 Pipeline on-bottom stability: design features of elastic sidebends

#### Calculating spring lateral forces

On-bottom stability analyses presented in "Recommended Practices" usually apply to straight sections of pipelines, or pipelines for which changes in horizontal direction are provided by using field cold bends or factory-made elbows.

In some cases, especially for offshore transmission and gathering pipelines, the changes in horizontal direction may be made by elastic sidebend. In accordance with B32.8 (Practices), the effect of prestressing, such as permanent curvature induced by installation, often referred to as elastic bends, shall be considered when they affect the serviceability of the pipeline. This residual stress should be considered in the operating design of the pipeline system.

The effect of friction force between the pipe and soil, which acts against the springing force, caused by the prestressing of pipeline, shall also be taken into consideration in determining the on-bottom stability of pipeline.

The paper presents an engineering method for determining the spring lateral force and, respectively, the required additional on-bottom weight for elastic sidebend sections of pipelines to be laid on the seabed. The paper also outlines a method for calculating the design parameters for elastic bend, such as, bend radius, bend angle, and layout of elastic curve. The procedure for the stability analysis of elastic sidebends is described and illustrated by calculation example.

### Lateral deflections

The entire curve, sidebend, consists of two parts of variable curvature located symmetrically about the angle. The curvature at the end of straight pipe, or at the beginning of curve, is equal to zero, and the radius of curvature at the axis of symmetry is equal to the designed radius.

As the concentrated lateral forces at the points where the curvilinear part connected to the straight parts are equal to zero, the friction forces are balanced. Thus, each part with variable curvature, in its turn, consists of the two equally spaced parts. The friction forces act in opposite directions.

Equations from a static condition of equilibration for equally spaced parts, based on the beam theory of relative small deflection, are: where:
x, y = rectangular coordinate system with the origin point at the beginning of curve, the axis x is directed as a tangent at the origin point,
(EI)c = bending stiffness of pipe or of composite section for concrete coated pipe,
FS = lateral soil resistance per unit length that is equal to lateral spring force per unit length.
For design purposes, the solution that proposed by A. Mousselli might be used to determine concrete coated pipe bending stiffness. This solution is based on the assumptions that in the compressive zone the concrete is an ideal elastic material and the tensile strength of the concrete is negligible. For the calculation example presented below this simplified method has been used.

The eight constants of integration of equations (1) may be found from boundary conditions: These first boundary conditions take into account that at the beginning of the curve, or the same at the end of the straight section of pipeline, the angle of turn, bending moment, and lateral concentrated force (reaction) are equal to zero. The second conditions describe conditions of continuity.

The solutions of differential equations (1) based on boundary conditions (2) are: The coordinate x of mid-bend may be found from the condition that at the mid-bend the curvature is designated by the minimum radius of elastic bending. This condition can be written as: where:
R = minimum radius of elastic bending. In most cases, the value of the radius is designated based on stress and/or buckling criteria.

From the second equation of (3) and condition (4) the coordinate of the mid-span point is: Based on the second equation of (3) and equation (5) the angle of turn on the mid-bend point, which is equal to half of the bend angle, is determined by the equation: ### Application to design

Finally, based on the equation (6), the spring lateral force per unit length, or required minimum lateral soil resistance for elastic sidebends with designated bend angle and minimum elastic radius is: It should be noted that in all equations the unit of angle is radian. Equation (7) allows the following to be determined:

• Minimum allowable bend angle with known values of minimum elastic bend radius and maximum lateral soil resistance, depending on pipe on-bottom weight,
• Minimum allowable elastic bend radius from on-bottom stability condition with known values of bend angle and maximum lateral soil resistance.
For design purposes, the layout parameters of sidebend, which is dictated by design minimum radius R and bend angle j may be determined from the above solution. To determine these sidebend parameters, it is especially important for the pipelines to be laid in the ditch, as changes in pipeline direction caused by elastic bending are required to conform to the contour of the ditch.

The tangent of curve (the distance from the beginning point of the curve to the point of the angle vertex of the route turn) is determined by the equation: The external distance (the distance from the point of mid-bend to the point of the angle vertex of the route turn) is determined by the equation: More detailed geometry of the curve may be obtained using equations (3), where the values of distance l and spring force Fs are determined by equations (5) and (7), respectively.

It should be particularly emphasized that elastic sidebend is a curvilinear section with a variable radius in the range from design elastic radius to infinity. Therefore, as may be seen from equation (8), the tangent of this parabolic curve is about two times greater than the tangent of the circular curve with the same angle.

### Stability analysis

To satisfy the on-bottom stability requirements of offshore pipelines, consideration shall be given to buoyancy, lift, and drag forces.

For elastic sidebend, lateral spring force induced by installation should be also considered. Therefore, the equation (5) in RP 1111 for calculation of on-bottom weight reduction may be rewritten as follows: where:
FL = drag force per unit length,
FD = lift force per unit length,
FS = spring lateral force per unit length ( = coefficient of friction between pipe and soil.

It should be noted that appropriate to equation (5), the equation in RP E305 also consists of the inertial force. To verify on-bottom stability (to determine the required submerged weight), two methods are usually used in engineering practice. According to the first method, the safety factor, often named as the specific gravity, is determined using the equation: API RP 1111 does not establish the value of this safety factor, while RP E305 recommends a value of 1.1. In this equation:

WP = pipe weight in air, including the weight of the concrete coating and/or the concrete weight,
WB = buoyant force to the pipe, including the force applied to the concrete coating and/or the concrete weight,
WR = on-bottom weight reduction due to the lift, drag and spring forces calculated by equation (10).

According to the second method, the stability safety factor, or the calibration factor, is determined using the equation: API RP 1111 does not establish the value of this factor, while RP E305 gives this value in the range of 1.2-1.6. In this equation:

WS = submerged weight of the pipe, including the weight of the concrete coating and/or concrete weight.

The procedure for stability analysis, based on API RP 1111, is illustrated in the example calculation below.

The existing recommended practices developed by API and DnV leave it up to the designers what methods they will use to design the elastic sidebend. This paper has been prepared as a suggested practical aid for an on-bottom stability analysis of the elastic sidebend sections of offshore pipelines.

### References:

API RP 1111 Design, Construction, Operating, and Maintenance of Offshore Hydrocarbon Pipelines.
RP E305 On-bottom Stability Design of Submarine Pipelines, Det Norske Veritas.
ASME B31.8 Gas Transmission and Distribution Piping Systems.
Mousselli A.H. Offshore Pipeline Design, Analysis, and Methods, PennWell Publishing Co., 1981.

### Author

Alexander Aynbinder is a senior project engineer at Fluor Daniel Company. Previously, he was in the civil engineering department of Gulf Interstate Engineering, and before that was a lead research scientist in the Russian State Research Institute for Pipeline Construction (VNIIST). He is a graduate of the Moscow Civil Engineering University and received a PhD in civil engineering from the Central Research Institute of Civil Structures in Moscow. He is a member of ASME.

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