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The beginning of Part 3 of the "Buckling Comments" document. The link is called "advice paper", which you can download.

Here is the first part of my message to Frank:

October 18, 2007

Dear Frank:

I've been thinking about your structural buckling problem while I lie abed very early in the morning. I have come up with the following suggestions.

First, let me introduce a few definitions applicable to a linear buckling formulation:

1. A linear buckling eigenvalue equation is:

[K1]{q} = lambda[K2]{q} (1)

in which [K1] is the stiffness matrix for the structure as loaded by “Load Set B”; [K2] = Load-geometric matrix for the structure as loaded by “Load Set A”; {q} = buckling eigenvector, that is, the buckling mode shape; lambda = buckling eigenvalue (buckling load factor). Example: suppose you want to find the buckling pressure of a cylindrical shell under uniform external pressure (“Load Set A”) and under uniform axial tension that has a “fixed” value, that is, axial tension that you know in advance and that is not to be multiplied by the eigenvalue, lambda (“Load Set B”).

2. “Load Set A” = loads from which [K2] is computed, that is, the loads corresponding to which you want to find a buckling load factor (eigenvalue). In the example “Load Set A” = uniform external pressure.

3. “Load Set B” = loads which affect [K1], that is, the loads that are “fixed” in the sense that they represent part of the structural system in the same way that stiffnesses and boundary conditions represent the structural system. In this example “Load Set B” = uniform axial tension.

The following suggestions are based on the assumption that all the loads are either in "Load Set A" (that is, all the loads are "eigenvalue" loads) or any loads that happen to be in "Load Set B" (that is, "non-eigenvalue" or “fixed” loads) are stabilizing. I'm assuming that there are no "fixed" pre-loads, such as fixed destabilizing thermal loads and/or fixed destabilizing pressure or other destabilizing "mechanical" loads. If it is hard to tell whether Load Set B loads are stabilizing or destabilizing, then do what you would do assuming that the loads in Load Set B are destabilizing. If there are significant destabilizing loads in Load Set B, then first perform a linear buckling analysis with all the loads in Load Set A set equal to zero and all the loads in Load Set B transferred to Load Set A. Check to see if this case yields any eigenvalues that are less than unity. If not (or if all eigenvalues less than unity are negative) then proceed as described below. If so, then the structure will buckle under Load Set B acting by itself and it must therefore be redesigned (or there might be something wrong with your modeling or with your specification of Load Set B).

Assuming that there are no "Load Set B" problems, proceed as follows:

(For the rest of this message to Frank, download the "Buckling Comments" document called "advice paper" from this website.)

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