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Prof. Sauvik BanerjeeCivil Engineering Department

Indian Institute of Technology BombayPowai, Mumbai 400076

Phone: (022) 2576 7343 Email: sauvik@civil.iitb.ac.in

web: www.civil.iitb.ac.in/~sauvik

Advanced Structural Dynamics with Applications to Structural Health Monitoring

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Can a structure behave a bit likehuman?

- feel pain- determine its location and severity- ask for service

A SHM system should:

Evaluate changes in critical structural parameters from baseline, if available.

Assess structural integrity. Recommend maintenance strategy

Continuous monitoring of the degradation of aerospace, mechanical and civilstructures in service using nondestructive techniques, with minimum manualintervention.

Structural Health Monitoring (SHM)

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Why Do We Need SHM?

An effective SHM system can: Improve safety and reliability

Detect and characterize growingand emerging hidden damage

Reduce life cycle cost Reduce labor intensive

inspection Prolong / extend service life

Reduce turnaround time Condition based inspection and

maintenance

SHM can be used throughout the life cycle of the structural system

Airbus failureClose-up of left center vertical stabilizer (tail fin) attachment point, at crash site of AA-587, Nov, 2001

Collapse of Inner containment dome of PHWR at Kaiga Atomic Power station during construction (May 1994)

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Transition to SHM

Traditional NDI:

Rapid NDI:

SHM:

Point by point manual inspection1 or 2 sensors Labor intensiveAccess required or disassembly of structureTime consuming

Large area inspectionDense network of sensors in a systemSystem mechanically driven, but can be costlyAccess required or disassembly of structureMinimal labor

Entire structure or component inspectionSparse network of low cost sensors embedded into the structureNo disassembly of structurePossibly wireless data collection and automated processing

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SHM Approaches and Limitations

Two general approaches are used in SHM Local approach - measures changes in wave propagation across

defects and infers defect characteristics directly--- Effective with small defects, but complexity in signal realization and

processing

Global approach - measures and analyzes damage-induced changes in the vibrational properties (e.g modal frequencies, mode shapes, frequency response) of a structure to detect and characterize the damage

--- Best suited for large defects

Identify locations with possible presence of damage and its severity instantaneously. NDI is still required to size the defect!

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Target Structures/DefectsMetal panels

(cracks, loose rivets) Laminated and Sandwich Plates

(delamination, disbond)

Concrete slabs(rebar separation, other voids)

Steel Structures(Joint defects)

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Structural Dynamics

- Single Degree of Freedom (SDOF) System

- Multi Degree of Freedom (MDOF) System

- Continuous System (with distributed mass)

Hamilton’s Principle

We Define Lagrangean by :Where T is the Kinetic energy and П is the potential energy.

TL

For an arbitrary time interval from t1 to t2, the state of motion of a body extremizes the functional

2t

1tLdtI

If L can be expressed in terms of generalized variables),...,,,,...,,( 2121 nn qqqqqq Where, dt

dqq ii

Then the equations of motions are given by

0.

iiqL

qL

dtd ni to1

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Kinetic Energy (T)

Nqu

dVuu21T T

V

T]w,v,u[u

The Kinetic energy is given by

where ρ is the density (mass per unit volume) of the materialis the velocity vector

Using FE discretization, u can be expressed in terms of the nodal displacement q, using shape function N :

Velocity vector is then given by: qNu

q]NρN[qe

TT dVT e 21

e

Te NNm dV

QMQqmq TT 21

21 e

e eeT T

q – localQ – global

Potential Energy П:

Bq

WP)potential(Work )(EnergyStrain U

V

T dVU 21

iTi

TT PuTufu S iV

-dSdVWP

= e

e dVU ee DT

21

iTi

TT

V

T PuTufuΠ --= S iV

-dSdVdV ees2

1

e

DBqBq dVTT

21 qkqT e

21

e

e DBB dVk T

Element Stiffness

KQQT

21

21U U

e ee qkq eT

Dεσ

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Work Potential

dVdVe

fNqfu TTT eT fq

edAdAeAeA

TqTNqTu TTT T dAe TNT T

dVfNf Te

FQPQ-T(fq Ti

ii

eeT )ee

WPWP

iTi

TT PuTufu S iV

-dSdVWP

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Truss element: Stiffness and Mass Matrices

The displacement function for the truss element

1 2u x a a x

L

f1, U1 f2, U2

U1

U2

Apply boundary condition

1 10u u a

2 1 2u L u a a L

2 12

u uaL

11 2

2

uu N N

u

Shape function for the truss element

1 1 xNL

2xNL

;

