Physics of Physics of BridgesBridges

ForcesForces

Before we take a look at bridges, we Before we take a look at bridges, we must first understand what are must first understand what are forces.forces.

So, what is a force?So, what is a force? A force is a push or a pullA force is a push or a pull

How can we describe forces?How can we describe forces? Lets a take a look at Newton’s law Lets a take a look at Newton’s law

Newton’s LawsNewton’s Laws

Sir Isaac Newton helped create the Sir Isaac Newton helped create the three laws of motionthree laws of motion

Newton’s First lawNewton’s First law When the sum of the forces acting on a When the sum of the forces acting on a

particle is zero, its velocity is constant. particle is zero, its velocity is constant. In particular, if the particle is initially In particular, if the particle is initially stationary, it will remain stationary.stationary, it will remain stationary.

““an object at rest will stay at rest unless an object at rest will stay at rest unless acted upon”acted upon”

Newton’s Laws Newton’s Laws ContinuedContinued

Newton’s Second lawNewton’s Second law A net force on an object will accelerate it—A net force on an object will accelerate it—

that is, change its velocity. The acceleration that is, change its velocity. The acceleration will be proportional to the magnitude of the will be proportional to the magnitude of the force and in the same direction as the force. force and in the same direction as the force. The proportionality constant is the mass, The proportionality constant is the mass, m,m, of the object. of the object.

““F = mass * acceleration”F = mass * acceleration”

Newton’s Laws Newton’s Laws ContinuedContinued

Newton’s Third lawNewton’s Third law The forces exerted by two particles on The forces exerted by two particles on

each other are equal in magnitude and each other are equal in magnitude and opposite in directionopposite in direction

““for every action, there is an equal and for every action, there is an equal and opposite reaction”opposite reaction”

So what do the laws tell So what do the laws tell us?us?

Looking at the second law we get Looking at the second law we get Newton’s famous equation for force: Newton’s famous equation for force: F=ma m is equal to the mass of the F=ma m is equal to the mass of the object and a is the accelerationobject and a is the acceleration Units of force are NewtonsUnits of force are Newtons

A Newton is the force required to give a A Newton is the force required to give a mass of one kilogram and acceleration of mass of one kilogram and acceleration of one metre per second squared (1N=1 kg one metre per second squared (1N=1 kg m/sm/s22))

So what do the laws tell So what do the laws tell us?us?

However, a person However, a person standing still is standing still is still being still being acceleratedaccelerated Gravity is an Gravity is an

acceleration that acceleration that constantly acts on constantly acts on youyou

F=mg where g is F=mg where g is the acceleration the acceleration due to gravitydue to gravity

So what do the laws tell So what do the laws tell us?us?

Looking at the third law of motionLooking at the third law of motion ““for every action, there is a equal and for every action, there is a equal and

opposite reaction”opposite reaction” So what does this mean?So what does this mean?

Consider the following diagramConsider the following diagram A box with a force due to gravityA box with a force due to gravity

So what do the laws tell So what do the laws tell us?us?

““for every action, for every action, there is an equal and there is an equal and opposite reaction”opposite reaction”

A force is being A force is being exerted on the ground exerted on the ground from the weight of the from the weight of the box. Therefore the box. Therefore the ground must also be ground must also be exerting a force on exerting a force on the box equal to the the box equal to the weight of the boxweight of the box Called the normal force Called the normal force

or For FNN

So what do the laws tell So what do the laws tell us?us?

From the first law:From the first law: An object at rest An object at rest

will stay at rest will stay at rest unless acted uponunless acted upon

This means that the This means that the sums of all the sums of all the forces but be zero.forces but be zero.

Lets look back at Lets look back at our diagramour diagram

The idea of equilibriumThe idea of equilibrium

The object is stationary, therefore all the The object is stationary, therefore all the forces must add up to zeroforces must add up to zero

Forces in the vertical direction: FForces in the vertical direction: FNN and F and Fgg

There are no horizontal forcesThere are no horizontal forces

The idea of equilibriumThe idea of equilibrium

But FBut FNN is equal to – F is equal to – Fg g (from Newton’s (from Newton’s third law)third law)

Adding up the forces we get FAdding up the forces we get FNN + F + Fg g = – F= – Fgg + F+ Fgg = 0 = 0

The object is said to be in equilibrium The object is said to be in equilibrium when the sums of the forces are equal to when the sums of the forces are equal to zerozero

EquilibriumEquilibrium

Another important aspect of being in Another important aspect of being in equilibrium is that the sum of equilibrium is that the sum of torques must be zerotorques must be zero

What is a torque?What is a torque? A torque is the measure of a force's A torque is the measure of a force's

tendency to produce torsion and tendency to produce torsion and rotation about an axis.rotation about an axis.

A torque is defined as A torque is defined as ττ=DF where D is =DF where D is the perpendicular distance to the force the perpendicular distance to the force F.F.

A rotation point must also be chosen as A rotation point must also be chosen as well.well.

TorquesTorques

Torques cause an Torques cause an object to rotateobject to rotate We evaluate torque We evaluate torque

by which torques by which torques cause the object to cause the object to rotate clockwise or rotate clockwise or counter clockwise counter clockwise around the chosen around the chosen rotation pointrotation point

But what if the force isn’t But what if the force isn’t straight?straight?

In all the previous diagrams, the forces In all the previous diagrams, the forces have all been perfectly straight or they have all been perfectly straight or they have all been perpendicular to the object.have all been perpendicular to the object.

But what if the force was at an angle?But what if the force was at an angle?

Forces at an AngleForces at an Angle

If the force is at an angle, we can think of If the force is at an angle, we can think of the force as a triangle, with the force the force as a triangle, with the force being the hypotenusebeing the hypotenuse

Forces at an AngleForces at an Angle To get the vertical To get the vertical

component of the component of the force, we need to force, we need to use trigonometry use trigonometry (also known as the (also known as the x-component)x-component)

The red portion is The red portion is the vertical part of the vertical part of the angled force the angled force (also known as the (also known as the y-componenty-component

Θis the angle Θis the angle between the force between the force and it’s horizontal and it’s horizontal partpart

■ To calculate the vertical part we take the To calculate the vertical part we take the sin of the forcesin of the force■ FFverticalvertical =F * sin ( =F * sin (Θ)Θ)

■ Lets do a quick sample calculationLets do a quick sample calculation■ Assume Θ=60Assume Θ=60oo and F=600N and F=600N■ FFverticalvertical = 600N * sin (60 = 600N * sin (60oo) = 519.62N) = 519.62N

Forces at an AngleForces at an Angle

■ Like wise, we Like wise, we can do the can do the calculation of calculation of the horizontal the horizontal (the blue) (the blue) portion by portion by taking the taking the cosine of the cosine of the angleangle■ FFhorizontalhorizontal= F * cos = F * cos

((Θ)Θ)■ FFhorizontalhorizontal= 600N * = 600N *

cos (cos (6060oo) =300N) =300N

BridgesBridges

Now that we have a rough Now that we have a rough understanding of forces, we can try understanding of forces, we can try and relate them to the bridge.and relate them to the bridge.

A bridge has a deck, and supportsA bridge has a deck, and supports Supports are what holds the bridge Supports are what holds the bridge

upup Forces exerted on a support are called Forces exerted on a support are called

reactionsreactions Loads are the forces acting on the Loads are the forces acting on the

bridgebridge

BridgesBridges

A bridge is held up by the reactions A bridge is held up by the reactions exerted by its supports and the loads exerted by its supports and the loads are the forces exerted by the weight are the forces exerted by the weight of the object plus the bridge itself.of the object plus the bridge itself.

Beam BridgeBeam Bridge

Consider the Consider the following bridgefollowing bridge

The beam bridgeThe beam bridge One of the simplest One of the simplest

bridgesbridges

What are the forces acting What are the forces acting on a beam bridge?on a beam bridge?

