Simulating Self Supporting Structures

A Comparison study of Interlocking Wave Jointed Geometry using Finite Element and Physical Modelling Methods

Shayani Fernando1, Dagmar Reinhardt2, Simon Weir3 1,2,3University of Sydney, Australia 1,2,3{shayani.fernando|dagmar.reinhardt|simon.weir}@sydney.edu.au

Self-supporting modular block systems of stone or masonry architecture are amongst ancient building techniques that survived unchanged for centuries. The control over geometry and structural performance of arches, domes and vaults continues to be exemplary and structural integrity is analysed through analogue and virtual simulation methods. With the advancement of computational tools and software development, finite and discrete element modeling have become efficient practices for analysing aspects for economy, tolerances and safety of stone masonry structures. This paper compares methods of structural simulation and analysis of an arch based on an interlocking wave joint assembly. As an extension of standard planar brick or stone modules, two specific geometry variations of catenary and sinusoidal curvature are investigated and simulated in a comparison of physical compression tests and finite element analysis methods. This is in order to test the stress performance and resilience provided by three-dimensional joints respectively through their capacity to resist vertical compression, as well as torsion and shear forces. The research reports on the threshold for maximum sinusoidal curvature evidenced by structural failure in physical modelling methods and finite element analysis.

Keywords: Mortar-less, Interlocking, Structures, Finite Element Modelling, Models

INTRODUCTION / CONTEXT OF RESEARCH used, consideration of relative position of each block Self-supporting stone or masonry architecture is the within the overall geometry, and joints of a block as- arrangement of modular elements that are struc- sembly become the driving factor that determine the turally performative and hold together through ver- architecture. tical force, without the need for mortar or connec- In mortar-less structures based on topological in- tors. This method of construction has potential to terlocking blocks, the mortar and connectors make be used in areas where eﬃcient assembly and disas- the structure stiﬀer, which reduces its resilience to sembly are required. As no secondary construction is vibrations and seismicity (Dyskin et al. 2012). In

FABRICATION - VIRTUAL AND PHYSICAL PROTOTYPING - Volume 2 - eCAADe 35 | 177 most dry stone structures, the self-weight and fric- compression: the Armadillo vault by Block Research tion hold modules in place. In addition, cantilevering Group (Rippmann 2016), however, the prototype was structures with planar joint typologies can be supple- based on planar joint typologies, with limited capac- mented by use of steel-reinforcement (Fallacara et al ity to integrate forces beyond compression. 2013). Consequently, this paper focuses on the further development of customised modules, in testing different three-dimensional joint typologies for the abil- Figure 1 ity to perform under various load scenarios, self- Interlocking base weight, cantilevering, and resulting shear forces. It is block geometry part of an ongoing research into six-axis robotic fab- with variations in The increased ability to analyse the line of thrust in a rication of multi-dimensional customised face joints sinusoidal and masonry arch or vault is contributing to the renewed in form and force fitting connections (Jung, Rein- catenary curvature interest in simulating stereotomic structures. hardt, Watt 2016), and an increased formal and structural complexity by robotic wave joints (Weir, Moult, Figure 2 Fernando 2016). Hence, the focus of this paper is Left: FEA the structural integrity of complex three dimensional visualizations of a joints and their capacity to interlock, and withstand catenary arch forces of shear and rotation, based on the amplitude Traditional methods of thrust line analysis performed structure modelled of their curvature. Specifically, the amplitude of vari- in by hand are often tedious and inaccurate. As has ations in sinusoidal and catenary curvature is inves- been argued, “historic masonry buildings fail due to ABAQUS/Standard tigated here. This is significant because the higher and Right: twisted instability rather than a lack of material strength be- the amplitude of the curvature, the greater the in- cause stresses in historical masonry are typically an catenary arch terlocking capacity of the blocks (Figure 1). While structure in order of magnitude lower than the crushing capac- the structural performance may vary based on the ro- ity of the stone” (Heymen 1995). In order to over- ABAQUS/Explicit tation of the blocks, boundary conditions, supports (Fernando 2017) come these problems, current practices apply struc- and location of loads, the foundational investigation tural analysis tools, such as Finite Element Modelling of structural performance for a singular joint, and its (FEM) techniques, and Finite Element Analysis (FEA) threshold in maximum curvature is essential for un- where an initial model is developed based on a de- derstanding the performance in macro geometries. signer’s concept, then analysing the model providing As can be seen in figure 1, variations were tested from a feedback loop informing what design decisions are a generic planar block (A) to a sinusoidal 30 and 45 to be made. The method of FEA has direct associa- degree curve (B) and (C), then towards an equivalent tions with predicting the behavior and performance 30 and 45 degree catenary curve (D) and (E) respec- of masonry structures in relation to factors such as in- tively. stability. Other modeling and analysis tool have ben customized, such as RhinoVault (Rippmann 2016). As Figure 3 a solution to this, Block at al (2006) developed an Rotational axis on equilibrium approach used for analysing the struc- twisted catenary tural behaviour of masonry buildings, using thrust arch structure with lines to visualize forces and predict potential collapse FEA mesh modes. The interactive and parametric nature of the models allow for the exploration of possible equilibrium conditions in real time. This was applied to recent self-supporting sandstone modules acting in

