# «On the extension of graphical statics into the 3rd dimension M. Schrems & Dr. T. Kotnik ETH, Zurich, Switzerland ABSTRACT: In the 19th century ...»

Structures and Architecture: Concepts, Applications and Challenges – Cruz (ed)

© 2013 Taylor & Francis Group, London, ISBN 978-0-415-66195-9

On the extension of graphical statics into the 3rd dimension

M. Schrems & Dr. T. Kotnik

ETH, Zurich, Switzerland

ABSTRACT: In the 19th century methods of graphical statics - systematized and published by

Carl Culmann in 1866 – were used to study in more detail the relationship between form and

force flow. Historically, they were superseded by analytical methods, due to the relative ease of numerical calculations and the increasing importance of experimentation the developing discipline of engineering with its need for precision. Nowadays, however, with the availability of digital tools and the possibility of visualization of geometry in 3D space there is a re-emerging interest within architecture in graphic statics. It is because of this that a visual dialogue about force flow based on graphic statics should be fostered because these methods offer a simple yet precise way to discuss the geometrical dependency of form and forces. The extended graphical statics (EGS) is part of an ongoing process of extension of the methods of graphic statics to the third dimension in order to overcome some of the limitations of Culmann’s approach.

Similar to the situation in 2D it is possible to show, that by using the EGS in 3D any finite constellations of forces in general position can be reduced to a couple of forces, which represent all forces in intensity, direction and position. With that method the construction remains always spatially without any need for projections and always independent of any global coordinate system, necessary for numerical approaches. Like in 2D there are two linked diagrams, a form diagram and a force diagram, which mean that modifications in one diagram force an adjustment in the other diagram. With that method the dependency of form and forces in 3D can be seen without using projections and an active and synthetic designing process could begin. In previous research it has already been shown, that EGS allows considering the internal forces of a complex surface by using static determinate space trusses, which can be calculated graphical. These trusses can be summarized and three dimensional stress fields can be generated using CAD programs like e.g. Rhino in combination with Python and Mathematica.

In continuation of this research it is shown how EGS can be used to explore the 3D equilibrium by constructing a (trial) funicular polygon-net, which has to be seen as the 3D likeness of the (trail) funicular polygon in 2D. This netlike structure works as a closed system, so that every single cable of the net is determined. As in 2D, this funicular system helps to find out the ‘neuralgic’ points of a structure, so that with a few information the entire system can be considered.

Using this method, important structural information of complex free forms can be examined that enables a geometrical and physical form finding process without using relaxation processes or FEM.

## 1 INTRODUCTION

The objective of a graphical approach to statics is to focus on geometric construction of the force flow in contrast to numerical calculation. That means the focus is on the logic of the inner force flow within a building structure rather than on the numeric evaluation of the building structure itself. Graphic statics, therefore, fosters a process oriented and intuitive approach to force-oriented design.Through the expansion of graphical methods of construction of force-flow into the third dimension, the foundation for a design methodology can be laid where architecture and engineering enter into an active and visual dialog. This is also the motivation for an ongoing research into a visual language for the flow of forces whose roots are to be found in the ‘graphic statics’ developed in the 19th century. ETH professor Karl Culmann, in particular, is considered one of the major advocates of this method and for him it was clear that ‘drawing is the language of the engineer’ (Mauerer 1998).

Architects primarily think and communicate using diagrams, sketches and plans that is drawing is the language of architects. The same is true for engineers. From this it is clear that graphic statics, as a method for the direct visual representation of the inner distribution of forces, is predestined to promote dialog between engineers and architects and can offer the foundation for fruitful discourse.

This is all the more valid since three-dimensional visualization is perfectly feasible using new CAD programs and so the method that was so far only used in 2D can intuitively be expanded to 3D. As a consequence, a direct relation between force and form can again be established. The first applications in this direction are already available for supporting frameworks subject to pure tensile and compression loading (Kilian 2005), and through them the experimental and elaborate design methods of Antonio Gaudi (Huerta 2006), Heinz Isler or Frei Otto become accessible for the drafting process. This creates an excellent basis for studying the interrelation-ships between form and the progression of its inner forces more directly and thus more effectively and provides the opportunity to develop a design methodology based on an intuitive and easily comprehensible language. By developing the usability of the method by implementing into CAD programs a synthetic - instead of an analytic - design process can get started.

**2 HISTORY**

The central element of all drawing methods for representing the inner flow of forces is the investigation of the dependence of equilibrium and force polygons, as well as their representation in two geometrically dependent diagrams with different units. Such a description of equilibrium and force polygons is already found e.g. in Varignon’s work ‘Nouvelle Méchanique ou Statique’ published in 1725 (Varignon 1725). For this research the most important roots are the projective correlation of the equilibrium and force polygon, which were mathematically investigated by Culmann as ‘graphic statics’ (Culmann 1866) based on the ‘newer geometry’ of Poncelet (Poncelet 1822) or Staudt’s ‘Geometrie der Lage’ published in 1847 (Staudt 1847).

Because of the duality of the planar equilibrium and force polygon, Culmann called these diagrams reciprocal. Maxwell proved almost simultaneously (1864) ‘that for non-central force systems two reciprocal figures arise if and only if one figure can be considered a projection of a polyhedron’ (Kurrer 2002). However, through the development of faster computing tools and the scientific processes (Westermann 2010) of the dawning 20th century, the graphical methods and their enormous potential as intuitive tools fell into oblivion, displaced more and more by analytical methods. Nowadays ‘graphic statics’ plays only more of a subordinate role in engineering practice and has almost completely been displaced by analytical statics since the latter is more conducive to numerical analysis and therefore to calculations.

