BUCKLING STUDIES INTO LARGE SCALE HYPERBOLIC PARABOLOID SHELL AND LATTICE STRUCTURES

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1 BUCKLING STUDIES INTO LARGE SCALE HYPERBOLIC PARABOLOID SHELL AND LATTICE STRUCTURES James Bernasconi 1 ABSTRACT: This paper covers some aspects of the bifurcation and elastic geometric non-linear buckling behaviour of hyperbolic paraboloid lattice and concrete shells up to 1 metres in plan size. The paper summaries the important design parameters discovered by earlier research and used variations in the and edge beam flexural stiffness to undertake a parametric study. The bifurcation buckling of lattices was found to be similar to that of shells. The elastic geometric non-linear analysis was always found to be unstable for shells and stable for rectangular lattices. However, diagonal lattices behaved in a similar fashion to the unstable shells. KEY WORDS: hyperbolic paraboloids, shells, bifurcation buckling, elastic geometric non-linear buckling 1. INTRODUCTION This conference paper presents some of the research results from a recent PhD thesis [3] into the design and parametric study of large-scale lattice hyperbolic paraboloids. the centre portion is a segment of a hyperbolic paraboloid, used on the aquatic centre for the same Olympic Games, Figure 2. WHAT ARE HYPERBOLIC PARABOLOIDS? A hyperbolic paraboloid ('hypar' for short and used hereafter) is the name given to a mathematical description of a well-known physical shape. The name gives away the principal elements, hyperbolas and parabolas. Hypars are anticlastic surfaces and unlike cups cannot hold a fluid no matter how they are orientated in space. As well as possessing maximum and minimum curvatures called principal curvatures, they also have the property of being surfaces of translation. The most obvious format is a saddle shape such as in the velodrome for the London Olympic Games in 212 shown below in Figure 1. Figure 2: Aquatic centre for the London summer Olympics, 212 [1] Not so obvious in the aerial view, Figure 2, but during construction of the centre portion, Figure 3 clearly shows the hyperbolic paraboloid shape. Figure 1: Velodrome for the London summer Olympics, 212 [1] However sometimes architectural demands require that only a portion of the basic shape be used. In this case, 1 James Bernasconi, consulting engineer, Brisbane, Australia, jimjgb@bigpond.com Figure 3: Construction of aquatic centre for London summer Olympics, 212 [2]

2 The same general saddle shape is presented again, but this time with a hole in the middle and the ends missing, as used in the Sydney Olympic Games in 2, see Figure GEOMETRY AND RESEARCH PROPOSAL HYPAR GEOMETRY All of the previous examples generate the shape in the same basic way - by sliding or translating vertical parabolas over each other. This is shown in Figure 5, taken from the out of print book by Schueller [5].. Figure 4: Artist impression of the stadium for the Sydney Olympics, 2 It will be obvious from viewing these buildings, that apart from the pleasing lines created by the hypar, the shape has an unique ability. The ability to span large areas with no intermediate columns and this feature was recognised as early as the 193s. However, it was Felix Candela who popularised the shape during the 196s in the new working material of the time, reinforced concrete, to create structural shells. The increasing use of the form, lead to a rise in research interest and most research was conducted with concrete in mind and involved the use of small shell models. In the beginning, the concrete was proportioned by a few simple formulae based on membrane theories all of which had some known inconsistencies in their theory. After some years, the shape fell out of favour along with the construction of large concrete shells. In recent times, the shape has been revived but in the form of lattices in steel or timber. However, there was little research into large-scale lattice hypars available, only the earlier, mainly concrete, work. In usual construction, shells fabricated generally conform to the geometric requirements for 'shallow shells' when there rise is small to their span. Vlasov [7] considered shells to be shallow if the ratio of the rise to the shorter side is less than or equal to.2. The research project described here has a range of rise to span ratios up to.33. Shells are considered to be 'thin shells' when they conform to Novozhilov's [4] requirement for if the ratio of thickness to radius of curvature is less than.5. Shells in civil construction always meet this requirement. According to Novozhilov [4] if these requirements are met, an engineering accuracy giving errors of about 5% is achieved. Figure 5: Hypar with parabolic shape together with straight line generators, Schueller [5] However there are other ways to look at things. The shape allows us to draw grids upon it and create something different by rotating the grid by 45 degrees. It is then possible to create a straight-sided rectangular or square plan shape of translation using either parabolas or straight lines. The illustration in Figure 6 is again taken from the book by Schueller and shows the square shape inscribed with parabolic lines, hyperbolic lines (obtained by cutting the figure in the horizontal plane) and straight lines. Figure 6: Relationships between parabolic, straight and hyperbolic curves, Schueller [5] The shapes shown in Figure 6 actually were the design basis for one of the main initial formats of the hypar.

3 The straight-line mathematical formulation extended very nicely into concrete construction. Here the curved shape of the hypar could be formed in concrete by using straight-line formwork. RESEARCH PROPOSAL The previous hypar shell research had been curtailed some decades ago and had been based on small laboratory bench models, with even the most ambitious up to about 3 metres in plan size. There was no point in conducting more small-scale research. It is also impossible to research a project specific shape such as those depicted in Figures 1 to 4. Therefore, a more generic hypar shape was chosen, such as those shown in Figure 6. This new research would be based on investigating square hypar models formed using straight-line generators in three sizes, based on the following plan dimensions: 18 metres 5 metres 1 metres There had been no research undertaken before into hypar models (shell or lattice) of this size. Of course, laboratory models of this size could not be physically built, so the logical modelling choice was using the computer finite element method. Many finite element programs now exist that provide convenient and reliable results in both linear and non-linear work. This research was undertaken using the program, Strand7 [6], developed by Strand7 Pty Ltd, Sydney, Australia. The research emphasis would be a parametric investigation into hyperbolic paraboloid shells and lattices to discover what implications become apparent for real structures of this size. RESEARCH SET UP AND MODELLING The large quantity of concrete hypar research available from the 196s would provide a platform of useful parameters that would form the starting point for this current research project. The main design parameters that would be worthwhile to investigate in research program would be: A comparison between concrete and lattice models to determine differences and similarities between the two groups Size effects that may come into play as the plan size increased to 1 metres Effects arising due to changes in the rise to span ratio, changing the apparent curvature of the models Changes in the orientation of lattice bars Changes in the flexural stiffness of the edge beam as this had been shown to be significant Effects arising from asymmetric loading Effects arising from induced defects in lattice construction Variations in the method of support offered to the corners and edge beams. Concrete provides a continuous surface, a continuum. Lattices however can come in a variety of forms and not just in one layer, sometimes two and three layer forms are used. For a research thesis, useful comparisons could be obtained between concrete and lattice if the configurations were similar. Therefore, only single plane lattices were considered. Lattice joints were considered fully rigid. Concrete and lattice edge beams were proportioned to give similar ratios of flexural stiffness. Lattice members were based on cold formed steel circular hollow sections (chs) in actual sizes available in the Australian market. Overseas steel markets would allow a greater choice in some cases. The concrete shell thickness is chosen to be close to a real life concrete thickness incorporating the usual allowances for cover and thus durability requirements are met. Models were created in two different arrangements: Corner supported (restrained in translation only about the 3 axes) Simply supported along the entire edge beam Loads were established firstly on the surface of the concrete models. Here the base load (15 Pa) was modelled on the projected area of the model. The load was set as 'global face pressure' on the surface and thus was independent of the change in curvature. To create the lattice models, the total equivalent load was placed at the nodes and the overall reactions checked between the concrete and lattice models to ensure compatibility. In all, the research program required the generation of over 4 computer models to investigate the various parameters. Figure 7 to Figure 11 show the different shell and lattice configurations.

4 Figure 7: A typical hypar shell (using plate elements) Figure 8: A typical hypar rectangular lattice grid (using beam elements) Figure 11: A typical hypar lattice with edge beam detail Strand7 offers the full range of analysis types including the static solvers: linear elastic, elastic geometric nonlinear and bifurcation analysis also known as linear buckling. In this paper only results from geometric non-linear and bifurcation, buckling will be considered, comparing concrete hypar shells with hypar lattices. This is a valid technique paralleling the technique called continuum modelling. Continuum modelling is most often used in preliminary stages as a structural analysis time saver, being much easier to construct a shell made of plate elements rather than building a lattice arrangement possibly containing tens of thousands of members. The plate model is then used to examine the macro effects in the structure. 3. SHELL AND LATTICE COMPARISONS This paper presents only some of the results obtained from bifurcation buckling and elastic geometric nonlinear buckling, comparing the response of hypar shells to hypar rectangular and diagonal lattices. BIFURCATION BUCKLING Figure 9: A typical hypar diagonal lattice grid (using beam elements) Bifurcation buckling modes were similar between shells and lattices in both corner supported versions and simply supported edge beam versions. The comparison illustrates that some useful macro effects can be observed in continuum modelling with plate models. Corner supported models Figure 1: A typical hypar shell with edge beam detail Figures 12 and 13 compare the buckling modes of similar sized shell and lattice models with a standard still within the definition of a shallow shell (<.2). Figures 14 and 15 compare bifurcation buckling modes of models with comparable edge beam flexural ratios.

