Damage of a 5-story framed structure was identified from two types of measured data, which are frequency response functions (FRF) and natural frequencies, using a finite element (FE) model updating procedure. In this study, a procedure to determine the appropriate weightings for different groups of observations was proposed. In addition, a modified frame element which included rotational springs was used to construct the FE model for updating to represent concentrated damage at the member ends (a formulation for plastic hinges in framed structures subjected to strong earthquakes). The results of the model updating and subsequent damage detection when the rotational springs (RS model) were used were compared with those obtained using the conventional frame elements (FS model). Comparisons indicated that the RS model gave more accurate results than the FS model. That is, the errors in the natural frequencies of the updated models were smaller, and the identified damage showed clearer distinctions between damaged and undamaged members and was more consistent with observed damage.
Monitoring integrity of infrastructures without interruption of its function is the primary motivation in developing vibration-based damage detection methods. This method relies on the fact that changes in structural properties, such as damage, affect the overall dynamic properties of the structure. The general approach to detecting damage in this method is to establish analytical models which represent the reference state and damaged state of the structure and then to investigate the change in structural properties between the two analytical models. Thus, a relevant parameter identification method yielding the optimal analytical model which represents the observed behaviors of a structure is essential. Several approaches have been explored to solve this inverse problem and are summarized by Doebling et al. [
In the FE model updating method, structural properties that represent the stiffness of the members in the analytical model, such as the modulus of elasticity or sectional property, are usually selected as the updating parameters. At this point, to obtain accurate results, it is important to establish a relevant analytical model and choose appropriate updating parameters so that the effects of damage on the dynamic properties can be accurately replicated. For example, if the flexural stiffness values of members were chosen as the updating parameters when the analytical model was not sufficiently discretized, it would be impossible to replicate the dynamic properties of a framed structure that has concentrated damage at only a few joints or portions of a few beam-column members. A typical problem arising from the use of an irrelevant model is that the identified damage tends to diffuse or spread out into other members [
Another issue for accurate damage detection is improvement of accuracy in parameter identification from measured data. Generally, including many kinds of data as observation is advantageous since more information can be supplied and the ill-posedness of the inverse problem can be mitigated. The modal data (the natural frequencies and mode shapes obtained from various system identification methodologies) are frequently used to represent the dynamic properties of the structure. Even though parameter identification based on the model data is simple, replication of behavior in higher modes is impossible because it is generally difficult to identify the modal properties of higher modes. Thus, it is beneficial to include the frequency response function (FRF) to the modal properties as an observation for parameter identification to provide more information. When more than two kinds of measured data are used, the relative contribution of each type should be properly adjusted because the amounts of data and noise in each are different. The weighting factors are used for this purpose. Previous literature indicates that the weighting factor should be adjusted based on the variance of noise contained in each type of measured data [
In this study, a model-updating-based damage detection procedure for framed structures subjected to seismic damage was proposed. For efficient detection of concentrated damage at the ends of a beam-column member, which is a typical damage pattern in framed structures subjected to major earthquakes, a modified frame element which includes the rotational springs at its ends was used in the analytical model. Both the FRF and natural frequencies were used as measured data for the FE model updating, and a procedure to obtain appropriate weighting factors is presented. The proposed procedure was applied to damage detection in a five-story one-bay reinforced concrete test structure subjected to earthquake damage. The accuracy of damage detection using the proposed procedure was compared with that of the conventional approach which uses conventional frame elements for the FE model and the flexural stiffness of the individual frame members as the updating parameters.
The basic premise of seismic design for framed structures is to ensure ductile behaviors even though some parts of the structures may behave nonlinearly. Under large earthquake excitations, typical damage in moment-resisting frames starts with the plastic hinge formulation (a concentration of flexural yielding) at the ends of the frame members. To properly replicate the nonlinear behavior of framed structures subjected to seismic damage, modified frame elements which include rotational springs at both ends were adopted in the analytical model which will be used in the FE model updating procedure.
The stiffness matrix of a frame element with rotational strings at both ends can be derived (
Frame element with rotational springs.
In (
Analytical models for framed structures are generally established using conventional frame elements (thus, the stiffness matrix for a single element does not contain the parentheses at each coefficient in (
The FE model updating procedure used in this study is based on a nonlinear least-squares method which minimizes the difference between the measured and analytical frequency response functions and the natural frequencies using the sensitivity matrix and residual vector appended by the side constraints [
The equation of motions of a system with
For updating, the transfer function
For model updating, the dynamic stiffness matrix
In the nonlinear least-squares solution scheme,
In addition, setting the norm of the error vector in (
The gradient of
The sensitivity matrix based on the natural frequency can be obtained in a similar manner. The norm of the error vector in the analytical natural frequencies
Similarly, using a Taylor series expansion truncated at the first order, the least-squares equation is obtained as
Similar to (
Combining the equations for the FRF and the natural frequency in (
The solution can be found using an iterative calculation until the incremental solution
Equation (
Three aspects must be considered in determining the weighting factors: (a) even though groups of data may be measured in different units, weighted quantities should be in the same units; (b) relatively accurate measurements are weighted more heavily than inaccurate measurements; and (c) the numbers of observations in each group should be considered to ensure the desired contributions.
