Structural Model Updating of Jacket Platform by Control Theory Using Vibration Measurement Approach

The identification of variations in the dynamic behavior of structures is an important subject in structural integrity assessment. Improvement and servicing of offshore platforms in the marine environment with constant changing, requires understanding the real behavior of these structures to prevent possible failure. In this work, empirical and numerical models of jacket structure are investigated. A test on experimental modal analysis is accomplished to acquire the response of structure and a mathematical model of the jacket structure is also performed. Then, based on the control theory using developed reduction system, the matrices of the platform model is calibrated and updated. The current methodology can be applied to prepare the finite element model to be more adaptable to the empirical model. Calibrated results with the proposed approach in this paper are very close to those of the actual model and also this technique leads to a reduction in the amount of calculations and expenses. The research clearly confirms that the dynamic behavior of fixed marine structures should be designed and assessed considering the calibrated analytical models for the safety of these structures.


Introduction
As the foundation platform of ocean engineering, offshore jacket platform is the main supporting part of fixed marine structures, which generally works within the depth of 10 m to 200 m. The offshore jacket platforms, especially the oil and gas production platforms play a very important role in the present world economy. Safety of offshore jacket platforms is commonly presumed to be acquired by design according to the established standards and approaches, for an expected design life. If a jacket structure is intended to be utilized beyond its design life, a thorough control of the structural safety must be performed. In particular, this will be significant with respect to fatigue and other continuous failure mechanisms. Regulations and principles may have been slightly changed since the initial design. The loading pattern and the environmental load may have altered, and the structure may have deteriorated to an uncertain extent during decades in harsh weather. Therefore, a correct design for jacket structures should be imposed. The design should be performed based on the performance evaluation of previous structures. For this purpose, close-toreality analyzes are needed. In this study, control-based Structural Model Calibration (SMC) of jacket platforms using vibration test data is provided. In engineering practice, Finite Element (FE) models are always considered to predict the response and dynamic properties of structural system. But the real dynamic behavior of jacket structures may differ from the designed case. The steel structures (such as fixed marine structures) are generally strengthened by brace elements to increase stiffness. Hereof, it may be difficult to predict the real behavior of jacket structures because of the fact that there are many connections on the beam-to-column joints and brace elements. To investigate the performance of the FE model, the concept of model calibration and model updating is presented by researchers (Dinh-Cong et al., 2020;Astroza et al., 2019). SMC is a method that minimizes the differences between analytical and experimental results. The most preferred methodology to calibrate initial mathematical models is to utilize the modal parameters. This requires that empirical results of real structures be properly identified. The matching between FE model and the empirical model is developed by modifying the physical parameters of the model employing measured dynamic test data. Need for acquiring accurate mathematical models has led to the improvement of technique of SMC that targets at reducing the inaccuracies presented in a mathematical model in the light of measured dynamic information. All uncertainties in modeling structures are considered as a calibrat-ing parameter in the model calibration process. In this study, the model calibration is used to update and refine dynamic properties of jacket platform model. In the 1980s, the basic theory of SMC began to form, and developed into many approaches. Mottershead and Friswell (1993) systematically summarized the improvement of classical SMC theory, and the detail will be skipped here. In summary, SMC approaches could be categorized according to the following properties: (1) techniques based on modifying the system matrix or the system parameters, which differ by the selection of updating variables; (2) techniques based on direct (sensitivity) process or iterative process, which differs by the calculation method; (3) techniques based on modal information, which differ by the objective functions. Direct techniques are one-step procedures that seek to make a minimum change in the structural matrices so that the measured dynamic test data are reproduced by the updated and calibrated model (Berman and Nagy, 1983). A number of variations of these techniques are improved depending upon whose data are treated as reference data and whose quantities are refined (Caesar, 1986). One of the concerns with refined models gained employing direct algorithms has been that they may show some spurious modes of vibration. Scheme proposed by Carvalho et al. (2007) to calibrate mass and stiffness matrix preserves the modes that are not updated and hence helps prevent appearance of spurious modes. Mao and Dai (2012) has also studied refined models showing spurious modes that have been referred as spillover phenomenon in SMC. The iterative techniques of SMC utilize either the experimentally identified modal information or the measured Frequency Response Functions (FRFs). The use of eigendata sensitivity for analytical SMC in an iterative scheme was first offered by Collins et al. (1974). The effect of including second order sensitivities in this scheme was investigated by Kim et al. (1983). Lin et al. (1995) proposed to consider both the analytical and the experimental modal information for assessing sensitivity