## 99 Effect of idealization on the analysis of open and closed section beams

The addition of direct stress carrying booms to open and closed section beams will clearly modify the analyses presented in the earlier part of this chapter. Before considering individual cases we shall discuss the implications of structural idealization. Generally, in any idealization, different loading conditions require different idealizations of the same structure. In Example 9.11, the loading is applied in a vertical plane. If, however, the loading has been applied in a horizontal plane the assumed stress distribution in the panels of the section would have been different, resulting in different values of boom area.

Suppose that an open or closed section beam is subjected to given bending or shear loads and that the required idealization has been completed. The analysis of such sections usually involves the determination of the neutral axis position and the calculation of sectional properties. The position of the neutral axis is derived from the condition that the resultant load on the beam cross-section is zero, i.e.

The area A in this expression is clearly the direct stress carrying area. It follows that the centroid of the cross-section is the centroid of the direct stress carrying area of the section, depending on the degree and method of idealization. The sectional properties, /vv etc., must also refer to the direct stress carrying area.

### 9.9.1 Bending of open and closed section beams

The analysis presented in Section 9.1 applies and the direct stress distribution is given by any of Eqs (9.6), (9.7) or (9.9), depending on the beam section being investigated. In these equations the coordinates (.v, r) of points in the cross-section are referred to axes having their origin at the centroid of the direct stress carrying area. Furthermore, the section properties /vv, /,.,. and /Vl. are calculated for the direct stress carrying area only.

In the case where the beam cross-section has been completely idealized into direct stress carrying booms and shear stress only carrying panels, the direct stress distribution consists of a series of direct stresses concentrated at the centroids of the booms.

### Example 9.12

The fuselage section shown in Fig. 9.49 is subjected to a bending moment of 100 kN m applied in the vertical plane of symmetry. If the section has been completely idealized into a combination of direct stress carrying booms and shear stress only carrying panels, determine the direct stress in each boom.

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