Mathematical Problems in Engineering

Volume 2015, Article ID 954926, 10 pages

http://dx.doi.org/10.1155/2015/954926

## Damage Localization of an Offshore Platform considering Temperature Variations

College of Engineering, Ocean University of China, Qingdao 266100, China

Received 14 June 2015; Revised 4 September 2015; Accepted 6 September 2015

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2015 Shuqing Wang 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

Modal parameters are sensitive indicators of structural damages. However, these modal parameters are sensitive not only to damage, but also to the environmental variations. Development of vibration based damage detection methodology which is robust to environmental variation is essentially important for the structural safety. The present paper utilizes a recently developed modal strain energy decomposition (MSED) method to localize the damage of an offshore structure. A progress of the present paper is to take the temperature variation into consideration and Monte Carlo simulation is introduced to investigate the effect of temperature variation on the robustness of damage localization. Numerical study is conducted on an offshore platform structure considering the temperature variation. Several damage cases, including single and double damage scenarios, are included to investigate the damage localization algorithm. Results indicate that the MSED algorithm is able to detect the damage despite the temperature variations.

#### 1. Introduction

Vibration-based damage assessment and structural health monitoring (SHM) have been widely investigated during the past decades [1, 2]. The essential feature of vibration-based damage detection is that the dynamic characteristics of the structure would change due to the physical properties alteration of the structures. Such variations can be detected by utilizing modal identification methods and by monitoring the changes in the structure on a global basis.

Among the vibration based damage assessment methods, one type of damage identification methods only uses modal frequency information for damage assessment. Cawley and Adams [3] were among the first to use an incomplete set of natural frequencies to identify the location of damage. A detailed discussion on the use of natural frequency as a diagnostic parameter in structural assessment procedure can be found in the review paper by Salawu [4]. More recently, Wang and Li [5] and Wang [6] proposed an iterative modal strain energy method for damage localization and severity estimation, only requiring the changes of a few lower natural frequencies. Gillich and Praisach [7] presented a new method, based on natural frequency changes, able to detect damages in beam-like structures and to assess their location and severity, by considering the particular manner in which the natural frequencies of the weak-axis bending vibration modes change due to the occurrence of discontinuities. The advantages of frequency based methods include availability of modal frequencies from only one sensor record, and the identified frequencies have higher accuracy and robustness than those of mode shapes. However, information from only structural frequency is not sufficient, because modal frequency is a global property of structure and changes of frequency due to local damages may not be reflected, especially for large-scale structures such as offshore platforms.

Another class of damage diagnosis methods uses mode shapes. Early studies focus primarily on modal assurance criteria (MAC) and coordinate MAC (COMAC) values to identify damage by determining the level of correlation between modes from the test of undamaged and damaged structures [8–10]. Later, mode shape derivatives, such as mode shape curvature and strain-based mode shapes, were used as an alternative to obtain spatial information about vibration changes [11, 12]. After that, many methods based on dynamically measured flexibility have been proposed for damage detection. Because of the inverse relationship to the square of the modal frequencies, the measured flexibility matrix is most sensitive to changes in the lower-frequency modes of the structure [13–15]. However, mass-normalized mode shapes are required when formulating the flexibility matrix. Among the damage identification methods using mode shapes, the modal strain energy based methods seem to be promising for damage evaluation.

The Stubbs damage index algorithm, based on the decrease in modal strain energy, is defined by the curvature of the measured mode shapes [12, 16]. This method requires that the mode shapes before and after damage be known, but the modes do not need to be mass normalized and only a few modes are required. More recently, modal strain energy decomposition (MSED) method has been developed for damage localization [17–19]. The MSED method defines two damage indicators, axial damage indicator and transverse damage indicator, for each member. Analyzing the joint information of the two damage indicators can greatly improve the accuracy of damage localization.

In vibration-based damage identification and SHM, dynamic modal parameters, such as natural frequencies and mode shapes, of a structure are usually used as damage-sensitive features. However, these modal parameters are not only sensitive to structural damage, but also to the environmental factors and operational conditions such as temperature, humidity, and traffic conditions [20–22]. The structural modal parameters may change along with the environmental factors and operational conditions, which could possibly mask the subtle structural changes caused by damage. Therefore, the influence of environmental effects should be recognized and eliminated before applying damage identification methods [23, 24]. Several researchers have emphasized the importance of removing environmental or/and operational variations and have proposed some data cleansing procedures for SHM [20, 25–30]. Generally, three classes of methodologies, theoretical derivation methods, trend analysis methods, and methods of quantitative models, have been developed to correlate the temperatures and vibration properties. The interested reader can find more information in the two recently published review articles [21, 22].

From the literature review, one can see that a lot of papers concerning variations in vibration properties of civil structures under changing temperature conditions have been published. However, most of these studies focus on variations in frequencies of bridge structures, with some studies on variations in mode shapes and damping of bridge structures. Offshore platforms, operated in harsh environment and under various external forces, constantly accumulate damage during their service life. Clearly the development of robust techniques for early damage identification is crucial to avoid the possible occurrence of a catastrophic structural failure. For offshore structures, factors that influence the modal parameters cover variations of temperature, tidal level, foundation scouring, marine growth, and deck mass, and so forth. Modal changes produced by environmental conditions can be equivalent or greater than the ones produced by damage [31]. Therefore, development of robust damage assessment methodology, where the damage indicator is insensitive to environmental variations, has become to be a key problem in the damage detection and structural health monitoring. Generally there are two approaches which are able to handle such cases. One way is to firstly correlate the environmental variation and the modal parameters and conduct the damage assessment after the effect of environmental variation has been removed. Another more promising way is to develop a robust damage indicator, which is sensitive to structural damage but insensitive to environmental variations.

