## 3 Probabilistic Numerical Model of SFRC Beam Flexural Behavior
### Stress-Strain Relations  
The effects of steel fibers on concrete mechanical behavior, such as the appearance of residual tensile strength, are accounted for by modifying the stress-strain relation of concrete (Figure 1). The stress-strain relations used in this study are based on those proposed by [27, 28]. Compressive strength, elastic modulus, and the tensile strength of SFRC are assumed to have the same values as for plain concrete. Assuming that the compressive strength of the concrete is known, the equations defined for plain concrete are used to estimate the tensile strength (Equation (25) and the elastic modulus (Equation (28)) [28].  
In compression, a bilinear stress-strain relation is used (Figure 1b). Peak compressive strength is estimated to be 85% of the characteristic compressive strength [29].  
A trilinear stress-strain relation is used in tension (Figure 1a). Strain corresponding to the peak tensile strength, \(\epsilon_{1}\), can be calculated by dividing the peak tensile strength by the elastic modulus. Residual tensile strength is calculated using Equation (26), assuming that the tensile failure is due to fiber pullout, not fiber fracture. Coefficient \(a\) is used to account for the effect that different types of fibers have on the residual tensile strength. For example, due to their geometry, hooked fibers have a higher pullout strength than straight fibers, thus resulting in higher residual tensile strength. The values of the coefficient \(a\) are adopted as 1 for straight fibers, 1.5 for irregular fibers, and 2 for hook-end fibers, based on [27]. Strain \(\epsilon_{2}\) (Equation (27)) (Figure 1a), after which residual tensile strength is assumed constant, is adopted from [11].  
\[f_{t}=0.56\cdot\sqrt{f_{ck}[MPa]} \tag{25}\]  
Figure 1: Stress–strain relation in (**a**) tension and (**b**) compression.  
\[f_{t,res}=0.3\cdot V_{f}\cdot a\cdot\frac{l}{d}\cdot\sqrt{f_{ck}[MPa]} \tag{26}\]  
\[\epsilon_{2}=\epsilon_{1}+0.0001 \tag{27}\]  
\[E_{c}=4730\cdot\sqrt{f_{ck}[MPa]} \tag{28}\]  
where:  
\(V_{f}\)--volumetric fiber content  
\(\frac{l}{d}\)--fiber factor format (i.e., aspect ratio)  
\(f_{ck}\)--characteristic compressive concrete strength  
\(\epsilon_{1}\)--strain at peak tensile strength  
\(f_{t,res}\)--residual tensile strength  
\(f_{t}\)--tensile strength  
\(a\)--coefficient accounting for different types of fiber (\(a=1\) for straight fibers, \(a=1.5\) for irregular fibers, \(a=2\) for hooked fibers)  
\(\epsilon_{2}\)--strain after which residual tensile strength is constant (Figure 1a)  
Concrete stress-strain relations implemented in OpenSEES could not appropriately represent the proposed SFRC stress-strain relations. Therefore, the Elastic Multi-Linear model, whose nonlinear stress-strain relation is given by a multi-linear curve defined by a set of points, is used [25]. Steel in longitudinal bar reinforcement is modeled using the Giuffre-Menegotto-Pinto model (i.e., Steel02 in OpenSEES). The values of yield strength and elastic modulus varied among experiments and are given in [2; 10; 12]. The hardening ratio was assumed to be 0.01, and the parameters that control the transition from the elastic to the plastic branch of the steel stress-strain relation were \(R_{0}=15\), \(cR_{1}=0.925\), \(cR_{2}=0.15\) for all specimens used for numerical model validation.