Novel Visualization of Building Earthquake Response Recorded by a Dense Network of Sensors

1. Introduction

We present a new method for visualizing, in space and in time, observed three-dimensional (3D) motion of a densely instrumented building, with an application to earthquake response recorded in a 50-story skyscraper (Figure 1) [1]. The method involves representation of the floor slab motion as a dynamic surface deformation defined by a two-dimensional (2D) biharmonic spline [2,3,4,5,6,7,8], with parameters determined from recorded motions by multiple triaxial accelerometers on that floor and additional constraints obtained from motions recorded at other floors. The method can be extended to other types of densely instrumented structures, like bridges and dams.

Earthquake records from instrumented structures have been essential for the evolution of earthquake engineering; a review of the progress in the sensor technologies and deployment since the 1930s can be found in [9]. The first comprehensive set of earthquake records in structures were generated by the San Fernando, California earthquake of 1971 and presented visually in terms of plots of time histories of recorded acceleration, velocity and displacement of the individual records and the corresponding Fourier and response spectra [10]. Such form of visual presentation has become the standard and is currently disseminated over the internet [11,12,13,14]. To maximize the coverage of different types of construction in different seismic zones, the strong motion arrays in buildings have been sparse, with only few instrumented floors and with uniaxial or biaxial accelerometers recording only horizontal response. Methods have been proposed to reconstruct the motion at the non-instrumented floors directly by interpolation [6,7,8], a model of the structure [15,16] or a combination of the two [17,18]. The number of densely instrumented buildings has been slowly but steadily growing [19,20,21,22,23,24,25,26,27,28,29]. Yet, even in these buildings, only the horizontal components of the floor responses are recorded, while vertical motions are typically recorded only in the basement.

Rare examples of densely instrumented buildings with sensors recording vertical floor motion are the Los Angeles 52-story building, instrumented with one triaxial accelerometer at practically every floor [25] and the case study in this paper, a 50-story building, with instrumented approximately every fifth floor above ground, but by multiple triaxial accelerometers, and two basement levels instrumented each with 7 triaxial accelerometers, which enables reconstruction of the motion of the floor slabs [1]. Analyses of vertical vibrations recorded in these two buildings can be found in [30,31].

The case study in this paper is also a rare example of a full-scale building permanently instrumented with rotational seismometers. One of them is in the basement and the other one is on the 48th floor (Figure 1), the latter being used to reconstruct the 48^th floor slab motion, as shown in the next section. Recent articles on rotational sensors and their applications to recording ground motion and structural response can be found in [28,29,30,31,32,33]. An analysis of earthquake records of rotations recorded in the 50-story case study can be found in [34]. Finally, unique elements of the sensor array in the case study are two borehole arrays installed close to the building, with one sensor at the ground surface and another at the depth reached by the piles (~50 m), the motions of which are also animated.

At the core of the proposed visualization methodology is biharmonic spline interpolation, which is particularly effective in reconstructing smooth, continuous surfaces from sparse and irregularly spaced data points. It has been previously applied, e.g., to interpolation of satellite altimeter data [2], surface reconstruction in computer graphics [3], solar radiation mapping [5] and reconstruction of horizontal floor motion in sparsely instrumented buildings [6,7,8].

In this paper, it is used to reconstruct the floor slab motion, separately for each component of motion, at a sequence of time instants, creating an animation. To the knowledge of the authors, this is the first case of visualization of floor deformations of buildings during an earthquake directly from observed data, not involving a model of the structure, and the first visualization of 3D response of a building directly from recorded response.

This paper is organized as follows. Section 2 presents a brief summary of the case study, the data and the methodology. The theoretical background of biharmonic spline interpolation for observations distributed spatially in 1D and 2D is reviewed and the specific steps of the application to the case study are deliberated. Section 3 presents the results in the form of snapshots of an animation for an earthquake and a pointer to a YouTube channel where animations can be viewed for several earthquakes. Finally, Section 4 presents the conclusions of this study.

