Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers

Abstract

The damping estimation of an eight-story steel building equipped with oil dampers is examined, carried out by adopting a proposed framework, which consists of an enhanced strain-energy method and an improved direct method for model updating. The building is located at Tohoku Institute of Technology and is equipped with a structural monitoring system that measures its seismic response, including floor acceleration and displacement and force of oil dampers. The enhanced strain-energy method is first developed and employed to assess the supplemental damping and stiffness provided by oil dampers, herein quantified in the form of equivalent damping ratios and natural frequencies. Then, modal characteristics extracted from the earthquake measurements are modified accordingly and utilized for the building model updating, in which mass and stiffness matrices are corrected by the improved direct method. The updated model accurately reproduces the target modal data, especially measured mode participation factors, and is further used for the building response predictions. Through prediction validations, the precision of the modified modal parameters is verified. Finally, a large earthquake event is chosen to demonstrate the effectiveness of the proposed framework for the damping estimation of the investigated building.

1. Introduction

In the past few decades, many damping devices have been developed and applied in civil engineering structures, especially in earthquake-prone regions [1,2,3]. Viscous dampers generally represent a broad class of passive energy dissipation devices and have emerged as one of the most popular ones. By providing additional damping and energy dissipation, they can efficiently suppress earthquake-induced vibrations and, therefore, limit the damage to structural and non-structural components [4,5,6]. To better understand the damping properties of such dampers, many experimental studies have been conducted, including damper element tests [7,8] and shaking table tests of small-scale and full-scale structures with viscous dampers [4,5]. However, studies focusing on in-service structures with such dampers when subjected to large earthquakes are relatively rare [9,10]. This practical issue requires accurate state quantification for dampers and structure itself, which holds a key position in structural health, safety, and risk assessments [11,12]. This paper presents a study on the damping estimation of an eight-story building structure equipped with oil dampers through a proposed framework, which consists of an enhanced strain-energy method and an improved direct method for model updating.

The strain-energy method is usually adopted at the design stage of passively controlled structures to assess the damping effect of added dampers quantitatively. Specifically, it incorporates the supplemental damping provided by dampers into modal damping ratios, which are frequently referred to as equivalent damping ratios and are defined as the ratios of energy dissipated by dampers and stored in structure system. Zhang and Soong [13] provided a procedure to quantify the damping effect of viscoelastic dampers and showed their effectiveness in attenuating structural seismic response. Chang et al. [14] carried out a comprehensive experimental program on a 2/5-scale five-story steel frame with viscoelastic dampers and applied this procedure to assess the equivalent damping ratios for measured modes. Ji et al. [15] conducted a series of shaking table tests on a full-scale five-story steel building with three types of passive dampers, including viscous dampers and oil dampers. Using the strain-energy method and system identification, they investigated the damping characteristics of those dampers and presented simplified estimations for their supplemental damping ratios.

In this study, the strain-energy method is adopted to estimate the supplemental damping ratios provided by oil dampers for the investigated building under earthquake excitations. Additionally, it is notable that, except for the supplement damping, dampers can contribute to the increased lateral stiffness of the entire structure [14,15,16,17]. This stiffness contribution, usually not large, may affect structural dynamic behavior, such as increasing natural frequencies. For evaluating such modal changes, an enhanced strain-energy method is proposed herein, quantifying the damper stiffness in the form of equivalent natural frequencies, i.e., the natural frequency changes of structures with and without dampers. Besides, note that the strain-energy method adopts assumptions of harmonic motion and mode-proportional deformation for dampers and structure itself, which violate their realistic situation during an earthquake event [15]. Consequently, the estimated equivalent damping ratios and natural frequencies may be questionable and, therefore, need to be verified under earthquake situations. In this study, it is achieved through model updating and prediction validations.

Model updating using dynamic measurements has been a topic of extensive focus for many engineering principles, including civil engineering [18]. In structural dynamics community, it is well-understood that the predictions from an initial structural model are frequently inconsistent with those observed from its real-world counterpart [19]. These predictions could be modal properties, time histories, or frequency response functions. The objective of model updating is to reduce such discrepancy, improving the correlation between numerical models and their real-world counterparts. Generally, model updating belongs to the family of inverse problems, and, in most practical applications, it can be approached from an optimization perspective. In recent decades, a significant number of model updating methods have been developed. Literature review for this topic is not presented here since is beyond the scope of this study. The interested reader is referred to references [19,20,21]. This paper is concerned only with the so-called direct updating methods because of their particular features, the resulting updated model reproduces exactly the target/measured modal data, including natural frequencies and mode shapes. By taking advantage of such features, the damper stiffness effect on structural dynamic behavior can be easily captured in model updating.