coefficients with the objective of developing convergence and widening the applicability of the technique to cases with a higher error magnitude. Kim and Park (2008) introduced SMC problem as a multi-objective optimization with eigenfrequency residual and modal strain energy residual as parts of the objective function. The use of measured FRFs in SMC instead of modal information has the advantage that the influence of modal analysis errors on the refining procedure is avoided. Lin and Ewins (1994) offered an iterative FRF technique in which the physical difference between the measured and analytical reacceptance is written as a linear function of the parameters to be refined. Recently, a technique for refining only the mass and stiffness matrix considering the complex FRF information has been presented (Pradhan and Modak, 2012). This refining scheme insulates the updating of the mass and stiffness matrix from influence of damping that is present in the measured complex FRFs. The uncertainty in the results between the FE analysis and the empirical modal analysis is because of the presumptions made in defining unsuitable element material feature and geometrical feature. The effects of errors due to insufficiency of information are analyzed employing FE analysis and developments must frequently be made to trim down the errors related with the FE model. The benefit of refined FE model is that, it is capable of modeling other loads and boundary conditions without going for any additional empirical testing (Chakraborty and Sen, 2014). The focus of model calibrating is created for analyzing the dynamics behaviors of a structure can be improved considering empirical test results measured on the actual structure of a platform. It becomes the most wanted and challenging applications for testing. An important need in dynamic analysis is to create an analytical model capable of reproducing the empirical information. In this regard, empirical modal analysis and FE model that characterize the behaviors of the structure in terms of frequencies and mode shapes were compared (Gopichand et al., 2013). Most offshore jacket platforms topsides are transported to the final site by barge. Offshore jacket platforms are normally moved long distances to install in the site. The transportation part can be critical for some permanent members and specifies the design of the sea fastenings and other temporary attachments. Large forces and vibration can be created, particularly by the roll and pitch rotations and accelerations and the heave acceleration of the barge. This is especially the case for offshore jacket platforms where the length of the legs results in large bending moments from roll and pitch motions. These factors and effects can cause important fatigue defect as well as possibly overstressing the legs. Parts of the structure which overhang the barge deck may endure buoyancy, drag and inertia loading and also, possibly more seriously, slam loading if the member penetrates the water surface. The sea area, barge size, season and duration of the tow should be taken into account when selecting the environmental conditions to be considered for the analysis. As a result, the step of loading is very vital and it has an important effect on the design of offshore jacket platforms. Thus, the physical model is established according to this loading condition. Accordingly, in this work, the model tests were accomplished in dry.
The present work for considering the incomplete empirical data uses only low-dimensional matrices, even though the FE model might be very large. For this aim, Developed Reduction Algorithm (DRA) is employed. It is worth mentioning, in the structural integrity assessment method using the characteristic parameters; the use of refined matrices based on empirical results is useful. On the other hand, in SMC of an offshore jacket structures considering empirical modal analysis, there are two major challenges ahead: (1) the mismatching of measurement sensors and degrees of freedoms of the numerical model, namely the spatial incom-pleteness and (2) the unavoidable corrupted measurements (Yuen, 2012;Wang, 2013). In dealing with spatially incomplete situations and the effects of noise to overcome uncertainty problem, we can employ DRA to implement of precise test. Furthermore, to overcome modeling uncertainty problem, the FE model refining method is applied by considering results of the experiment on physical model of the offshore jacket platforms, when data are limited, spatially incomplete modal information is available. In this research, the presented reduction approach eliminates the detrimental impact of model reduction method on the model refining procedure by adding a correction term (inertial effects) to the formula of the simplest reduction schemes. Therefore, the sketch of the present method would experience the effects of the predicted or unforeseen uncertainties which may be significant and have not been considered. Calibration is often used to tune unknown calibration parameters of a computer model. In this paper, the model calibration is done by adjusting the basic characteristics of the structure in the form of dynamic matrices to obtain a best fit between the real model and analytical model, and so the model calibration is employed instead of the model updating. On the other hand, in previous studies, simple models have been used. Extracting dynamic matrices of complex and large structures (such as offshore structures) is not easy and always difficult. In this paper, feature extraction is easily done. Also, this article is an underlying research so that its results can be used in other studies related to the analysis and recognition of the behavior of offshore structures. However, in similar previous studies, only one specific goal (problem) has been defined and solved. The results of the present study are wide-ranging and generalizable. In other words, If the corrected dynamic matrices are available, we do not need to re-test (re-laboratory work) to perform a new reliable numerical study (Knowing that modeling and laboratory work is time consuming and costly).

