This study proposes damage detection algorithms for multistory shear structures that only need the first two or three natural frequencies. The methods are able to determine the location and severity of damage on the basis of damage location indices (DLI) and damage quantification indices (DQI) consisting of the changes in the first few squared natural frequencies of the undamaged and damaged states. The damage is assumed to be represented by a reduction in stiffness. This stiffness reduction causes a shift in the natural frequencies of the structure. The uncertainty associated with system identification methods for obtaining natural frequencies is also carefully considered. The methods are accurate and cost-effective means only requiring the changes in the natural frequencies.

Structural health monitoring (SHM) systems are garnering attention as a way of maintaining building structures subject to natural hazards such as large earthquakes and strong winds [

Doebling et al. [

Level 1: determining that damage is present in the structure;

Level 2: determination of the location of the damage;

Level 3: quantification of the severity of the damage;

Level 4: prediction of the remaining service life of the structure.

The first three levels are most often related to structural dynamic testing and modeling issues. Level 4 is not addressed in the structural vibration or modal analysis literature. Hence, most of damage detection methods aim to classify damage into the first three levels.

Damage detection algorithms based on the modal properties of a structure, such as modal frequencies, mode shapes, curvature mode shapes, and modal flexibilities, have been studied in the SHM field for decades. However, most algorithms have difficulties in identifying the precise location and magnitude of the damage. Their accuracy and reliability are not considered sufficient, if not completely inadequate [

Zhao and DeWolf [

A damage detection method based on natural frequencies only needs two vibration sensors (or even one acceleration sensor) to obtain the modal frequencies. It is known that, of the various characteristics, the natural frequencies are the least contaminated by measurement noise and can generally be measured with good accuracy [

Although many previous studies concluded that the frequency cannot provide spatial information about structural changes, multiple frequency shifts may provide spatial information about structural damage in situations where many natural frequencies can be measured [

The purpose of this study is to devise a damage detection method to identify the existence, location, and amount of damage to multistory shear structures by using new damage indices consisting of the changes in the first two or three natural frequencies.

A multistory shear structure (

Simplified structural model with

The characteristic equation for such a structure is written as

Solving (

In addition, the squared natural circular frequency of the

Although the yielded structure is undamaged and may also have reduced stiffness sometime later, in this study, we will assume that the structural damage can be directly expressed as a stiffness reduction and that the structural mass remains unchanged. The difference between the squared natural frequencies of the

Here,

In [

The squared natural frequency change ratio is determined by dividing both sides of (

From (

Similarly, the frequency change ratio of the

The sum of these changes is

Thus, the ratio of the changes in the two natural frequencies of the

From (

Substituting (

This ratio depends on the location of damage

Let us formulate a simple algorithm for detecting structural damage, its location, and its amount. Note that the existence of damage is determined by a change in the first two or three natural frequencies. In previous, some researches [

The previous section derived the ratio of the changes in two squared natural frequencies of the

The number of obtained frequencies is usually smaller than the number of degrees of freedom

The

The modal parameters are often sensitive to various environmental conditions such as temperature, humidity, and excitation amplitude. The effect of environmental conditions or excitation amplitude is treated as “noise” in a simulation, so we should obtain a confidence interval on the modal parameters.

Denoting by

The standard deviation of the difference between the squared natural frequencies is

Moreover, the standard deviation of the changes in the

The standard deviation of the changes in the

The standard deviation of the sum of the changes in the first two frequencies is

The standard deviation of the

Statistically, we expect the

Now let us formulate a simple damage quantification technique. The change in the natural frequency can be used to interpolate the amount of damage at the detected location [

The

Because the sensitivity to a stiffness reduction depends on the modes, it would be a good idea to introduce the following averaging method. The damage quantification index (

The reliability band of

A four-story shear structure was measured to show the feasibility of the proposed method. It was modeled as a one-dimensional lumped mass shear model, as shown in Figure

The stories of the structure had the same mass,

Figure

Sensitivity to changes in square natural frequencies (

Figure

From Figure

The

The variances of

In this simulation, two modes were enough to detect the damaged story. After determining the location of the damage by using the

Comparison of

Next, we verified our method on an eight-story structure. The damping ratio was chosen to be 3% and the data sampling frequency was 200 Hz; the excitation loading is white-noise. The mass of each story was 1000 tons. The stiffness of the first story was assumed to be

The first three natural frequencies of the undamaged structure were calculated to be 0.99, 2.82, and 4.58 Hz. Table

Damage scenarios of eight-story structure.

Damage case number | Location of damage (damaged story) | Quantification of damages (percentage of the stiffness reduction) |
---|---|---|

1 | 1st | 23% |

2 | 4th | 14% |

3 | 7th | 6% |

Natural frequencies of 3 damage cases of eight-story structure.

Damage case number | The 1st natural frequency, |
The 2nd natural frequency, |
The 3rd natural frequency, |
DLI^{12} |
DLI^{23} |
---|---|---|---|---|---|

1 | 0.97 | 2.75 | 4.47 | 0.51 | 0.52 |

2 | 0.98 | 2.82 | 4.50 | 0.95 | 0.04 |

3 | 0.99 | 2.76 | 4.49 | 0.15 | 0.55 |

Eight cases of damage (from the 1st to 8th story) were simulated. For each case, the damage was simulated by reducing the stiffness of each story from 5 to 25%, and these 21 levels of damage were reflected in the changes in the natural frequencies. The

The next step is using the 2nd and 3rd frequency changes to get

After determining the location of the damage by using the

Damage indices are dependent on the mass and stiffness distribution. Thus, we need a prior knowledge of them to apply our method. If design drawing is not available, some system identification tools may be needed. This is the limitation of this method to be applied to a real building.

The presented method determines the location and amount of damage to shear structures by using the changes in the first few natural frequencies only. Natural frequencies decrease as a result of damage and the damage indices (

As we need only a few natural frequencies, two vibration sensors are enough to obtain the modal frequencies, one on the ground detecting an input and the other on the roof detecting an output. If the input lasts long and the spectrum is flat, we may identify those parameters using the output data without input information. Thus, in such a case, only one sensor is needed.

The uncertainty associated with system identification methods for obtaining natural frequencies was also carefully considered, and the confidence intervals of the

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work was supported in part by a Grant-in-Aid no. 22310103 (PI: Akira Mita) and Grant-in-Aid to the Global Center of Excellence Program for the “Center for Education and Research of Symbiotic, Safe and Secure System Design” from the Ministry of Education, Culture, Sport, Science and Technology of Japan.