Force network analysis using complementary energy

The method presented solves statically indeterminate force networks, which are graphical representations of forces (force polygons) of a structure, by using complementary energy. Statically indeterminate force networks, which have nodes where four or more members come together, have for each node multiple possible force polygons that make equilibrium. This holds for force networks of structures in one plane, such as trusses, as for geometric three-dimensional structures, such as shells. In the case of the latter the surface of thrust of the shell is discretized into a network of forces (equivalent to the thrust line of an arch), with discrete loads at its vertices.Building TechnologyArchitectur

Reverse engineering of free form shell structures: From point cloud to finite element model

Many free form shell structures that have been designed and build in previous decades are fascinating structures. We can learn from these structures by analysing them and studying their structural behaviour. However, in some cases the geometry of these structures is not available; most notably the shapes of shell structures designed and build by Heinz Isler, who has built over 1400 shells. The geometry of many of his scale models and build structures have been obtained by the authors by making use of 3D laser scanners which create point clouds.This paper presents a method for reverse engineering of free form shell structures from point cloud to finite element model. Since shape and force interact, special attention is given to the geometric accuracy. Every model must be sufficiently accurate. The method has been applied to data obtained by scanning Isler’s shells. Important aspects that influence the quality of the resulting finite element model are described.Structural Design & Mechanic

A form-finding method for membrane shells with radial basis functions

The equilibrium of a membrane shell is governed by Pucher's equation that is described in terms of the relations among the external load, the shape of the shell, and the Airy stress function. Most of the existing funicular form-finding algorithms take a discretized stress network as the input and find the shape. When the resulting shape does not meet the user's expectation, there is no direct clue on how to revise the input. The paper utilizes the method of radial basis functions, which is typically used to smoothly approximate arbitrary scalar functions, to represent C∞ smooth shapes and stress functions of shells. Thus, the boundary value problem of solving Pucher's equation can be converted into a least-squares regression problem, without the need of discretizing the governing equation. When the provided shape or stress function admits no solution, the algorithm recommends users how to tweak the input in order to find an approximate solution. The external load in this method can easily incorporate vertical and horizontal components. The latter part might not always be negligible, especially for the seismic hazard zones. This paper identifies that the peripheral walls are preferable to allow the membrane shells to carry horizontal loads in various directions without deviating from their original shapes. When there are no sufficient supports, the algorithm can also suggest the potential stress eccentricities, which could inform the design of reinforcing beams.Structural Design & Mechanic

Scanning in 3D and analysing the models of Heinz Isler, the preliminary results

During his live Heinz Isler built around 1400 shell structures, until he deceased in 2009. Heinz Isler is part of a Swizz tradition of structural art in the 20th century, which includes engineers such as Robert Maillart, Othmar Ammann und Christian Menn [1]. During his live Heinz Isler developed several methods for physical form finding of his shell structures [2, 3]. Methods such as hanging models, inflated membranes etc. The physical scale model where used for determining the strains and stresses in the shell structure. This was done by loading the scale models and measuring the strains and consequently calculating the stresses. The geometry of the scale models was used for the actually build shell structures by precisely measuring the scale models and scaling these up to the real size shell. Analysing Isler’s shells has always been impossible because Isler never published the precise geometry of his shell structures. Isler’s model where scanned for the first time ever in 2011, the results where used to construct NURBS (Non Uniform Rational B-spline) surfaces which describe the exact geometry of Isler’s scale models. The results are used for all kinds of analysis, such as finite element (FEM) calculations, curvature analysis etc. This means that for the first time a qualitative investigation can be made of Isler’s shell structures. This paper will present the first results. Hopefully it will give us a greater insight in the relation between geometry and the structural behaviour of shell structures.Architectural Engineering +TechnologyArchitecture and The Built Environmen

Teaching Structures with Models: Experiences from Chile and the Netherlands

This paper states the importance of using scaled models for the teaching of structures in the curricula of Architecture and Structural Engineering studies. Based on 10 years’ experience working with models for different purposes, with a variety of materials and constructions methods, the authors will address the advantages and possible approaches for a methodology of working with models.Architectural Engineering +TechnologyArchitecture and The Built Environmen

An approach on form-diversity of free-form shells generated from numerical hanging models

Structural Design & Mechanic

Form-finding of gridshells generated from hanging-chain models by using the Dynamic Relaxation method and the NURBS technique

Hanging models play an important role in shaping a structure since a very early age, and were favored by A. Gaudi, H. Isler, F. Otto and other architects or engineers. Nowadays, with the development of numerical analysis theory and computer technique, it is more accurate and convenient to simulate these physical models via numerical means. Based on the background, this paper presents a numerical form-finding method of gridshell structures generated from hanging-chain models by using Dynamic Relaxation method and the NURBS technique, which aims to obtain more complex structural forms with multiple control points.This method uses global NURBS surface interpolation to describe the initial cable-net model passing through the given target points, which serve as the fitting points of the NURBS surface. The cable elements of the cable-net are not allowed to elongate after form-finding, and clearly, this kind of cable-nets belongs to geometrically unstable system, whose form-finding process of it has a very strong nonlinearity. To solve this problem, it uses the Dynamic Relaxation method, which can complete the form-finding of geometrically unstable systems but with some special sets, to get the equilibrium form of the hanging cable-net under the gravity. However, this structural form may no longer pass through the given target points, and then it introduces the inverse iteration method to adjust the coordinates of the fitting points of the NURBS, which actually means to find the initial structural form which after form-finding can just right meet the target requirements. At last, some numerical examples are presented to demonstrate the validity of the proposed method in this paper.Structural Design & Mechanic

The Vector Form Intrinsic Finite Element method and several other form-finding methods for general networks

Discrete networks is a kind of form-active structural system which actively change its shape under varying load conditions. And for this kind of structural system, form-finding is the initial and essential part in their design process. Before the computer age, people complete the form-finding process using physical models, while with the advances in computational techniques, the research has focused on the numerical form-finding methods since the 1960s. A brief discussion on several numerical formfinding methods is presented in this paper. Firstly, two relatively mature numerical method, Dynamic Relaxation method and Force Density method, are introduced conceptually. And then, a newly developed numerical method, the Vector Form Intrinsic Finite Element method, is presented in more detail. At last, with a replacement of the calculation of the internal force of the element which obeys the Hooke's Law by the product of the force density and the length of the element, two derived methods based on the above three methods are proposed in this paper. Moreover, several numerical examples of hanging networks are shown to illustrate the validity and characteristic of the VFIFE method and the two newly proposed derived methods.Structural Design & Mechanic

Discretised Airy stress functions and body forces

This paper extends polyhedral Airy stress functions to incorporate body forces. Stresses of an equilibrium state of a 2D structure can be represented by the sec- ond derivatives of a smooth Airy stress function and the integrals of body forces. In the absence of body forces, a smooth Airy stress function can be discretised into a polyhedron as the corresponding structure is discretised into a truss. The differ- ence in slope across a creases represents the axial force on the bar, while the zero curvatures of the planar faces represent zero stresses voids of the structure. When body forces are present, the zero-stress condition requires the discretised Airy stress function to curve with the integrals of these body forces. Meanwhile, the isotropic angles on the creases still indicate concentrated axial forces. This paper discretises the integrals of body forces into step-wise functions, and discretises the Airy stress function into quadric faces connected by curved creases. The proposed method could provide structural designers (e.g. architects, structural engineers) with a more intuitive way to perceive stress fields.Structural Design & Mechanic