DSpace at VNU: Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification

DSpace at VNU: Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification tài liệu, giáo...

Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification Duc-Duy Ho1, Jeong-Tae Kim2,*, Norris Stubbs3 and Woo-Sun Park4 1Faculty of Civil Engineering, Ho Chi Minh City University of Technology, Vietnam of Ocean Engineering, Pukyong National University, Korea 3Department of Civil Engineering, Texas A&M University, College Station, USA 4Coastal Engineering & Ocean Energy Research Department, Korea Ocean Research & Development Institute, Korea 2Department Abstract: In this paper, a vibration-based method to estimate prestress-forces in a prestressed concrete (PSC) girder by using vibration characteristics and system identification (SID) approaches is presented Firstly, a prestress-force monitoring method is formulated to estimate the change in prestress forces by measuring the change in modal parameters of a PSC beam Secondly, a multi-phase SID scheme is designed on the basis of eigenvalue sensitivity concept to identify a baseline model that represents the target structure Thirdly, the proposed prestress-force monitoring method and the multi-phase SID scheme are evaluated from controlled experiments on a labscaled PSC girder On the PSC girder, a few natural frequencies and mode shapes are experimentally measured for various prestress forces System parameters of a baseline finite element (FE) model are identified by the proposed multi-phase SID scheme for various prestress forces The corresponding modal parameters are estimated for the model-update procedure As a result, prestress-losses are predicted by using the measured natural frequencies and the identified zero-prestress state model Key words: prestress concrete girder, presstress-loss, modal parameters, system identification, structural health monitoring INTRODUCTION The interest on the safety assessment of existing prestressed concrete (PSC) girders has been increasing For a PSC girder, typical damage types include loss of prestress-force in steel tendon, loss of flexural rigidity in concrete girder, failure of support, and severe ambient conditions Among them, the loss of prestressforce is an important monitoring target to secure the serviceability and safety of PSC girders against external loads and environmental conditions (Miyamoto et al 2000; Kim et al 2004) The loss of prestress-force occurs along the entire girder due to elastic shortening and bending of concrete, creep and shrinkage in concrete, relaxation of steel stress, friction loss and anchorage seating (Collins and Mitchell 1991; Nawy 1996) Unless the PSC girder bridges are instrumented at the time of construction, the occurrence of damage can not be directly monitored and other alternative methods should be sought Since as early as 1970s, many researchers have focused on the possibility of using vibration characteristics of a structure as an indication of its structural damage (Adams et al 1978; Stubbs and Osegueda 1990; Doebling et al 1998; Kim et al 2003) Recently, research efforts have been made to investigate the dynamic behaviors of prestressed composite girder bridges (Miyamoto et al 2000), and to identify the change in prestress forces by measuring dynamic responses of prestressed beams (Kim et al 2004) However, to date, no successful attempts have been made to estimate the relationship between the loss in prestress forces and the change in geometries, material properties, *Corresponding author Email address: idis@pknu.ac.kr; Fax: +82-51-629-6590; Tel: +82-51-629-6585 Advances in Structural Engineering Vol 15 No 2012 997 Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification and boundary conditions of the PSC girder bridges Hence, it is necessary to develop a system identification (SID) method that can identify the change in structural parameters due to the change in prestress forces An accurate finite element (FE) model is prerequisite for civil engineering applications such as damage detection, health monitoring and structural control For complex structures, however, it is not easy to generate accurate baseline FE models for the use of structural health monitoring, because material properties, geometries, boundary conditions, and ambient temperature conditions of those structures are not completely known (Kim and Stubbs 1995; Kim et al 2007) Due to those uncertainties, an initial FE model based on as-built design may not truly represent all the physical aspects of an actual structure Consequently, there exists an important issue that how to update the FE model using experimental results so that the numerically analyzed structural parameters match to the real experimental ones Many researchers have proposed model update methods for SID by using vibration characteristics (Friswell and Mottershead 1995; Kim and Stubbs 1995; Zhang et al 2000; Jaishi and Ren 2005; Yang and Chen 2009) Among those methods, the eigenvalue sensitivitybased algorithm has become one of the most popular and effective methods to provide baseline models for structural health assessment (Brownjohn et al 2001; Wu and Li 2004) The FE model update is a process of making sure that FE analysis results better reflect the measured data than the initial model For the vibrationbased SID, this process is conducted in the following steps: (1) measuring vibration data to be utilized; (2) determining structural parameters to be updated; (3) formulating a function to represent the difference between the measured vibration data and the analyzed data from FE model; and (4) identifying parameters to minimize the function (Friswell and Mottershead 1995; Kwon and Lin 2004) The objective of this paper is to present a prestressforce estimation method for PSC girders by using changes in vibration characteristics and SID approaches The following approaches are implemented to achieve the objective Firstly, a prestress-force monitoring method is formulated to estimate changes in prestressforces in a PSC girder by measuring changes in modal parameters Secondly, a multi-phase SID scheme is