Comparison between real-time dynamic substructuring and shake table testing techniques for nonlinear seismic applications

Transcription

1 EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2010; 39: Published online 11 March 2010 in Wiley Online Library (wileyonlinelibrary.com)..994 Comparison between real-time dynamic substructuring and shake table testing techniques for nonlinear seismic applications C.-P. Lamarche 1,,, R. Tremblay 1,P.Léger 1, M. Leclerc 1 ando.s.bursi 2 1 Département des Génies Civil, Géologique et des Mines, École Polytechnique de Montréal, Montréal, Canada 2 Department of Mechanical and Structural Engineering, University of Trento, Trento 38100, Italy SUMMARY Results from real-time dynamic substructuring (RTDS) tests are compared with results from shake table tests performed on a two-storey steel building structure model. At each storey, the structural system consists of a cantilevered steel column resisting lateral loads in bending. In two tests, a slender diagonal tension-only steel bracing member was added at the first floor to obtain an unsymmetrical system with highly variable stiffness. Only the first-storey structural components were included in the RTDS test program and a Rosenbrock-W linearly implicit integration scheme was adopted for the numerical solution. The tests were performed under seismic ground motions exhibiting various amplitude levels and frequency contents to develop first and second mode-dominated responses as well as elastic and inelastic responses. A chirp signal was also used. Coherent results were obtained between the shake table and the RTDS testing techniques, indicating that RTDS testing methods can be used to successfully reproduce both the linear and nonlinear seismic responses of ductile structural steel seismic force resisting systems. The time delay introduced by actuator-control systems was also studied and a novel adaptive compensation scheme is proposed. Copyright q 2010 John Wiley & Sons, Ltd. Received 17 July 2009; Revised 10 December 2009; Accepted 14 December 2009 KEY WORDS: real-time dynamic substructuring; hybrid testing; shake table testing; Rosenbrock; benchmark; delay compensation Correspondence to: C.-P. Lamarche, Département des Génies Civil, Géologique et des Mines, École Polytechnique de Montréal, Montréal, Canada. charles-philippe.lamarche@polymtl.ca Contract/grant sponsor: Natural Sciences and Engineering Research Council of Canada (Canada Research Chair Program) Contract/grant sponsor: Fonds Québecois de la Recherche sur la Nature et les Technologies (FQRNT) Contract/grant sponsor: Canadian Foundation for Innovation Copyright q 2010 John Wiley & Sons, Ltd.

2 1300 C.-P. LAMARCHE ET AL. 1. INTRODUCTION Real-time dynamic substructuring (RTDS) testing, often referred to as hybrid testing or hardware in the loop testing, was first introduced in the late 1990s [1 5]. RTDS involves splitting the studied structure into two parts: the physical substructure that contains a key region of interest and is experimentally tested, and a numerical substructure that contains the reminder of the structure that is numerically simulated. By the use of numerical schemes that consider the compatibility and equilibrium conditions, the two substructures are made to interact such that they can emulate the dynamic behaviour of the full structure subjected to earthquake ground motions. The increasing interest in RTDS testing is motivated by two major features of the method: (1) critical structural components of interest in large civil engineering structures can be tested independently at large scale in a laboratory environment, whereas the remainder of the linear or nonlinear system is modelled numerically, (2) velocity-dependent phenomena, such as strain rate effects on material properties or viscous damping forces, can be accounted for experimentally because the test is performed in real time. A key component for successful RTDS testing is a fast and stable real-time integration scheme that does not require the knowledge of the deformation and velocity state beyond the current time in the analysis. In the past decade, integration methods of the Newmark family have been used for RTDS and their performance has been investigated, notably in research performed by Bonnet et al. [5], Wuet al. [6, 7], and Chen et al. [8]. Rosenbrock-based methodologies have also been proposed by Bursi et al. [9] and Lamarche et al. [10]. As RTDS testing becomes more popular among the earthquake engineering community, it is of utmost importance to evaluate the quality of the results obtained from this experimental tool using benchmark testing. Because shake table testing provides true dynamic loading, direct comparison between the test results from techniques, such as RTDS, can be made. Similar comparative studies showed good agreement between the results from pseudo-dynamic tests [11, 12] and the effective force testing method (EFT) [13] when compared with shake table test results. In this paper, the results obtained from shake table tests carried out in the Structural Engineering Laboratory of École Polytechnique de Montréal on the two-storey half-scale building structure presented in Figure 1 are compared with RTDS test results. In the RTDS tests, the first-storey (a) (b) Figure 1. Shake table test set-up (diagonal brace not shown): (a) 3D representation and (b) test set-up in the laboratory.

