Seismic Damage Assessment of Instrumented Wood-frame Buildings: A Case-study
of NEESWood Full-scale Shake Table Tests111This paper is an extended version
of the paper titled ”Element-by-element seismic damage diagnosis and prognosis
in minimally instrumented wood-frame buildings” presented at Engineering
Mechanics Institute Conference 2018 (MS23-Advanced deep learning based SHM)
and participated in the EMI SHM and Control Committee Student Paper
Competition at MIT, Boston, MA, May 29-June 1, 2018.

Milad Roohi1, Eric M. Hernandez2, David Rosowsky3

1 Ph.D. Candidate, University of Vermont, Burlington, VT, USA

2 Associate Professor, University of Vermont, Burlington, VT, USA

3 Provost and Senior Vice President; Professor, University of Vermont,
Burlington, VT, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The authors propose a methodology to perform seismic damage assessment of
instrumented wood-frame buildings using response measurements. The proposed
methodology employs a nonlinear model-based state observer that combines
sparse acceleration measurements and a nonlinear structural model of a
building to estimate the complete seismic response including displacements,
velocity, acceleration and internal forces in all structural members. From the
estimated seismic response and structural characteristics of each shear wall
of the building, element-by-element seismic damage indices are computed and
remaining useful life (pertaining to seismic effects) is predicted. The
methodology is illustrated using measured data from the 2009 NEESWood Capstone
full-scale shake table tests at the E-Defense facility in Japan.

##  1 Introduction

Vibration measurements from structures can be used for a variety of purposes,
including but not limited to model calibration, code verification, structural
health monitoring, and vibration control. In the past two decades, researchers
and engineers have recognized the importance of seismic instrumentation for
quantitative and rapid performance assessment of buildings during and
immediately following earthquakes [1, 2]. Despite the immediate appeal, there
are technical, logistical, and economic challenges associated with seismic
instrumentation and performance assessment in the case of wood-frame
buildings. Building instrumentation and its maintenance can be expensive, and
budget constraints may not allow for exhaustive floor-by-floor or component-
level instrumentation. These buildings exhibit a high degree of nonlinear
behavior even during moderate ground motions, which makes it challenging to
interpret the measured seismic response and provide a quantitative measure of
the estimated level of damage caused by an earthquake.

One objective of this paper is to derive an extended model-based state
observer (EMBO) that be used for interpretation of the measured seismic
response from instrumented building structures. The EMBO is capable of
combining a detailed nonlinear structural model and noise contaminated
measured response of a structural system to estimate the complete dynamic time
history response at all degrees of freedom of the model. This nonlinear state
observer is an extension of the model-based state observer (MBO) that has been
derived and used for state estimation in real-world structural systems that
behave linear or mildly nonlinear under input motion [3, 4].

Specifically, the aim of this paper is to propose a methodology for estimation
of element-by-element seismic damage indices in minimally instrumented wood-
frame buildings. The methodology first employs the EMBO to estimate complete
seismic response of an instrumented wood-frame building. The estimated
response is then used as input to mechanics-based damage models to quantify
element level damage indices and perform seismic damage diagnosis and
prognosis of a wood-frame building. The proposed framework is verified and
validated using simulated and real measured data from a six-story wood-frame
instrumented building as part of the 2009 NEESWood Capstone building shake
tests conducted at the E-Defense facility in Japan.

The paper begins with a section describing the system of interest and
measurement model. The following section presents the derivation of the
proposed EMBO. This is followed by a section describing the 2009 NEESWood
Capstone full-scale tests conducted at the E-Defense shake table in Japan. The
numerical portion of the paper starts with verification of the proposed
methodology using simulated measurements from a nonlinear model of the
NEESWood Capstone building under input motions during the full-scale tests.
Finally, the proposed methodology is validated using real seismic response
measurements from the NEESWood Capstone tests.

##  2 System of Interest and Measurement Model

This paper focuses on typical building structures in which floor diaphragms
can be assumed to be rigid for in-plane deformations. For this type of
structures, the response to seismically induced ground acceleration can be
accurately modeled by the following simultaneous set of non-linear
differential equations

$\mathbf{M}\ddot{q}(t)+\mathbf{C}_{D}\dot{q}(t)+F_{r}(q(t))=-\mathbf{M}\mathbf{b}_{1}\ddot{u}_{g}(t)+\mathbf{b}_{2}w(t)$
(2.1)

