Fiber-reinforcement of concrete plates through topology optimization

The achievement of an optimal arrangement of fiber-reinforcement is a crucial issue for the efficient retrofitting of concrete elements. Numerical approaches to the automatic optimal design of fiber-reinforced polymers (FRPs) have been recently proposed in the literature to address in-plane loaded structural elements, such as beams and shear walls [1]. Combining topology optimization techniques [2] with methods typical of the so called free material design [3], a mathematical problem can be formulated that searches for the best distribution of a given amount of unidirectional FRP along with its optimal orientation, with the aim of maximizing any objective function. Dealing with concrete slabs subjected to lateral loads, the reinforcing layers to be applied on the two sides of the element are usually different. Modeling the slab through plate finite elements, the contribution of each layer to the local stiffness matrix is shown to depend on a density field and an orientation field. Accordingly, the proposed procedure defines the layout of the layers that minimizes the overall strain energy of the reinforced structure, computing the optimal values of four unknown design variable fields. Indeed, this choice for the objective function is suggested by technical codes when performing the automatic research of load paths for the design of reinforced structural elements, see the case of strut-and-tie methods [4]. A suitable set of stress constraints is introduced to penalize any compressed FRP element. This turns out to be an effective technique to achieve a distribution of no-compression reinforcement, as required in practice. The discrete setting is solved through mathematical programming [5]. Numerical investigations are presented to discuss the features of the computed optimal layouts, along with the possible application as preliminary design for the structural retrofitting of concrete slabs. The obtained layouts are compared with solutions suggested by limit analysis theory, which suggests to reinforce plates perpendicularly to the hinges arising in the unreinforced elements. The performances of the adopted computational procedure are investigated as well.