Response Surface Model Updating for Nonlinear Structures

Transcription

1 Response Surface Mode Updating for Noninear Structures Gonaz Shahidi a, Shamim Pakzad b a PhD Student, Department of Civi and Environmenta Engineering, Lehigh University, ATLSS Engineering Research Center, 117 ATLSS Drive, Imbt Labs, Bethehem, PA 1815 (sgs31@ehigh.edu) b Assistant Professor, Department of Civi and Environmenta Engineering, Lehigh University, ATLSS Engineering Research Center, 117 ATLSS Drive, Imbt Labs, Bethehem, PA 1815 (pakzad@ehigh.edu) ABSTRACT- This paper presents a procedure to update noninear finite eement modes in time. In the proposed method, accurate response surface modes are constructed and evauated to repace the finite eement mode at every time step of the anaysis. Then, the optimization probem of mode updating is formuated and soved iterativey eading to histograms of the updated mode parameters. This methodoogy is beneficia in extracting more information from measured signas and compensate for the error present in the regressed response surface modes. The proposed method was verified through a numerica case study of a stee frame with goba noninearity. Appropriate design and mode orders were successfuy estabished and the optimization in time performed we in the simuated scenarios under the assumption of noise free and noisy measurement data. Keywords: Finite eement mode; Mode updating; Noninear structure; Response surface mode; Optimization INTRODUCTION Finite eement (FE) modes are extensivey used for anaytica purposes in the engineering fied. These modes serve as the preiminary base for designing and anaysing behaviour of the actua structures. However, the anaysis resuts of these modes are not same as that obtained from testing the actua structures. This is mainy because of a number of simpifying assumptions used in the FE mode or/and deterioration of the structure with use and time. Modifying the FE mode to obtain better agreement with the experimenta records is termed as Finite Eement Mode Updating. Over the past decades severa computationa procedures have been deveoped to update parameters of anaytica modes based on experimenta resuts. Methods for inear mode updating are we-documented in the iterature. These techniques are mainy based on the sensitivity anaysis and inearization of the generay noninear reationship between measured outputs such as natura frequencies, mode shapes or dispacement responses and the parameters of the mode in need of correction [1]. However, iterative determination of oca gradient in such methods may cause not ony computationa intensive, but aso convergence difficuty [2]. Moreover, in the presence of noninearities in the structure these procedures wi fai to yied the parameters associated with noninear behavior of the mode and other measures are required to update the mode. One of the proposed approaches to overcome these probems is to repace the FE mode with a mathematica expression which approximates the reationship between pre-seected inputs and output of the FE mode and update the parameters of the mode by directy optimizing this surrogate mode. One of the commony used surrogate modes are poynomia functions constructed based on Response Surface (RS) methodoogy. This method has shown attractive potentia in modifying FE mode parameters. Guo and Zhang [3] found that, compared with the sensitivity-based mode updating, the RS-based method gave ikewise accurate predictions whie requiring much fewer number of FE anayses. Ren and Chen [2] compared the performance of RS-based and Sensitivity-based FE mode updating on a fu size precast continues box girder bridge and observed that the rate of convergence in RS-based updating is faster. Ren et a [4] concuded that for compex structures with arge number of uncertain parameters uniform design economized the computationa effort to construct the RS modes and the accuracy of the RS modes in such probems coud be improved by shrinking the design space and repeating the RS modeing and updating. Zhang et a [5] proposed a mode updating technique based on generic agorithm and RS methodoogy. Appication of the method on a numerica simuation of an antenna successfuy reached the goba optima.

