Deriving Partial Derivative Method Due to Flexural Strains

From our previous post, recall that we need an expression for the strain energy \(U\) to fit our general equation. We will focus first on strains due to flexural action.

This energy we need is familiar because we already have derived it in another post. In our discussion with real work strain energy due to flexural strains, we recall that strain energy is:

\(U=\frac{1}{2}\int\frac{M^2}{EI}dx\)

We highly recommend reading the said post if we need a refresher.

Key Idea: Partial Derivative Method Due to Flexural Strains

Moving on, with flexural strain energy derived, we can apply the general equation to it. As a result, we will obtain the following equations:

Partial Derivative Method - Flexural Strains - Translation: \(\Delta=\frac{\partial}{\partial{F}}\left[ \frac{1}{2}\int\frac{M^2}{EI}dx \right]\)

Partial Derivative Method - Flexural Strains - Rotation: \(\theta=\frac{\partial}{\partial{M_P}}\left[ \frac{1}{2}\int\frac{M^2}{EI}dx \right]\)

Here is the meaning of these variables:

• \(F\) or \(M_P\) refers to the placeholder loads.
• \(\Delta\) or \(\theta\) refers to the deflection component we would like to investigate.
• \(M\) refers to the bending moment due to the imposed loading and the placeholder.
• \(EI\) is flexural rigidity.

These equations are what we use to solve for the translation and rotation of a structure.

Simplified Approach

Let's move on and discuss how to evaluate the equations above. Generally, there are two ways:

• The first approach is to find an expression of strain energy \(U\) in terms of \(P\) (or \(M\)), then apply the equation.
• Another approach is to interchange the derivative and the integral. It is possible because these two are opposite operations. 

Between these two, we recommend the second one. We first start with our first expression:

\(\Delta=\frac{1}{2}\int\frac{\partial}{\partial{F}}\left[ \frac{M^2}{EI} \right]dx\)

\(\Delta=\frac{1}{2}\int2\times\frac{M}{EI}\times\frac{\partial{M}}{\partial{F}}dx\)

As a result, we have the following expressions:

\(\Delta=\int\frac{M}{EI}\times\frac{\partial{M}}{\partial{F}}dx\)

\(\theta=\int\frac{M}{EI}\times\frac{\partial{M}}{\partial{M_P}}dx\)

These equations are simpler to evaluate. Later, we shall use an example to demonstrate how to use these 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) \(\Delta=\int\frac{M}{EI}\times\frac{\partial{M}}{\partial{F}}dx\), and (2) \(\theta=\int\frac{M}{EI}\times\frac{\partial{M}}{\partial{M_P}}dx\).