Physics of Bridges

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Physics of Bridges

• Norman Kwong
• Physics 409D

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Forces

• Before we take a look at bridges, we must first
  understand what are forces.
• So, what is a force?
• A force is a push or a pull
• How can we describe forces?
• Lets a take a look at Newtons law

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Newtons Laws

• Sir Isaac Newton helped create the three laws of
  motion
• Newtons First law
• When the sum of the forces acting on a particle
  is zero, its velocity is constant. In
  particular, if the particle is initially
  stationary, it will remain stationary.
• an object at rest will stay at rest unless acted
  upon

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Newtons Laws Continued

• Newtons Second law
• A net force on an object will accelerate itthat
  is, change its velocity. The acceleration will be
  proportional to the magnitude of the force and in
  the same direction as the force. The
  proportionality constant is the mass, m, of the
  object.
• F mass acceleration

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Newtons Laws Continued

• Newtons Third law
• The forces exerted by two particles on each other
  are equal in magnitude and opposite in direction
• for every action, there is an equal and opposite
  reaction

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So what do the laws tell us?

• Looking at the second law we get Newtons famous
  equation for force Fma m is equal to the mass
  of the object and a is the acceleration
• Units of force are Newtons
• A Newton is the force required to give a mass of
  one kilogram and acceleration of one metre per
  second squared (1N1 kg m/s2)

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So what do the laws tell us?

• However, a person standing still is still being
  accelerated
• Gravity is an acceleration that constantly acts
  on you
• Fmg where g is the acceleration due to gravity

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So what do the laws tell us?

• Looking at the third law of motion
• for every action, there is a equal and opposite
  reaction
• So what does this mean?
• Consider the following diagram
• A box with a force due to gravity

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So what do the laws tell us?

• for every action, there is an equal and opposite
  reaction
• A force is being exerted on the ground from the
  weight of the box. Therefore the ground must
  also be exerting a force on the box equal to the
  weight of the box
• Called the normal force or FN

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So what do the laws tell us?

• From the first law
• An object at rest will stay at rest unless acted
  upon
• This means that the sums of all the forces but be
  zero.
• Lets look back at our diagram

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The idea of equilibrium

• The object is stationary, therefore all the
  forces must add up to zero
• Forces in the vertical direction FN and Fg
• There are no horizontal forces

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The idea of equilibrium

• But FN is equal to Fg (from Newtons third law)
• Adding up the forces we get FN Fg Fg Fg
  0
• The object is said to be in equilibrium when the
  sums of the forces are equal to zero

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Equilibrium

• Another important aspect of being in equilibrium
  is that the sum of torques must be zero
• What is a torque?
• A torque is the measure of a force's tendency to
  produce torsion and rotation about an axis.
• A torque is defined as tDF where D is the
  perpendicular distance to the force F.
• A rotation point must also be chosen as well.

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Torques

• Torques cause an object to rotate
• We evaluate torque by which torques cause the
  object to rotate clockwise or counter clockwise
  around the chosen rotation point

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But what if the force isnt straight?

• In all the previous diagrams, the forces have all
  been perfectly straight or they have all been
  perpendicular to the object.
• But what if the force was at an angle?

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Forces at an Angle

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

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Forces at an Angle

• To get the vertical component of the force, we
  need to use trigonometry (also known as the
  x-component)
• The red portion is the vertical part of the
  angled force (also known as the y-component
• Tis the angle between the force and its
  horizontal part

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• To calculate the vertical part we take the sin of
  the force
• Fvertical F sin (T)
• Lets do a quick sample calculation
• Assume T60o and F600N
• Fvertical 600N sin (60o) 519.62N

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Forces at an Angle

• Like wise, we can do the calculation of the
  horizontal (the blue) portion by taking the
  cosine of the angle
• Fhorizontal F cos (T)
• Fhorizontal 600N cos (60o) 300N

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Bridges

• Now that we have a rough understanding of forces,
  we can try and relate them to the bridge.
• A bridge has a deck, and supports
• Supports are what holds the bridge up
• Forces exerted on a support are called reactions
• Loads are the forces acting on the bridge

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Bridges

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

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Beam Bridge

• Consider the following bridge
• The beam bridge
• One of the simplest bridges

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What are the forces acting on a beam bridge?

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

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Forces on a beam bridge

• Here the red represents the weight of the bridge
  and the blue represents the reaction of the
  supports
• Assuming the weight is in the center, then the
  supports will each have the same reaction

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Forces on a beam bridge

• Lets try to add the forces
• Horizontal forces (x-direction) there are none
• Vertical forces (y-direction) the force from the
  supports and the weight of the bridge

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Forces on a beam bridge

• Lets assume the bridge has a weight of 600N.
• From the sums of forces Fy -600N 2 Fsupport0
• Doing the calculation, the supports each exert a
  force of 300N

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• To meet the other condition of equilibrium, we
  look at the torques (tDF) with the red point
  being our rotation point
• t (1m)(600N)-(2m)(600N)(3m)(600N) 0

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Limitations

• With all bridges, there is only a certain weight
  or load that the bridge can support
• This is due to the materials and the way the
  forces are acted upon the bridge

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What is happening?

• There are 2 more other forces to consider in a
  bridge.
• Compression forces and Tension forces.
• Compression is a force that acts to compress or
  shorten the thing it is acting on
• Tension is a force that acts to expand or
  lengthen the thing it is acting on

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• There is compression at the top of the bridge and
  there is tension at the bottom of the bridge
• The top portion ends up being shorter and the
  lower portion longer
• A stiffer material will resist these forces and
  thus can support larger loads

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Bridge Jargon

• Buckling is what happens to a bridge when the
  compression forces overcome the bridges ability
  to handle compression. (crushing of a pop can)
• Snapping is what happens to a bridge when the
  tension forces overcome the bridges ability to
  handle tension. (breaking of a rubber band)
• Span is the length of the bridge

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How can deal with these new forces?

• If we were to dissipate the forces out, no one
  spot has to bear the brunt of the concentrated
  force.
• In addition we can transfer the force from an
  area of weakness to an area of strength, or an
  area that is capable of handling the force

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A natural form of dissipation

• The arch bridge is one of the most natural
  bridges.
• It is also the best example of dissipation

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• In a arch bridge, everything is under compression
• It is the compression that actually holds the
  bridge up
• In the picture below you can see how the
  compression is being dissipated all the way to
  the end of the bridge where eventually all the
  force gets transferred to the ground

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Compression in a Arch

• Here is another look at the compression
• The blue arrow here represents the weight of the
  section of the arch, as well as the weight above
• The red arrows represent the compression

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Arches

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

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A Stronger Bridge

• Another way to increase the strength of a bridge
  is to add trusses
• What are trusses??
• A truss is a rigid framework designed to support
  a structure
• How does a truss help the bridge?
• A truss adds rigidity to the beam, therefore,
  increasing its ability to dissipate the
  compression and tension forces

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So what does a truss look like?

• A truss is essentially a triangular structure.
• Consider the following bridge (Silver Bridge,
  South Alouette River, Pitt Meadows BC )

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Trusses

• We can clearly see the triangular structure built
  on top of a basic beam bridge.
• But how does the truss increase the ability to
  handle forces?
• Remember a truss adds rigidity to the beam,
  therefore, increasing its ability to dissipate
  the compression and tension forces

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Trusses

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

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• Lets take a look at the forces here
• Assumptions all the triangles are equal lateral
  triangles, the angle between the sides is 60o

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• Lets see how the forces are spread out

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• Sum of torques (1m)(-400N)
  (3m)(-800N)(4m)E0
• E700N
• Sum of forces AY E - 400N - 800N
• Ay500N

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• Now that we know how the forces are laid out,
  lets take a look at what is happening at point A
• Remember that all forces are in equilibrium, so
  they must add up to zero

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• Sum of FxTAC TAB cos 60o 0
• Sum of FyTAB sin 60o 500N 0
• Solving for the two above equations we get
• TAB -577N TAC 289N

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Compression and Tension

• TAB -577N
• TAC 289N
• The negative force means that there is a
  compression force and a positive force means that
  there is a tension force

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• Lets take a look at point B

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• Sum of Fx TBD TBC cos 60o 577 cos 60o 0
• Sum of Fy -400N 577sin60o TBCsin60o0
• Once again, solving the two equations
• TBC115N TBD-346N

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Tension and Compression

• TBC115N
• TBD-346N
• The negative force means that there is a
  compression force and a positive force means that
  there is a tension force

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Forces in a Truss

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

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Limitations of a Truss

• As we can see from our demo, the truss can easily
  hold up weights, but there is a limitation.
• Truss bridges are very heavy due to the massive
  amount of material involved in its construction.

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Limitations of a Truss

• In order to holder larger loads, the trusses need
  to be larger, but that would mean the bridge gets
  heavier
• Eventually the bridge would be so heavy, that
  most of the truss work is used to hold the bridge
  up instead of the load

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Suspension Bridge

• Due to the limitations of the truss bridge type,
  another bridge type is needed for long spans
• A suspension bridge can withstand long spans as
  well as a fairly decent load.

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How Suspension Bridge Works

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

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Suspension Bridge

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

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Suspension Bridge

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

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Anchorages

• Each supporting cable is actually many smaller
  cables bound together
• At the anchorage points, the main cable separates
  into its smaller cables
• The tension from the main cable gets dispersed to
  the smaller cables
• Finally the tensional forces are dissipated into
  the ground via the anchorage

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Suspension Bridge Cable

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

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A Variation on the Suspension

• A cable stayed bridge is a variation of the
  suspension bridge.
• Like the suspension bridge, the cable stayed
  bridge uses cables to hold the bridge and loads up

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Comparison
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Forces in a Cable Stayed

• A cable stayed bridge uses the cable to hold up
  the deck
• The tension forces in the cable are transferred
  to the towers where the tension forces become
  compression forces

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Forces in a Cable Stayed

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

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Forces in a Cable Stayed

• The Lifting force holds up the bridge
• The higher the angle that the cable is attached
  to the deck, the more load it can withstand, but
  that would require a higher tower, so there has
  to be some compromise

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Limitations

• With all cable type bridges, the cables must be
  kept from corrosion
• If the bridge wants to be longer, in most cases
  the towers must also be higher, this can be
  dangerous in construction as well during windy
  conditions
• The bridge is only as good as the cable
• If the cables snap, the bridge fails