Shell structures can be either calculated with complex mathematics and differential geometry, or with the help of finite element method (FEM) software. Neither method gives the designer sufficient insight about the behaviour of the shell in the early design stage. The first method is for most designers too complex, it requires proficiency in at higher order mathematics and it can be only used for shells the geometry of which can be described by analytical equations. The second method is more accessible for most designers, and in combination with 3D modelling software is an attractive alternative to the cumbersome mathematics. However, the disadvantage of using FEM software is that the analytical relations between the different parameters important for obtaining insight into the structural behaviour of the shell is lost. The ideal tool for designers would take the best of both methods: the analytical insight of the mathematics and the relatively easy access of FEM and 3D modelling software. The aim of the research done presented in this thesis is to do precisely that. By extending the well-known beam-analogy and its relations for an arch to shell structures a construct of relations is available for providing the designer with insight and means to influence the geometry of the shell and the resulting stress state. The proposed hypotheses are based on classic analytical geometry and mechanics, especially analogies and methods developed in the past to elucidate the complex mathematics and mechanics for the purpose of insight. The theory of graphic statics, reciprocal diagrams, complementary and potential energy was used to develop the method of solving the thrust network. Analogies such as the moment-hill for out-of-plane loaded slabs and the static-geometric analogy for thin shells as well as the load path theorem and stress functions were used to develop the slab – shell analogy. Two approximate hypotheses are proposed in this thesis, the first is used to solve 3D indeterminate thrust networks by using complementary energy. The second hypothesis extends the beam – arch analogy in two directions to the slab – shell analogy; this method produces results in range of solutions found in classical shell theory. The result of the different examples has been checked with the help of well-known solutions of classical shell mechanics, FEM calculations or graphic statics. Examples with relatively basic analytical formulas have been used to elucidate the proposed method and for other examples simple purpose made tools based on the method have been used. Some simplifications have been made to avoid unnecessary complications in the derivation of the proposed method, such as only applying a uniformly distributed load. But most of the simplifications are not technically necessary; the conclusion and recommendations section include some suggestions have been added for extending the method. The inception of numeric methods for analyzing structures in the 1960s was the end of the development of analytical mechanics for shell structures. This thesis aims to continue this development by tying the used theories and analogies together and bridge the gap with the numeric methods and to increase the understanding of the structural performance of shell structures.@en

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.

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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.@en

This paper presents a detailed structural analysis of a bubble shell engineered by Heinz Isler. Through 3D scanning the geometry of this shell structure has become available to the authors. Structural analysis has not been possible before since the geometry of the shell was not available. The bubble shell was Isler’s most built type of shell. In the paperfirst the process of reverse engineering the geometry of the shell is described. Second, the effect of pre-stress in the edge beams is described. Third, the load distribution throughout the shell and the membrane behaviour relative to bending behaviour is assessed.@en

Membrane shells, which have minimized bending moments under certain load conditions, are regarded as ideal structural forms in terms of material efficiency. Most of the existing numerical form-finding methods are based on discretizing membranes into finite panels or funicular networks and focusing on gravitational loading only. In order to obtain smooth shells and to consider horizontal loads, this paper presents a method to find the equilibrium forms of the membrane shells by solving Pucher’s equation. Radial base functions (RBFs) is utilized to represent stresses and shapes of the membranes, and a least square method is applied to find the controlling coefficients which allow the functions to fit the boundary conditions (e.g. zero stresses at the free edges) and the governing equation. When all the parameters are carefully chosen, the stress and shape functions can achieve sufficient accuracy. The presented method has been preliminarily implemented to find shells on a triangle ground plan incorporating horizontal loads. The form-found geometries are then analyzed by finite element models. The result confirms that the form-found shells have the stress distributions similar to the prescribed stresses.@en

Shell structures generated from hanging models have structurally efficient forms. Form-control of these shells, which aims to obtain structural forms with single- and multiple target heights due to some architectural requirements, is discussed in this article. First, the vector form intrinsic finite element method is applied to generate the equilibrium form of hanging membranes and thus shell structures. Subsequently, the form-control problem is discussed, which aims to generate a structural form subject to given target height constrains. By introducing the Local Linearization Method to adjust Young’s modulus of the initial structural model, a form-control strategy to generate the equilibrium structural form with a single target height is proposed. By introducing the Inverse Iteration Method to adjust the geometry of the initial model, a form-control strategy to generate the equilibrium structural form with several target heights is proposed. Moreover, to verify the effectiveness of the vector form intrinsic finite element method and form-control strategies, structural analyses and shell behavior assessment of these shells are conducted. These strategies are effective and efficient, which can help architects or engineers to determine structurally efficient geometries in the design process much more easily.@en

