# Numerical Modeling of Non-Uniformly Reinforced Carbon Concrete Lightweight Ceiling Elements
###### Abstract  
The paper focuses on the specifics of macro-scale modeling of thin-walled textile-reinforced concrete shells. Application of layered shell finite elements requires systematic procedures for identification of material characteristics associated with the individual layers within the cross section. The identification of the material parameters describing the tensile behavior of a composite cross section is done using data obtained from the tensile test. Such test is usually performed only for a reference configurations with a simple layup of fabrics and a chosen thickness. The question is how to derive the strain-hardening response from the tensile test that is relevant for a changed cross-sectional configuration. We describe and discuss scaling and mixture rules that can be used to modify the material parameters for modified cross-sectional layups. The rules are examined in the context of the test results obtained on a shell that was reinforced non-uniformly, with varying types of textile fabrics and varying thickness within the shell surface. textile-reinforced concrete; thin-walled shells; cementitious composites; layered finite elements; mixture rules; model calibration +
Footnote †: journal: Applied Physics Letters  
_Applied Physics Letters_  
technologies calls for an effective support in terms of efficient and validated modeling approaches delivering a correct prediction of the structural behavior of thin-walled textile-reinforced concrete (TRC) shells. Efficient and realistic models of non-uniformly reinforced fabric reinforced shells are the assumption for the development of robust design and assessment concepts [12; 13]. Such modeling strategies must account for the specific features of the material and of the nonlinear structural behavior of TRC shells. The particular issues to be considered in the macro-scale modeling of TRC shells include:  
* Correct reproduction of the strain-hardening response of the shell cross section for a wide range of loading conditions including the tensile or bending loads. * Initial and damage induced anisotropy owing to the oriented crack pattern within the two-dimensional stress state in the shell plane. * Geometrical nonlinearity in interaction with imperfections requiring the buckling analysis as documented in experimental studies [4] and numerical investigations of the effect of imperfections on the structural behavior [14; 15]. The focus of the present paper is on the first item, i.e., a flexible reflection of the strain-hardening behavior in macro-scale simulations of TRC shells. The second aspect of damage-induced anisotropy is considered here as well; however, it does not have a significant influence on the structural behavior of the studied girder. The issue of geometrical nonlinearity and buckling behavior is not within the scope of the present paper. Considering a TRC element with sufficient reinforcement ratio loaded in tension, three phases of tensile response are distinguished:  
* Phase I: linear elastic behavior with the strain-stress relation characterized by the composite. elasticity modulus,
* Phase II: formation of the cracks at the level of the matrix tensile strength,
* Phase III: saturated crack state with steady matrix stress level between the cracks; the effective stiffness of the composite in this phase is approximately equal to the stiffness of the reinforcing fabrics. Existing approaches to modeling of this tensile behavior can be roughly classified into three categories summarized in Figure 1. The theoretical descriptions based on discrete representation of the matrix cracking process and debonding between matrix and reinforcement displayed in column (a) of Figure 1 considers an elastic-brittle behavior of fabrics and matrix material components. In this model, the nonlinear strain-hardening response results from the evolution of discrete cracks emerging along the specimen with random matrix strength and nonlinear bond stress--slip relation [16; 17].