Cantilever or not? 1

[RHTPE]

Good morning all.

The uploaded .PDF shows a condition for which I have to review the design. The dbl C5 is being suspended from an existing precast box beam and will support the bridge deck concrete placement.

I maintain that, since the dbl C5 cannot deflection upward between the hanger rods, the design model shown at the bottom is correct. The left hanger rod carries all gravity load.

If one were to assume a support about 1" to the right of the left hanger rod, then the shear between it and the left hanger rod becomes astronomical, as does the tension in the hanger rod.

Am I missing anything?


Ralph
Structures Consulting
Northeast USA

 

[Lion06]

The left hanger rod will carry all of the gravity load plus the additional load from the C5 bearing against the concrete girder.
You will have a tension reaction at the left rod, and a bearing reaction of the c5's against the girder. The net vertical sum of these reactions will obviously be the gravity load, and the net moment will be the moment in the c5's just to the left of the left hanger rod.

 

[rb1957]

won't the load in the left hanger exceed the applied loads ? maybe it's just something in your terminology that i'm not familiar with ??

it's easy to calculate the moment in the beam at the left hanger. this moment is reacted by the beam pushing up against the existing precast box beam, maybe a triangular distribution is appropriate. that gives you the moment arm of the reacting couple (2/3 the beam overlap), and so the total load in the left hanger.

 

[vandede427]

I agree with EIT, the tension in the left hanger rod will actually be more than the sum of P1, 2, 3, and 4.

If the right support weren't tension only, it would have compression reaction, but it's not so the dbl channel bears against the bottom of the concrete beam.

 

[bootlegend]

Is there some way to verify that the bottom of the box beam is flat enough to allow bearing.? If concave down just enough the system would act more like a simple beam overhanging one support.

 

[RHTPE]

BUT, if the slope of the dbl C5s very very close to (and to the right of) the left hanger rod is zero (it would have to be, as it cannot deflect upward against the bottom of the precast beam), what its real behaviour?

How can one ever achieve a "pure" cantilever?



Ralph
Structures Consulting
Northeast USA

 

[Lion06]

RHTPE-

There is no reason to say that the slope is zero at the left rod. The rod will elongate, which will provide some slope. Even a truly fixed cantilever is expected to have some moment-rotation response. Even if the rod had infinte axial stiffness, it has almost no flexural stiffness, therefore the beam can't be cantilevered from it. If you didn't extend the C5 under the girder then it would be unstable. Imagine that instead of this rod, you have a bar coming down with a single, frictionless pin that goes through the C5. I think you would agree that even if the bar were infinitely stiff (axially) and didn't elongate at all that this wouldn't be fixed. This is, for all intents and purposed, what you have.

To draw out the FBD, I would use a pin-pin-cant. I would put one pin at the left rod, the other pin at the resultant of the bearing against the girder. The moment in the C5 just to the left of the left rod is what it is, it doesn't matter if it's fixed or pinned-pinned. That being said, the reaction on the left rod will definitely be higher than the total gravity load.

 

[rb1957]

consider a simple beam with an overhang. the beam between the supports deflects upwards. in your case this upwards deflection is prevented by the support precast box; actually this gives you a more rational basis for the assumed moment reaction (in lieu of my triangular assumption)... the reaction by the precast box could be proprotional to the deflection of the simple beam ... more like a sinusodal distribution; you could also analyze with an elastic support distributed along the non-overhang portion of the beam.

 

[InDepth]

Don't foregt that the idealized model will be stiffer than a model that accounts for the anchor rods and bearing.

Is there a deflection criteria that needs to be met for the formwork?

 

[Lion06]

rb-

I was intrigued by your last post re: the sinusoidal bearing distribution. I believe this is only true if you have a second pin support where the right rod is. The right rod may be sufficient, but it may not. If, however, that rod is gone and there is no "pin support back there, I don't believe that the upward deflection has the opportunity to develop because of the lack of a single stiff support point, and that the "backspan" will move more like a rigid body and produce much closer to a triangular pressure distribution.

 

[spats]

I would simply assume a cantilevered beam with the right support at 0.85d, where d is the distance from the left support rod to the far side of the concrete beam.

 

[hokie66]

Agree with spats, for design purposes. I prefer to call this a propped cantilever. The only "true cantilever" is encastre, with an infinitely stiff support.

 

[BAretired]

I believe spats is being unduly conservative. It is a cantilevered beam with a tieback of d. If the moment at the left hanger is M, then the uplift at the right edge of the beam is M/d and the tension on the left hanger is P1 + P2 + P3 + p4 + M/d.

BA

 

[rb1957]

SEIT ... it was only a thought-du-moment ... from the OP's sketch it doesn't look like there's a continuous connection with the precast box. assume there's a 0.001" gap, then the beam would initially deflect like a simply supported beam, and bear up against the support and the new reaction (with the precast box) would relieve the hanger reactions.

IMHO, BA's approach is possibly unconservative for the left hanger load, 'cause he's maximised the couple arm (therefore minimised the couple).

also i think we're over-reacting to the word "cantilever". in my lexicon, a cantilever is distinguished by developing a moment that reacts the applied loads, having zero slope at the support. there are few true cantilevers, but many close approximations. in this case we have a beam, with an internal moment due to the applied loads, and a redundant support ... how does the beam interact with the pre-cast box ? the hangers and this distributed reaction to the box beam react the applied loads (by the equilibrium equations).

maybe the OP is concerned about the deflections of the beam (i've been focusing on the reactions). in which case, spat's post gives an approach for estimating the slope of the beam at the left hanger. another idea might be to look at the extension of the left hanger due to it's reaction. now you've got a gap between the beam and the pre-cast box and so you've got to adjust the bearing surface, etc.

 

[PMR06]

If the left hanger were pretensioned greater than P1 + P2 + P3 + P4 + M/d, then the channels would always remain smashed against the precast beam. Then wouldn't the slope be zero and the left side be a true cantilever?

 

[spats]

BAretired,

I'm not being overly conservative. The resisting couple is left rod tension balanced against concrete compression on the bottom of the beam. The centroid of the compression area is approximated based on a singley reinforced concrete beam analysis. Maybe 0.9d is OK, but not the full d.

 

[csd72]

RHTPE,

An interesting question for which you have had a surprising variety of answers.

In order to answer this we need to step back to first principles.

The loads on a cantilever induce a shear force and a moment in the beam.

In order to resist this, a classic cantilever support has a vertical reaction opposite to the shear force and a moment reaction opposite to the applied reaction.

Now, for your situation:
As the face of the support is not encastre and cannot resist moment at the face then it must resist this moment by a push/pull couple and therefore the reaction on the rod is always going to be greater than the shear force alone.

As you have surmised, it will bear on the underside of the concrete beam, but where will this occur.

The rod will deflect due to the reaction, but the section of beam under the concrete will be sloped from this deflection anf will be curved due to the bending moment from the cantilever. This will give you a set of simultaneous equations to calculate where the steel beam touched the underside of the concrete.

Add on sufficient length for bearing and you have your lever arm.

Alternatively you can just assume that at ultimate limit state the rod will deflect sufficiently that this intermediate bearing will not be an issue. In this case you would design it like a conc beam or a base plate.

 

[rb1957]

PMR06 makes a good suggestion, regarding preloading the hanger. we can quibble about M/d, M/(d/2), M/(0.9*d), ... and about how to ensure the actual preload exceeds the calculatd reaction; but that's all just "grist for the mill".

 

[miecz]

If you insert a shim between the channels and the box near the right hanger rod, this would ensure that the couple is the distance from the left hanger to the shim. Then the additional tension is surely M/d.

 

[BAretired]

I guess the answer is variable according to the axial stiffness of the left suspender. If it is axially stiff, the channels are not able to flex upward between hangers, so the right reaction moves very close to the left rod.

If the left support is axially flexible, the channels can flex upward and the right reaction moves very close to the right rod.

BA