Exact or Lower Bound Tapered Column Buckling Loads

The curves presented enable the buckling loads to be found for tapered columns with a uniform axial force P applied over all their length L or over all except $\beta L$ (for $0⩽\beta ⩽0.9$) of their length. The main curves cover six combinations of end conditions, $0.05⩽r⩽1$ and $h=2\text{,}3\text{,}$ or 4, where r is the ratio of the second moment of area at the smaller end to the second moment of area at the larger end and h is the power which governs the variation of second moment of area with length. An evaluation is also undertaken of the accuracy of a method which uses the curves to obtain lower bound approximations to the critical load factor for stepped distributions of axial force. These lower bounds are on the safe side, so that they can cause loss of economy but not loss of safety, and for most practical problems they are within 10% of the exact answer.