Help with purlin calculations

[anthonyl123]

I have a single 180x50mm purlin either side of my concrete tiled roof (battens & rafters).
The purlin is about half way down the roof and the plan view of the roof is around 3.4m from the ridge line to the eaves.

Supports for the purlins are around every 2m except some previous owner though it a good idea to remove a couple whilst re-doing the water tank and other works. The result is that one side of the roof has a slight sag due to the 5m unsupported span in this area.

I've commissioned a set of calculations which I am trying to follow and possibly rework with some different values. The reason being that the proposal is to put a single support block to reduce the large span to 3.3m and reinforce that part of the purlin with another 175x75 purlin bolted on. Now there is a possibility of using a different wall for the support block which involves moving/removing the water tank which I probably want to do anyway. So if I can understand the calcs involved I can punch in the different length.

The calcs are (and which all have very little meaning to me) - based on span 3.3m
w = 1.7 x 1.85 = 3.1 KN/m
M(max?) = 3.1 x (3.3 x 3.3)/9 = 3.8 KN/m

Z(xx) = 50 x (180x180)/6 = 270 x 10 to power 3
I(xx) = 50 x (180 to power 3)/12 = 24.3 x 10 to power 6

Therefore sigma = 3.8 x 10 to power 6/270 x 10 to power 3 = 14
and sigma (all) = 5.3 x 1.25 = 6.6 N/m squared

Then after trying the calcs with a 175 x 50 C16 purlin which shows insufficient (sigma = 7.2N/m square) which is too high they are redone with 175 x 75mm C16 giving a sigma of 5.8 which is deemed ok. All handwritten so not easy to read writing either.

Anyhow a bit of help with understanding the process above will enable me to play with the figures a bit. I did Mech Eng and awful long time ago so I assume there are some standard static and dynamic loads as well as standard beam bending moment figures in there somewhere.

Thanks

 

[tony1851]

Without going too far into the figures at this stage, I suspect the combined dead- and live loads quoted are a little on the low side. If your roof pitch is around 30 degrees (?), I would think the unit load would be more like 3.5-3.6 kN/m.

Also, maximum bending moment for a uniform load is WL/8, while he has gone a little lower at WL/9. Maybe that's because he's assuming some continuity of the purlin over a support, which reduces Mmax slightly, but should really play safe at WL/8.

The sigma bit is the maximum stress per mm² and it clearly is too high with the original purlin. But with an additional 7"x3" bolted on, I'd bet my pension fund that the stress will be below 5.3x1.25.

Having said that, do the calcs show maximum deflection? - that (rather than maximum bending stress) is usually the determining factor with longer-span timber beams.

 

[theprinceofdarkness]

The calcs are (and which all have very little meaning to me) - based on span 3.3m
w = 1.7 x 1.85 = 3.1 KN/m **** weight supported on purlin per metre? of roof***
M(max?) = 3.1 x (3.3 x 3.3)/9 = 3.8 KN/m**** L^2/9 X weight per metre of purlin this is the bending moment.***

Z(xx) = 50 x (180x180)/6 = 270 x 10 to power 3 *** this is the section modulus, its a number relating the "vertical stiffness" of the beam to its shape, notice the strange units its mm^4 ***

I(xx) = 50 x (180 to power 3)/12 = 24.3 x 10 to power 6 *** moment of inertia, another strange unit! ***

Therefore sigma = 3.8 x 10 to power 6/270 x 10 to power 3 = 14
**** rewriting in English:-
Sigma = bending moment /section modulus ***
and sigma (all) = 5.3 x 1.25 = 6.6 N/m squared *** where did these figures come from?***

Then after trying the calcs with a 175 x 50 C16 purlin which shows insufficient (sigma = 7.2N/m square) which is too high they are redone with 175 x 75mm C16 giving a sigma of 5.8 which is deemed ok.
***The first set of annotations are from here:- http://civilengineer.webinfolist.com/str/micalcr.php.
I have a problem here, the sigma calculated for a 50 X 180 mm purlin gives an answer 14, putting in the numbers for a 75 X 175mm , I make this to be 10 . I'll do it again For a 75 X 180, :- 9.3, makes sense 180 is a little stiffer the 175.
So the formula are about right but some where hidden is the strength of the wood, we have a weight for the roof, but nothing relating the strength of the wood, so there is nothing to decide what the absolute strength is, could be made of balsa wood or oak!

Can you find some more figures on your sheet of paper?
Frank

 

[tony1851]

@ Frank. The value for bending strength is buried in the figures - 5.3N/mm²; they are assuming C16 grade which is reasonable.

That can be increased by a factor of 1.25, because the live load is only a temporary applied load (eg snow), hence the figure 5.3x1.25.
The allowable stress can actually be increased a little further, firstly by 1.1 because there will be 2 timbers bolted together, and secondly by the 'depth factor', which for 7" deep beams is about 1.06, so his maximum allowable bending stress could be
5.3x1.25x1.1x1.06 = 7.7N/mm².

But I still suspect deflection rather than stress might be the issue.

 

[anthonyl123]

Thanks for your replies and I'll work through them in more detail but not till at least tomorrow afternoon. I might try and get some scans uploaded so that the workings can be seen. Some photos are at https://www.flickr.com/photos/79018230@N05/albums/72157657651956135 - the block support down the road I have since learned is how that part was already. My roof had the supports taken out at least over 11yrs ago and probably more like 15yrs by an Italian owner who gave little consideration to building requirements. The ceiling is sagging elsewhere as a wall between the dining and lounge areas has been removed. This is a 1968 build! Electrics and plumbing make no sense in parts either.

 

[jeds]

My guess is your calcs allow 1.0kN for live loads and 0.85 for dead loads, which seem reasonable to me. For a 3.3m span the purlin would need to be in the order 140x175mm. You need to consider how you are going to fit this purlin. Most people would jack the existing purlin up and fit the new one to it. That's fine but bear in mind the existing purlin will have become permanently deformed to some extent (creep) so will be acting against the new one trying to pull it into shape.

 

[anthonyl123]

I've scanned and uploaded the calcs for the roof which I hope I have correctly attached to this post and that helps make more sense.

 

[anthonyl123]

I have received two opinions on strengthening and supporting the purlin. Whilst both are similar one builder does not want to straighten the existing bow out, whilst the other says it shouldn't be a problem. Whilst I would have preferred to have the roof flat again I can understand the unwillingness of the first builder in jacking up the purlin and the potential for damaging the rafters and distrupting the roof. Any views please.

NB The existing sag is slight and has been there probably around 15+ years.

 

[tony1851]

Jed's post above mentioned long-term 'creep' of timber beams; this is a permanent deformation under load.
It's possible that you may not be able to get the bend out of it, without lifting it at the ends.
If you try this, you need to jack it very slowly and carefully, and to check frequently what's happening at the ends.

 

[anthonyl123]

The sag is slight, though noticeable now I know about it, but longstanding so I guess the timber has acquired a new shape that probably is best not interfered with so I'll opt for simply strengthening and supporting without inducing new stresses. Thanks.