Influence Lines
or Bridges Under the Influence
What are They?
At first glance they look like the shear and
Moment diagrams we use for finding stress
Maximums in beams.
What is Different?
Shear and Moment
Diagrams show the
Impact of “dead” or
Fixed loads across an
Entire beam or
Structure.
We use them to find out where peaks of shear or moment are located
Influence Lines
Influence lines show the impact of live moving loads
At a single point as the load moves across the beam.
They are important because sometimes we have moving traffic on bridges or beams that
Contribute to stress and are not accounted for by loads at fixed positions.
Methods of Producing Influence Lines
You take a moving load of one
unit weight.
You pick your point of interest
You place your moving load at
Various points and use statics
Principles to find the load at
Your point of interest.
Several Methods but the
Tabulation Method is
The easiest to understand
Make life easier – for statically
Determinate structures you get
Straight lines (although the line
Slope may change as the load
Passes over key points.
Why Do I Use a Unit Load of One for
an Influence Line?
1 is easy to multiply by the weight of any thing
Or any number of things I want.
Influence lines are popular for studying the impact of moving – variable loads on bridges
And other such structures.
Sign Conventions
A reaction acting upward is
positive.
On Shear
Right Side Up
Is Positive.
More Sign Conventions
Moments that bend upward are positive
Lets Do a Simple Tabulated Influence
Line
Create an influence line for the
Reaction at B as the unit load moves
Across the beam.
X
Try different values
Of x and compute
The result.
Our controlling statics equation is
That the sum of moments about
A must be 0.
Observations
The weight of our moving load is 1
(could be one anything)
2.5 M
1
10 M
M1 = 1*2.5 = 2.5
Reaction B
The points we picked to try for reaction
At B were fairly arbitrary though they
Follow a consistent pattern.
Sum of Moments = 0
2.5 = 10 * X
M2= 10* X = 10X
X= 0.25
We Now Use Our Tabulation to Draw
the Influence Line
Note the straight line shape typical of statically determinate
Beams.
Lets Throw in Another Twist
This time we will cantilever the
beam
Using a unit weight of 1
We calculate the reaction
At B needed to keep the
Sum of the moments at A
Equal to zero
And We Plot Our Influence Line
Lets Try Some Shear and Moment
Influence Lines
We’ll try a total Cantilever beam.
Lets try first for shear at point A
Known Facts at Point A
Therefore a load of 1
Anywhere on the beam
Must produce a reaction
Of 1 at A
So What is the Shear Influence Line at
Point A?
Lets Try for Shear at Point B
B
How much of a shear load is transmitted through
Point B when our load is between A and B?
What About When the Load is Beyond
B?
Now Lets Do the Shear Thing When
We Have Support at Both Ends
I want an influence line for shear at point c
What Happens When a Load is Parked
Smack Dab on C?
So How Did I Know the Shear was 0.5
in Each Direction?
If the load is in the middle how much
Support comes from each reaction?
7.5 M
7.5 M
I can always refer to my reaction diagram
For point B if I am unsure
Next Consideration
How much load is there
Transmitted through point
C when the load is totally at
Point B?
0.5
Working on our shear line
So What Happens In-between?
0.5
We know that for statically
Determinate structures
We should get straight lines
?
0
So Does Our Straight Line Hypothesis
Make Sense?
Remember our reaction at point B increases linearly until it has the full load?
That means the shear that has to pass through C to reach support
A is declining linearly - Yup it makes sense
We now also know why we wanted to know the
Reactions at A and B (they tell us how much
Shear has to pass through our point of interest).
The Moment has Come to Do
Moments
Lets think about the moment at A as our
Moving load of 1 unit Cow moves down
The cantilever.
Tabulate and Plot
X
Moment
1 1 cowmeter
2 2 cowmeter
3 3 cowmeter
The tabulation and equation
6 6 cowmeter
8 8 cowmeter
10 10 cowmeter
0
X
10
-10
So Lets Up the Toughness and Make it
Point B
What is the bending action at point B when our 1 unit Antelope
Is between point A and point B.
But What Happens When Our
Antelope Goes Past B
If we are 1 meter past B, what is the
Moment at B?
1 unit Antelope * 1 meter = 1 antelope*meter
(the cows and antelope illustrate that we do
Influence lines with a unit weight of 1 – what ever
That unit might be).
The Result
Now Lets Try it With Supports on both
sides
Here we take advantage of having computed the reactions at A
And B as our unit load moves across the beam.
The Computation
Reaction at B
We know
We know the reaction at B because we have
Already calculated it (or could have calculated it).
Let us suppose that X = 2.5 meters and calculate it out.
The moment at C has to cancel the
Reaction at B X its lever arm of 7.5 M
Referring to the Tabulation at B
At 2.5 meters the reaction at B is 0.167
Plug that into our equation.
We now solve for the moment at C
We Build Our Table Off of the Principle
We take our tabulated reactions at B and solve for
Moment at C at each point as the load moves across
And We Plot the Result
There is One Mystery
I calculated the increasing moment as
The load approached C using the reaction
At B. Then after the load passed I used
The reaction at A.
How did I know whether to use the reaction
At A or the reaction at B for my moment
Calculation at C?
Actually It Didn’t Really Make In
Difference to the Answer
What if I calculated
I would need
The moment
From reaction
At A
Reaction at A
Using reaction at A
But I would have to
Directly also compute
The opposing moment
From my load
The Answer Would Be the Same
The amount of work needed to get
The answer would not be the same.
If you get the point.
Assignment
Plot the influence lines for the reactions at A and B
Plot the influence line for shear at point C
Plot the influence line for the moment at point C