[ppt]physics of bridges - boy scout troop 8 - madison,...
TRANSCRIPT
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 the exerting a force on the box equal to 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 bridge certain weight or load that the bridge can supportcan 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
■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 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
■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 into forces are dissipated into the ground via the 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
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 in The tension forces in the cable are the cable are transferred to the transferred to the towers where the towers where the tension forces tension forces become compression become compression forcesforces
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 deck, attached to the deck, the more load it can the more load it can withstand, but that withstand, but that would require a 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 most If the bridge wants to be longer, in most
cases the towers must also be higher, cases the towers must also be higher, this can be dangerous in construction as this can be dangerous in construction as well during windy conditionswell during windy conditions
““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