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# Beam deflection by virtual work

## Beam deflection by virtual work

(OP)
I have a beam situation that doesn't match anything in my handbooks.  The equations for the bending and shear change at x=a.  Setting up
M(x) = xxxx if x<a
M(x) = yyyy if x>a
works for both bending and shear.  But when I try to find deflection by doing an integral of M(x)*m(x)dx/EI where,    m(x) is the moment due to a virtual load, the result is just a straight line.

Can a discontinuous function like this be integrated?  Am I missing something here?

Thanks,
DPA

### RE: Beam deflection by virtual work

Hi,

maybe you integrated two times over the WHOLE length instead of until x.

In my case it looks plausible:

Christian

### RE: Beam deflection by virtual work

(OP)
Thanks Chris,

I found a couple of loads in the steel handbook that I could calculate directly and integrated them as a check.  If you integrate piecewise up to the point where the moment formula (i.e. loading) changes and integrate the entire length rather than the length to the virtual load you get the same answer as the steel handbook equations.  That was kind of a surprise but I verified it a couple of ways.  Now that I finally have time I'll have to go back and study the derivations in my old structures book.

You have to place the virtual load at the point of max deflection i.e. point of zero shear.

Thanks,
DPA

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