# The Numerical Analysis of Textile Reinforced Concrete Shells: Basic Principles
## 3 Materials Models
### Trilinear Material Model  
An alternative concept for modelling the tensile behaviour of TRC composites is a solution introduced by [PERSON] et al. [23] in 1977 using a trilinear stress-strain curve characterized by three linear ranges similar to that of the ACK model [22]. However, the line in the second area is not horizontal but slightly ascending (Figure 4). The slope of the line in each range is calculated in a similar way using a rule of mixtures, combined with the appropriate efficiency factors. The determination of the factors is the objective for further material model calibration. Figure 4: Principle of the trilinear material model. Graphic: [PERSON]. The trilinear material model gives a similar representation for the uncracked zone to the ACK model. The reason for that similarity lies in the basic principles that describe the material according to the rule of mixtures and the joint assumption that until the initiation of a first crack in the composite, the compatibility of strain between the fibre and the matrix is satisfied [23]. Thus, the strain level \(\varepsilon_{p}\) for point \(p\) can be similarly calculated via Equation (10) with the respective levels of the matrix tensile strength \(\sigma_{mn}\) and elastic modulus \(E_{I,Tri}\) (MPa) for the uncracked range in State I. In the further description, the rules are based on Model Code 90 guidelines for tension-stiffening effects according to refs. [5; 24]. The next point \(q\) with \(\sigma_{q}\) and \(\varepsilon_{q}\) can be calculated using Equations (9) and (10) [28]:  
\[\sigma_{q}=1.3\cdot\sigma_{p} \tag{9}\]  
\[\varepsilon_{q}=\varepsilon_{f,q}-\Delta\varepsilon \tag{10}\]  
where \(\varepsilon_{f,q}\) is the strain of the reinforcement in a crack. For the calculation of the composite's strain in point \(q\), at first, the strain of the pure reinforcement is calculated for the loading at this load level \(\sigma_{q}\). The associated strains differ between embedded reinforcement and pure respective unembedded grid. The difference \(\Delta\varepsilon\) reflects the influence of the tension stiffening effect and may be considered via the following:  
\[\Delta\varepsilon=\beta_{t}\cdot\left(\varepsilon_{f,p}-\varepsilon_{p}\right) \tag{11}\]  
where \(\beta_{t}\) is the integration factor for the reinforcement strain along the transmission length; \(\varepsilon_{f,p}\) is the fibre strain at the point of zero slip under first crack forces reaching \(\sigma_{mn}\); and \(\varepsilon_{p}\) is the fibre strain in the crack under cracking stress reaching \(f_{\text{ctm}}(t)\)[24]. A factor of 1.3 is recommended as the difference between \(\sigma_{p}\) and \(\sigma_{q}\) for TRC samples according to [20]. In addition, the composite's stiffness \(E_{IIa}\) in the area of successive crack formation after the first crack can be described using the principle of first-order polynomial approximation as follows:  
\[E_{IIa,Tri}=k\cdot E_{m}\cdot V_{m}+E_{f}\cdot V_{f} \tag{12}\]  
where \(k<1\) is an empirical parameter accounting for the reduction in the effectiveness of the concrete matrix [29]. The value of \(k\) is dependent on the volume fraction of fibres \(V_{f}\) and can be determined, e.g., experimentally. Thus, the value \(\sigma_{p}\) may be alternatively determined via the intersection of the two first-order polynomial functions that describe State IIa with a slope of \(E_{IIa,Tri}\) and State IIb with a slope of \(E_{IIb,Tri}\). There are two close approaches: one based on Model Code 90, Equation (9), and the other based on the geometrical interpretation of Equation (12). Furthermore, the strain in the State IIa range can be estimated in practical way with Equation (13):  
\[\varepsilon=\varepsilon_{f,p}-\frac{\beta_{t}\cdot\left(\sigma-\sigma_{p} \right)+\left(\sigma_{q}-\sigma\right)}{\sigma_{q}-\sigma_{p}}.\left(\varepsilon _{f,p}-\varepsilon_{p}\right) \tag{13}\]  
with \(\sigma_{p}<\sigma\leq\sigma_{q}\) and \(\varepsilon_{p}<\varepsilon\leq\varepsilon_{q}\).