Journal of Pressure Vessel Technology, Vol.
126, No. 3, pp. 382390, August 2004
©2004 ASME. All rights reserved.
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Stability^{ }of Vertically Bent Pipelines Buried in Sand
Sahel N. Abduljauwad
Hamdan N. AlGhamedy
email: hghamdi@kfupm.edu.sa
Junaid^{ }A. Siddiqui
Ibrahim M. Asi
Naser A. AlShayea
Department of Civil Engineering, King Fahd University of^{
}Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Received:
May 10, 2002; revised: February 16,^{ }2004
This paper discusses the stability of
underground pipelines with preformed^{ }vertical bends buried in sandy
soil. More specifically, the minimum^{ }cover height required to prevent
the pipe from bowing under^{ }the action of forces due to temperature
change and internal^{ }pressure is estimated. The variables considered
include the pipe and^{ }soil materials, diameter, thickness, overburden
height, bend radius, bend angle,^{ }internal pressure, fluid specific
weight, and temperature variation. A comprehensive^{ }threedimensional
finite element analysis is carried out. The results are^{ }extracted
from the output obtained. These results are put in^{ }a database which
is used to develop general regression models^{ }to determine the
relationships among the different variables. Different buckling^{ }modes
are also considered. All of these results and models^{ }are entered into
a computer software program for ready access.^{ }©2004 ASME^{
}
Contributed by the Pressure Vessels and Piping Division
for publication^{ }in the JOURNAL OF PRESSURE VESSEL
TECHNOLOGY. Manuscript received by^{ }the PVP Division May 10,
2002; revision received February 16,^{ }2004. Associate Editor: C.
Jaske.
Contents
Introduction
Buried pipelines are very common in
industry; they^{ }may carry water, gas, petroleum products, or other
fluids. In^{ }certain situations, it is unavoidable, or at least more
economical,^{ }to have vertical bends in crosscountry pipelines. The
behavior of^{ }such bent pipelines is quite different from straight
ones, especially^{ }under temperature change. In order to make a
comprehensive investigation^{ }and end up with solid conclusions and
recommendations, several variables^{ }need to be considered in the
study. They include the^{ }different soil properties and the parameters
related to the pipe^{ }and the bend, such as pipe material properties,
diameter, thickness,^{ }overburden height, bend radius, bend angle,
internal pressure, fluid specific^{ }weight, and temperature variation.
^{}
In the literature, only a limited^{
}number of studies related to pipe bends have been carried^{ }out
or discussed. Bends are mentioned in some standards/codes. The^{
}American Society of Mechanical Engineers Code ASME B31.4 [1] recognizes^{ }the flexural behavior of pipe
bends by the use of^{ }what is termed a flexibility factor (k)
and a stress^{ }intensification factor (i) in which simple beam
theory is utilized.^{ }Karman [2] presented the first theoretical solution for smooth
unrestrained^{ }bends, after which several studies were carried out,
e.g., Vigness^{ }[3], Pardue and Vigness [4], Kafka and Dunn [5], Rodabaugh^{ }and George [6], and Findlay and Spence [7]. More recently,^{ }Thomson and Spence [8] presented some new analytical solutions. Thin^{
}shell theory was used by Whatham [9] who presented a^{ }solution without simplifying
assumptions. Gresnight and van Foeken [10] presented^{ }an analytical model for the
elastic/plastic design of pipe bends^{ }utilizing the minimum potential
energy theory; in that model, the^{ }soil load that acts on a buried
pipe bend was^{ }explicitly incorporated. ^{}
The finite element method was used by
Natarajan^{ }and Blomfield [11], Ohtsubo and Watanabe [12], and Weiß et al.^{ }[13] to develop different design aids. Natarajan and
Blomfield [11]^{
}examined several forms of end constraints for different parameters;
it^{ }was concluded that the significance of the tangent depends on^{
}the ratio of the bend angle to the radius. Weiß^{ }et al. [13]
demonstrated the use of the finite element method^{ }for the design of
pipe bends with respect to fatigue^{ }strength and load carrying
capacity. ^{}
In general, the proposed pipe^{
}bend elements can be broadly divided into two categories: beamshell^{
}and shellring elements. Beamshell elements are those in which shell^{
}type ovalizationdeformation is superposed on a curved beam element.