2. Materials and Methods

2.1. Building and Data

The building is known as Tongde Plaza Yue Center (TPYC) and is located in Kunming, Yunnan Province of China. It is a reinforced concrete (RC) construction, with 50 floors (238 m) above and 4 floors (17.8 m) below ground (Figure 1 and Figure 2), comprising a tower and a basement proper encompassing the tower below ground. The tower is 70.8×29 m² in plan, approximately symmetrical in both horizontal directions, while the basement is 105.6×58.9 m² in plan and 17 m deep. They are both supported by a raft foundation and 30 m long RC friction piles. The building use is commercial, and the basement is used for parking. Figure 1 shows the East and South elevations and Figure 2 shows typical floor layouts of the floors above ground (parts (a) to (d)) and of basements -B1 and -B4 (parts (e) and (f)). Figure 3 shows the distribution of mass and lateral stiffness along the height in the Nort-South (NS) and East-West (EW) directions; the latter was obtained from a detailed finite element model of the tower, fixed at the ground floor [30]. It can be seen that the tower is stiff near the base and flexible near the top.

Since January of 2021, more than 35 earthquakes were recorded in the building, with epicenters in Yunnan and Sichuan Provinces of China and neighboring Burma, Laos and India, magnitudes $2.1\le M_s\le 6.6$, epicentral distances between 10 and 1,008 km and focal depts between 9 and 128 km. The largest motions were recorded during M6.4 Yangbi, Dali, Yunnan earthquake of May 21, 2021, and M6.8 Kanding, Garze, Sichuan earthquake of September 5, 2022. The peak ground horizontal and vertical accelerations recorded on the ground surface near the South-East corner of the basement were $a_H^{\max }=$1.84×10^-3 g and $a_V^{\max }=$1.02×10^-3 g (Yangbi), and $a_H^{\max }=$1.30×10^-3 g and $a_V^{\max }=$0.68×10^-3 g (Kanding); 1g = 9.80665 m/s². The earthquakes caused small response, within 9×10^-3 g at the 48^th floor, and no damage. More details about the structure, soil, structural health monitoring system and recorded earthquakes can be found in [1,39,306].

The instrumentation includes 26 triaxial accelerometers distributed across eleven levels of the tower, located at floors 1F, 5F, 10F, 16F, 21F, 26F, 31F, 36F, 42F, 48F, and 50F. Additionally, fourteen triaxial accelerometers are installed at two basement levels (−1B and −4B). The two borehole arrays are equipped each with two accelerometers, one at the surface and the other one at 48.4 m depth. Complementing the accelerometer array are two rotational seismometers situated at −4B and 48F and a weather receiver located on Roof 4. The sensor layout can be seen in Figure 1 and Figure 2.

All accelerometers are Class B Micro-Electro-Mechanical Systems (MEMS) servo silicon sensors (EQR120) [1]. Those in the tower and basement have recording range of ±3g and a threshold sensitivity of 2.4×10⁻⁶ g while those of the borehole arrays have recording range of ±5g with a threshold of 3.9×10⁻⁶ g. The rotational seismometers are R2 angular velocity meters, with resolution 6×10^-8 rad/s at 1 Hz [30,38]. Data acquisition is performed at a sampling frequency of 200 Hz, corresponding to a Nyquist frequency of 100 Hz.

Figure 4 shows a schematic representation of the sensor locations, relative to the slabs of the instrumented levels, which is used to create the animations. On the left, a side view is shown and, on the right, a bird’s view. The locations of the sensors in the basement can be seen more clearly in Figure 2. Table 1 shows the coordinates of the sensor locations relative to the $X-Y-Z$ coordinate system, described in more detail in the next section. Figure 5 shows the convention for the six degrees of freedom motion (6DOF) recorded by collocated accelerometer and R2 rotational seismometer.

2.2. Slab Motion Representation by 2D Biharmonic Splines Using Data from Triaxial Sensors

Historically, the term "spline" referred to flexible strips used by draftsmen to draw smooth curves between points. Biharmonic splines represent an extension of this concept to higher dimensions and are valuable in applications where data may be sparse or noisy, and a smooth approximation is desired [2,3]. In one and two dimensions, the biharmonic splines are cubic polynomials that have minimum curvature and continuous first and second derivatives. It has been shown that, in all dimensions, the biharmonic splines have minimum curvature satisfying the constraints [40].

