Extended force density method and its expressions

The objective of this work can be divided into two parts. The first one is to propose an extension of the force density method (FDM)[2], a form-finding method for prestressed cable-net structures. The second one is to present a review of various formfinding methods for tension structures, in the relation with the extended FDM. In the first part, it is pointed out that the original FDM become useless when it is applied to the prestressed structures that consist of combinations of both tension and compression members, while the FDM is usually advantageous in form-finding analysis of cable-nets. To eliminate the limitation, a functional whose stationary problem simply represents the FDM is firstly proposed. Additionally, the existence of a variational principle in the FDM is also indicated. Then, the FDM is extensively redefined by generalizing the formulation of the functional. As the result, the generalized functionals enable us to find the forms of tension structures that consist of combinations of both tension and compression members, such as tensegrities and suspended membranes with compression struts. In the second part, it is indicated the important role of three expressions used by the description of the extended FDM, such as stationary problems of functionals, the principle of virtual work and stationary conditions using ∇ symbol. They can be commonly found in general problems of statics, whereas the original FDM only provides a particular form of equilibrium equation. Then, to demonstrate the advantage of such expressions, various form-finding methods are reviewed and compared. As the result, the common features and the differences over various form-finding methods are examined. Finally, to give an overview of the reviewed methods, the corresponding expressions are shown in the form of three tables.