Keywords: stiffness reduction factors, geometric and material nonlinearities, design codes, EC2, ACI.

The assessment of second-order effects according to Eurocode 2 [1] can be done according to three methods: the general method; a method based on nominal stiffness; and a method based in nominal curvature. The first is based on a nonlinear analysis taking into account both geometric and material nonlinearities in which the effects of material plasticity, cracking, creep and shrinkage are taken into consideration through adequate constitutive laws. The other two are simplified methods that aim to approximate such behaviour.

In the nominal stiffness method, second-order effects are estimated through a set of expressions that amplify the first order moments according to the relationship between the axial load and the critical buckling load. The assessment of the buckling load is made taking into consideration the combined effect of geometric and material nonlinearity on the structural element by means of an effective flexural stiffness. Eurocode 2 [1] proposes expressions for the estimation of this flexural stiffness by means of reduction factors that for short term loading depend primarily on the element's slenderness, applied reduced moment and the concrete characteristic compressive strength. The reduction factors are, however, estimated for columns with equal end eccentricities which do not necessarily represent the actual moment distribution of a column.

This paper compares the expressions presented in Eurocode 2 [1] with numerical trials conducted on a series of columns over a range of eccentricities of both equal, equal and opposite, and opposite and one half of each other [2].

In redundant structures, Eurocode 2 [1] also states that allowance should be made for the effect of cracking and creep in adjacent members, such as beams, suggesting as a simplification one can consider that the sections are fully cracked throughout the element. Since geometric nonlinearity is not usually an effect present in these elements, the stiffness reduction factors mentioned before cannot be applied.

This paper presents a series of numerical trials that aim to evaluate the appropriate stiffness reduction in the ultimate limit state of beams [2]. To this end, several types of sections, end restraints, and mechanical reinforcement ratios are modelled.

1

European Committee for Standardization, "Eurocode 2: Design of concrete structures Part 1-1 General rules and rules for buildings", Brussels, CEN. EN 1992-1-1:2004.

2

A.C. Sousa, "Aspects on Nonlinear Geometric and Material Analysis of Three-Dimensional Framed Structures", in MSc Dept. of Civil Engineering, Faculty of Engineering, University of Porto, Portugal, 2009.

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