It must be recognized that all drawing methods known thus far for calculating the inner flow of forces operate in two-dimensional space. Even in the most recent research such as the ‘Thrust Network Analysis’ of P. Block (Block 2009) investigations are carried out on projections, which is why the subject matter deals with planar, and are just valid for compression or tension only structures. If ‘graphic statics’ is expanded into the third dimension, then both the representation of the equilibrium polygon (form diagram) and the setup of the force polygon (force diagram) must be done using spatial and thereby physical models or in digital space.

## 3 EXTENSION OF GRAPHICAL STATICS

3.1 Number of resultants depending on the geometric position Planar force configurations can be consolidated into one equivalent force, of the resultant R, where an equivalent force is a force, which produces the same reaction forces as the sum of all individual forces using any static system. Different methods were developed for this purpose.

The equilibrium polygon method deserves special mention in this connection. It allows the combination of a finite number of intersecting forces and a finite number of parallel and/or intersecting forces into one resultant in two-dimensional space. Intersecting or parallel forces can be consolidated into one resultant even in three-dimensional space by linking coplanar cases. On the other hand, it is not clear how skew forces can be combined graphically.

**3.2 Generic case: skew forces**

For a skew and thus generic force configuration in R3 it should therefore be shown that it can be consolidated into a force pair (from many possible force pairs) through complete induction. This force pair is called (R1, R2 ). What is true here as well is that an equivalent force pair using any static system brings about the same reaction forces as the sum of all individual forces. Without

**loss of generality, the following conditions must apply to three forces F1, F2, F3 in general position:**

1. All three forces = 0

2. All three forces by pair are not in a plane

3. In the form diagram:

4. In the force diagram:

+ + + =0 + + =0

For the initialization there are three forces F1, F2, and F3 in whose action lines g, h and i are skewed against each other and whose direction and position are determined by two points each.

A is freely chosen as the source of the coordinate system. Analogous to two-dimensional graphic statics, a three-dimensional force diagram is developed whose source is freely selectable. It is obvious, that a straight line s can be found, that intersects all three action lines g, h and i. So it is possible to generate two planes Ejh and Eni which are intersecting in s (Figure 1).

By splitting F2 and F3 in two directions and F1 in three directions it is possible to figure out the point T which is one point on the action of R1. Because of the order of F1, F2, and F3 in the force diagram, it is in the generic case not possible to split the whole force F3 into the two directions. That means that a remainder of the force F3 is R2.

** Figure 1. Intersecting line s It is clear that the construction described creates only one of the infinite number of force pairs.**

Just like in the equilibrium polygon method it is completely irrelevant (cf. pitch) in which action line and at which point on the action line the construction is started but the geometric relationships of the form diagram must be transferred to the force diagram. In this respect the construction can be executed so that the remaining force lies on the action line of F1, F2 or F3 (Figure 2).

** Figure 2. Three skew forces subdivided into and In an iterative process three skew forces combined into one force pair can be linked to another force and so a finite number of skew forces can be reduced to one force pair through complete induction.**

This force pair corresponds in intensity and direction to the sum of all forces.

( + + + + + ) ( + )=0, and (, ) (, ) =0

## 4 SPATIAL FUNICULAR POLYGON NET

4.1 Most simple spatial net using R1 and R2 As shown in chapter 3, any configuration of skew forces can be consolidated into two equivalent forces. This pair of forces - using any statically determined system - brings about the same reaction forces as the sum of all individual forces. So it is clear, that working with these two forces helps to find out the ‘neuralgic’ points of a structure, so that out of these information the entire system can be considered. On the other hand it means, that the simplest net to get an overview is out of these two forces connected with any three supports.

4.2 Extension of the funicular line (2D) to a spatial polygon net (3D) All the research is about showing, that planar graphic statics is a special case of the 3D. Therefore, it is evident to discover the analog system of the 2D funicular system in 3D.

In 2D the difference by the force diagram between using the reslutant or the origin forces is, that the forcepolygon gets refined (Figure 3).

**Figure 3. Refinement by using polygonal approximation**

In planar graphic statics there are many conventions, so that the order of the forcepolygons defines the shape of the construction or vice versa. However from a numeric point of view in 2D e.g. by two forces 4 possible constellations of (force-)polygonlines with a freely selected sag are existing. In 3D by two forces 12 possible constellations of polygonlines with a freely selected sag are existing (Table 1). That shows that with increasing the number of forces the number of possible polygonlines increases too. That means that for a practical implementation it is necessary to define conventions.

Table 1. Comparison of the degrees of freedom __________________________________________________________

Number of possibilities for 2D 3D __________________________________________________________

starting with the polygon at any support 2 3 connecting the first line of action 2 2 connecting the second line of action 1 1 connecting the second support 1 2 connecting the third support - 1 __________________________________________________________

Total 4 12 __________________________________________________________

In the two-dimensional force diagram the line of action of the forces are always connected with one single line/forcevector. In 3D two lines/forcevectors – remember the plane – are necessary to connect the origin forces. This leads to deflections, which are – depending on the origin forces – visible as compactions or ‘vortexisations’. In planar graphic statics just the conventions prevent overlapping forces, the ‘vortexisation’ is unique in 3D.

All this polygonlines whether in 2D or in 3D are unstable and just valid for a singular force combination. For getting a system that resists any force combination in 2D a net - that means a statically determined truss - has to be constructed. In 3D the eqivalent would be a spatial truss out of tetrahedrons.