10.2.2.1 Overpressure

The allowable overpressure varies between design codes. Table 10.1 outlines the percentage of overpressure that is permitted.

10.4.1.1 Thick Cylinders

The full Lamé equations are simplified for the design of thick-walled pipelines. A pipeline with D/t < 20 is known as a thick-walled pipeline. Taking into consideration a thick-walled pipe, subjected to an internal pressure, P_i, with zero external pressure [10],

(10.5)

(10.6)

The two well-known conditions of stress that allow the Lamé constants A and B to be determined are

(10.7)

(10.8)

Therefore, from Equations (10.5–10.8), the radial and hoop stresses are calculated as

(10.9)

(10.10)

The maximum radial and hoop (circumferential) stresses occur at r = R₁ when σ_r = P_i. The negative sign indicates tension.

(10.11)

(10.12)

where σ_H is the hoop stress (MPa), σ is the radial stress (MPa), R₁ is the internal radius (m), R₂ is the external radius (m), r is the radius at point of interest (measured from the pipeline center), P_e is the external pressure (gauge) (MPa), and P_i is the internal pressure (gauge) (MPa).

10.4.1.2 Thin-Walled Pipeline

A pipeline with D/t > 20 is known as a thin-walled pipeline. A basic approach is known as thin wall hoop stress theory. Since the maximum hoop stress is normally the limiting factor, it is this stress that will be considered [10–12].

It is predictably accurate for D/t > 20. The hoop stress is then calculated as follows:

(10.13)

Hoop stress developed in the pipe wall at the internal design pressure is given by

(10.14)

where σ_H is the hoop stress (MPa), P_i is the internal design pressure (gauge) (MPa), P_e is the external pressure (gauge) (MPa), t is the design thickness (m), D₁ is the inside pipe diameter (m), and D₂ is the outside pipe diameter (m).

10.4.2.1 Fully Restrained Pipeline

A fully end constrained boundary condition can occur at an anchor block or pig trap and pipeline end manifold (PLEM) or pipeline end termination (PLET) sled. For a fully end constrained pipeline (Figure 10.1), the longitudinal strain (ε₁ = 0) and deflection (Δ = 0) components are 0, and the longitudinal stress response can be determined from Equation (10.15), assuming a constant uniform temperature field [10–12].

Piping in which soil or supports prevent axial displacement of flexure at bends is restrained. Restrained piping may include the following [2, 6]:

• straight sections of buried piping;
  
• bends and adjacent piping buried in stiff or consolidate soil;
  
• sections of above-ground piping on rigid supports.

  

The net longitudinal compressive stress in a restrained pipe is calculated based on Clause 419.6.4 of ASME B31.4 as

(10.15)

where σ_LR is the restrained longitudinal stress (MPa), α is the thermal linear coefficient of expansion (mm/mm/°C), T₂ is the operating temperature (maximum or minimum metal temperature) (°C), T₁ is the installation temperature (°C), v is the Poisson’s ratio (v = 0.30 for steel), σ_H is the hoop stress (MPa), and E is the modulus of elasticity (GPa).

Piping and equipment should be supported so as to prevent or reduce excessive vibration and should be anchored sufficiently to prevent undue strains on connected equipment. Supports, hangers, and anchors should be so installed as not to interfere with the free expansion and contraction of the piping between anchors. Suitable spring hangers, sway bracing, and so on should be provided where necessary [2].

Anchor blocks are used to stop axial movement of a pipeline. Anchors are normally required when the pipeline comes above ground, prior to pig hatches, other branches, or manifolds. Connection of the pipeline to the anchor block is normally done by the addition of slip-on flanges to the pipeline, which are fillet welded in place. A large concrete block is then constructed around the pipeline to resist the expansion forces.

Restrained portions are always prevented from moving by installing anchors and guides, but in a buried line, a large portion is fully restrained by soil friction only [12].

The axial compressive force required to restrain a pipeline can be calculated as follows:

(10.16)

(10.17)

where F is the axial force (N), E is the modulus of elasticity (GPa), T₁ is the installation temperature (°C), S_h is the hoop stress (MPa), v is the Poisson’s ratio (0.3 for steel), A is the cross-sectional area of the pipe wall (m²), α is the coefficient of thermal expansion (°C⁻¹), T₂ is the maximum or minimum metal temperature (°C), k is the ratio of the outside diameter to inside diameter, and P is the internal design pressure (gauge) (MPa).

