How to do the elastic analysis for concrete

Theoretical background

According to Eurocode 2nd order effects (EN 1992-1-1 §5.1.4 (1)P) and imperfections (EN 1992-1-1 §5.2 (2)P) must be taken into account.

• For the 2nd order effects (EN 1992-1-1 §5.8.5), you
  • either do the calculation according to the general method based on a non-linear 2nd order calculation (see EN 1992-1-1 §5.8.2 (2)P, §5.8.5 (1), §5.8.6 and §5.7).
  • either use one the two simplified methodes:
    • method based on nominal stiffness (see EN 1992-1-1 §5.8.7): not implemented in Diamonds
    • method based on nominal curvature (see EN 1992-1-1 §5.8.8)
      In this method the 2nd order effects are taking into account by increasing the 1st order bending moment M_Ed with a nominal second order moment equal to e₂ x N_Ed.

      M_{Ed.reinforcement} = M_Ed + e₂ x N_Ed

      e₂ [mm] is function of the effective length (syn. buckling length).

• The imperfections can be represented by an inclination θ_i. The inclination θ_i may be taken into account (EN 1992-1-1 §5.2 (7)):
  • either by translating the inclination θ_i to equivalent horizontal forces (= ‘Global imperfections’ in Diamonds)
  • either by translating the inclination θ_i to an accidendental eccentricity e_i
    In this method the imperfections are taking into account by increasing the 1st order bending moment M_Ed with an additonal moment equal to e_i x N_Ed.

    M_{Ed.reinforcement} = M_Ed + e_i x N_Ed

    The default value for e_i is set to 20mm in acccordance to EN 1992-1-1 §6.1(4). But this is a minimum value. You should check if a higher value might apply to your model (using EN 1992-1-1 equ. 5.2)

So if you would use the simplyfied method for both the 2nd order effects and the imperfections, you will end up with the following bending moment:

M_{Ed.reinforcement} = M_Ed + e₂ x N_Ed + e_i x N_Ed = M_Ed + N_Ed x (e₂ + e_i)

How is the theory implemented in Diamonds?

Depending on the selected options during the elastic analysis , you will have to (un)check some options in the Buckling parameters window. An overview:

Case 1: assume you do a 1st order calculation [Recommended and most often used option]

⇒

Then: ei = 20mm because global imperfections have NOT BEEN taken into account yet e2 ≠ 0 because 2nd order effects have NOT BEEN taken into account yet So, in the window you must: check the option ‘Verify buckling with an additional eccentricity’ set the buckling lengths to a meaningful value, because e2 = ft(buckling lengths). set the accidental eccentricity ei to 20mm

Case 2: assume you do a 1st order calculation  + global imperfections

⇒

Then: ei = 0 because global imperfections have been taken into account e2 ≠ 0 because 2nd order effects have NOT BEEN taken into account yet So, in the window you must: check the option ‘Verify buckling with an additional eccentricity’ set the buckling lengths to a meaningful value, because e2 = ft(buckling lengths) set the accidental eccentricity ei to 0mm

Case 3: assume you do a 2nd order calculation + global imperfections

⇒

Then: ei = 0 because global imperfections have been taken into account e2 = 0 because 2nd order effects have already been taken into account So, in the window you must: uncheck the option ‘Verify buckling with an additional eccentricity’. it is not necessary to calculate the buckling lengths, because only e2 = ft(buckling lengths), and e2 =0 because the option ‘Verify buckling with an additional eccentricity’ is unchecked. The entered value for the accidental eccentricity ei is irelevant because the option ‘Verify buckling with an additional eccentricity’ is unchecked.

Case 4: assume you do a 2nd order calculation

⇒

Then: ei = 20mm because global imperfections have NOT BEEN taken into account yet e2 = 0 because 2nd order effects have already been taken into account So, in the window you must: check the option ‘Verify buckling with an additional eccentricity’ set the buckling lengths in both directions to a small value (1cm for example, but not 0!), so that e2 = 0. Otherwise the 2nd order effect, will be taken into account twice! set the accidental eccentricity ei to 20mm

Notes:

• If your model contains multiple independent structures (for example: isolated columns), and you want to take global imperfections into account: make one Diamonds-file for each structure. Because Diamonds only remember the worst direction of the global imperfection, not the enveloppe. That direction can be the worst for structure 1, be not necessarily for structure 2.
• In Diamonds the reinforcement pattern “Optimal” is the default. However, this doesn’t always results in the most economic reinforcement. Read more about it.