Vibration-based damage detection for structural connections using incomplete modal data by Bayesian approach and model reduction technique

Engineering Structures 132 (2017) 260–277

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Vibration-based damage detection for structural connections using incomplete modal data by Bayesian approach and model reduction technique Tao Yin a,⇑, Qing-Hui Jiang a, Ka-Veng Yuen b a b

School of Civil Engineering, Wuhan University, Wuhan, PR China Civil and Environmental Engineering, University of Macau, Macao

a r t i c l e

i n f o

Article history: Received 10 March 2016 Revised 10 November 2016 Accepted 15 November 2016

Keywords: Structural connection Damage detection Bayesian probabilistic approach Finite element model reduction System modal parameters

a b s t r a c t Most of the existing damage detection methods focused on damage along members of the structure without considering possible damage at its connections. Under the Bayesian framework, the finite element (FE) model reduction technique and the system mode concept, this paper presents a practical method for structural bolted-connection damage detection using noisy incomplete modal parameters identified from limited number of sensors. Based on the incomplete modal identification results, the most probable structural model parameters, the most probable system eigenvalues and partial modes shapes together with the associated uncertainties can be identified simultaneously. There are several significant features of the proposed method: (1) it does not require computation of the system mode shapes for the full model due to the FE model reduction technique; (2) matching between measured modes and model predicted modes is avoided in contrast to most existing methods in the literature; and (3) an efficient iterative solution strategy is also proposed to resolve the difficulties arisen from the high-dimensional nonlinear optimization problem for the structural model parameters and the incomplete system modal parameters. Numerical simulations and experimental verifications of a four-storey two-bay boltconnected steel frame and a two-storey laboratory bolted frame, respectively, are utilized to demonstrate the validity and efficiency of proposed methodology. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Over the last few decades, there has been great effort in developing structural health monitoring (SHM) methodologies utilizing vibration measurements [1,2]. Most of the methods in the literature have been verified by various types of structural components or systems, such as truss-type structures [3–5], beam-type structures [6–9], railway sleepers [10,11], plates [12–15], frame structures [16–19], and shear building models [20–24]. The majority of these mentioned damage detection methods are based on modal parameters, where an objective function is usually defined in terms of the discrepancies between the experimental modal parameters and those calculated from a FE model by assuming that the influence of damage on structural mass can be neglected, and it is then minimized for the estimation of the stiffness parameter changes. However, this type of damage identification procedures generally ⇑ Corresponding author at: School of Civil Engineering, Wuhan University, No. 8 East Lake South Road, Wuhan 430072, PR China. E-mail address: [email protected] (T. Yin). http://dx.doi.org/10.1016/j.engstruct.2016.11.035 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

requires solving the eigensystem equation repeatedly to compute the model output and the objective function at least once for each iteration step, which is extremely time-consuming especially for large-scaled structural models. In addition, in the formation of the above objective function, it is necessary to ensure that the measured modes are matched well with the calculated ones by using the modal assurance criterion (MAC) technique but it is difficult in real application due to limited number of measured degrees of freedom (DOFs). Moreover, since damage might cause changes in the order of the modes, such mode matching becomes even more difficult. For this reason, some methods have been proposed to avoid the above-mentioned mode matching problem by introducing the concept of system mode shapes, which is distinct from those calculated from the structural model specified by some given model parameters. Some of these methods employ Rayleigh quotient frequencies, which are derived from the structural model and the system mode shapes, to avoid repeatedly solving the eigensystem equation during damage detection process. Then, the structural model parameters and the system mode shapes are identified

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T. Yin et al. / Engineering Structures 132 (2017) 260–277

simultaneously [20,25,26]. Later, an improved method has been proposed so that the system eigenvalues are also included as additional unknown parameters to be identified based on the incomplete modal data besides of the system mode shapes in order to represent the actual modal frequencies of the structural system [27,28]. It is noted that, in the above-mentioned methods utilizing the concept of system modes, the system mode shapes with respect to the full structural models are required. In real situations, mode shapes are usually measured with incomplete components or missing DOFs. In order to resolve the difficulties arisen from the limited number of measurement channels, the methods usually start from computing the missing components of the mode shapes through mode shape expansion procedure [27]. However, it has been revealed that this would aggregate the modeling error, experimental noise and other sources of uncertainties into the resultant mode shapes [29,30], affecting substantially the damage detection results. On the other hand, the dimension of full system mode shapes and, thus, the number of the unknown parameters becomes extremely large, resulting in unreliable or even unidentifiable damage detection results especially for large-scaled structures. In such circumstances, the FE model reduction method originally developed for the purpose of reducing the computation effort for large-scaled structural models [31– 34], particularly for the dynamic-reduction method [34], becomes a more practical alternative since it does not introduce any error in the transformation process within a certain frequency range [35,36]. Using the dynamic model reduction technique, the unknown full system mode shapes correspond only to the condensed structural model so the dimension of the inverse problem could be significantly reduced, especially efficient for large-scaled complex structural models with a huge number of DOFs. On the other hand, majority of existing damage detection methods concentrated on damage along members of the structure without consideration of connection damages, which frequently occur in structural frames. In this paper, by acknowledging this importance and the aforementioned difficulties, the system mode based damage detection method [27,28] is improved by involving the efficient dynamic model reduction based method [35,36] to become a more practical and workable method to detect structural connection damage for frame-type structures. The proposed method can handle the real situations with very limited number of sensors available and has the particular potential for large-scaled complex structures due to the model reduction strategy. In the proposed method, the uncertainty issues associated with the unknown structural model parameters and system modal parameters are well treated by the Bayesian probabilistic approach. In addition, the FE model reduction technique is also utilized to construct the prior PDF of the incomplete system modal parameters by evaluating the compatibility of the system modal parameters to the reduced structural model. One of the most attractive features of the proposed methodology is that the identification of full system mode shapes and, hence, mode matching are completely avoided. Furthermore, an efficient iterative solution method is also presented to solve the optimization problem for the most probable values of the structural model parameters and the incomplete system modal parameters. After presenting the theoretical development in detail, the proposed methodology is validated and demonstrated thoroughly by a comprehensive set of numerical case studies of a four-storey two-bay bolt-connected steel frame and experimental verification of a laboratory two-storey bolted frame with both single- and double-damage situations.

2. Theoretical developments 2.1. Dynamic reduction of the eigen-system equations Consider a class of structural models M discretized by FE method into N DOFs. Assuming that the change of mass matrix due to possible damage is negligible, the corresponding eigensystem equation is thus given by:

KðhÞ/j ¼ kj M/j

ð1Þ

where kj , /j 2 RN, j = 1, 2, . . . , Nt, are the jth eigenvalue and eigenvector, respectively. Nt is the number of measured modes. The global stiffness matrix K(h) is parameterized by the stiffness scaling parameter vector h ¼ ½h1 ; h2 ; . . . ; hNh T 2 RNh :

KðhÞ ¼ K 

Nh X

hi KðiÞ

ð2Þ

i¼1

where the stiffness scaling parameters h allow the nominal stiffness matrix given by h = h0 in Eq. (2) to be updated based on the identified modal parameters from the structure. K(i), i = 1, 2, . . . , Nh is the contribution of the ith member or substructure to the global stiffness matrix of the FE model; and Nh is the number of unknown stiffness scaling parameters to be identified. By using Eq. (2), Eq. (1) can be rearranged to partition the measured and unmeasured DOFs as follows:

"

Kmm 

PN h

i¼1 hi ½KðiÞ mm PN h Ksm  i¼1 hi ½KðiÞ sm



¼ kj

Mmm Mms Msm Mss

(

Kms 

PN h

i¼1 hi ½KðiÞ ms PN h Kss  i¼1 hi ½KðiÞ ss

f/m gj

)

f/s gj

#(

f/m gj

)

f/s gj ð3Þ

where {/m}j and {/s}j are the measured and unmeasured parts of full mode shape /j for the jth mode, with dimensions Nm and Ns, respectively, and Nm + Ns = N. The lower portion of Eq. (3) can be rewritten as:

f/s gj ¼ Dj f/m gj

ð4Þ

and

Dj ¼ F1 j Gj

ð5Þ

where Dj is the dynamic reduction matrix for the jth mode, and it is the function of eigenvalue kj and the stiffness scaling parameters h, and

