Heinz Isler's physical form finding of the HIB tennis shells

The Swiss engineer Heinz Isler (1926 – 2009) is among the most prominent figures in shell design. Thanks to a form-finding approach based on the use of physical models, he designed and built many shell projects in reinforced concrete. His unconventional structures still represent an important source of inspiration for today ’ s structural engineers. The paper reconstructs Isler ’ s experimental method by looking at the multiple physical form-finding models he developed for his tennis hall shells. Designed for the first time in 1977, they became one of Isler ’ s most successful shell typologies, promoted as “ HIB ” shells in Switzerland. Despite their apparently simple shape, Isler produced the largest number of physical form-finding models for this specific shell type. Their double symmetry challenged his design method: the highest precision was needed to avoid any irregularities in finding the appropriate geometry. By studying the original materials stored at the Heinz Isler Archive (gta Archive / ETH Zurich), details about Isler ’ s experimental approach to the conceptual design of his shell structures are revealed for the first time.


Introduction
Shells are among the most elegant structural systems thanks to the direct relation between their shape and the corresponding flow of internal forces.They are efficient because they optimize material use.Whenever self-weight is the predominant load case, they accommodate several different loads by an almost uniform membrane state due to the "double arch effect" [1].Such structures perform well over the years: if they are built correctly, no substantial problems linked to cracking or creep emerge.The structures of the Swiss engineer Heinz Isler (1926Isler ( -2009) ) are good examples in this sense: after many decades and despite the unprotected concrete surface, the shells are still in good condition.However, any slight deviation from the original design might cause serious problems.For reinforced concrete shells, for example, tension and bending should be avoided.In this context, the form definition of shells is one of the most relevant tasks in the conceptual design phase.
Form-finding methods help obtain the appropriate equilibrium shape in compliance with external and internal forces.Together with the German architect Frei Otto (1925Otto ( -2015)), and the Italian engineer Sergio Musmeci , Isler is considered one of the masters of physical form-finding techniques in the second half of the 20th century [2].Form-finding approaches ask for design parameters as the starting point of any design exploration.If external forces are linked to material choices (dead load) and to geographic location (wind and snow loads), the internal ones are dependent on numerous factors such as material characteristics (isotropy or anisotropy), plan geometry (symmetry or asymmetry), and potential prestressing forces applied to the system.Such methods help generate free-form shells in equilibrium with free edges (Fig. 1).The loading cases and the boundary conditions are controlled, while the three-dimensional geometry is unknown.Because of the common principles behind the technique adopted, physical form-finding methods have always been an important reference for structural designers.The foundations of computational form-finding tools were laid in these initial experiences [5].In the early 1970s, Linkwitz and Scheck followed the same approaches and developed the Force Density Method (FDM) [6,7], which is considered the first computational method for finding the shape of cable-net structures.

