MAXSURF | MOSES | SACS | OpenWindPower - Tow Fatigue with SACS

Introduction

A tow fatigue analysis may be performed in SACS by using the Tow and Fatigue programs. This post describes the methods used to generate a transfer function from the Tow program and the response spectrum and damage in the Fatigue program. Sample analyses have been provided to illustrate some of the methods described below.

Response Amplitude Operators

A response amplitude operator (RAO) represents a set of responses to a given range of wave frequencies. In Tow, Motion RAOs may be entered as displacements, velocities, accelerations, or g's. Each RAO point is defined by a period as well as an amplitude and phase for each of the six degrees of freedom of the system. An example set of Displacement RAO Amplitudes and Phases are shown below:

Displacement RAO Amplitude

Displacement RAO Phase

The RAO is defined with respect to a reference point. In SACS, the RAO reference point is entered as the Center of Motion on the tow option line (TOWOPT). The Center of Motion does not necessarily refer to the center of motion of the vessel. The Center of Motion is the point about which the rotational inertial loads will be calculated. The reference point is determined by the hydrodynamic analysis program and is typically defined as the center of gravity of the cargo.

Note that the moving the reference point will only modify the translational RAOs, the rotational RAOs are the same regardless of the reference point location.

When performing a Tow analysis, the RAO's may be used to generate responses in a few different ways. The user may define a series of waves and phases which are used to generate responses based upon the RAO information using the WAVDEF line. SACS will solve the inertial loading for each phase. For instance, in the attached example, there are 8 directions with 36 wave definitions and 4 phases each for a total of 1,152 load cases. Each load case is visible in the towoci file that is generated by the analysis.

Note that WAVDEF lines do not necessarily have to match the RAO periods. Responses will be interpolated between existing points.

Alternatively, the LCRAO option may be used to automatically generate load cases based upon the RAO data. If the NPT option is selected, a unit wave with the selected number of phases (points) for each RAO period is used to generate load cases. The output will be similar to the WAVDEF line where each load case corresponds to a particular wave period and phase. For instance, in the attached example, there are 8 directions with 36 RAO periods and 4 points each for a total of 1,152 load cases. The R+I option may be used to generate a real and imaginary load case for each period which are 90 degrees out of phase with each other. For instance, in the attached example, there are 8 directions with 36 RAO periods and 2 cases each for the real and imaginary components for a total of 576 load cases.

Phase angles used for NPT option

$\phi=\frac{2\pi}{NPT}$

Phase angles used for R+I option

$NPT=1,2$

$\phi=\frac{-(NPT-1)\pi}{2.0}$

where $NPT$ is the number of points

Note that the real and imaginary load cases are solved as 0 and -90 deg

Inertial Loading

When performing a tow analysis with RAO's, the RAO's are first converted into displacement RAO's. This allows SACS to calculate the maximum pitch and roll angles. The following equations are used to convert RAO's to displacement RAO's:

If input is gravity (translation only): $A_d = \frac{-A_g g}{\omega^2}$

If input is acceleration: $A_d = \frac{-A_a}{\omega^2}$

If input is velocity: $A_d = \frac{A_v}{\omega}$

where

$A$ is the RAO amplitude

and

$\omega$ is the angular frequency

After the RAO's are converted into displacement RAO's, the maximum pitch and roll are calculated:

$\theta_{Roll} = A_{Roll} H \cos(\phi + \phi_{Roll})$

$\theta_{Pitch} = A_{Pitch} H \cos(\phi + \phi_{Pitch})$

where

$A$ is the RAO Amplitude

$H$ is the Wave Height

and

$\phi$ is the Phase Angle

The accelerations for each degree of freedom can then be calculated:

$\alpha_{Surge} = -A_{Surge} H \omega^2 \cos(\phi +\phi_{Surge})+g\sin(\theta_{Pitch})$

$\alpha_{Sway} = -A_{Sway} H \omega^2 \cos(\phi +\phi_{Sway})+g\sin(\theta_{Roll})$

$\alpha_{Heave} = -A_{Heave} H \omega^2 \cos(\phi +\phi_{Heave})$

$\alpha_{Roll} = -A_{Roll} H \omega^2 \cos(\phi +\phi_{Roll})$

$\alpha_{Pitch} = -A_{Pitch} H \omega^2 \cos(\phi +\phi_{Pitch})$

$\alpha_{Yaw} = -A_{Yaw} H \omega^2 \cos(\phi +\phi_{Yaw})$

where

$A$ is the RAO amplitude

$H$ is the wave height

$\omega$ is the angular frequency

and

$\phi$ is the phase angle

Inertial loading for each element can then be calculated from the accelerations using the equations provided in section 3.1 of the Tow manual.

