**3.1 Modeling**

176 Modern Metrology Concerns

The terminating device may not react in the same way for microwave and DC power absorption, so the effective efficiency *ηe* is used to perform the correction. In equation (1), microwave power *PHF* absorbed by the terminating device is calculated by dividing the

> *DC e HF P P*

Since the effective efficiency is independent of the mismatch correction, the calibration

The calibration factor *K* is generally used at the time of calibration transfer from a reference standard to an unknown microwave power sensor. It is the focus in the following sections. The measurement uncertainty is a non-negative parameter characterizing the dispersion of the effective efficiency *ηe* or the calibration factor *K* being attributed to the standards. The uncertainty is evaluated using "law of propagation of uncertainty" following "Guide to the expression of Uncertainty in Measurement" (GUM) (JCGM 100:2008). Evaluation of measurement and calibration uncertainty by Monte Carlo Method (MCM) is to use Monte Carlo simulation in the uncertainty evaluation of output quantities based on "uncertainty probability distribution propagation" (JCGM 101:2008). The following sections will cover

The value of a primary standard is disseminated to a secondary standard through calibration or comparison. Then the reference standard and working power sensors will be calibrated for use. The measurement results through such relations as unbroken chain of calibrations establish the metrological traceability, each contributing to the measurement uncertainty. The traceability is illustrated in Fig. 1. Here for reference purpose we deliberately provide not only the hierarchy, but also the uncertainties typically related. The

real uncertainties depend on the frequency band and each laboratory conditions.

Fig. 1. Dissemination of primary standard to end user- Traceability Chart

*K - e* 2

(1)

(1 ) (2)

ડ

as follows:

factor *K* is used to describe both the effective efficiency *ηe* and mismatch

substituted DC power *PDC* by the effective efficiency *ηe*:

both methods for the measurement uncertainty evaluations.

The calibration of RF and microwave power sensor is to transfer the effective efficiency or calibration factor from a primary standard to a secondary standard or from a secondary standard to a reference standard or from a reference standard to a power sensor. The parameter transfer is through comparison, or calibration one against the other. The simplest and the most obvious method to calibrate a power sensor against a reference standard is to connect each in turn to a stable power source, as illustrated in Fig. 2.

Fig. 2. Calibration of power sensor by the method of simple direct comparison transfer

Generally power from a source, *Pi,* with reflection coefficient ડ*G*, incident to a load with reflection coefficient ડ*L* , can be expressed as follows (Agilent, 2003; Engen, 1993; Mial, 2007):

$$P\_i = P\_{Z\_0} \times \frac{1}{|1 \cdot \Gamma\_G \Gamma\_L|^2}$$

and a reflected power *Pr*

$$P\_r = P\_{Z\_0} \times \frac{\left\| \begin{array}{c} \Gamma\_L \\ \hline \end{array} \right\|^2}{\left\| \begin{array}{c} \Gamma\_G \Gamma\_L \end{array} \right\|^2}$$

For Fig. 2, the power dissipated to the reference power standard *PStd* , can be derived as

$$P\_{Std} = P\_i \text{ - } P\_{r\_r Std} = P\_{Z\_0} \times \frac{1 \text{ - } \left\| \begin{array}{c} \Gamma\_{Std} \end{array} \right\|^2}{\left\| \begin{array}{c} 1 \ -\Gamma\_G \Gamma\_{Std} \end{array} \right\|^2}$$

And the power dissipated to the power sensor to be calibrated (DUT), *PDUT*, is as

$$P\_{DUT} = P\_i - P\_{r, DUT} = P\_{Z\_0} \times \frac{\mathbf{1} \cdot \left| \begin{array}{c} \Gamma\_{DUT} \end{array} \right|^2}{\left| \mathbf{1} \cdot \Gamma\_G \Gamma\_{DUT} \right|^2}$$

where *PZ0* is the power available to a load with characteristic impedance *Z0*. When idealized source with available power *Pa* has no internal impedance *Z0*, *PZ0 =Pa (1-|*ડ*G|2)*.

As shown in equation (1), the effective efficiency of a power sensor is a ratio. *PDC* is DC or low frequency equivalent power, generating the same heat effect as the high frequency power being measured. For calibration of a power sensor, that is, to transfer the effective efficiency of a reference standard, *ηStd* , to a power sensor to be calibrated (DUT), *ηDUT*, it can be derived:

*DC DUT DC DUT DUT Z DUT RF DUT G DUT DC DUT Std G DUT Std DC Std DC Std DC Std DUT G Std RF Std std Z G std P P - P P P - - - P P P - - P - P -* 0 0 , 2 , 2 2 2 , , 2 2 , , , 2 , 2 1| | |1 | 1 | | |1 | 1 | | |1 | 1| | |1 |

So the calibration equation of effective efficiency for the method of Fig. 2 is expressed as:

$$\eta\_{\rm DLT} = \eta\_{Std} \times \frac{P\_{\rm DC,DIT}}{P\_{\rm DC,Std}} \times \frac{\mathbf{1} \cdot \left| \Gamma\_{Std} \right|^2}{\mathbf{1} \cdot \left| \Gamma\_{DUT} \right|^2} \frac{\left| \mathbf{1} \cdot \Gamma\_G \Gamma\_{DUT} \right|^2}{\left| \mathbf{1} \cdot \Gamma\_G \Gamma\_{Std} \right|^2} \tag{3}$$

where *ηDUT* is the effective efficiency of DUT sensor.

This calibration equation involves three factors:

$$
\eta\_{DUT} = \eta\_{Std} \times P\_{RATIO} \times M\_1 M\_2
$$

*ηStd* is the effective efficiency of a standard sensor. It comes from a national metrology institute, a calibration laboratory, or a manufacturer with traceability. *PRATIO* is an equivalent DC or low frequency power ratio, depending on the system setup. *M1M2* is the mismatch factor.

Similarly, calibration equation of calibration factor for the method of Fig. 2 is expressed as follows, considering equation (2):

$$K\_{DITT} = K\_{Std} \times \frac{P\_{DC,DITT}}{P\_{DC,Std}} \times \frac{\left|1 \cdot \Gamma\_G \Gamma\_{DITT}\right|^2}{\left|1 \cdot \Gamma\_G \Gamma\_{Shd}\right|^2} \tag{4}$$

And the calibration factor transfer equation can also be expressed as three factors:

$$\mathcal{K}\_{DUT} = \mathcal{K}\_{S\!\!\!M} \times P\_{R\!\!\!A\!\!T\!\!M\!\!O} \times M^{\!\!\!A}$$

Notice that *M* is the mismatch factor for the calibration factor calibration transfer.

#### **3.2 Uncertainty evaluation**

From calibration transfer equation of effective efficiency (3) and that of calibration factor (4), if DUT power sensor has identical reflection coefficient as that of reference power standard, which means �*DUT =* �*Std*, each of them absorbs exactly the same amount of power from the source. Then by power ratio measurement it transfers the effective efficiency, *ηe*, or calibration factor, *K*, from standard to DUT, to complete the calibration.

But the actual reflection coefficients of power sensors being compared usually differ significantly from one another (refer to Table 1). The signal generator reflection coefficient cannot dismiss easily. The complex items inside (3) and (4), i.e. the mismatch factors, cannot be neglected in accurate power sensor calibration with small uncertainty.

*DUT RF DUT G DUT DC DUT Std G DUT Std DC Std DC Std DC Std DUT G Std*

*P - - P - -*

*DUT Std RATIO P MM*1 2

*ηStd* is the effective efficiency of a standard sensor. It comes from a national metrology institute, a calibration laboratory, or a manufacturer with traceability. *PRATIO* is an equivalent DC or low frequency power ratio, depending on the system setup. *M1M2* is the mismatch

Similarly, calibration equation of calibration factor for the method of Fig. 2 is expressed as

*DC DUT G DUT*

2 , 2 , |1 | |1 |

*Std*, each of them absorbs exactly the same amount of power from the

(4)

*DC Std G Std*

*P -*

*K KP M DUT Std RATIO*

From calibration transfer equation of effective efficiency (3) and that of calibration factor (4), if DUT power sensor has identical reflection coefficient as that of reference power standard,

source. Then by power ratio measurement it transfers the effective efficiency, *ηe*, or

But the actual reflection coefficients of power sensors being compared usually differ significantly from one another (refer to Table 1). The signal generator reflection coefficient cannot dismiss easily. The complex items inside (3) and (4), i.e. the mismatch factors, cannot

2 2 2 , , 2 2 , , ,

*DC DUT Std G DUT*

2 2 , 2 2 ,

1 | | |1 | 1 | | |1 |

(3)

*DC Std DUT G Std*

1 | | |1 |

*P P - - - P P P - -*

2

So the calibration equation of effective efficiency for the method of Fig. 2 is expressed as:

*DC DUT*

1| |

1| | |1 |

,

*P*

*G std*

*P - K K*

And the calibration factor transfer equation can also be expressed as three factors:

Notice that *M* is the mismatch factor for the calibration factor calibration transfer.

calibration factor, *K*, from standard to DUT, to complete the calibration.

be neglected in accurate power sensor calibration with small uncertainty.

*DC DUT DUT Z*

0

2 ,

*P - P*

factor.

*P - P -*

*DUT Std*

where *ηDUT* is the effective efficiency of DUT sensor.