Fj ¼ Kss 

Nh X hi ½KðiÞ ss  kj Mss i¼1

Nh X Gj ¼ Ksm  hi ½KðiÞ sm  kj Msm

ð6Þ

i¼1

Thus, for the jth mode, the full mode shape can be represented only by the corresponding measured part as

"

f/m gj f/s gj

#

 ¼

INm Dj

 f/m gj ¼ Tj f/m gj

ð7Þ

where Tj 2 RNNm is the transformation matrix of the jth mode, and INm is the Nm  Nm identity matrix. Substituting Eq. (7) into Eq. (3), and pre-multiplying transpose of Tj to the both sides of the resultant equation set, the eigensystem equation of the reduced FE model corresponding to the Nm measured DOFs is obtained:

KRj f/m gj ¼ kj MRj f/m gj

ð8Þ

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and

KRj ¼ KRj 

Nh X

h

hi KRðiÞ

i¼1

KRj

i

ð9Þ

j

MRj

where and are the reduced global stiffness and mass matrices related to the jth mode, respectively:   Kmm Kms Tj ¼ Kmm þ DTj Ksm þ Kms Dj þ DTj Kss Dj KRj ¼ TTj KTj ¼ TTj Ksm Kss ð10Þ   Mmm Mms MRj ¼ TTj MTj ¼ TTj Tj ¼ Mmm þ DTj Msm þ Mms Dj þ DTj Mss Dj Msm Mss h i and KRðiÞ is the reduced matrix of K(i) corresponding to the jth j

mode:

h

KRðiÞ

i j

" ¼ TTj KðiÞ Tj ¼ TTj

½KðiÞ mm

½KðiÞ ms

½KðiÞ sm

½KðiÞ ss

#

2.2. Bayesian formulation It is assumed herein that there are Nt(Nt  N) measured modes, and these modes are referred to the system modes, i.e., system eigenvalue vector k ¼ ½k1 ; k2 ; . . . ; kNt T and partial mode shape vector h iT /m ¼ f/m gT1 ; f/m gT2 ; . . . ; f/m gTNt , to distinguish them from the corresponding modal parameters calculated from any reduced form of structural model specified by h in Eq. (8). In addition, due to the unavoidable modeling errors, the pair of measured system modal parameters (k and /m) do not satisfy exactly the eigen-system equation of any given reduced structural model specified by h. In order to quantify the uncertainty in the equation errors for each mode with respect to the reduced structural model given in Eq. (8), the eigensystem equation errors are assumed to follow the normal distribution and they are modeled as independent and identically distributed (i.i.d.) random variables. In addition, for simplicity T

purpose, an augmented parameter vector w ¼ ½kT ; /Tm  2 RNt ðNm þ1Þ is introduced for the system and experimental modal parameters. It should be pointed out that one significant merit of the proposed method is that the system mode shapes related to the Nt modes with only Nm measured DOFs are simply the full mode shapes with respect to any given reduced structural model provided in Eq. (8). Therefore, it is not necessary to identify the full system mode shapes corresponding to all N DOFs of the original full structural model. The prior PDF for both the system modal parameter pair w and the stiffness scaling parameters h can be written as [37]:

ð12Þ

and the prior PDF pðhjMÞ is defined as:

ð13Þ

where it is assumed herein that pðhjMÞ follows the normal distribution with mean h0 representing the nominal values of the stiffness scaling parameters with covariance matrix Rh. For the structural damage detection problem investigated in this paper, the prior covariance matrix Rh is taken to be a diagonal matrix with large variances, indicating a non-information prior of pðhjMÞ. On the other hand, the prior PDF for the system modal parameters pðwjh; MÞ is given as [27]:

  1 pðwjh; MÞ ¼ j1 exp  J 1 ðw; hÞ 2

ð15Þ

and Nj is the equation-error vector for the jth mode (j = 1, 2, . . . , Nt):

  Nj ¼ KRj  kj MRj f/m gj 2 RNmt

ð16Þ

In Eq. (15), the amplitude of these equation errors with respect to the reduced structural model are controlled by the prior covariance matrix RN ¼ r2N INmt , and Nmt = Nm  Nt, where INmt denotes the Nmt  Nmt identity matrix and r2N is a prescribed equation-error variance. In such situation, Eq. (15) can be further simplified as: Nt X kNj k2

ð17Þ

j¼1

ð11Þ

pðhjMÞ ¼ N ðh0 ; Rh Þ

h iT T T T J 1 ðw; hÞ ¼ NT R1 N N and N ¼ ðN1 Þ ; ðN2 Þ ; . . . ; ðNN t Þ

T 2 J 1 ðw; hÞ ¼ r2 N N N ¼ rN

Tj

¼ ½KðiÞ mm þ DTj ½KðiÞ sm þ ½KðiÞ ms Dj þ DTj ½KðiÞ ss Dj

pðw; hjMÞ ¼ pðwjh; MÞpðhjMÞ

where j1 is a normalizing constant for ensuring that the integration of the PDF over predefined domain is equal to unity, and the measure-of-fit function J1 is defined by

ð14Þ

where kk denotes the Euclidean norm. It is noted that by using r2N to represent the uncertainties of the equation errors, it is allowed to control the tradeoff between degree of agreement of the measured modal parameters and the modeling error [27]. The measurement data set D is defined as follows: def

D¼

N set n [

o N[ o set n ~ ðiÞ ¼ ~ ðiÞ ~ðiÞ ; / w k m m

i¼1

ð18Þ

i¼1

where Nset is the number of repeatedly measured modal parameter pairs, which can be identified from recorded time-domain responses with sufficient duration by proper modal parameter identification techniques. The identified modal parameters can be expressed as:

~ ¼wþe w

ð19Þ

where the measurement noise vector e 2 RNt ðNm þ1Þ is assumed to follow the normal distribution with zero mean and covariance matrix Re . Then, by employing the Bayes’ theorem, the posterior (or updated) PDF for all the unknown parameters is given as

pðw; hjD; MÞ ¼ j2 pðDjw; h; MÞpðw; hjMÞ

ð20Þ

where j2 is another normalizing factor which ensures that the integration of right hand side of Eq. (20) over the predefined domain is equal to unity; and pðDjw; h; MÞ is the likelihood function:

pðDjw; h; MÞ ¼ N ðw; Re Þ

ð21Þ

where N ðw; Re Þ represents the normal distribution with mean vector w and covariance matrix Re . Therefore, the posterior PDF pðw; hjD; MÞ in Eq. (20) is further assumed to take the following form:

  1 pðw; hjD; MÞ ¼ j3 exp  J 2 ðw; hÞ 2

ð22Þ

where j3 is also a normalizing constant similar to j1 and j2; and the measure-of-fit function J2(w, h) is defined by: T 1 T ~ ~ J 2 ðw; hÞ ¼ r2 N N N þ ðw  wÞ Re ðw  wÞ

þ ðh  h0 ÞT R1 h ðh  h0 Þ

ð23Þ

The most probable values of all the unknown parameters can be estimated by maximizing the posterior PDF pðw; hjD; MÞ in Eq. (22), which is equivalent to minimize the measure-of-fit function J2(w, h) in Eq. (23). In the present paper, instead of directly solving the challenging nonlinear optimization problem for minimizing the measure-of-fit function J2(w, h) in Eq. (23) in high-

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dimensional parameter space, an effect iterative solution strategy is developed for minimizing the function J2 through a sequence of linear and implicit nonlinear optimization problems. 2.3. Proposed solution methodology 2.3.1. Estimation of the most probable values The most probable values of stiffness scaling parameters, system eigenvalues, and partial system mode shapes are estimated one by one in the following. Firstly, the first-order partial derivative of J2(w, h) with respect to the stiffness scaling parameters h can be calculated as follows:

1 @J 2 T T 1 ¼ r2 N ðAh  bÞ C þ ðh  h0 Þ Rh 2 @h

ð24Þ

C ¼ Bh  Ah  A

ð25Þ



 @A @A @A Ah ¼ h; h; . . . ; h 2 RNmt Nh @h1 @h2 @hNh   @b @b @b 2 RNmt Nh Bh ¼ ; ;...; @h1 @h2 @hNh

ð26Þ

where the matrix A 2 RNmt Nh and vector b 2 RNmt are given below

2 h

i

KR f/ g 6 h ð1Þ i1 m 1 6 6 KR f/ g 6 m 2 ð1Þ 2 A¼6 6 .. 6 6h . 4 R i Kð1Þ f/m gNt Nt

h i KRð2Þ f/m g1 h i1 KRð2Þ f/m g2 2

h KRð2Þ

.. i . Nt

  ..

f/m gNt

.



h h

i

3

.. i.

