A Graphical Method for Determining Truss Stability

Abstract

Graphic statics has been used for over 150 years, having been pioneered by the likes of Maxwell, Cremona, Culmann and Rankine, and has recently seen a resurgence in popularity because of its use in design. However, it is only concerned with equilibrium; as any engineer will testify, whilst equilibrium is necessary, it is not sufficient and stability must also be obtained. This paper develops a novel graphical method for determining the stability and stiffness of prestressable structures. By considering the weighted sum of the Maxwell-Minkowski diagram, the stiffness and stability of the structural mechanisms can be determined. This work extends to cover structures with multiple mechanisms and has been compared to results obtained through experimentation and the finite element method. Furthermore, it extends the work on stiffness to provide a graphical method to estimate the natural frequency of a truss. Whilst this method accurately determines the stiffness of structures, it represents a significant development in the field of graphic statics as it allows an engineer to 'eye-ball' the stability of a given truss. Engineers can also manipulate the form and force diagrams, as desired, to adjust the stiffness of their structure accordingly, whilst being able to visualise the process. Much of the previous work in this area relies heavily upon large matrices, while this method allows a more intimate and hands-on alternative.

Stability of trusses by graphic statics

“…Thus if the apex angle between the tension wires across the columns (between c and q , say) is less than 90° then the system is unstable. This prediction was confirmed by simple experiments, as described in Millar et al [ 15 ].…”

“…The presentation here is restricted to the small displacements of two- or three-dimensional pin-jointed trusses with axially inextensional bars and with only a single kinematic freedom. The generalizations of this theory which describe the stiffness of trusses with numerous kinematic freedoms, with extensionable members and for large displacements are presented in companion papers [ 15 , 17 , 18 ], including the example application of the multimodal linearized theory to describe the stability of Robert Maillart’s iconic design for the shed roof at Chiasso [ 16 ].…”

“…This paper relates to others that the authors are currently in the process of publishing on this subject. It expands the initial ideas presented in the conference paper of Millar et al [15]. A detailed case study applying this approach to the stability of Robert Maillart's historically significant roof design at Chiasso, Switzerland is presented in McRobie et al [16].…”