Doing the Math: Analysis of Forces in a Truss Bridge

[00:00:04]Wow, what an amazing view! I’m  so glad that this bridge is here, though, because it would take me so long  if I had to go around this entire canyon!

[00:00:15]But, that does make me wonder… how do  engineers design bridges? They have to make sure they can hold up to forces,  which would mean knowing what all the forces are that are acting upon the  bridge, and probably a lot of math.

[00:00:30]I’m Anika, a first year student studying  Creative Technology and Design at CU Boulder, and let’s learn about how to  calculate forces on a bridge!

[00:00:40]Today, we’re going to learn how to calculate  the internal forces on a truss bridge. A truss bridge is a bridge built with straight elements  that are connected at their endpoints. Usually, trusses are made of triangles, and they tend  to be the easiest kind of bridge to analyze.

[00:00:58]There are lots of kinds of truss bridges. One is a  Warren, which is made of equally sized components, like equilateral triangles. They can also have  verticals in the middle of those triangles.

[00:01:10]Another is a Pratt, which has the elements –  the bars – arranged in right triangles that, along the bottom, have their hypotenuses  facing “inward”. And then there’s the Howe, which is like the Pratt but  facing “outward” instead.

[00:01:25]When it comes to calculating forces on truss  bridges, there are a few important things to note: one, we want bridges to be still, which means  the forces on the bridge must be in equilibrium.

[00:01:38]In other words, the sum of all the forces  must add to 0. Or, to break it down further, the sum of all the forces acting along the  x axis must add up to 0, and the sum of all the forces acting along the y axis must add up  to 0. Also, with truss bridges, we can assume that all the forces converge at the joints of  the structure, so we only have to look at the joints to determine all the acting force values. Speaking of joints, we’re going to use the joints method to determine these forces. This method  comes with a set of other assumptions we must make in order for it to work, which  you can pause here to read through.

[00:02:23]Let’s analyze a super simple Warren bridge today, made up of three equilateral triangles with  five nodes. Nodes 1 and 5 are the ones that the bridge stands on, and the triangles have  side lengths of 4 inches. Let’s say we apply a force of 10 lbf – pound-force, which is a  measurement of force – to nodes 2, 3, and 4.

[00:02:46]Now that we have everything set up,  we can start calculating the force.

[00:02:51]And the joint method breaks down into four  steps, so let’s jump into the first one!

[00:02:57]Our first step is to identify all  the forces acting on each element of the truss. To do this more easily,  we’re going to draw a little diagram.

[00:03:08]Now, let’s start filling in where forces are  occurring and giving them names. We know there’s a 10 lbf external force acting on nodes 2, 3, and 4,  so let's label those as F2, 3, and 4, to stand for force 2, 3, and 4. There are also forces between  each node - node 1 is putting a force on 2, and 2 is putting a force on 1. We’ll call these F12  and F21, to represent that it’s the force between nodes 1 and 2 and 2 and 1. One of the assumptions  we make when using the joint method is that the value of a force at one end of an element will be  reflected equally but in the opposite direction on the other end. So, rather than having these be  two different variables, let’s just call them both F12. We can do this for the forces between every  other node, and then we finish off by adding the reaction force – which I’ll explain in a minute  – to nodes 1 and 5, which we’ll call R1 and R5.

[00:04:17]That’s step one done – we know where  all of the forces are occuring! Now, step two: finding the value  of the reaction forces.

[00:04:27]Reaction forces are found where the structure’s  supports are. Reaction forces are forces created in reaction to external forces attempting to cause  a rotation elsewhere. Essentially, because F2 and F3 are pulling downward, and node 1 is locked in  place, the structure wants to rotate around node 1 clockwise. But node 5 stops this from happening  with a reaction force. This same thing happens around node 5, where the structure wants to rotate  counterclockwise around node 5, but can’t because node 1 is stopping it. In order to calculate these  reaction forces, we need to add up the moments of force acting on each node, which is all the forces  affecting the attempted rotation around the node.