eu N q

Strain displacement relationship

Te

V

k B E B dv Elemental stiffness matrix

Elemental mass matrix

edu B qdx

eE B q

1 11 1

e AEkL

Te

V

m N N dv 1 1 1 2

2 1 2 20

Le N N N N

m A L dxN N N N

2 11 26

e A Lm

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4

4 0d v

EIdx

The Governing equation

The General solution 3 21 2 3v x C x C x C x C

Where, C1, C2, C3, C4 are constant

Apply boundary conditions : , 0 1

2 2

(0) 0 (0) |

( ) , ( ) |

x

x L

dvvdx

dvv L v Ldx

L

V1, f1y V2, f2y

θ1, M1 θ2, M2

Typical beam element

31 2 1 23 2

21 2 1 2 1 12

2 1

3 1 2

v x v v xL L

v v x x vL L

Displacement inside the element

The displace at any point inside the element

1 1 2 1 3 2 4 2v x N v N N v N

ev x N q

Shape functions

1 2 3 4N N N N N

where 1 1 2 2

Teq v v

3 2 31 3

1 2 3N x x L LL

3 2 22 2

1 2N x x L xLL

3 23 3

1 2 3N x x LL

3 24 2

1N x x LL

Beam element: Stiffness and Mass Matrices

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2

2ed vydx

Strain displacement relationship

2

2e

e

d Ny q

dx

ee B q

1 2 3 4[ ]B y N N N N

Constitutive law

x xE

ex D B q

D E

Stiffness and mass matrices

0

TLek N D N Idx

0

TLek EI N N dx

1 1 1 2 1 3 1 4

2 2 2 3 2 4

3 3 3 40

4 4

Le

N N N N N N N NN N N N N N

k EI dxSym N N N N

N N

2 2

2

12 6 12 64 6 2

12 64

e

L LL L LEIk

Sym LLL

0

LTem m N N dx

2 2

2

156 22 54 134 13 3

156 224204

e

L LL L LmLm

Sym LL

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Lumped Mass Assumption

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Equations of Motion

QMQT 21 T FQKQQΠ TT

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Using the lagrangean L= T - П and applying Hamilton’s principle

FKQQM

Time Integration Technique:

Explicit (Forward Euler): Simple to implement, but conditionally stable

Implicit (Backward Euler): Unconditionally stable but lengthy computation

T.J.R. Hughes, The Finite Element Method — Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 07632, 1987.

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Continuous System: Analytical Solution

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Continuous System: Analytical Solution

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10

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MDOF System Approximation

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MDOF System Approximation

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MDOF System Approximation

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Effects of Damage on the Modal Response

Damage Level Mode 1, rad/s

Mode 2, rad/s

Mode 3, rad/s

Undamaged 73.506 294.040 661.620Damaged 72.918 291.510 661.450

MSC plot of Simply-supported beam

0

2

4

6

8

10

12

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0 0.2 0.4 0.6 0.8 1 1.2Length of beam, m

Mod

e Sh

ape

Cur

vatu

re

UndamagedDamage 1

Simply Supported Steel Beam (1000mmx50mmx5 mm) Damage: removal of 20x20mm area through thickness between Control point 6 & 7 (at a distance of 10 mm from CP 7)

Changes in modal frequencies and mode shapes are not significant. Change in MSC is highly localized

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FRF is a transfer function that describes the input-output relationship between two points of a structure as a function of frequency.

FRF is a measure of how much displacement, velocity, or acceleration response a structure has at an output point per unit of excitation force at an input point.

The FRF depends on the mass, damping and stiffness properties of the structure, and any changes in these properties produce changes in H.

Input Force Transfer Function Displacement Response

)(F )(H )(X

)(F)(H)(X )(F)(X)(H

12 KCiMH

Frequency Response Function (FRF)

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Frequency Response Function Curvature (FRFC) and Damage Index (DI)

2,1,,1

,

)()(2)()(

hjijiji

ji

j,i''

j,i''d

''j,i )()(

The absolute difference between the FRF curvatures of the damaged and undamaged structure summed over a range of frequency

DI:

FRFC:

A typical Displacement FRF Plot

Calculation of curvature at control point i, due to excitation at j

Excitation (input)

hj i

h

i+1i-1

ji , is the displacement FRF measured at location i for a force input at location j

The FRF curvature at each frequency is given by

In case of a plate, calculated DI at a control point along both the in-plane directions (x & y) are added to obtain the final damage index plot.

In all cases studied, frequency range upto the first mode is considered for DI calculation

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Excitation (input) Damage locations

B A C

100 mm 6 1 2 3 4 5 7 8 9

1000 mm

87

23

65

1

4

9

B

C

Damage Type Damage Location DistanceDL1 A (between CP 6 & 7) 20mm from CP 7

DL2 B (between CP 2 & 3) C (between CP 7 & 8)

20 mm from CP 220 mm from CP 7

Simply Supported Steel Beam

50 mm wide5 mm deep

Removal of material over an area of 20x20 mm

Damage Type: DL1 Damage Type: DL2

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Mode Analytical, Hz ABAQUS, HzUndamaged Undamaged DL 1 DL 2

1 11.697 11.695 11.601 11.5652 46.789 46.782 46.379 45.8093 105.256 105.264 105.237 104.203

Excitation (input) Damage locations

B A C

100 mm 6 1 2 3 4 5 7 8 9

1000 mm

Simply Supported Steel Beam: Damage Index Plots

Displacement FRF at control

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Damage Index (DI)

DL1 DL2

DL1 DL2

DI is pronounced at control points closer to the damage location