SENSITIVITY ANALYSIS OF ADAPTIVE MAGNITUDE SPECTRUM ALGORITHM IDENTIFIED MODAL FREQUENCIES OF REINFORCED CONCRETE FRAME STRUCTURES

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1 SENSITIVITY ANALYSIS OF ADAPTIVE MAGNITUDE SPECTRUM ALGORITHM IDENTIFIED MODAL FREQUENCIES OF REINFORCED CONCRETE FRAME STRUCTURES K. C. G. Ong*, National University of Singapore, Singapore M. Maalej, National University of Singapore, Singapore Z. Wang, National University of Singapore, Singapore 31st Conference on OUR WORLD IN CONCRETE & STRUCTURES: August 2006, Singapore Article Online Id: The online version of this article can be found at: This article is brought to you with the support of Singapore Concrete Institute All Rights reserved for CI Premier PTE LTD You are not Allowed to re distribute or re sale the article in any format without written approval of CI Premier PTE LTD Visit Our Website for more information

2 31 st Conference on OUR WORLD IN CONCRETE & STRUCTURES: August 2006, Singapore SENSITIVITY ANALYSIS OF ADAPTIVE MAGNITUDE SPECTRUM ALGORITHM IDENTIFIED MODAL FREQUENCIES OF REINFORCED CONCRETE FRAME STRUCTURES K. C. G. Ong*, National University of Singapore, Singapore M. Maalej, National University of Singapore, Singapore Z. Wang, National University of Singapore, Singapore Abstract This paper presents the sensitivity analysis of Hilbert-Huang transform (HHT) based adaptive magnitude spectrum algorithm (AMSA) identified modal frequencies of reinforced concrete frame structures in both undamaged and damaged states. Characterized by the a posteriori property, the recently proposed AMSA provides an alternative frequency identification approach without resorting to the a priori information about the frequencies to be identified which is needed for other HHT based identification techniques. Such a property makes AMSA especially appropriate for frequency identification and damage detection for civil engineering structures, as the a priori knowledge of the modal frequencies is usually unavailable. To study the sensitivities of AMSA identified modal frequencies, various acceleration signals generated by a series of reinforced concrete (RC) plane frame finite element models with different geometric and stiffness properties representing undamaged states are analyzed. Then the acceleration signals corresponding to progressive damage severities simulated by degradation in stiffness properties with two different lengths of damaged regions are investigated. The results also demonstrate the potential of applying AMSA in structural health monitoring. Keywords: Sensitivity analysis; Hilbert-Huang transform; Adaptive magnitude spectrum algorithm; Frequency identification; Damage detection 1. Introduction Sensitivity analysis of modal parameters is of paramount importance in such fields as seismic design and control of structures as well as retrofitting and rehabilitation of existing structures. Incorporating modal analysis, sensitivity analysis of modal parameters of structures can be carried out in either theoretical or experimental ways. In theoretical sensitivity analysis, with appropriate analytical or numerical models, the problem is usually converted to a generalized eigen-problem, and then standard procedures, such as Jacobi method, subspace iteration and Lanczos algorithm, could be invoked to solve it. As for experimental sensitivity analysis, however, special considerations for such issues as noise-to-signal ratios, nonlinearities involved and, in some cases, inverse problems, must be given to obtain useful results, which substantially complicates the experimental sensitivity analysis and makes it more challenging than its theoretical counterpart. In addition, with the aging of existing structures, structural repair and strengthening has become one of the key topics in structural engineering research. To develop a well behaved repair and strengthening scheme, it is of necessity to investigate the modal characteristics of the existing structures and their sensitivities.due to stiffness