The vector $q(t)\in\mathbb{R}^{n}$ represents the relative displacement (with
respect to the ground) of all stories. For most buildings of interest, this
results in three independent components per floor (two lateral displacements
and a rotation about the vertical axis). The number of degrees of freedom is
denoted as $n$, $\mathbf{M}=\mathbf{M}^{T}\in\mathbb{R}^{n\times n}$ is the
mass matrix, $\mathbf{C}_{D}=\mathbf{C}_{D}^{T}\in\mathbb{R}^{n\times n}$ is
the damping matrix, $F_{r}(\cdot)$ is the resultant global restoring force
vector which is obtained from the contribution of individual shear wall
restoring forces to the global diaphragm coordinates. The matrix
$\mathbf{b}_{1}\in\mathbb{R}^{n\times r}$ is the influence matrix of the $r$
ground acceleration time histories defined by the vector
$\ddot{u}_{g}(t)\in\mathbb{R}^{r}$. The matrix
$\mathbf{b}_{2}\in\mathbb{R}^{n\times p}$ defines the spatial distribution the
vector $w(t)\in\mathbb{R}^{p}$, which in the context of this paper represents
the process noise generated by unmeasured excitations and/or modeling errors.

To analyze the system model in Equation 2.2, the equation is re-written in
incremental form between $t$ and $t+\Delta(t)$ as follows

$\mathbf{M}\Delta\ddot{q}+\mathbf{C}_{D}\Delta\dot{q}+\Delta
F_{r}=-\mathbf{M}\mathbf{b}_{1}\Delta\ddot{u}_{g}+\mathbf{b}_{2}\Delta w$
(2.2)

where $\Delta\cdot=\cdot(t+\Delta t)-\cdot(t)$ and $\Delta F_{r}$ (the
increment in the global restoring force) is given by

$\Delta F_{r}=\mathbf{K}_{T}(t)\Delta q$ (2.3)

where $\mathbf{K}_{T}(t)$ is the global tangent stiffness matrix at time $t$,
as the contributing stiffness of each SDoF shear wall in the global stiffness
matrix is load (or displacement) history dependent.

This paper assumes that measurements $y(t)$ of the dynamic response of the
structure consist in horizontal accelerations measured in three independent
and non-intersecting directions. Vertical accelerations are typically also
measured, however, this paper focuses only on horizontal acceleration
measurements.  The vector of $m$ acceleration measurements $y(t)$ is modeled
as

$y(t)=-\mathbf{c}_{2}\mathbf{M}^{-1}\left[\mathbf{C}_{d}\dot{q}(t)+F_{r}(q(t))-\mathbf{b_{2}}w(t)\right]+\nu(t)$
(2.4)

where $\mathbf{c}_{2}\in\mathbb{R}^{m\times n}$ is a Boolean matrix that maps
the DoFs to the measurements, and $\nu(t)\in\mathbb{R}^{m\times 1}$ is the
measurement noise.

##  3 Extended model-based state observer (EMBO)

A nonlinear state observer for state estimation in nonlinear systems can be
written in state-space form as

$\displaystyle\dot{\hat{x}}(t)=\bm{f}(\hat{x}(t))+\textbf{G}(y(t)-\textbf{C}\hat{x}(t))$
(3.5)

where ${\hat{x}}(t)$ denotes mean state estimate, G is a feedback gain, C is
the measurement matrix and is the measurement. We expand nonlinear function
$\bm{f}(.)$ using a Taylor series around the current estimate of the state
vector and obtain a first-order approximation by dropping higher order terms
of power series as follows

$\displaystyle\dot{\hat{x}}(t)$
$\displaystyle=\mathbf{A}_{\hat{x}(t)}\hat{x}(t)+\textbf{G}[y(t)-\textbf{C}\hat{x}(t)]$
(3.6)
$\displaystyle=(\mathbf{A}_{\hat{x}(t)}-\mathbf{GC})\hat{x}(t)+\mathbf{G}y(t)$

We assume that estimates of velocity are equal to the derivative of the
estimates of displacement and choose the upper partition of feedback gain to
be zero and lower partition to be
$\mathbf{M}^{-1}\mathbf{c}_{2}^{T}\mathbf{E}$, where matrix $\mathbf{E}$ is a
matrix free to be selected in order to minimize the trace of the state error
covariance and maps the DoFs to the measurements [3]. This choice of G makes
the observer realizable as a modified finite element model of the system with
added grounded dampers and excited by corrective forces that are obtained from
velocity measurements scaled by added damper values. In second-order form, the
proposed extended model-based state observer (EMBO) estimate of displacement
response is given by

$\displaystyle\mathbf{M}\ddot{\hat{q}}(t)+(\mathbf{C}_{D}+\mathbf{c}_{2}^{T}\mathbf{E}\mathbf{c}_{2})\dot{\hat{q}}(t)+\textbf{F}_{r}(\hat{q}(t))=\mathbf{c}_{2}^{T}\mathbf{E}y(t)$
(3.7)

where $\hat{q}(t)$ is the time history of the estimated response at all DoFs
of the model and $y(t)$ is noise contaminated velocity measurements obtained
from acceleration measurements. The main advantage of the EMBO is that the
nonlinear state observer can be implemented in advanced structural simulation
software packages and this allows the state observer to be computationally
efficient in propagation of the state estimate through nonlinear dynamics of a
system.

###  3.1 Selection of feedback gain matrix

To determine the feedback gain matrix E the objective function to be minimized
is the trace of the estimation error covariance matrix. Since for a general
nonlinear multi-variable case, a closed-form solution for the optimal matrix
$\mathbf{E}$ has not been found, a numerical optimization algorithm is used.
To derive the optimization objective function, Equation 3.7 is linearized as
follows

$\displaystyle\mathbf{M}\ddot{\hat{q}}(t)+(\mathbf{C}_{D}+\mathbf{c}_{2}^{T}\mathbf{E}\mathbf{c}_{2})\dot{\hat{q}}(t)+\mathbf{K}_{0}{\hat{q}}(t)=\mathbf{c}_{2}^{T}\mathbf{E}y(t)$
(3.8)

where the elements of the stiffness matrix $\mathbf{K}_{0}$ are given by

$\displaystyle\mathbf{K}_{0_{i,j}}=\frac{\partial\mathbf{F}_{r,i}}{\partial
q_{j}}|_{q_{j}=0}$ (3.9)

By defining the state error as $e=q-\hat{q}$, the expression for the state
estimation error is given by

$\displaystyle\mathbf{M}\ddot{\hat{e}}(t)+(\mathbf{C}_{D}+\mathbf{c}_{2}^{T}\mathbf{E}\mathbf{c}_{2})\dot{\hat{e}}(t)+\mathbf{K}_{0}e(t)=b_{2}u(t)-\mathbf{c}_{2}^{T}\mathbf{E}\nu(t)$
(3.10)

To derive an expression for the state error covariance, we take Fourier
transforms of both sides of Equation 3.10 and obtain the following

$\displaystyle(\mathbf{M}\omega^{2}+(\mathbf{C}_{D}+\mathbf{c}_{2}^{T}\mathbf{E}\mathbf{c}_{2})i\omega+\mathbf{K}_{0})e(\omega)=b_{2}u(\omega)-\mathbf{c}_{2}^{T}\mathbf{E}\nu(\omega)$
(3.11)

with $\mathbf{H}_{o}$ defined as

$\displaystyle\mathbf{H}_{o}=\left(-\mathbf{M}\omega^{2}+\left(\mathbf{C}_{D}+\mathbf{c}_{2}^{T}\mathbf{Ec}_{2}\right)i\omega+\mathbf{K}_{0}\right)^{-1}$
(3.12)

From Equation 3.11, the expression for the state error estimate in the
frequency domain is

$\displaystyle
e(\omega)=\mathbf{H}_{o}(b_{2}u(\omega)-\mathbf{c}^{T}_{2}\mathbf{E}\nu(\omega))$
(3.13)

and the error spectral density matrix $\boldsymbol{\Phi}_{ee}$ is given by

$\displaystyle\boldsymbol{\Phi}_{ee}(\omega)=\mathbf{H}_{o}\mathbf{b}_{2}\boldsymbol{S}_{ww}(\omega)\mathbf{b}_{2}^{T}\mathbf{H}_{o}^{*}+\mathbf{H}_{o}\mathbf{c}_{2}^{T}\mathbf{E}\boldsymbol{S}_{vv}(\omega)\mathbf{E}^{T}\mathbf{c}_{2}\mathbf{H}_{o}^{*}$
(3.14)

where the matrices $\boldsymbol{\Phi}_{ww}(\omega)$ and
$\boldsymbol{\Phi}_{vv}(\omega)$ are the power spectral density of the
uncertain excitation on the system and measurement noise, respectively. To
select the $\mathbf{E}$ matrix, the following optimization problem must be
solved

$\displaystyle\begin{aligned}
&\underset{\mathbf{E}}{\text{minimize}}&&J=tr(\mathbf{P})\\\ &\text{subject
to}&&\mathbf{E}\in\mathbb{R}^{+}\end{aligned}$ (3.15)

where $\mathbf{P}$ is the displacement estimation error covariance matrix
given by

$\displaystyle\mathbf{P}=\mathbb{E}\left[(q(t)-\hat{q}(t))(q(t)-\hat{q}(t))^{T}\right]=\int_{-\infty}^{+\infty}\boldsymbol{\Phi}_{ee}(\omega)d\omega$
(3.16)

With this selection of the feedback matrix $\mathbf{E}$, the EMBO becomes a
modified nonlinear model of the system with added grounded dampers obtained
from a linearized model of the system in the measurement locations and excited
by forces that are linear combinations of the measurements proportional to the
added dampers.

