Using blurred images to assess damage in bridge structures?

Using blurred images to assess damage in bridge structures?

Dr Alessandro Palmeri

The School of Civil and Building Engineering

Loughborough University, UK Email: [email protected]

1

Structural Dynamics

Stochastic Mechanics

Seismic Analysis and Design

Wind Engineering

Train-Bridge Interaction

Random Vibration

Random Composites

Design Assisted by Testing

Performance-Based Design

Fast Dynamics (Blast Loading)

Isolators and Dampers

Structural Health Monitoring

Research Mind Map

2

Research Word Cloud (2000-date)

3

University of

Messina 2000-07

University of

Naples Federico II 2001

University of California at

Berkeley 2002

University of

Patras 2004

University of

Bradford 2008-09

Loughborough University 2010-date

Outline

•  Introduction

•  1. Coupled dynamics of composite bridges (Analytical formulation, 2009-10)

•  2. How sensitive is the envelope to damage?

(Numerical study, 2012-13)

•  3. How can we measure the envelope? (Experimental investigations, 2011-14)

•  Conclusions

4

Introduction

•  Composite steel-concrete beams are widely used in bridge engineering

5

Introduction

•  Their performances are strongly influenced by the flexibility of the shear connection

6

Introduction

Partial-interaction No Interaction (Non-composite bridge)

•  Steel-concrete shear interaction allows reducing deflections and mitigating the accelerations experienced by vehicles

7

Coupled Dynamics of Composite Bridges

Analytical formulation, 2009-10

8


•  Literature Review

•  The most popular approach for the mechanics of composite steel-concrete beam is due to Newmark et al. (1951), in which top slab and bottom girder are two beams continuously connected by a linear-elastic interface

However…

•  Non-rigid steel-concrete connection is ignored in the technical literature devoted to the coupled vibrations of bridges and vehicles

9


10

Moving force: Time-invariant equation of motion

Moving mass: Time-dependent inertia

Moving oscillator: Dynamic interaction (mass, stiffness and damping vary with time)


•  Computational Approach

•  Higher-order partial differential equations of motion for slender composite beams with partial interaction under a platoon of moving oscillators are cast in a novel state-space form with time-varying coefficients

•  Time-independent modifications in inertia and rigidity due to partial interaction between concrete and steel

•  Time-dependent modifications due to the dynamic interaction between composite beam and moving oscillators

Palmeri, 10th Int Conf on Rec Adv in Struct Dyn (RASD), 2010

11


v(1)mv(2)mvv( )nm L

v(1)kv(2)kvv( )nk

v(1)cv(2)cvv( )nc

(s)bf

bL

s s

s s

,,E

A Iρ⎧

⎨⎩

c c

c c

,,E

A Iρ⎧

⎨⎩

}i b,K dc

v

,( , ),( )

zw z tz t

b

v

,( , ),( )

yv z tv t

Figure 1. Simply-‐supported steel-‐concrete composite beam crossed by a platoon of moving oscillators.

Simply-supported steel-concrete composite beam crossed by a platoon of moving oscillators

12


!  Cascade'equations'of'motion:'

!  where:'

[ ](s)b

c c s sb

fA A Ag

ρ ρ ρ= + +

[ ]

[ ] [ ]

c c s sb(0)

2c s c sbb( ) b(0)

c c s s

EI E I E I

E E A AEI EI dE A E A∞

= +

= ++

[ ] [ ][ ]

[ ][ ]

[ ] [ ][ ]

22 i bb

b(0)b(0)

b( )

2b(0)2 2 i b

b bb( ) b(0)

b( )b( )

1

;

1

K dEI

EIEI

EI K dEI EI

EIEI

α

β α

∞

∞

∞∞

=⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠

= =⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠

Non$Composite$

Fully$Composite$

Composite$beam$transverse$deflections$

Concrete$slab$axial$displacements$

[ ] [ ]

[ ] [ ]

