Curved Beam with Straight portion 1

[pperlich]

This question is just for the sake of curiosity.

Lets say you a straight beam with an in-plane curve on each end.

How would the neutral axis and stress distribution look at the transition from straight to curved?

 

[rb1957]

like an arch ?

develop internal reactions based on section cuts (radial along the curves)

another day in paradise, or is paradise one day closer ?

 

[pperlich]

Like in the image I attached to the original post.

I think there would be a discontinuity in the way you describe the stress distribution near the transition that wouldn't be there in reality.

 

[rb1957]

yes, I saw that. I don't see a discontinuity at the transition.

another day in paradise, or is paradise one day closer ?

 

[pperlich]

I'm saying if one were to look at it in sections, the neutral axis would have a discontinuity, which wouldn't exist in reality.

 

[rb1957]

the NA either side of the transition should be much the same, IMO.

another day in paradise, or is paradise one day closer ?

 

[JStephen]

Yes, there would be discontinuities in the analysis at the transition.
Keep in mind that the normal equations for shear, bending, etc., all assume that you're not near a support. So the situation isn't that much different from other beams, you're just noticing an effect that is normally neglected.

 

[dhengr]

Pperlich:
You said..., “I'm saying if one were to look at it in sections, the neutral axis would have a discontinuity, which wouldn't exist in reality.” It does happen in reality, but I wouldn’t call it a discontinuity, I’d call it a transition from the straight beam condition, which we all know so well, to the curved beam condition. Look up Winkler-Bach’s formula for stresses in curved beams, and curved beams, hooks, chain links, hoops and the like in any number of good Advanced Strength of Materials and Theory of Elasticity text books. The N.A. moves toward the inside of the curve, toward the center point of the curve.

 

[rb1957]

that makes sense ... I was looking at the U.Washington article for curved beam stress and it just states an off-set between the centroid and the NA. But it makes sense (to me at least) that this is true for a continuous curve, and at the straight portions (at the start an end of the curve) we have the normal situation and a transition between the two.

another day in paradise, or is paradise one day closer ?

 

[pperlich]

I'll look into that reference.

Maybe I'm just not seeing it correctly. It seems however that if the NA "jumps" from inside the centroid, to the centroid, then the stress distribution would also have a "jump" in it. That "jump" just wouldn't exist. Am I wrong on that?

 

[dhengr]

Pperlich:
You said..., “It seems however that if the NA "jumps" from inside the centroid, to the centroid, then the stress distribution would also have a "jump" in it.” There is no jump, it is a gradual transition from the one to the other, over some finite length of beam. The structural member is continuous so the stresses must transition in some reasonable way, they can’t abruptly change, or you would be rippin things apart. Theory of Elasticity just won’t allow that. I used to do a lot of these calcs./designs, but it has been a long time since the last one, so I’d have to do some digging to refresh my thinking. I was comfortable with the approach I had developed and it was generally proven by strain gaging and FEA over some period of time. The N.A. shift is a function of the inner radius vs. the outer radius of the curved member. I would look at the straight beam condition near the shape change and the curved beam condition, and if I was happy with the stress picture in both, I didn’t much worry about the exact length or conditions in the transition length, that couldn’t be worse. The middle of the curved portion was usually the worst stress condition and you do have some complex shear stresses, radial stresses and normal bending stresses become circumferential stresses in t&b flanges, etc. At the tension flg., the flg. is pulled into the webs and at the compression flg., the flg. tries to rip away (is pushed away) from the webs, so the web to flg. welding becomes more complex.

 

[pperlich]

dhengr, that is exactly what I was thinking about. Thank you for the insights.