An extended formulation of the Force Density Method accounting for bending

The Force Density Method enables the form-finding of reticulated shells by exploiting the equilibrium conditions of bar structures. An extended formulation is introduced in this contribution, tailored to gridshells that exhibit bending resistance in the vertical plane. The concept of force density is expanded to encompass not only axial forces but also bending moments and shear forces. Under this framework, it is shown that the translational equilibrium of nodes remains linear with respect to vertical coordinates, provided the gridshell has fixed footprint. A min-max problem is formulated using both sets of force densities to identify reticulated shells that satisfy equilibrium within a prescribed plan projection, while minimizing the peak reaction forces at restrained nodes. Vertical equilibrium of the unrestrained nodes is inherently satisfied, whereas horizontal equilibrium, rotational equilibrium, and total member length are enforced via equality constraints. To solve the problem, a smoothed approach is employed using a constrained minimization algorithm. Numerical simulations demonstrate that the bending capacity of the members significantly influences the equilibrium-driven form-finding of compressive gridshells. Examples are provided showing bending-resistant reticulated shells that can transmit gravity loads without requiring horizontal reactions, across different types of geometry and connectivity of the grid.