The direct updating methods are essentially analytical methods which do not require iterations. In those methods, modal orthogonality, symmetry, and eigenvalue equation are usually imposed in correcting mass matrix or stiffness matrix or both, and minimum possible changes in updated matrices are achieved by using Lagrange multipliers or generalized inverse. Baruch and Bar-Itzack [22] first formulated a procedure using Lagrange multipliers to correct stiffness matrix to satisfy modal orthogonality and the eigenvalue equation with nominal mass matrix. Berman [23] questioned the assumption that nominal mass matrix is correct and developed a new procedure to modify mass matrix under modal orthogonality constraint. Later, Berman and Nagy [24] integrated these two procedures into the so-called analytical model improvement technique to update a long-exposure facility structure. Based on this research, Caesar and Peter [25] further described two direct methods for model updating using modal data. Wei [26] considered the interaction between mass and stiffness matrices in their updating processes and proposed a united procedure to modify them simultaneously using the vibration test data. Yue [27] developed an algorithm turning model updating to a forward problem, in which a transformation matrix is only required to update mass and stiffness matrices. Lee and Eun [28] employed the Moore–Penrose inverse, rather than Lagrange multipliers, to reformulate the model updating problem. But the resulting corrected stiffness matrices lose symmetricity. Yang and Chen [29] established a simple procedure to maintain the physical meanings in mass and stiffness matrices corrections, which allows the updated model to be compatible with the measured natural frequencies. All of these methods discussed above are computationally efficient and can be easily applied to large structural models [24,27,29]. However, for applications in earthquake engineering, a problem arises that the predicted mode participation factors by updated models may not be consistent with those identified from earthquake measurements. To address this problem, an improved Berman–Nagy method is developed, using a new procedure to correct nominal mass matrix under the constraints of both modal orthogonality and mode participation factors.

A large earthquake event is chosen to demonstrate the effectiveness of the proposed framework for the damping estimation of the investigated building. The enhanced strain-energy method is first employed to calculate the equivalent damping ratios and natural frequencies provided by oil dampers, which are modeled by a linear Maxwell model consisting of a spring element connected in series with a dashpot element [7,15]. Then, the modal characteristics extracted from earthquake measurements are modified accordingly. The modified parameters are ultimately the data utilized to update an initial finite element model of the building by using the improved Berman–Nagy method. Finally, through prediction validations, the precision of the modified parameters as well as the estimated equivalent damping ratios and natural frequencies are verified. The remaining of this paper is organized as follows: in Section 2, the investigated building and its structural monitoring system are introduced. Section 3 presents the details of the enhanced strain-energy method and the improved Berman–Nagy method. The results of damping estimation for the building during the selected earthquake are illustrated in Section 4. Section 5 provides the conclusions of this study.

2. Building Description and Its Structural Monitoring System

2.1. Overview of the Building

The investigated structure, as shown in Figure 1, is the administration building of Tohoku Institute of Technology, located in Sendai, Japan. It was constructed in 2003 and is steel framed with precast concrete slabs. The building has a rectangular plan of dimensions 48 by 9.6 m, while the total height is 34.2 m, with eight stories above the ground. The story height is 3.8 m, except for the first floor, which has a height of 8 m since the first two stories merge for a large public place.

The building was designed according to the Japanese Building Standard Law without considering the incorporation of damping devices. For enhanced seismic protection, a total of 56 oil dampers were additionally installed in the building. Each floor has 8 oil dampers equally distributed in the two orthogonal directions, as shown in Figure 1b. Two types of such oil dampers are fixed on floors and are connected with the adjacent upper floors through V-type steel braces. These dampers have different orifice specifications and stroke limits for the 1st floor (Type I) and 3rd to 8th floor (Type II), as described in Figure 2 using notation 1F and 3–8F.

The oil damper, designed by Kawamata et al. [30], consists of a central cylinder and a pair of pistons. The gaps between them are sealed by using viscoelastic polymers. The pistons are fixed on a U-type abutment on floors, and the central cylinder is attached to a V-type brace moving between the two pistons. Figure 2 shows the detailed layout and dimensions of the installed oil dampers. The relative movements between the cylinder and the pistons create the damper resisting force, which consists of two parts: the pressure resistance coming from the flowing infilled oil through a small orifice and the viscoelastic resisting force caused by the shearing deformations of the sealing materials.

Forced vibration tests were conducted after completing the building frame in June 2003, to investigate the dynamic characteristics of the structure with and without oil dampers. The identification results from vibration tests showed an apparent increase in natural frequencies, the fundamental frequencies increased from 1.05 to 1.12 Hz along the transversal direction and 1.02 to 1.10 Hz along the longitudinal direction. The increased natural frequencies stem from the additional stiffness provided by oil dampers, which motives this study to investigate such stiffness effect during earthquakes.

2.2. Structural Monitoring System

A structural monitoring system is installed in the building to monitor its seismic motions. The monitoring system consists of three two-directional acceleration sensors for measuring floor acceleration and four pairs of load cell and displacement transducer for recording the force and displacement responses of oil dampers. These accelerometers are placed near the central locations of the 1st, 4th, and 8th floor plans, as shown in Figure 1b. They are utilized to measure the floor translational motions along with the two horizontal directions. Four oil dampers, as representatives of the two-type dampers, were selected to be monitored. Specifically, dampers installed on the 1st and 8th floors were chosen, with each floor having two monitored oil dampers along with its two principal directions.

Figure 3 shows a typical set of measured seismic responses, including floor acceleration and force and displacement of oil dampers in the building translational direction. This data set was recorded during a large earthquake with a moment magnitude of 7.0, which happened on 26 May 2003. The force-displacement hysteretic curve of the oil dampers shows their damping and stiffness properties during this earthquake. The earthquake measurements obtained from this earthquake are utilized in this study to demonstrate the damping estimation process for the building.