designed on the basis of eigenvalue sensitivity concept to estimate a baseline model which represents the target structure Thirdly, the proposed prestress-force monitoring method is evaluated from controlled experiments on a lab-scaled PSC girder On the PSC girder, a few natural frequencies and mode shapes are 998 experimentally measured for various prestress forces System parameters of a baseline FE model are identified by the proposed multi-phase SID scheme for various prestress forces The corresponding modal parameters are estimated for the model-update procedure As a result, prestress-losses are predicted by using the measured natural frequencies and the identified zeroprestress state model THEORY OF APPROACH 2.1 Vibration-Based Prestress-Force Monitoring Method Based on the previous study by Kim et al (2004), an effective flexural rigidity model of a simply supported PSC beam with an eccentric tendon is schematized as shown in Figure The curved tendon is initially stretched and anchored to introduce prestressing effect Then, as shown in Figure 1(b), the structure is in axial compression due to the prestress loads applied at the anchorage edges The beam is also subjected to the upward distributed load, f(x), which is induced by the prestressed tendon That is, the structure is initially deformed in compression (e.g., up to the deformed span length Lr) and the tendon is still in tension due to the constraint after elastic stretching as shown in Figure 1(c) The tendon is also subjected to the downward distributed force, f(x) The initial deformation of the beam results in the reduction of span length, δL(= L − Lr), and the expansion in the cross-section by Poisson effect Concrete beam Steel tendon x Lr = L(1 − δL /L) y (a) Prestressed beam with a parabolic tendon T Anchor force T f(x) Lr (b) Upward distributed force and anchor force T on beam e(0) ε T y x e(x) e(Lr /2) f(x) Ls Tendon force T (c) Tension force T on pin-pin ended tendon of arc-length Ls Figure Effective flexural rigidity model of PSC beam with an eccentric tendon Advances in Structural Engineering Vol 15 No 2012 Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park The governing differential equation of the effective flexural rigidity model of the PSC beam with the curved tendon [as shown in Figure 1(a)] is expressed by: 2 ∂ y ∂  ∂ y + mr = EI 2 r r 2 ∂t ∂x  ∂x  (1) where Er Ir is the effective flexural rigidity of PSC beam section which is assumed constant along the entire length of the beam and mr is the effective mass per unit length of the beam The effective flexural rigidity of PSC beam can be evaluated as the combination of the flexural rigidity of concrete beam section and the equivalent flexural rigidity of tendon As shown in Figure 1, the effective flexural rigidity Er Ir and the effective mass mr of the PSC beam can be estimated, respectively, as follows: Er I r = Ec I c + E p I p (2) mr = ρc Ac + ρ p Ap (3) where Ec is the elastic modulus of concrete, Ic is the second moment of concrete beam’s cross-section area, Ep is the elastic modulus of steel tendon, and Ip is the second moment of tendon’s cross-section area Also, ρc Ac is the concrete mass per unit length and ρp A p is the tendon mass per unit length The equivalent flexural rigidity of tendon is derived from analyzing flexural vibration of tendon of arc-length Ls, as shown in Figure 1(c) The arc-length Ls is calculated as Ls = βLr, in which the geometric constant β is computed approximately as β ≈ (Lr /4ε) sin−1(4ε/Lr ) and ε = e(Lr /2) –e(0) with e(x) is the eccentric distance between the neutral axis of beam and the center of tendon section at x location By analyzing a pin-pin ended cable with the same span length Ls and the mass property ρp A p as the tendon, as shown in Figure 2(a), the cable subjected to tension force T leads the nth natural frequency ω nc By setting a corresponding beam with a span length Lr which produces the same nth natural frequency ω nc , as shown in Figure 2(b), the equivalent flexural rigidity EpIp to the tension force T is obtained as: ω nc  nπ  E p I p  nπ  T = =  β Lr  ρ p Ap  Lr  ρ p Ap T L  E pI p =  r  β  nπ  (4a) Advances in Structural Engineering Vol 15 No 2012 (4b) ρp Ap f (x) ωcn T Ls = βLr (a) Pin-pin ended cable of span-length Ls subjected to tension force T Ep Ip f (x) ρp Ap ωnc Lr (b) Equivalent beam of span-length Lr with flexural rigidity EpIp Figure Flexural rigidity model of tendon subjected to tension force T where n is mode number and T is tension force of cable On substituting Eqn 4(b) into Eqn and furthermore applying Eqn with appropriate boundary conditions to Eqn 1, the nth natural frequency of the effective flexural rigidity model of the PSC beam can be obtained as: ω n2  nπ   T  Lr   E I + =      c c  Lr  mr  β  nπ   (5) Once the nth natural frequency ω n of the PSC beam is known, the prestress force can be identified from an inverse solution of Eqn 5, as follows: 2   nπ    Lr  Tn = β  ω n mr   − Ec I c     nπ    Lr   (6) where Tn is the identified prestress force by using the nth natural frequency and structural properties By assuming the mass property (mr) and the span length (Lr) remain unchanged due to the change in prestress force, the first variation of the prestress force can be derived as: 2   nπ    Lr  δ Tn = β  δ ω n mr   − δ Ec I c     nπ    Lr   (7) where δTn is the change in the prestress force that is identified from the nth mode and δω 2n is the change in ω 2n due to the change in prestress-force From Eqns and 7, the relative change in the prestress force that can be identified from the nth mode is obtained as: δ Tn = Tn  nπ  L δ ω n2 mr  r  − δ Ec I c    nπ   Lr   nπ  L ω n2 mr  r  − Ec I c    nπ   Lr  2 (8) 999 Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification On dividing both numerator and denominator by mrL2r /(nπ)2 and by further assuming that the change in concrete beam’s flexural rigidity due to changes in the prestress force is negligibly small (i.e., δEc Ic ≈ 0), Eqn is simply rearranged as: δ Tn δ ω2 = n Tn ω n − ϖ n (9) where ϖn is the nth natural frequency of the beam with zero prestress force and is given by: where Zi