Mode Contribution Calculation

It is desirable to understand how the different pure modes of Eq. (8) for G, D, L, and O contribute in an all-mode or traditional FSM calculation, i. e. Eq. (1). This may be completed by transforming any displaced shape (buckled mode shape) into the eigenbasis created by the pure mode solution of Eq.

(8) . The eigenvectors d from the solution to Eq. (8) fully describe the pure mode solutions. Eq. (7) provides transformation from (or to) the pure mode space to the original DOF space.

The basis vectors must be normalized. Here, we select a normalization so that each base vector is associated with unit strain energy. In practice, let us consider again the eigenvalue problem of the member, as defined by Eq. (1). Any orthogonalized base vector satisfies Eq. (1), thus, we may write:

Kd0 =XKgd0 (9)

where d0 denotes the orthogonalized base vector. By pre-multiplying Eq. (9) with ^d0T:

where the left-hand side of the equation is the elastic strain energy, which by scaling the do vector can be set to unity. Any displacement vector can now be expressed as a linear combination of the basis vectors, by solving the linear matrix-equation as follows:

Doc = d

where Do is a square matrix constructed from the orthonormal do base vectors so that each column of Do would be a base vector; d is the given general displacement vector, while c is a vector containing the coefficients which are to be calculated. The contribution of any individual mode can be calculated as the ratio of the coefficient of that mode and the sum of all the coefficients, as follows:

(12)

Similarly, the contribution of a mode class can be defined as:

(13)

These definitions are preliminary, and imperfect. Currently the modal contributions defined in this manner are not unique. However, from a heuristic standpoint, they have value in allowing for an exploration of the various modal contributions.

Figure 6 provides modal contribution results for the cross-section of example (a), following Eq. (13). Generally the figure indicates the extent to which the G, D, L, or O classes contribute to the deformations at a given half-wavelength. For example, at the location of the distortional minimum (second minima in the curve of Figure 6a) D is the dominant class, but contributions from the other classes are observed. The defined contributions are imperfect, as they do not directly reflect the deformations impact on the buckling load. For example, at the local minima both D and O classes would appear to provide a significant modal contribution, but the analysis of Figure 5a indicates that the L class can provide the buckling load with only small error. While a robust modal contribution factor still remains a topic of future work, the value of such a metric is illustrated in Figure 6.

Figure 6. Mode contribution/identification for example (a)