3. A Framework for Damping Estimation of Passively Controlled Structures

This section proposes a general framework for the damping estimation of passively controlled structures. It consists of two newly developed methods, namely, an enhanced strain-energy method and an improved Berman–Nagy method. The former method aims to quantify the additional damping and stiffness provided by dampers in the forms of equivalent damping ratios and natural frequencies. In particular, simple estimation formulas are derived for the case added dampers modeled by linear Maxwell models. The latter method is devoted to correcting the nominal mass and stiffness matrices of passively controlled structures themself using measured modal data, including natural frequencies, mode shapes, as well as mode participation factors.

3.1. An Enhanced Strain-Energy Method

3.1.1. Estimation of Equivalent Damping Ratios

The traditional strain-energy method assumes passively controlled structure vibrates in harmonic motion with a frequency consistent with one of their natural frequencies and deflects proportionally to the corresponding mode shape. Based on these assumptions, the equivalent damping ratio provided by added dampers can be estimated by

$${\xi }_{eq,i}=\frac{1}{4\pi }\frac{E_{D,i}}{E_{S,i}}$$

(1)

where ξ_eq,i is the equivalent damping ratio for the ith mode, E_D,i is the energy dissipated by dampers in one cycle of motion, E_S,i is the corresponding maximum strain energy stored in the entire structure system, which is estimated by

$$E_{S,i}=\frac{1}{2}{\Delta }_i^2{\mathsf{\varphi }}_i^{T}K{\mathsf{\varphi }}_i=\frac{1}{2}{\Delta }_i^2{\omega }_i^2$$

(2)

where K is the stiffness matrix of the entire structure, ω_i and ϕ_i are, respectively, the ith circular frequency and mass-normalized mode shape, Δ_iϕ_i denotes the maximum structural deformation vector in one cycle of motion, Δ_i is an unknown factor determining the ith mode vibration amplitude.

Formulation of E_D,i depends on damper types. For the case added dampers modeled by linear Maxwell models, E_D,i is given by [15]

$$E_{D,i}=\pi {\displaystyle \sum _j\frac{{\omega }_i}{1+{\tau }_{d,j}^2{\omega }_i^2}}{c}_{d,j}{\Delta }_i^2{\varphi }_{i,dj}^2$$

(3)

where τ_d,j = c_d,j/k_d,j is the relax time for the jth damper, c_d,j and k_d,j are the damping and stiffness coefficients, respectively. ϕ_i,dj is the axial deformation of the jth damper in the ith mass-normalized mode shape.

Substitute Equations (2) and (3) into Equation (1) yields the ith equivalent damping ratio. For simplifying this formula, a coordinate transformation, Δ_iϕ_i,d = T_DΔ_iϕ_i, is introduced between damper deformation Δ_iϕ_i,d (local damper coordinate system) and structure deformation Δ_iϕ_i (global coordinate system). T_D is a predefined transformation matrix determined by damper configure details. By substituting such transformation into Equation (3), we can get

$$E_{D,i}=\pi {\omega }_i{\Delta }_i^2{\mathsf{\varphi }}_i^{\mathrm{T}}{\tilde{C}}_{D,i}{\mathsf{\varphi }}_i$$

(4)

where the matrix ${\tilde{C}}_{D,i}$ is given by

$${\tilde{C}}_{D,i}=T_D^{\mathrm{T}}diag\left\{\frac{c_{d,j}}{1+{\tau }_{d,j}^2{\omega }_i^2}\right\}{T}_D$$

(5)

It can be regarded as the equivalent damping matrix for the ith mode provided by added dampers. The equivalent damping ratio is ultimately expressed by

$${\xi }_{eq,i}=\frac{{\mathsf{\varphi }}_i^T{\tilde{C}}_{D,i}{\mathsf{\varphi }}_i}{2{\omega }_i}$$

(6)

Notably, such an expression follows the classical definition of modal damping ratios and assign strong physically meanings to the calculations of equivalent damping ratios.

3.1.2. Estimation of Equivalent Natural Frequencies

An energy approach is presented here to quantify damper stiffness by equivalent natural frequencies. It assumes that the added dampers of a structure do not affect its mode shapes, which can be justified by the fact that damper stiffness is usually minor compared with structure stiffness [15,31]. The basic idea of this approach is based on Equation (2), which establishes a relationship between strain energy and natural frequency, and the critical point in this approach is to determine the distribution of strain energy stored in the entire structure. Specifically, for the ith mode, the strain energy stored in structure itself may be expressed by

$$E_{S,i}^s=\left(1-\frac{E_{D,i}^k}{E_{S,i}}\right)E_{S,i}=\left(1-{\eta }_{D,i}\right)E_{S,i}$$

