Structural performance and damage prediction using a novel digital cloning technique

Abstract

This paper presents a novel, mechanics-based framework to create digital twins for earthquake-affected pinched structures, like reinforced concrete framed buildings. It uses robust and accurate structural health monitoring (SHM) results delivered by the hysteresis loop analysis method as an input to create digital twin models to predict nonlinear dynamic responses of previously damaged structures under potential future seismic events. Method validation is implemented using unique real-world data from the Bank of New Zealand (BNZ) building in Wellington, New Zealand, which experienced severe structural damage due to three earthquakes (Events 1, 2, and 3) between 2013 and 2016.Results show the digital twin derived from the SHM results of Event 1 can predict the inter-story displacement of the BNZ building for Events 2 and 3 with average correlation coefficients of ∼0.95 and ∼0.97 between predicted and measured responses, respectively. Moreover, the maximum difference between the measured and predicted peak values of inter-story displacements was 13 mm, a negligible difference in inter-story drift ratio (IDR). These results were accurate and consistent for all stories. Finally, the accuracy of this framework in capturing IDR values makes it a promising tool for assessing potential future structural collapse risk and its consequent financial risks using well-known incremental dynamic analysis methods.

Keywords

Digital twin predictive structural model basis function structural health monitoring hysteresis loop analysis incremental dynamic analysis

Introduction

Seismic events cause considerable damage with subsequent severe social and economic consequences. Structural damage significantly threatens the overall health and integrity of earthquake-affected structures, where deteriorated structural performance increases the risk of structural collapse and loss of lives due to the subsequent aftershocks or new seismic events. While retrofit can enhance the safety of damaged structures, the cost of retrofit to meet increasingly higher policy-driven safety standards may be uneconomic.

In addition to limited budgets, limited engineering experience, the lack of accurate information on the damage/status of earthquake-affected structures, and the inability to predict how damaged structures will perform in any remaining service life under possible future earthquakes—all these factors make it difficult for owners to make optimal and on-time decisions in the aftermath of devastating earthquakes.¹ These factors highlight the importance of advanced structural health monitoring (SHM) methods and instrumentation to significantly improve the robustness of retrofit decisions.

Effective SHM methods need to accurately identify, localize, classify, and quantify structural damage, and ultimately estimate remaining service life after a seismic event.^2,3 However, most SHM methods do not focus on predicting the future response of earthquake-affected structures.² Therefore, an accurate, predictive framework identifying structural damage and using this data to create forecast models able to predict subsequent seismic performance and collapse, or further damage risks due to possible future earthquakes, would be a highly valuable tool for post-event decision-making, saving lives and cost.¹

However, the highly nonlinear behavior of structures, which intensify as earthquake loads and structural damage increase, and the complexity of earthquake signals make it challenging to accurately predict future nonlinear structural performance.^4,5 The most conventional approach for predicting seismic structural response employs finite element modeling (FEM) to simulate seismic response using nonlinear time history analysis.^6,7 These approaches are computationally intensive, and thus cannot be employed for a real or near real-time post-event evaluations.^4,5,8 More importantly, the accuracy of these model-based methods is highly dependent on the fidelity of the FEM and the damage information used to inform model creation.^4,5,8 Furthermore, in general, their accuracy decreases as nonlinearity increases in structural responses,^5,8 which is further exacerbated in the presence of unknown or poorly modeled earthquake-induced structural damage.

Data-driven methods using artificial intelligence^8–23 and support vector machines (SVMs)^1,5,24 have been increasingly used for seismic response prediction.^4,8 However, the training process for these methods requires a lot of real-world seismic data over a wide range of earthquake scenarios, which is not necessarily available in practice. Furthermore, building-specific prediction with these methods would require building-specific data, which is not possible. Therefore, costly shake-table testing on scaled/full-scale structures or FEM simulations were alternatively employed to generate enough seismic response data,⁴ which is unrealistic and may lead to inaccurate and unreliable data-based predictive models for real structures.

Currently, data-driven models are black-box models and do not explicitly consider the underlying mechanical interactions between ground motion and seismic responses. However, this interaction is essential for accurate prediction, as nonlinearity means it can vary depending on the seismic input.⁴ Moreover, a black-box approach limits real-world application, where explicit procedures are more trusted and explicable to stakeholders engaging in post-earthquake evaluation, such as insurance firms and building owners.

To address these deficiencies, the model-free, mechanics-based hysteresis loop analysis (HLA)-SHM method was developed and validated for numerical^3,25–27 and experimental^25,28–30 structures covering a wide range of nonlinear dynamic behaviors and damage mechanisms. HLA is a rapid, real-time SHM method with an explicit design needing no prior training process, while returning coherent SHM results immediately after earthquakes. HLA can capture and track changes in critical physical, structural design parameters based on force-deformation hysteresis loop shapes from monitored structures. HLA performance estimating elastic stiffness degradation due to seismic loads has been recently validated for a real-world reinforced concrete (RC) structure subjected to multiple earthquakes.³¹ Further HLA study on the real RC structure,³² showing highly nonlinear pinched behavior, highlights the damage classification capability of HLA, which successfully distinguished elastic stiffness reductions and pinching stiffness changes jointly contributing to overall structural damage.

To move from SHM to prediction, HLA-SHM results were combined with the learning and classification features of SVM to create digital clones, or digital twin models, for structures.^1,5 Although these cyber-predictive models performed well in reconstructing time-series seismic responses, their black-box design and required prior training limit their efficiency and practical potential. Importantly, this prior work overcame the data issue by training the SVM approach using model-based responses, and demonstrated good performance, but was still a black-box approach lacking explicability.

Therefore, an alternative prediction framework using HLA-SHM results and predictive basis functions is introduced in this paper. Basis functions have already been used to develop virtual patient models for enhanced mechanical ventilation in a bioengineering application with similar hysteresis loop mechanics.^33–36 Thus, well-defined basis functions could create a meaningful connection between earthquake loads, structural seismic responses, and physical parameters identified accurately by the HLA-SHM method. Such digital twin models could predict structural responses and damage to the actual structure (physical twin) due to future seismic events, making it a promising post-event evaluation tool. In particular, accepted incremental dynamic analysis (IDA) methods^37–40 would provide accurate structural collapse and financial risk assessments using the HLA-derived predictive models.^38,41,42

This paper presents and validates this proposed predictive digital twin modeling framework using real-world data along the fully instrumented × direction of the BNZ building, which was damaged severely due to three earthquakes (Events 1, 2, and 3) between 2013 and 2016.^31,32,43 Peak values of measured and predicted inter-story displacements and forces are compared to show the accuracy of the proposed method in predicting nonlinear responses of a damaged structure to future earthquakes, which would be critical for optimizing design and retrofit.^44,45

Methods

To develop a digital twin for an earthquake-damaged building with highly pinched responses typically seen in RC frame structures,^46–49 a force-deformation hysteretic model is an essential first step. This model must efficiently simulate major nonlinear structural damage and deterioration mechanisms, such as pinching, stiffness degradation, and strength degradation.⁴⁶ Second, this model must avoid unnecessary complexity to maximize robustness and usability, while ensuring its physical parameters are identifiable using HLA-SHM results. Third, basis functions estimating the evolution of model-defining parameters must be identifiable from measured structural responses and physical parameter changes captured using HLA, and able to accurately predict physical parameter changes from measured structural responses. All three elements enable the digital twin to simulate and predict the damaged structure’s seismic response and damage due to subsequent future seismic events.

Simplified pinching model

Bouc–Wen models extended for pinched hysteretic systems^50–53 typically use a combination of three separate springs with different behaviors, as shown in Figure 1. Based on this configuration, an identifiable piecewise linear hysteretic model for pinched systems is proposed, as shown in Figure 2. Here, $f$ and $x$ stand for inter-story restoring force and inter-story displacement, respectively, which are called force and deformation for simplicity.

To simplify the pinching model (Figure 2), it is assumed the slip-lock spring only controls the middle pinching region length, while its slight contribution to the stiffness of the pinching region is ignored. The linear spring individually defines the stiffness values of pinching and post-yielding regions. Moreover, the parallel combination of linear and hysteretic springs defines the stiffness for the post-pinching elastic region. For any asymmetric behavior, as seen in real-world cases in practice, the simplified model is separately defined for positive and negative directions exactly in the same manner.

As shown in Figure 2, the loading branch of the ith hysteresis loop in the positive direction is modeled using $k_{p}^{i}$ , $k_{e}^{i}$ , $d_{p}^{i}$ , and $d_{y}^{i}$ values, while the unloading branch needs the values of $d_{p}^{i + 1}$ and $k_{p}^{i + 1}$ , which are the updated values of $d_{p}^{i}$ and $k_{e}^{i}$ since the system underwent new deformation demand level of $d_{m}^{i}$ at the end of the loading branch in the positive direction. However, these updated values might remain unchanged depending on the size of $d_{m}^{i}$ . This issue is discussed with full details within Subsection “Basis functions.” Moreover, once $(d_{m}^{i}, f_{m}^{i})$ and $(d_{p}^{i + 1}, f_{p}^{i + 1})$ breakpoints are obtained, the unloading stiffness of the pre-pinching unloading region, $k_{u}^{i + 1}$ , can be calculated as defined:

k_{u}^{i + 1} = \frac{f_{m}^{i} - f_{p}^{i + 1}}{d_{m}^{i} - d_{p}^{i + 1}}

(1)

where $f_{m}^{i}$ and $f_{p}^{i + 1}$ are the corresponding force values of $d_{m}^{i}$ and $d_{m}^{i}$ deformation breakpoints, respectively. From the hysteresis loop geometry in Figure 2(b), the value of $f_{p}^{i + 1}$ can be obtained:

f_{p}^{i + 1} = k_{p}^{i + 1} d_{p}^{i + 1}

(2)

