Whereas the equilibrium condition is linear with respect to the interior forces and moments, the yield condition for frame structures is non-linear [4]. In order to apply linear optimization methods, being faster and numerically easier to handle, piecewise linear approximations are needed.

Here, four approaches are presented, two simple ones with only one or two straight lines [1] and two iterative methods using tangents and secants, respectively, to the yield domain [5], and compared by means of numerical aspects and by a certain quality criterion [2,3,6,7].

After the theoretical consideration and comparison of the different approximation methods, we apply them to the robust design of plane frames in the case of uncertainty. We suppose the applied load to be stochastic which leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, the recourse problem with discretisation (RPD) is used [8,9]. After linearization of the yield condition and introducing the RPD, we can use efficient numerical solution procedures (LP-solvers).

The methods presented are finally compared by finding the optimal design of a two-storey frame. Here, as expected, the less restrictive methods yield the cheapest design.

1

Manual of Steel Construction, American Institute of Steel Construction, Inc., New York City, 1970.

2

L. Collatz, "Approximationstheorie und Anwendungen", In "Iterationsverfahren, Tagung über numerische Methoden der Approximationstheorie", Birkhäuser Verlage, Basel und Stuttgart, 1970.

3

P. Deuflhard, A. Hohmann, "Numerische Mathematik I", de Gruyter Lehrbuch, Berlin/New York, 2002.

4

S. Kaliszky, "Plastizitätslehre - Theorie und technische Anwendungen", VDI-Verlag, Düsseldorf, 1984.

5

K. Marti, "Expected total cost minimum design of plane frames by means of stochastic linear programming methods (SLP)", To appear in: "Coping with Uncertainty (CwU): Robust decisions", Lecture Notes in Economics and Mathematical Systems (LNEMS), Springer-Verlag, Berlin/Heidelbereg/New York, 2009.

6

G. Opfer, "Numerische Mathematik für Anfänger", Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 2001.

7

M.J.D. Powell, "Approximation theory and methods", Cambridge University Press, Cambridge, 1981.

8

G. Stöckl, "Optimaler Entwurf elastoplastischer mechanischer Strukturen unter stochastischer Unsicherheit", Fortschritt-Berichte VDI, Reihe 18, Nr. 278, VDI-Verlag GmbH, Düsseldorf, 2003.

9

S. Zier, "Optimal Design of Trusses Considering Uncertainty: A Comparison of Two Approaches", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 53, 2008. doi:10.4203/ccp.88.53

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £140 +P&P)