is the fractional change in the ith eigenvalues between two different structural systems (e.g., an analytical model and a real structure); M is the number of known eigenvalues Also, Sij is the dimensionless sensitivity of the ith eigenvalue ωi2 with respect to the jth structural parameter pj (Stubbs and Osegueda 1990; Zhang et al 2000) Sij = δ ω i2 p j δ p j ω i2 (13a) δ ω i2 ω i2 (13b) ϖ n2  nπ  E I =  c c  Lr  mr (10) From Eqn 9, the relative change (estimated by the nth mode) in prestress force between a reference prestress state (Tn, ref ) and a prestress-loss state (Tn, los) can be estimated as δ T n T n = (Tn ,ref − Tn ,los ) T n ,ref by measuring the corresponding nth natural frequencies ωn,ref and ωn,los, from which the reference eigenvalue is defined as ω n2 = ω n2,ref and the eigenvalue changes is computed as δω n2 = ω n2,ref − ω n2,los Unless measured at as-built state, the zero-prestress ϖn should be estimated from numerical modal analysis In most existing structures, its field measurement is almost impossible and we should rely on a baseline model updated from well-established SID process 2.2 Multi-Phase System Identification (SID) Scheme To identify a realistic theoretical model of a structure, Kim and Stubbs (1995) proposed a model update method based on eigenvalue sensitivity concept that relates experimental and theoretical responses of the structure (Adams et al 1978; Stubbs and Osegueda 1990) Suppose p*j is an unknown parameter of the jth member of a structure Also, suppose pj is a known parameter of the jth member of a FE model Then, relative to the FE model, the fractional structural parameter change of the jth member, αj ≥ –1, and the structural parameters are related according to the following equation: ( p*j = p j + α j ) (11) The fractional structural parameter change α j can be estimated from the following equation (Stubbs and Osegueda 1990): M Zi = ∑ Sijα j i =1 1000 (12) Zi = The term δ pj is the first order perturbation of pj which produces the variation in eigenvalue δωi2 The fractional structural parameter change of NE members may be obtained using the following equation: {α } = [ S ]−1 { Z } (14) where {α} is a NE × matrix, which is defined by Eqn 11, containing the fractional changes in structural parameters between the FE model and the target structure; {Z} is defined as Eqn 13(b) and it is a M × matrix containing the fractional changes in eigenvalues between two systems; and [S] is a M × NE sensitivity matrix, which is defined by Eqn 13(a), relating the fractional changes in structural parameters to the fractional changes in eigenvalues The sensitivity matrix, [S], is determined numerically in the following procedure (Stubbs and Osegueda 1990): (1) Introduce a known severity of damage (αj, j = 1, NE) at jth member; (2) Determine the eigenvalues of the initial FE model (ωi2o , i = 1, M); (3) Determine the eigenvalues of the damaged structure (ωi2 , i =1, M); (4) Calculate the 2 fractional changes in eigenvalues by Zi = ω i / ω io − ; (5) Calculate the individual sensitivity components from Sij = Zi / αj ; and (6) Repeat steps (2)−(5) to generate the M × NE sensitivity matrix If the number of structural parameters is much larger than the number of modes, i.e., NE >> M, the system is ill-conditioned and Eqn 14 will not work properly, which is a typical situation for civil engineering structures To produce stable solution, therefore, the number of structural parameters should be equal to or less than the number of modes, NE ≤ Μ In addition, for most complex structures, only a few vibration modes can be measured with good confidence and many substructural members are combined together with complex response motions in the vibration modes In order to ( ) Advances in Structural Engineering Vol 15 No 2012 Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park Perform model update phase-by-phase (K = 1, NP ) Select target structure: Select a model update phase (K ) Measure experimental modes: φ i,m, ω 2i,m (i = 1, M ) Compute numerical modal parameters of FE model φ ∗i,a, ω ∗2i,a (i = 1, M ) Establish initial FE model: Analyze numerical modes: φi,a, ω 2i,a (i = 1, M ) Compute sensitivity and fractional eigenvalue change Sij = Select NE model-updating parameters: δω 2i,a p ∗j δ pj ω∗2i,a & Zi = ω 2i,m =1 ω∗2i,a Group FE model into NE sub-structures Fine-tune structural parameters Analyze modal sensitivities of NE parameters up to M modes {α} = [S ]−1{Z } Check {α } ≅ Determine multi-phase for model update: Decide number of phases: NP = NE/M Arrange model-updating parameters (pj, j = 1, M ) for each phase No Yes No Update parameters p ∗ = pj (1 + αj ) j Check K = NP ( j = 1, M ) Yes Identify the baseline model Figure Multi-phase system identification (SID) scheme overcome these problems, a multi-phase model update approach is needed to be implemented for updating the FE models of the complex structures For a target structure which has experimental modal parameters , a multi-phase SID is designed as schematized in Figure First, an initial FE model is established to numerically analyze modal parameters Second, NE structural parameters ( pj, j = 1,NE ) are selected by grouping the FE model into NE sub-structures and analyzing modal sensitivities of the NE parameters up to M modes Third, the number of phases NP is determined by computing NP = NE/ M and arrange the M number of structural parameters ( pj, j = 1, M) for each phase Finally, the following five sub-steps are performed for phase K (i.e., K =1, NP): (1) Compute numerical modal parameters of a selected FE model; (2) Compute sensitivities of structural parameters and the fractional change in eigenvalue between the target structure and the updated FE model (i.e., M × {Z} matrix); (3) Fine-tune the FE model by first solving Eqn 14 to estimate fractional changes in structural parameters (i.e., NE × {α} matrix) and then solving Eqn 11 to update the structural parameters of the FE model; Advances in Structural Engineering Vol 15 No 2012 (4) Repeat the whole procedure until {Z} or {α} approach zero when the parameters of the FE model are identified; and (5) Estimate the