Performance alarming method for bridge expansion joints based on temperature displacement relationship model

Abstract

The present invention belongs to the technical field of health monitoring for civil structures, and a performance alarming method for bridge expansion joints based on temperature displacement relationship model is proposed. First, the canonically correlated temperature is proposed to maximize the correlation between bridge temperature field and expansion joint displacement; second, a temperature displacement relationship model for bridge expansion joints is established based on canonically correlated temperatures; then, a mean-value control chart is constructed to the error of temperature displacement relationship model; finally, reasonable control limits are determined for the mean-value control chart. A more accurate temperature displacement relationship model can be established based on canonically correlated temperatures, which is of important value to improve the performance alarming ability for expansion joint.

Claims

1. A performance alarming method for bridge expansion joints based on temperature displacement relationship model, wherein specific steps of which are as follows: step 1: calculate canonically correlated temperatures of a bridge structure acquire T and D of the bridge structure, T=[T.sub.1, T.sub.2, . . . T.sub.m].sup.T represents a measurement sample of m temperature measurement point in a structural health monitoring system, D=[D.sub.1, D.sub.2, . . . , D.sub.n].sup.T represents a measurement sample of n expansion joint displacements, calculate a covariance matrix and a cross-covariance matrix of temperature and displacement monitoring data as follows: $\begin{array}{c}\begin{array}{c}\begin{array}{c}{R}_{\mathrm{TT}}=\frac{1}{l-1}\underset{t=1}{\overset{l}{.Math.}}\left\{T\left(t\right)-\stackrel{\_}{T}\right\}{\left\{T\left(t\right)-\stackrel{\_}{T}\right\}}^T\\ R_{\mathrm{DD}}=\frac{1}{l-1}\underset{t=1}{\overset{l}{.Math.}}\left\{D\left(t\right)-\stackrel{\_}{D}\right\}{\left\{D\left(t\right)-\stackrel{\_}{D}\right\}}^T\end{array}\\ R_{\mathrm{TD}}=\frac{1}{l-1}\underset{t=1}{\overset{l}{.Math.}}\left\{T\left(t\right)-\stackrel{\_}{T}\right\}{\left\{D\left(t\right)-\stackrel{\_}{D}\right\}}^T\end{array}\\ R_{\mathrm{DT}}=\frac{1}{l-1}\underset{t=1}{\overset{l}{.Math.}}\left\{D\left(t\right)-\stackrel{\_}{D}\right\}{\left\{T\left(t\right)-\stackrel{\_}{T}\right\}}^T\end{array}$ where T(t) represents a tth temperature measurement sample; T represents a mean-vector of temperature data; D(t) represents a tth displacement measurement sample; D represents a mean-vector of displacement data; l represents a number of samples; R.sub.TT represents a covariance matrix of temperature data; R.sub.DD represents a covariance matrix of displacement data; R.sub.TD represents a cross-covariance matrix of temperature and displacement data; R.sub.DT represents a cross-covariance matrix of displacement and temperature data: determine a pair of base vectors, i.e., u.sub.1, and v.sub.1, to maximize a correlation between a linear combination of temperatures u.sub.1.sup.TT and a linear combination of displacements v.sub.1.sup.TD, their correlation coefficient is as follows: $\rho \left(u_1,v_1\right)=\frac{u_1^T{R}_{\mathrm{TD}}{v}_1}{\sqrt{u_1^T{R}_{\mathrm{TT}}{u}_1.Math.v_1{R}_{\mathrm{DD}}{v}_1}}$ solving a combination coefficients of the base vectors, which maximize the correlation coefficient, are described by the following optimization problem: $\hspace{1em}\{\begin{array}{cc}{\max }_{u_1,v_1}& u_1^T{R}_{\mathrm{TD}}{v}_1\\ s.t.& u_1^T{R}_{\mathrm{TT}}{u}_1=1\\ & v_1^T{R}_{\mathrm{DD}}{v}_1=1\end{array}$ solving combination coefficients of subsequent base vectors, and solving the combination coefficients of all base vectors a following eigenvalue decomposition:
R.sub.TT.sup.−1R.sub.TDR.sub.DD.sup.−1R.sub.DT=UΓU.sup.T
R.sub.DD.sup.−1R.sub.DTR.sub.TT.sup.−1R.sub.TD=VΓV.sup.T where Γ=diag(γ.sub.1, γ.sub.2, . . . , γ.sub.k) is a diagonal eigenvalue matrix; γ.sub.i=ρ.sup.2(u.sub.i, v.sub.i) is an ith eigenvalue; U=[u.sub.1u.sub.2, . . . , u.sub.k] and V=[v.sub.1, v.sub.2, . . . , v.sub.k] are eigenvector matrices; k=min(n,n) is a number of non-zero solutions; step 2: establish a relationship model between canonically correlated temperatures and displacements define an ith canonically correlated temperature, i.e., T.sub.c,i i=1, 2, . . . , k, as follows:
T.sub.c,i=u.sub.i.sup.TT establish a temperature displacement relationship model for bridge expansion joints, using canonically correlated temperatures, as follows: $\left[\begin{array}{c}{\hat{D}}_1\\ {\hat{D}}_2\\ \mathrm{.Math.}\\ {\hat{D}}_n\end{array}\right]=\left[\begin{array}{cccc}{\beta }_{1,1}& {\beta }_{1,2}& \mathrm{.Math.}& {\beta }_{1,k}\\ {\beta }_{2,1}& {\beta }_{2,2}& \mathrm{.Math.}& {\beta }_{2,k}\\ \mathrm{.Math.}& \mathrm{.Math.}& \ddots & \mathrm{.Math.}\\ {\beta }_{n,1}& {\beta }_{n,2}& \mathrm{.Math.}& {\beta }_{n,k}\end{array}\right]\left[\begin{array}{c}{T}_{c,1}\\ T_{c,2}\\ \mathrm{.Math.}\\ T_{c,k}\end{array}\right]+\left[\begin{array}{c}{\beta }_{1,0}\\ {\beta }_{2,0}\\ \mathrm{.Math.}\\ {\beta }_{n,0}\end{array}\right]$ where {circumflex over (D)}.sub.i represents an estimated displacement of the ith expansion joint, i=1, 2, . . . , n; β represents a regression coefficient; step 3: construct control chart based alarming method define an error of temperature displacement relationship model as follows:
E.sub.i={circumflex over (D)}.sub.i−D.sub.i where E.sub.i represents the error of temperature displacement relationship model of an ith expansion joint displacement, i=1, 2, . . . , n; let E(t) represent an error sequence of an expansion joint, t=1, 2, . . . , l, a mean-value and a standard variation of which are as follows: $\begin{array}{c}\stackrel{\_}{E}=\frac{1}{l}\underset{t=1}{\overset{l}{.Math.}}E\left(t\right)\\ {\sigma }_E=\sqrt{\frac{1}{l-1}\underset{t=1}{\overset{l}{.Math.}}{\left\{E\left(t\right)-\stackrel{\_}{E}\right\}}^2}\end{array}$ where Ē represents a mean-value of the error sequence; σ.sub.E represents a standard variation of the error sequence; construct a mean-value control chart to the error sequence to realize performance alarming of expansion joints, and three parameters of the mean-value control chart are as follows:
UCL=Ē+ασ.sub.E
CL=Ē
LCL=Ē−ασ.sub.E where UCL represents an upper control limit; CL represents a center line; LCL represents a lower control limit; α represents a scaling factor which can be determined according to a given significance level; step 4: determine reasonable control limit calculate an absolute value of the error sequence, and estimate its probability density function, then obtain a cumulative density function and an inverse cumulative density function: as a result, a control limit of the absolute value of the error sequence, i.e., L, is calculated as:
L=F.sup.−1(1−θ) where F.sup.−1(.Math.) represents an inverse cumulative density function of the absolute value of the error sequence; θ represents a significance level; a calculation formula for the scaling factor α is as: $\alpha =\frac{L}{{\sigma }_E}$ the upper and lower control limits can then he determined through the scaling factor; feed a newly acquired temperature and displacement monitoring data into the temperature displacement relationship model of bridge expansion joints, and a model prediction error of an expansion joint E can be obtained; criteria for judging performance degradation of expansion joints is as:
E>UCL
E<LCL determining that the performance of expansion joints degrades when E>UCL and/or E<LCL.

