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

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Discussion Overview

The discussion focuses on solving a three-moment equation for a continuous beam with three spans subjected to a uniform distributed load in the middle span. Participants are exploring methods for beam analysis, particularly in the context of statically indeterminate structures.

Discussion Character

  • Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant expresses uncertainty about how to apply the three-moment equation and seeks clarification on determining the values of centroids and areas of moment curves.
  • Another participant points out that there are six unknowns and only two equations provided, suggesting that additional equations are needed to solve the problem.
  • A different participant notes that the case is statically indeterminate and mentions the need for special methods, including the three-moment equation, while expressing difficulty in finding relevant examples for the specific loading condition described.
  • One participant advises defining terms and including specific parameters related to the load and its location to simplify the problem, while acknowledging the statically indeterminate nature of the situation.
  • Another participant indicates they have a sketch to illustrate the problem but still struggles with the analytical solution, despite having found a spreadsheet that can solve it.
  • There is a suggestion that defining terms may be helpful for others involved in the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on how to proceed with the problem. There are multiple competing views on the approach to take, and the discussion remains unresolved regarding the specific method to solve the three-moment equation.

Contextual Notes

Participants express uncertainty about the necessary equations and parameters needed for the analysis, highlighting the complexity of the statically indeterminate case and the specific loading conditions.

XaeroR35
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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.
 

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

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