Overview of second order effects (concrete column: EC2)

For 'isolated' columns and walls, EN1992-1-1 (EC2) allows for second order effects and member imperfections in a number of ways,

It specifies a minimum level of member imperfection along with a conservative value - see Clause 5.2 (7).
It provides for the additional moment due to slenderness (member buckling) using one of two methods. One method (the (Nominal) Stiffness Method) increases the first-order moments in the column using an amplifier based on the elastic critical buckling load of the member - see Clause 5.8.7.3. The second method (the (Nominal) Curvature Method) calculates the 'second-order' moment directly based on an adjustment to the maximum predicted curvature that the column section can achieve at failure in bending - see Clause 5.8.8.
The impact of the slenderness is increased or decreased depending upon the effective length factor for the member. For braced members this will be ≤ 1.0 and for unbraced (bracing) members it will be ≥ 1.0 see Clause 5.8.3.2.
Finally (and unrelated to second order effects), EC2 also requires consideration of a minimum moment based on the likelihood that the axial load cannot be fully concentric, see Clause 6.1 (4). The analysis moment (including the added effects of imperfection and second order effects) should not be less than this value.

Member imperfections (Clause 5.2 (7))

The imperfection moment is calculated using the eccentricity, e_i = l₀/400, and it is conservatively assumed that it increases the first-order moments irrespective of sign.

The imperfection moment is added to the analysis moment and if the member is slender a further second moment is added, if this total is < the 6.1(4) moment then the 6.1(4) moment applies.

In the case of the Stiffness Method the imperfection moment is added before the moment magnifier is applied. It is applied to both braced and bracing columns/walls.

Curvature Method (Clause 5.8.8)

This method is only applied to symmetrical, rectangular and circular sections and is equally applicable to columns and walls. The second-order moment, M₂ (= N_Ed e₂), is calculated but the resulting design moment is only used if it is less than that calculated from the Stiffness Method. It is applied in the same manner as that for the Stiffness Method to both braced and bracing columns.

Stiffness Method (Clause 5.8.7)

This method is applied to all columns and walls.

For braced columns the second-order moment M₂ is calculated from:

M₂ = M_e.1 x π² /(8 x (N_B /N_Ed - 1))

Where

M_e.1 = the maximum first-order moment in the mid-fifth

N_B = the buckling load of the column based on nominal stiffness and the effective length, hence

N_B = π² EI/l₀²

N_Ed = the maximum axial force in the design length

When a point of zero shear occurs inside the mid-fifth or does not exist in the member length, the value of M₂ is added algebraically to the first-order moments at the ends but only if this increases the first-order moment. At the mid-fifth position M₂ is always "added" in such a way as to increase the first-order mid-fifth moment.

When a point of zero shear occurs within the member length and is outside the mid-fifth, the second-order moments is taken as the greater of that calculated as above and that calculated as per Clause 5.8.7.3 (4) by multiplying all first-order moments by the amplifier,

1/(1 - N_Ed/N_B)

For bracing columns the second-order moments are calculated in the same way as braced columns except that in the determination of the amplifier, the buckling load is based on bracing effective lengths. These are greater than 1.0L and hence produce more severe amplifiers.

Second-order analysis

When second-order analysis is selected then both braced and bracing columns are treated the same as if first-order analysis were selected. If the second-order analysis is either the amplified forces method or the rigorous method then this approach will double count some of the global P-Δ effects in columns that are determined as having significant lateral loads. Also, when it is a rigorous second-order analysis there is some double counting of member P-δ effects in both braced and bracing columns.

Because global second order effects are introduced when second order analysis is selected, it is logical to set all members to "braced" so that only additional effects due to member slenderness are introduced.

Minimum moment (Clause 6.1 (4))

The minimum moment about each axis, M_min is calculated from Clause 6.1 (4).

For the specific circumstance of the moments from analysis being < the minimum moment for both axes, design codes generally require that the minimum moment need only be considered acting about one axis at a time. In Tekla Structural Designer the behaviour in this situaton is as follows:

For any section that is not circular or rectangular, the minimum moment is always applied about both axes, the rationale being:
- We believe the logical intention behind the requirement is that a cross section should be able to resist a minimum moment in every direction. However, if the local X and Y axes of a member are not aligned to the strongest and weakest axes (e.g. for an L-shape section) then applying the minimum in only one of the X or Y directions will not guarantee meeting the intention. Hence we consider applying the moment about both axes together for this contingency is a safe and conservative approach.
For rectangular (non-square) sections, the minimum moment is applied in the weaker direction.
For square and circular sections, the minimum moment is applied in the direction with the smaller analysis moment.