How to Use Virtual Work to Solve Shear and Moment?

Get Total Work

After deflecting the beam to its fictional shape, we need to get the work done by all forces corresponding to this deflected shape. 

There are two types of work: external work \(W_E\) and internal work \(W_I\).

• The work done by external loads is external \(W_E\).
• The work done by reactions and internal force (axial, shear, moment, torsion) is internal \(W_I\).

In our example, the work done by the concentrated load \(P\) is external, while the work done by the reaction force (as well as shear or moment) is internal.

\(W_E=R_A\times{\delta_A}\)

\(W_I=P\times{\delta_B}\)

The \(\delta\) components are imaginary translations from the beam's original state to its fictional deflected shape.

The sum of these \(W\) is the total work or "net" work \(W_{tot}\) done on the body.

\(W_{tot}=W_E+W_I\)

In our example, (W_{tot}\) is equal to:

\(W_{tot}=R_A\times{\delta_A}-P\times{\delta_B}\)

The work due to the external load is negative because the force is opposite to the vertical deflection (see negative work).

Apply Work-Energy Theorem and Equilibrium

In any Physics class, one will discuss the work-energy theorem: the work done by all forces \(W_{tot}\) acting on a body equals the change in the body's kinetic energy:

\(W_{tot}=\Delta{K}\)

To solve for the unknown component, we apply this equation; however, it is essential to note that this structure is in static equilibrium - meaning it is at rest. Therefore, there must be no kinetic energy acting on the beam \(\Delta{K}=0\). As a result, the work-energy theorem becomes:

Work-Energy Theorem (Static Equilibrium): \(W_{tot}=0\)

To illustrate this outline, let's consider an example.

Summary

Let's summarize:

We can use virtual work and the work-energy theorem to solve a structure's reactions and internal forces.

We first identify the component we want to solve: reaction, shear, or moment. Afterward, we let it do "work" on the beam and allow it to deflect. The purpose of this deflected shape is to provide the fictional \(\Delta\) or \(\theta\) we need later on. This imaginary shape serves as our set-up when using the virtual work method.

Afterward, we need to solve all external work \(W_E\) and internal work \(W_I\) on the beam and get the total work on the body: \(W_{tot}=W_E+W_I\)

Finally, we apply the work-energy theorem and static equilibrium to solve for the missing component: \(W_{tot}=0\)