Damage Assessment of the Z24 bridge by FE Model Updating. Anne Teughels 1, Guido De Roeck

Transcription

1 Damage Assessment of the Z24 bridge by FE Model Updating Anne Teughels, Guido De Roeck Katholieke Universiteit Leuven, Department of Civil Engineering Kasteelpark Arenberg 4, B 3 Heverlee, Belgium Anne.Teughels@bwk.kuleuven.ac.be Research assistant with National Fund for Scientic Research (FWO Flanders) Keywords: FE model updating, damage identication, modal parameters, damage functions, civil structures. Abstract. Damage to civil engineering structures can be identied with a nite element (FE) model updating method using experimental modal data. In such a procedure the uncertain properties (e.g. stiffness distribution) in the FE model are adjusted by minimizing the differences between the measured modal parameters and the numerical (FE) predictions. In civil engineering the differences in eigenfrequencies and mode shapes are minimized, mostly identied from ambient vibrations. Since the modal data are nonlinear functions of the uncertain properties, an iterative sensitivity-based minimization method is used to solve this inverse problem. In order to reduce the number of unknowns, damage functions are used. The FE model updating technique is applied to a prestressed concrete bridge with 3 spans whose girder is damaged by lowering one of the intermediate piers. The damage pattern is identied (localized and quantied) by updating the Young s and the shear modulus. Introduction Accurate condition assessment of civil engineering structures has become increasingly important. FE model updating provides a very efcient, nondestructive, global damage identication technique. The uncertain properties of the FE model are updated by minimizing the discrepancies between the measured modal data and those computed with the numerical FE model [, 2]. The damage identication procedure is performed in two updating processes. In the rst the initial FE model is tuned to a reference state, i.e. the undamaged structure. In the second process the reference FE model is updated to obtain a model which can reproduce the experimental modal data of the damaged state. The damage is identied by comparing the differences between the reference and the damaged FE model. The technique is applied to the Z24 bridge in Switzerland. It is a prestressed concrete bridge with three spans which is damaged by lowering one of the intermediate piers. A nonlinear least squares problem is solved. The residual vector contains the test/analysis differences of the rst 4 bending and/or torsion modes. Frequency residuals as well as mode shape residuals are minimized. Eigenfrequencies contain global, accurate information, whereas mode shapes provide important local, but more noisy information. Therefore, both types of residuals are weighted with an appropriate factor in the residual vector. The updating parameters are both the Young s and the shear modulus of all the girder elements. The least squares problem is solved with a sensitivity-based Gauss-Newton algorithm. In order to improve the condition of the sensitivity matrix the number of unknown parameters is reduced by using a limited set of damage functions [2]. The girder stiffness distribution is found by combining these damage functions, multiplied with the appropriate factors which are the actual variables of the minimization problem. Only linear damage functions are used, but the method can be extended by including higher order functions. With this approach always a realistic smooth result is obtained. A damage pattern is identied which resembles the observed one. The general updating procedure and the application to the Z24 bridge are presented in the paper.

2 General FE model updating procedure Objective function. In FE model updating an optimization problem is set up in which the differences between the experimental and numerical modal data have to be minimized by adjusting the uncertain model properties []. The experimental modal data, i.e. the eigenfrequencies ~ν j and mode shapes f ~ j, are obtained from measurements. In civil engineering, the measurements are often obtained in operational conditions (ambient vibrations), which means that the exciting forces (coming from wind, trafc,...) are unknown. As a consequence, an absolute scaling of the mode shapes is missing. Furthermore, only the translation degrees of freedom of the mode shapes can be measured. The minimization of the objective function is stated as a nonlinear least squares problem: min p f (p) = 2 kr(p)k2 = 2 r f (p) r s (p) 2 ; r f :IR n! IR m f r s :IR n! IR ms () in which k:k denotes the Euclidean norm. The residual vector r :IR n! IR m contains the frequency residuals r f and mode shape residuals r s (m = m f + m s ). The vector p 2 IR n represents the set of design variables. Both types of residuals can be formulated respectively as [2]: r f (p) = j(p) ~ j ~ j ; with j =(2ßν j ) 2 (2) r s (p) = fl j (p) f r j (p) ~ f l j ~f r j ; (3) where l and r denote respectively an arbitrary and a reference degree of freedom of mode shape f j (or ~ f j ) (Fig. ). Relative differences are taken in r f in order to obtain a similar weight for each frequency residual. In r s each mode shape component l is divided by a reference component r which is a component with a large amplitude, since the numerical and experimental mode shapes can be scaled differently. r φ [ ] l... Fig. : Mode shape f with its components l and one reference component r. Since the modal data (ν and f) are nonlinear functions of the uncertain model properties, Eq. is a nonlinear least squares problem. It is solved with an iterative sensitivity-based optimization method. The characteristics of the least squares problem can be exploited, namely the gradient and the Hessian of the objective function (Eq. ) have the following special structure: rf (p) = mx j= r 2 f (p) = J p (p) T J p (p) + r j (p)rr j (p) =J p (p) T r(p) (4) mx j= r j (p)r 2 r j (p) ß J p (p) T J p (p) (5)

3 with J p the Jacobian matrix (or sensitivity matrix), containing the rst partial derivatives of the residuals r j (r f and r s ) with respect to p. In the Gauss-Newton method [3], the Hessian is approximated with the rst order term in Eq. 5, which is equivalent with solving the following linear least squares problem in each iteration k: min p kr(p 2 k )+J(p k ) pk2 ; with p k+ = p k + p k : (6) In the paper, Matlab-software [3] is used to apply the Gauss-Newton method. Furthermore, we have chosen the trust region implementation in order to stabilize the optimization problem. Design variables p. One or more uncertain physical properties X (e.g. the Young s modulus) are updated in each element e of the numerical FE model. A dimensionless correction factor a e expresses the updated value of property X relative to its initial value X e in element e: a e = X e X e =) X e X e = X e ( ae ): (7) The correction factors can affect one element or may be assigned to an element group. If the uncertain physical property is linearly related to the stiffness matrix of the element (group), we have: K e = K e ( ae ) and K = K u + Xn e e= K e ( ae ) (8) where K e and K e are the initial and updated element stiffness matrix respectively, K is the global stiffness matrix and K u is the stiffness matrix of the element (group) whose properties remain unchanged. n e is the number of elements (groups) that are updated. Adjusting the model property of all the elements separately would result in a high number of updating variables fa e g, which causes the sensitivity matrix J to become ill-conditioned for the same residual vector r. Furthermore, a physically meaningful optimization result is not guaranteed since neighbouring elements can be adjusted independently. Therefore, the distribution of the correction factors fa e g which dene on their turn the distribution of the updated physical properties X over the FE model is approximated by combining a limited set of global damage functions N i [2]. For the correction factor in element e, wehave: a e = nx i= p i N i (x e ) with n: the number of damage functions N i (x) (n n e ); p i : their multiplication factor; and x e : the coordinates of the center of element e. In vector notation we have: a ne =[N ] ne n p n or 8 >< >: a a 2. a ne 9 >= >; = N (x ) N 2 (x ) ::: N n (x ) N (x 2 ) N 2 (x 2 ) ::: N n (x 2 ).. :::. N (x ne ) N 2 (x ne ) ::: N n (x ne ) In this paper piecewise linear functions are used (Fig. 2), varying between and, which results in a piecewise linear approximation of the continuous distribution of the physical properties. Analogously as shape functions in FE theory, the damage functions are dened on a mesh of damage elements, which on its turn is dened on top of the mesh of nite elements. The accuracy of the result is determined by the coarseness of the damage element mesh rather than its specic layout and it can be improved by rening the mesh, resulting in more linear pieces (damage elements) used By discretising the continuous distribution in the center points, we take the correction factor constant for each element. 3 8 >< 7 5 >: p p 2. p n 9 >= >; (9) ()

4 to approximate the continuous distribution. Alternatively, also higher order functions can be used to improve the accuracy. Both means result in more unknown parameters p i to be identied. Changing the layout of the mesh but keeping the same neness, on the other hand, results in similar overall approximations due to the equal level of discretisation and thus does not improve the accuracy. For damage identication, rst a coarse mesh can be used to locate the damage and simultaneously assess its severeness in broad outlines. If required, a more detailed damage pattern can then be identied in a second phase by updating only the elements at the damaged zone using a ner mesh. (a) A set of 9 damage functions (b) isolated damage function N i N Element n o Element n o Fig. 2: Piecewise linear damage functions to approximate the distribution of the element correction factors a e. Each set of multiplication factors p denes all the updating parameters a in a unique sense, with the linear combination of Eq.. Therefore, the only unknown variables of the optimization problem are these multiplication factors p. Hence, this approach reduces the number of unknowns considerably, resulting in a robust optimization method. Furthermore, it generates always a smooth distribution of the model properties, being a weighted sum of smooth global damage functions. Sensitivity matrix. The sensitivities of the residuals r j with respect to the correction factors a e e = s and e l j f r fl r j e (f r j : () e The modal sensitivities in Eq. are calculated using the formulas of Fox and Kapoor [4]. If only stiffness parameters have to be corrected, the formulas of Fox and Kapoor are simplied e = f f Eq:8 e j = f T j Ke f j j dx f e = (f q q=;q6=j j f e j ) Eq:8 dx f q = ( f T q K e f q=;q6=j j j ): (3) q Instead of the complete base (d is the analytical model order) a truncated base is used, which should be high enough in view of the condition of the sensitivity matrix. The expressions of Eq. are used to calculate the sensitivity matrix J a of the residual vector r with respect to the correction factors a. In the optimization procedure, however, we need the sensitivities of r with respect to the design variables p. Using the mutual dependency between a and p expressed by Eq., each component of the sensitivity matrix J p can be calculated i = Xn e e i Xn e e N i(x e ); (4) in which the expressions of Eq. have to be lled in. Equivalently, in matrix notation, we have: [J p ] m n =[J a ] m n e [N ] ne n (5) where J p and J a are the sensitivity matrices with respect to the design variables p and the element correction factors a respectively. N is the matrix containing the global damage functions.

5 Weighting. Eigenfrequencies are a good indicator for damage in general and can be measured quite accurately. However, it is difcult to detect zones of local damage using only eigenfrequencies. Mode shapes permit a more detailed prediction of the damage distribution, but measurements are more noisy. Therefore, in the objective function (Eq. ) the mode shape residual vector r s is scaled with a factor C, such that its squared norm is W sf times the squared norm of the frequency residual vector r f. r s;scaled = C r s ) kr s;scaled k 2 kr f k 2 = W sf : (6) Using Eq. 6 the scaling factor C can be dened in each iteration, given the desired weight ratio W sf. Additionally and before applying the scaling with W sf, the mode shape residual vectors r s;fj corresponding to each mode shape f j are scaled such that their norm is equal: [subscript scaled omitted] kr s;f k 2 =kr s;f2 k 2 = ::: =kr s;fmf k 2 : (7) Damage detection of bridge Z24 The FE model updating technique is applied to identify the damage of the Z24 bridge in Switzerland. It is a prestressed concrete bridge with three spans, supported by two intermediate piers and a set of three columns at each end (Fig. 3). Both types of supports are rotated with respect to the longitudinal axis which results in a skew bridge. The overall length is 58 m. In the framework of the Brite EuRam Programme CT SIMCES [5], one of the central piers (at 44 m) is lowered by 95 mm, inducing cracks in the bridge girder above this pier. The modal data are identied before and after applying the damage. It is the aim to detect, localize and quantify the damage pattern by adjusting the stiffness of the bridge girder. (a) Elevation (c) Box section (b) Top view Fig. 3: Highway bridge Z24, Switzerland. Experimental modal data. Measurements are performed in operational conditions. The Stochastic Subspace Identication technique [6] is used to extract the modal data from the ambient vibrations. The rst 4 eigenmodes are identied. Accelerometers are placed on the bridge deck along 3 parallel measurement lines: at the centerline and along both sidelines (front and back). The measured mode shapes of the undamaged and damaged bridge are plotted in Fig. 4. The eigenfrequencies are given in Table.

6 Undamaged bridge (Reference state) Damaged bridge Exper. Numerical FEM Exper. Numerical FEM Initial Updated Reference Updated Mode ~ν j [Hz] ν j MAC [%] ν j MAC ~ν j ν j MAC ν j MAC Table : Measured and calculated modal data of the undamaged and damaged bridge (a) Mode : pure bending undamaged:front undamaged:center undamaged:back damaged:front damaged:center damaged:back (b) Mode 2: coupled bending-torsion undamaged:front undamaged:center undamaged:back damaged:front damaged:center damaged:back (c) Mode 3: coupled bending-torsion undamaged:front undamaged:center undamaged:back damaged:front damaged:center damaged:back (d) Mode 4: pure bending undamaged:front undamaged:center undamaged:back damaged:front damaged:center damaged:back Fig. 4: Experimental mode shapes of the undamaged and damaged bridge. FE model. The bridge is modelled with a beam model. Equivalent values for the bending and torsion stiffness of the box section (Fig. 3c) are calculated. The bridge has higher stiffness values above the supporting piers (Fig. 5a,b) because of the increased thickness of bottom and top slab. The principal axes of the piers are rotated to model the skewness of the bridge (Fig. 3b). In order to account for the influence of the soil, soil springs are included at the pier foundations and at the end abutments. The stiffness values of the soil springs are determined by a prior updating process. FE model updating process. When applied to damage detection, two updating processes are performed: the rst to determine a reference (undamaged) state; the second to identify the damage pattern, with respect to the reference state. In both processes the procedure is analogous. The residual vector in the objective function (Eq. ) contains the test/analysis differences of the 4 bending and torsion modes. So there are 4 frequency residuals r f. For the mode shape residuals r s, the vertical displacements measured along the three lines (Fig.4) are used. The weight ratio W sf is 8.

