Deflection formula for Cantilever

[John Stevens]

I've a completed table top for a light-duty work bench, and now I'm ready to design the base. The thought occurred to me that, because the amount of deflection of the bench top will increase in proportion to the third power of the span between the base supports, I may be able to minimize sagging by reducing the span and allowing the ends of the top to overhang the base on the ends. However, I'm guessing that the cantilevered ends would deflect at an even greater rate than the middle part.

I realize that in practice, the deflection of the table top will depend on how the load is distributed both between the base supports and along the cantilever. However, the thing I'm mainly concerned with is how this type of a table-and base-combo will handle the hypothetical worst-case situation, a load that rests completely on the line defined by edge of the table top. Anyone know the formula to calculate the deflection of the cantilever under those conditions?

I realize I may be splitting hairs here, but I find this type of stuff fun to think about, and besides, it may matter for real in a future project.

Thanks in advance.

[Carl Eyman]

John, I'm not sure you can find a canned formula. There are too many factors undefined. For instance, the material your bench top is made from, its thickness, its moment of inertia, how it is fastened to the understructure, and its modulus of elasticity (will it take a permanent set ) are all items that need to be factored in. I doubt anyone has derived a formula for all those conditions. I'll be interested to read responses to your post. Perhaps I am confusing the issue with too many variables.

[Bobby Hicks]

John,

The deflection formulas are in the attached JPEG. The top formula is the deflection the section between supports. The bottom formula is for the ends.

W = Weight
c = Distance to center of weight outboard of supports
l = span between supports
E = Modullus of elatsicity
I = Moment of inertia

[John Stevens]

Thanks, Bobby, that's exactly what I was looking for. I got the first formula from Ken Horner's book Woodworkers Essential Facts, but not the second. The book also gives E for various materials and shows how to calculate I, so I should have enough rope to hang myself at this point ;-)

Regards,

John

[Bobby Hicks]

John,

For my future ref. what do they list for E for some common woods?

[Mark Singer]

John,

That is the formula. An interesting point is that if the supports are in .207 of the overall length, then the positive and negative moments will be equal. So if the table is 10' long and the cantilevers are 2'-I" the positive moment mid sapn and the cantilever moments will balance. This is also proportionatly balanced.

[John Stevens]

Bobby, I don't have the book handy right now, but I'll post again with the modulus values. Just off the top of my head, oak is ~1.6, pine is ~1.4, and MDF is ~0.5. If these numbers seem patently unreasonable, chalk it up to the fact that I have a memory like a sieve. (I just made a note to remind myself to look up and post the E values :-)

Mark, thanks for doing the "brain work" for me. I'll still probably try to solve the problem myself just for the sake of doing it, but it's been 25 years since I last did any calculus. When you say "proportionately balanced," I assume you're talking about aesthetics, and noting that the ratio of the total length and center span is very close to the golden ratio. Yeah, I find that very interesting, too, but after reading Matila Ghyka's book, I'm almost never surprised at where the golden ratio seems to pop up.

I find this particular use of these formulas to be kind of interesting in itself. Normally, I'd design a piece of furniture based on aesthetic proportions and design standards (human proportions), then calculate the deflection and adjust the materials, span, thickness and/or construction methods as necessary to bring deflection into acceptable limits. However, in this situation, I was using parts/materials supplied by Festool to build a two-meter-long version of their Multi Function Table, so the modulus of elasticity and thickness of the materials were "dictated" to me. I find the table too flexible, but I have several options for stiffening it--put an extra set of legs under the middle; add more lengthwise supports underneath the table top or aluminum rails; or change the span between the legs and allow some portion of the ends to cantilever out over the ends of the new base. Since other considerations have led me to build a new base, I figured I might as well design it to minimize deflection.

If I'm correct in believing that deflection between the supports is proportional to the third power of the span between them, then reducing the span so that there's an overhang on each end of .207 of the overall length will reduce deflection by around 80%...which should be good enough that I won't need to do more.

[Chris Padilla]

Cool...nice to see some physics nuts around here besides me. I like to toss some math around myself.

[John Stevens]

For my future ref. what do they list for E for some common woods?

Bobby, here are the values listed in Ken Horner's book, Woodworkers' Essential Facts Formulas and Short-Cuts. Just to note, Horner attributes these figures to Bruce Hoadley's book Understanding Wood. Units of measure are psi, and values must be multiplied by 10^6. No error brackets were given, and you can bet the rent that there's probably plenty of room for variation within each type of wood listed. I'm typing the figures verbatim, which is why I'm including the zeroes in the second decimal place, but the text doesn't say how many significant digits there are.

Alder 1.38
Ash 1.77
Beech 1.60
Birch 1.80
Cherry 1.50
Chestnut 1.23
Elm 1.40
Locust 2.00
Maple 1.70
Oak 1.65
Poplar 1.58
Walnut 1.68
Balsam Fir 1.23
Cedar 1.19
Douglas fir 1.95
Hemlock 1.40
Pine 1.40
Redwood (old growth) 1.34
Spruce 1.40

One last thing, for the benefit of Ken Horner, whoever/wherever he is--I can't say enough about his book. It seems like what he did was distill the most valuable info from the best woodworking books around, and publish a chapter on each topic--30 in all, 300 pages. Unlike almost every author who takes such an approach, he doesn't reduce the info to a superficial gloss. And although the "density" of the info is high, he makes it understandable to a lay reader who is willing to focus and put a little effort into understanding it. I've spent around $400 on various woodworking books currently in print, and this one is IMO an outstanding resource. However, I have to consider myself a beginner, so take this for what it's worth.

Regards,

John

[Tom LaRussa]

Cool...nice to see some physics nuts around here besides me. I like to toss some math around myself.

Me too!

E = MC Hammer (or something)