baseline model after the parameters are identified from phase K In each phase, the selection of structural parameters is based on the eigenvalue sensitivity analysis and the number of available modes Primary structural parameters which are more sensitive to structural responses will be updated in the prior phases It is also expected that the error will be reduced phase after phase, and, as a result, the accuracy of the baseline model will be improved consequently Note that numerical modal analysis is performed by using commercial FE analysis software such as SAP2000 (2005) VIBRATION TEST ON LAB-SCALED PSC GIRDER Dynamic tests were performed on a lab-scaled posttension PSC girder to determine the experimental modal parameters for a set of prestress cases The schematic of the test structure is shown in Figure The PSC girder was simply supported with the span length of m and installed on a rigid testing frame Two simple supports of the girder were simulated by using thin rubber pads as interfaces between the girder and the rigid frame The 1001 Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification Stressing jack Load cell Accelerometer 1m Anchor plate Sensor Sensor Sensor Sensor 0.2 m 0.95 m Sensor RC beam Tendon Wedge Impact Sensor Sensor Rubber support 6m 0.2 m (a) Experimental setup for PSC girder (b) Test girder (c) Stressing jack and load cell Figure Vibration test on the lab-scaled PSC girder T-section was reinforced in both longitudinal and transverse direction with 10 mm diameter reinforcing bars (equivalent to Grade 60) The stirrups were used to facilitate the position of the top bars A seven-wire straight concentric mono-strand with 15.2 mm diameter (equivalent to Grade 250) was used as the prestressing tendon The tendon was placed in a 25 mm diameter duct that remained ungrouted The structure was tested in Smart Structure engineering Lab located at Pukyong National University, Busan, Korea During the test, temperature and humidity in the laboratory were kept close to constant as 18−19oC and 40−45% by air conditioners, respectively, in order to minimize the effect of those ambient conditions that, if not controlled, might lead to significant changes in dynamic characteristics Recently, the interest on variability of dynamic properties of bridges (i.e., natural frequency, mode shape, damping ratio) caused by environmental effects (i.e., temperature, humidity, wind) has been increasing Cornwell et al (1999) reported that the natural frequencies of the Alamosa Canyon Bridge in southern New Mexico were varied by up to 6% over a 24hour period The results of almost one year monitoring of the Z24-Bridge located in Switzerland were presented by Peeters and De Roeck (2001) During the monitoring period, the frequency differences ranged from 14−18% due to normal environmental changes To study the environmental effects on modal parameters, a long term 1002 monitoring test was carried out during months on the Romeo Bridge which is a prestressed concrete box girder bridge located in Switzerland (Huth et al 2005) Due to the temperature change of 40oC, the variations of natural frequencies of the first three bending modes were 0.3 Hz, 0.35 Hz, and 0.5 Hz, respectively In addition, Kim et al (2007) proposed a vibration-based damage monitoring scheme to give warning of the occurrence, the location, and the severity of damage to a model plate-girder bridge under temperature-induced uncertainty conditions For the test bridge, natural frequencies went down as the temperature went up and bending modes were more sensitive than torsional modes As shown in Figure 4(a), seven accelerometers (Sensors 1–7) were placed on top of the girder with a constant m interval The impact excitation was applied in vertical direction by an electromagnetic shaker VTS100 at a location 0.95 m distanced from the right edge Seven ICP-type PCB 393B04 accelerometers with the nominal sensitivity of V/g and the specified frequency range (± 5%) of 0.06–450 Hz were used to measure dynamic responses with the sampling frequency of kHz The accelerometers were mounted on magnetic blocks which were attached to steel washers bonded on the top surface of the girder The data acquisition system consists of a 16-channel PXI4472 DAQ, a PXI-8186 controller with LabVIEW (2009) and MATLAB (2004) Advances in Structural Engineering Vol 15 No 2012 Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park Axial prestress forces were introduced into the tendon by a stressing jack as the tendon was anchored at one end and pulled out at the other A load cell was installed at the left end to measure the applied prestress force Each test was conducted after the desired prestress force has been applied and the cable has been anchored During the measurement, the stressing jack was removed from the girder to avoid the influence of the jack weight on dynamic characteristics of the test structure The prestress force was applied to the test structure up to five different prestress cases (i.e., T1−T5 as indicated in Table 1) The maximum and minimum prestress forces were set to 117.7 kN and 39.2 kN, respectively The force was uniformly decreased by 19.6 kN for each prestress-loss case Figure 5(a) shows acceleration response signals measured from Sensor when the prestress force was 117.7 kN Figure 5(b) shows frequency response curves measured from Sensor for the five prestress cases, T1–T5 Frequency domain decomposition (FDD) technique (Brincker et al 2001; Yi and Yun 2004) was implemented to extract natural frequencies and mode shapes from the acceleration signals For the five prestress cases, natural frequencies of the first two modes were extracted as summarized in Table Experimental natural frequencies of test structure for five prestress cases Prestress force Prestress case Natural frequency (Hz) (kN) T1 T2 T3 T4 T5 Mode Mode Mode Mode 23.72 23.60 23.39 23.23 23.08 102.54 101.70 101.65 101.39 98.73 – 0.51 1.39 2.07 2.70 – 0.82 0.87 1.12 3.72 117.7 98.1 78.5 58.9 39.2 10−4 0.1 Power spectrum Acceleration (g) −0.05 −0.1 T5 10−10 10−12 10 200 300 400 (a) Acceleration signal (b) Frequency responses T4 T3 T2 0.7 T1 0.5 