So what are the forces?So what are the forces? There is the weight of the bridgeThere is the weight of the bridge The reaction from the supportsThe reaction from the supports

Forces on a beam bridgeForces on a beam bridge

■ Here the red represents the weight of the Here the red represents the weight of the bridge and the blue represents the bridge and the blue represents the reaction of the supportsreaction of the supports

■ Assuming the weight is in the center, then Assuming the weight is in the center, then the supports will each have the same the supports will each have the same reactionreaction

Forces on a beam bridgeForces on a beam bridge

Lets try to add the forcesLets try to add the forces Horizontal forces (x-direction): there are noneHorizontal forces (x-direction): there are none Vertical forces (y-direction): the force from the Vertical forces (y-direction): the force from the

supports and the weight of the bridgesupports and the weight of the bridge

Forces on a beam bridgeForces on a beam bridge

Lets assume the bridge has a weight of Lets assume the bridge has a weight of 600N.600N.

From the sums of forces FFrom the sums of forces Fyy = -600N + 2 = -600N + 2 FFsupportsupport=0=0

Doing the calculation, the supports each Doing the calculation, the supports each exert a force of 300Nexert a force of 300N

To meet the other condition of To meet the other condition of equilibrium, we look at the torques equilibrium, we look at the torques ((ττ=DF) with the red point being our =DF) with the red point being our rotation pointrotation point

ττ= (1m)*(600N)-(2m)*(600N)= (1m)*(600N)-(2m)*(600N)+(3m)*(600N) = 0+(3m)*(600N) = 0

LimitationsLimitations

With all bridges, there is only a With all bridges, there is only a certain weight or load that the certain weight or load that the bridge can supportbridge can support

This is due to the materials and the This is due to the materials and the way the forces are acted upon the way the forces are acted upon the bridgebridge

What is happening?What is happening?

There are 2 more other forces to There are 2 more other forces to consider in a bridge.consider in a bridge.

Compression forces and Tension Compression forces and Tension forces.forces. CompressionCompression is a force that acts to is a force that acts to

compress or shorten the thing it is compress or shorten the thing it is acting on acting on

TensionTension is a force that acts to expand is a force that acts to expand or lengthen the thing it is acting onor lengthen the thing it is acting on

There is compression at the top of the bridge and There is compression at the top of the bridge and there is tension at the bottom of the bridge there is tension at the bottom of the bridge

The top portion ends up being shorter and the The top portion ends up being shorter and the lower portion longerlower portion longer

A stiffer material will resist these forces and thus A stiffer material will resist these forces and thus can support larger loadscan support larger loads

Bridge JargonBridge Jargon

Buckling is what happens to a bridge Buckling is what happens to a bridge when the compression forces when the compression forces overcome the bridge’s ability to overcome the bridge’s ability to handle compression. (crushing of a handle compression. (crushing of a pop can)pop can)

■ Snapping is what happens to a Snapping is what happens to a bridge when the tension forces bridge when the tension forces overcome the bridge’s ability to overcome the bridge’s ability to handle tension. (breaking of a handle tension. (breaking of a rubber band)rubber band)

■ Span is the length of the bridgeSpan is the length of the bridge

How can deal with these How can deal with these new forces?new forces?

If we were to dissipate the forces If we were to dissipate the forces out, no one spot has to bear the out, no one spot has to bear the brunt of the concentrated force.brunt of the concentrated force.

In addition we can transfer the force In addition we can transfer the force from an area of weakness to an area from an area of weakness to an area of strength, or an area that is of strength, or an area that is capable of handling the forcecapable of handling the force

A natural form of A natural form of dissipationdissipation

The arch bridge is The arch bridge is one of the most one of the most natural bridges. natural bridges.

It is also the best It is also the best example of example of dissipationdissipation

In a arch bridge, everything is under In a arch bridge, everything is under compressioncompression

It is the compression that actually holds the It is the compression that actually holds the bridge upbridge up

In the picture below you can see how the In the picture below you can see how the compression is being dissipated all the way to compression is being dissipated all the way to the end of the bridge where eventually all the the end of the bridge where eventually all the force gets transferred to the groundforce gets transferred to the ground

Compression in a ArchCompression in a Arch

Here is another Here is another look at the look at the compressioncompression

The blue arrow The blue arrow here represents here represents the weight of the the weight of the section of the arch, section of the arch, as well as the as well as the weight aboveweight above

The red arrows The red arrows represent the represent the compressioncompression

ArchesArches

Here is one more Here is one more look at the look at the compression lines compression lines of an archof an arch

A Stronger BridgeA Stronger Bridge

Another way to increase the Another way to increase the strength of a bridge is to add trussesstrength of a bridge is to add trusses

What are trusses??What are trusses?? A truss is a rigid framework designed to A truss is a rigid framework designed to

support a structuresupport a structure How does a truss help the bridge?How does a truss help the bridge?

