Need help with 3-moment equation (multi-span beams)

[XaeroR35]

It has been a few years since I have done beam analysis, and I really don’t remember how to solve this type of problem so I have been reading up on it. I have found similar situations but nothing that matches the problem I am having.

I have a continuous beam of 3 spans with a uniform distributed load centered in the middle span (not full width).

I believe I need to use the Equation of Three Moments to solve this with Matrices, but I am stuck.

Since my material and cross sections are the same for the beam, I believe my 3-moment equations reduce to:

(M1*La) + (2*M2*(La+Lb)) + (M3*Lb) = - (6*xa*Aa)/(La) - (6*xb*Ab)/(Lb)
(M2*Lb) + (2*M3*(Lb+Lc)) + (M4*Lc) = - (6*xb*Ab)/(Lb) - (6*xc*Ac)/(Lc)

How do I get xa, xb, xc, and Aa, Ab, Ac? I understand these to be the centroids and areas of the moment curves for each span, but I am really struggling to come up with their values.

 

[sardonicus]

It's been too long for me too. But it appears you have six unknowns and only two equations. Am I missing something? Or is that what you need, 4 more equations?

Sum of loading = sum of reactions
Deflection at supports = 0

That's 5 more?

S

 

[XaeroR35]

This is a statically indeterminate case, which requires special methods to solve, one of which is the three-moment equation.

I just can't find an example with this loading condition. I found point load, and uniformly loaded across the entire span, just not uniform load at a small section.

 

[sardonicus]

R35,

You need to define your terms, and you need to include w, the load, and the length of the load, it looks like it's about the center quarter of the middle span. If it's symmetric you can probably simplify it although it will still be statically indeterminate.

There are other ways to solve your beam problem, unless the 3 moment equation is your interest maybe try another method. S

 

[XaeroR35]

I know all the terms, I just don't know how to solve this problem. I sketched it quick to give an example of what I am trying to do.

I found a spreadsheet that solves this, but I would like to know how to do it by hand. I have not done beam analysis since school (8+ years ago)

 

[sardonicus]

It's helpful to others if you define the terms. S