Structural Engineering - Deflection

[AlenDK]

Hey Guys,

I wanted to know if anyone knew the proof for the Simply supported beam with central load formula; w_max = F*L^3 / (48*E*I). I have looked some soureces up, such as https://en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory , but i don't really understand how they ended up with that formula for the maximum deflection. Can anyone explain how they ended up with that formula?

 

[SteamKing]

Hey Guys,

I wanted to know if anyone knew the proof for the Simply supported beam with central load formula; w_max = F*L^3 / (48*E*I). I have looked some soureces up, such as https://en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory , but i don't really understand how they ended up with that formula for the maximum deflection. Can anyone explain how they ended up with that formula?

You can find derivations of this formula in most strength of materials texts:

Here is one reference:
ftp://ftp.ecn.purdue.edu/sozen/ICHINOSE/BEAM.pdf

Make sure your calculus skills are up to snuff.

 

[AlenDK]

Wow, Thanks! But is there multiple ways to end up with the formula? I found this text, which shows another way.. Is one of them more "right" than the other? http://www.google.dk/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&ved=0ahUKEwi_m8Oi497JAhWiEHIKHf6VAlYQFggdMAA&url=http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/bdisp.pdf&usg=AFQjCNHO4xDZ1a50d_rng1t5JfB1EJZm2Q&sig2=FMxF5Xrc9ggiS-KDOoxhCA&bvm=bv.110151844,d.bGQ or Google http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/bdisp.pdf and it should be the first link.

 

[SteamKing]

Wow, Thanks! But is there multiple ways to end up with the formula? I found this text, which shows another way.. Is one of them more "right" than the other? http://www.google.dk/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&ved=0ahUKEwi_m8Oi497JAhWiEHIKHf6VAlYQFggdMAA&url=http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/bdisp.pdf&usg=AFQjCNHO4xDZ1a50d_rng1t5JfB1EJZm2Q&sig2=FMxF5Xrc9ggiS-KDOoxhCA&bvm=bv.110151844,d.bGQ or Google http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/bdisp.pdf and it should be the first link.

There are several different methods which can be used.

For most students in a basic strength of materials course, usually the slope-deflection method and the double integration method are taught.

http://www.facweb.iitkgp.ernet.in/~baidurya/CE21004/online_lecture_notes/m3l14.pdf

The first part of the link you supplied uses the double integration method with singularity functions, which functions make it easier to represent quantities which contain several discontinuities in their mathematical representation.

https://en.wikipedia.org/wiki/Singularity_function

A slightly different version of the singularity function is called Macauley's method.

https://en.wikipedia.org/wiki/Macaulay's_method

Students taking more advanced strength or materials or structures courses are taught how energy methods can be used to determine deflections, which is what your link discusses in the second part.

There are multiple ways to derive these deflection formulas. The important thing is that the various methods all arrive at the same result for the same support conditions and loading.

 

[AlenDK]

Thank you! :)

 

[AlenDK]

You can find derivations of this formula in most strength of materials texts:

Here is one reference:
ftp://ftp.ecn.purdue.edu/sozen/ICHINOSE/BEAM.pdf

Make sure your calculus skills are up to snuff.

Hey Steamking, i have just one more question! - How and why is the Constant weird 0_A added after the integration? :)

 

[SteamKing]

Hey Steamking, i have just one more question! - How and why is the Constant weird 0_A added after the integration? :)

There are a number of articles discussed in the posts above. Is there one particular place where 0_A appears?

In general, indefinite integration always leads to a constant of integration which is tacked on at the end. Depending on external conditions, this constant may or may not have a non-zero value. It's better to include it and then try to evaluate it later, if given certain boundary conditions which must also be satisfied.

This is often the case in structural analysis where the deflection or slope of a member is established by the support conditions. For example, a cantilever beam must have its deflection and slope both equal to zero at the fixed end.