Moment Influence lines for statically determinate beams.
Consider this bridge.
It is designed to carry moving loads.
You are asked to determine vehicle locations on the bridge that lead to maximum bending moment in member CD.
With regard to bending moment, you know that the critical location in the beam is mid-span.
That is, bending moment at point G is generally larger than bending moment at any other point in the beam segment.
But note that the value of bending moment at G changes as the vehicle moves across the bridge.
So exactly where should the vehicle be for the bending moment at G to reach its maximum value?
We can answer the question by drawing what is called the moment influence line.
Here is the moment influence line for point G.
Before I explain how the diagram is drawn, let me explain how to interpret it.
The diagram shows bending moment values at G as a unit load moves across the bridge.
The diagram shows that when the unit load is at A, bending moment at G is zero.
When the unit load reaches point B, moment at G becomes a large negative value.
When the unit load is at C, moment at G becomes zero again.
Moment at G reaches its maximum positive value when the unit load is at G.
Then, the moment decreases in value as the load moves toward D.
When the load reaches D, moment at G reaches zero.
Moment at G continues to decrease as the unit load moves toward E.
When the load is at E, moment at G reaches a large negative value.
Then, the moment value starts moving towards zero as the unit load moves toward point F.
When the unit load reaches F, moment at G becomes zero again.
So where should the vehicle be located for bending moment at G to reach its maximum positive value?
The vehicle must be on top of point G for the moment to reach its maximum positive value.
And where should the vehicle be located for the moment at G to attain its maximum negative value?
The diagram shows two peak negative values: one at point B, and the other at point E.
This means bending moment at G becomes maximally negative when the vehicle is at point B while another vehicle is at E.
So, what is a moment influence line good for?
It facilitates the determination of critical load patterns for bending moment in beams.
Now, let's see how a moment influence line can be drawn qualitatively.
To draw the moment influence line for a point, we follow three steps:
One: Place a hinge at the point of interest.
Two: Apply a positive bending moment to the hinge.
Three: Draw how the beam displaces as a result.
Here the hinge is placed at C.
This means we want to draw the moment influence line for point C.
Note that a positive moment here means a pair of moments.
A counter-clockwise moment at the left end of the hinge, and a clockwise moment at the right end of the hinge.
How does the beam displace as a result of adding the hinge and moments at point C?
The right segment wants to turn clockwise because of the clockwise moment we placed at C.
The left segment wants to turn counter-clockwise as a result of the counter-clockwise moment at C.
For CB to turn clockwise, the left end of the segment has to move up, like this:
Similarly, for AC to turn counter-clockwise, the right end of the segment has to move up, like this:
So the beam is going to displace like this:
This displaced shape of the beam is the moment influence line for point C.
Please keep in mind that this is not the actual displacement of the beam as the hinge and the moments at C are not real.
They are placed there as part of the conceptual method for drawing the influence line.
Here is another example beam. What is the moment influence line for point B?
First, place a hinge at B.
Then apply a positive bending moment to the hinge.
Now draw the resulting displaced shape of the beam.
When drawing the displaced shape, make sure the beam segments, let's call them bars, remain straight.
That is, they can move up or down, and turn, but they cannot bend.
Here, bar BC wants to turn clockwise, and bar AB wants to turn counter-clockwise.
BC does turn clockwise, since the free end of the beam can move down to accommodate the rotation.
Bar AB however cannot turn, as neither A nor B is free to move vertically;
There is a pin at A, and a roller at B.
These support types prevent the support points from moving vertically.
So, this is the moment influence line for B.
The diagram indicates that bending moment at B is zero so long as the unit load is located to the left of B.
When the unit load moves to the right of B, a negative moment develops at B.
The moment reaches its maximum value when the load is at C.
Here is another example.
What is the moment influence line for point C?
In this beam, we have a real hinge at C.
Bending moment at a hinge is always zero, no matter where the unit load is located.
Therefore, the influence line for moment at C is a flat line, like this:
What is the influence line for moment at E?
Place a hinge at E.
Then subject the hinge to a positive moment.
Now, draw the displaced shape of the beam.
Bar AE wants to turn counter-clockwise, since the moment supplied to the bar at E is counter-clockwise.
For AE to turn counter-clockwise, point E has to move up.
But if E moves up, C has to move down as bar EC has to have a clockwise rotation.
Consequently, bar CD rotates like this:
So, here is the influence line for moment at E.
Let's consider another example.
Draw the moment influence line for point D.
Place a fictitious hinge at D.
Apply a positive moment to the hinge.
Now draw the displaced shape of the beam.
The beam is divided into three bars: AC, CD, and DF.
The clockwise moment at D wants to turn bar DF in the clockwise direction,
but it cannot, since neither E nor F can move vertically.
Therefore, bar DF does not undergo any displacement.
The counter-clockwise moment at D wants to turn bar CD counter-clockwise.
This could happen only if bar AC permits point C to move down.
AC indeed can accommodate a downward movement of C, as the bar can rotate about point B,
forcing A up and C down.
So, here is the moment influence line for point D.
It is important to note that for moment influence lines, we cannot readily determine diagram values qualitatively.
Further analysis is required when the diagram values are needed.
For that reason, we generally draw the qualitative moment influence line only for determining vehicle locations that lead to the maximum moment value.
We then have to analyse the beam to actually calculate the values.
Here is an example illustrating this process.
Suppose we are asked to determine the maximum negative bending moment at point G in this beam.
The beam is subjected to a moving load of 2 kN.
Here is the moment influence line for point G.
The diagram tells us that maximum negative moment at G happens when the moving load is at F.
To calculate the maximum negative moment at G,
we can place a unit load at F, and analyze the beam.
Here is the beam analysis.
When the load is just to the right of F, segment AF carries no portion of it.
That is, the entire load is carried by segment FD.
So, we can draw this:
We then calculate the support reactions.
Now cut the beam at G, and write the equation for internal bending moment.
Then, the magnitude of the maximum negative moment at G is 4 kN-m.
Here are a few exercise problems.
For each beam shown below, draw the influence line for the circled point,
then calculate the maximum positive and negative moments at that point
for a moving load that exerts a force of 50 kN on the beam.