Transcript Physics of Bridges - Boy Scout Troop 8

Physics of Bridges
Forces
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Before we take a look at bridges, we must first
understand what are forces.
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So, what is a force?
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A force is a push or a pull
How can we describe forces?
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Lets a take a look at Newton’s law
Newton’s Laws
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Sir Isaac Newton helped create the three laws of
motion
Newton’s First law
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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.
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“an object at rest will stay at rest unless acted upon”
Newton’s Laws Continued
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Newton’s Second law
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A net force on an object will accelerate it—that 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.
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“F = mass * acceleration”
Newton’s Laws Continued
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Newton’s Third law
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The forces exerted by two particles on each other are
equal in magnitude and opposite in direction
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“for every action, there is an equal and opposite
reaction”
So what do the laws tell us?
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Looking at the second law we get Newton’s
famous equation for force: F=ma m is equal to
the mass of the object and a is the acceleration
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Units of force are Newtons
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A Newton is the force required to give a mass of one
kilogram and acceleration of one metre per second
squared (1N=1 kg m/s2)
So what do the laws tell us?
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However, a person
standing still is still being
accelerated
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Gravity is an acceleration
that constantly acts on
you
F=mg where g is the
acceleration due to gravity
So what do the laws tell us?
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Looking at the third law of motion
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“for every action, there is a equal and opposite reaction”
So what does this mean?
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Consider the following diagram
A box with a force due to gravity
So what do the laws tell us?
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“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
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Called the normal force or FN
So what do the laws tell us?
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From the first law:
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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
The idea of equilibrium
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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
The idea of equilibrium
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But FN is equal to – Fg (from Newton’s 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
Equilibrium
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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 τ=DF where D is the
perpendicular distance to the force F.
 A rotation point must also be chosen as well.
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Torques
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Torques cause an object
to rotate
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We evaluate torque by
which torques cause the
object to rotate clockwise
or counter clockwise
around the chosen
rotation point
But what if the force isn’t straight?
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In all the previous diagrams, the forces have all been
perfectly straight or they have all been perpendicular to
the object.
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But what if the force was at an angle?
Forces at an Angle
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If the force is at an angle, we can think of the force as
a triangle, with the force being the hypotenuse
Forces at an Angle
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To get the vertical
component of the force,
we need to use
trigonometry (also
known as the xcomponent)
The red portion is the
vertical part of the
angled force (also known
as the y-component
Θis the angle between
the force and it’s
horizontal part
■
To calculate the vertical part we take the sin of the
force
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Fvertical =F * sin (Θ)
Lets do a quick sample calculation
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Assume Θ=60o and F=600N
Fvertical = 600N * sin (60o) = 519.62N
Forces at an Angle
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Like wise, we can do
the calculation of the
horizontal (the blue)
portion by taking the
cosine of the angle
■
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Fhorizontal= F * cos (Θ)
Fhorizontal= 600N * cos
(60o) =300N
Bridges
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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
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Forces exerted on a support are called reactions
Loads are the forces acting on the bridge
Bridges
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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.
Beam Bridge
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Consider the following
bridge
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The beam bridge
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One of the simplest
bridges
What are the forces acting on a beam
bridge?
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So what are the forces?
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There is the weight of the bridge
The reaction from the supports
Forces on a beam bridge
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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
Forces on a beam bridge
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Lets try to add the forces
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Horizontal forces (x-direction): there are none
Vertical forces (y-direction): the force from the supports and
the weight of the bridge
Forces on a beam bridge
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Lets assume the bridge has a weight of 600N.
From the sums of forces Fy = -600N + 2 Fsupport=0
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 (τ=DF) with the red point being our
rotation point
τ= (1m)*(600N)-(2m)*(600N)+(3m)*(600N) = 0
Limitations
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With all bridges, there is only a certain weight or
load that the bridge can support
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This is due to the materials and the way the
forces are acted upon the bridge
What is happening?
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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
Bridge Jargon
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■
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Buckling is what happens to a bridge when the
compression forces overcome the bridge’s ability
to handle compression. (crushing of a pop can)
Snapping is what happens to a bridge when the
tension forces overcome the bridge’s ability to
handle tension. (breaking of a rubber band)
Span is the length of the bridge
How can deal with these new forces?
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If we were to dissipate the forces out, no one
spot has to bear the brunt of the concentrated
force.
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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
A natural form of dissipation
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The arch bridge is one
of the most natural
bridges.
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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
Compression in a Arch
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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
Arches
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Here is one more look at
the compression lines of
an arch
A Stronger Bridge
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Another way to increase the strength of a bridge
is to add trusses
What are trusses??
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A truss is a rigid framework designed to support a
structure
How does a truss help the bridge?
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A truss adds rigidity to the beam, therefore,
increasing it’s ability to dissipate the compression
and tension forces
So what does a truss look like?
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A truss is essentially a triangular structure.
Consider the following bridge (Silver Bridge, South
Alouette River, Pitt Meadows BC )
Trusses
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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?
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Remember a truss adds rigidity to the beam, therefore, increasing it’s
ability to dissipate the compression and tension forces
Trusses
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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
■
Sum of torques = (1m)*(-400N) + (3m)*(-800N)+(4m)*E=0
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■
E=700N
Sum of forces = AY + E - 400N - 800N
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Ay=500N
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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 Fx=TAC + TAB cos 60o = 0
Sum of Fy=TAB sin 60o +500N = 0
Solving for the two above equations we get
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TAB = -577N TAC= 289N
Compression and Tension
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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
■ Sum of Fx = TBD + TBC cos 60o + 577 cos 60o= 0
■ Sum of Fy = -400N + 577sin60o –TBCsin60o=0
■ Once again, solving the two equations
■ TBC=115N
TBD=-346N
Tension and Compression
■
■
■
TBC=115N
TBD=-346N
The negative force
means that there is a
compression force
and a positive force
means that there is a
tension force
Forces in a Truss
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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
Limitations of a Truss
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As we can see from our demo, the truss can
easily hold up weights, but there is a limitation.
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Truss bridges are very heavy due to the massive
amount of material involved in its construction.
Limitations of a Truss
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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
Suspension Bridge
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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.
How Suspension Bridge Works
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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.
Suspension Bridge
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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.
Suspension Bridge
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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.
Anchorages
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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
Suspension Bridge Cable
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Here is a cross
section picture of
what a main cable of
a suspension bridge
looks like
A Variation on the Suspension
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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
Comparison
Forces in a Cable Stayed
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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
Forces in a Cable Stayed
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Lets take a quick look at the forces at one of the
cable points.
Forces in a Cable Stayed
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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
Limitations
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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”
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If the cables snap, the bridge fails