Consider a truss bridge.
The structure is subjected to a moving load.
So, if the system is to be designed properly,
we need to determine the effect of the moving load on each truss member.
What do I mean by that?
I mean, how does the axial force in the member change over time as the load moves across the bridge?
For example, consider member DJ.
How does the force in the member change, as the unit load moves from A to F?
We can show the change using a line graph, like this:
We call this an influence line.
So, what does this influence line tell us?
It tells us that axial force in DJ is zero when the unit load is at A.
Then the force gradually increases as the unit load moves to the right, until it reaches point D.
At that time, the member would be in tension, carrying a force of 6/5 square root of 2.
How do we know the member is in tension?
Because the point is above the x-axis.
A positive value for the axial force means the member is in tension, and a negative value means the member is in compression.
Then, the member force starts decreasing until the unit load reaches point E.
By then the diagram has dropped below the x-axis,
indicating the presence of a compressive force in the member.
As the unit load moves towards point F, the axial force starts approaching zero.
The member becomes free of any internal force when the load reaches F.
It is important to note that an influence line is always drawn for a unit moving load.
But once drawn, we can use the diagram for member force calculations for any load magnitude.
For example, if the moving load has a magnitude of 50 kN,
to determine the maximum tensile force in DJ due to the load,
we simply multiply 50 by 6/5 square root of 2.
In addition, an influence line gives us a visual tool to better understand the effect of a moving load on the member.
For example, using this diagram we can easily determine the exact location of the moving load that makes DJ a zero-force member.
See if you can figure it out on your own.
So,now we know how to interpret an influence line.
But how do we draw one?
Since there is a moving load involved, first we need to determine the path of travel for the load.
For example, for this bridge, the path of travel is along the bottom chord.
But for this bridge, the load moves along the top chord.
This allows us to identify the truss joints that the moving load can visit.
In this case, the moving load visits these joints:
But here, the load visits these joints:
To draw the influence line for a specific member,
we find the axial force in the member every time the unit load visits a joint.
For this bridge, we need to do the calculation 4 times.
One time when the load is placed at B.
Another time when the load is at C.
One more time when the load reaches D.
And a final time when the load is at E.
Let's draw the influence line for member DJ.
We know that when the load is at A or F, all member forces are zero.
So, our line graph starts at zero and ends at zero.
To determine the height of the graph at B,
We place the unit load at B.
Then we calculate the axial force in member DJ.
The fastest way to determine the force is to use the method of sections.
So,first we are going to calculate the right support reaction, like this:
Then we can cut the truss vertically, like this:
Now we draw the free-body diagram for the right segment of the truss.
Since we are only looking for F_dj,
we simply set the sum of the forces in the y-direction to zero.
Then, solve for the force.
This gives us F_dj equals 2/5 square root of 2.
Now we plot the point on the line graph,
and connect these two points using a straight line.
Then, we place the unit load at C and calculate F_dj in a similar manner.
This time we get:
We plot the point and connect it to the previous point on the graph using a straight line.
We then place the unit load at D and determine F_dj.
Now the graph evolves into this:
Finally, we place the unit load at E and calculate F_dj.
This time we get: -2/5 square root of 2
This means that when the load is at E, the member undergoes compression.
We plot the point below the x-axis and complete the graph.
We call this the influence line for member DJ.
Before we end the lecture, let's draw another influence line.
Say, for member EJ.
A close examination of the truss and the member tells us that when the unit force is at any joint other than E,
the member carries no force, EJ acts as a zero-force member.
But, when the unit load is at E, the member carries an axial tensile force of 1.
Of course, we can always analyze the truss repeatedly, just like the previous example.
But in this case, that would not be necessary.
Let's plot the axial load values.
We have: zero,
zero,
one,
and zero.
Now we connect the points using straight lines, like this:
This is the influence line for member EJ.
Here are a few exercise problems.