Damage identification based on incomplete modal data and constrained nonlinear multivariable function

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Damage ientification base on incomplete moal ata an constraine nonlinear multivariable function To cite this article: S S Kourehli 215 J. Phys.: Conf. Ser View the article online for upates an enhancements. Relate content - Corrigenum: Multivariable statistical regression moels of the areal extent of hypoxia over the Texas Louisiana continental shelf Davi R Forrest, Robert D Hetlan an Steven F DiMarco - The AN-1- an BN-Calogero moels Akinori Nishino an Hieaki Ujino - An Improve Metho to Control the Critical Parameters of a Multivariable Control System P. Subha Hency Jims, S. Dharmalingam an G. Jims John Wessley This content was ownloae from IP aress on 4/12/217 at 14:39

2 Damage ientification base on incomplete moal ata an constraine nonlinear multivariable function S. S. Kourehli Department of Civil Engineering, Ahar Branch, Islamic Aza University, Ahar, Iran Abstract. Damage etection an estimation in structures using incomplete moal ata is presente. In the propose approach, amage location an severity is etermine by solving an optimization problem using the constraine nonlinear multivariable function of Matlab (socalle fmincon) to perform constraine minimization. The feasibility of the presente metho is valiate with a three-story plane frame as numerical example containing one or several amages with ifferent value of amage severity. The obtaine results inicate that propose metho is effective an robust in etection an estimation of amage. 1. Introuction Damage etection is one of the branches of structural health monitoring (SHM) which has recently attracte many scientific efforts. Health monitoring refers to a process of measuring an interpreting ata from a system of sensors istribute about a structural system to objectively quantify the conition of the structure [1, 2]. Damage etection techniques have been successfully applie to several real-worl problems. Base on the performance of structures, amage etection methos can be categorize into four levels [3]. The first level is evote to etection of existing amages in a structure, an the secon an thir levels focus on the etermination of the location an severity of amage in structures, respectively. The last level is a complete stuy that inclues the estimation of the resiual life of a structure, reaching a point that requires more information from fracture mechanics an structural reliability. Among numerous methos, approaches that are base on the observation of the ynamic behavior of a structure heve been evelope [4-7]. Many of these techniques use the ientifie moal parameters like moe shapes an natural frequencies for structural amage etection an estimation. The ientification of alteration in moe shapes an natural frequencies at the amage system in comparison with the unamage system is one of the popular methos in the structural amage etection. Carvalho et al. [8] presente a irect metho for moel upating with incomplete moal ata. The propose metho uses an algorithmic way without requiring any moel reuction or moal expansion techniques. Huajun et al. [9] extene the CMCM metho to simultaneously upate the mass, amping an stiffness matrices of a finite element moel when only few spatially incomplete, complex-value moes are available. The results reveal that applying the CMCM metho, along with an iterative Guyan reuction scheme can yiel goo amage etection in general. Also, Chen [1] presente an approach for etecting local amage in large scale frame structures by utilizing Content from this work may be use uner the terms of the Creative Commons Attribution 3. licence. Any further istribution of this work must maintain attribution to the author(s) an the title of the work, journal citation an DOI. Publishe uner licence by Lt 1

