(1)*delta = integral from 0 to L ( m * M)/EI dx

if the moment equation for the real structure and virtual are from 2 different coordinate system (ex: one taken from the left and one taken from the right), can it still be used for the above equation?

OR DOES BOTH MOMENT EQUATION have to come from the same coordinate system, (ex: all the equations are derived by takin a cut from the left side)

i assume you mean castiglianos second theorem
if you decide to use left to right convention then do right to left at the end, you have to integrate right to left for the last part
when you change direction your dx changes to a something else, perhaps d xbar

If you swap coordinate systems you have to start over from scratch. SCRATCH.

Worked an example for you. Kinda just made up my own example since I didn't understand yours.

The traditional coordinate system I got from the book (pre-derived), but you can derive it the same way that I derived the reverse. In the end I checked against the published deflection at the end of the beam to confirm my equation.

Depending on the type of beam and loading, some coordinate systems are definitely more convoluted than others. You almost ALWAYS start from the left, though. There is little reason to go from the right, but sometimes if you have to derive it, it does make it easier (depends on loading, like I said).



This post was edited by Dontrunaway on Nov 11 2014 12:31am

yea i guess thats the theorem, my prof never mentioned its name lol
now we are doin virtual work - visual integration method, where area of (1)*delta = M/EI * value of m/EI at centroid of M/EI



im not sure if im askin applies to "Deflections by double integration method"

like for example, if a beam has various loads ( UDL, point load, triangular, etc.) there would different moment equation at each discontinuity of the loading,

and in the formula of "(1)*delta = integral from 0 to L ( m * M)/EI dx" I would need moment equation of each discontinuity on the real strucutre.

on the virtual strucutre it may be easier to use left to right because theres 1 virtual load on it, but on the real structure it may be easier to use right to left. (This is to get the moment eequation)... so idk if that works.

It does work, but you need a NEW moment equation from the right. You can't just copy pasta your moment equation from the left because if your loading isn't 100% symmetric it will be a completely different moment equation. That's what I tried to show.

TLDR: A moment equation is only correct for the coordinate system in which it was made.

This post was edited by Dontrunaway on Nov 12 2014 09:10pm

so what ur saying is, u cant multiiply an equation that was from "left to right" by another equation that was from "right to left" then integrate?

--> this formula
"(1)*delta = integral from 0 to L ( m * M)/EI dx

where m is the moment equation from the virtual structure and M is the moment equation from the real structure

I have only ever worked with Castigliano's so I can't tell you with 1000% certainty for virtual structures because I haven't worked with that method obviously... but for anything symmetric such as an end point load on a cantilever beam as shown above your moment equations from either direction are symmetric.

Just think worst case scenario...is there possibly something you could multiply moment equations to flop directions if you have things that are uncentered such as a UDL, a DL (not uniform) a few point loads, etc...there's no way in hell. Unless you have something very simple that is symmetric I think you're boned on the "transforming" term as far as your question is concerned.

yea i am boned, wat if my prof gives us a complicated loading wit a bunch of trinagular loads, and i have to get the equation of the moment which is a cubic.

You should only be boned if you don't know how to derive the moment equations. Do you need help with that?

no. just saying its gonna turn out to be a ugly lookin integral. cubic polynomial * cub polynomial