How To Apply Reciprical Theorem To Beams

[EngNoob]

I am trying to learn and figure our the reciprical theorem when applied to beans and the deflection on a beam when a load is applied.

The beam would be x long, say 10m and would have a weight/load applied at 5m, or dead center. There would be point A (Dead Center) point B (Half way between dead centre and end) or 1/4 beam. Load applied to A and deflection at a and b needed, and then load applied at B and deflection and A and B.

I need to create a graph with a slope of load cases to varify the theorem.

I am a little lost on this one, i know what the theorem is, but i am not sure how to apply it to beam deflection and how to create a graph.

Does anyone know any good links to learn further or some examples.

This is needed for a test in 48 hours, and i am lost on it completely, the graph part especially

Thanks for any help

 

[KataKoniK]

!

Hi there,

The reciprocal theorem is a fundamental principle in structural mechanics that allows us to relate the effects of a load applied at one point on a structure to the effects of a load applied at another point. In your case, we are looking at the deflection of a beam when a load is applied at different points along its length.

To apply the reciprocal theorem, we need to understand the concept of virtual work. Virtual work is the work done by a load on a structure when it is moved a small distance in the direction of the load. The reciprocal theorem states that the virtual work done by a load at one point is equal to the virtual work done by the same load at another point.

In your example, we have a beam of length 10m and a load applied at the center (point A) and at a quarter of the beam's length (point B). To apply the theorem, we need to consider the virtual work done by the load at these two points. This can be calculated by multiplying the load by the deflection at that point.

For example, if we have a load of 1000N applied at point A and the deflection at point A is 5mm, then the virtual work done by the load at point A would be 1000N x 0.005m = 5J. Similarly, if we have the same load applied at point B and the deflection at point B is 3mm, then the virtual work done by the load at point B would be 1000N x 0.003m = 3J.

Since the reciprocal theorem states that these two virtual works should be equal, we can set up the following equation: 1000N x 0.005m = 1000N x 0.003m. This allows us to solve for the deflection at point A when the load is applied at point B and vice versa.

To create a graph, we can plot the load cases (1000N at point A and 1000N at point B) on the x-axis and the corresponding deflections at points A and B on the y-axis. This will give us a linear relationship between the two points, with a slope of 1 (since the two virtual works are equal).

I would recommend looking for some examples of the reciprocal theorem being applied to beam deflection to get a better understanding. You can also try using a structural analysis software to simulate and visualize

 

[piercas]

I understand your struggle in trying to apply the reciprocal theorem to beams. The reciprocal theorem is a powerful tool in structural analysis, as it allows us to determine the deflections and reactions of a structure by considering the effects of a different loading configuration.

In order to apply the reciprocal theorem to beams, we first need to understand the basic principle behind it. The theorem states that the deflection at a point due to a unit load applied at another point is equal to the deflection at the second point due to a unit load applied at the first point. This means that we can interchange the load and deflection points and still get the same result.

In your case, you are trying to determine the deflection at points A and B due to a load applied at point A and B respectively. To apply the reciprocal theorem, you can consider a unit load applied at point A and determine the deflection at point B. This will give you the deflection at point B due to a load applied at point A. Then, you can apply the same process with a unit load applied at point B and determine the deflection at point A. This will give you the deflection at point A due to a load applied at point B.

To create a graph with a slope of load cases, you can plot the deflections at points A and B for different load cases. The slope of the graph will represent the ratio of the deflections at points A and B, which should be equal according to the reciprocal theorem. This will help you verify the theorem and understand its application in beam deflection analysis.

As for further learning resources, I suggest checking out textbooks on structural analysis or consulting with a professor or tutor for more guidance. I also recommend practicing with different examples to gain a better understanding of the concept. I wish you the best of luck on your test.