A truss adds rigidity to the beam, A truss adds rigidity to the beam, therefore, increasing it’s ability to therefore, increasing it’s ability to dissipate the compression and tension dissipate the compression and tension forcesforces

So what does a truss look So what does a truss look like?like?

A truss is essentially a triangular A truss is essentially a triangular structure.structure.

Consider the following bridge (Silver Consider the following bridge (Silver Bridge, South Alouette River, Pitt Bridge, South Alouette River, Pitt Meadows BC )Meadows BC )

TrussesTrusses

We can clearly see the triangular structure built We can clearly see the triangular structure built on top of a basic beam bridge.on top of a basic beam bridge.

But how does the truss increase the ability to But how does the truss increase the ability to handle forces?handle forces? Remember a truss adds rigidity to the beam, therefore, Remember a truss adds rigidity to the beam, therefore,

increasing it’s ability to dissipate the compression and increasing it’s ability to dissipate the compression and tension forcestension forces

TrussesTrusses

Lets take a look at a simple truss and how Lets take a look at a simple truss and how the forces are spread outthe forces are spread out

Lets take a look at the forces hereLets take a look at the forces here Assumptions: all the triangles are equal Assumptions: all the triangles are equal

lateral triangles, the angle between the lateral triangles, the angle between the sides is 60sides is 60oo

Lets see how the forces are spread Lets see how the forces are spread outout

■Sum of torques = (1m)*(-400N) + (3m)*(-Sum of torques = (1m)*(-400N) + (3m)*(-800N)+(4m)*E=0800N)+(4m)*E=0

■E=700NE=700N

■Sum of forces = ASum of forces = AY Y + E - 400N - 800N+ E - 400N - 800N■AAyy=500N=500N

Now that we know how the forces are laid Now that we know how the forces are laid out, lets take a look at what is happening out, lets take a look at what is happening at point Aat point A

Remember that all forces are in Remember that all forces are in equilibrium, so they must add up to zeroequilibrium, so they must add up to zero

■ Sum of FSum of Fxx=T=TACAC + T + TABAB cos 60 cos 60oo = 0 = 0■ Sum of FSum of Fyy=T=TABAB sin 60 sin 60oo +500N = 0 +500N = 0■ Solving for the two above equations we getSolving for the two above equations we get

■ TTABAB = -577N T = -577N TACAC= 289N= 289N

Compression and Compression and TensionTension

■TTABAB = -577N = -577N

■TTACAC= 289N= 289N

■The negative force The negative force means that there is means that there is a compression a compression force and a force and a positive force positive force means that there is means that there is a tension forcea tension force

Lets take a look at point BLets take a look at point B

■Sum of FSum of Fxx = T = TBDBD + T + TBCBC cos 60 cos 60o o + 577 cos 60+ 577 cos 60oo= 0= 0

■Sum of FSum of Fyy = -400N + 577sin60 = -400N + 577sin60oo –T –TBCBCsin60sin60oo=0=0

■Once again, solving the two equationsOnce again, solving the two equations■TTBCBC=115N T=115N TBDBD=-346N=-346N

Tension and Tension and CompressionCompression

■TTBCBC=115N =115N

■TTBDBD=-346N=-346N

■The negative force The negative force means that there means that there is a compression is a compression force and a force and a positive force positive force means that there means that there is a tension forceis a tension force

Forces in a TrussForces in a Truss

If we calculated the rest of the forces If we calculated the rest of the forces acting on the various points of our truss, acting on the various points of our truss, we will see that there is a mixture of both we will see that there is a mixture of both compression and tension forces and that compression and tension forces and that these forces are spread out across the these forces are spread out across the trusstruss

Limitations of a TrussLimitations of a Truss

As we can see from our demo, the As we can see from our demo, the truss can easily hold up weights, but truss can easily hold up weights, but there is a limitation.there is a limitation.

Truss bridges are very heavy due to Truss bridges are very heavy due to the massive amount of material the massive amount of material involved in its construction.involved in its construction.

