# How to analyse a beam considering P-delta effect?

I want to do some analysis of an axially loaded simply supported beam considering P-delta effect. I set up a non-linear load case and apply axial load to it. Subsequent analysis of live load and dead load of the beam is done linearly using stiffness matrix from the non-linear case. However, the non-linear analysis results in the beam hogging upwards, instead of sagging downwards, which mitigate effects of live load and dead load, resulting in a reduced bending moment on the beam.

I am not sure if above analysis is done correctly, or is there a way to force the beam to sag downwards in the non-linear case? Why does it cause beam hogging rather than sagging given that the beam is doubly symmetric with axial load applied to its centroid? I am using SAP 2000 v18 software for above analysis.

TL;DR - Your current results are incorrect and unsafe to use. Carry out the analysis again, including all load in the P-delta analysis, not just the axial load.

If you apply axial load "P", alone, to a perfectly straight beam, then there is no "delta" for the P-delta effect to occur.

There are various methods for creating an initial "delta"; I have never used SAP 2000 so have no idea whether there is a built-in method, or whether you have to do it manually to some extent. In software I have used, I would carry out an eigenvalue analysis, pick the first mode-shape, and use it with a code-specified magnitude for imperfections as the initial shape for my P-delta analysis.

The issue with this is selecting an appropriate mode shape. If you pick a hogging mode shape, then P-delta will tend to increase the hogging; if you pick a sagging mode shape then it will increase sagging. Numerically, for a perfectly straight beam, the software could easily choose either hogging or sagging, as they are both as numerically valid as each other.

We now come to your case. An important thing to learn about non-linear analysis is that you cannot carry out linear superposition - it isn't valid. You need to have all loads in the non-linear case.

Going back through the previous methodology, if you carry out an eigenvalue analysis with axial load and with transverse load causing sagging, then the mode shape will be a sagging one (caused by the transverse load). In the P-delta analysis, again including all load, the transverse load will increase the delta still further, giving an even greater P-delta effect.

Your current results showing the axial load reducing the bending moment are incorrect and unsafe to use.

• "Numerically, for a perfectly straight beam, the software could easily choose either hogging or sagging, as they are both as numerically valid as each other." For an eigenvalue analysis, they are always "both as numerically valid as each other" whether or not the beam is perfectly straight. The sign of the eigenvector is arbitrary, unless your software has some way to choose it in the input - for example by prescribing the displacement of one degree of freedom in the vector as either positive or negative. Jun 21, 2017 at 13:29
• @alephzero - I'll be honest, my understanding of the mathematical theory behind eigenvalue analysis is lacking. But if I had a beam with an initial sag imperfection I would think there was an error if the first eigenvalue mode shape was not sagging. This is based on experience of doing this sort of analysis in finite element software. Jun 21, 2017 at 13:44

In a nonlinear analysis, the reason the beam displaced "the wrong way" is probably because you didn't define all the loads acting on it.

For example, if the beam was horizontal and you included its self-weight as well as the axial load, it would be very strange if it deflected upwards rather than downwards.

I'm not a civil engineer, but as a general principle, trying to use the stiffness from a nonlinear analysis as the basis for further "pseudo-linear" analyses seems theoretically "just plain wrong" unless it is justified by some civil engineering codes, or assumptions about the relative sizes of the loads, that I don't know about.