Abstract

A number of theoretical studies of the stability of structures subjected to follower loads have been reported in the literature. The classic example is due to Beck who, in 1952, investigated the stability of a cantilevered beam-column subjected to a load applied to its free end that remained exactly tangent to the deflection curve (Atanackovic, 1997 and Timoshenko, 1961). To obtain the critical load it was necessary to perform a dynamic analysis of the problem. In problems such as these, instability comes in the form of flutter rather than divergence.

The purpose of this study was to assess the stability of the lumbar spine under loading conditions similar to those in Patwardhan, et al., 1998. In their experiments, a follower load was applied to the lumbar spine using guided cables that remained tangent to the deformed shape of the lumbar curve. In this study we adopted a model similar to Crisco and Panjabi (1992a) but with important differences.

Crisco and Panjabi (1992a) investigated the Euler stability of the lumbar spine using a lumped parameter model consisting of five (rigid) vertebral bodies connected by pin joints and torsional springs. The loading was applied vertically at L1. The model predicted buckling loads similar to the ones reported by Lucas and Bresler (1961). Both the predicted and the measured buckling loads of Crisco and Panjabi (1992a,b) are far below the load levels seen in vivo.

In our model, the loads were applied at each joint instead of at L1 only. Also, the direction of the applied loads depended upon the deformation. The direction of the load at one joint was proportional to the angular displacement of the segment below it. That is, the loads did not exactly follow the direction of the tangent to the deformed curve. This allowed us to assess the stability of the model spine when it was subjected to an “imperfect” follower load. An interesting feature of the problem was that it required only a static analysis.

The results of the study demonstrated the sensitivity of the buckling load to the “misalignment” of the follower path. As the direction of the loads approach the follower path, the buckling load increases beyond the Euler critical load. This was not an unexpected result. The increased buckling load predicted in this analysis was consistent with the experimental observations of Patwardhan, et al. (1998).

However, there was a surprising result. The analysis showed that there is a mechanism for a smooth transition between the buckled shapes which is not seen in the classical Euler analysis (Timoshenko, 1961).

These results may have implications for how imperfections in muscle coactivation can affect the stability of the lumbar spine. Also, they may help explain some of the phenomena seen in progression of diseases of the spine such as scoliosis.

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