Typically, the size of the sensitivity matrix based on the FRF is much larger than one based on natural frequencies. Thus, a larger weighing factor should be used for the natural frequencies to ensure an equivalent contribution. Otherwise, the natural frequencies will have very little impact on the solution. From statistical theory, it is known that the best weighting matrix for the least-squares problem is the inverse of the covariance matrix of the measurement uncertainty [
In this study, the weighting factors were determined by considering the three aspects mentioned above. The objective function for the weighted least-squares problem in (
Change in average error norm with scaling factor
The parameter identification problem is generally an ill-conditioned problem. Typically, in the solution of an ill-conditioned problem using the least-squares equation, a few parameters exhibit large changes from their initial values. Ill-conditioning occurs when two or more parameters have very similar effects on the response of the system and the measurements are contaminated by noise. The regularization method [
Equation (
The proposed FE model and model updating procedure were applied to identify seismic damage in a test structure using the acceleration data measured during the shaking table test. The test structure was a one-bay, five-story, reinforced-concrete wall-slab building. Figure
Test structure.
After each stage of the test, cracks in the structure were inspected and recorded. With increased ground acceleration amplitude, some members of the test structure suffered significant damage from concentrated cracks at the ends of the members. Figure
Crack patterns observed after each stage of the tests: (a) the right end of the third-floor slab after 0.20 g shaking, (b) the third- and second-floor slabs after 0.30 g shaking, (c) the base of the first-story wall after 0.50 g shaking, and (d) locations of observed cracks.
The natural frequencies and modal damping ratios of the test structure at each stage of shaking were identified from the measured acceleration data using N4SID (subspace state-space system identification), which is a time domain system identification method [
Natural frequencies and damping ratios of the test structure.
Natural frequencies (Hz) | Damping ratios (%) | |||||||
---|---|---|---|---|---|---|---|---|
1st | 2nd | 3rd | 4th | 1st | 2nd | 3rd | 4th | |
El Centro 0.06 g | 4.01 | 13.08 | 25.15 | 41.60 | 2.56 | 2.78 | 3.87 | 5.68 |
El Centro 0.12 g | 3.78 | 12.65 | 24.85 | 40.80 | 3.60 | 3.42 | 3.88 | 6.51 |
El Centro 0.20 g | 3.00 | 11.06 | 22.42 | 39.73 | 10.43 | 5.19 | 7.20 | 11.16 |
El Centro 0.30 g | 1.90 | 8.49 | 19.64 | 34.67 | 16.00 | 7.09 | 4.89 | 10.17 |
El Centro 0.40 g | 1.59 | 8.16 | 18.55 | 33.40 | 15.90 | 5.72 | 5.30 | 10.83 |
El Centro 0.50 g | 1.24 | 7.57 | 17.49 | 32.32 | 30.99 | 7.80 | 5.59 | 9.65 |
As indicated in the table, modal properties up to the fourth mode could be identified. With increased vibration amplitude, the fundamental natural frequency gradually decreased and finally reached about 30% of that of the 0.06 g shaking. The damping ratio of the fundamental mode of the 0.50 g shaking was increased by about eight times that of the initial test, which is believed to be the result of hysteretic damping due to the nonlinear behavior of the test structure. The table shows that the decrease in the natural frequency at the 0.30 g shaking was as large as 40% compared with the previous step, which indicates that the change in the overall stiffness distribution occurred somewhat abruptly during the 0.30 g shaking.
The initial model used to determine the reference model was composed based on the information obtained from the material test results and the measured dimensions of the members. Young’s modulus for concrete was evaluated with the compressive strengths from core sample tests using the equation in ACI
The average Young modulus of concrete was found to be 24.77 GPa. The stiffness of the members of the initial model was determined based on the moment of inertia of a gross (uncracked) section. The story mass was obtained from the self-weight of the members and the mass blocks attached to the test structure having values of 108.87 kg for the roof floor and 138.08 kg for the other floors. The resulting natural frequencies of the initial FE model were 3.63, 11.92, 22.76, 35.89, and 47.74 Hz for the first to fifth modes. The basic assumptions used in the initial model were that (i) the mass of the structure was concentrated at floor level, (ii) the floor was rigid in in-plane, and (iii) the damping properties of the structure were those of classical damping.