2 2 2 4 2b b

b b2 2 2 2 2 4 2 2b b( )b b b

2 2 42c

b i b b b b2 2 2 4b b(0)b b c c

1 ( , ) 1 ( , ) 11 1 ( , ) 1 ( , ) ,

( , ) 1 ( , ) ( , ) ( , ) ( , ) ;

v z t v z tA EI R z t f z tz t z z z

w z t f z t K d v z t A v z t EI v z tz r E A z t z

ρβ α β

ρβ

∞

⎧ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫∂ ∂ ∂ ∂ ∂− + − + = −⎪ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬∂ ∂ ∂ ∂ ∂⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎪⎪⎨⎪ ⎛ ⎞∂ ∂ ∂ ∂⎪ = + − −⎜ ⎟⎪ ∂ ∂ ∂ ∂⎝ ⎠⎩

13 Vibration of the beam: Governing equation


!  Transverse(loading:(

!  Classical(modal(analysis:(

( )v

(s)b b b v( ) v( ) v( )

1( , ) ( ( )) ( ) ( )

n

i i ii

f z t f z t f t z z tχ δ=

= + −∑

Dynamic(loads(due(to(moving(oscillators(

Static(load(b b( ) ( ) ( )

1 , Within the beam0 , Outside the beam

z U z U z Lχ = − −

⎧= ⎨⎩

b(s) (s) T

b b b( ) b( ) b b b1

( , ) ( ) ( ) ( ) ( ) ( ) ( )n

j jj

v z t v z z q t v z z tφ=

= + = + ⋅∑ qφ

Static(contribution(

Modal(contributions(

14 Vibration of the beam: Loads and displacements


!  Modal&equations&of&motion:& qb(t)+Ξb ⋅ qb(t)+Ωb

2 ⋅qb(t) = Jb ⋅Qb(t)

b

1

b bin

−⎡ ⎤= + Δ⎣ ⎦J I m

1 22b b b( ) bi

b b b2ζ

−

∞⎡ ⎤⎡ ⎤= ⋅ + Δ⎣ ⎦⎣ ⎦

=

J kΩ Ω

Ξ Ω

bb( ) b( ,1) b( ,2) b( , )Diag nω ω ω∞ ∞ ∞ ∞⎡ ⎤= ⎣ ⎦LΩ

[ ][ ]

2b( )

b( )b b

j

EIjL Aπω

ρ∞⎛ ⎞

= ⎜ ⎟⎝ ⎠

[ ]

6b( )

bi b2b bb

[ ]EIA L

πα ρ

∞ ⎡ ⎤Δ = ⎢ ⎥

⎣ ⎦k N

[ ]b bDiag 1 2 n=N L

2

bi b2b b

1Lπ

β⎡ ⎤

Δ = ⎢ ⎥⎣ ⎦

m N

v

b b v( ) v( ) b v( ) b v( )21 b

1( ) ( ( )) ( ) ( ( )) ( ( ))n

i i i ii

t z t f t z t z tχβ=

⎧ ⎫′′= −⎨ ⎬

⎩ ⎭∑Q φ φ

15 Vibration of the beam: Modal analysis


!  Absolute)displacements)are)used:)!Impulsive!terms!in!the!equations!of!motion!can!be!avoided!)(Muscolino,)Palmeri)&)Sofi,)2009))

mv(i ) vv(i ) (t) = −cv(i ) vv(i ) (t) − vw(i ) (t)( )− kv(i ) vv(i ) (t) − vw(i ) (t)( ) ,

vw(i ) (t) = rv(i ) (t) + χb (zv(i ) (t))abv(i )T (t) ⋅qb (t)

vw(i ) (t) = sv(i ) (t) + χb (zv(i ) (t)) abv(i )T (t) ⋅ qb (t) + bbv(i )

T (t) ⋅qb (t)( )Static!contributions! Dynamic!contributions!

16 Vibration of the oscillator: Governing equation


!  Equations*of*motion:*Mv ⋅ vv (t) +Cv ⋅ vv (t) +Kv ⋅vv (t) =Cv ⋅sv (t) +Kv ⋅rv (t)

+Cvb ⋅ qb (t) + Kvb (t) +Lvb (t)⎡⎣ ⎤⎦ ⋅qb (t)

Mv = Diag mv(1) mv(2) mv(nv )⎡⎣⎢

⎤⎦⎥

Cv = Diag cv(1) cv(2) cv(nv )⎡⎣⎢

⎤⎦⎥

Cvb(t) = Cv ⋅Xv (t) ⋅AbvT (t)

