Why does the top shelf slump forward?

[Minhtran1092]

I'm trying to figure out a way to fasten a [book] shelf onto a wall without using corbels. I decided to use L-brackets for support but I positioned them horizontally instead of vertically.

In the attached image, there are two designs:
1. A single shelf fastened to drywall using two drywall anchors, each rated for 70-lb.
2. Two shelves, as described above, screwed together using two boards stood on their edges. The two vertical boards are not fastened to the wall.

When attached to the wall and under no load, design 1 rotates forward under its own weight.
Design two, despite being fastened to the wall the same way as design 1, could hold at least 70kg (I did not find its failing point - I hung my weight off of it).

Despite being fastened to the wall the same way, why did the wall-shelf connection in design 1 fail with no load while design 2 was able to support much more?

 

[Minhtran1092]

I'm trying to figure out a way to fasten a [book] shelf onto a wall without using corbels. I decided to use L-brackets for support but I positioned them horizontally instead of vertically.

In the attached image, there are two designs:
1. A single shelf fastened to drywall using two drywall anchors, each rated for 70-lb.
2. Two shelves, as described above, screwed together using two boards stood on their edges. The two vertical boards are not fastened to the wall.

When attached to the wall and under no load, design 1 rotates forward under its own weight.
Design two, despite being fastened to the wall the same way as design 1, could hold at least 70kg (I did not find its failing point - I hung my weight off of it).

Despite being fastened to the wall the same way, why did the wall-shelf connection in design 1 fail with no load while design 2 was able to support much more?

I modified design 1 going off intuition: if a load were to be placed on the top shelf of design 2, the bottom shelf and the two standing boards would effectively act as corbel/gusset. If a load were to be placed on the bottom shelf of design 2, it would stress the design the same way as a load would stress design 1 except in design 2, the top shelf would effect act as a fastener to pull the weight up.

I'm not satisfied with this intuition though. I'd like to verify this intuition using physics formulas but I don't know how to analyze this system.

 

[jack action]

It is a question of moment. Not only forces must equal each other, but moments too.

A moment is defined by a force times a distance.

Let's say your shelf is 10" wide and 1" thick.

If you hang a weight of 70 lb on the shelf, it acts on the middle, so 5" from the wall. That creates a 350 lb.in moment (= 70 lb X 5 in).

The screw must withstand that moment by a pulling force, which will be counteract by an equal pushing force at the base of the shelf thickness, i.e. 0.5" lower than the screw location. So the force that the screw must support is 700 lb (= 350 lb.in / 0.5 in). If you have 2 screws, it is then 350 lb per screw. Not only your screws will deform, causing the shelf to slump, your drywall anchors won't support this force and break.

By putting a vertical board, you increase the effective thickness of the shelf. Say the vertical board is 10" high. Now the counteracting pushing force is 10" apart from the screw pulling force, so both of these forces have a magnitude of 35 lb (= 350 lb.in / 10 in). If you have 2 screws, it is then 17.5 lb per screw.

Now, you can appreciate the power of leverage!

 

[Minhtran1092]

It is a question of moment. Not only forces must equal each other, but moments too.

A moment is defined by a force times a distance.

Let's say your shelf is 10" wide and 1" thick.

If you hang a weight of 70 lb on the shelf, it acts on the middle, so 5" from the wall. That creates a 350 lb.in moment (= 70 lb X 5 in).

The screw must withstand that moment by a pulling force, which will be counteract by an equal pushing force at the base of the shelf thickness, i.e. 0.5" lower than the screw location. So the force that the screw must support is 700 lb (= 350 lb.in / 0.5 in). If you have 2 screws, it is then 350 lb per screw. Not only your screws will deform, causing the shelf to slump, your drywall anchors won't support this force and break.

By putting a vertical board, you increase the effective thickness of the shelf. Say the vertical board is 10" high. Now the counteracting pushing force is 10" apart from the screw pulling force, so both of these forces have a magnitude of 35 lb (= 350 lb.in / 10 in). If you have 2 screws, it is then 17.5 lb per screw.

Now, you can appreciate the power of leverage!

Thanks for your reply. I never thought about the counter-moment as acting at the base off the thickness of the shelf! Your analysis makes sense but I do have a follow up question about the screw load-capacity.

How do we apply moments to analyze the load-capacity of the screw itself? The 70 lb "load capacity" of the screw indicates that a load of about 70-lb can be hung from the screw before the screw comes loose. Suppose the screw protrudes 5mm from the wall when set (during testing), and the load applied acts at 2.5mm (half the length of protrusion) from the wall. Does this mean that the "70-lb load capacity" of the screw has the equivalent supporting-moment "rating" of 175 lb.mm (70 lb x 2.5mm) or ~1779 lb.in?

This doesn't seem right ... because if a 10" deep, 1"-thick shelf is connected to that screw, then a load of 355-lb (1779/5) acting 5" from the wall should be supported.

 

[jack action]

The load capacity that you were given is most probably the pulling force, parallel to the screw. Or the weight it can support if you put the screw & anchor on the ceiling and hang a light fixture or a plant for example.

The situation you are describing (the pulling force perpendicular to the screw) is known as shearing force. The screw has a limit in that regard as well. In the case of a drywall anchor, it is probably much higher than the 70 lb rated load capacity. Also, assuming you had, say 100 lb shearing capacity, you couldn't have both a 70 lb pulling force and a 100 lb shearing force at the same time on the screw; It's one or the other, or a mixture of both (and it is not simple as adding them up).

There is no way for a screw manufacturer to determine the «moment capacity» as it depends on how the screw is used. The only thing you can do is to calculate the moments, convert them to pulling and shearing forces on the screw and compare them to the manufacturer's data.

 

[Minhtran1092]

Reviewing your analysis on the supposition that the vertical board is 10" high, I'm a little confused.

If the vertical board is 10" high and 10" wide, and the force acts 5" from the wall (on the top board), how is it possible that "the pushing force is 10" apart from the screw pulling force"?

Perhaps a diagram would clarify things.

 

[jack action]

Sorry, I didn't specify that I imagined putting the screws at the top part of the shelf: