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1
Buried Hybrid-Segmented Pipes Subjected to Longitudinal Permanent Ground
Deformation
C. A. Davis1 and B. P. Wham2
1Resileince Program Manager, Los Angeles Department of Water and Power, 111 N. Hope Street,
Room 1345, Los Angeles, CA, 90012; PH (213) 367-2319; email: craig.davis@ladwp.com
2Research Faculty, Center for Infrastructure, Energy, and Space Testing, University of Colorado
Boulder, 1111 Engineering Dr. UCB 428 ECOT 428 Boulder, CO, 80309; PH (303) 492-6716;
email: brad.wham@colorado.edu
ABSTRACT
A new family of hybrid-segmented pipes is emerging into the international market which
allow the joints to displace axially and deflect a prescribed quantity before locking up and
restraining further movement. These pipe systems are mainly targeting improvement to
earthquake-induced ground deformations, but also apply to other types of ground-induced strain
such as landslides, flooding-induced scour, and differential subsidence. As the segmented pipe
joints lock up in response to ground movement, the system behaves similar to a continuous pipe
along the regions where ground movement is sufficient to engage the locking mechanisms. Other
than employing finite element modeling, there is no methodology allowing engineers to evaluate
the stresses and strains that develop along segmented pipes consistently with those of continuous
pipes at equivalent levels of potential ground movement. This paper presents an evaluation
methodology based on a single set of equations that allow continuous and segmented pipes, of any
defined material, to be evaluated consistently for block ground deformations moving parallel to
the longitudinal pipe axis.
INTRODUCTION
To improve the resilience of water distribution networks, new technologies to accommodate
ground deformation associated with seismically induced fault rupture, landslides, liquefaction-
induced lateral spreading and settlement, and ground movements caused by other natural hazards,
construction, mining, and regional dewatering activities are being developed and utilized. One
technology-type emerging in the industry are hybrid-segmented pipelines which allow the joints
to displace axially and deflect a defined quantity before locking up and restraining further
movement. As the segmented pipe joints lock up in response to ground movement, the system
behaves like a continuous pipe along the regions where ground movement is sufficient to engage
the locking mechanisms. Continuous pipe solutions exist (e.g., O’Rourke and Nordburg, 1992),
but cannot account for the hybrid-segmented pipe behavior prior to the joints locking up. There is
a need for methodologies to assess in a consistent manner the response of both continuous and
segmented pipelines, constructed of various elastic materials, to permanent ground deformation.
This paper presents an evaluation methodology based on a single set of equations which allows
continuous and segmented pipes, of any defined material, to be evaluated consistently for block
2
ground deformations moving parallel to the longitudinal pipe axis. The equations reduce to
previously recognized solutions by O’Rourke and Nordburg (1992) for continuous pipes as the
segmented pipe joints reach their allowable displacement (i.e., converges to a continuous pipe
transferring axial force through the joints). The performance of segmented hybrid pipe is compared
to prior solutions by reference to O’Rourke and Nordburg (1992). This newly proposed pipe model
will improve the engineering design selection of continuous and segmented pipes for use in hazard
prone regions.
HYBRID-SEGMENTED PIPE ANALYTICAL SOLUTION
The segmented pipeline under consideration can be a system composed of any material with bell
and spigot joints. Currently available systems commonly employ push-on style joints sealed with
rubber gaskets. The joints are assumed to provide capacity to deform axially a distance
in both
tension and compression, and lock into place once the joint moves an amount
. Figure 1 shows
an illustration of a hybrid-segmented pipe system. Each pipe segment has a length h and provides
an average strain capacity along each segment,
p, defined as
p =
/h (1)
As shown in Figure 1, the pipe is assumed to be parallel to the ground surface (i.e., constant
burial depth). The ground surface in Figure 1 is shown as horizontal but may be taken at a sloped
angle. The pipe response parallel to the slope, or along the pipe axis (horizontal axis in Figure 1),
is evaluated. Any movement perpendicular to the pipe axis (i.e., heave or settlement along on the
vertical axis) is neglected.