K v = Diag kv(1) kv(2) kv(nv )⎡⎣⎢

⎤⎦⎥

K vb(t) = K v ⋅Xv (t) ⋅AbvT (t)

Lvb(t) = Cv ⋅Xv (t) ⋅BbvT (t)

Xv (t) = Diag χb (zv(1) (t)) χb (zv(2) (t)) χb (zv(nv ) (t))⎡⎣⎢

⎤⎦⎥

Abv (t) = abv(1) (t) abv(2) (t) abv(nv ) (t)⎡⎣⎢

⎤⎦⎥

Bbv (t) = bbv(1) (t) bbv(2) (t) bbv(nv ) (t)⎡⎣⎢

⎤⎦⎥

17 Vibration of the oscillator: Matrix equations


!  The$generic$oscillator/beam$interaction$force$is$given$by:$

$Leading$to:$

fv(i ) (t) = mv(i ) g

+cv(i ) vv(i ) (t) − sv(i ) (t) − χb (zv(i ) (t))abv(i )T ⋅ qb (t) − χb (zv(i ) (t))bbv(i )

T ⋅qb (t)( )+kv(i ) vv(i ) (t) − rv(i ) (t) − χb (zv(i ) (t))abv(i )

T ⋅qb (t)( )qb (t) + Ξb + ΔCb (t)⎡⎣ ⎤⎦ ⋅ qb (t) + Ωb

2 + ΔKb (t)⎡⎣ ⎤⎦ ⋅qb (t)

=Cbv (t) ⋅ vv (t) +Kbv (t) ⋅vv (t) + Qb (t)

ΔCb (t) = Tbv (t) ⋅Cvb (t)

ΔKb (t) = Tbv (t) ⋅ Kvb (t) +Lvb (t)⎡⎣ ⎤⎦Cbv (t) = Tbv (t) ⋅CKbv (t) = Tbv (t) ⋅Kv

Qb (t) = Tbv (t) ⋅ gMv ⋅τ v −Cv ⋅sv (t) −Kv ⋅rv (t){ }

τ v = 1 1 1{ }T

Tbv (t) = Jb ⋅ Abv (t)+ ΔAbv (t)⎡⎣ ⎤⎦ ⋅Xv (t)

ΔAbv (t) = − 1βb

2 ′′φb(zv(1) (t)) ′′φb(zv(2) (t)) ′′φb(zv(nv ) (t))⎡⎣⎢

⎤⎦⎥

18 Vibration of the oscillator: Matrix equations


!  Matrix'equations'of'motion'can'be'obtained'for'the'platoon'of'moving'oscillators:'

qv (t) +Ξv ⋅ qv (t) +Ωv2 ⋅qv (t) = Qv + µv ⋅Cvb ⋅ qb (t) + µv ⋅ Kvb (t) +Lvb (t)⎡⎣ ⎤⎦ ⋅qb (t)

Ξv = 2Diag ζ v(1) ζ v(2) ζ v(nv )⎡⎣⎢

⎤⎦⎥⋅Ωv

Ωv = Diag ω v(1) ω v(2) ω v(nv )⎡⎣⎢

⎤⎦⎥

Qv (t) = µv ⋅ Cv ⋅sv (t) +Kv ⋅rv (t){ }

v( )v( )

v( )

v( )v( )

v( ) v( )2

ii

i

ii

i i

km

cm

ω

ζω

=

=

1 2v v

−=Mµ{ }b

Tv v(1) v(2) v( )( ) ( ) ( ) ( )nt q t q t q t=q L Dimensional*

consistency*

19 Coupled vibration


!  Finally'the'two'matrix'equations'are'rewritten'in'an'enlarged'modal'space:'

q(t) + c0 + Δc(t)⎡⎣ ⎤⎦ ⋅ q(t) + k0 + Δk(t)⎡⎣ ⎤⎦ ⋅q(t) =Q(t)

v b

b v

v v

v

0b

v vb

bv v b

( )( )

( ) ( )

n n

n n

n n tt

t t

×

×

×

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

− ⋅⎡ ⎤Δ = ⎢ ⎥− ⋅ Δ⎢ ⎥⎣ ⎦

Oc O

O Cc

C C

Ξ

Ξ

µ

µ[ ]

v b

b v

v v

2v

20b

v vb vb

bv v b

( ) ( )( )

