Analysis of Crack Propagation Rate in Steel Pipeline

Analysis of Crack Propagation Rate in Steel Pipeline

International Journal of Engineering Research & Technology (IJERT)

ISSN: 2278-0181

Vol. 3 Issue 1, January – 2014

Analysis of Crack Propagation Rate in Steel Pipeline

Prince O. Agbainor1 and S. Odi-Owei2

1Department of Mechanical Engineering, Faculty of Engineering, University of Port Harcourt,

P.M.B. 5323, Choba, Port Harcourt, Nigeria.

2Department of Mechanical Engineering, Faculty of Engineering, Rivers State University of Science and Technology, Port Harcourt, Nigeria.

Abstract

For the stability assessment in piping components, it is important to determine the point of initiation of a crack, and to monitor and diagnose the subsequent crack propagation behavior as well as the failure patterns, to ensure pipeline structural integrity. This research analytically determines the crack propagation rate by the use of Improved crack tip opening area, (CTOA), and limiting crack speed,( LCS), models and subsequently predict the crack propagation behavior under fatigue loading. With reference to steel pipelines, inspection of sample data from Murtagians work is simulated and compared with analytically developed model and the results compared and eventually used to determine the depth of crack. Levenberg-Marquardt Algorithm, (LVA), was developed as an optimization tool to determine crack growth evolution. Also, the probability of pipeline failure caused by crack growth and uncertainties related to loads and material properties of the structure is estimated using a new Monte Carlo simulation technique.

Keywords: Crack Driving Force, Crack length, Limiting Crack speed, Crack tip opening area, Dynamic material toughness.

1. Introduction

   Oil and Gas transmission pipelines usually have a good safety record. This is due to a combination of

   good design, materials and operating practices. However, like any engineering structure, pipelines do occasionally fail, and such failures do have economic implications and are sometimes catastrophic, leading to loss of human life. The performance of an engineered system or product is often affected by unavoidable uncertainties [1]. The most common causes of damage and failures in onshore and offshore oil and gas transmission pipelines are external interference (also known as mechanical failure), corrosion and sabotage. Cracks occurring during the fabrication of a pipeline are usually assessed against recognized and proven control limits. Pipeline failures are often related to a breakdown in a system such as the corrosion protection system has become faulty and a combination of ageing coating, aggressive environment and rapid corrosion growth may lead to corrosion induced failure.

   Fatigue plays a very important role in piping systems and may lead to crack initiation from either the highly stressed regions or the flaws. The crack initiation and subsequent propagation may be avoided in any piping system. These pipe installations occasionally experience high amplitude vibration (e.g. seismic vibration) which may initiate/extend the existing cracks [10]. The monitoring of the cracks becomes more significant in the installation of a pipe carrying hazardous fluid. The design for through-wall- cracked pipe (TWC) is leak-before-break (LBB), based on fracture mechanisms concepts have also been adopted for fail-safe design criteria.

   The LBB demonstration which is based on failure mechanisms requires information on the initial size of a defect, initiation of crack growth from the inherent defect and subsequent crack growth rates. The nature of crack will grow and penetrate the wall thickness under fatigue loading. Thereafter, the crack will grow in a circumferential direction under cyclic loading.

   Generally, fatigue crack growth depends on the initial crack length, materials properties, dimensions and loading conditions, etc.

   In the past, analyzing the theoretical and technological aspects of pipeline bursting tests in which fracture propagation and arrest behavior were determined through measurement of fracture extension vs. time, depressurization, and strain at several points were expensive and time consuming, but they rendered useful and instrumental information on pipeline behavior. The battelle- curves semi-empirical analysis was one of the first methods used by the industry to predict the arrest of dynamic ductile propagating fractures. The method is based on the correlations of full scale testing with results of the CVN (Charpy Vee Notch) test [6]. More recently, a method based on the CTOA (crack tip opening angle) was proposed for characterizing the toughness of steel pipelines by the use of two- specimen CTOA test [5]. These specimens are three-point bend specimens similar to the drop- weight tear test (DWTT) specimen, and the methods based on dynamic fractures [8] which are relatively simple and inexpensive, give partial information on actual pipeline behaviour, and in many cases do not allow a proper assessment of it.

   Shashi [10] proposed the gamma model (- model) to account for a tremendous amount of uncertainty and difficulty in predicting and measuring the crack growth. The gamma model was used to determine the next depth of crack by

   introducing a non-dimensional number and crack fitting or varying the exponent values. The -model was also applied on single edge notch tension (SENT) specimen.

