PIAS Manual  2021
Program for the Integral Approach of Shipdesign
Motions: ship motions computation
With this module the ship motions can be predicted in the frequency domain using one of the following methods:

  • Jensen et al, for quick calculations in the concept design phase.
  • [to be implemented in 2021] A strip theory based method for more robust calculations.

Overview and applicability of the calculation methods

Please take the warning in the note of Overview and applicability of the calculation methods. — on the attitude with respect to empirical/statistical prediction methods — heartily.

Jensen et al

This method is based on Jensen, J. J., Mansour, A. E., & Olsen, A. S. (2004). Estimation of ship motions using closed-form expressions. Ocean Engineering, 31(1), 61-85. They developed closed-form semi-analytical expressions for the frequency response function for heave, pitch and roll and the vertical (relative) motion, velocity and acceleration of monohull ships.

The formulas predict the variation of the motions and accelerations with the main dimensions, frequency, heading and speed fairly accurately, with the exceptions of:

  • Heave is too small for wavelengths larger than the ship length.
  • Pitch is too large where the wave length is around the same as the ship length for Froude numbers larger than 0.2.
  • Roll is too large around the resonance frequency.


A strip theory method will be implemented in the summer of 2021.

Wave spectra

In order to calculate the probability of exceedance of a motion, a wave spectrum is required. Six wave spectra have been programmed, according to the formulation in Stansberg, C. T., Contento, G., Hong, S. W., Irani, M., Ishida, S., Mercier, R., ... & Kriebel, D. (2002). The specialist committee on waves final report and recommendations to the 23rd ITTC. Proceedings of the 23rd ITTC, 2, 505-551.

These spectra are discussed below.


The JONSWAP spectrum defines seas with finite fetch. It requires input of a peak frequency and a peak enhancement factor. The approximations for this spectrum are believed to be correct for a range of the peak enhancement factor between 1 and 7. When the wind speed U and the fetch F are known, the peak frequency of the spectrum can be calculated according to \(f_p=\frac{g\hat{F}^{-1/3}}{U}\) where \(\hat{F}=\frac{gF}{U^2}\).

Spectra of the generalized Pierson Moskowitz form

The other spectra are of the generalized Pierson Moskowitz or Bretschneider form. These spectra define fully developed seas. The spectra and their required input are listed below:

  • One-parameter Pierson-Moskowitz: requires input of either wind speed, peak frequency or significant wave height. When one of these values is inputted into the program, the corresponding values for the other two parameters are automatically updated.
  • Two-parameter Pierson-Moskowitz: requires input of the peak frequency and the significant wave height.
  • ISSC: requires input of the mean frequency and the significant wave height.
  • ITTC: requires input of significant wave height and one of the energy, peak, mean or zero-crossing periods.
  • Liu: requires input of the wind speed and the fetch.

Main menu

Input data for ship motion analysis

In this window the parameters which are needed for the various methods are given. The list below now only lists those for the Jensen et al method. For additional detail it is advised to consult the source publication.

  • Method: the chosen calculation method.
  • Name of the vessel: can be taken from the PIAS hullform if a project name is defined in Hulldef (see section /secref{hulldef,hulldef_input_general_maindimensions})
  • Starting frequency, frequency Interval and end frequency: The frequency range for which the motions are calculated is defined here. Should the frequency interval be zero or larger than (end frequency – starting frequency), the calculation shall only be performed for the starting and end frequency.
  • Main hullform parameters: length waterline, breadth, draft, block coefficient, waterline coefficient: if a PIAS hullform is available, these parameters can be derived from it.
  • Calculate roll RAO: choice to calculate roll or not. Requires additional input
  • Metacentric height: GM. If unknown, it can be estimated from the hullform. In this case VCG is assumed to be equal to VCB.
  • Natural roll period: if the natural period is not known it can be estimated. For this the IMO A.685(17) resolution is used, which uses the length, breadth, draft and metacentric height parameters as input.
  • Length ratio: The Jensen method simplifies the vessel into two prismatic beams for the roll calculation, with the same draft but different breadth and cross-sectional areas. The length ratio δ defines the length of these two prismatic beams, as seen in Figure 1. The value of the length ratio must be between 0 and 1, but not greater than the waterline coefficient.
    Length ratio delta (Jensen, Mansour, & Oslen, 2004)
  • Critical damping: With the Jensen et al method the viscose roll damping is approximately accounted for by adding a percentage of critical damping to the calculated inviscid damping. It is inputted as a percentage.
  • Wave spectrum: select one of the wave spectra mentioned in 1.2. Each spectrum requires different input as specified under section Wave spectra.
  • Number of speeds: a maximum of 50 can be selected.
  • Number of wave headings: a maximum of 100 can be selected.

Specify output

In these menus the points of interest and the specific output can be specified.

Specify points of interest

In this menu the so-called ‘points of interest’ can be defined. These are the points that output can be specified for in the next menu. A point is defined by a name, abbreviation and location coordinates. For the limited Jensen method, this location is only defined by a longitudinal position, the transverse position is always on the centerline and the vertical position is always on the waterline. For the strip theory based method all three coordinates are required.

Specify output at point of interest

In this menu the output for a point of interest is specified. First, after a new line has been created, a point of interest must be selected from a popup menu containing all of the points of interest defined in the previous menu. Alternatively, in the second column an abbreviation of one of the points of interest can be manually entered.

The third column opens a menu to select all the output for the point of interest. The heave, pitch and roll transfer function are always around the center of gravity of the vessel., which for the Jensen et al method is always at 0.5Lwl. The (relative) motions, velocities and accelerations are at the location specified for the point of interest.

As only three of the six motion transfer function can be calculated using the Jensen method, and information about the phase angles of these motions are not calculated, only vertical motions on the centerline of the vessel can be calculated. A more complete calculation, where also the transverse and longitudinal motions, velocities and accelerations are calculated will be introduced with the strip theory method.

If a wave spectrum is defined in menu /secref{motions,motions_main_input_vessel}, and only a single type of output is selected, then a probability of exceedance of a defined threshold value of the output type against the wave spectrum can be calculated. When this option is selected, a threshold value must be defined. An acceptable probability of exceedance can optionally be defined, which is used to clarify the graph output. The unit of the threshold value naturally depends on the selected type of output, and is shown in the cell comment when the cell is selected. The acceptable probability is given in percentages (0%-100%).

Calculate and print output

Using the parameters given in the first menu option, the output specified in the second menu option is calculated and printed as specified.


With this option design versions can be managed, with a mechanism as described in Data storage and backups.