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A problem about virtual work principle for continuous system

  1. Sep 25, 2014 #1
    dear all, the virtual work pinciple can be used to derive the equilibrium equations for the mechanical systems. however, when I want to apply it to a continuous system, I found it can not give out the simple equilibrium equations. there should be something wrong with my thinking. I expect some expert could give me some advice. thanks very much.

    the problem detail is shown in the pictures:

    Attached Files:

  2. jcsd
  3. Mar 1, 2015 #2
    The virtual work needed to compress (or elongate) the small element is the stress times the amount of the compression. So,
    &TA(\delta u+\frac{d \delta u}{dx} dx - \delta u)=TA\frac{d \delta u}{dx} dx,\\
    &\delta W=\int_0^L TA\frac{d \delta u}{dx} dx=TA\delta u|_{x=0}^{x=L}-A \int_0^L \delta u\frac{dT}{dx}dx =TA\delta u|^{x=L}-A \int_0^L \delta u\frac{dT}{dx}dx,
    where we use integration by parts.
    As [itex]\delta u[/itex] is arbitrary, we have [itex]\frac{dT}{dx}[/itex] anywhere other than the open end[itex]x=L[/itex]. Happily we know that [itex]T=P[/itex] at [itex]x=L[/itex]. We get equation (1).
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