Control theory and structural model refining algorithms
Control-based model calibrating approaches, such as the state-space Eigen-Structure Assignment (ESA) technique, the Quadratic Pencil Technique (QPT) are explained in Section 2. All of the techniques stated in this section are taken employing control theory, which provides the ability to calibrate a subset of modal parameters. This ability is useful since it is very difficult to extract the complete eigen-structure in large scale jacket structures.
The basic idea underlying control-based SMC technique is the use of the concepts of feed-back control to simultaneously assign the system eigenvalues and eigenvectors to desired locations. Feed-back in terms of control theory is when a controller (gain matrix) is considered to adjust the characteristics of a defined system.
A dynamical system in state-space given by:ẋ (1) where A, B, x(t) and C stand for the state matrix, input matrix, state vector and output matrix, respectively. Also, u(t) and y(t) represent an external force vector (or the feed-back force) and the response vector. Furthermore, B regulates the locations of the feedback forces denoted by u(t).
The basic idea of the control-based techniques is to specify the control vector u(t), which is fictitious, that will place the eigenvalues and eigenvectors of A in their desired locations, a procedure known as ESA.
For structural systems, the state space matrix is created considering the stiffness (K), mass (M), and damping (D) matrices given by: The characteristic parameters for state-space are set up similar to that of the quadratic case (Andry et al., 1983). The eigenvalues are determined by solving the following equation: In order to perform the eigen-structure assignment the form of the control vector u(t) is first chosen; (a) full-state feed-back with u(t) of the form, u(t) =−Kx(t), where K and x(t) are a gain matrix and a state vector. Since all the states are considered in x(t), it is known as full-state feed-back, (b) output feed-back, with u(t) = −Ky(t).
In the work, only case (a) will be imposed. In this regard, the state equations in Eq. (3) become: A − BK and the calibrated state matrix is equal to . The purpose is to acquire a gain matrix (K), such that the calibrated state matrix has the same eigen-structure as the measured modal data.

State−space ESA approach
The use of ESA in civil engineering is relatively new. In general, control-based approaches can be categorized as techniques that operate directly on the state matrices and those that operate on the system properties directly. For the state space technique, the basic equations can be presented as: where x, u, and y represent the state, control, and output matrices; A, B, and C represent real constant matrices; and the rank of B and C is not 0. It is also important that the system is controllable (Andry et al., 1983), is described as follows: n × n where the size of A is . As mentioned, the state matrix 98 Farhad HOSSEINLOU China Ocean Eng., 2021, Vol. 35, No. 1, P. 96-106 (A) contains the system information, and B, is the feed-back gain matrix that needs to be calculated. Given 2n measured eigenvalues and 2n measured eigenvectors, the full-state problem computes a feedback gain matrix K that updates the state matrix: such that the closed loop eigenvalues and eigenvectors correspond precisely to the 2n experimental eigenvalues and eigenvectors. Moore (1976) presented a technique that establishes a matrix K that will update the state matrix (A), to precisely conform the experimental eigenvalues and eigenvectors. In the first step, R λ S λ are presented, where the columns of from a basis are for the nullspace of .
The eigenproblem can be obtained by: 11) for the state space equation, and rearranged to present: It follows from the assessment of Eqs. (10) and (12) that v i spans N λ . In this regard, there is a vector z i such that (13) By combining Eq. (10) along with Eq. (12) and then by multiplying the vector z i , we have: and (15) Substituting Eq. (13) into Eq. (15) leads to: (16) By utilizing the parallels between Eqs. (14) and (16) the feedback matrix can now be substituted into Eq. (19) to create a new state matrix with the empirical eigenstructure.

