Result Post-Processing

1. Introduction

Results from static and dynamic analyses are stored on results files. The contents on the result files can be retrieved, further analysed and results presented by means of the program module OUTMOD.

The results from static analysis and most of the dynamic analysis can be retrieved and presented as tables or graphs. Some results require further analysis. In the following these analyses are described.

2. Curvature Time Series Calculated from Dynamic Nodal Displacements

The curvature at a nodal point, i, can be calculated, provided that the position time series for points

i-1, i, and i+1 are stored. Thus, i can not be an end node of a line.

Referring to Figure 1 the calculation is as follows:

  1. Define a plane P through the 3 points.

  2. Specify a second order polynomial in P through the points. \(\mathrm {y=Ax^2+Bx}\)

  3. Calculate the line curvature, \(\mathrm {\kappa}\) , as the second derivative of the polynomial at i.

\[\kappa=\frac{\partial ^2y}{\partial x^2}(1+(\frac{\partial y}{\partial x}²))^\frac{3}{2}\]
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Figure 1. The curvature at a nodal point i

3. Support Force Calculations

(to be completed)

4. Element Angle Time Series

The element angle can be calculated relative to

  • a global axis \(\mathrm {(x,y\textrm{or}z)}\)

  • a support vessel axis \(\mathrm {(x_v,y_v\textrm{or}z_v)}\)

  • another element

In all cases the angle is calculated by:

\[\alpha =\cos^{-1}(\frac{\boldsymbol{v}_1\cdot \boldsymbol{v}_{\mathrm {ref}}}{|\boldsymbol{v}_1|\cdot |\boldsymbol{v}_{\mathrm {ref}}|})\]

where

\(\mathrm {\boldsymbol{v}_1=\begin{bmatrix}x_{2,i}-x_{1,i}\\y_{2,1}-y_{1,i}\\z_{2,i}-z_{1,i}\end{bmatrix}}\)

\(\mathrm {x,y,z}\) are nodal point coordinates of element i. Indexes 1 and 2 refers to end 1 and end 2 of the element. \(\boldsymbol{v}_{\mathrm {ref}}\) is a reference vector: one of the global axes, one of the vessel axes, or another element, according to the selected reference.

5. Distance Time Series

This option calculates the distance between two specified line segments.

Segment 1: A specified segment on a line.

Segment 2: A second specified segment, or a globally fixed line segment, or a line attached to a support vessel.

Figure 2 illustrates the two line segments. The calculation of distance is as follows.

  1. Calculate a common normal vector to the two line segments by

\[\boldsymbol{a}=\boldsymbol{e}_1\times \boldsymbol{e_2}\]

  1. Define 2 parallel planes through the two lines

\[\begin{array}{l}D_1=\displaystyle \sum_{\mathrm {i=1}}^{3}\,a_\mathrm {i}x_{1,i}\\\\D_2=\displaystyle \sum_{\mathrm {i=1}}^{3}\,a_\mathrm {i}x_{2,j}\end{array}\]

  1. Calculate the distance between the two planes by

\[d=|D_2-D_1|\]

  1. Check that the point of shortest distance is between the nodal points, i, i+1 and j, j+1 on the two segments. If not, carry out similar check for next element. This check is carried out for all combinations of i and j until the point of smallest distance is within the elements under consideration.

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Figure 2. Distance between two segments

The above procedure implies:

  • Only the absolute value of distance is calculated.

  • Crossing is not certainly identifiable. A snapshot plot must be used in addition.

  • The exact location of minimum distance is not known.

  • The procedure is time consuming, so the number of nodes in segments to be checked should be kept moderate.

6. Stress Calculations/Stress Time Series

Thick wall theory for linear elastic materials is implemented and described in the following:

Torsional stress:

\[\tau_{\mathrm {tmax}}=\frac{T\cdot r_y}{J_P}\]

where:

  • \(\mathrm {T}\) - torsional moment

  • \(\mathrm {r_y}\) - outer radius of pipe

  • \(\mathrm {J_P}\) - polar moment of inertia = \(\mathrm {(\pi /2)(r_y^4-r_i^4)}\)

  • \(\mathrm {r_i}\) - inner radius of pipe

Axial stress:

\[\sigma _a=\frac{F}{A}\]

where:

  • \(\mathrm {F}\) - axial force

  • \(\mathrm {A}\) - cross sectional area of pipe

Bending stress:

\[\sigma _b=\frac{M}{I}Y\]

where:

  • \(\mathrm {M}\) - bending moment

  • \(\mathrm {I}\) - moment of inertia = \(\mathrm {(\pi /4)(r_y^4-r_i^4)}\)

  • \(\mathrm {Y}\) - distance from pipe center to point in pipe wall \(\mathrm {(r_y,r_i)}\)

Shear stress:

\[\tau_Q=\frac{r_y^2}{I}Q\]

where:

  • \(\mathrm {Q}\) - shear force

Hoop stress:

\[\sigma _n=\frac{(p_i-p_e)(\displaystyle \frac{y}{r_y})^2+p_i(\displaystyle \frac{r_i}{r_y})^2-p_e}{I-(\displaystyle \frac{r_i}{r_y})^2}\]

where:

  • \(\mathrm {p_i}\) - internal pressure

  • \(\mathrm {p_e}\) - external pressure

Radial stress:

\[\sigma _r=\frac{(p_i-p_e)(\displaystyle \frac{y}{r_y})^2-p_i(\displaystyle \frac{r_i}{r_y})^2-p_e}{I-(\displaystyle \frac{r_i}{r_y})^2}\]

A three-dimensional stress calculation is performed utilizing a von Mises formula:

\[\sigma _{eq}=\sqrt{\frac{1}{2}[(\sigma _x-\sigma _y)^2+(\sigma _y-\sigma _z)^2+(\sigma _z-\sigma _x)^2]+3\tau^2}\]

The different components of this formula are given by:

\(\mathrm {\begin{array}{l}\sigma _x=\sigma _a-\sigma _b\\\sigma _y=\sigma _n\\\sigma _z=\sigma _r\\\tau=\tau_t-\tau_Q\end{array}}\)

The maximum \(\sigma _{\mathrm {eq}}\) is found at a point in the cross section by varying the \(\mathrm {y,(r_i,r_y)}\)

7. Time Domain Fatigue Analysis

(To be completed)