Question about bathroom vanity

[ukalumni]

I have a 50" x 24" 3/4" thick piece of glass that I'm going to use as a top on a bathroom vanity. The only question is what kind of vanity to build for it to sit on.

I'm thinking about making a frame and bolting it to the wall and setting the glass on top of it. Something like the top part in this picture:

question is how would you make the frame? I don't know what the glass weighs, but it is pretty heavy. I'm guessing around 100 pounds or so. The sink is going to sit on top of it, so another 20 pounds or so.

I was thinking about buying some kind of hardwood (cherry maybe?), making a rectangle out of it a little smaller than the glass, and bolting it to the studs in the wall. At 50", I should have 3 studs to bolt it to, correct?

What kind of thickness for the cherry? I'm assuming I'll have to fasten some boards together to make the frame thick enough to hold that kind of weight.

Would you countersink bolts through the cherry into the wall studs? Is there some other kind of fastener that would work better?

How wide should the board against the wall be? How many bolts per stud?

If you see something obvious I'm missing or some reason I shouldn't do this, please speak up.

Thanks,

--Michael

[Evodawg]

My only question is, what will you be looking through and seeing from the top of the vanity? A vanity structure? Some kind of backing material? Frame piece's? What?

RV

[ukalumni]

I think from the top, you would just see the surrounding frame along the outside of the glass, and your feet on the floor. I'm just talking about a shelf, basically.

[Josh]

I think you'll be limited to two "arms" holding the glass top out from the wall, given that a third one in the middle would likely line up with the sink. Also, it seems unlikely that the final placement of the vanity will just happen to line up exactly with the studs you'd be screwing into. You may need to temporarily take off a section of drywall and add in a horizontal 2x10 or 2x12 to screw into, or else add a cherry backer board to your frame, which can then screw into three studs, like you said.

If the depth of the glass top is 2 feet, and the weight is roughly 120 lbs, that should be roughly 120 ft-lbs of torque total, or 60 ft-lbs per arm. Supposing that you screwed the arms into wall studs with only one screw apiece, each arm would act as a lever trying to pivot on the fulcrum formed by the bottom corner of the arm where it butts up against the wall. The mechanical advantage of the lever would then be

1/D, where D is the distance (in feet) from the fulcrum to the screw. Thus, if your screw entered the wall 6 inches (0.5 feet) from the bottom of the arm, the mechanical advantage would be 2:1. In other words, the screw would be experiencing 60 x 2 = 120 lbs of force pulling it outward. If the distance was only 4 inches, the advantage would be 3:1 and the screw would "feel" 180 lbs of force.

Obviously you'd want to size your screws to accomodate a lot more force than just the torque from the weight of glass and sink. If someone was to lean heavily on it or even sit on it, especially on the very edge, they could conceivably add another 300 or 400 ft-lbs of torque on each arm. Bear that in mind when sizing the screws or bolts. Also, keep in mind that the wall framing, itself, experiences roughly the same torque and needs to withstand it. Studs shouldn't have a problem, but if you add horizontal framing to nail into, the taller it is the better (hence the suggestion of a 2x10 or bigger). Here, the mechanical advantage is

1/D, where D is the distance between the top nail and the bottom nail holding the wood in place.

Lastly, keep in mind that the wood frame itself needs to withstand all that torque without deforming at the joints. If there's any way you can make it look nice with more of a truss design, you'd be much better off. The picture you linked to had plane old rectangles for the arms. I'm guessing they're welded steel or aluminum. With wood, even really hard wood like cherry, there will be a tremendous strain on the joints. If you added a single diagonal brace to the rectangle, or a V or W or a solid wood panel to the interior, it would be MUCH stronger.

[ukalumni]

Wow, that is good information. Thank you very much.

You mentioned a backer board. That is what I was imagining. So if I were to use boards 8" wide and put about 3 bolts per stud through that backer board, how would that affect the numbers you gave me? You said with 1 bolt, "the mechanical advantage of the lever would then be 1/D." How does 3 bolts change the equation?

[Josh]

Hi, Michael. I reread your post and realize now that you'd likely be using solid planks for the arms, not rectangular frames like the picture showed. And, since you'd be wrapping it around four sides, yes, you can screw directly into the studs, like you said. The force calculation remains the same, except, as you said, the force on the wall is spread out over three studs. Also, of course multiple screws per stud will lighten the load on any one screw, but keep in mind that the top-most screw will still experience the most force. If you put two screws per stud, one above the other, that doesn't mean that the force on the top one will be cut in half.

[ukalumni]

Thanks again.

BTw, is this basic mechanical engineering, or is there some resource for information like you're providing?

