S Accounting for secondorder PA effects

Section 4 of EN 1998-1 requires taking into account second-order (P-A) effects in buildings, when for the vertical members of the storey, these exceed 10% of the first-order effects in the aggregate. The criterion is the value of the interstorey drift sensitivity coefficient, 9, defined for storey i as the ratio of the total second-order moment in storey i to the change in the first-order overturning moment in that storey:


Nm ; is the total gravity load at and above storey i in the seismic design situation, i.e. as determined according to Section 4.4.2.

Adi is the interstorey drift at storey i, i.e. the difference of the average lateral displacements at the top and bottom of the storey, d; and d-: _if linear elastic analysis is

used on the basis of the design response spectrum, (i.e. the elastic spectrum for 5% damping divided by the behaviour factor q), then the values of the displacements to be used for di and di_1 are those from the analysis multiplied by the behaviour factor q; the value of the interstorey drift is determined at the centre of mass of the storey (at the master node, if one is used). 8 Vm i is the total seismic shear at storey i. • h; is the height of storey i.

Second-order effects may be neglected, provided that the value of 6>; does not exceed 0.1 in any storey; however, they should be taken into account for the entire structure, if at any storey the value of Qi exceeds 0.1. If the value of 0t does not exceed 0.2 at any storey, Section 4 of EN 1998-1 allows P-A effects to be taken into account approximately without a second-order analysis, by multiplying all first-order action effects due to the horizontal component of the seismic action by 1/(1 - Although it is the value 9i of the individual storeys that can be used in this amplification, the use for the entire structure of the maximum value of 6l in any storey is safe-sided and maintains force equilibrium in the framework of first-order analysis. In the rather unlikely case that a value of 9i exceeds 0.2 in any one of the storeys, an exact second-order analysis is required. This analysis may be performed with the modelling described in the next paragraph for buildings without rigid diaphrams.

If the vertical members connect floors considered as rigid diaphragms, P-A effects can be accounted for sufficiently according to the previous paragraph. If there are no such floors, or if floors cannot be taken as rigid diaphragms, then P-A effects may be considered on an individual column basis, by subtracting from the column elastic stiffness matrix its linearized geometric stiffness matrix. If the analysis is elastic on the basis of the design response spectrum, the linearized geometric stiffness matrix of each column should be multiplied by the behaviour factor q, to account for the fact that P-A effects should be computed for the full inelastic deformations of the structure and not for the elastic ones which incorporate division by the behaviour factor q. Within the framework of elastic analysis, column axial forces in the geometric stiffness matrix may be considered as constant and equal to the value due to the gravity loads included in the seismic design situation according to Section 4.4.2.

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