Biaxial tension-compression on continuous slab 1

[Italo01]

Hello,

I have to design a small composite steel-concrete bridge for 24 ton trucks and due to the high live load i need to design the slab with continuity over the transverse beams.
My question is: since the continuity will create tensile stresses on the top part of the slab on the slab direction and this portion of concrete is compressed on the beam direction, isn't unsafe to use the concrete fck for the beam strength without consider the reduction due to biaxial stress?

I suppose that is acceptable, since the codes do not cite any reduction and the books(at least the ones that i saw) do not treat this question, but i cannot see how i can neglect this effect.

Thank you.

 

[centondollar]

Your question is unclear. Perhaps a sketch might help.

Are you describing what is essentially a "wide beam" (one-way slab) supported by several transverse beams (idealized as non-settling supports)? In that case, there is no biaxiality, and I am not quite sure what you mean when you write about "beam direction" and "slab direction".

If you use a plate model, there will of course be bending moment in two orthogonal directions (Mxx, Myy) and a twisting moment (Mxy) everywhere in the slab, and reinforcement should be designed (e.g., using Wood-Armer method) to resist those moments and the shear force. A slab with more plate action (ratio of side lengths closer to 1 than some large number) and less symmetrical support conditions will lead to a more siginificant effect from the twisting moment component.

 

[Italo01]

I'm attaching a sketch.

Are you describing what is essentially a "wide beam" (one-way slab) supported by several transverse beams (idealized as non-settling supports)?

Yes, i'm talking about a steel deck slab acting as one-way slab with reinforcement for hogging moment over the beams. If There's a hogging moment, tensile stress σx will develop on the top of the concrete(Although we consider onle the steel tensile stress, the concrete is under tension). Also, since the beam is under a sagging moment and act compositely with the slab, the same top part of the concrete will develop compressive steress σy. So i don't understand how the concrete is not under biaxial stress.

Could you explain me why there's no biaxiality?

Thank you.

 

[BAretired]

That is not biaxial bending by definition. The concrete under tension may crack, but it is not being relied upon to resist any load. Mohr's circle indicates a higher shear stress at 45^o to the principle axis, but as soon as the concrete cracks, the high shear is partially relieved.

BA

 

[Italo01]

It makes total sense.When the concrete cracks, the tensile stresses are relieved and the concrete will behabe as block between the cracks with the nominal compressive strength, correct?

 

[BAretired]

Yes, that's a good way of putting it.

BA

 

[dik]

Can you put a joint centered on the beam? Also, I'm not sure of using steel deck for bridges.

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

-Dik

 

[centondollar]

There is no significant bending in the transverse direction ("B-B") if the slab is one-way (i.e., length of slab along "B-B" is much larger than length along span direction "A-A"), and certainly none at all right above the beam in that direction. Furthermore, if the load is supposed to move primarily along direction of section "A-A", you have placed the beams in the wrong way - rotate them 90 degrees.

The design formulas for composite steel-concrete slabs (steel profile sheeting, concrete slab on top, shear studs and hot rolled steel profiles) in e.g., Eurocode are based on simplified analyses in which orthogonal axis bending moments are analyzed in one direction at a time. By using Wood-armer and plate equations, you will receive equivalent moments in the two orthogonal directions "x" and "y"; after that, you can design the slab accordingly.

 

[le99]

Since the edges of the deck, parrel to the traffic, are not restrained, theoretically, there is no stress in the direction parallel to the beam. The negative bending is resisted by the rebars in the direction of the traffic (assuming the concrete is cracked). However, the transverse beam must be supported on its ends, thus the beam will sag, which causes compression on the deck above. Practically, the compressive force is assumed to be resisted by the concrete above through composite action, the resistance of the rebars in the direction of the beam is usually ignored. So there is no two-way action, which usually describes the phenomenon of bending of an element in both directions.

 

[Italo01]

Also, I'm not sure of using steel deck for bridges.

It's a small bridge that will be built in a small city to connect two neighbourhood only during rainy times when the clay roads become flooded. The bridge will have only one lane with about 11 ft width, so there are two longitudinal beams supported by columns and these beams supports the transverse beams(depicted on the sketch) that support the slab.

Since the bridge will be traveled only by cars and small trucks, the Brazilian code allows the load to be based on a truck with 6 tires(9 kips each). I checked and Shear and punching shear will not govern. The bending is high so i decided for this layout, since it will allow me the continuity of the slab with the hogging moment reinforcement and the continuity of the deck when acting a shuttering(This continuity cannot be provided if the slab is rotated since there are only two longitudinal beams)

 

[Italo01]

le99, you described exactly the system.

 

[le99]

So, there are two tasks in your design process. In the direction of the traffic, you should design the oneway continuous slab as usual. In the transverse direction, you shall design the composite T-beam (steel beam plus effective concrete deck) to make sure the compressive stress in the concrete is kept within the allowable, and the steel beam is good for tension.

 

[Italo01]

Thank you le99.

I'm good with the design process, i asked the question because i wanted a understanding of why the design models did not take into account the biaxial stress effect. It seemed unconservative to me but now i understand.

 

[centondollar]

Italo01,

You should place the beam in the direction of the traffic and join it with shear studs into the slab. That is how the most important composite effect in the main direction (1-way slab) is created in a bridge girder; you may google "steel beam concrete deck composite bridge" for the concept. In the transverse direction, you should place rebar in the slab to control cracking.

If you place the beams as shown (transverse to the primary load-carrying direction), you will not achieve the composite effect you desire in the direction of traffic. Furthermore, those beams will not act as intermediate supports in your current configuration unless they are extremely stiff (and thus uneconomical); each beam will bend significantly during loading, unevenly, and will not provide the effect of intermediate supports. The current design is basically an unsymmetrical two-way plate with two stiffening ribs (the beams) in the less stressed direction (the "long" direction), and is far from optimal for a bridge structure expected to carry heavy loading.

 

[Italo01]

Thanks Centondollar, i'll check the issue of stiffness but i don't think that it will have the problem that you are suggesting because this transverse beams will have a small length(Approximately 10'8") and the load is not so heavy(6 nominal loads of 9kips), and since the stiffness is proportional to 1/L^4, the adequate beam to the bending moment will probably have a big stiffness.

 

[centondollar]

You will not achieve non-settling supports (essentially 100% rigid transverse beams) with any reasonably sized steel profile. The most straightforward design is to place the steel profile in the direction of the loading (short span direction in this case), and to size the slab aspect ratio so that one-way action is a reasonable estimate of actual behavior; this provides a simply supported beam (steel profile + stiffening concrete slab flange) with a large rigidity and bending capacity.

Design of a two-way slab is much more complicated, and the idea of adding intermediate transverse beams to provide supports (which would need to be supported by very rigid walls or columns at each beam end) is even more difficult to realize in both theory and practice. My advice is to keep it simple.

 

[Italo01]

Centondollar, i really appreciate your advice, but since this layout would help me a lot because of the continuity of the slab, i cannot discard it for now and would like to ask two questions regarding your position.

1 - One of the reasons that you cited is that since the aspect ratio is low, the behaviour of the slab won't be close to the one-way model and i understand it, but supposing that the slab has a ductile behaviour and that LBT applies, isn't one corollary of the LBT that any additional restraint will increase the failure load(Someone correct me if i'm wrong)?
In this case, isn't the one-way slab model conservative for the slab?
Also, the restraints created by the Girders will not reduce the load on the transverse beams, meaning that the model is conservative for theses beams?
I don't think that the model will be unconservative for the girders since on this model the loads will be transmited to the girders and concentrated on points away from the girder supports, what, in my view shouldn't matter, as long as the system has ductility, because of the LBT.

2 - With respect to the stiffness of the transverse beams. What criteria would you think sufficient to consider the beams stiff enough, because clearly at some point it can be(The Girders also don't have infinite stiffness, so i understand tha by your logic they wouldn't be considered a good support).

 

[centondollar]

"In this case, isn't the one-way slab model conservative for the slab?"
It is, if you place the beams in the direction of the load and support the slab only at two ends (wall or closely spaced columns at the two ends of the span), which I suggested.

"Also, the restraints created by the Girders will not reduce the load on the transverse beams, meaning that the model is conservative for theses beams?"
I do not understand what you mean. The transverse steel beams (oriented in the wrong direction: they are transverse to the direction of main line loads (from wheels) in your current design) are not useful unless you support them on walls or columns at each end, at which point you have created a two-way slab.

"I don't think that the model will be unconservative for the girders since on this model the loads will be transmited to the girders and concentrated on points away from the girder supports"
If you assume one-way action (i.e., support the slab only on the two edges where the transverse beam does not terminate) in the direection that is 90 degrees rotated from the girders (as you have now done), the slab will be taking almost all the load, which is unfavorable.

"2 - With respect to the stiffness of the transverse beams. What criteria would you think sufficient to consider the beams stiff enough, because clearly at some point it can be(The Girders also don't have infinite stiffness, so i understand tha by your logic they wouldn't be considered a good support). "
No reasonably sized profile will provide you with a "four-span beam" (what you have until now proposed) consisting of a slab and two intermediate beams transverse to the main load bearing direction.

Furthermore, you cannot design the slab assuming one-way action and simultaneously support it on all edges (along edges in direction of load, and on the edges where the beams terminate) - therefore, the current design is not wise.

I suggest that you once again google "steel beam to concrete composite connection, shear studs" or something to that effect. That way, you will understand that the way to place the beams is in the direction of load (90 degrees rotated from your current configuration), that you need shear studs, and that you should support the slab only on two edges. Anything else is a two-way slab, for which bending and deflection cannot be calculated by any of the standard "unit-width beam with rectangular stress block + steel" methods!

 

[Settingsun]

Are the top flanges of the transverse beams at the same level as the top flanges of the longitudinal beams, or do the transverse beams sit on top of the longitudinal beams?

In either case, a computer structural analysis would likely be fairly quick and answer these questions about stiffness. This discussion is fairly abstract without knowing all dimensions and proposed section sizes.