STRESS DISTRIBUTION IN BEAM WITH SMALL 'TEETH' 1

[NAFTALI-HAKOHEN]

hi,

had an argument with my boss today about stress distribution in a beam with small 'teeth' sticking out in tension area, bottom of beam:

i said there will not be tension stress in the teeth , obviously depending on the dimensions an ratio of width/depth etc, somehow as a gut feeling that is has to do with the flow of shear tau into this small area ,

i then did a FEM model to confirm this and i think this is correct,

can someone explain the theory behind this or post some links?

thanks

Naftali

 

[GeorgeTheCivilEngineer]

There would be tension stress in the teeth - it is continuous material from the region above which is in bending.

There would be a short development length so the stress wouldn't be immediately transferred to the deeper cross section and they are too short to really contribute anything to global capacity.

 

[BridgeSmith]

There may be less stress in the bottom of the teeth, if the section modulus increases, but there will still be tension stress in the teeth. If the teeth were really narrow in width and very tall, the tension at the bottom would be greatly reduced, and might go to zero, depending on the material properties of the beam. At the depth to width ratio shown, the tension stress would be slightly lower at the bottom of the teeth than for a beam that is the depth of the teeth for the full length. At the free corners, the stress would go to zero, but the shear flow would distribute the stress to the bottom in the middle of the teeth.

Rod Smith, P.E., The artist formerly known as HotRod10

 

[MotorCity]

Sorry but I side with your boss on this one. There will be tension in the teeth. Can you post your FEM results that confirm your opinion? And just to be clear, I think we are all assuming the beam is loaded on the top only. This looks like a concrete beam, so I am also assuming that the beam is one homogenous, integrally cast member.

 

[BridgeSmith]

I suppose if it's a concrete beam, and you ignore the concrete in tension, as we typically do for design, then there would be no tension in the concrete whatsoever, and the only tension would be in the reinforcing steel. Of course, that is only an assumption made for design modeling, and doesn't reflect reality, where concrete can carry some tensile stress, until it cracks.

Rod Smith, P.E., The artist formerly known as HotRod10

 

[canwesteng]

If the beam is shallow enough that it behaves as a classic bernoulli beam (plane sections remain plane) there is of course tension in the bottom. This beam theory doesn't apply to the teeth themselves and the stress distribution is going to be pretty irregular, and possibly 0 at the bottom of the tooth, but always some tension at the top of the tooth.

 

[rb1957]

I think there'd be some sort of "shear lag" as stress shears into (and out of) the teeth.

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

 

[Brad805]

Post a picture of your stress diagrams. I see a pin/pin in the model.

 

[WARose]

I side with your boss (assuming this is something like steel). I would think there would also be stress concentrations as well. (Which may/may not be a issue depending on the material, loading ,etc.)

 

[driftLimiter]

Reality is unfortunately grey. Classic beam theory where plane sections remain plane says if there is a curvature of a beam at any point, then there will be a linear strain distribution of the cross section at that point which would indicate there is tension there. Good chance that those notches are a point of weakness from stress concentrations. From a beam design standpoint though If the segment of the beam that is continuous and prismatic is sufficient to carry the loading, then wouldn't any protrusions (teeth) just be kind of going along for the ride? I guess the biggest concern is that the stress concentration at the re-entrant corner will be so large that a crack can form and compromise the beam. Material type plays a big part in this if its concrete with bottom reinforcement the crack might not be consequential, if its wood or steel there is nothing there to take the load once it cracks.

 

[Tomfh]

There will be some tension in the teeth.

The bottom of the beam cannot stretch without stretching the teeth.

 

[dik]

I would design it based on the lower least depth with added 'hangers' for the toothed part. Most of the tension, I would suspect would occur at the re-entrant corners of the teeth, not at the bottom of them. Your FEM model likely shows that.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Do you feel any better?

-Dik

 

[human909]

can someone explain the theory behind this or post some links?

Sounds like you want to understand this and also explain it to your boss. I'd say the best approach would be from a strain point of view. You likely can both agree that the there is strain on the bottom of the beam. The continuity material the adjacent to the area experiencing strain also experience a similar strain. This will decrease the further away from the agreed point of strain you go.

If there is strain then there is stress. Unless of course there is cracking and the material then becomes discontinuous.

Or the quicker answer:

The bottom of the beam cannot stretch without stretching the teeth.

 

[driftLimiter]

I think we are all interested what type of material, loading, and if there is a design concern or if this is just academic. It seems to me a FEA solution with good meshing near the corners would pretty much answer your question. Me thinks there is stress there, it doesn't just magically stop at the point of discontinuity. Like others have said perhaps not the same stress as you might have at the same location if it were made of a 'deeper' and continuous section. But there is still stress. How to deal with the stress practically is frankly more interesting

 

[GeorgeTheCivilEngineer]

If there is strain then there is stress

Could make a engineering 'motivational' poster...


"Where there be stress, there be strain. Where there be strain, there be stress."

 

[ALK2415]

@ op

i then did a FEM model to confirm this and i think this is correct,

could you take screenshot of stresses ?
sure there will tensile stress in bottom ("on every sections deep or shallow it was")
theory is simply that flexural stress Inversely related to section modulus Sx (depth in this case)

 

[NAFTALI-HAKOHEN]

hi all,

thanks for the responses,

it is a concrete beam loaded from above, and ill try post the FEM when i get to office tomorrow,

there seems to be some agreement here that due to the shear lag around these corners, the tension will certainly be less than at continuous part of beam, even though some posters(correctly imo) are saying there is strain and therefore stress at bottom of beam,

dont have a clear picture from the responses

thanks

 

[rb1957]

concrete, presumably reinforced, makes a difference. The re-bar won't follow the steps, no?

you can picture this by thinking if I want zero (or negligible, for there is no zero) stress in these steps, make them separate pieces. Attach these pieces to the main beam with two pins, put oversize holes in the step piece to further reduce the load into the steps.

if the beam was a continuous material, like steel, then the shear lag story applies.

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

 

[le99]

My understanding of this matter is that the tensile stress at the bottom of the slab can't make a sharp turn at the corners but undergo a smooth transition. The slope of the transition, the depth of its reach (with respect to the teeth), and the intensity of the stress are influenced by the geometry (width and depth of the teeth) and the ductility of the material.