0.3 0.4 0.3 0.2 0.1 Sensor location T5 T4 T3 T2 T1 Sensor location 500 0.1 −0.1 −0.3 −0.5 Mode 100 Frequency (Hz) 0.5 Time (s) Mode value Mode value 10−8 0.6 T5 T4 T3 T2 T1 10−6 0.05 0.7 Variation of frequency (%) −0.7 Mode 2 (c) Bending mode shapes Figure Acceleration signal, frequency responses and mode shapes from experimental measurement Advances in Structural Engineering Vol 15 No 2012 1003 Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification Table Also, the variation of natural frequencies with respect to the maximum prestress force T1 as the reference are given in Table The corresponding mode shapes of the first two bending modes were extracted as shown in Figure 5(c) Note that mode shapes were not changed significantly due to the change in prestress forces From the frequency response plots, Figure 5(b), there are several peaks between first and second bending modes These modes are torsional modes, axial modes and horizontal bending modes However, only vertical bending modes were considered in this study As shown in Figure 4(a), seven accelerometers were placed on top of the girder with a constant m interval Also, the impact excitation was applied in vertical direction by an electromagnetic shaker VTS100 For this reason, only vertical bending modes were extracted exactly from the experimental setup SYSTEM IDENTIFICATION OF PSC GIRDER WITH VARIOUS PRESTRESSFORCES 4.1 Initial FE Model and Model-Updating Parameters A structural analysis and design software, SAP2000 (2005), was used to model the PSC girder As shown in Figure 6, the girder was constructed by a threedimensional FE model using solid elements For analysis purpose, we divided the girder into 11,264 block elements The dimensions of the FE model were described in Figure For the boundary conditions, spring restraints were assigned at supports: horizontal and vertical springs for the left support and vertical spring for the right support Initial values of material, geometric properties and boundary conditions of the FE model were assigned as follows: (1) for the concrete girder, elastic modulus Ec = × 1010 N/m2, the second moment of area Ic = 4.9 × 10−3 m4, mass density ρc = 2500 kg/m3, and Poisson’s ratio vc = 0.2; (2) for the steel tendon, elastic modulus Ep = × 1011 N/m2, the second moment of area Ip = 1.9 × 10–5 m4, mass density ρp = 7850 kg/m3, and Poisson’s ratio vp = 0.3 ; and (3) the stiffness of vertical and horizontal springs kv = kh = 109N/m Numerical modal analysis was performed on the initial FE model and initial natural frequencies of the first two bending modes were computed as 23.65 Hz and 97.77 Hz, respectively Figure shows mode shapes of the two modes analyzed from the FE model Choosing appropriate structural parameters is an important step in the FE model-updating procedure All parameters related to structural geometries, material properties, and boundary conditions can be potential choices for adjustment in the modelupdating procedure For the PSC girder, therefore, structural parameters which were relatively uncertain in the FE model due to the lack of knowledge on their properties were selected as model update parameters Also, structural parameters which are relatively sensitive to vibration responses were considered as prior choices As shown in Figure 8, for the present PSC girder, six model update parameters were selected as follows: (1) flexural rigidity of concrete girder (EcIc) in the simple-span domain, (2) flexural rigidity of steel tendon (EpIp) in the overall structure, (3) flexural rigidity of the left overhang zone (EloIlo), (4) flexural rigidity of the right overhang zone (EroIro), (5) vertical spring stiffness (kv) at the left and right supports, and (6) horizontal spring stiffness (kh) 71 cm cm cm cm 27 cm cm cm cm 27 cm Springs 32 cm Concrete Tendon 14 cm cm Springs 18 cm Figure Initial FE model of the PSC girder 1004 Advances in Structural Engineering Vol 15 No 2012 Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park (a) Mode (b) Mode EcIc EloIlo kh EroIro kv EpIp kv 0.2 m 0.07 m 0.6 m Figure Numerical mode shapes of initial FE model 6m 0.2 m Figure Six model update parameters for the PSC girder Table Eigenvalue sensitivities of six model update parameters Mode No anchors and concrete sections on dynamic responses under varying prestress forces On estimating the initial FE model, the initial values of the six model update parameters were assumed as follows: EcIc = 9.81 × 107 Nm2, EpIp = 5.73 × 106 Nm2, EloIlo = EroIro = 9.81 × 107 Nm2, and kv = kh = 109 N/m Then, the eigenvalue sensitivity analysis for the six model update parameters was carried out, as summarized in Table From the results, the flexural rigidity of concrete girder was the most sensitive parameter for both mode and mode The flexural rigidity of steel tendon was the second sensitive parameter Those high sensitive parameters were expected to contribute more intensively on the model update The stiffness of overhang zones and the stiffness of support springs were relatively less sensitive parameters That is, those less sensitive parameters were expected to contribute less intensively on the model update Due to the availability of the two modes, three model update phases were chosen to treat the six model update parameters In each phase, two structural parameters were chosen for adjustment Based on their sensitivities as listed in Table 2, the order of model update was arranged as follows: (1) Phase I: flexural rigidities of concrete girder (EcIc) and steel tendon (EpIp); (2) Phase II: flexural rigidities of left overhang (EloIlo) and right overhang (EroIro); and (3) Phase III: vertical spring stiffness (kv) and horizontal spring stiffness (kh) Eigenvalue sensitivities Eclc Eplp Elollo Erolro kv kh 0.8855 0.8817 0.1039 0.0537 0.0064 0.0223 0.0034 0.0105 0.0029 0.0150 0.0007 0.0268 at the left support Note that the left overhang zone includes stressing-jack, load-cell, tendon anchor, and 0.2 m girder section at the left edge, as shown in Figure 4(a) Also, the right overhang zone includes tendon anchor and 0.2 m girder section at the right edge Both overhang zones were selected