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Reliability

The application of reliability concepts to offshore structures began as an American story. In 1979, under the direction of Dr. Fred Moses, the American Petroleum Institute began a series of studies to implement reliability design procedures for fixed offshore platforms. This would (a) offer greater uniformity in platform component reliability; (b) enable a more effective material utilisation; (c) directly account for randomness and uncertainties in engineering parameters; (d) be capable of consistent modifications to account for different location, platform type, and life; (e) offer a logical basis for incorporating new information; and (f) help to focus on research activities to emphasise areas of greatest uncertainty and have results impact reliability factors.

PRAC 79-22 Project recommended that API RP-2A standard should change its current method of checking component safety known as working stress design (WSD) format to a multiple safety factor format. This concept came to be known as the Load Resistance Factor Design or LRFD.

To illustrate this, in WSD, a component is checked with an equation of the form:

Nominal Strength / Safety factor > Sum of Nominal Loads

Notice only one safety factor is used in the above, and hence the reliability would depend on the range of design application. Whereas, in the LRFD approach, partial safety factors are used in the format:

Component factor * Nominal Strength > Sum of Factored Loads

The research that went into producing this method included the following steps:

  1. Gather statistical data on load effects and component strengths.
  2. Review present performance criteria and experience.
  3. Establish target reliability level for each component type, based on performance experience.
  4. Calibrate load and resistance factors for tabulation in the standard.

PRAC 80-22 Project defined the reliability model for a structure as one that should incorporate safety margins and uncertainties in evaluating risk to a component or system.

Risk = 1 - Reliability = P[failure] = [Strength < Load]

The probability of component failure is as shown in Figure below.

Probability of failure

The (extreme) load frequency curve overlapping the strength curve represents the risk. The model is viewed as a situation in which the probabilities correspond to the worst loading case — annual or lifetime, as appropriate. This overlap (i.e., probability of failure) would decrease if either (i) the mean margin of safety increases, or (ii) the uncertainty (σ) in load or resistance reduces.

The analysis of reliability is carried out by defining a failure function, g, such that g < 0 denotes failure, or:

\begin{aligned} g = R - E \end{aligned}

where, R is the resistance or capacity, and E is the load effect. An exact solution for the probability of failure, Pf, could be obtained if R and E are both assumed as normal distributions with respective mean values, R and E, and coefficients of variations (cov),1 VR, and VE. Using these assumptions, Pf can be written as:

\begin{align} P_f &= \Phi\left( \frac{-\overline{g}}{\sigma_g} \right) \newline &= \Phi(-\beta) \end{align}

where,

\begin{aligned} \Phi(x) = \int_{-\infty}^{\infty} {\frac{1}{2\pi}}e^{\left({\frac{-1}{2}}z^2\right)dz} \end{aligned}

However, normal distributions are not always applicable to both load and strength variables, and both R and E in turn depend on several other random variables, such as load type (gravity, wave), beam-column stability, combined hydrostatic, axial and bending strengths, etc. Therefore, a generalisation is needed to carry out realistic reliability analyses. These results are often described in terms of a safety index, $\beta = \bar{g} / \sigma_g$, i.e., the ratio of mean safety margin to uncertainty level. In other words, $\beta$ is the distance in terms of standard deviations ($\sigma_g$) from the mean ($\bar{g}$) of the safety margin to the failure region (g < 0). When expanded, the expression becomes:

\begin{aligned} \beta = \frac{\ln\left({\frac{R_m}{E_m} \cdot \sqrt{\frac{1 + V_E^2}{1 + V_R^2}} } \right)}{\sqrt{\ln\left[ (1 + V_R^2) \cdot (1 + V_E^2) \right]}} \newline \end{aligned}

The LRFD method caught-on. Shell engineers extended the concept to numerous geographical areas2 to develop the following table:

Parameters of lognormal distribution of extreme 20-year load in various geographical areas (E=1.0 corresponds to E100)
Area MPM (E20) Mean (E20m) $\sigma$E COV (V20m)
GoM 0.68 0.79 0.25 0.32
NNS 0.75 0.81 0.21 0.265
CNS, SNS 0.80 0.84 0.18 0.212
AUS 0.67 0.78 0.26 0.33

where, E20 is the most probable value, and E20m is the mean value. In addition to $V_E$ provided above, uncertainty in hydrodynamic areas, volumes and marine growth, must be accounted for. This effect is covered by using COV = 8% and a bias factor of 1.0 (= ratio of mean value to characteristic value). This COV must be combined with COV values in table above.

