Three-dimensional beam theory

In the previous chapter, Euler-Bernoulli theory is developed for beams under axial and transverse loads. The analysis is limited, however, to deformations of the beam in plane (ī₁, ī₂). This behavior can be observed, for instance, when the cross-section of the beam presents a plane of symmetry and the only applied loads are acting in this plane.

In numerous practical applications, the beam's cross-section presents no particular symmetries and is instead of arbitrary shape. In addition, the applied loads may act along several distinct directions and not just in plane (ī₁, ī₂). Consider an aircraft wing: the cross-section is of a complex shape involving curved skins and two or more spars, and the wing is subjected lift and drag forces. In the case of a helicopter blade, large centrifugal forces generated by the rotation of the blade are also present. Similarly, machine components often operate in a complex, three-dimensional loading environment.