Transfer Function Generation

When the inertia load cases are solved, SACS can generate a transfer function for each connection location. These transfer functions correspond to the solved load cases from the tow analysis. If the R+I option is selected, the stresses are calculated as follows:

$\sigma=2\sqrt{\sigma_R^2 + \sigma_I^2}$

note: that the factor of 2 is used to double the stress so that the stress range represents the max and min of the response.

Otherwise, the transfer function is calculated using the maximum and minimum response from the phases that were chosen.

An example of a transfer function is shown below:

Transfer Function

Fatigue Environment

The transfer function just represents the behavior of the structure in response to different waves. The fatigue environment must be provided to determine fatigue damage. In SACS, the user may define the fatigue environment in a number of ways.

Wave Spectra

SACS provides a number of standard wave spectra such as JONSWAP, Pierson-Moskowitz, and Ochi-Hubble. Each spectrum has been developed to represent certain environments and should be used accordingly.

The spectrum type, fraction of design life, significant wave height and dominant period are entered on the WSPEC line along with any other data that is required for that type of spectrum. For each direction a number of periods and significant wave heights will be used to represent the fatigue environment. An example fatigue environment example is shown below:

Peak Period	Significant Wave Height
Peak Period	0-2	2-4	4-6	6-8	8+
0-2	0.02	0.03	0.04	0.02	0.01
2-4	0.05	0.06	0.08	0.04	0.02
4-6	0.05	0.05	0.10	0.04	0.03
6-8	0.04	0.04	0.12	0.05	0.02
8-12	0.02	0.02	0.05	0.01	0.01

The wave spectra information would be entered in SACS using the lines below:

FTCASE   1      0.35       1.0 SPC                                                                                      
WSPEC   1 PM   1.00    1.0   0.02                                                                                       
WSPEC   1 PM   1.00    3.0   0.05                                                                                       
WSPEC   1 PM   1.00    5.0   0.05                                                                                       
WSPEC   1 PM   1.00    7.0   0.04                                                                                       
WSPEC   1 PM   1.00   10.0   0.02                                                                                       
WSPEC   1 PM   3.00    1.0   0.03                                                                                       
WSPEC   1 PM   3.00    3.0   0.06                                                                                       
WSPEC   1 PM   3.00    5.0   0.05                                                                                       
WSPEC   1 PM   3.00    7.0   0.04                                                                                       
WSPEC   1 PM   3.00   10.0   0.02                                                                                       
WSPEC   1 PM   5.00    1.0   0.04                                                                                       
WSPEC   1 PM   5.00    3.0   0.08                                                                                       
WSPEC   1 PM   5.00    5.0   0.10                                                                                       
WSPEC   1 PM   5.00    7.0   0.12                                                                                       
WSPEC   1 PM   5.00   10.0   0.05                                                                                       
WSPEC   1 PM   7.00    1.0   0.02                                                                                       
WSPEC   1 PM   7.00    3.0   0.04                                                                                       
WSPEC   1 PM   7.00    5.0   0.04                                                                                       
WSPEC   1 PM   7.00    7.0   0.05                                                                                       
WSPEC   1 PM   7.00   10.0   0.01                                                                                       
WSPEC   1 PM  10.00    1.0   0.01                                                                                       
WSPEC   1 PM  10.00    3.0   0.02                                                                                       
WSPEC   1 PM  10.00    5.0   0.03                                                                                       
WSPEC   1 PM  10.00    7.0   0.02                                                                                       
WSPEC   1 PM  10.00   10.0   0.01

Note that the fraction of design life will be multiplied by the frequency of occurrence on the WSPEC line. A warning message will be printed if the total design life fractions do not sum very nearly to unity.