(7)

where η_D,and is a strain-energy ratio, and $E_{D,i}^k$ is the maximum strain energy stored in dampers in one cycle of motion. It has a closed-form solution for oil dampers modeled by linear Maxwell models as

$$E_{D,i}^k=\frac{1}{2}{\displaystyle \sum _j\frac{{\tau }_{d,j}^2{\omega }_i^2}{1+{\tau }_{d,j}^2{\omega }_i^2}}{k}_{d,j}{\Delta }_i^2{\varphi }_{i,dj}^2=\frac{1}{2}{\Delta }_i^2{\mathsf{\varphi }}_i^T{\tilde{K}}_{D,i}{\mathsf{\varphi }}_i$$

(8)

where the matrix ${\tilde{K}}_{D,i}$ is given by

$${\tilde{K}}_{D,i}=T_D^{T}diag\left\{\frac{{\tau }_{d,j}^2{\omega }_i^2}{1+{\tau }_{d,j}^2{\omega }_i^2}{k}_{d,j}\right\}{T}_D$$

(9)

It can be regarded as to the equivalent damping matrix for the ith mode provided by added dampers. Combining Equations (2) and (8) yields the ith strain-energy ratio as

$${\eta }_{D,i}=\frac{{\mathsf{\varphi }}_i^T{\tilde{K}}_{Dk,i}{\mathsf{\varphi }}_i}{{\omega }_i^2}$$

(10)

Then, substitute Equation (2) into Equation (7), we can get

$${\omega }_{s,i}=\sqrt{1-{\eta }_{D,i}}{\omega }_i$$

(11)

where ω_s,i is the ith natural frequency of structure itself.

Finally, the equivalent natural frequency provided by dampers is given by

$${\omega }_{eq,i}={\omega }_i-\sqrt{\left(1-{\eta }_{D,i}\right)}{\omega }_i$$

(12)

It should be noted that Equations (6) and (12) require modal characteristics of passively controlled structures, including natural frequencies and complete mode shapes. However, in practice, civil engineering structures are often sparsely instrumented. As a result, their measured mode shapes are incomplete and correspond to a few monitored DOFs only. In this study, the mode shape expansion technique [19,24,27] is adopted to obtain the complete mode shapes for the investigated building, as detailed in Section 4.

3.2. An Improved Berman–Nagy Method

An improved direct updating method, called improved Berman–Nagy method, is presented here to correct the initial models of passively controlled structures themself. First, the traditional Berman–Nagy method [24] is utilized to demonstrate the incomplete modal matching issue, i.e., the predicted mode participation factors by updated models might not be consistent with those measured from real structures. Subsequently, a new procedure is proposed for mass matrix updating, in which the constraint of mode participation factors is considered.

3.2.1. Problem Description

Let M_a and K_a represent, respectively, the nominal mass and stiffness matrices of a structure with n degrees of freedom. As discussed in the introduction part, its analytical modal characteristics frequently deviate from those measured from the real structure, with mass and stiffness matrices denoted as M and K. In the traditional Berman–Nagy method, the updated mass matrix, denoted as M_B, and stiffness matrix, denoted as K_B, are

$$M_B=M_a+M_a\Phi {\left({\Phi }^{\mathrm{T}}{M}_a\Phi \right)}^{-1}\left(I-{\Phi }^{\mathrm{T}}{M}_a\Phi \right){\left({\Phi }^{\mathrm{T}}{M}_a\Phi \right)}^{-1}{\Phi }^{\mathrm{T}}{M}_a$$

(13)

$$K_B=K_a+M_B\Phi \left({\Phi }^{\mathrm{T}}{K}_a\Phi +{\Omega }^2\right){\Phi }^{\mathrm{T}}{M}_B-K_a\Phi {\Phi }^{\mathrm{T}}{M}_B-M_B\Phi {\Phi }^{\mathrm{T}}{K}_a$$

(14)

where Ω is the m × m diagonal matrix of measured natural frequencies, m is the number of measured modes, Φ is the n × m matrix of measured mode shapes, and I is the m × m identity matrix. Notably, such updated mass and stiffness matrices accurately replicate the measured natural frequencies and mode shapes, while they also generate predictions for mode participation factors as

$$P_B={\Phi }^{\mathrm{T}}{M}_B{I}_n={\left({\Phi }^{\mathrm{T}}{M}_a\Phi \right)}^{-1}{\Phi }^{\mathrm{T}}{M}_a{I}_n$$

(15)

where P_B denotes the vector of predicted mode participation factors, I_n is the identity vector of order n. As discussed earlier, these byproduct predictions may differ from those measured from real structures. For applications in earthquake engineering, mode participation factors participate in determining mode response amplitudes, therefore, any error in their predictions will lead to flaws in updated structural models. To address this problem, a new procedure for mass matrix updating is developed as below.

