modeling a simply supported bridge complex eigenvalue

[assimi]

I tried to model a simply supported bridge, but I can't reproduce the results in the literature. The description of the truss is as follows,
"To demonstrate the flexibility of the approach, a simply supported bridge is optimized for its weight minimization with several cases of frequency constraints. Members on the lower chord are represented by beam elements with fixed rectangular cross sections B = 8 cm and H = 5 cm. Others are modeled as bar elements with initial sectional areas A = 1 cm2 . Young’s modulus is E = 2.1 × 1011 Pa, and the material density is ρ = 7800 kg/m3 for all elements. The initial configuration of the structure is shown in Fig. 6. A nonstructural mass m = 10 kg is attached at each of the nodes on the lower chord. The natural frequencies of the truss are about 20, 40 and 60" with some variations in the third or fourth floating point.
The schematic of the truss is depicted in : https://image.ibb.co/fkBQCa/Ground_Structure.png
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First I modeled all elements as bar elements and it resulted in natural frequencies as "20.35 40.00 60.27". I believe that It came to a correct K matrix but the M matrix somehow was wrong. I surveyed the literature and I found out that the lower chord elements should be beam elements and I modified my model. But it still gives me very wrong Frequencies that are mentioned at the end. I also supposed that Iz in the beam element command of opensees should be (bh/12)*(b^2+h^2) and cosistent mass matrix as (rho*cross-section). I will appreciate if your help me to modify the model to repeat the literature results.

My tcl code is as follows:
model BasicBuilder -ndm 2 -ndf 3
node 1 0.0000 0.0000
node 2 1.0000 0.0000
node 3 1.0000 0.9392
node 4 2.0000 0.0000
node 5 2.0000 1.3270
node 6 3.0000 0.0000
node 7 3.0000 1.5063
node 8 4.0000 0.0000
node 9 4.0000 1.6086
node 10 5.0000 0.0000
node 11 5.0000 1.6679
node 12 6.0000 0.0000
node 13 6.0000 1.6086
node 14 7.0000 0.0000
node 15 7.0000 1.5063
node 16 8.0000 0.0000
node 17 8.0000 1.3270
node 18 9.0000 0.0000
node 19 9.0000 0.9392
node 20 10.0000 0.0000
fix 1 1 1 1
fix 20 1 1 1
uniaxialMaterial Elastic 1 2.1e+11
mass 1 10 10 0
mass 2 10 10 0
mass 4 10 10 0
mass 6 10 10 0
mass 8 10 10 0
mass 10 10 10 0
mass 12 10 10 0
mass 14 10 10 0
mass 16 10 10 0
mass 18 10 10 0
mass 20 10 10 0
geomTransf Linear 1
element elasticBeamColumn 1 1 2 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 2 1 3 0.0002983800 1 -rho 2.33 -cMass 1
element Truss 3 2 3 0.0001109800 1 -rho 0.87 -cMass 1
element elasticBeamColumn 4 2 4 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 5 3 4 0.0001009100 1 -rho 0.79 -cMass 1
element Truss 6 3 5 0.0002595500 1 -rho 2.02 -cMass 1
element Truss 7 4 5 0.0001261000 1 -rho 0.98 -cMass 1
element elasticBeamColumn 8 4 6 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 9 5 6 0.0001197500 1 -rho 0.93 -cMass 1
element Truss 10 5 7 0.0002426400 1 -rho 1.89 -cMass 1
element Truss 11 6 7 0.0001358800 1 -rho 1.06 -cMass 1
element elasticBeamColumn 12 6 8 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 13 7 8 0.0001477100 1 -rho 1.15 -cMass 1
element Truss 14 7 9 0.0002564800 1 -rho 2.00 -cMass 1
element Truss 15 8 9 0.0001129500 1 -rho 0.88 -cMass 1
element elasticBeamColumn 16 8 10 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 17 9 10 0.0001319900 1 -rho 1.03 -cMass 1
element Truss 18 9 11 0.0002921700 1 -rho 2.28 -cMass 1
element Truss 19 10 11 0.0001000400 1 -rho 0.78 -cMass 1
element elasticBeamColumn 20 10 12 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 21 10 13 0.0001319900 1 -rho 1.03 -cMass 1
element Truss 22 11 13 0.0002921700 1 -rho 2.28 -cMass 1
element Truss 23 12 13 0.0001129500 1 -rho 0.88 -cMass 1
element elasticBeamColumn 24 12 14 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 25 12 15 0.0001477100 1 -rho 1.15 -cMass 1
element Truss 26 13 15 0.0002564800 1 -rho 2.00 -cMass 1
element Truss 27 14 15 0.0001358800 1 -rho 1.06 -cMass 1
element elasticBeamColumn 28 14 16 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 29 14 17 0.0001197500 1 -rho 0.93 -cMass 1
element Truss 30 15 17 0.0002426400 1 -rho 1.89 -cMass 1
element Truss 31 16 17 0.0001261000 1 -rho 0.98 -cMass 1
element elasticBeamColumn 32 16 18 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 33 16 19 0.0001009100 1 -rho 0.79 -cMass 1
element Truss 34 17 19 0.0002595500 1 -rho 2.02 -cMass 1
element Truss 35 18 19 0.0001109800 1 -rho 0.87 -cMass 1
element elasticBeamColumn 36 18 20 0.0040000000 2.1e+11 2.9666e-6 1 -mass 31.20 -cmass 1
element Truss 37 19 20 0.0002983800 1 -rho 2.33 -cMass 1
puts "eigen values: [eigen -fullGenLapack 3]"