178 | eCAADe 35 - FABRICATION - VIRTUAL AND PHYSICAL PROTOTYPING - Volume 2 The Finite Element Method rors” (Ip, 1999). In particular, this is important in the Analyzing the structural action of complex domes context of non-Cartesian geometries, such as cate- and vaults as a series of two-dimensional arches nary structures, arches and domes, as it helps to enables the additional structural integrity resulting forecast structural behavior. To illustrate this, Fig- from the three-dimensional aspect of vaults and ure 2 shows a generic catenary arch, and variations domes which in turn provides a further margin of of structural complexity with a 90 degree twist at safety (Ochsendorf et al 2012). both footings. The use of structural FEA software ABAQUS/Explicit is used to solve and visualize the Figure 4 compressive forces in the more complex twisted arch Hydro-Stone test system, due to increased complexity in geometry and specimens. 6 sets of mesh resolution. This is an example of a macro- each geometry A modelled catenary arched structure acting as one (planar blocks), B & continuous part. Benefits of this type of modelling C(sinusoidal include simpler mesh structures, which contribute to curvature), D & overall efficiency of the analysis. In contrast Figure 3 E(catenary Structural analysis and modelling undertaken as ‘fi- starts to reveal the arch structure combined with de- curvature) nite element modelling’ (FEM) and ‘finite element tailed wave joints. Micro-modelling is used to simu- analysis’ (FEA), are methods of numerical analysis. late the interactions between contact surfaces. Here, a process of ‘discretization’ is used with sets of Figure 5 simultaneous equations (Ip 1999). These equations MTS Criterion are utilized to connect the internal forces with the ex- Machine ternal applied loads. Different standard software sets methods(I)Vertical area available for this, such as Simulia ABAQUS (Das- (II)Topside rotated sault Systèmes), ANSYS LS-DYNA and Karamba3D. 90 degrees, (III)Side FEM can be applied to macro-modelling and micro- rotated 90 degrees modelling of complex geometries. The first FEM application, macro-modelling, does not consider the use of joints or details between units and connections of the model. The second FEM application, micro-modelling, is used to simulate more detailed aspects of masonry units including mortar gaps and contact surfaces. It requires intensive computational The assessment of the ‘reliability of numerical non- effort and is suitable for small structural elements linear analysis as a versatile tool’ for the simulation with strong heterogeneous states and direct consid- of masonry structures are often compared to physi- eration of stress and strain. cal experimental tests (Tahmasebinia & Remennikov However, the accuracy of the set of equa- 2008). Relationship between digital modeling in FEA tions is reliant on the linear matrixes. Masonry or programs such as ABAQUS, and physical modeling stone blocks typically behave in a non-linear man- analysis for different structures such as for exam- ner, therefore FEA programs such as ABAQUS adopt ple reinforced concrete and masonry have been dis- a ‘Newton-Raphson procedure’, whereby a conver- cussed in a comparison study (Yong 2011, Tahmase- gence criterion is established and so simulation er- binia 2008), where ABAQUS was chosen as basis for rors are minimized, because “if the simulation were comparison studies, due to its versatility and level of to be non-linear, this simple matrix solution will give control in all aspects of modeling and analysis us- rise to incorrect accumulation of computational er- ing both Standard and Explicit solvers. This allows a