7 7 7 7 7; 7 7 7 5

KRðNh Þ f/m g1 i1 KRðNh Þ f/m g2 2

h

KRðNh Þ

Nt

f/m gNt

b¼

 8  R > K1  k1 MR1 f/m g1 > > >   > > > < KR2  k2 MR2 f/m g2

9 > > > > > > > =

By setting oJ2/oh to be zero, the optimal solution h is readily obtained:

  1 2 T 2 T R1 h  rN C A h ¼ Rh h0  rN C b

ð29Þ

It should be noted that, Eq. (29) is a set of implicit nonlinear equations of h due to the utilization of FE model reduction technique and it can be solved efficiently by a simple yet efficient iterative method [35,36]. Secondly, by calculating the first-order partial derivative of J2(w, h) with respect to the system eigenvalue vector k, the following expression is obtained:

 T   1 @J 2 T  1 T  m  / ÞT R1   ð/ ¼ r2 m e 21 N s  k U  ðk  kÞ Re 11 2 @k

ð30Þ

where the block matrix form for the inverse of covariance matrix Re is partitioned as:

"

1

Re ¼

½R1 e 11  1 Re 21

 

R1 e 1

Re

#

12 ;



R1 e

11

and U 2 RNt Nt is a diagonal matrix, and its the jth (j = 1, 2, . . . , Nt) diagonal element is:

Uj ¼ f/m gTj MRj Rj f/m gj

2 RNt Nt ;

ð33Þ

where the matrix Rj 2 RNm Nm , for j = 1, 2, . . . , Nt, are given in Appendix A. Similarly, the optimization solution of the system eigenvalues k can be obtained by setting @J 2 =@k to be zero, leading to the following implicit equation set:





1 1 2 r2 N U  ½Re 11 k ¼ rN S  Re



11

    R1 ð/ k e 12 m  /m Þ

ð34Þ

and it can also be solved efficiently by the same method used for Eq. (29). Finally, calculating the first-order partial derivative of J2(w, h) with respect to the partial system mode shapes /m gives

 1 @J 2 T  1 T  m  / ÞT R1  ¼ r2  ð/ m N /m V  ðk  kÞ Re e 22 12 2 @/m

ð35Þ

where V is a Nmt dimensional block diagonal matrix, having main diagonal blocks square matrices while off-diagonal blocks are zero matrices. The jth (for j = 1, 2, . . . , Nt) block submatrix is given as:

 2 Vj ¼ KRj  kj MRj

/m ¼

ð28Þ

> > .. > > > > > > .  > >  > > > : KR  k MR f/ g > ; Nt m Nt Nt Nt

ð32Þ

ð36Þ

Setting oJ2/o/m = 0 in Eq. (35), the partial system mode shapes /m is given by the following explicit form:

ð27Þ and

sj ¼ f/m gTj KRj Rj f/m gj



where the mean of Nset pairs of repeatedly identified modal param m ) is utilized as the initial estimates to replace the  and / eters (k unknown system modal parameters.

and

In Eq. (30), the jth component of vector s 2 RNt , for j = 1, 2, . . . , Nt, is given by:

 1 Re 22 2 RNmt Nmt ;

22

ð31Þ





1 r2 N V þ Re

1  1      kÞ þ R1 / Re 21 ðk e 22 m 22

ð37Þ

At this point, the optimization results with respect to the stiffness scaling parameters h, the system eigenvalues k, and the partial system mode shapes /m are obtained through Eqs. (29), (34) and (37), respectively. However, it is still insufficient to obtain the most probable values of the entire set of unknown parameters. In order to get the optimal values for all parameters, a simple yet very efficient solution methodology is proposed to sequentially update the stiffness scaling parameters, system eigenvalues, and partial system mode shapes in an iterative manner. It is worth highlighting that the efficiency of the iterative optimization procedure is due to the analytical expressions of the derivatives with respect to the unknown parameters. This will be demonstrated in the numerical and experimental verification sections. To summarize, the proposed iterative solution strategy for determining the most probable values of stiffness scaling parameters and the system modal parameters is given below. (1) The first step is to assign the initial values for the unknown  m , and K = K(hopt),  /opt ¼ / parameters, i.e., hopt = h0, kopt ¼ k, m where h0 can be set to zeros, representing an undamaged initial stage. Furthermore, the initial values of the system modal parameters are set to be their measured values. (2) The updated stiffness scaling parameters hopt is obtained by solving Eq. (29). (3) By solving Eq. (34) with the same procedure as for the stiffness scaling parameters, the system eigenvalues kopt is updated with the updated stiffness scaling parameters and partial system mode shapes. (4) The updated partial system mode shapes /opt m is achieved by using Eq. (37). (5) Repeat steps (2)–(4) until the incremental change of stiffness scaling parameters falls within the prescribed tolerance. Finally, the most probable values for all the unknown parameters are obtained.

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T. Yin et al. / Engineering Structures 132 (2017) 260–277

2.3.2. Estimation of the uncertainties With above-obtained most probable values for all unknown parameters, the posterior PDF in Eq. (22) can be well approximated by a normal distribution as

pðw; hjD; MÞ ¼ N

h

T

ðwopt Þ ; ðhopt Þ

i T T

the present study, the computational effort of the finite difference method for calculating the Hessian matrix would be very significant, and making this method impractical, if not impossible, for real applications. Therefore, another important feature of the proposed methodology is that the explicit expression of Hessian matrix associated with the most probable parameters is obtained by matrix calculus without resorting to the finite difference method. This will be further verified in the followed verification sections. By calculating the second-order partial derivative of 1/2J2(w, h) defined in Eq. (23) with respect to the system eigenvalue vector k, partial system mode shape vector /m, and stiffness scaling parameters h, respectively, the covariance matrix is given as:

 ð38Þ

; Ropt

which is centred at the optimal parameters [(wopt)T, (hopt)T]T with covariance matrix Ropt equal to the inverse of the Hessian of the function  ln pðw; hjD; MÞ / 1=2J 2 ðw; hÞ evaluated at the corresponding optimal parameters [27], and the covariance matrix Ropt reflects the uncertainties associated with the previously identified most probable parameters. In general, the finite difference technique is the most common method employed to calculate the approximated Hessian matrix of the measure-of-fit function J2 given in Eq. (23), and the inverse of the Hessian matrix evaluated at the optimal solution gives the corresponding covariance matrix for assessing the corresponding uncertainties. However, for high-dimensional parameter space in

71/34

91/51

Ch.4

2 6 Ropt ¼ 4

H11

31 7 H23 5 H33

H12

H13

H22 sym

92/52 34/18 93/53

ðk;/;hÞ¼ðk

74/35

;/

opt

;h

opt

Þ

94/54

17/9

51/27

D16

70

72

Ch.11

16

D17 D18

73

75

Ch.12

33

15/8

49/26

31

D12

64

65/32

66

D19 50

32/17

14

Ch.3

ð39Þ opt

48

D13 D14

67

68/33

69

D15

13/7

47/25 87/47

88/48 30/16 89/49

Ch.9

12 11/6

29

90/50

Ch.10

46

28/15

10

45/24 44

27

D8

Ch.2

58

59/30

60

D9 D10

62/31

61

9/5

63

D11 43/23

83/43

84/44 26/14 85/45

Ch.7

8

Ch.8

25

FE node markers

7/4 6

86/46 42

Semi-rigid connections of beam-column

24/13

41/22

23

D4

Ch.1

52

53/28

54

360×4=1440 cm

FE node numbers

40

D5

D6

56/29

55

57

D7

5/3

39/21

Ch.5

4

82/42

80/40 22/12 81/41

79/39

Ch.6

21 20/11

3/2

Semi-rigid connection of column-base

2 76/36

D1 1/1

38 37/20

y 19

36

77/37

O

D2

18/10

x

D3

78/38 35/19

480×2=960 cm Fig. 1. Two FE models for the four-storey portal frame (FEM1 (nodes marked with blue circles) for generating simulated measurement data; FEM2 (nodes marked with pink squares) for damage identification). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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T. Yin et al. / Engineering Structures 132 (2017) 260–277

and the matrices @Bh =@hi 2 RNmt Nh and @Ah =@hi 2 RNmt Nh are:

where

H11 ¼

1 @ 2 J2 1 @ 2 J2 1 1 ¼ r2 ¼ r2 N W11 þ ½Re 11 ; H12 ¼ N W12 þ ½Re 12 T 2 @k@k 2 @k@/Tm

