Technical Note Bridge Influence Line Identification Based on Regularized Least-Squares QR Decomposition Method Downloaded from ascelibrary.org by UNIVERSITY OF NEW SOUTH WALES on 08/24/19. Copyright ASCE. For personal use only; all rights reserved. Xu Zheng1; Dong-Hui Yang2; Ting-Hua Yi, M.ASCE3; Hong-Nan Li, F.ASCE4; and Zhi-Wei Chen5 Abstract: The bridge inﬂuence line is an important tool to study the bridge response under moving loads and contains tremendous structural information. Structural deterioration during the bridge’s service life will induce variation in the bridge inﬂuence line, which makes it essential to identify the inﬂuence line exactly from the ﬁeld measurement data under moving vehicle loads. This article establishes the underdetermined inﬂuence line identiﬁcation model for a bridge under multiaxle vehicle excitation, in which the bridge inﬂuence line is identiﬁed based on the regularized least-squares QR decomposition method. Then an example of a four-span girder bridge is provided to prove the validity of the method, and the accuracy and computational complexity of different methods in this condition are compared. This study thus provides an effective method for the identiﬁcation of the bridge inﬂuence line with very high computational efﬁciency and identiﬁcation accuracy. DOI: 10.1061/(ASCE)BE.1943-5592.0001458. © 2019 American Society of Civil Engineers. Author keywords: Inﬂuence line; Performance evaluations; Weigh-in-motion; Least-squares QR (LSQR); Tikhonov regularization. Introduction In the last 30 years, many large infrastructures have emerged worldwide, especially in China. However, after a long time of service, it is vital to determine the deterioration condition of these infrastructures. Long-span bridges are the pivotal projects of many transportation systems. With increased concern about the service performance and condition of such bridges (Asadollahi and Li 2017; Yang et al. 2017), getting bridge information in real-time becomes an important research ﬁeld. The inﬂuence line is a static property of the bridge and is directly related to bridge ﬂexibility; it also has important applications in bridge weight-in-motion system (Moses 1979), damage detection (Zeinali and Story 2017), model correction (Strauss et al. 2012), and bridge evaluation (Hirachan and Chajes 2005). However, bridge inﬂuence lines will change when bridge stiffness and constraint conditions change, and the identiﬁcation of the bridge inﬂuence line from direct measurement data becomes an important issue in these ﬁelds. The inﬂuence line can be extracted from structural response data induced by moving loads in several ways. OBrien et al. (2006) ﬁrst presented a least-squares method to extract the inﬂuence line from measurement data. Ieng (2015) presented a more direct 1 Ph.D. Candidate, School of Civil Engineering, Dalian Univ. of Technology, Dalian 116023, China. Email: [email protected] .edu.cn 2 Assistant Professor, School of Civil Engineering, Dalian Univ. of Technology, Dalian 116023, China. Email: [email protected] 3 Professor, School of Civil Engineering, Dalian Univ. of Technology, Dalian 116023, China (corresponding author). Email: [email protected] 4 Professor, School of Civil Engineering, Dalian Univ. of Technology, Dalian 116023, China. Email: [email protected] 5 Associate Professor, Department of Civil Engineering, Xiamen Univ., Xiamen 361005, China. Email: [email protected] Note. This manuscript was submitted on October 29, 2018; approved on March 27, 2019; published online on May 20, 2019. Discussion period open until October 20, 2019; separate discussions must be submitted for individual papers. This technical note is part of the Journal of Bridge Engineering, © ASCE, ISSN 1084-0702. © ASCE identiﬁcation method using the maximum-likelihood estimator principle, which generalizes the matrix method by simultaneously taking into account measurements available from as many calibration trucks as needed and leads to a more robust estimation. Frøseth et al. (2017) present a new method to extract the inﬂuence line in the frequency domain. The method reduces the complexity of inﬂuence line extraction by several orders of magnitude compared with the conventional matrix method. It can be summarized that the previous methods for the identiﬁcation of the inﬂuence line have already made great breakthroughs, but more development is needed. First, all of the matrix-based methods need to take the inverse of the vehicle information matrix. That method results in calculation complexity, and the calculation also needs a large amount of additional storage space. Second, the response equation is actually an underdetermined equation. In the previous methods, the inﬂuence lines are extracted through an overdetermined equation. In some cases, these methods will give rise to inaccuracy of the inﬂuence line. To solve the problems related to the accuracy and