Therefore, four main physical parameters are required to simulate pinching responses using the proposed simplified model, including pinching stiffness $k_{p} (t)$ , elastic stiffness $k_{e} (t)$ , and the breakpoints of pinching deformation $d_{p} (t)$ and yielding deformation $d_{y} (t)$ . All these model-defining parameters and their evolutions are identifiable using the HLA-SHM method. Hence, the proposed hysteretic model (Figure 2) can capture and simulate four structural damage mechanisms typical to RC buildings with pinched hysteretic seismic response, as shown in Figure 3⁴⁶: (1) Increase in the middle pinching region length; (2) Cyclic stiffness degradation; (3) In-cycle stiffness degradation; and (4) Cyclic strength degradation.

All these damage mechanisms are quantitatively identifiable using the HLA-SHM method. To predict changes of the model-defining parameters ( $d_{p}$ , $d_{m}$ , $k_{e}$ , and $k_{p}$ ) under external excitation, appropriate basis functions are proposed based on HLA results and measured response for the next step. These basis functions create a digital twin model for the building, able to predict its nonlinear response and damage state under future seismic events.

Basis functions

To develop basis functions predicting the evolution of model-defining parameters of $d_{p}$ , $d_{y}$ , $k_{p}$ , and $k_{e}$ under new loads, it is assumed these parameter values in the current hysteresis loop loading branch ( $d_{p}^{i}$ , $d_{p}^{i}$ , $k_{p}^{i}$ , and $k_{e}^{i}$ ) fundamentally depend on the maximum deformation the system sustained at the end of the previous loading branch ( $d_{m}^{i - 1}$ ). Therefore, two different types of basis function are required to predict: (1) changes of $d_{p}$ and $d_{y}$ ; and (2) $k_{p}$ and $k_{e}$ evolutions, as shown schematically in Figure 4 for the positive direction, where $d_{M}^{i}$ is the maximum deformation the system sustained form the beginning of the vibration to the current step, and defined:

d_{M}^{i} = \max (d_{m}^{i} (t))

(3)

From Figure 3(a) and 4(a), as $d_{M}$ increases, $d_{p}$ and $d_{y}$ breakpoints move further away from the origin. In this study, the linear correlation observed between the HLA-identified changes of $d_{p} / d_{y}$ and measured $d_{M}$ values for the BNZ building due to Event 1 leads using a simple linear basis functions to predict the evolution of these model-defining breakpoints due to earthquake excitations:

d_{p}^{i} = a_{p} d_{M}^{i - 1} + b_{p}; d_{p} \geq d_{p}^{0}

(4)

d_{y}^{i} = a_{y} d_{M}^{i - 1} + b_{y}; d_{y} \geq d_{y}^{0}

(5)

Moreover, Figure 4(b) shows deformation beyond the pinching region ( $d_{m} > d_{p}$ ) causes stiffness degradation in the simplified pinching model. The decaying evolution of elastic, $k_{e}$ , and pinching, $k_{p}$ , stiffness values observed for the BNZ building due to Event 1 confirms the studies^54,55 used power function for modeling stiffness degradation in nonlinear hysteresis models, which are used in this study to predict the changes of $k_{p}$ and $k_{e}$ due to further, unknown ground shaking:

k_{p}^{i} = A_{p} {(S_{mp}^{i - 1})}^{- B_{p}}

(6)

k_{e}^{i} = A_{e} {(S_{mp}^{i - 1})}^{- B_{e}}

(7)

S_{mp}^{i - 1} = \sum_{j = 1}^{j = i - 1} (d_{m}^{j} - d_{p}^{j}) + S_{mp}^{0}; \forall d_{m}^{j} > d_{p}^{j}

(8)

Figure 5 schematically shows these basis functions. For an instrumented building under a damaging earthquake, deformation responses $(d_{m}, d_{M})$ are measurable, and changes of model parameters ( $k_{p}$ , $k_{e}$ , $d_{p}$ , and $d_{y}$ ) are identifiable using the HLA-SHM method, which can be used to identify the basis functions in Equations (4) to (7). When there is no structural damage in building, there would be no pinching region in building responses, and thus $d_{p}^{0}$ and $S_{mp}^{0}$ are equal to zero, and the value of $d_{y}^{0}$ is obtained using HLA and the identified basis functions. However, for each seismic event, these initial values must be updated using the known final values of $d_{p}$ , $d_{y}$ , and $S_{mp}$ obtained at the end of the previous event.

