# Numerical Modeling of Non-Uniformly Reinforced Carbon Concrete Lightweight Ceiling Elements
###### Abstract
The ambition of this modeling approach is to predict the strain-hardening behavior of a particular cross-sectional design using the parameters of the material components and of the bond between the concrete and fabrics [18]. Even though the discrete crack modeling approaches provide a valuable insight into the process of matrix cracking and debonding, a realistic and reliable prediction of the composite response \(\sigma_{\text{c}}(\varepsilon_{\text{c}})\) is still not possible. The reason is the complexity of interaction mechanisms governing the bond behavior. Moreover, the effect of material heterogeneity and imperfections at the levels of the fiber bundles, of the fabrics and of the cross-sectional thickness introduces further challenging tasks for the development of micro- and meso-scale modeling approaches. At this level of modeling, further effort is necessary to capture the inelastic interaction effects within a multi-scale model with a broader range of validity than available so far. As a result, the strain-hardening response of the composite is usually experimentally characterized using a composite tensile test. The measured response provides the input for an efficient calibration of phenomenological numerical material models. These macro-scale material models, applicable in finite element simulations, represent the strain-hardening response phenomenologically in terms of inelastic state variables, i.e., damage and plastic strains. An approach representing the composite with a uniform material behavior over the cross section depicted in column (b) of Figure 1 has been applied by the authors for the simulation of with laminated carbon TRC shells [14]. In this simulation concept, the material parameters are specific to a particular cross-sectional layup and thickness. If the cross-sectional configuration changes, the material model is not valid anymore and its material parameters must be recalibrated. Due to the plane stress state within the shell surface, the inelastic effects, i.e., damage or yielding, need to be captured only in the in-plane direction of the shell. The application of this model is limited to cross sections with uniform fabric reinforcement over their height or to structures with prevailing membrane stresses. As discussed later on, the reason is that the position of the fabrics within the cross section is not distinguished in the model so that the bending behavior of sparsely reinforced cross sections cannot be correctly reproduced. In this paper, we focus on the modeling approach depicted in column (c) of Figure 1 with a resolved representation of the matrix and fabric layers within the cross section. In contrast to the discrete cracking models, the matrix cracking represented in a smeared way along the specimen length but is ascribed to a resolved layer of concrete within the shell cross section. Such a distinction between the effective fabrics and matrix layers allows for the rescaling of material parameters for a changed  
Figure 1: Classification of modeling approaches to tensile behavior of textile-reinforced concrete shells including the resolution of stresses along a uniaxial specimen, resolution of stress and strain components in a material point and examples of inelastic material characteristics. cross-sectional configuration including e.g., reinforcement ratio or layup of the fabrics within a cross section. Such scaling is needed to enhance the validity range of the macroscopic models for layered TRC shells so that a smaller amount of calibration experiments is required for the identification of the material parameters. The model classification in Figure 1 is focused on the representation of a the composite cross section. Finite element models based primarily on solid finite element discretization with an explicit resolution of material interfaces including also the localization of individual concrete cracks, presented e.g., in [19; 20], have a different focus and different purpose, i.e., local stress effects in structural details of the simulation of TRC sandwich panels.