The definition of curved geometry for widespan enclosures

Chris J K WilliamsDepartment of Architecture and Civil EngineeringUniversity of BathUK

1 Introduction

If an enclosure is to be constructed of curved lines and surfaces rather than straightlines and flat planes, the questions arise as to how the geometry is first to be chosenand then how it can be defined with sufficient accuracy for the structure to be builtand clad.

There are clearly many ways that the geometry can be chosen and defined, but theyfall into three broad categories and the methods used on any one project may fall intomore than one of these categories. The categories are:

Sculptural in which a model is sculpted by hand or a computer model isconstructed that can be deformed interactively.

Geometric in which the form is defined in terms of geometrical objects whichmight be simple spheres, cylinders or cones, or much more complicatedobjects which can only be visualised using computers.

Physical in which the shape is controlled by some physical process such as asoap film or a hanging chain. The physical process may be modelled by anactual physical model or a mathematical model which may by analytic ornumerical in a computer.

An example of a mixed sculptural and physical approach would be bending a piece ofwire by hand (the sculptural part) and then dipping it into soap solution andwithdrawing it to form a soap film (the physical part).

An example of a mixed geometric and physical approach would be forming a soapfilm between two parallel circular rings so that the rings are simple geometric entities.In this case the soap film forms a catenary of revolution so that one might say that it isa relatively simple geometric object formed by a physical process.

The methods used for any one project will depend upon many factors. Perhaps themost important of these is the relative importance of structural, architectural and otherconstraints. Another is the experience of the design team in using various techniques,especially since the technology may have evolved in other disciplines such assculpture, medicine or automobile, aerospace or ship design.

In the following I shall discuss some recent experience using a number of methods totry and illustrate the possibilities of the three approaches.

2 Sculptural

Traditionally large sculptures or even car bodies were first made as small clay modelsor maquettes which were measured and enlarged. Now much of this work is doneusing computers employing software written for the automobile and aerospace

industries. Frank O. Gehry & Associates use aerospace software, but the starting pointis still physical models.

Such software is expensive and time is needed to learn how it can be used. Curvedlines are divided up into a series of spline curves which fit together with anappropriate continuity of orientation and curvature. Curved surfaces are constructedfrom curved patches. These patches, developed for Computer Aided Design are verysimilar to the finite elements developed for the analysis of shell structures.

Figure 1 Body Zone sculptureArchitect: Branson Coates ArchitectureEngineer: Buro Happold

Figures 1 to 5 show the Body Zone in the Millennium Dome. The shape was definedby the small physical model in the photographs in figure 1. A structural grid wasdrawn on the model and this was measured using a standard CAD package fromscanned images of the photographs. This data was used to construct the computermodel shown in figure 2.

Figure 2 First computer modelArchitect: Branson Coates ArchitectureEngineer: Buro Happold

Figure 3 Sections through sculptureArchitect: Branson Coates ArchitectureEngineer: Buro Happold

Figure 4 Computer model from sectionsArchitect: Branson Coates ArchitectureEngineer: Buro Happold

Following this work it was decided to concentrate on defining the figures in terms ofparallel cross-sections. A copy of the physical model was sliced using a saw and theresulting cross-sections were scanned and 'traced' to produce figure 3. These cross-sections were joined to produce the three dimensional image shown in figures 4 and 5.

Figure 5 Rendered imageArchitect: Branson Coates ArchitectureEngineer: Buro Happold

The software used to produce figures 2, 3 and 4 was specially written for the project.

3 Geometric

The limit of what can be done using geometry is the mathematical knowledge andimagination of the individual. As an example let us consider Le Corbusier's Modulor1

shown in figure 6b. Modulor is based upon the Fibonacci series, examples of whichare

1, 1, 2, 3, 5, 8, 13, 21,34. . . . .

and

1, 3, 4, 7, 11, 18, 29, 47, . . . .

In both cases each term is the sum of the two preceding terms. Thus if xn is the nth

term,

xn = xn-1 + xn-2 .

Figure 6a Figure 6b Le Corbusier's sketchComputer imageThis equation has the general solution

xn =

lf 2 -1( )f n + l - f 2( ) -1( )n

f n

l -1( ) f 2 +1( )x0

where f =

1+ 5

2 is the Golden Section and

l =1

1-x0

x1

. As n gets larger, the ratio of

successive terms becomes closer to f regardless of the starting two terms of the series,

x0 and x1 . If

x1

x0= f , then l = f2 and xn = f nx0 which is the basis of Modulor. The

computer program below was used to produce the dxf file that is plotted in figure 6a,replicating Modulor.

dxf files are text files which can be read by CAD programs such as AutoCAD,MicroStation or MiniCAD.

