Mitigating effects of temperature variations through probabilistic-based machine learning for vibration-based bridge scour detection

Abstract

This paper presents a novel approach to mitigating the effect of temperature variations on the bridges’ dynamic modal properties for more reliably detecting scour damage around bridge piles based on the vibration-based measurements. The novelty of the presented approach lies in its ability to reasonably remove the impacts on the modal properties of bridges, particularly caused by changes in material properties and structural boundary conditions due to temperature variations without explicitly modeling these complex effects. The main idea is to adopt the probabilistic-based machine learning method, Gaussian Process Model, to learn the correlation between the changes of modal properties of a monitored bridge and the corresponding temperature variations from in situ sensor measurements, and probabilistically infer the bridge scour based on the modified vibration measurements, which have mitigated the identified impacts of temperature variations, by applying Bayesian inference through the Transitional Markov Chain Monte Carlo simulation. The proposed approach and its applicability are presented and validated through the numerical simulation of a prototype bridge, demonstrating its potential for practical application for mitigating effects of temperature variations or other environmental impacts for vibration-based Structural Health Monitoring. The limitation of the presented study and future research needs are also discussed.

Appendices

Appendix 1: Verification of the basic assumption for the proposed framework through numerical simulation

To verify the assumption that the changes of the modal parameters of an undamaged bridge are not significantly different from those of the same bridge with scour occurrence under the same temperature variation, the impact of temperature variations on modal parameters of undamaged bridge and scoured bridge $\Delta \Phi^{u}$ and $\Delta \Phi^{s}$ can be rearranged into two one-dimensional vectors V^u and V^s in the form of a row vector $\left[ {\begin{array}{*{20}c} {\Delta \omega^{T} } & {\Delta \varphi_{1}^{T} } & \cdots & {\Delta \varphi_{j}^{T} } & \cdots & {\Delta \varphi_{{N_{s} }}^{T} } \\ \end{array} } \right]_{{1 \times m\left( {N_{{\text{s}}} + 1} \right)}}$, respectively. The correlation between the two vectors V^u and V^s can be measured in terms of the correlation coefficient defined by Asuero et al. [1] as follows:

$$r_{{{\text{us}}}} = \frac{{\sum \left( {V_{k}^{u} - \overline{V}^{u} } \right)\left( {V_{k}^{s} - \overline{V}^{s} } \right)}}{{\sqrt {\sum \left( {V_{k}^{u} - \overline{V}^{u} } \right)^{2} \left( {V_{k}^{s} - \overline{V}^{s} } \right)^{2} } }},$$

(18)

where $V_{k}^{u}$ and $V_{k}^{s}$ are the kth element of vectors V^u and V^s, respectively; $\overline{V}^{u}$ and $\overline{V}^{s}$ are the mean of V^u and V^s, respectively. The correlation coefficient of r_us closer to 1 indicates a high correlation between V^u and V^s. Based on the defined scour parameters and the analytical model of a prototype bridge as delineated in the subsequent section, analytical results obtained from the computational simulation presented subsequently in Sect. 3.5 show that the impact of temperature variations $\Delta \Phi^{u}$ on the undamaged structure and these $\Delta \Phi^{s}$ on scoured bridge structure are very close to each other, indicating that the adopted assumption is reasonable and may be practically applicable for the model-based damage identification method. Besides, any errors or uncertainties caused by adopting the above assumption are quantified in terms of the confidence level of identified scour damage based on probabilistic results obtained from implementing Bayesian inference within the presented framework presented subsequently.