( ) ( )

n n

n n

n n t tt

t t

×

×

×

⎡ ⎤Ω⎢ ⎥=

Ω⎢ ⎥⎣ ⎦⎡ ⎤− ⋅ +

Δ = ⎢ ⎥− ⋅ Δ⎢ ⎥⎣ ⎦

Ok

O

O K Lk

K K

µ

µ

{ }TT Tv b( ) ( ) ( )t t t=q q q Q(t) = Qv

T (t) QbT (t){ }T

“Small”'modifications'

20 Coupled vibration: Proposed model


!  Single'step+(unconditionally+stable)+numerical+integration:+

!  Reference+transition+matrix+without+dynamic+interaction:+

!  Dynamic+modification+matrix:+

[ ] [ ] [ ]{ }0 01 01 02( ) ( ) ( ) ( ) ( ) ( )t t t t t t t t+ Δ = ⋅ + ⋅ Δ ⋅ + ⋅ ⋅ + ⋅ ⋅ + Δx E D x V Q V QΘ Γ Γ Γ

v b

1

2( ) 02( ) ( )n nt t t−

+⎡ ⎤= − ⋅Δ + Δ⎣ ⎦E DΙ Γ

v b

v b

10 0 2( ) 0

101 0 0 0

102 0 2( ) 0

1

1

n n

n n

t

t

−+

−

−+

⎡ ⎤= − ⋅⎣ ⎦⎡ ⎤= − ⋅⎢ ⎥Δ⎣ ⎦⎡ ⎤= − ⋅⎢ ⎥Δ⎣ ⎦

L I D

L D

L I D

Θ

Γ Θ

Γ

[ ]0 0exp t= ΔDΘ

21 Coupled vibration: Proposed numerical scheme


Time histories of beam’s transverse deflection at midspan for different levels of partial interaction

22


Time histories of oscillator’s absolute acceleration for different levels of partial interaction

23

Part-1 Conclusions (from 2010)

•  A novel method of dynamic analysis has been proposed and numerically validated for studying the dynamic interaction phenomenon in composite steel-concrete beams subjected to a platoon of single-DoF moving oscillators •  Time-independent modifications arise in the composite beam because of the

partial interaction between concrete slab and steel girder •  Beam-oscillators dynamic interaction is represented by a set of time-

dependent functions, playing the role of time-varying stiffness and damping coefficients

•  A single-step numerical scheme of solution has been formulated, based on the observation that the dynamic modifications are small

•  Further studies: •  Effect of roughness in the beam-oscillators’ contact •  Sensitivity of the dynamic response of the subsystems to the degree of PI in

the supporting beam

24

Part-1 Conclusions

Bending Moment M Shear Force V

Mean value µ (top) and standard deviation σ (bottom) of the internal forces M and V due to a single moving oscillator at midspan of a simply-supported solid beam with rough surface

Muscolino, Palmeri & Sofi, 10th Int Conf on Struct Safety & Reliability (ICOSSAR), 2009

25

How sensitive is the envelope to damage?

Numerical study, 2012-13

26



•  Conventional approaches of damage detection (including ultrasonic, thermal, eddy current and X-ray testing) were termed as cumbersome and expensive

•  Vibration-based damage methods have emerged, as they allow identifying meaningful changes in the dynamic characteristics of the composite beam

•  Accelerometers have been extensively employed, BUT their application to large structural systems may be difficult because of long cabling, number of sensors and installation time

•  Laser doppler vibrometers can be used as a viable non-contact alternative, especially when targets are difficult to access, BUT the simultaneous acquisition of vibration at multiple points would make very expensive the dynamic testing

Therefore…

•  The idea of using the envelopes profile of deflections and rotations induced by a moving load has been investigated

•  That’s radically different than recording and analysing multiple time histories

Kasinos, Palmeri & Lombardo, Structures, In press

27


•  Key Assumptions

1.  Linear-elastic constitutive law 2.  Finite element model built with SAP2000, using:

•  Beam elements for top concrete slab and bottom steel girder •  Elastic springs for the shear connectors