The motion of the building is described relative to a global reference coordinate system $X-Y-Z$, with origin at the center of the first floor, the $X$ and $Y$ axes pointing East and North and the $Z$-axis pointing upward. Let $x-O-y$ be local coordinate systems with origin at a particular floor that is at elevation $Z_{floor}$ from the ground surface (Figure 4). Then, in the global coordinate system, the points on the slab have coordinates $X=x$, $Y=y$ and $Z=Z_{floor}$. Both coordinate systems are fixed and are defined relative to the position of the building at rest. Let $r=\left(x,y\right)$ be a position vector describing the location of an arbitrary point on the slab in the local coordinate system and let $\mathit{u}(r,t)=\left({\mathit{u}}_x,{\mathit{u}}_y,{\mathit{u}}_z\right)(r,t)$ be a three-dimensional vector describing the displacement of that point at time $t$ relative to its position at rest. We wish to express $\mathit{u}(r,t)$ at an arbitrary time instant $t$ as a smooth function of the spatial coordinates such that satisfies the constrains that at a given set of $N$ points on the slab, it has values

$$\mathit{u}(r_j,t)=\left({\mathit{u}}_{j,x},{\mathit{u}}_{j,y},{\mathit{u}}_{j,z}\right)(t),j=1,\dots ,N.$$

(1)

Assuming small displacements, we represent each component $\mathit{u}(r,t)$ individually as a biharmonic spline satisfying the corresponding component of the constraints.

Let $w(r)$ be any one of the components of $\mathit{u}(r,t)$ at time $t$ and let $w(r_i)=w_i$ be its constraints. Then, $w(r)$ must satisfy the biharmonic equation and the constraints

$$\begin{array}{c}{\nabla }^{4}w\left(r,t\right)={\displaystyle \sum _{j=1}^N{\alpha }_j\delta (r-r_j)};{\nabla }^4\equiv {\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right)}^2\\ w(r_j)=w_j,j=1,\dots ,N\end{array}$$

(2)

in which ${\nabla }^4$ is the biharmonic operator and $\delta (r-r_j)$ is the Dirac delta function (Sandwell, 1997). The solution can be represented as a linear combination of the Green’s functions $\varphi \left(r-r_j\right)$ of the biharmonic operator

$$w\left(r,t\right)={\displaystyle \sum _{j=1}^n{\alpha }_j\varphi (r-r_j)}.$$

(3)

The Green’s functions are impulse response functions of the operator and satisfy

$${\nabla }^4\varphi (r-r_j)=\mathsf{\delta }(r-r_j).$$

(4)

In two dimensions [2]

$$\varphi (r-r_j)={\left|r-r_j\right|}^2\left(\ln \left|r-r_j\right|-1\right).$$

(5)

Both $\varphi (r)$ and its gradient are continuous everywhere, including at $r=0$. The unknown coefficients ${\alpha }_j$ can be obtained by substituting into the constraints the representation of $w(r)$ from Eqn (3), which leads to a linear system of equations

$${{\displaystyle \sum }}_{j=1}^N{\mathsf{\alpha }}_j\varphi (r_i-r_j)=w_i,i=1,\dots ,N.$$

(6)

The system of equations in (6) can be written in matrix form as

$${\Phi }_{N\times N}{\alpha }_{N\times 1}=w_{N\times 1}$$

(7)

in which $\Phi$ has entries ${\Phi }_{ij}=\varphi (r_i-r_j)$, $\alpha ={\left({\alpha }_1,\dots ,{\alpha }_N\right)}^T$ and $w={\left(w_1,\dots ,w_N\right)}^T$, and is solved for $\alpha$. The system has a stable solution if no two rows or columns of $\Phi$ are the same, which requires that the points are well distributed on the slab. In case of many available measurement points that are “noisy”, it is desirable to reduce the order of the system using singular value decomposition.

The Green’s functions matrix $\Phi$ is the same for all three components of motion and does not depend on time. So, it is precomputed and used repeatedly. After the coefficients for all three degrees of freedom, ${\alpha }_x$, ${\alpha }_y$ and ${\alpha }_z$, have been computed at a time instance $t$, the displacement components are computed at that instance on a dense grid of uniformly spaced points on the slab, $r_k=(x_k,y_k)$

$$\begin{array}{l}{u}_x(r_k;t)={{\displaystyle \sum }}_{j=1}^N{\mathsf{\alpha }}_{x,j}(t)\varphi (r_k-r_j),k=1,\dots ,K\\ u_y(r_k;t)={{\displaystyle \sum }}_{j=1}^N{\mathsf{\alpha }}_{y,j}(t)\varphi (r_k-r_j),k=1,\dots ,K\\ u_z(r_k;t)={{\displaystyle \sum }}_{j=1}^N{\mathsf{\alpha }}_{z,j}(t)\varphi (r_k-r_j),k=1,\dots ,K\end{array}$$

(8)

Then, the position of the points on the grid at time $t$ in the absolute coordinate system is computed

$${\mathit{R}}_k(t)=(x_k+{\mathit{u}}_x(x_k,y_k;t),y_k+{\mathit{u}}_y(x_k,y_k;t),Z_{floor}+{\mathit{u}}_z(x_k,y_k;t)),k=1,\dots ,K$$

(9)

plots of which show how the slab deforms in time during the earthquake shaking.