7 The unknown model properties are the bending and torsion stiffness of the bridge girder. They will be updated both, by adjusting the Young s and shear modulus (E;G) of all (82) girder elements: E e upd = Ee ref ( ae E ) and Ge upd = Ge ref ( ae G ): (8) Both properties are corrected with respect to a reference value (E ref ;G ref ), which is the initial FE value in the rst updating process; in the second updating process it is the identied value obtained as the result of the rst updating process. In order to reduce the number of unknowns a (2 82), 2 sets of 9 linear damage functions are used (Fig. 2), which results in 8 design variables p (p E;i and p G;i ): a E = 9X i= p E;i N i and a G = 9X i= p G;i N i : (9) The initial values for p in both processes are set to zero, corresponding to zero initial correction factors a over the whole girder length. During the updating p E;i and p G;i are constrained to upper and lower bounds, namely :99 and :99 respectively (for i =2;:::;8) and : and : (for i =; 9). Results. The stiffness distribution for bending as well as for torsion of the bridge girder are plotted in Fig.5a,b. The initial and also the updated values for reference and damaged state are shown. In the reference state only minor changes of the stiffnesses are necessary at the side spans of the bridge. In Table the initial and updated eigendata are given. The updated numerical eigenfrequencies correspond globally better with the experimental values. For the mode shapes, a clear improvement of jf T j f ~ j j 2 the correlation can be observed, which is quantied by the MAC-values, MAC =, (f T j f j )( f ~ T ~ j f j ) [ f:experimental ~ value]. Especially both bending-torsion modes (nos. 2,3) are improved considerably. For the damaged state, a decrease in stiffness above the pier at 44 m, is clearly visible. This decrease is due to the lowering of the pier, which induced cracks in the beam girder at that location. The corresponding identied damage pattern, dened by the reduction factors a, is plotted in Fig.5c,d for bending and torsion stiffness respectively. It is a realistic smooth damage pattern, located in the expected cracked zone. But for the torsion stiffness, also a non-physical stiffness increase is obtained above the other pier (at 4 m). This anomaly might be due to measurement errors and modelling assumptions. In fact we have used a beam model, which is not able to model the structural behaviour of the box girder exactly (no modelling of restrained warping, shear lag effects,...). Also for the damaged bridge, the correlation between the numerical and experimental eigendata is improved very clearly with the updated FE model, and this for both, eigenfrequencies and mode shapes (Table ). In Fig. 6 the initial and updated fourth mode shape are shown. Conclusions A FE model updating method using modal data is presented. The updating procedure can be regarded as a parameter estimation technique which aims to t the uncertain parameters of an analytical model such that the model behaviour corresponds as closely as possible to the measured behaviour. The method is applied to identify damage in a highway bridge in Switzerland. The damage is represented by a reduction in bending and torsion stiffness of the bridge girder. For both properties a realistic damage pattern is identied with the updating method. Furthermore, a good correlation between the experimental and the updated numerical modal data is obtained. References [] M. I. Friswell and J. E. Mottershead. Finite Element Model Updating in Structural Dynamics. Kluwer Academic Publishers, Dordrecht, The Netherlands, 995.

8 (a) Bending stiffness EI EI :initial FE EI :reference FE ref EI :damaged state upd (b) Torsion stiffness GI t GI :initial FE GI :reference FE ref GI upd :damaged state EI y [ Nm 2 ] GI t [ Nm 2 ] (c) Correction factors a E;damaged (d) Correction factors a G;damaged a E Correction factor n o a G Correction factor n o Fig. 5: Bending and torsion stiffness, EI and GI t, and their correction factors for damaged state. (a) Experiment and reference FE model Exp: 2.3 Hz Ref FE: 2.9 Hz Experimental Reference FE (b) Experiment and updated FE model Exp: 2.3 Hz Upd FE: 2.37 Hz Experimental Updated FE Fig. 6: Mode shape 4 (damaged bridge): Experimental and numerical values at centerline. [2] A. Teughels, J. Maeck, and G. De Roeck. Damage assessment by FE model updating using damage functions. Computers and Structures, 8(25): , October 22. [3] MATLAB. Matlab Optimization Toolbox User s Guide. Version 2. (Release 2.), The Mathworks, 2. [4] R. Fox and M. Kapoor. Rate of change of eigenvalues and eigenvectors. AIAA Journal, 6: , 968. [5] J. Maeck and G. De Roeck. Description of Z24 benchmark. Mechanical Systems and Signal Processing, 7():27 3, January 23. [6] B. Peeters and G. De Roeck. Reference-based stochastic subspace identication for output-only modal analysis. Mechanical Systems and Signal Processing, 6(3): , 999.