eigen valuses changes after applying gravity loads.why?

[madarshahian]

Dear silvia and friends
I found very amazing problem! In following simple model I need eigen values.
After applying gravity load I obtained eigen values which are different from initial eigen values before applying gravity loads. It is more interesting that when I use Linear transformer, another results were obtained!! I don’t know if there is a bug in PDelta transformer or the results are rational? If there are rational which eigen values give me natural periods of model?

Please take a look at my model.

wipe
# Steel three story, four bay frame
# Units: KN, m
#
# Static pushover analysis with Fiber Nonlinear beam column element
#
# ________ ________ ________ ________ _
# | | | | |
# | | | | | 3.2
# | | | | |
# |_ ____|________|____ ___|______ _| _
# | | | | |
# | | | | |
# | | | | | 3.2
# |________|________|______ _|__ ____| _
# | | | | |
# | | | | |
# | | | | | 3.2
# | | | | | _
# === === === === ===
# | 6.0 | 6.0 | 6.0 | 6.0 |
#

model BasicBuilder -ndm 2 -ndf 3
#supports
# tag X Y
node 1 0 0
node 2 6 0
node 3 12 0
node 4 18 0
node 5 24 0
#Nodes and # mass (D+0.2L)/9.81 KN/9.81=ton
set m 0.0001
#first storey
# tag X Y
node 6 0 3.2 -mass 19.26605505 19.26605505 $m
node 7 6 3.2 -mass 14.67889908 14.67889908 0.
node 8 12 3.2 -mass 14.67889908 14.67889908 0.
node 9 18 3.2 -mass 14.67889908 14.67889908 0.
node 10 24 3.2 -mass 19.26605505 19.26605505 0.
#second storey
# tag X Y
node 11 0 6.4 -mass 19.26605505 19.26605505 0.
node 12 6 6.4 -mass 14.67889908 14.67889908 0.
node 13 12 6.4 -mass 14.67889908 14.67889908 0.
node 14 18 6.4 -mass 14.67889908 14.67889908 $m
node 15 24 6.4 -mass 19.26605505 19.26605505 0.
#roof
# tag X Y
node 16 0 9.6 -mass 16.20795107 16.20795107 0.
node 17 6 9.6 -mass 14.67889908 14.67889908 0.
node 18 12 9.6 -mass 14.67889908 14.67889908 0.
node 19 18 9.6 -mass 14.67889908 14.67889908 0.
node 20 24 9.6 -mass 16.20795107 16.20795107 0.

# node DX DY RZ
fix 1 1 1 1
fix 2 1 1 1
fix 3 1 1 1
fix 4 1 1 1
fix 5 1 1 1
# Constraint nodes in each floor
equalDOF 8 6 1 2
equalDOF 8 7 1 2
equalDOF 8 9 1 2
equalDOF 8 10 1 2

equalDOF 13 11 1 2
equalDOF 13 12 1 2
equalDOF 13 14 1 2
equalDOF 13 15 1 2

equalDOF 18 16 1 2
equalDOF 18 17 1 2
equalDOF 18 19 1 2
equalDOF 18 20 1 2

## Define beam and column property variables KN/m^2
set E 200000000.0
set fy 240000.0
set b 0.03
# tag E fy post hardening
uniaxialMaterial Steel01 1 $fy $E $b

source Wsection.tcl

# Columns sections ... H...B==> 1.H400B 2.H300B 3.H280B 4.H240 5.H220 10.H340
# tag matID d tw bf tf nfdw nftw nfbf nftf
Wsection 1 1 0.4 0.0135 0.3 0.024 20 2 5 4
Wsection 2 1 0.3 0.0110 0.3 0.019 20 2 5 4
Wsection 3 1 0.28 0.0105 0.28 0.018 20 2 5 4
Wsection 4 1 0.24 0.0100 0.24 0.017 20 2 5 4
Wsection 5 1 0.22 0.0095 0.22 0.016 20 2 5 4
Wsection 10 1 0.34 0.0120 0.30 0.0215 20 2 5 4

# Columns sections ... H...B==> 6.H400A 7.H360A 8.H320A 9.H300A
# tag matID d tw bf tf nfdw nftw nfbf nftf
Wsection 6 1 0.39 0.011 0.3 0.019 20 2 5 4
Wsection 7 1 0.35 0.010 0.3 0.0175 20 2 5 4
Wsection 8 1 0.31 0.0090 0.3 0.0155 20 2 5 4
Wsection 9 1 0.29 0.0085 0.3 0.014 20 2 5 4