3 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1301 of the structure is tested in the laboratory whereas the remainder is modelled numerically. Time integration is performed using a Rosenbrock-W-based methodology [10]. Inelastic response occurs in the form of plastic hinging at the column bases. Inelastic response due to yielding in tension and buckling in compression is also present in some of the test cases where a slender diagonal steel brace is present at the bottom storey. With an added brace, the system exhibits an unsymmetrical nonlinear behaviour with inelastic strain accumulation dependency. The test program involved seven experiments using three excitation signals including the 1940 El Centro Imperial Valley earthquake record, a high frequency motion typical of Eastern North America, and a chirp-type signal. The amplitude of the excitations was varied such that both the linear and nonlinear structural responses could be investigated. In cases where the shake table test structure exhibited a nonlinear response in both storeys, the numerical model used in the RTDS tests also included nonlinear modelling capabilities. The results from purely numerical simulation are also compared with test results. Compensation methods to overcome the problems caused by the delayed response of actuatorcontrol systems are also investigated and a novel adaptive compensation scheme is proposed. This project represents a unique opportunity to compare, on a sound basis, the RTDS and shake table testing techniques as the two test programs were conducted in the same laboratory environment involving the same physical structural components. 2. TEST SET-UP The test structure used in this study represented a half-scale model of a two-storey steel building. At each level, the structural system consisted of a single cantilevered column rigidly anchored at its base and pin-connected at its upper end. The total height of the building was 3.0 m, i.e. 1.5 m per storey. A 3D representation of the shake table test set-up is shown in Figure 1(a), including the shake table, the gravity load supporting frame, and the strong floor of the laboratory. The W columns used to provide lateral stiffness and strength were made of ASTM A992 steel and were installed such that bending occurred about their strong axis. The steel shape had the following nominal properties: A =2470mm 2, I x = mm 4, Z x = mm 3.The yield and tensile strength values from mill test certificates are, respectively, F y =447MPa and F u =581MPa. The measured yield strength is 49% higher than the expected nominal yield strength (F y =345MPa). All column specimens used in the test program came from the same bin and were assumed to have essentially identical properties. The columns were anchored at their base with heavy steel plates carefully designed using a 3D nonlinear finite element model to ensure an optimal fixed end condition. The bottom-storey column was anchored to the shake table, whereas the top column was anchored to the first-storey mass. Each column was replaced after strong ground shaking producing significant yield. The masses at levels 1 and 2 were equal to 7250 and 6500 kg, respectively. Each mass was made of two reinforced concrete blocks that slid horizontally on low-friction rollers supported by the steel frame shown in Figure 1(a) and (b). At each storey the mass was connected to the top end of the column by the use of a stiff pinned-pinned horizontal steel strut. A load cell was mounted on each strut to measure the inter-storey shear at each level. The weight of the concrete masses was transferred to the laboratory strong floor by an independent two-storey braced steel frame located outside of the shake table. Consequently, no P Δ effects were induced in the columns that acted as simple cantilevers. An idealized 3D representation of the physical model considered in the RTDS test program is shown in Figure 2(a). The physical structure only included the

4 1302 C.-P. LAMARCHE ET AL. (a) (b) Figure 2. RTDS test set-up (diagonal brace not shown): (a) 3D representation and (b) test set-up in the laboratory. first-storey column, whereas the remainder of the building was modelled numerically. The column was anchored to the strong floor of the laboratory and horizontal displacements were applied through a high-performance hydraulic actuator. A counter weight system was used to maintain the actuator s effective weight near zero during the tests, thus avoiding axial loads to be induced in the tested columns. This counter weight device included a steel cable, a strut, and two pulleys, as depicted in Figure 2(b). In Tests #6 and #7, the shake table and RTDS tests were performed introducing a slender diagonal tension-only bracing member made of a 19.1mm 3.2mm flat steel bar at the firststorey. The measured yield and tensile strength properties of the steel bars were F y =340MPa and F u =452MPa, respectively. The tensile yield strength of the brace was equal to 20.8 kn. The brace had fixed end condition (K =0.5) and overall brace slenderness ratio KL/r 439, giving an estimated elastic buckling load of 0.2 kn. Owing to its very small buckling capacity, the brace can be considered as only acting when stretched in tension. This led to an unsymmetrical system with highly variable stiffness during the ground motions, depending on whether the brace was acting in tension or compression. 3. TEST PROGRAM Table I summarizes the dynamic tests performed in the course of this study. Unless stated otherwise, new (undamaged) structural elements were used in the nonlinear tests. Tests #1 and #2 were performed using a modulated chirp signal. Test #1 aimed at computing the experimental frequency response functions (FRFs) of the structure. Test #2, using the same input signal as in Test #1, was carried at higher amplitude to obtain repeated nonlinear excursions at both storeys. Test #3 was performed using the scaled 1988 Saguenay earthquake record (Chicoutimi North, S00E). This signal has relatively high-frequency content and small displacements, which are typical for earthquakes expected in Eastern North America. This ground motion record aimed at exciting the

