structural instability of nonlinear plates modelling ... · stability in suspension bridge models...

Structural instability of nonlinear platesmodelling suspension bridges

Elvise BERCHIO

Dipartimento di Scienze Matematiche, Politecnico di Torino

Workshop in Nonlinear PDEs, Brussels September 7-11, 2015

Elvise BERCHIO Structural instability of nonlinear plates modelling suspension bridges

References

E. B., A. Ferrero, F. Gazzola, Structural instability of nonlinear platesmodelling suspension bridges: mathematical answers to somelong-standing questions, arXiv:1502.05851

E. B., F. Gazzola, C. Zanini, Which residual mode captures the energy ofthe dominating mode in second order Hamiltonian systems?,arXiv:1410.2374


Stability in suspension bridge models

The spectacular collapse of the Tacoma Narrows Bridge has attracted theattention of engineers, physicists, and mathematicians in the last 74 years.

The collapsed Tacoma Narrows Bridge (1940).

The crucial event in the collapse was the sudden change from a vertical to atorsional mode of oscillation.

O.H. Ammann, T. von Kármán, G.B. Woodruff, The failure of the TacomaNarrows Bridge, Federal Works Agency (1941)


Stability in suspension bridge models

Many others bridges manifested similar uncontrolled torsional oscillations.

Brighton Chain Pier (1836).

Why do torsional oscillations appear suddenly?

There have been many attempts to answer but up to nowadays an "opinionon the exact cause of the Tacoma Narrows Bridge collapse is even today notunanimously shared."

R. Scott, In the wake of Tacoma. Suspension bridges and the quest foraerodynamic stability, ASCE Press (2001)

Full theoretical answers to the above questions are not available.

P.J. McKenna, Oscillations in suspension bridges, vertical and torsional,DCDS (2014)


Facts:

The only torsional mode which developed under wind action on the bridge oron the model is that with a single node at the center of the main span.

F.C. Smith, G.S. Vincent, Aerodynamic stability of suspension bridges:with special reference to the TNB, Univ. Washington Press (1950)

...the motions, which a moment before had involved a number of waves (nineor ten) had shifted almost instantly to two.

O.H. Ammann, T. von Kármán, G.B. Woodruff, The failure of the TacomaNarrows Bridge, Federal Works Agency (1941)

Further questions:

Why do torsional oscillations appear with a node at midspan?

Are there longitudinal oscillations which are more prone to generatetorsional oscillations?


PDE’s approach: a continuous model for suspension bridges

We view the bridge as a plate Ω = (0,L)× (−`, `) (` << L) which ishinged on the small edges and free on the large edges:

⇓

PDE APPROACH

Together with Alberto Ferrero (UniPMN) we set up a continuousmodel for suspension bridges.

We view the bridge as a plate

= (0,) (`, `) (` )

which is hinged on the small edges and free on the large edges:

• A. Ferrero, F. Gazzola, A partially hinged rectangular plate as amodel for suspension bridges, Disc. Cont. Dynam. Syst. A 2015

F. Gazzola - DipMat - PoliMi Maths & Bridges

L


The nonlinear hyperbolic problem

mutt +δut + E d3

12(1−σ2)∆2u+h(y ,u)= f (x , y , t) ∈ Ω×(0,T )

u(0, y , t)=uxx (0, y , t)=u(L, y , t)=uxx (L, y , t)=0 (y , t)∈(−`, `)×(0,T )

uyy (x ,±`, t)+σuxx (x ,±`, t)=0 (x , t)∈(0,L)×(0,T )

uyyy (x ,±`, t)+(2− σ)uxxy (x ,±`, t)=0 (x , t)∈(0,L)×(0,T )

u(x , y ,0)=u0(x , y) , ut (x , y ,0)=u1(x , y) (x , y)∈Ω

Physical constants: m mass density, δ > 0 damping, h restoring forcedue to the cables, f external forces, d thickness of the plate, E Youngmodulus and σ Poisson ratio


Our assumptions

X bridge = isolated system (no dissipation, no interactions);

X The suspended structure of the TNB consisted of a “mixture” ofconcrete and metal =⇒ σ = 0.2;

X TNB data: L = 2800 ft. ≈ 853.44 m , 2` = 39 ft. ≈ 11.89 m=⇒ 2`

L = 392800 ≈ 1

75 =2π150π =⇒ by scaling L = π and ` = π

150 ,

Xh(y ,u) = Υ(y)