The dynamic properties (FRF and identified natural frequencies) at 0.06 g shaking were used to determine the reference FE model. In the model updating for the reference FE model, the connection percentages of all rotational springs were assumed to be one, which represents a rigid connection. These values were not selected as updating parameters. Both of the flexural stiffness and connection percentage cannot be used as updating parameters at the same time because selecting many parameters, some having similar influences on the dynamic behavior of the structure, is a source of ill-conditioning. For the undamaged state, an assumption of rigid connections was appropriate because no cracks were observed at this stage.
The total of ten updating parameters consisted of five for the columns and five for the slabs and was used as shown in Figure
Updating parameters for reference model.
Parameters | Value | Parameters | Value |
---|---|---|---|
Wall 1S | 1.30 | Slab 2F | 1.14 |
Wall 2S | 1.39 | Slab 3F | 1.31 |
Wall 3S | 1.19 | Slab 4F | 1.16 |
Wall 4S | 1.34 | Slab 5F | 1.03 |
Wall 5S | 1.32 | Slab RF | 1.19 |
Natural frequencies of reference model.
1st | 2nd | 3rd | 4th | ||
---|---|---|---|---|---|
Measured (0.06 g shaking) | Freq. (Hz) | 4.01 | 13.08 | 25.15 | 41.60 |
Initial model | Freq. (Hz) | 3.63 | 11.92 | 22.76 | 35.89 |
Percent error | −11% | −9% | −11% | −15% | |
Updated model | Freq. (Hz) | 4.01 | 13.03 | 25.38 | 40.92 |
Percent error | 0% | 0% | 1% | −1% |
Updating parameters for initial analytical model.
After obtaining the reference model, which corresponds to the state of the structure after 0.06 g shaking, model updating for the rest of the shaking stages was performed, and the damage at each stage was evaluated using the updated parameters. However, the connection percentages of the rotational springs, instead of the effective flexural stiffness
The number of rotational springs used in the damaged model was 30, which is twice the number of frame members. The number of parameters was reduced using a similar grouping scheme. The influence of the spring located at the bottom of one wall is almost identical to the spring located at the bottom of the other wall. The two bottom springs of a story were grouped together. Likewise, the two top springs of a story and the two springs located at both ends of a beam were grouped together. Therefore, a total of 15 parameters, three parameters per story, were used for the damaged model. Figure
Updating parameters for damaged model.
Table
Comparison of measured and analytical natural frequencies.
Stage | Mode | Measured Freq. (Hz) | RS model | FS model | ||
---|---|---|---|---|---|---|
Freq. (Hz) | Percent error | Freq. (Hz) | Percent error | |||
PGA 0.12 g | 1st | 3.78 | 3.78 | 0.0% | 3.78 | 0.0% |
2nd | 12.65 | 12.73 | 0.6% | 12.68 | 0.2% | |
3rd | 24.85 | 24.82 | −0.1% | 24.99 | 0.6% | |
4th | 40.80 | 40.60 | −0.5% | 40.13 | −1.6% | |
|
||||||
PGA 0.20 g | 1st | 3.00 | 2.90 | −3.3% | 3.13 | 4.3% |
2nd | 11.06 | 11.02 | −0.4% | 11.17 | 1.0% | |
3rd | 22.42 | 22.61 | 0.8% | 22.14 | −1.2% | |
4th | 39.73 | 37.98 | −4.4% | 36.00 | −9.4% | |
|
||||||
PGA 0.30 g | 1st | 1.90 | 1.83 | −3.7% | 2.03 | 6.8% |
2nd | 8.49 | 7.84 | −7.7% | 8.79 | 3.5% | |
3rd | 19.64 | 17.73 | −9.7% | 17.80 | −9.4% | |
4th | 34.67 | 31.67 | −8.7% | 29.38 | −15.3% | |
|
||||||
PGA 0.40 g | 1st | 1.59 | 1.68 | 5.7% | 1.82 | 14.5% |
2nd | 8.16 | 7.27 | −10.9% | 8.04 | −1.5% | |
3rd | 18.55 | 16.80 | −9.4% | 16.10 | −13.2% | |
4th | 33.40 | 30.55 | −8.5% | 26.45 | −20.8% | |
|
||||||
PGA 0.50 g | 1st | 1.24 | 1.32 | 6.5% | 1.45 | 16.9% |
2nd | 7.57 | 6.31 | −16.6% | 6.98 | −7.8% | |
3rd | 17.49 | 15.65 | −10.5% | 14.90 | −14.8% | |
4th | 32.32 | 29.41 | −9.0% | 24.78 | −23.3% |
Table
Updated parameters normalized with those of undamaged model.