5 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1303 Table I. Test program. Test Type Record Scale factor PGA (g) Freq. content #1 Linear Modulated chirp Hz #2 Nonlinear Modulated chirp Hz #3 Linear 1988 Saguenay High #4 Linear 1940 El Centro Low #5 Nonlinear 1940 El Centro Low #6 Nonlinear 1940 El Centro Low #7 Nonlinear 1940 El Centro Low Test involving the slender, tension-only, diagonal steel bracing member at the first-storey. second mode of the structure in the linear regime. Subsequent tests were performed under scaled versions of the 1940 Imperial Valley earthquake record (El Centro, S90W), which mainly excited the fundamental mode of the structure. In Test #4, the El Centro record was scaled to excite the structure in the linear regime. In Test #5, the unscaled El Centro record was used to induce large inelastic excursions at the bottom storey. Tests #6 and #7 were performed using a slender diagonal tension-only bracing member at the first-storey, as described in Section NUMERICAL MODEL AND TIME INTEGRATION An idealized two DOF model of the test building structure is presented in Figure 3. In the shake table test set-up case, the steel frame carrying the masses was supported by the strong floor of the laboratory whereas the first storey column was anchored to the shake table. Hence, two reference systems had to be used to describe the equation of motion, rather than the relative to ground formulation used in standard practice. In this mixed reference formulation, characterized by Equation (1), the displacement and the acceleration outputs from the shake table were used as inputs to properly model the frictional forces and adequately compare the shake table test results with the purely numerical simulations and RTDS test results. Mü(t)+C u(t)+r S (u(t),t)+r F (u t (t),sign( u t (t)),t)= Mzü g (t) (1) In Equation (1), M and C are the mass and damping matrices, respectively. Vector r S is the relative displacement-dependent restoring force vector, e.g. in our case, the shear forces developed in the columns. The force vector r F aims at modelling the frictional forces induced by the rollers supporting the masses. Kinematic vectors u, u, ü, respectively, represent the displacements, velocities, and accelerations relative to the ground. The vectors with t as superscript are absolute kinematic quantities and ü g (t) is the ground acceleration with respect to the absolute reference system. The two reference systems are related by u t =u +u g (2) and its time derivatives. In the numerical model, the pseudo-static vector z was assumed to be a unit vector as in the case of classical ground motion dynamic analyses. This assumption is made because frictional forces r F affect the shape of the pseudo-static vector only at a very low amplitude of vibrations. The adequateness of this assumption is verified in the subsequent sections comparing

6 1304 C.-P. LAMARCHE ET AL. Figure 3. Finite element model representation. numerical simulations and RTDS test results with shake table test results. The numerical model as well as the integration scheme were implemented using MathWorks Simulink R and XPC target R Selection of model parameters Prior to carrying out the RTDS tests, quasi-static cyclic test were performed to assess the friction induced by the rollers supporting the concrete masses and the hysteretic behaviour of the steel columns. These properties were used in the numerical substructure of the RTDS tests as well as in purely numerical simulations. Frictional forces were measured by imposing constant velocity displacement cycles on the full shake table model. In these characterization tests, it was found that the beams of the braced frame supporting the rollers had a slight geometrical inclination. This defect was also included in the numerical model. These combined effects were modelled with the inverted parallelogram hysteretic law shown in Figure 4. In Figure 4, a comparison between a quasi-static cyclic test and the numerical model is presented for both storeys. F F,1 and u t 1, respectively, correspond to the frictional force and the total displacement at storey 1. The friction model slopes were calibrated using a least-square curve fitting method. The amplitudes were determined so that the amount of energy dissipated in a full cycle in the model was equal to the measured value. A quasi-static cyclic test was performed using the RTDS test rig to characterize the behaviour of the steel columns. The test protocol used, presented in Figure 5(a), was adapted from Chapter S6.2. of the AISC seismic provisions for structural steel building [14] (loading sequence for beam-to-column moment connections). The inelastic flexural behaviour of the steel columns was modelled using a modified Giuffré--Menegotto Pinto hysteretic model [15]. In the model, the following parameters based on the nomenclature employed in Mazzoni et al. [16] were used: F y =36kN, E =700kN/m, b =5%, R 0 =15, cr 1 =0.8, cr 2 =0.15, a 1 =a 3 =2.0%, a 2 =a 4 =1.0. In Figure 5(b), the complete measured and predicted hysteretic flexural responses of the column are

7 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1305 (a) (b) (c) Figure 4. Comparison between the friction test results and the numerical model: (a) storey 1; (b) storey 2; and (c) response from preliminary numerical simulations and shake table test results. compared. The hysteretic model in Figure 5(b) and (c) is in excellent agreement with the test data. In Figure 5, V 1 and Δ 1 are the inter-storey shear and inter-storey displacement at the first-storey, respectively. The stiffness k 1 and k 2 used in the purely numerical analyses, based on the shake table hysteretic results, are summarized in Table II. Following the calibration of the numerical model, preliminary numerical simulations were performed. These numerical results suggested that the energy dissipated by friction in the dynamic range was in fact less than predicted by the numerical model. Therefore, the friction amplitudes at both storeys were reduced by 50%, keeping the negative slope of the hysteretic laws intact. In addition, viscous Rayleigh damping was used with ξ=1.5% of critical damping in both modes. A comparison between the dynamic responses obtained with the two aforementioned energy dissipation mechanisms is presented in Figure 4(c). Numerical simulation and test results in the time domain and frequency domain are presented in Section 6 that also support the adequacy of the chosen energy dissipation parameters Numerical integration The integration scheme chosen for this study is the Rosenbrock-W variant proposed in [10]. Prior to testing, the effective mass matrix M 0 is calculated from M 0 =[M+γΔtC 0 +γ 2 Δt 2 K 0 ] (3) In Equation (3), γ is a constant of the method that controls the numerical energy dissipation, Δt is the integration time step, C 0 and K 0 are estimates of the initial damping and stiffness matrices, and M is the mass matrix. The discrete time integration process begins with the resolution of the

8 1306 C.-P. LAMARCHE ET AL. (a) (b) (c) Figure 5. Hysteretic response of a column: (a) cyclic displacement protocol; (b) complete hysteretic response; and (c) isolated inelastic cycles at different displacement amplitudes. Table II. Properties determined from the shake table tests. Tests # k 1 (kn/m) k 2 (kn/m) f 1 (Hz) f 2 (Hz) 1, , 4, linear system in Equation (4), for ẽ, and to calculate the intermediate displacement and velocity vectors u k+1/2 and u k+1/2. ẽ = M 1 0 Δt(p k r (n) k r (e) k γδt(k 0 u k )) u k+1/2 = u k + Δt 2 ( u k +γẽ) u k+1/2 = u k + 1 2ẽ (4) In Equation (4), p k is the external force vector at pseudo-time k, andr (n) k and r (e) k are the numerical and experimental restoring force vectors, respectively. The kinematic quantities calculated from Equation (4) are imposed to the test specimen and the restoring force vector r (e) k+1/2 is assembled from the experimental force feedbacks. It is noted that the inverse of the effective mass matrix M 1 0 is computed only once, prior to testing, and remains constant throughout the duration of the

9 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1307 RTDS test. Then, the linear system in Equation (5) is solved for e and the updated displacement and velocity vectors u k+1 and u k+1 are calculated. e = M 1 0 Δt(p k+1/2 r (n) k+1/2 r(e) k+1/2 +(γδtβ 0K 0 +γc 0 )ẽ) u k+1 = u k +Δt( u k +β 1 ẽ+γe) u k+1 = u k +e (5) In Equation (5), the constants β 0 and β 1 are equal to: β 0 =2γ 1 2 and β 1 = 1 2 γ. The updated kinematic quantities u k+1 and u k+1 are imposed to the test specimen and the updated experimental restoring force vector r (e) k+1 is assembled. Pseudo-time index k is incremented and the linear system in Equation (4) is solved. This procedure is repeated until the end of the test. The digital controller used in this study had a minimum control time step of Δt = s. Therefore, the integration time step was set equal to Δt = s to perform the two sub-steps inherent to the method. The parameter γ was taken as γ= 1 2, corresponding to the unconditional stable Rosenbrock-W in the nonlinear regime [10]. The initial matrix estimates C 0, K 0,andM, given in Equation (6), were used in all tests, including the tests where the structure was braced. If the components of the initial matrix estimates C 0 and K 0 are greater than the actual corresponding true damping and stiffness values, the scheme is known to be dissipative. However, for small integration time steps and small natural frequencies, as in the present study, the energy dissipation is very small [10]. The main advantages of the approach, provided that the experimental mass is null or assumed to be so and that the damping and stiffness matrices are in a suitable norm larger than those of the tested substructure, are: (1) its capability to combine positive properties of implicit and explicit methods, i.e. it needs only the solution of a linear system of equations at each time step, (2) its unconditional stability, (3) second-order accuracy maintained also in the nonlinear regime without involving any matrix inversion during testing, but only once at the beginning, (4) suitable for large classes of stiff problems through controllable numerical energy dissipation, (5) explicit evaluation of target displacements and velocities. Moreover, the Rosenbrock-W scheme has been developed not only for RTDS applications but is also suitable for pseudo-dynamic testing. These properties have been analysed in depth in Lamarche et al. [10]. [ ] [ ] [ ] K 0 = kn/m, C 0 = kns/m, M= kg (6) ACTUATOR-CONTROL AND DELAY COMPENSATION The hydraulic actuator used for the RTDS tests had a 100 kn capacity and a ±127mm dynamic stroke. The 227 l/min two-stage servo-valve of the actuator was driven using a real-time MTS R structural PID control system connected to a real-time PC via Scramnet R shared memory. The integration scheme was implemented using MathWorks Simulink R and XPC Target R. The chosen PID gains, after careful tuning of the actuator-control system, were: P =4.5, I =0.1, and D =0. The derivative gain (D) was purposely set to zero to avoid control instabilities. The aforementioned gain values were used for all RTDS experiments.

10 1308 C.-P. LAMARCHE ET AL. (a) (b) Figure 6. Bode plots of the transfer function of the actuator-control system with η=( s) (1024s 1 ): (a) transfer functions obtained experimentally and theoretically in the 0 10 Hz bandwidth and (b) theoretical transfer functions in the 0 50 Hz bandwidth Constant delay compensation The delay of the actuator-control system was evaluated using a Hz band limited pink noise command signal with a standard deviation σ values of 5 mm, resulting in a maximum absolute displacement of 15 mm. From the phase plot of the experimental transfer function of the actuatorcontrol system presented in Figure 6(a), the average group delay was estimated to be τ=19ms. The contribution of natural vibration mode signatures to the total response in the displacement of a building structure excited at its base typically decreases in importance for higher natural modal frequencies. Pink noise is therefore a good test signal candidate because its Fourier spectral amplitude F( f ) is proportional to the inverse of the square root of the frequency: F( f ) = S( f ) 1 (7) f where S( f ) is the power spectral density function of the signal. To compensate for the measured delay of the actuator-control system, two approaches were initially considered. The first approach, commonly referred to as feed forward compensation [17, 18], is u(kδt c +τ c )=u k +τ c u k =(1+η)u k ηu k 1 (8) In Equation (8), τ c is the experimentally determined actuator-control system s delay value (τ c =19ms) and η=τ c /Δt c is a dimensionless constant parameter where Δt c is the control timestep. In Equation (8), the velocity u k at pseudo-time index k is approximated by the first-order numerical derivative: u k = u k u k 1 Δt + O(Δt) (9)