3.  Planar motion (no twisting moment) 4.  Moving force F (massless) with constant velocity V 5.  Damage simulated as stiffness reduction in the shear springs

28

Application Programming Interface


•  Governing equations

•  Dynamic response of interest (displacement or rotation)

•  Envelope of the dynamic response

•  Damage measure (DM)

29


Dynamic amplification factors of midspan deflection δM and right support rotation φR for different levels of concrete-steel partial interaction

= θ due to gravitational loads

= θ when the moving force is applied statically

Amplification factors:

30


Normalised envelope of midspan deflection δM and right support rotation φR for different levels of concrete-steel partial interaction

31


Damage sensitivities fi,j for the natural frequencies associated with the first six flexural modes of vibration in case of medium (left) and stiff (right) partial interaction

32


Damage sensitivities di,j for the displacements’ envelope Eδi in case of medium (left) and stiff (right) partial interaction

V= 250 km/h

V= 300 km/h

33

0.6


Damage sensitivities ri,j for the rotations’ envelope Eφi in case of medium (left) and stiff (right) partial interaction

V= 250 km/h

V= 300 km/h

34

0.6


Different damage sensitive features (f= modal frequency; d= displacement’s envelope; r = rotation’s envelope; q= curvature’s envelope) in case of medium (left) and stiff (right) partial interaction

V= 250 km/h

V= 300 km/h

35

Part-2 Conclusions

•  The envelope of deflections and rotations induced by moving loads has been suggested as damage sensitive feature for composite steel-concrete bridges

•  The envelope of the dynamic response tends to increase when damage occurs in the shear connectors

•  The envelope enjoys:

•  High sensitivity to the damage (higher than frequency shifts, at least for the first few modes of vibration)

•  The sensitivity tends to increase closer to the ends of the bridge, where damage in the shear connectors is more likely to happen

•  Ordered sets of results, that can potentially enhance the predictiveness of damage-detection algorithms

36

How can we measure the envelope?

Experimental investigations, 2011-14

37



•  Advantages of photogrammetric monitoring techniques includes:

•  Simultaneous measurement of many points

•  Non-contact •  Small and inexpensive

targets •  Relatively less expensive •  Scalable

Ronnholm et al., The Photogrammetric Record, 2009

Albert et al., 2nd Symposium on Geodesy for Geotechnical and Structural Engineering, 2002 38


Control points

Monitoring points

39

Displacement

Image number


•  Some studies have used the same approach for monitoring vibration, by increasing the rate at which images are taken (temporal resolution) to many per second.

Displacement

Image number

40


•  Current sensor hardware requires a compromise between image resolution and temporal resolution (rate at which images are taken).

•  Real-time monitoring only possible at reduced image resolution

Image resolu?on

Tempo

ral resolu?

on

Consumer)DSLR)

16#MP#

<5#fps#

££#

Consumer)Camcorder)

2"MP"(1080p)"

30/60"fps"

££"

Specialist*Sensors*

15+$MP$ 0.5$MP$

30$fps$ 1000$fps$

£££££$41


•  Literature Review Vehicle speed

detection

Lin, Li, & Chang, Image and Vision Computing, 2008

Measuring vibration of computer circuits

Wang et al., Pattern Recognition Letters, 2007

Measuring motion of sports balls

Caglioti & Giusti, Computer Vision and Image Understanding, 2009

Blurred images for…

42

Spacecraft guidance systems

Xiaojuan & Xinlong, Acta Astronautica, 2011


High speed imaging

Proposed: Long-exposure image, deliberately blurred

•  Advantages •  Allows measuring the

envelope of the dynamic response

•  Higher image resolutions •  No temporal resolution

limitation •  Less image data •  Frequency independent 43


0

50

100

150

200

250

0

255

35

80

30

35

40

45

50

55

60

65

70

75

80

0

50

100

150

200

250

0

255

diameter = d vibra?on < d vibra?on > d

How does a blurred target look like?

