While this is about a home improvement project, I figure the core part is interesting physics.
My situation is this: I am hanging 500 lbs of drywall (two layers weighing about 250 lbs each) on 5 rows of metal furring channels (hat channels) which are mounted onto RSIC (sound insulation clips) which are screwed into studs with drywall screws. There are 25 total clip assemblies across all 5 rows of channels. This wall is suspended (ie, mounted solely on channels and none of the edges touch the walls, floor, or ceiling). Here's a diagram of a single clip assembly:
Below is a diagram of my actual clip assembly array. The wall is 13' 10" long and 8' tall. The vertical black lines represent the studs and are spaced apart at about 16" (it's not exact because I had to add a couple studs and the frame joins up with another frame at one point in the wall), the red dots are the clip assemblies, and the horizontal grey lines are the hat channels that are mounted onto the clips. Ignore the green dots and yellow arrows.
The problem: Since drywall screws are not as strong as wood screws which I should have used, I am worried that the 25 screw/clip assemblies may not be strong enough to bear the weight of 500 lbs. To test the strength of a single screw, I mounted a single clip into a dummy stud and ended up being able to hang 120 lbs of dumbells on it. This mock assembly has been up for over a month and shows no signs of being on the verge of snapping. I could probably hang another 30 lbs on it before it gave away.
So, using simple math, it seems I would be able to say, "Well, if one screw can support 120 lbs then 25 screws can support 3,000 lbs!" Of course, I am sure the math is not so simple. I am sure there is a curve there somewhere where adding more screws certainly allows for greater weight capacity, but the weight capacity deteriorates the more screws and more weight you add even if the number of screws and amount of weight was proportionate.
So is it really as easy as using simple math to solve this problem or does it require something more advanced?
Best Answer
As others have pointed out in the comments, it is not really trivial to apply any model to reality, especially since we don’t know much about reality. However, we can make a few educated guesses and estimations and see how well everything fits together.
First, assuming perfect weight distribution, we see that five screws would probably be sufficient to support your wall. And while the assumption is probably wrong, we can be relatively sure that a safety margin of 500% is a good start.
Second, we can look at how much a single screw has to support according to your diagram. An upper boundary for the area of the wall supported by a single screw seems to be four ‘rectangles’, corresponding to about $$\frac{4}{44} \approx 0.09 \hat{=} 45.5\textrm{ lb }\hat{=} \frac{1}{3}\textrm{ screwweight}_{\textrm{max}}\quad.$$ That still looks rather good, doesn’t it?
Third, we could check if there are any screws that, if they are removed, leave another screw with many more ‘neighbouring’ tiles. As far as I can see, the maximum would still be about six (corresponding to approx. $\frac{1}{2}\textrm{ screwweight}_{\textrm{max}}$).
I would hence tend to say that you are fine, but there are many, many problems that could possibly arise (not to mention that I am not a construction engineer and roughly followed http://xkcd.com/793/).