Fixed end moments with a beam with udl

[date.chinmay]

I came across a reference where there is a beam (lets say length L)
both ends fixed... with a UDL of say "W" per unit length... i want to know how the end moments are W L^2/12 ...shudnt they be W L^2/4??


Its pretty simple mechanics. but it got me wondering,,,,

 

[SteamKing]

In the fixed end beam, both the slope and deflection at the ends must equal zero. In a simply supported beam, there are different end conditions and the moments are typically zero, unless there is also an applied couple.

 

[date.chinmay]

nice,,,
i wondered if u cud provide a solution or a link to the actual stepwise solving of one...

 

[SteamKing]

Here is an excerpt from a Schaum's Outline of Strength of Materials (prob. 11.6):

This problem shows a beam which is partially loaded with a UDL. The technique can be used when the UDL is extended the full length of the beam.

 

[DeeNos]

I can provide a response to this content. The end moments in a beam with both ends fixed and a uniform distributed load (UDL) of W per unit length are indeed W L^2/12. This is because in this scenario, the beam experiences a negative bending moment at both ends, which cancels out the positive bending moment created by the UDL. This results in a net end moment of W L^2/12 at each end of the beam.

On the other hand, if the beam had only one fixed end and one pinned end, the end moments would be W L^2/8. This is because in this case, the fixed end would experience a larger negative bending moment due to the UDL, while the pinned end would experience a smaller positive bending moment. The difference between these two moments results in an end moment of W L^2/8 at the fixed end.

In summary, the end moments in a beam with both ends fixed and a UDL of W per unit length are W L^2/12, while the end moments in a beam with one fixed end and one pinned end are W L^2/8. This is a result of the distribution of bending moments along the beam, which is influenced by the type of support at each end. I hope this explanation helps to clarify your understanding.