Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1089645, 11 pages

https://doi.org/10.1155/2017/1089645

## Damage Detection in Grid Structures Using Limited Modal Test Data

^{1}State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China^{2}Jiangsu Key Laboratory of Environmental Impact and Structural Safety in Engineering, School of Mechanics and Civil Engineering, Xuzhou 221116, China

Correspondence should be addressed to Bei-dou Ding

Received 12 January 2017; Revised 13 March 2017; Accepted 2 April 2017; Published 1 June 2017

Academic Editor: Michele Betti

Copyright © 2017 Bei-dou Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The detection of potentially damaged elements in grid structures is a challenging topic. By using limited measured test data, damage detection for grid structures is developed by the modal strain energy (MSE) method. Two critical problems are considered in this paper in developing the MSE method to detect potential damage to the grid structure by using limited modal test data. First, an updated mode shape expansion method based on the modal assurance criterion is adopted to ensure that the modal shape obtained from the reference baseline model is reliable and has explicit physical meanings. Second, after identifying the location of the element damage by the element MSE method with expanded mode shapes, multivariable parameters denoting element damage severity are simultaneously determined. These parameters are included in the column vector and matched with the corresponding element stiffness matrix while the error tolerance value of the Frobenius norm of the column vector is undercontrolled. Finally, a three-dimension numerical model of the grid structure is used to represent different damage cases and to demonstrate the effectiveness of the present method. The application of the three-dimension physical model to a full-scale grid structure is also verified. Analysis results demonstrate that the presented damage detection method effectively locates and quantifies single- and multimember damage in grid structures and can be applied in engineering practice.

#### 1. Introduction

The grid structures have the advantages of light weight, high bearing capacity, high rigidity, simple and reliable connection, easy processing, and good comprehensive technical and economic indexes. They have been widely used in all kinds of public constructions such as large stadiums, gymnasiums, exhibition centers, gas stations, and industrial plants. However, grid structures in hostile environments are prone to damage because of extreme loads, fatigue, and corrosion. Grid structures continually accumulate damage throughout their service lives as a result of various environmental effects. Thus, the damage inspection and assessment of a grid structure in service are required in order to detect potential damage that may lead to a structural collapse. At present, vibration-based methods as damage detection tools instead of static measurement methods have been increasingly adopted by structural engineers because of their measurement flexibility, measurement information accessibility and easiness, and cost-effectiveness [1–3]. The basic principle of vibration-based methods is that the changes in the physical properties of the structure will alter the dynamic characteristics of the structure, such as natural frequencies, modal damping, and mode shapes. Hence, changes in these parameters or in their combinations may be considered for localization and quantification, which are important components of damage detection [4–6].

Alvandi and Cremona [7] reviewed and summarized a few common vibration-based methods for the detection and evaluation of structural damage. Compared with previously evaluated vibration-based methods, the modal strain energy (MSE) method is more sensitive to changes in response than natural frequencies and mode shapes. MSE method is highly stable in handling noisy signals and can easily locate local damage. Jaishi et al. [8, 9] also reported that MSE is more sensitive to local damage than other mode shape-based indices. Stubbs et al. [10–12] first presented a damage localization method on the basis of changes in the MSE method using MSE as a damage indicator and mode shapes measured from the actual structure. Li et al. [13, 14] developed an improved damage localization method, specifically for the three-dimension frame structure, called the modal strain energy decomposition (MSED) method. The MSED method has two damage indicators, namely, axial and transverse damage indicators for each member of the structure. By analyzing and comparing information from both damage indicators simultaneously, the location of the damaged element can be accurately determined. Hu and Wang [15] developed a cross MSE (CMSE) method to estimate damage severity. This method is based only on the product terms obtained from two same/different modes, which are associated with mathematical and experimental models. The CMSE method does not require data on the mass distributions of the baseline and damaged structures. Implementing this method requires only the information measured by a few modes from the damaged structure. Wang et al. [6, 16, 17] developed an iterative modal strain energy (IMSE) method to locate and quantify the damage for three-dimension frame structures especially effective when incomplete modal data are available. The merits of this new method are that both the modal frequencies and spatially incomplete mode shapes can be used. Also, the modal frequencies do not need to pair the mode shapes one by one.

MSE based methods for damage detection have been successfully applied in one-dimension or two-dimension structures. However, damage detection cases are often restricted to limited element damage stimulation and planar structure. Numerical studies and experimental verifications for a three-dimensional grid structure based on synthetic data of the frequencies and mode shapes generated from the finite element models or extracted directly from the actual structure are relatively few. Grid structures are three-dimension structure and typically possess numerous elements and are geometrically symmetrical; the natural vibration frequencies and mode shapes of grid structures are highly intensive. Usually, the traditional MSE methods for the damage detection are required to achieve mass normalized mode shapes or spatially complete mode shapes. Considering that the grid structure possesses many degrees of freedom (DoFs) and only a limited number of sensors can be placed in it, hence, the modal expansion technique [15, 16] is used to get the spatially complete mode shapes of grid structure from limited measuring sensors applied to the grid.

