Introduction to Work Methods

Elastic and Plastic Regions

To adequately explain, say that we are to gradually increase force \(P\) continually and plot the results using a force-displacement \(P-\Delta\) diagram:

At first, there is a direct linear relationship between \(P\) and \(\Delta\), as evidenced by the straight line. During this phase, the material returns to its original unstressed state whenever we remove \(P\). In Strength of Materials, we say that the bar is at the elastic region.

As \(P\) continually increases, there comes the point where the deformations it experiences becomes permanent. Once we release \(P\) at this state, the bar will never return to its original unstressed state. At this point, we say that \(P\) and \(\Delta\) observe a nonlinear relationship. We say the bar has reached the plastic region.

Between these two regions lies a point we call the yield point. It marks the transition between the two areas.

Because of these different regions, the answer to how to compute strain energy is not precisely direct. However, we can interpret it as the area under a \(P-\Delta\) graph.

If the work applied causes the structure to deform in the elastic region, then the area under the \(P-\Delta\) diagram is:

Strain Energy - Elastic Region: \(U=\frac{1}{2}P\Delta\)

This expression is what we used to solve for strain energy \(U\) (if the object is still in the elastic region). It means you can only use it until the object has reached its yield point.

Superposition

• Superposition allows you to add reactions, internal forces, and deflection of similar structures with different loading conditions to get a combined similar model.
• Superposition can also "break down" an existing structure into similar structures that can be combined again.

You can learn more about these concepts in this post.

Work Methods

We are now ready to discuss the different work methods for solving deflections of structures. There are two work or energy methods approaches (1) Virtual Work and (2) Real Work.

The second method, real work, is limited to certain instances, as we will discover soon. To break its limitations, there is the Partial Derivative Method.

From these approaches, there is real work and fictional (virtual) work. To expound, recall that work is force \(F\) times displacement \(\Delta\) or moment \(M\) times rotation \(\theta\). If both are real quantities, then it is real work. On the other hand, if any of those quantities is imaginary, then it is virtual work.

Summary

Virtual and real work methods are essential procedures in solving translations and rotations.

Also known as energy methods, we investigate the work or energy done by the structure rather than analyzing it in terms of its loads.

One massive benefit of the work methods is that they allow you to solve for the deflection of plane beams, frames, and trusses.

As preparation, we recall and introduce several concepts: work, energy, the Law of Conservation of Energy, strain energy, and superposition.

Work \(W\) happens if a force or moment causes something to move: \(W=P{\times}{\Delta}\) or \(W=M{\times}{\theta}\).

Energy is the ability to do work.

The Law of Conservation of Energy states that nothing can create or destroy energy. Generally speaking, we can express this law as \(W=U\)

Strain Energy \(U\) is potential energy due to the deformation of an object.

Strain energy is the area under a \(P-\Delta\) graph. It can either be elastic or plastic.

If the work applied causes the structure to deform in the elastic region, then the area under the \(P-\Delta\) diagram is \(U=\frac{1}{2}P\Delta\).

Superposition allows you to add reactions, internal forces, and deflection of similar structures with different loading conditions to get a combined similar model.

Superposition can also "break down" an existing structure into similar structures that can be combined again.

There are two work or energy methods approaches (1) Virtual Work and (2) Real Work.

An expansion of the Real Work method exists which is the Partial Derivative Method.

If both \(F-\Delta\)/\(M-\theta\) are actual quantities, then it is real work. On the other hand, if any of those quantities is imaginary, then it is virtual work.