To numerically verify the above basic assumption for the proposed framework as mentioned in Sect. 2.1, the modal parameters of the prototype bridge under both undamaged and scoured conditions at eight different temperatures − 25 °C, − 5 °C, 10 °C, 20 °C, 40 °C, 55 °C, 65 °C, and 75 °C were simulated based on the bridge structural model as presented in Sect. 3.1 to obtain the temperature-induced change of the modal parameters $\Delta \Phi^{u}$ and $\Delta \Phi^{s}$ as defined in Eqs. 1 and 2. For scoured conditions, three different scenarios of scour, i.e., slight, medium, and severe scour, θ = [0.4 0.4 0.4 0.4], [0.6 0.6 0.6 0.6], and [0.8 0.6 0.5 0.7], were considered. The correlation coefficients of the $\Delta \Phi^{u}$ and $\Delta \Phi^{s}$ at these different temperatures are calculated based on Eq. 18 and presented in Fig. 

Fig. 7

Correlation coefficients of temperature effect before and after scour damage

7 for these three scour scenarios. Figure 7 demonstrates that the two vectors $\Delta \Phi^{u}$ and $\Delta \Phi^{s}$ at different temperature variations have strong correlations for the three scour scenarios, indicating that the impacts of temperature variations on modal parameters of the bridge before and after scour occurs are very close to each other and the adopted assumption is reasonably accepted.

Appendix 2: Detailed procedures in Bayesian inference of scour damage

The Bayesian inference as proposed by Beck and Katafygiotis [3] is employed to quantify uncertainties associated with scour damage identification. This approach has been extensively applied in structural damage identification, model updating, and parameter estimation, including identification of scour by Zheng and Yu [33] and proven to be robust for quantifying uncertainties associated with unknown parameters and models. Within Bayesian inference, uncertainties associated with scour identified through a structural model M at the reference temperature T^r can be described by the PDFs of unknown structural model parameter vector θ, which is the indexes of scour damage at piles of the bridge as described in Sect. 3.3. Because the extent of scour is unknown or uncertain, the indexes of scour damage θ may have many different candidate values. As a result, the structural model M actually represents a class of possible models specified by different structural model parameter vectors θ. Thus, M is referred to as a model class in general.

For the given model class M at the reference temperature, Bayesian inference is used for identifying bridge scour damage through updating the PDFs of unknown parameter vector x = {θ^T, σ₂}^T, which includes the structural model parameter θ, as well as an additional parameter σ₂ that is defined in the likelihood function in Eq. 21, based on the prior knowledge and the measurement data $\Phi$, i.e., the modified modal parameters with the removal of the impact of temperature variations as specified in Eq. 3 in this application.

Based on Bayes’ rule, the posterior PDF of unknown parameter vector x = {θ^T, σ₂}^T after updated is given by Cheung and Beck [4] as follows:

$$p({\mathbf{x}}|\Phi ,M) = c^{ - 1} p(\Phi |{\mathbf{x}},M)p\left( {{\mathbf{x}}|M} \right),$$

(19)

where p($\Phi$ | x, M) is the likelihood function specified in the same sense as defined in Sect. 2.3, and measures the probability for obtaining the observed output $\Phi$ from the structure model M specified by the structural model parameters θ; p(x |$\Phi$) is the prior PDF of unknown model parameters that can be estimated from engineering judgment, and $c = p\left( {\Phi {|}M} \right) = \smallint p(\Phi |{\mathbf{x}},M)p({\mathbf{x}}|M){\text{d}}{\mathbf{x}}$ is referred to as the model evidence, which actually is a normalizing constant that ensures the integral of the posterior PDF as specified in Eq. 19 over the parameter space to be unit.

