Suppose we want to build a truss bridge,
that provides a passage for vehicles across a canal.
The structure is going to be constructed in such a way that the vehicle load transfers
to the two side trusses via a series of beams supporting the bridge deck.
The bridge is expected to support the maximum load of a truck having three axles.
For the sake of our analysis,
this can be viewed as a series of concentrated loads
spaced according to the distance between the axles.
So, we wish to determine the maximum impact
that the moving load series could have on a given truss member.
More specifically,
we want to determine the truck location on the bridge
that induces maximum axial compressive or tensile force in the member.
We can solve this problem using influence lines.
But before we get into the analysis details,
let's visualize how the load transfers from the truck to the truss joints.
In this visualization,
I am going to assume that the entire load of the truck is concentrated at its center of gravity.
As shown here,
the bridge deck rests on seven beams,
each attached to two truss joints, one on either side of the bridge.
Here are the beams supporting the deck:
And here are the exact same beams shown attached to the truss joints.
When the truck enters the bridge,
when the load is directly above the outermost beam,
the entire load transfers to that beam.
The load is then distributed to the two truss joints supporting the beam.
When the truck moves forward,
when it is somewhere between the first beam and the second beam,
the load is distributed between the two beams in proportion to its distance from them.
In turn, the beam loads are transferred to their respective end truss joints, like this:
So, as the truck moves forward,
this pattern of load transfer shifts forward
from joint to joint until the vehicle leaves the bridge.
We use this moving load pattern to construct the influence line for a truss member.
However, in drawing the influence line,
we only need to consider the loading cases in which the load is directly on top of a beam.
All the in between cases would be automatically captured by the straight lines
that we draw when constructing the influence line.
Let's draw the influence line for one of the truss members:
Say, for member AB.
To do so, we start by placing a unit load at this joint and calculate the support reactions.
Then, we cut the truss through member AB,
draw the free-body diagram for right segment of the structure,
and sum the forces in the y-direction in order to determine the axial force in AB.
We then plot this value as a point in our line graph,
and use a straight line to connect it to the starting point of the graph.
Now, we move the load to the next truss joint,
we then calculate the resulting support reactions
and determine the force in AB, just like the previous step.
This time we get negative square root of 13 divided by 6.
We plot the point and connect it to the previous point using a straight line:
We continue moving the unit load from joint to joint,
calculating the resulting force in member AB at each step,
and plotting the value on our line graph until the diagram is complete, like this:
We are now in a position to determine the maximum effect
that the moving load series would have on member AB.
Let's start by assuming that the truck is at this position on the bridge.
This means, the three concentrated loads are bearing down on the truss here,
here,
and here.
Let's refer to these points as X, Y and Z.
The value of the influence line at Y and Z are already known.
And we can determine the value at X using simple geometry, like this:
But what do these values represent?
This value means that if we place a unit load at Z,
the resulting axial force in member AB will be this:
Similarly, if we place a unit load at Y,
the resulting axial force in AB would be this:
And, if the unit load is placed at X, we get this for the axial force in AB:
But what if instead of a unit load, we place a load of 60 kN at Z?
What would be the resulting axial force in AB?
We can determine the resulting force by multiplying 60 by negative square root of 13 over 6.
This gives us negative 36 kN.
And what would be the resulting axial force if we placed a load of 70 kN at Y?
It would be negative 21 kN.
Further, if we placed a load of 65 kN at X,
the axial force in AB would become negative 13 kN.
Therefore, if all three loads are present at the same time,
the total axial force in AB becomes negative 70 kN.
Now, one can ask, is this the largest possible
compressive force that can develop in member AB?
To answer this question,
we need to consider the other potential truck locations
that create a large compressive force in AB.
A visual inspection of the influence line
suggests at least one more position for the truck
that could produce a large member force.
If the truck drives forward by three meters,
we get this configuration for the load series.
This position results in a compressive force of 129 kN in AB.
So, are we done here?
Did we find the largest possible compressive force in AB?
Not quite.
What if the truck is moving in the opposite direction?
Would that cause different load patterns,
and perhaps a larger compressive force in the member?
Let's investigate.
If we change the truck's direction of travel and place the vehicle here,
we get this configuration for the load series:
Under this loading scenario,
the compressive force in AB comes out to be 145 kN.
As the truck moves forward, like this,
we can deduce that the axial force in AB becomes smaller and smaller,
since the influence line values associated with the load series
become smaller and smaller in magnitude.
Therefore, we can safely conclude that the largest compressive force in AB is 145 kN.
This force develops when the truck is in this direction and location.
Now, let's turn our attention to calculating the maximum tensile force in AB.
For this, we focus on the right side of the influence line,
where the values are all positive, indicating the presence of a tensile force in the member.
By visually inspecting the diagram,
we can conclude that the maximum tensile force in AB
occurs under one of two loading scenarios.
This scenario, where the truck is facing to the right and the rear axle is on top of joint B:
Or this scenario, where the truck is facing to the left and the front axle is on top of B.
For scenario one, we can calculate the tensile force in AB, like this:
For scenario two, we get a tensile force of 70 kN.
Since there is no other loading scenario that could produce a larger tensile force in AB,
we can, therefore, conclude our analysis by stating that the maximum tensile force in AB is 86 kN.
This force develops when the truck is in this direction and location:
In summary, when we have a moving load series acting on a truss structure,
we need to investigate all the loading scenarios that potentially could produce
the largest axial force in the member under consideration.
This means we need to move the load series in both directions
up and down the bridge in search of the critical locations.
Here are a couple of exercise problems: