As you may be able to imagine a waterline does not stay at a constant depth.
Because it is under pressure the pipe goes where it must so the calculation for dead
loads and live loads should be done over a range of values.
Thanks to spreadsheets the task of calculating the loading on the pipe is just a matter of
copying formulas down the spreadsheet at whatever depth is needed.
The calculation for horizontal deflection is given by Spangler - a former student of
Anton Marston and later, also a professor of engineering at Iowa State University.
The equation was further modified by Watkins, who was a student of Spangler's at Iowa State,
who then became a professor of engineering at Utah State University until he retired
in the early 2000's.
I bring this up because over the course of nearly 100 years the tools that we use here
were developed and continue to evolve as we have a better understanding of buried structures.
The equation to calculate horizontal deflection is given as follows.
Here Delta-x is the horizontal deflection measured
in inches.
Dl is the deflection lag factor, which ranges from 1 to 1.5
K is the Bedding constant which is typically specified as 0.1,
E prime is the modulus of soil reaction measured in PSI.
This value comes from a geotechnical analysis of the soils where the pipeline is being installed.
Without a specific study it is possible to make conservative estimates from past studies
in the region.
W is the summation of the dead and live loads.
Because we are calculating the deflection measured in inches, then we need to convert
the load from pounds per linear foot into pounds per linear inch of pipeline.
All this means is that we divide the load values by 12 before using it in the Spangler-Watkins
deflection equation.
Moving on ... r is the mean radius of the pipe in inches.
The mean radius is the average of the radius measured to the inside pipe wall - shown as
ID - and the outside pipe wall - shown here as OD.
From this section view of a cement mortar lined and coated pipeline we can see the inside
and outside diameters here.
The pipe wall section displays a steel conduit that is coated in cement mortar as well as
being lined with cement mortar.
There are other wall profiles that are possible, but this one will help us understand how the
equations handle a composite pipe wall.
The next term that needs some explaining is the product of the E and I terms: The modulus
of elasticity of the steel or mortar as well as the transverse moment of inertia.
The EI term is the summation of the EI products for each of the individual materials used
to construct the pipe wall.
The moment of inertia for a hollow cylinder is calculated as b, which is the distance
from the center of the pipe to the centroid of the given material,
times the cube of the thickness of the specific material section,
divided by 12.
Remember that I is calculated for each material separately, then the EI term is calculated
for each material, and then the sum of the EI's is used in the deflection equation.
These terms are not a function of depth, so they can be calculated in a single cell within
your spreadsheet.
This allows us to plug in the equation for the deflection at each depth, then calculate
the percent deflection compared with the total diameter of the pipeline.
We do this because the percent deflection must be below 2% for Mortar lined and coated
steel pipes, It must be less than 3% for mortar coated
pipes with a flexible lining, and the deflection must be less than 5% for
a steel pipeline with a flexible lining and coating.
As mentioned before the deflection analysis is typically run over a wide range of cover
depths, because a pipeline typically does not stay at a given depth over the entire
alignment.
Therefore the deflection analysis is complete when you identify the range of depths over
which the pipeline does not exceed the allowable deflection.