on the dynamics of suspension bridge decks with...

On the Dynamics of Suspension Bridge Decks with Wind-induced Second-order Effects

Gianfranco Piana

Department of Structural and Geotechnical Engineering - Sapienza University, Rome (Italy)

Department of Structural, Geotechnical and Building Engineering - Politecnico di Torino (Italy)

MMPS 2015 Convegno “Modelli Matematici per Ponti Sospesi”

Politecnico di Torino – Dipartimento di Scienze Matematiche – 17-18 Settembre 2015

• Motivations and objectives

• Reference model and basic equations

• Free vibrations and stability under steady aerodynamic loads• Governing equations of motion• Antisymmetric modes • Numerical example

• Implications in flutter analysis

• Final remarks

Outline

G. Piana | MMPS 2015 2 / 48

Motivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

OutlineMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks





• Final remarks

G. Piana | MMPS 2015 3 / 48

Motivations

4 / 48 G. Piana | MMPS 2015


The prediction of the critical wind speed for dynamic aeroelastic instability requires the knowledge of the natural frequencies and mode shapes of the structure.

They are generally obtained by a modal analysis conducted with respect to the deformed configuration under permanent loads, and therefore used in both experimental tests and numerical computations.

Motivations

5 / 48 G. Piana | MMPS 2015


The interaction between bridge vibration and wind is usually idealized as consisting of two kinds of forces: motion-dependent and motion-independent.

According to this schematization, the equation of motion in presence of the aerodynamic forces can be expressed in the following general form:

[M], [C], and [K] are the generalized mass, damping, and stiffness matrices, respectively; {δ} is the displacement vector, {F}md is the motion-dependent aerodynamic force vector, and {F}mi is the motion-independent wind force vector.

[ ]{ } [ ]{ } [ ]{ } ( ){ } { },mimd

M C K F Fδ δ δ δ δ+ + = +

Motivations

6 / 48 G. Piana | MMPS 2015


Sometimes, the cable geometric stiffness matrix is added:

[ ]{ } [ ]{ } [ ]( ){ } ( ){ } { },cg mimd

M C K K F Fδ δ δ δ δ⎡ ⎤+ + + = +⎣ ⎦

Causes dynamic instability!

Cause deformation!

is usually neglected in dynamic stability analyses, even if it affects the bridge global stiffness.{ }mi

F

Objectives

7 / 48 G. Piana | MMPS 2015


1 To investigate the effects of steady aerodynamic loads on stability and natural frequencies of long-span suspension bridges through a simplified analytical model.

2 To put these effects in relation to the classic linear flutter analysis.

► Taking into account the second-order effects induced by a constant transverse wind in the bridge equations of motion, we will derive a generalized eigenvalue problem.

► We will show that the natural frequencies of a suspended deck-girder dependupon the average wind loading.






• Final remarks

G. Piana | MMPS 2015 8 / 48

Reference modelMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 9 / 48

The single (central) span suspension bridge model is considered, and the linearized integro-differential equations describing the flexural-torsionaldeformations of the bridge deck-girder are adopted as starting point.

• Deck-girder: modeled as an elastic beam of constant cross-section, deformable in flexure and torsion, but inextensible and not deformable in shear;

• Main cables: modeled as purely extensible elastic elements;

• Hangers: assumed to be inextensible.Single span suspension bridge model

Reference modelMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 10 / 48

Flexural-torsional deformation of a suspended deck-girder subjected to a distributed vertical load, p(z), and a distributed torque moment, m(z)

Basic equationsMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 11 / 48

According to the linearized theory, the integro-differential equations governing the flexure and torsion of suspension bridges subjected to a vertical load p(z) and a torque moment m(z), distributed along the deck-girder, are:

v(z) is the vertical deflection and ϑ(z) is the torsion of the deck-girder; l is the deck-girder/main cables span; f is the main cables sag; qg is the bridge weight per unit length; μg the bridge mass per unit length; H is the horizontal component of the tension in both cables due to the bridge weight qg (H = qg l 2 /8 f); Ec Ac is the extensional stiffness of both main cables; EI the flexural stiffness (about the x-axis) of the deck-girder; G Itand E Iω are the primary (St. Venant’s) and warping torsion stiffness of the deck-girder, respectively.