Hibett^{ }[14], Bathe and Almeida [15], and Mackenzie and Boyle [16]^{ }presented such a type. On the other hand,
the shellring^{ }type of elements are wholly based on the thin
shell^{ }theory. Ohtsubo and Watanabe [12]
proposed such an element. De^{ }Melo and De Casto [17] presented a pipe element, derived^{ }from the
arch bending theory, for the analysis of inplane^{ }bending of curved
pipes. ^{}
The restraint offered by soil
against^{ }the movement of buried pipes, termed subgrade reaction, has
been^{ }studied and modeled by many researchers. The first pioneer
who^{ }introduced the concept of elastic subgrade reaction was Winkler
[18],^{ }followed by Hetenyi [19]. Vesi [20] computed the uplift capacity^{ }of cylinders
on the basis of the pressure required to^{ }expand a surface cavity.
Audibert and Nyman [21] performed tests^{ }on the horizontal movement
of pipes. There have also been^{ }some studies to quantify soil
restraint against the oblique motion^{ }of pipelines such as that by
Nyman [22] and Hsu^{ }[23]. Trautmann et al. [24][25] carried out an extensive laboratory study^{
}of the uplift and lateral movement of buried pipes. They^{
}compared the results with that of Vesi [20],
Row and^{ }Davis [26], Ovesen [27], and Audibert and Nyman [21].
Dickin^{ }[28] and Poorooshasb et al. [29] carried out centrifuge model studies,^{ }while
Hsu [30] studied velocity effects on the lateral soil^{
}restraint of pipelines. Utilizing the finite element method, Yin et
al.^{ }[31], Altaee and Boivin [32], and Altaee et al. [33] performed^{ }some analyses of different soils.
For restrained underground pipes, several^{ }other studies, such as that
of Peng [34], Goodling [35],^{ }and Ng et al. [36], have been carried out. In the^{ }oil
industry, Saudi Aramco, the biggest oil company in the^{ }world in terms
of production, in its standard SAESL051 [37]^{ }specifies a simplified method for
calculating the required soil cover^{ }over bent buried pipes using an
"in house" computer program.^{ }It is based on the idealistic column
buckling with distributed^{ }transverse loading, which represents the
soil weight. ^{}
Description of the Problem and Need for the^{
}Research
Temperature variation and Poisson's
effect due to internal pressure may^{ }cause significant longitudinal
deformations in buried pipe bends. The earth^{ }pressure of the
confining soil at the bend contributes in^{ }resisting the movement (It
offers resistance to the moment.); thus,^{ }the strength of the soil is
important to keep the^{ }buried pipe bend adequately restrained against
excessive deformation. Methods based^{ }on classical theories have been,
and are still being, used^{ }for such problems; however, they have
proven to be inadequate^{ }in modeling the actual field behavior of the
pipesoil system.^{ }Numerical methods based on improved modeling
techniques are occasionally used,^{ }but their application is limited
for practical purposes due to^{ }the effort required in modeling the
complex pipesoil composite system.^{ }In particular, the finite element
method (FEM) has proven to^{ }be capable of modeling buried pipelines
satisfactorily; the major work^{ }on the subject has been summarized
above. In this study,^{ }a very comprehensive investigation on the
stability/soil cover requirement of^{ }vertically bent pipelines is
carried out utilizing threedimensional finite element^{ }analyses.
Several variables including soil properties, pipe material properties,
diameter,^{ }thickness, internal pressure, fluid specific weight, bend
radius, bend angle,^{ }temperature variation, and overburden height are
all considered. No such^{ }complete research has been done previously.