For future earthquakes, the already-identified basis functions can be used to estimate online $k_{p}$ and $k_{e}$ reduction from the building’s seismic responses. More importantly, the already-identified basis functions can also be combined with the simplified pinching model, as defined previously, to develop a digital twin predicting building responses and likely structural damage due to future seismic events. This novel predictive model is formulated in Subsection “Digital twin for pinched hysteretic structures.”

Digital twin for pinched hysteretic structures

Once basis functions estimating model-defining parameter changes are obtained, the building motion under a ground acceleration of ${\overset{\cdot\cdot}{x}}_{g}$ is simulated using:

M \overset{\cdot\cdot}{X} (t) + C \overset{\cdot}{X} (t) + K (t) X (t) = M {1}_{n} {\overset{\cdot\cdot}{x}}_{g}

(9)

where, the vector of ${1}_{n}$ dimension is equal to the building degree of freedom number, $n$ , and $X$ , $\overset{\cdot}{X}$ , and $\overset{\cdot\cdot}{X}$ are the displacement, velocity, and acceleration vectors with respect to the ground. The constant matrices of $M$ and $C$ stand for mass and damping of the building, respectively, while the time-varying matrix of $K (t)$ is the tangent stiffness of the building, defined:

K (t) = {\begin{matrix} k^{+} (t), x (t) \geq 0 \\ k^{-} (t), x (t) < 0 \end{matrix}

(10)

where $x$ is the inter-story displacement, and $k$ is the inter-story tangent stiffness for the corresponding two adjacent stories. To consider any asymmetrical responses, the already-identified basis functions are used to model nonlinear stiffness changes of the building for the positive and negative directions separately, yielding:

k^{+} (t) = {\begin{matrix} k_{p}^{+} (t), & ‖ x (t) < d_{p}^{+} (t) \\ k_{e}^{+} (t), & x (t) \dot{x} (t) \geq 0 ‖ x (t) \geq d_{p}^{+} (t) x \leq d_{y}^{+} (t) \\ k_{p}^{+} (t), & x (t) \dot{x} (t) \geq 0 ‖ x (t) \geq d_{p}^{+} (t) x \geq d_{y}^{+} (t) \\ k_{u}^{+} (t), & x (t) \dot{x} (t) < 0 ‖ x (t) > d_{p}^{+} (t) \end{matrix}; \forall x (t) \geq 0

(11)

k_{u}^{+} (t) = \frac{f_{m}^{+} - k_{p}^{+} (t) d_{p}^{+} (t)}{d_{m}^{+} (t) - d_{p}^{+} (t)}

(12)

k^{-} (t) = {\begin{matrix} k_{p}^{-} (t), & ‖ x (t) < d_{p}^{-} (t) \\ k_{e}^{-} (t), & x (t) \dot{x} (t) \geq 0 ‖ x (t) \leq d_{p}^{-} (t) x \geq d_{y}^{-} (t) \\ k_{p}^{-} (t), & x (t) \dot{x} (t) \geq 0 ‖ x (t) \leq d_{p}^{-} (t) x \leq d_{y}^{-} (t) \\ k_{u}^{-} (t), & x (t) \dot{x} (t) < 0 ‖ x (t) < d_{p}^{-} (t) \end{matrix}; \forall x (t) < 0

(13)

k_{u}^{-} (t) = \frac{f_{m}^{-} - k_{p}^{-} (t) d_{p}^{-} (t)}{d_{m}^{-} (t) - d_{p}^{-} (t)}

(14)

where the values of $d_{p}^{\pm}$ and $d_{y}^{\pm}$ are obtained using Equations (4) and (5), and $k_{p}^{\pm}$ and $k_{e}^{\pm}$ values are obtained using Equations (6) to (8). Equations (12) and (14) are used to model the stiffness of the building for the pre-pinching unloading region $k_{u}^{\pm}$ .

To clarify the novel proposed method, a flowchart schematically summarizes the process of predictive model development in Figure 6. The figure shows basis functions can have two applications: (1) online estimation of structural damage from measured responses; and (2) developing digital twins predicting dynamic responses of buildings under future earthquakes.