5 Simply supported edge beam models Figures 16 and 17 compare similar sized shell and lattice models with a standard within the definition of a shallow shell. Figures 17 and 18 compare buckling modes with comparable edge beam flexural ratios. Figure 12: 5 m shell model,.16 Figure 16: 5 m shell model,.16 Figure 13: 5 m lattice model,.16 Figure 17: 5 m lattice model,.16 Figure 14: 5 m shell model, 5.56 edge beam flexural stiffness ratio Figure 18: 5 m shell model, 5.56 edge beam flexural stiffness ratio Figure 15: 5 m lattice model, 51.8 edge beam flexural stiffness ratio

6 normalised load intensity normalised load intensity normalised load intensity rectangular lattice bifurcation analysis - normalised load intensity vs m lattice 5 m lattice 1 m lattice Figure 19: 5 m lattice model, 51.8 edge beam flexural stiffness ratio It is also possible to examine the bifurcation buckling carrying capacity of shells and lattices as the rise to span ratios change (space does not allow for edge beam charts). It is also possible to compare corner supported models with simply supported edge beams. Figures 2 and 21 show a similar buckling capacity improvement for both changes in rise ratio with little effect due to changes in the plan size of the model. shell bifurcation analysis - normalised load intensity vs Figure 21: Corner supported lattice bifurcation analysis, Figure 22 for a simply supported lattice shows a large variation in buckling capacity with a variation in lattice plan size simply supported perimeter rectangular lattice bifurcation analysis - normalised load intensity vs m lattice 5 m lattice 1 m lattice m shell 5 m shell 1 m shell Figure 2: Corner supported shell bifurcation analysis, Figure 21 for rectangular lattices demonstrates some buckling capacity improvement and more size variation than similar shells. Figure 22: Simply supported lattice bifurcation analysis, rise to span For the variation in the edge beam flexural stiffness, concrete shells recorded normalised buckling capacity improvements of only 1. (18 m) to 1.25 (1 m). However rectangular lattices recorded improvements of 2. (1 m) to 25. (18 m), a vastly difference response to the shells. ELASTIC GEOMETRIC NON-LINEAR BUCKLING Shells The elastic geometric non-linear behaviour is depicted by use of load vs. deflection curves. Here the structures were loaded until instability stopped the solver progressing. All hypar shells exhibited instability as the load increased. This instability was independent of the changes in support from corner supported to edge supported. The instability was also independent of changes in (effectively curvature) and independent of changes in edge beam

7 flexural stiffness, see Figures 23 and 25. Figures 24 and 26 show the computer plots of the unstable shell behaviour m and 1 m corner supported shell geometric non-linear response load intensity vs deflection/span for various s (all curves truncated at 2 (kn/m 2 ) 5m rise to span.5 5m rise to span.8 5m rise to span.12 5m rise to span.16 5m rise to span.2 5m rise to span.33 1m rise to span.5 1m rise to span.8 1m rise to span.12 1m rise to span.16 1m rise to span.2 1m rise to span deflection/span (/L) (x1 4 ) Figure 23: Corner support shell elastic geometric analysis with variation in rise ratio Figure 26: 5 m shell, 5.65 edge beam ratio, 1st mode buckling In the case of simply supported shells, low rise to span ratio models (.5) exhibited stability that was independent of shell size, see Figure 27. However once the rise ratio increased (.16 for example), buckling again occurred, Figure 27 and the computer plot, Figure 28. Variations in edge beam flexural stiffness always allowed instability to develop as shown in Figures 29 and the buckled shell shown in Figure 3. Figure 24: 5 m shell,.16 rise ratio, 1st mode buckling m and 1 m simply supported shell geometric non-linear response load intensity vs deflection/span for various s 5m rise to span.5 5m rise to span.8 5m rise to span.12 5m rise to span.16 5m rise to span.2 5m rise to span.33 1m rise to span.5 1m rise to span.8 1m rise to span.12 1m rise to span.16 1m rise to span.2 1m rise to span m and 1 m corner supported shell geometric non-linear response 5m eb stiffness 1. 5m eb stiffness 1.1 5m eb stiffness m eb stiffness m eb stiffness 1. 1m eb stiffness 1.1 1m eb stiffness m eb stiffness deflection/span (/L) (x1 4 ) Figure 27: Simply supported shell elastic geometric analysis with variation in rise ratio deflection/span (/L) (x1 4 ) Figure 25: Corner support shell elastic geometric analysis with variation in edge beam