The main objective of the present paper is to develop MSE method for damage localization and severity estimation of the grid structures of numerical and actual models by using limited modal test data. First, a physical model of a three-dimension grid structure is built, modal testing is conducted, and the damage localization detection indicator and damage severity detection based on MSE methods are validated with numerical data. Second, the incomplete spatial mode shapes measured from the dynamic test of grid structure due to quantitative limitation of sensors are then expanded to the complete mode shapes using the modal expansion technique. Third, a full-scale grid structure with different corrosion rates of the elements is used to demonstrate the effectiveness of the damage localization and severity detection of three-dimensional structure. This paper is arranged as follows. First, a mode shape expansion method based on the modal assurance criterion (MAC) is adopted to ensure that the measured incomplete mode shapes from the limited modal test data in the lab test are expanded to the complete mode shapes, so that the modal shapes obtained from the analytical model are reliable and have explicit physical meanings. Second, damage localization and severity estimation of the grid structures based on MSE method are developed with analytical modal frequencies and expanded mode shapes. At the same time, both single and multiple damage cases for numerical models of the grid structures are set up with different levels of severity to verify the effectiveness of damage localization. Finally, to demonstrate the capabilities of the developed damage severity estimation algorithm, a grid structure physical model is constructed and tested with different corrosion rates of the members. Modal parameters are identified and used for the damage detection methods. Results indicate that the MSE method is effective on damage localization and severity detection of three-dimensional grid structure and can be applied in engineering practice.

#### 2. Methodology

##### 2.1. Mode Shape Expansion

The measured mode shapes of the grid structure due to quantitative limitation of sensors are incomplete and they should be expanded to finite element model DoFs. Before the mode shape expansion, the comparison between the analytical modal shapes from the finite element model analysis and the measured mode shapes from the experimental modal analysis is performed by calculating the MAC value. Equation (1) is the formula for calculating the MAC value of the th modal shape.where denotes the th analytical modal shape and subscript “” indicates modal shape associated with the damaged structure. High MAC values indicate greater correlation between the analytical modal shapes and the measured incomplete mode shapes. If the MAC value is greater than or equal to 0.97, the analytical modal shapes can then be used directly without any further improvement. Then, the measured incomplete mode shapes could be expanded to complete the mode shapes by using the mode shape expansion method suggested by Ren and Chen [9]. In this method, the measured modes are assumed to be a linear combination of the analytical modes, and the unmeasured modes can be expanded by using a transformation coefficient, which is given by the following relationship:where is the number of DoFs, is the number of measured DoFs, is the number of identified mode shapes, is the number of mode shapes used in the expansion, and is the transformation coefficient. As long as , coefficient can be obtained by using a least-squares technique

The mode shapes of the unmeasured DoFs, that is, , can then be easily computed by using (2). After the measured mode shapes are expanded according to the analytical mode shapes of the undamaged structure, the updated mode shapes can be used in the process of damage localization and severity detection based on MSE method with explicit physical meaning and small iteration error.

##### 2.2. Damage Localization Indicator

The indicator initially comes from the Stubbs index, which is based on traditional MSE method and developed by Stubbs et al. (1995). Yang (2003), Li et al., and (2006) Wang et al. (2014) further developed two damage indicators, the axial damage indicator and the transverse damage indicator based on the modal strain energy decomposition (MSED) method for three-dimensional frame structures. In this section, the damage localization indicator based on MSE method is introduced for the damage localization of the grid structure.

Consider that the grid structure mainly bears the axial force, and the th modal strain energy of the structural member can be expressed aswhere is the axial stiffness matrix terms of member . The whole modal strain energy of the grid structure should be the sum of all the members, so for a linear, the th modal strain energy of the undamaged grid structure based on the finite element model is given bywhere is axial stiffness matrix of the grid structure. The fraction of the th modal energy that is concentrated in the th member (i.e., the sensitivity of the th member to the th mode) is defined as

If the subscript “” is used to represent the modal parameters in (4) to (6) for a damaged structure, the damaged sensitivity of the th member to the th mode can be written asin which the quantities and are given by

Assume that the damage only changes the stiffness of the member, but not the mass of the structure, and the modal strain energy for each th mode before and after damage remains constant: that is, . Meanwhile, assume that the modal sensitivity for the th mode and the th location is the same for both undamaged and damaged structures: that is, .

