How to Calculate Shelf Sag for an Acrylic Hot Air Dryer?

AI Thread Summary
The discussion revolves around calculating the sag of an acrylic shelf designed to support a distributed load of 50N at 60°C over a lifespan of 8000 hours. The user seeks guidance on the correct approach to determine the deflection caused by creep, referencing a graph of creep modulus and expressing concern about their initial calculations yielding an unrealistic sag of 6.1m. Key points include the need to connect the creep modulus to stress and strain, and to accurately apply the relevant formulas for deflection. The user realizes a mistake in unit conversion from GPa to MPa, which corrects the deflection calculation to a more plausible 6.1cm. The conversation highlights the importance of verifying calculations and understanding the relationship between stiffness, force, and deflection in beam theory.
peleus
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Hi all,

I'm trying to crack what is admittedly a homework question. I don't necessarily want you to just spit out the right answer for me, but if you could point me in the right direction it would be appreciated.

A shelf for a hot air dryer is to be made from acrylic sheet. The shelf is simply
supported as shown in Figure 1, and has width w = 500mm, thickness t = 8mm and
depth b = 200mm. It must carry a distributed load of 50N at 60oC with a design life of 8000 hours of continuous use. How much will the shelf sag in that time?

Essentially Figure 1 simply show's that the force is evenly distributed across the entire shelf, it's not a point load.

We also have a graph of Creep Modulus (GPa) vs Time (s).

8000 hours * 60 seconds/hour = 480000s

Reading the graph of 4.8x10^5 seconds, we have a Ec of ~2.5 GPa.

Can anyone give me some pointers of a direction to go in from here?

Thanks.
 
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Here's where I would start:

1. Find the definition of "creep modulus."

2. Find the relevant state of stress in the beam (hint: shelf problems are often bending problems).

3. Connect the creep modulus to the stress to get strain.

4. Relate strain to the amount of sag.
 
Any chance you can help me out with finding the stress in the shelf?

I'm using

defl = Force * Length^3 / Second moment of Area * Creep Modulus * Loading constant.

I don't think this is right, because I'm getting a deflection of 6.1m (obviously wrong) and I don't think this method takes into account the fact the creep modulus is changing.

Any chance of giving a bit more of a hint for which formula to use?

Thanks.
 
It's likely there's a calculation or units problem somewhere. Recheck your calculations carefully.
 
I = 0.5m * 0.008m^3 / 12

I = 2.13333x10^-8 m^4
defl = 50 N * 0.5m / (384/5) * 2.5x10^6 Pa * 2.13333x10-8 m^4Pascals cancels a m^2 down the bottom and N up the top leaving m / m^2

This gives m^-1 which I suppose can't be right, any idea where I've gone wrong though?
 
Your equations in #3 and #5 seem to be different, and neither looks quite right. I suggest checking a textbook or reference book for the exact equation, and check what a GPa is again.
 
Thanks for spotting the mistake. Stupid me.

The equation is definitely right in the following sense.

Stiffness = C1 * E * I / L^3

Stiffness = force / deflection.

So force / deflection = C1 * E * I / L^3

Therefore, deflection = L^3 * F / C1 * E * I

I changed 2.5 GPa to MPa in my outline of the data, labelling it as 2500 MPa, But stupid me only said 2.5x10^6 (MPa) instead of 2.5x10^8 (GPa) in the working.

That gives a deflection now of 6.1cm, much more reasonable.
 
OK, cool. (And I was wrong about your #3 equation being wrong; I was thinking that the distributed load was in N/m, but it wasn't.)
 

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