3.2.2. A New Procedure for Mass Matrix Updating

This new procedure considers the constraints of symmetry, modal orthogonality, as well as mode participation factors in mass matrix updating. Its formulation process follows the guidance in reference [23]. Specifically, the cost function to be minimized is defined as

$$\epsilon =\Vert M_a^{-1/2}(M-M_a)M_a^{-1/2}\Vert$$

(16)

subjected to constraints as follows:

$$M=M^{\mathrm{T}}$$

(17)

$${\Phi }^{\mathrm{T}}M\Phi =I$$

(18)

$${\Phi }^{\mathrm{T}}M{I}_n=P$$

(19)

where P is the vector of measured mode participation factors. Using the Lagrange multiplier method, the Lagrangian function may be formulated as

$$J=\epsilon +{\displaystyle \sum _{ij}{\Psi }_{ij}{\left(M-M^{\mathrm{T}}\right)}_{ij}}+{\displaystyle \sum _{ij}{\lambda }_{ij}{\left({\Phi }^{\mathrm{T}}M\Phi -I\right)}_{ij}}+{\displaystyle \sum _i{\gamma }_i{\left({\Phi }^{\mathrm{T}}M{I}_n-P\right)}_i}$$

(20)

where Ψ_ij, λ_ij, and γ_i are the undetermined multiplier coefficients, corresponding to the constraint equations above, and their matrix forms are denoted as Ψ, Λ, and Γ, respectively. Differentiating Equation (20) with respect to each element in M and setting these results equal to zero, it leads to

$$M=M_a-\frac{1}{2}{M}_a\left(\Phi \Lambda {\Phi }^{\mathrm{T}}+\Phi \Gamma I_n^{\mathrm{T}}+\Psi -{\Psi }^{\mathrm{T}}\right)M_a$$

(21)

Substituting Equation (21) into Equation (17) and arranging the result, the solution for Ψ − Ψ^T is

$$\Psi -{\Psi }^{\mathrm{T}}=\frac{1}{2}\left(\Phi {\Lambda }^{\mathrm{T}}{\Phi }^{\mathrm{T}}+I_n{\Gamma }^{\mathrm{T}}{\Phi }^{\mathrm{T}}-\Phi \Lambda {\Phi }^{\mathrm{T}}-\Phi \Gamma {\mathrm{{\rm I}}}_n^{\mathrm{T}}\right)$$

(22)

Inserting Equation (22) into Equation (21) to update mass matrix, it follows that

$$M=M_a-\frac{1}{4}{M}_a\Phi \left(\Lambda +{\Lambda }^{\mathrm{T}}\right){\Phi }^{\mathrm{T}}{M}_a-\frac{1}{4}{M}_a\left(\Phi \Gamma {{\rm I}}_n^{\mathrm{T}}+I_n{\Gamma }^{\mathrm{T}}{\Phi }^{\mathrm{T}}\right)M_a$$

(23)

Substitute Equation (23) into Equation (18), we can get

$$\Lambda +{\Lambda }^{\mathrm{T}}=-4m_a^{-1}\left(I-m_a\right)m_a^{-1}-\Gamma P_a^{\mathrm{T}}{m}_a^{-1}-m_a^{-1}{P}_a{\Gamma }^{\mathrm{T}}$$

(24)

where $m_a={\Phi }^{\mathrm{T}}{M}_a\Phi$, $P_{\mathrm{a}}={\Phi }^{\mathrm{T}}{M}_{\mathrm{a}}{I}_n$. Eliminating the item Λ + Λ^T in Equation (23), mass matrix is rewritten as

$$M=M_B+\frac{1}{4}{M}_a\Phi \Gamma \left(P_a^{\mathrm{T}}{m}_a^{-1}{\Phi }^{\mathrm{T}}-I_n^{\mathrm{T}}\right)M_a+\frac{1}{4}{M}_a\left(\Phi m_a^{-1}{P}_a+I_n\right){\Gamma }^{\mathrm{T}}{\Phi }^{\mathrm{T}}{M}_a$$

(25)

Substitute Equation (25) into the last constraint, Equation (19), the solution for Γ is

$$\Gamma =\frac{4}{c}{m}_a^{-1}\left(P-P_B\right)$$

(26)

where $c=\left(P_a^{\mathrm{T}}{m}_a^{-1}{\Phi }^{\mathrm{T}}-I_n^{\mathrm{T}}\right)M_a{I}_n$ is a constant.

Finally, using Equation (26) to eliminate the matrix Γ in Equation (25), the new, updated mass matrix is given by

$$M=M_B+\Delta M+\Delta M^{\mathrm{T}}$$

(27)

where the matrix ΔM is

$$\Delta M=\frac{1}{c}{M}_a\Phi m_a^{-1}\left(P-P_B\right)\left(P_a^{\mathrm{T}}{m}_a^{-1}{\Phi }^{\mathrm{T}}-I_n^{\mathrm{T}}\right)M_a$$

(28)

In addition, Using M to replace M_B in Equation (14), the new, updated stiffness matrix is

$$K=K_a+M\Phi \left({\Phi }^{\mathrm{T}}{K}_a\Phi +{\Omega }^2\right){\Phi }^{\mathrm{T}}M-K_a\Phi {\Phi }^{\mathrm{T}}M-M\Phi {\Phi }^{\mathrm{T}}{K}_a$$

(29)

Equations (27)–(29) form the improved Berman–Nagy method for mass and stiffness matrices updating. Notably, comparing Equations (13) and (27), ΔM is the additional matrix rendering the new, updated mass matrix consistent with the measured mode participation factors. However, if P_B equal to P, ΔM will equal a zero matrix. In this instance, the improved Berman–Nagy method turns to the traditional Berman–Nagy method.

