D.2.1 Combinations and envelopes

Recall that using a load combination consists of accumulating the structural effects of different loadings by assigning weighting coefficients to the different loadings as defined in the standards.

Strictly speaking, the codes require all combinations to be verified. In the case of a building, the number of these combinations remains small, so the software can calculate all of them. However, for more complex cases, and particularly in the case of rolling loads, the theoretical number of combinations becomes unmanageable. In such cases, force envelopes are used.

An envelope contains several load cases and only records the extreme values of the individual components (with the concomitant components).

Remember that accumulating unit cases calculated by a non-linear approach (NL) has no physical meaning. However, some software allow non-linear calculations on "combinations". In this case, the software will create a new case (the "combination") from the unit load cases and perform the NL calculation on this sum of loads. If the software does not allow it, it will be necessary to create combinations by manually grouping the unit loads. In this case, it is, once again, fundamental to understand what the software does.

Illustration of the above text using an example

Illustration of the structure

The following two unit cases are defined:

• self-weight + earth weight and thrust,

• hydrostatic pressure.

The following results show that although the software can conduct non-linear calculations using all the unit cases, the combination of the two cases is recalculated integrally and independently...

Results of the unit cases: bending moments - gravity loads (left) and water pressure loads (right)

Results of the combination of the two cases: bending moments - software combination (left) and cumulative loads in a new case created manually from the unit cases (right) - identical results

... since the accumulation of the unit cases does not lead to the results of the combination.

The diagram of the pressures on the soil speaks for itself: the water pressure forces the vault to rest on the ground "upwards" at the top (right figure) ...

... but once accumulated in the case of gravity loads, the top of the vault no longer pushes upwards, which can be seen on the results of the combination made by the software:

Soil pressures from the combination of gravity loads and pressures

Note: without springs on the arch, the second case would not converge.

D.2.2 Be careful when using the envelope results

When using envelopes, one must record in a database the displacements, extreme stress values, strains, or reaction forces at the supports.

Most calculation software offer the possibility of storing the extreme values of stresses and strains, either alone or together with the values of concomitant stresses and strains.

Before using the results of the envelopes for further post-processing, one should clearly understand whether the stresses and strains are concomitant.

For example, if one wants to reconstruct the most unfavorable stress state of a section, it should be verified that the extreme stresses selected for the upper and lower fibers are indeed concomitant.

Also, it is important to base the analysis of the results not only on the most unfavorable stresses and their concomitances but also on the concomitant stresses that generate the most unfavorable stress states. Thus, a maximum normal stress associated with a small concomitant moment can generate less unfavorable effects than a slightly smaller normal stress associated with a larger moment.

To verify a section, it is acceptable in upstream phases to perform the verification with all the extreme stresses in the same torsor. However, in the execution phase, for optimization reasons, it is advised to recover the torsors of concomitant stresses.

D.2.3 Beware of automatic combinations!

The combinations of loadings are used differently for building or other civil engineering structures.

For buildings, elementary loads induce a very large number of possible positions, all of which must be explored to determine the maximum effects on each structural element. This multiplicity of loads and configurations leads very naturally to the use of automatic combination modules.

In general, the automatic combination modules proposed by most software should be used with great precautions because it is a frequent source of errors. Some modules are black-box modules and not all software are allowed to know what are the elementary load cases used for the envelope combinations.

Moreover, the verification and coding of the combinations is a tedious exercise for which it is difficult to detect the error.

One of the most effective methods to prevent errors related to combinations and envelopes is to dissect the design stresses and strains. For some key stresses or strains (maximum bending moment, extreme stresses), it is a matter of finding the participation of each design elementary load in the overall stress or strain. Thus, one can verify that there are no errors in the accumulated values and coefficients and that the "logical" load cases are design loads.

For buildings, the same exercise can be done with the reaction forces at the supports.