On how were span-depth ratios derived. 12

[IJR]

My friends

I need a reference text on this subject: How were the popular span-to-depth ratios for steel and reinforced concrete beams derived. I prefer the derivations, not general discussion.

Thanks in advance.
and for making this forum a great place.

ijr

 

[CANPRO]

I don't think you'll find a neat and tidy derivation that at the end says "therefore, span to depth shall not exceed L/20". No doubt you could crunch the numbers on a million different scenarios to show that exceeding popular span to depth ratios is impractical. I think the span to depth ratios are a product of experience and someone saying "why did I just waste my time checking such a shallow beam for this span? Maybe next time I should start at L/20 and go from there". If you're trying to understand where these ratios came from, I think a general discussion may actually be more useful than derivations.

 

[mtu1972]

Early in my career I learned the following rule of thumb for steel beams. Half the span in feet is the depth in inches. It’s a good starting point. Then depending on loads, etc. the beam size can be finalized. For instance, a 30’ span would suggest a 15” beam. As there are no W15’s I would start with W16’s and W14’s as my selections.

For concrete - no rules come to mind.

gjc

 

[IJR]

The steel design book by Salmon and others, and the r.c design book by Salmon and Wang have detailed derivations of L/d ratios.

thanks
ijr

 

[jike]

Early in my career (46 years ago) I wondered the same thing. I remember deriving the one for steel. I know that it was based on assuming L/360 deflection and full stress based upon a uniform load although I cannot remember if the max. stress was 22 ksi or 24 ksi. You can write two equations: one for deflection and one for stress. wl^4/384 EI = L/360 and wl^2/8/S = 24 ksi Change S to I/d/2, solve the 2nd equation for I and substitute into the first equation and solve for d. I will let you do the rest. Concrete derivations should be similar.

Good luck!

 

[BAretired]

A steel beam with span L under constant moment M will deflect Δ = ML[sup]2[/sup]/8EI at midspan.

The maximum fiber stress f at d/2 will be f = M.d/2I
so M/I = 2f/d where d is the depth of the beam.

Thus Δ = f.L[sup]2[/sup]/4dE
Setting Δ as L/360 and re-arranging, gives L/d = E/90f
Setting Δ as L/240 gives L/d = E/60f

For a uniformly distributed load, Δ = 5wL[sup]4[/sup]/384EI or 5ML[sup]2[/sup]/48EI and the L/d ratio can be determined for any specified deflection and permissible fiber stress.

BA

 

[lsmfse]

I remember back in the day an older steel manual suggested Fy/800 x span (inches)
as a starting point to limit vibrations in floors and Fy/1000 x span for roof members
to limit ponding issues. I think it was in the "blue" steel manual.

 

[WARose]

The 50b (i.e. lateral support requirement; where b= width of the section) in reinforced concrete was meant to preclude any sort of LTB. (Assuming no torsion/localized affect is going on. How they word that section has always made me check things manually where I have any doubts.)

 

[BAretired]

I remember back in the day an older steel manual suggested Fy/800 x span (inches)
as a starting point to limit vibrations in floors and Fy/1000 x span for roof members
to limit ponding issues. I think it was in the "blue" steel manual.

I don't remember that suggestion and I don't agree with it. I do remember "Half the span in feet is the depth in inches." cited by mtu1972 which made sense for the low yield steels of the day. When using higher strength steel efficiently, d should increase and L/d should decrease because E is virtually identical for all steels.

Edit:
Now I agree with it.
Fy is expressed in ksi, so for an A36 beam spanning 28', 36/800 * 28*12 = 15.1".
For Fy = 50 ksi, 50/800 * 28*12 = 21".
AISC was correct.

BA

 

[Hokie93]

The F[sub]y[/sub]/800 and F[sub]y[/sub]/1000 depth "suggestions" are in the Commentary to the Specification in the 7th, 8th, and 9th edition AISC Steel Construction Manuals. Interestingly, the older AISC Specifications (see 4th edition steel manual) codified the "half the span in feet is the depth in inches" rule of thumb for rolled floor beams by requiring L/d < 24, where L is the beam span in inches.

 

[BAretired]

Hokie93, I don't have those AISC references and do not believe they were ever in CISC, but I may be mistaken. For A36 steel where Fy = 36,000 psi, L/d for a roof beam could be 36 to limit ponding whereas for Fy = 50,000 psi, L/d could be 50? Am I interpreting that correctly? No...see below! Neither of these limits seems to make any sense, keeping in mind that for both of those yield points, the modulus of elasticity, E = 29,000,000 psi.