Figure 1: Summary of the proposed EMBO for state estimation in nonlinear
structural systems

##  4 Method of Approach

Our proposed methodology based on EMBO uses sparse response measurements,
typically in the form of accelerations, to reconstruct the complete dynamic
response of the structure. From the estimated response, a Park and Ang type
low-cycle cumulative damage index can be constructed for every structural
element in the building by [5]

$DI=\dfrac{\Delta_{m}}{\Delta_{u}}+\dfrac{\psi}{F_{ey}\Delta_{u}}\int{dE}$

where $\Delta_{m}$, $\Delta_{u}$ and $F_{ey}$ are estimated maximum
deformation during the earthquake, ultimate deformation before collapse
failure under monotonic loading determined experimentally and the equivalent
yield force of the wall; and $\int{dE}$ is estimated incremental hysteretic
energy absorbed by the wall during the earthquake. Also, $\psi$ is calibration
parameter for the desired damage-based limit-state given by

$\qquad\psi=\beta_{0}+\beta_{1}x_{NS}^{2}+\beta_{2}x_{NS}^{2}x_{WH}$

where $x_{NS}$ and $x_{WH}$ are nail spacing of the shear wall and width-to-
height ratio of the shear walls; and $\beta_{0}$, $\beta_{1}$ and $\beta_{2}$
are regression coefficients calibrated from NEESWood full-scale shake table
tests on a two-story light-frame wood building [6] which are presented in
Table 1. Figure 2 shows a summary of the methodology for element-by-element
seismic damage index estimation.

Table 1: Regression coefficients calibrated from NEESWood full-scale shake table tests [6] Regression coefficient | $\beta_{1}$ | $\beta_{2}$ | $\beta_{3}$
---|---|---|---
Value | 1.121 | 0.014 | 0.026
Figure 2: Summary of the proposed methodology for element-by-element seismic
damage index estimation using EMBO

##  5 Case study: NEESWood full-scale tests

We implemented the proposed methodology on a six-story wood-frame Capstone
building tested in a series of full-scale seismic tests in the final phase of
the NEESWood project. The building was tested with various hazard levels
including (1) Test 3 (hazard level 50% in 50 years), (2) Test 4 (hazard level
10% in 50 years) and (3) Test 5 (hazard level 2% in 50 years) and was
instrumented with over 300 channels consisted of acceleration, displacement,
strain and optical tracking measurements [8, 7]. First, we verify the
application of EMBO for state estimation using simulated response
measurements. Then, we validate the proposed methodology using actual measured
data from full-scale test of the building.

###  5.1 Verification using simulated response

In the verification step, a nonlinear 3D model of the building in OpenSEES
(verified with M-SAWS model in [7]) is used as a surrogate model and simulated
data is generated by subjecting the model to the measured ground motion. The
model includes every structural wall idealized as a pure shear element capable
of resisting horizontal forces in its plane. The force-displacement
relationship in each wall is modeled using the SAWS 10-parameter hysteretic
model (Figure 3).

Figure 3: Schematic of shear wall modeling using SAWS 10-parameter hysteretic
model  Figure 4: Instrumentation locations: accelerometers in every floor
(left) and optical tracking lasers (right)

The OpenSEES model response during time history analysis under ground motion
is assumed as the ”real building response”. The proposed EMBO model is
implemented using a nonlinear 3D model of the building in OpenSEES with added
grounded dampers at measured locations (Figure 5). We compared the observer
estimates with the simulated building response and demands for every shear
wall. Various measurement feedback scenarios were tested, and the results show
that acceleration measurements at two floors (story three and roof with 3
measurements per floor) are enough to reconstruct the complete nonlinear
dynamic response with high accuracy. The proposed observer provides very good
tracking capabilities in demand estimates including nodal displacements,
inter-story drifts and force-displacement hysteresis of shear walls using a
relatively small number of measured seismic responses from the simulated
building. Figure 6 and Figure 7 show a comparison of EMBO estimates of
displacement and force-deformation for node 4 at story 5 (location 54 in
Figure 5) with results from time history analysis under the ground motion from
test 3.

Figure 5: OpenSEES Nonlinear 3D model with measurement locations (left) and
OpenSEES EMBO model with added dampers in feedback locations and applied
corrective forces (right) Figure 6: Displacement comparison between EMBO
estimation and time history analysis under ground motion (simulation) Figure
7: Force-deformation comparison between EMBO and time history analysis under
ground motion (simulation)

###  5.2 Validation using real data measured during shake table tests

In the validation case, we used measured data from the instrumented building
as feedback to the same OpenSEES EMBO model as in the verification case.
Complete demand estimates were computed for the structure and the results were
compared with measured seismic responses from the tests. Figure 8 and Figure 9
show comparison of displacement and velocity estimates from EMBO and recorded
data during test (location 54 in Figure 5). Finally, from the estimated
dynamic response and structural characteristics of each wall, an element-by-
element damage index is computed. Figure 10 shows estimated element-by-element
damage indices at story 5 during various tests (hazard levels).

Figure 8: Comparison of measured and EMBO estimated velocity at story 5
(location 54 in Figure 5) during test 5 Figure 9: Comparison of measured and
EMBO estimated velocity at story 5 (location 54 in Figure 5) during test 5
Figure 10: Estimated element-by-element damage indices at story 5 during
various tests: (left) test 3 and (right) test 4

##  6 Conclusion

This paper presents a methodology for element-by-element seismic damage
diagnosis and prognosis of minimally instrumented wood-frame buildings.
Seismic demands are computed using an extended model-based observer (EMBO).
The EMBO combines a nonlinear structural model of the building and response
measurements to reconstruct the complete dynamic response at all degrees of
freedom of the model. The algorithm was successfully verified and validated
using simulated and real data from a six-story wood-frame instrumented
building as part of the NEESWood Capstone building shake test conducted at the
E-Defense facility in Japan. Seismic damage index for every element of the
structure was computed. Future work will focus on validating the estimated
damage indices using pictures from the tests and on a methodology to assign
uncertainty to the estimated responses and damage indices.

##  7 Acknowledgement

The authors would like to thank Professor John van de Lindt (Colorado State
University) for his valuable support and his assistance with the acquisition
and interpretation of the experimental data, Professor Bruce Ellingwood
(Colorado State University) and Professor Shiling Pei (Colorado School of
Mines) for the valuable discussion and suggestions. The authors would also
like to thank Mr. Kazuki Tachibana (Tsukuba Research Institute) for his
assistance in acquiring experimental data and photographic records. The
research presented in this paper was partially funded by the National Science
Foundation award No. 1453502. This support is gratefully acknowledged.

## References

  * [1] Porter, K., Mitrani‐Reiser, J. and Beck, J.L., 2006. Near-real-time loss estimation for instrumented buildings. The Structural Design of Tall and Special Buildings, 15(1), pp.3-20.
  * [2] Celebi, M., Sanli, A., Sinclair, M., Gallant, S. and Radulescu, D., 2004. Real-time seismic monitoring needs of a building owner—and the solution: a cooperative effort. Earthquake Spectra, 20(2), pp.333-346.
  * [3] Hernandez, E.M., 2013. Optimal model‐based state estimation in mechanical and structural systems. Structural control and health monitoring, 20(4), pp.532-543.
  * [4] Hernandez, E., Roohi, M. and Rosowsky, D., 2018. Estimation of element‐by‐element demand‐to‐capacity ratios in instrumented SMRF buildings using measured seismic response. Earthquake Engineering & Structural Dynamics, 47(12), pp.2561-2578.
  * [5] Park, Y.J., Ang, A.H.S. and Wen, Y.K., 1985. Seismic damage analysis of reinforced concrete buildings. Journal of Structural Engineering, 111(4), pp.740-757.
  * [6] Park, S. and van de Lindt, J.W., 2009. Formulation of seismic fragilities for a wood-frame building based on visually determined damage indexes. Journal of Performance of Constructed Facilities, 23(5), pp.346-352.
  * [7] Pang, W., Rosowsky, D.V., Van de Lindt, J.W. and Pei, S., 2010. Simplified direct displacement design of six-story NEESWood capstone building and pre-test seismic performance assessment. MCEER.
  * [8] Van de Lindt, J.W., Pei, S., Pryor, S.E., Shimizu, H. and Isoda, H., 2010. Experimental seismic response of a full-scale six-story light-frame wood building. Journal of Structural Engineering, 136(10), pp.1262-1272.