11 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1309 The experimental transfer function corresponding to Equation (8) is presented in Figure 6(a) where delay is well compensated in the 0 5 Hz range, but overshoots the target displacement as the frequency increases above 5 Hz. A better overall delay compensation performance was obtained by using the compensation scheme proposed by Bonnet et al. [19]: u(kδt c +τ c ) = ( η+η η3) u k ( 3η+ 5 2 η η3) u k 1 +( 32 η+2η η3) u k 2 ( 13 η+ 1 2 η η3) u k 3 (10) Equation (10) is an adaptation of a scheme proposed by Horiuchi et al. [1] based on Lagrange interpolation polynomials. The experimental transfer function related to Equation (10) is also presented in Figure 6(a) for comparison. Excellent performance was achieved both in magnitude and phase in the 0 10 Hz range. Notwithstanding these desirable properties in the 0 10 Hz range, the scheme exhibited spurious low-amplitude high-frequency oscillations when performing RTDS tests. This can be explained by studying the theoretical transfer functions of the scheme. Applying the Z-transform in Equation (11) to the delay compensation difference equation in Equation (10), δ[k 0 k]=z k (11) where δ is the Kronecker delta function. The frequency response can be obtained by replacing z with the complex exponential: z =e i2π f (12) with i= 1. This leads to the Bode plots of the theoretical transfer functions presented in Figure 6(a) and (b). In Figure 6(b), the theoretical curve corresponding to Equation (10) significantly overshoots the displacements for high-frequency values compared with the curve obtained from Equation (8), thus explaining the aforementioned spurious low-amplitude high-frequency oscillations observed during RTDS tests. In the theoretical analyses, the actuator s transfer function was assumed to be a linear constant delay system with a unit magnitude (perfect delay system). The slight systematic difference in phase between the experimental and the theoretical results in Figure 6(a) is due to a degradation of the actuator-control system s performance above f =5Hz. In order to obtain a compromise between the overshooting phenomenon as the frequency increases and proper delay compensation, a novel compensation scheme was derived using the well-known constant acceleration equation in a way similar to what has been proposed by Horiuchi and Konno [20] and Ahmadizadeh et al. [21]: u(kδt +τ)=u k +τ u k + τ2 2 ü (13) where for the present case, the velocity u k was approximated using Equation (9) and the acceleration was taken as the following first-order second numerical derivative: ü k = u k 2u k 1 +u k 2 Δt 2 + O(Δt) (14) Introducing Equations (9) and (14) into Equation (13) leads to compensation Equation (15). u(kδt c +τ c )=u k +η(u k u k 1 )+η 2 ( uk u k 1 +u k 2 2 ) (15)

12 1310 C.-P. LAMARCHE ET AL. (a) (b) (c) (d) Figure 7. (a) Measured displacement-dependent actuator-control system s delay values; (b) command vs feedback without delay compensation; (c) response with constant and adaptive delay compensation; and (d) control delay values. Equation (15) can also be written in the same format used for Equations (8) and (10): ) ) ( u(kδt c +τ c )= (1+η+ η2 u k (η+ η2 η 2 ) u k 1 + u k 2 (16) The transfer function associated with this scheme is presented in Figure 6(b) for comparison. This constant delay compensation scheme was used in all RTDS tests Adaptive delay compensation The constant delay compensation approach resulted in a less accurate compensation for very small displacement as the delay increases exponentially when approaching zero oscillation amplitudes. Figure 7(a) shows the delay of the actuator-control system evaluated using pink noise at nine different standard-deviation amplitudes ranging from 0.25 to 7 mm. Based on these data, the delay can be estimated from the following regression analysis: 34.5ms, σ 0.25mm ( τ c = ) (17) σ 0.75 ms, σ>0.25mm This phenomenon caused the system to exhibit a self sustained, small amplitude harmonic steadystate response at the beginning and at the end of the tests when the external excitation input is absent and the energy fed by the delay is dissipated by the damping of the system. This had a negligible impact on the quality of the test results due to the flexibility of the test structure. In contrast, a small displacement error can lead to large undesirable energy input in the case of stiff physical substructures. In this case, an adaptive delay compensation technique would be more