2 Studies of Cundy [6] and Fang and Perera [7] found that appication of RS-based mode updating in damage detection performs we in ocating damage and quantifying its severity to some extent in numerica and experimenta case studies. There are few exampes of appication of RS-based mode updating in the iterature for structures with noninearities. Schutze et a [8] appied this method to seect significant parameters to update a mode consist of a cyindrica stee impactor and a foam ayer assembed on a mounting pate attached to a drop tabe under impact on a concrete foor. Zhang and Guo [9] proposed a mode updating procedure based on Principa Component Decomposition and RS method to update a mode of frame with thin wa components showing strain-rate-dependence noninearity under impact test. In this paper a procedure is proposed to update noninear FE modes in time. For this purpose, ow computationa effort associated with RS modeing is used to formuate and sove the optimization probem of mode modification in the ength of time domain data iterativey. This approach is beneficia in extracting more information from the measured experimenta signas as opposed to the traditiona approaches in which the whoe measured signas are summarized into one or more response features. Another advantage of this method is that it is not imited to the type of mode behavior or anaysis. It can be appied to inear or noninear modes under static or dynamic anaysis. Since the procedure of finding an appropriate design to buid accurate RS modes requires a number of trias and errors with different designs and subset modes, a procedure is aso proposed to design the eves of input parameters and construct the RS modes prior to mode updating. This procedure resuts in RS modes capabe of generating the resuts of FE anaysis in a specific domain of input variabes. In the foowing sections of the paper the proposed procedure is expained and the resuts of appication of this method on a numerica case study are demonstrated. NONLINEAR MODEL UPDATING USING RESPONSE SURFACE MODELS To update noninear FE modes through time history of measured responses, in every time step of the anaysis a RS mode is constructed to produce the response of the FE mode at that time step. To do so, the experimenta input force is used to generate the equivaent responses of FE mode at different eves of the mode parameters. These eves of the mode parameters, corresponding responses of the FE mode and east square estimation techniques are used to find the best poynomia mode which can repace the FE mode at every time step of the anaysis. th Eq. (1) denotes the RS mode at the time step of the anaysis, where h is the poynomia surrogate mode in that time step and Θ represents a vector of mode parameters seected for modification. RS h (Θ) (1) = By competing this process for every response, an objective function is formuated to minimize a function of residuas of RSbased and experimenta response features at every time step. Eq. (2) represents this minimization probem which is soved inside the domain of mode parameters. min θ f ( H ( Θ), Y exp ) (2) In Eq. (2) time step. H and Y exp are vectors containing a the surrogate modes and corresponding experimenta responses at the th Prior to RS modeing, the appropriate design and mode order shoud be found so that the regressed RS modes are accurate at the associated time steps. First, an initia region for the pre-seected uncertain parameters of the FE mode shoud be chosen. This region, in which the FE mode is repaced by the RS mode, is caed RS domain. To regress the poynomia RS modes, a number of points are samped in the RS domain based on fu factoria design of the mode parameters. The RS

3 mode construction starts with a fu factoria design with three eves for each parameter and incuding inear terms of the updating parameters in the RS modes. Initiay the performance of the RS modes is checked at the design points based on the residuas of the RS and FE modes. Then the overa adequacy of the RS modes is evauated by adjusted R 2 statistics. If R 2 adj is cose to one, it impies a perfect regression. Therefore, when R 2 adj is much smaer than one, the RS mode is not accurate in estimating the FE responses at the design points. After competing R 2 adj cacuation through the time domain data, if the regressed RS modes are not fitted we to the design points, higher order terms of the mode parameters shoud be added to the RS modes and the mode evauation repeated to find the appropriate mode order. After finding the suitabe mode order, the prediction quaity of the RS modes shoud be checked. For this purpose, residuas are cacuated at points in the RS domain that did not contribute in the regression. These points, which are caed intermediate points, are samped from RS domain in different sets. Each set represents the intermediate eves for one parameter. To sampe a set of new points corresponding to a parameter, one of the origina data points is