# Experimental Testing of Reinforced Concrete Deep Beams Designed by Strut-And-Tie Method
## 2 Determination of Strut and Tie Models Using Software \"ST Method\"  
This chapter summarizes the procedure for determining the Strut-and-Tie models and is explained in detail in [14; 16]. Strut-and-Tie models are suitable for the design of discontinuity regions in structural concrete elements, where the conventional analysis methods are incorrect, as the main modeling assumptions deviate significantly from the actual behavior (i.e., plane sections remain plane). Discrete topology optimization [17; 18], is implemented in the software \"ST method\", which has the ability to analyze a reinforced concrete member in the plane. By discrete topology optimization, a member can be modeled with the truss system, such as the Strut-and-Tie model. One of the most common optimization conditions is the smallest amount of reinforcement. In the software \"ST method\", according to discrete topology optimization, the geometry of a member is represented with an initial network of nodes, which are interconnected by line elements (rods). The optimal density of this initial network is analyzed in detail in [16]. The analysis for the determination of the Strut-and-Tie models is iterative, where the stiffness recalculation of individual elements is performed during each step, based on the analysis results from the previous step. The correction of cross-sectional stiffness depends on the sign and the intensity of the truss element force. The stiffness element is changed indirectly through changes in the cross-sectional area. The stiffness of the individual elements of the truss model is determined by the expression [19; 20]:  
\[K_{ij}=\beta_{i}\cdot(E_{d}\cdot N_{ij\,-\,1})/(L_{i}\cdot f_{d}) \tag{1}\]  
The cross-sectional area of individual truss elements is determined with:  
\[A_{ij}=\beta_{i}\cdot N_{ij\,-\,1}/f_{d} \tag{2}\]  
where \(\beta_{i}\) is the reduction factor which depends on the \"desired\" reinforcement layout in the member; \(N_{ij\,-\,1}\) is the force in the \(i^{\text{th}}\) simple element from \(j-1\) iteration; \(E_{d}\) is the modulus of elasticity for concrete (compression) or steel (tension); \(f_{d}\) is the design strength of the concrete or reinforcement depending on the nature of the axial force in the truss element (compression-concrete or tension-steel), and \(L_{i}\) is the length of the element. The value of the \(\beta\) coefficient depends on the angle of the \"desired\" reinforcement inclination. The values of the \(\beta\) coefficient for the compressed struts are always 1. For tensioned truss elements, according to the desired reinforcement, they can be 0 or 1. Adopting the value of 1 for one of the previously mentioned angles implies the \"desired\" reinforcement inclination. Usually, for the tensioned directions at an angle of 0\({}^{\circ}\), 45\({}^{\circ}\), or 90\({}^{\circ}\), the \(\beta\) coefficient has the value 1, and for other angles the value is 0, because in this way the simplest reinforcement layouts are obtained. According to the axial force sign (compression-concrete or tension-steel), the modulus of elasticity of individual truss elements shall be adopted. In the zero iteration, a mesh of truss elements is created with boundary conditions and the external loads. All model elements have the same mechanical and geometrical characteristics of the cross-sections. The axial forces in the elements are determined on the established model. Based on the sign of the axial forces, the area of the cross-sections is determined for all the elements according to Equation (2). Simultaneously, the modulus of elasticity for each truss element is corrected. In this way, the truss model is formed for the next iteration. Iterations are repeated until the change of the cross-sectional areas of the truss model elements in two consecutive iterations is small enough. Recommendations for completing the iterative procedure and determining the Strut-and Tie model, as a truss model from the last iteration, are given in [14; 16].