3 regularization methos with incomplete noisy ata. A system of linear basic equations for etermining the amage inicators has been evelope by irectly aopting the measure incomplete moal ata. In this research, a new metho for localizing an estimating the severity of structural amage is introuce. The amage ientification is carrie out through fmincon to minimize an objective function erive from incomplete ynamic characteristics of amage structure. Numerical example shows that the propose metho can be consiere as a flexible an robust approach in amage ientification of structures. 2. Problem formulation The moal characteristics of a structure without amage are escribe by the following equations: where, K u an M are unamage stiffness an mass matrices, respectively; λ i is the square of the natural frequency corresponing to the moe shape Φ i ; an n is the total number of obtaine moe shapes. One of the simplest techniques to etermine amage-inuce alteration stiffness is the egraation in Young's moulus of an element as follows: E j (1) u E 1 ) (2) j ( j where, E j an E j u are the amage an unamage Young's moulus of the jth element in the finite element moel, respectively; an j inicates the amage severity at the jth element in the finite element moel whose values are between for an element without amage an 1 for a rupture element. Moreover, it is assume that no change woul occur after amage in the mass matrix, which seems to be reasonable in most real problems. Thus, as it was mentione above, the eigenvalue equations for a amage structure became: where, K u is the amage stiffness matrix; λ i an Φ i are the square of the ith natural frequency an the ith moe shape of the amage structure, respectively. As the number of sensors use to measure moal ata is normally limite an usually is less than the number of DOFs in the finite element moel, either the moel reuction metho shoul be use to match with incomplete measure moe shapes or the measure moe shapes must be expane to the imension of the analytical moe shapes. Because of no convergence in the propose optimization metho using the moal expansion, the first option has been aopte using the Guyan [11] static reuction metho. This metho is employe to conense the mass an stiffness matrices. In this metho, the mass an stiffness matrices, an the isplacement an acceleration vectors in Eq. (1) are partitione into a set of master an slave DOFs: In which, x anx are the isplacement an acceleration vectors, respectively an the subscript m an s are the master an slave coorinates, respectively. To eliminate the slave DOFs, the inertia terms for the secon set of equations are neglecte that leas to: ([ K u ] i [ M]) Φ i i 1, 2,..., n ([ K ] [ M]) Φ i i i1,2,..., n Mmm Mms xm Kmm Kms xm Msm M ss x s Ksm K ss xs (3) (4) 2

4 where: Substitution of Eq. (5) into Eq. (4), followe by permultiplication by [T] T an using incomplete moe shapes, yiels the reuce eigenproblem as: where: xm xs T T 1 in which, K r an M r are the reuce stiffness an mass matrices of amage structure, respectively; λ i,r an Φ i,r are the square of the ith natural frequency an the ith incomplete moe shape of the amage structure in the reuce state, respectively. Applying the incomplete moe shapes an natural frequencies of amage structure to Eq. (7) leas to form the inverse problem of etermining the amage severity parameter. The efinition of a local amage severity parameter in the finite element moel allows estimating amage quantity an location together, since amage ientification is then carrie out at the element level. The problem can be formulate as optimization problem of objective function, while using some transforms as a irect inversion to obtain solution is impossible most of the time. The general statement for the objective function is: In the process of substituting the incomplete measure moal parameters of the amage structure in Eq. (1), a ynamic resiue vector can be efine over each measure moe as follows: where, λ i,r m an Φ i,r m are the square of the ith natural frequency an the ith incomplete moe shape from measurements, respectively; an m is the number of available moe shape for amage etection. Then, if structural amages are etermine correctly, the resiue vector woul be next to in Eq. (11). Therefore, the problem of amage etection can be formulate as an optimization problem. So, the first objective function can be formulate as follows: x I K ss K ([ Kr] i r[ Mr]) Φi r i 1,2,..., n,, m sm m 1 ( ) ) i1 K r T K T T M T T MT r F f ( 1, 2,..., N e ) m m Ri Kr i, r Mr Φi, r ( ) ([ ] [ ]) i1,2,..., m 2 R i ( (5) (6) (7) (8) (9) (1) (11) f (12) 1, 1,..., N e where, represents the Eucliean length of R i (). 3

