An Approach to the Optimization of a Thin-walled Z-beam Nina Andelic* - Vesna Milosevic Mitic - Tasko Maneski University of Belgrade, Faculty of Mechanical Engineering, Serbia One approach to the optimization of a thin-walled open section Z-beam subjected to bending and to the constrained torsion is considered. For given loads, material and geometrical characteristics the problem is reduced to the determination of minimum mass i.e. minimum cross-sectional area of a structural thin-walled beam of a chosen shape. The area of the cross section is assumed to be the objective function. The stress constraints are introduced. A general case when bending moments about two centroidal axes and the bimoment are acting simultaneously is derived, and then some particular loading cases are considered. A method of solving the optimal relation of the parts of the considered cross-section is described. Applying the Lagrange multiplier method, the equations, whose solutions represent the optimal values of the ratios of the parts of the chosen cross-section, are formed. The obtained results are used for numerical calculation. © 2009 Journal of Mechanical Engineering. All rights reserved. Keywords: optimization, thin-walled beams, optimal dimensions, stress constraints, saved mass 0 INTRODUCTION Many modern metal structures (motor and railroad vehicles, naval structures, turbine blades) are manufactured using thin-walled elements (shells, plates, thin-walled beams) which are subjected to complex loads. In most structures it is possible to find the elements in which, depending on loading cases and the way they are introduced, the effect of constrained torsion is present and its consequences are particularly evident in the case of thin-walled profiles. Investigations of the behaviour of thin-walled members with open cross-sections have been carried out extensively since the early works of S. P. Timoshenko [1], who was among the first to publish a number of books on materials strength, the theory of elasticity and the theory of stability, and who also developed the theory of beams and plates bending. V. Z. Vlasov [2] contributed largely to the theory of thin-walled structures by developing the theory of thin-walled open section beams. Kollbruner and Hajdin [3] and [4] expanded the field of thin-walled structures by a range of their works. Exceptionally valuable among them are two monographs [3] and [4] which with a series of contributions originally describing the authors' area of interest, constitute a unique work in terms of content,. Also, Murray [5] and Rhodes et al. [6] should be mentioned as the authors who introduced the theoretical aspects of the behaviour of thin-walled structures. Due to their low weight, thin-walled open section beams are widely applied in many structures. Thin-walled beams have a specific behaviour, which is the reason why their optimization is a particular problem. Analyzing the process of designing various types of structures, it can be observed that the classical procedure of defining dimensions of a structure based on the theory of the strength of materials provides sufficient, however, not the optimum geometric parameters. In general, the optimization is a mathematical process through which a set of conditions, which give the maximum or minimum value of a specified function as a result, are obtained. In the ideal case the perfect solution for the considered design situation is supposed to be obtained but in reality only the best solution although not the perfect one, can be found. Among the authors who developed theoretical fundamentals of the optimization method, Fox [7], Brousse [8], Prager [9] and Rozvany [10] should be given the most prominent place. Many studies have been conducted on optimization problems, treating the cases where geometric configurations of structures are specified and only the dimensions of structural members and the areas of their cross-sections are determined in order to attain the minimum structural weight or cost [11] and [12]. Tian et al. [13] present a combined theoretical and *Corr. Author's Address: Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade 35, Serbia, nandjelic@mas.bg.ac.rs experimental study on the minimum weight and the associated optimal geometric dimensions of an open-channel steel section. Many authors, including Farkas [14], applied mathematical problems to the conditional extreme of the function with more variables onto the cross-sectional area of the structure and defined optimum cross-section from the aspect of load and consumption of the material. Then, a series of works appear where the problem of optimization of various cross-sections, such as triangular cross-section [15], I-section [16] and [17] or channel-section beams [18] are solved by using the Lagrange multiplier method. The main purpose of this paper is to present one approach to the optimization of a thin-walled Z-section beam. 1 PROBLEM FORMULATION The starting points during the formulation of the basic mathematical model are the assumptions of the thin-walled-beam theory, on one hand, and the basic assumptions of the optimum design on the other. During the process of structure dimensioning, apart from defining the requested dimensions necessary to permit the particular part of the structure to support the applied loads, it is often of significance to determine the optimal values of the dimensions. The Z - cross section as a very often used thin-walled profile in steel structures is in this paper considered as the object of the optimization. The determination of its optimal dimensions is a very important process but not always the simplest one. The aim of the paper is to determine the minimum mass of the whole beam, i.e. the minimum area A of the cross-section of the considered beam for the given loads and material properties. The cross-section of the considered beam (Fig. 1) with principal centroidal axes X (i = 1, 2) has the centre and not the axis of symmetry. It is assumed that its flanges have equal widths b = b3, and thicknesses tx = t3, and that its web has the width b2 and thickness t2. The ratios of thicknesses and widths of flanges and web are treated as non-constant quantities. Fig. 1. Cross-section It is also assumed that the loads are applied in two longitudinal planes, parallel to the centroidal axes x and y at the distances £ b (i = 1, 2) (Fig. 1). If applied in such a way, the loads will cause the bending moments acting in the above mentioned two planes parallel to the longitudinal axis of the beam, and consequently the effects of the constrained torsion will occur in the form of the bimoment, causing the stresses that depend on the boundary conditions [3] and [19]. The aim of the paper is to determine the minimal mass of the beam or, in other words, to find the minimal cross-sectional area A = Amin (1) for the given loads and material and geometrical properties of the considered beam, while satisfying the constraints. The formulation of the structural design optimization problem plays an important role in the numerical solution process [6]. A particular choice of the objective function and constraints affects the final solution, and the efficiency and robustness of the solution process. 1.1 Objective Function The process of selecting the best solution from various possible solutions must be based on a prescribed criterion known as the objective function. In the considered problem the cross-sectional area will be treated as an objective function and it is evident from the Fig. 1 that A = ^b,t,, i = 1, 2, 3 or (because b1 = b3) A = A(bj, b2) = 2 b t1 + b2 t2 . (2) The thickness-length ratios of the cross-sectional walls are assumed to be non-constant variables (Fig. 1) bi ■ const, i = 1, 2, 3 , (3) where ti and bi are thickness and widths of flanges and web. 1.2 Constraints The formulation is restricted to the stress analysis of thin-walled beams with open sections. Only normal stresses will be taken into account in the consideration that follows and the constraints treated in the paper are the stress constraints. The expressions (4) and (5) for equivalent bending moments [20] taking into account the influence of the bending moments around centroidal axes x and y, denoted as Mx and My respectively, will be used Mx =- 1 - My f I ) * xy V !y) I2 xy Ix Iy (4) and My - Mx xy My =- V !x 1 — xy Ix y (5) where !x, !y are the moments of inertia of the cross-sectional area about the centroidal axes x and y, and Ixy is the product of inertia. The normal stresses are caused by the bending moments M x and M y and by the bimoment B that appears in the case of constrained torsion, and they will be denoted as ax and ay and a, respectively [3] and [19]. Bimoment is not a static value, and can not be defined by static equilibrium conditions. When the bending moments act in planes parallel to the longitudinal axis (Fig. 1) at the distances £1bi (i = 1, 2) the bimoment will appear as a consequence, and it can be expressed as the function of the bending moments and the eccentrities of their planes £ibi (i = 1, 2) in the following way [3] and [19] B = bMx + ^2 M y (6) For the allowable stress <0 the constraint function can be written as P = P<)=