13 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1311 appropriate. For this purpose, the constant delay scheme presented in Equation (16) was modified and tested experimentally, updating τ c during RTDS tests according to aprioriknowledge of the delay vs standard-deviation curve of the actuator-control, and specimen trio as the one described by Equation (17). For this adaptive scheme, the standard deviation σ of the displacement at each time interval T c was chosen over of the RMS value because it accounts for displacement offsets typically encountered during nonlinear tests. When the mean displacement equals to zero, σ=rms. Figure 7(b) presents the command vs feedback response without delay compensation under a displacement-controlled logarithmically decaying displacement signal. The response under the same signal is presented in Figure 7(c) when using Equation (16) with η= s/ s as well as its adaptive counterpart. The control delay values τ c corresponding to Figure 7(c) are presented in Figure 7(d) where as expected in the adaptive case, τ c increases as the vibration amplitude decreases. In this proposed novel adaptive delay implementation, T c is a control parameter. The smaller the value of T c, the more instantaneous the scheme becomes. On the other hand, the larger the value of T c, the more reliable the standard-deviation estimate is. Based on experiments, taking a value of T c equal to half of the estimated fundamental period of the tested structure, i.e. the lowest natural frequency, was proven to be a valuable starting point to choose a value for T c.theseismic behaviour of building structures is typically governed by their fundamental mode. It is thus natural to relate T c to the fundamental period T a.whent c 0.5T a, the delay compensation scheme is updated twice in every fundamental cycles of vibration. The adaptive scheme proposed herein is based on an aprioriknowledge of the performance of the system. Other adaptive delay schemes have been proposed in the past that do not necessarily need knowledge on the performance of the actuator-control system, e.g. the Darby estimator [22], the modified Darby estimator [19], the adaptive forward prediction (AFP) scheme [23], or the learning gain approach [21]. For most of these schemes, the gains and tuning parameters involved are not always easy to tune because their effect on the response is not necessarily easy to interpret. Conversely, in the case of the adaptive scheme proposed herein, control parameter T c has a clear physical interpretation. 6. COMPARISON BETWEEN TEST RESULTS 6.1. FRF of the structure (Test #1) The test structure was subjected to a modulated chirp signal on the shake table to compare the resulting FRF with the FRFs obtained from a numerical simulation and from an RTDS test. The maximum amplitude of the excitation base acceleration signal was limited to 0.1 g such that the structure remained in the elastic range. Figure 8(a) and (c) present the inter-storey response at both levels. Excellent correlation is observed between the three simulation techniques. The FRFs corresponding to Figure 8(a) and (c) are presented in Figure 8(b) and (d), respectively. At both storeys, the magnitude and the phase plots of the FRFs are all in agreement. As expected, resonances and anti-resonances coincide with a major phase shift at θ= 90. The natural frequencies identified from the FRFs are presented in Table III, along with the equivalent viscous damping ratios evaluated using the half power bandwidth method. For both vibration modes, the natural frequencies and equivalent modal damping ratios evaluated from the numerical simulations and RTDS test data correspond well with the shake table test results.

14 1312 C.-P. LAMARCHE ET AL. (a) (b) (c) (d) Figure 8. Chirp signal (PGA=0.1g): (a) inter-storey displacement at storey 1; (b) FRF at storey 1; (c) interstorey displacement at storey 2; and (d) FRF at storey 2. Table III. Dynamic properties from the FRFs. Test method f 1 (Hz) ξ eq,1 (%) f 2 (Hz) ξ eq,2 (%) Shake table Numerical RTDS Nonlinear response to the chirp signal (Test #2) The test structure was subjected to the same modulated chirp input as in Test #1, except that it was scaled to a PGA value of 0.3 g such that the structure would exhibit cyclic inelastic excursions at both storeys. The nonlinear time-based and hysteretic responses at both storeys are presented in Figure 9. Again, at both storeys, the inter-storey displacements are in good agreement for all testing methods, except for a slight first-storey inelastic overshoot during the RTDS test at time t =21.9s (Figure 9(a)). That inelastic overshoot resulted in a permanent displacement offset that remained until the end of the test. This offset is observable in Figure 9(b) where the thin continuous curve is shifted to the left of the origin. The absolute maximum and the standard deviation of the inter-storey displacement values as well as the corresponding errors relative to the shake table test results obtained for each test method

15 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1313 (a) (b) (c) (d) Figure 9. Chirp signal (PGA= 0.3g): (a) inter-storey displacement at storey 1; (b) hysteretic response at storey 1; (c) inter-storey displacement at storey 2; and (d) hysteretic response at storey 2. Table IV. Response in displacement: chirp signal (Test #2, PGA = 0.3g). Storey 1 Storey 2 Inter-storey Δ 1 max e rel,1 σ 1 e σ,1 Δ 2 max e rel,2 σ 2 e σ,2 displ. (mm) (%) (mm) (%) (mm) (%) (mm) (%) Shake table Numerical RTDS are presented in Table IV. The maximum inter-storey displacement values aim at quantifying local errors in the responses, whereas the standard deviation σ aims at quantifying the global errors due to systematic overshooting/undershooting errors. In Table IV, all relative errors remain below 11%. In the case of the RTDS test, e rel,1 =9.8% is higher due to the aforementioned inelastic overshoot Earthquake simulations in the linear regime (Tests #3 and #4) The test structure was subjected to scaled historical earthquake records. The first earthquake test in the linear regime, referred to as Test #3, was performed using the 1988 Saguenay earthquake record (Chicoutimi North, S00E) scaled to a PGA=2.62 g. This record has a relatively high frequency content and small displacements that predominantly excited the second mode of the