   Within the last three decades, several other models for the analysis of crack propagation were also developed, since the determination of the crack initiation. The research work by Murtagian and others [7] proposed the CTOA (crack tip opening angle) for characterizing the toughness of steel pipeline by the use of the two-specimen CTOA test [8]. They used three-point bend specimens, and the method requires a minimum CTOA for a given material to prevent crack propagation.

   Murtagian and others [7] failed to consider non- classical tools in their approach. Thus, the work seeks to address the knowledge gap in Murtagians investigation. A non-classical approach is adopted in this work.

2. Materials and Methods

   The proposed use of the material dynamic toughness on the governing equation of the crack motion accounts for the combination of effects of the finite rate of energy transfer by the elastic waves and the static material toughness. Inertia effects are considered by computing the momentum of the effective mass that is moving at the crack speed. This mass acts as an additional material resistance to crack growth when the crack is accelerating, and as an additional crack driving force when the crack is decelerating, extending the crack beyond the point that would be expected in a static analysis. The schemes employed are shown diagrammatically in figures 1 and 2.

   Sample Mechanical Properties

   Fluid Properties

   Fluid Flow assessment

   Fracture area

   Decompression

   assessment

   Sample geometry

   Japp

   Figure 1. Input/output scheme for crack driving force estimation

   Japp

   Crack

   extension

   Fig 2 below shows the process, variable dependence and the loop scheme adopted for the numerical solution.

   Initial Crack length

   Burst pressure

   Japp

   JRD

   Crack Speed

   Depressurization

   Fluid Flow rate

   Depress Coef.

   Crack area

   Crack area

   Figure 2. Numerical loop scheme for solving crack growth problem

   The model here developed is based in the momentum conservation principle shown in Eq. (2.1)

   1

   v

   m0 = 2

   Lim

   a0

   p>. J app da (2.6)

   0

   d(m.v) = F.dt (2.1)

   where v is the crack speed, m the mass in motion, F the motion driving force, and t time. As a function

   Replacing Eqs. (2.3) (2.5) into Eq. (2.2) the differential equation of the dynamic crack

   propagation is obtained as shown in Eq. (2.9)

   of crack length this equation becomes

   2

   (J J )da

   dm . v2 + m.v. dv = F (2.2)

   v + v .

   app RD

   (J

   .

   dv

   – 1 =

   da da

   Lim

   2

   app

   app

   J RD ) da

   v

   v

   It is postulated that the motion driving force is given by the difference between the crack driving force (Japp) and the dynamic material toughness (JRD) as shown in Eq. (2.3)

   F = Japp – JRD (2.3)

   By physical constraints, v cannot exceed vLIM. Therefore for a steady state crack propagation, when

   v vLIM Eq. (2.2) becomes Eq. (2.4)

   0 (2.7)

   By solving Eq. (2.7) it is possible to find the instantaneous values of the crack tip position and its velocity. It should be noted that Japp, JRD and v are unknowns and strongly coupled. For the limit case when v vLIM, it can be seen that dv/da 0.

   1. Crack driving force Japp

      dm . v 2

      da L Im

      = Japp – JRD

      (2.4)

      To obtain the exact analytical solution for the crack driving force, spatial fluid pressure evolution,

      From Eq. (2.4), the (variable) effective specimen mass as shown in Eq. (2.5), can be defined. It is assumed that this effective mass is

      located at the crack tip [9]

      body inertial forces, specimen geometry changes and stress wave reflections must be considered [5]. A simplified analysis was used here to compute the crack driving force based on the works of Erdogan

      1

      m = 2

      a

      .

      Japp –

      a

      J RD

      da (2.5)

      and Kilber [2] and Folias [3], for a pressurized cylinder with an axial through-the-wall crack including the crack tip plasticity and geometry

      v LIm 0

      a 0

      Note that in Eq. (2.5) the lower limit of

      integration varies for Japp and JRD. For Japp this limit is zero, i.e. the process starts at a = 0, while for JRD

      corrections to the plate solution. Considering the relation shown in Eq. (2.8) the solution for the crack driving force when plasticity is present is given by Eq. (2.9)

      K 2

      it is the initial crack length. An effective initial

      Japp =

      app

      E

      (2.8)

      mass m0 can be obtained for the onset of crack propagation as

      Where E is the Elastic modulus of the material

      o 2 a 2

      V(R,L )

      2 R3 L

      Japp = h . 1 + h

      . (1 + 1.252 )

      p =

      E t

      (2.15)

      E 0

      0< 1 (2.9a)

      o 2 a 2

      Japp = h . 1 + h .

      E 0

      (0.6 + 0.9 )2 1< 5 (2.9b)

      Where

      h

      To simplify the problem, it is assumed that the there is no pressure gradient in the sample and that the fluid flow, and consequently the depressurization, is dependent on a depressurization coefficient () function of the crack speed with the form expressed in Eq. (2.17). If v vfw it is assumed that no depressurization takes place,

      a

      =

      R th

      (2.10)

      = 1

      v

      v fw

      ; 0< v vfw

      The hoop stress in Eq. (2.9) is dependent on the variable fluid pressure on the sample. In this

      =0; v < vfw (2.16)

      The mass fluid flow (Qm) is given by Eq. (2.17)

      problem the crack length and the applied load are coupled variables; the scheme to compute the crack

      Qm = dm = A v dt fl

      (2.17)

      driving force is shown in figure 2 above. Fracture area (A), fluid and sample mechanical properties are necessary inputs for the computation of the crack driving force. The fracture area evolution is assumed to grow proportionally to the crack length. The initial and final crack lengths are obtained analytically.

      Conservation of mass and energy was utilized

      Where is the system hydraulic discharge coefficient, assumed to be of unitary value for all the experiments, A is the crack opening area, and vfl the fluid flow velocity through the fracture area, which can be computed using Bernoullis theorem, p being the hydrostatic pressure and the fluid density inside the specimen.

      for the calculation of depressurization. The total fluid mass that can flow through the crack area is

      v fl =

      2. p

      (2.18)

      given by:

      • elastic volumetric sample deformation, and

      • the change of specific volume of the fluid inside the sample as shown in Eq. (2.12)

      dm v

      = + v (2.11)

      dp p p

      Using Eqs. (2.11), (2.15), (2.17) and (2.18),

      disregarding the variations of R, , and V with pressure by using their initial values, and making two variable substitutions, it can be deduced that the hoop stress drop by crack extension can be expressed as

      In Eq. (2.11) the term

      is the specimen

      p

      d h =

      o h A

      R 2

      th

      2.19

      mechanical volume variation and

      is the fluid

      p

      da v

      V 2 R

      E t

      +

      h

      compressibility (), V being the sample internal

      volume, and the fluid density. For a cylindrical specimen of length L

      Eq. (2.19) has four strongly coupled variables: hoop stress, crack speed, crack opening area, and

      depressurization coefficient. The remaining terms

      V(R,L )

      p

      = 2 R L R 2

      • R + R 2

        P

      • L

      P

      of the equation are constant for a given test. A typical parameterized decompression curve can be seen in Figure 3 below.

      (2.12)

      from linear elasticity

      R R 2 v

      To find the crack length evolution, the differential system of Eqs. (2.7) and (2.19) has to

      be solved. Hence, a Matlab program was developed

      P = E t

      1

      2

      (2.13)

      to accomplish this task.

      h

      And

      L = R L 1 v

      (2.14)

      Table 1. Specimen Geometry/Material Properties

      2

      Radius	Th (mm)	Length (m)	Hoop Tension (MPa)	Hoop Stress (MPa)
      50	10	2	1500	1200
      50	10	2	1500	1200

      P E th

      For steels 0.3, then [(2 – ) + (1 2. )] 2, giving

      Then,

      dthb da

      = A . .k v

      (2.24)

      Thoop / Tburst

      But is defined from Eq. (2.16) and we then

      obtain Eq. (2.25) as below;

      A.1

      v

      v

      .k

      dt fw

      hb

      da

      =

      v

      (2.25)

      If v fw = 1 since vlim < v fw , and since crack

      a\sample length

      Figure 3. Typical parameterized decompression curve

   2. Dynamic material toughness (JRD)

      growth is on dependent on k we set the other parameter of k to unity, i.e k = 1, we now obtain the expression for the change in hoop and burst tension with respect to the crack extension as,

      dthb = A.(1 v).k , dthb = A 1 v k

      By extending the work of Kanazawa and others

      [4] to the case of elastoplastic dynamic fracture, the functionality shown in Eq. (2.20) is proposed for

      da v

      da

      (2.26)

      . .

      v

      the material dynamic toughness

      J = R (a )

      J

      RD M

      1 v

      vLim

      (2.20)

      Taking an assumption based on inference for

      the crackspeed with a range between 0 1500 [m/s] and scaling to between zero and one (for realistic uniform random distribution), we define v as 0.5[m/s], then if we plunge it into Eq. (2.26) with k = 1, we have;

      The exponent M is considered to have a value of 0.5, vLIM is a temperature dependent parameter

      dthb = A

      da

      (2.27)

      obtained empirically, and JR(a) is the material resistance curve obtained from static fracture tests, which can also be expressed as Eq. (2.21). The material parameters JIC, , and n, are obtained by testing.

