MARCO SODANO Master’s Thesis Project
DIGITAL BIOMIMETIC MORPHOGENESIS OF HIGH-RISE BUILDING WITH RESPECT TO STRUCTURAL SHAPING OPTIMIZATION Category: Master’s Thesis Project Location: Pisa, Italy Year: 2018 Tutor: Prof. Eng. Maurizio Froli, Eng. Francesco Laccone The thesis ‘Digital Biomimetic Morphogenesis of High-Rise Building’ presents the planning of a skyscraper through a method based on biomimetic, which is defined as the imitation of models, systems and elements of nature aimed to solve complex human problems. This particular planning process utilised in-depth analysis including parametric models which permitted the optimisation of the structural elements. The biomimetic analogies, including the mechanical and functional properties of bamboo stem, were transposed through a mathematical and analogic process, within the morphological and structural configuration of a high-rise building, in order to obtain advantages both in terms of
static performances and optimisation of the use of materials. Comparing in fact, the bamboo stem with a high-rise building, it can be noticed how the performance of the biological model are similar to the chosen structural system. Therefore, an eventual shape of the tower, conceived through the differentiation principle of the bamboo laws, can provide an appropriate reaction to the lateral loads, which are preponderant compared to the gravitational actions. The parametric planning, integrated with new software and methods, gave the opportunity to face the complexity of the project and resulted fundamental for the planning management of this type of building.
Morphogenesis Process - Biomimetic design
Parametric Process
Using basic parameters, the mathematical rules in the process are: • The relation between the number of internodes and the length of the internode (L(i,1),L(i,2)); • The relation between the number of internodes and the diameter of the internode (D(i,1),D(i,2)). The parameters used here have been manipulated to give a geometric shape to the building. Therefore, the length of the internodes was transformed into the total height of the building, while the internode diameter was converted into the base radius. Subsequently, ascertaining that a complete use of the proportions of the bamboo would have led to an excessive reduction of the lengths and diameters of the last internodes, the most logical solution consists in imposing a percentage limit on the relations, resulting in a linear scaling of the values of the functions on the two axes. The percentages range was identified through a visual estimate of the suitable areas and involved values between 55% and 60% of the equations. 100 90
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Thus, the Grasshopper algorithm has been structured in three distinct parts: 1. Defining input parameters: • Percentage of exploitation of bamboo relations; • Total number of internodes; • Building height; • Base radius of the building.
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2. Defining the functions: • The relation of the number of internodes to the internode length of bamboo; • The parabolic relationship of the number of internodes and the internode diameter of bamboo. 3. The geometric extrapolation: • Creating a series of circles through equations D(i,1),D(i,2). As a result, a variable domain was obtained in which the building exists according to the mathematical relations. Recalling that one of the principles of this project is the collection of wind flow from all directions, the geometric model was then modified, factoring in a series of internal circles with a constant diameter, extending from the base to the top. Defining these new empty spaces, intended to represent the space occupied by the wind turbines, has directly influenced the height profile of the three towers.