This calibration equation involves three factors:

follows, considering equation (2):

**3.2 Uncertainty evaluation**

�*DUT =* �

which means

*RF Std std Z*

0

*DUT Std*

2 ,

In recognition of the standards adopted internationally, the GUM (JCGM 100:2008) is considered as the basic technique to evaluate the uncertainty of measurement. The method proposed in the GUM is based on the law of propagation of uncertainty which is essentially the first-order Taylor series approximation of calibration equation, such as equation (4) of calibration factor. The partial differentiation of the output estimate with respect to the input estimates, which is termed sensitivity coefficient, is interpreted as a description of how the output estimate varies with changes in the values of the input estimates. The following equation is the definition of combined standard uncertainty *uc(y)* which includes both type A (*ua(x)*) and type B (*ub(x)*) uncertainties when the mathematical model is of the form *y = f(x1, x2,…)*:

$$\mu\_c(\mathbf{y}) = \sqrt{\mu\_a^2(\mathbf{x}) + \sum\_{i=1}^{N} \left(\frac{\partial f}{\partial \mathbf{x}\_i}\right)^2 \mu\_b^2(\mathbf{x}\_i)}\tag{5}$$

Considering that the setup in Fig. 2 is the simple direct comparison transfer method, we present the uncertainty evaluation equations in different implementations, aiming at providing laboratories more realistic choices. The realistic magnitudes for reflection coefficients of instruments are used for the calculation and comparison. The specifications of reflection coefficients of the instruments quoted from different products at some frequency points are listed in Table 1.

Table 1. Typical reflection coefficients of power sensors and signal generators

#### **3.2.1 The simplest evaluation of measurement uncertainty**

The simplest way in the calibration transfer of calibration factor from standard to DUT sensor is to simplify equation (4) to the following equation:

$$K\_{DIT} = K\_{Std} \times \frac{P\_{\underline{U}}}{P\_S} = \text{Ratio Factor} \tag{6}$$

In which only the ratio factor is considered and set the mismatch factor *M* equal to 1. The real values of reflection coefficients are considered for the uncertainty budget only. In this case, the mismatch factor *M* for calibration factor *K*DUT is expressed as:

$$\mathbf{M} = \frac{\mathbf{M}\_{\mathbf{U}}}{\mathbf{M}\_{\mathbf{S}}}$$

where *MU* and *MS* are given by:

$$\mathbf{M}\_{\mathrm{II}} = \left| \mathbf{1} \cdot \Gamma\_{\mathrm{G}} \Gamma\_{\mathrm{D}LT} \right|^2 \quad \mathbf{M}\_{\mathrm{S}} = \left| \mathbf{1} \cdot \Gamma\_{\mathrm{G}} \Gamma\_{\mathrm{S}\mathrm{H}} \right|^2$$

The associated uncertainties are calculated with the following equations (Agilent, 2003; Shan et al., 2010a):

$$\mu\left(M\_X\right) = \sqrt{2\left\|\Gamma\_G\right\|\left\|\Gamma\_X\right\|}; \qquad \chi = Std,\\
\text{DUT} \tag{7}$$

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiations with respect to the individual variables in equation (6):

$$\begin{aligned} \frac{\partial \mathcal{K}\_{DUT}}{\partial \mathcal{K}\_{Std}} &= \frac{P\_{\mathcal{U}}}{P\_{\mathcal{S}}} = \frac{\mathcal{K}\_{DUT}}{\mathcal{K}\_{Sdd}} \\\\ \frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{\mathcal{U}}} &= \frac{\mathcal{K}\_{Sdd}}{P\_{\mathcal{S}}} = \frac{\mathcal{K}\_{DUT}}{P\_{\mathcal{U}}} \\\\ \frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{\mathcal{S}}} &= -\frac{\mathcal{K}\_{Sdd} \times P\_{\mathcal{U}}}{P\_{\mathcal{S}}^2} = (-1) \frac{\mathcal{K}\_{DUT}}{P\_{\mathcal{S}}} \end{aligned} \tag{8}$$

In which the expression is prepared for relative uncertainty representation since the combined standard uncertainty *uc(y)* can be expressed as an estimated relative combined

variance 2 2 2 1 ( ) ( ) *N c i i i i u y u x P y x* when the mathematical model is of the form 1 2 1 2 ... *P P PN Y cX X XN .*

The uncertainty budget is listed in Table 2 at frequency 18 GHz with type N connector. The calculations of mismatch uncertainties are based on Table 1 best and worst specifications.

It can be seen from Table 2 that the simplest evaluation of measurement uncertainty method is not an accurate method to obtain a small uncertainty. But it is useful for calibration laboratories with simple measurement set up as illustrated in Fig. 2. When this uncertainty value meets the demand, it is acceptable for industrial applications. In some evaluation experiments, it is also a practical method. Note that the value and uncertainty of a reference standard comes from the calibration of national metrology institutes or other calibration laboratories if the laboratory provides the calibration service using the method.

The associated uncertainties are calculated with the following equations (Agilent, 2003; Shan

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients

*K P K K PK*

*DUT U DUT Std S Std*

*DUT Std DUT USU K K K P PP*

*DUT Std U DUT S S S*

when the mathematical model is of the form

*K KP K - - P P P*2 ( 1)

In which the expression is prepared for relative uncertainty representation since the combined standard uncertainty *uc(y)* can be expressed as an estimated relative combined

The uncertainty budget is listed in Table 2 at frequency 18 GHz with type N connector. The calculations of mismatch uncertainties are based on Table 1 best and worst specifications.

It can be seen from Table 2 that the simplest evaluation of measurement uncertainty method is not an accurate method to obtain a small uncertainty. But it is useful for calibration laboratories with simple measurement set up as illustrated in Fig. 2. When this uncertainty value meets the demand, it is acceptable for industrial applications. In some evaluation experiments, it is also a practical method. Note that the value and uncertainty of a reference standard comes from the calibration of national metrology institutes or other calibration

Quant. Estim. Standard probability Sensitivity Uncertainty

*X*i *x*i *u*(*x*i) *c*i *u*i*(y) K*S 0.9894 0.0012 normal 1.0137 0.0012 *M*S 1.0000 0.0098 U-shaped 1.0000 0.0098 *M*U 1.0000 0.0195 U-shaped 1.0000 0.0195 *P*U 1.0158 0.0018 normal 0.9873 0.0018 *P*S 1.0021 0.0004 normal -1.0008 -0.0004 *y=K*U 1.0029 0.0219

uncertainty distribution coefficient contribution

laboratories if the laboratory provides the calibration service using the method.

18 GHz based on best specifications

2 2 2 1 ( ) ( ) *N c i i i i*

*u y u x P y x*

are partial differentiations with respect to the individual variables in equation (6):

*X Std,DUT u MX GX* 2 ; (7)

(8)

et al., 2010a):

variance

1 2 1 2 ... *P P PN Y cX X XN .*

Table 2. Uncertainty budget at 18 GHz for simplest evaluation of uncertainty for Fig. 2 measurement setup. Uncertainties *ui* are at one standard deviation. Powers are measured in mW.

#### **3.2.2 Measurement uncertainty improvement with mismatch correction**

To improve the calibration accuracy and uncertainty evaluation, we have to perform a mismatch correction, i.e., the complex reflection coefficients have to be considered in the calibration model. The complex value is either representing in term of magnitude and phase or its real and imaginary components. In the majority of engineering applications, the magnitude and phase representation (Polar coordinates) is generally preferred because this representation bears a direct relationship to physical phenomena affecting the measurement process (Ridler & Salter, 2002). For example, phase relates directly to the electrical path length of a signal, and magnitude relates directly to the signal attenuation. The same cannot be said for the representation of real and imaginary components (Cartesian coordinates). If the scales are used to depict the different representations, the real and imaginary axes in the complex plane extend infinitely (±∞). It is the same as the scale is used to depict all real numbers which is routinely for the arithmetic operations. While the scales are used to represent magnitude and phase each possess a significant difference. The magnitude has a lower bound of zero below which values cannot exist, and phase is represented convertionally on a cyclical scale, either from -180⁰ to +180⁰ or from 0⁰ to 360⁰.

With mismatch correction added in computing the calibration factor using equation (4), the calculation of sensitivity coefficients involves the partial differentiations with respect to complex reflection coefficients. In the following sections, we separately provide the derived sensitivity coefficients in both Cartesian and Polar coordinate representations for practical measurement uncertainty solution and application; and also examples for their associated uncertainties are included. The derived sensitivity coefficients in both representations have been numerically appreciated by making use of MATLAB version R2010a, symbolic differentiation in Math Toolbox.

#### **<IP_ADDRESS> Cartesian representation of sensitivity coefficients for equation (4)**

Representing calibration factor *K*DUT of equation (4) with magnitude and phase components, it becomes

$$\mathbf{K}\_{\mathrm{DJT}} = \mathbf{K}\_{\mathrm{Sfd}} \times P\_{\mathrm{RATIO}} \times M = \mathbf{K}\_{\mathrm{Sfd}} \times \frac{P\_{\mathrm{DJT}}}{P\_{\mathrm{Sfd}}} \times \frac{\mathbf{1} + \|\boldsymbol{\Gamma}\_{\mathrm{DJT}}\|^2 \left\|\boldsymbol{\Gamma}\_{\mathrm{G}}\right\|^2 \cdot \mathbf{2} \left\|\boldsymbol{\Gamma}\_{\mathrm{DJT}}\right\| \left\|\boldsymbol{\Gamma}\_{\mathrm{G}} \left|\cos(\boldsymbol{\theta}\_{\mathrm{DJT}} + \boldsymbol{\theta}\_{\mathrm{G}})\right\rangle}{\mathbf{1} + \|\boldsymbol{\Gamma}\_{\mathrm{Sfd}}\|^2 \left\|\boldsymbol{\Gamma}\_{\mathrm{G}}\right\|^2 \cdot \mathbf{2} \left\|\boldsymbol{\Gamma}\_{\mathrm{G}}\right\| \left\|\boldsymbol{\Gamma}\_{\mathrm{G}} \left|\cos(\boldsymbol{\theta}\_{\mathrm{Sid}} + \boldsymbol{\theta}\_{\mathrm{G}})\right\|\right\|}\tag{9}$$