Quadratic pencil technique
This form of eigenstructure assignment was presented by Datta et al. (2000) and is similar to the one defined in Andry et al. (1983). The process is similar; however, the basic difference is that the adjustments are accomplished directly on a second order differential Eq. (20). This leads to the quadratic eigenvalue problem, which is described by Eq. (20) where and stand for an eigenvector matrix and a eigenvalue matrix, respectively.
This dynamic equation can be modified by utilizing a control force Bu(t), where B denotes an matrix and u(t) denotes an vector which signifies the feed-back force presented in Eq. (24), where F and G denote matrices, called state feedback control matrices. If these characterizations are presented into the eigenvalue problem, it results in Eq. (24). So, we have: (24) This will provide the foundation of the eigenstructure assignment technique. Hence, give where and are an matrix of measured eigenvectors and an matrix of the remaining unmeasured eigenvectors, respectively. Also, denotes an matrix of measured eigenvalues, and denotes an of the remaining unmeasured eigenvalues.B

TFTGT
The values of , , and must be determined to satisfy (Datta et al., 2000): (26) Datta et al. (2000) presented: where denotes an matrix of original eigenvalues that are to be reassigned and also, stands for an matrix of original eigenvectors that are to be reassigned, and In order to solve B, F, and G, singular value decomposition (SVD) must be accomplished on H. The compact SVD creates three matrices; U, Σ, and V and arises as such (32) From which, B is extracted to be the product of U and Σ, and the first n rows of V are considered to be F, and the last n rows are considered to be G.

DRA because of incomplete modal data
One of the simplest reduction algorithms is static or Guyan reduction. The full scale model may have certain nodal freedoms indicated as master freedoms and the remaining freedoms are slave freedoms. For dynamic analysis theory, the mass, stiffness and loading on the slave freedoms are condensed and summarized to these master freedoms. In matrix notation the overall matrices may be separated and partitioned into master, slave and cross coupling terms.
"m" "s" VV where, the subscripts and stand for the master and slave coordinates, respectively and also and denote the nodal displacement and acceleration vectors of the structure.
The technique then disregards the inertia terms in the second set of equations. Disregarding the inertia terms for the second set of equations we have: (34) By disregarding the slave degree of freedom, we obtain: T s I and stand for Guyan transformation matrix and identify matrix, respectively.
The reduced Guyan mass and stiffness matrices are then given by Guyan (1965): For larger offshore jacket structures, where it is essential to reduce many slave degrees of freedoms, this procedure will not be as accurate as some of the more improved algorithms. Accordingly, DRA is probably the best practical process for solving large dynamic problems. Only the smallest frequencies are generally excited and for a typical jacket no more than 30 degrees of freedom (DOF) would normally be required.
The skill known as the DRA was offered by O"Callahan in 1989 and Friswell and Mottershead (1995). This skill is an expansion over the Guyan static reduction system through introducing a term that includes the inertial effects as pseudo static forces. A transformation matrix is used to reduce the mass and stiffness matrices. It is presented as: Hereof, and denote the statically reduced mass and stiffness matrices.
The new reduced mass and stiffness matrices can be ac-quired by: In this regard, the rows and columns corresponding to the slave coordinates are ignored from the mass and stiffness matrices one at a time; this allows the mass and stiffness matrices to adapt to the elimination of a slave, and can possibly alter the degree of freedom that will be disregarded. After each reduction, the degree of freedom with the lowest term is the slave which will be disregarded next (Barltrop and Adams, 1991).
For the case of an un-damped system, the quadratic pencil reduces to Υ Ψ According to Eq. (42), the eigenvalues and eigenvectors are intricately related to the system features K and M. Hence, in traditional modal analysis and model refining, the primary approach is to adjust K and M to reflect and observed utilizing test information.
Υ Ψ Most of the model refining techniques employ a twostep algorithm. First, M and K matrices are refined utilizing the measured and , then, the damping matrix D is computed using M and K. In this study, the mass and stiffness matrices are calibrated. Then, if necessary, the damping matrix can be reproduced using the following equations: (44) Accordingly, the damping matrix will also be affected via reduction of the mass and stiffness matrices.