due to the uncertainty in the stiffness due to the effect of tendon 4.2 System Identification Results for Various Prestress-Forces After selection of vibration modes and model-updating parameters, an iterative procedure schematized in Figure was carried out for model update It should be noted that three phases were performed phase-afterphase and two model-updating parameters were updated iteratively at each phase Consequently, the analytical natural frequencies determined at the end of iterations gradually approached those experimental values For prestress case T1 (117.7 kN), SID results are summarized in Table and also shown in Figure Table Natural frequencies (Hz) during model update iterations for prestress case T1 (117.7 kN) Updated frequencies (Hz) at each iteration _ Initial Phase I Phase II Phase III Target Mode Freqs _ (Girder & Tendon) (Overhang zones) (Spring supports) Freqs No (Hz) 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th (Hz) 23.65 97.77 24.70 104.08 23.22 23.80 23.95 24.00 24.01 24.01 24.04 24.01 23.99 23.97 23.97 23.94 23.92 98.16 100.47 101.05 101.22 101.26 101.28 102.29 102.02 101.79 101.52 101.47 102.15 102.01 Advances in Structural Engineering Vol 15 No 2012 23.91 101.80 23.72 102.54 1005 Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification 5.0 Phase I (Girder & Tendon) Error (%) 4.0 Phase III (Support springs) Phase II (Overhang zones) 3.0 Mode Mode 2.0 1.0 0.0 Initial 10 11 12 13 14 15 Iteration Figure Convergence errors of natural frequencies for prestress case T1 (117.7 kN) Table shows natural frequencies during 15 iterations of multi-phase model update Figure shows convergence errors of updated natural frequencies with compared to target natural frequencies which were experimentally measured at the prestress force of 117.7 kN Natural frequencies were converged with 1.2% error at Phase (when concrete girder and steel tendon members were updated), 1.0% error at Phase (when overhang members were updated), and less than 0.8 % error at the end of Phase (when support spring members were updated) Meantime, the flexural rigidities of concrete girder and steel tendon were identified, respectively, as EcIc = 1.12 × 108 Nm2 and EpIp = 5.68 × 105 Nm2 The flexural rigidities of overhang zones were identified as EloIlo = 4.15 × 108 Nm2 and EroIro = 1.38 × 106 Nm2, respectively Also, the stiffness parameters of support springs were identified as kv = 3.38 × 108 N/m and kh = 2.12 × 1012 N/m System identification results for all five cases (i.e., T1–T5) are summarized in Table and Table Table shows natural frequencies of updated FE models with compared to those of the target structure For all five prestress cases, natural frequencies were converged with 0.1−1.2% error range Meanwhile, the six modelupdating parameters were identified as listed in Table As listed in Table 5, the updated model parameters were changed as the prestress forces were changed from T1 (117.7 kN) to T5 (39.2 kN) Figure 10 shows the relative changes in updated model parameters (with respect to the maximum prestress force T1 as the Table Natural frequencies (Hz) of updated FE models and target structures for five prestress cases Prestress case T1 T2 T3 T4 T5 1st Frequency (Hz) 2nd Frequency (Hz) Prestress force (kN) Experiment FEM Error (%) Experiment FEM Error (%) 117.7 98.1 78.5 58.9 39.2 23.72 23.60 23.39 23.23 23.08 23.91 23.74 23.62 23.50 23.09 0.80 0.59 0.98 1.15 0.06 102.54 101.70 101.65 101.39 98.73 101.80 101.11 100.64 100.13 98.54 0.72 0.58 0.99 1.24 0.19 Table Identified values of model update parameters for five prestress cases Prestress case T1 T2 T3 T4 T5 1006 Prestress force (kN) Updated model parameter EcIc (Nm2) EpIp (Nm2) 117.7 98.1 78.5 58.9 39.2 1.12E+8 1.11E+8 1.10E+8 1.09E+8 1.06E+8 5.68E+5 4.78E+5 3.79E+5 2.88E+5 1.92E+5 EloIlo (Nm2) 4.15E+8 4.14E+8 4.14E+8 4.14E+8 4.13E+8 EroIro (Nm2) kv (N/m) kh (N/m) 1.38E+6 1.37E+6 1.37E+6 1.36E+6 1.35E+6 3.38E+8 3.32E+8 3.28E+8 3.20E+8 3.17E+8 2.12E+12 2.12E+12 2.12E+12 2.12E+12 2.12E+12 Advances in Structural Engineering Vol 15 No 2012 Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park 1.2 δ EcIc = 7.0 × 10−4 T + 0.921 Relative change of EpIp Relative change of EcIc 1.2 1.1 1.0 0.9 0.8 39.2 58.9 78.5 98.1 Prestress force (kN) δ EpIp = 8.45 × 10−3 T + 0.007 1.1 1.0 0.9 0.8 39.2 117.7 (a) Concrete girder’s EcIc 1.2 δ EloIlo = 5.73 × 10−5 T + 0.993 Relative change of EroIro Relative change of EloIlo 117.7 (b) Steel tendon’s EpIp 1.2 1.1 1.0 0.9 0.8 39.2 58.9 78.5 98.1 Prestress force (kN) δ EroIro = 2.88 × 10−4 T + 0.967 1.1 1.0 0.9 0.8 39.2 117.7 (c) Left overhang’s EloIlo 117.7 1.2 Relative change of kh δ kv = 8.07 × 10−4 T + 0.904 1.1 1.0 0.9 0.8 39.2 58.9 78.5 98.1 Prestress force (kN) (d) Right overhang’s EroIro 1.2 Relative change of kv 58.9 78.5 98.1 Prestress force (kN) 58.9 78.5 98.1 Prestress force (kN) 117.7 (e) Vertical spring’s kv δ kh = 1.63 × 10−5 T + 0.998 1.1 1.0 0.9 0.8 39.2 58.9 78.5 98.1 Prestress force (kN) 117.7 (f) Horizontal spring’s kh Figure 10 Relative changes in updated model parameters due to changes in prestress forces reference) due to the changes in prestress forces For both concrete girder and steel tendon, their flexural rigidities (δEcIc and δEpIp) changes almost linearly as the prestress force changes The steel tendon’s stiffness was greatly influenced by the prestress forces, but the change in concrete girder’s stiffness was relatively small The change in the right overhang’s stiffness (δEloIlo and δEroIro) was relatively larger than the left overhang which remained almost unchanged The change in the vertical spring’s stiffness (δkv) was relatively small and the horizontal spring’s stiffness (δkh) was remained nearly unchanged as the prestress Advances in Structural Engineering Vol 15 No 2012 force changes To estimate the relationships between the prestress-forces and the relative changes in the six structural parameters (i.e., δEcIc, δEpIp, δEloIlo, δEroIro, δkv, and δkh), six empirical equations were established as follows: Ec I c = 1.12 × 108 × δ Ec I c = 78.62T + 1.03 × 108 Nm (15a) E p I p = 5.68 × 105 × δ E p I p = 4.8T + 3.99 × 103 