THE ELASTIC AXIS

In the preceding Section, the shear center of a straight, cylindrical beam is defined. Since the analysis is based on St. Venant’s theory of bending and torsion, the exact boundary conditions at the ends of the beam cannot always be satisfied. The shear center so defined is a section property. For each cross-sectional shape, there associates a shear-center location.

The concept of shear center can be extended to a curved beam. One imagines that, at any point on the beam axis, a cylinder tangent to the given beam, and having the same normal cross section, is prescribed. The shear center of this cylinder is then taken as the shear center of the curved beam at that point.

The locus of shear centers of the cross sections of a beam is called the elastic axis of that beam.[1] The elastic axis is a natural reference line in describing the elastic deformation of the beam; for the resulting differ­ential equations are the simplest.

By using the elastic axis as defined above, it is possible to derive the classical Bernoulli-Euler equations of bending for a curved beam in analogy with Eq. 1 of § 1.1. The theory of a curved beam whose elastic axis coincides with the centroidal axis of the beam has been developed completely in Chapter 18, § 289 of Love’s book.12 If the shear centers do not coincide with the centroids of the cross sections, it is necessary to define a set of local coordinate axes parallel to the generalized principal torsion-flexure axes defined by Love, but with the origin located at the shear center. The “torsion” and curvature of the elastic axis can then be determined in the same manner as those of the centroidal axis, and the relation between the bending moments and torque and the change of curvatures and “torsion” at a section can be derived. The resulting equations are useful for “thin” beams, the cross-sectional dimensions of which are, by definition, much smaller than the radius of curvature of the beam.

The practical application of the elastic axis to aircraft structures is, however, subject to severe limitations. Owing to the effects of sweep angle, shear lag, restrained torsion, cutouts, buckling of skin panels, etc., the shear center often cannot be defined without ambiguity. In other words, the concepts of shear center and elastic axis lose their simplicity or usefulness when a structure other than a simple cylindrical beam is con­sidered. In any case of doubt, it is better to use influence functions which describe the deformation pattern of the structure due to a unit load acting on the structure. An influence function is a well-defined quantity. It can be measured on an existing structure, and it may be calculated by refined structural theories in special cases.

In the present book, the elastic axis will be used only when it is a straight line and is associated with a structure that behaves like a simple beam.

In engineering literature, the terms flexural center, center of twist, and flexural line are often used. They are defined as follows: Consider a

p

Fig. 1.7. Flexural center of a cantilever beam.

cantilever beam of uniform cross section. If a load P is applied at a point A on the free end (see Fig. 1.7), the section АС В will rotate counter­clockwise in its own plane. If P is applied at It, the section АС В will rotate clockwise. Somewhere between A and В is a point (say C) at which the load P can be applied without causing rotation of the section АС В in its own plane. This point is called the flexural center of the section ACB.

More generally, for a slender, curved, cantilever beam, the flexural center of a cross section is defined as a point in that section, at which a shear force can be applied without producing a rotation of that section in its own plane. Closely associated is the center of twist which is a point in a cross section that remains stationary when a torque is applied in that section. If the supporting constraint of the beam is perfectly rigid, the flexural center coincides with the center of twist. (See Ref. 1.15, or p. 29 of Ref. 1.9.) It is important to notice that these definitions are referred to particular sections. For example, when a force is applied at the flexural center of a section, that section will not rotate; but, unless the beam is homogeneous, cylindrical, and clamped in a plane normal to its
axis, other sections of the beam may rotate. In other words, whether other sections of the beam rotate or not is irrelevant to the definition of a flexural center. This fact makes the locus of flexural centers useless except when it is a straight line; then it coincides with the elastic axis.

For a given loading, a flexural line is defined as a curve on which that loading may be applied, so that there results no twist at any section of the beam. In general, different load distributions correspond to different flexural lines, and there exist load distributions that do not have a flexural line. A simple example is a concentrated load acting on a curved beam. This property renders the flexural line useless in aeroelasticity except in the simplest cases.