# Optimal Design of Earthquake-Resistant Buildings Based on Neural Network Inversion
## 1 Introduction  
The design of earthquake-resistant reinforced-concrete buildings is based on the concepts of ductility, strength hierarchy, and capacity design. Under exceptional seismic events buildings should deform plastically, exhibiting a deformation capacity greater than the demand in elements (typically beams) that are delegated for developing plastic hinges, while keeping more fragile elements (typically columns) safe. To assure this dissipative ductile behavior the code guidelines impose the fulfillment of ductility and strength hierarchy rules, which entail an iterative process for the more suitable choice of the geometrical and reinforcement design parameters. Commercial design programs help the designer to meet building code provisions and to check the fulfillment of rules, but an optimal cost-effective design is not usually ensured. To design earthquake-resistant buildings, the present paper proposes a procedure, which is based on the inversion of an artificial neural network and on an optimization algorithm to achieve the minimum total cost under code constraints. The procedure is tested on a multistory reinforced-concrete building assumed to be built in a medium seismicity Italian area. Reference to the Italian Building code [1] and to the European Code EC8 [2] was made. Three design parameters are adopted as input variables to obtain about a thousand suitable combinations for the training dataset, while seven code checks are chosen as constraints to obtain the output matrix. A three-dimensional finite-element model of the building was developed and non-linear dynamic analyses under a suite of seven spectrum-consistent earthquakes were carried out. Among the methodscovered by the design regulations, the non-linear dynamic analysis is the most accurate to assess the seismic performance of ductile dissipative structures, although it requires higher computational burden and designer skill [34]. It can be successfully applied to both the design of new buildings [5] and the seismic retrofit of existing structures [6,7]. Artificial neural networks (ANNs) have been successfully applied for different purposes in Civil Engineering, from the early warning of ground settlements in urban areas [8] to the control of dynamic effects in bridges and the development of decision support systems for industrialized manufacturing of buildings [9]. ANNs have been also used to predict the seismic response of buildings [10,11] and even to perform an optimal seismic design [12]. A method based on an ANN is proposed in this paper for the code-compliant seismic design of reinforced concrete buildings. The novelty of the present approach is that of adopting an inversion of the ANN, which allows one to solve an inverse problem, that is to find the combination of design variables that meet given code-based performance constraints. Based on the ANN inversion procedure, an optimization algorithm is also presented in this paper which leads to a design configuration that minimizes the cost function. The neural network inversion is a non-standard tool which was exploited for pollution evaluation [13, 14, 15, 16] and for electric capacitance tomography in nondestructive testing [17]. It was never applied, however, for structural design optimization. To test the proposed method, a multistory reinforced-concrete building will be considered as a case-study. Based on non-linear dynamic analyses carried out under a suite of seven spectrum compatible earthquakes, an input matrix made of many combinations of three design parameters (corresponding to different building design configurations) and a target matrix containing seven performance variables calculated for each of the building configurations, will be obtained. The ANN will be trained on the input and the target datasets and the inversion algorithm will be then applied to find a design solution that falls within the intersection between the input and the output feasibility domains.