The problem of optimal design of a steel plated girder according to the Eurocode 3 is considered. Code regulations admit the Finite Element Analysis (FEA) in designing plated structures with variable cross-sections. A technique of determining an approximate solution to the optimization problem is presented. It is determined a solution of a control theory optimization task, in which Eurocode requirements regarding the Ultimate Limit State (bearing capacity, local and global stability) as well as Serviceability Limit State (flexural rigidity) are used as appropriate inequality constraints. Static analysis is performed within the framework of linear elasticity and Bernoulli-Euler beam theory making an account for second-order effects due to prescribed imperfections. Obtained solutions, after regularization, may be used for direct verification with the use of FEA or as the first guess for iterative topology optimization algorithms. Code requirements governing the determination of optimal shape are visualized in the constraint activity diagram, which is a proposed tool for analysis of optimization process.

This study investigated the optimal shape of glue-laminated timber beams using an analytical model of a slender beam, taking into account the anisotropy of its strength properties as well as boundary conditions at the oblique bottom face of the beam. A control theory problem was formulated in order to optimize the shape of the modeled beam. Two optimization tasks were considered: minimizing material usage ($V_{min}$) for a fixed load-carrying capacity (LCC) of the beam and maximizing load-bearing capacity ($Q_{max}$) for a given volume of the beam. The optimal solution was found using Pontryagin’s maximum principle (PMP). Optimal shapes were determined using Dircol v. 2.1 software and then adjusted according to a 3D finite element analysis (FEA) performed in Abaqus. The final shapes obtained through this procedure were used in the CNC-based production of three types of nine beams: three reference rectangular beams, three $V_{min}$ beams, and three $Q_{max}$ beams. All specimens were subjected to a four-point bending test. The experimental results were contrasted with theoretical assumptions. Optimization reduced material usage by ca. 12.9% while preserving approximately the same LCC. The maximization of LCC was found to be rather unsuccessful due to the significant dependence of the beams’ response on the highly variable mechanical properties of GLT.

The paper presents a beam theory for composite girders consisting of two beams joined together with an adhesive layer. The height of the bottom beam is considered variable. The governing equations are suitable for formulation of a shape optimization problem in terms of control theory. The use of...