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 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{·}, while the initial stiffness matrices(unit: MN/mm) are

$$K_a=\left[\begin{array}{ccccccc}1.076\hfill & \hfill -1.081& 0.290\hfill & \hfill -0.047& 0.007\hfill & \hfill -0.001& 0.000\hfill \\ -1.081\hfill & \hfill 1.895& -1.163\hfill & \hfill 0.313& -0.052\hfill & \hfill 0.008& -0.001\hfill \\ 0.290\hfill & \hfill -1.163& 1.732\hfill & \hfill -1.101& 0.296\hfill & \hfill -0.047& 0.006\hfill \\ -0.047\hfill & \hfill 0.313& -1.101\hfill & \hfill 1.652& -1.055\hfill & \hfill 0.271& -0.035\hfill \\ 0.007\hfill & \hfill -0.052& 0.296\hfill & \hfill -1.055& 1.565\hfill & \hfill -0.963& 0.201\hfill \\ -0.001\hfill & \hfill 0.008& -0.047\hfill & \hfill 0.271& -0.963\hfill & \hfill 1.357& -0.625\hfill \\ 0.000\hfill & \hfill -0.001& 0.006\hfill & \hfill -0.035& 0.201\hfill & \hfill -0.625& 0.453\hfill \end{array}\right]\mathrm{for}\mathrm{longitudinal}\mathrm{direction}$$

(30)

$$K_a=\left[\begin{array}{ccccccc}1.084\hfill & \hfill -1.077& 0.278\hfill & \hfill -0.043& 0.006\hfill & \hfill -0.001& 0.000\hfill \\ -1.077\hfill & \hfill 1.091& -1.160\hfill & \hfill 0.293& -0.047\hfill & \hfill 0.007& -0.000\hfill \\ 0.278\hfill & \hfill -1.160& 1.756\hfill & \hfill -1.101& 0.278\hfill & \hfill -0.044& 0.005\hfill \\ -0.043\hfill & \hfill 0.293& -1.101\hfill & \hfill 1.679& -1.058\hfill & \hfill 0.262& -0.033\hfill \\ 0.006\hfill & \hfill -0.047& 0.278\hfill & \hfill -1.058& 1.592\hfill & \hfill -0.965& 0.193\hfill \\ -0.001\hfill & \hfill 0.007& -0.044\hfill & \hfill 0.262& -0.965\hfill & \hfill 1.372& -0.631\hfill \\ 0.000\hfill & \hfill -0.001& 0.005\hfill & \hfill -0.033& 0.193\hfill & \hfill -0.631& 0.466\hfill \end{array}\right]\mathrm{for}\mathrm{transversal}\mathrm{direction}$$

(31)

The corresponding condensed models are referred to as the initial models, which are utilized in the following analysis. The equation of motion of the building in each of the two translational directions may be expressed by

$$M\ddot{x}+C\dot{x}+Kx+T_D^{\mathrm{T}}{f}_D\left(x,\dot{x},c_d,k_d\right)=MI{\ddot{x}}_g$$

(32)

where M, K, and C are, respectively, the real mass, stiffness, and inherent damping matrices, x represents the vector of floor displacement relative to the structure base, ${\ddot{x}}_g$ is the acceleration input, corresponding to the acceleration of the ground floor of the building, I is the vector of earthquake influence coefficients, f_D(·) denotes the vector of damper forces, T_D is the transformation matrix, relating movement across the ends of each damper to x, and c_d, and k_d are the vectors of damping coefficients and stiffness parameters of linear Maxwell models, respectively, composed of the respective characteristics of the oil dampers for each floor. The damper force, for example, for the ith floor damper is expressed as

$$\frac{{\dot{f}}_{d,i}}{k_{d,i}}+\frac{f_{d,i}}{c_{d,i}}={\dot{x}}_{d,i}$$

(33)

where f_d,i and x_d,i are, respectively, the damper force and the damper deformation between its two ends. For the investigated building, the supporting brace stiffness is estimated based on their design information and is much larger than the identified damper stiffness. Since these two components are connected in series, the damper deformation is taken as the corresponding inter-story drift. Besides, note that a representation of the form of Equation (32) holds separately for each of the horizontal directions, though for notational simplicity, this distinction is not explicitly noted herein.

4.1. Damping Estimation of the Building

Implementations of the enhanced strain-energy method require the building modal data and the oil damper parameters. These characteristics are identified by adopting a standard time-domain identification technique [7,35] and using the selected earthquake measurements. The identification is separately performed in each horizontal direction, which facilitates a better consideration of the impact of the excitation/response amplitude for oil dampers. Precisely, for oil damper identification, the input-output earthquake data corresponds to the input displacement and output force data, with the governing equation given by Equation (33), while, for modal identification, the data is the acceleration of the ground floor (input) and two monitored floors (output), with modal-superposition models as governing equation [35].

Table 1. The identified model parameters for monitored oil dampers.

Parameters	Longitudinal Direction		Transversal Direction
	Type I	Type II	Type I	Type II
cd (kN/mm·s−1)	0.696	0.685	0.876	0.516
kd (kN/mm)	4.258	2.562	3.395	3.295

lists the identification results for the four monitored oil dampers, in which I and II represent the damper types. Recall that Type I oil dampers were installed on the ground floor, while Type II oil dampers were placed on other upper floors. It is evident from the results that a discrepancy exists between the identified parameters of same-type dampers, verifying their anticipated amplitude-dependent characteristics.