10.4.2.2 Unrestrained Pipeline

An end-free boundary condition can occur at locations where no physical longitudinal restraint exists, for example, at a riser bend extending from the seabed to the production platform. Since the pipeline is not restrained axially, the Poisson effect and thermal expansion component do not produce stress in the line pipe. The pipeline longitudinal stress is only due to the end cap effect [2–5].

Piping that is free to displace axially or flex at bends is unrestrained. Unrestrained piping may include the following:

• aboveground piping that is configured to accommodate thermal expansion or anchor movements through flexibility;
  
• bends and adjacent piping buried in soft or unconsolidated soil;
  
• an unbackfilled section of an otherwise buried pipeline that is sufficiently flexible to displace laterally or that contains a bend;
  
• pipe subject to an end cap pressure force.

For unrestrained sections of a pipeline, the longitudinal tensile stress should be calculated as follows (refer to BS 8010, Part 2, Section 2.8, and Clause 2.9.3.2):

(10.18)

For a thin-walled pipeline, use k = 1.

When the internal pressure is greater than the external pressure, the stress will be positive and tensile, as expected.

The unrestrained longitudinal stress can generally be expressed as

(10.19)

Expanding Equation (10.19), the expression for an approximate thin wall is given as

(10.20)

where σ_LU is the unrestrained longitudinal stress (MPa), M_b is the bending moment (N m), D₁ is the inside diameter (m), D₂ is the outside diameter (m), Z is the pipe section modulus (m³), σ_E is the end cap stress (MPa), i is the stress intensification factor (see Section 10.6.1), and k is the ratio D₂/D₁.

10.4.2.3 Partially Restrained Pipeline

At the free end of a pipeline, the total friction force acting on the pipeline increases as one progresses from the free end. This friction is passive and acts against the forces that try to displace the line [2, 4, 5].

For a constant frictional force per unit length of the line, the rate of change of longitudinal stress is constant. The expression for the longitudinal stress experienced by a pipeline that is partially restrained is given as

(10.21)

(10.22)

Expanding the above, the expression for an approximate thin wall is given as [5]

(10.23)

where μ is the seabed coefficient of friction, W_s is the submerged weight of the pipeline, X is the distance from the free end of the pipeline (m), A_s is the area of steel (A_s = πDt) (m²), and L_x is the length between anchors (m).

Note

10.4.2.4 Bending Stress

A pipe must sustain installation loads and operational loads. In addition, external loads such as those induced by waves, current, uneven seabed, trawl-board impact, pullover, and expansion due to temperature changes need to be considered. A pipe subjected to increasing bending may fail due to local buckling/collapse or fracture, but it is the local buckling/collapse limit state that usually dictates the design [13].

The pipe (beam) bending is analyzed using the engineer’s theory of bending (ETB) and simple beam theory. The theory associates the bending stress at a point to the moment imposed on the section or the curvature experienced. It is used to calculate the bending stress at any point in a pipe.

Assumptions

• Transverse planes before bending remain transverse after bending, with no warping.
  
• The pipe material is homogenous and isotropic and obeys Hooke’s law with E the same in tension or compression.
  
• The pipeline (beam) is straight and has a constant or slightly tapered cross-section.
  
• Load does not cause twisting or buckling. This is satisfied if the loading plane coincides with the section’s symmetry axis.
  
• The pipeline is subject to pure bending. This means that the shear force is 0 and that no torsional or axial loads are present.

The equation is valid for elastic pure bending and does not take into account the geometrical deformations that occur, particularly in hollow cylinders (ovality) under bending. The equation is expressed as

(10.24)

This is called the engineer’s theory of bending. The standard form of this equation is

(10.25)

(10.26)

where σ_x is the bending stress (0 ≤ σ_x ≤ σ_y) (MPa), M is the bending moment (N m), y is the distance from the neutral axis (0 ≤ y ≤ OD/2) (m), I is the moment of inertia (kg m³), E is the Young’s modulus of elasticity (GPa), and R is the radius of curvature (m).

Therefore, knowing the applied bending moment and the location of the centroid, the second moment of area and the stresses along the depth of the pipe section can be calculated.

In the simple theory of bending of beams, it is assumed that no appreciable distortion of the cross-section takes place so that there is no displacement of the material either toward or away from the neutral axis. For a thin-walled pipe subjected to bending, movement of the fibers toward the neutral axis does occur [14].