repicated, and then the seected parameter is repaced by the average of one pair of its origina eves. Intermediate points which resut in arger residuas than the origina design points indicate that athough the RS mode has been fitted we to the origina data, it cannot predict the FE responses for new points. Therefore, the design of eves of parameters shoud become finer and new RS modes shoud be regressed. By repeating this procedure, an RS mode with high quaity in regression and prediction is constructed for every time step of the data. Upon competion of this procedure for every response feature, the optimization probem of mode updating is soved for every time step which resuts in histograms of the updated mode parameters. The optimization step can be repeated in a smaer region for mode parameters centred on the mean vaue of the resuts of the first cyce of optimization. Using the design and mode order estabished in the first cyce, ony the foowing steps are needed: (1) generate the FE responses for new eves; (2) fit the new RS modes through the time history; and (3) optimize the new objective function iterativey. CASE STUDY: NONLINEAR STEEL FRAME The case study presented here is a stee frame with noninear materia properties under dynamic oading. The frame consists of one span with overa ength of cm supported by coumns that are 83.8 cm ong. The cross section of the beam and coumn members is uniform hoow 5.8 cm tube, with.21 cm wa thickness. The stee has biinear behavior with the yied stress of MPa. Moduus of easticity (E) and post yieding stiffness ratio of stee (b) were chosen as the updating parameters. To simuate the experimenta data, these parameters were set to GPa and.18 for E and b respectivey. The oading is a concentrated harmonic atera oad with ampitude 22.2 kn and 5 sec period, appied at the beam coumn joint. The ampitude of the oad is seected so that under atera oading the stress in the coumns and beam exceeds the yied stress. To update the seected parameters, simuated time histories of dispacement at two ocations on the frame were assumed as the experimenta data. Fig. 1 shows the configuration of the stee frame, oading and the responses used in the updating procedure. u 2 f u 1 y x Fig.1: Configuration of the noninear stee frame

4 A 2-D FE mode was deveoped by Opensees software using fiber section procedure and Stee1 uniaxiamateria properties. The initia domain of the updating parameters was set to to GPa for E and.5 to.25 for b. The RS mode construction starts with fu factoria design of parameters, each having three eves. RS modes incuding the inear terms of E and b were regressed to the data at every time step. The arge residuas associated with the regressed modes indicate that the RS modes are not accurate to repace the FE mode. Consequenty, quadratic terms were added to the poynomia modes and regression was repeated. Adding the quadratic terms to the inear modes significanty improves the accuracy of the RS modes at the design points. Fig.2 and 3 compare the maximum normaized residuas at origina and intermediate points of the 3ä3 design for the quadratic RS modes through the time data. The RS modes generate u 2 with smaer residuas; however, they are not successfu in predicting both u 1 and u 2 at the intermediate eves corresponding to b. Therefore, the eves associated with stiffness ratio, b, in the RS domain shoud be finer. Maximum normaized residua (%) Origina Design Points D. P. (1): b=.5& E(GPa)=186.2 D. P. (2): b=.15& E(GPa)=186.2 D. P. (3): b=.25& E(GPa)=186.2 D. P. (4): b=.5& E(GPa)=26.8 D. P. (5): b=.15& E(GPa)=26.8 D. P. (6): b=.25& E(GPa)=26.8 D. P. (7): b=.5& E(GPa)=227.5 D. P. (8): b=.15& E(GPa)=227.5 D. P. (9): b=.25& E(GPa)=227.5 D. P. (1): b=.1& E(GPa)=186.2 D. P. (11): b=.2& E(GPa)=186.2 D. P. (12): b=.5& E(GPa)=196.5 D. P. (13): b=.5& E(GPa)= Fig.2: Normaized residuas of origina and intermediate design points using 3ä3 design: u 1 Maximum normaized residua (%) Origina Design Points D. P. (1): b=.5& E(GPa)=186.2 D. P. (2): b=.15& E(GPa)=186.2 D. P. (3): b=.25& E(GPa)=186.2 D. P. (4): b=.5& E(GPa)=26.8 D. P. (5): b=.15& E(GPa)=26.8 D. P. (6): b=.25& E(GPa)=26.8 D. P. (7): b=.5& E(GPa)=227.5 D. P. (8): b=.15& E(GPa)=227.5 D. P. (9): b=.25& E(GPa)=227.5 D. P. (1): b=.1& E(GPa)=186.2 D. P. (11): b=.2& E(GPa)=186.2 D. P. (12): b=.5& E(GPa)=196.5 D. P. (13): b=.5& E(GPa)= Fig.3: Normaized residuas of origina and intermediate design points using 3ä3 design: u 2 RS mode construction and evauation were repeated with 4ä3 design and observed that the RS modes are not accurate at intermediate eves of parameter b for this design. Fig.4 and 5 dispays the maximum normaized residuas of RS and FE mode for a 5ä3 design. The RS modes contain terms up to order 4 and 2 for stiffness ratio, b, and moduus of easticity, E, respectivey. These figures show that the RS modes perform we at both origina and intermediate eves. Therefore, the RS modes are accurate for the optimization procedure.