Results and discussion

SHM-HLA results capturing changes in the model-defining parameters of the BNZ building due to the 2013 Seddon earthquake (Event 1) are used to identify basis functions. These identified basis functions from Event 1 are then used to predict the model-defining parameter evolution (Figure 6) using the BNZ building’s responses to the 2013 Lake Grassmere and 2016 Kaikoura events (Events 2 and 3). In short, a digital twin is created for the BNZ building to predict nonlinear structural responses and damage due to Events 2 and 3 (Figure 6). Importantly, predicting Event 3 is two forward predictions of nonlinear response, from Event 1 to Event 2, and further, without re-identification, to Event 3. These analyses validated both the evolution and overall predictive capability of the digital twin modeling method presented.

Basis functions and online structural damage estimation

Linear basis functions estimating pinching, $d_{p}$ , and yielding, $d_{y}$ , points are shown in Figure 7. They show the identified linear basis functions to SHM-HLA results for the BNZ building under Event 1 accurately predict pinching region length increase and yielding point motion seen in measured responses due to Events 2 and 3. Furthermore, the results show larger deformation demand, $d_{M}$ , increases the pinching contribution to seismic response, and thus delays elastic and post-yielding behavior. Figure 7 shows there are several data samples for Event 1 and Event 3, but there is only one for Event 2. It shows Event 2 was not significantly damaging to the BNZ building and, based on Equations (3) to (5), Event 2 increased the maximum deformation, $d_{M}^{i}$ , once due to only one large half-cycle in the BNZ building’s responses. For Events 1 and 3, numerous large half-cycles contributed to structural damage.

Although the pinching region length increase can potentially delay the yielding point and thus reduce the rate of elastic stiffness, $k_{e}$ , degradation,³² more pinching also deteriorates overall seismic performance.⁴⁶ In particular, buildings with more severe pinching behavior have less hysteretic energy dissipation capacity and are expected to experience larger inelastic deformation, increasing the risk of structural collapse.^{46,48,56–58}

Figure 8 shows the power basis functions identified from SHM-HLA results for elastic, $k_{e}$ , and pinching, $k_{p}$ , stiffness reduction in the BNZ building due to Event 1. They show the identified basis functions accurately predict elastic and pinching stiffness reduction in the measured responses of the BNZ building under subsequent Events 2 and 3. Results show pinching stiffness degrades at a higher rate than elastic stiffness. The average values of the power are ${\bar{B}}_{p} = - 0.43$ and ${\bar{B}}_{e} = - 0.24$ in pinching and elastic stiffness basis functions, respectively, meaning the BNZ building is more vulnerable to pinching stiffness reduction due to earthquake loads, when deformation, $d_{m}$ , exceeds the pinching deformation, $d_{p}$ , and damage index, $S_{mp}$ , accumulatively increases. In particular, this vulnerability is more considerable at the initiation of structural damage during Event 1, where the damage index is less than 1 ( $S_{mp} \leq 1$ ).

Figure 9 compares the elastic and pinching stiffness evolution identified by HLA with the values estimated using the basis functions from the building’s responses (Figure 6). Results show a good match between the HLA-identified and basis-function-estimated values across the three earthquakes. The absolute difference between the HLA-identified and predicted elastic stiffness values at the end of Event 3 to the initial stiffness at the beginning of Event 1 are 1%, 2%, 3%, 2%, and 2% for Stories 1–5, respectively. These differences for pinching stiffness are 1%, 3%, 1%, 1%, and 3%. For Story 6, the difference value is 10% and 20% for elastic and pinching stiffness due to the failure of the sensor installed at this level during Event 3.^31,32

The between-event continuity seen in the HLA-SHM results for changes in pinching point, yielding point, elastic stiffness, and pinching stiffness are expected for the BNZ building being accumulatively damaged by the three earthquakes. To our best knowledge, the building received no significant structural retrofit affecting the building’s seismic performance which would have led to potential error if not identified by HLA in smaller events between the major events used.³¹ Such between-event consistency confirms the robustness and accuracy of the HLA-SHM method, which is essential for a reliable SHM method needed for rapid post-earthquake evaluations.^3,31,32

However, more importantly, such between-event continuity can also be predicted in real-time using basis functions identified from HLA-SHM results, as shown in Figures 7 to 9. These figures show all basis functions identified from the HLA-SHM results for the BNZ building using only results from Event 1 can successfully estimate the model-defining physical parameters of the building due to subsequent nonlinear Events 2 and 3. The basis functions merely need seismic responses of the BNZ building for Event 1 to instantly deliver the SHM results of interest, as illustrated in Figure 6.

