# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 4 Application to the Investigated Building  
The proposed framework is now implemented for the damping estimation of the investigated building using its earthquake measurements from the selected earthquake, as shown earlier in Figure 3. The finite element (FE) modeling of the building is first introduced to provide preliminary information for the following analysis. Subsequently, using the enhanced strain-energy method, the equivalent damping ratios and natural frequencies provided by oil dampers are estimated, and the modal parameters extracted from the earthquake measurements are modified accordingly. Finally, the improved Berman-Nagy method is utilized to update the initial FE model using the modified modal parameters. Moreover, the accuracy of these estimates is demonstrated through prediction validations. Figure 4 shows a high-fidelity FE model of the building, initially established in the SAP2000 modeling environment based on the available architectural and structural drawings. The structure was assumed to be fixed on the ground floor without considering any soil-structure interaction effect. All steel columns, beams, and damper braces were modeled by using frame elements, while concrete floor slabs were modeled using flexible shell elements [32]. Following design guidance, the connections between structural components were assumed to be fully constrained. The oil dampers were modeled by linear Maxwell models with stiffness and damping parameters identified from their earthquake measurements, as detailed later. To reduce the computational complexity of this FE model, simplifications are adopted by invoking common assumptions used for building structures in earthquake engineering applications [33; 34]. Note that the translational and torsional modes of this FE model are uncoupled due to the uniform distribution of mass and stiffness on floor plans, as illustrated in Section 2, which allows the analysis through planar models for each horizontal direction. This facilitates damping estimation and model updating of each direction separately, using the earthquake measurements for the corresponding motions of the building. Additionally,  
the building mass is assigned as lumped mass applied to beam-column joints at floor levels, while the column deformation in the vertical direction is not considered [34]. Therefore, rotational DoFs have zero mass, while vertical DoFs are neglected. The resulting planar FE models have 7 translational DoFs (with mass) and 279 (zero-mass) rotational DoFs for the beam-column joints. Static condensation [33] is further performed to remove the zero-mass rotational DoFs. The resulting nominal mass matrix is \(\mathbf{M}_{a}=diag\{446\ 348\ 337\ 334\ 328\ 324\ 311\}\)/ton for the two horizontal directions, where _diag_ stands for a diagonal matrix with elements included in the bracket\(\{\cdot\}\), while the initial stiffness matrices(unit: MN/mm) are  
\[\mathbf{K}_{a}=\left[ \begin{array}{cccccc}1.076&-1.081&0.290&-0.047&0.007&-0. 001&0.000\\ -1.081&1.895&-1.163&0.313&-0.052&0.008&-0.001\\ 0.290&-1.163&1.732&-1.101&0.296&-0.047&0.006\\ -0.047&0.313&-1.101&1.652&-1.055&0.271&-0.035\\ 0.007&-0.052&0.296&-1.055&1.565&-0.963&0.201\\ -0.001&0.008&-0.047&0.271&-0.963&1.357&-0.625\\ 0.000&-0.001&0.006&-0.035&0.201&-0.625&0.453\\ \end{array} \right] \tag{30}\]  
\[\mathbf{K}_{a}=\left[ \begin{array}{cccccc}1.084&-1.077&0.278&-0.043&0.006&-0. 001&0.000\\ -1.077&1.091&-1.160&0.293&-0.047&0.007&-0.000\\ 0.278&-1.160&1.756&-1.101&0.278&-0.044&0.005\\ -0.043&0.293&-1.101&1.679&-1.058&0.262&-0.033\\ 0.006&-0.047&0.278&-1.058&1.592&-0.965&0.193\\ -0.001&0.007&-0.044&0.262&-0.965&1.372&-0.631\\ 0.000&-0.001&0.005&-0.033&0.193&-0.631&0.466\\ \end{array} \right] \tag{31}\]  
The corresponding condensed models are referred to as the initial models, which are utilized in the following analysis.