damage identification of Euler Bernoulli beams using static responses

7/27/2019 damage identification of Euler Bernoulli beams using static responses

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Faouzi Ghrib, PhD, PEng.

Associate Professor

University of Windsor

Department of Civil and Environmental Engineering

Ontario, Canada,

REPRODUCED BY:

MD RIYAZ ALI

ROLLNO:11CE65RO1

STRUCTURAL ENGINEERING

IITKGP



• ABSTRACT• INTRODUCTION

• PHOTOGRAMMETRIC MEASUREMENT OF BEAMDEFLECTIONS

• THE EQUILLIBRIUM GAP FORMULATION

• DATA-DISCREPANCY BASED FORMULATION

• EXPERIMENTAL PROGRAM VALIDATION

• CONCLUSION• REFERENCES



presents two computational procedures to reconstruct the stiffness

distribution and to detect damage in Euler-Bernoulli beams. The principle of the equilibrium gap along with a finite element

discretization.

The solution is obtained by minimizing a regularized functional using

a Tikhonov Total Variation (TTV) scheme.

The minimization of a data discrepancy functional between

measured and model-based deflections.

The optimal solution is obtained using a gradient based minimization

algorithm and the adjoint method to calculate the Jacobian.

Three beams with predefined damage scenarios were tested.

In each case, the location and the damage levels were reconstructed

with good accuracy.



Damage is usually related to the change in the internal material

structure for ductile behaviour and/or the appearance of cracks ingeomaterial.

Damage accumulation is associated with the stiffness reduction of

structural members, and generally expressed as a damage variable

measuring the remaining stiffness to a reference value.

Based on used response data, damage identification methods can be

divided into two categories: dynamic or static based.

While some dynamic based methods have had limited success, they

faced serious challenging problems when applied to real structures.

One way to classify static based damage identification methods is toconsider whether the problem is formulated to locate and quantify

localized damage in a form of discrete cracks, or to reconstruct a

spatially distributed damage field.



•Successful damage identification and stiffness recovery require

efficient measurement techniques of the structural response.

•Traditional measurement techniques using Linear Variable Differential

Transformers (LVDT) and strain gauges allow only localized

measurements along the beam and therefore provide reduced amount

of information.

•Recent progresses in image processing techniques and digital cameras

permit a quasi-continuous deflection profile measurements of beams .

•This progress offers new possibilities in structural parameter

identification, damage localization.

•In the present paper, we propose a methodology for damage

identification in Euler-Bernoulli beams using static deflections

measurement.



Photogrammetry is a technique of measuring objects (2D or 3D)

from digital photographs.

A continuous measurement can be obtained in the field of view.

Edge-detection based close-range digital photogrammetry. A mollification based procedure is used to filter the measured data

and reduce them to the finite element model nodes.

The mollification technique is also used to approximate the

evaluation of the rotations.

The numerical convolution and the optimal selection of the

mollifier’s radius using cross-validation are used to filter the data

and compute numerically the derivatives



•. Within the Euler-Bernoulli beam theory, the rotation angle is the

derivative of the deflection,

•The reconstruction of the derivative of the mollified data function

is stable with respect to the noise existing in the measurements.

•The rotation at a given node xi ,

•The elasticity modulus of the tested beam was identified to be

70.14 Mpa.



Restricted to the elasticity domain and in the absence of volumetricloading, the continuous format of the equilibrium equation is given by

div (σ(u))=0

Cauchy stres σ is a function of the displacement field gradient, Eand μ.

Above equ as a set of discrete equations in the form of [σ.n]=0.

The stress jump represent an equilibrium gap can also be given as σin-σ jn=0.

where n denotes the normal of an interface between the two sidesof a section

The principle of equilibrium gap formulation can be applied specificallyfor beams where the equilibrium equations are :

FL+ Fr= Plr ; ML+ Mr = Mlr



•Within the context of FE formulation of Euler-Bernoulli beams, the

force-displacement relations of an Euler-Bernoulli beam are given by

(v1,,ө1,v2, ө2)are the nodal degrees of freedoms (vertical deflection,

rotation angle), and (V1,M1,V2,M2) are the dual variables (shear,

moment).

•The stiffness matrix of the element at the damaged state is denotedkd =(1-D)k0, D is a damage parameter, 0≤D ≤1

EI0(x) is reference stiffness can correspond to the stiffness of the

undamaged beam.



• EI(x) is the actual stiffness of the cross-section at a

position x

damage variable, D(x)=1-

• For each pair of two adjacent elements, e and e+1 the

following equilibrium equations can be written

(1-De)(K03)e de,r + (1-De+1)(K0

1)e +1 de+1,l =(V2)l +(V1)r =Plr

(1-De)(K04)e de,r + (1-De+1)(K0

2)e +1 de+1,l =(M2)l +(M1)r =Mlr

• Writing these equilibrium equations for each pair of adjacent

elements,



• Finally, the problem of identifying the damage variable is written

as:

find θ=(1-D1 1-D2 .... 1-DNel)T

such that G θ=R

where θ is the set associated to the damage variables corresponding

to eachelement and R=(P1M1 .....PIMI....)T is the external forces ,nodal

displacements(d 1 ,d 2 ,.....d n).

• In the inverse problem matrix G encapsulates both the model and

the data.

• model errors and measurement errors lead to the ill-posedness of the inverse problem.

• A Tikhonov-Total Variation (TTV) regularization scheme is used to

solve system



• The problem is treated in the least-square sense augmented

with the TTV regularization leads to

where α is the Tikhonov parameter ,

ɸ(θ) regularization term

•The function ɸ(θ) is the total variation (TV) functional of the

vector θ

• Minimization of the TTV regularized functional yields an

efficient scheme to solve the discrete inverse problem.



• for calculating ɸ(θ) , the total variation (TV) of a function f(x) on

the interval [0,1]

•In the one-dimensional case, the smooth form TV functional defines

the total variance function ϕ as:

•The final smooth total variation functional

•The discretized version of above equation is



•Within the presented framework, both statically determinate and

indeterminate beams can be analyzed with the formulation as

long as the applied load and displacements measurements are

available



Using a measured deflection of a beam, um, the problem of reconstructing the stiffness of a beam can be formulated as an

optimization problem using a misfit function between um and u(ϴ)

displacements:

where J(u,ϴ) is the data-discrepancy functional augmented with a

TTV functional

The constraint equations appearing in the minimization problem is

the static finite element direct problem



• Evaluation of gradient of the cost functional, J.

• An adjoint formulation is used to compute the gradient vector of

the cost functional to the unknown stiffness distribution

parameters θ

• The constraint equations in residual form

R(θ,u(θ)) = K(θ)u-R = 0

the residual is an implicit function of the unknown internal variables θ • The Lagrangian multipliers, λ , to convert the constrained problem

to an unconstrained optimization.

L(θ,u) = J(θ,u)-λT R(θ,u)



additional equation ,adjoint equaton

(u-um)= k(θ)

( ) θ ═

•The global stiffness matrix k is the classical assembly of

the elements’ stiffness matrices

ki.

•For the Euler-Bernoulli bending element



•In conclusion, the adjoint method leads to an efficient computational

method to calculate the gradient of the cost functional as it consists of

two consecutive steps:

1) solve the adjoint equation for λt , then insert λt to calculate the gradient

of the Lagrangian.

2) With the gradient calculated, any efficient gradient-based optimizationtechnique can be used to find the optimal solution



An aluminum cantilever beam with pre-defined damage locations istested and the associated inverse problems are solved for

validation

The beam’s cross-section is a HSS of 25.4 x 25.4mm and the

thickness of the wall section plate is1.58mm.

E=70.1GPa

In all four cases, the deflection profile is obtained using a close

range photogrammetric method.

The reference frame used to measure the deformation was the

image of the beam under its self-weight.



Test # 1: Step-wise damage detection



Equillibrium gap method

Data discrepancy method



performance of the adjoint based optimization algorithm



Test # 2: Single saw-cut damaged beam

Equillibrium gap method Data discrepancy method



Test #3: Combination of discrete and step cuts damaged beam

Equillibrium gap method Data discrepancy method



Two inverse problem based formulations are proposed to identify

damage in Euler-Bernouilli beams.

The static deflection profile of the beam is obtained from close

range photogrammetric technique.

We describe a technique to measure the displacement of beams

based on an edge detection algorithm The first inverse problem formulation uses the equilibrium gap

principle along with a finite element forward problem solver.

An over-determinate algebraic system is obtained and solved in the

least squares sense with a TTV regularization scheme.

The second formulation is based on a data discrepancy expression

of the measured and model based deflection.

The minimization of the functional is obtained through an adjoint

method and a TTV regularization.



•Four tests were conducted on beams with different damage

scenarios

•The two methodologies were validated and showed overall goodperformance within a laboratory conditions.

•In each test, the location and level of damage were identified with a

good level of accuracy

•A validation on more complex systems, such as rehabilitated

reinforced concrete beams is underway.

•Two aspects need to be addressed in future work:

(i) develop a reliable methodology for deflection measurements

using digital image correlation, and(ii) include error uncertainty in the formulation of the inverse

problems.

•These two enhancements will allow practical use of the proposed

methodologies for the health monitoring of structures



REFERENCES

•Claire, D., Hild, F., and Roux, S. (2004). “A finite element formulation to identify damage

fields: the equilibrium gap method.” International Journal for Numerical Methods in

Engineering, 61(2), 189-208.

• Aster, R.C., Borchers, B., and Thurber, C.H. (2005). “Parameter estimation and inverse

problems”, Elsevier Academic Press, Burlington, MA, USA. •Banan, M. R., Banan, M. R., and Hjelmstad, K. D. (1994). “Parametric estimation of

structures from static response: I. computational aspects.” ASCE Journal of Structural

Engineering, 120(11), 3243-3258.

•Bicanic, N., and Chen, H.P. (1997). “Damage identification in framed structures using

natural frequencies.” International Journal for Numerical Methods in Engineering, 40(23),

4451-4468.

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damage identification of Euler Bernoulli beams using static responses