Journal of Failure Analysis and Prevention

, Volume 18, Issue 6, pp 1643–1651 | Cite as

Probabilistic Fracture Mechanics for Analysis of Longitudinal Cracks in Pipes Under Internal Pressure

  • Belaïd Mechab
  • Nadji ChioukhEmail author
  • Boubaker Mechab
  • Boualem Serier
Technical Article---Peer-Reviewed


In this paper, we present a probabilistic fracture mechanics methodology to analyze elastic and elastic–plastic fracture of semi-elliptical longitudinal cracks in pipes under internal pressure. Numerical results are acquired using three-dimensional finite element simulations. Analytical expressions are proposed with unknown coefficients obtained by nonlinear fitting to the numerical results. For the elastic case, results of the shape function using the newly proposed expression are found to be in a good agreement with those found in the literature. In the elastic–plastic case, estimates of the J-integral are presented for various ratios including crack depth to pipe thickness (a/t), reference stress to material yield stress (σref/σy) and mean pipe radius to its thickness (Rm/t). It is found that the range of applicability of the proposed expressions is extended even beyond those found in the literature. Finally, failure probability is accessed by a statistical analysis for uncertainties in loads and material properties, and structural reliability. The probability density function is estimated by the Monte Carlo Method. It is shown from the present results that the crack size is an important factor influencing the distribution function of (J/Je), failure and reliability rates.


Pipes Fracture mechanics Elastic–plastic analysis Finite element method Analytical expressions Probabilistic analysis Failure Reliability 

List of symbols


Crack depth


Crack half length


Young’s modulus


Shape function of the stress intensity factor


Total J-integral, including elastic and plastic components


Elastic component of the J-integral


Plastic component of the J-integral


Stress intensity factor


Strain hardening index in the Ramberg–Osgood (R-O) stress–strain model

Ri, Rm, Ro

Inner radius, mean radius and outer radius of the cylinder


Internal pressure


Pipe thickness


Coefficient for the Ramberg–Osgood (R–O) law


Yield strain


Reference strain


Angle reflecting the location along the semi-elliptical crack front


Half of total crack angle in the part circumferentially cracked cylinder


Non-dimensional factor relating the limit load and the optimized reference load


Poisson’s ratio


Applied stress


Reference stress


Ultimate stress


Yield stress


Finite element method


Enhanced reference stress method


Elastic–plastic fracture mechanic


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Copyright information

© ASM International 2018

Authors and Affiliations

  • Belaïd Mechab
    • 1
  • Nadji Chioukh
    • 2
    Email author
  • Boubaker Mechab
    • 3
  • Boualem Serier
    • 1
  1. 1.LMPMUniversity Djillali LiabesSidi Bel AbbèsAlgeria
  2. 2.Laboratory of Aero-HydrodynamicsUSTO MBOranAlgeria
  3. 3.Laboratory of Statistics and Stochastic ProcessesUDLSidi Bel AbbèsAlgeria

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