3 degradation and crack propagation during service life of a reinforced concrete (RC) structure, it is usually not justifiable to simply use the design material properties and follow the procedures of theoretical sensitivity analysis of modal parameters. Instead, experimental sensitivity analysis has to be resorted to. That is, modal parameters and their sensitivities have to be studied through an identification scheme based on the measured response time histories, an indispensable component of which is signal analysis techniques Traditionally, by converting signals from time domain into frequency domain, Fourier transform dominated the field of signal analysis. Effective as it is, Fourier spectral analysis requires the signals to be linear and stationary [1]. To analyze nonlinear and nonstationary signals, Huang et al. [1-4] have recently proposed a time-frequency domain analysis technique designated as Hilbert-Huang transform (HHT), which consists of two major steps: empirical mode decomposition (EMD) and Hilbert spectral analysis (HSA). By EMD, a signal is decomposed to a series of mono-component intrinsic mode functions (IMF), following which Hilbert transform is applied to obtain the instantaneous frequencies and the instantaneous magnitudes of the IMFs. Characterized by its broad applicability, HHT has been demonstrated to be efficacious in many branches of structural engineering. As for modal frequency identification, Koh et al. [5] applied HHT to develop a structural health monitoring scheme for RC beams, and Yang et al. [6, 7] proposed an HHT-based system identification method for linear structures. Unlike these established HHT-based frequency identification techniques which rely on some a priori information about the frequencies to be identified, an innovative a posteriori approach designated as adaptive magnitude spectrum algorithm (AMSA) has just been developed by the authors. In this paper the AMSA is used to implement the identification procedures and thus provide a new and effective way for experimental sensitivity analysis. In the following sections, the fundamentals and procedures of HHT and AMSA are summarized, and then sensitivity analysis of AMSA identified modal frequencies of RC frame structures is implemented based on simulated data. 2. Hilbert-Huang Transform and Adaptive Magnitude Spectrum Algorithm Proposed by Huang et al. [1-4], HHT essentially consists of EMD and HSA. By EMD, a signal is expressed as the sum of a series of mono-component signals defined as IMFs and a residue. An IMF has characteristics such that the number of zeros and the number of extrema of the function differ at most by one and that the difference between the mean of the upper envelope and the lower envelope of the function and zero is within a pre-selected tolerance. In contrast with classic Fourier series representation of a signal in which the base functions are always trigonometric functions, the EMD procedures result in a group of adaptive base functions, i.e. IMFs. Besides, by Fourier series representation, a signal is generally written as a sum of an infinite number of trigonometric functions, while the number of IMFs for a signal is finite. Then during HSA, Hilbert transform is invoked to calculate the instantaneous frequency and the instantaneous magnitude for each IMF, leading to a Figure 1. 2-bay-by-2-story benchmark frame used for sensitivity analysis (Total span: 2 1.5=3m; total height: 2 1.5=3m) Figure 2. Typical random excitation force time histories at (a) the first story, and (b) the second story

4 Table 1. Geometric and material properties of the benchmark frame Property Value Number of stories 2 Number of bays 2 Story height 1.5m Span 1.5m Beam dimension 110mm 200mm Column dimension 160mm 160mm Young's modulus 28GPa Density 2400kg/m 3 time-frequency domain representation of the original signal. More detailed information about HHT can be found in References [1-4]. Different from the established HHT-based frequency identification techniques in which some a priori information about the modal frequencies must be known before these techniques can be effectively implemented, an innovative approach, i.e. AMSA, is proposed by the authors. In AMSA, HHT-based adaptive magnitude spectrum (AMS) is developed by such procedures as banded frequency sweep, forward weighted averaging and backward weighted averaging, and the modal frequencies are then identified from the AMS. Characterized by the a posteriori properties, AMSA is especially appropriate for experimental modal analysis and sensitivity analysis in that frequency information of existing structures is usually unavailable before identification procedures are carried out. A thorough presentation about AMSA is beyond the scope of this paper and will be given in a subsequent publication. 3. Sensitivity Analysis with Respect to Geometric and Stiffness Properties: Undamaged Frames To perform the sensitivity analysis, a two-bay-by-two-story RC frame shown in Figure 1 is selected as the benchmark structure. The geometric and material properties of the benchmark frame are listed in Table 1. In this study, the RC frames involved are modeled by finite element method (FEM) as linear elastic structures. With an approximate element size of 0.3m and consistent mass matrixes, two-node plane frame elements with 3-degree-of-freedom at each node are used to discretize the frames. Each frame is excited by two random forces applied at the floor level and the roof level, respectively, and typical random excitations are shown in Figure 2. For sensitivity analysis of AMSA identified modal frequencies, only one acceleration signal is needed. The acceleration signals used in this study are extracted from the roof level loading point of each frame. To simulate practical experimental conditions, noises are superimposed on the acceleration signals. Gaussian pulse processes with root mean Figure 3. Typical acceleration time history at the second story Figure 4. Adaptive magnitude spectrum of the acceleration time history corresponding to a representative value of span (Span: 3.75m)

5 Table 2. AMSA identified frequencies of the frames with different spans Span (m) Frequency (Hz) AMSA identified Theoretical square (RMS) 10% of that of the original acceleration signals are used to model noises. The typical resulting noisy acceleration time history is shown in Figure 3. Covering both undamaged and damaged states of the frames, parameters chosen in this study are geometric and stiffness properties of the frames, i.e. span, story height and moment of inertia of members, as well as damage severity modeled as progressive stiffness degradation. The sensitivities of AMSA identified frequencies with respect to geometric and stiffness properties are investigated in this section, and those with respect to progressive stiffness degradation are presented in the next section. 3.1 Sensitivities of AMSA Identified Frequencies with Respect to Geometric Properties Ten span values ranging from 0.75m to 7.5m are selected as the data points, and the corresponding story-height-to-span ratios vary from 2 to 0.2. For each span value, the noisy acceleration time history collected from the roof level is processed by AMSA. A typical resulting AMS is shown in Figure 4, from Figure 5. Sensitivity analysis of AMSA identified frequencies with respect to span: (a) AMSA identified frequencies, and (b) average sensitivities