Table 2. The identified modal parameters for the building.

Parameters	Longitudinal Direction		Transversal Direction
	1st Mode	2nd Mode	1st Mode	2nd Mode
ωi (rad/s)	5.608	18.132	5.062	16.552
ξi	0.031	0.061	0.036	0.053
ϕi,2	0.014	−0.026	0.013	−0.026
ϕi,6	0.025	0.016	0.026	0.016
pi	47.421	−15.392	48.952	−15.324

shows the identified (translational) modal parameters of the first two modes of the building. The remaining higher modes are of small importance, and identification of their properties suffers to a much larger degree by measurement noise [35]. Notably, the identified mode shapes and mode participation factors in Table 2 are determined by using the so-called mode scale factor, which is commonly adopted to (mass) normalize measured mode shapes. Furthermore, to obtain complete mode shapes, an expansion technique suggested in Reference [27] is adopted, though note that other alternatives exist for resolving this issue [19]. The implementations of this expansion approach are omitted here since it is a common practice. For clarity of explanation, the expanded mode shape is denoted hereinafter as ϕ_i, for the ith mode of the building.

Table 3. The estimated equivalent natural frequencies and equivalent damping ratios provided by oil dampers. The corresponding parameters of the building itself also shown.

Parameters	Longitudinal Direction		Transversal Direction
	1st Mode	2nd Mode	1st Mode	2nd Mode
ωs,i (rad/s)	5.495	17.721	4.951	16.108
ωeq,i (rad/s)	0.113	0.411	0.111	0.444
ξs,i	0.012	0.055	0.017	0.045
ξeq,i	0.019	0.006	0.019	0.008

shows the estimated equivalent damping ratios (ξ_eq,i) and natural frequencies (ω_eq,i) by the enhanced strain-energy method. Results of the inherent damping ratios (ξ_s,i) and natural frequencies (ω_s,i) for the building itself are also reported in this table. They are obtained by subtracting those equivalent estimates from the corresponding identified modal parameters in Table 2. It can be seen from Table 3 that the estimated equivalent damping ratios have a sharp decrease when mode order increases and the second equivalent damping ratios are negligible. It indicates that the energy dissipated by oil dampers concentrates in the seismic motion of the fundamental modes. On the other side, for equivalent natural frequencies, an increasing trend is observed. Overall, they are relatively small compared with the identified natural frequencies of the building. However, it does not indicate that the estimated equivalent natural frequencies are unimportant. Instead, they can significantly affect the prediction accuracy of updated models, as will be shown in the next section.

Besides, the enhanced strain-energy method also allows quantifying the contribution from a specific damper to the equivalent damping ratios and natural frequencies, as evident from Equations (3) and (8). Figure 5 plots the damping and stiffness contribution in percentages from the oil dampers of different floors along with the two horizontal directions. The dampers on the first floor dominate the contribution, up to 70% for damping and up to 50% for stiffness, which is attributed to its larger story drift. Because the damping and stiffness contribution from oil dampers is proportional to the square values of ϕ_i,dj, as indicated in Equations (3) and (8). Higher floor dampers contribute more to the damping of the second mode, but the total damping contribution in that mode is negligible, as shown in Table 3.

It is important to stress once more that the enhanced strain-energy method is built based on the assumptions of harmonic motion and mode-proportional deformation of structure and added dampers, which violate their real situations during earthquakes. Therefore, it is necessary to verify the method accuracy, which is achieved through model updating and prediction validations.

4.2. Model Updating of the Building

The model updating of the building itself is separately performed for its two horizontal directions. Specifically, for each updating case, the natural frequencies (ω_s,i), expanded mode shapes (ϕ_i), and mode participation factors (p_i) of the first two modes are regrouped and utilized to correct mass and stiffness matrices by the improved Berman–Nagy method, i.e., Equation (27) for mass matrix updating and Equation (29) for stiffness matrix updating. Besides, for inherent structural damping, the damping matrix is modeled through a common modal damping assumption and is adjusted to match the estimated damping ratios (ξ_s,i), as listed in Table 3. Damping ratios for the remaining higher modes are generally large and are constrained to a predefined value (taken in the case study later as 10%). Note that using a large damping ratio for these modes, though not critical, can efficiently suppress higher-frequency components in the building response predictions.

First, mass and stiffness matrices are updated to demonstrate the effectiveness of the improved Berman–Nagy method. Two updating cases of using different modal data are considered to illustrate the impact of additional matrix ΔM on updated mass matrix. Specifically, the regrouped modal parameters of the first mode and the first two modes are separately utilized to update the nominal mass matrix M_a. Figure 6 presents the new, updated mass matrix M by Equation (27). For comparisons, the updated mass matrix M_B by the traditional Berman–Nagy method is also reported. The results in this figure correspond to the transversal direction FE model.

