# Nonlinear Time History Analysis Using the Example of a Tower Structure with Prestressed Cables

### Technical Article

New

001569

24 April 2019

There can be numerous nonlinearities in a structural system. In order to model them realistically in a dynamic analysis, the RF-DYNAM Pro - Nonlinear Time Histoey add-on module was developed. To explain how the add-on module works, the procedure is described below with an example.

#### Structural System and Preliminary Studies

The structural system is a tower consisting of a pipe cross-section RO 355.6 x 10 with the steel strength S235. In the middle of the tower, it is supported by two prestressed cables and fixed at the footing. The prestress in the cables amounts to 100 kN.

Before the time history analysis can be started, a modal analysis is performed to analyse the dynamic behavior. Since a modal analysis is always linear, the effect of cables cannot be considered. The cables are replaced by linear trusses. However, the RF-DYNAM Pro add-on module offers options to model the behaviour of cables more realistically. Use the "Stiffness Modifications" function in the "Natural Vibration Cases" tab. This option allows you to change the geometric stiffness matrix used to determine the eigenvalues. Therefore, for this example, it is possible to consider the prestress of the cables by importing the relevant load case.

Considering the prestress significantly leads to higher natural frequencies compared to an analysis without taking into account the prestress, and to a realistic display of the natural frequency of the structure. It is important to know the natural frequencies so that you can convert the damping and understand the behaviour of the structure. The values for the two governing mode shapes are as follows:

 Mode Shape No. Angular Frequency ω [rad / s] Natural Frequency f [Hz] Effective Modal Mass Factor [-] 1 12.926 2.057 0.281 6 81.310 12.941 0.345

#### Input in RF-DYNAM Pro - Nonlinear Time History

The situation to be analysed is the action of a horizontal wind load with a size of 10 kN, which acts as a single load at the upper end of the tower. The motion of the wind is greatly simplified in this example and represented by means of a transient time diagram.

By taking into account all nonlinearities in the "Nonlinear Time History" add-on module, the characteristics of the cable are considered. This includes the failure of the cables under pressure and the influence of the prestress.

In this case, the "Nonlinear implicit Newmark analysis" is selected. For this solver, a sufficiently small time step is necessary to get precise results. For this purpose, a time step convergence study can be carried out. For this example a time step of 0.001 s was selected. Smaller time steps do not lead to more precise results.

Furthermore, the prestress of the cables is imported as "condition". The condition should act as a stationary state, which means that the prestress is present over the entire time course.

A Lehr's damping measure of 0.02 is assumed as damping. Since the implicit Newmark analysis requires the definition of the Rayleigh damping, Lehr's damping has to be converted. This happens within the program by defining the angular frequencies of the two governing mode shapes. The following formula is used here:

${\mathrm D}_{\mathrm i}\;=\;\frac{\mathrm\alpha}{2\;\cdot\;{\mathrm\omega}_{\mathrm i}}\;+\;\frac{\mathrm\beta\;\cdot\;{\mathrm\omega}_{\mathrm i}}2$

#### Evaluation of Results

Numerous functions are available to evaluate the results. On the one hand, the motion of the structural system can be displayed graphically, either for each saved time step, for the dynamic envelope, or as animation over the time course. On the other hand, it is possible to use the time course monitor for evaluation where you can select any node or member and so the results are visible over the time.

For the tower, the deformations and accelerations at the top node are governing. Based on the accelerations, the root mean square can be additionally determined. These results can be compared with the required values.

The axial force in the cables shows the effect of the prestress. The axial force starts at a value of 87.7 kN. As a result, both cables remain in the tension area all the time and do not fail.

Another option is to export the results into load cases (the results of single time steps are exported) or result combinations (the results of the dynamic envelopes are exported). By means of these results, further verifications can be carried out in the design modules.

#### Reference

 [1] Manual RF-DYNAM Pro. Tiefenbach: Dlubal Software, October 2018.