(k1 u + k2 u3

),

Υ characteristic function of (− π150 ,− 3π

500 ) ∪ ( 3π500 ,

π150 )

Plaut-Davis J. Sound and Vibrations 2007

X change of variable (not renamed)

u(x , y , t) =

√k1

k2u

(πxL,πyL,

√k1

mt

), γ =

E d3

12k1(1− σ2)

π4

L4


Dimensionless and isolated problem.

utt +γ∆2u+Υ(y)(u + u3)=0 in Ω×(0,∞)

u(0, y , t)=uxx (0, y , t)=u(π, y , t)=uxx (π, y , t)=0 for (y , t)∈(− π150 ,

π150 )×(0,∞)

uyy (x ,± π150 , t)+0.2 · uxx (x ,± π

150 , t)=0 for (x , t)∈(0, π)×(0,∞)

uyyy (x ,± π150 , t)+1.8 · uxxy (x ,± π

150 , t)=0 for (x , t)∈(0, π)×(0,∞)

u(x , y , 0)=u0(x , y) , ut (x , y , 0)=u1(x , y) for (x , y)∈Ω

where Ω = (0, π)× (− π150 ,

π150 ) and σ = 0.2.

The ibv problem is isolated, its energy is constant in time:

E(u) =

∫Ω

u2

t2 + γ

2 (∆u)2 + 4γ5 (u2

xy − uxxuyy ) + Υ(y)(

u2

2 + u4

4

).


The associated eigenvalues problem∆2w = λw (x , y) ∈ Ω

w(0, y) = wxx (0, y) = w(π, y) = wxx (π, y) = 0 y ∈ (−`, `)wyy (x ,±`) + σwxx (x ,±`) = wyyy (x ,±`) + (2− σ)wxxy (x ,±`) = 0 x ∈ (0, π)

A. Ferrero, F. Gazzola, A partially hinged rectangular plate as a modelfor suspension bridges, Disc. Cont. Dynam. Syst. A 2015

TheoremThe set of eigenvalues may be ordered in an increasing sequenceλk of strictly positive numbers diverging to +∞;Any eigenfunction belongs to C∞(Ω) and the set of eigenfunctions isa complete system in

H2∗(Ω) :=

w ∈ H2(Ω); w = 0 on 0, π × (−`, `)

.

Moreover, all the eigenvalues and eigenfunctions belong to one of thefollowing families:


(i) for any m ≥ 1, there exists a unique eigenvalueλ = µm,1 ∈ ((1− σ)2m4,m4) with corresponding eigenfunction[

Am cosh(

y√

m2 + µ1/2m,1

)+ Bm cosh

(y√

m2 − µ1/2m,1

)]sin(mx) ;

(ii) for any m ≥ 1, there exist infinitely many eigenvaluesλ = µm,k > m4 (k ≥ 2) with corresponding eigenfunctions[

Am cosh(

y√µ

1/2m,k + m2

)+ Bm cos

(y√µ

1/2m,k −m2

)]sin(mx) ;

These are the longitudinal eigenfunctions, of the kind cm sin(mx).


(iii) for any m ≥ 1, there exist infinitely many eigenvaluesλ = νm,k > m4 (k ≥ 2) with corresponding eigenfunctions[

Am sinh(

y√ν

1/2m,k + m2

)+ Bm sin

(y√ν

1/2m,k −m2

)]sin(mx) ;

(iv) for any m ≥ 1 satisfying `m√

2 coth(`m√

2) >( 2−σ

σ

)2there exists

an eigenvalue λ = νm,1 ∈ (µm,1,m4) with corresponding eigenfunction[Am sinh

(y√

m2 + ν1/2m,1

)+ Bm sinh

(y√

m2 − ν1/2m,1

)]sin(mx) .

These are the torsional eigenfunctions, of the kind cmy sin(mx).

the destructive oscillations observed in several suspension bridgesare of the kind: y sin(2x) (one node at midspan).


The eigenvalues solve explicit equations, we numerically obtained thefollowing values:

eigenvalue λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8kind µ1,1 µ2,1 µ3,1 µ4,1 µ5,1 µ6,1 µ7,1 µ8,1√

eigenvalue ≈ 0.98 3.92 8.82 15.68 24.5 35.28 48.02 62.73

eigenvalue λ9 λ10 λ11 λ12 λ13 λ14 λ15 λ16kind µ9,1 µ10,1 ν1,2 µ11,1 µ12,1 µ13,1 µ14,1 ν2,2√

eigenvalue ≈ 79.39 98.03 104.61 118.62 141.19 165.72 192.21 209.25

Approximate value of the least 16 eigenvalues.