      The changes for the hoop and burst tension with respect to the crack extension is obtain as the slope from the typical parameterized decompression

      curve in figure 3, where the coefficient of the curve using the curve fit tool (cftool) in MATLAB for the

      J R (a )

      = J IC

      + (a)n

      (2.21)

      cubic is thus,

      To validate the effectiveness of the model, the

      dthb

      =P t 3 + P t 2 + P t + P

      (2.28)

      optimal values of the test variables are compared

      da 1

      2 3 4

      with the existing models of Murtagian and others [7]. The overall numerical results are thus summarized.

      Dividing through by t we obtain Eq. (2.29) which is now equivalent to our crack growth area A . Therefore,

      A =P t 2 + P t + P + P t 1

      (2.29)

      1 2 3 4

   3. New Model based on Levenberg- Marquart Algorithm

   JIC (kJ/m2)		N	vLIM (m/s)
   1000	250	0.5	1000
   1100	250	0.5	6000
   1200	250	0.5	8000
   1300	250	0.5	3000
   1400	250	0.5	2000
   1500	250	0.5	1000

   From Eq. (2.19), we develop a function which will be use to fit set of data to obtain optimum solution for our crack model.

   So A is the crack exponent growth function that that is used to fit the Levenberg-Marquart algorithm. The LM is to come up with the optimum coefficient of the given function that gives the least square error.

   Table 2 and 3 below provides analytical tables for working;

   d h

   If

   o h

   = Thoop Tburst

   = dthb

   (2.22)

   Table 2. Specified Specimen Properties

   And k =

   R 2

   th

   (2.23)

   V 2 R

   E t

   +

   h

   Table 3. Crack Area at Specified Conditions

   tn	An
   0	0
   100	15.6798
   200	15.6405
   400	15.6210
   600	15.6415
   1000	15.6092

3. Results and Discussion

In this paper, a computer program was developed using the model developed, and a numerical simulation was carried out to determine the crack growth evolution in steel pipeline.

Figure 4. Plot of crack extension vs. Time

From the graph above, there was a gradual increase in the crack growth within the first 100 [secs], and thereafter the crack reaches it terminal where no further crack growth can occur (i.e. the limiting crack speed), where it maintain a linear growth after the first 100 [secs], the crack growth was now constant.

Figure 5. A plot showing the comparison be tween Murtagians and Developed model

From figure 5 above, it can be seen that unlike the Murtagians model where the constant crack extension value of about 275 [mm] was attained rapidly within the first 100 [secs], the developed model displayed a uniform crack extension with respect to time(crack speed) before the constant

crack extension was attained at 800 [sec]. So, the developed model could thus attain a constant crack extension rate at 275 [mm] progressively within 800 [sec]. The model showed a reasonable agreement in crack extension vs. time between Murtagians and the developed model. It thus clearly deduced the model developed as a working model which can serve as a guide to the actual if operational data are used.

Figure 6. A graph of crack speed [m/s] vs.

Time [sec]

From the graph, the acceleration tends to linearity after 600 [sec] at a speed range of approximately 1300 [m/s]. Thus, as an increase in time leads to a corresponding increase in the crack speed.

Figure 7. A graph of crack length [mm] agai nst critical material toughness [kJ/m2]

The critical material toughness, between 1000

1100 [kJ/m2] maintains linearity between 2 12 [mm] and thereafter form a linear curve as it increases and attains its peak at a crack length of 17 [mm]. Thus, the critical material toughness must have value over 990 [kJ/m2] for prevention of crack propagation.

Figure 8. A graph plot of Speed [m/s] again st

critical material toughness [kJ/m2]

The crack speed is observed to negligible for critical material toughness less than 1000 [kJ/m2]

Hence after 1000 runs, the estimates for the crack growth produces good approximate to the mean from 160 values of the crack growth value for randperm (selecto-random permutation). Any optimize design will take the least size/memory with the least cost. Hence, Monte-Carlo optimized the best possible solution with the least possible data set.

Also, Monte-Carlo shows the possibility of reducing the amount of work by limiting the selection of points to regions where the sample mean (i.e. Crack growth values) was changing most rapidly.

Table 4 below shows the error that resulted after those numbers of simulations were carried out.

Table 4. Optimize values of Monte Carlo wit

h error function

S/N Runs Mcarl Error

Figure 9. A graph plot of crack length [mm] against crack speed [m/s]

From figure 9 the minimum crack length was attain at the crack speed of 400 [m/s], and thereafter embarked on continuous in crack length as the crack speed increases.