" # @Bh @2b @2b @2b ¼ ; ;...; @hi @h1 @hi @h2 @hi @hNh @hi

H13 ¼

1 @ 2 J2 1 @ 2 J2 1 ¼ r2 ¼ r2 N W13 ; H22 ¼ N V þ ½Re 22 2 @k@hT 2 @/m @/Tm

and

2

H23 ¼

" #     @Ah @2A @A @2A @A ¼ hþ ;...; hþ @hi @h1 @h1 ith col @hi @hNh @hNh ith col @hi

2

1 @ J2 1 @ J2 1 ¼ r2 ¼ r2 N W23 ; H33 ¼ N W33 þ Rh 2 @/m @hT 2 @h@hT ð40Þ

The diagonal elements of the covariance matrix in Eq. (39) give the posterior variance for the corresponding unknown parameters. In Eq. (40); W11 is a Nt  Nt diagonal matrix; and its jth (j = 1, 2, . . . , Nt) diagonal element is expressed as:

   @R  j f/m gj ½W11 jj ¼ f/m gTj ðRj Þ2 þ KRj  kj MRj @kj

W12

w12 1

0 w12 2 ..

.

3 7 7 7 7 7 5

ð42Þ

w12 Nt

0 where

h     i T R R w12 þ KRj  kj MRj Rj j ¼ f/m gj R j Kj  kj Mj

ð43Þ

In addition, the ith columns (for i = 1, 2, . . . , Nh) of the matrices W13 2 RNt Nh and W23 2 RNmt Nh are given as, respectively:

h   i 1 f/m g1 f/m gT1 Xi1 R1 þ KR1  k1 MR1 @R @hi h   i T i R @R2 R f/m g2 X2 R2 þ K2  k2 M2 @hi f/m g2

2

½W13 ith col

6 6 6 6 ¼6 6 6 4

f/m gTNt

h

XiNt RNt

.. .   @R i þ KRNt  kNt MRNt @hNT f/m gNt

3 7 7 7 7 7 7 7 5

2

½W23 ith col

6 6 6 6 ¼6 6 6 4h

i

h

    i Xi1 KR1  k1 MR1 þ KR1  k1 MR1 Xi1 f/m g1 h     i Xi2 KR2  k2 MR2 þ KR2  k2 MR2 Xi2 f/m g2 .. .

3

   i XiNt KRNt  kNt MRNt þ KRNt  kNt MRNt XiNt f/m gNt

7 7 7 7 7 7 7 5

ð45Þ Xij

where the matrices and oRj/ohi (i = 1, 2, . . . , Nh, j = 1, 2, . . . , Nt) are given in Appendix A. Nh Nh

Furthermore, the expression of the matrix W33 2 R by:

W33 ¼ CðBh  AÞ þ Y

is given

Y 2 RNh Nh is:

@CT @A ðb  AhÞ  CT h @hi @hi

T @Bh @Ah @A @A ¼   ðb  AhÞ  ðBh  Ah  AÞ h @hi @hi @hi @hi

In this section, a four-storey two-bay plane frame (shown in Fig. 1) with standard 10# I-steel columns and beams is employed for verification purpose. The column-base and beam-column connections of the frame are treated as semi-rigid and their rotational stiffness is considered. The four-storey frame is discretized by plane beam elements (each node with two translational DOFs and one rotational DOF) and two FE models with different number of nodes are considered. The first FE model (denoted as FEM1) with 94 nodes and 273 DOFs is utilized only for data generation while the second one (denoted as FEM2) with 54 nodes and 153 DOFs is utilized for connection damage detection using modal data from FEM1. This is done to purposely introduce modeling error to mimic the reality. It’s noted that the various numbers next to the markers shown in Fig. 1 denote the FE node numbers, and two numbers separated by a slash represent the corresponding nodes belonging to FEM1 and FEM2, respectively.

Parameter name

Values

Young’s modulus (E) Mass density (q) Cross-sectional area (A) Moment of inertia (I) Overall height of cross section (H) Width of top and bottom flanges of cross section (b) Web thickness of cross section (tw)

2.00  1011 N/m2 7.85  103 kg/m3 1.43  103 m2 2.45  106 m4 100 mm 68 mm 4.5 mm

Table 2 Cases considered for the four-storey portal frame. Cases

Description

Damaged connection (reduction of rotational stiffness)

1 2

Single damage on CBC Double damage on adjacent BCC Multi-damage on all CBC Multi-damage on all BCC in one storey Multi-damage on CBC and BCC in one column Multi-damage on BCC in adjacent columns Few modes (Mode 6 is removed) Fewer measured DOFs (Ch. 12 is removed) Larger noise (NL = (1%, 4%))

D2 (20%) D5 (15%) & D6 (20%)

3 4 5

ð46Þ

where the ith (i = 1, 2, . . . , Nh) column of the square matrix

½Yith col ¼

3. Numerical case studies

Table 1 Sectional and material properties of the four-storey portal frame.

ð44Þ and

where i = 1, 2, . . . , Nh, and the derivation of the second-order partial derivatives o2b/ohiohk and o2A/ohkohi (k = 1, 2, . . . , Nh) are also given in Appendix A.

3.1. Introduction of the four-storey two-bay frame example

The matrix W12 2 RNt Nmt in the submatrix H12 is given by

6 6 6 ¼6 6 4

ð49Þ

ð41Þ

where the expressions of Rj and @Rj =@kj are provided in Appendix A.

2

ð48Þ

6 7 8

ð47Þ

9

D1 (20%) & D2 (15%) D4 (20%) & D5 (15%) (25%) D2 (20%) & D5 (15%) (20%) & D17 (25%) D4 (20%) & D5 (15%) (20%) & D12 (15%) & Same as Case 6

& D3 (25%) & D6 (20%) & D7 & D9 (25%) & D13 & D8 (25%) & D9 D13 (20%)

Note: CBC and BCC represent column-base and beam-column connection, respectively.

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In this paper, the semi-rigid connections for both column-base and beam-column connections are simulated by beam elements of very short length with smaller flexural rigidity as compared to the regular beam and column components, and the connection damage is simulated by reducing the flexural stiffness of the short beam elements. Referring to Fig. 1, there are 19 potential damaged connections, denoted by D1, D2, . . . , and D19. It is noted that the semi-rigid column-base connections (D1–D3) are simulated by three short beams, respectively, and lower end of each short beam is fixed to the base. Thus, although the DOFs associated with these base nodes are eliminated from the global system matrices to consider the boundary conditions for both FEM1 and FEM2, the semirigid properties of D1–D3 are kept for possible damage detection of each column-base connection. The sectional and material properties of the frame are shown in Table 1, and the flexural stiffness of the short beam elements with length 0.01 m for representing the beam-column and column-base connections is assumed to be 1.47  104 N m2. The measurement points are the horizontal DOFs

D17 D18

D16

D19

D17 D18

D16

(a)

for each floor and the vertical DOFs for middle of beams as shown in Fig. 1, and the number of measurement DOFs is 12. The measurement includes the eigenvalues and partial mode shapes at the measured DOFs of the first six modes under both healthy and damaged situations. They are obtained from FEM1 with measurement error assumed to be an i.i.d. Gaussian white noise of zeromean. The noise level is 1% and 2% for the eigenvalues and partial mode shapes, respectively. Subsequently, damage detection will be proceeded with FEM2. Nine damage cases are considered for the four-storey steel frame example and they are shown in Table 2. The configurations of the first six cases are shown in Fig. 2, and damage configurations of the last three cases are the same as Case 6 except other control parameters. Specifically, Case 1 is a single-damage case, in which the middle column-base connection is the only damaged member with 20% flexural rigidity reduction for the short beam element D2. Case 2 is a double-damage case, in which the adjacent beamcolumn connections D5 and D6 are simulated as damaged ones

D19

31

D17 D18

D16

(b)

D19

(c) 31

31

D12

D13 D14

D15

D12

D13 D14

D15

D12

D13 D14

D15

D8

D9 D10

D11

D8

D9 D10

D11

D8

D9 D10

D11

D4

D5

D7

D4

D5

D7

D4

D5

D7

D6

D2

D1

D17 D18

D16

D3

D19

(d)

D6

D2

D1

D17 D18

D16

D3

D19

D2

D1

D17 D18

D16

D3

D19

(f)

(e) 31

D6

31

31

D12

D13 D14

D15

D12

D13 D14

D15

D12

D13 D14

D15

D8

D9 D10

D11

D8

D9 D10

D11

D8

D9 D10

D11

D4

D5

D7

D4

D5

D7

D4

D5

D7

D1

D6

D2

D3

D1

D6

D2

D3

D1

D6

D2

D3

Fig. 2. Damage cases considered for the four-storey portal frame (damaged connections are marked with red circles): (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; (f) Cases 6, 7, 8 and 9. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 3. Identified results obtained by the proposed method for all cases: (a)–(i) for Case 1–Case 9.