Limitations of a TrussLimitations of a Truss

In order to holder larger loads, the In order to holder larger loads, the trusses need to be larger, but that trusses need to be larger, but that would mean the bridge gets heavierwould mean the bridge gets heavier

Eventually the bridge would be so Eventually the bridge would be so heavy, that most of the truss work is heavy, that most of the truss work is used to hold the bridge up instead of used to hold the bridge up instead of the loadthe load

Suspension BridgeSuspension Bridge

■Due to the Due to the limitations of the limitations of the truss bridge type, truss bridge type, another bridge type another bridge type is needed for long is needed for long spansspans

■A suspension bridge A suspension bridge can withstand long can withstand long spans as well as a spans as well as a fairly decent load.fairly decent load.

How Suspension Bridge How Suspension Bridge WorksWorks

■A suspension bridge uses the tension of A suspension bridge uses the tension of cables to hold up a load. The cables are cables to hold up a load. The cables are kept under tension with the use of kept under tension with the use of anchorages that are held firmly to the anchorages that are held firmly to the Earth.Earth.

Suspension BridgeSuspension Bridge

■The deck is suspended from the cables and The deck is suspended from the cables and the compression forces from the weight of the compression forces from the weight of the deck are transferred the towers. the deck are transferred the towers. Because the towers are firmly in the Earth, Because the towers are firmly in the Earth, the force gets dissipated into the ground.the force gets dissipated into the ground.

Suspension BridgeSuspension Bridge

■The supporting cables that are connected The supporting cables that are connected to the anchorages experience tension to the anchorages experience tension forces. The cables stretch due to the forces. The cables stretch due to the weight of the bridge as well as the load it weight of the bridge as well as the load it carries.carries.

AnchoragesAnchorages

■ Each supporting cable is Each supporting cable is actually many smaller actually many smaller cables bound togethercables bound together

■ At the anchorage points, At the anchorage points, the main cable separates the main cable separates into its smaller cablesinto its smaller cables

■ The tension from the The tension from the main cable gets main cable gets dispersed to the smaller dispersed to the smaller cablescables

■ Finally the tensional Finally the tensional forces are dissipated forces are dissipated into the ground via the into the ground via the anchorageanchorage

Suspension Bridge CableSuspension Bridge Cable

■Here is a cross Here is a cross section picture section picture of what a main of what a main cable of a cable of a suspension suspension bridge looks likebridge looks like

A Variation on the A Variation on the SuspensionSuspension

■A cable stayed bridge is a variation of the A cable stayed bridge is a variation of the suspension bridge.suspension bridge.

■Like the suspension bridge, the cable stayed Like the suspension bridge, the cable stayed bridge uses cables to hold the bridge and bridge uses cables to hold the bridge and loads uploads up

ComparisonComparison

Forces in a Cable StayedForces in a Cable Stayed

■A cable stayed A cable stayed bridge uses the bridge uses the cable to hold up the cable to hold up the deckdeck

■The tension forces The tension forces in the cable are in the cable are transferred to the transferred to the towers where the towers where the tension forces tension forces become become compression forcescompression forces

Forces in a Cable StayedForces in a Cable Stayed

■Lets take a quick look at the forces Lets take a quick look at the forces at one of the cable points.at one of the cable points.

Forces in a Cable StayedForces in a Cable Stayed

■The “Lifting force” The “Lifting force” holds up the bridgeholds up the bridge

■The higher the angle The higher the angle that the cable is that the cable is attached to the attached to the deck, the more load deck, the more load it can withstand, but it can withstand, but that would require a that would require a higher tower, so higher tower, so there has to be some there has to be some compromisecompromise

LimitationsLimitations

With all cable type bridges, the cables With all cable type bridges, the cables must be kept from corrosionmust be kept from corrosion

If the bridge wants to be longer, in If the bridge wants to be longer, in most cases the towers must also be most cases the towers must also be higher, this can be dangerous in higher, this can be dangerous in construction as well during windy construction as well during windy conditionsconditions

““The bridge is only as good as the The bridge is only as good as the cable”cable” If the cables snap, the bridge failsIf the cables snap, the bridge fails