Lumped Parameter Model for Stability Analysis

The problem of column instability due to axial compressive loading is treated frequently. For most simple cases an exact analytical solution is possible. However, a column with in-span conditions, such as varying axial forces, changing cross-sectional properties, multiple supports, or foundations, is quite complex and requires a numerical technique. A common approach, especially for computer programs available commercially, employs tranfer matrices. This is inefficient because repetitive analyses of the beam are required in the eigenvalue search, and it is difficult to automate since root estimates are necessary. This note suggests using a modeling scheme with the axial force as the discretized parameter. This is particularly convenient since experience is so great with the similar approach of lumping the masses (Myklestadt method) of vibrating members. It is shown that use of the lumped parameter model in a transfer matrix analysis leads to an efficient computational technique that requires neither root estimates nor repetitive member analyses. Although the beams are considered in depth, the technique can be extended to other members, such as plates and symmetric shells of revolution. These members can have arbitrary changes in geometry, axial load distribution, and in-span occurrences such as supports or foundations.