Table 3 Comparison of COV values between proposed method and finite difference technique for all cases.

h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 h17

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

Case 9

0.2993 (0.2979) 0.1343 (0.1343) 0.3006 (0.2947) 0.1556 (0.1531) 0.2144 (0.2152) 0.2184 (0.2174) 0.1551 (0.1551) 0.1409 (0.1397) 0.1860 (0.1860) 0.1863 (0.1854) 0.1399 (0.1395) 0.2278 (0.2277) 0.2585 (0.2585) 0.2586 (0.2584) 0.2273 (0.2272) 0.1363 (0.1353) 0.1742 (0.1738)

0.2970 (0.2946) 0.2170 (0.2176) 0.2947 (0.2929) 0.1641 (0.1550) 0.1876 (0.1889) 0.1807 (0.1814) 0.1533 (0.1542) 0.1400 (0.1384) 0.1826 (0.1821) 0.1831 (0.1816) 0.1384 (0.1376) 0.2256 (0.2257) 0.2508 (0.2513) 0.2432 (0.2422) 0.2136 (0.2129) 0.1312 (0.1302) 0.1689 (0.1685)

0.2655 (0.2604) 0.1932 (0.1962) 0.2645 (0.2516) 0.1597 (0.1585) 0.2143 (0.2144) 0.2134 (0.2114) 0.1561 (0.1562) 0.1397 (0.1382) 0.1812 (0.1811) 0.1819 (0.1811) 0.1381 (0.1378) 0.2232 (0.2232) 0.2510 (0.2510) 0.2493 (0.2490) 0.2198 (0.2197) 0.1324 (0.1312) 0.1686 (0.1683)

0.3415 (0.3366) 0.2350 (0.2360) 0.3387 (0.3362) 0.1495 (0.1432) 0.2005 (0.2036) 0.2048 (0.1935) 0.1362 (0.1359) 0.1537 (0.1522) 0.2010 (0.2001) 0.2000 (0.1985) 0.1510 (0.1504) 0.2349 (0.2344) 0.2636 (0.2632) 0.2667 (0.2664) 0.2386 (0.2384) 0.1445 (0.1434) 0.1810 (0.1807)

0.3016 (0.3000) 0.1360 (0.1364) 0.3028 (0.2969) 0.1590 (0.1524) 0.1927 (0.1929) 0.2209 (0.2200) 0.1568 (0.1570) 0.1441 (0.1359) 0.1522 (0.1507) 0.1867 (0.1852) 0.1401 (0.1400) 0.2243 (0.2207) 0.2165 (0.2148) 0.2558 (0.2556) 0.2267 (0.2262) 0.1361 (0.1328) 0.1401 (0.1393)

0.3083 (0.3065) 0.2166 (0.2182) 0.2979 (0.2954) 0.1454 (0.1405) 0.1980 (0.1994) 0.2262 (0.2251) 0.1658 (0.1656) 0.1165 (0.1125) 0.1560 (0.1554) 0.1936 (0.1922) 0.1470 (0.1461) 0.2030 (0.1995) 0.2190 (0.2171) 0.2601 (0.2598) 0.2305 (0.2298) 0.1386 (0.1376) 0.1764 (0.1760)

0.3747 (0.3729) 0.2662 (0.2660) 0.3651 (0.3631) 0.5790 (0.5767) 0.6521 (0.6501) 0.7238 (0.7210) 0.6725 (0.6698) 0.2837 (0.2812) 0.3301 (0.3273) 0.4079 (0.4072) 0.3717 (0.3710) 0.2457 (0.2418) 0.2605 (0.2582) 0.3074 (0.3070) 0.2792 (0.2783) 0.2445 (0.2431) 0.2549 (0.2546)

0.3511 (0.3479) 0.2390 (0.2400) 0.3383 (0.3349) 0.1518 (0.1465) 0.2023 (0.2077) 0.2289 (0.2323) 0.1679 (0.1698) 0.1198 (0.1155) 0.1548 (0.1607) 0.1906 (0.1968) 0.1473 (0.1490) 0.2097 (0.2060) 0.2288 (0.2270) 0.2798 (0.2789) 0.2513 (0.2501) 0.2007 (0.1985) 0.2124 (0.2031)

0.5589 (0.5490) 0.3761 (0.3800) 0.5360 (0.5297) 0.2461 (0.2288) 0.3065 (0.3121) 0.3522 (0.3487) 0.2666 (0.2652) 0.1961 (0.1846) 0.2412 (0.2409) 0.3025 (0.2983) 0.2383 (0.2353) 0.3590 (0.3503) 0.3746 (0.3702) 0.4499 (0.4495) 0.4104 (0.4087) 0.2339 (0.2309) 0.2745 (0.2739) (continued on next page)

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Table 3 (continued)

h18 h19

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

Case 9

0.1741 (0.1736) 0.1348 (0.1343)

0.1704 (0.1701) 0.1335 (0.1330)

0.1685 (0.1680) 0.1306 (0.1301)

0.1805 (0.1801) 0.1425 (0.1420)

0.1751 (0.1745) 0.1370 (0.1361)

0.1739 (0.1734) 0.1354 (0.1350)

0.2520 (0.2511) 0.2363 (0.2350)

0.2265 (0.2246) 0.2566 (0.2567)

0.2751 (0.2742) 0.2342 (0.2327)

Note: the COV values of finite difference method are given in the parenthesis.

Fig. 4. Marginal cumulative distribution of stiffness scaling parameter for the connection D1.

at the same time, with 15% and 20% reduction of flexural rigidity, respectively. This case is designed to investigate the feasibility of the proposed methodology in detecting the adjacent damage with respect to the same column. Case 3 is a triple-damage case, and all column-base connections D1, D2 and D3 in the base floor are considered as the damaged ones simultaneously, with 20%, 15% and 25% damage extents, respectively. This case is intended to test the possibility of the proposed methodology to identify all the column-base connection damages with different extents at a time. Case 4 and Case 5 are both multi-damage cases, and they are utilized to verify the proposed methodology for identifying connection damage in the same beam and column, respectively. Four connections (D4–D7) on the first floor are considered to be damaged in Case 4, and five connections (D2, D5, D9, D13 and D17) along the middle column are considered as damaged ones for Case 5. Case 6 is also a multi-damage case, where six damaged beamcolumn connections are located in two adjacent columns. As compared to the previous damage cases, this case is employed to verify the ability of proposed methodology in a more difficult situation for detecting larger number of damaged connections with different severity. The damage configuration of the last three cases, i.e., Cases 7, 8 and 9, are the same as that of Case 6 except some control parameters. Among these four cases, Case 6 is considered as a baseline with its default parameters to serve as a comparison with the last three cases. More specifically, referring to Table 2, the only difference between Case 6 and Case 7 is that only the first five modes are used in the latter so as to investigate the effects of number of modes. Compared with Case 6, the last measurement channel (i.e., Ch. 12) in Case 8 is removed. Since the measured mode shapes are identified with lower accuracy in practice than the natural frequencies, Case 9 is designed to investigate the effects of higher noise levels on mode shapes from 2% to 4%.

Fig. 5. Iteration history of the proposed iterative solution approach and fminsearch algorithm for Case 6: (a) iteration history of proposed solution approach; (b) iteration history for each stiffness scaling parameter; and (c) iteration history of fminsearch algorithm.