Figure 10. Monte Carlo estimate of crack ar ea evolution

10 1000 14.99777 0.005

Figure 11. A plot of Y input against time

Figure 11 shows the plot of the generated curve for crack growth for Levenberg-Marquardt algorithm with crack growth (A) as the initial guesses from developed model.

Table 5. Optimum values that satisfies the

nonlinear criteria for model parameters

X1 -5.4472

X2 -8.4472

X3 7.9528

X4 4.0528

Figure 12. A plot of Y Estimate against Tim e

The plot above shows the estimate value against time. At the stage the curves smoothened out.

he results for the various values of X1, X2, X3 and X4 for the given y-input values, that satisfies the non-linear least criteria as developed in the Matlab source code are provided below:

The LM optimization thus showed the Levenberg-Marquardt algorithm as a good iterative procedure and illustrates the use of software to solve nonlinear least squares curve-fitting problem. The projected Levenberg-Marquardt method guarantee local fast convergence under suitable assumptions and shows that it is perfectly appropriate and good to use the best available estimate of the desired parameters as the initial guess, fitting the algorithm to converge to different local minima.

0

1.561

3.122

6.244

9.366

15 .61

Fig 13 below shows the Simulink implementation of crack growth evolution. The limiting crack speed vss and slope gain have been defined using constant Simulink blocks, while an embedded Matlab function block is used to compute the crack propagation area A and a constant k which defines crack growth over time. The crack growth model is controlled by using a multi port switch and a stopping criteria is enforced in the scaling subsystem [11].

slope

slope

fcn

Constant

-0.02875

k

k

Acrack A1

0.015454702288752

Rate

v

A

A2

Embedded

MATLAB Function

vss Constant 1

uT

Math Function

0 1.561 3.122 6.244 9.366 15 .61

Display

3.122

In1

In2

Out 1

1

Out 1

1

In 1

Crack Area Subsystem Logic

An

dA

tn

Out1

dt

2

Out 2

Constant 2

In1

In2

Out 1

Rate Calculator Subsystem

2

In 2

Timing Subsystem

200

Rate

Rn

tn

Figure 13. The model developed for crack growth evolution in Simulink

1. Conclusion

   A software program is developed for the estimation of cracks based on crack tip opening area, CTOA and limiting crack speed, LCS models for steel cracked pipeline as a check of the experimental results of the samples considered. Thus, critical values for the test were generated using a Monte Carlo simulation procedure. To ensure pipe safety, such prediction becomes necessary for both simple and complex cracks in pipelines to determine the failure patterns for such pressure vessels. The model allows for quantitative prediction of crack length evolution with time.

   With this approach, the effects of sudden failure associated with cracks in the pipe lines can be minimized.

   The algorithm is needed to help site engineers solve relative problems of this nature.

2. References

1. G. Apostolakis, The concept of probability in safety assessments of technological systems, Vol. 250, No. (4986): 1990, pp. 1359-64, Battelle Press, Columbus, OH.

2. F. Erdogan and J.J Kibler, Cylindrical and spherical shells with cracks, Int J Fract Mech, vol (5), 1969, pp. 229-3.

3. E.S. Folias, An axial crack in a pressured cylindrical shell. Int J Fract Mech, Vol (1), 1965, pp. 104 13.

4. T. Kanazawa and S. Machida, Fracture dynamics analysis on fast fracture and crack arrest experiments, In: Kanazawa T, Kobayashi AS, Lido K, editors. Fracture Tolerance Evaluation, Toyoprint, 1982.

5. M.F, Kanninen, T.S Grant and G. Demofonti, The development and validation of a theoretical ductile fracture model, In: Eighth symposium on line pipe research, paper 12, Arlington, VA: Pipeline Research Committee of AGA, 1993.

6. W.A. Maxey, R.J. Eiber and R. J. Podlasek, Observations on shear fracture propagation behavior, In: Symposium on crack propagation in pipelines, Institute of Gas Engineers, London, 1974.

7. Murtagian and G. R. Ernst, D. H. Johnson and H. A. Ernst, Dynamic Crack Propagation In Steel line Pipe, Engineering Fracture Mechanics, 2005.

8. D.L. Rudland, G. M. Wilkowski and Z. Feng, Experimental investigation of CTOA in linepipe steels, 2003, pp. 567-77.

9. D.K. Robert, A.A. Wells, The velocity of brittle fracture, Engineering 1954, 178:820-1.

10. K. Shashi, Prediction of Crack Propagation using gamma-model for through wall cracked pipes, Mechanical Engineering National Institute of Technology, Rourkela, 2011.

11. N.E. Osegi, An Introduction to mathematical modeling using simulink. 1st Ed. United State: Lulu publisher, 2011.