5 Maximum normaized residua (%) D. P. (1): b=.5& E(GPa)=186.2 D. P. (2): b=.1& E(GPa)=186.2 D. P. (3): b=.15& E(GPa)=186.2 D. P. (4): b=.2& E(GPa)=186.2 D. P. (5): b=.25& E(GPa)=186.2 D. P. (6): b=.5& E(GPa)=26.8 D. P. (7): b=.1& E(GPa)=26.8 Origina Design Points D. P. (8): b=.15& E(GPa)=26.8 D. P. (9): b=.2& E(GPa)=26.8 D. P. (1): b=.25& E(GPa)=26.8 D. P. (11): b=.5& E(GPa)=227.5 D. P. (12): b=.1& E(GPa)=227.5 D. P. (13): b=.15& E(GPa)=227.5 D. P. (14): b=.2& E(GPa)=227.5 D. P. (15): b=.25& E(GPa)=227.5 D. P. (16): b=.75& E(GPa)=186.2 D. P. (17): b=.125& E(GPa)=186.2 D. P. (18): b=.175& E(GPa)=186.2 D. P. (19): b=.225& E(GPa)=186.2 D. P. (2): b=.5& E(GPa)=196.5 D. P. (21): b=.5& E(GPa)= Fig.4: Normaized residuas of origina and intermediate design points using 5ä3 design: u 1 Maximum normaized residua (%) D. P. (1): b=.5& E(GPa)=186.2 D. P. (2): b=.1& E(GPa)=186.2 D. P. (3): b=.15& E(GPa)=186.2 D. P. (4): b=.2& E(GPa)=186.2 D. P. (5): b=.25& E(GPa)=186.2 D. P. (6): b=.5& E(GPa)=26.8 D. P. (7): b=.1& E(GPa)=26.8 Origina Design Points D. P. (8): b=.15& E(GPa)=26.8 D. P. (9): b=.2& E(GPa)=26.8 D. P. (1): b=.25& E(GPa)=26.8 D. P. (11): b=.5& E(GPa)=227.5 D. P. (12): b=.1& E(GPa)=227.5 D. P. (13): b=.15& E(GPa)=227.5 D. P. (14): b=.2& E(GPa)=227.5 D. P. (15): b=.25& E(GPa)=227.5 D. P. (16): b=.75& E(GPa)=186.2 D. P. (17): b=.125& E(GPa)=186.2 D. P. (18): b=.175& E(GPa)=186.2 D. P. (19): b=.225& E(GPa)=186.2 D. P. (2): b=.5& E(GPa)=196.5 D. P. (21): b=.5& E(GPa)= Fig.5: Normaized residuas of origina and intermediate design points using 5ä3 design: u 2 Fig. 6(a) shows the histogram of the updated parameters resuted from soving the optimization probem in every time step for the first time. This histogram shows where the updated parameters ocate in the RS domain. The updated mode parameters are distributed in a consideraby narrower region than the initia region used in the RS mode construction. To decrease the variation of the updated parameters, the design and mode order estabished in the previous section for E and b are used to repeat the optimization probem. The new domain for E and b is centered on the mean vaue of the updated parameters in the first round of optimization. Fig. 6(b) shows the resut of the second round of mode updating in terms of the distribution, mean and coefficient of variation of the updated parameters.