5 The optimization technique use in this paper was a constraine nonlinear minimization, fmincon, which is available in the MATLAB Optimization Toolbox. This routinely implement Sequential Quaratic Programming (SQP) to minimize the nonlinear cost function was subjecte to linear an nonlinear equality an inequality constraints. SQP converts a nonlinear minimization to a linear minimization using a Hessian matrix of cost function an graient of nonlinear constraints. In aition to the use of fmincon, multiple start points were use to ensure that the global minimum was reache. The global minimum can be seen with repeate answers in the final output. 3. Numerical stuy Consier a three-story plane steel frame for which the finite-element moel consists of nine elements (six columns an three beams) an six free noes, as shown in Fig. 1. For the steel frame consiere, the material properties of the steel inclue Young s moulus E=2 GPa, mass ensity ρ=785 kg/m3. The mass per unit length, moment of inertia, an cross-sectional area of the columns are: m= kg/m, I= m4 an A= m2, respectively; for the beams are: m= kg/m, I= m4 an A= m2. Also, the amage severity in each element is given by the reuction factor liste in Table 1. Fig. 1 Three-story plane frame with the finite element moel In this example, two amage scenarios are represente as the elements with reuction in Young s moulus. The amage severity in each element is given by the reuction factor liste in Table 4.1. In this case, only 6 translational DOFs are selecte as the measure DOFs in the process of amage etection an quantification. Table 1. Damage scenarios for three story plane frame Scenario 1 Scenario 2 Element 2 2% Element 2 2% Element 4 35% Element 9 25% Different initial values of amage severities in the propose metho have been teste to check its convergence. Figures 2 an 3 show the results of amage ientification in the three story plane frame for two amage patterns with zero an 5% initial values, respectively. It epicts that the propose 4

6 metho is a robust an effective metho in etecting an quantifying various amage patterns with ifferent initial values of the amage severities Damage Pattern (1) Damage Pattern (2) Fig. 2 The obtaine results for two amage patterns of the three story frame with % initial values Damage Pattern (1) Damage Pattern (2) Fig. 3 The obtaine results for two amage patterns of the three story frame with 5% initial values 5

7 4. CONCLUSIONS In this paper, a metho has been evelope for etection an estimation of amage in structures base on the incomplete moal ata of the amage structure using an optimization problem. In this metho, constraine nonlinear minimization, fmincon, is use to etermine the amage in structures by optimizing a cost function. For amage etection an estimation, this propose metho was applie to a three story plane frame with one or several amage patterns. The obtaine results inicate that the propose metho is a strong an viable metho to the problem of etection an estimation of amage in the structures. The results reveale high sensitivity of the propose metho to the amage in spite of incomplete measurements. References [1] Johnson E A, Lam H F, Katafygiotis L S an Beck J L 24 Phase I IASC-ASCE structural health monitoring benchmark problem using simulate ata J. Eng. Mech. 13(1) [2] Zingoni A 25 Structural health monitoring, amage etection an long-term performance J. Eng. Struct. 27(12) [3] Rytter, A 1993 Vibration base inspection of civil engineering structures PhD Dissertation Aalborg University, Aalborg. [4] Doebling SW, Farrar CR, Prime MB, Schevitz DW 1996 Damage ientification an health monitoring of structural an mechanical systems from changes in their vibration characteristics: A literature review Technical Report LA-137-MS, Los Alamos National Laboratory, Los Alamos. [5] Kourehli, S S 215 Damage assessment in structures using incomplete moal ata an artificial neural network International Journal of Structural Stability an Dynamics Vol. 15 No DOI: 1142/S [6] Kourehli S S, Ghorati Amiri G, Ghafory-Ashtiany M., Bagheri A 213 Structural amage etection base on incomplete moal ata using pattern search algorithm Journal of vibration an control Vol.19 No.6 pp [7] Kourehli S S, Bagheri A, Ghorati Amiri, G, Ghafory-Ashtiany, M 213 Structural amage etection using incomplete moal ata an incomplete static response KSCE journal of civil engineering Vol.17 No.1 pp [8] Carvalho J, Datta BN, Gupta, A an Lagaapati M 27A irect metho for moel upating with incomplete measure ata an without spurious moes Mechanical System an Signal Processing, Vol. 21 No. 7 pp [9] Huajun L, Fushun L, James HS 28 Employing incomplete complex moes for moel upating an amage etection of ampe structures Sci China Ser E-Tech Sci. Vol. 51 No. 12 pp [1] Chen H 28 Application of regularization methos to amage etection in large scale plane frame structures using incomplete noisy moal ata Engineering Structures Vol. 3 No. 11 pp [11] Guyan RJ 1965 Reuction of stiffness an mass matrices AIAA Journal, Vol. 3, No. 2, pp