16 1314 C.-P. LAMARCHE ET AL. (a) (b) Figure 10. First-storey response in the linear regime under: (a) Saguenay 1988 record (PGA= 2.62g) and (b) El Centro 1940 record (PGA= 0.174g). Table V. Seismic displacement response at storey 1 in the linear regime (Tests #3 and #4). Test #3 Test #4 Saguenay 1988 (PGA=2.62g) El Centro 1940 (PGA =0.174g) Inter-storey Δ 1 max e rel,1 σ 1 e σ,1 Δ 1 max e rel,1 σ 1 e σ,1 displ. (mm) (%) (mm) (%) (mm) (%) (mm) (%) Shake table Numerical RTDS structure. The first-storey response is presented in Figure 10(a). Good agreement is found between the displacement responses from the numerical simulation, the RTDS test, and the shake table test. The good match is confirmed in Table V where the absolute maximum and the standarddeviation errors for RTDS Test #3 are 8.5% or less. The second earthquake simulation test in the linear regime, referred to as Test #4, was performed using the 1940 Imperial Valley earthquake (El Centro, S90W) scaled to a PGA=0.174g that mainly excited the fundamental mode of the structure. The response obtained from the numerical simulation is compared with the RTDS and shake table test results in Figure 10(b). In all cases, the correspondence is excellent except for slight overshoots in the numerical simulations Earthquake simulations in the nonlinear regime (Tests #5, #6, and #7) Figures 11(a) and (b), respectively, present the nonlinear time history and hysteretic first-storey response of the unbraced structure to the El Centro 1940 unscaled record (PGA=0.348g). Compared with the shake table, the numerical simulation and the RTDS test predicted higher peak positive displacement values at t =3.3s, which affected the response for the remaining of

17 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1315 (a) (b) (c) (d) (e) (f) Figure 11. First-storey time history and hysteretic responses in the nonlinear regime under the scaled El Centro record: (a) and (b) unbraced structure (PGA = 0.348g); (c) and (d) braced structure (PGA= 0.178g); and (e) and (f) braced structure (PGA=0.348g). the ground motion. Those overshoots resulted in absolute maximum and standard-deviation differences of 10% or less compared with the shake table test results in Table VI. They also resulted in greater energy dissipation by the bottom-storey column compared with the shake table test case, as obtained from the area enclosed by the hysteretic curve in Figure 11(b). This is illustrated by the energy plot of the experimental substructure presented in Figure 12(a), as proposed by Mosqueda et al. [24], based on Uang and Bertero [25]. The slight nonlinear overshoot that occurred at t =3.3s in Figure 11(a) yielded to the ΔE 1 =0.5kNm energy offset in Figure 12(a) between the RTDS and the shake table tests. This offset corresponds to 10% of the total energy dissipated by the bottom-storey column up to time t =15s. Except for the differences in peak displacements at the beginning of the earthquake, all three methods gave comparable responses. In Figures 11(a) and (b), the time history and the corresponding hysteretic response of a purely numerical simulation using the Newmark implicit constant average acceleration method with Newton iterations to reach convergence are presented to investigate if the overshot that occurred at t =3.3s is due to the semi-implicit nature of the Rosenbrock-W scheme that does not involve any Newton iteration. As seen on Figures 11(a) and (b), the nonlinear response from the implicit Newmark scheme is

18 1316 C.-P. LAMARCHE ET AL. Table VI. Nonlinear first-storey response (unbraced structure). Δ 1 max e rel,1 σ 1 e σ,1 Inter-storey displacement (mm) (%) (mm) (%) Shake table test Numerical (semi-implicit Rosenbrock-W) Numerical (implicit Newmark with iterations) RTDS test (semi-implicit Rosenbrock-W) (a) (b) (c) Figure 12. Strain energy plots of the first-storey: (a) unbraced structure (PGA = 0.348g); (b) braced structure (PGA = 0.174g); and (c) braced structure (PGA = 0.348g). very similar to the one obtained from the purely numerical simulation using the linearly implicit Rosenbrock-W method. This fact is supported by the similar differences presented in Table VI. The strain energy curve corresponding to the implicit Newmark scheme, presented in Figure 12(a), is also in good agreement with those obtained using the Rosenbrock-W method. Tests #6 and #7 were performed on the structure that included a slender diagonal tensiononly bracing member at the first-storey leading to an unsymmetrical system with highly variable stiffness. Figure 11(c) and (d), respectively, present the nonlinear time histories and hysteretic first-storey responses of the braced structure to the El Centro1940 record scaled to a PGA=0.174g (Test #6). During that test, inelastic behaviour essentially developed in the form of buckling and tensile yielding of the bottom-storey brace. In Figure 11(d), three brace tension yielding excursions can be depicted at t =3.75, 4.75, 5.75 s. The columns remained elastic except for a slight flexural yielding event of the first-storey column at peak negative deformation at t = 4.9s during the shake table test. In Figure 11(c), the RTDS and shake table test results agree well except that the RTDS test predicted larger peak displacements in early high amplitude cycles and at the end of the seismic ground motions. Despite these differences, the energy plots of the first-storey column and brace system are in good agreement in Figure 12(b). In Table VI, the absolute maximum and standard-deviation differences between the two testing techniques are 2.4 and 11%, respectively. In Test #7, the structure in the condition left after Test #6 (damaged components not replaced) was subjected to the El Centro1940 unscaled record (PGA = 0.348g). Plastic rotation developed at the column bases at the two storeys and additional brace buckling and stretching occurred in the bottom-storey brace. The time domain and hysteretic responses at the first-storey are presented in Figure 11(e) and (f), respectively. In the RTDS test, displacement overshoots were observed in the

