The Method of Virtual Work
In previous lectures, The Work Energy principle was used to relate external work
to internal energy in a structure.
See lectures SA19 and SA20.
For example, for a simply supported beam subjected to a concentrated load,
the work energy equation can be written as:
1/2 P times Delta equals: 1/2 times the integral of M, times d theta.
The left side of the equation represents external work done by the load.
The right side of the equation represents the resulting internal energy stored in the beam.
On the left side, P is the applied load and Delta is displacement under the load.
On the right side, M is bending moment in the beam,
and theta is slope of the elastic curve, both due to the beam's deformation.
We can use this equation to calculate Delta, the displacement under the load,
as was demonstrated in lecture SA20.
The work energy principle can also be used for trusses.
For example, in the case of a simple truss like this,
we can write: 1/2 P times Delta, equals the sum of Fi delta_i, over 2.
Again here, the left side of the equation represents external work,
and the right side represents internal energy.
On the left side, P is the applied load, and Delta is the horizontal displacement under the load.
Fi is the axial force in member i, and delta_i is the elongation of the member.
So if we were interested in calculating the horizontal displacement at C, we could write:
Delta equals to the sum of Fi delta_i over P.
Since the truss is statically determinate
we can easily calculate member forces and elongations,
as was demonstrated in lecture SA19.
But what if we needed to calculate vertical displacement at B?
Since Delta B does not appear in the work energy equation,
we cannot solve for it using the work energy principle.
We need to use a more general method, like the virtual work method to accomplish this.
The method of virtual work is based on the work energy principle,
but can be used to calculate displacement at any arbitrary point in the structure.
Let's start by examining the basis of the method.
Suppose we had a structure subjected to a set of external loads.
Consequently, the structure is going to displace like this:
Let's refer to the horizontal displacement under P as Delta H,
and call the vertical displacement under load Q, Delta V.
Now suppose we add an imaginary vertical load to point B.
Since this load here denoted by P* is imaginary, we'll call it a virtual load.
The structure can be easily analyzed under P*
in order to determine the virtual member forces.
Let's refer to these forces as f*1, f*2, and so on.
Since there are five members in the truss, there are five virtual member forces.
Note that P* is applied in the direction of Delta V.
Since this imaginary load travels through the real displacement, it does imaginary or virtual work.
The work done equals P* times Delta V.
We call this an external virtual work, since it is done by an external virtual load.
Since the real loads P and Q cause real member deformations,
then it is also true that the virtual load causes internal virtual work.
For member 1, internal virtual work equals f*1 times delta_1.
Where f*1 is the virtual member force due to P*,
and delta_1 is the real elongation of the member due to P and Q.
Similarly, from member 2, internal virtual work can be written as:
F*2, times delta_2.
So in this case the total internal virtual work equals:
The principle of virtual work states that:
External virtual work equals the internal virtual work, or:
Now since P* is imaginary, we can give it any value we wish.
To simplify our calculations, let's make it a unit load.
So we get: Delta v equals the sum of f*i, times delta_i.
Here, delta_i is the real elongation in member i due to the real loads,
and f*i is the internal force in the member due to the unit virtual load.
Using this equation we can easily calculate Delta V,
the vertical deflection at B due to the real loads.
In fact, using this method we can calculate displacement at any truss joint.
For example, to calculate horizontal displacement at D,
we place a horizontal unit virtual load at D,
and we calculate member forces due to the load.
Then,
horizontal displacement at D equals to the sum of f*i, times delta_i,
where f*i is the virtual force in member i,
due to the unit virtual load at D,
and delta_i is the elongation of the member due to the real loads.
Let's work through an example to better illustrate the method.
Consider the following truss structure.
The members are made of structural steel,
with a modulus of elasticity of 200 GPa.
All the members have the same cross-sectional area, 20 cm^2.
The truss is subjected to two concentrated loads as shown.
We wish to determine:
the horizontal displacement at C,
the vertical displacement at B,
and the horizontal displacement at D.
We can start by calculating the member forces in the structure.
This can be done easily since the truss is statically determinate.
The support reactions are:
Ax=500 N
Ay=-250 N
And Dy equals 450 N.
The member forces are:
F1=307 N.
F2=321 N
F3=200 N
F4=321 N
F5=-553 N
Now let's determine the axial elongation of each member due to the real loads.
According to Hooke's Law, axial stress equals to modulus of elasticity of the material times axial strain,
where axial stress equals axial force divided by the cross-sectional area of the member,
and axial strain equals to the member elongation divided by its length.
Therefore we can determine axial elongation of each member using equation (FL)/(EA).
For member 1, axial elongation can be written as:
delta_1 equals (F1)(L1)/(EA).
We calculated F1 to be 307 N.
L1, the length of member 1, is 8.6 m.
E, modulus of elasticity of steel, is 200x10^9 N/m^2.
And the member's cross-sectional area is 20x10^-4 m^2.
So we get: 6.6x10^-6 m for delta_1.
We calculate the elongation of each of the remaining members in the same manner.
delta_2 equals 4.0x10^-6 m.
delta_3 equals 3.5x10^-6 m.
delta_4 equals delta_2, which we calculated to be 4.0x10^-6 m.
And delta_5 equals -12x10^-6 m.
The negative sign indicates that the member actually shortens, it is in compression.
Now that we have determined the elongation for each member due to the real loads,
we can start calculating the desired displacements using the method of virtual work.
To determine the horizontal displacement at point C,
place a virtual unit load in the direction of the displacement at C.
Now determine the member forces due to this load.
A simple truss analysis gives us the following virtual member forces:
A negative force value indicates the member is in compression.
The external virtual work for this truss is: 1 times Delta.
The internal virtual work equals to the sum of f*i, times delta_i.
This gives us:
So we can write: external virtual work equals to internal virtual work, or:
One times Delta equals 20x10^-6.
Therefore the horizontal displacement at B equals 0.02 mm.
To determine vertical displacement at B,
place a unit virtual load in the vertical direction at B,
and determine the resulting member forces.
So the internal virtual work equals:
9.67x10^-6
The virtual work equation then becomes:
one times Delta equals 9.67x10^-6,
where Delta is the vertical displacement at B.
So, Delta equals 0.00967 mm.
Finally, to determine the horizontal displacement at D,
place a virtual unit load at D in the horizontal direction.
The analysis of this truss gives us the following member forces:
Therefore the expression for internal virtual work can be written as:
Then the horizontal displacement at D equals: 0.008 mm
We are going to examine the use of the virtual work method
for calculating beam displacements in the next lecture.