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Simulating Self Supporting Structures 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 dif- ferent 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 struc- tural 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 equi- librium 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 ap- plication, macro-modelling, does not consider the use of joints or details between units and connec- tions 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
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