A rigid block ground deformation pattern displacing an amount
downslope over a block
length Lb is assumed. This results in a crack of width
on upslope end of movement and
compression ridge over distance
on downslope end a distance Lb from the crack. By comparing
various idealized ground strain patterns, O’Rourke and Nordburg (1992) showed that the rigid
Figure 1. Hybrid segmented pipe system response to block movement.
Lb
Block
Movement h
H
fr
fr
(b) Neutral position,
initial condition
(a) Joint elongation,
tensile capacity (c) Full compressive
displacement capacity
Compression Bulge
Tension Crack
3
block is expected to result in the largest axial pipe strains of the five patterns considered. The
model assumes total length of pipe of at least 2Lb and there are no bends, tees, elbows, or other
discontinuities along this length.
The pipe has a nominal diameter D, buried at a uniform depth H in cohesionless soil with
internal angle of friction
and unit weight
. Slippage along the pipe is assumed to follow the
O’Rourke and Nordburg (1992) rigid spring/slider model.
The coordinate system is established with x=0 at the head of the sliding block. The ground
displacement Ug is defined as follows (O’Rourke and Nordburg, 1992):
Ug = 0 for x < 0, Region I in Figure 2
Ug =
for 0 ≤ x ≤ Lb, Regions II and III in Figure 2 (2)
Ug = 0 for x > Lb, Region IV in Figure 2
The problem is represented and defined by the following differential equations:
(3)
(4)
(5)
Where UT is the total pipe displacement, E is the modulus of elasticity of the pipe material, A is
the cross-sectional area of the pipe, fr is the resistance force per unit of pipe length, and C1 and C2
are integration constants. Equation 3 is developed by force equilibrium from summation of forces
per unit length. Equations 4 and 5 are developed through integration of Equation 3.
Figure 2 shows plots of pipe strain and displacement relative to the ground movement and
identifies Regions I to IV. As indicated by Equation 4 and Figure 2, pipe strain is assumed to be a
linear function between dUT/dx =
p having a peak of dUT/dx = ±( +
p). Similarly, UT is an
exponential function ranging between UT = 0 and a peak value of UT = d + Uj, where Uj is the
displacement taken up in the joints.
Tensile strain is taken as positive and compressive strain as negative. The average
resistance force fr is positive when pointing in positive x direction; therefore, as indicated in Figure
2, fr is positive in Regions II and III and negative in Regions I and IV.
Figure 1 shows that as the block of length Lb moves down slope, the joints expand on each
side of the tension crack (Figure 1a) and contract on each side of the compression bulge (Figure
1c) until they lock into place after moving an amount
. Once joints lock into place, the pipe
system behaves as a continuous pipe. This means that the hybrid segmented pipe system will have
portions of its length behaving as a continuous pipe if the block moves an amount
>
. The
length of hybrid-segmented pipe behaving as a continuous pipe depends on the amount of block
displacement, block length, and the total number of engaged joints N over length 4ls. ls is defined
as the total length of engaged pipe resulting in an axial pipe strain
. The number of joints engaged
over length ls is:
(6)
4
From Equations 1 and 6:
(7)
The total displacement from fully engaged joints in length ls is n
. n is an integer because
once engaged, ls increases by entire segment lengths h. Figure 2 shows the pipe strain and
displacement for the hybrid segmented pipe. The total strain and displacement in Figure 2 are the
sum of the joint strains
p and displacements
and the strain and displacement within the pipe
barrel.
and d are variables representing the peak pipe strain and displacement, respectively,
within the pipe barrel, and do not include
p and
. Additionally, a variable dT is introduced as the
total maximum displacement defined as
dT = d + 2n
(8)
As shown in Figure 2, the pipe displacement is symmetric, and strain is antisymmetric
about a point x = Lb/2. The results will be presented for x ≤ Lb/2 and they apply to the region
x ≥ Lb/2 with a sign change for displacement and strain.
Figure 2. Plot of hybrid segmented pipe strain and displacement in relation to
ground movement (Condition I).
Lb
ls
Block
Movement
CL
(+)
(-)
x
+p
(+)
(-) x
U(ground) = Ug
U(joint) = Uj
U(pipe) = Up
Displacement, u(x)
h
H
pls
Strain,
U(Total) = UT
-(+p)
d
(a)
(b)
(c)
-p
Region I Region II Region III Region IV
0Lb
dT
5
The solution to Equations 4 and 5 are found by applying the boundary conditions in Tables
1 and 2. The boundary conditions assume displacements in the leading or trailing joints, at x = - ls,
and x = Lb + ls, have reached
. This is neither conservative or unconservative because once a
segment is engaged and fr mobilized, ls increases by an amount h, and the force does not increase
until the next joint is fully engaged. As seen in Figure 2, the displacement is a step function at
x = -ls, and x = Lb + ls, having Uj = 0 or Uj =
. For this case it is acceptable to allow the boundary
condition to be Uj = 0 at x = -ls, and x = Lb + ls because ls is to be solved as a variable from the
formulations and the number of segments within ls is determined using Equation 6. Assuming
Uj =
simply adds an additional joint and segment length, which is automatically solved for when
calculating ls regardless of the boundary condition Uj = 0 or Uj =
.
There are two sets of boundary conditions applicable to this solution. Table 3 presents the
resulting integration constants for Region I, which were solved by applying the boundary condition
sets using negative fr. The integration constants are the same for Region II, using positive fr, except
for Equations 11 and 12 which are
and
, respectively. The integration constants can be substituted into
Equations 4 and 5 to derive the equations governing the total curve shapes in Figure 2, making
note to properly change the signs in Regions III and IV.
Table 1. Boundary conditions for Region I, x < 0, hybrid segmented pipe.
x
UT
Boundary
Condition
-ls
p
0
(a), (c)
0
+
p
dT/2 ≤
/2
(b), (d)
Table 2. Boundary conditions for Region 2, hybrid segmented pipe.
x
UT
Boundary
Condition
+ls
p
dT ≤
(e), (g)
0
+
p
dT/2 ≤
/2
(f), (h)
Table 3. Integration constants for Region I, hybrid segmented pipe.
Integration
Constant
Equation
Notes
(9)
Inserting B.C. (a) into Eq. 3
(10)
= EA(
+
p)
inserting B.C. (b) into Eq. 3
(11)
inserting B.C. (a) into Eq. 4 and
B.C. (c) into Eq. 5
(12)
+
inserting B.C. (b) into Eq. 4 and
B.C. (c) into Eq. 5
(13)
inserting B.C. (d) into Eq. 5 for
any C1 function
6
Pipe Strains and Displacements. Since the integration constants must be equivalent, the
following results are found. The proper signs for fr were used to solve for each of the following
equations, for which the absolute value of fr is to be used from hereon.
(14)
Equation 14 is found by equating Equations 9 and 10 or 11 and 12. The axial pipe strain
in Equation 14 results from the equilibrium of applied force frls and resulting force within the pipe
AE.
(15)
Equation 15 is found by equating Equations 11 and 13 and applying Equation 7.
(16)
Equation 16 is found by equating Equations 12 and 13 and substituting Equation 15; this
applies for both Regions I and II. Solving for ls from Equation 15:
(17)
and substituting into Equation 14:
(18)
The above equations are in the same form found for the continuous pipe solution
(O’Rourke and Nordburg, 1992). This demonstrates that the hybrid segmented pipe behaves as a
continuous pipe for those segments having fully engaged joints. From Equations 7 and 16:
(19a)
(19b)
Equation 19b essentially describes the two strain components that result in the total relative
displacement, dT. The
term is a result of the equal and opposite strain that develops on either side
of the ground crack, while
p is doubled to account for the displacement driven joint opening that
occurs along sections of pipe on both sides of the ground crack. Inserting Equation 13 into
Equation 5 and evaluating at x = 0 gives,
(20)
7
which is the same as Equation 8. Substituting Equations 7 and 15 and rearranging gives:
(21)
Equation 21 can also be derived by substituting Equation 14 into Equation 19b, and
rearranging. Solving for ls using the quadratic equation results in:
(22)
Rearranging Equation 21 and substituting Equation 14 gives:
Solving for
using the quadratic equation
(23)
Pipe stains
and displacements d, and dT, increase or decrease as a function of increasing
or decreasing Lb and
. For design, the conditions resulting in maximum pipe strains
max and
displacements dmax and dT,max need to be determined. The total pipe displacement cannot exceed
the ground displacement at x = ls, resulting in dT,max =
, as indicated by boundary conditions (d),
(g), and (h) in Tables 1 and 2.
