The Collapse of Tacoma Narrows Bridge: The Ordinary Differential Equations Surrounding the

Bridge Known as Galloping Gertie

Math 6700 Presentation

Qualitative Ordinary Differential Equations

Laura Lowe and Laura Singletary

December 15, 2009

Construction on the Tacoma Construction on the Tacoma Narrows Bridge began in Narrows Bridge began in November 1938 and was November 1938 and was completed on July 1, 1940. completed on July 1, 1940.

The structure of the bridge The structure of the bridge was characterized by was characterized by lightness, grace, and lightness, grace, and flexibility.flexibility.

The completion of the bridge The completion of the bridge was heralded as “a was heralded as “a triumph of man's ingenuity triumph of man's ingenuity and perseverance”and perseverance”

The Tacoma Narrows Bridge

Video

Galloping Gertie’s Gallop

During construction, the During construction, the roadbed flexed or displayed roadbed flexed or displayed vertical oscillationsvertical oscillations

This raised questions about the This raised questions about the bridge’s stability.bridge’s stability.

A light wind of 4 mph could A light wind of 4 mph could cause these oscillationscause these oscillations

Additional engineers were Additional engineers were contracted to “fix” these contracted to “fix” these oscillationsoscillations

People came from hundreds of People came from hundreds of miles away to experience miles away to experience driving on Galloping Gertiedriving on Galloping Gertie

Galloping Gertie

Galloping Gertie:

On the fateful morning, On the fateful morning, the center span the center span experienced vertical experienced vertical oscillations of 3 to 5 oscillations of 3 to 5 ft under wind speeds ft under wind speeds of 42 mphof 42 mph

The Bridge was closed The Bridge was closed by 10:00 AM on by 10:00 AM on November 7, 1940November 7, 1940

Galloping Gertie:

The motion changed to a two-The motion changed to a two-wave torsional motion wave torsional motion causing the roadbed to tilt as causing the roadbed to tilt as much as 45 degreesmuch as 45 degrees

This motion continued for 30 This motion continued for 30 minutes before a panel from minutes before a panel from the center span broke offthe center span broke off

By 11:00 am, the By 11:00 am, the center span of center span of the Tacoma the Tacoma Narrows Bridge Narrows Bridge had fallen into had fallen into Puget SoundPuget Sound

Galloping Gertie’s Gallop

The Collapse:

Reporters, engineers, and passersby witnessed the collapse at 11:00 AM

The only fatality was a dog abandoned in a car on the bridge

Eventually called the “Pearl Harbor of Engineering”

"Just as I drove past the towers, the bridge began to sway violently from side to side. Before I realized it, the tilt became so violent that I lost control of the car... I jammed on the brakes and got out, only to be thrown onto my face against the curb."Around me I could hear concrete cracking. I started to get my dog Tubby, but was thrown again before I could reach the car. The car itself began to slide from side to side of the roadway."On hands and knees most of the time, I crawled 500 yards or more to the towers... My breath was coming in gasps; my knees were raw and bleeding, my hands bruised and swollen from gripping the concrete curb... Toward the last, I risked rising to my feet and running a few yards at a time... Safely back at the toll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows."

Eyewitness Report by Leonard Coatsworth :

"A few minutes later I saw a side girder bulge "A few minutes later I saw a side girder bulge out on the Gig Harbor side, due to a failure, out on the Gig Harbor side, due to a failure, but though the bridge was buckling up at an but though the bridge was buckling up at an angle of 45 degrees the concrete didn't break angle of 45 degrees the concrete didn't break up. Even then, I thought the bridge would be up. Even then, I thought the bridge would be able to fight it out. Looking toward the Gig able to fight it out. Looking toward the Gig Harbor end, I saw the suspenders -- vertical Harbor end, I saw the suspenders -- vertical steel cables -- snap off and a whole section of steel cables -- snap off and a whole section of the bridge caved in. The main cable over that the bridge caved in. The main cable over that part of the bridge, freed of its weight, part of the bridge, freed of its weight, tightened like a bow string, flinging tightened like a bow string, flinging suspenders into the air like so many fish lines. suspenders into the air like so many fish lines. I realized the rest of the main span of the I realized the rest of the main span of the bridge was going so I started for the Tacoma bridge was going so I started for the Tacoma end."end."

Purpose:

Our project plans to explore the events surrounding this catastrophe:

We will explore a complex model proposed by Dr. P. J. McKenna that takes into account the potential and kinetic energy involved with the oscillations of the Bridge.

