Hello,

I am going to reinforce a lightweight concrete floor from the underside with a steel beam. If we assume that the beam is loaded with the entire weight resulting from the floor and construction materials placed on the floor, it totals 3000 kg including the weight from the steel beam. The columns will be under 2400 mm long and will be mounted at each end of the beam. When I calculated this, I was puzzled by the small dimensions I got.

Therefore, it would be very helpful if someone could perform the calculation and suggest the minimum I needed to prevent buckling, including a safety factor.
 
3000kg isn't that much load, what dimensions did you get?
 
You seem to have only accounted for the load of the dead weight. Don't forget about the live load! It is usually between 2 and 5 kN/m2, (that is, about 200 - 500 kg/m2). Google live load and you should be able to find an appropriate value.
How is the beam dimensioned? The beam calculation should provide the column loads.
 
Thank you for the relevant points.

The reason I want to reinforce the intermediate ceiling is that the old pipes were sunken into the lightweight concrete, weakening the strength. I do not want to risk any settlements in the new floor that would lead to higher costs and problems to fix than relieving it with a steel beam.

I made a conservative estimate of the load on the beam, but in light of the feedback, I have done it a bit more thoroughly.
The columns (Steel) will be built into the walls on the lower floor, so the freely supported span will be 4.2 m.
Between the outer wall and the heart wall is 4 m.
On the upper floor, this area is covered by a bathroom that is 2.9 x 2.9 m, part of a bedroom, and a corridor.

40 mm EPS cement, 50 mm self-leveling compound, and 20 mm adhesive and surface layer provide about 1470 kg load from the bathroom construction.
The remaining floor is floating wood flooring on sparse panels, which I estimated to 30 kg/m2, resulting in a 420 kg load.
A useful load of 200 kg/m2 gives 3360 kg and the ceiling on the lower floor about 336 kg.
This totals 5580 kg. The heart wall and outer wall take half the load, giving 2790 kg on the beam.

Estimate of the deflection becomes Delta(1/2)=(5/384)*(QL^3/EI)(Another elementary case for the correct deflection is needed)
I was planning to use HEB 180.

The above load gives a high estimated column force of about 15 kN, including the steel beam (for simplicity, the columns take the same load in the example, but the actual load will need to be calculated with moment equilibrium later).
The column attachment will be a combination of Euler 1 and 2.
Calculating for case 1, I find that I need to be 167000 mm4, and with a safety factor of 2, it becomes 334000 mm4.
According to the attached link above, VKR 60x60x3, which has I = 36.2 10^4 mm4 and a cross-sectional area of 674 mm2, would be sufficient.
Does this seem correct?
 
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Feels a bit like you're mixing things up in your calculation. I recommend asking someone to calculate the beam. The pillars won't be large at all; the size of the pillars might be determined more by the execution. How do you position an HEB180 on a VKR60, for example? Are wooden pillars of construction timber preferable?
 
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