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1. Physics of Bridges Norman Kwong
2. 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
3. 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
4. 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
5. 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
6. So what do the laws tell us? Looking at the second law we get Newtons famous equation for force: F=ma 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 (1N=1 kg m/s2)
7. 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
F=mg where g is the acceleration due to gravity
8. 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
9. 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
10. 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
11. 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
12. 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
13. 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 t=DF where D is the perpendicular distance to the force F.
A rotation point must also be chosen as well.
14. 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
15. 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?
16. 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
17. 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
18. To calculate the vertical part we take the sin of the force
Fvertical =F * sin (T)
Lets do a quick sample calculation
Assume T=60o and F=600N
Fvertical = 600N * sin (60o) = 519.62N
19. 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
20. 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
21. 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.
22. Beam Bridge Consider the following bridge
The beam bridge
One of the simplest bridges
23. 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
24. 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
25. 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
26. Forces on a beam bridge 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
27. To meet the other condition of equilibrium, we look at the torques (t=DF) with the red point being our rotation point
t= (1m)*(600N)-(2m)*(600N)+(3m)*(600N) = 0
28. 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
29. 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
30. 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
31. 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
32. 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
33. A natural form of dissipation The arch bridge is one of the most natural bridges.
It is also the best example of dissipation
34. 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
35. 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
36. Arches Here is one more look at the compression lines of an arch
37. 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
38. 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 )
39. 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
40. Trusses Lets take a look at a simple truss and how the forces are spread out
41. Lets take a look at the forces here
Assumptions: all the triangles are equal lateral triangles, the angle between the sides is 60o
42. Lets see how the forces are spread out
43. Sum of torques = (1m)*(-400N) + (3m)*(-800N)+(4m)*E=0
Sum of forces = AY + E - 400N - 800N
44. 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
45. Sum of Fx=TAC + TAB cos 60o = 0
Sum of Fy=TAB sin 60o +500N = 0
Solving for the two above equations we get
TAB = -577N TAC= 289N
46. Compression and Tension TAB = -577N
The negative force means that there is a compression force and a positive force means that there is a tension force
47. Lets take a look at point B
48. Sum of Fx = TBD + TBC cos 60o + 577 cos 60o= 0
Sum of Fy = -400N + 577sin60o TBCsin60o=0
Once again, solving the two equations
49. Tension and Compression TBC=115N
The negative force means that there is a compression force and a positive force means that there is a tension force
50. 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
51. 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.
52. 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
53. 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.
54. 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.
55. 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.
56. 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.
57. 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
58. Suspension Bridge Cable Here is a cross section picture of what a main cable of a suspension bridge looks like
59. 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
61. 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
62. Forces in a Cable Stayed Lets take a quick look at the forces at one of the cable points.
63. 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
64. 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