0.12 g | 0.20 g | 0.30 g | 0.40 g | 0.50 g | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|
RS | FS | RS | FS | RS | FS | RS | FS | RS | FS | ||
Wall 1S | Bot. | 1.00 | 0.99 | 0.81 | 0.82 | 0.36 | 0.52 | 0.29 | 0.45 | 0.21 | 0.35 |
Top | 1.00 | 1.00 | 0.79 | 0.79 | 0.78 | ||||||
|
|||||||||||
Wall 2S | Bot. | 1.00 | 1.02 | 1.00 | 0.90 | 0.78 | 0.64 | 0.78 | 0.62 | 0.78 | 0.56 |
Top | 1.00 | 0.85 | 0.80 | 0.71 | 0.67 | ||||||
|
|||||||||||
Wall 3S | Bot. | 0.88 | 0.92 | 0.88 | 0.82 | 0.78 | 0.59 | 0.70 | 0.59 | 0.67 | 0.56 |
Top | 0.95 | 0.95 | 0.83 | 0.74 | 0.69 | ||||||
|
|||||||||||
Wall 4S | Bot. | 1.00 | 1.04 | 1.00 | 0.92 | 0.75 | 0.66 | 0.67 | 0.64 | 0.67 | 0.58 |
Top | 1.00 | 1.00 | 1.00 | 0.99 | 0.98 | ||||||
|
|||||||||||
Wall 5S | Bot. | 1.00 | 1.04 | 1.00 | 0.94 | 1.00 | 0.69 | 0.96 | 0.66 | 0.96 | 0.57 |
Top | 1.00 | 0.98 | 1.00 | 1.00 | 1.00 | ||||||
|
|||||||||||
Slab 2F | 0.89 | 0.91 | 0.57 | 0.55 | 0.20 | 0.22 | 0.17 | 0.14 | 0.07 | 0.07 | |
Slab 3F | 0.91 | 0.78 | 0.46 | 0.40 | 0.14 | 0.13 | 0.12 | 0.07 | 0.02 | 0.03 | |
Slab 4F | 0.93 | 0.89 | 0.61 | 0.55 | 0.23 | 0.23 | 0.19 | 0.15 | 0.11 | 0.08 | |
Slab 5F | 1.00 | 0.90 | 0.84 | 0.59 | 0.44 | 0.32 | 0.39 | 0.24 | 0.27 | 0.17 | |
Slab RF | 1.00 | 0.90 | 0.92 | 0.79 | 0.57 | 0.58 | 0.50 | 0.57 | 0.34 | 0.51 |
Change in stiffness ratios: (a) FS model, and (b) RS model.
The distribution of stiffness ratios for RS model enabled a clear distinction between damaged and undamaged members. The range of stiffness ratios increased (0.98 to 0.02 in the case of 0.50 g shaking). The stiffness ratios of the rotational springs of the walls indicated that the stiffness drop at the bottom of the first-story wall was most significant, while the stiffness ratios of the other rotational springs did not decrease considerably.
As mentioned previously, the stiffness of a frame member with nonuniform damage along its length cannot be simulated accurately by adjusting the flexural stiffness
In this paper, damage of a 5-story framed structure induced by shaking table testing was identified and quantified using an FE model updating procedure. The model updating method in this study used the frequency response functions and natural frequencies as observations, which were obtained from acceleration measurements during the test. Generally, it is known that using many kinds of observations is advantageous, since more information can be supplied and the ill-posedness of the inverse problem can be mitigated. However, for more accurate results, appropriate weighting and scaling between the groups of observations are needed. In this study, a procedure to determine the rational weightings of different groups of observations was proposed and applied. In addition, considering the typical features of earthquake-induced damage in framed structures, an FE model for updating was constructed using a modified frame element which includes rotational springs at both ends, and the rotational springs were chosen as updating parameters. When conventional frame elements were used and the flexural stiffness of individual members was chosen as updating parameters, the effects of concentrated damage at the members’ ends were not replicated with sufficient accuracy. Therefore, damage detection based on this type of model updating may contain large inaccuracy. In this paper, model updating using conventional frame elements (FS model) and modified frame elements (RS model) was conducted separately. Then, the degrees of damage evaluated using the results of the two model updating methods were compared. The comparison showed that the errors in the natural frequencies of the updated FE models at 0.50 g shaking were about 8–23% in the FS model and 6–17% in the RS model, which indicated that the RS model is capable of representing the dynamic behavior of the structure more accurately. The location and severity of damage were identified from changes in the updated parameter values in each case. The damage identified by the FS model showed somewhat diffused; that is, the stiffness ratios of severely damaged members were not reduced sufficiently, and some undamaged members showed large stiffness drops although no major cracks were observed. On the other hand, the damage identified by the RS model showed clear distinctions between damaged and undamaged members, and were more consistent with observed damage.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by a Grant (13AUDP-B066083-01) from Architecture and Urban Development Research Program funded by Ministry of Land, Infrastructure, and Transport of the Korean Government.