( )

24 2

4 2 0

24 22 2

4 2 0

d d d ,d d

d d d .d d

lg c c

lg c ct

q E Av vEI H p v zH lz z

q E AEI GI H b m b z

H lz zωϑ ϑ ϑ

⎛ ⎞− = − ⎜ ⎟⎜ ⎟

⎝ ⎠

⎛ ⎞− + = − ⎜ ⎟⎜ ⎟

⎝ ⎠

∫

∫






• Final remarks

G. Piana | MMPS 2015 12 / 48






• Final remarks

G. Piana | MMPS 2015 13 / 48

Governing equations of motionMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 14 / 48

We will study the free oscillations of a suspension bridge deck-girder subjected to a transverse constant wind (mean wind).

We shall analyze the small flexural and torsional oscillations about the original(rectilinear) configuration, taking into account the second-order effectsinduced by the aerodynamic loads into the equations of motion of the suspended deck-girder.

As is well known, the mean wind loading consists of the quasi-static load arising from the wind flow past the bridge, and is determined by its size and shape, the air density, the square of the mean wind speed, and its angle of inclination to the structure (angle of attack).


G. Piana | MMPS 2015 15 / 48

The mean wind, blowing with velocity U, induces the aerodynamic loads of lift, drag, and moment on the deck-girder (here the lift is neglected).

We assume that the deck-girder is horizontal in its original configuration and that the wind, constant in space and time, invests the girder with zero angle of attack (i.e., it has horizontal direction).

Single span suspension bridge subjected to steady aerodynamic drag, , and moment, xpzm , uniformly distributed along the deck-girder


G. Piana | MMPS 2015 16 / 48

We have the following expressions for the steady aerodynamic loads:

ρ is the air density; U is the mean wind velocity; B is the deck width; and CD (0) and CM (0) are the aerodynamic coefficients of drag and moment, respectively, evaluated for zero angle of attack.

( ) ( )

( ) ( )

2

2 2

10 0 ,2

10 0 .2

x s D

z s M

p D U BC

m M U B C

ρ

ρ

= =

= =

drag

moment


G. Piana | MMPS 2015 17 / 48

induces a bending of the deck-girder in the horizontal plane (xz-plane), and therefore produces the bending moment about the vertical axis:

xpym

( ).2y xzm p l z= −

Considering a deformed configuration of the deck-girder characterized by v(z) and ϑ(z), the following destabilizing distributed actions arise:

ym( ϑ )" vertical load ym v" torque moment


G. Piana | MMPS 2015 18 / 48

Moreover, the steady torsion ϑ(z) produces also an additional distributed aerodynamic moment.This can be expressed in the following form, linearized in a neighborhood of ϑ = 0:

being (0) = (dCM / dϑ)ϑ = 0 , and where we have set

( ) ( ) ( )

( )

2 21 02

,

z M

s

m z U B C z

z

ρ ϑ

μ ϑ

′Δ = =

=

( )2 21 0 .2s MU B Cμ ρ ′=MC′


G. Piana | MMPS 2015 19 / 48

Now, with the aim of analyzing the free bending-torsional oscillations of the bridge, let us replace the distributed vertical load p and torque moment m with the forces of inertia:

where is the bridge mass per unit length and Iϑ denotes the bridge polar mass moment of inertia per unit length, about the z-axis.

2 2

2 2d d, ,d dg

vp m It tϑ

ϑμ= − = −

gμ

( )

24 2

4 2 0

24 22 2

4 2 0

d d d ,d d

d d d .d d

lg c c

lg c ct

q E Av vEI H p v zH lz z

q E AEI GI H b m b z

H lz zωϑ ϑ ϑ

⎛ ⎞− = − ⎜ ⎟⎜ ⎟

⎝ ⎠

⎛ ⎞− + = − ⎜ ⎟⎜ ⎟

⎝ ⎠

∫

∫


G. Piana | MMPS 2015 20 / 48

Thus, taking into account the second-order effects induced by the aerodynamic drag and moment, the bending-torsional oscillations of the bridge are governed by the following partial differential equations:

hR and hL being the additional horizontal components of the cables tension:

( )( ) ( )

( ) ( ) ( )

24 2 2 2

4 2 2 2 2

4 2 2 2 22

4 2 2 2 2

,

,

yg R L

t y s R L

mv v v yE I H h t h tz z z t z

v yE I GI H b m I b h t h tz z z t zω ϑ

ϑμ

ϑ ϑ ϑμ ϑ

∂∂ ∂ ∂ ∂ ⎡ ⎤− + = − + +⎣ ⎦∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ⎡ ⎤− + + − = − − −⎣ ⎦∂ ∂ ∂ ∂ ∂

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0

2 2, , d , , , d .

l lg gc c c cR L

q qE A E Ah t v z t b z t z h t v z t b z t z

l H l Hϑ ϑ= − = +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∫ ∫

torque moments

vertical load






• Final remarks

G. Piana | MMPS 2015 21 / 48

Antisymmetric modesMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 22 / 48

In the case of antisymmetric oscillations of the deck-girder, the functions v(z,t) and ϑ(z,t) are such that the additional forces hR and hL are identically zero.