Tested frame and empirical work
To achieve the aims of this research, results of experimental work on 2D steel frame model structure of recently designed and installed jacket in the Persian Gulf have been used. The experimental 2D frame structure has been constructed for laboratory vibration testing, as presented in Fig. 1. The empirical model is fabricated of steel tubes, considering sections of 34 mm×3.5 mm (outer diameter and thickness) as two legs and sections of 21 mm×2.0 mm as all braces, and box cross section at the top with dimension of 40 mm×20 mm and thickness of 2 mm to simulate the upside structure. The 2D platform model is fixed into a rigid steel base plate and has four storeys with 29 structural members at all storeys, i.e. columns, beams and braces. The 2D steel frame model structure has dimensions of 560 mm (on the bottom) and 480 mm (on the top) in width and has heights of 280 mm, 285 mm, 270 mm and 270 mm at the 1st, 2nd, 3rd and 4th storey, respectively, giving a total height of 1105 mm. The FE model demonstrated in Fig. 2 has 15 nodes and 29 elements with a total number of 45 degrees of freedoms. The material properties of Young's modulus of and density of 7850 are considered in calculations for all beam, column and brace elements since the tested platform model is built from steel. In the laboratory vibration testing for the 2D jacket structure, a total number of 15 uni-axial accelerometers are placed at the beam-column joints to measure translational displacements, i.e. measuring displacements in y direction and in x direction on each nodes of structure. Three types of uniaxial sensors are adopted in the tests and installed at beam-column joints to record the response of the 2D frame structure excited by means of an electro-dynamic exciter (type 4809) with a force sensor (AC20, APTech) driven by a power amplifier (model 2706), all made by Bruel & Kjaer. The excitation signal to the power amplifier could be imposed using a time series generated by a personal computer. The exciter is rigidly fixed on a support frame close to the model and loads are used, as presented in Fig. 3.
The white noise signals are imposed as the input exciting signal. As the considered feature sets of this work are with the natural frequencies which are independent of the excitation types, the proposed technique of current research is adaptable to each type of input forces and the inherent structural dynamic output response must be considered more carefully which is the significant key point of the study. In other words, there are different considerations when performing the empirical modal analysis. For in-stance, the test set-up needs to be selected such that all the modes of interest are excited by the external forces over the frequency range to be considered. Otherwise, all the modes will not be excited adequately in order to acquire good measurement and therefore the plotted coherence function is seen to deteriorate as well as FRF over the frequency range. For this target, empirical modal analysis is performed in the high frequency range. Laboratory facilities should also be considered such as the sensor sensitivity. The frequency sampling of the test setup is selected to be 16.385 kHz and for frequency range to be 0−800 Hz based on the expected natural frequencies obtained from the initial FE modal analysis. Artificial excitation of jacket structures is not always practicable and ambient excitation because of waves, tidal currents and winds may not be adequate for collecting the natural vibration modes. Alternate approaches can be considered for the same aim, for example, the application of an impact or a sudden relaxation of an imposed force for exciting the structure. For jacket structure, impact can be employed by gently pushing the structure at the fender while relaxation can be done by pulling the structure and then suddenly releasing it using a tug or a supply vessel in both cases (Mangal et al., 2001). Also, there are different considerations when performing the experimental modal analysis. For instance, the test set-up needs to be selected such that all the modes of interest are excited by the external forces over the frequency range to be considered. Otherwise, all the modes will not be excited adequately in order to obtain good measurement and therefore the plotted coherence function is seen to deteriorate as well as frequency response function over the frequency range. In this regard, experimental modal analysis is performed in the high frequency range (Hosseinlou et al., 2017). Laboratory facilities should also be considered such as the sensor sensitivity.
The experimental and the numerical modal analysis for finding the structural dynamic characteristics of system have been utilized in this article. One main aspect of the experimental modal analysis is extraction of FRFs. In the first step of an experimental modal analysis, the elements of at least one full raw or one full column of the FRF matrix should be measured and then the natural frequencies can be   Farhad HOSSEINLOU China Ocean Eng., 2021, Vol. 35, No. 1, P. 96-106 determined utilizing a variety of different techniques such as Rational Fraction Polynomial technique (Richardson and Formenti, 1985). The single input and multiple output test process is imposed for the vibration testing for the 2D frame jacket structure, and the curve fitting procedure is used on reference set of FRFs to extract modal parameters by applying ME'Scope VES modal analysis software. The software covers three built-in curve-fitting methods: Peak Fit, Quadrature Fit and Polynomial Fit. The Polynomial Fit has been utilized to extract the accurate natural frequencies of the model.