The objective of the present paper is to investigate the robustness of environmental variation on damage localization of offshore platform structures. A recently developed modal strain energy decomposition method is used for damage localization of an offshore platform structure. A progress of this paper is to take the temperature variation into consideration and Monte-Carlo simulation is introduced to investigate the effect of temperature variation on the damage localization. Different temperature variations for structural members which are located in air and beneath water are assumed. Both single damage and double damage scenarios are considered. To this end, the paper is structured as follows. In Section 2, the modal strain energy decomposition method for damage localization is briefly described. The model of a platform structure and simulations of environmental variation are introduced in details in Section 3. Section 4 reports and discusses the results of environmental variation on the damage localization. Finally, the paper is concluded with some final remarks in Section 5.

#### 2. Modal Stain Energy Decomposition Method for Damage Localization

In this section, the modal strain energy decomposition method is briefly reviewed and will be used for damage localization [17–19].

Stubbs et al. [12] proposed a damage index for damage localization. The damage indicator, named Stubbs’ damage index, is defined aswhere and are Young’s modulus for the th element before and after damage, respectively; is the number of modes being used; the elemental and global modal strain energy for the th mode shape is, respectively, defined asin which and are the th mode shape of the undamaged and damaged structure, respectively; the superscript “” is the transpose operator and superscript “” is used to indicate a damage version. , where is the stiffness matrix of the th element defined in global coordinates for the undamaged system. is the global stiffness matrix, which is constituted of for all the finite elements.

For a more robust classification criterion of damage location, a normalized damage indicator is defined as where and represent the mean and the standard deviation of , respectively.

The major concept of the modal strain energy decomposition method is to separate the structural modal strain energy into two groups according to local element coordinates. One is axial modal strain energy and the other is transverse modal strain energy. Then two damage indicators can be defined [18]. For an elemental stiffness matrix , it can be decomposed intowhere superscripts , , , and stand for axial, transverse, rotational, and transverse-rotational terms, respectively. That is, is the matrix containing axial stiffness terms only, is associated with the transverse stiffness terms only. It is recognized that the measurements associated with rotational coordinates are difficult to obtain practically, so most damage detection methods use mode shapes that include only translational coordinates.

Following the formulation of Stubbs index, the* axial damage index* can be obtained as follows:In (8), where is defined as matrix transformed from local to global coordinates. And the matrix involves only geometric quantities. Equation (8) can be viewed as the counterpart of (1) while only the axial modal strain energy is under consideration.

By the same token, if only the* transverse modal strain energy* is considered, one should obtain the corresponding index asIn (9), is the geometric part of stiffness matrix corresponding to transverse parts in global coordinates.

Following the same normalization procedure as in (6), one can define the* axial damage indicator* asand the* transverse damage indicator* as where the* overline* represents the mean value and represents the standard deviation of the corresponding variable.

The damage pattern is classified via a statistical-pattern-recognition technique using hypothesis testing. Then the decision rule for damage location is selected as follows [32]. The structure is not damaged at the th location when and is damaged when in which is a number that reflects the level of significance of the test (e.g., if , then the confidence level is 97.7%). It should be mentioned that is used in the following numerical study for damage location.

A typical 3D frame structure generally consists of vertical columns, horizontal beams, and diagonal braces. If the vibration modes used in damage detection are mainly lateral (horizontal) motion, the modal strain energy of vertical members would be dominated by the transverse modal strain energy. On the other hand, the axial modal strain energy would contribute significantly to the modal strain energy of horizontal members and diagonal braces. Therefore, analyzing the joint information of the two damage indicators can greatly improve the accuracy of damage localization. For detailed explanation and comparison, the reader may refer to two published articles for more information [18, 19].

#### 3. Numerical Example

##### 3.1. Description of the Structure

An offshore platform structure described in [33] is used here for numerical study. The structure consists of 36 steel tubular members that comprise 12 leg members, 12 horizontal brace members, and 12 diagonal brace members in vertical planes. All members have uniform outer diameter 17.8 cm and wall thickness 0.89 cm. The heights of the three stories are all 9.14 m, and the side lengths of the bottom and top floors are 12.19 m × 10.97 m and 4.88 m × 3.66 m, respectively. The essential material properties of the steel tubular members are elastic modulus N/m^{2}, volumetric mass density kg/m^{3}, and Poisson’s ratio . The finite element model of the platform structure is synthesized, where each structural member is modeled as a three-dimensional uniform beam element, and is distinguished by assigning an element number. Also shown in Figure 1 are the node and element numbering, respectively. Modal analysis is carried out by developing a program in MATLAB environment to get the modal frequencies and mode shapes. The first three modal frequencies are 7.72, 8.31, and 8.87 Hz, respectively. Also the first two mode shapes are exhibited in Figure 2, where the first mode is vibrated dominantly in the -(short-span) direction and the second mode in the -(long-span) direction. And the third mode is a torsion mode around -axis. For simplicity, the still water level (SWL) is assumed to be exactly located on the second floor from the top, as shown in Figure 1.