The predictive digital twin

The basis functions update the physical parameters defining the simplified pinching model to create a digital twin for the BNZ building damaged due to Event 1. This digital twin model can predict BNZ building responses and structural damage for subsequent future earthquakes Events 2 and 3. Figures 10 and 11 compare the measured and predicted force-deformation hysteresis loops for all stories of the BNZ building under Events 2 and3, respectively, where failure of the Story 6 sensor during Event 3 is evident and these measures are ignored.^31,32

Superficially, the largest mismatches are in Stories 2–3 under Event 3, where stiffness deterioration with negative stiffness is apparent. The absolute difference between the measured and predicted values of the maximum inter-story displacement are 18% and 23% for Stories 2 and 3, while are less than 5% for the other stories. Stiffness deterioration typically occurs due to significant deformation and cyclic loading effects (in-cycle strength degradation).⁴⁶ The simplified pinching model proposed in this paper cannot completely simulate this damage mechanism, which can lead to comparatively larger mismatches between the predicted and measured hysteresis loops for Stories 2–3 of the BNZ building under Event 3. Importantly, the prediction of Event 3 is two predictions forward in time, events, and damage.

Figures 12 and 13 compare measured and predicted inter-story displacement responses for all stories of the BNZ building under Events 2 and 3, respectively, as predicted from the model identified at the end of Event 1. Results show the average correlation coefficient is 0.95 and 0.97 for Events 2 and 3, confirming a very good match between the measured and predicted responses. Failure of the sensor installed at Story 6 during Event 3 led to outlying measurements,^31,32 as can be seen in Figures 10 and 11. Thus, the results of Story 6 under Event 3 are not considered further here.

The other measures adopted in this paper to evaluate the accuracy of the digital twin in predicting seismic responses of the BNZ building are the peak values of the inter-story displacement and force signals, in both positive and negative directions, and a comparison between measured and predicted inter-story drift ratio (IDR) values. Tables 1 and 2 list the measured and predicted values of inter-story displacement, the absolute prediction error, and IDR values for Events 2 and 3, respectively. Results show the maximum errors take place in Stories 2–3 during Event 3, as expected due to inability of the simplified pinching model in simulating highly nonlinear stiffness deterioration.

Table 1.

Measured and digital-twin-predicted values of IDR and inter-story displacement extrema for all stories of the BNZ building subjected to Event 2.

Story	IDR (%)		Predicted (mm)		Measured (mm)		Difference (mm)		$\| Error \|$ (%)
	Predicted	Measured	$D_{max}^{-}$	$D_{max}^{+}$	$D_{max}^{-}$	$D_{max}^{+}$	$D_{max}^{-}$	$D_{max}^{+}$	$D_{max}^{-}$	$D_{max}^{+}$
1	0.83	0.74	−22	20	−19	20	3	0	16	0
2	0.8	0.86	−31	29	−33	32	−2	3	6	9
3	0.73	0.9	−28	26	−34	32	−6	6	18	19
4	0.58	0.64	−22	20	−25	21	−3	1	12	5
5	0.44	0.41	−19	16	−17	15	2	−1	12	7
6	0.57	0.57	−24	18	−24	18	0	0	0	0
									Average = 9

Table 2.

Measured and digital-twin-predicted values of IDR and inter-story displacement extrema for all stories of the BNZ building subjected to Event 3.

Story	IDR (%)		Predicted (mm)		Measured (mm)		Difference (mm)		$\| Error \|$ (%)
	Predicted	Measured	$D_{max}^{-}$	$D_{max}^{+}$	$D_{max}^{-}$	$D_{max}^{+}$	$D_{max}^{-}$	$D_{max}^{+}$	$D_{max}^{-}$	$D_{max}^{+}$
1	1.47	1.58	−37	39	−42	40	−5	1	12	3
2	1.35	1.69	−52	50	−65	61	−13	11	20	18
3	1.37	1.6	−52	44	−61	57	−9	13	15	23
4	1.1	0.96	−42	35	−37	34	5	−1	14	3
5	0.61	0.51	−26	23	−22	22	4	−1	18	5
6	0.59	6.2	−25	21	−264	231	−239	210	91	91
									Average = 13

Although the average errors are 9% and 13% for Events 2 and 3, the maximum difference between measured and predicted inter-story displacement responses is 13 mm, which is comparatively small, considering construction variability. This level of inaccuracy in inter-story displacement response prediction has a negligible effect on the predicted IDR values compared with the measured IDR values in Tables 1 and 2 and Figure 14, where both predicted and measured IDR values deliver similar post-earthquake evaluations for damage level in all stories of the BNZ building under Events 2 and 3 using HAZUS.⁵⁹ Thus, this digital twin can potentially be employed to assess structural collapse and associated financial risks by IDA using IDR values as one of the main evaluation measures.^30,37,38

Tables 3 and 4 compare the peak values of measured and predicted inter-story restoring forces, both in positive and negative directions, and their differences and errors for Events 2 and 3, respectively. While the results show average errors of 6% and 4% for Events 2 and 3, the maximum absolute difference is $0.15 \times 10^{7} N$ , which is only ∼2% of the total weight of the BNZ building’s Unit 3.^31,32. Thus, the digital twin also predicts maximum inter-story forces for the damaged BNZ building under future earthquakes, which can be useful for structure design and retrofit,^44,45 and is thus useful for guiding decision-making in its own right.