Based on the GoM database, the following were derived, viz.,

\begin{aligned} R_m &= 1.85 \newline \gamma_E &= 1.35 \newline cov_R &= 0.05 \end{aligned}

Using these above together with $E_m$ and $V_E$ = (0.322 + 0.082)0.5, the safety index, $\beta$ (GOM) was computed to be 2.77, for a reference period of 20 years — based on a characteristic 100-y design event. (From $\beta$, Pf could be determined as 4E-4/annum for the GoM database.)

Further, it was noted that the mean RSR is proportional to $\gamma_E$, which resulted in the expression:

\begin{aligned} R_m = \frac{1.85}{1.35} \cdot \gamma_E = 1.37 \cdot \gamma_E \end{aligned}

Once this was done, calibrating other locations to Gulf of Mexico safety level became easier. To extend this to a target probability of failure, first a target needed to be defined for the platform type. The industry now generally recognises it to be 3E-5/annum for a manned new installation, which would be 6E-4 for 20 years (= 20 * 3E-5/year).

For the target Pf = 3E-5/year (which corresponds to $P_{f20}$ = 6E-4, for 20 years), $\beta_{20}$ could be determined as equal to 3.239. Using this safety index, probability density function plots could be generated for all tabulated regions — as I’ve done below.

For exposure level L1 ($P_f=3\cdot10^{-5}/y$)

Probability density for exposure level: L1

For exposure level L2 ($P_f=5\cdot10^{-4}/y$)

Probability density for exposure level: L2

Also, using this new safety index and the table above, the ISO 19902 has tabulated the mean RSR (Rm) and $\gamma_E$ for the Australian Northwest Shelf and the North Sea in A.9.9.3.

So, in a gist, that’s the general theory to develop mean RSR and $\gamma_E$, which could be applied to new locations, if Emean and VE could be determined from Metocean and Structural response studies like Shell did in the 90s for its NS and AU assets, and later for its Southeast Asian assets. Of course, statistically extending a dataset of one region’s safety index entirely to another depends very much on how well the means and covs are computed for the new region — and in general feels both like a novel idea as well as a hack. But for our generation, this will do.

Probability density plot code, now updated to include exposure levels L1 as well as L2, is as follows:

#!/usr/bin/env python
# -*- encoding: utf-8 -*-
"""Load and Resistance probability density and Safety Margins per ISO 19902:2007
2018 ckunte

Usage: python pdf.py ( --gom | --nns | --cns | --aus ) [--exp=L]
       python pdf.py --help
       python pdf.py --version

Options:
  -h, --help  Show help screen
  --gom       Plot for Gulf of Mexico
  --nns       Plot for Northern North Sea
  --cns       Plot for Central North Sea (also applicable to SNS)
  --aus       Plot for Australian Northwest shelf
  --exp=L     Exposure level: 1 for L1, 2 for L2 [default: 1]
  --version   Show version
"""
from scipy.stats import norm
import numpy as np
import matplotlib.pyplot as plt
from docopt import docopt

args = docopt(__doc__, version='Load and Resistance probability density and Safety Margins per ISO 19902:2007, ver:0.3')
cat = int(args['--exp'])
"""
Legend:
  Em -- Mean load for a reference return period
  Ve -- Load COV
  Rm -- Mean resistance
  Vr -- Resistance COV

COV = [0.212, 0.265, 0.32, 0.33] # For CNS & SNS, NNS, GOM, AUS
Ve = map(sqrt(x^2 + 0.08^2), COV) => [0.2266, 0.2768, 0.3298, 0.3396]
Rm = RSR_mean = [1.73, 1.92, 1.85, 2.18] for target Pf = 3E-5/annum (used in these plots)
Rm = RSR_mean = [1.40, 1.49, 1.60, 1.60] for target Pf = 5E-4/annum
"""
def ER(Em,Ve,Rm,Vr):
    mu_e = np.log(Em / np.sqrt(1 + Ve**2)) # mean
    mu_r = np.log(Rm / np.sqrt(1 + Vr**2)) # mean
    sigma_e = np.sqrt(np.log(1 + Ve**2))
    sigma_r = np.sqrt(np.log(1 + Vr**2))
    #
    s_e = np.random.lognormal(mu_e, sigma_e, 1000)
    s_r = np.random.lognormal(mu_r, sigma_r, 1000)
    count_e, bins_e, ignored_e = plt.hist(s_e, 100, normed=True, align='mid', alpha=0.09, color='r')
    count_r, bins_r, ignored_r = plt.hist(s_r, 100, normed=True, align='mid', alpha=0.09, color='g')
    x_e = np.linspace(min(bins_e), max(bins_e), 10000)
    x_r = np.linspace(min(bins_r), max(bins_r), 10000)
    #
    pdf_E = (np.exp(-(np.log(x_e) - mu_e)**2 / (2 * sigma_e**2)) / (x_e * sigma_e * np.sqrt(2 * np.pi)))
    pdf_R = (np.exp(-(np.log(x_r) - mu_r)**2 / (2 * sigma_r**2)) / (x_r * sigma_r * np.sqrt(2 * np.pi)))
    #
    plt.plot(x_e, pdf_E, linewidth=1, color='r')
    plt.plot(x_r, pdf_R, linewidth=1, color='g')
    plt.ylabel('Probability density')
    misc()
    pass