Hanging membranes
According to Isler, the shapes of his shells are based on the same structural principles as those found in natural shells.As advertised in the brochure of the tennis shell projects (Fig. 2), his design goal was to reproduce natural forms through a physical form-finding experiment [9].By exploiting multiple physical models, he controlled "new shapes for shells" [3], whose geometry could not be expressed with analytical equations at that time.At the beginning of his career, he developed three form-finding methods that were all related to physical concepts with static meanings: the "freely shaped hill", the "membrane under pressure", and the "hanging cloth reversed".
The most complex shapes were found through the "hanging cloth reversed" methodalso known as "hanging membrane" (Fig. 1, right), which expresses a relationship between a flexible hanging membrane and a shell.The first one assumes a shape in tension without bending under gravity, representing the (inverted) dominating load case (selfweight).If it is made rigid and inverted, it results in a funicular form under pure compression, for that same loading case, which is particularly appropriate for materials with low resistance for tensional forces such as masonry or concrete.The technique expresses an inverse formulation of equilibrium.It was based on Hooke's principle of the inverted hanging line [10] and Isler was not the first one who had adopted it.Early examples can be traced to Giovanni Poleni's analysis of       St. Peter's dome in Rome, Christopher Wren's studies for the dome of St. Paul Cathedral [5] and Antoni Gaudí 's combinations of catenary curves [11] or the spatial equilibrium for the crypt of the Colònia Güell [12].Isler extended Hooke's principle to continuous shells.If in his well-known public demonstrations, he used hanging cloths saturated with resin, in real projects he employed a rubber membrane clamped by a timber frame, onto which gypsum was poured.The isotropic material helped control the hanging procedure.During the experiment, the membrane would deform and find its equilibrium form due to the force of gravity [13].Even though Isler adopted other form-finding techniques, the "hanging membrane" was considered "the best method for design" [3: 2].Most of Isler's surviving models followed this principle.For the tennis hall projects, the highest number of models were made.
In the tennis structures, Isler and his team mastered the control of the relation between form and force in a system of adjoining buildings (Fig. 3).Their iconic form became the reference image for any tennis hall facility in Switzerland, as had happened for Isler's industrial buildings (Fig. 4, left).
The series was designated "HIB" shells, from the names of the three actors involved in their promotion (Haus + Herd), design (Isler), and construction (W.Bösiger AG).Between 1977 and 1993, fourteen projects were built following the "HIB" typology, for a total of sixty reinforced-concrete shells.After the industrial structures, they became the most prolific shell typology in Isler's oeuvre.Isler's early studies looked back at his well-known shell series, either as a roof componentlike the industrial bubble shells -or as a stand-alone buildinglike the garden centre typology [14] (Fig. 4).
However, it became clear that the requirements of a tennis courta longitudinal dimension of approximately 40 m, an inner height of a minimum of 5 m, and a maximum of 9 m, and diffuse lighting conditions asked for a new shell typology (Fig. 3).Following Isler's statement that "the shell is the supporting structure and the space enclosure at the same time" [15: 149], the tennis halls guaranteed an elegant structural solution to the architectural program.

Form-finding apparatus
The "HIB" structures showed the purest application of Isler's hanging membrane form-finding method since they constitute a shell with a rectangular plan sitting on four supports at equal height.However, despite the shape's apparent simplicity, they represented Isler's most challenging project (Fig. 5).
After ten years of experience with hanging membrane models, extensive refinement of the form-finding technique became necessary.At first glance, this aspect is quite surprising.Other prominent projects by Isler, like the Sicli SA factory in Geneva (1968-69), asked for highly accurate apparatuses to find the unconventional doubly curved form [16,17].However, it was easier to find a completely asymmetric shape than the doubly symmetric one of the tennis shells.While formally complex shells allowed tolerances in the form-finding phase, the "HIB" shells demonstrated the "need of highest accuracy" [18: 222] in the physical modelling technique.The equal thickness of the gypsum layer and symmetry in both directions required many iterations until the "[geometric] solution [was] found" [19].The fifteen iterations of form-finding models for the first tennis shell project in 1977 underline these difficulties (Fig. 5).Isler spent 38 days of intensive laboratory work on them.It was the biggest exploration he ever did on form-finding methods for shell design.Through the twelve surviving models, the authors traced back his experimental procedure for the first time.Unlike Otto's research laboratory, where sophisticated machines controlled his shape explorations at the Institute of Lightweight Structures (IL) at the University of Stuttgart [20], Isler's form-finding tools were constituted of economic materials borrowed from daily life: a membrane hanging from a timber frame was used in the basement of his office in Lyssachschachen [2].However, despite the appearance, his engineering practice was the outcome of extremely accurate techniques that were refined over the course of his career [21].The hand-crafted experimental devices allowed easy and fast repetitions of the same test.This became fundamental for the physical models of the "HIB" shells.At different stages, Isler worked on the position and prestressing of the hanging membrane (Section 3.1), the gypsum mixture (Section 3.2), the experimental equipment (Section 3.3), and the overall methodology (Section 4).Such elements represented design parameters that could be modified to achieve symmetric results in a pure hanging membrane test.Specific protocol sheets collected the experimental outcomes through qualitative data [22].They indicated the date, time, project, the person responsible, and the progressive number of the experiment.A series of comments enriched such precious documentation.They highlighted the positive and negative results of each study in terms of the ratio of the ingredients and setting time, description of the experimental procedure, sketches, and geometric considerations (position of the supports and curvatures of the shell).The following sections will describe Isler's explorations by focusing on the different elements constituting the physical form-finding device (Fig. 6).Their changes throughout the tests helped derive the most appropriate shape to accomplish the requirements of geometry, symmetry, structural behaviour, and architectural appearance.

Membrane
In its undeformed state, the membrane followed the plan dimensions of the initial design idea at the scale of the physical model.Its material, Fig. 6.Elements constituting the physical form-finding device for Isler's tennis shells: (a) timber anchorage, (b) timber frame, (c) frame cut, (d) securing mechanism, (e) rubber membrane, (f) rubber bands, (g) timber bottom plate.The device is symmetrical along the "frame cut".The drawing shows a cut section.(Drawing by Giulia Boller).
shape and elasticity influenced the found form.Even though Isler's public shows always displayed a hanging textile saturated with resin, the actual form-finding experiment for his projects employed a 0.3 mmthick rubber membrane onto which gypsum was poured (Fig. 7).The isotropic material guaranteed a uniformly stressed funicular shape, thanks to the constant stiffness of the membrane and without the influence of textile threads.If the membrane was too small and stiff, the resulting shape would be too flat; if too large and flexible, wrinkles would develop during the deformation.Different options for cutting and fixing it to the wooden board were explored [23].
Reference lines were drawn with a black pencil and ruler, on the membrane side towards the poured material.Since they were automatically transferred onto the model's gypsum surface during the formfinding process, they are still visible on the surviving objects (Fig. 8).On the one hand, such guidelines helped evaluate the geometric features of each shape by comparing different physical models belonging to the same study.On the other hand, they provided a precise picture of the membrane's behaviour during the experiment: a denser measurement grid showed higher curvature, while a flatter one emphasized a gentler transition in its three-dimensional geometry.
Additional circular marks highlighted the reference points to detect the geometry: main axes, peak, and support points.Once the model hardened, Isler annotated the number of the study with Roman numerals on its surface (Fig. 8).These physical traces, together with Isler's hand annotations, help reconstruct the changes applied to the rubber membrane after each form-finding study.
The membrane was anchored to the experimental device using nails and small timber elements (Fig. 7, left).While the hanging shape coincided with that of the final configuration of a numerical form-finding study, the membrane's anchorage points in the physical study showed a feature related to its manufacturability: the support points required membrane strips that were longer and wider than in real conditions to ensure proper anchorage to the timber frame.The archival documents and the original physical models do not explain clearly how Isler managed to have straight open edges in the long direction.Additionally, from the available photo documentation, it cannot be concluded if the membrane was initially straight on the form-finding apparatus or whether it had a longitudinal edge slightly curved outwards (Fig. 7, left).It might be that Isler's collaborators adjusted the model's edges afterwards by shaving off the excess material.Since the documentation about the fifteenth and last form-finding study shows that attention was paid only to the height difference between the shell's peak and the longitudinal edges, it is not possible to compare the overall geometric definition with the construction drawings of the built shell.
The membrane's position was fundamental for the experimental output, but it was not the only design parameter.Membrane prestressing was introduced to better control the hanging process [24].The applied force highly influenced the final shape: the more the membrane was  prestressed, the more symmetric the shape was, but the lower the longitudinal curvature was.Isler explored tests with a range of membrane prestressing levels, measured as strains between 0% and 7% of the membrane's initial, unstressed length.Markers with metric units on the board helped control the membrane prestressing geometrically and annotate the different values during the studies.Experimental results showed that a prestressing of 2.8% helped achieve the longitudinal curvature meeting the design requirements for the tennis shell projects [23].