6 Figure 6. Adaptive magnitude spectrum of the acceleration time history corresponding to a representative value of story height (Story height: 1.2m) which the fundamental modal frequency can be clearly identified. The results along with the theoretical values based on generalized eigen-analyses are summarized in Table 2. As observed from Table 2, the frequencies can be faithfully identified by AMSA from the simulated noisy signals. The identified frequencies are also plotted against span values in Figure 5(a), based on which the average sensitivities, defined in this case as the increment in identified frequency divided by the increment in span, can be calculated and are shown in Figure 5(b). It can be observed from Figure 5 that both the AMSA identified modal frequencies and the absolute values of their sensitivities decrease monotonically with the increasing span values. One practical indication is that removal of damaged columns which results in an increase in span, as used in some structural retrofitting projects, might lead to a decrease in the modal frequencies, especially when the frames have high story-height-tospan ratios. Figure 7. Sensitivity analysis of AMSA identified frequencies with respect to story height: (a) AMSA identified frequencies, and (b) average sensitivities

7 Figure 8. Adaptive magnitude spectrum of the acceleration time history corresponding to a representative value of moment of inertia of columns (Moment of inertia: m 4 ) To investigate the sensitivities with respect to story height, eight different story heights are chosen. With a representative AMS shown in Figure 6, the results are presented in Table 3 and Figure 7. One can find that, similar to the case of span increase, the identified frequencies as well as the absolute values of sensitivities decrease when story heights increase. However, as indicated by the greater absolute values of sensitivities with respect to story height, the identified fundamental frequencies respond more sensitively to the variations in story height than to those in span. Hence, for the current configuration, in terms of variations of the member length needed to be introduced, it is more efficient to change the story height than to change the span to achieve the desired fundamental frequency. 3.2 Sensitivities of AMSA Identified Frequencies with Respect to Stiffness Properties With the moment-of-inertia ratios of beams to columns covering a range from 0.5 to 1.75, five different values of moment of inertia of columns are used to implement the sensitivity analysis. The dimensions of the column cross-sections are varied to achieve the specified moments of inertia, while the crosssectional area of the columns remains the same value as that of the benchmark frame. Figure 8 shows the AMS of the acceleration signal extracted from the loading point at the roof level of a frame with a moment of inertia of columns of m 4, from which one can easily find the fundamental frequency of the frame. Following the same procedures as in the previous subsections, the identified results are presented in Table 4 and Figure 9(a), and the calculated sensitivities are shown in Figure 9(b). From Table 4, one can see that the AMSA identified frequencies agree well with the theoretical values obtained from generalized eigen-analyses. As observed from Figure 9, with the increase in moment of inertia of columns, the identified frequencies also increase, while their sensitivities decrease. For the frames under investigation, the sensitivities corresponding to the moment-of-inertia ratios of beams to columns ranging from 0.5 to 1, which, from the point of view of flexural stiffness, is Table 3. AMSA identified frequencies of the frames with different story heights Story height (m) Frequency (Hz) AMSA identified Theoretical

8 Figure 9. Sensitivity analysis of AMSA identified frequencies with respect to moment of inertia of columns: (a) AMSA identified frequencies, and (b) average sensitivities indicative of a strong column, weak beam structural configuration, are lower than those corresponding to the ratios ranging from 1 to 1.75, in which the beams are flexurally stronger than the columns. 4. Sensitivity Analysis with Respect to Progressive Stiffness Degradation: Damaged Frames Monitoring stiffness degradation is among the most important applications of experimental modal analysis and sensitivity analysis. To execute the experimental sensitivity analysis with respect to progressive stiffness degradation, and to demonstrate the capability of AMSA in the fields of damage detection and structural health monitoring, damage modeled as stiffness degradation is introduced to the benchmark RC frame defined in the previous section. With the intention to simulate damage due to cracking arising from lateral loading, e.g. seismic loading, the stiffness degradation is introduced at both ends of each beam or column which are assumed to be the cracked regions, and the rest of the beam or column is assumed undamaged. The stiffness degradation is achieved in the finite element model by a reduction in the Young s modulus. The damage state is established by two parameters, i.e. stiffness ratio defined by the ratio of adjusted Young s modulus to original Young s modulus, and uncracked length ratio defined by the ratio of uncracked length to the total length of the beam or column. In this study, two levels of uncracked length ratio, i.e. 60% and 20%, are modeled, and for each level of uncracked length ratio, five different stiffness ratios ranging from 1 to 0.1 are used to carry out the sensitivity analysis. Figure 10 illustrates a damaged frame with an uncracked length ratio Table 4. AMSA identified frequencies of the frames with different moments of inertia of columns Moment of inertia of columns Frequency (Hz) (m 4 ) AMSA Theoretical 1.47E E E E E