X the 10 lowest eigenfunctions are all longitudinal

X the first torsional eigenfunction is the 11th which corresponds to theeigenvalue ν1,2 and to the eigenfunction ≈ Cy sin x

X the second torsional eigenfunction is the 16th which corresponds to theeigenvalue ν2,2 and to the eigenfunction ≈ Cy sin 2x

X Federal Report of 1941: in the TNB the motions, which a moment beforehad involved ten waves had shifted almost instantly to two (having onenode at midspan)

Why sin(10x)→ y sin(2x) ?


Back to the evolution equation

utt +γ∆2u+Υ(y)(u + u3)=0 in (0, π)× (− π150 ,

π150 )×(0,∞)

u(0, y , t)=uxx (0, y , t)=u(π, y , t)=uxx (π, y , t)=0 for (y , t)∈(− π150 ,

π150 )×(0,∞)

uyy (x ,± π150 , t)+0.2 · uxx (x ,± π

150 , t)=0 for (x , t)∈(0, π)×(0,∞)

uyyy (x ,± π150 , t)+1.8 · uxxy (x ,± π

150 , t)=0 for (x , t)∈(0, π)×(0,∞)

u(x , y , 0)=u0(x , y) , ut (x , y , 0)=u1(x , y) for (x , y)∈(0, π)× (− π150 ,

π150 )

u ∈ C0(R+; H2∗(Ω)) ∩ C1(R+; L2(Ω)) ∩ C2(R+;H(Ω))

is a solution if it satisfies the initial conditions and if

〈u′′(t), v〉+γ(u(t), v)H2∗

+(h(y , u(t)), v)L2 = 0 ∀v ∈ H2∗(Ω) and ∀t ∈ (0,T ) ,

H2∗(Ω) :=

w ∈ H2(Ω); w = 0 on 0, L × (−`, `)

with H(Ω) its dual space.

Theorem

For all u0 ∈ H2∗(Ω) and u1 ∈ L2(Ω) there exists a unique global solution u.


Finite dimensional analysis: Galerkin procedure

um(t) =m∑

i=1

gmi (t)wi and gm(t) := (gm

1 (t), . . . , gmm (t))T

(wi eigenfunctions) as m→∞ we have

um → u in C0([0,T ]; H2∗(Ω)) ∩ C1([0,T ]; L2(Ω))

where(gm(t))′′ + γ Λmgm(t) + Φm(gm(t)) = 0 ∀t ∈ (0,T )

gm(0) = ((u0,w1)L2 , . . . , (u0,wm)L2 )T , (gm)′(0) = ((u1,w1)L2 , . . . , (u1,wm)L2 )T

Λm := diag(λ1, . . . , λm) and Φm : Rm → Rm

Φm(ξ1, . . . , ξm) :=

(h(

y ,m∑

j=1

ξjwj

),w1

)L2, . . . ,

(h(

y ,m∑

j=1

ξjwj

),wm

)L2

T

.


Stability analysis

The Federal Report and the fact that among the least 16 modes, there are 14longitudinal modes and 2 torsional modes, suggest to fix m = 16.

The system becomesϕ′′k (t) + γ µk,1ϕk (t) + Φk

(ϕ1(t), ..., ϕ14(t), τ1(t), τ2(t)

)= 0

τ ′′k (t) + γ νk,2τk (t) + Γk(ϕ1(t), ..., ϕ14(t), τ1(t), τ2(t)

)= 0

ϕk (0) = φk0 , ϕ′k (0) = φk

1 , τk (0) = ηk0 , τ ′k (0) = ηk

1

with

ϕk (coefficients of the longitudinal eigenfunctions) for k = 1, ..., 14,

τk (coefficients of the torsional eigenfunctions) for k = 1, 2.


We isolate each longitudinal mode ϕk by setting to 0 all the remaining 15components and we solve

ϕ′′k (t) + (γµk,1 + ak )ϕk (t) + bkϕ3k (t) = 0 ∀t > 0

ϕk (0) = φk0 , ϕ′k (0) = φk

1 .

For 1 ≤ k ≤ 14, we call k -th longitudinal mode at energy E(φk0, φ

k1) > 0 the

unique (periodic) solution ϕk of the above Cauchy problem.Since the problem is nonlinear, the period of ϕk depends on the energy.