Mathematical Models

Mathematical Models

Vertical Kinetic Energy:

Torsional Kinetic Energy:

Total Kinetic Energy:€

KET = 12 mv 2

€

KE r = 124 mL2ω 2

€

KETotal = 12

m˙ y 2 + 124

mL2 ˙ θ 2

Mathematical Models

Mathematical Models

Vertical Potential Energy:

Torsional Potential Energy:

Total Potential Energy:€

PET = −mgy

€

PETotal = 12 k y − L

2 sinθ( )+

( )2

+ y + L2 sinθ( )

+

( )2 ⎛

⎝ ⎜

⎞

⎠ ⎟− mgy

€

PER = 12 k y − L

2 sinθ( )+

( )2

+ y + L2 sinθ( )

+

( )2 ⎛

⎝ ⎜

⎞

⎠ ⎟

Mathematical Models

Lagrangian Equations:

€

d

dt

€

∂L∂ ˙ θ

⎛

⎝ ⎜

⎞

⎠ ⎟=

∂L

∂θ

€

d

dt

€

∂L∂˙ y

⎛

⎝ ⎜

⎞

⎠ ⎟=

∂L

∂y

L = KETotal – PE Total

Mathematical Models

€

˙ ̇ θ = 6kmL cosθ y − L

2 sinθ( )+

+ y + L2 sinθ( )

+

[ ] + f (t) +δ ˙ θ

€

˙ ̇ y = − km y − L

2 sinθ( )+

+ y + L2 sinθ( )

+

[ ] + g + δ ˙ y

After a bunch of math…

is the forcing term

are the dampening terms

€

f (t)

€

δ ˙ θ , δ ˙ y

Mathematical Models

Assuming the cables never lose tension, let

and

€

y + L2 sinθ( )

+= y + L

2 sinθ

€

y − L2 sinθ( )

+= y − L

2 sinθ

€

˙ ̇ θ = −6km sinθ cosθ + f ( t) + δ ˙ θ

€

˙ ̇ y = − 2km y + g + δ ˙ y

Then simplify and we get:

Mathematical Models

•The mass of the center span was approximately 2500 kg/ft, and 12m wide.

•The bridge deflected approximately 0.5m per 100kg/ft.

Hooke’s Law: F = -kx

So k ≈ 1000

Mathematical Models

Let

Then, since the bridge oscillated at a frequency of 12-14 cycles per minute,

And let€

f ( t) =λ sin μt

€

μ =1.2 →1.6

€

0 ≤ λ ≤ 3o = 0.052 radians

€

δ =0.01

Mathematical Models

Put it all together and we get:

€

˙ ̇ θ = −2.4sinθ cosθ + λ sin μt( ) − 0.01˙ θ

€

˙ ̇ y = −0.8y − 9.8 − 0.01˙ y

Mathematical Models

System of ODE’s:

€

˙ θ = ϕ

˙ ϕ = −2.4sinθ cosθ + λ sin μt( ) − 0.01ϕ

˙ y = x

˙ x = −0.8y − 9.8 − 0.01x˙ t =1

Mathematical Models

In 1940, the technology was not available to solve a non-linear system. Instead, the engineers of the time used a linearized ODE believing that would give them the information they needed.

In the following slides we will compare these two ODE’s.

€

˙ ̇ θ = −2.4θ + λ sin μt( ) − 0.01˙ θ

Linear vs. Non-linear

Linear model with initial conditions

and

(large push)

€

θ 0( ) =1.2, ˙ θ 0( ) = 0

€

μ =1.2, λ = 0.05

Linear vs. Non-Linear

Non-linear model with initial conditions

and

(large push)

€

θ 0( ) =1.2, ˙ θ 0( ) = 0

€

μ =1.2, λ = 0.05

Linear vs. Non-linear

Linear model with initial conditions

and

(large push)

€

θ 0( ) =1.2, ˙ θ 0( ) = 0

€

μ =1.3, λ = 0.05

Linear vs. Non-linear

Non-linear model with initial conditions

and

(large push)

€

θ 0( ) =1.2, ˙ θ 0( ) = 0

€

μ =1.3, λ = 0.05

Linear vs. Non-linear

Non-linear model with initial conditions

and

(large push)

€

θ 0( ) =1.2, ˙ θ 0( ) = 0

€

μ =1.25, λ = 0.05

Linear vs. Non-linear

Non-linear model with initial conditions

and

(large push)

€

θ 0( ) =1.2, ˙ θ 0( ) = 0

€

μ =1.26, λ = 0.05

The unforced, non-linear model with The unforced, non-linear model with

Unforced Model

€

θ 0( ) =1.2

Unforced Phase Portrait

Conclusions

Here we focused on the torsional motion of the bridge with specific emphasis on the forcing term.

We assumed the cables did not lose tension and that the torsion was symmetric and independent from the vertical motion.

Areas of Further Research

Broughton Bridge

Angers Bridge

Millennium Bridge

References

Hobbs, R. S. (2006). Catastrophe to triumph: Bridges of the Tacoma Narrows. Pullman, WA: Washington State University Press.

McKenna, P. J. (1999). Large torsional oscillations in suspension bridges revisited: Fixing an old approximation. The American Mathematical Monthly, 106, 1-18.