T. Yin et al. / Engineering Structures 132 (2017) 260–277

3.2. Probabilistic damage detection results Fig. 3 shows the identified damage results by the proposed methodology. In general, the identified stiffness reductions for all cases by the proposed methodology are very close to the actual damage extents. For example, referring to the results of Case 1 shown in Fig. 3(a), the identified damage extent of the single damage at column-base connection D1 is about 0.19 which is very close to the actual damage extent 0.2. For the double-damage case, the identification results of Case 2 are shown in Fig. 3(b), it is seen that the locations of two damaged connections are well identified, and identified stiffness reductions at the two locations are about 0.125 and 0.162, respectively. They are also close to the corresponding pre-defined actual damage extents, i.e., 0.15 and 0.20. For the multi-damage cases, i.e., Case 3–Case 6, the identified damage locations together with corresponding extents for either adjacent or separated damaged connections are all very close to the actual values. As for the last three cases, i.e., Cases 7–9, it is observed that although the damaged connections are successfully identified, the degree of agreement between identified stiffness reductions and corresponding actual values is less than that of Case 6. This is not surprising since these last three cases are intended to investigate

269

the feasibility of proposed methodology in more difficult situation, such as fewer modes, fewer measured DOFs, and larger noise level, as compared to the nominal case (i.e., Case 6), and the corresponding identification accuracy for the last three cases is expected to be relatively lower. In conclusion, the feasibility and accuracy of the proposed methodology is clearly verified. Next, the coefficient of variation (COV) is adopted to illustrate the uncertainties associated with the identified results. Table 3 shows the identified COV values associated with all damage cases for the four-storey bolt-connected frame. For comparison purpose, the finite difference technique is employed to calculate the approximated Hessian matrix of the objective function J2 (i.e., Eq. (23)), and the inverse of the Hessian matrix at the optimal solution gives the corresponding covariance matrix, which is then compared with the covariance matrix provided by the proposed methodology in Eq. (39). As shown in Table 3 that, the COV values calculated by the proposed methodology are extremely close to those obtained by the finite difference approximation, which are shown in the parenthesis in the same table, and the accuracy of the proposed methodology for estimating the uncertainties is verified. It should be point out that for calculating the Hessian matrix at the optimal values, the computational cost for the finite difference technique is

Fig. 6. Experimental two-storey bolt-connected steel frame: (a) bolt-connected frame; (b & c) beam-column connection detail (middle & top); (d) column-base connection detail; (e) tightening bolts by the wrench.

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much more expensive than the proposed method, and computational time of the former is about 100 times as that of the latter. In addition, it can be observed from all cases from Table 3 that, the COV values associated with the column-base connections are relatively larger than those of the beam-column connections. It implies that for the frame-type structure with semi-rigid connections, the stiffness parameters of column-based connections are usually more uncertain than those of the beam-column ones. Furthermore, it can be seen from this table that, for the same floor, the COV values of beam-column connections at either side of the middle column and those at the left and right columns are all close to each other, moreover, the COV values of the former are relatively larger than the latter. For instance, taking the four beam-column connections, i.e., D12–D15, in Case 1 as an example, COV values of h13 and h14 are 0.2585 and 0.2586, respectively, and they are very close to each other, so are h12 and h15. The COV values of h13 and h14 are relatively larger than those of h12 and h15. This means that stiffness scaling parameters of beam-column connections in the two sides of the middle column are more uncertain than those on the two side columns. The same phenomenon can also be observed for other floors in other cases. Fig. 4 shows the marginal cumulative distribution of the identified stiffness scaling parameter h1 for all considered cases. It is clearly seen from this figure that, the uncertainties associated with h1 in Case 9 and Case 3 are the largest and smallest ones, respectively, while those in the rest of seven cases are similar and falls somewhere between the previous two cases, and this phenomenon can also be verified from the COV values in the first row of Table 3. 3.3. The convergence of the proposed methodology

D5

Ch.4

23

24

D6 18 30

29

Ch.7

8

Ch.8

17

Ch.3 7

16

6

15

D3

Ch.2

20

19

21

D4

5

14 28

27

Ch.5

4

Ch.6

13

Ch.1 3

12

y

11

2 25

4. Experimental verifications

22

9

115×2=230 cm

In addition, the convergence of the proposed iterative solution algorithm presented in this paper is investigated, and the predefined convergence criterion is set to be 1.0  103 for the present numerical case studies. Taking Case 6 as an example again, Fig. 5 shows the convergence history of the proposed iterative solution strategy. In addition, for comparison purpose, iteration history for minimization of the measure-of-fit function given in Eq. (23) with the Nelder-Mead algorithm, a very efficient unconstrained optimization method, is also presented, as shown in Fig. 5(c). For the iterative solution method proposed in the present paper, it is very clear from Fig. 5(a) and (b) that the convergence procedure is very fast, and the number of iterations required to achieve the solution is about ten for this case. Moreover, the estimated stiffness scaling parameters after the first iteration are already very close to the solution, as observed from Fig. 5(b). Although not shown in this paper, the similar phenomena are also observed for the other cases. On the other hand, for using the conventional numerical optimization algorithm, the convergence rate is extremely slow especially for large number of iteration steps as illustrated in Fig. 5(c). In spite of the large number of iterations used, the found optimal solution is still far from the actual value, since the objective function given in Eq. (23) for all unknown parameters are highly nonlinear in high-dimensional parameter space, making the ordinary optimization procedure for this type of problem to be, if not impossible, extremely inefficient.

frame. The span of the frame is 1.6 m and the height is 1.15 m for each storey. Both the beams and columns are built with standard 10# I-steel as in the numerical case studies, and the sectional and material properties of the two-storey frame are also shown in Table 1. Fig. 6(b) and (c) shows the connection details of the middle and top beams to the column, respectively. The beams and columns are connected through two pairs of angle steel with different sizes. The pair of larger-sized angle steel (with 4 cm leg length) is used to fix the top and bottom flanges of beam to the inner flange of column, respectively, and the pair of relatively smaller-sized angle steel (with 3 cm leg length) is employed to connect the beam web with the column inner flange. Fig. 6(d) illustrates the columnbase connection detail, and each column is firstly welded to a small steel plate and then fixed to the ground by four bolts. All bolts in beam-column and column-base connections are tightened by a wrench as shown in Fig. 6(e). The column-base and beamcolumn connections of the frame are considered as semi-rigid and they are treated in the same way as in the previous numerical simulations. Fig. 7 shows the FE model of the experimental two-storey boltconnected frame utilized for damage detection. By using the same plane beam element as in the previous numerical example, the frame is discretized into a FE model of 30 elements and 30 nodes with 84 DOFs. Six potential damaged connections, i.e., two column-base connections and four beam-column connections denoted by D1, D2, . . . , and D6, respectively, are also shown in Fig. 7. In addition, as shown in the same figure, eight measurement channels are utilized, and the first and last four sensors are placed on the column and beam, respectively. Chs. 1–4 are used to measure the horizontal vibration of the left column, while Chs. 5–8 are employed to monitor the vertical motion of the two beams. The initial FE model representing the healthy status would be utilized to identify the connection damage with experimental modal data. Two cases of connection damage are considered for the laboratory two-storey bolt-connected frame. As shown in Fig. 8, the schematic diagrams for both cases are shown in Fig. 8(a) and (c), while the associated practical configuration before and after damage (i.e.,

D1

O

x

26

D2 10

1

4.1. Introduction of the laboratory two-storey frame In this section, a laboratory two-storey bolt-connected steel frame (Fig. 6) is utilized to verify the proposed methodology. Fig. 6(a) shows the configuration of the target laboratory steel

160 cm Fig. 7. FE model of the experimental two-storey bolt-connected steel frame for connection damage detection.