Edit: Fy = 36 ksi. Beam depth d = 36/1000 *L = 0.036L or L/27.8
If Fy = 50 ksi, d = 50/1000 * L = 0.05L or L/20
BA

 

[Ingenuity]

The Fy/800 and Fy/1000 depth "suggestions" are in the Commentary to the Specification in the 7th, 8th, and 9th edition AISC Steel Construction Manuals

From: AISC 7th Edition Manual:

1[sup]st[/sup] to 7[sup]th[/sup] Edition Manual are a free download at AISC for members.

 

[BAretired]

Thanks Ingenuity. I see it now but it still makes no sense. Hopefully, no one ever used AISC Article 1.13.1 as a means to control deflections.

Edit: It does make sense. See previous edits.

BA

 

[kipfoot]

you might be interested in the AISC webinar:

Rules of Thumb for Steel Design

rules based on observation, collected from peers, and derived

 

[BAretired]

kipfoot,

I skipped quickly through the video but did not find any reference to the problem we are discussing. At 18:50, an approximate weight of beam is calculated by a rule of thumb with a span of 32' and an assumed depth of 18". At 20:00 a similar calculation is made for a 32' span and a 16" depth. No mention is made of how the depth was chosen, but 1/2" per foot of span seems to be the rule used, also a rule which I have used for many years as a preliminary choice of beam depth. If I missed the part of the video you are referencing, please indicate the time on the video where it starts.

I cannot see how any rule of thumb to limit deflection could suggest that L/d should be based on Fy/k where k is a constant. As I showed in an earlier post, the L/d ratio should vary inversely as the stress in the outer fiber. This means it should vary inversely as Fy. Moreover, Fy/k has units of stress whereas L/d is dimensionless, so the whole concept seems invalid.

If anyone thinks otherwise, please show me why you think so.

Edit: Please ignore the text in red and see earlier edits.

BA

 

[CANPRO]

I think trying to "derive" a rule of thumb is almost a pointless task. How can such a general rule be proven with a precise set of calculations? I could start specifying W14x176's for 40' floor beams and "disprove" the rules of thumb. I might not have a job for very long, but that's my point - these rules of thumb are meant to get us in the right ballpark economically...there is an unlimited number of absurd options we could use to make a structure work and blow away the rules of thumb at the same time. As I mentioned before, I'm sure you can crunch the numbers to prove a rule of thumb works for typical conditions, but I doubt (I could be wrong here) that that is how most of these rules were initially developed.

Bottom line, I think we're over complicating a simple concept with no benefit. For example, BA's derivation above showing L/d = E/90f - using that derivation, I'm starting with a 36" beam for a 20' span. I mean no disrespect to BA - I've been reading his responses to questions for years now and have learned a lot, I have a ton of anonymous internet respect for BA. But I think he missed the mark with this derivation, when as he stated above "1/2" per foot of span seems to be the rule used, also a rule which I have used for many years as a preliminary choice of beam depth". Why make things more complicated than they need to be?

 

[BAretired]

Bottom line, I think we're over complicating a simple concept with no benefit. For example, BA's derivation above showing L/d = E/90f - using that derivation, I'm starting with a 36" beam for a 20' span.

E/90f applies to constant moment over the entire span and deflection of L/360. Using f = 20,000 psi, it results in an L/d of 16.1; for a 20' span, d would be 14.9", not 36".

I need a reference text on this subject: How were the popular span-to-depth ratios for steel and reinforced concrete beams derived. I prefer the derivations, not general discussion.

I would think that L/d ratios were derived on the basis of a uniform load with a fiber stress of 20,000 psi which was fairly prevalent at the time the rule of thumb was proposed and a deflection between L/360 (live) and L/240 (dead), say L/300 (combined).

so Δ = 5wL[sup]4[/sup]/384EI or 5ML[sup]2[/sup]/48EI
which is equivalent to 5L[sup]2[/sup]/48E * 2f/d

Equating this to L/300 with E = 29,000,000 psi and f = 20,000 psi gives L/d = 23.2 (close enough to 24)

I don't know, but I am guessing this is where the 1/2" per foot of span came from.

I think I will leave the reinforced concrete beam deflections to others.






BA

 

[CANPRO]

BA, my apologies. Your derivation didn't work with my LRFD mindset - I used almost the full yield strength in your equation, which gave me a much deeper beam. When you equated deflection to strength I should have known that comparison had to be done with unfactored loads and allowable stress. I would like to change my statement "I think he missed the mark with this derivation" to "he was bang on with his derivation".

 

[BAretired]

Thanks CANPRO.

BA

 

[dhengr]

IJR:
Do 4000 beam designs, and keep track of the L/d ratio and types of three or four or five different loading conditions and you will be twice as smart/knowledgeable about that rule of thumb, as you were after you had done 2000 beam designs. Someone didn’t wake up one morning and say, ‘I’m gona formulate a simple rule of thumb, and bang, it happened.