The problem of buried^{ }pipeline bends is commonly encountered in the
field, especially in^{ }the oil industry; thus, such a study is
necessary in^{ }order to arrive at an optimum design which incorporates
safety^{ }as well as economy. ^{}
Research Methodology
In order to carry out the research and
achieve^{ }its objectives, the following steps need to be executed:
^{}
1^{ }Review the literature on the
subject; this has been summarized^{ }above. ^{}
2 Select suitable software that is
capable of modeling^{ }the system discussed above including a
nonlinear/inelastic material model for^{ }soil behavior. ^{}
3 Set up and validate a threedimensional
FEM^{ }model that is capable of modeling a soilvertical pipe bend^{
}system. ^{}
4 Carry out a complete analysis of the
system^{ }for all possible combinations of the parameters that influence
the^{ }behavior of vertical buried pipe bends. ^{}
5 Develop tables, graphs,^{
}and/or charts, which may be used as design aids, utilizing^{ }the
results obtained by the FEM analyses to study the^{ }relationship
between various variables. Similarly, regression models, which correlate
the^{ }variables stated above, are to be formulated. ^{}
Material Models
Since it^{ }is always desirable,
and most of the time required, to^{ }keep the working stress in the pipe
below the yield^{ }strength, it is assumed that the pipe behavior will
be^{ }within the linear elastic range and that the material of^{
}the pipe is steel. With regard to local soil, sand^{
}predominates, and it is always used as the trench backfill^{
}without compaction. Sand was thus considered in this study;
therefore,^{ }the MohrCoulomb failure criterion was used. The steel
properties (for^{ }different grades) are known, while the strength
parameters of the^{ }local sand were determined experimentally by
triaxial and direct shear^{ }tests. The angle of friction for the soil,
, came^{ }out to be 35°, while the cohesion,
c, was zero.^{ }An interface (joint) element was also used and
will be^{ }discussed in the FEM model section. ^{}
Computer Program and Validation Checks
There are many FEMbased software^{
}packages available in the market. Among other factors, the
availability,^{ }the need/nature of the problem at hand, and the
cost^{ }should be considered when selecting a program for a study^{
}such as this. Accordingly, the Structure Medium Analysis Program
(SMAP3D)^{ }[38] was selected because it has special features which
met^{ }our needs. ^{}
In order to validate the program and
the^{ }models used, especially in the absence of previous studies
(experimental^{ }and analytical) in the same field, several runs were
carried^{ }out to study and compare individual structural phenomena.
They included^{ }the load distribution or arching in the soil around
the^{ }pipe, the soil resistance to the uplift movement of a^{
}straight pipe, and centrifuge modeling of buried bent pipes; details^{
}are given next. ^{}
To check the arching effect of
flexible^{ }and rigid pipes, several problems were analyzed. The
diameter chosen^{ }was 1219 mm (48 in.), while the elastic moduli
and^{ }the thicknesses were 200 GPa (29,000 ksi) and 152 mm^{
}(6 in.) for the rigid material, representing steel, and 690^{ }MPa
(100 ksi) and 6.35 mm (0.25 in.) for the^{ }flexible material, which
represents plastic. These were chosen in order^{ }to have distinct
properties for the two different pipes. Cover^{ }depths of 762 mm (30
in.), 1067 mm (42 in.),^{ }1524 mm (60 in.), and 2286 mm (90 in.)
were^{ }selected. Compared with the formulas of Marston and Anderson [39],^{ }the overall trend and behavior are
similar; however, more accurate^{ }results were expected using the
FEM than with the formulas,^{ }due to their crude approximation and
assumptions. Deformations as well^{ }as stress contours obtained were as
expected for both types^{ }of pipes. Details can be obtained in
Abduljauwad et al. [40][41].^{ }^{}
In continuation of the validation
process, the uplift movement of^{ }buried pipes was analyzed. The data
used for the comparison^{ }and verification were taken from the
Trautmann et al. [24]
study^{ }in which fullscale laboratory tests were carried out. That
investigation^{ }is widely recognized, and the use of its findings
in^{ }design has been recommended in various publications such as
ASME^{ }B31.1 [42] and CGL [43]. When the results of this^{ }study were
compared with the experimental values, good agreement was^{ }obtained
for small cover depths. As the cover depth increased,^{ }the FEM results
started to deviate and became noticeably different^{ }for loose sand
with the largest cover depth (52 in.).^{ }The same discrepancy was
observed with other studies, e.g., [20][26].^{
}Trautmann et al. [24]
mentioned that a punching mechanism develops during^{ }the uplift of a
deeply buried pipe in loose sand.^{ }They described the reason for this
discrepancy as the inability^{ }of analytical models to account for the
contractive behavior during^{ }shear; the high porosity of loose sand
results in large^{ }volume change, and this effect was not taken into
account^{ }by the analytical model. The original reference [24]
can be^{ }referred to for more details. As stated in that study,^{
}the uncertainty in deeply buried pipes is higher than that^{ }of
the shallow ones. Nevertheless, the results obtained here are^{ }better
than those of the previously published work, which was^{ }cited above.
^{}
Since it was not feasible to carry
out^{ }fullscale testing, centrifuge modeling was utilized to simulate
field conditions.^{ }The main concept behind the centrifuge modeling is
to amplify/scale^{ }the small model at hand by increasing the
gravitational force^{ }by "n" times such that full scale testing
is simulated.^{ }By doing so, the benefits of full scale testing
are^{ }obtained, and, on the other hand, the disadvantages of
normal^{ }laboratory experiments (small, idealized, etc.) and full scale
testing (cost,^{ }time, etc.) are eliminated. The complete theory behind
this is^{ }beyond the scope of the paper. For readers who are^{
}not familiar with centrifuge modeling concepts, many references on
the^{ }subject, including [40],
are available. The experiments were done using^{ }the centrifuge of the
University of Colorado, Boulder, U.S.A. For^{ }the reason stated below,
a 50.8 mm (2 in.) plastic^{ }pipe, with 1.93 MPa (280 psi) maximum
pressure (ASTM D^{ }1785), was used to prepare both 90° and 45°
bends.^{ }The bends had an internal diameter of 50.8 mm (2^{
}in.) and a thickness of 4.2 mm (0.165 in.). The^{ }model
properties were selected to represent AP1 60 carbon steel^{ }pipe with
1218 mm (48 in.) outer diameter and 19^{ }mm (0.76 in.) thickness using
a scale factor of 20.^{ }The same setting was idealized by a
threedimensional FEM mesh^{ }for each bend. Reasonably good agreement
between the centrifuge model^{ }measurements and the finite element
predictions was observed. More details^{ }can be found in Abduljauwad et
al. [40][41].
^{}
FEM Idealization and Analysis
Virtual Achor. The finite^{
}element analysis constituted the major and most demanding task in^{
}this work. Before elaborating on the threedimensional behavior of
buried^{ }bent pipes, some words about boundary conditions and pipe
anchors^{ }are warranted. A typical buried pipe bend is shown in^{
}Fig. 1. When a straight pipe connected to a bend^{
}expands (or contracts) under temperature change and/or internal pressure,
it^{ }causes the bend apex to move vertically, and this movement^{
}is resisted by the surrounding soil. The friction between the^{
}pipe and the soil restrains the longitudinal movement of the^{
}straight pipe relative to the soil. The maximum movement occurs^{
}at the end of the pipe where the bend is^{ }connected and starts
to be reduced from there to a^{ }point beyond which there is no movement
of the pipe^{ }relative to the soil. This point is called the
virtual^{ }anchor. The location of the virtual anchor is required
to^{ }provide appropriate boundary conditions for the threedimensional
mesh of a^{ }buried pipe bend. The location of the virtual anchor
is^{ }thus calculated using the method given in ASME B31.1 Appendix^{
}VII [42].
The following equation is used for calculating the^{ }virtual anchor
location, L_{va}:
where ^{}
Figure
1.
= AE/k is an effective^{ }length parameter
^{}
F_{max} is the maximum axial
force in^{ }pipe ^{}
f is the unit soil friction force
along^{ }the pipe ^{}
A is the crosssectional area of the^{
}pipe ^{}
E is the modulus of elasticity of
the^{ }pipe material ^{}
represents the pipesoil system characteristics
^{}
^{ }k is the soil modulus of the
subgrade reaction ^{}
The^{ }value of the influence
length L_{inf}, which is the length^{ }at which the
hyperbolic function in Hetenyi's equation [19]
approaches^{ }unity, is calculated using the equation
^{}
The uplift movement of^{ }a
vertical pipe bend is resisted by the overburden soil^{ }pressure _{s} (as shown in Fig. 1)
and the shear^{ }strength of the soil _{s}, as illustrated in Fig. 2.^{ }In addition, the movement of a buried pipe is
counteracted^{ }by the weight of the pipe and its contents. All^{
}of this is taken care of in the FEM idealization.^{
}^{}
Figure
2.
