Modal Analysis of a cantilever beam in SAP2000 vs theory

[edufranca13]

Hello,

first of all the model I will present here is in Newton and meters (SI). Well, I am trying to solve a modal analysis myself and compare with the results of SAP2000. In Analyze > Set Analysis Option > Solver Options > Analysis Case Name > MODAL I can get the results of the stifness matrix and the mass matrix of my model. Let's go to the problem:

Modal Analysis of a concrete cantilever beam
Length = 5 m
E = 2,486E+10 (default for 4000psi concrete)
G = E/2(1+ni) = 1,036E+10
Dimensions: b = 0,2 m and h = 0,3 m therefore I = 2xE-4 m4 and I = 4,5E-4 m4, J = 4,695E-4
Mass source: concetrated load at the top of the structure of 100 N

After run analysis I get the following results:

Stifness matrix:

1 1 475038.127376544
2 1 0.000000000000000E+000
3 1 0.000000000000000E+000
4 1 0.000000000000000E+000
5 1 -1187595.31844136
6 1 0.000000000000000E+000
2 2 1062742.46799371
3 2 0.000000000000000E+000
4 2 2656856.16998427
5 2 0.000000000000000E+000
6 2 0.000000000000000E+000
3 3 298266939.417185
4 3 0.000000000000000E+000
5 3 0.000000000000000E+000
6 3 0.000000000000000E+000
4 4 8879142.47058955
5 4 0.000000000000000E+000
6 4 0.000000000000000E+000
5 5 3963211.42749402
6 5 0.000000000000000E+000
6 6 972538.429347940

Mass matrix:

1 1 10.1971619222242
2 2 10.1971619222242
3 3 10.1971619222242
4 4 0.000000000000000E+000
5 5 0.000000000000000E+000
6 6 0.000000000000000E+000
If I compare this results with theory here is small differences (someone can tell me why?) but nothing that would change my results, as you can see:

k11 = 12EI/l³ = 477312
k22 = 12EI/l³ = 1073952
k33 = EA/l = 298320000
k51 = 6EI/l² = 1193280
k42 = 6EI/l² = 2684880
k44 = 4EI/² = 8949600
k55 = 4EI/l = 3977600
k66 = GJ/l = 972804

The lumped mass matrix is pretty clear 100 N/g.

Finally, when I try to solve the eigenvalue problem which is [K-omega²M] = 0 I get to following equation:
(475038.1-10.19x²)(1062742.4-10.19x²)(298266939.4-10.19x²)*8879142.4*3963211.4*972538.4-(-1187595.3)²*(2656856.1)²*(298266939.4-10.19x²)=0
Which gives me three non-negative roots:

x1 = 215.912
x2 = 322.944
x3 = 5410.23

but in SAP2000 I get to:
x1 = 108,1
x2 = 162,04
x3 = 5408,3

which, as you can see, with exception of the third value, half of the values I get.

Can someone please explain me where I making a mistake? Sorry if there's already some post like this one, but I didn't find anything.

Thank you very much

Bye