Minimum mass design of tensegrity bridges with parametric architecture and multiscale complexity
Introduction
The subject of form-finding of tensegrity structures has attracted the attention of several researchers in recent years, due to the special ability of such structures to serve as controllable systems (geometry, size, topology and prestress control, cf, e.g., Tilbert and Pellegrino, 2011), and also because it has been recognized that the tensegrity architecture provides minimum mass structures for a variety of loading conditions, including structures subject to cantilevered bending load; compressive load; tensile load (under given stiffness constraints); torsion load; and simply supported boundary conditions (e.g. a bridge), without yielding and buckling (refer, e.g., to Skelton and de Oliveira, 2010a, Skelton and de Oliveira, 2010b, Skelton and de Oliveira, 2010c, Skelton and Nagase, 2012, and references therein). Other additional advantages of tensegrity structures over more conventional control systems are related to the possibility to integrate control functions within the design of the structure: in controlled tensegrity systems the mechanics of the controller and the structure can naturally cooperate, through the change of the configurational equilibrium of the structure, as opposed to traditional control systems, where often the control pushes against the equilibrium of the structure. It is also worth noting that it is possible to look at a tensegrity structure as a multiscale sensor/actuator, which features highly nonlinear dynamical behavior (geometrical and/or mechanical nonlinearities), and can be controlled in real time (Skelton and de Oliveira, 2010c, Fraternali et al., 2012).
Particularly interesting is the use of fractal geometry as a form-finding method for tensegrity structures, which is well described in Skelton and de Oliveira, 2010a, Skelton and de Oliveira, 2010b, Skelton and de Oliveira, 2010c. Such an optimization strategy exploits the use of fractal geometry to design tensegrity structures, through a finite or infinite number of self-similar subdivisions of basic modules. It looks for the optimal values of suitable complexity parameters, according to given mechanical performance criteria, and generates admirable tensegrity fractals. The self-similar tensegrity design presented in Skelton and de Oliveira, 2010a, Skelton and de Oliveira, 2010b, Skelton and de Oliveira, 2010c is primarily focused on the generation of minimum mass structures, which are of great technical relevance when dealing, e.g., with tensegrity bridge structures (refer, e.g., to Bel Hadj Ali et al., 2010). The ‘fractal’ approach to tensegrity form-finding paves the way to an effective implementation of the tensegrity paradigm in parametric architectural design (Sakamoto et al., 2008, Rhode-Barbarigos et al., 2010, Phocas et al., 2012).
The present work deals with the parametric design of tensegrity bridges, through self-similar repetitions, at different scales of complexity. Michell (1904) derived the minimal mass topology when superstructures is only allowed above the roadbed. Deck design requires structure below the roadbed. Here we integrate the two to minimize mass of the total bridge. The design variables consist of two complexity parameters and two aspect angles, which rule the geometry of the superstructure and the substructure. The iterative procedure proposed in Nagase and Skelton (2014) is employed to generate minimum mass shapes under yielding and buckling constraints, for varying values of the design variables. We begin by formulating the present bridge model in Section 2, and summarizing the employed minimization strategy in Section 3. Next, we present a variety of numerical results, which illustrate the potential of the proposed design strategy in generating minimum mass shapes of tensegrity bridges in association with different search landscapes (Section 4). A key result that we observe is that the minimum mass topology of the tensegrity bridge features two different (discrete–continuous) structural scales, which are related to the two employed complexity parameters. We end by presenting the main conclusions of the present study and future work in Section 5.
Section snippets
Tensegrity bridge model
In a famous work dated 1904, A.G.M. Michell examines the problem of finding the minimum volume network of fully stressed truss elements, which transmit a vertical force applied at the middle point C of a given segment AB to two fixed hinge supports applied at A and B (Michell, 1904). On pages 594–597 of this work, Michell deals with a truss network spanning a 2D continuous domain including the points A, B and C along its boundary (centrally loaded beam), and assumes that the material of such a
Mass minimization algorithm
We deal with the minimum mass design of the fractal bridge presented in Section 2 through the iterative linear programming procedure extensively presented in Nagase and Skelton (2014) that we briefly summarize hereafter. Let denote the yield stress of the material. We enforce the following yield constraint in the generic stringwhere denotes the maximum admissible stress in such an element. Concerning the bars, we assume that the maximum admissible compressive stress
Numerical results
In this section we present a collection of numerical results, which aim to illustrate the potential of the minimum mass design under consideration. We use the symbols μ*, α* and β* to denote the minimum mass and the optimal aspect angles of the tensegrity bridge under combined yielding and buckling constraints, respectively, and the symbols , and to denote the optimal values of the same quantities under simple yielding constraints. In all the examples, we search for a global minimum
Concluding remarks
We have presented a design methodology for tensegrity bridges, which is aimed to the generation of minimum mass shapes through parametric self-similar iterations. It makes use of basic units consisting of Michell trusses carrying a central point load (Michell, 1904, Baker et al., 2013, Sokóf and Rozvany, 2012); compressed arches above the deck level; and tensile cords below the deck. The proposed design procedure is ruled by two complexity parameters (n and p), two aspect angles (α and β), and
Acknowledgement
GC, FF, and RES acknowledge financial support from the Province of Avellino.
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