Deriving Real Work Due to Axial Strains

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

\(U=\frac{1}{2}P\Delta\)

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

Equation 1: \(dU=\frac{1}{2}Sd\Delta\)

• \(S\) refers to the internal bar force caused by actual loads (tension or compression)
• The variable \(\Delta\) refers to the axial strain caused by actual loads (elongation or compression)

We've derived the latter variable \(\Delta\) during our discussion on virtual work; hence, we can substitute such expression to Equation 1 to get:

Real Work - Axial Strain Energy: \(U=\frac{1}{2}\int\frac{S^2}{AE}dx\)

Key Idea: Real Work Due to Axial 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 - Axial Strains - Translation: \(F\times\Delta=\int\frac{S^2}{AE}dx\)

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

• \(F\) (or \(M\)) represents the actual load applied on joint.
• \(\Delta\) represents the translation of joint.
• \(\theta\) represents the rotation of a member.
• \(S\) is the internal bar force caused by actual loadings.
• \(L\) is the length of the member.
• \(AE\) is axial rigidity.

Later, we shall use these equations to demonstrate how to use the real work method.

Before we move on, we must make some remarks on these equations. First, these expressions consider any variation in terms of \(A\) and \(E\). That is, these variables must be a function of \(x\); however, if \(A\) and \(E\) are constant throughout the length \(L\) of the member, then the equations will become:

\(F{\times}\Delta=\sum_{i=0}^{n}\frac{S_i^2L_i}{A_iE_i}\)

\(M{\times}\theta=\sum_{i=0}^{n}\frac{S_i^2L_i}{A_iE_i}\)

Lastly, when we're dealing with truss members, the \(\theta\) of these equations refers to the rotation of the member and not the joint. It is because joints are hinges; hence, we cannot apply the moment at joints. It is only applicable if a moment on the member is a pair of two equal and parallel forces applied at the ends of the joint.

Summary

Let's summarize:

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

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

If \(S\), \(u\), \(A\), and \(E\) are constant, we can further simplify such equations as (1) \(F{\times}\Delta=\sum_{i=0}^{n}\frac{S_i^2L_i}{A_iE_i}\) and (2) \(M{\times}\theta=\sum_{i=0}^{n}\frac{S_i^2L_i}{A_iE_i}\).