# BRIDGES

## The 3 ways in which a material is strong

The three ways in which materials are strong are pulling, pushing and sliding.
Engineers and scientists use the term tension for pulling, compression for pushing and shear for sliding.
These three ways to be strong are expressed in BATS in different ways - so let's look now at each one in turn.

## Tension

Imagine a tug of war between two teams with say 5 people in each team.
Each team is pulling on a fairly substantial rope and there is a tag on the rope right in the middle.
The referee of the contest watches the tag because the team that pulls it towards them a measured distance will win.

Imagine that we are looking at the rope at the moment when both teams are pulling equally hard so the result as to which team will win is in balance - the tag on the middle of the rope is not moving either way.

We want to understand the strength of the rope in tension - being pulled.

In the diagram, the top picture shows a tug of war between 2 matchstick men.
Now we want to think about what is happening inside the rope at that point where the tag is attached.
One way to do this is to carry out a thought experiment - in other words to mentally do something to the rope and think what would happen as a consequence.

So what we'll do is imagine we can cut the rope at the tag and separate the two halves of the rope.

What would happen?

Both matchstick men might stumble but the tug of war teams would probably collapse in a heap!

They would suddenly be pulling against nothing - just as if the rope had snapped.
So to prevent our teams from falling we would have to get the two halves back together and replace what the internal fibres were doing before we cut the rope.
To do that we would have to pull with a force equal to that produced by the two teams in both directions.
We would have to pull against one team one way and against the other team the other way at the same time.
Imagine doing that yourself, you would have to get hold of both of the cut ends and pull them towards you to balance the pull of the two teams - just like the matchstick man in the middle of the second picture.

Engineers and scientists call the force that you are now providing as a substitute for the fibres of the rope, an internal force.
The third picture in the diagram shows arrows for the forces - rather than matchstick men.
This internal force is a response to the external force from the teams and is shown dotted in the third picture. In the bottom picture the forces are shown as a bridge builder would show them.

This distinction between internal and external forces is essential to an understanding of the way bridges work.

When the internal forces balance with the external forces then the rope is said to be in equilibrium - everything equalises out.
If the teams pull so hard that the internal force gets so very big that you have to let go (or else your arms may be pulled out of their sockets) then that defines the breaking strength.
Of course that's your breaking strength.
You could find the real breaking strength of the rope by pulling it until the internal force gets so big that the fibres snap.
Of course the rope will be too strong for you to do this manually but you could do the same thing with a piece of cotton.
In reality engineers and scientists use a special testing machine in a laboratory to apply varying tensions big enough to break lengths of rope and pieces of steel or other materials used in a real bridge to find out how strong they are.

The internal force is acting all along the length of the rope from one of the teams to the other.
We could have made our cut anywhere along its length and used the same argument.
So we call the force an axial tension - it is acting axially along the length of the rope.
The cross section of the rope is an end view of the cut.
The area of the cross section of your rope is quite small.
In a real bridge with lengths of steel or timber in tension the area of a cross section will be much larger and the internal force may not be exactly along the axis of the member.
Another way of describing the action of the axial force is to say that the rope has just one 'degree of freedom'.
In other words it has just one way of changing. Click here for more on degrees of freedom.

Just as we measure length in metres so we measure force in Newtons (usually abbreviated to N).

The tension might not act exactly along the axis of the rope so scientist's talk about the force on each little bit or little element of the cross section.
They call this force, on a particular small element, a stress.
They then examine how the stress varies across the cross section.
The little bits of the cross section are usually described in terms of a unit of area.
Commonly we use square millimetres which we abbreviate to mm2.
A square rod which is 10 mm. by 15 mm. has an area of 150 mm2.

Imagine a rod is pulled by an axial force of 15,000 N (or 15 kiloNewtons = 15 kN).
The stress will then be 15,000/150 = 100 Newtons per square millimetre (usually abbreviated to N/mm2).

So far we haven't said anything about how much the rope stretches when it is pulled.
Imagine that your rope was made from a gigantic elastic band or a length of coiled spring.
Clearly when you pull - the band or the spring would stretch quite a lot.

In fact all materials stretch when pulled - it's just that some stretch more than others.

The amount of stretch is very visible for an elastic band or a spring but it is so very small for a piece of wood or steel that you need a special measuring instrument to detect it.

The amount of stretch of a material is crucially important in bridge building because it contributes to two things - how much the bridge might deflect and how much it might vibrate as the wind blows or as heavy lorries pass over it.

This stretching is not the only factor in deciding the amount of deflection or vibration but it is an important one.
Engineers are interested in the amount of stretch for every unit length of a piece of material and they call it strain.
So if a piece of string 1 metre long (i.e. 1,000 mm.) stretches 10 mm. the strain is defined as 10/1000 or 0.01.
Notice that strain has no units - it is dimensionless.