The torsional part of the linearized system around (0, ..., ϕk , ..., 0) reads

ξ′′(t) +(γνl,2 + al + dl,kϕ

2k (t))ξ(t) = 0 , (l = 1, 2) .

Fix 1 ≤ k ≤ 14 and l = 1, 2. We say that the k -th longitudinal mode ϕk atenergy E(φk

0, φk1) is stable with respect to the l-th torsional mode if the

trivial solution of the above Hill equation is stable.


Theorem

Fix 1 ≤ k ≤ 14, l ∈ 1,2. Then there exists E lk > 0 and a strictly

increasing function Λ such that Λ(0) = 0 and such that the k -thlongitudinal mode ϕk at energy E(φk

0, φk1) is stable with respect to

the l-th torsional mode provided that

E ≤ E lk

or, equivalently, provided that

‖ϕk‖2∞ ≤ Λ(E l

k ) .

It is not a perturbation result (Floquet theory), we have explicitvalues for E l

k .

• N.E. Zhukovskii (1892), V.A. Yakubovich, V.M. Starzhinskii (1975)


Numerical experiments confirm the existence of an energythreshold where stability is lost.

Plot of the solution of the linearized equation for k = 14 and l = 1 (ϕ14(0) =

0.79 (left) and 0.8 (right), ϕ′14(0) = 0 and ξ1(0) = ξ′1(0) = 1).

k 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Ak1 1.44 0.91 0.9 0.88 0.82 0.69 >10 >10 >10 >10 0.2 0.44 0.63 0.8

Ak2 1.87 2.91 1.86 1.85 1.82 1.77 1.69 1.54 1.3 0.76 >10 >10 >10 >10

Instability thresholds for the initial data of the k -th longitudinal mode withrespect to the first and the second torsional modes: Ak

1 and Ak2


Our theoretical (and numerical) results suggest the followingexplanations of the origin of torsional instability:it is due to an internal resonance which generates an energytransfer between different oscillation modes. When the bridge isoscillating longitudinally with a suitable amplitude, part of the energyis suddenly transferred to a torsional mode giving rise to widetorsional oscillations.Furthermore, from the computed energies the tenth longitudinalmode seems the most prone to generate the second torsionalmode (on the day of the TNB collapse the motions were considerablyless than had occurred many times before).

what happens for large energies?which is the criterion governing the transfer of energy?


Prototype problem: µ, λ1, λ2 > 0, x0 ∈ R \ 0 and ε > 0,ϕ+ µ2ϕ+ Uϕ(ϕ, τ1, τ2) = 0 ϕ(0) = x0, ϕ(0) = 0τ1 + λ2

1τ1 + Uτ1 (ϕ, τ1, τ2) = 0 τ1(0) = εx0, τ1(0) = 0τ2 + λ2

2τ2 + Uτ2 (ϕ, τ1, τ2) = 0 τ2(0) = εx0, τ2(0) = 0 ,

where

U(ϕ, τ1, τ2) =ϕ2τ2

1 + ϕ2τ22 + τ2

1 τ22

2.

For ε = 0, unique solution (x0 cos(µt),0,0) with conserved energy

E :=ϕ2

2+µ2

2ϕ2 =

µ2

2x2

0 .

For ε small, we linearize the τi equations around the above solutionand we obtain the following family of Mathieu equations

w + (αi + 2q cos(2t)) w = 0

where

αi = αi (q) =λ2

iµ2 + 2q and q = q(x0) =

x20

4µ2 =E

2µ4 .


Euristic idea

The instability regions in the Mathieu diagram become more narrow asa increases. Since the parametric lines associated to each torsionalcomponent take their origin when a = λ2

i /µ2, it would be desirable that

λi >> µ.

The torsional stability of a suspension bridge depends on the ratiosbetween torsional and vertical frequencies; the larger they are, morestable is the bridge.


Coming back to the initial questions:

X Why do torsional oscillations appear with a node at midspan?

X Are there longitudinal oscillations which are more prone to generatetorsional oscillations?

The 10th longitudinal eigenvalue minimizes the ratio 2nd torsional eigenvaluelongitudinal eigenvalue

and is therefore the most unstable.

In fact, the first mode does not exist, instead of moving to y sin x , the cableforces the motion to move to y sin(2x).

P. Bergot, L. Civati, Dynamic structural instability in suspension bridges,Master Thesis, Civil Engineering, Politecnico of Milan, Italy (2014)

To be continued.....

F. Gazzola’s talk (Wednesday 14:00 p.m.)


structural instability of nonlinear plates modelling ... · stability in suspension bridge models...

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