Radius of curvature of steel rod under stress

[n00beng]

Hi,

I have a problem which you guys probably could help me solve or at least advise how to approach.

I am building a mechanical system that consists of 2 steel rods acting as rails and a platform that travels along. I need to find radius of curvature of a steel rod under stress to see by how much platform might go down.

The diagram kind of has the info, this is a static problem. Points A and B of course moving along the rod but the distance between them is constant. Two forces applied are equal. I have stated initial value of 50 N, but it would be useful to get idea on radius of curvature as function of force (but this is a later issue).

It will probably be sufficient to look at specific cases:

1. Platform near one end
2. Platform in the middle

I would really appreciate any advice.

 

[timthereaper]

First off, draw a FBD and solve for the reaction forces. Then, you can apply Castigliano's theorem to solve for deflections at any point. Now, you can then either take a bunch of deflections and spline/fit a curve to them in a region you care about, or you can try to use Castigliano's to get a relationship between your dummy force's position and the deflection so you can get an equation out of it, then solve for curvature at a point.

 

[n00beng]

First off, draw a FBD and solve for the reaction forces. Then, you can apply Castigliano's theorem to solve for deflections at any point. Now, you can then either take a bunch of deflections and spline/fit a curve to them in a region you care about, or you can try to use Castigliano's to get a relationship between your dummy force's position and the deflection so you can get an equation out of it, then solve for curvature at a point.

Thanks for the reply, timthereaper. I will have a long "before bed" think about it tonight. I think my main problem lies in the second end being fixed as well, which introduces tension in the rod during bending. To be honest I have no clue how to even begin to tackle this.

 

[nvn]

n00beng: Are you really interested in the radius of curvature of the steel rod, as you said in post 1? Or are you instead interested in midspan deflection of each steel rod?

Because there are two steels rods, half of F1 = 50 N, and half of F2 = 50 N, will go to each steel rod, which, on one steel rod, will be 25 N at point A, and 25 N at point B, correct? Is this what you intended, in post 1?

If so, then when your platform centerline (point C) is located at the steel rod (beam) midspan, the beam midspan deflection will be y2 = 0.261 mm, downward.

(The induced axial tension you mentioned is essentially negligible. I currently would recommend ignoring it, unless you have a very special requirement that requires it, in which case it would make the problem much more complicated.)

Regarding item 1 in post 1, when your platform is at one end of the beam, then the beam midspan deflection will be y1 = 0.261 mm, downward (the same as when the platform centerline is located at the beam midspan).

These deflections are a linear function of the applied loads. Therefore, if you, e.g., double both F1 and F2, then it will double y1 and y2.

 

[n00beng]

n00beng: Are you really interested in the radius of curvature of the steel rod, as you said in post 1? Or are you instead interested in midspan deflection of each steel rod?

Because there are two steels rods, half of F1 = 50 N, and half of F2 = 50 N, will go to each steel rod, which, on one steel rod, will be 25 N at point A, and 25 N at point B, correct? Is this what you intended, in post 1?

If so, then when your platform centerline (point C) is located at the steel rod (beam) midspan, the beam midspan deflection will be y2 = 0.261 mm, downward.

(The induced axial tension you mentioned is essentially negligible. I currently would recommend ignoring it, unless you have a very special requirement that requires it, in which case it would make the problem much more complicated.)

Regarding item 1 in post 1, when your platform is at one end of the beam, then the beam midspan deflection will be y1 = 0.261 mm, downward (the same as when the platform centerline is located at the beam midspan).

These deflections are a linear function of the applied loads. Therefore, if you, e.g., double both F1 and F2, then it will double y1 and y2.

Thank you, nvn. Could you please state in brief how you calculated it.

Effectively I am interested by how much the platform goes down. If it goes down considerably (~> 1 mm) then I would also need to calculate the angle to horizontal of platform. This can be done quite easily knowing rad. of curvature or midspan deflection. 50N is the weight already halved between 2 rods so the answer (based on yours) is y = 0.522 mm.