VCUPDATE, A NUMERICAL TOOL FOR FINITE ELEMENT MODEL UPDATING Andrea Mordini VCE Vienna Consulting Engineers, Helmut Wenzel VCE Vienna Consulting Engineers, Austria Austria mordini@vce.at Abstract Due to the constant aging of civil engineering structures, the request of structural reliability evaluation and remaining life assessment is assuming a major importance. In this optic, the use of non-destructive dynamic testing in conjunction with advanced numerical techniques can provide a valid answer to these problems. In this field, the Finite Element Model Updating method is one of the most promising techniques. It provides the possibility to minimize the differences between the numerical Finite Element model and the real structure with respect to the dynamic response. One of the possible applications of this technique is the damage detection. In this contribution, the VCUPDATE framework for Finite Element Model Updating is presented. VCUPDATE is a Scilab code performing the iterative updating algorithm interfaced with a Finite Element code (OpenSees or ANSYS) executing the numerical analysis. At first, the theoretical basics of the code are explained. Later on, some representative applications to practical cases are reported and discussed. Introduction The use of advanced numerical techniques in conjunction with the results from non-destructive experimental data is an interesting possibility for Structural Health Monitoring. In particular, the Finite Element Model Updating (FEMU) is a powerful and well-established numerical method which is used to modify the Finite Element (FE) model properties until an adequate agreement between numerical and experimental results is achieved. The resulting property distributions can provide useful information about the possible structural damage. In this contribution, the calculation technique has been implemented in VCUPDATE, an iterative updating algorithm developed in Scilab [] and interfaced with the FE codes OpenSees [2] and ANSYS [3]. The code has been already applied to several different structures [4] [5]. The use of two different FE codes provides VCUPDATE with a wider flexibility. In fact, the commercial code ANSYS has very powerful modeling capabilities and allows the graphical visualization of the model and of the results, including the outcomes from the updating procedure. On the other hand, the updating process using OpenSees is much faster. Moreover, OpenSees is an open source code, which can be used for commercial purposes also. Therefore, when possible (i.e. when the model is simple), the use of OpenSees is preferred.
2 Theoretical basics The FEMU can be performed using direct or iterative methods [6]. In this work, the authors present a sensitivity-based iterative method. This means that a non-linear penalty function is minimized through subsequent linear steps. In Figure, the main steps of the code are shown. Scilab (updating code) OpenSees / ANSYS (FE code) 2. Sensitivity analysis The analysis is performed to select the most sensitive parameters for the FE model. A perturbation technique using the forward difference of the function with respect to each parameter is implemented. This allows obtaining the sensitivity matrix using the results from multiple FE analyses without knowledge of the system matrices. In particular, if the problem has n parameters, n+ FE runs are required: the first one with the starting values for the parameters and n different runs perturbing one parameter only in each of them. The forward difference method has the form [7] yes START computing the sensitivity matrix updating parameters convergence? computing the sensitivity matrix END no FE run (with starting parameters) n FE runs (with modified parameters) FE run (with updated parameters) n FE runs (with modified parameters) Figure VCUPDATE flow chart. d Δd d p p d p S = = p Δp Δp ( + Δ ) ( ), () where Δp is the perturbation in the parameters and Δd is the difference between measured and numerical data. The perturbation in the i th parameter is computed as a fraction of the difference between the upper and lower bound for the i th parameter. Every vector Δd / Δp gives one column of the sensitivity matrix. 2.2 Sensitivity matrix and mode shape scaling Using different structural properties with very different values as parameters may lead to numerical problems, e.g. ill-conditioning of S, in the iterative solution [8]. An appropriate scaling of the sensitivity matrix can improve the stability and speed up the convergence. In this work, the parameters and the data are normalized by their initial values. Experimental and numerical mode shapes should be scaled consistently to properly perform the updating algorithm. Moreover, experimental and numerical mode shapes can be 8 deg out of phase. To solve these problems, within VCUPDATE, the modal scale factor (MSF) is used according to [6] and [9].
2.3 Updating algorithm The updating problem can be represented as the minimization of a penalty function J(Δp) subjected to the constraint Δd = SΔp [6]. Minimizing J(Δp) with respect to Δp and inserting the weighting matrices W d and W p for data and parameters respectively, the updated parameter values can be written as: ( ) T T j+ = j + j d j + p j d j p p S W S W S W d d at j th iteration. (2) The sensitivity matrix should be computed at each iteration, but this can lead to an excessive computational time. VCUPDATE allows the user to choose the frequency with whom the sensitivity matrix is updated. For some extremely time consuming cases, the initial matrix can be used for the entire analysis. According to experience, not all the experimental data are measured with the same accuracy. In order to express the user confidence in measured data, weighting matrices are used. Parameters can also be weighted separately. Using weighting matrices is very powerful, but engineering insight is required. Five different ways to create the W p matrix according to [8] are implemented in VCUPDATE. Several different tests were performed in order to verify the methods: in particular some of them should be used carefully since they provide, in some cases, an increased convergence speed but, in other cases, the obtained results are worse. 2.4 Convergence criteria The iterative scheme requires a convergence criterion which can be based on parameters or on data. In VCUPDATE, the convergence is checked using a control based on the frequency deviation (CC abs ) or on the frequency and MAC deviation (CC tot ) []: CC abs, j n f, i f n exp j, i =, fexp, i f j, i CCtot, j = Wf + W ( MACi ) Φ n i = fexp, i 2nWmax i= f exp, i The convergence criterion is shown in Figure 2. The convergence is achieved if CC ε (point A). If the convergence is not attained after the maximum number of iterations, the information relative to the iteration with the minimum value of CC (point B) is used as output. CC tot / CC abs A B. (3) convergence limit ε maximum iteration Figure 2 Convergence criterion. iterations
3 Application to a Prestressed Reinforced Concrete Beam A prestressed reinforced concrete beam tested in laboratory is investigated by VCUPDATE. In general, the damage in prestressed elements is difficult to be detected since, after the load is removed, the cracks tend to close and this leads to relatively small changes in modal data. The experimental data as well as the test description are taken from a previous study []. The results obtained by VCUPDATE are compared, in terms of damage distribution, with those obtained in this study. 8 The investigated beam is 7.6 m long and m high. Along the member, the cross section has an I-shape while, at the ends, it has a solid square section (Figure 3). The beam is prestressed by an inclined post-tensioned cable. 8 8 8 5 Figure 3 Beam cross-sections. The beam was artificially damaged by increasing static load levels in the configuration shown in Figure 4-a. The external load Q is applied in six different steps with an increasing amplitude of 45, 67.5, 25, 4, 5 and 54 kn respectively. The first visible crack on the beam corresponds to Q=67.5 kn while the reinforcement yielding is achieved for Q =5 kn. After each step, the beam was dynamically tested in a free-free configuration (Figure 4-b). The structure is excited by the impact of a falling weight at its first edge. During the dynamic test procedures, 74 sensors were placed on the structure, at both longitudinal upper sides of the beam with a distance of.5 m between them. In most load steps, the dynamic behavior of the beam was found to be pure flexural with small torsional components. Therefore, the values from two sensors placed at the same longitudinal distance are averaged in order to obtain 37 values only. In this way, the torsional components of the mode shapes are neglected. The beam is modeled in OpenSees by means of a Beam model. 36 uniaxial beam elements with five integration points and three degrees of freedom (DOF) for each node are used. In order to simulate the free-free boundary conditions, two springs with very 4 (a) (b) 22 impact 2 2 2 8 2 43 396 43 22 4 Q 76 Figure 4 Static (a) and dynamic (b) test configuration with experimental crack pattern. Q
