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.
Concepts from isotropic geometry, Timoshenko’s shell theory, Airy stress functions and Maxwell’s reciprocal diagrams are combined in the design of plane-faced funicular gridshells. The notions of self-Airy and mixed-Airy gridshells are introduced, with an emphasis on self-tied gridshells. This paper extends the work of J.C. Maxwell for 2D pin-jointed trusses to 2.5D gridshells with the addition of a new reciprocal figure called the slope diagram. The form, force and slope diagrams are combined by a mixed area calculation to produce another new figure called the Maxwell-Mondrian diagram. A powerful new design process leveraging the relationship between the gridshell geometry and the Airy stress function is presented.
This paper presents a graphical method for determining the linearized stiffness and stability of prestressed trusses consisting of rigid bars connected at pinned joints and which possess kinematic freedoms. Key to the construction are the rectangular areas which combine the reciprocal form and force diagrams in the unified Maxwell–Minkowski diagram. The area of each such rectangle is the product of the bar tension and the bar length, and this corresponds to the rotational stiffness of the bar that arises due to the axial force that it carries. The prestress stability of any kinematic freedom may then be assessed using a weighted sum of these areas. The method is generalized to describe the out-of-plane stability of two-dimensional trusses, and to describe three-dimensional trusses in general. The paper also gives a graphical representation of the ‘product forces’ that were introduced by Pellegrino and Calladine to describe the prestress stability of trusses.
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