2.3. Slab Motion Representation by a Plane Using Data from 6DOF Sensor

At the 48th floor, the motion was recorded by a 6DOF sensor (collocated accelerometer and R2 rotational seismometer), located at the center of the floor, i.e., at the origin of the local coordinate system. In the local coordinate system, the position of the points on the slab is defined by vector $r={(x,y,0)}^T$. Let ${\mathit{u}}_0(t)=({\mathit{u}}_x,{\mathit{u}}_y,{\mathit{u}}_z)(t)$ and ${\Psi }_0(t)=({\mathbf{\psi }}_x,{\mathbf{\psi }}_y,{\mathbf{\psi }}_z)(t)$ be the displacements and angles of rotation recorded at the origin, $r=0$. Then, the slab motion can be represented by a plane that translates and rotates.

The displaced position of points on the plane $(x,y)$ can be computed by adding to the translation at $r=0$, $\mathbf{u}(t)$, displacements resulting from a sequence of rotations about the $x$-, $y$- and $z$-axes. The latter are linear transformations represented respectively by rotation matrices

$${\mathit{R}}_x=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\mathbf{\psi }}_x& -\sin {\mathbf{\psi }}_x\\ 0& \sin {\mathbf{\psi }}_x& \cos {\mathbf{\psi }}_x\end{array}\right]$$

(10)

$${\mathit{R}}_y=\left[\begin{array}{ccc}\cos {\mathbf{\psi }}_y& 0& \sin {\mathbf{\psi }}_y\\ 0& 1& 0\\ -\sin {\mathbf{\psi }}_y& 0& \cos {\mathbf{\psi }}_y\end{array}\right]$$

(11)

$${\mathit{R}}_z=\left[\begin{array}{ccc}\cos {\mathbf{\psi }}_z& -\sin {\mathbf{\psi }}_z& 0\\ \sin {\mathbf{\psi }}_z& \cos {\mathbf{\psi }}_z& 0\\ 0& 0& 1\end{array}\right]$$

(12)

Then, the position of the points on the slab at time $t$ in the local coordinate system are

$$\mathbf{u}(r;t)={\mathbf{R}}_x(t){\mathbf{R}}_y(t){\mathbf{R}}_x(t)r+{\mathbf{u}}_0(t).$$

(13)

2.4. Estimating Additional Constraints from 3D Motions Recorded by Sensors at Other Floors

The accuracy and stability of biharmonic spline interpolation in representing the slab deformation depend on the number of the measurement points and their spatial distribution. The number of measurements in our case study, however, is limited to two or three in the tower above ground and to 7 in the basement, and, at the 48th floor, there is only one 6DOF measurement at the center. Consequently, to ensure realistic displacements of the slabs at the unconstrained corners, we attempt to create additional constraints, based on information from the triaxial accelerometers at the same or other floors and the fitted plane at the 48th floor (see section 2.3) and considering the fact that the displacements from the slab deformations are relatively small compared to the overall deformation of the building.

For the floors with three sensors, two of which are at the opposite corners of one of the diagonals (see Figure 6), we construct constrains at the corners of the other diagonal using the measurements at the same floor, as follows. Let $r_1=\left(x_1,y_1\right)$ and $r_2=\left(x_2,y_2\right)$ be the positions of the existing sensors and $\mathit{u}\left(r_1,t\right)$ and $\mathit{u}\left(r_2,t\right)$ be their displacements at time $t$. Then the midpoint position and displacement are

$${\mathit{r}}_m={\displaystyle \frac{{\mathit{r}}_1+{\mathit{r}}_2}{2}}$$

(14)

$$\mathit{u}\left({\mathit{r}}_m,t\right)={\displaystyle \frac{\mathit{u}\left({\mathit{r}}_1,t\right)+\mathit{u}\left({\mathit{r}}_2,t\right)}{2}}.$$

(15)

Assuming small rotations and displacements, the rotational angles at $r_m$, ${\mathbf{\psi }}_x\left(t\right)$, ${\mathbf{\psi }}_y\left(t\right)$, and ${\mathbf{\psi }}_z\left(t\right)$, can be approximated using finite differences as [41]

$$\begin{array}{ccc}{\mathbf{\psi }}_x\left(t\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\mathsf{\Delta }{\mathit{u}}_z}{\mathsf{\Delta }y}}-{\displaystyle \frac{\mathsf{\Delta }{\mathit{u}}_y}{\mathsf{\Delta }z}}\right)& {\mathbf{\psi }}_y\left(t\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\mathsf{\Delta }{\mathit{u}}_x}{\mathsf{\Delta }z}}-{\displaystyle \frac{\mathsf{\Delta }{\mathit{u}}_z}{\mathsf{\Delta }x}}\right)& {\mathbf{\psi }}_z\left(t\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\mathsf{\Delta }{\mathit{u}}_y}{\mathsf{\Delta }x}}-{\displaystyle \frac{\mathsf{\Delta }{\mathit{u}}_x}{\mathsf{\Delta }y}}\right)\end{array}$$

(16)

which enables construction of the rotation matrices ${\mathit{R}}_x$, ${\mathit{R}}_y$, and ${\mathit{R}}_z$, defined in Eqns (10) to (12), and the combined rotation matrix

$$\mathit{R}\left(t\right)={\mathit{R}}_x\left({\mathbf{\psi }}_x\left(t\right)\right){\mathit{R}}_y\left({\mathbf{\psi }}_y\left(t\right)\right){\mathit{R}}_z\left({\mathbf{\psi }}_z\left(t\right)\right)$$

(17)

Then, the motions at the non-instrumented corners, at $r_k=\left(x_k,y_k\right)$, $k=3,4$, can be estimated as

$$\mathit{u}\left(r_k,t\right)=\mathit{R}\left(t\right)\left(r_k-r_m\right)+\mathit{u}\left(r_m,t\right)$$

(18)

providing additional constraints $\mathit{u}(r_j,t)$ in Eqn (1).

For floors with two sensors, we construct additional constraints using measurements at the other floors along the same vertical, using 1D spline interpolation, as follows. Let $w\left(z\right)$ represent any one of the displacement components (${\mathit{u}}_x$, ${\mathit{u}}_y$, or ${\mathit{u}}_z$) as a function of elevation $z$, with known values $w\left(z_j\right)=w_j$, $j=1,\dots ,N$. The biharmonic equation in one dimension and the associated constraints can be written as [2]

$$\begin{array}{cc}{\displaystyle \frac{d^{4}w\left(z\right)}{d{z}^4}}& ={\displaystyle {\sum }_{j=1}^{N}6{\alpha }_j\delta (z-z_j)}\\ w\left(z_j\right)& =w_j,j=1,\dots ,N\end{array}$$

(19)

and has solution that is a linear combination of the 1D biharmonic operator Green's functions $\varphi \left(z-z_j\right)$

$$w\left(z\right)={{\displaystyle \sum }}_{j=1}^N{\mathsf{\alpha }}_j\varphi \left(z-z_j\right)$$

(20)

in which

$$\varphi \left(z-z_j\right)={\left|z-z_j\right|}^3$$

(21)

Then, the displacement at another height $z_i$ along the vertical, where there is no sensor, can be estimated by solving the linear system of equations

$${{\displaystyle \sum }}_{j=1}^N{\mathsf{\alpha }}_j\varphi \left(z_i-z_j\right)=w_i,i=1,\dots ,N$$

(22)

By applying this approach to each displacement component, we estimate the displacements at the non-instrumented corners providing additional constraints for the representation of the slab motion at the floors with two sensors.

At the 48th floor, where measurement of 6DOF motion is available but only at one point, we showed in section 2.3 how the slab motion can be approximated by a translating and rotating in time plane. The motion of the slab can also be approximated by a 2D spline interpolation, by using the plane to construct constraints at the four corners (Figure 6). Then, a biharmonic spline can be fitted though the five points, representing a more natural slab displacement for the visualization.

It is noted that, using 1D spline interpolation along vertical lines, where motion is measured at some levels, the motion at the non-instrumented floors can be reconstructed, and can then be used for rapid assessment of the structural health during or immediately after the end of the shaking.

2.5. Steps

First, the data are prepared. The displacements are computed by double integration of the recorded accelerations and the angles of rotation are computed by integration of the recorded angular velocities, after performing instrument correction with manufacturer provided sensor specific transfer-functions. Then, the data is high pass filtered, to remove low frequency noise, low pass filtered at 10 Hz and subsampled by a factor of 6 to reduce the computation time to generate the animation. This results in an animation with about 33 frames per second.