8 and 32. Figures 33 and 34 show the simply supported lattice behaving in a stable fashion with the changes in the rise ratio and the edge beam stiffness as well. Figures 35 and 36 show typical computer models for corner supported lattices, but these responses were also typical for simply supported as well, as the load increased the lattice remained stable. Figure 28: 5 m simply supported shell,.16 rise ratio, 1st mode buckling , 5, 1 m corner supported lattice geometric non-linear response load intensity vs deflection/span for various s 18m rise to span.5 18m rise to span.8 18m rise to span.12 18m rise to span.16 18m rise to span.2 18m rise to span.33 5m rise to span.5 5m rise to span.8 5m rise to span.12 5m rise to span.16 5m rise to span.2 5m rise to span.33 1m rise to span.5 1m rise to span.8 1m rise to span.12 1m rise to span.16 1m rise to span.2 1 m rise to span m and 1 m simply supported shell geometric non-linear response 5m eb stiffness 1. 5m eb stiffness 1.1 5m eb stiffness m eb stiffness m eb stiffness 1. 1m eb stiffness 1.1 1m eb stiffness m eb stiffness deflection/span (/L) (x1 2 ) Figure 31: Corner support lattice elastic geometric analysis with variation in rise ratio deflection/span (/L) (x1 4 ) Figure 29: Simply supported shell elastic geometric analysis with variation in edge beam , 5, 1 m corner supported lattice geometric non-linear response 18m eb stiffness 1. 18m eb stiffness m eb stiffness m eb stiffness m eb stiffness 1. 5m eb stiffness 9.9 5m eb stiffness m eb stiffness 9.6 1m eb stiffness 1. 1m eb stiffness 2. 1m eb stiffness 3.9 1m eb stiffness deflection/span (/L) (x1 2 ) Figure 32: Corner support lattice elastic geometric analysis with variation in edge beam , 5, 1 m simply supported lattice geometric non-linear response load intensity vs deflection/span for various s 18m rise to span.5 18m rise to span.8 18m rise to span.12 18m rise to span.16 18m rise to span.2 18m rise to span.33 5m rise to span.5 5m rise to span.8 5m rise to span.12 5m rise to span.16 5m rise to span.2 5m rise to span.33 1m rise to span.5 1m rise to span.8 1m rise to span.12 1m rise to span.16 1m rise to span.2 1m rise to span.33 1 Figure 3: 1 m shell, 5.65 edge beam ratio, 1st mode buckling Rectangular lattices In contrast to shells, rectangular lattices exhibited complete stability as the loads increased. The curves were very predictable. Changes in and edge beam flexural stiffness for corner supported lattices did not induce any instability, see Figures deflection/span (/L) (x1 2 ) Figure 33: Simply supported lattice elastic geometric analysis with variation in rise ratio

