# Analysis of second order effect in concrete

## Abstract

This article deals with members and structures in which the structural behaviour is significantly

influenced by second order effects. The EN 1992-1-1 defines several methods for analysis of second order effects with axial load (general method, simplified method based on nominal stiffness, simplified method based on nominal curvature...). The comparison of general method and one simplified method is given and both methods are applied to the same example in order to compare the results.

## Introduction

When an member is subjected to an axial load combined with a moment, it will deflect. This deflection will increase the moment at any section in the element. This is normally referred to a second-order effects. This phenomenon is also commonly referred to as the “P-Δ effect“, because axial forces P will contribute a further bending moment and increase the lateral sway and deflection Δ.

The code EN 1992-1-1 provides four basic approaches for calculation of second order effect:

• A general method based on nonlinear analysis of the structures including geometric nonlinearity
• A method based on the nominal stiffness’s of the column and structure
• The moment magnification factor method, where the design moment including second order effect is obtained by multiplying the first order moment by a factor
• The nominal curvature method , in which ultimate deflection is estimated and from this, an estimate of the second order moment is made and added to first order moment.

Within this article, the calculation of second order effect is examined with two different method available in SCIA Engineer:

• The simplified method based on nominal curvature method (EN 1992-1-1, clause 5.8.8)
• General method (EN 1992-1-1, clause 5.8.2(2))

## The nominal curvature method

This simplified method for analysis second order effect should be used after linear calculation. This method is based on nominal curvature according to EN 1992-1-1, clause 5.8.8.

The second order effect may be ignored, if slenderness of the column is lesser than limit slenderness

## The general method

The general method is based on non-linear analysis, including geometric non-linearity i.e.

second order effects (PNL module in SCIA Engineer). Equilibrium and resistance for this method is verified in the deformed state, where deformations shall be calculated taking into account the relevant effects of cracking, non-linear material properties and creep.

The geometrical nonlinearity and initial deformation can be taken into account too.

Second order effects may be ignored if they are less than 10 % of the corresponding first order effects. This condition can be analysed in SCIA Engineer by stability calculation. It means that if critical load factors for all stability combinations are greater than 10, then the structure is thus not sensitive for second order effects.

## The comparison of results

The method described above was calculated for column of the single frame. The biggest moment in column B2 without second order effect and imperfection is in head of the column and the value is -69,14 kN/m. For general method, the bending moment in the head with second order effect and imperfection is -96,2 kN/m, it means 1,4 multiple increasing . For simplified method, the bending moment in the head of column with second order effect and imperfection is -135,32, it means 1,95 multiple increasing .

With using of batch analysis in SCIA Engineer it is possible to create P-Δ diagram.

## Conclusion

Within this article the calculation of second order effect according to EN 1992-1-1 has been described using two methods.

The simplified method based on nominal curvature produces a quite high value of the second order moment and very slender member(columns) is not possible to design with using this method.

The general method allows the engineer to calculate very slender members (columns) and using this method leads to more economical design.

## References

[1] EN 1992-1-1: 2004 Eurocode 2 : design of concrete structures – Part 1: General rules and rules for building

[2] R.S. Narayanan & A. Bebby : Designers guide to EN 1992-1-1 and EN 1992-1-2 Eurodcode2: design of concrete structures. General rules and rules for building and structural fire design.

[3] Diese Stützenuntersuchung wurde im Jahre 1965 an der TH Darmstadt durchgeführt

[4] Mehmel, A., Schwarz, H., Kasparek, K.-H. und Makovi, J.: Tragver-halten ausmittig beanspruchter Stahlbetondruckglieder. DafStb Heft 204. Verlag Wilhelm Ernst & Sohn, Berlin 1969

[5] Design of column. Part 1: Global and local setting, slenderness and internal forces – Theoretical background, SCIA 2012

Pavol Valach

05/02/2013