore, we neglect squares and higher powers oft in the computation of sectional properties and take the section to be represented by the mid-line of its wall. As an illustration of the procedure we shall consider the channel section of Fig. 9.9(a). The section is singly symmetric about the x axis so that Ixy = 0. The second moment of area Ixx is then given by
which reduces, after powers of t2 and upwards are ignored, to
The second moment of area of the section about Cy is obtained in a similar manner. We see, therefore, that for the purpose of calculating section properties we may regard the section as being represented by a single line, as shown in Fig. 9.9(b). Thin-walled sections frequently have inclined or curved walls which complicate the calculation of section properties. Consider the inclined thin section of Fig. 9.10. Its second moment of area about a horizontal axis through its centroid is given by
We note here that these expressions are approximate in that their derivation neglects powers of t2 and upwards by ignoring the second moments of area of the element ds about axes through its own centroid.
The open section beam of arbitrary section shown in Fig. 9.18 supports shear loads Sx and Sy such that there is no twisting of the beam cross-section. For this condition to be valid the shear loads must both pass through a particular point in the cross-section known as the shear centre (see also Section 11.5).
Since there are no hoop stresses in the beam the shear flows and direct stresses acting on an element of the beam wall are related by Eq. (9.22), i.e.