The maximum bending moment and hence the bending stress at any point in the cross-section can be determined by assuming a beam configuration.

Beam Configurations

A fixed–fixed beam under a uniformly distributed load gives a maximum bending moment (M_ff) at the fixed ends of the beam.

(10.27)

A beam with pinned supports under a uniformly distributed load gives a maximum bending moment (M_pp) at midspan.

(10.28)

It is commonly accepted that a real beam configuration for an unrestrained pipeline resting on the seabed is somewhere between the fixed–fixed and the pinned–pinned cases and is given a value as follows:

(10.29)

where M_ff is the fixed–fixed bending moment, M_pp is the pinned–pinned bending moment, W is the uniformly distributed load (per meter), L is the length of pipeline span, and M_fp is the bending moment halfway between the fixed–fixed and the pinned–pinned cases.

The stages of longitudinal stresses connected with bending are as follows:

• Elastic bending up to yield at the outer fiber. At this point, the bending moment is equal to the first yield moment.
  
• Elastoplastic bending up to the development of a full plastic hinge. When the plastic hinge is developed, it is known as the full plastic moment.

On the other hand, this can be solved by drawing the bending moment diagram, that is, if the value of the bending moment, M, is determined at various points of the beam and plotted against the distance x measured from one end of the beam. It is further facilitated if a shear diagram is drawn at the same time by plotting the shear, V, against x.

This approach facilitates the determination of the largest absolute value of the bending moment in the beam.

10.4.2.5 Stress due to Sustained Loads

Sustained loads are the sum of dead weight loads, axial loads caused by internal pressure, and other applied axial loads that are not caused from temperature and accelerations [3].

In accordance with ASME Boiler and Pressure Vessel Code, Section III, Subsections NC and ND, the calculated stresses due to pressure, weight, and other sustained mechanical loads must meet the allowable 1.5S_h, that is,

(10.30)

where T is the temperature derating factor, P is the internal design pressure (MPa), D₀ is the outside diameter of the pipe (m), Z is the section modulus of the pipe (m³), M_A is the resultant moment loading on the cross-section due to weight and other sustained loads (N m), S_h = 0.33S_uT, at the maximum installed or operating temperature (MPa), and S_u is the specified minimum ultimate tensile strength (N/m²).

10.4.2.6 Stress due to Occasional Loads

Occasional loads are loads such as wind, earthquake, breaking waves or green sea impact loads, and dynamic loads such as pressure relief, fluid hammer, or surge loads. In accordance with ASME Boiler and Pressure Vessel Code, Section III, Subsections NC and ND, the calculated stress due to pressure, weight, other sustained loads, and occasional loads must meet the allowable stress as follows [3]:

(10.31)

where P_max is the peak pressure (MPa) and M_B is the resultant moment loading on the cross-section due to occasional loads, such as thrusts from relief and safety valves, loads from pressure and flow transients, and earthquake, if required. For earthquake, use only one-half of the range. Effects of anchor displacement due to the earthquake may be excluded if they are included under thermal expansion. kS_h = i.85S_h for upset condition but not greater than 1.5S_y, 2.25S_h for the emergency condition but not greater than i.85S_y, and 3.0S_h for the faulted condition but not greater than 2.0S_y. S_h = 0.33S_uT, at the maximum installed or operating temperature (MPa), S_u is the specified minimum ultimate tensile strength (N/m²), and S_y is the material yield strength at a temperature consistent with loading under consideration.

10.4.2.7 Stress due to Thermal Expansion

Thermal expansion may be detrimental for the pipe itself, flanges and bolts, branch connections, pipe supports, and connected equipment such as pumps and compressors. Sufficient pipe flexibility is necessary to prevent such detrimental loads [2].

Stresses due to expansion for those portions of the piping without substantial axial restraint shall be combined in accordance with the following equation (refer to Clause 833.8 of ASME B31.8):

(10.32)

(10.33)

The cyclic stress range S_E ≤ S_A, where

(10.34)

If Equation (10.32) is not met, the piping may be qualified by meeting the following equation:

(10.35)

where S_E is the stress due to expansion (MPa), S_A is the allowable stress range for expansion stress, f is the stress range reduction factor, M_E is the range of resultant moment due to thermal expansion (N m), also includes moment effects of anchor displacements due to the earthquake if anchor displacement effects were omitted from occasional loadings, S_c = 0.33S_uT, at the minimum installed or operating temperature (MPa), S_h = 0.33S_uT, at the maximum installed or operating temperature (MPa), S_u is the specified minimum ultimate tensile strength (N/m²), S_L is the unrestrained longitudinal stress (N/m²), T is the temperature derating factor, i is the stress intensification factor (see Section 6 or Appendix E of ASME B31.8), M_i is the in-plane bending moment (N m), M_t is the torsional moment (N m), M_o is the out-of-plane bending moment (N m), i_o is the out-of-plane stress intensification factor (refer to Appendix E of ASME B31.8), and i_i is the in-plane stress intensification factor (refer to Appendix E of ASME B31.8).

