A pipeline restrained by fixed anchors will experience a series of stresses including longitudinal, bending, and axial. Virtual anchor lengths are taken as the distance required for the frictional force provided by the soil surrounding the pipe to equal the forces applied by thermal/ pressure expansion and the soil’s resisting friction per unit length of pipe.
Anchors are placed strategically along the pipeline to help prevent movement. Unrestrained pipeline movement can cause damage to the connecting piping and equipment. The length of piping required to form the virtual anchor is known as the active length. It must be noted that the force required to fully restrain a pipe is not a function of its length. Factors that influence the required force to restrain pipe include temperature, pressure and percentage strain within the pipe.
Soil provides a constant frictional force along a buried steel pipeline. The magnitude of the frictional force depends on the burial depth, pipe weight, soil density, and coefficient of friction between the soil and steel. Therefore, starting from a free end, the total restraint exerted by the soil on the pipe gradually increases until it reaches the fully restrained load at the virtual anchor. At this point, the naturally occurring forces are balanced with a restraint point. Similarly, moving along the pipe away from the virtual anchor, the pipe expansion becomes gradually minimized until a point of zero expansion is reached indicating the pipe is fully held in place.
Axial expansion is calculated by taking the average of the full axial restraint at one end of the pipe and zero restraint at the opposite end of the pipe. When compared to above ground piping, the total axial expansion at the free end of a buried pipeline is half of the calculated value for similar scenario involving above ground pipe.
In reality most pipes do not have a totally free end but have some resistance due to soil restraint as the pipe exits the ground and from the connection to above ground piping. This acts to reduce the expansion at the ‘free’ end. Soils with lower friction resistance or pipes with less depth of cover have longer active lengths and thus have greater expansion at the free end.
Do = Outside Diameter of Pipe
Di = Inside Diameter of Pipe
D = Mean Diameter of Pipe = Do – TNom
TNom = Nominal Wall Thickness of Pipe
TAc = Actual Thickness of Pipe =TNom -CA
v = Poisson’s Ratio
a = Coefficient of Thermal Expansion
ᵨP = Pipe Density
CA = Corrosion Allowance
E = Modulus of Elasticity of Pipe
ᵨF = Pipe Fluid Density
L = Length of Pipe
A = Cross Section Area of pipe
WP = Full Weight Pipe and its fluid.
Design & Other Conditions
P = Design Pressure
TD = Design Temperature
TI = Installation Temperature
D T = TD – TI
µ = Coefficient of Friction between Pipe & Soil
ᵨS = Soil Density
H = Depth of Burial
Sh = Hoop Stress
Sa = Axial Stress
e = Strain
FEx = Total Force Due to Expansion
FEx(T) = Expansion Force Due to Temperature Change
FEx(P) = Expansion Force Due to Pressure
FP = Force due to Pressure
FF = Frictional Force
La = Anchor Length
Theoretically, there will be pipe movement from entry point due to thermal expansion. Also an expansion will be there due to the pressure. Opposing these two is the frictional force between pipe and soil. Let us find these factors first:
Expansion Force due to Temperature Change:
Expansion Force due to Change in Temperature will be:
FEx(T) = E × A×a × DT ………………………………..(1)