Let *MN* represent the numerator and *MD* denominator of mismatch *M*:

$$M\_{\rm N} = \left| \mathbf{1} \cdot \Gamma\_{D\rm MT} \Gamma\_{G} \right|^{2} = \mathbf{1} + \left| \Gamma\_{D\rm MT} \right|^{2} \left| \Gamma\_{G} \right|^{2} \cdot 2 \left| \Gamma\_{D\rm MT} \right| \parallel \Gamma\_{G} \parallel \cos(\theta\_{\rm DIT} + \theta\_{\rm G})$$

$$M\_{\rm D} = \left| \mathbf{1} \cdot \Gamma\_{S\rm d} \Gamma\_{G} \right|^{2} = \mathbf{1} + \left| \Gamma\_{S\rm d} \right|^{2} \left| \Gamma\_{G} \right|^{2} \cdot 2 \left| \Gamma\_{S\rm d} \right| \parallel \Gamma\_{G} \parallel \cos(\theta\_{\rm S\rm d} + \theta\_{\rm G})$$

$$\begin{array}{llll} \textbf{Let} & A = \left| \begin{array}{c} \Gamma\_{G} \end{array} \right|\_{\prime} & B = \left| \begin{array}{c} \Gamma\_{D\rm MT} \end{array} \right|\_{\prime} & \mathbf{C} = \left| \begin{array}{c} \Gamma\_{S\rm d} \end{array} \right|\_{\prime} & D = \cos(\theta\_{\rm DIT} + \theta\_{\rm G}) & \mathbf{E} = \cos(\theta\_{\rm S\rm d} + \theta\_{\rm G}) & \mathbf{and} \end{array}$$

Then,

$$\begin{aligned} \mathcal{K}\_{\text{DLT}} &= \mathcal{K}\_{\text{Sfd}} \times P\_{\text{RATIO}} \times M\\ &= \text{Ratio Factor} \times M = \text{Ratio Factor} \times \frac{1 + A^2 B^2 \cdot 2ABD}{1 + A^2 C^2 \cdot 2ACE} \end{aligned} \tag{10}$$

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiations with respect to the individual variables in equation (10), total 9 items. The derived sensitivity coefficients are as follows:

1. The sensitivity coefficient for KStd

*Std*

*P* .

*Std*

$$\frac{\partial \mathbf{K}\_{DUT}}{\partial \mathbf{K}\_{Std}} = \frac{\mathbf{K}\_{DUT}}{\mathbf{K}\_{Std}} \tag{11}$$

2. The sensitivity coefficient for PDUT

$$\frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{DUT}} = \frac{\mathcal{K}\_{DUT}}{P\_{DUT}} \tag{12}$$

3. The sensitivity coefficient for PStd

$$\frac{\partial \mathcal{K}\_{DUT}}{\partial P\_{Std}} = -\frac{\mathcal{K}\_{DUT}}{P\_{Std}} \tag{13}$$

4. The sensitivity coefficient for |ΓStd|(=C)

$$\frac{\partial K\_{DIT}}{\partial \left| \Gamma\_{Std} \right|} = \frac{K\_{DIT}}{M\_D} \times 2A \text{(AC - E)}\tag{14}$$

5. The sensitivity coefficient for θStd

$$\frac{\partial \mathcal{K}\_{DUT}}{\partial \,\theta\_{Std}} = -\frac{K\_{DUT}}{M\_D} \times \text{2AC}\sin(\theta\_{Std} + \theta\_G) \tag{15}$$

6. The sensitivity coefficient for |ΓDUT|(=B)

$$\frac{\partial K\_{DUT}}{\partial \left| \Gamma\_{DUT} \right|} = \text{Ratio Factor} \times \frac{1}{M\_D} \times \text{2A} \{AB \text{ -} D\} \tag{16}$$

7. The sensitivity coefficient for θDUT

182 Modern Metrology Concerns

*M - N DUT G DUT G DUT G DUT G -* 2 22 |1 | 1 | || | 2| || |cos( )

*M - D Std G Std G Std G Std G -* 2 22 |1 | 1 | || | 2| || |cos( )

Let *A=|ΓG|, B=|ΓDUT|, C=|ΓStd|, D=cos(θDUT+θG), E=cos(θStd+θG),* and

*A B - ABD Ratio Factor M Ratio Factor*

According to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiations with respect to the individual variables in equation (10), total 9

> *DUT DUT Std Std*

*DUT DUT DUT DUT*

*DUT DUT Std Std*

*- A AC - E M* 2( ) | |

> *M* 2 sin( )

*M* 1 2( ) | |

*K K K K*

*K K P P*

*K K - P P*

*- AC*

*DUT D*

*K Ratio Factor A AB - D*

*DUT DUT Std D*

*K K*

*DUT DUT*

*K K*

*DUT*

*Std D*

*A C - ACE*

(11)

(12)

(13)

(14)

*Std G*

(16)

(15)

2 2 2 2 1 2 1 2 (10)

Let *MN* represent the numerator and *MD* denominator of mismatch *M*:

*DUT*

*Std*

*K KP M DUT Std RATIO*

items. The derived sensitivity coefficients are as follows:

*Std*

*P* .

1. The sensitivity coefficient for KStd

2. The sensitivity coefficient for PDUT

3. The sensitivity coefficient for PStd

5. The sensitivity coefficient for θStd

4. The sensitivity coefficient for |ΓStd|(=C)

6. The sensitivity coefficient for |ΓDUT|(=B)

*P Ratio Factor K*

Then,

$$\frac{\partial K\_{DIT}}{\partial \theta\_{DIT}} = \text{Ratio Factor} \times \frac{2AB}{M\_D} \text{sin} (\theta\_{DIT} + \theta\_G) \tag{17}$$

8. The sensitivity coefficient for |ΓG|(=A)

$$\frac{\partial \mathcal{E}\_{\text{DUT}}}{\partial \left| \, \Gamma\_{\text{G}} \right|} = \text{Ratio Factor} \times \frac{2B}{M\_{\text{D}}} \times \text{(AB - D)} \cdot \frac{M\_{\text{N}} \times 2\text{C}}{\text{(M}\_{\text{D}}\text{)}^{2}} \times \text{(AC - E)}\tag{18}$$

9. The sensitivity coefficient for θG

$$\begin{aligned} \frac{\partial K\_{\text{DIT}}}{\partial \theta\_{\text{DIT}}} &= \text{Ratio Factor} \times \{ \frac{2AB}{M\_D} \sin(\theta\_{\text{DIT}} + \theta\_{\text{G}}) \cdot \frac{M}{M\_D} \times 2AC \sin(\theta\_{\text{Sd}} + \theta\_{\text{G}}) \} \\ &= \text{equation} \text{(17)} + \text{equation} \text{(15)} \end{aligned} \tag{19}$$

#### **<IP_ADDRESS> Polar representation of sensitivity coefficients for equation (4)**

Representing calibration factor *K*DUT of equation (4) with real and imaginary components, the following denotation is used to denote the real and imaginary components of reflection coefficient:

 $a = \text{Re}\{\Gamma\_{\text{Sid}}\} = \Gamma\_{\text{Sid} \cdot \text{Re}\,\prime\,\prime\,\prime}$   $b = \text{Im}\{\Gamma\_{\text{Sid}}\} = \Gamma\_{\text{Sid} \cdot \text{Im}\,\prime\,\prime\,\prime\,\prime} = \text{Re}\{\Gamma\_{\text{DIT}}\} = \Gamma\_{\text{DIT} \cdot \text{Re}\,\prime\,\prime\,\prime\,\prime\,\prime}$   $d = \text{Im}\{\Gamma\_{\text{DIT}}\} = \Gamma\_{\text{DIT} \cdot \text{Im}\,\prime\,\prime\,\prime}$   $e = \text{Re}\{\Gamma\_{\text{G}}\} = \Gamma\_{\text{G-Im}}$   $\text{and } f = \text{Im}\{\Gamma\_{\text{G}}\} = \Gamma\_{\text{G-Im}}$

Then the calibration model becomes:

$$\begin{split} K\_{\text{DIT}} &= K\_{\text{Sfd}} \times P\_{\text{RATIO}} \times M1 \\ &= \text{Ratio Factor} \times \frac{1 + 2df \cdot 2ce + c^2e^2 + d^2e^2 + c^2f^2 + d^2f}{1 + 2bf \cdot 2ae + a^2e^2 + b^2e^2 + a^2f^2 + b^2f^2} \end{split} \tag{20}$$

Let *M1N* represent the numerator and *M1D* denominator of mismatch *M1* in real and imaginary format:

$$\begin{aligned} M\mathbf{1}\_N &= \|\mathbf{1} \cdot \boldsymbol{\Gamma}\_{DLT}\boldsymbol{\Gamma}\_G\|^2 = \mathbf{1} + 2df \cdot \mathbf{2}ac + a^2e^2 + d^2e^2 + c^2f^2 + d^2f^2 \\ M\mathbf{1}\_D &= \|\mathbf{1} \cdot \boldsymbol{\Gamma}\_{Std}\boldsymbol{\Gamma}\_G\|^2 = \mathbf{1} + 2bf \cdot \mathbf{2}ac + a^2e^2 + b^2e^2 + a^2f^2 + b^2f^2 \end{aligned}$$

Similarly, according to GUM for the law of propagation of uncertainties, the sensitivity coefficients are partial differentiation with respect to each input quantities in equation (20), total 9 items. The derived sensitivity coefficients are as follows:

$$\frac{\partial \,\,\mathrm{K}\_{\mathrm{DUT}}}{\partial \,\,\mathrm{T}\_{\mathrm{Std\cdot Re}}} = \frac{\mathrm{K}\_{\mathrm{DUT}}}{M \mathbf{1}\_{D}} \times \mathrm{(2e-2ae^{2}-2af^{2})}\tag{21}$$

3. The sensitivity coefficient for Γ*Std-Img* (= *b*)

$$\frac{\partial \, K\_{\text{DIT}}}{\partial \, \Gamma\_{S\text{tf}\text{-}\text{Im}\, \text{g}}} = \frac{K\_{\text{DIT}}}{M \mathbf{1}\_D} \times \{ (2f + 2be^2 + 2bf^2) \} \tag{22}$$

4. The sensitivity coefficient for Γ*DUT-Re* (= *c*)

$$\frac{\partial \, K\_{DIT}}{\partial \, \Gamma\_{DIT-\text{Re}}} = \text{Ratio Factor} \times \frac{2e + 2ce^2 + 2cf^2}{M1\_D} \tag{23}$$

5. The sensitivity coefficient for Γ*DUT-Img* (= *d*)

$$\frac{\partial \, K\_{DIT}}{\partial \, \Gamma\_{DIT-\text{Im}\,g}} = \text{Ratio Factor} \times \frac{2f + 2de^2 + 2df^2}{M1\_D} \tag{24}$$

6. The sensitivity coefficient for Γ*G-Re* (= *e*)

$$\frac{\partial \text{ } \mathbf{K}\_{\text{DITT}}}{\partial \text{ } \Gamma\_{G\text{-Re}}} = \text{Ratio Factor} \times \frac{2c + 2c^2e + 2d^2e + \left(M1\right)\left(2a - 2a^2e - 2b^2e\right)}{M1\_D} \tag{25}$$

7. The sensitivity coefficient for Γ*G-Img* (= *f*)

$$\frac{\partial \, K\_{\text{DLT}}}{\partial \, \sigma \, \Gamma\_{G-\text{Im}\,\mathcal{g}}} = \text{Ratio Factor} \times \frac{2d + 2c^2 f + 2d^2 f \cdot \text{(M1)} \left(2b + 2a^2 f + 2b^2 f\right)}{M \mathbf{1}\_D} \tag{26}$$

#### **<IP_ADDRESS> Example**

With the same data as used in Table 2, the uncertainty budget is listed in Table 3 with mismatch corrections considered in the calibration equation.

) *DUT DUT*

{(2 2 2 } 1

2 2

*M*

*M*

2 2 2 2

2 2 2 2

*M*

*M*

uncertainty distribution coefficient contribution

1

2 2 2 12 2 2

(26)

2 2 2 12 2 2 1 (25)

22 2 1 (24)

22 2 1 (23)

(22)

2 2

2 2

*K K f be bf M*

*DUT - D K e ce cf Ratio Factor*

*DUT - g D K f de df Ratio Factor*

*G- D*

*G- g D*

18 GHz based on best specifications

*DUT*

*DUT*

*K d c f d f- M b a f b f Ratio Factor*

With the same data as used in Table 2, the uncertainty budget is listed in Table 3 with

Quant. Estim. Standard probability Sensitivity Uncertainty

*X*i *x*i *u*(*x*i) *c*i *u*i*(y) K*Std 0.9894 0.0012 normal 0.9996 0.0012 *P*DUT 1.0158 0.0018 normal 0.9737 0.0018 *P*Std 1.0021 0.0004 normal -0.9870 -0.0004 DUTmag 0.0600 0.0120 normal -0.4613 -0.0055 DUTPhase 3.1416 1.5709 normal 0.0000 0.0000 Stdmag 0.0300 0.0060 normal 0.4581 0.0027 StdPhase 3.1416 1.5709 normal 0.0000 0.0000 Gmag 0.2300 0.0460 normal -0.0606 -0.0028 Gphase 3.1416 1.5709 normal 0.0000 0.0000 *y=K*U 0.9890 0.0071

*K c c e d e M a- a e- b e*

*Std- g D*

Im

*DUT*

*DUT*

*Ratio Factor*

mismatch corrections considered in the calibration equation.

Im

Re

3. The sensitivity coefficient for Γ*Std-Img* (= *b*)

4. The sensitivity coefficient for Γ*DUT-Re* (= *c*)

5. The sensitivity coefficient for Γ*DUT-Img* (= *d*)

6. The sensitivity coefficient for Γ*G-Re* (= *e*)

Re

7. The sensitivity coefficient for Γ*G-Img* (= *f*)

Im

**<IP_ADDRESS> Example**

Table 3. Uncertainty budget at 18 GHz for measurement uncertainty improvement with mismatch correction for Fig. 2 measurement setup. Uncertainties *ui* are at one standard deviation. Powers are measured in mW.

Compare the result in Table 3 with that in Table 2, the uncertainty is improved by mismatch correction. In the calculation, the magnitude uncertainty of reflection coefficient is assumed to be 40% of its value and the phase uncertainty is assumed to be 180⁰ and value is *π* for all. In terms of computation cost, it is the same for both Cartesian and Polar coordinate representations. Note that additional uncertainties should also be included such as the connector repeatability, noise, cable flexibility, drift, linearity and frequency error when they are not negligible in practical application.


**Abstract**

Bangladesh, a country with a size of 147,570 km2 , has the largest delta in the world with one-third of the country residing a costal shoreline. The livelihood of the inhabitants is exposed to a series of tribulations causing radical setbacks, of which natural disasters like tropical cyclones bring large-scale salinity intrusion. Occurring due to three main causal factors including climate change, sedimentation, and low water flow, salinity poses challenges in agriculture and overall food security, hence impeding the health and livelihood of marginalized women, children, and elderly and explicably the overall vulnerable population at large. Against that backdrop, this chapter will delineate three broad spectra of Bangladesh's approach to "living with salt": (1) the assessment of salt intrusion in water and soil in Bangladesh's coastal zones due to climate change, (2) understanding the vulnerabilities to salinity within marginalized population, and (3) coping or adapting strategies to combat and live with salt. The chapter also includes the findings of a recent case study conducted on Bangladesh coastal zone to demonstrate the current livelihood conditions under salinity, conditions of the actions taken by the government and nongovernment organizations, gaps, and recommendations for a more resilient coastal community.

**Keywords:** coastal salinity, climate change, adaptation, Bangladesh, resilience


6. Pipeline vibration limiting

Pipeline vibration limiting regulations can be divided into the following categories:

The requirements [design documentation] state that "the maximum allowable amplitude of vibrations of the process pipelines is 0.2 mm at vibration frequency of max 40 Hz" [2].

Offshore pipeline specifications do not provide either limitations for the pressure pulsations or vibration limitations.

Low-frequency vibrations of the pipelines under principal modes, when such vibrations are close to be harmonic, can be easily evaluated on the basis of the amplitude of vibration displacement since in this case they are proportional to the stresses induced in the pipelines and can be regarded as a strength factor of the pipelines.

We get the following expression for k-form of the vibrations using formula [5], for the root mean square value of vibrations:

$$\sqrt{\sigma\_k^2(z)} = \left[ \int\_0^\infty \mathbb{C}\_k(\omega) \frac{\mathfrak{d}\_{\mathbb{Q}\mathbb{Q}}(\omega) d\omega}{\sqrt{\left(\omega\_0^2 - \omega^2\right)^2 + \left(2\beta\alpha\omega\_0\right)^2}} \right] \frac{ED}{2}$$

In the event of random vibrations, the combined stress in the pipeline is the following:

$$
\overline{\sigma^2}(z) = \sum\_{k=1}^N \overline{\sigma\_k^2}(z).
$$

However, in the regulatory documents for offshore pipelines, there are not only restrictions on pressure pulsations but also restrictions on vibrations. Low-frequency oscillations of pipelines along lower forms, when these oscillations are close to harmonic, can be conveniently estimated from the amplitude values of the vibrational displacement because in this case they are proportional to the stresses arising in the pipelines and are indicators of the strength of the pipelines.

Vibration velocity can be written as:

$$V\_l = \int\_0^T a(t)dt = \int\_0^T B\_0 \sin\omega t dt = B\_0 n / 2\pi f\_K \tag{20}$$

Development of the standardized vibration limits is complicated due to a wide variety of the

Vibration Strength of Pipelines

31

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However, for determination of the pipeline system reliability exposed to vibrations, it is required to consider the effect of vibrations on failures and malfunctions on the basis of certain assumptions regarding damage accumulation in the structures and failure occurrence. If stresses in the elastic elements of pipelines (supports) are considered failure criteria, the vibration strength of the supports shall be evaluated using calculation models of support structures. The private oilextracting companies apply burying of pipelines. It is just concealment of a problem. In the

During the analysis of the first fundamental forms of vibrations which lie within 20 Hz,

The following stages are included into the obligatory vibration tests of the pipeline: study of the operating conditions of the system and analysis of the dynamic loads acting on the pipeline; determination of the potential failure patterns; and selection of failure occurrence criteria due

According to the regulations of the Ministry of Gas Industry, emergency vibration level [2] is measured using vibration velocity Ve = 18 mm/s, and the warning vibration level is estimated

During the pipeline vibration analysis, it is necessary to know rigidity characteristics of the system components. The rigidity of pipe of permanent round section is characterized by the

Up to the present moment, frequency ranges have not been yet defined for the offshore pipelines where one or another vibration parameter shall be used for vibration limitation purposes.

Vmax ¼ vbasekη Vmax:

Amplitude of vibration stresses at different frequencies is determined during calculation of the

The main criterion of the pipeline vibration strength is detuning of natural frequencies fj from

Let us determine safe vibration velocity for the pipeline kinematically excited on the movable base plate. Amplitudes of vibration voltage at different frequencies are determined by the results of calculation of forced oscillations of the pipeline. The way to ensure the vibration

Vibrations of marine pipelines with a protective coating to decrease the interaction of the pipeline with the coating result in cracking in the coating of corrosion and damage to the coating.

discrete frequencies of the excitatory loads fip, defined as described in 2.2 [6].

resistance of the pipeline is to detune the natural frequencies fj of the structure [7].

.

requirements to the characteristics of the vibration capacity of various equipments.

seismic phenomena, vibrations arise in the thickness of the Earth.

vibration displacement is frequently used as a test parameter.

to vibrations.

by the exceedance Ve = 41 mm/s [9, 10].

forced vibrations of the pipeline [8].

following parameters: EI is bending rigidity, N�m2

The allowable amplitude of vibrations is defined as follows:

where Vl is vibration velocity (mm/s); a is vibration acceleration (mm/s2 ) with amplitude В0; fk is frequency of multiplicity factor k.

During the evaluation of the vibration strength, the maximum amplitude of equivalent vibration stresses shall be determined for each representative section of the pipeline. This amplitude is obtained as a result of various modal superpositions. Dimension of the transfer function depends on the type of disturbance and response against which the transfer function is determined.