Condition I, Lb ≥ 2ls. As the block length increases such that Lb > 2ls, as shown in Figure 2, the
block becomes large relative to the length of pipe being strained. The maximum pipe displacement
equals the ground displacement as shown in Figure 2 over a distance Lb - 2ls within the sliding
block. From this it is clear how dT ≤
, and dT,max =
and the following is obtained from Equation
19b
+2
(24)
From Equation 22:
(25)
And from Equation 23:
(26)
8
In Equations 25 and 26, only the + sign is of significance to the problem solution and thus they
may be re-written as
(25a)
(26a)
Using Equations 24 to 26, the problem can be solved by finding the length of engaged pipe and
maximum strain. Substituting dT,max =
into Equation 8 and utilizing Equation 7:
(27)
When
= 0 (i.e.
p = 0), Equations 19 to 27 resolve to that of continuous pipe as presented
by O’Rourke and Nordburg (1992). The strain
p relieved by joint displacement reduces the stress
in the pipe by an amount E
p. As a result, for all other properties being equal, the length ls ≤ lc;
that is, the length of engaged hybrid segmented pipe is always less than or equal to the length of
continuous pipe having the same resistance force applied. Additionally, the pipe strain for hybrid-
segmented pipe is always less than or equal to that of continuous pipe under identical conditions.
Condition II, Lb ≤ 2ls. Figure 3 shows the strains and displacement distributions for Condition II.
The length ls cannot exceed Lb/2. When 2ls = Lb the pipe segments within the block will all lock
up. This requires the ground displacement
to exceed the total amount of joint deflection plus
pipe elongation over the pipe length Lb/2:
(28)
Where the number of fully engaged joints is
over half the block length. Substituting
Equation 15 into Equation 28 for 2ls = Lb:
(29)
Equation 29 identifies the amount of block displacement needed to create the condition
2ls = Lb. If the block displacement is less than that given by Eq. 29, then the solution is defined by
Condition I. This indicates that for Condition II the hybrid-segmented pipe solution is conditional
upon a minimum block displacement relative to the pipe properties; which differs from a
continuous pipe solution (O’Rourke and Nordburg, 1992) where Condition II is completely
independent of
. From Eq. 21:
(30)
9
From Equation 16:
(31)
As with Condition I, when
= 0 (i.e.
p = 0), Equations 28 to 31 resolve to that of
continuous pipe as presented by O’Rourke and Nordburg (1992). Equations 28 and 31 indicate the
hybrid segmented pipe system, having all other properties equal to a continuous pipe (O’Rourke
and Nordburg, 1992), can accommodate an increase in strain of 2
p and ground movement of an
amount Lb
p. As previously described ls ≤ Lb/2. The case Lb = 2ls is the transition between a short
block and long block solutions, relative to the length of strained pipe, for which the following is
obtained from Equation 24:
(32)
Figure 3. Plot of hybrid segmented pipe strain and displacement in relation to
ground movement (Condition II).
Lb
ls
Block
Movement
CL
(+)
(-)
x
+p
(+)
(-) x
U(ground) = Ug
U(joint) = Uj U(pipe) = Up
Displacement, u(x)
h
H
pls
Strain,
U(Total) = UT
-(+p)
d
(a)
(b)
(c)
0Lb
Region I Region II Region III Region IV
p
dT
10
from Equation 14:
(33)
From Equation 7:
(34)
PROCEDURE FOR CALCULATING PIPE STRAIN AND DISPLACEMENTS
Figure 4 presents a flow diagram for the elastic design of hybrid segmented pipelines subjected to
longitudinal permanent ground deformations. First check ls by calculating the length using
Equation 25, then compare it to Lb/2. If ls is less than Lb/2 then the total peak displacement is
and the maximum pipe strain, determined using Equation 14 or 26, results from fr acting over the
distance ls on each side of the tension or compression crack (Condition I). The pipe displacement
is determined from Equation 15 or 27. On the contrary, if ls is greater than Lb/2 then it is set equal
to Lb/2, dT <
, and the block displacement can exceed the values determined using Equation 29
(Condition II). Additionally, the maximum pipe strain and displacements result from fr acting over
the pipe length of ls = Lb/2 and are determined using Equations 33 and 15, respectively. The
maximum total pipe displacement is determined from Equation 30 using ls = Lb/2.
Figure 4. Flow diagram for elastic design of hybrid segmented pipe subjected to
longitudinal permanent ground deformations.
(Eq. 25)
(Eq. 14)
or
(Eq. 26)
ls < Lb/2ls
ls = Lb/2
(Eq. 33)
(Eq. 15)
Y
N*
(Eq. 15)
or
(Eq. 27)
(Eq. 30)
*Requires
(Eq. 29)
Condition I
Condition II
11
The process in Figure 4 can be simplified if the most conservative result of dT =
and ls =
Lb/2 is used. This provides the largest pipe strains for either path in Figure 4, which can be
calculated using Equation 32 or 33. This results in the same equation as for the continuous pipe
when Condition II is met. For hybrid segmented pipes, as
p increases, and/or E decreases, the
probability of Condition II being met (i.e. ls > Lb/2) decreases. It is more accurate to calculate pipe
strain for Conditions I and II individually. For Condition I Equations 32 and 33 may be highly
conservative for hybrid-segmented pipes, depending on the values of
p.
In all cases, and all equations described above for segmented pipe, the solution reduces to
the continuous pipe solution when
= 0, that is when the joints have no displacement but still are
locked sufficiently to transfer strain.
The maximum strain values determined using Figure 4 need to be checked against
allowable critical limit states for inelastic behavior, which are governed by the yield strains
y in
tension and the buckling or
y strains in compression.
JOINT CAPACITY
The design limit strain will vary for different pipe materials and, for segmented pipelines with joint
capacity force less than the pipe barrel, the limiting strain is controlled by the axial strength of the
joint. From Figure 4, the maximum axial force Fmax considering all conditions is determined from:
(35)
However, as described above, for hybrid segmented pipes it is more accurate to calculate
joint force for Conditions I and II individually. For Condition I Equation 35 may be highly
conservative. When ls < Lb/2 the maximum force can be determined from:
(36)
The minimum joint resisting force required to ensure the piping functions as a hybrid segmented
system without pulling apart can be determined using Equations 35 and 36. Alternatively, the joint
resisting force can be determined by assuming a minimum distance L the pipe must pull through
the soil, which can be correlated with potential block lengths Lb and displacements
, as is done in
ISO 16134 (2006);
(37)
Where Fs is a factor of safety applied to ensure the design force is not exceeded.
CONCLUSIONS
A new methodology for assessing hybrid-segmented and continuous pipes in a common way, using
a single set of equations for axial strains caused by block ground movement, has been presented.
Results show how the segmented pipes with joint locking mechanisms have lower axial strains
12
than continuous pipes, for the same amount of block deformation. The model can also be used to
identify the limited capability of segmented joints which do not lock up (i.e., allow the spigot to
pull out of the bell) to withstand permanent ground deformations. This newly proposed pipe model
will improve the engineering design of continuous and segmented pipes in hazard prone areas.
ACKNOWLEDGEMENTS
The authors want to thank Prof. M. O’Rourke, Renssalaer Polytechnic Institute, for valuable
discussions and input. Comments from the ASCE Task Committee on Seismic Design of Buried
Pipelines during the March 2018 meeting are acknowledged.
REFERENCES
O’Rourke, M.J., and C. Nordburg. (1992). “Longitudinal Permanent Ground Deformations Effects
on Buried Continuous Pipelines.” National Center for Earthquake Engineering Research,
NCEER-92-0014.
International Organization for Standards (ISO) 16134. (2006) Earthquake- and subsidence-
resistant design of ductile iron pipelines.
