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This work presents a design framework for the selection of the topology and cross-section size of elastic timber gridshells, taking as constraints the shape of the structure and the maximum value of bending stress that can be reached in a given area of the gridshell. For this purpose, a parametric design environment and a genetic optimisation algorithm are used, which provides a set of solutions (optimal and near-optimal) that can be examined by the designer before adopting the final solution. The construction of the parametric mesh model is based on a geometric approach using an original adaptation of the Compass Method by developing two algorithms. The first one plots geodesic curves on a surface given a starting point and a direction. The second algorithm adapts the accuracy of the Compass Method to the local curvature of the surface, substantially minimising the computation time. The results show that the optimisation process succeeds in significantly reducing the initial bending stresses and offers an interesting solution space, consisting of a set of solutions with sufficiently diverse topologies and cross-section sizes, from which the final solution can be chosen by the Decision Maker, even according to additional non-programmed structural or aesthetic requirements. The design framework has been successfully applied and verified in the design of the PEMADE gridshell, an innovative elastic timber gridshell recently realised by the authors. Finally, the most relevant details of its construction process carried out to ensure the exact position of the timber laths are presented. Full article