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

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Discussion Overview

The discussion revolves around calculating the sag of an acrylic shelf used in a hot air dryer under a distributed load. Participants explore the theoretical and mathematical aspects of shelf deflection due to bending, considering factors such as creep modulus and material properties. The context includes homework-related queries and technical reasoning.

Discussion Character

  • Homework-related
  • Technical explanation
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant seeks guidance on calculating the sag of an acrylic shelf under a distributed load, providing specific dimensions and conditions.
  • Another participant suggests steps to approach the problem, including definitions and relationships between stress, strain, and deflection.
  • A participant expresses difficulty in calculating stress and deflection, indicating a potential misunderstanding of the formulas used.
  • Concerns are raised about calculation errors and unit conversions, with a participant questioning the validity of their results.
  • One participant provides a formula for deflection, but later realizes a mistake in unit conversion from GPa to MPa, leading to a more plausible deflection result.
  • Another participant acknowledges a misunderstanding regarding the nature of the load, clarifying their earlier comments on the equations presented.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach or formulas for calculating shelf sag, with multiple competing views and unresolved calculations. Some participants correct or refine earlier claims without establishing a definitive solution.

Contextual Notes

Participants express uncertainty regarding the appropriate equations and the implications of changing creep modulus over time. There are also unresolved issues related to unit conversions and the nature of the load applied to the shelf.

Who May Find This Useful

This discussion may be useful for students or professionals interested in material science, structural engineering, or mechanics, particularly those dealing with deflection calculations and material properties under load.

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.)