# Calculating joist strength



## renoman

Hi everyone this is my first post.
Is this true?  "bending goes by power of 3, so a 2by6 is 1.6 times thicker than a 2by4 so its going to bend 4X less. Also a 2by8 on edge can carry more than a 6by6." That info I found on the internet but I lost the site and I have been searching and searching to find another site to confirm that formula. If that info is true then I'll have to use 2by8's for the floor in my attic(presently 2by4's). Because of the ceiling height I would prefer to use 2by6's instead of 2by8's . The max span I have is 13'-6". One idea is to use 2-2by6's with 3/4" plywood sandwiched in between as my joists(all screwed,glued or bolted together). On top of this would be 5/8" T&G plywood screwed and glued to the built up joists. And lastly instead of having a drywall ceiling below why not have a plywood ceiling. Would that add more strength to the floor structure? Any feedback would be appreciated. Thanks


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## travelover

Here is the formula - you can calculate it using on-line stress or deflection calculators.

Structural Beam Bending Equations / Calculation Supported on Both Ends Uniform Loading* - Engineers Edge

A simple calculator:

Beam Deflection Calculators - Solid Rectangular Beams, Hollow Rectangular Beams, Solid Round Beams

Having plywood glued and screwed to the top and bottom of the ceiling joists would add a great deal of strength. you are essentially creating a series of interconnected  I beams.


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## Nestor_Kelebay

renoman said:


> Is this true?  "bending goes by power of 3, so a 2by6 is 1.6 times TALLER than a 2by4 so its going to bend 4X less.



The formula for the maximum deflection of a rectangular beam under a uniformly distributed load in Travelover's link is given as:

deflection = (5/384) * (W*l*l*l)/(EI)

The constant out in front changes depending on what units for force and distance you use.

What you need to do to answer your question is to look at the term, I, which is the "Moment of Inertia" of the beam, and is dependant on the SHAPE of the beam.

For a solid rectangular beam, I = (b*h*h*h)/12
or, the width of the beam times the cube of it's height over 12.

So, all things being equal, the deflection of a 2X4 and a 2X6 under a given uniformly distributed load will be proportional to the calclulated values for I for the 2X4 and the 2X6.  And, since the width of the beams is going to be the same (1 1/2 inches) in both cases, the ratio of the beam deflections will be the same as the ratio of the cube of their heights:

or (5 1/2)*(5 1/2)*(5 1/2) / (3 1/2)*(3 1/2)*(3 1/3) = 166.4/42.9 = 3.88

So, the 2X6 will be about 4 times are rigid as the 2X4, or, the 2X4 will deflect about 4 times as much as the 2X6.



> Also a 2by8 on edge can carry more than a 6by6." That info I found on the internet but I lost the site and I have been searching and searching to find another site to confirm that formula.



I don't know if it's fair to say that a 2X8 can "carry more" than a 6X6 because that would be determined by the point at which one of them actually breaks.  But, it's EASILY possible to determine which would bend less under the same distributed load:

I for a 2X8 would be proportional to  (1.5)*(7 1/2)*(7 1/2)*(7 1/2) = 632.8

I for a 6X6 would be proportional to  (5.5)*(5.5)*(5.5)*(5.5) = 915.1

So, a 6X6 made of the same material as a 2X8 would actually be about 50% stiffer.  That is, under the same loading, the 2X8 would deflect about 50% more than the 6X6.



> If that info is true then I'll have to use 2by8's for the floor in my attic(presently 2by4's). Because of the ceiling height I would prefer to use 2by6's instead of 2by8's . The max span I have is 13'-6". One idea is to use 2-2by6's with 3/4" plywood sandwiched in between as my joists(all screwed,glued or bolted together). On top of this would be 5/8" T&G plywood screwed and glued to the built up joists.



You CAN sandwich some plywood between your joists, but you can also start thinking outside the box.  The formula for beam deflection involves both E and I in the denominator.  I, the Moment of Inertia is determined entirely by the size and shape of the beam, so if you use a bigger beam you get less deflection.  E is the "Modulus of Elasticity", and it depends of the rigidity of the material the beam is made of.  For example, the Modulus of Elasticity for your average garden variety No. 1 grade 2X4 is 1,400,000 psi.

Design Value Comparisons for 2x10 | Southern Pine Council&#8482; | 1-504-443-4464

Now, compare that with a Modulus of Elasticity for steel of 30,000,000 psi.

So, let's figure out which would be stiffer; a 2X8 or a pair of 2X4's with a 1/2 inch thick steel plate sandwiched between them:

E for the 2X8 and the 2X4's would be 1,400,000.

