Virtual Work Method for Beam and Frame

Virtual work method was developed by John Bernoulli in year 1717. This method is used to determine the displacement and slope for beam, frame and truss.

To apply virtual work method to a structure, we first need to solve for its support reactions, and subsequently the internal forces in all members. These internal forces, namely moment, axial and shear should be expressed as a function of local ordinate. We take the same procedure for the virtual force case. For this scenario, we apply either a unit load or couple moment to the point we interested in. A unit load helps us determine the elastic curve displacement of a point, and a couple moment aids us in calculating the elastic curve slope.

After the internal force functions are ready, we can apply virtual work equation to determine the displacement or slope of the interested point. Note that, the deformation of beam and frame elastic curve is mainly contributed by moment. Both shear and axial usually pose negligible effect on it.

In this worked example, we have a 7m steel beam subjected to uniformly distributed load of 16kN/m. We are required to determine the beam deflection at node C i.e. 2m from support A. For this example, we will be using virtual work method.

Based on the deformation we want to find, we need to establish a virtual force case. 1 unit load is applied to node C in downward direction.

Then, we proceed with calculating the support reactions, and subsequently the functions of moment and shear under actual load case. These functions should be in terms of ordinate, x. Next, we repeat the steps for virtual force case.

Once the moment and shear functions for both cases are in place, we can proceed with the determination of elastic curve deflection. From the result, 98.9% of the total deflection is contributed by moment, and merely 1.1% of it is due to shear. The result from virtual work method can be validated using readily available beam deflection formula.