Thermal stress with 2 objects

[themanonthemo]

Hello everybody,

Here is a problem I have.

QUESTION:

The main span of a bridge has a length of 473 m. On each end there are the expansion joints like the one on the photo below. One day the city changed from -4 to +15 degrees between 6 in the morning and 2 in the afternoon.

At 6 in the morning a piece of tire rubber fell into one of the cracks filling it completely. The rubber was 10 cm long and had a cross section of 4 cm2. What was the stress in the rubber at 2 PM? Clearly state all the assumptions, which you made, while solving this problem.

Data:
Young modulus of steel 200 GPa
Young modulus of rubber 7 kPa
Linear Expansion coefficient of steel 13·10-6 K-1
Linear Expansion coefficient of rubber 77·10-6 K-1

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How I decided to approach it
(dL/L)steel expansion + (dL/L)rubber expansion + (dL/L)steel stress + (dL/L)rubber stress = 0
a_sdT + a_rdt + (F/A)/Ys + (F/A)/Yr = 0
F/A must be equal for both
F/A = -dT(a_s + a_r)/(1/Ys + 1/Yr)
I only get about 6Pa... a number way to low.
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The answer key solves it the following way:
∆L=∆Lbridge+∆Lrubber = 58mm (rubber expansion is much smaller and can be neglected)
F/A = Y(dL)/L
S=7kPa *58mm/100mm = 4kPa

Assumptions:
The span warmed up uniformly to the air temperature
Most construction is steel
The span expanded uniformly in both directions
Only the rubber is compressed by the stress
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Firstly, I there is something wrong with the way the solution key solved it. They obtained that number assuming that all the stress goes into the rubber. It even says so in the assumptions but it still doesn't make sense.

Secondly, I don't see why the number I solved for is wrong. Could someone please explain what incorrect assumption I made?
Thanks so much!

 

[haruspex]

As you did not post the picture I'm not completely clear on the set up, but I would have thought that the expansion of the bridge would be entirely taken up by the expansion joints until they are both closed. Only then would the rubber and the span feel any pressure. What did you calculate for the total expansion of the span?

I see that you have tried to allow for the amount of strain taken by the span. Because the span is very much longer than the rubber is thick you are right not to assume straight off that the rubber takes virtually all of the strain. However, that treats the span as only having the same cross sectional are as the rubber, which is clearly incorrect. So I think you need to assume all the strain is taken by the rubber.

 

[themanonthemo]

Sorry I don't think I fully understand. I assumed that the Stress (pressure) would be equal, isn't that a correct assumption even if the areas are different? Why would this mean that we have to assume all the strain is taken by the rubber?

 

[haruspex]

Sorry I don't think I fully understand. I assumed that the Stress (pressure) would be equal, isn't that a correct assumption even if the areas are different? Why would this mean that we have to assume all the strain is taken by the rubber?

The force is equal, and locally, over the area of contact between rubber and steel, the pressure is equal. But the steel will take that force as a strain over a much broader area, so the deformation will be minute. You cannot calculate what it is because you do not know how broad the bridge span is (i.e. its cross section).

 

[Nickie]

I would like to address both the solution provided and the assumptions made in solving this problem.

Regarding the solution provided, it is important to consider all factors that may contribute to the stress in the rubber. In this case, the temperature change and resulting expansion of the bridge may also contribute to the stress. Therefore, it is not accurate to assume that all of the stress is solely on the rubber. This could explain the discrepancy between the two solutions provided.

In terms of assumptions, it is important to consider the real-life conditions in this scenario. It is unclear whether the bridge actually warmed up uniformly or if there were any other factors that may have affected the expansion. It is also important to consider the composition of the bridge and whether there are any other materials involved that may have different expansion coefficients. Additionally, it is important to consider the size and shape of the rubber and how it may interact with the bridge and the other materials involved.

In order to accurately solve this problem, it is important to consider all factors and make realistic assumptions based on the real-life conditions. This may require further data and analysis. It is also important to double check calculations and assumptions to ensure accuracy.