44


Accuracy ≅ 1 pixel Sub-pixel accuracy

45


model structure

shake table

control points

measurement points accelerometer

signal amplifier

input/output device

laser displacement gauge

46


•  50 points/image •  horizontal scale x15

5 Hz

1st Mode: 5 Hz 2nd Mode: 8 Hz 3rd Mode: 12 Hz

47


110 120 130 140 150 160 170

-20

-15

-10

-5

0

5

10

15

20

25

Dis

plac

emen

t (m

m)

Time (s)

•  Full-scale case study •  Wilford bridge, Nottingham •  ~70m span suspension footbridge

Laser Doppler vibrometer (courtesy of Polytec Ltd)

Proposed image processing

48


0.20.4

0.6

0.2

0.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

0.20.4

0.6

0.20.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

0.20.4

0.6

0.2

0.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

1st mode (5 Hz) 2nd mode (8 Hz) 3rd mode (12 Hz)

Unm

odifi

ed st

ruct

ure

(a)

Mod

ifica

tion

1 (b

) M

odifi

catio

n 2

(c)

0.5 0.6 0.7 0.80

0.10.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

0.5 0.6 0.7 0.80

0.10.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

0.5 0.6 0.7 0.800.1

0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

0.5 0.6 0.7 0.800.1

0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xyz

0.5 0.6 0.7 0.80

0.10.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

0.5 0.6 0.7 0.800.1

0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xy

z

•  3D effects of structural modifications

Added mass

Standard member

Reduced stiffness member (back of model)

Unmodified Modification 1 Modification 2

49

Part-3 Conclusions

•  Novel approach to identifying vibration patterns in civil engineering structures using long-exposure images

•  Targets appear blurred because of the motion of the structure

•  The vibration envelope is recorded, not the instantaneous deformed shapes •  Sensors with higher image resolutions can be used

•  High-quantity measurements achieved in both laboratory and field tests

•  The approach can also be used to detect structural changes

•  The proposed frequency-independent approach expands the capabilities of existing sensors

•  Otherwise restricted by their imaging frequency

McCarthy, Chandler & Palmeri, Photogrammetric Record, Under review

50

Final Remarks

•  Part-1: Concrete-steel partial interaction and vehicle-bridge dynamic interaction can be represented efficiently with the proposed analytical formulation for the dynamics of composite bridges

•  Part-2: A numerical study has shown a promising level of sensitivity to damage for the envelope of the dynamic response of composite bridges subjected to moving loads

•  Part 3: Experimental investigations have confirmed that long-exposure digital images can be used to measure the envelope of 2D and 3D structural vibrations with good accuracy (sub-pixel)

51

Acknowledgments

52

Stavros Kasinos Loughborough University

Dr Mariateresa Lombardo Loughborough University

David McCarthy Loughborough University

Prof Jim Chandler Loughborough University

References

•  J Albert, HG. Maas, A Schade & W Schwarz, Pilot studies on photogrammetric bridge deformation measurement, 2nd Symp on Geodesy for Geotechnical and Structural Engineering, Berlin, May 2002

•  L Frýba, Vibration of Solids and Structures Under Moving Loads, 3rd Ed., Thomas Telford,1999 •  S Kasinos, A Palmeri & M Lombardo, Using the vibration envelope as damage-sensitive feature in

composite beam structures, Structures, In press •  DMJ McCarthy, JH Chandler & A Palmeri, Monitoring dynamic structural tests using image deblurring

techniques, 10th Int Conf on Damage Assessment of Structures, Dublin, July 2013 •  DMJ McCarthy, JH Chandler & A Palmeri, Monitoring 3D vibrations in structures using high resolution

blurred imagery, The Photogrammetric Record, Under review •  G Muscolino, A Palmeri & A Sofi, Absolute versus relative formulations of the moving oscillator problem,

Int Journal of Solids and Structures 46: 1085-1094, 2009 •  G Muscolino, A Palmeri & A Sofi, Random fluctuation of internal forces in rough beams under moving

oscillators, 10th Int Conf on Structural Safety and Reliability, Osaka, September 2009 •  NM Newmark, CP Siess & IM Viest, Test and analysis of composite beams with incomplete interaction.

Proc of the Society of Experimental Stress Analysis 9: 75-92, 1951 •  A Palmeri, Vibration of slender composite beams with flexible shear connection under moving oscillators,

10th Int Conf on Recent Advances in Structural Dynamics, Southampton, July 2010 •  P Rönnholm, Comparison of measurement techniques and static theory applied to concrete beam

deformation, The Photogrammetric Record, 24: 351-371

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