10.4.3.1 Shear Stress due to Torsion

Torsion is the twisting of a straight bar when it is loaded by twisting moments or torques that tend to produce rotation about the longitudinal axes of the bar. When subjected to torsion, every cross-section of a circular shaft remains plane and undistorted, and the bar is said to be under pure torsion [15].

The shear stress on a uniform cylindrical shaft that is under a uniform torsion is given by

(10.37)

For a thin cylindrical shaft (or thin-walled tube) with t < R/10,

(10.38)

(10.39)

The maximum shear stress due to torsion can be calculated as follows:

(10.40)

where τ is the shear stress (N/m); T is the torque or twisting moment (N m); R is the radial distance from the longitudinal axis (m); I_x and I_y are the moments of inertia about the x– and 33 y-axis, respectively (kg m³); Z is the section modulus (m³); and J is the polar moment of inertia.

10.4.3.2 Shear Stress due to Spanning

The shear stress due to spanning is composed of vertical shear and a longitudinal shear due to the bending. The maximum shear force acting on a simple span is equal to the maximum support reaction. This is in turn equal to the change in shear force at the reaction. The maximum vertical shear stress is defined as the force per unit area. The maximum vertical shear stress is calculated as follows [15]:

(10.41)

where F_s is the shear force applied to the pipeline (N), A_s is the cross-sectional area of the pipe (m²), and R_max is the maximum vertical reaction on the pipe (N).

10.4.5.1 Allowable Hoop Stress

The allowable hoop stress may be calculated using the following equation (refer to Clause 805.234 of ASME B31.8 and Clause 201.4.1 of API RP 1111):

(10.44)

where σ_aH is the allowable hoop stress, S_y is the specified minimum yield strength, f is the design factor, e is the weld joint factor, and T is the temperature derating factor.

The allowable hoop stress for the cold-worked pipe is 75% of the above value (Clause 201.4.4 of API RP 1111). The design factor is 0.72 for pipelines and liquid risers, 0.60 for gas risers, and 0.50 for gas platform piping [2].

10.4.5.2 Allowable Equivalent Stress

(10.45)

In accordance with Clause 833.4 of ASME B31.4, the maximum allowable equivalent stress is 90% of the SMYS.

(10.46)

where σ_ae is the allowable equivalent stress and S_y is the specified minimum yield strength.

10.4.5.3 Limits of Calculated Stress due to Sustained Loads

The sum of the longitudinal stresses due to pressure, weight, and other sustained external loads shall not exceed 0.72S_A, where S_A = 0.75S_y (S_y is the specified minimum yield strength) [6].

10.4.5.4 Limits of Calculated Stress due to Occasional Loads

The sum of the longitudinal stresses produced by pressure, live and dead loads, and other sustained loadings and of the stresses produced by occasional loads, such as wind or earthquake, may be as much as 1.33S_h [6].

10.4.5.5 Limits of Calculated Stress due to Expansion Loads

The computed displacement stress range (expansion stress range) S_E in a pipeline should not exceed the allowable displacement stress range S_A [6].

(10.47)

When S_h is greater than S_L, the difference between them may be added to the term 0.25S_h; in that case, the allowable stress range is calculated as

(10.48)

where S_E is the expansion stress range = (MPa), M_E is the resultant bending stress =[(i_iM_i)² + (i_oM_o)²]^1/2/Z (MPa), S_t is the torsional stress = M_t/2Z (MPa), M_i is the in-plane bending moment (N m), M_o is the out-of-plane bending moment (N m), M_t is the torsional moment (N m), i_i is the in-plane stress intensification factor, i_o is the out-of-plane stress intensification factor, Z is the section modulus of the pipe (m³), S_c = 0.33S_uT, at the minimum installed or operating temperature (MPa), S_h = 0.33S_uT, at the maximum installed or operating temperature (MPa), S_u is the specified minimum ultimate tensile strength (MPa), and f is the stress range reduction factor (obtained from Table 302.3.5 of ASME B31.3) or calculated as follows:

(10.49)

where N is the equivalent number of full displacement cycles during the expected service life of the piping system.