Mesh Generation. Since a soil
system comprises a semiinfinite domain extending^{ }a large distance in
the horizontal direction and downwards, one^{ }of the important aspects
in making an FEM mesh is^{ }to truncate the mesh in the semiinfinite
domain of the^{ }soil at a place where the geostatic condition exists.
The^{ }limits used to truncate the mesh and specify the free^{
}field condition are shown in Fig. 3. These limits were^{ }conservatively established
based on the recommendations in the literature (e.g.,^{ }[26][33])
and utilizing the observation made during the two and^{
}threedimensional validation and trial runs using the SMAP [38]
and^{ }CANDE [44] programs. ^{}
Figure
3.
The task of generating the
threedimensional mesh^{ }of the buried pipe bend system for a given
problem^{ }is accomplished utilizing the finite element modeling and
postprocessing, FEMAP^{ }[45][46] program. The basic strategy used in FEMAP is to^{
}first generate a twodimensional mesh along the pipe crosssection.
The^{ }twodimensional mesh is then extruded along the pipeline
longitudinal axis^{ }to get the full threedimensional mesh.
^{}
To model the system,^{ }continuum
elements characterized by the MohrCoulomb failure criterion were used^{
}for the soil, shell elements were utilized to model the^{ }pipe,
while joint elements were assumed to represent the pipesoil^{
}interface. Since the thickness of the joint element occupies a^{
}region that is physically taken up by the soil, it^{ }is,
therefore, desirable to keep its thickness as small as^{ }possible.
However, it was found during the trial and validation^{ }runs that the
solution did not converge if a very^{ }small value for the thickness of
the joint element was^{ }used. Each of the validation runs was,
therefore, solved a^{ }number of times by changing the value of the
joint^{ }element thickness until a stable solution was obtained for
the^{ }smallest possible value of the joint thickness. Thus, it was^{
}concluded that a suitable value for the joint element's thickness^{
}was D/40 where D is the outer diameter of the^{
}pipe. Apart from the thickness, the value of the joint^{ }element
shear parameter, G, was also found out to be^{ }significant in
achieving stable results because of the longitudinal movement^{ }of the
pipe relative to the soil. Stable and converged^{ }results are obtained
when the value of G does not^{ }exceed a certain limit. A value
of 172 kPa (25^{ }psi) emerged as the most appropriate for a cover
depth^{ }of 305 mm (12 in) or more. A smaller value^{ }for
G needs to be used in some cases where^{ }a very small cover
depth is used. ^{}
A typical twodimensional^{
}mesh, which is used to generate the threedimensional mesh, is^{
}shown in Fig. 4 in which 24 shell elements are^{ }used to model
the circle of the pipe; due to^{ }symmetry, only half of the domain is
shown. The aspect^{ }ratio of the soil continuum elements is kept as
close^{ }to 1 as possible within a width of 1.5D on^{
}each side of the pipe center. Beyond that width, the^{ }element
aspect ratio is increased gradually up to the geostatic^{ }condition
when it becomes 3, as a maximum. This scheme^{ }allows for satisfactory
mesh density near the pipe while keeping^{ }the problem size relatively
manageable. This conclusion was reached after^{ }many trial runs were
carried out for different meshes, ranging^{ }from very fine with square
or almost square elements to^{ }relatively coarse with
rectangular/slender elements. ^{}
Figure
4.
The extrusion of twodimensional
meshes^{ }to threedimensional meshes is quite lengthy and geometrically
complex, due^{ }to the nature of the problem and boundary conditions.
However,^{ }a typical threedimensional generated mesh is shown in Fig.
5^{ }in which symmetry is taken advantage of so that
only^{ }one quarter of the domain is considered with appropriate
boundary^{ }conditions. Details can be found in Siddiqui [47]. ^{}
Figure
5.
