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(c) Rectongular (b)I-section (c) Channel (d)T-section section section

Fig. 1.1. Skew loading of sections containing one axis of symmetry.

perpendicular to it are then principal axes and the term skew loading implies load applied at some angle to these principal axes. The method of solution in this case is to resolve the applied moment Ma about some axis A into its components about the principal axes. Bending is then assumed to take place simultaneously about the two principal axes, the total stress being given by

With at least one of the principal axes being an axis of symmetry the second moments of area about the principal axes Iu and Iv can easily be determined.

With unsymmetrical sections (e.g. angle-sections, Z-sections, etc.) the principal axes are not easily recognized and the second moments of area about the principal axes are not easily found except by the use of special techniques to be introduced in §§1.3 and 1.4. In such cases an easier solution is obtained as will be shown in §1.8. Before proceeding with the various methods of solution of unsymmetrical bending problems, however, it is advisable to consider in some detail the concept of principal and product second moments of area.

### 1.1. Product second moment of area

Consider a small element of area in a plane surface with a centroid having coordinates (x, y) relative to the X and Y axes (Fig. 1.2). The second moments of area of the surface about the X and Y axes are defined as

Similarly, the product second moment of area of the section is defined as follows:

Since the cross-section of most structural members used in bending applications consists of a combination of rectangles the value of the product second moment of area for such sections is determined by the addition of the Ixy value for each rectangle (Fig. 1.3), i.e.

where h and k are the distances of the centroid of each rectangle from the X and Y axes respectively (taking account of the normal sign convention for x and ;y) and A is the area of the rectangle.

12. Principal second moments of area

The principal axes of a section have been defined in the introduction to this chapter. Second moments of area about these axes are then termed principal values and these may be related to the standard values about the conventional X and Y axes as follows.

Consider Fig. 1.4 in which GX and GY are any two mutually perpendicular axes inclined at 6 to the principal axes GV and GU. A small element of area A will then have coordinates (u, v) to the principal axes and (x, y) referred to the axes GX and GY. The area will thus have a product second moment of area about the principal axes given by uvdA. .-. total product second moment of area of a cross-section

Unsymmetrical Bending J (xy cos2 6 + y2 sin 0 cos 0 — x2 cos 6 sin 6 — xy sin2 9) dA

Principol axis v dA

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