Extended force density method and its expressions

TL;DR

This paper extends the force density method (FDM) to handle prestressed structures with tension and compression members, reviews related form-finding methods, and highlights the importance of different expressions in form-finding analysis.

Contribution

It introduces a generalized FDM functional for combined tension and compression structures and reviews various form-finding methods emphasizing their expressions.

Findings

Generalized FDM enables form-finding of tensegrities and suspended membranes.

The extended FDM overcomes limitations of the original method for prestressed structures.

Comparison of form-finding methods highlights common features and differences.

Abstract

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)(H.J. Schek, 1974), a form-finding method for prestressed cable-net structures. The second one is to present a review of various form-finding 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 Nabla 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.