When exposed to random vibrations, root-mean-square value of the maximum stress in the pipeline can be found using the transfer function for the maximum stress with respect to acceleration of the pipeline supports:

$$|H\_{\sigma}(\omega, z)| = \frac{\mathsf{C}\_{k}(\omega)\frac{\mathsf{ED}}{2}y\_{k}''(z)}{\sqrt{\left(\omega\_{0}^{2} - \omega^{2}\right)^{2} + \left(2\beta\omega\omega\_{0}\right)^{2}}}\tag{21}$$

where D is outside diameter of pipe.

Pipeline response to broadband random vibrations can be defined as a combined effect of several narrowband random vibrations. The narrowband vibrations of the pipelines occur as a response to the broadband excitation under low damping. The mean frequency of the narrowband vibrations can be calculated from the Rice's formula:

$$
\omega\_0^2 = \frac{\int\_{-\infty}^{\infty} \omega^2 \Phi\_{YY}(\omega) d\omega}{\int\_{-\infty}^{\infty} \Phi\_{YY}(\omega) d\omega} = \frac{R\_y^{\cdot}(0)}{R\_y(0)} = \frac{\sigma\_{\dot{y}}^2}{\sigma\_y^2} \tag{22}
$$

Root mean square value of the pipeline movement to be subject to vibrations can be calculated from the following formula:

$$
\sigma = \left[ \int\_{f\_1}^{f\_2} \eta\_f^2 \Phi(f) df \right]^{1/2}, \quad \text{here } \sigma = \sqrt{\sum\_{i=1}^N \eta\_{f\_i}^2 \Phi(f\_i) \Delta f\_i}. \tag{23}
$$

where ηf is a dynamic response factor-relation between displacement amplitude of the anchor points for the pipeline and relative displacement amplitude of the pipeline sections at specified frequency; Ф(f) is a spectral density of the disturbed random vibration in the frequency band f1 and f2; Δfi is an interval of the frequency band segmentation f1, f2; N is a number of intervals of the frequency band segmentation.

Development of the standardized vibration limits is complicated due to a wide variety of the requirements to the characteristics of the vibration capacity of various equipments.

However, for determination of the pipeline system reliability exposed to vibrations, it is required to consider the effect of vibrations on failures and malfunctions on the basis of certain assumptions regarding damage accumulation in the structures and failure occurrence. If stresses in the elastic elements of pipelines (supports) are considered failure criteria, the vibration strength of the supports shall be evaluated using calculation models of support structures. The private oilextracting companies apply burying of pipelines. It is just concealment of a problem. In the seismic phenomena, vibrations arise in the thickness of the Earth.

During the analysis of the first fundamental forms of vibrations which lie within 20 Hz, vibration displacement is frequently used as a test parameter.

The following stages are included into the obligatory vibration tests of the pipeline: study of the operating conditions of the system and analysis of the dynamic loads acting on the pipeline; determination of the potential failure patterns; and selection of failure occurrence criteria due to vibrations.

According to the regulations of the Ministry of Gas Industry, emergency vibration level [2] is measured using vibration velocity Ve = 18 mm/s, and the warning vibration level is estimated by the exceedance Ve = 41 mm/s [9, 10].

During the pipeline vibration analysis, it is necessary to know rigidity characteristics of the system components. The rigidity of pipe of permanent round section is characterized by the following parameters: EI is bending rigidity, N�m2 .

Up to the present moment, frequency ranges have not been yet defined for the offshore pipelines where one or another vibration parameter shall be used for vibration limitation purposes.

The allowable amplitude of vibrations is defined as follows:

Vibration velocity can be written as:

is frequency of multiplicity factor k.

acceleration of the pipeline supports:

where D is outside diameter of pipe.

from the following formula:

σ ¼

of the frequency band segmentation.

ðf 2 f 1 η2 f Φð Þf df " #1=2

band vibrations can be calculated from the Rice's formula:

ω2 0 ¼ Ð ∞

determined.

30 System of System Failures

Vl ¼ ðT 0

a tð Þdt ¼

where Vl is vibration velocity (mm/s); a is vibration acceleration (mm/s2

ðT 0

During the evaluation of the vibration strength, the maximum amplitude of equivalent vibration stresses shall be determined for each representative section of the pipeline. This amplitude is obtained as a result of various modal superpositions. Dimension of the transfer function depends on the type of disturbance and response against which the transfer function is

When exposed to random vibrations, root-mean-square value of the maximum stress in the pipeline can be found using the transfer function for the maximum stress with respect to

> 2 y00 k ð Þz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> > yð Þ0 Ryð Þ0 ¼

> > > X N

i¼1 η2 f i Φ f i � �Δf i : vuut (23)

σ2 y\_ σ2 y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0 � ω2 � �2 þ 2βωω0

j j Hσð Þ ω; z ¼ Ckð Þ ω ED

ω2

Pipeline response to broadband random vibrations can be defined as a combined effect of several narrowband random vibrations. The narrowband vibrations of the pipelines occur as a response to the broadband excitation under low damping. The mean frequency of the narrow-

�∞ ω2ΦYYð Þ ω dω

�∞ ΦYYð Þ ω dω ¼ R}

Root mean square value of the pipeline movement to be subject to vibrations can be calculated

, here σ ¼

where ηf is a dynamic response factor-relation between displacement amplitude of the anchor points for the pipeline and relative displacement amplitude of the pipeline sections at specified frequency; Ф(f) is a spectral density of the disturbed random vibration in the frequency band f1 and f2; Δfi is an interval of the frequency band segmentation f1, f2; N is a number of intervals

Ð ∞

B0 sin ωtdt ¼ B0n=2πf K (20)

� �2 q (21)

) with amplitude В0; fk

(22)

$$\mathbf{V}\_{\text{max}} = \upsilon\_{\text{base}} k \eta \text{ V}\_{\text{max}}$$

Amplitude of vibration stresses at different frequencies is determined during calculation of the forced vibrations of the pipeline [8].

The main criterion of the pipeline vibration strength is detuning of natural frequencies fj from discrete frequencies of the excitatory loads fip, defined as described in 2.2 [6].

Let us determine safe vibration velocity for the pipeline kinematically excited on the movable base plate. Amplitudes of vibration voltage at different frequencies are determined by the results of calculation of forced oscillations of the pipeline. The way to ensure the vibration resistance of the pipeline is to detune the natural frequencies fj of the structure [7].

Vibrations of marine pipelines with a protective coating to decrease the interaction of the pipeline with the coating result in cracking in the coating of corrosion and damage to the coating.

#### Example.

#### Vibration strength of pipelines.

Low-frequency vibrations of the pipelines under principal modes, when such vibrations are close to be harmonic, can be easily evaluated on the basis of the amplitude of vibration displacement because in this case they are proportional to the stresses induced in the pipelines and can be regarded as a strength factor of the pipelines.

σ�1 value can be defined either using reference data or Manson formula [4]:

σ�1 ¼ 1, 75σB=N0:12, here N is a number of loading cycles.

Let us consider the following example of calculating the allowable stress amplitudes in the pipeline wall.

Researches were made for vibration speed (response characteristic: root mean square value of Vmax = 0.0103 cm/s) and pressure pulsation amplitude of ΔP = 0.5 MPa for the landfall section of the offshore pipeline with rated pressure of 17.5 MPa. Outside diameter of the pipeline is D = 406 mm, wall thickness is 17.5 mm. Material grade is X52 (σв = 455 MPa, σ�1 ≈ 916 MPa, E = 0.20457�106MPa, σe = 358 MPa). Pipe section modulus is W = 0.00199 m3 .

strength of the pipeline steel (see Table 1), stress amplitude σΔt ¼ 38.322 MPa in the pipeline

Let us review the seismogram of seismic intensity 6 to MSK-64 scale for the vibration strength analysis of the landfall section of the offshore pipeline. The seismic impact is characterized by the following parameters: maximum acceleration amax = 0.94485 cm/s2 at t = 0.12 s (Figure 2).

> ¼ 2π N∙Δt

Periodogram results can be interpreted as dispersed data at frequencies given in Table 2 and

Ser.no. Frequency Period Periodogram Density Spectral density in hamming window

ð Þ rad=s

Vibration Strength of Pipelines

33

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wall. Fatigue strength factor of the pipeline shall be at least п = 2.0 [3].

Figure 2. Accelerogram recording used during calculations.

We get the line spectrum of the signal consisting of the harmonics:

 0.00000 0.0 1488.55 0.035714 0.02174 46.0 2836.21 1828.18 0.241071 0.04348 23.0 1695.297 1839.68 0. 446429 0.06522 15.3333 1456.8 1528.67 0.241071 0.08696 11.5 1341.92 1355.41 0.035714

 0.1087 9.2 1255.43 1258.18 0.13044 7.66667 1174.63 1174.32 0.15217 6.67143 1092.56 1091.35 0.17391 5.75 1007.15 1005.84 0.19585 5.11111 918.245 917.192 0.21739 4.6 826.556 825. 941

where Тmin is a minimum period.

Figure 3.

Δω ¼ 2π Tmin

Kt≔1

σBB�β�k=806.08

$$\text{norm1} \cdot \left(1 + \frac{\sigma \text{BB}}{\sigma \text{B}} \cdot \frac{1 + \text{rq}}{1 - \text{rq}}\right) = 64.214$$

If length l of the pipeline section is 30 m, displacement y0 is determined by the formula (7) and is equal to 0.026 m, considering the formula (9) vmax = 0.035 m

If pressure pulsation ΔР is 0.5 MPa and work pressure is 15.7 MPa (β = 0.8), stress variation on the pipeline wall is 6.386 MPa. Stress ratio (17) qr is 0.938. Stress amplitude in the pipeline wall is equal to σΔq ¼ 12.553 MPa considering steel tensile strength and fatigue strength (refer to Table 1).