--Michael

[Josh]

Being too lazy (and busy with other things) to try to derive the relationship for sure, intuition tells me the force distribution on the screws is as follows:

Let's say you have three screws per stud and you place them at 3", 5", and 7" from the bottom. I think the force distribution is 3/(3+5+7) =

20% of the total force on the bottom screw, 5/(3+5+7) = 33% on the middle screw, and 7/(3+5+7) = 47% on the top screw.

I think (anyone else please tell me if you agree or disagree), the total force referred to above is still the same as if there was only the top screw. Adding the other two screws essentially cuts the force on the top screw in half by absorbing the other half between the two of them.

By the way, we're referring to the outward (tensile) force on the screws caused by the torque. There's also a downward (shear) force divided equally among them which is simply the total weight of the vanity and anyone sitting one it divided by the total number of screws. That force shouldn't be much of an issue.

Lastly, regardless of how many studs you screw into, there will still only be two wood arms that reach out to hold up the vanity. Assuming that the arms are held onto the backer board with screws from the back side, the force calculation on those screws is just the same as before, except that the total force will only be divided between the two arms. Having three or three hundred studs to screw the backer board into doesn't help for those screws.

[Josh]

It's mostly stuff you'd learn in Physics 101. If you can get your hands on a basic physics book, it'll be in the section with all the statics problems (trusses, pulleys, etc.). I'm sure a good mechanical engineering text on statics and strengths or the like would have some good info. I looked briefly for a good online reference but didn't see one. Perhaps someone else reading this group knows one?

Here's a really brief out-of-my-head description.

When an object is at rest, the sum of all forces on it must be zero. Some forces want to push or pull an object in one direction, such as gravity pulling down on an apple hanging from a tree. Given that the apple is stationary, there must be an equal and opposite force pulling up on the apple from the stem. Other forces are rotational, such as the torque experienced by your vanity. If it's not falling off the wall, then the screws that fix it to the wall must be applying an equal and opposite torque.

The calculation of torque is pretty simple; it's merely force (usually expressed in lbs or newtons) times perpendicular distance from the point about which the rotation occurs.

T = F*D_perpendicular

When I say perpendicular distance, think of it this way. Draw a line along the direction of the force you're applying (straight up and down in the case of the gravitational force on your vanity) passing through the point at which you are applying the force. Then draw another line perpendicular to that one which passes through the point about which the rotation occurs. Depending on the direction of the force, this second line will not necessarily pass through the point where it is being applied. The perpendicular distance is the distance from your original line (but not necessarily the point where the force is applied) to the center of rotation along the perpendicular line. This becomes important if the force you're applying is directed toward or away from the pivot-point, rather than tangential to it. Imagine trying to spin a merry-go-round by pushing straight in toward the middle. It wouldn't happen. You'd be applying a lot of force several feet from the center, but the perpendicular distance (and hence the torque) would be zero.

In the case of your glass-top vanity, the downward force would not be directed at a single point, but would be spread out evenly from the wall out to the outer edge. You could find the total torque using integral calculus, but in this case we can do it more simply. If we ignore the weight of the sink for simplicity's sake, we can say that the 100 lbs of glass is spread out evenly over the span from the wall to a distance 2 feet away (the outer edge of the glass). This is effectively the same as if the full 100 lbs of force was applied in the middle of the glass, at a distance of 1 foot from the wall. Granted, the weight of the sink might not be spread symetrically, but we can probably ignore that and just say that the two together are applying ~120 lbs of force at a perpendicular distance of about 1 foot, giving us 120 ft-lbs of torque. If a 200 lb person sat on the edge of the glass, they would be applying 200 lbs of downward force at a distance of 2 feet from the pivot point, and hence another 800 ft-lbs of torque.

Meanwhile, assuming your screws hold and the vanity doesn't fall down, the screws must be applying 120 ft-lbs of force in the opposite direction (or 920 ft-lbs if the 200 lb guy is still sitting on the edge). If the screw is 1/2 ft from the pivot point, it must be applying 240 lbs of force in order to come out to 120 ft-lbs of torque (or a whopping 1840 lbs of force with the person sitting on the edge). Of course there will be multiple screws spread out over multiple studs, so no one screw will have to survive that much force.

Given your three-screw-per-stud design where the top screw is, say, 7

1/2 " from the pivot point at the bottom of the backer board, you'd probably want to use screws that can support around 300 lbs of tensile force. I don't think that's a really big deal; you won't have to use 3/4" diameter lag bolts or anything like that.

You can always do a mockup out of cheap lumber, screw it to a wall out in your garage or somewhere, and sit on it or jump up and down on it or whatever to convince yourself that it'll hold.

Josh

[Josh]

In my third from last paragraph above, I wrote "the screws must be applying 120 ft-lbs of force in the opposite direction." I meant to say "torque", not "force".

[ukalumni]

Great information. Thanks very much.