#include <fstream.h>

#include <iostream.h>#include <math.h>int i,j,k,m,nhalfycles,finish;float PI,x,y,z,U,V,W,alpha,beta, A,C,phi;ofstream Julia("Modulor.dxf");int main(void){PI=4.0*atan(1.0);Julia<<"0\nSECTION\n2\nENTITIES\n";nhalfycles=18;m=20*nhalfycles;phi=(1.0+sqrt(5.0))/2.0;C=216.0;for(j=1;j<=4;j+=1){if(j==4)finish=90.0;else finish=60;for(k=0;k<=finish;k+=3){A=1.2*cos((PI*k)/180.0);for(i=0;i<=m;i+=1){beta=-(1.0*i*nhalfycles)/(2.0*m);alpha=beta;if(j<=2)alpha-=0.5;y=C*pow(phi,alpha);if(j==1||j==3)x=A*y*fabs(sin(beta*PI));if(j==2||j==4)x=A*0.5*y*fabs(sin(2.0*beta*PI));if(j<=2)x=-x;x=x/8.0;z=0.0;if(i!=0){Julia<<"0\nLINE\n8\n0\n";Julia<<"10\n"<<U<<"\n20\n"<< V<<"\n30\n"<<W<<"\n";Julia<<"11\n"<<x<<"\n21\n"<< y<<"\n31\n"<<z<<"\n";}U=x;V=y;W=z;}}}Julia<<"0\nENDSEC\n0\nEOF\n";Julia.close();cout<<"DXF file written\n";return 0;}

Figure 7 Golden section log spiral

Figure 8 Two spiralsThe program will run on any computer (Macintosh, PC etc.) with a C++ compiler andvery little work would be required to convert the program to Basic or Fortran.

An unlimited variety of curves and surfaces can be produced by such programs. Forexample, figures 7, 8 and 9 were produced using the formulae

x = fa cos2ap

y = fa sin2ap

¸ ˝ ˛

,

x = fa cos2ap +f -a cosap

y = fa sin2ap -f -a sinap

¸ ˝ ˛

and

x = 1 + r( )fa + 1 - r( )f-a cosap

y = 1- r( )f -a sinap

¸ ˝ ˛

respectively.

Figure 9 Spirals to lines

Figure 10 Rendered imageIn each case a is varied to draw a curve and in the case of figure 9, a different valueof r is used for each curve.

The surface in figure 10 was obtained from the curves in figure 9 by giving eachcurve a different value of z.

Figure 11 Bridge study

Figure 12 Bridge studyThe bridge studies in figures 11, 12 and 13 were also produced by purelymathematical methods as was the shell study in figure 14. In each case the wholeobject is defined by the just one set of mathematical formula so that there is completecontinuity of all derivatives, orientation, curvature, rate of change of curvature etc.

Figure 13 Bridge study

Figure 14 Shell study for Stuttgart railway stationArchitect: Ingenhoven Overdiek Kahlen und PartnerConsultant Architect: Professor Frei OttoEngineer: Buro Happold

Figure 15 Millennium Dome Rest Zone - system geometryArchitect: Richard Rogers PartnershipEngineer: Buro Happold

Figure 16 Rest ZoneArchitect: Richard Rogers PartnershipEngineer: Buro Happold

Figures 15 and 16 show the Rest Zone in the Millennium Dome which was producedby deforming a torus. Again there is complete continuity of all derivatives.

4 Physical

In the membrane theory of shell structures the geometry of the structure and the loadsare assumed to be known and the three membrane stresses - two tensile orcompressive and one shear are unknown.

There are three equations of equilibrium, one in the direction normal to the surface,

sab bab + p = 0 ,

and two in the plane of the surface,

sab

:a + pb = 0 .

The notation here is similar to that in Green and Zerna2. In these equations thegeometry of the shell is determined by the components of the metric tensor, gab , and

of the curvature tensor, bab . The components of load are pa and p , and the

unknown membrane stress components are s11 , s

22 and s21 = s 12 .

The fact that there are three equilibrium equations and three unknown membranestress components means that shells are essentially statically determinate if the overallshape and boundary supports permit. An inappropriate shape or lack of support maymean that a shell is a mechanism.

It is not at all obvious which shapes and support conditions lead to mechanisms andwhich do not. Spivak3 discusses this issue in purely geometric terms, for example theCohn-Vossen theorem states that any complete convex surface with positive Gaussiancurvature is not a mechanism if membrane strains are prevented.

A cooling tower on the ground is not a mechanism, but a spherical shell with a hole inthe top is.

A shell may also be a mechanism if it is made of masonry, so that the principalmembrane stresses must be compressive, or if it is made of fabric in which case theymust be tensile.

A mechanism can carry certain loads if it has the correct shape. In the case of amasonry structure, the dominant load is the dead load and in the case of a fabricstructure it is prestress upon which wind and snow are added.

Form finding is the process of establishing a structural geometry for amechanism to carry a particular load.