Description

FIGURE ILLUSTRATION

(1) The sole FIGURE describes the solution process of canonically correlated temperatures.

DETAILED DESCRIPTION

(2) The following details is used to further describe the specific implementation process of the present invention.

(3) The monitoring data of temperatures and expansion joint displacements, acquired during 14 months, from a long-span bridge is used to verify the validity of the present invention. The monitoring data acquired during the first 12 months is used as training dataset, which represents the intact state of expansion joints; whereas the monitoring data acquired during the last 2 months is used as testing dataset, which represents the unknown state of expansion joints.

(4) The detailed implementation process is as follows:

(5) (1) Obtain canonically correlated temperatures from the training dataset (the solution process can be seen in the sole FIGURE), and then establish temperature displacement relationship model for bridge expansion joints using canonically correlated temperatures.

(6) (2) Construct mean-value control chart to the modelling error of the temperature displacement relationship model, and calculate the corresponding upper and lower control limits of the control chart.

(7) (3) Simulate performance degradation of expansion joints in the testing dataset; feed the testing data into the temperature displacement relationship model to obtain the prediction error of the expansion joint displacement; Compare the prediction error with the upper and lower control limits, and trigger a performance alarm when the error falls beyond the control limits; results show that the alarming rate achieves more than 99%, when the performance degradation of expansion joints achieves a severity of 8 mm.