How weight affects surface bending over time

[nigels]

Very dumb classic mechanics question here:

The other day I caught sight of a trivial objects arrangement: a basketball placed on top of a 6-sided cardboard box on the floor, and I wondered how the weight of the hollowed sphere could cause bending on the supported, flat top surface of the box. However, despite having studied physics in undergrad, I could not work out the mechanisms involved: 1) Could time be a contributing factor to the bending (i.e, the box gradually cave in over time) or if the surface fails to bend at t₀, it won't ever bend? 2) does bending depend on the top surface area of the box, e.g., whether increasing/decreasing the top surface area 100x while maintaining the same four sides would make a difference? 3) how does the material (e.g., rigidity) and weight of either the basketball (~700g) or the cardboard box factor into the their opposing forces and latter's bending?

Any intuition or explanatory equation is much appreciated!

 

[Chestermiller]

Would you know how to solve a beam bending problem under a distributed load?

 

[SergioPL]

1) Could time be a contributing factor to the bending (i.e, the box gradually cave in over time) or if the surface fails to bend at t0, it won't ever bend?

When you initially place the basketball, the cardboard box will bend "more and more" until it reaches the equilibrium position.

3) how does the material (e.g., rigidity) and weight of either the basketball (~700g) or the cardboard box factor into the their opposing forces and latter's bending?

The basketball makes a force towards the ground that causes the cardboard box to bend, when bended, the cardboard region just under the basket ball suffers a stress from the adjacent points that are no so bend as the center and this stress has a net component upwards that cancels the force done by the basketball weight.

2) does bending depend on the top surface area of the box, e.g., whether increasing/decreasing the top surface area 100x while maintaining the same four sides would make a difference?

I cannot answer you this point confidently, I guess the cardboard box will "shrink" the same heigh at the center causing a lesser inclination.

 

[nigels]

@Chestermiller Hm.. no idea..

 

[nigels]

@SergioPL Thank you for the detailed explanation! I find your response to (3) especially useful where you described the upward force due to adjacent stress. As you mentioned, this cancels out the downward force exerted by the basketball, which I infer to mean "no bending" as long as the cardboard material is dense enough, i.e., provides sufficient stress with minimal weight on top. Is that correct?

On a related note, in the original scenario, once I remove the basketball, what factor determines whether or not the top box surface will return to its initial shape (i.e., un-bend)? Is it even possible? Based on daily observations, feeble box surfaces obviously crease due to weights placed above. However, can structural restoration result from a much denser box material? Thanks!

 

[Chestermiller]

@Chestermiller Hm.. no idea..

OK. We engineers solve beam and plate bending problems all the time. As @SergioPL indicated, the deformations in these structures are determined by applying the 3D version of Hooke's law in conjunction with the stress equilibrium equation. For a plate like the lid of a box, the lid is in a state of "plane stress," in which the stress in the thickness direction is much smaller than the stresses in the horizontal direction. Get yourself a book on Strength of Materials, which goes into detail on how to solve beam and plate problems.

 

[Chestermiller]

@SergioPL Thank you for the detailed explanation! I find your response to (3) especially useful where you described the upward force due to adjacent stress. As you mentioned, this cancels out the downward force exerted by the basketball, which I infer to mean "no bending" as long as the cardboard material is dense enough, i.e., provides sufficient stress with minimal weight on top. Is that correct?

On a related note, in the original scenario, once I remove the basketball, what factor determines whether or not the top box surface will return to its initial shape (i.e., un-bend)? Is it even possible? Based on daily observations, feeble box surfaces obviously crease due to weights placed above. However, can structural restoration result from a much denser box material? Thanks!

This is related to the yield behavior of the material. Once the lid deforms beyond the elastic limit of the material, it experiences the phenomenon of yield which prevents it from returning to its original shape.

 

[nigels]

@Chestermiller Thanks for the new knowledge! The phenomenon makes more sense to me now. :)

 

[Nidum]

Heavy ball made from rigid material on flexible box . Static analysis .

 

[Chestermiller]

View attachment 195493

Heavy ball made from rigid material on flexible box . Static analysis .

Was this FEM calculation from a membrane model, a plate model, or a 3D model?

Pretty impressive!

 

[Nidum]

It is a 3D model .