I for the 2X8 = bh^3/12 = (1.5)*(7.5)*(7.5)*(7.5)/12 = 52.73

I for each 2X4 = (1.5)*(3.5)*(3.5)*(3.5)/12 = 5.36

I for the steel plate = (0.5)*(3.5)*(3.5)*(3.5)/12 = 1.79

so, EI for the 2X8 would be (1,400,000)*(52.73) = 73,820,000

EI for each 2X4 would be (1,400,000)*(5.36) = 7,500,000

EI for the steel plate would be (30,000,000)*(1.79) = 53,700,000

So, EI for the 2X4/steel plate sandwich would be:
53.7 million plus 2 times 7.5 million = 68.7 million psi   (pretty close to that of a 2X8)

Had we used a 3/4 inch thick steel plate, I for the plate would be 50% larger, or 2.68 and EI for the 2X4/steel sandwich would have been 95.4 million psi.

So, if you went to a machine shop and had them drill holes in a piece of 5/8 inch thick steel plate, and you used bolts through those holes to make a 2X4 steel sandwich, you could have more strength than 2X8 floor joists without sacrificing any headroom at all.  Also, you're not going to get 13 foot long steel plates.  You'd either need to weld shorter plates together end-to-end, or to have a short third steel plate straddling the joint between steel plates.

What you would need to do, however, is discuss this game plan with any structural engineering or architectural firm to ensure that the design is suitable and to advise you on the number and size of the bolts needed to hold the sandwich together.



> And lastly instead of having a drywall ceiling below why not have a plywood ceiling.



Mostly because of cost.  Drywall is cheaper than plywood.



> Would that add more strength to the floor structure?



It would help to strengthen the floor, but I have to respectfully disagree with the previous post that said it would help a lot.  It's true that glueing and screwing the plywood to the undersides of the joists WOULD effectively seem to make a bunch of wooden I beams all connected together.  But, at the joints between the plywood panels, you'd effectively have only the joist strength.  The plywood would have to be continuous all along the length of the joists in order to get that "I" beam effect, and since you'll end up with a bunch of separate 4X8 panels up there, you wouldn't have a single continuous sheet to carry the load the way the bottom flange of an I beam does.  (Imagine if you cut the bottom flange off of an I beam every 8 feet or so.  It would severely weaken the I beam.  So too would the joints between the plywood panels weaken the "I beams" created by the plywood ceiling.  That's just my understanding.  I don't know how much it would weaken it because we never studied that sort of stuff when I was in university.  But, to make a series of wooden I beams, the bottom flange of those I beams has to be continuous, from one end of the joist to the other, and you can't really do that with joints every 4 and 8 feet.

(Understanding why a continuous sheet of plywood glued and screwed to the bottoms of the joists would greatly strengthen the floor is easy to explain.  Imagine bending a deck of cards.  It's fairly easy to do.  In fact, it's about 52 times harder than bending a single card.  Now, imagine that you glue all the cards together.  If you try to bend the deck now, you'll find it very much harder.  The reason why is that now, instead of each card bending the same amount, the outer cards actually have to STRETCH to accomodate the new shape of the deck.  In your case, a continuous sheet of plywood would have to stretch before the floor joists could bend, and that would very much prevent the floor joists from bending.  In the case of a steel I beam, the beam is very strong because the bottom flange of the I beam has to stretch before the I beam will bend, and it's steel's high strength in tension that is what keeps the I beam rigid and prevents it from bending.)

But, just nailing or drywall screwing the plywood ceiling to the floor joists would add little strength because it would still allow for some relative movement between the joists and the plywood just like there is between the individual cards of an unglued deck.  Plywood would add some strength because the plywood panels would also have to bend along with the joists, but unless you could bond them together so the plywood would have to stretch for the joists to bend, you wouldn't get the same increase in strength as is provided by the flange at the bottom of an I beam.

If you're wanting to save headroom, then I'd say the greatest potential lay in using some steel to strengthen your joists.

As you can see from the previous EI calculation, as soon as you start putting steel in those joists, the strength of the steel pretty well dominates the calculation, and most of the rigidity comes from the steel, not the wood.  I'm thinking that if you could have steel plates made up and merely glue them to the sides of your existing joists with a very strong adhesive, like epoxy of LePage's PL Premium construction adhesive, that would provide the strength you need without sacrificing head room.  Alternatively, have holes drilled into the steel plates and use screws to glue and screw the steel to your existing joists.

