Fatigue Analysis

1. Introduction

VIVANA will calculate the fatigue damage by assuming that the cross section is circular with a diameter defined from the cross section data, see RIFLEX User manual, input to INPMOD. Stresses will be calculated on the outer surface according to the cross section data, see Figure 1. Stress concentration factors and SN curves are given directly to VIVANA.

Figure 1. Simple circular cross section.

The method for calculating fatigue damage will vary depending on the actual analysis option. An overview of analysis options and methods is given in Table 1.

Table 1. Overview of methods for fatigue damage calculation
Fatigue analysis VIV Analysis option	Pure IL	Pure CF	Combined IL and CF
Simultaneously acting frequencies	Generation of time histories, rainflow counting, see Section 2.	Generation of time histories, rainflow counting, see Section 2.	Generation of time histories, rainflow counting, see Section 2.
Time sharing	Closed form solution, see Section 3.	Closed form solution, see Section 3.	Closed form solution, Gouyang (1989), see Section 4.

2. Fatigue analysis in time domain, multi-frequency response

When the VIV response analysis is finished we are left with \(\mathrm {N_f}\) complex response vectors, \(\mathrm {\boldsymbol{x}_r}\), \(\mathrm {\:r=1,2,...,N_f}\). These are used in combination with the element stiffness matrices and cross section properties to arrive at time series of stress at the element ends. Stress cycles are counted by the rainflow counting method [ref.] and fatigue damage is then found by Miner-Palmgren summation.

The procedure is as follows:

2.1. 1. Complex vectors of element forces for all elements at all response frequencies are calculated by

\[\boldsymbol{s}_{r,iel}=\boldsymbol{k}_{iel}\boldsymbol{x}_{r,iel},\quad r=1,2,...,N_f\quad iel=1,2,...,N_{el}\]

where \(\mathrm {N_{el}}\) is the total number of elements, \(\mathrm {\boldsymbol{k}_{iel}}\) is the element stiffness matrix and \(\mathrm {\boldsymbol{x}_{r,iel}}\) contains the components from \(\mathrm {\boldsymbol{x}_r}\) that correspond to the local degrees of freedom for the element. The dimension of \(\mathrm {\boldsymbol{s}_{r,iel}}\) and \(\mathrm {\boldsymbol{x}_{r,iel}}\) is \(\mathrm {12}\), and \(\mathrm {\boldsymbol{k}_{iel}}\) is \(\mathrm {12\times 12}\).

2.2. 2. The stress at the element ends are found as

\[\begin{array}{r}\sigma _{r,iel,iend,ipt}=\boldsymbol{p}_{iel,ipt}\boldsymbol{s}_{r,iel,iend},\quad r=1,2,...,N_f\quad iel=1,2,...,N_{el}\\iend=1,2\quad ipt=1,2,...,N_{pt}\end{array}\]

where \(\mathrm {\boldsymbol{s}_{r,iel,iend}}\) is a vector of dimension \(\mathrm {6}\) (the first \(\mathrm {6}\) components in \(\mathrm {\boldsymbol{s}_{r,iel}}\) if \(\mathrm {iend=1}\), and the last \(\mathrm {6}\) components if \(\mathrm {iend=2}\)), \(\mathrm {\boldsymbol{p}_{iel,ipt}}\) is a vector of dimension \(\mathrm {6}\) containing the cross section properties of the element, \(\mathrm {N_{pt}}\) is the number of places on the cross section that is to be checked for fatigue.

\[\begin{array}{l}\displaystyle \boldsymbol{p}_{iel,ipt}=\begin{bmatrix}\displaystyle \frac{SCFAX}{A_{st,iel}}&0.0&0.0&0.0&\displaystyle \frac{SCFMY}{W_{y,eil,ipt}}&\displaystyle \frac{SCFMZ}{W_{z,iel,ipt}}\end{bmatrix}\\\\\displaystyle W_{y,iel,ipt}=\frac{W_{y,iel}}{\sin(\alpha )}(\frac{2\pi \:ipt}{N_{pt}})\\\\\displaystyle W_{z,iel,ipt}=\frac{W_{z,iel}}{\cos(\alpha )}(\frac{2\pi \:ipt}{N_{pt}}),\quad iel=1,...,N_{el}\quad ipt=1,2,...,N_{pt}\end{array}\]

where \(\mathrm {A_{st,iel}}\) is the cross section steel area, \(\mathrm {W_{y,iel}}\) is the section modulus for bending about the y-axis and \(\mathrm {W_{z,iel}}\) is the section modulus for bending about the z-axis. \(\mathrm {\alpha }\) is the angle between the local y-axis to the point on the cross-section.

