Damage detection from static measurement
5.2 Constrained minimization
Damage can be defined as a structural change affecting the performance of the struc-ture. The damage results in a loss of functionality. For a structural system, the loss of functionality means a reduction in the load-carrying capacity or a reduction in its ability to control motion under imposed loads. With this definition of damage, changes in certain properties of the structure between two time-separated inferences must be considered. Static-based methods allow damage identification by measuring changes in the static structural response. The measured quantities are typically displacements or strains under environmental and applied loads.
5.2.1 O u t p u t e r r o r f u n c t i o n
The first method to describe the changes in the static response is the output error function. This is the difference between the analytical and measured static responses.
The parameters are identified by minimizing an objective function formulated by the output error function (Hjelmstad and Shin, 1997) defined as the difference between the computed and measured displacements. A recursive quadratic programming algorithm was used to solve the constrained nonlinear optimization problem. Sanayei et al. (1997) presented the output error function based on the static displacements and strains.
Both static displacement and strain measurements are used to estimate the element stiffness parameters. The method is extended to identify the element properties of a truss structure using static strains from a series of concentrated forces (Liu and Chian, 1997), and the effect of measurement errors on the identified results was studied using the perturbation technique. Shenton and Hu (2006) presented a static procedure based on the idea that the dead load is redistributed when damage occurs in the structure. The static strain measurements due to dead load only are used for the damage identification.
5.2.1.1 D i s p l a c e m e n t o u t p u t e r r o r f u n c t i o n
Consider a discrete structural system with nd degrees-of-freedom (DOFs), having a stiffness matrix, K, characterized bynpconstitutive parameters, p. Let the structure be separately subject tonf static load cases. For a linear elastic structure, the force–
displacement relationship based on the finite element model is
fi=Kui (5.1)
wherefiis the vector of applied forces for theith load case anduiis the corresponding displacement vector.
The output error function is defined as the difference between the analytical and measured displacements for theith load case as follows
ei(p)=K−1fi−ui (5.2)
whereK−1is the inverse of the stiffness matrix.
Since the displacements are taken only at a subset of DOFs, matrixui may be par-titioned into uai and ubi, which are the measured and unmeasured displacements, respectively. Giving
fai
fbi
=
Kaa Kab
Kba Kbb
uai
ubi
(5.3) Using the static condensation
fai=(Kaa−KabK−bb1Kba)uai+KabK−bb1fbi (5.4) With the applied forces and measured responses, the output error function can be defined as
ei(p)=(Kaa−KabK−bb1Kba)−1(fmai−KabK−bb1fmbi)−umai (5.5)
where umai, (fmai,fmbi) are the displacements at the measured DOFs and the vector of applied forces for theith load case.
For all the load cases, the output error function can be written in matrix form as e(p)=(Kaa−KabK−bb1Kba)−1(Fma −KabK−bb1Fmb)−Uma (5.6) where e(p)= {ei(p),i=1, 2,. . .nf}T, Fma = {fmai,i=1, 2,. . .nf}T, Fmb = {fmbi,i=1, 2,. . .nf}T andUma = {umai,i=1, 2,. . .nf}T are obtained from the test data.
If the stiffness parameters are unchanged, thene(p) will be zero. Otherwise, it will not be zero. To adjust the parameter,p, with a small increment, a first-order Taylor series expansion is used to linearize the vector,e(p), that is a nonlinear function of the parameters
e(p+p)=e(p)+S(p)p (5.7)
whereS(p)=∂e(p)∂p is the sensitivity matrix.
A constrained nonlinear optimization problem is solved for the optimal parameters by minimizing the objective function of the output error as
Minimize
α∈Rnα J(p)=e(p+p)TWe(p+p) (5.8)
where W is a weighting factor. The unknown parameters, p, are obtained from Equation (5.8).
