# Nonlinear Dynamic Response of a Precast Concrete Building to Sudden Column Removal
## 2 Case Study and Modelling Technique
### FE Modelling and Analysis Technique
#### 2.2.2 Moment-Rotation Relationship for Rotational Spring  
To consider the rotational behavior of structural connections between beam and adjacent column members, the moment-rotation relationships for both negative and positive branches are required. The positive branch was calculated through the formulation by [PERSON] et al. [50], see Figure 4a. The rotational equilibrium of the connection relies on the stress block of concrete in compression and on dowel strength. The connection skeleton curve is represented by the yield moments and rotations, and no ultimate resistance calculation is provided. For this reason, the residual strength of the connection was kept constant and equal to the yield value until large connection rotations were achieved. The negative moment branch was calculated according to [PERSON] et al. [51; 52], see Figure 4b, which is able to consider the contributions of dowel and/or continuous reinforcement. As reported in Figure 4b, the rotational equilibrium of the connection relies on the stress block of concrete in compression and on dowel and/or continuous reinforcement forces. The connection skeleton curve is represented by the yield and ultimate moments and rotations. The residual strength of the connection is assumed to be negligible after the achievement of the ultimate rotation. Figure 4: Schematic representation of moment–rotation relationship: (**a**) positive branch according to [PERSON] et al. [50]; (**b**) negative branch according to [PERSON] et al. [51; 52]. Regarding the positive moment-rotation relationship (Figure 4a) the connection yield moment \(M_{Rcy}\) considering only the dowel strength was defined as follows:  
\[M_{Rcy}=V_{Rd}z_{p} \tag{8}\]  
where \(z_{p}\) is equal to (\(h_{b}-0.5y_{p}\)). The height of the compression zone \(y_{p}\) was calculated through the following equation:  
\[y_{p}=\frac{V_{Rd}}{f_{c}b} \tag{9}\]  
where \(b\) is the beam width and \(f_{c}\) is the concrete compressive strength. The rotational stiffness \(k_{p}\) of the connection was calculated through moment equilibrium as follows:  
\[k_{p}=\lambda\big{(}h_{b}-0.5y_{p}\big{)}^{2} \tag{10}\]  
where \(\lambda\) is the stiffness associated with the shear deformation of the dowel:  
\[\lambda=\frac{V_{Rd}}{0.1\phi_{d}} \tag{11}\]  
The rotation \(\theta_{Rcy}\) was then calculated by the moment \(M_{Rcy}\) to stiffness \(k_{p}\) ratio. The negative moment-rotation diagram (Figure 4b) was constructed by assuming the connection yield moment \(M_{Rcy}\) to be associated with yielded tying reinforcement \(A_{ST}\) as follows:  
\[M_{Rcy}=f_{y}A_{ST}(d_{T}-0.45X) \tag{12}\]  
The neutral axis depth \(X\) was defined through the following equation:  
\[X=\frac{f_{y}A_{sT}}{0.67f_{c}0.9b}=\frac{F_{T}}{0.67f_{c}0.9b} \tag{13}\]  
where \(b\) is the beam width and \(f_{c}\) is the concrete compressive strength. The term \(f_{y}A_{sT}\) can be replaced by the dowel strength \(V_{Rd}\) if no tying reinforcements are considered [52]. The rotation at yield of the connection \(\theta_{Rdy}\) was calculated by summing three contributions:  
1. The joint opening at the interface due to yield elongation of continuous reinforcing bars (or the achievement of the shear force of the dowel): \[\theta_{Rcy1}=\frac{f_{y}l_{e}}{E_{s}d_{T}}\] (14) \[\theta_{Rcy1}=\frac{V_{Rd}}{\lambda d_{d}}\] (15)
2. The beam end rotation related to its curvature in the vicinity of the joint: \[\theta_{Rcy2}=\frac{M_{Rcy}l_{p}}{E_{c}l_{beam}}\] (16) where \(l_{e}\) is assumed equal to the column width, the length \(l_{p}\) is equal to the sum of the corbel length \(l_{supp}\) and the total beam height \(h\)[52] (refer to Figure 4b) and \(l_{beam}\) is the second moment of inertia of beams. 3. The column rotation on top and bottom of the joint zone: \[M_{Rcy}=V_{Rd}z_{p} \tag{8}\]0 (17) where \(h_{col}\) is equal to the column height and \(l_{col}\) is the second moment of inertia of columns [52]. The last two contributions were assumed to be constant [52].