Deriving Real Work Due to Flexural Strains

Let's consider our discussion on strain energy. It is equal to:

\(U=\frac{1}{2}M\theta\)

From this equation, we can say that differential strain energy is:

Equation 1: \(dU=\frac{1}{2}Md\beta\)

• \(M\) is the bending moment caused by real loads.
• The variable \(d\beta\) refers to the flexural strain caused by actual loads.

In our discussion of virtual work for flexural strains, we have already derived \(d\beta\) as:

\(d\beta=\frac{M}{EI}dx\)

Hence, we can express Equation 1 as:

Real Work - Flexural Strain Energy: \(U=\frac{1}{2}\int\frac{M^2}{EI}dx\)

Key Idea: Real Work Due to Flexural Strains

With the previous equation, we can expound on the general real work equation to formulate the different equations we will use to solve for translation and rotation:

Real Work - Flexural Strains - Translation: \(F\times\Delta=\int\frac{M^2}{EI}dx\)

Real Work - Flexural Strains - Rotation: \(M\times\theta=\int\frac{M^2}{EI}dx\)

• \(F\) (or \(M\)) represents the load (or couple) applied at position \(x\).
• \(\Delta\) (or \(\theta\)) represent the components we would like to investigate at position \(x\)
• \(M\) is the bending moment equation caused by real loadings.
• \(EI\) is flexural rigidity.

We'll use these equations to solve for deflections where bending is the main form of stress, such as beams and frames.

Later, we shall use an example of applying this set of equations.

Summary

Let's summarize:

The expression for the strain energy due to flexural strains is \(U=\frac{1}{2}\int\frac{M^2}{EI}dx\).

The equations we will use to solve for the deflections are the following: (1) \(F\times\Delta=\int\frac{M^2}{EI}dx\), and (2) \(M\times\theta=\int\frac{M^2}{EI}dx\).