Gypsum mixture
A layer of poured gypsum represented the load applied to the membrane for the form-finding process to start: gravity defined a doubly curved tensile surface between the supports.The uniform gypsum layer represented the dead load of the reinforced concrete shell of uniform thickness.The quantity and choice of material mixture were as crucial as the membrane for the success of the experiment.In Isler's first formfinding studies with the hanging membrane method at the end of the 1960s, the focus was on the amount of gypsum in terms of material thickness on the experimental apparatus.For example, the study for the architectural office Grob in Sargans used a constant thickness of 1 cm at the beginning of the test [24], while the one for the company building Sicli SA explored different options by varying the material thickness between 4 and 6 mm [25].
For the "HIB" project a more advanced investigation was conducted: the ratio between water and gypsum powder was explored.This parameter influenced the material workability during the hanging process: the more water was put in, the easier its manipulation was, but the less strong the hardened material would become.On the contrary, if the material was too dry, it cracked easily.Five different material tests were performed in Isler's model workshop.A specific devicethe "gypsum machine" -was manufactured to control the mixture's properties (Fig. 9) [26].
A white plastic band was placed around two plastic cylinders rotating around their longitudinal axes through a rudimentary electric mechanism.Each time, a sample of the gypsum mixture was poured onto the band with a metal ladle.Isler observed the way the material was progressively poured during the machine's rotation and carefully annotated the minutes and time when cracks appeared in the material sample.On the one hand, the ratio between water and powder was increased to 0.67, which is above average for gypsum mixtures.On the other hand, drops of lemon juice were added.Experimentally, citric acid produced a retarding action.This double operation avoided cracks on the surface caused by an early drying mechanism.The final recipe of the most appropriate gypsum mixture was constituted of 3 kg of gypsum, 2 l of water, and 60 drops of lemon juice for making a model at a scale of 1:75 [27].

Equipment
In most cases, two people were involved in the model-making phase.Like the material choices of both membrane and mixture, the dimensions of the form-finding equipment influenced the experimental results.Typically, the scale of the test was chosen to obtain a model's longitudinal length between 50 and 60 cm.This size was appropriate for the object's manipulation: if too small or too big, it would not have been possible to achieve the necessary precision for scaling up the results for  the further design phases.As a consequence, the form-finding studies for the shells of the projects Gips Union AG (plan dimensions of 31.5 ×25 m) and Grob (plan dimensions of 24 ×20 m) were performed at a scale of 1:50, while the ones for the Sicli shell (plan dimensions of 57.5 ×34.5 m) were at a scale of 1:100.Since the longitudinal dimension of the "HIB" shells was close to the one of the Sicli project, the first nine "HIB" studies were at a scale of 1:100, which resulted in a model with longitudinal length of 47 cm.The increase in model size to 1:75 from the tenth test onwards, with a resulting longitudinal length of 63 cm, helped reduce incorrect outcomes due to tolerance [28].
Guaranteeing double symmetry in the physical form-finding procedure of the "HIB" shells did not only ask for an increase in model size.Each structure constituted a module that needed to be placed next to each other.While the straight prestressed edge beams of Isler's industrial shells had to assure a perfect match between the modules while equilibrating the spatial structure [29], the tennis shells had to respect an accurate definition of both longitudinal and cross curvatures to match perfectly.As for the material choices, the form-finding equipment for the "HIB" experiments became more sophisticated than in Isler's early hanging studies.A few changes to the experimental device became necessary (Fig. 10) [26].
Small upturned edges stiffened the shell's open edge in the longitudinal direction.This detail was understandably difficult to achieve in a scale model.It was further complicated by the fact that symmetry in the transverse direction had to be maintained.One additional rubber band was placed on each side to control the cross-section of the experiment.Rubber tests were performed to define the type, size, and prestressing level of these additional bands [30].Bands with a width of 1.33 cm and a prestressing of 2% were preferred to threads (Fig. 7, left).The symmetry of the longitudinal curvature was controlled through equal membrane prestressing.In the studies for the Grob project, this procedure had already been tested by two people simultaneously pulling the membrane on both short sides of the form-finding apparatus [24].However, difficulties in the form-finding led to asymmetric outcomes for the first five models of the tennis shells.The sixth study introduced a transversal cut in the timber frame, which worked as a pre-tensioning device: the membrane was first anchored to the supports, and only at a later stage, stretched by moving the two, symmetric U-shaped wooden elements outwards (Fig. 10, left).Once the needed tension force was applied to the membrane, the mechanism was secured [31].
While the longitudinal and cross-sectional curvatures were controlled through elements that affected the behaviour of the rubber membrane, the overall form was improved thanks to a change in the equipment.In the form-finding studies of previous projects, once the membrane was fixed to the timber frame, it was placed on a flat bottom plate.This support allowed pouring a controlled layer of gypsum onto it.The fifth form-finding study for the tennis shell projects resulted in a model with a central area that was too flat.To remedy this, a slightly curved bottom plate was used (Fig. 10, right), with a sag of 7 mm [32].Despite this small height difference, the change in the device helped modify the form in its central part.During the pouring process, the material distribution was not uniform due to the bottom plate's transversal section: more gypsum in the central area increased the cross-sectional curvature of the hanging membrane, while reducing the horizontal forces towards the supports.This aspect is immediately visible by comparing the longitudinal side views of the fifth and sixth models (Fig. 5).If in the form-finding process more material was placed towards the centre of the model, the opposite was featured on the full-scale shell.The material thickness was increased towards the supports.In the first case, the additional material helped correct the geometry.In the second case, it was placed to sustain the stress concentration.Changing the weight distribution during form finding offers to obtain different doubly curved geometries.However, this affects the shell's structural behaviour.Because of the different load case used for the form finding, differing from the self-weight distribution to be expected in the real structure, the central surface of the built shell would not be in a uniform membrane state.This would be relevant in the theoretical case of an infinitesimally thin shell.Practically, though, because of both the finite thickness of the shell that allows deviations for the theoretical solutions, and the double curvature, slightly different equilibrium states can be found that stay very close to the central surface.Additionally, the shell can also find an equilibrium with non-uniform stress states.