# Data: Em, Ve, Rm, Vr
d = [
    0.79,0.3298,1.85,0.05, # L1: GOM
    0.79,0.3298,1.60,0.05, # L2: GOM
    0.81,0.2768,1.92,0.05, # L1: NNS
    0.81,0.2768,1.49,0.05, # L2: NNS
    0.84,0.2266,1.73,0.05, # L1: CNS and SNS
    0.84,0.2266,1.40,0.05, # L2: CNS and SNS
    0.78,0.3396,2.18,0.05, # L1: AUS
    0.78,0.3396,1.60,0.05  # L2: AUS
    ]

def gom():
    plt.title('Gulf of Mexico')
    if cat == 1:
        ER(d[0],d[1],d[2],d[3])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[0],d[2]))
        plt.axvline(x=d[0],color='r',linestyle=':')
        plt.axvline(x=d[2],color='g',linestyle=':')
        plt.savefig('GoM_L1.png')
    if cat == 2:
        ER(d[4],d[5],d[6],d[7])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[4],d[6]))
        plt.axvline(x=d[4],color='r',linestyle=':')
        plt.axvline(x=d[6],color='g',linestyle=':')
        plt.savefig('GoM_L2.png')
    pass

def nns():
    plt.title('Northern North Sea')
    if cat == 1:
        ER(d[8],d[9],d[10],d[11])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[8],d[10]))
        plt.axvline(x=d[8],color='r',linestyle=':')
        plt.axvline(x=d[10],color='g',linestyle=':')
        plt.savefig('NNS_L1.png')
    if cat == 2:
        ER(d[12],d[13],d[14],d[15])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[12],d[14]))
        plt.axvline(x=d[12],color='r',linestyle=':')
        plt.axvline(x=d[14],color='g',linestyle=':')
        plt.savefig('NNS_L2.png')
    pass

def cns_sns():
    plt.title('Central North Sea (also applicable to Southern North Sea)')
    if cat == 1:
        ER(d[16],d[17],d[18],d[19])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[16],d[18]))
        plt.axvline(x=d[16],color='r',linestyle=':')
        plt.axvline(x=d[18],color='g',linestyle=':')
        plt.savefig('CNS-SNS_L1.png')
    if cat == 2:
        ER(d[20],d[21],d[22],d[23])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[20],d[22]))
        plt.axvline(x=d[20],color='r',linestyle=':')
        plt.axvline(x=d[22],color='g',linestyle=':')
        plt.savefig('CNS-SNS_L2.png')
    pass

def aus():
    plt.title('Australian Northwest Shelf')
    if cat == 1:
        ER(d[24],d[25],d[26],d[27])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[24],d[26]))
        plt.axvline(x=d[24],color='r',linestyle=':')
        plt.axvline(x=d[26],color='g',linestyle=':')
        plt.savefig('AUS_L1.png')
    if cat == 2:
        ER(d[28],d[29],d[30],d[31])
        plt.xlabel('Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$' %(d[28],d[30]))
        plt.axvline(x=d[28],color='r',linestyle=':')
        plt.axvline(x=d[30],color='g',linestyle=':')
        plt.savefig('AUS_L2.png')
    pass

def misc():
    plt.rcParams['grid.linestyle'] = ':'
    plt.rcParams['grid.linewidth'] = 0.5
    plt.grid(True)
    plt.xlim(0.25,2.75)
    plt.ylim(0,5)

if __name__ == '__main__':
    if args['--gom']:
        gom()
    elif args['--nns']:
        nns()
    elif args['--cns']:
        cns_sns()
    elif args['--aus']:
        aus()
    else:
        print "No option was selected. For help, try: python pdf.py -h"

