Form-finding Method Research Articles

Structural optimization and form finding of light weight structures

There exist three different lines of computational form finding methods: (i) simulation of hanging models, (ii) numerical simulation of soap films, and (iii) structural shape optimization. The aim of the present paper is to show their differences and common aspects when applied to the special task of maximizing stiffness. The inverse problem of simulation of soap films is formulated in more detail on the basis of differential geometry. The effect of internal pressure is discussed. Several illustrative examples are given.

A form function method for form finding of tension structure

A new form function involving parametersβ i is presented. On the basis of the form function, an initial form of tension structure was found by interpolating through the control points on boundary of the structure. The form function can be controlled by changingβ i according to the pre-tension and the boundary of the structure.

Aβ as a homeostatic negative regulator of synaptic function?

This paper proposes a form-finding method of free-form aluminum alloy reticulated structures with semi-rigid gusset joints. The vertical coordinates of the control points of the B-spline surface are selected to be the design variables adjusting the shape of the structure. The joint gap index, which is proposed for characterizing the manufacturing and assembling convenience of aluminum alloy structures, the strain energy, and the safety factor under short-term snow load, are chosen as the objective functions of the multi-objective optimization problem. The Pareto optimal solutions of the numerical example indicate that slight differences in the structural shape can result in quite different mechanical behavior, and the NSGA-II algorithm is effective in providing a wide range of optimal structures as compromise solutions of the multi-objective problem. Finally, the same optimization problem is solved for rigid jointed structures to study the influence of the joint rigidity on the form-finding result. By comparison of the internal force distribution, it is found that the joint semi-rigidity will influence the optimal structural shape due to the difference in the load-bearing mechanism. Therefore, it is suggested that the actual joint rigidity should be considered in the structural optimization process of reticulated structures.

Shape of Deployable Membrane Reflectors

The shape and stress distribution in the new Collapsible Rib-Tensioned Surface reflector are investigated both analytically and experimentally. A methodology is established for the preliminary design of symmetric reflectors of given aperture, focal length, and target root-mean-square (RMS) error, by extending form-finding methods originally developed for membrane roof covers. The concepts of a reference surface and of an equilibrium surface are introduced, and algorithms are developed to compute these surfaces and their associated RMS error. Then, the cutting pattern for making the membrane is computed and the RMS error of the actual surface is predicted. Estimates are made of the RMS error of reflectors with apertures of 1, 3, 5, and 10 m, with 6, 12, and 24 ribs. Measurements of prestress and shape in a one-sixth sector of a 1 m diameter reflector with 6 ribs are compared with predictions obtained from the computational study. In the central part of the gore, about half of the total surface, the average...

Structural Morphology Of Tensegrity Systems

Tensegrity systems are structures in which morphological and mechanical aspects are closely related by selfstress requirements. In the first part, this paper describes the birth of the idea which led to these systems. Snelson's work seems to be the main contribution for their initial design, even if Fuller created the word tensegrity and contributed with other searchers, like Emmerich to the popularization of these systems. The second part is devoted to form-finding methods which are necessary to reach a morphology compatible with mechanical requirements. References for main computation methods are given. Among them numerical methods like force density method could constitute an useful design tool.

遺伝的アルゴリズムの空間造形形態解析への応用

Applicability of genetic algorithms (GAs) to the form optimization problem of structures is discussed in this paper. The form optimization problems on truss structures under the prescribed displacement mode are solved by using GAs. These problems have strong nonlinearity, and the layout of the members is deeply concerned with the existence condition of solutions. In the present study, the form finding method in which not only the positions of the objective nodes but also the layout of the members are encoded to strings in the GAs as the design parameters of the structure is proposed, and the effectiveness of the method is examined. Moreover, the form finding method is extended to the minimal surface problem, in which several control points of a spline function are introduced in order to sescribe the shape of the membrane surface.

Form finding of shells by structural optimization

Shell structures are known to be extremely parameter sensitive; even small changes of the initial design, e.g., to the shape of the shell, may drastically change the internal stress state. The ideal case for concrete shells is a pure membrane stress state in compression for all loading conditions. Since in many realistic situations the solution for an ‘optimal’ shape is not obvious, the need for form finding methods is evident. This paper presents computational methods of structural optimization as a general tool for the form finding of shells. The procedure as a synthesis of design modelling, structural analysis and mathematical optimization is discussed with special emphasis on the modelling stage. Several examples show the power of the approach and the similarities to experimental solutions.

On shape finding methods and ultimate load analyses of reinforced concrete shells

Due to considerable imperfection sensitivity the initial shape of a shell can play a dominant role for the quality of the load carrying behaviour. In particular, this is important if large deformation effects and material failure may interact unfavourably, as can be the case for thin reinforced concrete shells. The present contribution discusses different form-finding methods for the shape of free form shells. Among these are the well-known experiments with pressurized or dead-loaded membranes which can also be simulated by a nonlinear membrane analysis or the application of shape optimization schemes using stress levelling or strain energy minimization. The second part is devoted to ultimate load analyses of RC shells. After a brief review of the material models used, selected examples are described, for instance conical shaped cooling towers and a free form model shell which is also experimentally investigated.

Tensegrity Systems and Geodesic Domes

The name Buckminster Fuller is associated with the development of geodesic domes and with tensegrity systems. The principle of the latter is that rigidity is obtained by selfstressing alone. The shapes of domes constructed using tensegrity systems are not generally known a priori; these domes belong to the class of systems with indeterminate shape, such as cable systems. A form-finding method using the dynamic relaxation technique with attenuation of kinetic energy is presented here. The results given by this general method are compared with known solutions in certain cases and make it possible to quantify the deviations between geometries of tensegrity systems and polyhedral geometries. An example is given to illustrate the possibilities of the combination of the principle of tensegity systems with that of geodesic domes. It is a double-layer, double-curvature system whose shape respects the selfstressing criterion. Industrial construction of the system can be envisaged thanks to the design of an assembly node which permits variations in curve radius.

The force density method for form finding and computation of general networks

A new method for network analysis, the “force density method”, is presented. This concept is based upon the force-length ratios or force densities which are defined for each branch of the net structure. It is shown that the force densities are very suitable parameters for the description of the equilibrium state of any general network: the associated node coordinates of the structure are obtained by solving one single system of linear equations. A Gaussian transformation of the topological branch-node matrix yields the equation matrix. A method based on force densities renders a simple linear “analytical form finding” possible. If the free choice of the force densities is restricted by further conditions, the linear method is extended to a nonlinear one. A damped least squares approach is then introduced. In this version the method may be applied to the detailed computation of networks. Moreover, an interesting cross-connection between geometrical minimum way nets and prestressed unloaded structures is derived using the force densities.