Experience shows that positive temperature fluctuation and internal pressure create longitudinal compression force, which can cause significant bulging of buried elements, as far as surfacing above ground. Therefore, overall stability of straight and bent buried elements must be checked. Trench backfill soil weight has a positive effect on stability, since it prevents elements from bulging upward.
Let's look at a buried pipeline with initial bend (fig. 1) under equivalent compression force S caused by temperature fluctuation and internal pressure. The correlation of soil resistance to transverse displacement is modeled as shown in fig. 2.
Fig. 1
Fig. 2
All analyses are performed with A.B. Ainbinder's methods, described in [1]. Some allowances and hypotheses are used mainly toward safety margins.
An approximate energy method based on the analysis of the total system energy is used; therefore, the result varies significantly depending on input bifurcation shape.
Initial bending shape is considered to corresponding to bifurcation shape.
With vertical bulging, it is assumed that bending shape corresponds to infinite stiffness of soil base under the pipe. In this case, bulging shape is:
initial bending - 
, additional bending - 
.
With horizontal bulging, it is assumed that base stiffness is the same on both sides of the pipeline. In this case, the formulas are:
initial bending - 
, additional bending - 
.
This model can be used with a small bend angle (acute angle α between straight elements adjacent to the bend) – up to 10 degrees.
Soil elastic properties for critical longitudinal force are ignored. Soil is considered to function along the length of bulging in an elastic stage, characterized by allowable soil resistance to transverse displacement qlong and pressure-balance factor Cpb. So additional stability analyses are used assuming soil elastic properties.
If initial bending plane is slanted horizontally, initial skew is independent of vertical and horizontal planes. START-PROF uses the slant angle from the horizontal plane φ for this (angle between the bend plane and and the plane passing through the curve peak and perpendicular to the vertical plane).
Stability is checked using 4 conditions:
 ; |
(1) |
 ; |
(2) |
 ; |
(3) |
 . |
(4) |
where m – operation conditions factor (for analyses with RD 10-400-01 - equal to 0.9; for analyses with SNIP 2.05.06-85 - depends on pipeline category);
,
- equivalent forces taking into account
"self-compensation" using the curved axis shape, calculated
independent of vertical and horizontal planes;
- equivalent from not taking into
account "self-compensation" using the curved axis shape;
,
- critical forces, calculated taking
into account initial pipeline bending and non-linear correlation of soil
resistance to transverse displacement in vertical and horizontal planes;
,
- critical forces, calculated as for
a straight compressed rod in a linear-elastic environment.
When
calculating,
soil stiffness is considered to be
the same on both sides; while for
backfill soil stiffness is significantly
lower than base soil stiffness. 
is the critical force calculated using RD 10-400-01. If pipeline bend angle exceeds 10 degrees, the methodology [1] cannot
be used and analyses are done only with the last 3 conditions.
If bend angle exceeds 10 degrees, bend stability check using formula (1) is not run (since methodology is not applicable here). Analysis using formulas (2), (3) and (4) can be performed.
1. Aynbinder A.B., Kamerstein A.G. Transmission pipelines stress and buckling analysis. Moscow, 1982