Maths solution for partially restrained cantilever beam

[Roger44]

Hello

Can somebody guide me to a mathematical solution for a cantilver beam which is totally restrained on only 2/3 of its depth.

Thanks in advance

 

[Studiot]

By mathematical solution I presume you mean an analytical solution as opposed to a numerical (FE?) one.

Do you have an FE one?

I don't think there is a single mathematical solution since analysis depends in part on keeping the dimensions, or their ratios with certain limits.

Are you saying the cantilever is so short that the bearing conditions are significant?

You haven't shown any down fixings, are you confusing fixity with encastre?

 

[Roger44]

Hello Studiot, thanks again for taking interest in my questions.

Do you have an FE one? - FE ,I think you mean finite solution, no?

Are you saying the cantilever is so short that the bearing conditions are significant? -No, just the usual conditions for which the standard solution is sufficiently valid BUT with the exception that only the lower 2/3 of the extremity fulfils the condition of being restrained (in french encastré)

This may be 2/3 embedded in the top of a concrete wall, or 2/3 held by a perfect wall hanger.

 

[Studiot]

I'm puzzled.

If the standard cantilever formulae are applicable why does it matter that only the bottom is built in?

Diving boards, for instance, are near perfect cantilevers but are only fixed on the underside.

FE = finite element (model solution)

 

[Roger44]

"If the standard cantilever formulae are applicable why does it matter that only the bottom is built in?"

The standard cantilever formulae are applicable when all the horizontal fibres are locked at one end, therefore they cannot be appilicable in all other cases, such as when an upper third are not locked.

To take an extreme example, I can't help feeling that a steel cantilever I beam bolted to a framework via its lower flange would exert a different couple on the framework if it were bolted by its upper flange too.

Do you see what I'm getting at Studiot?

 

[Roger44]

Maybe the I beam would exert the same couple. For a cantilever beam of length L, at any point x along the beam, only the moment of distance x to L needs to be considered.

 

[Studiot]

To take an extreme example, I can't help feeling that a steel cantilever I beam bolted to a framework via its lower flange would exert a different couple on the framework if it were bolted by its upper flange too.

Do you see what I'm getting at Studiot?

Yes I get your meaning, but you are suffering under a misundestanding of structural action.

The moment that has to be applied at the support does not depend on the support conditions.
It only depends on the loading.
This is a matter of basic mechanics.

The way this moment is applied depends upon the support configuration. This may lead to local stress variations in the cantilever at the support zone but the projecting length will always act in the same fashion.

I asked in my first post about down fixings. This was because the cantilever will tend to tip about the line of the wall vertical face, lifting the back edge off the wall top, like a see saw.

Thus the back needs to be bolteds down or otherwise fixed.
The holddown force times this lever arm equals ( and provides) the reaction moment for the cantilever.
It is obviously equal and opposite to the total moment on the projecting part of the cantilever, including self weight.

 

[Roger44]

OK, I get you. The basic mechanics is that moments, like forces, have to be equated in static mechanics.

Am I therefore right in saying that if I have a horizontal board bolted along a wall, it doesn't matter a hoot how AND WHERE the floor beams are perfectly rigidally fixed to the side surface of this board, be it nailed, glued, with hangers or whatever you like, fixed towards the top or towards the bottom of, there will always be the same traction force trying to pull the bolts out of the wall?

Thanks a lot

 

[Studiot]

A perfect cantilever has a vertical reaction and a moment reaction, but no horizontal reaction, at the support.

Since the lever arm that provides this moment reaction is likely to be short the forces and therefore local stresses are likely to be high in the region of the support.

So if you are cantilevering a timber member, make sure any fixings are designed to spread the load.

 

[Roger44]

A perfect cantilever has ... a moment reaction...at the support. Since the lever arm that provides this moment reaction is likely to be short (it is, the 200 mm depth of the horizontal wall board) ...local stresses ..(pulling out the bolts holding the board to the wall) ...are likely to be high. Make sure any fixings are designed to spread the load.

they are very high. But you haven't confirmed that the mode of rigid fixation of the cantilever to the horizontal wall board won't change the moment reaction between the wall board and the masonry wall.

 

[Studiot]

Look again at the second paragraph of post#7

The only agents which determine the moment reaction are the loadings.

This moment has to be generated by any fixing method or the cantilever will fail.

It is impossible to have partial fixity of a simple cantilever in the same way you can with multispan bridges.