Your house is your biggest investment and most valuable asset.  If you can afford it, I'd suggest you talk to a structural engineering or architectural firm about using steel to strengthen your existing joists so that you don't have to sacrifice any headroom.  You're going to need a technical help to design the size and number of the screws to hold the steel to the joist as well as design the way of fastening the steel plates together so they act as a single plate running the whole length of the joist.


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## renoman

Hi Travelover
Thanks very much for your reply. I hired an engineer and he said that a 2by4 on flat glued and screwed to a 2by4 would form a Tee. On top and bottom of the Tee 5/8"T&G Fir Plywood would be glued and screwed and that would be a strong enough floor system. The deflection would only be about 1/4".
Thanks from Renoman


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## renoman

Thanks very much Nestor for your reply. I was having trouble understanding the formulas and you explained them really well to me. Because of conflicting opinions I hired an engineer to do an analysis of different floor systems. The top choice was a 2by4 on flat glued and screwed to the existing 2by4's to form a Tee (not perpendicular) with 5/8" Tongue and Groove Fir plywood glued and screwed to the top and bottom of the Tee which forms a sandwich. A thin layer of drywall could be attached to the bottom layer of plywood to form a nice ceiling below. I mentioned to the engineer that you felt that the plywood wouldn't act as one continous piece. He seemed to feel it would, provided the tongue and the groove were glued together and the plywood glued and screwed to the joists. This is what he wrote to me" Plywood may be laid on top of the 2by4 Tee running E-W in the long direction, not running across the joists, for fewer joints that need to be glued. The full 8' sheet should be centered on the span between supports and the ends added to as required to fill out the room. That way, the critical center tension flange (plywood) is not spliced right at the middle. Any pot lights are usually located in the corners of the room, about 30" in from each corner. They should be placed so that they do not intrude on the main central 8' sheets, which provide for eighty percent of the strength and stiffness of the member." I've run out of space. Thanks from Renoman


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## renoman

Hi Nestor
The idea of using steel is a good one. There are only two drawbacks that I can see and they are weight and cost. Each 13'-6" peice of 5/8" X 3.5" steel weighs 100lbs and I haven't priced the steel yet ( I will ) but I think that it will be quite expensive. If the wood sandwich floor system works I think that I'll be using it. Easier, lighter and faster to build. With the formula that you gave me I calculated that a 2by12 will deflect 1.9 times less than a 2by10. Right now in my basement the house is held up with 2by10's 16"oc with span of 9'3" due to 6by6 posts and beams. I am planning on taking the six posts out and having a center wall so now my spans will be 14' and 13'-6". I was told that I could sister on new 2by10's to the old ones and that would be fine. Comparing the two for strength, I don't think 2-2by10's spanning 14' equals 1-2x10 spanning 9'3". If that is the case I'll be sistering on 2by12's against the existing 2by10's. ( the house is going to be lifted with the old rubble foundation taken out and a new foundation poured so I can get an 8' ceiling and a basement suite)
Thanks from Renoman


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## Nestor_Kelebay

Using the 8 foot dimension of the plywood to span the middle section of the joists is important to deriving the most strength from it as possible, and I'm glad to see that's what your engineer wants to do.  I hadn't thought of that.

But, be aware that the plywood you add to the underside of the floor on either side of the sheets of T&G plywood located at mid-span are primarily decorative.  You need that plywood at mid-span where the greatest bending stresses on the joists are.  You really don't need that plywood close to the walls supporting that floor.  (But, if it makes for a nicer ceiling, then I guess that's reason enough to put up plywood all over the underside of that floor.)

Yep, steel is heavy and can be expensive and hard to work with.  But, is strong like bull.


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## Franky

I have a  bouncy I-joist floor that I'm trying to cure and I see some ideas in this thread. Although I was told screwing plywood on the bottom  that it should cross  at a 90degrees to the joists to spread the tension across as many joists as possible. Although I get the theory of going length ways. I was also told that T& G wouldn't really make a difference and you could use regular plywood or OSB because once it's glued to the joists it doesn't matter. Although I guess it makes sense if you're trying to make a stress skin panel. Reactions? 
But if I have to put a beam under my joists (20 feet across and the joists are spanning 20 feet the other way) how much could you reduce the beam size by using a piece of metal sandwhiched between two LVL's or 2X12's.  If I use LVL's on their own I need at least 14" depth and 3 plys. I was told that even the shear strength of a thick piece of galvanized metal sheeting between two 2X12 can make a very stiff beam. Ideas?


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## nealtw

to bad we don't know how it worked.


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