Table 3.

Measured and digital-twin-predicted values of inter-story force extrema for all stories of the BNZ building subjected to Event 2.

Story	Predicted ( $\times 10^{7} N$ )		Measured ( $\times 10^{7} N$ )		Difference ( $\times 10^{7} N$ )		$\| Error \|$ (%)
	$F_{\max}^{-}$	$F_{\max}^{+}$	$F_{\max}^{-}$	$F_{max}^{+}$	$F_{max}^{-}$	$F_{max}^{+}$	$F_{max}^{-}$	$F_{max}^{+}$
1	−1.71	1.57	−1.58	1.59	0.13	0.02	8	1
2	−1.51	1.38	−1.51	1.47	0	0.09	0	6
3	−1.27	1.14	−1.42	1.27	−0.15	0.13	11	10
4	−1.13	0.97	−1.25	1.05	−0.12	0.08	10	8
5	−0.89	0.78	−0.99	0.79	−0.1	0.01	10	1
6	−0.61	0.46	−0.64	0.49	−0.03	0.03	5	6
							Average = 6

Table 4.

Measured and digital-twin-predicted values of IDR and inter-story displacement extrema for all stories of the BNZ building subjected to Event 3.

Story	Predicted ( $\times 10^{7} N$ )		Measured ( $\times 10^{7} N$ )		Difference ( $\times 10^{7} N$ )		$\| Error \|$ (%)
	$F_{max}^{-}$	$F_{max}^{+}$	$F_{max}^{-}$	$F_{max}^{+}$	$F_{max}^{-}$	$F_{max}^{+}$	$F_{max}^{-}$	$F_{max}^{+}$
1	−2.17	2.04	−2.25	2.02	−0.08	−0.02	4	1
2	−1.99	1.86	−2.07	1.83	−0.08	−0.03	4	2
3	−1.8	1.6	−1.75	1.57	0.05	−0.03	3	2
4	−1.44	1.27	−1.35	1.25	0.09	−0.02	7	2
5	−1.03	0.89	−0.93	0.93	0.1	0.04	11	4
6	−0.61	0.52	−0.51	0.7	0.1	0.18	20	26
							Average = 4

Figure 15(a) compares the elastic stiffness evolution identified by HLA with the elastic stiffness values predicted by the digital twin of the BNZ building under Events 2 and 3. Figure 15(b) does the same for the pinching stiffness evolution. Results show a good match between the HLA-identified and digital-twin-predicted values for both elastic and pinching stiffness of the BNZ building. For elastic stiffness, there are 1%, 2%, 3%, 5%, and 1% absolute difference between the HLA-identified and predicted stiffness values at the end of Event 3 to the initial stiffness at the beginning of Event 1 for Stories 1–5, respectively. These differences for the pinching stiffness evolutions are 0%, 4%, 2%, 1%, and 2%. For Story 6, the difference values are 5% and 35% for elastic and pinching stiffness due to the sensor failure during Event 3.^31,32

Results show the digital twin derived using the HLA-SHM results for the BNZ building under Event 1 can successfully predict the seismic performance of the building under future major earthquakes causing sever structural damage and highly nonlinear responses. The created digital twin can predict structural damage with good accuracy. Furthermore, it shows very good capability in predicting inter-story displacement responses, which can be combined with IDA methods to predict structural collapse risk and its subsequent financial risks.

The choice of basis function is not limited to those used in this work. However, in particular, identifiability is a key element in selecting a basis function, where robust determination of its parameters is critical to good prediction results. A more complex basis function, which is theoretically or practically not identifiable, is not of use.^60,61 Thus, this work considered limited complexity and identifiable functions, and future work should consider a more robust examination of the full set of potential basis functions.

The basis functions and the resultant digital twin can be readily updated after each damaging earthquake. However, in this specific case study, Event 2 did not cause significant changes (structural damage) in the model-defining parameters of the BNZ building (Figures 7 –9), and thus updating after Event 2 led to negligible changes in the identified basis functions and the digital twin accuracy. Therefore, in this study, predicting Event 3 is two forward predictions of nonlinear responses, from Event 1 to Event 2, and further, without re-identification, to Event 3. For earthquakes after Event 3, due to the severity of structural damage, the updated basis functions and digital twin are recommended and it should be investigated in future work. Equally, such forward prediction over multiple events shows the potential full capability of the approach presented, where updating after each event is also possible for even better results.