19 COMPARISON BETWEEN RTDS AND SHAKE TABLE TESTING TECHNIQUES 1317 Table VII. Nonlinear first-storey displacement response of the braced structure. Test #6 El Centro 1940 (PGA =0.174g) Test #7 El Centro 1940 (PGA=0.348g) Inter-storey Δ 1 max e rel,1 σ 1 e σ,1 Δ 1 max e rel,1 σ 1 e σ,1 displacement (mm) (%) (mm) (%) (mm) (%) (mm) (%) Shake table RTDS There is an accumulation of damage from Test #6 to Test #7. first two cycles between t =2.4s and t =3.8s. A comparison between the peak displacements is presented in Table VII where a 15% maximum relative difference is reported for the first-storey. It is noted that a part of this error is due to different initial damage conditions between the two physical test structures. In the rest of the ground motion, the responses from the two tests are in good agreement if the permanent offset difference due to previous dissimilarities in peak displacements is ignored. The RTDS strain energy plot associated with the first-storey column and diagonal bracing member is also comparable to the shake table test case in Figure 12(c). In the RTDS test case, the first and the only plastic excursion in the bracing member is shifted to the right on the hysteretic plot in the first quadrant of Figure 11(e) due to the accumulation of permanent elongation during Test #6. The response in the third quadrant of Figure 11(f) suggests that the column steel yield strength in the RTDS case was slightly higher than for the shake table test. In the shake table test series, slight inelastic action occurred in Test #6, as discussed earlier. In addition, a small inelastic excursion occurred in Test #7 prior to the largest inelastic excursion depicted in the third quadrant of Figure 11(f). No such column yielding occurred during the RTDS test series. Therefore, for the RTDS test case, strain hardening and the Baushinger effect had not been initiated when the largest inelastic excursion depicted in Figure 11(f) took place, leading to a sharper yield response developing at a lower strength value. Differences between numerical, RTDS, and shake table simulations in the nonlinear regime essentially exist only in the prediction of peak displacement values in the large deformation cycles, when yielding takes place. These differences also resulted in variations in the response offset observed after large inelastic excursions. Otherwise, excellent correlation between these three simulations was observed in the overall response during the lower amplitude cycles as well as in terms of phase response over the entire duration of the imposed earthquake signals. Although every effort was made to have consistent modelling assumptions in the three techniques used, slight differences were still present between the models and it is believed that those led to the differences observed in peak responses. For instance, the use of the Coulomb friction model and the constant Rayleigh proportional damping in the purely numerical and RTDS methods likely not adequately represented the damping and frictional properties of the shake table test model subjected to dynamic loading, especially upon yielding. Small differences in the initial kinematic conditions at the onset of an inelastic excursion can also result in significant differences in the peak yielding response. Moreover, it is known that slight dissimilarities in hysteretic properties, either between two different physical specimens or between physical and numerical models, can have consequences on the extent of yielding that is predicted (e.g. [26]). In this test program, inherent variations in the yield strength between test specimens as well as approximations in the

20 1318 C.-P. LAMARCHE ET AL. numerical modelling of strain hardening response or the Bauschinger effect may have contributed to the differences between the responses that were obtained. The fact that the purely numerical and RTDS simulations led to similar predictions of the shake table test results also confirm the key importance of proper modelling for accurate earthquake simulation and that the proposed RTDS approach is suitable for the prediction of the nonlinear seismic response of structures provided that both the physical and the numerical models are representative of the studied structures. 7. SUMMARY AND CONCLUSIONS A study was conducted to directly compare the results obtained from shake table tests and RTDS testing technique using the Rosenbrock-W integration method. The results from purely numerical simulations were also compared with the test predictions. The experiments were conducted on a half-scale model of a two-storey steel building structure. The RTDS physical specimens tested in a laboratory environment comprised the first-storey structural components, whereas the remainder of the structure was simulated using a nonlinear numerical model. Inelastic response developed in the form of plastic hinging in the columns. In two of the tests, a slender diagonal tension-only steel bracing member was used at the first-storey to create an unsymmetrical system with highly variable stiffness depending on the direction of motion. The simulations were performed under modulated linear chirp base acceleration inputs and seismic ground motions exhibiting various amplitude levels and frequency contents. This permitted to examine both first and second modedominated responses in the frequency domain and in the time domain. Attention was devoted to adequately compensate for the delay of the actuator-control system in the RTDS test program. A novel constant coefficient polynomial-based compensation procedure that does not excessively overshoot the displacements at high frequencies, as compared with other compensation schemes proposed in the literature, was used in all tests. In addition, an adaptive standard-deviation-driven adaptive delay compensation scheme was proposed and discussed on the basis of performance tests. In this project, special efforts were devoted to ensure uniformity in the models used in the three simulation techniques that were examined and very consistent results were obtained from these methods in all the tests performed, demonstrating that the proposed RTDS and delay compensation methods described in this paper can adequately predict the seismic response of ductile structural steel seismic force resisting systems, both in the linear and nonlinear regimes. Nearly perfect match was obtained when the test structure responded in the linear range. In tests involving inelastic response, differences in peak displacement predictions between numerical, RTDS, and shake table simulations methods were observed during large deformation cycles, when yielding took place. The difference between RTDS and shake table test peak displacement results was generally less than 10% except that it reached 15% in a test where the structures were in a different initial damaged state due to the previous testing (cumulative effects). These dissimilarities are mainly attributed to inherent variations that likely existed between the models used in each simulation, not to the simulation techniques themselves, emphasizing the key importance of using representative models in earthquake simulation studies. The test structure examined in this paper was relatively simple and exhibited a ductile inelastic response, without abrupt strength degradation and subsequent force redistribution or collapse. As RTDS testing becomes more popular among the earthquake engineering community to assess the seismic performance of structures up to failure or collapse, it

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