1 - backfill soil, 2 - expansion cushion, 3 - foundation soil

Fig. 1

Cross-section of pipes with industrial polyurethane foam (PUR) insulation in a polyethylene shell is a three-layered ring, at the center of which is a steel pipe. Since insulation and pipe work as a single structure (peeling is not permitted), longitudinal stretching of the steel pipe, polyurethane layer and polyethylene shell are equal. This allows the deducation of a fairly simple formula for normal stress in PUR insulation along a buried pipeline axis:

where *σ* and *Е* - normal
stress and elastic modulus of the steel pipe, and
*Е _{ппу}* - PUR elastic
modulus.

Maximum shear stress along the pipeline axis due to friction of the polyethylene shell against soil occurring on the surface of the PUR layer attached to the external steel pipeline surface equal:

Where q_{тр}
- frictional force per pipeline length
unit.

To determine stress in the cross-sectional (circumferential) direction of the PUR layer, a 1cm ring cut out of the pipe in a flat deformed state is analyzed (fig. 2).

1 - foundation soil, 2 - polyethylene casing, 3 - foamed polyurethane, 4 - steel pipe

Fig. 2

Soil pressure is divided into two parts: vertical and horizontal. Vertical load is equal to soil layer pressure for the corresponding depth. This applies only to the top half of the ring, while base elastic properties determined by analysis act on the lower half. Horizontal load is distributed along the trapeze and is equal to vertical pressure multiplied by the lateral pressure factor (fig. 4).

Fig. 4

The
following factors are taken into account: soil pressure from the top and
side (for radial σ_{r} and
tangent σ_{τ}
loads at each surface point of insulation
casing), soil detachment due
to cross-section ovalization, thrust force from internal pressure preventing
ovalization. Elastic restraints
modeling soil apply only for compression. Soil elastic resistance is calculated
through analysis. Detachment
zone is determined through iteration analysis. At
each step, radial restraints with stretching are disabled.

Stress analysis for PPM insulation is the same as for PUR insulation.

Fig. 5

6 stress tensor components are calculated (fig. 5). Equivalent stress is determined based on the theory of greater sheer stress as a difference between the first and third main stress:

Allowable stress for PUR insulation layer:

Where

– allowable stress during stretch-compression,

– allowable stress for tangent shift (circumferential),

– allowable stress for longitudinal shift (along the pipe axis).

Allowable stress for PPM insulation layer:

Where

– allowable stress during stretch-compression,

– allowable stress for shift.

During analysis, a non-linear equation system is solved. Non-linearity is due to accounting for detachment of the elastic base modeling soil in the top part of the pipe and the presence of internal pressure (considering geometric stiffness due to pressure). In some cases, iteration may not produce results. There can be two causes:

Insufficient pipe wall thickness. In this case, wall thickness stability may be lost due to pressure of the surrounding soil. A warning regarding insufficient wall thickness for stability conditions will be given.

Insufficient base soil carrying capacity. If the base soil stiffness is insufficient for the given model, pipe balance in soil may be lost since the system becomes instantly-changing. This is due to the fact that elastic springs without non-linear properties model soil. In this case, turning on the function to increase base soil stiffness is recommended to ensure pipe stability.

1. Alexandrov A.V., Magalif V.Y., Matveev A.V., Stress analysis of foamed polyurethane insulation, «Pipelines and ecology», 2003, №3, ZAO «NPO Stroypolimer»

2. GOST 30732-2006, Steel pipes and fittings with foamed polyurethane insulation in polyethylene casing, Gosstroy of Russia, 2006