The equations of motion are therefore reduced to the following coupled equations:

where

( )

( )

24 2 2

4 2 2 2

4 2 2 22

4 2 2 2

,

,

yg

t y s

mv v vEI Hz z z t

vE I GI H b m Iz z z tω ϑ

ϑμ

ϑ ϑ ϑμ ϑ

∂∂ ∂ ∂− + = −

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− + + − = −

∂ ∂ ∂ ∂

( ).2y xzm p l z= −

Solution for antisymmetric modesMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 23 / 48

Substituting the expression of in the previous equations, we obtain:

The solution to the previous system can be found in the following variable-separable form:

so that the boundary conditions η(0)=η(l)=η"(0)=η"(l)=ψ(0)=ψ(l)=ψ"(0)==ψ "(l)=0 are satisfied.

ym

( ) ( )

( ) ( ) ( )

4 2 2 2

1 4 2 2 2

4 2 2 22

2 4 2 2 2

, : 2 0,2

, : 0.2

x g

t x s

v v z vE z t E I H p l zzz z z t

z vE z t E I GI H b p l z Iz z z tω ϑ

ϑ ϑϑ μ

ϑ ϑ ϑμ ϑ

⎡ ⎤∂ ∂ ∂ ∂ ∂= − + − + − + + =⎢ ⎥

∂∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

∂ ∂ ∂ ∂= − + + − − + =

∂ ∂ ∂ ∂

( ) ( ) ( ) ( ) ( ) ( ), , ,v z t V t z z t t zη ϑ ψ= = Θ with ( ) ( ) 2sin ,n zz zlπη ψ= =

Solution for antisymmetric modesMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 24 / 48

We apply the Galerkin Method by imposing the following integral conditions:

and we finally obtain the following differential system in matrix form:

( )

( )( )

2 2 2 2

2 2

2 2 2 22

2 2

2 2

2 2

4 4 00

0 4 40

10 3 4 0 0 012 ,1 0 1 03 4 012

g

t

x s

n n EI Hl l VV

I n n EI GI H bl l

n V Vp n

n

ϑω

π πμ

π π

πμ

π

⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟

⎧ ⎫⎡ ⎤ ⎧ ⎫⎝ ⎠⎢ ⎥+ +⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥ ΘΘ ⎛ ⎞ ⎩ ⎭⎣ ⎦ ⎩ ⎭ ⎢ ⎥+ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤+⎢ ⎥ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫

− − = ∈⎢ ⎥ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥Θ Θ⎩ ⎭ ⎣ ⎦ ⎩ ⎭ ⎩ ⎭⎢ ⎥+⎢ ⎥⎣ ⎦

( )0

2, sin d 0, 1, 2,l

in zE z t z ilπ⎛ ⎞ = =⎜ ⎟

⎝ ⎠∫

Linear eigenvalue problemMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 25 / 48

Rewritten in symbolic form:

Looking for a general solution in the form , where ω represents the angular frequency of free oscillation, we obtain:

and therefore:

through which can be obtained as a function of and .

[ ] { } [ ] { } { } { } { }0 .px g s gM q K q p K q K qμμ⎡ ⎤ ⎡ ⎤+ − − =⎣ ⎦ ⎣ ⎦

[ ] [ ]( ) { } { }20 0 ,p

x g s gK p K K M qμμ ω⎡ ⎤ ⎡ ⎤− − − =⎣ ⎦ ⎣ ⎦

[ ] [ ]( )2det 0,px g s gK p K K Mμμ ω⎡ ⎤ ⎡ ⎤− − − =⎣ ⎦ ⎣ ⎦

xp sμ2ω

{ } { } i0 e

tq q ω=

Purely dynamic problemMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 26 / 48