Temperature fluctuations Δt = 5� result in stress variation of 12.36 MPa in the pipeline wall and are equal to rt = 0.817 as shown in the formula (19). Allowing for tensile strength and fatigue

Table 1. The main physical characteristics of steel pipe [2].

Figure 2. Accelerogram recording used during calculations.

Example.

32 System of System Failures

pipeline wall.

σBB�β�k=806.08

Table 1).

Vibration strength of pipelines.

and can be regarded as a strength factor of the pipelines.

σ�1 ¼ 1, 75σB=N0:12, here N is a number of loading cycles.

σ�1 value can be defined either using reference data or Manson formula [4]:

0.20457�106MPa, σe = 358 MPa). Pipe section modulus is W = 0.00199 m3

nm1 � 1 þ

is equal to 0.026 m, considering the formula (9) vmax = 0.035 m

Name of the characteristics Steel

Table 1. The main physical characteristics of steel pipe [2].

Low-frequency vibrations of the pipelines under principal modes, when such vibrations are close to be harmonic, can be easily evaluated on the basis of the amplitude of vibration displacement because in this case they are proportional to the stresses induced in the pipelines

Let us consider the following example of calculating the allowable stress amplitudes in the

Researches were made for vibration speed (response characteristic: root mean square value of Vmax = 0.0103 cm/s) and pressure pulsation amplitude of ΔP = 0.5 MPa for the landfall section of the offshore pipeline with rated pressure of 17.5 MPa. Outside diameter of the pipeline is D = 406 mm, wall thickness is 17.5 mm. Material grade is X52 (σв = 455 MPa, σ�1 ≈ 916 MPa, E =

Kt≔1

1 þ rq 1 � rq

¼ 64:214

σBB σB �

If length l of the pipeline section is 30 m, displacement y0 is determined by the formula (7) and

If pressure pulsation ΔР is 0.5 MPa and work pressure is 15.7 MPa (β = 0.8), stress variation on the pipeline wall is 6.386 MPa. Stress ratio (17) qr is 0.938. Stress amplitude in the pipeline wall is equal to σΔq ¼ 12.553 MPa considering steel tensile strength and fatigue strength (refer to

Temperature fluctuations Δt = 5� result in stress variation of 12.36 MPa in the pipeline wall and are equal to rt = 0.817 as shown in the formula (19). Allowing for tensile strength and fatigue

Yield strength, МPа 358 448 Strength limit Rm, МPа 455 530 Permissible stresses in metal pipes σ, МPа 255.6 292.8

.

Х52∅406 Х65∅711

strength of the pipeline steel (see Table 1), stress amplitude σΔt ¼ 38.322 MPa in the pipeline wall. Fatigue strength factor of the pipeline shall be at least п = 2.0 [3].

Let us review the seismogram of seismic intensity 6 to MSK-64 scale for the vibration strength analysis of the landfall section of the offshore pipeline. The seismic impact is characterized by the following parameters: maximum acceleration amax = 0.94485 cm/s2 at t = 0.12 s (Figure 2).

We get the line spectrum of the signal consisting of the harmonics:

$$
\Delta w = \frac{2\pi}{T\_{min}} = \frac{2\pi}{N \cdot \Delta t} \left(\text{rad/s}\right),
$$

where Тmin is a minimum period.

Periodogram results can be interpreted as dispersed data at frequencies given in Table 2 and Figure 3.

Mean mathematic amplitude Вк over a period of observation Т shall be written as:

B0 ¼ 1 T ðT 0

where BKð Þt is a behavior of k-х components of vibration spectrum.

a tð Þdt ¼

where Vl is vibration velocity (mm/s); a is vibration acceleration (mm/s2

ðT 0

According to the standards of the Ministry of Gas Industry [6], an emergency vibration level is equal to the vibration speed Ve = 18 mm/s, and an alarm vibration level exceeds Ve = 41 mm/s. For pipeline sections more than 0.5 m, the vibration displacement span is restricted to 0.5 mm (refer to the standards of the National Compressor Engineering Association "Souzcompre-

Based on the allowable strength from [2], the allowable amplitude of vibrations is calculated for the pipeline of 408 mm diameter and 17.5 mm wall thickness, and allowable stress is 255.6 MPa. The effective span length of the pipeline section under review is 30 m. Amplitude of forced vibrations Y0 is 0.061 m. Safe vibration speed of the pipeline, which is excited kine-

Offshore pipelines can be regarded as new constructions in Russia. Industry-Specific Construc-

None of the private oil companies is interested in studying vibration strength of the pipelines; they make references to the existing standards. Offshore pipeline vibration analysis is very important and requires government involvement. Such researches should be performed by the

Private companies use a pipe burial method. The pipelines can be buried; however, it is not a

Vibrations of marine pipelines with a protective coating to decrease the interaction of the

way to solve the problem. Seismic impacts come from under the ground.

Vl ¼ ðT 0

Vibration velocity can be written as:

is frequency of multiplicity factor k.

matically on the moving platform, is 0.035 m/s.

tion Standard specifies dynamic vibrations of the pipeline.

pipeline with the coating result in cracking in the coating.

ssmach") [2].

7. Conclusion

large scientific institutes.

BKð Þt dt, (24)

B0sinωtdt ¼ B0n=2πf K (25)

) with amplitude В0; fk

Vibration Strength of Pipelines

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Table 2. Vibration impact periodogram.

Figure 3. Vibration impact periodogram.

Mean mathematic amplitude Вк over a period of observation Т shall be written as:

$$B\_0 = \frac{1}{T} \int\_0^T B\_K(t)dt,\tag{24}$$

where BKð Þt is a behavior of k-х components of vibration spectrum.

Vibration velocity can be written as:

Ser.no. Frequency Period Periodogram Density Spectral density in hamming window

 0.23913 4.18182 733. 242 733.16 0.26087 3.83333 639.703 640.204 0.28261 3.53846 547.464 548.566 0.30435 3. 28571 458.096 459.796 0.32609 3.06667 373.16 375.439 0.34783 2.875 294.168 296.99 0.36957 2.70588 222.536 225.855 0.3913 2.55556 159.563 163.32 0.41304 2.42105 106.391 110.52 0.43478 2.3 63.992 68.417 0.45652 2.19048 33.139 37.781 0.47826 2.09091 14.398 19.172

Table 2. Vibration impact periodogram.

34 System of System Failures

Figure 3. Vibration impact periodogram.

$$V\_{l} = \int\_{0}^{T} a(t)dt = \int\_{0}^{T} B\_{0} \sin \omega t dt = B\_{0} n / 2\pi f\_{K} \tag{25}$$

where Vl is vibration velocity (mm/s); a is vibration acceleration (mm/s2 ) with amplitude В0; fk is frequency of multiplicity factor k.

According to the standards of the Ministry of Gas Industry [6], an emergency vibration level is equal to the vibration speed Ve = 18 mm/s, and an alarm vibration level exceeds Ve = 41 mm/s. For pipeline sections more than 0.5 m, the vibration displacement span is restricted to 0.5 mm (refer to the standards of the National Compressor Engineering Association "Souzcompressmach") [2].

Based on the allowable strength from [2], the allowable amplitude of vibrations is calculated for the pipeline of 408 mm diameter and 17.5 mm wall thickness, and allowable stress is 255.6 MPa. The effective span length of the pipeline section under review is 30 m. Amplitude of forced vibrations Y0 is 0.061 m. Safe vibration speed of the pipeline, which is excited kinematically on the moving platform, is 0.035 m/s.

### 7. Conclusion

Offshore pipelines can be regarded as new constructions in Russia. Industry-Specific Construction Standard specifies dynamic vibrations of the pipeline.

None of the private oil companies is interested in studying vibration strength of the pipelines; they make references to the existing standards. Offshore pipeline vibration analysis is very important and requires government involvement. Such researches should be performed by the large scientific institutes.

Private companies use a pipe burial method. The pipelines can be buried; however, it is not a way to solve the problem. Seismic impacts come from under the ground.

Vibrations of marine pipelines with a protective coating to decrease the interaction of the pipeline with the coating result in cracking in the coating.


**3. Materials and Methods**

## *3.1. General Experimental Procedures*

HRESIMS data were obtained on a Waters Synapt G2 Q-TOF mass spectrometer (Waters Corporation, Milford, MA, USA). Optical rotations were measured on a Rudolph Research Analytical Autopol III polarimeter (Rudolph Research Analytical, Hackettstown, NJ, USA). 1D and 2D NMR spectra were acquired using a Bruker 600 MHz spectrometer (Bruker BioSpin GmbH, Rheinstetten, Germany). IR spectra were measured on a JASCO FT/IR-4100 spectrophotometer (JASCO Corporation, Tokyo, Japan). UV-visible spectra were measured by a Shimadzu UV-1650PC spectrophotometer. HPLC was carried out with a PrimeLine Binary pump (Analytical Scientific Instruments, Inc., El Sobrante, CA, USA) and a RI-101 detector (Shoko Scientific Co. Ltd., Yokohama, Japan). Semi-preparative HPLC was conducted using an ODS column (YMC-Pack-ODS-A, 250 × 10 mm i.d, 5 μM). Analytical HPLC was performed with an ODS column (YMC-Pack-ODS-A, 250 × 4.6 mm i.d, 5 μM). All the used reagents were purchased from Sigma-Aldrich (Merck KGaA, Darmstadt, Germany).


**4.1. Pipe deformation, buckling and ovality**

A bare pipe with no FJCs was tested first in the bend rig as a control specimen, whereupon it buckled at the first bend to the reel former, as shown in **Figure 14a**; this early onset buckle can be attributed to the stiffness mismatch between the full linepipe coating and the bare steel pipe causing strains to concentrate within the bare steel. It can also be seen that the point of initiation of the buckle is located to the left of the weld. The strain field predicted by the numerical model is shown in **Figure 14b**, where the strain concentration can indeed be observed in the uncoated region of the pipe. It can be seen that the pipe was also predicted to buckle after the first bend to the reel, albeit with the point of initiation of the buckle located closer to the weld. Given the high imperfection sensitivity of cylindrical shells in compression, this discrepancy in buckle location is likely down to a localised thinning of the pipe wall in the area around the buckle due to corrosion. It can be seen that, in areas in the steel pipe away from the buckle, the tensile strain is approximately 2.5%, in keeping with analytical predictions.

Despite the presence of the polymer coating, one of the field joints on the pipe with the thin hourglass FJC also buckled on the first bend to the reel. During initial simulations prior to the test campaign, ovalities in excess of 10% were expected; based on previous experience [9] this level of ovality is a strong predictor of the occurrence of buckling. The buckled field joint is shown in **Figure 15a**; as can be seen, there is noticeable lift-off from the reel former. In **Figure 15b**, the equivalent numerical prediction of the stress field is shown, with rippling observable in the compression zone.

The pipes with the thick hourglass and full FJCs did not buckle throughout the five bending cycles; the numerical models also predicted that no buckling would occur. In **Figure 16**, the ovalities recorded along the length of the pipe with the thick hourglass FJC after the first bend

**Figure 14.** Bending of field joint with no coating; (a) experimental observation and (b) numerical prediction of strain field.

to the reel are compared with numerical predictions, with the ovalities calculated using Eq.(1). The two peaks in the ovality distributions coincide with the location of the field joints. As can be seen, there is particularly good agreement in the region around the field joint furthest from the anchor end, while the predictions of ovality are underestimated at the field joint closest to the anchor end. This can be attributed to some relaxation of contact stress towards the anchor end during the simulation as the point of contact progresses along the pipe. Similar accuracy was obtained across all the tested pipes, thus increasing confidence in the ability of the numerical model to predict pipe deformations and ovalities but most importantly whether the field joints can withstand five full bending cycles without buckling. The results of the experimental investigations and the numerical analysis showed that, provided the correct FJC thickness is applied, reeling of pipelines with 100 mm-thick MLPP coatings is indeed achievable.

### **4.2. Stress whitening**

During testing of the specimens with FJCs applied, a phenomenon known as stress whitening was observed. Stress whitening occurs when the molecule chains within a polymer become damaged due to excessive tensile stresses causing plastic deformation, with holes and tears forming as the molecular structures are altered [10]. Light incident upon the affected zones is then diffused and scattered more readily, appearing as white discolorations on the surface of the polypropylene. In the following discussion, comparisons are made after the first bend to the reel former and holding in position overnight, i.e., after the field joint has cooled down fully.

of stress whitening in the thick hourglass FJC is compared to the numerical prediction of the

**Figure 16.** Comparison of measured and predicted ovality distributions in the pipe with the thick hourglass FJCs after

**Figure 15.** Pipe with thin hourglass FJC; (a) experimental observation, (b) simulated stress field with rippling observable.

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the location of areas where plastic deformation has occurred agree very well with the location

In the case of the thin hourglass FJC, as can be seen in **Figure 18a**, the level of stress whitening was not as prevalent or as noticeable, indicating that much less plastic deformation has occurred than in the thick hourglass FJC. This corroborates with the predicted stress field

it can be seen that

principal stress field. Given that the yield stress is approximately 16 N/mm2

of stress whitening above the chamfer toes observed onsite.

the first bend to the reel.

Owing to the inherent difficulty of installing stress measuring instrumentation into the coating and retrieving it afterwards, visual identification of stress whitening was used as an indicator of tensile stress concentrations within the IMPP material. In **Figure 17**, the appearance Numerical Analysis of Hot Polymer-Coated Steel Pipeline Joints in Bending http://dx.doi.org/10.5772/intechopen.72262 91

**Figure 15.** Pipe with thin hourglass FJC; (a) experimental observation, (b) simulated stress field with rippling observable.

to the reel are compared with numerical predictions, with the ovalities calculated using Eq.(1). The two peaks in the ovality distributions coincide with the location of the field joints. As can be seen, there is particularly good agreement in the region around the field joint furthest from the anchor end, while the predictions of ovality are underestimated at the field joint closest to the anchor end. This can be attributed to some relaxation of contact stress towards the anchor end during the simulation as the point of contact progresses along the pipe. Similar accuracy was obtained across all the tested pipes, thus increasing confidence in the ability of the numerical model to predict pipe deformations and ovalities but most importantly whether the field joints can withstand five full bending cycles without buckling. The results of the experimental investigations and the numerical analysis showed that, provided the correct FJC thickness is

**Figure 14.** Bending of field joint with no coating; (a) experimental observation and (b) numerical prediction of strain field.

90 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

applied, reeling of pipelines with 100 mm-thick MLPP coatings is indeed achievable.

During testing of the specimens with FJCs applied, a phenomenon known as stress whitening was observed. Stress whitening occurs when the molecule chains within a polymer become damaged due to excessive tensile stresses causing plastic deformation, with holes and tears forming as the molecular structures are altered [10]. Light incident upon the affected zones is then diffused and scattered more readily, appearing as white discolorations on the surface of the polypropylene. In the following discussion, comparisons are made after the first bend to the reel former and holding in position overnight, i.e., after the field joint has cooled down fully. Owing to the inherent difficulty of installing stress measuring instrumentation into the coating and retrieving it afterwards, visual identification of stress whitening was used as an indicator of tensile stress concentrations within the IMPP material. In **Figure 17**, the appearance

**4.2. Stress whitening**

**Figure 16.** Comparison of measured and predicted ovality distributions in the pipe with the thick hourglass FJCs after the first bend to the reel.

of stress whitening in the thick hourglass FJC is compared to the numerical prediction of the principal stress field. Given that the yield stress is approximately 16 N/mm2 it can be seen that the location of areas where plastic deformation has occurred agree very well with the location of stress whitening above the chamfer toes observed onsite.

In the case of the thin hourglass FJC, as can be seen in **Figure 18a**, the level of stress whitening was not as prevalent or as noticeable, indicating that much less plastic deformation has occurred than in the thick hourglass FJC. This corroborates with the predicted stress field

In the full FJC, stress whitening was observed after testing as shown in **Figure 19a**; stress concentrations were also predicted in the numerical model above the chamfers, as shown in **Figure 19b**. Although the agreement is not as clearly observable as in the hourglass FJCs, the level of stress predicted in the areas of stress whitening is commensurate with the yield stress of the material. Finally, in addition to the three pipes with hot tie-in field joints, another pipe with full FJCs was tested after it had fully cooled down to the ambient temperature. As can be seen in **Figure 20a**, there was no evidence of stress whitening visible in this specimen, and as shown in **Figure 20b**, the numerical model also predicted a uniform stress field with no concentra-

Bending of the steel when it is still hot and weakened leads to higher strains and deformations around the field joint as a whole. The thin hourglass field joint would have cooled quicker than the thick hourglass field joint, allowing the steel to regain relatively more strength and thus limiting the amount of strain. The increased deformation in the thick field joint has led to higher stresses in the pipe as well as the field joint coating, which can be observed upon

In the each of the three hot tie-in field joints, it can be seen that stress concentrations and areas of stress whitening occur above the chamfers, particularly above the chamfer toe. One explanation is that there is a stiffness mismatch either side of the chamfer that causes stress to accumulate at this point; however, the transition between the IMPP field joint coating and the MLPP linepipe coating occurs quite gradually over the length of the chamfer (approximately 175 mm), and so sudden peaks in stress would not be expected. Upon inspection of the evolution of the distribution of ovality along the length of the pipe specimens during the bend test, a correlation between the ovality gradient and the peaks in stress was apparent. In **Figure 21**, the distribution of ovality gradient along the length of the thick hourglass FJC specimen, as predicted by the numerical model, is shown. The locations of the chamfers and field joints are also overlaid on the graph. It can be seen that areas of peak ovality gradient coincide with the toes of the chamfer, where

**Figure 19.** Full hourglass FJC; comparison of (a) stress whitening observed during test with (b) numerical prediction of stress.

in the IMPP.

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tions, with maximum stresses of approximately 15 N/mm2

comparison of **Figures 17b** and **18b**.

**Figure 17.** Thick hourglass FJC; comparison of (a) stress whitening observed during test with (b) numerical prediction of stress.

shown in **Figure 18b** where the maximum stress is 13.85 N/mm2 , suggesting that the material has not yielded yet (although in practice some polymer chains may have been damaged already when the material was hotter and therefore less strong, as evidenced by the material curves shown in **Figure 10**). While it may be counter-intuitive that less stress whitening has occurred in the thin hourglass FJC rather than in the thick hourglass FJC, this can be explained by considering the reduction in longitudinal strain on the outer surface of the thin hourglass FJC since it is closer to the neutral axis of the section than the outer surface of the thick hourglass FJC. It can be seen that the areas of peak stress above the chamfer toe coincide with the lightest areas on the external surface of the tested pipe.

**Figure 18.** Thin hourglass FJC; comparison of (a) stress whitening observed during test with (b) numerical prediction of stress.

In the full FJC, stress whitening was observed after testing as shown in **Figure 19a**; stress concentrations were also predicted in the numerical model above the chamfers, as shown in **Figure 19b**. Although the agreement is not as clearly observable as in the hourglass FJCs, the level of stress predicted in the areas of stress whitening is commensurate with the yield stress of the material.

Finally, in addition to the three pipes with hot tie-in field joints, another pipe with full FJCs was tested after it had fully cooled down to the ambient temperature. As can be seen in **Figure 20a**, there was no evidence of stress whitening visible in this specimen, and as shown in **Figure 20b**, the numerical model also predicted a uniform stress field with no concentrations, with maximum stresses of approximately 15 N/mm2 in the IMPP.

Bending of the steel when it is still hot and weakened leads to higher strains and deformations around the field joint as a whole. The thin hourglass field joint would have cooled quicker than the thick hourglass field joint, allowing the steel to regain relatively more strength and thus limiting the amount of strain. The increased deformation in the thick field joint has led to higher stresses in the pipe as well as the field joint coating, which can be observed upon comparison of **Figures 17b** and **18b**.

