Case Study: The Fort Atkinson Truss Bridge – Iowa.
In the previous post we introduced the concept of a load path and highlighted some of the differences between structural models and the actual behaviour of structures. In this post we’ll tease out the load path concept a little further. From here on we’ll base the discussion on the The Forth Atkinson Truss Bridge (Fig 1.). Built in 1892 this is a classic example of what is known as a Pratt truss pattern. The Pratt truss was invented by Thomas Pratt in 1844 and is still a common form of truss. However it was first popularised in the design of railway bridges in the late 19th and early 20 century. A Pratt truss is identified by its diagonal members sloping down and towards the centre of the truss.
Our bridge has a span of just over 40 meters and deck width of just under 5 meters. The bridge actually consists of 2 parallel Pratt trusses. This bridge has enough complexity to demonstrate load paths and truss analysis. However it’s not so complex that we get buried in the details. For anyone who wants a detailed look at the bridge construction, take a look at this site which hosts plenty of survey photos. To get a feel for how the bridge is put together you can explore an interactive 3D model of the bridge below.
1.1 The External Loading
We will start of by considering a simple scenario; a truck is parked at the centre of the roadway at the mid-span of the bridge. The truck weight is 16,500 kg, evenly distributed between the the front and rear axles and the axles are 6.4 m apart. This will equate to an axle force of 16,500 kg 9.81 m/s2 = 161.9 103 N. So lets say 81 kN per axle or 40.5 kN per wheel contact point (assuming 4 wheels).
Our task is to work out how that force is transmitted through the structure back to the four supporting piers. We can logically say now that because the loading is symmetrically placed on the structure and because the structure itself is symmetrical, the support reactions at the piers should be 40.5 kN. Let’s see how that number arises. For the purposes of this discussion we will ignore the self-weight of the bridge and just track the imposed truck loading.
1.2 The Load Path – From Deck to Truss
Starting at the point of load application we can see that the wheels impose forces on the timber deck, Fig. 2. The deck beams span between the main bottom members of the trusses on each side of the bridge. In this case we have modelled the deck beams as simply supported beams. The pin and roller support represent the support provided by the bottom chords of the trusses. So we can say that the wheel loads are transmitted from the deck to the supporting trusses via bending and shear in the deck beams, as per Fig. 3.
Because the deck beams are supported by the bottom chord of the trusses, and because the loading is symmetrical, 40.5 kN is transmitted into the bottom chord at each bearing point, Fig 4. With reference to Fig. 4, the force labelled ‘A’ represents the force imparted on the deck, by the truss chord whereas the force labelled ‘B’ is an equal magnitude, opposite direction force (with thanks to Sir Isaac Newton) representing the force imparted on the truss by the deck beam. This is force transmission in action.
Now that we’ve clearly established how the wheel loads make their way into the bottom chords of the trusses, we need to pay attention to how the loads make their way into the nodes of the truss. Remember, we’re currently dealing with a case of inter-nodal loading. Before we can think about axial load transmission within the truss, we have to account for how the loads make their way into the nodes for onward transmission.
1.3 The Load Path – From Inter-Nodal to Nodal Loading
You will have noticed that due to the positioning of the wheel loads, we have inter-nodal loading of each truss. As such, there will be bending induced in the bottom chords. Because the road deck bears directly onto the bottom chord of the truss, there will always be inter-nodal loading of the bottom chord. It would be important to consider this in the design of these truss elements.
In order to model the influence of this inter-nodal loading on the bottom chord, it would be reasonable to model it as a continuous beam; as it essentially runs continuously over the top of the transverse beams positioned below the deck, Fig. 5. The bending moments determined from this analysis would be used to calculate normal stresses due to bending in the chord. These normal stresses would then be superimposed on top of the axial stresses arising from the truss analysis.
It’s important to appreciate that the primary mechanism through which forces are transmitted back to the support piers is not via the bottom chord behaving as a continuous beam. This model was simply a means of approximating the bending stresses that develop due to inter-nodal loading. In terms of force transmission along the load path, the continuous beam model describes how the force is transmitted into the adjacent nodes of the truss. Ask yourself the question, ‘what supports the transverse beams onto which the bottom chord sits?’. Considering this it should become apparent that the complete truss (of which the vertical hangers and bottom chord are an integral part) is the primary structure that’s spanning the 40 m.
At this point we have described the path the load has taken and the mechanism of transmission from the deck into the nearest truss nodes. Note that the force transmitted into the adjacent nodes will not be exactly 40.5 kN, due to the continuous nature of the beam model. Nevertheless, this is a reasonable approximation for this analysis. Now we’re ready to focus on how the loading is transmitted through the truss itself. This involves working out the magnitude of internal force induced in each truss member. In order to do this, in the next instalment of this series I’ll demonstrate two methods of truss analysis, the Joint Resolution Method and the Method of Sections.
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