Optimal archgrids: a variational setting
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RESEARCH PAPER
Optimal archgrids: a variational setting ´ 1 R. Czubacki1 · T. Lewinski Received: 19 August 2019 / Revised: 17 February 2020 / Accepted: 27 February 2020 © The Author(s) 2020
Abstract The paper deals with the variational setting of the optimal archgrid construction. The archgrids, discovered by William Prager and George Rozvany in 1970s, are viewed here as tension-free and bending-free, uniformly stressed grid-shells forming vaults unevenly supported along the closed contour of the basis domain. The optimal archgrids are characterized by the least volume. The optimization problem of volume minimization is reduced to the pair of two auxiliary mutually dual problems, having mathematical structure similar to that known from the theory of optimal layout: the integrand of the auxiliary minimization problem is of linear growth, while the auxiliary maximization problem involves test functions subjected to mean-square slope conditions. The noted features of the variational setting governs the main properties of the archgrid shapes: they are vaults over a subregion of the basis domain being the effective domain of the minimizer of the auxiliary problem. Thus, the method is capable of cutting out the material domain from the design domain; this process is built in within the theory. Moreover, the present paper puts forward new methods of numerical construction of optimal archgrids and discusses their applicability ranges. Keywords Prager-Rozvany archgrids · Michell structures · Least-volume design
1 Introduction Funiculars are planar frameworks which, albeit subject to a transverse and transmissible load, do not undergo flexure, i.e. they remain bending-free and, consequently, are not subject to transverse shear forces, while the axial stress resultants are characterized by a fixed sign: all the bars are in tension (or- in compression). The load is called transmissible, if it follows the design, yet keeping the given (usually vertical) direction and keeping the value of its intensity q(x) measured per unit length of the line orthogonal to the direction of the load [q] = [N/m], cf. Fuchs and Moses (2000). Of the funiculars being fully and uniformly stressed (up to a given limit (equal, say, −σC , σC being the permissible stress in compression)) one can find one of
Responsible Editor: Mehmet Polat Saka R. Czubacki
[email protected] T. Lewi´nski [email protected] 1
Faculty of Civil Engineering, Warsaw University of Technology, al. Armii Ludowej 16, 00-637, Warsaw, Poland
the least volume. Its rise cannot be too high but also cannot be too small, since big values of thrusts (i.e. horizontal reactions at the pin supports) lead to an inevitable increase of areas of the cross-sections, resulting in the increase of the volume. Thus, the rise should be appropriately chosen. Let z = z(x) represent the shape of the funicular pin supported at A and B at the levels z(0) = hA , z(l) = hB . The optimal rise of the least-volume funicular is determined by the condition 1 l
l
dz dx
2 dx = 1 + 2
h
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