Differences between considering either the forces of individual load hypotheses and or the force envelope

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To explain the difference between considering the forces of the individual load hypotheses defined and considering the force envelope to dimension the necessary reinforcement both in the x and the y direction of the slab, it is convenient to clarify the concepts of integrated moment in a cross section of a specific width and reinforcing moment.

• Moment integrated in cross sections (mx, my, mxy): it is the sum of the results obtained in a cross section as defined with a specific width.
• Reinforcing moment (maxb, maxt, mayb, mayt): these are the moments obtained by CEDRUS to dimension the necessary reinforcements in each of the directions. These moments are obtained in the following way:

 

Looking at the formulae presented above it can be observed that to obtain the reinforcing moments, CEDRUS takes the integrated moments for each of the directions and then combines them.

With all of the above it can be concluded that to dimension the reinforcements with CEDRUS it will always be necessary to take the values of the reinforcing moments since those ones will include the integrated moments in the x and y directions and the torsional moments that each of said moments induces in the perpendicular direction.

It is interesting to include in this article an explanation of the sign criteria used by CEDRUS to define the moment nomenclature. For that matter, we have section B 13.2.2 from the CEDRUS-7 manual as a base:

 

It can be seen that the subscript of each of the forces indicates the direction for which CEDRUS dimensions the reinforcements, that is, mx will be the moment used to dimension the reinforcements in the x direction, my will be the moment used to dimension the reinforcements in the y direction and mxy will be the torsional moment that each of the moments induce in the perpendicular direction.

To numerically illustrate the difference between working with enveloping forces (and its consequent reinforcing moments) and working with individual hypotheses or results combinations (and their corresponding moments mx, my and mxy), we see the case of a slab, in which we have a vertical section of 4.5m wide, in which we are integrating the moments of a combination of results CP:

 

At the lower part, we see a value of 77.86 kNm.

Now, let’s see the results for an identical enveloping force and with the same magnification coefficients. The difference is that now the program throws in results of isolines of reinforcing moments and their integration gives other values.

 

The user might be surprised by that value of 138.23 kNm, but it all becomes clear when we see the isolines of torsional moments mxy.

 

The results are correct because, as we mentioned, if we analyse the difference between the results: 138-78=60 kNm, and we divide that by the width (4.5 m), we obtain a difference of 13.3 kNm/m => 13 kN, which are there due to mxy. This value, as we can see, coincides with the value that we have in the isoline torsions diagram above (we can see the zero isoline -dashed lines- and the 10, but not the 20, so the values of mxy on the yellow zones are between 10 and 20: 13.3 -> OK).