T. Yin et al. / Engineering Structures 132 (2017) 260–277

271

Fig. 8. Damage cases considered for the laboratory two-storey bolt-connected steel frame: (a and b) Case 1 and damage configuration; (c and d) Case 2 and damage configuration.

connection bolt loosening) is demonstrated in Fig. 8(b) and (d). Specifically, the first damage case is a single connection damage on D3, where the two bolts (marked in blue1 circle shown in Fig. 8(b)) within the top part of the larger-sized angle steel pairs are loosen. To consider a more realistic but difficult situation, the latter case is a double-damage case, where both connections denoted by D3 and D5 are chosen as damaged ones simultaneously (referring 1 For interpretation of color in Figs. 3, 8 and 13, the reader is referred to the web version of this article.

to Fig. 8(c)), and the bolts marked in blue circles shown in Fig. 8(d) with regarding to the lower and upper beam-column connections are loosen by the wrench. 4.2. Modal parameter identification and FE model updating The main experimental instruments utilized for the laboratory two-storey frame are shown in Fig. 9. The impact hammer is employed to excite the frame as shown in Fig. 9(a), and the excitation point is fixed and chosen to avoid the stationary nodes of the

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T. Yin et al. / Engineering Structures 132 (2017) 260–277

Fig. 9. Main experimental equipments: (a) Impact hammer; (b) MPS-140801 data acquisition software and ERA modal parameter identification program; (c) MPS-140801 dynamical signal acquisition box with 8 channels; (d and e) sensors attached on steel beam and column with magnetic bases, respectively.

first few modes. The dynamic responses are acquired by the accelerometers attached to both column and beam through magnetic bases as shown in Fig. 9(d) and (e), and then collected by the 8-channeled MPS-140801 dynamical signal acquisition box together with data acquisition software (referring to Fig. 9 (b) and (c)). It should be pointed out that excitation force time history is not measured, and collected ambient free vibration responses are analyzed by the ERA method programmed by MATLAB. The first four identified modes under healthy and damaged status are employed for finite element model updating and subsequent damage identification procedures. A series of repeated free vibration testing are carried out for the laboratory two-storey bolt-connected steel frame under both healthy and damaged situations, where the ERA method is employed for modal parameter identification. Ten sets of measured modal parameters are finally obtained for each case, and the associated standard deviations estimated from the ten modal data sets are within the range of 1% and 5% for the eigenfrequencies and mode shapes, respectively. However, it should be realized that the estimated standard deviation varies from method to method used for modal parameter identification and it is also significantly affected by many other practical factors including the operator’s experience. Hence, the obtained standard deviation results presented here are only confined to the forgoing collected modal data with respect to the particular experimental configuration, and they are adopted for demonstration purpose only. The standard deviations of the measured eigenfrequencies and mode shapes for the present experimental case studies are set to be 1% and 5% accord-

ingly for the measured eigenfrequencies and mode shapes; meanwhile, the mean values of estimated from these sets of repeatedly measured modal parameters are also utilized in the following illustration. Fig. 10 shows the measured mode shapes of the first four modes under healthy and two damaged situations, with the undeformed shape of the two-storey frame plotted for reference. As expected, the damage-induced changes due to the connection bolt loosening in mode shape are very small, and it is very difficult, if possible, to directly identify the connection damage from the measured changes in mode shapes. This is especially difficult if the temperature effects are considered for real applications [38]. Table 4 shows the natural frequencies calculated from the initial FE model (see Fig. 7) and identified from the modal tests and the numbers in the parenthesis give the percentage difference between the FE calculated and experimental natural frequencies in the healthy status. It is clear that the frequency differences for the first four modes are obvious, and the maximum difference in the first mode is nearly 30%. It implies that the initial FE model was insufficient to capture the dynamic behavior of the target frame in its healthy situation, and FE model updating is necessary. Six model parameters are selected for adjustment and they are the Young’s modulus, mass density, sectional area and second moment of inertia of the steel I-beam, and rotational stiffness of beamcolumn and column-base connections, respectively. The rotational stiffness of all beam-column connections is treated as a single parameter and so is that of the column-base connections. The initial structural FE model is updated by utilizing the measured

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2.5

2.5

(b)

2

2

1.5

1.5

y-axis

y-axis

(a)

1 0.5

1 0.5

0 -0.5

0

0.5

1

1.5

0 -0.5

2

0

x-axis

2.5

2.5

1.5

2

1.5

2

(d)

2

y-axis

2

y-axis

1

x-axis

(c)

1.5

1.5

1

1

0.5

0.5

0 -0.5

0.5

0

0.5

1

1.5

2

0 -0.5

x-axis

0

0.5

1

x-axis

Fig. 10. Experimental mode shapes of the two-storey bolt-connected frame (red line with circle marker: healthy state; green line with triangle marker: Case 1; blue line with square marker: Case 2): (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 4 Experimental and FE natural frequencies (Hz) for the laboratory two-storey bolt-connected steel frame. Mode

1 2 3 4

FE model

Experiment

Initial (Error %)

Updated (Error %)

Healthy

Case 1

Changes (%)

Case 2

Changes (%)

22.85 (29.83) 90.44 (16.29) 163.02 (9.50) 177.08 (10.81)

17.82 (1.25) 77.63 (0.18) 148.93 (0.04) 159.75 (0.03)

17.60 77.77 148.87 159.80

16.70 77.43 148.76 157.01

5.11 0.44 0.07 1.75

16.16 75.05 145.80 156.16

8.18 3.50 2.06 2.28

modal parameters under healthy status, and after that, the natural frequencies are recalculated from the refined FE model and listed in the 3rd column of Table 4. It is obvious that the model predicted natural frequencies are very close to the measured ones in the healthy status, and the maximum discrepancy is only about 1% for the first mode. The refined FE model would be consequently used for identifying the connection damage of two-storey boltconnected frame. 4.3. Probabilistic damage detection based on measured modal data By utilizing the measured modal parameters on damaged cases, the optimal values of stiffness reduction of connections and associated uncertainties for both cases identified by the proposed methodology are shown in Fig. 11. It is clearly found from Fig. 11 (a) and (c) that the connection damages in both cases are successfully identified by the proposed methodology and the identified stiffness reductions for connection D3 in both cases are close to each other, implying the consistency of the measured modal parameters for the two damage cases. In addition, the uncertainties represented by COV values associated with the optimal values are given in Fig. 11(b) and (d). The uncertainties of two column-base

connections are relatively larger than those of the beam-column connections, which is consistent with the previous numerical simulation results. The COV values calculated by the finite difference technique are also provided and the two sets of results are very close. Similar to the numerical simulation, the finite difference technique employed for calculating the Hessian matrix are extremely time-consuming as compared to the proposed method for directly calculating the Hessian matrix by Eq. (39). In addition, Fig. 12 shows the convergence history of the proposed iterative solution algorithm for both cases, and it is clear that the convergence is very fast, and the number of iterations required to achieve the solution are both within a few steps for Cases 1 and 2, respectively, implying the efficiency of the proposed iteration solution strategy in laboratory situation. Furthermore, for the laboratory two-storey bolt-connected steel frame, besides of finding the optimal model parameters (i.e., damage identification results), taking Case 1 as an example, comparison of the variance between identified most probable system mode shapes and originally measured mode shapes corresponding to all measured DOFs are also shown in Fig. 13. It is interesting that, for all the first four modes involved in the damage detection procedure, the red dash line representing the variance of the most prob-

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Fig. 11. Identified connection damage results by the proposed methodology: (a and b) Identified connection damages associated COV values for Case 1; (c and d) identified damages and associated COV values for Case 2.

Fig. 12. Iteration history of the proposed iterative solution approach: (a) Iteration history for Case 1 and (b) iteration history for Case 2.

able system mode shapes are all below the blue solid line denoting the variance of the measured mode shapes, indicating that the uncertainties associated with the obtained most probable mode shapes stay at a relatively lower level. This is expected from the Bayesian point of view. As mentioned in the previous section, the purpose of introducing the concept of system mode shapes is that

they can be used to represent the actual mode shapes of the structural systems for avoiding mode matching between measured and model-calculated mode shapes. Referring to Eq. (19), the measured mode shapes are considered as observed components of the system mode shapes involving the measurement noise effect, and thus they could be reasonably utilized as the initial estimation of

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T. Yin et al. / Engineering Structures 132 (2017) 260–277

Fig. 13. Variance of the identified most probable system mode shapes for Case 1: (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4.