Application of Loads
The loads^{ }considered in this
investigation are gravity, which includes the weight^{ }of the soil and
pipe and its contents, internal pressure,^{ }and temperature. When
calculating the weight of fluid inside the^{ }pipe, the elevation of the
vertical bend was taken into^{ }account at different nodes. Due to the
nonlinearity of the^{ }problem, these loads were incremented, and within
each increment iterations^{ }were performed until convergence of the
solution was reached. After^{ }many numerical tests, 20 load steps were
found to be^{ }the optimum for most runs. ^{}
Parametric Study
After the preliminary,^{ }but
necessary, work presented above, a full and comprehensive parametric^{
}study was carried out. An extensive numerical analysis program,
utilizing^{ }the FEM and considering all variables and factors of
concern,^{ }was run, and large outputs and results were obtained.
Only^{ }a brief description and sample results are presented here,
and^{ }further explanations and presentation can be found in Abduljauwad
et al.^{ }[40][41].
^{}
The parameters, along with their ranges,
considered in this^{ }work are shown in Table 1. The values used for^{ }the FEM analysis were
carefully selected within these ranges, with^{ }more emphasis on
critical values and limits and intermediate points^{ }so that the
results could be used to develop regression^{ }models which are general
and reliable, as discussed later. The^{ }values of these parameters are
varied within their limits, and^{ }various combinations were considered
in order to obtain the effect^{ }of each of the parameters individually
as well as the^{ }interaction among them. ^{}
To define the capacity of the buried^{
}pipe vertical bend due to temperature changes (in addition to^{
}gravity loads and internal pressure), two criteria are possible. The^{
}first one, termed by the authors as the ultimate temperature^{
}method (UTM) defines the point when the soil above the^{ }pipe is
on the verge of shear failure. This means^{ }that the pipe would have
moved "some distance" up before^{ }failure, which implies that the soil
would have "flowed" beneath^{ }the pipe. This action is regarded as
completely undesirable by^{ }some oil companies, including Saudi Aramco;
thus, it is not^{ }presented here even though it is more economical. The
second^{ }method, named by the authors as the installation condition
method^{ }(ICM), requires that the upward movement of the bend
under^{ }the combined applied loads is restricted to the installation
condition,^{ }which is defined as the state of the trench before^{
}the pipe is laid. After the installation of the pipe,^{ }the whole
system settles down under the weight of the^{ }pipe and soil cover.
Therefore, according to the ICM, the^{ }allowed upward movement of the
bend apex is equal to^{ }the settlement caused by the weight of the soil
cover^{ }and pipe before applying the loads. Care has to be^{
}taken in the FEM analysis regarding the total settlement. The^{
}contribution from the mesh below the pipe under its own^{
}weight (before laying the pipe and filling the trench) should^{
}be subtracted from the total settlement of the pipe bend^{
}extrados apex in order to get the allowed uplift movement^{
}according to this method. Due to space limitation, details of^{
}the two methods cannot be fully presented here; e.g., see^{
}Siddiqui [47].
^{}
The results obtained by the FEM analysis,
which^{ }are of concern here, are best summarized in a tabular^{
}form. Since the list is very long, only a partial^{ }list of the
results is presented in Table 2. They^{ }have been extracted from the huge output
of the threedimensional^{ }analyses which took several months to run on
the latest^{ }Pentium processor. Generally, each single run took several
hours to^{ }complete. As sample representatives, some of the results are
presented^{ }graphically in Figs. 6 to 8. ^{}
Figure
6. Figure
7. Figure
8.