Nm2 (15b) 1007 Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification Elo I lo = 4.15 × 108 × δ Elo I lo = 23.79T + 4.12 × 108 Nm2 (15c) Ero I ro = 1.38 × 106 × δ Ero I ro = 0.4T + 1.33 × 106 Nm2 (15d) kv = 3.38 × 108 × δ kv = 272.61T + 3.05 × 108 N/m (15e) kh = 2.12 × 1012 × δ kh = 34, 594T + 2.12 × 1012 N/m (15f) From the empirical equations, values of the six model parameters can be identified with respect to the required amount of prestress force That is, the updated FE model represents the six model parameters corresponding to dynamic responses for a certain prestress-force state of the PSC girder The identified baseline model can be the role of the reference structure to make diagnosis and prognosis on the structure of interest PRESTRESS-FORCE MONITORING OF PSC GIRDER 5.1 Estimation of Natural Frequencies for Various Prestress Forces Natural frequencies of the PSC girder are predicted by an updated FE model, from which a zero-prestress state model was also identified for prestress-force estimation By noticing that the updated FE model is represented by the six model parameters (i.e., EcIc, EpIp, EloIlo, EroIro, kv, and kh), natural frequencies for various prestress forces can be estimated by using the six empirical equations So the FE model parameters corresponding to certain prestress-forces can be estimated from Eqns 15(a) to 15(f), in which values of the six model parameters are linearly related to required amounts of prestress forces As plotted in Figure 11 and also listed in Table 6, the FE analysis produced natural frequencies of the PSC girder for the five prestress-force cases and a zeroprestress (i.e., T = 0) state Compared to the experimental results, the natural frequencies of the FE model, fn,f, show very small estimation errors: 0.5−0.9% 104 25 Mode Natural frequency (Hz) Natural frequency (Hz) Mode 24 23 22 102 100 98 96 20 40 60 80 Prestress force (kN) 100 120 Experiment 20 40 60 80 Prestress force (kN) 100 120 FE model by empirical equations Figure 11 Prediction of natural frequencies of the PSC girder by two zero-prestress models Table Estimation of natural frequencies for five prestress cases Prestress case T1 T2 T3 T4 T5 Zero-Prestress 1008 Prestress force (kN) Experiment fn,e (Hz) Mode Mode 117.7 98.1 78.5 58.9 39.2 0.0 23.72 23.60 23.39 23.23 23.08 N/A 102.54 101.70 101.65 101.39 98.73 N/A FE model by empirical equations fn,f (Hz) Mode Mode 23.94 23.76 23.57 23.39 23.20 22.81 101.94 101.20 100.46 99.71 98.94 97.38 Advances in Structural Engineering Vol 15 No 2012 Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park 5.2 Evaluation of Prestress-Force Monitoring The relative change in prestress force with reference to the full prestress force is identified by measuring the relative change in the nth natural frequency with reference to the frequency margin between the full prestress state and the zero prestress state In order to predict the prestress-loss for the PSC girder, Eqn is rewritten in a convenient form as follows:  δT  fn2,ref − fn2,los δ fn2 = =   2 fn2,ref − ζ n2  Tref  n fn ,ref − ζ n (16) From Eqn 16, the relative change in prestress force, δT / Tref = (Tref – Tlos) / Tref, between a reference prestress state (Tref) and a prestress-loss state (Tlos) can be estimated by measuring the corresponding nth natural frequencies of the reference state fn,ref and the prestressloss state fn,los In this study, we selected Tref = 117.7 kN as the reference state, and f1,ref = 23.72 Hz for mode and f2,ref = 102.54 Hz for mode 2, accordingly, as listed in Table or Table Next, we selected two zero-prestress models as follows: the initial FE model and the updated FE model by the six empirical equations Then, the nth natural frequency for the zero-prestress state, ζ n, were estimated from the two zero-prestress models The first two natural frequencies ζ1 and ζ2 were estimated as follows (also listed in Table 7): (1) ζ1 = 23.65 Hz and ζ2 = 97.77 Hz for the initial FE model and (2) ζ1 = 22.81 Hz and ζ2 = 97.38 Hz for the updated FE model for T = state Here, all five prestress cases in Table were examined to detect the prestress-loss Table shows prestress-loss prediction results for the PSC girder using Tref = 117.7 kN and the two zero-prestress models The predicted prestress-loss results from the updated FE model were compared with the measured experimental prestress-losses The predicted prestresslosses versus the inflicted experimental prestress-losses were plotted in Figure 12 Depending on the accuracy of the experimental natural frequencies, the prediction by mode was more accurate than by mode In mode 1, the correlation between those sets was good; however, in mode 2, the correlation was relatively low due to measurement errors which might affect the experimental natural frequencies By substituting the reference state Tref = 117.7 kN into the prestress-loss prediction results (i.e., Table 7), prestress forces were predicted by the updated FE model As shown in Figure 13, compared to the experimental results, the accuracy of prestress-force prediction was relatively high in the updated FE model 1.0 Predicted prestress-loss in mode and 0.2−1.7% in mode It is observed that the updated FE model shows very accurate estimation results of natural frequencies 0.8 0.6 0.4 Mode 0.2 Mode 0.0 0.0 0.2 0.4 0.6 0.8 Experimental prestress-loss 1.0 Figure 12 Predicted prestress-losses versus inflicted prestresslosses for updated FE model Table Prestress-loss prediction using Tref = 117.7 kN and two zero-prestress models Initial Experiment Prestress case T1 T2 T3 T4 T5 Updated FE model FE model ζ1 = 23.65 Hz ζ = 97.77 Hz ζ1 = 22.81 Hz ζ2 = 97.38 Hz T (kN) δT Tref f1 (Hz) f2 (Hz)  δT   T   δT   T   δT   T   δT   T  117.7 98.1 78.5 58.9 39.2 0.0 0.17 0.33 0.50 0.67 23.72 23.60 23.39 23.23 23.08 102.54 101.70 101.65 101.39 98.73 0.0 N/A N/A N/A N/A 0.0 0.18 0.19 0.25 0.80 0.0 0.13 0.37 0.54 0.71 0.0 0.17 0.18 0.23 0.74 Advances in Structural Engineering Vol 15 No 2012 1009 Prestress force (kN) 120 Experiment Mode Mode 100 80 60 40 20 T2 (98.1 kN) T3 (78.5 kN) T4 (58.9 kN) T5 (39.2 kN) Prestress case Figure 13 Predicted prestress forces