Desktop computer generation of membrane structure cutting patterns*

IAN L I D D E L L and D A V I D W A K E F I E L D Minilec Lid, 14 Gay Street, Bath

PETER H E P P E L University o./" Camhridfle {formerly Minitec IJ~l)

I N T R O D U C T I O N

The current trend towards aesthetically pleasing, free- form, shapes for membrane structures necessitates the use of numerical modelling for determination of the equilib- rium form and generation of fabrication information. Recent developments in both analytic techniques and computer hardware have meant that it is now practical to perform such numerical modelling on a desktop com- puter. This makes the techniques described in this paper an economically viable proposition for the design office, a particularly relevant feature when membrane structures are being utilised as a low-cost building solution.

The report describes a computer program, MAS/CP, which is used for the generation of form and cutting patterns for membrane structures. The program is moun- ted on a Hewlett-Packard 9845 desktop computer, and produces full cutting pattern information for both tents and air-supported structures in a format suitable for direct use by membrane fabricators.

C O M P U T A T I O N E N V I R O N M E N T

The program, MAS/CP, has been designed to fit within the general philosoph.y and structure of the MAS System as described in detail elsewhere in these proceedings ~. The overall objective of.this system is the efficient application of computer techniques in the design office. System features particularly relevant to MAS/CP include ease of data input and validation, efficient data management through the MAS-EDIT facility and the production of output formatted as fully titled and numbered A4 calcu- lation sheets.

The current generation of desktop computers have sufficient user memory capacity to perform the non-linear analysis of equilibrium forms. This process is enhanced by the availability of graphic display facilities, essential for assimilation of the visual implications of the resulting form and for providing ready validation of data. Hardcopy output of text and graphics enables production of information suitable for direct transmission to the fabricator.

The cutting pattern program is currently mounted on the following hardware configuration:

Computer - - Hewlett Packard 9845T desktop com- puter with 187K bytes of user memory, graphics VDU

* Paper taken from the Proceedings of the Second International Conference and Exhibition on Engineering Software April 24th to 26th, 1981.

and integral thermal printer capable of reproducing graphic display.

Disc Drive - - Hewlett Packard 9895M twin 8" floppy disc drive with a gross capacity of 1.2M bytes per disc.

Optional Peripherals - - Incremental plotter and digi- tizer tablet.

ILLUSTRATIVE PROBLEM

The principal features of the cutting pattern program, MAS/CP, are illustrated in this paper by reference to the Rotork Tent at Poole, Dorset (Figs. 1 and 2). This 40 m by

b Ficture la,b. Rotork Tent, Poole

0141-1195/81/030115-0452.00 ©1981 CML Publications Ado. Eng. Software, 1981, Vol. 3, No. 3 115

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20 m tent is a prime example of the modern membrane structure described in the introduction.

Project: Location: Client: Architect: Tent Designers: Consulting Engineers:

Rotork Tent Poole, Dorset Rotork Marine Ltd Peter Hale Future Tents Ltd, New York Buro Happold

THE NUMERICAL MODEL

The numerical processes fall into two distinct phases; analysis to determine the equilibrium form of the mem- brane surface as coordinates of points along geodesic lines, followed by the development of cutting patterns for strips of fabric which can be made up into the surface.

The formfinding problem is formulated as a search for an equilibrium geometry compatible with given mem- brane stresses and boundary conditions (together with specified inflation pressure for pneumatic structures). The assumption of uniform membrane stress results in a form equivalent to that established by a soapfilm within the given boundary conditions. Computer methods for numerical formfinding are well established, with both implicit, Newton-Raphson 2, and explicit, Dynamic Relaxation 3, solution techniques currently in use. The latter, with its low storage requirements through sepa- ration of the equilibrium and compatibility conditions and relative simplicity of implementation and physical interpretation, is particularly suited to the desktop com- puter in a design office environment.

The main effort in the development of acceptable cutting patterns lies in the location of geodesic lines across the surface.

The cutting pattern program MAS/CP employs the method of simultaneous equilibrium form determination and geodesic location developed by Chris Williams of the Surface Stressed Flexible Structures Group, University of Bath 4.

Discretisation of membrane structures Complex membrane surface structures are generally

discretised into fields bordered by ridge and edge cable boundaries. Within these fields a given cloth orientation is followed, as illustrated by the plan on the Rotork Tent in Figure 2. This provides a convenient sub- division for the form-finding process. Individual fields are analysed separately, whilst overall equilibrium of the structure is ensured by relaxation of the ridge lines.

Field analysis The field idealisation employed enables simultaheous

form-finding and location of geodesic lines on the surface. Cloth seam lines are defined by the geodesics, with the membrane between idealised by uniform stress triangular finite elements (Fig. 3). Once these geodesics have been determined individual cloth geometry is readily estab- lished by unfolding the triangular facets into a plane (Fig~ 4).

A non-uniform stress distribution may be introduced by specification of additional tensile or compressive forces along geodesic lines. Thus the principal stresses in the membrane coincide with the warp and weft directions of the fabric. The equilibrium geometry of the field nodes is established by an iterative process of sequential coor- dinate adjustment.

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\ Figure 3. SmJ~we idealisation

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uniform stress membrane finite elements

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Figure 4. Ul!/bhled cloth pattern

116 Adv. Eng. Software, 1981, Vol. 3, No. 3

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(1) Motion normal to the membrane surface at the node is governed by Dynamic Relaxation of the residual membrane forces in that direction.

(2) In-plane motion perpendicular to the geodesics such that (0~ +02+03)=(04+0s+06) in order to ensure geo- desics on the surface (Fig. 3). Because of surface curvature, (0~ + 02 + 0 a + 0¢ + 05 + 06) does not generally equal 360 ~ .