Buckling of Buried Pipes
Since buckling^{ }of pipes can
occur, it needs to be checked, along^{ }with the analysis above; it
could be critical, especially in^{ }large diametersmall thickness
pipes. The buckling of shelltype structures is^{ }quite involved, while
the buckling of buried and relatively flexible^{ }pipes is even more
complicated. There are many buckling modes^{ }and "exact" theories do
not exist for some of them.^{ }Due to this, certain theories with
specific assumptions and limitations,^{ }supported with some
experimental results, if available, are utilized in^{ }the current
study. Without elaboration, the following buckling modes are^{
}considered: ^{}
1 Buckling of cylindrical shells under
the action of^{ }uniform axial compression (axial buckling by warping)
(Timoshenko and Gere^{ }[48], Antaki [49], Ellinas [50], and Watashi and Iwata [51]).^{ }^{}
2 Buckling of cylindrical shells under
the action of uniform^{ }external pressure (ring buckling) (Farshad [52], Timoshenko and Gere [48],^{
}Antaki [49],
AWWA C150 [53], and Moore and Booker [54][55]).^{ }^{}
3 Pure bending buckling (winkling due to
longitudinal bending) (Farshad^{ }[52],
Antaki [49],
Murray [56], Chiou and Chi [57], Hobbs^{ }[58][59], Taylor and Gan [60][61][62], Reddy [63], and Stephens [64]).^{ }^{}
4 Lateral beam/shell buckling
(beamcolumn/shell) (Antaki [49],
Yun and Kyriakides^{ }[65][66], Deutsch and Weston [67], Shaw and Bomba [68] Choiu^{ }and Chi [69], and Zhou and Murray [70]). ^{}
5 Buckling^{ }of buried
initiallybent pipes (Croll [71], Allan [72], and Raoof^{ }and Maschner [73]). ^{}
6 Buckling due to the combined
effect^{ }of the stress components (API RP 1102 [74], Farshad [52],^{
}and German Code DIN 18800 Part 4 in Jullien [75]).^{ }^{}
These checks were carried out utilizing
the results obtained from^{ }the FEM analysis. This was done by a
comprehensive computer^{ }program written for this project. If any of
the buckling^{ }modes occurs, then a message is given indicating the
mode^{ }of buckling, meaning that there is instability; i.e., the
stability^{ }of the system cannot be maintained. This leads to
problem^{ }redesign (especially pipe thickness), then analysis, and then
check. ^{}
Regression Models
The design variables used in^{
}developing the regression equations to predict the ultimate temperature,
as^{ }the dependent variable, that the pipe can withstand in the^{
}presence of a vertical pipe bend are pipe diameter, pipe^{
}thickness (or D/t ratio), depth of cover, radius and
angle^{ }of bend, internal pressure, and specific gravity of the
transported^{ }material. These are the variables which were varied in
the^{ }finite element runs to generate a database. As an
alternative,^{ }the cover height can be made the dependent variable.
^{}
To^{ }check the relationships
among the variables used in the development^{ }of the regression model,
first a correlation matrix is obtained.^{ }Second, on a further study of
the trend of the^{ }data, different groups of such data are created
according to^{ }the behavior of the buried pipe bend. ^{}
A regression analysis^{ }was
performed utilizing the software package STATISTICA (release 6.1). The^{
}resulting regression models for the different groups of data are^{
}shown in Table 3. The results of the finite element^{ }analysis
were utilized to develop the correlation coefficients of the^{ }models.
The coefficient of determination, R^{2}, and the significance
levels^{ }of the generated models are also presented in that table.^{
}The R^{2} values for all developed models are higher
than^{ }0.88. Moreover, the confidence levels for all models are
higher^{ }than 99.99%. Two forms of equations are presented. One is^{
}used to calculate the maximum allowable temperature change, T, as^{ }a function of the other
variables. The second form is^{ }to determine the required (minimum)
cover height, H_{c}, needed for^{ }specific values
of the other variables. The first form is^{ }suitable for checking
existing problems/applications, while the second one is^{ }appropriate
for the actual design (at the beginning). For values^{ }falling between
two groups, interpolation is utilized; this is done^{ }automatically in
the computer program written for this purpose. In^{ }addition, or as an
alternative, figures and charts can be^{ }plotted utilizing the data
generated. However, this is a lengthy^{ }process and is not presented
here. ^{}
The results of entire^{ }research
program discussed above were programmed into a computer code.^{ }The
result is a userfriendly software package called "Analysis and^{
}Design of Buried Pipelines" (ADBP) which is capable of making^{
}all necessary checks, analysis, and design (Abduljauwad et al. [76][77]). It^{ }is worth mentioning that the original
database used and the^{ }analyses carried out were in FPS/U.S. customary
units as shown^{ }in the table; thus, the coefficients and the variables
in^{ }the models must be in such units. The conversion factors^{
}from these units to the SI units are written at^{ }the bottom of
the table; however, such conversion factors are^{ }programmed in the
computer so that the user can select^{ }the SI units, and the program
automatically converts the SI^{ }units into the appropriate units at the
beginning of the^{ }analysis and at the end to show the results in^{
}the standard SI units. The SI units' user does not^{ }"feel" it.
