Time-lag effect of thermal displacement and its compensation method for long-span bridges

Abstract

The time-lag effect between temperature and thermal displacement may induce the displacement-based safety assessment results of long-span bridges to derivate from the truth. In this paper, the typical characteristics of the time-lag effect between temperature and thermal displacement are firstly investigated by using the synchronously monitored temperature and displacement data from a long-span steel box-girder arch bridge. And then, the inherent reasons of the time-lag effect are found out by employing the Kendall correlation coefficient. Following that, a general method derived from the Bayesian function registration model and the Z-mixture preconditioned Crank-Nicolson algorithm is proposed to compensate the time-lag effect. Finally, the proposed compensation method is verified by data from three bridges and compared with the traditional method achieved through shifting a fixed time interval. The results show that thermal displacement may be ahead of or lag behind temperature, depending on the temperature and thermal displacement of concern. The lag time varies from a few minutes to several hours with temperature and displacement variables, as well as time instants. The time-lag effect between temperature and thermal displacement is caused by the asynchronous change of the dominant temperature for the specific thermal displacement and other temperatures because of different material thermodynamic parameters and geometric characteristics of different bridge components. The developed compensation method can completely eliminate the time-lag effect between temperature and thermal displacement of various long-span bridges without any pre-correlation analysis and prior knowledge. The correlation between temperature and thermal displacement compensated by the method proposed in this paper is much stronger than that compensated by the traditional method.

Appendices

Appendix A

The SRSF of the warping function $\gamma (t)$ is

$$\psi \left(t\right)=\sqrt{\dot{\gamma }(t)}$$

(A.1)

Since $\gamma \left(0\right)=0$, this mapping operation is invertible. One typical feature of $\psi \left(t\right)$ is that it has unit ${{\text{L}}}_{2}$ norm as follows

$$\| \psi \left(t\right){\| }^{2}={\int }_{0}^{1}\psi (t{)}^{2}dt={\int }_{0}^{1}\dot{\gamma }(t{)}^{2}dt=\gamma (1)-\gamma (0)=1$$

(A.2)

The resulting space ${\mathbb{Q}}$ ($\psi \left(t\right)\in {\mathbb{Q}}$) is the positive orthant of the unit sphere in the Hilbert space ${{\text{L}}}_{2}$. On the space, the distance can be defined by the arc-length, and the mapping operation also has the property of isometry. Although the SRSF transformation simplifies the complicated geometry of $\Gamma$, the space ${\mathbb{Q}}$ is yet to be linear and cannot meet the requirement of the Bayesian function registration. As a result, the further transformation is required.

The space ${\mathbb{Q}}$ is mapped onto a tangent space. The identity element ${\gamma }_{{\text{id}}}\left(t\right)\in\Gamma$ accordingly maps to a constant function $1\in {\mathbb{Q}}$. The tangent space at this specified point is

$${\rm T}_{1}\left({\mathbb{Q}}\right)=\left\{g:\left[\mathrm{0,1}\right]\to {\mathbb{R}}\left|\langle {\text{g}},1\rangle \right.={\int }_{0}^{1}g(t)dt=0\right\}$$

(A.3)

where $g$ is the representation of the warping function $\gamma (t)$ in the tangent space. One way to connect ${\mathbb{Q}}$ and ${\rm T}_{1}\left({\mathbb{Q}}\right)$ is via the exponential map and inverse exponential map, which are defined as

$${\rm T}_{1}\left({\mathbb{Q}}\right)\to {\mathbb{Q}}\mathit{ }{\mathit{exp}}_{1}(g)=\mathit{cos}\left(\Vert g\Vert \right)+\frac{\mathit{sin}\left(\Vert g\Vert \right)}{\Vert g\Vert }g, {g\in {\rm T}}_{1}\left({\mathbb{Q}}\right)$$

(A.4)

$${\mathbb{Q}}\to {\rm T}_{1}\left({\mathbb{Q}}\right) {\mathit{exp}}_{1}^{-1}\left(\psi \right)=\frac{\theta }{\mathit{sin}\left(\theta \right)}\left[\psi -\mathit{cos}\left(\theta \right)\right], \theta =arccos\left(\langle 1,\psi \rangle \right),\psi \in {\mathbb{Q}}$$

(A.5)

where $\theta$ is the vectorial angle between 1 and $\psi$.

Appendix B

In Eq. (15), the prior and posterior distributions of $\left(g,{\sigma }_{1}^{2}\right)$ are probability measures defined on the product space ${\rm T}_{1}\left({\mathbb{Q}}\right)\times {\mathbb{Q}}$. Specifically, the prior measure is the product measure ${\mu }_{0}\equiv \text{Gaussian }\left(0,{\mathcal{C}}_{g};\,{I}_{A}\right)\times IG(a, \, b)$ and the posterior measure $\mu$ is absolutely continuous with respect to the prior measure. By using Bayesian formula, the Radon-Nikodym derivative of the posterior measure $\mu$ is

$$\frac{d\mu }{d{\mu }_{0}}\left(g,{\sigma }_{1}^{2}\right)\propto L\left(g,{\sigma }_{1}^{2}\mid {f}_{1},{f}_{2}\right)$$

(B.1)

where $L\left(\bullet ,\bullet \mid {f}_{1},{f}_{2}\right)$ is the likelihood function given by

$$L\left(g,{\sigma }_{1}^{2}\mid {f}_{1},{f}_{2}\right)\propto {\left(\frac{1}{{\sigma }_{1}^{2}}\right)}^{N/2}\mathit{exp}\left\{-\frac{1}{2{\sigma }_{1}^{2}}\mathit{SSE}(g)\right\}$$

(B.2)

$$SSE\left(g\right)=\sum_{i=1}^{N}{\left(q\left({f}_{1}\right)\left({t}_{i}\right)-q\left({f}_{2}\right)\left({\int }_{0}^{{t}_{i}}{exp}_{1}^{2}\left(g\right)\left(s\right){\text{ds}}\right)\times {exp}_{1}\left(g\right)\left({t}_{i}\right)\right)}^{2}$$

(B.3)

The Metropolis in Gibbs algorithm could be adopted to search the posterior distribution given by Eq. (B.1). At each iteration, the component ${\sigma }_{1}^{2}$ is updated with values extracted from the conditional distribution as follows:

$${\sigma }_{1}^{2}\mid g,{f}_{1},{f}_{2}\sim IG\left(\frac{N}{2}+a,\frac{1}{2}\mathit{SSE}(g)+b\right)$$

(B.4)

While the functional component $g$ is updated by the Metropolis–Hastings step. In this step, the functional component $g$ could be obtained by the Z-mixture pCN algorithm, which is

$${g}^{\prime}\sim \sum_{z=1}^{Z}{p}_{z}Gaussian\left(g\sqrt{1-{\beta }_{z}^{2}},{\beta }_{z}^{2}{\mathcal{C}}_{g};\,{\text{I}}_{\text{A}}\right)$$

(B.5)

where ${g}^{\prime}$ is the updated functional component $g$.