10.8.1.1 Deflection

Pipe stiffness and passive soil resistance of backfill play a significant role in predicting deflection. M.G. Spangler [21] of Iowa State University published the Iowa formula in 1941.

(10.57)

Deflection of a pipeline is calculated using the modified Iowa deflection formula as follows:

(10.58)

where Δx is the horizontal deflection of the pipe (m), D₁ is the deflection lag factor (1.0–1.5), K is the bedding constant (0.1), R is the pipe radius (m), (EI) is the pipe wall stiffness per meter of pipe length (in./lb, m/kg), E ′ is the modulus of soil reaction (GPa), and W is the external load per unit length of the pipe (dead load (earth load) + live load).

Some data on loading are presented in Table 10.4.

Spangler hypothesized that if the lateral movement of various points on the pipe ring was known, the distribution of lateral pressures could be determined by multiplying the movement of any point by the modulus of passive resistance, E′. For mathematical convenience, this lateral pressure was assumed to be a simple parabolic curve embracing only the middle 100° arc of the pipe (see Figure 10.3).

He also assumed that the total vertical load was uniformly distributed across the width of the pipe, and the bottom vertical load was distributed uniformly over the width of the pipe bedding [20].

The following terms can be introduced to describe the three separate factors that affect the pipe deflection:

• load factor (D₁KW);
  
• ring stiffness factor (EI/R³);
  
• soil stiffness factor (0.061 E ′). Values of E ′ are listed in Table 10.5.

The modified Iowa formula can be represented as a load factor

(10.59)

Information on the load factor, ring stiffness factor, soil stiffness factor, bedding constant, and deflection lag factor can be obtained from standards of American Concrete Pipe Association [20–22].

The steel pipe is designed as a flexible conduit; considerable deflection can occur without damaging the pipeline. Deflection limitations are a function of the rigidity of the specific lining and coating being used.

The calculated deflection is limited to 5%, although larger deflections may not affect pipe performance. Limits for pipe deflection for various forms of coating or lining should be obtained from governing standards and codes [20].

10.10.4.1 Pipeline Located Offshore

On the other hand, for a pipeline located offshore, the minimum bend (curvature) in order to prevent slippage is calculated by the following equation:

(10.75)

where R_sl is the minimum bend to prevent slippage (limiting slippage curvature), T_o is the normal on bottom tension, W_s is the submerged unit weight, and F is the lateral friction coefficient or the on-bottom friction coefficient.

where T₀ is the maximum tension, H is the water depth, and T_i is the tension at the inflexion point.

10.11.1.1 Check for Buckling

Buried pipelines supported by a well-compacted, granular backfill will not buckle due to vacuum (i.e., when the gauge pressure is below atmospheric pressure) in the pipeline system. To confirm the stability of a pipeline, an analysis of the external loads relative to the pipe stiffness can be performed.

If the soil and surface loads are excessive, the pipe cross-section could buckle. The ring buckling depends on limiting the total vertical pressure load on the pipe [20, 22].

(10.76)

where σ_rb is the allowable buckling pressure (MPa), FS is the factor of safety, C is the depth of soil cover above the pipe (m), D is the outside diameter of the pipe (m), R_w is the water buoyancy factor = 1 − 0.33(h_w/C), 0 < h_w < C, h_w is the height of the water surface above top of the pipe, (EI)_eq is the equivalent pipe stiffness, E ′ is the modulus of soil reaction (GPa), and B ′ is the empirical coefficient of elastic support.

(10.77)

In steel pipelines, buckling occurs when the ovality reaches about 20%.

The sum of the external loads should be less than or equal to the pipe’s allowable buckling pressure, σ_rb.

Confirming the resistance of a pipeline to bucking involves an analysis of the external loads relative to the pipe stiffness [20]. The total external load acting on a pipe must be less than or equal to the allowable buckling pressure of the pipe and can be calculated using the following equation:

(10.78)

where σ_rb is the allowable buckling pressure (MPa), γ_W is the specific weight of water (0.0361 lb/in.³), h_w is the height of water above the pipe (in., m), R_w is the water buoyancy factor, W_c is the vertical soil load on the pipe per unit length (lb/in., kg/m), D is the outside diameter (m), and P_v is the internal vacuum pressure (MPa).

The total live load acting on a pipe must be less than or equal to the allowable buckling pressure of the pipe. When analyzing the possibility of potential buckling of a pipeline, there is the need to determine the total live loads acting on a pipeline, using the following equation:

(10.79)

where W_L is the live load on the pipe per unit length (lb/in., kg/m).

When the allowable buckling pressure is not sufficient to resist the buckling loads, the soil envelope should first be investigated to increase the allowable E ′ [20].

10.11.2.1 Collapse Criterion

When a pipeline installed above ground is subjected to vacuum, the wall thickness must be designed to resist collapse due to the vacuum. Analysis should be based on pipe functioning in the open atmosphere, absent of support from any backfill material [20]. Collapse pressure, p_c, is the pressure required to buckle a pipeline.

The collapse pressure should be calculated using Timoshenko’s theory for collapse of a round steel pipe as follows [20]:

(10.80)

where P_c is the collapsing pressure (MPa), t_s is the steel cylinder wall thickness (m), t_I is the cement coating thickness (m), d_n is the diameter to the neutral axis of the shell (m), E_s is the modulus of elasticity for steel (30 × 10⁶ psi, 207 × 10³ MPa), E_I and E_c are the moduli of elasticity for cement mortar (4 × 10⁶ psi, 27.6 × 10³ MPa), v_s is the Poisson’s ratio for steel (0.30), and v_I and v_c are the Poisson’s ratios for cement mortar (0.25).

The mode of collapse is a function of D/t ratio, pipeline imperfections, and load conditions. A safety factor of 1.3 is recommended. When a pipeline is designed using the collapse criterion, a good knowledge of the loading conditions is required.

10.11.2.2 Propagation Criterion

Propagating pressure, P_p, is the pressure required to continue a propagating buckle. A propagating buckle will stop when the pressure is less than the propagating pressure (refer to DNV 2012, Clause 501). The recommended formula for calculating the propagation criterion is the latest given by AGA [18].

(10.81)

The nominal wall thickness (t_nom) should be determined such that

(10.82)

where P_p is the propagation pressure (MPa) and P_e is the external pressure (MPa).

The recommended safety factor of 1.3 is to account for uncertainty in the envelope of data points used to derive Equation (10.82).

Determination of Pipe Diameter

From process and hydraulic analysis, the flow rate, pipe diameter, operating temperature, and design pressure are as shown in Tables 10.A.1–10.A.3.

Wall Thickness Design

Based on the requirement of the ASME B31.8 code,

From API 5L Standardized Pipe Schedule Chart, at 12 in. outer diameter and calculated t = 14.9 mm, t_nom = 15.9 mm is selected.

All parameters are defined in Table 10.A.1.

Determination of Components of Stress

The structural analysis of pipeline systems is concerned with the determination of stress or strain state of a pipeline and the subsequent check of the stress or strain state against a fatigue limit or allowable stress. The general term of structural analysis is indeed very wide, and this chapter considers the more important aspects in relation to pipeline systems in detail.

Hoop Stress

Hoop stress is calculated using the Barlow’s formula (Equation 10.4) as

Thin-Walled Pipeline

With reference to Section 10.4.1.2, the standard dimension ratio is

Therefore, the offshore pipeline is a thin-walled pipeline.

Compressive Hoop Stress

For a deep-water application, the external hydrostatic pressure should be accounted for by using ∆P instead of P for hydrostatic conditions,

where, H is the pressure head and ɤ is the unit weight of water. All other parameters are as defined in Sections 10.4.1.1 and 10.4.1.2.

Longitudinal Stress

Restrained Longitudinal Stress

A fully end constrained boundary condition occurs at pipeline end manifold (PLEM) or pipeline end termination (PLET) sled of the offshore pipeline, 2-km section buried and restrained by soil friction. In accordance with section 833.2 of ASME B31.8, the net longitudinal stress is calculated as

Longitudinal Stress due to Internal Pressure

The longitudinal stress due to internal pressure in a fully restrained pipeline is calculated as

Longitudinal Stress due to Thermal Expansion

The net longitudinal stress due to thermal expansion in a fully restrained pipeline is calculated as

All parameters are as defined in Section 10.4.2.

Bending Stress

The bending stress is calculated using the engineer’s theory of bending in Section 10.4.2.4 and Equation (10.26).

Radius of curvature of the bent pipe, R

Bending stress

Maximum bending stress

All parameters are defined in Section 10.4.2.4.

Total Longitudinal Stress for the Restrained Section

Axial Compressive Force

The axial compressive force required to restrain a thin-walled pipeline can be calculated Equation (10.16) as follows:

Cross-Sectional Pipe Wall Area

The cross-sectional wall area or area of the piping material can be calculated as

Axial compressive force

All parameters are defined in Section 10.4.2.1.

Unrestrained Longitudinal Stress

An end-free boundary condition can occur at locations where no physical longitudinal restraint exists; for example, at a riser bend extending from the seabed to the production platform or PLET (see Figure 10.4). The unrestrained longitudinal stress in an operational pipeline can generally be expressed as

Longitudinal Stress due to Internal Pressure

The longitudinal stress due to internal pressure in an unrestrained pipeline is calculated as

Bending Stress

The bending stress is calculated using the engineer’s theory of bending in Equation (10.26) as

Radius of curvature, R

R = radius of curvature of the bent pipe, m

Bending stress,

With reference to Section 10.4.2.4 and Equation (10.25),

Maximum bending stress,

All parameters are defined in Section 10.4.2.4.

Total Longitudinal Stress for the Unrestrained Section

In accordance with Section 10.4.2, the total longitudinal stress is calculated as

where

Equivalent Stress

In accordance with Section 10.4.4, and Equation (10.43), equivalent stress in the pipeline is

All parameters are defined in Section 10.4.4.

Limit of Calculated Stress

Design factor, F = 0.72

Specified minimum yield strength for X65M, S_y = 450 MPa

Pipe material grade = API 5L X65M PSL2

Allowable Longitudinal Stress

Restrained Section

For a restrained pipeline, the allowable longitudinal stress is 0.9 S_y × T, (refer to Equation (10.43) and Section 10.4.5)

Therefore, the requirements of the code are met with respect to longitudinal stress in the restrained section.

All parameters are defined in Section 10.4.5.

Unrestrained Section

Based on Clause 833.6 of ASME B 31.8, for an unrestrained pipe, the allowable longitudinal stress is σ_AL ≤ 0.75 S_y × T, where S_y is the specified minimum yield strength, (MPa), and T is the temperature derating factor

Therefore, the requirements of the code are met with respect to longitudinal stress in the unrestrained section.

Allowable Hoop Stress

In accordance with Equation (10.41),

The calculated hoop stress σ_H = 160.49474 MPa

Therefore, the requirements of the code are met with respect to hoop stress

All parameters are defined in Section 10.4.5.1.

Allowable Equivalent Stress

The maximum allowable equivalent stress is 90% of the SMYS (refer to Section 10.4.5, and Equation (10.45))

Therefore, the requirements of the code are met with respect to equivalent stress.

All parameters are defined in Section 10.4.5.2.

Flexibility Analysis

As per clause 833.7 of ASME B 31.8, no formal analysis is required in systems which are of uniform size, have no more than two points of fixation, no intermediate restraints, and fall within the empirical equation below.

where

• D = outside diameter of the pipe
  
• Y = the resultant of total displacement strains to be absorbed by the piping system
  
• L = developed length between anchors
  
• U = anchor distance, straight line between anchors
  
• Note:

Calculation of L and U is based on Figure 10.4.

K = 208.3 For SI units or K = 0.03 for FPS units

Displacement Strain

Calculate displacement due to thermal expansion. The entire offshore system is considered, i.e., both riser and the entire offshore pipeline system.

Thermal Strain, ∆L

The change in length of the thin-walled pipeline is determined from the thermal strain.

Flexibility Check

The flexibility criterion is less than 208.3; therefore, no further or formal flexibility analysis is required. That is,

External Pressure check

The ASME code does not provide a formula to check for collapse resistance; thus, the API RP-1111 is used.

Collapse Criterion

Yield Pressure and Collapse

Elastic Collapse Pressure

Collapse Pressure

Propagation Criterion

The recommended formula for calculating the propagation criterion is the latest given by AGA (1990)

Bending Buckling Check

During installation,

During operation,

Where,ε is bending strain, P₀ is external pressure, P_i is internal pressure, f_o is collapse factor, (0.7 for seamless or ERW pipe), v is Poisson’s ratio (0.30 for steel), ɤ is sea water density, and P_p is propagation pressure.