[00:05:17]We calculate a moment of force by multiplying  the magnitude of the force by the perpendicular distance between the point of rotation and the  line of action of the force, which we represent with M equals F times d. The line of action of a  force is an imaginary line extending infinitely in the direction of the force, and our distance is  from the point on that line that is the closest to the rotational point to the rotational point. Because our bridge must be in equilibrium, we know that the moments of force  around a point must add up to 0.

[00:05:55]Let’s find all the moments of force around node 1.

[00:05:59]We start with R1. R1 is coming  from node 1 and acting on node 1, and so the distance – d – is 0. We  can put this in as our first piece.

[00:06:12]Next is F2. We draw a line of action of the force, and find that the closest point on  that line to node 1 is 2 inches away, which means our next piece is -F2 * 2. And F2 is  negative here because F2 is a force pulling down.

[00:06:33]We do the same thing with F3, F4, and R5,  and end up with this equation. We can also build a similar equation around R5. Then,  because we know the value of F2, 3, and 4, we plug that in, and then can use algebra  to find the values of R1 and R5 – 15 lbf.

[00:06:56]That’s step 2 done. Now that we have the  reaction forces, we can move onto step three: finding the equations for the other forces.

[00:07:06]Because we know our bridge is in equilibrium,  we know that the forces along x must add up to 0 at every joint, and so do the forces  along y. We’re going to use this knowledge to find the values of all the other  forces; but before we can do that, we need to sort out what forces are acting upon  each node in the x direction and the y direction.

[00:07:30]Let’s draw some free body diagrams!

[00:07:34]We’re going to take each node and turn it into a  graph, where the node is the origin. Now, we’re going to draw all of the forces acting on the node  as arrows. Luckily, some of our forces are already along just x or just y. But others are diagonal. We can use our knowledge of trigonometry to separate these diagonal forces into  vertical and horizontal forces!

[00:07:59]Take F12 acting on node 1. We want to know  what F12 is along the x axis. Do you see what I see? It’s a right triangle! We can thus use  trigonometry and cosine to define this force as F12 times the cosine of 60. We have to also get  its vertical piece, which we can use sine for.

[00:08:24]And then, once every force on every node is broken  down into its vertical and horizontal pieces, we are armed with an arsenal of equations that we can  move onto step four with: solving for each force!

[00:08:42]What we have now is a giant system of equations,  made up of ten equations and seven variables, which we can solve using substitution.  Most giant systems of equations like this one aren’t actually solvable. But  this one is, which makes our little truss a statically determinate one, meaning it  can be solved with just these equations.

[00:09:07]We can start by solving for F12,  then use that to find F13 and F23, then use F12 and F23 to find F24, and continue  dominoing our way through all the other variables.

[00:09:24]And just like that, we know the values of all the  forces within our truss after applying a 10 lbf force to three joints. Any of these values that  are negative are considered compression forces, while values that are positive  are considered tension forces.

[00:09:42]Today, we learned how to find the values of all  the active forces within a truss. And to do that, we used a very simple bridge. But  even with such a simple bridge, we still ended up with a system of ten  equations with seven variables to find!

[00:09:59]So imagine how much math you would need  to do for a giant bridge, like this one.

[00:10:05]When it comes to bigger bridges, engineers can  use tools like spreadsheets to automatically do the calculations for them, making their  lives easier when designing bridges, and our lives easier as a  result of having those bridges!

[00:10:19]But, being able to do the calculations  yourself and actually understanding the math that’s going on is a lot of  fun, and a good skill. Now you, too, know how to calculate tension and  compression on a truss, just like an engineer!

[00:10:36]Check out the full lesson on Teach Engineering,  then use what you learned about calculating forces in truss bridges in the associated activity,  Trust in the Truss: Design a Wooden Bridge.