4 Point Load Beam Deflection 1

[wsg1]

Hello,

I am looking at designing a pipe support using a universal beam. I believe I have the correct calculations for bending stress, shear stress etc. But I am struggling to find an equation for deflection. At the moment I am currently estimating the combined loads as a single point load (worst case scenario. Does anyone have an equation for this type of loaded beam (see attachment). I believe my other equations are correct but it has been a good few years since I have done these type of calcs.

Any help would be much appreciated

Cheers,

wsg1

 

[electricpete]

Great spreadsheet Doug. That's a creative approach to use cubic splines.

To put in my own words:

you apply a unit displacement at location of a given force and allow cubic spline to determine the displacement all along the beam (it matches the true function which is a cubic within each region between discrete loads).

Knowing the displacement polynomial on each side, we can differentiate 3 times and multiply by E*I to get shear. Difference in shear computed at the point using left-side formula and right-side formula is applied force to create the unit displacement. Multiply unit displacement function by ratio of actual force over force associated with unit displacement to get actual displacement

Use superposition to sum all applied loads (it is not required to sum reaction loads associated with zero displacement).

The existing spreadsheet handles fixed or simply-supported end conditions. It would be nice if it could handle "free" end condition also which should be a fairly straightforward alternate boundary condition. If I get a chance I might study your code to see if I can modify it to do that.

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[IDS]

Thanks Pete, and a nice summary of how it works.

Yes, it should be fairly easy to allow end cantilevers. It's on the "to do" list, but I don't know when I'll get round to it.

If you do do it yourself I'll be interested to see what you come up with.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

is fitting cubic splines any easier than applying the general solution ...
M(x) = Pb/Lx x<a Pb/Lx-P(x-a) x>a
EIv(x) = Pb/Lx^2/2+C1 Pax-Pa/Lx^2/2+C2
EId(x) = Pb/L^3/6+C1x+K1 Pax^2/2-Pa/Lx^3/6+C2x+K2

four boundary conditions d(0)=0, d(L)=0, v(0)=0, v(L)=0
C1 = K1 = 0
C2 = Pa/L*L^2/2 - PaL = -PaL/2
K2 = Pa/L*L^3/6-PaL^2/2+PaL^2/2 = PaL^2/6

so that
EIv(x) = Pb/Lx^2/2 Pax-Pa/Lx^2/2-PaL/2
EId(x) = Pb/L^3/6 Pax^2/2-Pa/Lx^3/6-PaLx/2+PaL^2/6

i think ...

 

[IDS]

rb1957 - I don't know if it's easier than other approaches, but if you have a general purpose cubic spline interpolation routine it is very easy to adapt it to finding deflections and forces in a continuous beam, and I find the concept quite neat.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

but there's no need ! you don't need to interpolate anything, you know the solution already

 

[electricpete]

You’re right that it is not interpolation... it’s exact. The Euler beam solution for discrete loads is a 3rd order polynomial in each region, so the cubic spline solution matching the specified boundary conditions will be identical to an algebraic solution matching those same boundary conditions

But the spline is also easily adapatable to handle more than 2 supports as shown in examples in the spreadsheet example. And a change in boundary condition (simple support, fixed, free) will just change 2 equations out of n+1, not the whole solution as it would for a an algebraic solution of a single predefined geometry.


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[rb1957]

i picked up on the word "interpolation" without looking at the method. if it does match the boundary conditions then it'll be the right solution. could you generalise the method to 4th order equations (it include distributed loads) ?

let's just leave it as "there are many ways to skin cats"

 

[electricpete]

no - could not generalize to distributed loads.


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[rb1957]

i realise a mistake with the solution i posted ... i've assumed fixed ends, for pinned ends v1(a) = v2(a) and d1(a) = d2(a) instead of the two slope BCs

 

[IDS]

Interpolation - it is interpolation. The method calculates the deflection at the supports (zero or specified deflection) and the loaded points (using standard slope-deflection methods), then fits a cubic spline through those points to find the deflection at intermediate points. The cubic spline is an approximation to the actual shape followed by a real beam, but the point is that it is exactly the same approximation as used in standard beam bending theory, so the results will be exactly the same as any other method using the standard theory.

Distributed loads - You would need to fit a fourth order curve to deal with distributed loads directly, but you could convert the distributed load to a point load and a moment at each loaded node position, then use the same method with a cubic spline by applying a unit rotation as well as a unit deflection at the load positions. This is how standard frame analysis programs do it, so the results will still be exactly the same as comes out of a standard program, or any other equivalent procedure assuming a cubic profile for the beam deflected shape in between node positions.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[electricpete]

Attached is my attempt to adapt Doug's approach to my own purposes.

I added capability for free boundary conditions as discussed.
I added the capability to put a spring under each intermediate support (along with existing capability to specify an alignment deviation of the support).

Altering someone else's program was too tedious for me so I rebuilt it from scratch using matrix algebra (but sticking very close to the algorithm that Doug used).

There is an instructions tab, a math tab (explaining the math approach and defining the conventions/symbols used in the vba program) and 3 example tabs.

To me this approach seems to provide tremendous flexibility over trying to solve the Euler problem algebraically (by hand) for more complex problems including those that are statically indeterminate. I shudder at the thought of trying to solve the "misalignment" example algebraically.

If any comments, suggestions or corrections, let me know. Thanks again to Doug for his idea.

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[electricpete]

Attached is a revised version "EP_BeamR1.xls" with a few corrections to the instructions and math tab as noted in "rev_history" tab.

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[IDS]

Thanks for posting that Pete. I have been struggling with springs, so I'll be interested to see what you have come up with.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

"springs" ... where did that come from ... i reviewed the thread and couldn't see anything ... is the beam on elastic foundations ??

 

[IDS]

"springs" ... where did that come from ... i reviewed the thread and couldn't see anything ... is the beam on elastic foundations ??

No the springs are an optional extra introduced along the way.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[electricpete]

"springs" ... where did that come from ... i reviewed the thread and couldn't see anything ... is the beam on elastic foundations ??

As I said, it was adapted “to my own purposes” which should signal that it was not a direct response to original question, but does relate to some of the followup discussion. If you are intrested, my purpose in adding a spring was for attempting to estimate effects of shaft misalignment in rotating machines (dynamic effects neglected).

I felt it was warranted to post the spreadsheet back in this thread because:
1 – Doug asked me to share what I came up with.
2 – You suggested the cubic spline technique was inferior (22 Jun 10 8:52). After the equivalence was clarified, you suggested it was no better than (22 Jun 10 10:11 cat skinning comment) solving the Euler beam equations in traditional algebraic fashion. These methods are certainly mathematically equivalent, but it is a heck of a lot easier to cover a wide range of problems very easily by spline method imo. It is a new useful tool for my toolbox and I think it might be for others also. Please feel free to use it or ignore it as you see fit.
=====================
Here is one more rev of the spreadsheet with a few very minor improvements corrections. The changes are again shown in RevHistory tab.

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