Result of the code:
FullGenEigenSolver::solve() - the eigenvalue 9 is complex with magnitude 2.41214e+15
FullGenEigenSolver::solve() - the eigenvalue 10 is complex with magnitude 2.41214e+15
FullGenEigenSolver::solve() - the eigenvalue 13 is complex with magnitude 1.90904e+08
FullGenEigenSolver::solve() - the eigenvalue 14 is complex with magnitude 1.90904e+08
FullGenEigenSolver::solve() - the eigenvalue 16 is complex with magnitude 9.83875e+07
FullGenEigenSolver::solve() - the eigenvalue 17 is complex with magnitude 9.83875e+07
FullGenEigenSolver::solve() - the eigenvalue 20 is complex with magnitude 8.9533e+06
FullGenEigenSolver::solve() - the eigenvalue 21 is complex with magnitude 8.9533e+06
FullGenEigenSolver::solve() - the eigenvalue 25 is complex with magnitude 842310
FullGenEigenSolver::solve() - the eigenvalue 26 is complex with magnitude 842310
eigen values: -3.007715e+15 -2.412142e+15 -2.412142e+15

[selimgunay]

Did you make sure all your units are consistent? You should calculate Iz as bh^3/12 and not as (bh/12)*(b^2+h^2).

[assimi]

Dear selimgunay, All units are consistent and in SI. I modified the code as you suggested and replaced the Iz 2.9666e-6 with 8.3333e-07, but no improvement in the result can be seen. The output of the code is as follows,
FullGenEigenSolver::solve() - the eigenvalue 9 is complex with magnitude 1.38843e+15
FullGenEigenSolver::solve() - the eigenvalue 10 is complex with magnitude 1.38843e+15
FullGenEigenSolver::solve() - the eigenvalue 13 is complex with magnitude 1.89559e+08
FullGenEigenSolver::solve() - the eigenvalue 14 is complex with magnitude 1.89559e+08
FullGenEigenSolver::solve() - the eigenvalue 16 is complex with magnitude 9.71949e+07
FullGenEigenSolver::solve() - the eigenvalue 17 is complex with magnitude 9.71949e+07
FullGenEigenSolver::solve() - the eigenvalue 21 is complex with magnitude 9.00654e+06
FullGenEigenSolver::solve() - the eigenvalue 22 is complex with magnitude 9.00654e+06
FullGenEigenSolver::solve() - the eigenvalue 25 is complex with magnitude 791442
FullGenEigenSolver::solve() - the eigenvalue 26 is complex with magnitude 791442
eigen values: -2.771764e+15 -1.388434e+15 -1.388434e+15

[assimi]

It got solved, there is a problem in support nodes and it's ambiguously mentioned in the literature.

[selimgunay]

Great! Let us know if you have other problems or questions.

[ahmadbsr]

assimi wrote:
> It got solved, there is a problem in support nodes and it's ambiguously
> mentioned in the literature.
how did you fix it? i have same problem