Characteristics extraction of structural system
Modal testing is an experimental technique considered to gain the modal parameters of a model of a linear time-invariant vibration system. The theoretical basis of the technique is secured by creating the relationship between the vibration response at one location and excitation at the same or other location, depending on the excitation frequency. In brief, empirical model analysis consists of three component phases: test preparation, measurement of frequency responses, and detection of modal parameters. The preparation includes selection of a structures support, type of excitation force(s), location(s), hardware to measure force(s) and responses; designation of a structural geometry model, which involve of points of response to be measured; and identification of mechanisms, which could lead to inaccurate measurement. However, it is unrealistic to expect such a FE model to be fully representative of faith because of the complexity and uncertainty of the structure. A fundamental methodology is to take a measurement of the structure, derive its modal model and use it to correlate with the existing FE model in order to refine it. The results of the empirical model analysis or measured frequencies are presented in Table 1. Empirical model analysis is presented as a technique for describing the dynamic features of a structural model containing frequencies, damping, and mode shapes. Natural frequencies and mode shapes of examined jacket have been extracted considering ME' Scope software. A result of the empirical model analysis for the studied model is shown in the form of FRF in Fig. 4. The mode shapes of first three modes are indicated in Fig. 5.
In the present study, FE model based on the empirical 2D platform model is produced employing the ANSYS software package. Modal analysis basically investigate the natural properties of a jacket structure. For the implementation of the proposed technique, initially the mass and stiffness matrices are extracted by AN-SYS software under SUBSTRUTUR analysis (see Fig. 6). The Block Lanczos method of the ANSYS software package was also applied for solving the analytical modal analysis and analytical modal shapes are also identified by AN-SYS software. Extracting dynamic matrices in ANSYS is easier and more accurate than MATLAB (Due to the complexity of offshore jacket structures). Then the matrices entered the MATLAB software environment and all calculations based on the proposed algorithm are performed with  MATLAB software. After correcting and adjusting the matrices based on the measured data, the frequencies are reproduced using these corrected matrices and these calculations are done with MATLAB.
The results of the numerical modal analysis are presented in Table 2 and Fig. 7.  The refining of the initial FE model is necessary to minimize the numerical model error according to the experimental signatures. The concepts of the "Modal Assurance Criterion ( )" method can be applied for this target. Concisely, it can be presented as follows: The parameters in Eq. (45) are described below: where and denote the i-th eigenvalue and mode shape, respectively, and the subscripts "a" and "e" present the analytical and corresponding experimental modal data. Using the first-order Taylor series, we find the following: where and are the experimental and analytical function vectors, T is the design sensitivity matrix of , are the changes in for the least squares minimization, and ε is a residual vector. The least squares solution for to minimize is: where the design sensitivity matrix modal functions of the eigenvalue and the eigenvector can be defined as follows: Eq. (34) is rearranged as follows: and . The state space ESA scheme is applied to calibrate the dynamic matrices. Since this approach works directly in state space form, there is no need to manipulate the process in any way. The natural frequencies of the calibrated model and obtained frequencies of the tested physical model are presented in Table 3. According to Fig. 8, results indicate that the MAC values between the mode shapes of the calibrated model and the mode shapes of the tested physical model are all larger than 0.99, which represents that an effective refined model has been created. The scheme can recreate the desired ei- Fig. 8. MAC values between the measured mode shapes and extracted mode shapes of the calibrated model based on ESA. Farhad HOSSEINLOU China Ocean Eng., 2021, Vol. 35, No. 1, P. 96-106 genvalues exactly. But, the main drawbacks are that the eigenvalues refined are not chosen by the user and may not correspond adequately to the position of the transducers; the state matrix is completely filled and does not resemble a physical representation of the structural system (see Fig. 9). Unlike the state space ESA technique, the QPT can only refine the mass and damping matrices. This needs the user to ensure that the initial structural system is given in terms of the mass, stiffness, and damping matrices. The refined structural system can be reorganized into a state matrix. The calibrated state matrix utilizing the QPT emerges more similar to the analytical state matrix than the refined state matrix presented by the state space ESA technique; this is because the mass and damping matrices are refined individually. The natural frequencies of the calibrated model and measured