The above likelihood function p($\Phi$ | x, M) is usually established based on the PDF of the error or discrepancy between the observed output $\Phi$ and the predicted output from the model M specified by structural model parameter θ. In this application, this error may result from the inaccurate predictions caused by the model error, the field measurement error, and the numerical error in the process of mitigating the impact of temperature variations. Assume that for a certain given structural model parameter vector θ, predicted output from the model M is denoted as $\Phi$_p(θ) = {${ }\omega_{i} \left( \theta \right)$,${ }\varphi_{ij} \left( \theta \right)$, i = 1, …,m, j = 1,…,N_s}. The error between the field measured modal parameters Φ and the predicted modal parameters $\Phi$_p from the model M can be specified in terms of natural frequency and mode shape components for the ith mode, respectively, as follows:

$$J_{\omega i} \left( {\varvec{\theta}} \right) = \frac{{\omega_{i} \left( {{\varvec{\theta}},M} \right) - \omega_{i} }}{{\omega_{i} }}\;{\text{and}}\;J_{\varphi i} \left( {\varvec{\theta}} \right) = \frac{{\beta_{i} \varphi_{i} \left( {{\varvec{\theta}},M} \right) - \varphi_{i} }}{{\varphi_{i} /N_{{\text{s}}} }},$$

(20)

where ||·|| means Euclidian norm, and $\beta _{i} = \mathbf{{\varphi }}_{i}^{T} \mathbf{{\varphi }}_{i} (\mathbf{{\theta }},M)/\mathbf{{\varphi }}_{i} \left( {\mathbf{{\theta }},M} \right)^{2}$ is a normalizing constant that guarantees the measured mode shape $\varphi_{i}$ closest to the predicted model shape $\beta_{i} \varphi_{i} \left( {\theta ,M} \right)$ for the given structural model parameter vector θ. Note that θ is a statistical variable. Thus, the above errors are also statistical variables. According to the maximum information entropy principle [14], those errors are assumed to be independent and identically (i.i.d) distributed Gaussian variables and also have a shared standard deviation σ₂, so that the established likelihood function can contain maximum uncertainties associated with the error. It should be noted that the parameter σ₂ is a shared standard deviation in the likelihood function specified below for predicting the index of scour damage and determined through Bayesian inference with TMCMC as described subsequently. It should be noted that the parameter σ₂ here is different from σ in Eqs. 9–17, which is for predicting the label of AE event for training the classifier and determined through optimization of the right side of Eq. 17. Thus, the likelihood function is a Gaussian joint PDF and can be written as:

$$p\left( {\Phi {|}{\mathbf{x}}, M} \right) = \left( {\sqrt {2\pi } {\upsigma }_{2} } \right)^{{ - m\left( {N_{{\text{s}}} + 1} \right)}} {\exp}\left( {\frac{{\mathop \sum \nolimits_{i = 1}^{m} \left[ {J_{\omega i} \left( {\varvec{\theta}} \right)} \right]^{2} + w\mathop \sum \nolimits_{i = 1}^{m} \left[ {J_{\varphi i} \left( {\varvec{\theta}} \right)} \right]^{2} }}{{2\sigma_{2}^{2} }}} \right),$$

(21)

where x = {θ^T, σ₂}^T as defined in the above, and w is a weight factor reflecting the relative contribution of errors of the mode frequencies and mode shapes to the likelihood function. The different values of w could significantly affect the accuracy and efficiency of the presented framework for scour damage identification. Goller et al. [11] showed that the optimum weight factor w can be determined effectively for Bayesian updating at the model class level.

In general, the posterior PDF of uncertain parameters specified in Eq. 19 can be hardly calculated, since the integrals involved in the model evidence in Eq. 19 cannot be analytically evaluated for high-dimensional spaces. To address such a problem, numerical methods based on probabilistic sampling simulation have been proposed as alternative ways to statistically estimate the integrals and have been successfully incorporated into Bayesian inference framework [2, 4,5,6]. Among the developed sampling methods, the Transitional Markov Chain Monte Carlo (TMCMC) algorithm proposed by Ching and Chen [5] has been demonstrated to be able to efficiently draw samples from some difficult PDFs (e.g., multimodal PDFs, very peaked PDFs, and PDFs with flat manifold) and estimate the model evidence. Thus, TMCMC algorithm was used by Zheng and Yu [33] and is also employed here to infer the posterior PDFs of the uncertain parameters of scour damage indexes. The key idea of TMCMC is to sample uncertain parameters from a series of intermediate PDFs that can be easily sampled and gradually converge to the target PDF p(x | $\Phi$, M). Readers interested in details of theoretical formulation and practical implementation steps of TMCMC algorithm may refer to Ching and Chen [5], and Zheng and Yu [33].