Partial action factors and reserve strength ratios

Extending the above concept to derive partial action factors and mean reserve strength ratio appropriate to the probability of failure of interest, Shell engineers documented indicative constants to represent key environments.3

Indicative constants representing key environments
Environment A E0
GOM 2.13 0.187
NNS 11.9 0.1411
CNS, SNS 180 0.102
AUS 1.342 0.2041

Using these above, Pf could be calculated as follows:

\begin{aligned} P_f = A \cdot e^{\left(\frac{-Rm}{E_0}\right)} \cdot e^{\left(\frac{(V \cdot Rm)^2}{2 \cdot E_0^2}\right)} \newline \end{aligned}

The easiest way to use the above equation is to pick a range of RSR values in terms of min. and max. (start with, say, an Rm = 1.4) and generate corresponding Pf at every small Rm interval until it exceeds 3E-5 (33,333 years = 1 / 3E-5/year). In the plots below, the green (dashed) line indicates 2,000 years, which corresponds to Pf = 1E-4/y (L2), and the blue (dashed) line marks 33,333 years, which corresponds to Pf = 3E-5/y (L1). Here’s a summary of results.

$\gamma_E$ RSR mean
Region/RP 2,000-y 33,333-y 2,000-y 33,333-y
GOM 1.17 1.59 1.60 2.17
NNS 1.08 1.39 1.47 1.90
CNS, SNS 1.00 1.23 1.37 1.69
AUS 1.21 1.65 1.66 2.27

One may notice some values in the table above that are either similar or close to those listed in §A.9.9.3.3, ISO 19902:2007.

Return period v. Partial action factor Return period v. Reserve strength ratio

Here’s the code to generate these plots.

#!/usr/bin/env python
# -*- coding: UTF-8 -*-
"""Partial action factor and corresponding reserve strength ratio per EP97-5050
2018 ckunte

Usage: python pra.py [--typ=L]
       python pra.py --help
       python pra.py --version

Options:
  -h, --help  Show help screen
  --typ=L     1 for partial action factor, 2 for RSR [default: 1]
  --version   Show version
"""
import numpy as np
import matplotlib.pyplot as plt
from docopt import docopt

args = docopt(__doc__, version='Partial action factor and corresponding RSR per EP97-5050, ver 0.1')
cat = int(args['--typ'])

def main():
    # -- begin inputs --
    lbl = ["Australian NWS", "Gulf of Mexico", "Northern North Sea", "Central & Southern North Sea"]
    A = [1.342, 2.13, 11.9, 180.]
    E0 = [0.2041, 0.187, 0.1411, 0.102]
    VE = 0.07 # Cd, Cm
    VR = 0.05
    # -- end inputs ----
    V = np.sqrt(VE**2 + VR**2)
    # RSR range (min., max.)
    x = np.linspace(1.4, 2.4)
    # Partial action factor (gamma_e)
    gamma_e = (x / 1.37)
    # Plot all regions
    for i, j, k in zip(A, E0, lbl):
        # Probability of failure (Pf)
        pf = i * np.exp(-x / j) * np.exp((V * x)**2 / (2.0 * j**2))
        # Return period
        rp = (1 / pf)
        # Select plot type
        if cat == 1:
            # Plot Partial action factor v. Return period
            plt.plot(rp, gamma_e, linewidth=2, label=k)
            plt.ylabel('Partial action factor, $\gamma_e$')
            plt.ylim(1.0,1.7)
        if cat == 2:
            # Plot RSR v. Return period
            plt.plot(rp, x, linewidth=2, label=k)
            plt.ylabel('Reserve strength ratio (RSR) mean')
            plt.ylim(1.4,2.4)
    plt.xscale('log')
    plt.axvline(x=2000, color='g', linestyle=':') # 2,000-y ret.period (=1/1E-4)
    plt.axvline(x=33333, color='b', linestyle=':') # 33,333-y ret.period (=1/3E-5)
    plt.xlabel('Return period (years)')
    plt.legend(loc=0)
    plt.show()
    pass    

if __name__ == '__main__':
    main()

  1. The cov is the standard deviation divided by mean value. 

  2. Efthymiou, M., et al., Reliability-based Criteria for Fixed Steel Offshore Platforms, Transactions of the ASME, Vol. 119, May 1997. 

  3. Efthymiou, M., van de Graaf, J.W., Reliability Based Design and Re-assessment of Fixed Steel Platforms, EP97-5050, February 1997.