Then, the visualization is created according to the following steps, each one consisting of adding a particular element to the animation. The first four steps are illustrated in parts a) to d) of Figure 7.

Step 1: Add the slabs at levels F48, F42, F36, F31, F21, F26, F16, F10, F5, F1, B1, and B4, which have been sufficiently instrumented to reconstruct the slab motion. The slab motion is defined at a 1×1 m2 grid of points.

Step 2: Add the physical boundaries of the structure, represented by vertical lines, to facilitate the visualization of the sensor placements relative to the physical layout.

Step 3: Add the locations of the sensors. The locations of the accelerometers are marked as red dots and those where 6DOF motion is recorded are marked by blue dots. The motion of these points is added as a check for the reconstructed motions and to show the particle motions at locations at which no interpolation was applied (at F51 and at the borehole arrays).

Step 4: Add lines connecting sensors on neighboring floors and add labels indicating the level.

Step 5: Construct hodograms showing the particle motion of the soil recorded by the borehole sensors at the surface (SURF-NW and SURF-SE) and at depth (BOR-NW and BOR-SE). The particle motions are shown in the vertical plane passing through the radial direction, relative to the earthquake epicenter. The purpose of these plots is to visualize the particle motion during the passage of Rayleigh waves. Figure 8 shows an example of a hodogram at a particular instant of time. The thick fading line shows the trajectory of the particle motion during the past 2 s, with the color fading in the direction of the past.

Step 6: Construct a plot of time histories of the $X$, $Y$ and $Z$ components of the displacement at the center of the 4th basement (-B4C5), with a slider showing the current time in the animation. This plot illustrates the entire time history of the effective input motion into the building and the current stage of the animation.

Step 7: Choose the magnification factor for the displacements to enable viewing the features of small amplitude response. The ratio between the peak displacement at the top and the half width of the basement can be used as a guideline.

Step 8: Decide what views of the building to include in the animation. Examples are shown in the Results section.

3. Results

The methodology is illustrated on the data from the $M_s$5.1 Shuangbai, Chuxiong, Yunnan earthquake, which occurred on June 10, 2021 at 19:46 local time, at epicentral distance $\mathit{R}=116$ km south-west of the building (back-azimuth $\Phi =$228°). Figure 9 shows traditional plots of the time histories of accelerations recorded along column line C5, approximately at the center of the floors. The video of the animation is provided as supplementary material S1. Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 show selected screenshots at different time instances for magnification factor of 4×10⁴. The animations show four views of the building: from top, from South, from South-West and from West, as well as the time history of the displacement at the basement and hodograms of the particle motion of the soil recorded by the surface and downhole sensors (see section 2.4). It can be seen that this earthquake excited the higher modes and significant torsional vibrations of the tower, that the basement deforms during the passage of the earthquake waves, and that tower moves relative to the basement, which we termed “tower-basement” interaction, confirming the findings of our previous studies on the effects of soil-structure interaction (SSI) on the building’s response using a detailed finite element model [39]. Additionally, the hodograms show that the particle motion of the surface waves is more complex than the theoretical retrograde ellipse predicted for Rayleigh waves in uniform half-space.

Animations of the response to five earthquakes, including Shuangbai, can be viewed at the @TPYC-seismic YouTube channel. The other four earthquakes are: Yangbi, Yunnan of May 21, 2021 ($M_s$5.1, $\mathit{R}=295$ km); Burma of July 29, 2021($M_s$5.7, $\mathit{R}=730$ km); Kanding (Luding), Sichuan of January 26, 2023 ($M_s$5.6, $\mathit{R}=512$ km); and Xishan, Yunnan of January 27, 2023 ($M_s$2.1, $\mathit{R}=10$ km).

4. Discussion

The new model-free method for visualization of the building 3D response, entirely from recorded response, effectively presented the nature of the response of the building, both in terms of computational efficiency and information conveyed. The method can be generalized to reconstruct the response at all floors and then used for raid assessment of the structural health during or soon after the shaking, based, e.g., on published interstory drift-damage relationships [42]. An advantage of such assessment, as opposed to assessment based on a model, is the computational efficiency and true representation of reality.

The insight into structural response to adverse events, such as earthquakes, gained from animations like those produced for the 50-story case study, are valuable for testing existing methods and developing new methods for structural health monitoring of buildings.