•

from which we obtain the natural frequencies of the unloaded structure:

flexural torsional

2 2

2

2 4 ,vng g

n n EI Hl lπ πω

μ μ= +

2 22

2

2 1 4 .n tn n EI GI H bl I lϑ ω

ϑ

π πω⎛ ⎞

= + +⎜ ⎟⎝ ⎠

0x sp μ= =

2 2 2 22

2 2

2 2 2 22 2

2 2

4 4 0004 40

g

t

n n EI Hl l V

n n EI GI Hb Il l ω ϑ

π π ω μ

π π ω

⎡ ⎤⎛ ⎞+ −⎢ ⎥⎜ ⎟

⎧ ⎫ ⎧ ⎫⎝ ⎠⎢ ⎥ =⎨ ⎬ ⎨ ⎬⎢ ⎥ Θ⎛ ⎞ ⎩ ⎭ ⎩ ⎭⎢ ⎥+ + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Purely dynamic problemMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 27 / 48

First antisymmetric torsional mode shape

Purely static problemMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 28 / 48

•

• If :

torsional divergence lateral-torsional buckling

0xp =

2 2 2 22

2 2

4 4sc n t

n n EI GI H bl l ω

π πμ⎛ ⎞

= + +⎜ ⎟⎝ ⎠ ( )

2 2

22 2

24 43 4xc n sc n

n np EI Hll n

π π μπ

⎛ ⎞= +⎜ ⎟

+ ⎝ ⎠

0g Iϑμ = =

( )

( )

2 2 2 22 2

2 2

2 2 2 22 2 2

2 2

4 4 3 412

det 0.4 43 4

12

x

xt s

pn n EI H nl l

p n nn EI GI H bl l ω

π π π

π ππ μ

⎡ ⎤⎛ ⎞+ − +⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥ =⎢ ⎥⎛ ⎞⎢ ⎥− + + + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

• If : 0sμ =

Purely static problemMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 29 / 48

0sμ ≠• If and (interaction between torsional divergence and lateral-torsional buckling):

0xp ≠

Nondimensional moment versus nondimensional drag ( )

2

1 1

1s x

sc xc

pp

μμ

⎛ ⎞= − ⎜ ⎟⎜ ⎟

⎝ ⎠

0, 1g I nϑμ = = =

Complete characteristic problemMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 30 / 48

•

For each n, we obtain two characteristic surfaces (one for the flexural and one for the torsional natural frequencies).

0, 0, 0, 0x s gp Iϑμ μ≠ ≠ ≠ ≠

( )

( )

2 2 2 22 2 2

2 2

2 2 2 22 2 2 2

2 2

4 4 3 412

det 04 43 4

12

xg

xt s

pn n EI H nl l

p n nn EI GI Hb Il l ω ϑ

π π ω μ π

π ππ μ ω

⎡ ⎤⎛ ⎞+ − − +⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥ =⎢ ⎥⎛ ⎞⎢ ⎥− + + + − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦


G. Piana | MMPS 2015 31 / 48

Characteristic surface plotting the nondimensional flexural frequency squared in terms of the nondimensional drag and moment loads (n = 1)

Characteristic surfaces

Characteristic surfacesMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 32 / 48

Characteristic surface plotting the nondimensional torsional frequency squared in terms of the nondimensional drag and moment loads (n = 1)

Limit casesMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 33 / 48

Nondimensional flexural (left) and torsional (right) frequencies squared versus nondimensional drag for (n = 1)0sμ =

flexural torsional

Limit casesMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 34 / 48

flexural torsional

Nondimensional flexural (left) and torsional (right) frequencies squared versus nondimensional drag for (n = 1)0xp =






• Final remarks

G. Piana | MMPS 2015 35 / 48

Numerical exampleMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 36 / 48

Case study: long-span suspension bridge (Lmain = 1400 m)

All data are taken from: L. Salvatori, C. Borri / Computers and Structures 85 (2007) 675–687


G. Piana | MMPS 2015 37 / 48


G. Piana | MMPS 2015 38 / 48

Aerodynamic properties: streamlined cross-section with semicircular fairings and width-to-height ratio B/D = 14.3

0.71, 5.59, 1.23D L MC C C′ ′= = = , 75.8 m s , 0.236 Hzcr fl flU f= =


G. Piana | MMPS 2015 39 / 48

Natural frequencies of the unloaded bridge (present model):

1,0

1,0

0.099 Hz0.358 Hz

vffϑ

=

=

(1st vertical antisymmetric)(1st torsional antisymmetric)

Critical wind speeds:

,

,

,

145.1m s86.4 m s

75.8m s , 0.236 Hz

cr div

cr lat

cr fl fl

UU

U f

=

=

= =

(present model)(present model)

(Salvatori & Borri)

Static instability

Dynamic instability


G. Piana | MMPS 2015 40 / 48

Natural frequencies under steady wind (present model):

