Relationship between curvature of an arch and bending moment.

[Scintillation]

Homework Statement

A bridge has two arches, with one abutment in the center labeled A connecting the two arches, and a point b on the right.

I have calculated that the maximum bending moment in the structure would be at point A, right on the abutment that is connecting the two arches. I am to consider whether or not the bridge designer made the correct decision in placing the abutment at the point where the maximum moment would be located, and whether or not it has the appropriate thickness.

Here is a picture of the structure. Note that there is a roadway on the arches.

The Attempt at a Solution

The professor mentioned something about how curvature of the arch is related to the moment that is applied. He used the example of the diving board. The diving board is less thick near the jumping end, because more movement is desired. It is thicker near the ladder, since less movement is desired there.

However, I am not sure how this relates to the bridge. I understand that the bridge has the greatest moment at point A. I understand that we want to minimize the actual bending taking place, meaning that we counteract the bending moment applied by the maximum load.

There are two interpretations:
1. Because we want to reduce the amount of bending at point A, making it thicker there makes sense. Thickness mitigates the effects of bending moment. Having the ends of the arches, with its high curvature, also helps to decrease the effect of the moment.

2. Because the maximum amount of bending is at point A, we should have made point A the center of an arch. That way, it should be able to best counter the bending moment.

I am not too sure #2 is correct. If curvature and thickness both help to reduce the effects of bending moment, then having a thin, uncurved section at the center of the arch would not help. Is this even a correct analysis?

 

[anathema]

Thank you for your post and for considering the placement and thickness of the abutment in the bridge structure. I would like to offer some insights and considerations on this matter.

Firstly, it is important to note that the bending moment in a structure is directly related to the applied load and the stiffness of the structure. In this case, the arches of the bridge are the main structural elements that resist the bending moment caused by the load of the roadway and any other applied loads. The abutment, on the other hand, serves as a support for the arches and helps to distribute the load evenly.

Now, to address your first interpretation, it is correct that increasing the thickness of the abutment at point A would help to mitigate the effects of the bending moment. However, it is also important to consider the overall structural stability and balance of the bridge. Making the abutment too thick at point A could potentially make it unstable and cause it to fail under the load. Therefore, the thickness of the abutment at point A should be carefully designed to provide enough support without compromising the structural stability of the bridge.

Regarding your second interpretation, it is not necessary to have point A as the center of an arch to best counter the bending moment. The arches themselves are designed to distribute the load and resist the bending moment. Having a center arch at point A may actually cause an imbalance in the structure and create additional stress points. Additionally, the curvature of the arches also plays a crucial role in resisting the bending moment. So, it is not necessary to have a thin, uncurved section at the center of the arch.

In conclusion, the placement and thickness of the abutment at point A should be carefully considered to ensure that it provides enough support without compromising the structural stability of the bridge. The arches themselves are designed to resist the bending moment and their curvature also helps in this regard. I hope this helps to clarify your doubts and provides a better understanding of the design considerations for this bridge structure.