n. of mode experimental freq. f [Hz] frequency f [Hz] before update fi f f i i NMD frequency f [Hz] after update fi f f NMD initial-to-undamaged step.32.54 2.%.49 9-5%.49 2 29.49 29.9.4%.32 29.49.%.38 3 57.77 59.84 3.59%.39 57.77.%.25 CC abs.237 CC abs.9 undamaged-to-damaged step 6.83 9 5.7%.82 6.55-4.8%.27 2 22.66 29.49 3% 5 22.63 -%.56 3 48.45 57.77 9.25% 9 48.44 -.2%.79 4 7.99 8 4.55%.395 7.3.5% CC abs.354 CC abs.7 Table : Updating procedure results for the OpenSees analysis. low stiffness are placed at the supports. The material reinforced concrete is modeled as homogeneous and isotropic. The section properties are: area A=.3 m 2 and 4 m 2, moment of inertia J=.2246 m 4 and.343 m 4 for the I- and the rectangular shape respectively. Adopting a mass per unit volume M=25 kg/m 3, a mass per unit length m=752.5 kg/m and 6 kg/m is obtained for the two sections respectively. The value of the elastic modulus for each element is used as updating parameter with a starting value of E =375 MPa. Therefore, a total number of 36 parameters is used. Two different updating procedures are performed: with the first one, the initial FE model is updated to the undamaged state. Then, this undamaged model is updated to the damaged state corresponding to the post reinforcement yielding step (Q=54 kn). The first four frequencies and flexural mode shapes are used in the second updating process while, due to the poor quality, the fourth mode shape is not used in the first updating process. The outcomes from the numerical procedures in terms of frequencies and mode shapes correlation are reported in Table. Since the mode shapes are highly correlated, the NMD is used instead of the MAC. The effect of the updating procedures is smaller for the initial-to-undamaged step. This means that the initial model fits quite well the undamaged one, both in terms of frequencies and mode shapes. In the undamaged-to-damaged step, the effect of the updating procedures is much higher as can be seen by comparing the CC abs values. The biggest frequency deviation is recorded in the first mode shape, probably due to the presence of a torsional component in the experimental data. The effect of the updating algorithm on the mode shapes can be clearly seen also in Figure 5 where the outcomes of the second step are shown. It can be noted that, starting from the undamaged state, the mode shapes are transformed to the damaged state which is always closer to the experimental results. The damage distribution is reported in Figure 6 in terms of the ratio between the elastic modulus values before and after the updating process. In the same figure, the VCUPDATE outcomes are compared with the distribution taken from the already mentioned previous study. It should be noted i i
- 2 3 4 5 6 7 8 9 2 3 4 5 6 7 - - - - - 2 3 4 5 6 7 8 9 2 3 4 5 6 7 - - - - mode shape mode shape 3 Exp. damaged VCUPDATE initial VCUPDATE updated - 2 3 4 5 6 7 8 9 2 3 4 5 6 7 - - - - - 2 3 4 5 6 7 8 9 2 3 4 5 6 7 - - - - mode shape 2 mode shape 4 Figure 5 Experimental and numerical mode shapes for undamaged-to-damaged step. that even if the static load is symmetrical, the corresponding damage distribution is not symmetrical. In this case, the mode shapes are fundamental for the damage location. From the undamaged-to-damaged step graph (Figure 6-b) four damaged zones can be recognized: two near the beam center with a higher damage and two closer to the beam ends with a lower damage level. The position of these points corresponds to the loading points in the static test. The two main plastic hinges are situated close to the beam center where the reinforcements yielded. Between the loading points, no damage is revealed, probably because between the plastic hinges the steel is not yielded and therefore, the cracks tend again to close after the load removal. The same structure is investigated with VCUPDATE by means of ANSYS. A single step analysis is considered: the starting model is directly updated to the damaged beam. This is, in fact, the standard way to assess the existing structures when no test data of the initial state are available. The use of ANSYS provides the user with an increased modeling power. In fact, the FE mesh can be freely chosen and there is no necessity to place a node of the FE model in the same location of a.4.2.4.2 Previous study VCUPDATE Eund / E Previous study VCUPDATE Eund / Edam 2 3 4 5 6 7 8 9 2 3 4 5 6 7 (a) 2 3 4 5 6 7 8 9 2 3 4 5 6 7 Figure 6 Damage distribution for initial-to-undamaged (a) and undamaged-to-damaged (b) step. (b)