The Proportions Definition
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The parametric model required imposing some criteria to define the variable parameters and consequently obtain a definite shape. Thus, it seemed appropriate to impose a solution from a structural point of view to solve this problem. In detail, a linear elastic calculation was chosen in order to compare the displacement of two equivalent models of beams, which have the same characteristics in terms of stiffness. It seems clear, therefore, that the lateral displacement, in this first phase, was to be prioritized over criteria based on resistance, since in a tall building the former is the main problem. Thus, the research was based on a comparison of the upper displacement of two models that have three cantilevered pillars connected at a distance of n_i meters by infinitely rigid diaphragms, where: • δ1 is the upper displacement of the equivalent model building that has the same volume as the counterpart, in terms of stiffness, and a constant division into the internodes; thus following the traditional construction theory; • δ2 is the upper displacement of the morphological model, with the geometric characteristics taken from the output of the parametric program, such as the radii of the diameters over the whole height and the length of the internodes, thus giving the position of the infinitely rigid diaphragms. The respective stiffness of each model depends on a cross-section with a circular base and constant thickness (to reproduce the diagrid structure, or rather the main body resistant to lateral loads). The upper displacements of the two models will be defined using the stiffness method. Wanting to find the best H/B ratio, the process was repeated several times, replacing:
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Different values of slenderness ratio (1: 3, 1: 4 and 1: 5); • Different numbers of internodes (5, 6, 7 and 8); • The two percentages of the bamboo relation, respectively 55% and 60%. Furthermore, in order to examine the actual advantage of relating elevation to bamboo, two distinct cases were created, with and without tapering. The results confirmed the positive effect of the proportions dictated by bamboo in the different configurations. In fact, we can deduce that the ratio δ1/δ2 is always greater than 1. Final dimensions were chosen by analyzing the δ/H ratio of each tapered combination. A first consideration can now be made concerning the percentage of exploitation of the bamboo relation: a higher value of this parameter negatively influences the lateral displacement, probably due to the excessive reduction of the cross-section of the upper internodes. Secondly, it becomes evident that the configuration based on the highest number of bamboo internodes (8) determined the lowest displacement in each analyzed aspect ratio, occupying in any case the first position of a hypothetical ranking that contains all configurations at a certain height, derived from the aspect ratio.
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Considering also aesthetic factors, the final choice was related to the configuration based on a building of 320 m of height, with a base diameter of 80 m, and a ratio of about 1:10 for each tower with 8 internodes.
Conception of Structural Elements
The three towers comprise a mixed system consisting of an external steel diagrid structure and an internal core of reinforced concrete (a. and b.). These two structures are connected to each other through post-tensioned reinforced concrete slabs, which guarantee an adequate diaphragm effect for horizontal stiffening and distribution of stresses between the two systems. This choice is justified by the desire to obtain an efficient structure, in which each element functions to the best of its ability, in order to optimize the structure and consequently reduce the associated costs. Furthermore, the presence of the reinforced
concrete core is fundamental for the functional and distributive organization of the building. From a purely structural point of view, the static schema allows us to delegate: • the global flexural strength, related to lateral loads, to the external diagrid; • global shear strength for both systems, exploiting the excellent shear stiffness of the concrete core; • the resistance to gravitational loads, acting on the slabs, is distributed equally between the central core and the diagrid. Subsequently, in order to increase the performance
of the three buildings and to bring the principles of the structural morphogenesis of bamboo back into the technological project, seven connecting stiffening rings were designed (c.) in order to create a hybrid system which, in its entirety, is completely new. These elements, initially modeled as infinitely rigid diaphragms, were designed as stiffening rings, composed of an interconnection of radial and concentric beams arranged on different levels. Each ring has been designed with a height of 3 meters. The structural design phase of the tall building
was concluded with the development of the system connecting the seven wind turbines (d.) and the three towers. The two guidelines for this project are: • Adequate aerodynamics, in order not to influence the wind flow between the towers; • A low visual impact so as not to create a stark contrast to the design of the bamboo.
Horizontal diaphragms - Spatial Truss
Turbine connection
Numerical model Geometry
F.E.M. Model
Structural Analysis
The analysis of the stresses was carried out using a finite element analysis model. This model was created according to a parametric process starting from the geometric model of Grasshopper. The geometric model of the entire building was created using the GeometryGym plug-in. Using GeometryGym, it was possible to assign materials, sections, external loads and create load combinations automatically, according to the Eurocode requirements. The model thus created was imported into the CSi Sap2000 structural analysis software, using the iFc format, where it was possible to obtain the diagrams relating to axial stress, shear stress and bending moment for each element, in addition to the horizontal and vertical displacements of the structure, for each load combination. In this way the surfaces that formed the walls of the central nucleus in the parametric model were converted into shell elements, the lines of the external diagrid into frame elements and the points of intersection between the different elements into nodes and eventually into constraints, where necessary. The post-tensioned slabs have been simulated as diaphragms, inserted on each tower independently at each hi. The diaphragm will contain all the nodes belonging to the elements of the system which are resistant to lateral loads, including the shell and diagrid nodes. In this perspective of the design, the diaphragms must be able to absorb the horizontal inertia and transmit them to the vertical systems (core and diagrid structure), behaving as much as possible as rigid bodies. Therefore, they must have the necessary resistance, and the stiffness in their own plane must be larger than the stiffness against horizontal loads of vertical systems. Instead, the horizontal spatial truss have been defined as finite elements of beams, which all the rods, defined as frames, are connected to each other by an internal hinge. At this point, the diagrams of internal stresses on the structure can be shown either through a graphical representation, or through a representation of the values of the normal stress of each rod.