\(\mathrm {W_{y,iel}=W_{z,iel}}\) in the present version of the program. \(\mathrm {SCFAX}\), \(\mathrm {SCFMY}\) and \(\mathrm {SCFMZ}\) are stress concentration factors for axial tension, bending about local y-axis and bending about local z-axis, respectively.

2.3. 3. Stresses at the element ends are represented by the complex number

\[\sigma _{r,iel,iend,ipt}\,,\quad r=1,2,...,N_f\quad iel=1,2,...,N_{el}\quad iend=1,2\quad ipt=1,2,...,N_{pt}\]

Time series of the stress can be written as a summation of the contributions from all response frequencies,

\[\sigma _{iel,iend,ipt}\,(t)=\sum_{r=1}^{N_f}\sigma _{r,iel,iend,ipt}\:e^{i(\omega _rt+\varepsilon _r)}\]

where \(\mathrm {\omega _r}\) is the response frequency for frequency no. \(\mathrm {r}\) and \(\mathrm {\varepsilon _r}\) is the phase angle for response at this frequency. The latter is drawn randomly from an even distribution between \(\mathrm {0}\) and \(\mathrm {2\pi }\).

Writing the complex number as one real component and one imaginary component we get,

\[\sigma _{r,iel,iend,ipt}=\sigma _{\mathrm {Re},r,iel,iend,ipt}+i\sigma _{\mathrm {Im},r,iel,iend,ipt}\]

We then have

\[\sigma _{iel,iend,ipt}\,(t)=\sum_{r=1}^{N_f}\sigma _{a,r,iel,iend,ipt}\:\cos(\omega _rt+\theta _{r,iel,iend,ipt}+\varepsilon _r)\]

where

\[\sigma _{a,r,iel,iend,ipt}=\sqrt{\sigma _{\mathrm {Re},r,iel,iend,ipt}^2+\sigma _{\mathrm {Im},r,iel,iend,ipt}^2}\]

\[\begin{array}{r}\displaystyle \theta _{r,iel,iend,ipt}=\arctan(\frac{\sigma _{\mathrm {Im},r,iel,iend,ipt}}{\sigma _{\mathrm {Re},r,iel,iend,ipt}}),\quad r=1,2,...,N_f\quad iel=1,2,...,N_{el}\\\quad iend=1,2\quad ipt=1,2,...,N_{pt}\end{array}\]

2.4. 4. A time series of length T (sec.) is generated

A number of stress ranges, (\(\mathrm {\Delta \sigma _i,\:i=1,2,...,N_{\Delta \sigma }}\)), must be defined, and the rainflow counting method is used for finding the number of occurrences for each cycle, \(\mathrm {n_{i,iel,iend,ipt}\,}\).

Total number of cycles for stress range \(\mathrm {i}\) during one year is calculated from,

\[n_{i,year,iel,iend,ipt}=\frac{n_{i,iel,iend,ipt}\cdot 365\cdot 24\cdot 60\cdot 60}{T}\]

and the accumulated fatigue damage is found from,

\[D_{iel,iend,ipt}=\sum_{i=1}^{N_{\Delta \sigma }}\frac{n_{i,year,iel,iend,ipt}}{N_i}\]

where \(\mathrm {N_i}\) is the number of cycles to failure for stress cycle \(\mathrm {i}\).

\(\mathrm {N_i}\) is found from;

\[\log(N_i)=\log(C)+m\cdot \log(\Delta \sigma _i\cdot (\frac{t_{iel}}{t_{ref}})^k)\]

where \(\mathrm {\log(C)}\), \(\mathrm {m}\), \(\mathrm {t_{ref}}\) and \(\mathrm {k}\) are SN-data for the curve segment used, \(\mathrm {t_{iel}}\) is the wall thickness for the cross section and \(\mathrm {\Delta \sigma _i}\) is the stress range.

3. Fatigue analysis based on closed form solution; time sharing for pure IL or pure CF response

According to the time sharing concept each possible response frequency will dominate for a given share of time, but the amplitude will become Rayleigh distributed by a low frequency envelope process. The process is illustrated in Figure 2.

Figure 2. Illustration of time sharing combined with low frequency envelope.

