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Comparison of the first- and second-order analysis and the possible effects of the deformation on the analysis results
Posted on in MasterFrame

Modern design codes provide several analysis options to determine the structural deflections and internal forces for dimensioning. These codes define certain usable boundaries, but between them, it is more or less the choice of the engineer which method is the most suitable to the actual design situation.

In engineering practice, for building structures, the only analysis types that are likely to be used are first-order (linear static) analysis and second-order (P-delta static) analysis.

First-order (linear static) analysis

In first-order (linear static) analysis, the stiffness of the structure is assumed to be constant and unaffected by changes in the geometry of the structure when it is loaded.

The moment at the column base can be calculated as follows:

Second-order (P-delta static) analysis

In second-order analysis, the effective stiffness of the structure is changed by the action of the loads upon it. This means that the arising deformations have an influence on the internal forces.

The moment at the column base can be calculated as follows:

Comparison of the results

The results from the first- and second-order analysis can be significantly different if our structure is sensitive to these deformations.

Eurocode 3 defines a criterion which helps to determine the behaviour of our steel structures. In our previous blog post, we showed how you can classify the structure’s sensitivity to the effects of deformed geometry and determine the critical load factor. Click here to read more.

Basically Eurocode said, if the critical load factor (αcr) of the structure is higher than 10, then the structure is not sensitive to effects of deformed geometry; thus the second-order effects can be neglected.

Example

In our following example, we will show how the results change using first- and second-order analysis on three different slenderness structures.

We have a goal post with the following dimensions. The columns are fixed at the bottom and the beam is pinned. The vertical load is -100 kN on the top of the columns, and we have a 60 kN horizontal load. The cross-section of the column is changed in the three frame variations.

Frame 1 – analysis results

Comparing the analysis results from the three frame variations, we see that if our structure’s critical load factor is far from the Eurocode’s limit, then the differences between the first- and second order results are negligible (less than 3%). However, if we are under the limit, the differences became more and more significant and can not be neglected. Especially, if we are around 2-3, which is the normal value of a well restrained, optimised portal frame, where the second-order increment is higher than 50%.