Form-finding using the Force Density Method: Existence of solutions, singularities, and an analogy to electric circuits

The Force Density Method is the oldest computational method for form-finding of funicular structures. The method is based on solving a linear system of equilibrium equations. Linearity ensures existence and uniqueness of the solution, provided that the coefficient matrix is non-singular. In the opposite case, the method often fails to deliver a feasible solution. In this study, we highlight a mathematical analogy between force density networks and electric circuits. Motivated by standard reduction steps in networks of electric resistors, inductors, or capacitors, we develop a new sequential reduction technique applicable to networks of force density elements. Network reduction enables analytic detection of singularity in many cases. Several examples illustrate the feasibility of the proposed singularity test, furthermore the role of singularities in classification of emergent structural forms is discussed.