Pipelines · ground movement
Buried pipeline under ground movement
Beam-on-springs model of a buried steel pipe. Set the pipe and steel, the soil springs, the alignment and ground movement, then run the analysis. Soil springs are bilinear per metre of pipe for drained soil (Su = c = 0), from the ALA/PRCI formulas.
Steel stress–strain
The uniaxial curve is applied to the von Mises equivalent stress with the hoop stress held constant, so pressure makes the axial response stiffer in tension and softer in compression. The dot marks the fully restrained operating state before the slide.
Drag the points on your uniaxial tension curve, or type values. Compression mirrors tension, and the curve with hoop stress follows. Beyond the last point the final slope continues. Points stay in order, never steeper than E. The custom curve replaces Ramberg–Osgood; E still sets the elastic slope. The dashed reference and the reset button use the Ramberg–Osgood form chosen last.
| # | Strain [%] | Stress [MPa] |
|---|
Soil springs
Each soil has its own properties, cover depth and springs. Soil 1 applies wherever the pipe isn’t given another soil under Soil sections in the slide definition.
Soil springs
Elastic–perfectly plastic per metre of pipe; displacements are pipe relative to ground. Values are calculated from each soil’s inputs. Type over any of them to set your own, and use ↺ to go back to the calculated value. Select a soil’s name to edit its inputs.
Qu = Nqvγ′HD with Nqv = tan(0.9φ)·H/D, capped at 2.5 for φ ≤ 20°, 4 at 25°, 8 at 30°, 13 at 35°, 20 at 40°, 28 at 45° and 37 for φ ≥ 50° (linear between), or from DNV (per soil); δu = 0.4·min(0.2D, 0.015H); Qd = γ′HNqD + ½γD²Nγ; δd = 0.4·0.15D; Tu = πDγ′H(1+K)/2·tan δ with δ = f·φ; Xu from the soil inputs.
Force–displacement
For the soil selected in the tabs above. Each spring on its own scale; the downward spring is far stiffer and stronger than the upward one.
Upward (pipe moves up)
Downward (pipe bears into soil)
Axial
Slide definition
Pipe alignment and ground movement. The pipe is either straight or read from a file of X, Z points. The ground movement is linear (values at the head and toe interpolated linearly in between, each transition ramping from zero to full value), a cosine shape over a zone, or read from a file of X, DX, DZ.
Set the start and end of the X axis to look at a section; the range applies to the pipe profile and existing bends below, the ground movement profiles, and the pipe profile in the results.
Pipe profile and existing bends
Existing bending strain
The bending strain the alignment’s own shape corresponds to, κ0·D/2, with D the outside diameter: κ0 is the change of direction at each point of the alignment divided by the mean length of the segments either side. Positive in a sag (concave up), negative in an overbend. The thin line is the value at each point, which shows rounding noise and errors in the file; the bold line is averaged over 1 m of pipe, the useful measure of how the pipe is bent. For information only: the model takes the alignment as stress-free, so its strain results do not include this. It shows where the pipe is already bent, which is where ground movement adds extra bending as it partly straightens those bends.
Head is the left end of the slide, toe the right. Shading marks the slide; lighter shading marks the transitions. For a pipe from a file, X is measured from its first point.
Solution
Beam elements following the pipe alignment, with a 32-fibre pipe section, solved for horizontal and vertical displacement and rotation together, on bilinear soil springs acting along and perpendicular to the pipe. With geometric nonlinearity on, the axial strain includes the stretching of the pipe as it rotates (ε = u′ + ½w′², in each element’s own axes, with ½w′² averaged over each element), so a deflected pipe develops cable tension, and axial force changes its bending stiffness (P-delta). Internal pressure and temperature are applied first; the ground movement then follows in increments, with Newton iterations in each.
Both model ends fixed: no axial or vertical displacement, no rotation. Ground movement in the number of load steps set, halved automatically if a step doesn’t converge; with geometric nonlinearity on, more steps are used if needed to keep the ground movement to 20 mm per step. A run stops after 3 minutes.
Pipe profile
Elevation of the pipe before and after the full ground movement. The vertical scale is exaggerated, and the pipe’s displacements in the “after” line are magnified by the factor chosen.
The X range is shared with the pipe profile and existing bends in the inputs above.
Displacement normal to the pipe
Pipe and ground at full ground movement, component perpendicular to the original pipe, positive upward. This is what the up and down springs see; for a straight pipe it is the vertical displacement.