Form-finding procedure
The form-finding experiment was based on a trial-and-error approach.If nowadays a computer program adopts an iterative procedure to find the most appropriate shape for a specific design problem, Isler's approach led to multiple physical models that were refined based on the experiment's observations.Each model was generated in approximately 20 min (preparation, pouring, and drying process).Therefore, it could be easily replicated multiple times.For the "HIB" shells, thirteen tests were conducted before the correct procedure for the form-finding study was defined [33].The edges and the supporting points represented the most critical areas.They were carefully controlled.Additionally, the prestressing of the membrane and the longitudinal bands were checked through visual observation and simple tension tests [30].After these preliminary steps, the experiment could start.Two timber trestle legs and a stool supported the hand-crafted equipment (Fig. 11).The form-finding device consisted of the timber frame and the rubber membrane was placed on the curved timber bottom plate.Water and lemon drops were mixed, right before gypsum powder at the defined ratio was added.The mixture was poured onto the membrane and distributed over the surface at minute four from the start of the experiment.A plexiglass ruler helped remove the excess of material.Air bubbles were removed by lightly tapping the form-finding apparatus.At minute nine, the cut-out timber frame was carefully separated from the bottom plate.As a consequence, the membrane started deforming, hung from its supports, adapting its shape to the gypsum's load during the hardening phase.The hanging procedure needed a gradual and symmetric movement.In earlier projects, the form-finding apparatus was lifted to reach higher supports on the trestle legs.Two draughtsmen facing each other at the short sides of the timber frame were in charge of this operation.For the tennis shell studies, instead, the approach was reversed.The apparatus' bottom plate was placed on the stool in between the trestle legs.The form-finding device was kept untouched in the same position, while its bottom plate was gradually lowered thanks to the stool's screw mechanism (Fig. 11).
The mixture setting time contributed to the success of the physical experiment, together with the mixture choice.If the operator had waited just a minute longer before the start of the hanging process, cracks would have appeared on the surface.The experiment ended as soon as the material was fully hardened, at minute twenty.Additional gypsum was used at a later stage to fix small steel elements that formally reproduced the prestressing cables that would be installed for the fullscale shell.
Once the gypsum hardened, the rubber membrane was removed.According to the multiple parameters implied, a significant number of solutions was found.The form-finding models were collected on a table.Since they were the physical output of a hanging membrane procedure, they all represented shapes in equilibrium under the defined constraints.However, experimental iterations showed that not all of them were satisfied at the same time.The interaction with the designer was important for deriving the most appropriate shape that accomplished the requirements of geometry, symmetry, structural behavior, and architectural appearance.Their shapes were compared by examining the reference lines on the gypsum surface.The geometry of the smallscale physical model supported the preparation of the drawings for manufacturing the timber falsework at full scale on the construction site.

Results
The original archival materials stored at gta Archives made it possible to understand Isler's iterative procedure that helped refine his form-finding technique based on the membrane features, the ratio of the materials in the gypsum mixture, the experimental equipment, and the overall procedure.Each time, Isler and his team changed these parameters to find the optimal solution meeting the aesthetic, functional, and structural requirements for a tennis sports hall (Table 1).
Since there is no surviving material (document or physical model) about the first test Isler developed for the tennis hall shells, the starting point of his investigations is not clear.The second form-finding study employed a prestressed membrane and a ratio between water and gypsum powder of 0.71.The result was a symmetric shape with a low crosssection.Although there is no information on the membrane, the surviving physical model shows bulging longitudinal edges, which suggest a wider membrane than its dimensions in the plan view.Since the third and fourth models show straight longitudinal edges, it is plausible to believe that the cut shape was reduced, in addition to adjusting the ratio of the gypsum mixture to 0.61.If the fifth test introduced a new membrane cut which was again wider in the transversal direction, it is with the sixth test that major changes were introduced to increase the cross curvature and have uniform thickness.Such modifications affected the experimental equipment (Section 3.3), while the gypsum mixture and the membrane cut remained the same as before.The result was a stronger curvature, but an asymmetric shape.This led to small variations in the membrane prestressing in the seventh and the eighth tests, until the extreme case of the ninth test which did not consider any membrane prestressing.The tenth test applied new changes to the rubber membrane, including the introduction of additional rubber bands (Section 3.1).It increased the model size from a scale of 1:100 to a scale of 1:75.Such variations were intended to better control the symmetry and the counter curvature along the longitudinal edges.Additionally, the tenth test reconsidered the gypsum mixture and developed five material tests to evaluate the proportion between the ingredients and the timing of the hardening process.Lemon drops were put into the gypsum mixture to better control the experimental procedure due to their retarding action in the hardening phase (Section 3.2).The effect of adding citric acid to the mixture can be seen by the minute when the form-finding apparatus was lifted to make the membrane hang down under the load of the gypsum mixture.The duration of the experimental procedure was about 10 min longer compared to previous tests.Unfortunately, the physical model resulting from the tenth test did not survive.The eleventh test reduced the waiting time before the hanging procedure from 19 to 15 min since the previous one showed cracks on the model surface as a sign of an excessively long waiting time.The twelfth and thirteenth tests changed the number of lemon drops to define the most appropriate experimental procedure in terms of quantities and timing.They showed that the retarding action did not increase according to the percentage of the lemon drops, but according to the time they remained in the mixture before the start of the experiment.The fourteenth test consolidated the experimental procedure in terms of the mixture, the equipment, and the overall timing in the single operations.The cross curvature was increased in the fifteenth model, which was considered the final model, thanks to additional prestressing in the rubber bands along the longitudinal edges.The geometric data of the fifteenth model served as the basis for the construction drawings of the built shell.

Discussion
The study of Isler's original materials revealed details about his experimental approach to the conceptual design of his shell structures.For the first time, it provided relevant information and new insights to understand the complexity behind his form-finding practices: the membrane prestressing (Section 3.1), the gypsum mixture (Section 3.2), the equipment (Section 3.3), and in general the experimental procedure that was developed in the office (Section 4).Such a method helped find a solution that met the design requirements.

