Catenary vs. Flexure

[bowlingdanish]

Hi there,

This is a random theoretical I've been thinking about - what exactly, fundamentally characterizes a catenary/'tension only' member? If I have a pin supported 10mm cable spanning 5 metres with a certain sag, and apply load at midspan I can calculate a resulting tensile force in the cable. If I have pin supported 300mm deep I beam spanning 5 metres I can calculate a bending moment. What if it's supported over 40 metres? Then I have a member with a certain sag that I can apply a load to - Where does the flexure stop and tension start? I suppose the beam could buckle laterally at some point and become unstable, but it wouldn't yield under self weight - perhaps at this point?

No idea if that makes any sense.

 

[dcarr82775]

Its a lot like porn. I can't define exactly what it becomes catenary, but I know it when I see it.

 

[JoshPlumSE]

When small deflection theory no longer holds. When the deflection is negligible compared to the properties of the cross section.

For example a 10 mm cable that sags 20mm over it's length would violate small deflection theory because its deflection is greater than it's depth.

A 300mm beam that deflections 3mm would NOT violate small deflection theory. If it deflects 300mm then it probably would.

 

[MotorCity]

If I understand your question, it is at what point does a beam act like a cable (or vice versa). I am not sure what the theoretical definition is, but here is one way to think of it: a cable has no (or very little) axial stiffness when loaded only by its own weight. In other words, you can't push on a rope. Of course things change a bit if the cable is prestressed.

 

[JStephen]

You can develop some criteria if it's really an issue. In the case of plates in bending, the criteria becomes that deflection is less than the plate thickness.

A major consideration in a lot of practical cases is whether the end conditions actually allow that tension to be established.

If you're interested in actual numbers, Roark has load cases for "Beams restrained against horizontal displacement at the ends".

 

[racookpe1978]

Look at four cases:

Cable or chain, pinned at both ends, spanning a gap, no "point" loads on the cable, only the distributed weight/length of the cable itself. So, a perfect catenary -> the "uniform load" of the cable's weight is carried BY the two pins at each end which are in tension, right?

Cable or chain, pinned at both ends, with two "heavy" point loads on the chain: Doesn't matter what the loads actually are, nor where they are along the chain. The chain is "straight" from left pin to load_1, then "straight" again (at a different angle) from load_1 to load_2, then "straight" again from load_2 to pin_2. Not a catenary at all, and not easy to calculate, but "pure" tension loads on the cable and the two pins, right?

Thin plate or flat bar, spanning a gap. The flat bar (say 2 inch wide x 1/8 thick) bends across the gap, but the two ends bend "up" so the two contact points support all of the load of the bar. Can calculate the bending as if it were a pin? Nope, you need to use a "roller" type of equation => NO tension is transmitted to the surfaces under both contact points, but the flat bar in the center does has the tension in its cross-section due to the weight, and it does form sort of a catenary. Get too far a gap, too thin a flat bar cross-section, the "bridge" fails because the tension pulls both ends down into the canyon, right?

Heavy beam supported at both ends. Similar theorectically to the catenary of the flat bar, but the "sag" is meaningless.