## 5 Finite Element Modeling
### Model Calibration  
The numerical models are developed based on a validated tie-wired rebar cage model presented in [7] with the addition of U-bolt connections. Through initial model simulations, it was observed that the model predictions are more flexible than the experimental results, as shown in Figure 13. Upon careful investigation of the possible sources of discrepancy, it was concluded that the torsional stiffness of the U-bolt connectors is not modeled accurately and is the possible source of discrepancies between the model-predicted and -measured deformations. To address this issue, a model-updating process is employed to estimate the  
Figure 12: Local coordinate system at a cross-bar intersection of a rebar cage. torsional stiffness of the U-bolt connectors using a limited set of experimental results (one of the test scenarios for three cages). Then, the results of the calibrated modeling process are cross-validated with other experiments. Further details are provided in this section. Based on the sensitivity analysis performed by [PERSON] et al. [1], the stiffness of connections in the \(Ry\) and \(Rz\) directions (see Figure 12) does not considerably impact the behavior of the rebar cage models. Instead, the stiffness in the \(Rx\) direction (i.e., the torsional stiffness of connection) has significant effects. To estimate the rotational stiffness of U-bolt connections in the \(Rx\) direction, i.e., \(K_{Rx}\), a simple optimization-based model-updating process is pursued. The objective function (also known as the error function) is defined to quantify the discrepancies between the measured and FE-predicted deformations, as shown below. \[e=\sqrt{\sum_{i=1}^{3}\frac{\sum_{i=1}^{n}\left(y_{exp,\ i}^{j}-y_{m,\ i}^{j} \right)^{2}}{\sum_{i=1}^{n}\left(y_{exp,\ i}^{j}\right)^{2}}} \tag{1}\]  
where \(n\) is the number of measurement points along the length of the cage, \(y_{exp,\ i}^{j}\) is the experimentally measured deflection of the \(j\)th cage at the \(i\)th measurement location, and \(y_{m,\ i}^{j}\) is the model-predicted deflection of the \(j\)th cage at the \(i\)th measurement location. The results of test scenario 3 for Cage #3, Cage #4, and Cage #5 (i.e., specimens with U-bolts and no internal stiffening elements) are used to estimate the rotation stiffness of the U-bolt connection. The experimental and numerical data are interpolated by utilizing a spline interpolation method to obtain three consistent deflection measurement points per meter, which results in a total of \(n=39\) measurement points along the length of each cage. The initial value of rotational stiffness is set to \(K_{Rx,0}=2400\) N\(\cdot\)m/rad. The optimization process is implemented using an iterative steepest descent method with the following correction equation to update the \(K_{Rx}\) value at each iteration [17]. \[K_{Rx,k+1}=K_{Rx,k}-\gamma_{k}\cdot\
abla e(K_{Rx,k}),\ k\geq 0 \tag{2}\]  
where \(\gamma\) is the step size, which is allowed to change at every iteration; \(\
abla e\) is the gradient of the objective function, \(e\); and \(k\) is the number of steps. The algorithm continues to iterate until the change in the objective function is smaller than the tolerance criterion (i.e., \(\left|\frac{\gamma_{k+1}-\epsilon_{k}}{\epsilon_{k}}\right|<0.05\)). The optimization process converges after eight iterations, and the value of the estimated rotational stiffness of the U-bolt connections is found to be \(K_{Rx}=5170\) N\(\cdot\)m/rad. Figure 13 compares the deflection of the cage during the C3-S3 test with the initial and updated model predictions. Using the updated value of \(K_{Rx}\), the experimental measurements of the cage displacement profiles are compared with the FE-predicted results for all test scenarios for Cage #1 to Cage #5 in Figures 14-18. It should be noted that the DIC measurement for Cage #1 was carried out using only seven points, resulting in roughly measured displacement profiles.