Free edges
The attention paid to the definition of curvature, which has an important impact on the shell's structural performance, is highly visible in the protocol sheets during Isler's experimental procedures.In particular, the free edges represented the most critical areas from the perspective of both symmetry and structural behaviour.In early formfinding studies, such as the one for the Grob shell, a membrane reinforcement along the perimeter was adopted.This additional detail and restraint in the model translated to a hidden edge beam in the real construction [24], which was discovered during the shell's demolition in 2021 [34].In contrast, the "HIB" studies showed that a more elegant alternative was possible.If the membrane was cut a bit bigger than the actual hole, the excess of flexible material would fold over, locally forming a negatively doubly curved shape, which automatically resulted in a stiffened edge.Moreover, the upturned longitudinal edges, a reminiscence of Isler's garden centre series, prevented the thin shell from buckling.A closer look at three models from the third, the sixth, and the eleventh "HIB" studies shows how changes to the membrane and the experimental equipment helped control the curvature in both directions (Fig. 12).
For the same amount of material, the sixth model (Fig. 12, middle) shows a more pronounced longitudinal curvature than in previous studies, as, for example, the third model (Fig. 12, top), precisely because of the curved bottom plate used in the form-finding apparatus, and the resulting non-uniform distribution of gypsum (Section 3.3).In Fig. 12, bottom, the use of rubber bands along the longitudinal edges is visible on the left side view of the model.The upturned edges were controlled in the longitudinal direction with these bands.The model's right side shows a problem that occurred during the experimental procedure, breaking off part of the upturned edge.The model was lifted when the material was still fluid, but because the rubber band did not have enough stiffness to keep the gypsum load, it slipped away, damaging the edge.It might be that this was caused by an excessive amount of lemon drops being put into the mixture, as subsequent experiments showed a change only in this design parameter (Table 1).

Physical and numerical form-finding approaches
Physical form-finding approaches such as the one described in this paper represented precursors of computational form-finding methods [35].Since the found shape was the result of an iterative procedure, the use of computers for this was an understandable choice in the transition toward digital approaches.For this reason, Isler can still be seen as an important reference for today's structural designers.In this section, similarities and differences between a physical and a numerical study of such free-form shapes will be compared.
The design parameters that contributed to the final shape in the "HIB" physical studies can be implemented in a computational framework.In its initial state, Isler's rubber membrane can be numerically translated as an undeformed plane membrane.Unlike the isotropic material adopted in the model workshop, the numerical membrane consists of a mesh.The hanging procedure starts as soon as a load case is applied to it.In the past and as witnessed by this paper, intuition gained over years of experience supported the evaluation of the form.On the contrary, the same physical experiment can be simulated nowadays with finite element procedures and further studies help correlate the materials used on the real shell structure with the found form: numeric data linked to the found shape can be processed in a subsequent structural analysis using Finite Element Methods (FEM).This procedure is much quicker than a study with physical models: within a few seconds of calculation, the result of the analysis can be obtained, whereas, in Isler's case, it took weeks to translate the shape found into a measurement model and perform detailed tests on its structural behaviour.If progressive iterations with physical models helped find the equilibrium form meeting the design requirements, numerical shape optimization techniques represent a synthesis of geometrical design, structural analysis, sensitivity analyses, and mathematical optimization [36].