computational efﬁciency of the existing methods for inﬂuence line identiﬁcation, this study investigated an improved time-domain model of the problem of bridge inﬂuence line identiﬁcation and a new algorithm focused on this model. The new mathematical model is suitable for general test conditions, and the model can be directly used with measured data instead of establishing a mathematical model for each test condition. The use of the regularization method makes the identiﬁed bridge inﬂuence line more accurate, and the least-squares QR decomposition (LSQR) method makes real-time identiﬁcation of the inﬂuence line possible. The article is organized as follows: First, the complete forms of the vehicle information matrix and response equation are established, and the relationship of the vehicle information matrix and bridge response is illustrated. Then the normal equation of Tikhonov regularization is presented, and the regularization coefﬁcient is properly chosen. After that, the iterative LSQR method is introduced to extract the inﬂuence line. Finally, an example of inﬂuence line identiﬁcation in a four-span girder bridge is presented to prove the efﬁciency of the proposed method. 06019004-1 J. Bridge Eng., 2019, 24(8): 06019004 J. Bridge Eng. between the ﬁrst axle and the ith axle. The formula of Ci is as follows: Theoretical Formulations for Influence Line Identification Ci ¼ Complete Vehicle Information Matrix and Response Equation Downloaded from ascelibrary.org by UNIVERSITY OF NEW SOUTH WALES on 08/24/19. Copyright ASCE. For personal use only; all rights reserved. N X Ai w ðkCi Þ (2) where f = sampling frequency of the sensor; and v = velocity of the calibration vehicle. The sampling frequency depends on the velocity of the vehicle and the required accuracy of the inﬂuence line. Then the response equation can be written in matrix form as follows: When a calibration vehicle with known axle weight Ai and known axle distance Di passes over a bridge, it results in a series of measurements of the load effect RðkÞ. For each sampling point k, the corresponding response caused by the calibration vehicle is given by r ðkÞ ¼ Di f v R ¼ LU (1) (3) i¼1 where R = response vector; U = inﬂuence coefﬁcient vector; and L = vehicle information matrix. The following matrix shows the detail of L: where N = number of vehicle axles; w ðkCi Þ = inﬂuence coefﬁcient corresponding to the ith axle; and Ci = sampling point difference 2 6 6 6 6 6 L¼6 6 6 6 6 4 A1 0 A2 0 A3 Ak 0 0 A1 0 A2 0 A2 Ak 0 .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 0 0 0 A1 0 A2 0 This vehicle information matrix has very explicit physical meaning. Each row of the matrix represents a relationship between N inﬂuence coefﬁcients and one bridge response. If there is a total of m sampling points for bridge response, there will be m þ CN 1 inﬂuence coefﬁcients involved in the response. Therefore, the complete response equation is a kind of underdetermined equation. However, this equation can be converted to an overdetermined equation in some test conditions. In the inﬂuence line test of shortspan bridge, the whole bridge is tested. In this condition, the inﬂuence coefﬁcients in the starting points and ending points can be zero. If the contribution of these coefﬁcients is ignored, the equation can be converted to an overdetermined one. However, the inﬂuence coefﬁcients in the starting points and ending points are not always zero in ﬁeld tests of bridge inﬂuence lines. In this condition, ignoring these coefﬁcients will cause instability of the identiﬁed bridge inﬂuence line. Therefore, to retain more inﬂuence line information, the complete vehicle information matrix should be used. Inverse Problems and the Regularization Method In the real test condition, the response equation [Eq. (3)] is always ill-posed because a small ﬂuctuation in R may result in considerable errors in the solution of U. Fluctuations in response are inevitable because of the dynamic effect, rail irregularity, and other factors. Therefore, the real measurement R can be expressed as the sum of the following two parts: R ¼ Rr þ e A2 3 7 0 7 7 7 .. 7 .. . . 7 7 7 .. 7 .. . . 7 5 Ak mðmþC (4) N 1Þ Commonly used regularization methods include Tikhonov regularization, sparse regularization, elastic net regularization, and truncated singular value decomposition. Among these regularization methods, the Tikhonov regularization method can give the most acceptable results. The expression of the Tikhonov regularization method is as follows: h i (6) min kR LUk22 þ λkTUk22 This means that to solve the ill-posed equation, a vector U that can minimize the sum of two Euclidean norms should be chosen. In Eq. (6), λ represents the regularization coefﬁcient, and T represents the regularization matrix. In this article, the regularization matrix is chosen as follows (Reichel and Rodriguez 2013): 1 0 1 2 1 C B C B 1 2 1 C B C (7) T¼B .. .. .. C B C B . . . A @ 1 2 1 ðn2Þn To solve this problem, the ﬁrst-order derivative of the Euclidean norms with respect to vector U is taken. After that, the Tikhonov regularization can be expressed by ðLT L þ λ2 TT TÞU ¼ LT R (5) where Rr = ideal static response of the bridge; and e = errors. To reduce these errors, the regularization method is commonly used. © ASCE 0 (8) Then the regularization coefﬁcient λ is determined. To ﬁnd the optimal regularization coefﬁcient, the L-curve method (Morigi and 06019004-2 J. Bridge Eng., 2019, 24(8): 06019004 J. Bridge Eng. Sgallari 2001) can be used. It is shown that the plot of the curve (logkLU Rk; logkTUk) is shaped roughly like an L. In inﬂuence line identiﬁcation problems, the identiﬁed inﬂuence line is very smooth when the regularization parameter corresponds to the L-corner. However, some important vertex information is ﬁltered. To avoid this phenomenon, another regularization parameter needs to be selected in the transition area between the horizontal and the vertical segment. (a) Downloaded from ascelibrary.org by UNIVERSITY OF NEW SOUTH WALES on 08/24/19. Copyright ASCE. For personal use only; all rights reserved. Influence Line Identification by LSQR Method The LSQR method is an iterative method for solving Ax ¼ b and mink Ax bk2 , where the matrix A is large and sparse. In this article, the LSQR method and Tikhonov regularization method are combined, which means the method can be used to solve Eq. (8). Compared with the least-squares method and frequency-domain method, the LSQR method can solve underdetermined response equation [Eq. (3)] with the complete vehicle information matrix [Eq. (4)] because the method can return the minimal normal leastsquares solution with good numerical stability. This feature gives the LSQR method good identiﬁcation results in different test conditions. At the same time, as an iteration method, the LSQR method saves storage space and has very fast convergence speed in the calculation (Paige and Saunders 1982). The algorithm of LSQR is shown as follows (the iteration coefﬁcients in the method will not be introduced in detail): 1. Initializing (b) 1 ¼ b 1; b 1 u1 ¼ b; a1 v1 ¼ AT u1 ; w1 ¼ v1 ; x0 ¼ 0; f (c) r 1 ¼ a1 2. Bidiagonalization processing b iþ1 uiþ1 ¼ Avi ai ui ; aiþ1 viþ1 ¼ AT uiþ1 b iþ1 vi 3. Orthogonal transformation processing 1=2 r i ¼ r 2i þ b 2iþ1 ; ci ¼ r i = r i ; si ¼ b iþ1 = r i ; iþ1 ¼ si f i; f i u iþ1 ¼ si aiþ1 ; r iþ1 ¼ ci aiþ1 ; f i ¼ ci f (d) 4. Updating x, w xi ¼ xi1 þ f i = r i wi :; wiþ1 ¼ viþ1 u iþ1 = r i wi 5. Convergence testing Stop the iteration when the stopping criterion is satisﬁed. Illustrative Example of a Four-Span Girder Bridge Description of Bridge and Calibration Truck Condition The four-span girder bridge is widely used in the engineering practice. A simpliﬁed model of a four-span girder bridge is presented in Fig. 1(a). For the convenience of discussion, uniformly distributed stiffness is assumed in this article. The calibration truck has ﬁve axles, and the axle loads of the truck are 210, 220, 125, 125, and 115 kN, respectively. The axle distances from the ﬁrst axle are 0.5, 2, 2.5, and 4 m, respectively. The calibration truck will go through the beam with constant velocity v ¼ 1 m/s, and the bending moment of the point is collected in the sampling rate of 10 Hz. At this © ASCE (e) Fig. 1. Inﬂuence line identiﬁcation for partial bridge test condition: (a) partial bridge test condition; (b) bending-moment response of bridge; (c) inﬂuence line identiﬁed by least-squares/RLSQR methods and baseline calculated by virtual work principle; (d) inﬂuence line identiﬁed by FD/RLSQR methods and baseline calculated by virtual work principle; and (e) inﬂuence line identiﬁed by ﬁltered FD/RLSQR methods and baseline calculated by virtual work principle. 06019004-3 J. Bridge Eng., 2019, 24(8): 06019004 J. Bridge Eng. Downloaded from ascelibrary.org by UNIVERSITY OF NEW SOUTH WALES on 08/24/19. Copyright ASCE. For personal use only; all rights reserved. velocity, the dynamic effect of the truck can be ignored. The bending-moment responses of the point K are calculated by Eq. (3). To simulate the responses in direct measurement, Gaussian white-noise disturbance with a signal-to-noise ratio of 15 is added in the bending-moment responses before calculation. The calculation accuracy will be compared among the baseline calculated by the virtual work principle, least-squares method, frequencydomain (FD) method, and LSQR method. Table 1. Comparison of computational complexity among three methods Method Complexity Least squares Frequency domain LSQR O(m2) O(m log2m) O(im) Identified Influence Line Comparison among Different Methods In this section, the bridge inﬂuence line identiﬁed by different methods is compared in a partial bridge test condition. In this condition, the back axles of the calibration truck are already on the bridge when the test starts, and the