Firefly Algorithm (FA, for short) is inspired by the social behavior of fireflies and their phenomenon of bioluminescent communication. Based on the fundamentals of FA, two improved strategies are proposed to conduct size and topology optimization for trusses with discrete design variables. Firstly, development of structural topology optimization method and the basic principle of standard FA are introduced in detail. Then, in order to apply the algorithm to optimization problems with discrete variables, the initial positions of fireflies and the position updating formula are discretized. By embedding the random-weight and enhancing the attractiveness, the performance of this algorithm is improved, and thus an Improved Firefly Algorithm (IFA, for short) is proposed. Furthermore, using size variables which are capable of including topology variables and size and topology optimization for trusses with discrete variables is formulated based on the Ground Structure Approach. The essential techniques of variable elastic modulus technology and geometric construction analysis are applied in the structural analysis process. Subsequently, an optimization method for the size and topological design of trusses based on the IFA is introduced. Finally, two numerical examples are shown to verify the feasibility and efficiency of the proposed method by comparing with different deterministic methods.@en

Vector form intrinsic finite element is a recently developed and promising numerical method for the analysis of complicated structural behavior. Taking the cable-link element as example, the framework of the vector form intrinsic finite element is explained first. Based on this, a constant strain triangle element is introduced, and relevant required equations are deduced. Subsequently, the vector form intrinsic finite element is successfully applied to carry out form-finding of shells generated from physical models, such as hanging models, tension models, and pneumatic models. In addition, the resulting geometries are analyzed with finite element method, thus demonstrating that a dominant membrane stress distribution arises when the shell is subjected to gravitational loading.@en

Due to its wide range of related research contents and diversified research approaches, the term ‘Structural Morphology’ has not been clearly defined by the Structural Morphology Group (SMG) of the International Association for Shells and Spatial Structures (IASS), founded in 1991, although some scholars have given their own viewpoints. This paper presents a different way to understand the meaning of “Structural Morphology” and its connotations. Nowadays, numerical techniques have become the most important means to do research in the field of structural engineering, and they can assist in the design, analysis and optimization of structures by handling a large number of parameters. In this paper, we present a common conceptual scheme for these numerical analysis methods. The scheme classifies the parameters of the initial structural system into five categories and, with the aid of numerical analysis methods, leads to the structural performance of the final structure. Two simple numerical examples are shown to verify the rationality of the scheme. On this basis, a conceptual formula to describe 'Structural Morphology' is proposed, which contains the whole numerical analysis process, shows the goal of structural morphology and also suggests a suitable methodology. Moreover, since numerical form-finding and computational morphogenesis have become two main research foci of structural morphology, a basic introduction, methodology and some achievements related to each research focus are presented in this paper.@en

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.@en

Arches are usually calculated using the Finite Element Method (FEM). However, they can be calculated using graphic statics as well. In the graphic statics method, the complementary energy is used to calculate the flow of forces in the structure. The correct solution can only be found through trial-and-error: by changing the horizontal reaction forces, the equilibrium changes, providing a different distribution of load-carrying through normal forces and bending moments. The distribution which results in the lowest amount of complementary energy is the correct way the forces flow. This method is only applied in an iterative way. If the total amount of energy is described in a mathematical way and the derivative of that equation is set equal to zero, the correct solution can be calculated directly. However, the extensiveness of this mathematical description prevents this method from being applied in a direct way. This paper proposes a new method in which the complementary energy due to normal forces is neglected and only the complementary energy due to bending moments is minimized. This simplification allows for a derivation of the equation for the complementary energy which changes the calculation method from an iterative to a direct one. The derivation is shown for a simple three-bar system, showing how such a direct method can be developed. The new method is compared to indirect calculations and FEM calculations, showing only small deviations. The proposed method can be extended to more complex arch structures and can function as a starting point for finding a graphic method for calculating shell structures. @en

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.@en

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.@en