If the provided spectra are not adequate, a user-defined spectra may be defined using SPEC lines. The wave spectra for each environment can then be defined using the WSPEC lines and specifying the user-defined spectrum.

Scatter Diagram

Alternatively, a scatter diagram may be used to define the fatigue environment using the SCATD, SCOFAC, SCWAV, and SCPER lines.

The same spectral data would be entered in SACS using the wave scatter diagram lines below:

FTCASE   1 .35      1.0        SPC                                                                                      
SCATD D                        PM                                                                                       
SCWAV          1.0   3.0   5.0   7.0  10.0                                                                              
SCPER    1.0  0.02  0.03  0.04  0.02  0.01                                                                              
SCPER    3.0  0.05  0.06  0.08  0.04  0.02                                                                              
SCPER    5.0  0.05  0.05  0.10  0.04  0.03                                                                              
SCPER    7.0  0.04  0.04  0.12  0.05  0.02                                                                              
SCPER   10.0  0.02  0.02  0.05  0.01  0.01

Response Spectrum

With the transfer function and the fatigue environment spectra, the statistical cyclic stress range can be calculated:

$\sigma_{RMS_i} = \sqrt{\int_0^{\infty}{H_i^2(f)S_h(f)df}}$

where

$H_i$ is the stress transfer function for point $i$

$S_h$ is the wave spectrum

and

$f$ is the frequency

Fatigue Damage

For every RMS stress there exists an average time, $T_z$ , between zero crossings with a positive slope for a stationary Gaussian process with zero mean. This period called the zero crossing period is given by:

$T_z = \frac{\sigma_{RMS_i}}{\sqrt{\int_0^{\infty}{f^2H_i^2(f)S_h(f)df}}}$

The expected number of cycles can then be calculated:

$n=\frac{mL}{T_z}$

where

$m$ is the fraction of the design life

$L$ is the design life

and

$T_z$ is the zero crossing period

Standard references show that if a linear system is excited by a Gaussian random process, then the response will also be a Gaussian process, thus in our case, having assumed system linearity and Gaussian excitation, the stress time histories are Gaussian (at least to the order of our approximations). We further assume that each response is narrow banded, that the spectral density of the response is significant only over a narrow range of frequencies. Under these conditions the stress range is a Rayleigh distributed random variable having a probability density function given by:

$p(s) = \frac{s}{\sigma_{RMS}^2} \exp\left[-\frac{s^2}{2\sigma_{RMS_i}^2}\right]$

where

$s$ is the stress range

and

$\sigma_{RMS}$ is the RMS value of the stress range

The damage for that stress range can then be calculated:

$D = \frac{n}{\sigma_{RMS_i}^2}\int_0^{\infty}\frac{s}{N_F(s)}\exp\left[-\frac{s^2}{2\sigma_{RMS_i}^2}\right]ds$

And the total damage can be calculated by summing the expected damages from each seastate.

SACS Example

Attached are three different Tow Fatigue Analyses, each with the same parameters except that the load cases generation method is different. The LCRAO-R+I analysis is solved using the LCRAO line with the R+I option, the LCRAO-NPT analysis is solved using the LCRAO line with the NPT option and the WAVDEF analysis is solved using the WAVDEF line.

The transfer functions for brace 2013 - 1000 at the top position for fatigue load case 1 for each method are shown below:

Transfer Function Comparison

The WAVDEF and LCRAO-NPT analyses yield the exact same transfer function. This is because the same waves and phases were selected for both methods. Modifying the selected waves and/or number of points will modify the results (increasing the number of waves and number of points will typically yield better results). Note that the transfer function is constant from 0 to the lowest frequency at 0.025 1/s (40 s) since there is no data before that point.

The LCRAO-R+I method envelops the WAVDEF and LCRAO-NPT methods. Increasing the number of points will cause those methods to approach the LCRAO-R+I method. It is recommended that the LCRAO-R+I method be used as it is more efficient in terms of load cases analyzed and more accurate.

communities.bentley.com/.../SACS_5F00_Examples.zip

communities.bentley.com/.../Tow-Acceleration-Verification.zip