Appendix 3: Simulation of the impacts of temperature variations on the bridges’ materials properties and boundary conditions

Since the presented numerical simulation is to examine the extent to which the impact of temperature variations, no matter how temperature variations may cause changes in the material properties and boundary conditions of the bridge, can be mitigated from the measured data, a reasonable model of temperature distributions over the bridge may be adopted for this numerical study. For simplicity, the linear distribution of temperature along three directions is adopted in this study. It is also reasonable to assume that the temperature variation within an individual element of the FE model of the bridge is identical, because the size of the individual element is relatively small in an FE model. Therefore, the temperature variation within any element of the bridge structural model can be expressed as △T = (T-T^r) (x + y + z), where T is environmental temperature, T^r is the reference temperature, and x, y, and z are the coordinates at the center of the element. The temperature variations within elements cause changes of elastic modulus of materials, leading to changes in the stiffness of elements and structural vibration measurements. The elastic modulus of concrete can be taken as E = (1-ζ_T△T) E₀, where E₀ is the modulus at the reference temperature, and ζ_T = 4.5 × 10^–3 denotes the temperature coefficient of the elastic modulus of concrete material [31].

For the same reason, the accuracy in modeling the boundary condition change caused by temperature variations is not essential for this numerical study as long as the model is reasonable. Thus, the zero-length link elements in SAP2000 are adopted at the two ends of the bridge deck to model the joint restraint to the bridge deck due to its expansion caused by temperature variations. The link element has six degrees of freedom, in which K_f1 is the axial stiffness along the longitudinal direction of the bridge, and K_f2, K_f3 are the shear stiffness in the horizontal and vertical directions transversal to the longitudinal direction of the bridge, and K_m1, K_m2, K_m3 are the rotational stiffness about three axes along the longitudinal, transversal, and vertical directions, respectively. In this numerical simulation, only three DOFs of the zero-length link element are considered and assigned with different stiffness, i.e., K_f1, K_f2, and K_m3. The assigned stiffness simulates the joint restraints to the bridge deck from expanding in the longitudinal direction, side-sliding along the horizontal direction, and rotating around the axis along the vertical direction caused by temperature variations, respectively.

It is noted that as environmental temperature arises, the actual restraint from the thermal expansion joints tends to increase due to the increased contact of the joints at the boundary. When the environmental temperature reaches a certain value and the thermal expansion joints at the deck ends may be completely close or have full contact, the restraint from thermal expansion joints may attain their maximum values, and the further increase of temperature may not be the augment of the joint restraint stiffness. Thus, the restraint stiffness K_f1, K_f2, K_m3 under changing temperature T can be reasonably modeled in the form of the logistic sigmoid function, respectively, as follows:

$$K_{l} = \frac{{K_{d} }}{{1 + e^{{\frac{{aT^{r} - T}}{b}}} }}\left( {l = {\text{f}}1,\;{\text{f}}2,\;{\text{m}}3,{\text{ and}}\;d = 1, \, 2, \, 3} \right),$$

(22)

where a and b are constant coefficients that control the shape of the curve, and K_d are the reference stiffness that can be determined from the axial, shear, and rotational stiffness of a segment of the bridge deck. As an example, Fig. 

Fig. 8

K_f1 variation along with environmental temperature

8 presents the variation of K_f1 with temperature T $\in$ [− 10, 70] for different combinations of constant coefficients a and b. It can be noted that a is actually related to the abrupt point of the stiffness, while b controls the steepness of the curve. In this numerical study, a = 2 and b = 3 are adopted for simulating the above three types of stiffness.