The fraction indicator of the th modal energy could be divided by zero possibly; to overcome the situation, according to the above assumption, the axis of the modal sensitivity should be shifted by one: that is, ; then

Substituting (4), (5), and (8) into (9), a damage indicator for th mode of the th member is obtained by

If the mode of the actual test structure has order measured modes obtained in the dynamic test of the grid structure, a damage indicator for the th member is defined by

Then, after normalization of the damage indicator , the damage location index can be used to detect the damage localization of the grid structure members.

##### 2.3. Damage Severity Detection

After the damaged elements are identified with damage localization indicator, a modified MSE method is developed to determine damage severity of the elements in the grid structure. The modal eigenvalue analysis of the baseline model and damaged model of the grid structure can be written as follows:where and are the mass and stiffness matrices of the undamaged structure, respectively, denotes the th analytical modal eigenvalue, and subscript “” indicates the values associated with the damaged structure.

Structural stiffness is generally decreased because of section reduction of local elements under corrosion environment. Assume that the mass of elements is unchanged: that is, , and the structural element damaged stiffness for the th element can be expressed as a decrease of the undamaged stiffness. Therefore, in the local coordinate system, the local stiffness matrices of the damaged elements are assumed as linear combination between undamaged and damaged stiffness matrices; that is,where and are the th stiffness matrices of the damaged and undamaged local element stiffness matrix, respectively, and represents the damage extent of the th damaged element. The global stiffness matrix of the damaged structures can then be written as follows:where is the total number of damaged members, and represent the damage extent and the number of damaged elements, respectively, and is the stiffness matrix of the th element. By using (15), damage severity can be estimated by an available damage detection method.

By premultiplying (12) by and (13) by and considering the symmetry of the mass and stiffness matrices, the following can be determined:

Substituting (15) into (16) yields the following:where, , and represent, respectively, the corresponding element damage severity, the MSE column vector in stiffness matrix , and the total structural MSE for structural stiffness between the undamaged and damaged structures for the th mode.

Equation (17) can then be simplified as follows:

equations can be derived from (17) when modes are available for the undamaged and damaged structures. In (19), is a matrix, and and are column vectors of size and , respectively. When is greater than or equal to , a least-squares approach can be conducted to solve . The estimate of , denoted as , is written as follows:

To solve (20), the mode shapes at the full coordinates of the damaged structures are required. These mode shapes can be calculated by the mode shape expansion technology with high MAC values, described in Section 2.2. To estimate the damage extent of a structural element, an iterative approach is adopted. In this approach, the mode shapes of the damaged structure are associated with damage severity by the following:

The errors of the damage severity column vector between two iterations are space vectors and expressed in the form of a Frobenius norm function, shown in (22), and are used as the judgment condition in the process of the iteration.

##### 2.4. Implementation Process of Damage Detection

Then, the damage localization and damage severity detection can be estimated iteratively as follows based on the methods in Sections 2.3 and 2.4.

*Step 1. *A finite element mode shape and modal eigenvalue of the undamaged structure are obtained by finite modal analysis. The measured incomplete mode shapes are expanded into complete mode shapes according to the finite element mode shape by using (2). can then be used in the subsequent process of damage localization and severity detection.

*Step 2. *A damage localization indicator is calculated by using (11). After normalization of the damage localization indicator, the normalized indicator is used to identify the damaged elements in the grid structure.

*Step 3. *After the damage elements are identified according to the damage localization indicator, the vector of the estimated damage severity is calculated by using (20) with the expanded . Subsequently, the damaged mode shape for is computed by using (21) with the vector of estimated damage severity . Damage severity is estimated by using (20) on the basis of (given ) sequentially.

*Step 4. *The iteration termination condition is set. Step 4 is repeated until the Frobenius norm , where tol is a predetermined threshold.

#### 3. Numerical Studies on Damage Detection in Grid Structures

##### 3.1. The Finite Element Model of Grid Structures

To verify the foregoing damage method, a 2 × 2 grid structure that consists of 32 steel tubular members with a length of 3.0 m, a width of 3.0 m, and a height of 1.0 m was numerically analyzed. This structure is shown in Figure 1. The members are assumed to be axial tension-compression members that have been pin-ended and pin-braced against the supports in the ground, and all the members have similar tube cross-sections 42 mm × 3.25 mm. The material properties of the steel tubular members include elastic modulus = 210 GPa, Poisson’s ratio , mass density kg/m^{3}, and yield strength = 235 MPa. Loads are applied to the nodes of the top chord. The design parameters of the test grid structure are listed in Table 1.