Computer-Aided Civil and Infrastructure Engineering 00 (2012) 1–15 Structural Analysis with Fuzzy Data and Neural Network Based Material Description W. Graf, ∗ S. Freitag, J.-U. Sickert & M. Kaliske Institute for Structural Analysis, Technische Universit¨ at Dresden, Dresden, Germany Abstract: In the article, a new approach is presented utilizing artificial neural networks for uncertain time- dependent structural behavior. Recurrent neural net- works (RNNs) for fuzzy data can be trained by uncertain experimental data to describe arbitrary stress–strain–time dependencies. The benefit is a generalized formulation, which can be applied to describe the behavior of several materials without definition of a specific material model. Model-free material descriptions can be used as numer- ical efficient material formulations within the finite ele- ment method. To perform fuzzy or fuzzy stochastic fi- nite element analyses, a new approach is introduced. An α-level optimization is utilized for signal computation and training of RNNs for fuzzy data. The applicability is demonstrated by means of examples. 1 INTRODUCTION Artificial neural networks are powerful tools to learn functional dependencies in data. They are applied for several tasks in civil engineering, see for example, Adeli (2001). In structural engineering and mechan- ics, neural networks are utilized, for example, for re- sponse surface approximation (Pannier et al., 2009; Papadrakakis and Lagaros, 2002), parameter identi- fication (Kuˇ cerov ´ a et al., 2007; Nov´ ak and Lehk´ y, 2006), system identification (Adeli and Jiang, 2006; Jiang and Adeli, 2005), damage detection (Arangio and Bontempi, 2010; Jiang and Adeli, 2007), lifetime predic- tion (Freitag et al., 2009), time series prediction (Reuter and M¨ oller, 2010), earthquake prediction (Adeli and Panakkat, 2009), structural control (Adeli and Jiang, 2007; Jiang and Adeli, 2008), etc. Applications of ar- ∗ To whom correspondence should be addressed. E-mail: wolfgang. graf@tu-dresden.de. tificial neural networks in structural analysis are pre- sented, for example in Graf et al. (2011) and Sickert et al. (2011). The focus of this article is to develop neural network concepts for constitutive modeling (Ghaboussi et al., 1991) taking uncertain time-dependent material behav- ior into account. The terminology model-free material description is selected to indicate that neural networks are utilized as material formulation in computational mechanics instead of physically based material models. The benefit is a wide applicability for several materi- als without restrictions to specific material characteris- tics. Additionally, the computation of neural networks is numerically efficient. Dependencies between stresses and strains can be identified from data obtained from real or numerical experiments. In Hashash et al. (2004), feed forward networks are used to describe nonlinear stress–strain dependencies. Current and previous stress and strain states can be considered, if the material be- havior is path dependent, see for example, Ghaboussi and Sidarta (1998). In Jung and Ghaboussi (2006), the stress and strain rates are used to describe vis- coelasticity with feed forward networks. The neural net- works can be utilized as material formulation for struc- tural analysis. An application within the finite element method (FEM) is presented in Hashash et al. (2004). In Haj-Ali et al. (2001), feed forward networks are used as material description at the macrolevel within a multi- scale approach. Whereas the discussed approaches are based on feed forward networks, recurrent neural network (RNN) concepts are developed in this article. RNNs are suit- able to consider time-history effects in data series, especially long-term dependencies, see for example, Panakkat and Adeli (2009), Puscasu et al. (2009), and Sch ¨ afer et al. (2008). Special RNNs are universal ap- proximators of dynamical systems in form of state C  2012 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/j.1467-8667.2012.00779.x