Keywords: discrete sizing optimization, joint flexibility, mixed-integer nonlinear programming, shape optimization, timber, truss.

The paper presents the shape and discrete sizing optimization of timber trusses with the consideration of joint flexibility. The optimization is performed using the mixed-integer nonlinear programming (MINLP) approach. In the optimization model an economic objective function for minimizing the structure's self-manufacturing costs is defined. The design conditions in accordance with Eurocode 5 [1] design code are considered as optimization constraints. The internal forces and deflections are calculated using finite element equations. The flexibility of joints is modelled by the procedure proposed in Eurocode 5, and consequently the structure's stifness matrix is composed by considering fictitiously decreased cross-sectional areas of all the flexibly connected elements.

The present paper represents a continuation of the research work introduced by Šilih et al. [2]. In the preliminary work, the optimization of timber trusses with special attention to joint flexibility was carried out by using the non-linear programming (NLP) approach. Sizing optimization of pitched trusses with a fixed slope of the top chord was performed for different span/depth (S/D) ratios and different numbers of intermediate members (diagonals, verticals).

The present study additionally includes the shape optimization of timber trusses. In this way the S/D ratio as well as the slope of the top chord are optimized simultaneously with the cross-sectional dimensions and joint connections. In addition, discrete sizing optimization is also performed. Alongside continuous variables (shape varibles, stresses, deflections, etc.), discrete sizing variables are also included. The problem thus corresponds to a mixed discrete-continous type of optimization problems, which can be solved with the use of the mixed-integer non-linear programming (MINLP) optimization approach. For the solution of the defined problem, the modified outer approximation / equality relaxation (OA/ER) algorithm [3] was applied.

The efficiency and the applicability of the proposed optimization approach is presented using a numerical example. The results obtained proved the importance of the consideration of joint flexibility. Deflections due to slips in connections represent over 40% of the total deflections. In general it is preferable to design higher timber trusses with a lower span/depth ratio and with a smaller number of diagonal and vertical elements. By increasing the number of elements the deflections cannot be reduced as effectively as this is possible in the case of for example steel trusses with rigid connections. While, by adding flexibly connected intermediate elements, the deflections due to the deformability of the elements decrease, the deflections due to the flexibility of the joints increase and the total deflections remain at the same level.

1

"EN 1995-1-1, Eurocode 5: Design of Timber Structures, Part 1-1 General rules and rules for buildings", European Comitee for Standardization, Brussels, 2005.

2

S. Šilih, M. Premrov, S. Kravanja, "Optimum design of plane timber trusses considering joint flexibility", Engineering Structures, 27, 145-154, 2005. doi:10.1016/j.engstruct.2004.10.001

3

Z. Kravanja, I.E. Grossmann, "New developments and capabilities in PROSYN - an automated topology and parameter synthesizer", Computers in Chemical Engineering, 18, 1097-1114, 1994. doi:10.1016/S0098-1354(94)85027-5

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