OPTIMAL CONFIGURATIONS OF ARCHES WITH VARIABLE CURVATURE AND TAPERED CROSS-SECTIONS

Arches have proven to be efficient load bearing structures for centuries. Today structural designers utilize long-span arches to support bridges for their stability, efficiency, high load capacity and their architectural and urban form-making potential. The efficiency of the arch structure is strongly related to its shape: it is difficult to find another type of structure where the connection between geometry and internal loading is so pronounced. The selection of the best arch geometry is key in reducing the structural volume because a good arch shape allows for the predominance of axial internal actions with low or no eccentricity under a wide range of loading combinations. In this master’s degree thesis, an original analytical and numerical formulation is presented as a solution for a system of static and kinematic ordinary differential equations for curved beams. Starting from the differential equations for plane linear elastic curved beams, a sixth order differential equation is derived via simplification after taking into consideration a specific set of assumptions. This equation, together with the relevant boundary conditions, is used to evaluate displacements, internal forces and stresses for elastic arches having variable curvature and tapered cross section. The numerical solution of the basic equation is obtained by an ad-hoc implementation of the finite-difference method in custom MATLAB code. This approach enables the evaluation of the best solution sets accounting for 1) various arch shapes; 2) different loading combinations and 3) cross-sections varying along the arch span. The proposed approach is then validated by comparison with Finite Element models. Following, the thesis discusses the implementation of a machine learning algorithm for the calculation of the geometrical parameters that allow to minimize the quantity of materials that constitute the arch structures. This type of algorithm requires the definition of variables, an objective function and a set of problem constraints. For the implementation described in this thesis, we defined a vector of variables that are used to define the geometry of the arch; the selected optimization objective function to be minimized is the volume of the arch; finally, the method used to calculate the stress values has been selected as a constraint function, to reduce the range of solutions to the only ones able to bear the design loads. The effectiveness of the optimization is then verified by comparing the maximum Von Mises stresses, that are exerted on structure by the application of external loads, with yield strength of the steel that constitutes the arch. To validate the results of the optimization, a comparison was performed between the optimization method coded in MATLAB and a different approach based on the Finite Element Method for the calculation of the constraint function for the Genetic Algorithm. This different approach has been coded in Grasshopper, a visual programming software. It allows for the creation of a tridimensional model of the arch where the geometry is defined by the parameters calculated with the Genetic Algorithm. The presented approaches are useful for the preliminary design of the arches since they allow the designer to determine the geometry of the arch that can support the design loads and requires the minimum volume of material. Moreover, the designer is automatically provided with the tridimensional model of the optimized structure.