The authors thought that this is the easiest/best^{ }way of doing it for
two main reasons. First, it^{ }is not worth changing all the units in
the database,^{ }regression analysis, etc. since the program accepts
either of the^{ }two systems of units and make the appropriate
conversion without^{ }the user's interference. Second, some
societies/associations/individuals still use the U.S.^{ }Customary
units, or at least they allow their usage. ^{}
Summary and Conclusions
The stability and cover height
requirements for^{ }buried pipelines with vertical bends were
investigated. Based on preliminary^{ }trial tests and laboratory
experiments, comprehensive finite element analyses were^{ }carried out,
and the required data were obtained. These results^{ }were utilized to
develop regression equations considering different variables including^{
}pipe and soil properties, diameter, thickness, overburden height, bend
radius,^{ }bend angle, internal pressure, fluid specific weight, and
temperature variation.^{ }The developed models gave good estimates for
the required cover^{ }height needed to prevent the pipe from bowing.
Moreover, the^{ }suggested models are easy to understand and apply by
practicing^{ }engineers. ^{}
Acknowledgment
The support^{ }of the Saudi
Arabian Oil Company (Saudi Aramco) is very^{ }much appreciated. The
utilization of facilities of King Fahd University^{ }of Petroleum and
Minerals in general and the Civil Engineering^{ }Department and the
Research Institute in particular is also acknowledged.^{ }The assistance
of the University Editing Board is also appreciated.^{
}^{}
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FIGURES
Full
figure (19 kB)
Fig. 1 Typical vertical^{ }buried pipe bend: (a) Perspective
sectional view, (b) side view^{ }showing key parameters First
citation in article
Full
figure (7 kB)
Fig. 2 Soil reaction against movement of buried vertical bend First
citation in article
Full
figure (9 kB)
Fig. 3 Location^{ }of mesh boundaries: (a) limits for pipe under
gravity loading;^{ }and (b) limits for pipe moving under uplift
forces First
citation in article
Full
figure (17 kB)
Fig. 4 Twodimensional mesh^{ }made to extrude a threedimensional
vertical bend mesh First
citation in article
Full
figure (21 kB)
Fig. 5 Buried pipe vertical^{ }bend mesh (threedimensional): (a)
perspective view; (b) plan; and (c)^{ }side view First
citation in article
Full
figure (17 kB)
Fig. 6 Effect of cover height First
citation in article
Full
figure (18 kB)
Fig. 7 Effect of pipe diameter First
citation in article
Full
figure (14 kB)
Fig. 8 Effect of bend^{ }radius First
citation in article
TABLES
Table 1. Range of parameters considered in^{
}the study 
Factor 
Minimum 
Maximum 
Comments 
Pipe outer^{ }diameter, D 
305 mm (12 in.) 
1524 mm (60 in.) 
This range^{ }is common in the oil industry

Height of overburden from surface to^{ }pipe
crown, H_{c} 
As required 
As required 
This is usually^{ }the needed variable 
Pipe bend radius, R_{b} 
15.2 m (50 ft) 
213.4^{ }m (700 ft) 
This range is common in the oil industry 
Pipe^{ }bend angle, 
1° 
20° 
This range is common in the^{ }oil industry

Diameter/thickness ratio, D/t 
50 
150 
This range is common^{ }in the oil industry

Internal pressure, p 
0 
^{*} 
^{*} The^{ }maximum the pipe can carry
before reaching the maximum allowable stress 
Specific gravity^{ }of pipe content,
G_{f} 
0 
1 
0 (Gas), 0.56 (LPG), 0.86 (Crude^{ }Oil), 1
(Water) 
Temperature change, T 
0 
66.7°C (120°F) 
This range^{ }is common in the oil industry

Pipe allowable stress 
^{*} 
^{*}^{ } 
^{*} Any grade of steel with an appropriate
safety factor 
Safety factor^{ } 
^{*} 
^{*} 
^{*} As specified by the
used code/standard etc. 
Modulus^{ }of soil reaction, E^{} 
^{*} 
^{*} 
^{*} