Why Tapered Beams Used for Roof Support?

[hyper]

I have a statics question. In my book under a picture it says.

"To save material the beams used to support the roof of this shelter where tapered since the roof loadings will produce a larger internal moment at the beams' centers than at their ends."

picture:

http://img228.imageshack.us/img228/8010/bilde001sp4.jpg

But why will the roof produce larger internal moments at the beams ends?

 

[tiny-tim]

Hi hyper!

It seems a daft way of putting it, but I think they just mean that the end of the beam only has to support the roof above it, but the centre of the beam has to support the end of the beam also, and therefore the weight of the whole of the roof.

 

[hyper]

Hi hyper!

It seems a daft way of putting it, but I think they just mean that the end of the beam only has to support the roof above it, but the centre of the beam has to support the end of the beam also, and therefore the weight of the whole of the roof.

Wow thanks, that's smart!

 

[hyper]

I have thought some more about this and I think you are wrong tiny-tim. Why would the center have to support mroe? And what does this have to do with the moment?

 

[tiny-tim]

I have thought some more about this and I think you are wrong tiny-tim. Why would the center have to support mroe? And what does this have to do with the moment?

Hi hyper!

Because the moment is the turning force that tends to make the beam bend (or break).

For example, the same force twice as far from the centre has twice the moment, and therefore need twice the strength to counter it.

And the centre of the beam needs to be thicker than the ends because ultimately all the force passes through the centre.

 

[rdbateman]

I have a statics question. In my book under a picture it says.

"To save material the beams used to support the roof of this shelter where tapered since the roof loadings will produce a larger internal moment at the beams' centers than at their ends."

picture:

http://img228.imageshack.us/img228/8010/bilde001sp4.jpg

But why will the roof produce larger internal moments at the beams ends?

That is not what it is saying. The quote says that beam center will hold the most internal moment, which is not it's end. To illustrate, draw a cantilever beam of length L (i.e. a beam with a fixed support at one end and free at the other end). Furthermore, place a distributed load w along the length of the beam. With this structure in static equilibrium and within the elastic range, the largest internal moment will be located at the fixed end and will equal in magnitude to (w)(L^2)/2. The magnitude of the internal moment at the free end will be 0, and moments in between will gradually decrease parabolic as you travel away from the fixed end. Now let's talk design. The cross sectional area "needed" to handle the internal moment must be bigger at the fixed support than at the end because a larger cross sectional area must be needed to support a larger internal moment. If the beam was chosen to be perfectly rectangular, material would be wasted because you now have a large cross sectional area at the free end of the beam supporting a theoretical zero moment. That is why the beam's area becomes smaller as it travels away from the fixed support in a parabolic fashion.

Hope this helps! :)