Numerical studies on Isler's tennis hall shells
In the early 1990s, the geometry of the tennis hall shells was explored with digital tools at the University of Stuttgart [37].Interestingly, Isler's physical models and these computational ones showed comparable results (Fig. 13).Even with different media, both were based on optimization processes obtained through iterative procedures.The physical form-finding method of the hanging membrane considered gravity as the governing design load.On the contrary, computational tools combined different load cases to define the most appropriate shape, even with deviations from the pure membrane state.Changes to the initial configuration improved the shape in terms of structural efficiency and optimized the material use.In both the physical and numerical outputs, the shell showed a boundary stiffened by a distinct negative curvature.
If the physical hanging procedure needed the full geometry to produce the gypsum model during the experimental phase, only one-quarter of the shell was considered to run the optimization process digitally: the double symmetry helped reduce the calculation time and the complexity of the numerical procedure.This furthermore reduced the risk of inaccurate results, which were the main concern during the physical experiment.While digital form-finding methods confirmed Isler's experimental results, recent FEM studies on the tennis hall shells highlighted that they helped find shapes with appropriate structural behaviour also from the point of view of stiffness and concrete allowable stresses [38].Additionally, the shell's formal clarity reflected the shape's efficiency linked to the appropriate use of concrete as building material [39].
Isler's career took place during the generation shift from physical to digital approaches for the design of structures.While the studies by Ekkehard Ramm and his group were developed, and most likely after seeing their research outputs, Isler acknowledged that computer programs could simulate the physical form-finding procedures in the near future, even though their results had not yet been compared to the built shells [40].He understood the potential of digital methods to easily explore different design options.However, in his opinion, "the computer only answers the question we have asked it.It is not in a position to point out questions that we have not yet asked, if we have not yet recognized or experienced the problems.The comprehensive [physical] model, on the other hand, can do that."[40: 61, translation by the author].Despite Isler's known aversion towards digital methods, recent discoveries in Isler's archive showed evidence of the extensive use of early computational tools in his engineering practice [21].They were primarily adopted to automate repetitive calculation processes and to deliver the required documentation to the foreign authorities, where the standards did not accept the use of physical models to verify the structural behaviour of his shells.In the tennis shell projects, such approaches played a crucial role in the refinement of the shape, after deformations occurred on the first built complex in Düdingen.A follow-up paper will be devoted to this and will explain how Isler modified the overall design for his subsequent tennis halls.However, regardless of these experiences, Isler's physical form-finding models remained the most important aspect of his unconventional engineering practice.

Conclusion
The paper reconstructed Isler's famous form-finding approach based on the hanging membrane method by researching the original documents, sketches, drawings, physical objects, and pictures that Isler's office produced during the early-design phase for the tennis shell projects.After an introduction that provided a framework for Isler's work within the field of structural design and form finding (Section 1), the hanging membrane method was explained by making references to Isler's built and published work (Section 2).The form-finding apparatus was then extensively described, in all its parts (Section 3).The explanation of the form-finding procedure helped us understand the different phases (Section 4).The experimental results were commented on by comparing the changes to the membrane, the gypsum mixture, the equipment, the procedure, the plan, and height dimensions (Section 5).A further discussion part focused on specific design features and opened up the reflection on contemporary computational form-finding methods (Section 6).This paper was possible thanks to the extensive documentation stored at the Heinz Isler Archive (gta Archive / ETH Zurich).On the one hand, Isler was the first person to archive himself.He carefully filed every document in his office and private house with specific codes that are similar to the common record-keeping methods used by archivists.On the other hand, the gta Archive kept most of his documents untouched, which is a very rare circumstance especially when we speak about the archives of an engineer.Thanks to such a rich collection, the paper demonstrated the importance of high-quality original materials of great master builders of the past as a basis to fully understand their intuitions, methods, and practices.This research does not only inform potential future contemporary digital form-finding applications.It can serve as a basis to consciously intervene in the existing built structures by knowing the complexity behind Isler's unconventional design methods.
Over his career, Isler consolidated an extremely accurate engineering practice that linked a form with its internal stress state.The fifteen physical models for the "HIB" shells embodied his approach to the conceptual design of shell structures.Being the output of high craftsmanship, the paper traced back to the form-finding procedures adopted by Isler's office for the tennis shell projects.
The "HIB" structures will remain among the most elegant shells that Isler ever designed.A few models of the project were prominently displayed in David Billington's exhibition "The Art of Structural Design: A Swiss Legacy" at the Princeton Art Museum in 2003 (Fig. 14).
While two adjoined tennis modules were shown on the table, the form-finding model and its wooden apparatus hung from the ceiling.However, the three objects were not original: they were approximated reproductions manufactured by a graduate student.From this viewpoint, it is possible to understand the negative reaction of the Swiss engineer, especially towards the form-finding model [41]: on a piece of paper, Isler annotated that "the real shape needs a delicate proceeding in several laborious steps [,] and this procedure is not yet given free by myself.and nobody else knows them" [42].
The paper filled in this gap of knowledge by providing details about Isler's form-finding procedures in the tennis shell projects.It showed how complex Isler's engineering practice was for this apparently simple project, despite the basic structural principles that were used as a starting point.G. Boller et al.

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Table 1
Summary table of the experimental results for each physical model, based on the archival documentation.