5.2.1.2 S t r a i n o u t p u t e r r o r f u n c t i o n
Equation (5.1) shows the relationship between the forces and displacements based on the finite element model. To utilize strain measurements, a mapping matrix between the displacements and strains can be derived from the geometric relationship of nodal displacements to elemental strains (Sanayei and Saletnik, 1996). The mapping matrix is defined as
ε=BU (5.9)
whereεis the vector of strains andBis the corresponding mapping matrix. Substituting Equation (5.1) into Equation (5.9), gives the static equation for a constrained structural system as
εi=BK−1fi (5.10)
Similar to Equation (5.2), the strain output error function can be defined as
ei(p)=BK−1fi−εi (5.11)
Similar to Equation (5.3), the strains from multiple load tests can be grouped into εai
εbi
= Ba
Bb
K−1fi (5.12)
and the strain output error function is written as
ei(p)=BaK−1fmi −εmai (5.13)
whereεmai,fmi are the strains at the measured DOFs and the vector of applied forces for theith load case, respectively. Combining all the load cases, Equation (5.13) can be rewritten as
e(p)=BaK−1Fm−εma (5.14)
whereFm= {fmi ,i=1, 2,. . .,nf}T,εma = {εmai,i=1, 2,. . .,nf}T.
Similar to Equation (5.8), a constrained nonlinear optimization problem can be constructed and solved for the optimal parameters.
5.2.2 D a m a g e d e t e c t i o n f r o m t h e s t a t i c r e s p o n s e c h a n g e s
The changes in the static response of a structure are characterized as a set of nonlinear simultaneous equations that relates the changes in the static response to the location and severity of damage. A structural component is considered to be damaged if any of its stiffness parameters has been reduced. The static loads applied at a subset of DOFs and the measured static response at another subset of DOFs are used to detect the damage in structural elements. These subsets can be overlapped, partially overlapped or independent. Though structural damage is usually associated with a non-linear type of behaviour, this method uses small magnitude loads that cause structures to behave only in their elastic range. Bakhtiari-Nejad et al. (2005) presented an algorithm for damage identification based on the changes in the static displacements. Zhu et al.
(2007) studied the effect of the load carried by a reinforced concrete beam on the assessment result of a crack damage, and they concluded that accurate assessment can only be obtained when a load close to the one that creates the damage is used in the identification. Zhu and Law (2007b) introduced the scalar damage parameters to characterize the changes in the interface between concrete and steel. The damage is identified using changes in the static responses of the beam.
The force–displacement relationship of a damaged structure can be expressed as
F=KU=(K−K)(U−U) (5.15)
whereF=Pis the force vector, in whichP= {P1,P2,. . .,PNp}Tis the vector of static loads; = {12. . . l. . . Np} is a2(N+1)×Np shape function matrix; Np is the number of loads; U and U are the vectors of nodal deformation without and with damage, respectively; and U=U−U is the vector of deformation difference due to the damage. When under the same applied load, Equation (5.15) indicates qualitatively that a reduction in the stiffness matrix corresponds to an increase in the vector of deformations.
VectorU can be estimated from Equation (5.15) as
U= −K−1KK−1F+K−1KU≈ −K−1KK−1F (5.16)
by neglecting the second-order terms. Substituting the force–displacement relationship of the intact structure and the stiffness matrix of the damage structure into Equation (5.16), the analytical vector of deformation due to damage is obtained as
U≈
N i=1
αiK−1ATiKiAiK−1P= N
i=1
αiUˆi (5.17)
whereUˆi=K−1ATi KiAiK−1PandAiis the transformation matrix for theith element for assembling the global stiffness matrix from the constituting elemental stiffness matrix.
ForNsmeasuring points at{xs,s=1, 2,. . .,Ns}, Equation (5.17) can be written as
Us=sU (5.18)
whereUs=0
u1,u2,. . .,uNs
1T
ands=0
φ1,φ2,. . .,φNs
1T
.
The error between the vectors of difference between the calculated and measured deformations of the structure is obtained from Equations (5.16) and (5.18) as
e(α)=ΦsU−Us (5.19)
whereUsis the vector of differences between the measured displacement of structures with and without damage.
The algorithm to identify the damage is based on minimizing the least-squares error function in Equation (5.19) as
MinimizeJ(α)=1
2e(α)2= 1 2 ,,,, ,
N i=1
αiΦsUˆi−Us
,,,, ,
2
subject to 0≤αi≤1, (i=1, 2,. . .,N) (5.20) This model can be cast into the following quadratic programming problem (Banan and Hjelmstad, 1994) for determining the damage indices
MinimizeJ(α)= 1
2αTKTTssαK−UTssαK+1
2UsTUs
subject to 0≤αi≤1, (i=1, 2,. . .,N) (5.21) whereα= {αi}T. The algorithm presented by Goldfarb and Idnani (1983) is used to solve this quadratic programming problem. Details of the iterative algorithm used to solve the nonlinear optimization problem are as follows:
1) Calculate the matrixs=0
φ1,φ2,. . .,φNs
1T
,= {12. . . l. . . ..Np}.
2) Initially assume that there is no damage in the beam, i.e.α0= {0, 0,. . ., 0}T. 3) When the deformation of the intact structure under a given load is not available,
the baseline deformation vector of the structure without damageUsis calculated and used instead of the measured deformation vector from the intact structure,
and the initial vector of measured deformation difference from the intact and damage structures is obtained asUs0=Us−Us.
4) Identify the damage indexαj using Equation (5.15). j=1 for the first cycle of iteration.
5) Calculate U using Equation (5.16) with the updated α=αj+α0 and UReconstructed=Us−sU.
6) Calculate the following criteria of convergence:
Error1= Us−UReconstructed
Us ×100%
Error2= αj+1−αj
αj ×100%
(5.22)
αj,αj+1 are the identified damage indices in two successive iterations. Con-vergence is achieved when both errors are less than the pre-defined tolerance values.
7) When the computed error does not converge, calculateUs=UReconstructed−Us
andα0=α. Repeat Steps 4 to 6 until convergence is reached.
5.2.3 D a m a g e d e t e c t i o n f r o m c o m b i n e d s t a t i c a n d d y n a m i c m e a s u r e m e n t s
In a finite element model, the characteristics of a structure are defined in terms of the stiffness, damping and mass matrices. Any variations in these matrices affect the dynamic response of the structure. Some researchers have studied algorithms using both the static and dynamic test data. A modified error function for the statistical parameter estimation was presented (Oh and Jung, 1998) to combine the curvature or slope of the mode shapes and the static displacements. This combination gives rise to a promising damage detection and assessment algorithm. Later Wang et al. (2001) used both the static displacements and changes in natural frequencies in a structural damage identification algorithm.
Theith eigenvalue,λi, and the corresponding eigenvector,ϕi, of ann-DOFs system are obtained by solving the characteristic equation
Kϕi=λiMϕi (5.23)
whereKis the structural stiffness matrix andMis the structural mass matrix. Combin-ing Equations (5.1) and (5.23), the relationship between the modal parameters, static responses and structural parameters is expressed in terms of a first-order of Taylor series expansion as
=
Pp
+S(P−Pp) (5.24)
where P is the vector of structural parameters; vector Pp is the prior estimate of the structural parameters; the vector with subscriptPpdenotes the responses of the system
when the parametersP=Pp; is the vector of measured eigenvalues; andandare vectors of dynamic and static error responses, respectively. Letnrbe the total number of responses for the target structure, andSbe the partial derivative or sensitivity matrix of the eigenvalues and dynamic and static error responses. It can be written as
S= ∂
∂pj
∂
∂pj
∂
∂pj
T
(5.25)
The different types of responses in Equation (5.24) are combined to form the error function. To prevent the case where the contribution of a special response type is reduced due to a relatively small magnitude of the response, an adjusted dynamic response should be defined. Therefore, when the combined data of static and modal tests are used in the identification procedure, the measured mode shapes are weighted so that the maximum values of each type of response are the same.
It is noted from Equation (5.24) that the number of structural parameters is not the same as the number of measured responses, such that the inverse matrix ofSdoes not always exist. Therefore, an estimateTon such an inverse can be defined as
p=TR (5.26)
whereR=
−
pp
.
The objective of this procedure is to find the best unbiased estimator,p, based on measured values of ,andand prior estimates of the structural parameters,pp. A scalar performance error function in the Bayesian estimation theory is defined for the derivation of the statistical system identification formula as
E=(R−Rk)TWr(R−Rk)+(P∗k−Pk)TWp(P∗k−Pk) (5.27)
whereWrandWpare the weighting matrices for the measured responses and the struc-tural parameters, respectively, which can be assumed as the inverse of the covariance matrices of the errors on the measured responses and the initial structural parameters, respectively;Ris the measured responses;Rk,Pkare predicted vectors of responses and structural parameters, respectively, at thek-th step; andP∗kis the vector of objective parameters.
Other error functions can also be used to estimate the structural parameters. The changes in curvatures are local in nature and hence can be used to detect and locate the damage in the structure. The differences in the curvatures of mode shapes can be used to form the error response vector as well as for estimating the uncertainty of the structural parameters. Certainly, the slope of mode shapes is also another possible candidate for formulating the error response vectors.
5.3 Variation of static deflection profile with damage