  We investigate the buckling under compression of a slender beam with a
distributed lateral elastic support, for which there is an associated cost. For
a given cost, we study the optimal choice of support to protect against Euler
buckling. We show that with only weak lateral support, the optimum distribution
is a delta-function at the centre of the beam. When more support is allowed, we
find numerically that the optimal distribution undergoes a series of
bifurcations. We obtain analytical expressions for the buckling load around the
first bifurcation point and corresponding expansions for the optimal position
of support. Our theoretical predictions, including the critical exponent of the
bifurcation, are confirmed by computer simulations.