If the amount of stretching is important in bridge design then even more important is the amount of stretching produced by a particular level of force.
The amount of force required to create an amount of stretch is called the stiffness of the rope or rod.

So if, as before, a force of 15 kN stretches a 1 metre rod by 10 mm then the stiffness of the rod is 15/10 or 1.5 kN/mm.
This is distinguished from the stiffness of the material which is defined as the amount of stress required to create an amount of strain in that material.
That is called the elastic modulus.

Thus if a stress of 100 N/mm2 makes the 1 metre length of rope stretch by 10 mm (which as we calculated above is a strain of 0.01) then the stiffness of the material is the stress divided by the strain or 100/ 0.01 = 10,000 N/mm2.
Notice that because the strain is dimensionless then the units of elastic modulus are the same as the units of the stress.

For many materials the elastic modulus remains the same for various loads.
We can show this by plotting a graph of stress against strain.
For many materials the result is a straight line and its slope is the elastic modulus.
Such a material is said to be linear elastic.

## Compression

Now let's turn our attention to the opposite of tension - the effect of pushing, squashing or compression.
If our tug of war teams were to push on the rope rather than pull on it the rope would just fold - you can't push on the end of a rope - it has no stiffness in compression.

So what could we do?
We could decide to replace the rope with a wooden rod or pole and hold a 'push of war' competition!

However the pole would have to be quite long.
The teams could push on it to some extent but unless the pole was very thick and chunky it would soon buckle and break.

Long thin materials like rope, string and long thin poles are ok in tension but soon buckle in compression.
In order to generate the same force in compression as in tension i.e. two teams of 5 people all pushing on a wooden rod together, you would need a massively thick piece of timber like a battering ram.
Thus we can immediately see that it is much more difficult for a material to resist a pushing force - a compressive force.

Two main factors determine the strength of a rod in compression
• its length and
• the shape of its cross section.

• The way it is held at its ends is also influential.
The longer the rod the more likely it is to buckle.
A very short rod will not buckle at all - it will just squash.

Just imagine standing on a single brick - it can carry a very big load before it squashes by crumbling.
Indeed we usually think of a single brick as a rigid block - meaning that the strain is so small before the final crumbling that we can neglect it.
Arches are one of the oldest forms of bridge and they rely on materials like masonry that are strong in compression.

## Shear

The last way in which structures have to be strong is in shear.

Shears, like scissors, are used to cut, so shearing is a cutting or slicing action.
In a bridge structure a shear force is a force that resists slicing or sliding.

Think of a block or brick sitting on a relatively rough flat surface and push it horizontally.
At first there is a resistance but if you push hard enough eventually it slides as you overcome the friction.
Now think of two blocks, one on top of the other but we stop the bottom block from moving by putting some kind of solid obstruction in its way and we push against the top block.
The top block will slide over the lower block in just the same way.

Then replace the two blocks by one new solid block, as shown in the upper left of the diagram. It is made of the same material and is the size of the two blocks together.
Again the obstruction prevents the lower part of this new block from moving.
When you push against the top part of the new block (shown by the arrow) you are doing exactly the same thing as when there were two blocks - except they are now joined together. The resistance to your push is shown by the bottom arrow.

The block will deform into a lozenge shape as shown in an exaggerated way for illustration in the upper right of the diagram. The upper part of the block won't move freely because an internal shear force has been created at the interface between the two previously separate but now joined up blocks as shown in the lower left of the diagram. Again the internal forces (this time shear forces) are shown dotted.
Of course the solid block could be separated into two blocks at many different levels so the same argument can be used to show that a shear force is created at every possible level of the block.

When we were considering tension we defined a force on a small piece or element of the rope as a stress.
For shear the situation is a little more complicated.

We need to think about a small piece or element of the block, say a small cube with sides of 10 mm.
If the cube is in equilibrium the horizontal internal shear force acting on the top of it has to be balanced by a shear force of the same size on the bottom.
However although these two forces may balance each other horizontally the two together would create a tendency for the piece to rotate.

So if there is to be no rotation and equilibrium is to be maintained then an internal shear force has also to be generated on the vertical faces of the piece - one up and one down - as shown in the lower right of the diagram.

It follows that shear force acts both horizontally and vertically on our small elemental cube.
If the block were to be subjected to a twisting motion then there would also be shear forces on the other faces of the cube.

It's worth just noting that so far we have, perhaps somewhat arbitrarily, just been talking about a cube with horizontal and vertical sides.
However there is a set of shear forces acting on any element of any orientation we may care to define within the block.

Just as earlier we replaced the stiff rope with an elastic band or coiled spring in order to make the tensile strain visible so now we can replace the block by an elastic rubber block if we want to see some shear strain as in the top right of the diagram.

As the top of the block moves visibly compared to the bottom you get the lozenge shape as the top moves and the bottom stays still.
Consequently one diagonal lengthens and the other shortens so one diagonal is in tension and the other in compression.