Buried pipe

In this blog I will discuss underground pipe stress calculation and different approaches for underground pipe modeling. During pipe thermal expansion, friction force from pipe to soil will occur resisting a pipe thermal expansion. Longitudinal forces from internal pressure must also be taken into consideration as well as forces as from Poisson shrinkage. At one point these forces will be balanced and that location is called virtual anchor. Virtual anchor is the location where all forces acting on underground pipe are in the balance resulting displacement at that point equal to zero. Please see the figure below.
Based on Eq.1 virtual anchor length (L) can be derived and it is given by Eq.2. Please see the figure below.
Soil friction force is shown on the figure below.
Virtual anchor for a short pipe run (L<Lp) is located at the middle of pipe span and for long pipe run (L>Lp) virtual anchors are located from the ends of pipe. Passive soil pressure has been neglected, but if we take passive soil pressure into consideration length of virtual anchor (La) will be somewhat smaller than previously calculated (L). Please see the figure below.
Using pipe sleeves or tunnels at the pipe elbow will eliminate passive soil pressure, which can be economical in the case of long buried vertical pipe section or in the case of high thermal load. Realistically, the location of virtual anchors is somewhat uncertain since buried depth and soil friction can vary. Also, uneven cooling and heating rate will contribute uncertainty of virtual anchor location. Now, let’s consider long pipe run shown on the figure below.
Calculation of restrained pipe portion is shown on figure below.
It should be noted that piping code B31.4 and piping code B31.3 has different approach. By piping code B 31.4, equivalent longitudinal stress is calculated using maximum shear equation, where hoop stress is added to longitudinal thermal stress. Calculated equivalent longitudinal stress must be equal or less than 0.9SMYS. By piping code B 31.3 stress that comes from sustain load is equal to zero. All stresses are secondary in nature and therefore only pipe flexibility criteria must be met – displacement stress range must be equal or less than allowable displacement stress range. Stress calculation (Liang-Chaun Peng) for restrained pipe complying B 31.4 is shown on the attachment ‘Underground-1’ and it can be downloaded from here. In the case of railway or road crossing external loads must be taken into consideration and stress analysis must be conducted as per API RP 1102 (Steel pipelines crossing railroad and highways). Although stress analysis per API RP 1102 is beyond this scope, it is worth to mention that API RP 1102 utilize Von Mises stress as failure criteria. Since restrained pipe portion is loaded with compression, bowing of restrained pipe portion must be checked, and equations are shown below.
Location of virtual anchor (active length) and stresses for unrestrained pipe portion can be calculated as per Liang-Chaun Peng method, where piping system is modeled as guided cantilever and passive soil pressure is taken into consideration. Stress calculation for unrestrained pipe portion complying B 31.4 is shown on attachment ‘Underground-2’ and it can be downloaded from here. Pipe stress at the elbow of 114 095 psi greatly exceed allowable stress of 37440 psi (see Undergroung-2 attachment by Liang-Chaun Peng). In this case, installing physical anchor will alleviate stress at the elbow. Please see the figure below.
It is evident that the physical anchor installed at 20 feet from the vertical run will greatly reduce end force (Q) and therefore bending stress at elbow (without stress intensification factor) will be much lower than allowable stress. Liang-Chaun Peng use approach where first approximation is made neglecting lateral pipe movement and second approximation is made by considering piping system as guided cantilever. These approximations greatly simplify stress calculation and for the most cases, stress calculation will be adequate. In the case of critical lines, sophisticated analysis is required and more realistic pipe modeling must be made. Enhanced pipe modeling involves using simulated soil supports with spring rate. Please see the figure below.
In the old days calculation was carried out by program made in Fortran or another programming language. If the calculated force (at the soil lateral restrains) is greater than the ultimate soil resistance, than second iteration was needed taking constant force instead of spring, since soil yield displacement is reached and soil is in plastic region. Please see the figure below.
Caesar II has adopted the similar approach; spring supports are placed on the pipe and stresses are calculated by balancing all forces using iterations. It should be noted that Caesar II takes soil yield displacement as 1.5% of burial depth (same as Liang-Chaun Peng). Below is the screenshot of the spring support placement.
Soil parameters are entered by starting buried modeler.
Soil parameters should be obtained from soil report and the soil yield displacement factor shall be taken 0.015 as per discussion above. Overburden compaction multiplier (OCM) shall be taken:
1. Common Backfill (OCM=4-5)
The backfill material shall be consolidated with the wheels or tracks of
excavating equipment.
2. Compacted common backfill (OCM=6-7)
Compacted common backfill shall be placed in layers no greater than 300 mm and
each layer shall be compacted to a density equivalent of native material.
3. Compacted select granular backfill (OCM=8)
Selected granular backfill shall be placed in layer no greater than 150 mm thick
and compacted to 98% of maximum Standard Proctor Dry Density.
4. OCM smaller than 4 can be used for loose material without any compaction.
OCM greatly influences calculated stress and having wrong values for OCM will give you incorrect result. (Larger OMC, soil is stiffer) Tutorial on utilizing Caesar II for stress analysis of underground piping can be found on the site What is piping.