Figure 7 Elastic modulus distribution for the Beam model with element size and. m. sensor. In this way, the best FE model can be chosen without taking care of the sensor positions. The code requires the position of each sensor and subsequently, the numerical mode shapes are computed in the sensor positions as a function of the mode shapes of the entire FE model and they can be directly compared with the experimental mode shapes. Both a Beam and a Plane model are used. The first model contains the 2D elastic beam BEAM3 element. It is a uniaxial element with tension, compression and bending capabilities having three DOF for each of its two nodes. Furthermore, the Plane model is based on the 2D structural solid PLANE42 element. It is a plane stress element having two DOF for each of its four nodes. Several different meshes are used in the calculations, for both the Beam and the Plane analysis. In Figure 7, the analysis outcomes in terms of elastic modulus distribution for the Beam model with two different meshes are shown. Analogously, for the Plane model, the results are shown in Figure 8 for an element size of and m. The damage distribution of these figures can be compared with the one from the OpenSees analysis (Figure 6). It can be noted that the agreement is very high. Moreover, for the ANSYS analyses, the outcomes are not depending on the used mesh as long as the mesh can effectively describe the structural response. The results in terms of eigenfrequencies and mode shapes are as well very good but they cannot be extensively reported here due to lack of space. Figure 8 Mesh and elastic modulus distribution for the Plane model with element size and m.
4 Conclusion In this work, the software VCUPDATE for Finite Element Model Updating was presented. The updating procedures were implemented in a Scilab code interfaced with the FE codes OpenSees and ANSYS. The code was applied to the study of a prestressed reinforced concrete beam tested in laboratory by combining eigenfrequencies and mode shapes. In this case, even if the structure configuration was symmetrical, it was found that the damage distribution is not symmetrical and therefore, the use of mode shapes is fundamental in order to evaluate the damage distribution. The experimental structural behavior could be captured with very high accuracy by using different FE codes, different model dimensionality and different meshes. After a detailed evaluation of the obtained results, it can be concluded that this approach can be a valuable tool for Structural Health Monitoring. 5 References [] Scilab Consortium: Scilab, A Free Scientific Software Package, http://www.scilab.org. [2] Pacific Earthquake Engineering Research Center: OpenSees, Open System for Earthquake Engineering Simulation, http://opensees.berkeley.edu. [3] SAS IP Inc.: ANSYS., http://www.ansys.com. [4] Mordini, A., Savov, K. and Wenzel, H.: Damage detection on stay cables using an open source-based framework for Finite Element Model Updating, accepted in September 26 for publication in SHM Structural Health Monitoring. [5] Mordini, A., Savov, K. and Wenzel, H.: The Finite Element Model Updating: a powerful tool for Structural Health Monitoring, submitted for publication in January 27 in Structural Engineering International. [6] Friswell M. I. and Mottershead J. E.: Finite Element Model Updating in Structural Dynamics, Kluwer Academic, Dordrecht, The Netherlands, 995. [7] Jaishi B. and Ren W.: Structural Finite Element Model Updating Using Ambient Vibration Test Results, ASCE Journal of Structural Engineering, 4, 67-628, 25. [8] Dascotte, E., Strobbe, J. and Hua H.: Sensitivity-based model updating using multiple types of simultaneous state variables, Proceedings of the 3th International Modal Analysis Conference (IMAC), Nashville, Tennessee, February 995. [9] Allemang, R. J. and Brown, D.L.: A Correlation Coefficient for Modal Vector Analysis, Proceedings of the st International Modal Analysis Conference (IMAC), Orlando, Florida, November 982. [] Dascotte E. and Vanhonacker P.: Development of an automatic mathematical model updating program, Proceedings of the 7th International Modal Analysis Conference (IMAC), Las Vegas, Nevada, February 989. [] Teughels A.: Inverse modeling of civil engineering structures based on operational modal data, Ph.D. thesis, Katholieke Universiteit Leuven, December 23.