# BRIDGES

## How trusses work

Whenever you see a bridge, or indeed any structure, made of lengths of timber or steel joined together to form triangles then you are looking at a truss.

Many small bridges around the world are of the form shown in the diagram below. The colours show the 3 main chapters of a Truss Bridge.

• Light blue for the truss girder.
• Red for the main deck
• Green for the foundations

• Timber truss girders are commonly used for the roofs of houses and small buildings.
Trusses can be clever highly interconnected structures.

The first truss was probably a simple 'King Post Truss' as in (a) in the diagram to the right. They are basically frames shaped as an upside down V but with a horizontal beam across the bottom at eaves level and a vertical timber from the middle of the tie to the apex at the top.

The other 3 trusses, shown in the diagram on the right, are just some of the many kinds you may see. Diagram (c) is a standard roof truss used in many buildings. Diagram (d) is known as a Warren Truss and is very efficient when used with tubes. Diagram (d) is an N-frame and was used for lots of older bridges.

The small circle in the diagrams at the joints represents a pinned or hinged joint.

Why is this important?

A pinned joint allows the members framing into it to rotate freely about each other. There are no restraints against free rotation and so no internal bending moments.

In other words a pin jointed truss contains no internal bending moments because the joints have no stiffness.

In the diagrams there are potentially three external reactions at the supports resisting the estimated applied loads - the two vertical reactions at each end and the one horizontal reaction at the left hand support.

When this is so the support conditions are said to be statically determinate.

We can then calculate the forces by balancing them vertically; then horizontally; then with respect to their turning effect - their moment.
In that way we get three mathematical equations that can be solved to find the three unknown reactions.

Let's make three cuts around joints in the King Post truss above - see the next diagram on the right. Start by looking at the left hand joint. We have two internal forces acting at that joint and one external force - the reaction (W/2). They must balance.

We can find the values of the internal forces C and T using the triangle of forces.
C and T must act axially along the length of the members.
Their directions determine the angles in the triangle.
We can therefore draw the triangle to scale and measure off the values.

Now it is quite a long winded process to draw a triangle of forces for each joint in a truss.

So instead bridge builders usually rely on a technique called 'resolving forces'.

The technique relies on the idea that a force in any direction can be expressed as two separate forces - one horizontal and one vertical.

These two forces then replace the single force.

But why would we want to make life more difficult by replacing one force with two?

The answer is that it actually will make the overall calculations simpler.

Where a number of forces in different directions come together at a joint we need to find the resultant i.e. how they all combine together.
This is a lengthy process when they are all in different directions. So we shorten the process by finding all of the horizontal components and all of the vertical components.

We can then simply add them up.

The result is a total horizontal force and a total vertical force - a pair of components - which represent the total effect of all of the forces combined.

They can then be converted back into a single force if we wish.

Now if a joint is in equilibrium then these resulting component forces in the horizontal and vertical directions will each be zero.
We can use this knowledge then to find forces at the joint which are unknown - as long as only two are unknown.

We form two equations - one by making the sum of the horizontal components equal to zero and one by making the sum of the horizontal components to zero.

We now have two equations and two unknown forces.
The algebra of simultaneous equations allows us to solve them.

When we have calculated all of the internal forces in the truss we then have to make sure that each member is strong enough in tension or compression.

The strength in tension is simply the cross sectional area of the member times a chosen level of limiting stress. The strength of a structural member in compression is not so straight forward. It depends on a number of complex factors.

Very short members just squash in a process which is more or less a reverse of pulling apart in tension.
The determining factors are the material, the elastic limit and the ultimate squash load when the material can no longer carry any load at all.

Longer struts are more of a problem however because they bow out and buckle.
A very slender strut buckles very easily, especially when not perfectly straight.
On the other hand a strut is stronger if the ends aren't allowed to rotate.
The standard end condition, against which others are compared, is a pin at both ends of the strut so that they can rotate freely. This is how most modern trusses are designed to behave.

The slenderness of a strut not only depends on its length and initial straightness but also on the shape of its cross section.
There are two main forms of slenderness that are important - longitudinal and torsional.

Longitudinal slenderness is the extent to which a strut is long and thin.
Torsional slenderness is its susceptibility to twisting.

When a strut buckles it takes the line of least resistance - it finds the easiest way to move out of the way of the squeezing load, either by moving sideways or by twisting.

The best shape for the cross section of a strut therefore is a circular tube because the distribution of material is equal in every direction.

This is so for both longitudinal and torsional buckling.
A deep I beam makes the worst kind of strut because although it is strong in one direction it is very weak in the other.

Just what is the difference between a truss and a beam? A truss with pin joints has only struts and ties - so where have the bending moments in a beam gone to? We can see the different ways in which trusses and beams work by looking at an N frame as shown in the diagram to the left. I have imagined a cut to expose some of the internal forces shown as the three dotted arrows.

The horizontal axial internal forces in the top and bottom members (often called chords) are resisting the turning effect just as the distributed stresses did within the beam.

The internal tensile force in the diagonal member has a vertical and horizontal component which is resisting the vertical and horizontal shear.

In summary the truss and the beam are balancing the same external forces but with different distributions of internal forces. They are just two ways of doing the same thing.

The first arch built with a material other than stone or masonry was truss like - the cast iron Ironbridge at Coalbrookdale in Shropshire in 1779. You can see the cast iron rings that make up the voussoirs but the structure is basically a framework or truss.

Notable examples of truss arches trusses are the two pinned the Garabit Viaduct and the iconic Sydney Harbour Bridge.

A development in the use of truss arches was the balanced truss arch such as the Viaur Railway Viaduct see picture below and diagram lower right.  The thrust at the centre pin of the arch is very much reduced when the outer trusses under dead load balance the inner trusses forming the arch.

Then came the idea that it is possible to suspend a simple truss between truss cantilever arms.

Perhaps the most famous example is the Forth Railway Bridge.  But there are many others around the world such as the Story Bridge in Australia - see picture on the right.

The principles are illustrated in the diagram below. Here the truss cantilevers balance under dead load but also carry the reactions from the suspended span in the middle.

Trusses are very efficient forms of structure for smaller spans, such as footbridges - see below the jetty truss bridge at the London Eye.

This is a common form of truss, called a Warren Truss after Captain Warren who suggested the shape, and built using welded tubes which form robust struts and ties.

Modern computer controlled cutting and welding equipment makes it much less costly to form the complex shapes of the joints. 