frequencies of the tested physical model are shown in Table 4. Hereof, results show that the MAC values between the mode shapes of the calibrated model and the mode shapes of the tested physical model are all larger than 0.99, which denotes that an effective refined model has been produced (see Fig. 10). The aspect of this technique that is most advantageous is that the stiffness and damping matrices are calibrated directly, meaning that there is a better physical representation of the calibrated structural system. Actually, modal data measurements always contain corruptions and errors. The measurement uncertainties are resulted from sensor noise and measurement corruptions. Mild noise environment and fluctuations caused by the impairments in the measuring instruments are the origins of theseΨ corruptions. The modern tools can reduce these uncertainties but they can never be removed. Hereof, the measurement of the polluted i th mode shape of the tested structure at the j-th DoF, denoted by , has been simulated by adding a Gaussian random error to the corresponding true value . So, we have . Here represents a noise level. Also, is a Gaussian random number with zero mean and unit standard deviation and has been generated via RANDN method of MATLAB software. In the numerical study, the results are always gained from taking a 500 repeated Monte Carlo simulations (MCS). So, to acquire statistical knowledge about the identification result, 500 simulations are done.
A factor called Correct Refinement Index (CRI) is considered in order to measure the noise effect on the accuracy of the suggested strategy. If is used to represent the number of MCS for a given level of noise and the number of realizations that an actual frequency is reproduced, the CRI will be given by . Furthermore, if the differences between analytical and experimental results are smaller than 10%, it is not required to refine the model (Bayraktar et al., 2011). Accordingly, reproduced frequencies with an error smaller than 10% are considered as the correct result. By increasing the level of noise from 1% to 4%, the percentage of success index is given in Fig. 11 for model calibration, as plotted against the noise level.
In this work, the percentage of success index for model calibration is 100% for a 1% noise level and 99.74% for a 2% noise level. Since errors are applied to slave coordinates through the utilization of transformation matrix, it seems useful to use the model reduction methodology, rather than the modal expansion technique. According to Fig. 11, results denote that the suggested strategy is executable and efficient to calibrate model with a reasonable percentage for noisy data.

Conclusions
In practice, there are always differences between an FE model of a structural system and the actual structural sys- tem. The differences could be caused by simplification of the model structure, extra facilities in the actual hardware, uncertainty in the physical characteristics, discretization corruption and mistakes, and so on. The area known as model refining is concerned with the correction of FE models by processing records of dynamic responses from physical structures. Calibrating and refining of the FE model are also needed to detect and correct the uncertain parameters of an FE model.
In this study the ability of practical examination of the complex steel jacket platform model refining is expressed for calibrating uncertain model parameters. A real condition approach is expressed with incomplete modal information, which uses tested data in order to progress the correlation between the empirical and computational models based on control theory.
An example with incomplete modal information of a reduced scale 2D frame of the offshore jacket structure is accomplished showing that the approach can correctly calibrate dynamic matrices and reproduce correctly the measured data. The mode shapes are not necessary to be acquired at all degrees of freedom. The suggested skill eliminates the detrimental impact of model reduction method on the model refining procedure.
The offered method in the current work is computationally efficient since it does not need iterations. It calibrates the dynamic matrices such that they are compatible with the modal information of the measured modes. The ESA method reproduces the measured eigen-system, however, the results are not physically meaningful, or in other words cause the refined structural system to lose its physical representation. This is a potential problem for situations where the stiffness and/or mass of a specific degree of freedom are required, such as in damage/or failure diagnosis. But, the calibrated models based on the ESA method can be employed in the earthquake, dynamic and seismic analysis of jacket structures for best design. In addition to the content listed above, instead, the results of the QPT are useful for application in the damage detection process, because the method provides a desirable physical nature of the structural behavior (i.e. physical representation of the dynamic matrices).
The FE model calibrating brings a practical and less expensive way for structural behavior assessment of fixed marine structures. The work clearly confirms that the dynamic behavior of jacket structures should be designed and assessed considering the calibrated analytical models for the safety of these structures.