#### Installation — business terrible - 1 part

September 8th, 2015

By the application of constraints associated with the strain assumptions of the various modes as described in Table 1, the problem of pure buckling mode calculation can be solved. Instead of solving

the generalized eigenvalue problem of the member, as given in Eq. (1) one solves the reduced (or constrained) problem:

Krd = ^Kgrd (8)

where Kr = RTKR and Kgr = RTKgR are the elastic and geometric stiffness matrix of the reduced DOF problem, respectively. R may be associated with any combination of the G, D, L, and O spaces or may include as little as one deformation field, for example D1 of Figure 4, and thus reduce the problem to as little as one DOF.

space, one constrained to the D space, and one constrained to the G space. The L space, spanned by the 8 deformation modes of Figure 4c requires solution of an 8×8 eigen problem. The D space, spanned by the last two deformation modes of Figure 4a and b requires solution of a 2×2 eigen problem, but actually the anti-symmetric D2 mode does not contribute in this loading, so the solution can be simplified to a simple algebraic equation involving only the D1 mode. The G space, spanned by the second through the fourth deformation modes of Figure 4a and b requires solution of a 3×3 eigen problem, and may be reduced to an algebraic equation for weak-axis flexural buckling and a 2×2 equation for flexural-torsional buckling. The constraint conditions successfully separate the modes.

The difference between the conventional FSM analysis and the D mode analysis is the most striking feature of the comparison. In Figure 5b the contribution of the L deformation modes and O deformation modes to this difference is demonstrated. Based on the mechanical definitions assumed here, the distortional minima in a conventional FSM analysis is not a pure mode; rather it includes interaction with additional mode classes; most notably, local buckling. Analysis of the combined LD space (9 DOF) provides a solution within %% of the conventional FSM analysis (24 DOF). In this example the O modes must also be included (an additional 10 DOF) to close the error to zero. In other cross-sections distortional buckling may have a stronger interaction with global modes and thus the G space may be more important to include than the L space. No general conclusion can currently be made with regard to the significance of D modes which have interaction with L or G modes, but it is reasonable to assume that such a situation does impact the nature of the post-buckling response.

(a) FSM all mode analysis compared with constrained (b) FSM all mode analysis compared with constrained analyses in the L, D, and G mode classes analysis in the D, D+L, and D+L+O mode classes

Figure 5. Comparison of conventional FSM analysis results with constrained models for example (a)