It is evident from the results in Figure 6 that ΔM, the difference between M and M_B, has an increasing impact on the updated mass matrix when more modal data is utilized. In the first updating case, the predicted mode participation factor by M exactly matches the measured one (48.952), while the prediction by M_B is 46.884. This prediction error is large but causes small changes in the two updated mass matrices, as shown in Figure 6a. When the regrouped modal data of the first two modes is utilized, M and M_B differ significantly in both diagonal and off-diagonal elements. The predicted mode participation factors are listed in

Table 4. Comparisons of the predicted mode participation factors by updated mass matrices.

Parameters	Longitudinal Direction		Transversal Direction
	1st Mode	2nd Mode	1st Mode	2nd Mode
pB,i	45.8910	−12.0991	45.9156	−14.0551
pI,i	47.4219	−15.3921	48.9527	−15.3241

, where the improved method results are denoted as p_I,i. Again, M reproduces exactly the measured mode participation factors, while M_B fails in the first two modes cases. Figure 7 shows the new, updated stiffness matrices by the improved method and using the regrouped modal data of the first two modes. Unlike updated mass matrices, changes in updated stiffness matrices are small. Notably, the updated mass and stiffness matrices by the improved method exactly possess the modal data, including natural frequencies, mode shapes, and mode participation factors.

Then, the precision of the estimated inherent damping ratios (ξ_s,i) and natural frequencies (ω_s,i) is considered. The updated models with new, updated mass and stiffness matrices are utilized for the building response prediction, according to Equation (34). The linear Maxwell models with identified parameters shown in Table 1 are assigned to the same-type oil dampers, i.e., the parameter configures of Type I for the ground-floor oil dampers and those of Type II for the other floors oil dampers. Two updated models are utilized for this response prediction: In the first updated model, the damper stiffness is not considered, the modal data utilized for correcting mass and stiffness matrices is {ω_i, ϕ_i, p_i, ξ_s,i, i = 1,2}, while, in the second updated model, the data is {ω_s,i, ϕ_i, p_i, ξ_s,i, i = 1,2}. To distinguish them, they are labeled by updated model I and II, respectively.

Figure 8 and Figure 9 present the measured and predicted floor acceleration and damper force, respectively. The results in these figures correspond to the longitudinal direction updated models. For clarity of illustration, the response predictions during the strong shaking phase (30 s~60 s) are shown, and the results of the two updated models are plotted separately. The normalized root mean squared error (NRMSE) between the measured and predicted response, defined as

$$e=\sqrt{\mathrm{mean}\left[{\left(y-\tilde{y}\right)}^2\right]}/\sqrt{\mathrm{mean}\left[{\tilde{y}}^2\right]}$$

(34)

is utilized to quantify the goodness of fit between them, in which $\tilde{y}$ and y are the measured and predicted response, respectively. The results for the two updated models are listed in

Table 5. Normalized root mean squared errors (NRMSEs) of the response predictions of updated models.

Updated Models	Longitudinal Direction				Transversal Direction
	Acceleration		Damper Force		Acceleration		Damper Force
	4F	8F	1F	8F	4F	8F	1F	8F
Model I	0.468	0.466	0.597	0.709	0.461	0.308	0.536	0.641
Model II	0.278	0.225	0.267	0.392	0.236	0.118	0.268	0.444

.

It can be seen from Figure 8 and Figure 9 that the predicted responses by updated model II match very well the measured data with NRMSE values close to 0.2, which indicates a high precision in response prediction. On the other side, the updated model I has non-negligible phase-lags errors in all response predictions with larger NRMSE values up to 0.7. A similar trend of calculated NRMSEs is also observed for the transversal direction updated models, as shown in Table 5. It verifies that, at least in this instance, the damper stiffness or the estimated equivalent natural frequencies, though small, can not be omitted in model updating for the building itself. Moreover, the high precision predictions of updated model II also show that the estimated equivalent damping ratios and natural frequencies are accurate, which addresses the concern at the end of the last section.

5. Conclusions

This paper examined the damping estimation of an eight-story steel building structure equipped with oil dampers, located at Tohoku Institute of Technology. It was carried out within a proposed framework that consists of an enhanced strain-energy method and an improved Berman–Nagy method for model updating. The former method was first employed to quantify the additional damping and stiffness provided by oil dampers in the form of equivalent damping ratios and natural frequencies. Subsequently, the improved Berman–Nagy was implemented to update an initial model of the building by using its estimated modal parameters, whose accuracy were further verified through prediction validations. The case study demonstrated the effectiveness of the proposed framework. The main conclusions and findings obtained from this study are the following: (1) The updated models show a high precision in prediction for both floor acceleration and damper force, showing the estimated natural frequencies and damping ratios are accurate. Besides, the damper stiffness, though not large, can significantly worsen the model prediction precision and, therefore, should be taken into consideration in model updating. (2) The updated model by the improved Berman–Nagy method reproduces exactly the measured modal data, including mode participation factors. However, the non-diagonal elements of updated mass matrix are not negligible and become large when more modal data utilized. (3) The supplemental damping ratios for the first modes are about 0.02, while the damping contributions for second modes are negligible, indicating that most energy dissipated by oil dampers concentrates in the seismic motion of fundamental modes. (4) The proposed framework and methods are computationally efficient and have good potential in the post-earthquake health and safety assessment for passively controlled building structures.