by updated FE model and Tref = 117.7 kN (i.e., 0.1−54.6% error) It is also observed that the accuracy of the prestress-loss monitoring depends on the accuracy of measured experimental frequencies and the accuracy of the baseline modeling of the zero prestress state as well For the vibration test, acceleration signals were measured in t = 10 seconds with the sampling frequency of fs = kHz, as shown in Figure 5(a) The number of data points used for the fast Fourier transform (FFT) was N = 213 = 8192 As a result, the frequency resolution was ∆f = fs / N = 1000/8192 = 0.122 Hz The accuracy of measured experimental frequencies depends on frequency resolution The more accurate natural frequencies were obtained as the smaller frequency resolution was used for the data process Furthermore, as described in section 2.1, the natural frequencies with zero prestress force are required in order to predict prestress-loss for the PSC girders However, in reality, the field measurement of most existing structures is almost impossible; therefore, they should be estimated from a baseline model updated from well-established SID process Due to the uncertainties in structural and environmental parameters, the initial FE model may not truly represent all the physical aspects of an actual structure For this reason, it must be updated by using experimental results As listed in Table 8, the errors in Table Effect of error in zero-prestress state’s natural frequencies (ζ) on prestress-loss prediction accuracy for prestress case T5 Error in ζ Mode Mode (%) ζ1  δT   T  ζ2  δT   T  1.0 5.0 10.0 25.0 22.81 22.58 21.67 20.53 17.11 0.71 0.57 0.32 0.21 0.11 97.38 96.40 92.51 87.64 73.03 0.74 0.63 0.39 0.27 0.15 1010 Predicted prestress-loss Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification 0.8 Inflicted prestress-loss (δ T/ Tref) = 0.67 0.6 Mode Mode 0.4 0.2 0 10 25 Error inζ (%) Figure 14 Effect of error in zero-prestress state’s natural frequencies (ζ) on prestress-loss prediction accuracy for prestress case T5 zero-prestress state’s natural frequencies were simulated for five error levels (e.g., 0, 1, 5, 10 and 25%) Figure 14 shows the effect of error in zeroprestress state’s natural frequencies (i.e., ζ ) on the accuracy of prestress-loss prediction for prestress case T5 From the results of mode and mode 2, it is observed that the more accurate prediction was obtained as the smaller error was inflicted in ζ Note that the zero-prestress state’s natural frequencies of the updated FE model (i.e., ζ1 = 22.81 Hz and ζ2 = 97.38 Hz) were set as the error-free baseline state Note also that the errors in natural frequencies were simulated by percentage reduction In reality, the mechanical properties of concrete vary with time and also by temperature, which would also cause the change in natural frequencies Besides, typical damage types of PSC girder bridges are not only tendon damage but also stiffness-loss in concrete girder and failure of support or connection Hence, the proposed method using the change in vibration characteristics alone may not be able to distinguish or isolate the change in prestress-loss from other damage types in the PSC girder To detect multiple damage types such as tendon damage and girder damage, Kim et al (2010) proposed the combined global vibrationbased and local impedance-based methods In their approach, the multiple damage types were classified into either tendon or girder damage by recognizing patterns of impedance features To deal with the real damage situations, therefore, global and local damage detection methods should be applied in conjunction with the presented method to identify the type, damage, and severity of damage in the PSC girder structures SUMMARY AND CONCLUSIONS In this study, a vibration-based method to estimate prestress-forces in a PSC girder by using vibration characteristics and SID approaches was presented The following approaches were implemented to achieve the Advances in Structural Engineering Vol 15 No 2012 Duc-Duy Ho, Jeong-Tae Kim, Norris Stubbs and Woo-Sun Park objective Firstly, a prestress-force monitoring method was formulated to estimate the change in prestress-force by measuring the change in modal parameters of a PSC beam Secondly, a multi-phase SID scheme was designed on the basis of eigenvalue sensitivity concept to identify a baseline model that represents the target structure Thirdly, the proposed prestress-force monitoring method and the multi-phase model update scheme were evaluated from controlled experiments on a lab-scaled PSC girder On the PSC girder, a few natural frequencies and mode shapes were experimentally measured for various prestress forces, 117.7−39.2 kN The corresponding modal parameters were analyzed from a baseline FE model, from which structural parameters were identified with respect to prestress forces from the proposed SID method From multi-phase SID, good correlations of natural frequencies between updated FE models and the target PSC girder were obtained for the various prestress forces From linear regression analysis of the results, the linear relationships between updated model parameters and prestress forces were established to estimate the influence of prestress forces on the performance of structural subsystems (and also to identify values of the six model parameters with respect to the required amount of prestress forces) Natural frequencies of the PSC girder under the various prestress forces were estimated by FE models, from which zero-prestress state models of the PSC girder were 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Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification On dividing both numerator and denominator by mrL2r /(nπ)2 and by further assuming that the change in concrete.. .Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification and boundary conditions of the PSC girder bridges Hence, it is necessary to develop a system identification. .. between the girder and the rigid frame The 1001 Prestress-Force Estimation in PSC Girder Using Modal Parameters and System Identification Stressing jack Load cell Accelerometer 1m Anchor plate Sensor