Analysis of horizontal displacements
Using the SAP2000 software, we were able to carry out a more detailed analysis of the wind load acting on the structure, using the calculation methods of the CNR DT 207 standard. This allowed a much more detailed calculation of the stresses and less conservative, determining the overall loads acting on each floor, the overall displacement of the building and the relative inter-stories displacement. Wind stresses were determined for both main directions of the building. The deformation limits under the load combination Q.P. are widely verified, both as regards the global displacement, and as regards the inter-stories displacement, resulting in: δ(i,max)= 0.36 m < H/500 = 0.64 m â&#x2C6;&#x2020;δ(i,MAX)= 0.004 m <hi/400 = 0.01 m The introduction of the complete model of the spatial reticular structures, compared to the numerical model with only the vertical structure, has caused a further lateral displacement reduction.
The reason can easily be found in the form of the stiffening rings which led to an implementation of the previous model. First, in fact, the parts of the structure which are characterized by the two infinitely rigid diaphragms at the level of the two ring beams, of each bamboo internode, presented a limited stiffness compared to the subsequent horizontal translations. In other words, the diaphragm constraints in the old model were only supporting a redistribution effect of the internal stress, which resulted in a noticeable lack of lateral containment due to displacement. As a result, the overall displacements of the structure have been considerably reduced, with a final value of 0.36 m in the upper part of the building, which corresponds to a reduction of 43% of the drift. Above, the graph shows a comparison between the previous model (infinitely rigid diaphragms) and the complete one (stiffening rings), obtained through FEM analyses of the inter-stories drifts.
Optimizing the Diagrid sections using a Genetic Solver Having ascertained, through the tests, that the lateral displacement of both the upper part of the building and of each floor does not exceed the limits stated in Eurocode 3, the project has been taken to the phase of optimizing material by reducing the transversal dimension of the diagrid elements. Given the severity of the restrictions related to the design of tall buildings, we decided to dedicate this phase to the criteria of displacement. In detail, the presence of a certain gap between the maximum allowed deflection and the actual displacement, has been interpreted as an excess of material. Therefore, implementing a progressive reduction of the cross-sections of the diagrid, through an increase in controlled displacement, seemed to be a mandatory choice given the biomimetic idea intrinsic to the project. The optimization procedure was developed with the help of Galapagos, a genetic algorithm solver available in Grasshopper. Thus, the fitting function has been related to the inter-story drift. The fitting function has been formalized as:
In which â&#x2C6;&#x2020;i is the drift and Hi the height of each plane. The fitness value has been studied to be minimized by the solver. The absolute value has been implemented in order to introduce a limit which is lower than the sum and to avoid exceeding the individual displacement value allowed for by the regulation. The optimal configuration was found in the twelfth generation and the final result led to a significant reduction of 25.51%. The structural analysis was considered valid thanks to the use of SAP2000, through which the displacements were calculated. Each iteration calculated by Galapagos begins with the definition of a geometric configuration of the model, determined by the parameters set by Grasshopper. Therefore, this geometric model was automatically exported to SAP, using the GeometryGym plug-in, in order to be analyzed statically.
MARCO SODANO
sodano.mrc@gmail.com +39 3804612934 Architect and Structural Engineer