(3) In-plane motion parallel to the geodesics such as to make AB=BC. Although not necessary for equilibrium, this eliminates bunching of nodes along the geodesics.

Therefore at each field node, one degree of freedom is adjusted according to equilibrium and the remaining two according to geometrical requirements. Figure 5 shows the idealisation for a field of the example structure that has attained equilibrium.

Boundary analysis Boundary lines, having extremities fixed in space, are

assigned elastic properties and lengths between node points and analysed for equilibrium under load by Dynamic Relaxation in the three global coordinate directions. Here the loading is defined by membrane forces from the adjacent field (edge lines) or fields (ridge lines).

Control of Dynamic Relaxation analysis The explicit numerical integration of the necessary

Dynamic Relaxation analyses is controlled by kinetic damping and optimised fictitious mass components to ensure stable and rapid convergence to solution with the need for prior determination of control parameters s.

MAS/CP - - PRINCIPAL FEATURES

Input data Having initially discretised the membrane structure

into uniquely labelled surface fields and boundary lines, the prime data input requirement concerns the geometry of the latter. Initial coordinate information is required at discrete intervals along all of the boundary lines. This can be determined from physical models or sketches of the appropriate desired form.

The topology of the surface field idealisation is then defined, with the number of geodesics in each field such as to give approximately the desired cloth width. Positions of geodesic ends, such as to give the desired cloth orientation, are defined by a single reference coordinate, with the remaining coordinates set on the relevant boundary line by cubic spline interpolation through the

basic boundary defintion data points. Initial surface coordinates for field nodes are then automatically estab- lished by linear or quadratic interpolation between the defined geodesic end points.

Thus data are essentially limited to geometry definition at boundary lines and the initial location of geodesic ends on those lines. Only the membrane stress level and outline topology need to be set for the surface fields. Dynamic Relaxation will cope naturally with any gross out of balance forces generated by approximations in the initial surface geometry.

Analysis sequence Figure 6 shows an outline flowchart for the sequence of

operations within MAS/CP. The field and boundary analysis subsequent to the initial data input yield and equilibrium solution for both the individual elements and the overall structure, with the actual number of analysis cycles dependent upon the accuracy of the initial boun- dary information. At this stage the boundary location may be considered to be finalised, with effort now concentrated on refinement of the cloth patterns through adjustment of the geodesic line locations. When this has been complete, a final analysis of boundary lines may be undertaken for confirmation of the equilibrium of the complete structure. Any coordinate changes indicated by this relaxation may then be returned to the field analysis at the discretion of the engineer.

Graphic display At any stage of the analysis, the current geometry of

individual lines, fields and combinations thereof may be

I INPUT DATA: BOUNDARY COORDINATES I FIELD TOPOLOGY

t INTERPOLATE SURFACE COORDINATES INITIALISE ANALYSIS

REVISE GEODESIC END POINTS

SURFACEFIELD I--[UNFOLDCLOTH FORMANALYSIS I-[PATTERNS

CHECK CLOTH WIDTH

BOUNDARY RELAXATION & EQUILIBRIUM CHECK

Figure 6.

OUTPUT DATA:

M AS/CP analysis sequence

CUTTING PATTERNS

Adu. Eng. Software, 1981, Vol. 3, No. 3 117

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displayed on the VDU with any desired projection. This will enable visual confirmation of the current form, and rapid data validation facilities•

Cutting pattern control and output The information produced by MAS/CP for the cutting

of individual cloths is illustrated by the sample output of Figures 7, 8 and 9. The use of geodesics as seam lines ensures patterns that are evenly distributed about the cloth centre lines (Fig. "7). In order to make best use of available cloth width, MAS/CP incorporates automatic adjustment of geodesic end points along boundary lines. Cubic spline interpolation again ensures that such adjust- ments are compatible with the current boundary geo-

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metry. Where necessary, whole geodesics may be added or deleted.

The pattern information submitted to fabricators (Figs. 8 and 9) has been adjusted to allow for stretch in the warp and weft directions, and MAS/CP permits decompen- sation of these effects adjacent to boundary elements to ensure compatibility of strain.

SUMMARY

A program suite, MAS/CP, for the generation of form and cutting patterns for membrane structures has been pre- sented. Recent developments in analytical techniques have enabled MAS/CP to be mounted on a desktop computer and provide an economically viable solution in a design office environment. The program has been designed to be compatible with the general philosophy and structure of the MAS System of programs implemen- ted by Minitec Ltd. The features of MAS/CP have been illustrated by sample output from the definition of the Rotork Tent at Poole, Dorset.

REFERENCES

1 Kelly, P. and Watson. A. Structural Analysis using Desktop computers. Proc. 2ml Int. Cm!J~,rence on E,.qineeriml So.[hrare 1981, 346-359

2 Haug, E. Numerical Design and Analysis of Lightweight Structures. Proc. Institution ~![' Strm'tm'al Engineers Symposium 'Air Supported Structm'es: The State o.f the Art' 1980

3 Barnes, M. R. Formfinding and Analysis of Tension Space Structures by Dynamic Relaxation. PhD Thesis, The City University, London, 1977,,

4 Williams, C. J. K. Formfinding and Cutting Patterns for Air Supported Structures. Proc. Institution of Structural Engineers Symposium 'Air Supported Structures: Tire State ty'tlre Art' 1980

5 Wakefield, D. S. Dynamic Relaxation Analysis of Pretensioned Networks supported by Compression Arches. PhD Thesis. The City University. London• 1980

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