Free Vibrations

1. Cable-Stayed Bridge

The lowest eigenfrequencies and their corresponding eigenmodes of a cable-stayed bridge are calculated. A global FE model of the structure, with shell and beam elements, has been established. Although the problem of cable-stayed structures is geometrically non-linear it is approximated by replacing the cable by beams with a given prestress.

Cable-stayed bridge: Mesh

Figure 61. Cable-stayed bridge: Mesh


The analysis with B2000++ is controlled with the MDL commands adir and case :

case 1
  analysis free_vibration
  nmodes 20
  ebc 1
end

adir
  case 1
end

The cases option of adir specifies which cases B2000++ should process (here: process case 1). In its turn, the case 1 description in the cases command specifies the analysis type and other options related to that particular type of analysis:

The analysis option of case indicates the type of analysis to be performed. free_vibration will launch the B2000++ free vibration analysis solver (see also comment below).
The nmodes option specifies the number of eigenmodes or eigenvalues to be computed. In the absence of a shift the smallest nmodes eigenvalues around 0.0 are computed.

By default B2000++ calculates the real symmetric eigenvalue problem with the Implicitly Restarted Lanczos Iteration variant of the Implicitly Restarted Arnoldi Iteration algorithm implemented in the ARPACK library. In addition, the model mass and modal stiffness for each mode are computed. Results are summarized below, printed with the b2print_modes tool (see Makefile, target print):

Number of computed free-vibration modes for case 1: 20
Mode Eigenvalue     Frequency      Omega          Stiffness      Mass          
   1 +7.2163123e+00 +4.2754102e-01 +2.6863195e+00 +1.2808003e+07 +1.7748682e+06
   2 +7.6339518e+00 +4.3973885e-01 +2.7629607e+00 +4.6736148e+07 +6.1221435e+06
   3 +1.6204356e+01 +6.4067241e-01 +4.0254635e+00 +2.9925187e+07 +1.8467372e+06
   4 +1.9129024e+01 +6.9609183e-01 +4.3736740e+00 +2.3560813e+07 +1.2316788e+06
   5 +1.9398214e+01 +7.0097254e-01 +4.4043404e+00 +2.3556654e+07 +1.2143723e+06
   6 +2.7513887e+01 +8.3482627e-01 +5.2453682e+00 +4.1424456e+07 +1.5055835e+06

Results can be rendered directly with the baspl++ viewer. Eigenmodes are visualized either with baspl++'s built-in modes viewing tool. The viewer script view.py produces the plot shown below:

Free vibration mode 10

Figure 62. Free vibration mode 10