9 Figure 10. Schematic representation of a damaged frame (Solid: uncracked regions; dots: cracked regions; total span: 2 1.5=3m; total height: 2 1.5=3m; uncracked length ratio: 60%) of 60%. As in the previous section, the benchmark frame in each damage state is also subjected to two random forces applied at the floor level and the roof level, and the acceleration signal extracted from the loading point at the roof level, polluted by a noise signal with RMS 10% of that of the original signal is used for AMSA-based identification procedures. Representative adaptive magnitude spectra corresponding to uncracked length ratios of 60% and 20% are shown in Figure 11 and 12, respectively, from which the fundamental frequencies can be obviously identified. The complete identification results are listed, and compared with theoretical values based on generalized eigen-analyses in Table 5. Again, one can find that good agreement is achieved. It can also be observed that with a decrease in stiffness ratio indicating progressive damage severities, the AMSA identified fundamental frequencies decrease, as shown in Figure 13(a), which illustrates the potential application of AMSA in developing vibration-based structural health monitoring schemes. Actually, by comparing AMSA identified modal frequencies with the correlation between modal parameters and stiffness properties, one would be able to assess the damage severities during the service life of the structures. Another observation from Figure 13(a) is that although the identified frequencies corresponding to the uncracked length ratios of 60% and 20% both decrease as stiffness degradation becomes more severe, their reductions are different, or the two configurations respond to damage with different sensitivities. The sensitivities are then calculated and shown in Figure Figure 11. Adaptive magnitude spectrum of the acceleration time history corresponding to a representative value of stiffness ratio and an uncracked length ratio of 60% (Stiffness ratio: 0.35) Figure 12. Adaptive magnitude spectrum of the acceleration time history corresponding to a representative value of stiffness ratio and an uncracked length ratio of 20% (Stiffness ratio: 0.8)

10 Table 5. AMSA identified frequencies corresponding to progressive damage severities of the benchmark frame Uncracked length ratio of 60% Uncracked length ratio of 20% Stiffness ratio Frequency (Hz) Frequency (Hz) AMSA Theoretical AMSA Theoretical (b), which gives a quantitative presentation. From Figure 13(b), it can be found that for both the levels of uncracked length ratio, the sensitivities with respect to stiffness ratio increase as damage becomes more severe. Compared with those corresponding to the uncracked length ratio of 20%, the sensitivities corresponding to the uncracked length ratio of 60% are lower when the stiffness ratios are high denoting minor damage, but become relatively higher when the stiffness ratios are low indicating severe damage. 5. Conclusions An innovative HHT-based technique AMSA proposed by the authors is used in this paper to implement the experimental sensitivity analysis of modal frequencies of RC frames based on simulated data. Compared with the established HHT-based frequency identification methods, AMSA basically does not require the a priori information about the frequencies to be identified, and thus is an a posteriori approach. In the sensitivity analysis, the parameters investigated are span, story height and moment of inertia for undamaged frames, and stiffness ratio for damaged frames. Good performances of AMSA in frequency identification and experimental sensitivity analysis are demonstrated by comparisons of the AMSA identified values with theoretical ones based on generalized eigen-analyses. The sensitivities are then calculated based on the identified frequencies. The sensitivity analysis with Figure 13. Sensitivity analysis of AMSA identified frequencies with respect to stiffness ratio: (a) AMSA identified frequencies, and (b) average sensitivities

11 respect to progressive stiffness degradation also shows the potential of applying AMSA in damage detection and structural health monitoring. References [1] Huang, N.E. et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of the Royal Society of London, Series A 1998; 454: [2] Huang, N.E., Long, S.R., Shen, Z. Frequency downshift in nonlinear water wave evolution. Advances in Applied Mechanics 1996; 32: [3] Huang, N.E., Shen, Z., Long, S.R. A new view of nonlinear water waves: the Hilbert spectrum. Annual Review of Fluid Mechanics 1999; 31: [4] Huang, N.E. et al. A confidence limit for the empirical mode decomposition and the Hilbert spectral analysis. Proceedings of the Royal Society of London, Series A 2003; 459: [5] Koh, S.J.A, Maalej, M., Quek, S.T. Damage quantification of flexurally loaded RC slab using frequency response data. Structural Health Monitoring 2004; 3(4): [6] Yang, J.N., Lei, Y., Lin, S., Huang, N. Hilbert-Huang based approach for structural damage detection. Journal of Engineering Mechanics (ASCE) 2004; 130(1): [7] Yang, J.N., Lei, Y., Pan, S., Huang, N. System identification of linear structures based on Hilbert-Huang spectral analysis. Part 1: normal modes. Earthquake Engineering and Structural Dynamics 2003; 32(9):