FABRICATION - VIRTUAL AND PHYSICAL PROTOTYPING - Volume 2 - eCAADe 35 | 179 better understanding of how modules form a struc- The structural performance of catenary structures ture, and how that structural unity then behaves un- and the concept of equilibrium have a long his- der force influence from relative rotation to collapse. tory. Robert Hooke stated in 1676 “as hangs a flex- ible cable, so inverted, stand the touching pieces of an arch”, with consecutively the first general the- Figure 6 ory on the stability of arches published by Coulomb Hydro-stone results in 1773 (Roca et al. 2010). This followed on from of vertical an understanding of the arch as a modular struc- compression ture; a “conjunction of rigid bodies which could experience relative displacements” (Roca et al. 2010). Whereas these catenary arches remain stable while experiencing singular direction of force impacts, a collapse occurs when sections are experiencing relative rotation, so as to ‘become a mechanism’ (Hey- men 1995). This is particularly significant in systems where more complex geometries (such as shown in Figure 3) with varied sections would weaken the overall structure. Yet Coulomb’s theory further provides a mathematical base offering different possible Figure 7 modes of collapse considering both relative rotations Vertical and sliding between parts. This has a direct corre- compression results lation with the plastic theory or limit analysis as de- for 5 geometry scribed by Heymen (1995) with the following condi- variations tions; that the compression strength of the material is infinite; that sliding between parts is impossible; and that the tensile strength of masonry is null (Heymen 1995). Further developments in limit analysis utilizes Coulomb’s law characterised by friction angles and contact interfaces. This leads to the factor of plas- ticity range defined by a Mohr-Coulomb law characterised by friction angle and cohesion of contact surfaces. This becomes useful as the material between the joints (stone or brick masonry) is described as ei- ther linear elastic or non-linear plastic homogeneous material. In addition to the aforementioned FEA and FEM methods, another method should be mentioned here; the discrete element method (DEM). Here, material is modelled as an assemblage of distinct blocks with interactions along boundaries. This particular methodology combines FEM with multi-body dy- namics, with capacity to analyse fractures, crack pat- terns and deformed parts of a model and reveal them

180 | eCAADe 35 - FABRICATION - VIRTUAL AND PHYSICAL PROTOTYPING - Volume 2 in real time. For this paper DEM is not utilized due or sinusoid describing smooth repetitive oscillation. to this comparison study being focused on finite el- Catenary curvature is based on a hyperbolic cosine ement and physical methods only. In the follow- or catenoid. This particular study pursued the ques- ing, the paper discusses physical test studies for a se- tion of how the geometry behaves under compres- ries of modular joint prototypes from planar joints to sion and shear forces. varying sinusoidal and catenary curvatures. Then, it These geometries were then 3D printed in PLA explains the micro-modeling of these geometries in plastic so that a silicon mould was cast for each ge- FEA by using ABAQUS software to provide stress and ometry variation to accommodate for plaster and strain analysis and data of each set of block variations. hydro-stone casting. In total 6 sets of the Hydro- Then, the paper concludes with results and research stone simulation specimens were prepared (Figure trajectories. 4). It will test the material behaviour and points of deflection in the overall joint assembly. The failure cri- Figure 8 teria include reasonable cracking and fractures which a) Base blocks b) are sometimes only visible in the deflection graph Geometry output by the compression machine (MTS Criterion) developed for load and software analysis program named TestWorks. testing, c) Initial Hydro-stone gypsum cement has a dry compres- FEM and FEA sive strength of 10000 psi or 69MPa. This is com- simulations parable to sandstone with a compressive strength (Abaqus Standard) of 59MPa (ASTM C170-87 test method). The spec- based on applied imens were 60x60x180mm blocks to the Australian top and bottom Standard test method ASTM C170 which specifies the loads of 50kN testing of equidimensional specimens with a mini- mum 50mm specimen size. The machines used for the experiment include MTS Criterion load testing machines (Figure 5) with both compression disk attachments and a specially PHYSICAL COMPRESSION TESTS FOR IN- configured tensile stress testing machine with at- tached steel plate to simulate a linear load. TERLOCKING WAVE JOINTS Figure 9 This section presents a comparison study into two FEA micro-models specific methods of structural analysis. One based with distributed on a finite element method and the other based on load applied to physical compression testing. Osteomorphic blocks middle block with based on pure sinusoidal geometry has been dis- constrained ends cussed in detail (Yong 2011) with a similar method comparing block C to be adopted for this process. Yong (2011) utilizes (sinusoidal) and ABAQUS FEA to both model and analyse the com- block E (catenary). pression test mechanism and the individual blocks. The initial ruled surface geometry (modelled in Rhinoceros 5 with Grasshopper plugin) is based on both sinusoidal and catenary curvature with a step The reason for the rotation is in order for the joint or linear section to assist in assembly of contact sur- typology to aggregate over larger structural assem- faces. Sinusoidal curvature is based on a sine wave blies such as arched structures as shown in Figure