Horizontal displacement
Pipe against ground, horizontal (X) component, at full ground movement.
Longitudinal strain, top and bottom
Strain at the top and bottom of the pipe: axial plus bending. Axial strain alone dashed. Set the start and end of the X axis to look at a section; the range applies to both strain charts.
Bending and axial strain
The two parts of the strain: axial strain, the same across the whole section, and bending strain = (bottom strain − top strain) / 2, positive in a sag bend (pipe curving up, concave upward) and negative in an overbend (pipe curving down, over a crest). Top strain = axial − bending; bottom strain = axial + bending.
Maximum strains against ground movement
Recorded at every load step, anywhere along the pipe: largest bending strain |κD/2|, and largest tensile and compressive strain (axial plus bending). The first point is the operating state under pressure and temperature.
Maximum stresses against ground movement
Recorded at every load step, anywhere along the pipe: largest longitudinal stress in tension and in compression, and largest von Mises equivalent stress, combining the longitudinal stress with the hoop stress pD/2t of the local wall. Yield stress fy dashed. The first point is the operating state.
Limitations
What this solver does not cover. Read the results with these in mind.
- Small strains, moderate rotations
- Strains are assumed small. With geometric nonlinearity off (the default), strains and equilibrium use the undeformed pipe, so a deflected pipe develops no cable (catenary) tension and axial force does not affect bending stiffness: at large ground movements, bending strains tend to be overestimated and axial tensile strains underestimated. With it on, the cable effect and P-delta are included through the von Kármán strain ε = u′ + ½w′² in each element’s own axes, which is accurate while each element rotates less than about 0.2 rad from its original direction; the solver reports the largest rotation and warns above that.
- Local buckling not modelled
- With geometric nonlinearity on, global effects of axial compression (P-delta, upheaval of a hot pipe) are included; with it off, compressive axial force does not amplify deflection. Local buckling, wrinkling and ovalization of the wall are not modelled in either case; compare compressive strains against a separate strain limit.
- Pressure thrust optional; ends always fixed
- Internal pressure always sets the hoop stress pD/2t used in the steel model (and its Poisson effect). With pressure thrust off (the default), that is all it does: the outward thrust at changes of direction (p·Ai/R along a curve of radius R, Ai = π(D−2t)²/4) is not applied, and with geometric nonlinearity on, P-delta uses the wall axial force N rather than the effective force N − p·Ai, understating the destabilising effect of pressure. With it on, pressure acts on Ai of each element’s own wall, as Abaqus pipe elements with closed-end pressure: thrust 2p·Ai·sin(Δθ/2) at each change of direction Δθ and p·ΔAi at changes of wall thickness; inside a straight pipe of one thickness it has no effect. Ai stays constant (no change of internal area as the pipe expands), and with geometric nonlinearity off the thrust keeps the original direction of each element. The end-cap force at the two fixed ends goes into the supports, so a free or closed pipe end can’t be modelled yet.
- Alignment and spring directions
- A pipe read from a file is modelled as straight elements between its points, and points closer than 0.1D to the previous one are merged. Soil springs act along and perpendicular to the pipe at each node, in their original direction as the pipe deforms. Their capacities come from the ALA/PRCI formulas for a horizontal pipe, with one cover depth H, measured vertically, for the whole pipe; on steep sections this is an approximation.
- Unloading depends on the steel model
- Nonlinear elastic (total strain): the steel curve (Ramberg–Osgood or custom) is used as a nonlinear elastic law (J2 deformation theory). A fibre that unloads goes back down the loading curve instead of unloading elastically, so permanent plastic strain is not modelled; results are valid for loading that increases monotonically. Nonlinear plastic (incremental strain): J2 flow theory with isotropic hardening; unloading is elastic and leaves permanent strain, but reversed loading has no Bauschinger effect, and results depend slightly on the number of load steps.
- Loading stages
- Added stages continue from the end of the stage before, with the same pipe, steel, soils and settings; their ground movement is added to the movements before. A stress relief removes springs at the nodes inside its span, so the span is in effect rounded to the nodes (within half an element; the stage status gives the X of the end nodes), releases their forces gradually, then attaches the new springs with zero force at the relieved position. Where only the axial and upward springs are removed, the downward spring keeps its state and the new upward spring acts separately from it. Use incremental plasticity for stages that unload the pipe.