9 , 5, 1 m simply supported lattice geometric non-linear response 18m eb stiffness 1. 18m eb stiffness m eb stiffness m eb stiffness m eb stiffness 1. 5m eb stiffness 9.9 5m eb stiffness m eb stiffness 9.6 1m eb stiffness 1. 1m eb stiffness 2. 1m eb stiffness 3.9 1m eb stiffness deflection/span (/L) (x1 2 ) Figure 34: Simply supported lattice elastic geometric analysis with variation in edge beam stiffness, Figure 39. Figures 38 and 4 show typical buckling effects in the lattice. For the cases of simply supported edges, the result was slightly different. The lower the (lower effective curvature), the more stable and predictable the result. As the rise ratio increased to the shallow shell limit,.2 (and just beyond.33), instability became a factor, see Figure 41. Figure 42 shows the buckled lattice as this very high degree of curvature. In the models possessing increasing edge beam flexural stiffness, the extra beam stiffness reintroduces stability to the lattice as Figure 43 displays. Figure 44 shows the steady deflection in the centre of the lattice, more load provides more deflection. The change in boundary condition support to simply supported has a beneficial effect in reducing or eliminating buckling under both increasing rise to span ratio (within reasonable limits). Increasing the edge beam flexural stiffness has a beneficial effect in diagonal lattice arrangements where the bar orientation parallels the principal axes of the hypar shell. 18 m corner supported diagonal lattice geometric non-linear response load intensity vs deflection/span for various s 8 rise to span.5 rise to span.8 rise to span.12 rise to span.16 rise to span.2 rise to span.33 Figure 35: 5 m lattice typical deflection for variation of rise ratio and edge beam deflection/span (/L) (x1 4 ) Figure 37: Corner diagonal lattice elastic geometric analysis with variation in rise ratio Figure 36: 1 m lattice typical deflection for.33 rise ratio Diagonal lattices However if the bar arrangement was changed from rectangular to diagonal, the load deflection behaviour of the lattices changed. It was found that the corner supported lattices buckled to an increasing extent as the increased, Figure 37. This effect also occurred in the case of increasing edge beam flexural Figure 38: Corner supported 18 m diagonal lattice,.16 rise ratio, 1st buckling mode

10 m corner supported diagonal lattice geometric non-linear response eb stiffness 1. eb stiffness 1.1 eb stiffness 35.7 eb stiffness deflection/span (/L) (x1 4 ) Figure 39: Corner diagonal lattice elastic geometric analysis with variation in edge beam Figure 42: Simply supported 18 m diagonal lattice,.33 rise ratio, 1st buckling mode m simply supported diagonal lattice geometric non-linear response eb stiffness 1. eb stiffness 1.1 eb stiffness 35.7 eb stiffness deflection/span (/L) (x1 4 ) Figure 4: Corner supported 18 m diagonal lattice, edge beam ratio, snap buckling m simply supported diagonal lattice geometric non-linear response load intensity vs deflection/span for various s rise to span.5 rise to span.8 rise to span.12 rise to span.16 rise to span.2 rise to span.33 Figure 43: Simply supported diagonal lattice elastic geometric analysis with variation in edge beam deflection/span (/L) (x1 4 ) Figure 41: Simply supported diagonal lattice elastic geometric analysis with variation in rise ratio Figure 44: Simply supported 18 m diagonal lattice, 5.7 edge beam ratio, steady deflection 4. CONCLUSIONS The bifurcation buckling modes are similar between shells and lattices for both corner supported and simply supported arrangements. The bifurcation buckling capacity of shells and corner supported rectangular

11 lattices was similar. Simply supported lattices demonstrated more variation in the bifurcation buckling capacity result across the model sizes. Shells buckle under elastic geometric non-linear analysis independently of shell size, support conditions and changes in rise ratio and edge beam flexural analysis. In contrast to hypar shells, lattice models exhibited stability as the load increased. The stability of hypar lattices was independent of the parametric changes in and edge beam flexural stiffness. Corner supported diagonal hypar lattices behaved in a similar unstable fashion to hypar shells. However, the introduction of complete edge beam support returned the lattice to a more stable behaviour. ACKNOWLEDGEMENTS This research was conducted in conjunction with the School of Civil Engineering, University of Queensland, Brisbane, Australia as a thesis submitted for the degree of Doctor of Philosophy in 212. The principal advisor was Associate Professor Faris Albermani. REFERENCES [1] London Olympics photographs, [2] London aquatic centre roof construction photograph, photograph by Helene Binet [3] Bernasconi, JG 212, A design and parametric study in large scale hyperbolic parabolic shell and lattice structures, PhD thesis, University of Queensland [4] Novozhilov,VV 1964, Thin Shell Theory, 2nd Edition, Groningen, P.Noordhoff. [5] Schueller, W 1983, Horizontal-Span Building Structures, Wiley-Interscience Publication [6] Strand7, 25, Using Strand7, 2nd Edition, Strand7 Pty Ltd, Sydney [7] Vlasov, VZ 1964, General Theory of Shells and its application in engineering, Vol TTF 99,NASA