FABRICATION - VIRTUAL AND PHYSICAL PROTOTYPING - Volume 2 - eCAADe 35 | 181 3. The test method suggests three test variations I, II ulate stone with a Youngs Modulus of 118MPa and and III for three specific angles explained in figure 5. Poisson’s ratio of 0.15. These parameters can vary Test methods (II) and (III) utilizes a linear distributed as more tests are undertaken with various materials. load only applied to the middle block to test shear Defining boundary conditions, constraints and loads strength and failure points of the joints. are kept as close as possible to the physical load testing method. Vertical Compression Testing For the vertical compression tests as shown in Fig- Figure 10 ure 5 test method (I), the MTS Criterion 50kN ma- Shear compression chine with fixed lower compression disk is used. Two tests on rotated sets of the 5 geometry variations were tested under blocks this method of compression. The test settings for all vertical compression specimens include pre-load speed and test speed of 1mm/min, a platen separa- tion of 50.8mm and a pre-load of 44.48 kN. The resultant data gathered included the load (kN), time (s), extension/deflection (mm), stress (MPa) and strain (mm/mm). The results are shown in Figure 6 with ac- companying graphs in figure 7. Whilst planar block A showed the optimum re- sponse, block D (low amplitude catenary curve) re- sponded best out of the series closely followed by block B (lower amplitude sinusoidal curve). This is shown in figure 7 where the lighter blue line repre- senting the planar block A for both test experiments are much higher in comparison to the catenary and sinusoidal wave blocks shown by the green and blue lines. The point of failure of the planar block occurs approximately 417 seconds with a load of 34kN in experiment 1, while experiment 2 yields the failure point at 313 seconds with a load of 45 kN. The planar block in this case is the control or base case block. The The mesh results as shown in Figure 8(c) reveal vari- actual test subjects block B,C,D and E all have sinu- ations which are not consistent due to varying mesh soidal or catenary curvature. The physical test speci- resolution. The interactions between the contact sur- mens after they are compressed are shown in Figures faces are further analysed through both these meth- 6 and 7. Block D (low amplitude catenary curve) takes ods. the highest load (green line in Figure 7) of 19 kN be- Figure 11 fore it fails at approximately 500 seconds. Hydro-Stone lateral The same geometry (Figure 1) is imported into test results the Finite Element Analysis program ABAQUS/Stan- dard. A consistent workflow is set up involving defining parts of an assembly and material properties. In These results shown in Figure 8 illustrate the poten- this case the density of 2000kg/m-³ is utilised to sim- tial inconsistencies where a change in mesh resolu-

182 | eCAADe 35 - FABRICATION - VIRTUAL AND PHYSICAL PROTOTYPING - Volume 2 tion can affect the overall outcome of the simulation. tated and the load is applied to the top (Figure 12), The specific ABAQUS/Standard solver was not suffi- the catenary joint 45 degrees block E takes the high- cient to analyse the 45 degree high amplitude blocks est load before it fails. The overall results of the physi- as there were mesh convergence issues. In order to cal test experiments reveal that whilst the high ampli- resolve this, ABAQUS/Explicit was used after some tude catenary and sinusoidal interlocking blocks do partitioning and refinement of the generated mesh. not perform well in relation to failure under vertical With the current settings as described above the compression, they take the highest loads before fail- failure points or most amount of deflection occurs ure when rotated laterally. The FEA results indicate where the blocks experience sharp changes in ge- the points of stress visualised both externally and in- ometry. In vertical compression for both the physi- ternally within the structure. cal and FEM methods the successful block is the pla- Figure 12 nar block which can take the highest load. However Shear compression when the block is rotated 90 degrees, the amplitude load tests rotated of the wave enables the block to extend its structural blocks topside capacity (Figure 9). The results of the FEA and physical testing reveal variations and inconsistencies. An accurate method of verifying whether the physical results match the FEA is overlaying the graph output from the physical compression blocks with the FEA resultant graph. However the resultant deflection visu- alisation as shown figures 8 and 9 reveal points of stress and strain which have a tendency to be lo- cated at the at the joints, especially where there are sharp changes in geometry. The linear ‘step’ elements seem to fail first in the physical compression tests and are also shown in red colour in the FEA results. Figure 10 shows the shear failure points of the joints under compression. Here the middle block is loaded with a linear distributed load. It is important to note that the graphs do not show the location of failure in relation to the physical specimen. Therefore it is integral to have both the FEA and physical test specimens. The graphs do however reveal the points As these results were only based on 2 sets of test ex- of failure in relation to load and time. The red and periments, the accuracy varies due a number of er- purple lines indicates high amplitude 45 degree si- rors. These include material inconsistencies due to nusoidal (C) and catenary (E) curve respectively. The hand made blocks; machine faults and errors with blue and green lines represent the low amplitude 30 the test method. For example the clamps caused the degree sinusoidal (B) and catenary (D) curved blocks. high amplitude blocks to lift off their fixed position When the load is applied to the front face of the on the steel side bars. This will have created more middle block, the sinusoidal joint 45 degrees block C axial forces due to rotation of the blocks. However takes the highest load (Figure 10). However when ro- these material errors and inconsistencies reveal the nature of construction and how material performs

