Reaction influence line for statically determinate beams.
Suppose we are asked to design column BG in this bridge.
In order to correctly design the column,
we first need to determine the maximum axial force that it will be carrying.
This means we need to calculate the maximum support reaction at B.
I say “maximum”, because here the support reaction is not constant;
it changes as the load moves across the bridge.
We can determine this maximum force with rather ease by drawing what is known as an influence line.
An influence line is a diagram.
It shows how a specific force in the structure changes due to a unit moving load.
Here is the influence line for reaction at B.
Before I explain how the diagram is drawn, let me explain how to interpret it.
The diagram shows the values for reaction at B, as a unit load moves across the bridge.
For example, when the unit load is at point A, the reaction force at B is 0.
When the unit load reaches point B, the reaction force becomes 1.
When the load is over the hinge at C, the reaction force at B equals 2.
The reaction force becomes 0, when the unit load reaches point D.
When the unit load is placed at E, the reaction force becomes -4.
By the way, the negative sign means the reaction is acting downward.
Finally, the reaction at B becomes 0 again when the unit load reaches point F.
A visual inspection of the diagram reveals that the absolute maximum reaction at B occurs when the load is at E.
Since the influence line is drawn for a unit moving load,
we must multiply the values given by the diagram by the actual magnitude of the load in order to obtain the correct reaction values.
For example, if the moving vehicle exerts a force of 5 kN on the bridge,
then the maximum negative reaction force at B equals 4 x 5, or 20 kN.
And the maximum positive reaction at B is 2 x 5, or 10 kN.
So, column BG has to be designed to carry a compressive force of 10 kN,
and a tensile force of 20 kN.
Now let’s turn our attention to the process of drawing the reaction influence line for statically determinate beams.
If the beam has no internal hinges, if it is a cantilever or a simply supported beam,
then drawing the reaction influence line is rather straightforward.
Here’s a cantilever beam.
The vertical reaction at A due to a unit moving load is 1.
The reaction does not change as the unit load moves across the bridge.
So, a diagram representing reaction at A due to a unit moving load looks like this:
As you can see, the diagram is a straight horizontal line,
as if we have taken the beam and pushed it up by one unit at A.
This idea of pushing the beam up by 1 unit at the support & drawing the shape that results is a convenient way of drawing reaction influence lines qualitatively.
Let’s put this qualitative approach to use.
Here is a simply supported beam.
Let’s draw the influence line for reaction at B.
To do so, we push the beam up by one unit at B and draw the displaced shape that results.
Here is the influence line.
It tells us that reaction at B reaches its maximum value when the moving load reaches point B.
It is instructive to compare the two influence lines that we have drawn so far,
the one for the cantilever beam, and the one for the simply supported beam.
If both diagrams were drawn by pushing the support point upward,
how come one is a straight line and the other an inclined one?
What makes the shape of the influence lines different in these two cases is the support condition at the other end of the beam segment.
In the case of the cantilever beam, the other end of the segment is free.
That is, it is free to move up as we push the left end of the beam up.
So the two ends move up together.
In the case of the simply supported beam, the left end of the beam is pin connected.
It cannot move up or down, but it can rotate.
So, as we push the right end of the beam up, the left end stays in place, but it rotates,
causing the beam to form an inclined line.
Drawing reaction influence lines this way becomes even more interesting when the beam has one or more internal hinges.
For example, consider this beam.
This is a statically determinate beam with a hinge at B.
Let’s say, we wish to draw the beam’s influence line for reaction at A.
We’re going to push point A up by one unit, and draw the resulting displaced shape of the beam.
The hinge divides the beam into two segments.
For drawing the influence line, we treat each segment as a straight bar.
So we have bars AB and BD.
When we push A up, the left end of AB goes up, but the right end stays in place.
So the influence line looks like this:
One may ask: But why doesn’t B move up with A?
Why does B stay in place?
Isn’t an internal hinge allowed to move up or down?
Although an internal hinge is free to move up or down,
in this case, it is not going to move since the supports at C and D for segment CD remain in place.
And, since CD is part of bar BD, the entire bar has to remain in place.
That is to say, the roller at C and the pin at D prevent the hinge at B from moving up.
Now let’s draw the influence line for reaction at C.
We need to draw the displaced shape of the beam when point C is pushed up by a unit.
Bar BD is pin connected at D, and has a hinge at B.
So if we push C up, the bar is going to displace like this,
but only if bar AB permits it.
AB does permit point B to move up as the bar can rotate counterclockwise to accommodate the displacement imposed by segment BD.
Why does AB rotate?
Because it is pin connected at A. Pins and rollers allow rotation.
So the influence line for the reaction at C looks like this:
What does the influence line for the reaction at D look like?
We need to figure out the displaced shape of the beam when point D is pushed up by a unit.
First let’s figure out how segment BD wants to displace, if permitted by the adjacent segment.
If we push D up, given that C cannot move, it can only rotate, then B has to move down, like this:
But is this configuration possible? Does the adjacent bar permit B to move down?
The answer is yes!
Since A can rotate, bar AB turns clockwise to accommodate the downward movement of B.
So, here is the influence line for reaction at D.
Let’s look at one more example.
Here we wish to draw the influence line for reaction at A.
Push A up while keeping in mind that bar CB is not able to rotate since it is fixed at B.
This means C cannot move.
Therefore, bar AC can only assume one displaced shape, this:
It is important to note that when drawing influence lines for statically determinate beams,
we treat the segments as straight bars, unable to bend.
That is, it might be tempting to draw the influence line for reaction at A like this:
arguing that segment CB can bend to accommodate the support conditions.
but this is not allowed, since here we are not really drawing the actual displaced shape of the beam.
Rather, we are using this qualitative approach to show how the reaction force changes.
In statically determinate beams, this change is always linear.
What about the influence line for reaction at B?
What does it look like?
Push B up a unit, what do we get?
When we push B up, C is going to go up as well, unless the adjacent segment limits the movement.
But bar AC is free to move up at C, since A can rotate to accommodate the movement.
See if you can draw the following influence lines: