The basis of the Virtual Work Method
and its use for calculating joint displacement in trusses
were discussed in lecture SA21.
In this lecture, I'm gonna to show you how this method can be used to calculate deflection and slope in beams.
Let's start by reviewing the method using a truss structure.
Consider this truss:
Under the applied load it is going to displace like this:
To determine a joint displacement, say horizontal displacement at C,
we place a virtual load at the joint.
In this case, a horizontal load at C.
Note that the direction of the load corresponds to the direction of the target displacement.
If we are seeking to find a horizontal displacement,
the virtual load should be horizontal.
If we wish to calculate a vertical displacement,
the virtual load must be vertical.
In this case, external virtual work equals p* times Delta,
and the internal virtual work equals:
Here, f*i is the force in member i, due to the virtual load.
and delta_i is the elongation of the member due to the real load.
According to the method of virtual work,
external virtual work equals to internal virtual work.
p*, the applied virtual load, and f*i is virtual member force due to p*.
This is displacement due to the real load,
and this is the member elongation due to the real load.
Now let's turn our attention to beams.
Say, we have this simple beam:
Here is how the beam is going to deflect under the load:
Suppose we want to determine the vertical displacement at mid-span.
To do so, we should place a virtual load at the midpoint of the beam,
where the deflection is denoted by Delta.
Then the expression for external virtual work can be written as:
p* times Delta.
To determine the expression for internal virtual work,
we need to know two things:
The internal member force due to the virtual load p*,
and the internal member deformation due to the real load P.
What is the main internal force in beams?
Bending moment.
When dealing with beams, we generally consider bending moment
to be the only internal force that contributes to the structure's deflection,
but since bending moment is not constant throughout the beam,
we need to express it using an equation.
Here, the virtual load is placed at the midpoint of the beam,
therefore the beam's virtual internal bending moment can be written as:
Now we need to determine the internal deformation in the beam.
To do so, let's examine a tiny segment of the deformed beam.
Since the beam has been deformed under the real load,
the segment also has a deformed shape.
We can treat it as an arc length of a circle.
The angle facing the arc length, this angle,
represents the internal deformation of the segment due to bending.
So internal virtual work in the segment can be written as:
m* times d theta.
To determine internal virtual work for the entire beam,
We need to integrate the expression over the entire length of the beam, like this:
But before we can use this equation to calculate internal virtual work,
we need to express d theta in terms of dx,
since m* is a function of x.
Let's start by referring to the length of this segment as dx.
This is the infinitesimal length of the tiny beam segment.
Also, let's refer to the radius of the circle containing the arc length as rho.
And keep in mind that this deformation is caused by the bending moment produced by the real load.
I'm gonna show the bending moment as big M.
Using simple geometry we can write:
d theta times rho equals dx, or,
d theta equals 1 over rho dx.
Now we need to express rho using more familiar terms.
There is an interesting relationship between rho and M.
When M increases, rho decreases.
When M decreases, rho increases.
Interestingly, the product of rho and M is constant.
More specifically, Rho times M equals E times I,
where E is the material's modulus of elasticity,
and I is the moment of inertia about the axis of bending.
Of course, here we are assuming linear elastic material.
Then we can write, d theta equals M/EI dx.
Finally, we can write the expression for internal virtual work for the entire beam as:
It is important to keep in mind that m* is bending moment equation due to the virtual load,
and M is bending moment equation due to the real load.
The virtual work principle can now be written as:
If we assume a unit virtual load, we can determine Delta using equation:
Delta equals the integral of m* (M/EI)dx,
where both m* and M are in terms of x.
Let's calculate Delta at the beams mid-span.
We determined m* for this beam earlier.
Here it is:
Note that m* and M equations are defined over different intervals.
To make the integration work, we need to divide it into three parts, as shown below:
If we assume a constant EI for the entire beam, we get:
So the vertical displacement at the beam's mid-span is:
Let's go ahead and calculate vertical displacement under load P as well.
Here is the moment equation due to P.
It is the same equation we used previously.
Now replace P with a virtual unit load.
Write the moment equation due to this unit load:
This is going to be the same as M equation where P is replaced with 1.
Therefore, the virtual work equation becomes:
Here, since the moment equation intervals match, we can write:
Or:
This gives us, Delta equals:
Here's another example.
A cantilever beam subjected to a uniformly distributed load.
Let's calculate its vertical deflection at A.
Assume constant EI.
We start by writing the moment equation for the beam.
Now remove the real load and place a unit virtual load at A.
Write the moment equation due to this unit load.
The virtual work method equation is:
So the vertical displacement of the cantilever beam at A is:
Suppose we are also asked to determine the slope of the elastic curve at A.
This can be done easily using the method of virtual work.
Here, since we are looking for slope, denoted by theta,
we are going to place a virtual unit moment at A.
This way, the product of the rotation and virtual moment is virtual work.
So the expression for external virtual work becomes:
1 times Theta.
The expression for internal virtual work remains to be:
However now, m* is the moment equation due to a unit virtual concentrated moment applied at A.
So the cantilever's rotation at A is:
We will examine the use of the method of virtual work for calculating deflection in frames in the next lecture.