Advanced Structural Dynamics with Applications to ... · 4 7 Structural Dynamics - Single Degree of...
Transcript of Advanced Structural Dynamics with Applications to ... · 4 7 Structural Dynamics - Single Degree of...
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Prof. Sauvik BanerjeeCivil Engineering Department
Indian Institute of Technology BombayPowai, Mumbai 400076
Phone: (022) 2576 7343 Email: [email protected]
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
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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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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
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2
4
6
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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
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65
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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