Continuing from the naming convention, we'll explore the general equation of the virtual work method - setting the variables and deriving the expression.

Continuing our previous post on Virtual Work, let's derive the general equation for solving structural deflections using the virtual work method.

If you need a refresher on work and energy concepts, please check out the fundamentals first.

Consider a simple beam \(AB\) loaded with a point load \(F\). Using the virtual work method, we aim to determine the vertical translation at point \(C\).

Unit Load

To use the work methods, we must represent the deflection with something. In the case of virtual work, we will use a unit load. It's a load with a magnitude of one.

There are two things we need to remember about this load:

This unit load doesn't exist in real life! It is simply a representation of the deflection component. In our example, this represents the vertical translation at point \(C\).

The direction of the unit load corresponds to our assumption of what we think is the direction of the translation. In this case, it's downward.

If we're dealing with rotation, we will use a unit moment if we wish to represent it instead. It can be either a clockwise or counterclockwise direction.

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The following section shows how we will derive the general virtual work equation. We will set up the expressions needed first.

Virtual Structure

The first step of the virtual work method is to create a structure similar to our example without the imposed structural loads. This structure will have the unit load we made earlier.

This beam is what we call the virtual structure. It acts as a placeholder in solving deflections, which will be more evident when we solve examples.

An important thing to note is that the beam remains within the elastic region. To accomplish this, we must place this unit load gradually.

Now, let's investigate the \(P-\Delta\) diagram of the virtual beam and observe the following:

When the unit load was gradually applied, it also caused displacement \(\delta_{C_{v}C_{v}}\). The unit load caused a fictional displacement.

As a result of this displacement, there was fictional work: \(W_{\delta}\)

This fictional work is the entire work for this system \(W_{\delta}\). From the Law of Conservation of Energy, there must be a transfer of this external work \(W_\delta\) to strain energy \(U_\delta\). Some things to note about \(U_\delta\):

Since an imaginary unit load caused this \(U\), this strain energy is also fictional.

The strain energy is flexural (more on this soon).

From this idea, the entire virtual structure's total strain energy equals \(U_\delta\).

We can equate the total external work and strain energy of the virtual structure:

Equation 1: \(W_\delta=U_\delta\)

Superimposed Structure

After loading our virtual structure with our unit load, we will gradually superimpose the actual structural loads of our original beam. In our example, it is the point load \(F\) at \(C\).

When we superimpose, we combine the responses of both beams. Note that the beam is still in the elastic range since we gradually superimposed the loads.

One essential item to note is that the fictional unit load on the beam is constant throughout the superimposition; hence, it is still there, as seen in the \(P-\Delta\) diagram.

Speaking of such, let's investigate it and observe the following:

The gradual application of the actual loads has caused real deflection \(\Delta\),

This application caused additional external work \(W_{s_\Delta}\).

Since the virtual unit load is constant when the actual load is superimposed, there is also additional work \(W_{s_\delta}\) due to the virtual force and the actual deflection \(\Delta\).

Overall, the total external work \(W_T\) done on this system is:

Again, from the Law of Conservation of Energy, there must be strain energy \(U\). The superimposition also caused an additional \(U\). Some things to take note of this \(U\):

The actual load's gradual application has caused the application of \(U_{s_\Delta}\).

Because there is \(W_{s_\delta}\), there is also an equivalent constant transfer of energy in the form of \(U_{s_\delta}\).

Consequently, the \(U\) of the system from these additions is now:

Let's put the superimposed structure aside and go back and investigate the original beam. In this example, we applied the loads gradually again. In this way, the beam remains within the elastic range.

Let's observe the \(P-\Delta\) diagram of the real beam:

When the actual loads were gradually applied, it also caused displacement \(\Delta_{C_v}\).

As a result, there was real work: \(W_{\Delta}\).

The entire work for this system is equal to \(W_{\Delta}\). From energy conservation, there must be a transfer of this external work \(W_\Delta\) to strain energy \(U_\Delta\).

Since actual loads caused this \(U\), we also have non-fictional strain energy.

The strain energy is flexural (more on this soon).

The total strain energy is equal to \(U_\Delta\).

As a result, we can equate the total external work and strain energy:

Equation 5: \(W_\Delta=U_\Delta\)

Key Concept: General Virtual Work Equation

Let's look back to our superimposed structure and derive the general virtual work equation.

Consider Equation 4: the expression of total superimposed work equals total superimposed strain energy.

From Equations 1 and 5, we can cancel some of the terms of Equation 4. Note that \(W_{s_\Delta}=W_\Delta\) and \(U_{s_\Delta=U_\Delta}\). It is true because the superimposed loads are the original loads of the structure.

As a result, we have the general equation of virtual work:

General Equation of Virtual Work: \(W_{s_\delta}=U_{s_\delta}\)

This equation is what we use to find the deflection of a structure. Generally, this equation tells us that virtual work \(W_{s_\delta}\) is equal to the strain energy \(U_{s_\delta}\).

To further explain what the equation means, let's look at the \(P-\Delta\) diagram of the superimposed structure. If you look at the rectangular parts, the external work and strain energy must be equal.

From this graph, we can say that \(W_{s_\delta}\) is the product of fictional unit load (or couple) and actual translation (or rotation); hence, we can represent the general virtual work equation as:

We still have another question about the general virtual work equation. What is \(U_{s_\delta}\)? How do we find the area of the rectangular part in the superimposed \(U\)-diagram?

From the graph of the Figure, \(U_{s_\delta}\) is the product of fictional internal forces and actual internal displacements or strains; however, this will entirely depend on the primary type of stress. For example:

Let's summarize the derivation of the general virtual work equation:

We represent the unknown deflection with a fictitious unit load. Its direction is the same as the assumed direction of our deflection.

We create a virtual structure. It acts as a placeholder in solving deflections. We gradually place the unit load on the virtual structure so that it still behaves in the elastic range.

Afterward, we gradually superimpose the actual loads on the virtual structure. As a result, the work and energy of the superimposed structure is \(W_\delta+W_{s_\Delta}+W_{s_\delta}=U_\delta+U_{s_\Delta}+U_{s_\delta}\).

Ultimately, the general virtual work equation is equal to \(W_{s_\delta}=U_{s_\delta}\).

\(W_{s_\delta}\) is the product of fictional unit load (or couple) and actual translation (or rotation); hence, the general virtual work equation is \(1\times(\Delta\space{or}\space\theta)=U_{s_\delta}\).

\(U_{s_\delta}\) is the product of fictional internal forces and actual internal displacements or strains; however, this will entirely depend on the primary type of stress: axial, flexural, or torsional.