A shape optimization method for geometrically non-linear structural mechanics based on a sensitivity gradient is proposed. This gradient is computed by means of an adjoint state equation and the structure is analysed with a total Lagrangian formulation. This classical method is well understood for regular cases, but standard equations 1 have to be modiÿed for limit points and simple bifurcation points. These modiÿcations introduce numerical problems which occur at limit points. 2 Numerical systems are very sti and the quadratic convergence of Newton-Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables. 3 A geometrically non-linear curved arch is implemented with a ÿnite element method via a formal calculus approach. Thickness and=or shape for di erentiable costs under linear and non-linear constraints are optimized. Numerical results are given for linear and non-linear examples and are compared with analytic solutions. ? 1998 John Wiley & Sons, Ltd.Example 2.1. For a geometrically non-linear shallow beam in 2-D with a small initial curvature 11 and linear change of curvature: ' = (v; w) ∈ V , ' = ( v; w) ∈ V; depending on the boundary conditions, V may be H 1 0 (0; L) × H 2 0 (0; L) for a beam clamped at both ends; V= {(v; w) ∈ H 1 (0; L) × H 2 (0; L) = v(0) = 0; w(0) = 0; w (0) = 0} for a cantilever beam clamped at the left end. V= {(v; w) ∈ H 1 (0; L) × H 2 (0; L) = v(0) = 0; w(0) = 0; v(L) = 0; w(L) = 0} for a simply supported beam.