Gaudi used hanging models which, when inverted, defined the shape of masonryarches and vaults for the Colonia Guell and the Sagrada Familia. Professor Frei Ottopioneered the use of physical models for fabric structures, cable nets and grid shells.

Figure 17 Tree of the FutureArchitect: Mark Fisher AssociatesEngineer: Atelier OneForm finding: Lynne Mabon, University of Bath

Now much of this work is done with numerical models, although physical models areindispensable for initial studies. Form finding a fabric structure with a soap film orminimal surface is done by setting the membrane stress components

sab = Tgab

where T is the surface tension. In addition a geodesic co-ordinate system forgenerating the cutting the pattern is obtained by imposing the conditions g12 = 0 and

g22 = constant .

An equal mesh net is produced by writing

s12 = 0 and g11 = g22 = constant ,

if there is no elastic extension, otherwise g11 and g22 increase with tension. Equalmesh nets are more difficult to form find than fabric structures due to the adjustmentof cable lengths at the boundary.

Figure 18 Mannheim Bundesgartenschau 1.5m gridArchitect: Mutschler & PartnersConsultant Architect: Atelier Warmbronn (Professor Frei Otto)Form finding: Büro LinkwitzEngineer: Ove Arup & Partners (Ted Happold and Ian Liddell)

Figure 19 Mannheim erectionFigure 17 shows one of the equal mesh nets of the Tree of the Future intended for theCentral Show in the Millennium Dome. In this case a computer program wasspecially written by Lynne Mabon of Bath University which automatically generatedthe boundary data.

Figure 18 shows the hanging chain model for the Mannheim Bundesgartenschau. Thisis a computer generated model by Büro Linkwitz, based upon Frei Otto's accuratephysical model. Figures 19 and 20 show the erection and load testing of the shells.

Figure 20 Mannheim load testFigures 21 and 22 show computer generated models of the Weald and DownlandMuseum. In this case the mathematical model had to contain bending stiffness duringform finding, otherwise compressive stresses produced wrinkling.

Figure 21 Weald and Downland MuseumArchitect: Edward Cullinan ArchitectsEngineer: Buro Happold

Figure 22 Weald and Downland MuseumArchitect: Edward Cullinan Architects Engineer: Buro Happold

5 The British Museum Great Court Roof

Figures 30 is an image of the computer model of the British Museum Great CourtRoof. It was generated by a mixed approach.

Figure 23 British Museum Great Court Roof - first functionArchitect: Foster and PartnersEngineer: Buro Happold

The shape of the surface was defined analytically by weighting and summingfunctions based on those shown in figures 23, 24 and 25. The weightings also variedwith position to satisfy architectural, planning, structural and clearance requirements.

Figure 24 British Museum Great Court Roof - second functionArchitect: Foster and PartnersEngineer: Buro Happold

The positions of nodes on the surface were obtained from the starting grid shown infigure 26. A displacement was calculated for each interior node of this grid to makeits x, y and z co-ordinates the weighted average of the current co-ordinates of the foursurrounding nodes. However, before moving a node, the component of displacementnormal to the surface (see figure 27) was removed so that the node remained on thesurface. This relaxation procedure was repeated thousands of times for the wholestructure until the geometry settled down to that in figure 28.

Figure 25 British Museum Great Court Roof - third functionArchitect: Foster and PartnersEngineer: Buro Happold

The weighting of the surrounding nodes was varied at different points on the surfaceto control the distribution of the nodes, in particular in relation to the sizes of the glasspanels.

Figure 27 British Museum Great Court Roof - surface normalsArchitect: Foster and PartnersEngineer: Buro Happold

Figure 26 British Museum Great Court Roof - original gridArchitect: Foster and PartnersEngineer: Buro Happold

The spiraling members were obtained by joining points in the form finding grid asshown in figure 29 to produce figures 30 and 31.

Figure 28 British Museum Great Court Roof - relaxed gridArchitect: Foster and PartnersEngineer: Buro Happold

Figure 29 British Museum Great Court Roof - steel members on gridArchitect: Foster and PartnersEngineer: Buro Happold

Figure 30 British Museum Great Court Roof - steel membersArchitect: Foster and PartnersEngineer: Buro Happold

Figure 31 British Museum Great Court Roof - steel membersArchitect: Foster and PartnersEngineer: Buro Happold

6 Conclusion

This paper discusses some of the ways in which curved forms can be generated. It isnot possible to say that any one method is the optimum, because there are so many

possibilities and the architectural, structural and environmental constraints will neverbe the same on two projects.

References

1 Le Corbusier, The Modulor, translated by Peter de Francia and Anna Bostock,Faber and Faber, London 1961.

2 Green, A.E. and Zerna, W., Theoretical elasticity, 2nd edition, Oxford UniversityPress, 1968.

3 Spivak, Michael, A comprehensive introduction to differential geometry, volume5, 2nd edition, Publish or Perish Inc., Delaware, 1975, 1979.