In the each of the three hot tie-in field joints, it can be seen that stress concentrations and areas of stress whitening occur above the chamfers, particularly above the chamfer toe. One explanation is that there is a stiffness mismatch either side of the chamfer that causes stress to accumulate at this point; however, the transition between the IMPP field joint coating and the MLPP linepipe coating occurs quite gradually over the length of the chamfer (approximately 175 mm), and so sudden peaks in stress would not be expected. Upon inspection of the evolution of the distribution of ovality along the length of the pipe specimens during the bend test, a correlation between the ovality gradient and the peaks in stress was apparent. In **Figure 21**, the distribution of ovality gradient along the length of the thick hourglass FJC specimen, as predicted by the numerical model, is shown. The locations of the chamfers and field joints are also overlaid on the graph. It can be seen that areas of peak ovality gradient coincide with the toes of the chamfer, where

shown in **Figure 18b** where the maximum stress is 13.85 N/mm2

92 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

lightest areas on the external surface of the tested pipe.

rial has not yielded yet (although in practice some polymer chains may have been damaged already when the material was hotter and therefore less strong, as evidenced by the material curves shown in **Figure 10**). While it may be counter-intuitive that less stress whitening has occurred in the thin hourglass FJC rather than in the thick hourglass FJC, this can be explained by considering the reduction in longitudinal strain on the outer surface of the thin hourglass FJC since it is closer to the neutral axis of the section than the outer surface of the thick hourglass FJC. It can be seen that the areas of peak stress above the chamfer toe coincide with the

**Figure 18.** Thin hourglass FJC; comparison of (a) stress whitening observed during test with (b) numerical prediction of stress.

**Figure 17.** Thick hourglass FJC; comparison of (a) stress whitening observed during test with (b) numerical prediction of stress.

, suggesting that the mate-

**Figure 19.** Full hourglass FJC; comparison of (a) stress whitening observed during test with (b) numerical prediction of stress.

varying the thickness of the FJCs and the amount of cooldown time. Bend tests on pipes with FJC thicknesses from 20 mm up to a full FJC and cooldown times from 1 hour up to 24 hours were simulated and assessed to ascertain whether five full bend cycles could be completed without the pipe buckling. The results of the parametric study showed that an optimal thickness and cooldown time existed that represented significant material and time savings compared to current practice.

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Finite element models simulating the thermal and mechanical behaviour of hot tie-in field joints during coating application and bending have been developed. Experimental investigations recording the temperature evolution profiles within the field joint coatings after pouring and the mechanical behaviour of the field joints during bend testing were used as a basis for calibration and validation of the numerical models. Three separate field joint coating geometries were tested in order to examine the influence of coating thickness on the overall behaviour of the field joint. Thermal modelling in COMSOL Multiphysics employed temperature-dependent thermal properties obtained from material fingerprinting. It was found that it was necessary to model internal air cooling in the test specimens, which would not normally be required when modelling longer pipeline sections. Close agreement was observed upon comparison of temperature evolutions recorded onsite with those predicted by the numerical models, allowing for the

Numerical models were developed in Abaqus to simulate bend testing of the various pipe specimens, employing temperature-dependent material models obtained from material fingerprinting. Temperature fields obtained from the thermal numerical models were mapped onto the mechanical models and the process of bend testing over five full bend cycles simulated. The numerical predictions for pipe ovality and coating stress distributions were compared with the experimental results, with close agreement observed. It was also found that ovality gradient can be used as a predictor of the occurrence of stress whitening in the coating materials. The numerical analysis, coupled with the results of the experimental investigation,

The successful validation of the numerical models allowed for an extensive parametric study to be conducted, varying the field joint coating thickness and the cooldown times provided after application of the IMPP. It was found that an optimal FJC thickness existed that balanced the quicker cooldown times associated with thinner FJCs with the material strength benefits of thicker FJCs. The results of the study showed that use of this optimal FJC thickness can

The authors wish to thank the team at the Innovation Department of Heerema Marine Contractors for their assistance during the planning and execution of the experimental campaign, and also to David Haldane and the technical staff at Heriot Watt University for their

result in significant time savings when conducting reeling operations in practice.

predicted temperature fields to be subsequently applied to mechanical models.

showed that reeling of pipes with 100 mm-thick coating is possible.

**5. Conclusions**

**Acknowledgements**

support throughout testing.

**Figure 20.** Full hourglass FJC bent after being fully cooled down; comparison of (a) stress whitening observed during test with (b) numerical prediction of stress.

stress whitening was also observed very clearly in the test pipe, as shown in **Figure 17a**. In addition, although stresses close to yield were predicted by the numerical model in the full FJCs of the pipe that was bent after fully cooling down, since the stress distribution and also ovality distribution were quite uniform, i.e., with a small gradient, there was no stress whitening visible. This agreement between peak ovality gradient and areas of stress whitening can be attributed to the higher levels of strain associated with sudden large deformations in the steel pipe, which would cause associated large strains, and hence stresses, in the coating materials.

### **4.3. Parametric study**

As demonstrated by the comparisons between the experimental observations and the numerical results, the thermal and mechanicals models are capable of accurately predicting the behaviour of hot tie-in field joints in bending. With the model validated, a parametric study was conducted

**Figure 21.** Ovality gradient along the length of the pipe with the thick hourglass FJC with chamfer locations overlain.

varying the thickness of the FJCs and the amount of cooldown time. Bend tests on pipes with FJC thicknesses from 20 mm up to a full FJC and cooldown times from 1 hour up to 24 hours were simulated and assessed to ascertain whether five full bend cycles could be completed without the pipe buckling. The results of the parametric study showed that an optimal thickness and cooldown time existed that represented significant material and time savings compared to current practice.


*2.2. Direct Numerical Simulation*

In order to study the departure from the laminar Sexl–Womersley flow and the dynamics of intermittent localised turbulence, we perform direct numerical simulations (DNS) using our open-source pseudo-spectral simulation code nsPipe [22]. In nsPipe, the governing Equation (2) are treated in cylindrical coordinates (*r*, θ, *z*) and discretised using a Fourier–Galerkin ansatz in θ and *z* and high-order finite differences in *r*. No-slip boundary conditions are imposed at the solid pipe wall and periodic boundary conditions in θ and *z*. The discretised NSEs are integrated forward in time using a second-order predictor–corrector method with variable time-step size; details are given in López et al. [22] and the references therein. We have modified nsPipe to account for a time-dependent driving force that maintains a pulsating flow rate according to Equation (1) and an additional volume force *Fp* to perturb the flow locally.

We use a computational domain of 100*D* in length, and the number of radial grid points and Fourier modes used in our DNS is (*Nr* × *N*θ × *Nz*) = (96 × 192 × 2400). After dealiasing, this results in a spatial resolution of Δθ*R*+ = 3.1 and Δ*z*+ = 3.8, whereas radial grid points are clustered towards the pipe wall such that 0.06 ≤ Δ*r*+ ≤ 1.4, and 14 points lie within the buffer layer based on the shear Reynolds number for the statistically steady case (*A* = 0). By comparison to the resolution used in other contemporary DNS studies in the literature and the fact that we consider only moderate *A* (the instantaneous Reynolds number is never > (1 + *A*)*Re*), we expect our choice of resolution to be sufficient for all set-ups considered here. The adaptive time-step size is roughly Δ*t* = 2 × 10−3 *D us* .


*3.2. Geometric Morphometrics*

To avoid measurement error in our results, we calculated a Procrustes ANOVA showing that the mean square for individual variation exceeds the measurement error for wing shape (MS centroid size individuals: 0.000002 < 0.000107 MS centroid size error; and 7.0284 × 106 MS shape individuals <7.428 × 105 MS shape error), so we can retain the following results. A multivariate regression analysis was performed before all the subsequent statistical analyses, discarding any allometric effect on the data (% predicted: 0.8033%).

The PCA of the hindwing shape showed an accumulation of the shape variation in a very few number of dimensions. The first three PCs accounted for 51.246% (PC1 = 21.12%; PC2 = 17.18%; PC3 = 12.93%) of the total shape variation and provided an approximation of the total amount of hindwing shape variation. After averaging the shape variation between the different populations, the population with Cry34/35Ab1 toxin was localized at the left of the PCA closer to the wing shape phenotype of the Cry3Bb1 but far away from the resistant and non-resistant populations where the latter was similar to the population of the combination Cry3Bb1\_Cry34/35 (Figure 7).

**Figure 7.** Principal component analysis of the hindwing average shape between different populations: resistant to the toxins, adapted to crop rotation, and non-resistant *Diabrotica virgifera virgifera.* Color and sign code: red triangle: Cry34/35Ab1 resistant population; green square: CryBb1 resistant population; pink star: population adapted to crop rotation (RR); black circle: CryBb1—Cry34/35Ab1 resistant population; and blue rhomboid (NON): non-resistant population.

Procrustes ANOVA showed clear significant differences between the hindwings size and shape between populations (Table 4).

In order to graphically visualize the differences, the CVA maximized the variance between groups, finding similar results with the genetic type in which the population of Cry34/35Ab1 separated from the non-resistant populations (Figure 8). Finally, significant differences (using the different morphometric distances) were found between populations after a permutation was run (Table 5).

**Table 4.** Procrustes ANOVA for both centroid size and wing shape of *Diabrotica virgifera virgifera*, Sums of squares (SS) and mean squares (MS) are in units of Procrustes distances (dimensionless).

**Figure 8.** Canonical variate analysis of the hindwing shape between different populations resistant to the toxins: adapted to crop rotation and non-resistant population in *Diabrotica virgifera virgifera*. Color and sign code: red Cry34/35Ab1 resistant population; green CryBb1 resistant population; pink population adapted to crop rotation (RR); black CryBb1-Cry34/35Ab1 resistant population; and blue (NON): non-resistant population.

**Table 5.** Mahalanobis and Procrustes distances between groups obtained from canonical variate analysis. \*: *p* < 0.05; \*\*: *p* < 0.001.