This post will expand the general real work equation to consider deflections due to axial strains.
Deriving Real Work Due to Axial Strains
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Let's consider our discussion on strain energy. It is equal to:
From this equation, we can say that the differential strain energy per member is:
Equation 1:
refers to the internal bar force caused by actual loads (tension or compression)- The variable
refers to the axial strain caused by actual loads (elongation or compression)
We've derived the latter variable
Real Work - Axial Strain Energy:
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:
Real Work - Axial Strains - Rotation:
(or ) represents the actual load applied on joint. represents the translation of joint. represents the rotation of a member. is the internal bar force caused by actual loadings. is the length of the member. 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
Lastly, when we're dealing with truss members, the
Summary
Let's summarize:
The expression for the strain energy due to axial strains is.
The equations we will use to solve for the deflections are the following: (1), and (2) .
If, , , and are constant, we can further simplify such equations as (1) and (2) .
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