One of the advantages of the proposed prediction digital twin modeling framework is it does not need large datasets and numerous damaging events to be trained, in contrast to data-driven methods proposed to date.⁴ One structurally damaging earthquake triggering all contributing structural damage mechanisms discussed in this paper is sufficient to identify the basis functions from the measured responses and HLA-SHM results and create the predictive digital twin. However, re-identification and updating the basis functions and the resultant digital twin after each damaging earthquake can help improve the accurate predictive performance of the framework for future earthquakes.

This paper is a primary study investigating the use of the digital twin modeling framework, successfully employed in bioengineering applications,^{33–36,62–66} to predict structural damage and performance under earthquakes. The proposed framework covers all possible SHM levels, including damage identification, localization, classification (diagnosis) and damage assessment, and lifetime prediction (prognosis).² Therefore, the method proposed here is not comparable to other SHM methods found in the literature in terms of the delivered SHM information layers and its explicit mechanics-based design. In future work, the proposed method’s accuracy should be investigated by further experiments, in a controlled environment with fewer uncertainties, to obtain better insights into basis functions and their relationship with the geometrical and mechanical properties of the monitored structure.

Conclusions and recommendations

This paper presented a novel, fully mechanics-based, and explicitly coherent framework using robust and accurate HLA-SHM results to develop digital twins for earthquake-damaged buildings showing highly nonlinear pinched behavior. A simplified pinching hysteresis model whose skeleton is defined by four physical parameters of pinching point ( $d_{p}$ ), yielding point ( $d_{y}$ ), elastic stiffness ( $k_{e}$ ), and pinching stiffness ( $k_{p}$ ) is introduced to simulate pinched seismic responses of structures. The basis functions updating these four model-defining parameters as a function of the structure’s deformation demand create a digital twin for structures in a combined operation with the simplified hysteresis pinching model. The digital twin developed for earthquake-damaged structures can predict structural responses and damage to the building (the physical twin) due to future seismic events. The proposed method has been validated using real-world data of the BNZ building, delivering the following outcomes:

The basis functions obtained from HLA-SHM results of the BNZ building under Event 1 accurately translate the measured responses of the BNZ building under Events 2 and 3 into structural damage in an online manner. The accurate online information on elastic ( $k_{e}$ ) and pinching ( $k_{p}$ ) stiffness reduction due to earthquakes can be vital for a rapid safety warning system that can save lives, particularly when strong ground motions strike a vast urban region and major aftershocks increasingly threaten already-affected structures.

The digital twin developed for the BNZ building using the basis functions extracted from the HLA-SHM results of Event 1 successfully predict and re-simulates force-deformation hysteresis loops of the BNZ building under Events 2 and 3. Thus, the proposed digital twin approach can successfully simulate the pinching, elastic, and post-yielding behavioral regime typically seen in the pinched hysteresis responses of structures. However, the predictive model could not simulate highly nonlinear stiffness deterioration behavior in Stories 2 and 3 under Event 3, which can be tackled in future work using more complex pinching hysteresis models.

Results show that the difference between the measured and predicted values for IDRs is negligible, leading to similar post-earthquake evaluations of damage severity across all stories of the BNZ building due to Events 2–3. The accuracy of this predictive model in capturing IDR values makes it a promising tool to assess structural collapse and its consequent financial risks using IDA.

The similarity between the coefficients of the identified basis functions can potentially lead to generalized forms of basis functions obtained from geometry and mechanical characteristics of structures, which will be investigated in future work. Furthermore, such generalized basis functions help consider retrofit effects on digital twins and, importantly, create predictive digital twins for healthy structures not developing any damage mechanisms yet.

This digital twin modeling approach enables forward prediction of damaged structure responses, which has not been previously feasible. Accurate forward prediction, capturing the nonlinear evolution of damage, further enables a wide range of analyses, such as IDA and loss estimation analysis, to optimize decision-making. The capability of automating the entire process in this approach ensures rapid availability of useful information to optimize decision-making and its timeframes.

Overall, this paper highlights the importance of efficient SHM methods and instrumentation. The rapid and accurate information on the current status of structures and their future safety and seismic performance, delivered by the proposed HLA-based predictive framework, can significantly lead to on-time and optimal decisions in the aftermath of earthquakes, potentially saving lives and money, and compensating all expenses that went on SHM instrumentation.

Footnotes

Acknowledgements

The authors acknowledge GNS Science of New Zealand for the monitoring data, enabling us to conduct this study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Mohammad Rabiepour

James Geoffrey Chase

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