Most HDD crossings are comprised of a combination of straight sections and bends, and Equations 7-4 and 7-5 have to be applied to each straight section and bend in the bore hole. The corresponding loads may be estimated by the following equations. For calculating the estimated peak force on the product pipe the load is calculated at four points, as shown in Figure 7-1. The greatest load will be experienced at point 4. The loads may be estimated by using the following equations:
T1 = exp ( Va * a) * ( Va * w (L1 + L2 + L3 + L4 )) Equation 7-6
T2 = exp(vb * a) * (t1 + vb * |wb| * L2 + wb * H - va * wa * L2 *exp (va * a))
Equation 7-7
T3 = T2 + vb * |wb| * L3 - exp (vb * a) * (va * wa * L3 *exp (va * a))
Equation 7-8
where: Equation 7-9
T1 = pull force at point 1 in pounds per foot T2 = pull force at point 2 in pounds per foot T3 = pull force at point 3 in pounds per foot T4 = pull force at point 4 in pounds per foot L1 = additional pipe length required outside of bore hole in feet L2 = horizontal distance to desired depth in feet
L3 = additional horizontal distance required to transverse at depth in feet L4 = horizontal distance to rise to surface in feet H = depth of bore in feet exp(x) = natural logarithm base (e = 2.71828)
va = coefficient of friction applicable at the surface before the pipe enters the bore hole vb = coefficient of friction applicable in the bore hole wa = weight of empty pipe in pounds per foot wb = net upward buoyant force on the pipe in the bore hole in foot-pounds per foot a = bore-hole angle at pipe entry in radians/degrees P = bore \-hole angle at pipe exit in radians/degrees
The above equations do not completely account for the resistance caused by the pipe's stiffness as it moves through the curves in the bore hole. This can be greatly reduced by having larger radius bends and sufficient clearance in the bore hole. The exponential factors correspond to the capstan effect, reflecting increased bearing pressure caused by the pipe being pulled against the inside surface of the bend. In addition, the equations above do not account for the resistance resulting from pipe stiffness at the bends along the bore-hole path. This effect can be greatly reduced by using sufficiently large radius bends and maintaining adequate clearance within the bore hole.
The coefficient of friction used in these calculations depends on the characteristics of the surfaces bearing against each other, the presence of any lubrication, and whether there is relative motion between the surfaces. The degree of friction immediately prior to slippage is generally greater than the level during subsequent sliding. Although brief interruptions in the HDD installation process are necessary during the removal of the drill rods during the pullback operation, it is important to attempt to complete the operation without extensive interruptions, which may allow the bore hole to collapse or the pipe to become stuck in the surrounding soil. The value for vb represents the pipe in the bore hole, surrounded by drilling fluid and mud slurry, and assuming minimal interruptions. It is recommended that the pipe external to the bore hole be supported to provide as low a coefficient of friction va as possible. The typically suggested design value for the frictional coefficients between plastic pipe and the wet bore hole (vb) is 0.3 and the value between the plastic pipe and the ground (va) is 0.5. For pipe supported on rollers, va is typically considered equal to 0.1. The friction is highest just before the product pipe starts to move and decreases during movement. When the pipe is stopped, the drilling-fluid viscosity will increase if left undisturbed, which will result in an increase in frictional and drag forces due to the thixotropic nature of the drilling fluid. If multiple pipes or a bundle of small-diameter pipes are pulled simultaneously into the bore hole, higher overall loads will result due to the greater weight or buoyancy of the combination as well as an effectively amplified coefficient of friction within the hole.
The required pulling force depends upon whether the product pipe is empty or filled with water to reduce buoyancy. Buoyant force will push the product pipe against the top of the bore hole, resulting in increased frictional drag. Filling the pipe with water will reduce this effect. The weight of the empty pipe or conduit may be obtained from the manufacturer or calculated by:
176 Chapter 7 ■ HDD Pipe Stress Analysis for Plastic Pipe
Typically assumed average wall thickness equals 1.06, and the minimum wall thickness is:
Equation 7-10a where:
pipeweight = weight of empty pipe in foot-pounds per inch
Ya = specific gravity of pipe material (for example, 0.955 for PE) pw = weight density of water in foot-pounds per inch.3 D = outside diameter of pipe in inches DR = pipe-wall dimension ratio
The net upward buoyant force on empty pipe surrounded by mud slurry may be calculated by:
Equation 7-11 (pipe empty)
Or the effective submerged weight per foot of the pipeline plus internal contents, lbs/ft , X2
:y Equation 7-12
Displaced mudweight v2/
Equation 7-13 waterweigM = 62.4~
plpevolume and weight of water equals pipevolume * water bf ft3
weight w = pipe + water .,, - Displaced s * r avg weight * f mudweight
Equation 7-14
where:
D = outside diameter of pipe in inches yb = specific weight of the mud slurry in foot-pounds per foot3 t = pipe wall thickness in inches
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