2nd order effects

The 2nd order effects induce additional bending moments due to the influence of structure deformations and/or imperfections.

An important distinction needs to be made between a linear analysis and a nonlinear analysis with geometrical non-linearities. The selected path affects how the 2nd order effects are taken into account in the composite column code check. Further info on the design routine may be found in Designers’ Guide to EN 1994-1-1, Fig. 6.36.

 

Linear analysis

This path is triggered in general in all cases when the 2nd order analysis is not triggered. There is a difference in the routine, if there are bending moments present or not.

Pure compression

Only compression force present possibly accompanied with shear force, no design bending moments from FEM analysis.

In case of pure compression the composite column code check verifies, whether the composite setup setting "Simplified method using European buckling curves" is activated or not. If it is activated, the program will calculate the compression resistance using the mentioned method (see further in "Compression").

In case of pure compression and the check box is not activated, depending on α, the additional moments from imperfections or 2nd order effects will be applied. The compression and bending moment resistance(s) will be based on interaction diagram (see further in "Interaction curve").

Bending moment(s) present

In case there are bending moments present besides compression, an evaluation of the α value for y-y and z-z for each direction is needed, as described in EN 1994-1-1 art. 5.2.1(3).

with:

N_Ed - design axial compression force

K₀ - calibration factor taken as 0,9

K_e,II - correction factor taken as 0,5

E_a - E-modulus of structural steel material

I_a - second moment of area of the structural steel for the considered bending plane

E_s - E-modulus of reinforcement material

I_s - second moment of area of the reinforcement for the considered bending plane (based on the 1st section, since reinforcement should be uniform)

E_c - E-modulus of the concrete material (= E_cm)

I_c - second moment of area of the un-cracked concrete section for the considered bending plane

E_c,eff - E-modulus of concrete material (long term effects)

|N_G,Ed|/|N_Ed|- ratio of normal force that is permanent (input value in composite setup)

φ_t - creep factor (input value in composite setup)

 

Depending on the value of α, the 2nd order effects are either needed or not.

If α ≥10, the 2nd order effects are not needed and initial bending moments M_Ed are increased only by adding a member imperfection moment M_imp.

• M_{Ed, mod} = M_Ed + M_imp

with:

M_imp = |N_Ed| x e₀

N_Ed - acting compression force

e₀ - member imperfection given by EN 1994-1-1 Table 6.5

In case the reinforcement ratio surpasses the upper limit of 6% we will still use buckling curve b, and a warning message is printed.

In case user modifies the buckling curves in the cross-section dialog which is different from the given table, warning is printed.

In use case user modifies the buckling curves in the cross-section dialog to either a₀ or d, which are not covered by table 6.5, an error is triggered and unity is set to 999.

If α <10, the 2nd order effects are needed. The initial bending moments are increased by adding a member imperfection moment M_imp and applying multiplication factors k₁ and k₂.

• M_{Ed, mod} = k₁ x M_Ed + k₂ x M_imp (for positive M_Ed)

• M_{Ed, mod} = k₁ x M_Ed - k₂ x M_imp (for negative M_Ed)

The multiplication factors k₁ and k₂ are given by EN 1994-1-1, 6.7.3.4(5) in which k₂ deviates from that equation by replacing the moment factor β (EN 1994-1-1 Table 6.4) by 1.

In case |N_Ed| > N_cr,eff a negative multiplication factor k is obtained, which is considered as error and unity is set to 999.

 

Nonlinear analysis with geometrical nonlinearity

This path is triggered in case a non-linear combination is checked and the “Geometrical nonlinearity” functionality is activated. All other cases are consider as linear analysis.

This is considered as a "real 2nd order analysis" or a "more accurate approach". In this case it is assumed, that the user has defined imperfections through nonlinear combination and thus the 2nd order effects are, by replacing the stiffnesses EI of the composite columns in the solver with the reduced stiffness EI_eff,II (EN 1994-1-1 eq. 6.42), already included in the internal forces during the analysis.

Depending on design internal forces present, the further compression and bending moment resistances(s) are based on interaction diagram.

 

In case of a tensile N_Ed, no second order effects are needed, note will be displayed and content of the output limited.