front axles are also still on the bridge when the test ends. Therefore, the bridge responses collected by the sensor are inﬂuence coefﬁcients. This kind of condition is used in the inﬂuence line test of part of a long-span bridge. The arrangement of the test is presented in Fig. 1(a), and the bending-moment responses collected by the sensor are shown in Fig. 1(b). In this case, the least-squares method tends to be unstable in the two ends of the inﬂuence line. The reason for this instability is that the contribution of the inﬂuence coefﬁcients in the two sides of the bridge is ignored. This simpliﬁcation will cause large ﬂuctuation in the boundary. For the FD method, the identiﬁcation result is similar to the least-squares method because these two methods use a similar mathematical model. After considering the regularizing ﬁlter, the boundary and other parts of the inﬂuence line tend to be stable, but the sharp vertex of the inﬂuence line is reduced. Conversely, the LSQR method gives an accurate prediction of the inﬂuence line in that condition because all of the contributions of the inﬂuence coefﬁcients are carefully considered. A comparison of these three methods is presented in Figs. 1(c, d, and e). Regarding the computational complexity of different algorithms for the identiﬁcation of the inﬂuence line, the FD method mainly uses fast Fourier transformation to solve the bridge inﬂuence line, and the computational complexity is approximately Oðm log2 mÞ. The least-squares method can be quickly solved by a spare matrix-vector algorithm in Oðm2 Þ. There will be 2Nm þ 4m þ ið4Nm þ 8mÞ total multiplications in the LSQR algorithm, where i represents the total iteration steps, and N represents the axle number. Therefore, the computational complexity of the LSQR method is approximately OðimÞ. When the stopping criterion has been met, the number of iteration steps is usually less than 20 in inﬂuence line identiﬁcation problems. The computational complexity of the three methods is summarized in Table 1, and a comparison of these three methods when the signal size varies from 103 to 107 is presented in Fig. 2. Conclusions This study brieﬂy introduces the development of inﬂuence line identiﬁcation technology and subsequently discusses the deﬁciencies of the current methods, based on which the regularized LSQR method is proposed for inﬂuence line identiﬁcation when the moving vehicle goes through the bridge. Finally, a four-span continuous girder bridge is presented to illustrate the validity of the method. The following conclusions can be drawn: 1. A new model for underdetermined inﬂuence line identiﬁcation is established. Compared with traditional inﬂuence line identiﬁcation models, the new mathematical model can consider all of the inﬂuence coefﬁcients involved in the inﬂuence line test, which © ASCE Fig. 2. Computational complexity comparison. can increase the accuracy of the inﬂuence line prediction. The model is then solved by the regularized LSQR method. 2. The results of the illustrative example show that the new inﬂuence line identiﬁcation model and method have especially good accuracy in the condition when only a part of the bridge is tested. Compared with the least-squares method, the regularized LSQR method gives more inﬂuence line information and a more accurate inﬂuence line boundary. Compared with the FD method, the regularized LSQR method can better predict the peak value of the inﬂuence line. In general, the new method can better preserve the inﬂuence line information. 3. The new method has very low computational complexity compared with other time-domain methods. The regularized LSQR method is at least 102 times faster than the traditional matrix method. With the improvement of sampling precision and enlargement of signal size, the advantage will be more obvious. Moreover, the computational complexity of the method is on the same order as that of the FD method. Acknowledgments This research work was jointly supported by the National Natural Science Foundation of China (51625802 and 51708088), the 973 Program (2015CB060000), and the Foundation for High-Level Talent Innovation Support Program of Dalian (2017RD03). References Asadollahi, P., and J. Li. 2017. “Statistical analysis of modal properties of a cable-stayed bridge through long-term wireless structural health monitoring.” J. Bridge Eng. 22 (9): 04017051. https://doi.org/10.1061 /(ASCE)BE.1943-5592.0001093. Frøseth, G. T., A. Rønnquist, D. Cantero, and O. Øiseth. 2017. “Inﬂuence line extraction by deconvolution in the frequency domain.” Comput. Struct. 189: 21–30. https://doi.org/10.1016/j.compstruc.2017.04.014. Hirachan, J., and M. Chajes. 2005. “Experimental inﬂuence lines for bridge evaluation.” Bridge Struct. 1 (4): 405–412. https://doi.org/10.1080 /15732480600578485. 06019004-4 J. Bridge Eng., 2019, 24(8): 06019004 J. Bridge Eng. 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