the system mode shapes at the beginning of proposed iterative solution strategy. By using the proposed Bayesian methodology, the updated or most probable system mode shapes are expected to be more representative of the actual structure system than their initial estimation, mainly reflected by the reducing of associated uncertainties. 5. Conclusions This paper develops a probabilistic methodology based on FE model reduction technique and Bayesian inference for bolted connection damage detection of steel frame structures. In the proposed method, only noisy incomplete modal parameters with limited number of sensors are utilized, and the main feature is that the method does not require the full system mode shapes and mode matching. Meanwhile, an efficient iterative solution strategy is also developed for calculating the most probable model as well as system modal parameters with associate confidence level. Numerical simulations are carried out for a four-storey two-bay bolt-connected steel frame with a comprehensive set of cases. The obtained results show that, for various configurations and extents of connection damage, the proposed methodology successfully estimates the structural model parameters, the system eigenvalues and partial modes shapes with associated uncertainties efficiently. Note that, the importance of the numerical case studies is to demonstrate the capability of the proposed methodology in detecting various connection damage situations and estimating associated uncertainties efficiently. On the other hand, experimental case studies with both single- and double-damage situations are also conducted for a laboratory two-storey bolt-connected steel frame to further verify the proposed methodology. The damage is simulated by loosening only small part of bolts for beam-column

connections. The identified results are very encouraging, and fabricated connection damage and associated uncertainties in both cases is successfully identified. This implies that the proposed methodology is capable to detect connection damage due to bolt loosening. In addition, results show that the bolt loosening has substantial influence on the connection stiffness of the bolted joint for the frame-type structure so attention should be paid in the design and service stages. Acknowledgments The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 51208390) and National Basic Research Program of China (973) Program (Grant No. 2011CB013506). Appendix A Defining Aji and bj (i = 1, 2, . . . , Nh, j = 1, 2, . . . , NT) as below

h i   Aji ¼ KRðiÞ f/m gj and bj ¼ KRj  kj MRj f/m gj j

ðA1Þ

the partial derivatives of A and b with respect to the ith stiffness scaling parameters hi (i = 1, 2, . . . , Nh) in Eq. (26) are given as:

@Aji ¼ @hi and

@bj ¼ @hi

h i @ KRðiÞ @hi

j

f/m gj ¼

  @ KRj  kj MRj @hi

! @TTj @Tj f/m gj KðiÞ Tj þ TTj KðiÞ @hi @hi

f/m gj ¼

! @KRj @MRj f/m gj  kj @hi @hi

ðA2Þ

ðA3Þ

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T. Yin et al. / Engineering Structures 132 (2017) 260–277

where

and

@KRj @TTj @MRj @TTj @Tj @Tj ¼ KTj þ TTj K and ¼ MTj þ TTj M @hi @hi @hi @hi @hi @hi

ðA4Þ

The matrix Rj 2 RNm Nm (j = 1, 2, . . . , Nh) in Eqs. (32), (33), (41), (43) and (44), is given as

@MRj Rj ¼  kj  MRj @kj @kj @KRj

ðA5Þ

and

with respect to the jth eigenvector kj are given by

Nh @KRj X @TTj @Tj ¼  hi KðiÞ Tj þ TTj KðiÞ @kj @kj @k @kj j i¼1

@KRj

!

ðA7Þ

2 R @ 2 MRj @MRj @Rj @ Kj ¼  k  2 j @kj @kj @k2j @k2j

ðA8Þ KRj ,

KRj

and

MRj

h i @ 2 KRðkÞ

@k2j

¼

Nh X @ 2 TTj @TTj @Tj @ 2 Tj hi KðiÞ Tj þ 2 KðiÞ þ TTj KðiÞ 2 2 @kj @kj @kj @kj i¼1

@ 2 MRj @k2j

@ 2 TTj

¼

@k2j

KTj þ 2

@ 2 TTj @k2j

¼

T @ 2 TTj @TTj @Tj @Tj @Tj @ 2 Tj KðkÞ Tj þ KðkÞ þ KðkÞ þ TTj KðkÞ @kj @hi @kj @hi @hi @kj @kj @hi

ðA16Þ

! @ 2 KRj @ 2 MRj f/m gj  kj @hi @hk @hi @hk h i @ 2 KRðiÞ @2A j ¼ f/m gj @hi @hk @hi @hk

@2b ¼ @hi @hk

@ 2 KRj @ 2 TTj @TTj @Tj @TTj @Tj @ 2 Tj ¼ KTj þ K þ K þ TTj K @hi @hk @hi @hk @hi @hk @hk @hi @hi @hk

! ðA9Þ

@TTj @Tj @ 2 Tj K þ TTj K 2 ; @kj @kj @kj

MTj þ 2

@TTj @Tj @ 2 Tj M þ TTj M 2 @kj @kj @kj

@MRj Xij ¼  kj @hi @hi @KRj

ðA11Þ

where Nh @KRj X @TTj @Tj ¼  hk KðkÞ Tj þ TTj KðkÞ @hi @hi @h @hi i k¼1

@KRj

!

h i  KRðiÞ

j

ðA12Þ

h i @ 2 KRðnÞ

j

¼

T @ 2 TTj @TTj @Tj @Tj @Tj @ 2 Tj KðnÞ Tj þ KðnÞ þ KðnÞ þ TTj KðnÞ @hi @hk @hi @hk @hk @hi @hi @hk ðA19Þ

partial derivatives of Tj, including @Tj =@kj , oTj/ohi, @ 2 Tj =@k2j , @ 2 Tj =@kj @hi and o2Tj/ohiohk are required beforehand. These expressions of Tj can be related to those of Dj by Eq. (7), and the corresponding derivatives of Dj is given below. The partial derivatives are (for i = 1, 2, . . . , Nh, j = 1, 2, . . . , NT):

 @Dj @Fj @Gj ¼ F1 ¼ F1 Dj þ j j ðMss Dj þ Msm Þ; @kj @kj @kj

    @Dj @Fj @Gj ¼ F1 ¼ F1 Dj þ KðiÞ ss Dj þ KðiÞ sm j j @hi @hi @hi

and the second-order ones are given by (for k = 1, 2, . . . , Nh):

 @Fj @Dj @Dj 1 ¼ F M  ¼ 2F1 ss j j Mss @kj @kj @kj @k2j



 2  @Dj  @Dj @ Dj @Fj @Dj @Fj @Dj ¼ F1 ¼ F1 þ KðiÞ ss þ KðkÞ sm j j @hi @hk @hi @hk @hk @hi @hk @hi



 2  @ Dj @Dj @Fj @Dj @Dj @Dj ¼ F1 ¼ F1 Mss  Mss þ KðiÞ ss j j @kj @hi @hi @hi @kj @hi @kj

ðA13Þ

ðA21Þ

where the second partial derivatives of the reduced system matriKRj ,

MRj

and with respect to both kj and hi (for i = 1, 2, . . . , ces Nh, j = 1, 2, . . . , Nt) are given as:

h

i

h

@ 2 KRðkÞ @ KRðiÞ Nh X @ 2 KRj j ¼  hk  @kj @hi @kj @hi k¼1 @kj @hi @kj @ 2 KRj

ðA20Þ

@ 2 Dj

In addition, the partial derivative of Rj with respect to the ith stiffness scaling parameter hi in Eq. (44) being a little more complicated than @Rj =@kj previously provided in Eq. (A8) is given by:

@ 2 KRj @ 2 MRj @MRj @Rj ¼  kj  @hi @kj @hi @kj @hi @hi

ðA18Þ

Finally, in the above formulations, the various first and second

ðA10Þ

And then, the detailed expression of matrix Xij 2 RNm Nm (for i = 1, 2, . . . , Nh, j = 1, 2, . . . , Nt) in Eqs. (44) and (45) is provided as

KRj ,

ðA17Þ

and for n = 1, 2, . . . , Nh,

@hi @hk

@k2j

j

T @MRj @ 2 TTj @TTj @Tj @Tj @Tj @ 2 Tj ¼ MTj þ M þ M þ TTj M @hi @hi @hk @hi @hk @hk @hi @hi @hk

@ 2 KRj

and

@ 2 KRj

(for

where for i, k = 1, 2, . . . , Nh, j = 1, 2, . . . , Nt,

where the second partial derivatives of the matrices with respect to kj are given as:



ðA15Þ

Furthermore, in Eqs. (48) and (49), the second partial derivatives of A and b with respect to both hi and hk are given below:

Consequently, below is the partial derivative of Rj with respect to kj in Eq. (41):

¼

j

ðA6Þ

@KRj @TTj @MRj @TTj @Tj @Tj ¼ KTj þ TTj K and ¼ MTj þ TTj M @kj @kj @kj @kj @kj @kj

@k2j

h i and the corresponding expression for @ 2 KRðkÞ @kj @hi

@kj @hi

and

@ 2 KRj

@ 2 MRj @ 2 TTj @TTj @Tj @TTj @Tj @ 2 Tj ¼ MTj þ M þ M þ TTj M @kj @hi @kj @hi @kj @hi @hi @kj @kj @hi

k = 1, 2, . . . , Nh) is:

where the partial derivatives of the reduced system matrices KRj , KRj MRj

@ 2 KRj @ 2 TTj @TTj @Tj @TTj @Tj @ 2 Tj ¼ KTj þ K þ K þ TTj K ; @kj @hi @kj @hi @kj @hi @hi @kj @kj @hi

i

j

ðA14Þ

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