## Test, Numerical Model, and Validation of Reinforced Concrete (RC) Waffle-Flat-Plate (WFP) Structure
### Experimental Validation of the Numerical Model  
The numerical model was subjected to the same combination of loads as the specimen tested in laboratory using OpenSees software. Firstly, the gravity loads were applied in a static analysis; afterwards, non-linear time history analyses (NLTHAs) were carried out following the same sequence of seismic simulations used for the shake-table tests (from simulations T10 until C300). A mass-proportional damping model was defined and calibrated with experimental test data, in order to derive a classical damping matrix that prevents spurious damping [30; 31; 32; 33]. The damping matrix, \(\mathbf{C}\)--classical--is given by \(\mathbf{C}=a_{0}\mathbf{M}\), where \(a_{0}\) is a parameter and \(\mathbf{M}\) is the mass matrix. In turn, \(a_{0}\) is defined through the expression \(a_{0}=2\xi_{i}a_{i}\), where \(\xi_{i}\) and \(\omega_{i}\) are the damping ratio and the angular frequency--corresponding to the frequency, \(f_{i}=\omega_{i}\)/2\(\pi\)--of the \(i\)-th vibration mode. The reference  
Figure 3: Overview of the numerical model.  
Figure 2: Solid zone discretization in interior and exterior column-slab connections.  
damping ratio and frequency used to calculate \(a_{0}\) were 0.03 and 3.16 Hz, respectively; these values are close to the one obtained experimentally under elastic deformations, 0.024 [7]. This gives \(a_{0}=1.19\). When the numerical model collapsed, the frequency of the fundamental mode was 1.3 Hz; using \(a_{0}=1.19\) the corresponding damping ratio at failure is \(\xi_{1}=a_{0}/(2\omega_{1})=a_{0}/(2f_{1}2\pi)=0.072\), a value close to the one measured at the end of the tests, 0.092 [7].  
Next, the response of the numerical model was compared with the experimental results. Figure 4 shows the history of displacements at the top of the structure. The response in X and Y directions for the successive simulations--C35, C50, C100, C200i, C200, and C300--is depicted in the two upper graphs of the figure. In addition, the detail corresponding to the response in X and Y directions of C100 is shown at the bottom of the figure. A good agreement is observed between the numerical prediction and experimental response, especially for the lower and moderate intensity levels, i.e., until the onset of C200. However, for higher intensity levels--C200 and C300--the large strength degradation observed in the tests, especially at the ends of columns and at the edge of the column-slab connection, was not properly reproduced in the numerical model, which led to the observed differences.  
Figure 5 shows the history of total energy input in the specimen by the bidirectional seismic loading, \(E_{I}\). The numerical model is seen to predict the input energy until the onset of simulation C200i very well. From this point on, the numerical model underestimates the input energy measured experimentally only for C200i and part of C200. It is important to note that the response history in terms of input energy provides a more convenient and accurate criterion to assess the goodness-of-fit between model and test, preferable to displacement or force histories. Displacements or forces are vectors defined at specific parts of the structure; in contrast, energy is a scalar quantity that synthesizes the overall response of the entire structure [23]. Moreover, variations in the input energy history are better distinguished by using a logarithmic scale.  
Figure 4: Top displacement history: (**a**) for the successive simulations in X and Y directions; (**b**) for C100 in X direction; (**c**) for C100 in Y direction.