FABRICATION - VIRTUAL AND PHYSICAL PROTOTYPING - Volume 2 - eCAADe 35 | 183 under various loading conditions. For example the Engineering BCCE 2013, Albania, pp. 667-675 inconsistency in loads during a storm or earthquake Heyman, J 1995, The stone skeleton: structural engineer- scenario will cause the failure of a structure based on ing of masonry architecture., Cambridge University Press, Cambridge its behavior and flexibility to external conditions. This Ip, P 1999, Compressive Strength and Modulus of Elasticity can potentially open up possible research trajectories of Masonry Prisms, Master’s Thesis, Faculty of Grad- for seismic analysis of masonry structures. uate Studies and Research, Department of Civil and Environmental Engineering, Carlton University, Ot- tawa, Ontario, Canada CONCLUSION / FUTURE TRAJECTORIES Jung, A, Reinhardt, D and Watt, R 2016, ’RBDM- Com- Mortar-less structures based on interlocking blocks plex Curved Geometries with Robotically Fabricated provide a level of structural stability due to its flexi- Joints’,in Reinhardt, D, Burry, J and Saunders, R (eds) bility and movement allowed in the joints compared 2016, Robotic Fabrication in Architecture, Art and De- to traditional mortar structures which are more brit- sign 2016, Springer International Publishing Switzer- tle. The ease of assembly and disassembly make land, Vienna, pp. 178-189 Ochsendorf, J.A. and Dahmen, J.F.D 2012, ’Earth ma- using these mortar-less interlocking blocks for the sonry structures: arches, vaults and domes’, in Hall, construction of arched and vaulted spaces more vi- M, Lindsay, R and Krayenhoff, M (eds) 2012, Mod- able. The process of utilizing FEA and physical testing ern Earth Buildings: Materials, Engineering, Construc- methods contribute to an effective feedback loop be- tions and Applications, Woodhead Publishing, Lon- tween the initial design, method, material, machine don, pp. 427-459 and outcome. Through the analysis of two specific Rippmann, M 2016, Funicular Shell Design: Geometric Ap- proaches to Form Finding and Fabrication of Discrete methods we were able to make initial assumptions Funicular Structures, Ph.D. Thesis, Department of Ar- on both the validity of the results and the most effi- chitecture ,ETH Zurich cient joint geometry using interlocking assemblies. Roca, P,Cervera, M, Gariup, G and Pela, L 2010, ’Structural Analysis of Masonry Historical Constructions. Clas- sical and Advanced Approaches’, Archives of Com- ACKNOWLEDGEMENTS putational Methods in Engineering, 17, Springer, p. We would like to thank the University of Sydney 299–325 Architecture and Civil Engineering Schools for the Tahmasebinia, F and Remennikov, A 2008, ’Simulation resources in the research laboratories and work- of the reinforced concrete slabs under impact load- shops, software use (ABAQUS) and physical compres- ing’,in Gad, E.F. and Wong, B (eds) 2008, Australasian sion testing facilities. We acknowledge the support Structural Engineering Conference (ASEC), The Meet- ing Planners, Melbourne: , pp. 1-10 of Sydney university students Xavier Lian, Joshua Weir, S, Moult, D and Fernando, S 2016, ’Stereotomy of Grasso and ETH Zurich student Lukas Sigrist for the Wave Jointed Blocks - Toward a wave-jointed stone assistance with the test experiments. construction using wire cutter toolpath generation’, in Reinhardt, D, Saunders, R and Burry, J (eds) 2016, Robotic Fabrication in Architecture, Art and Design REFERENCES 2016, Springer International Publishing Switzerland, Block, P, Ciblac, T and Ochsendorf, J 2006, ’Real-time Vienna, pp. 284-293 limit analysis of vaulted masonry buildings’, Comput- Yong, H. D. T. 2011, Utilisation of topologically- ers and Structures, 82(29), pp. 1841-1852 interlocking osteomorphic blocks for multi-purpose Dyskin, A, Pasternak, E and Estrin, Y 2012, ’Mortarless civil construction, Ph.D. Thesis, The University of structures based on topological interlocking’, Fron- Western Australia tiers of Structural and Civil Engineering, 6 (2), pp. 188- 197 Fallacara, G, Bagneris, M and Calabria, C 2013 ’The structural tree: new experimentation with re-inforced stone’, 2nd International Balkins Conference of Civil

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