PDelta transformation affects on eigen analysis

[theaspekt]

Hello
I found that PDelta transformation affects on eigen analysis result. When I apply only gravity loads, the first period is increasing by 16%! And if I use linear transformation than the period is nearly the same. The questions is:
1) Am I right and Pdelta can really affect that much on natural periods?
2) Which period should I use as true period to define spectral acceleration?

[fmk]

typically if you have done everything correctly the dfference in eigenvalues after a gravity analysis for a normal building (in which typically there is almost 0 lateral displacements) is negligable .. suggest you check the direction of the gravity loads and look at the results of the gravity.

[Tas]

If you apply gravity loads and then perform an eigen-analysis, if your elements are defined through fiber sections it would be normal to obtain different eigen-values because the stiffness of the structure isn't the same comparing with the case that there are no loads at all in the structure...
When you use PDelta transformation then lateral displacements are not exactly zero, after the application of gravity loads and that may cause (slightly) different results (this also depends on the tolerance that you have defined in the test command and generally the accuracy of the analysis).

Furthermore, you have to be carefull to verify if the equilibrium condition is fulfilled in all directions (i.e to compare the applied loads with the base reactions to see if they are equal...) because sometimes this not obvious... (see for instance: http://opensees.berkeley.edu/community/ ... =2&t=61381) and if that happens then everything will be different...

[theaspekt]

My model is a 5-storey steel frame (h=3.5 m, total load ≈ 17200 kN). It consists of elastic beam-column elements joined with zerolength elements (using bouc-wen material) and equalDOF command. Notify in advance, that BW nonlinearity participates slightly in this effect, but it's input is not essential comparing with the PDelta transformation input.

1) eigen vectors, recorded by corresponded command, don't change, changes are only in periods got by 'eigen' command
2) Gravity loads are vertical as they should be, and results are sensible. Maximum horizontal displacement is 0.5 mm
3) base reaction is also sensible - sum of shear reactions equals zero, sum of vertical reactions equals total load.

[fmk]

just how large are your axial forces? .. the geometric term added to the stiffness matrix is a function of P/L.

to see if the BoucWen model is the culprit use an elastic material and see how it effects the eigenvalue comparision.

[theaspekt]

Approximatly axial force at 1 floor level is 17200/5=3440 kN in each column.
As I wrote above I found that it is PDelta transformation that affects. To find this I erased BW from the model and there were near the same big difference in values.
May be there is some command to show or write stiffness or element matrix? I tried 'printA' command, but OpenSees didn't recognise it.

[fmk]

printA requires a FullGeneral matrix

[theaspekt]

Frank, do you mean 'system FullGeneral'? Then OpenSees print "invalid command name <<printA>>".

[theaspekt]

ok, I found sensible reason. Total vertical load is 25 % of critical load (Euler's load), so stiffness is reduced by 25%, that's why the period changes.
The only question is - why this does not affect on eigenvectors?

[fmk]

on printA (you might not have the most upto date version of the .exe)
http://opensees.berkeley.edu/wiki/index.php/PrintA

if K is a factor of original that is simply seen:
matrix eigen value problem:
K 0 = lambda M 0 (K stifness, M mass, and 0 being matrix of eigenvectors)
clearly if modify K by a factor a only thing that change will be eigenvalues:
i.e. (a K) 0 = (a lambda) M 0