# Coordinate transformation
geomTransf PDelta 1
#geomTransf Linear 1

#DEFENITION of columns
#1th storey
# tag ndI ndJ nIntPts sec transfTag
element nonlinearBeamColumn 1 1 6 5 2 1
element nonlinearBeamColumn 2 2 7 5 1 1
element nonlinearBeamColumn 3 3 8 5 10 1
element nonlinearBeamColumn 4 4 9 5 1 1
element nonlinearBeamColumn 5 5 10 5 2 1
#2nd storey
element nonlinearBeamColumn 6 6 11 5 3 1
element nonlinearBeamColumn 7 7 12 5 3 1
element nonlinearBeamColumn 8 8 13 5 3 1
element nonlinearBeamColumn 9 9 14 5 3 1
element nonlinearBeamColumn 10 10 15 5 3 1
#3th storey
element nonlinearBeamColumn 11 11 16 5 4 1
element nonlinearBeamColumn 12 12 17 5 5 1
element nonlinearBeamColumn 13 13 18 5 5 1
element nonlinearBeamColumn 14 14 19 5 5 1
element nonlinearBeamColumn 15 15 20 5 4 1

#DEFENITION of beams
# tag ndI ndJ nIntPts sec transfTag
element nonlinearBeamColumn 16 6 7 5 6 1
element nonlinearBeamColumn 17 7 8 5 7 1
element nonlinearBeamColumn 18 8 9 5 7 1
element nonlinearBeamColumn 19 9 10 5 6 1

element nonlinearBeamColumn 20 11 12 5 8 1
element nonlinearBeamColumn 21 12 13 5 8 1
element nonlinearBeamColumn 22 13 14 5 8 1
element nonlinearBeamColumn 23 14 15 5 8 1

element nonlinearBeamColumn 24 16 17 5 9 1
element nonlinearBeamColumn 25 17 18 5 9 1
element nonlinearBeamColumn 26 18 19 5 9 1
element nonlinearBeamColumn 27 19 20 5 9 1

puts "[eigen 2]"


# Constant gravity loads
pattern Plain 1 Constant {
# node FX FY MZ
load 6 0.0 -49.5 0
load 10 0.0 -49.5 0
load 11 0.0 -49.5 0
load 15 0.0 -49.5 0

load 16 0.0 -16.5 0
load 20 0.0 -16.5 0
# -ele eleTag1? -type -beamUniform Wz? <Wx> w=1.1*(D+.25L)
eleLoad -ele 16 17 18 19 -type -beamUniform -53.625
eleLoad -ele 20 21 22 23 -type -beamUniform -53.625
eleLoad -ele 24 25 26 27 -type -beamUniform -53.625

}


# Create a recorder to monitor nodal displacements

#recorder Node roof.out analyze -time -node 7 -dof 1
recorder Node -time -file node18.out -node 18 -dof 1 disp

# Create recorders to monitor section forces and deformations
# at the base of the left column

recorder Node -time -file node1.out -node 1 -dof 1 reaction
recorder Node -time -file node2.out -node 2 -dof 1 reaction
recorder Node -time -file node3.out -node 3 -dof 1 reaction
recorder Node -time -file node4.out -node 4 -dof 1 reaction
recorder Node -time -file node5.out -node 5 -dof 1 reaction

##drift and displacement
recorder Drift -file drift.out -time -iNode 2 4 6 -jNode 4 6 8 -dof 1 -perpDirn 2
recorder Node -time -file displacement.out -node 3 5 7 -dof 1 disp

#integrator LoadControl 0.1 3 0.001 1
integrator LoadControl 1
test EnergyIncr 1.0e-6 10 1

algorithm Newton
numberer Plain
constraints Plain
system BandGeneral
puts "[eigen 3]"
analysis Static

# Perform the gravity analysis
analyze 1
puts "Gravity load analysis completed"

loadConst -time 0.0
puts "[eigen 3]"

[endryus]

With PDelta transformation you take into account the stiffness variation due to the loads. Obviously the eigenvalues change.
A reference:
http://www.csiberkeley.com/Tech_Info/11.pdf

andrea

[madarshahian]

thank you very much but please let me know your idea about my second question too,I want do pushover and I need eigen value in order to get performance point .using which eigen value is more rational?

[fmk]

thre is nothing wrong with your model .. it is nonlinear and tangent matrix is changing after gravity .. if you want them to be the same, use elastic materials and linear transformation ..
which eigenvalues you choose to use is up to you.