how can we get the unbalanced moment if we have the moment for the column at section above level of slab and at the section just below the level of the slab ?
What is the sign rule when taking the sum or the subtract of these moments ?
You should take the algebraic sum of these two which means that if they are both acting in the same direction they will add up and that quantity will act opposite to them at the slab but that is an extremely rare case. On the other hand the more common case is that they are opposite to each other and the difference will act in the slab at the same direction of the smaller one to guarantee joint equilibrium.
Hossam: Mohamed provides an excellent answer, and I just want to provide a little more background. Unbalanced moment refers to the bending moment that must be transferred by a slab to a column in flat slab buildings. This moment arises from slab moments on either side of the column that are not "in balance", that is, there is a net difference that must be transferred by the slab-column connection to the column for equilibrium. These unbalanced moments result from lack of symmetry in loading and/or structural geometry. Because of the uncertainty associated with the magnitude and placement (distribution) of live loading, it is necessary to address unbalanced moment during design such that the slab-column connections have sufficient shear strength for the combination of "punching" and "eccentric" shear stresses. From your question regarding the sign rule, as well as the column moments above and below the slab, I take it that you are using results from a computer analysis of the flat slab structure. The sign convention is defined by the program that you are using, so I recommend that you identify the sign convention that is used by the program. Once you have done that, a free-body diagram of the slab-column joint will help you determine if the moments in the column above and below the slab are additive or if they have to be subtracted.
Unbalanced moment, transferred between columns and slabs, produces additional shear stress and torsion that increase the vulnerability of slab-column connections to punching failure.
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