Flexural U = 0 U = 0.5 Ucr,fl U = 0.6 Ucr,fl U = 0.7 Ucr,fl U = 0.8 Ucr,fl U = 0.9 Ucr,fl

fv1, Hz 0.0991 0.0969 0.0944 0.0899 0.0821 0.0684

Diff., % 0.00 -2.18 -4.73 -9.29 -17.18 -30.99

Torsional U = 0 U = 0.5 Ucr,fl U = 0.6 Ucr,fl U = 0.7 Ucr,fl U = 0.8 Ucr,fl U = 0.9 Ucr,fl

fθ1, Hz 0.3583 0.3465 0.3416 0.3361 0.3303 0.3243

Diff., % 0.00 -3.30 -4.67 -6.20 -7.83 -9.50






• Final remarks

G. Piana | MMPS 2015 41 / 48

Implications in flutter analysisMotivations and objectives Reference model … Free vibrations and stability … Implications in flutter analysis Final remarks

G. Piana | MMPS 2015 42 / 48

In linear finite element analyses, the general aeroelastic motion equationsof bridge systems are usually expressed in terms of the generalized modal coordinate vector {δ}:

[M], [C], and [K] are the generalized mass, damping, and stiffness matrices, respectively; [C *] and [K *] are the generalized aerodynamic damping and aerodynamic stiffness matrices, respectively, that are functions of the flutter derivatives.

By assuming harmonic oscillations in the form , the flutter speed, UCF, and the flutter frequency, ωF, are obtained from the nontrivial solution to Eq. (*).

[ ]{ } [ ] { } [ ] { } { }21 1 0 .2 2

M C U C K U Kδ ρ δ ρ δ∗ ∗⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤+ − + − =⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠

{ } { } i0 e

tωδ δ=

(*)


G. Piana | MMPS 2015 43 / 48

Based on what we have shown, Eq. (*) can be modified as follows:

where [Kg] is the generalized geometric stiffness matrix.

Therefore, the flutter speed, UCF, and the flutter frequency, ωF, can be obtained from the nontrivial solution to Eq. (**), which is given by the following complex eigenproblem:

(**)[ ]{ } [ ] { } [ ] { } { }2 21 1 1 0 ,2 2 2gM C U C K U K U Kδ ρ δ ρ ρ δ∗ ∗⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤⎡ ⎤+ − + − − =⎜ ⎟ ⎜ ⎟⎣ ⎦⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠

[ ] [ ] [ ]2 2 21 1 1det i 0.2 2 2gM C U C K U K U Kω ω ρ ρ ρ∗ ∗⎛ ⎞⎛ ⎞⎡ ⎤ ⎡ ⎤⎡ ⎤− + − + − − =⎜ ⎟⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎝ ⎠⎝ ⎠

Implications in flutter analysis






• Final remarks

G. Piana | MMPS 2015 44 / 48

Final remarks

G. Piana | MMPS 2015 45 / 48


We presented a simplified analytical model through which we showed that the natural frequencies of a suspension bridge deck-girder are affected by the mean wind loading.

We focused our analysis on the antisymmetric oscillations and we obtained the corresponding characteristic surfaces by solving a generalized eigenvalue problem in which the geometric stiffness matrix modifies the global stiffness of the system.

We presented some numerical results and we suggested the possibility of modifying the classic linear equations used in flutter analysis, in order to take into account the effect of motion-independent wind loads on the bridge global stiffness.

Final remarks

Future developments can regard:

► adding of self-excited forces

► implementation in a finite element code

► comparison to more sophisticated nonlinear models(*)

G. Piana | MMPS 2015 46 / 48


(*)Abdel-Ghaffar, A.M. Suspension bridge vibration: continuum formulation. Journal of Engineering Mechanics-ASCE, 108:1215–1232 (1982).

Abdel-Ghaffar, A.M., and Rubin, L.I. Nonlinear free vibrations of suspension bridges: theory. Journal of Engineering Mechanics-ASCE, 109:313–345 (1983).

Lacarbonara, W. Nonlinear structural mechanics: theory, dynamical phenomena and modeling, Springer, New York (2013).

Final remarks

G. Piana | MMPS 2015 47 / 48


G. Piana, A. Manuello, R. Malvano, A. Carpinteri, “Natural Frequencies of Long-Span Suspension Bridges Subjected to Aerodynamic Loads”, F.N. Catbas (ed.), Dynamics of Civil Structures, Volume 4: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, Conference Proceedings of the Society for Experimental Mechanics Series, 2014.

Thanks for your attention

[email protected]

G. Piana | MMPS 2015 48 / 48

on the dynamics of suspension bridge decks with...

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