6 Occurrence Occurrence 1 5 b: mean=.1844 C.O.V= b E: mean=192.81(gpa) C.O.V= E(GPa) (a) Occurrence Occurrence b: mean=.18 C.O.V= b E: mean=193.8(gpa) C.O.V= E(GPa) (b) Fig.6: (a) Resuts of the first optimization round, (b) Resuts of the second optimization round Furthermore, to evauate the performance of the proposed procedure in the presence of noisy measurement data, different eves of Gaussian noise were introduced into the simuated experimenta responses and the updating procedures were reiterated. The resuts were summarized in Tabe 1. Noise eve in this tabe represents the ratio of the root mean square of the simuated noise signa to the root mean square of the origina signa. It can be seen that the procedure shows robustness to different eves of noise. Tabe.1: Comparison of the performance of the proposed procedure (Simuated data with different noise eves) b:.5 to.25 E(GPa):186.2 to True mode parameters: b=.18 & E(GPa)=193.1 Updated Parameters Noise Leve (%) b E Reative Error (%) The advantage of using the proposed procedure for updating noninear FE modes is that this method successfuy finds a smaer region for the mode parameters and has corrective information for the initia estimate of the RS domain. Moreover, it is computationay efficient and shows robustness to moderate and high eve noise. CONCLUSIONS This paper presents a procedure for updating noninear Finite Eement modes using time domain data based on Response Surface methodoogy. The proposed procedure was vaidated through to a numerica case study of a stee frame with goba noninearity. The optimization in time domain performed we in the numerica case study. The first round of optimization resuted in a consideraby narrower bounds for the uncertain parameters of the mode than the initia boundaries set at the beginning of the procedure. Repeating the RS mode construction with known order and design for the new bounds of parameters and soving the optimization probem resuted in the true mode parameters. To verify the robustness of the resuts, the numerica case study were repeated assuming ow, moderate and high eve noise in the experimenta data and it was observed that the proposed procedure show robustness in a these scenarios.

7 ACKNOWLEDGMENT Research funding is partiay provided by the Nationa Science Foundation under Grant No. CMMI by Sensors and Sensing Systems program, and by a grant from the Commonweath of Pennsyvania, Department of Community and Economic Deveopment, through the Pennsyvania Infrastructure Technoogy Aiance (PITA). Their financia support is gratefuy appreciated. REFERENCES [1] Mottershead, J. E., Link, M., and Friswe, M. I., The sensitivity method in finite eement mode updating: a tutoria, Mech. Syst. Signa Pr., 25(7), , 21. [2] Ren, W., and Chen, H., Finite eement mode updating in structura dynamics by using the response surface method, Eng. Struct., 32(8), , 21. [3] Guo, Q. T., and Zhang, L. M., Finite eement mode updating based on response surface methodoogy, Proc. IMAC- XXII: Conference and Exposition on Structura Dynamics, 24. [4] Ren, W., Fang, S., and Deng, M., Response surface-based finite eement mode updating using structura static responses, J. Eng. Mech. (ASCE), 137(4), , 211. [5] Zhang, L. M., Fei, Q., and Guo, Q. T., Dynamic finite eement mode updating using meta-mode and genetic agorithm, Proc. IMAC-XXIII: Conference and Exposition on Structura Dynamics, 25. [6] Cundy, A. L., Use of response surface metamodes in damage identification of dynamic structures, M. S. Thesis, Virginia Poytechnic Institute and State University, 22. [7] Fang, S. E., and Perera, R., A response surface methodoogy-based damage identification technique, Smart Mater. Struct., 18(6), 659, 29. [8] Schutze, J. f., Hemez, F. M., Doebing, S. W., and Sohn, H., Appication of non-inear system mode updating using feature extraction and parameter effect anaysis. Shock Vib., 8, , 21. [9] Zhang, L. M., and Guo, Q. T., A case study of mode updating and vaidation of a frame structure with highy non-inear component, Proc. IMAC-XXV: Conference and Exposition on Structura Dynamics, 27.