How to Solve Deflections Using Real Work?

Setting Up the General Real Work Equation

Say we have a simple beam $AB$ loaded with a point load $F$ at point $C$. We want to investigate the translation at $C$.

From energy conservation, when there is an applied load $F$ that caused $\mathrm{\Delta }$, there is total external work $W$ and strain energy $U$:

Equation 1: $W=U$

In this situation, we'll assume that F was applied gradually, as seen in the $P-\mathrm{\Delta }$ graph in the figure. It makes the structure behave within the elastic region.

The total external work $W$ is then equal to:

Equation 2: $W=\frac{1}{2}F{\mathrm{\Delta }}_v$

Key Idea: General Real Work Equation

We equate Equations 1 and 2 to formulate the general equation used to find deflections using real work:

General Equation of Real Work - Translation: $\frac{1}{2}\times F\times \mathrm{\Delta }=U$

General Equation of Real Work - Rotation: $\frac{1}{2}\times M\times \theta =U$

• $F$ (or $M$) refers to the actual load (or couple) at position x.
• $\mathrm{\Delta }$ (or $\theta$) refers to the deflection we would like to investigate at the point where $F$ (or $M$) is located (at position $x$).
• $U$ is strain energy.

Real Strain Energy

We need to expound on the general real work equation due to strain energy $U$ (similar to what we did with the general virtual work equation). This value would entirely depend on the primary stress the structure is experiencing, whether flexural, axial, or torsional.

Summary

Let's summarize:

Real work uses actual forces and displacements to find translations and rotations.

The general equation for real work are $\frac{1}{2}\times F\times \mathrm{\Delta }=U$ and $\frac{1}{2}\times M\times \theta =U$

One drawback is that you can solve only one deflection component where the point load or couple is applied.

We need to expound on the general real work equation due to strain energy $U$. This value would entirely depend on the primary stress the structure is experiencing, whether flexural, axial, or torsional.