For this case it is possible to find an analytical solution for the Miner-Palmgren summation valid for each period with constant frequency and Rayleigh distributed amplitude. If the stress interval for which \(\mathrm {n_i}\) in Equation (11) is taken is of infinitesimal magnitude, the sum becomes an integral, and the fatigue damage can be found as

\[D=\sum_{i=1}^\infty\frac{n_i}{N_i}=\int_0^\infty\frac{\mathrm {d}n}{N}\]

Expected damage will be a function of expected number of cycles. Hence we have:

\[E(D)=\int_0^\infty\frac{1}{N}E(\mathrm {d}n)\]

The expected number of cycles with a given magnitude \(\mathrm {\Delta \sigma }\) can be found from the probability distribution

\[E(\mathrm {d}n)=N_{tot}\cdot f_{\Delta \sigma }(\Delta \sigma )\mathrm {d}(\Delta \sigma )\]

By introducing Equation (12) as the relation between \(\mathrm {N}\) and \(\mathrm {\Delta \sigma }\), the expected damage can now be found from the integral

\[E(D)=N_{tot}\cdot \int_0^\infty\frac{f_{\Delta \sigma }(\Delta \sigma )}{K(\Delta \sigma )^m}\mathrm {d}(\Delta \sigma )\]

If the SN curve is given by the above equation in the total definition range and no fatigue limit exists, and the Rayleigh distribution is valid for the stress ranges, the above integral has an analytical solution. Expected fatigue damage will then be given by:

\[E(D)=\frac{N_{tot}}{K}(2\sqrt{2}\sigma )^m\cdot \Gamma(\frac{m}{2}+1)\]

where

\(\mathrm {N_{tot}\quad }\) Total number of cycles for the actual sea state
\(\mathrm {\sigma \quad }\) Standard deviation of the stress process
\(\mathrm {\Gamma\quad }\) The Gamma function, cfr. mathematical tables
\(\mathrm {K,m\quad }\) Parameters in the SN curve

4. Fatigue analysis based on closed form solution; time sharing for combined IL and CF response

For this case two frequencies will act simultaneously. Hence, the method shown in Section 3 becomes invalid. Bending stress at any point on the circumference of the pipe at node \(\mathrm {j}\) can be found by considering the two components:

\[\sigma _{j,m}(t)=\frac{M_{CF}(t)}{W}\cos(\alpha _m)\sin(\omega _{CF}t)+\frac{M_{IL}(t)}{W}\cos(\beta _m)\sin(2\omega _{CF}t+\varepsilon _j)\]

where \(\mathrm {M_{CF}(t)}\) and \(\mathrm {M_{IL}(t)}\) are the modulated amplitudes of the bending moments at the CF and IL frequencies, \(\mathrm {W}\) is the section modulus of the pipe cross section, \(\mathrm {\alpha _m}\) is the angle between the CF moment and the point on the cross section and \(\mathrm {\beta _m}\) is the angle between the IL moment and the point. \(\mathrm {\varepsilon _j}\) is the phase between the CF and IL bending moments. The contribution to stresses from moments is illustrated in Figure 3. The red circle indicates the location on the pipe cross section where the bending stress is calculated. \(\mathrm {Y_L}\) and \(\mathrm {Z_L}\) are the coordinates in the local cross section coordinate system.

Figure 3. Stress calculation from moment components.

Since the modulation processes \(\mathrm {M_{CF}(t)}\) and \(\mathrm {M_{IL}(t)}\) are Rayleigh distributed, the stress process \(\mathrm {\sigma _{j,m}(t)}\) will appear as a sum of two narrow banded Gaussian processes which are considered to be statistically independent. That implies that the phase between the IL and CF stress components will become random and vary in time. This variation is observed in experiments with flexible beams, and may be attributed the complex, apparently chaotic, time sharing process. No attempts have been made to establish a dynamic analysis method that could describe this type of behavior. Instead, this feature is taken into account when the fatigue damage is calculated.

Fatigue damage from two narrow banded Gaussian processes was investigated by Guoyang Jiao (1989). He found an approximate closed form solution based on the standard deviations of the two processes. This method has been compared to rainflow counting and used by Guoyang Jiao (1989) to calculate fatigue from a combined wave frequency and low frequency process.

By applying this method, fatigue damage at each point on the cross section can easily be calculated when the IL and CF stress amplitudes and the variance of the modulation process is known. Fatigue accumulation for all pairs of response frequencies can then be performed using Equation (19)

\[D_{tot}=\frac{\displaystyle \sum_{i=1}^n\mu_iD_i}{\displaystyle \sum_{i=1}^n\mu_i}\]

where \(\mathrm {D_{tot}}\) is total fatigue damage during a specific time period, \(\mathrm {D_i}\) is fatigue damage due to response frequency pair \(\mathrm {i}\) for the same period, and \(\mathrm {\mu_i}\) is the ranking parameter.

The stress at each location is calculated for each pair of CF and IL response frequencies. The standard deviation of the harmonic stress process \(\mathrm {\sigma _{j,m}(t)}\) is assumed to represent the standard deviation of a narrow banded Gaussian process in the present model.