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Integrating Shear Along the Length of a Beam
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Joined: 04/10/2024(UTC) Posts: 3

Hello. I'm a long time lurker on this forum, but this is my first post. I'm trying to solve for shear, moment, and deflection along a beam via direct integration and superposition. The piecewise equation for shear is pretty simple and I am able to plot V(x). But I am running into trouble integrating it to get M(x). I know that the builtin integration tool sometimes has trouble with units, maybe that's the error that I'm running into? I appreciate any help! For reference, this is what I'm trying to do: V(x) from beam equations > M(x) = ∫ V(x) dx > θ(x) = 1/EI ∫ M(x) dx > Δ = ∫ θ(x) dx I am using mkraska's unoffical portable distribution with Maxima (v99.7579), if that matters. Updated Simple Span Beam Calculator.sm (131kb) downloaded 4 time(s).




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The Maxima plugin can't translate boolean expressions and the cases() function. Therefore, piecewise continuous functions need to be defined using the sign() function, as demonstrated in the attached document (showing shear and bending moment for piecewise continuous distributed load). You can, however, use the native numeric integrator, yet cascading it is not recommended due to performance reasons. The noise at the right is a sampling artifact due to the standard 100 integration intervals. Increase the value under Tools> Options to reduce the noise and increase computation time. Updated Simple Span Beam Calculator (1).sm (134kb) downloaded 5 time(s). Piecewise.pdf (359kb) downloaded 3 time(s). 




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SMath doesn't seem to like doing more than one integration here! However, it's easy to do a simple numerical quadrature for theta and delta: Updated Simple Span Beam Calculator_2.sm (138kb) downloaded 4 time(s).




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Joined: 04/10/2024(UTC) Posts: 3

mkraska, when I open your file MUDL doesn't plot and it gives an error: "x' is undefined". All the plugins are enabled and dependencies resolved, but there must be a back compatibility issue with opening your file in an older version. : ( StvMath, I get a similar error with your file: "X is undefined". Also your plots for theta and delta don't match up with what you would expect. For the uniformly distributed load entered at the top of the spreadsheet, delta should be roughly parabolic. Numerical integration gives the values for total area under the curve along the beam, which isn't a helpful value for this use case. I guess this example gets at my frustration with SMath (at least in an a structural engineering application). I see "what you see is what you get" as being incredibly value for performing "hand calcs" and showing your work, but when something as straightforward as double/triple integration requires setting up loops and/or other nontrivial functions, "what you see" gets obscured from "what you get" and you end up with something like Excel spreadsheets which are intuitive to the person that made them, but impossible to understand for anyone else.




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Hi. There are some long discussions here about issues handling piecewise functions for beam calculations. BTW, this could be a way to solve the problem: Simple Span Beam Calculator.sm (125kb) downloaded 6 time(s).Further, you can use interpolation for define functions for M, θ and Δ. Best regards. Alvaro.

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Razonar, what is that one sided bracket with the initial delta values in it? I am working with an older version of Smath (and older ODE plugin) that doesn't support that element. Trying to find out if there is a way around the unsupported element.




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1 user thanked Razonar for this useful post.



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@willmccartney
StvMath, I get a similar error with your file: "X is undefined". Also your plots for theta and delta don't match up with what you would expect. For the uniformly distributed load entered at the top of the spreadsheet, delta should be roughly parabolic.
"X is undefined" Can't explain this! It works for me. An SMath idiosyncrasy?
Plot for delta. Perhaps we're looking at different curves, as it looks pretty parabolic to me!




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