# How weight affects surface bending over time

• nigels
In summary, the heavy ball made from a rigid material on a flexible box will cause the box to bend, but the box will not bend if the ball is removed. The box will return to its original shape if the ball is removed and the box is made from a more rigid material.
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!

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

nigels said:
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.

nigels said:
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.

nigels said:
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.

@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!

nigels said:
@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.

nigels said:
@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.

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

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

Chestermiller
Nidum said:
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!

It is a 3D model .

## 1. How does weight affect the surface bending over time?

The weight of an object can cause the surface to bend over time due to the force of gravity. The heavier the object, the more force is exerted on the surface, causing it to gradually bend or deform.

## 2. Is there a limit to how much weight can be placed on a surface before it starts bending?

Yes, every surface has a weight limit before it starts to bend or deform. This limit depends on the material, thickness, and structural integrity of the surface. Exceeding this limit can cause permanent damage to the surface.

## 3. What factors can accelerate the bending of a surface due to weight?

Aside from the weight of the object, other factors that can accelerate the bending of a surface include the duration of the weight being applied, the temperature, and the humidity. Higher temperatures and longer periods of weight application can speed up the bending process.

## 4. How can we prevent or slow down the bending of a surface over time?

To prevent or slow down the bending of a surface over time, it is important to distribute weight evenly and avoid placing heavy objects on one specific area. Increasing the thickness or adding support structures can also help distribute the weight and prevent bending.

## 5. Can the bending of a surface be reversed?

In some cases, the bending of a surface can be reversed by removing the weight and allowing the surface to naturally regain its original shape. However, if the surface has been permanently deformed or damaged, it may not be possible to reverse the bending.

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