Calculating spring lateral forces
Onbottom 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 factorymade 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 onbottom stability of
pipeline.
The paper presents an engineering method for determining the spring
lateral force and, respectively, the required additional onbottom 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 deflectionsThe 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 midbend may be found from
the condition that at the midbend 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
midspan point is:
Based on the second equation of (3) and
equation (5) the angle of turn on the midbend point, which is equal to
half of the bend angle, is determined by the equation:
Application to designFinally, 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
onbottom weight,
 Minimum allowable elastic bend radius from onbottom 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 midbend 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 analysisTo satisfy the onbottom 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 onbottom 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 onbottom 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 = onbottom 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 1.21.6. 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
onbottom 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 Onbottom 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.
Copyright 1998 Oil & Gas Journal. All Rights Reserved.
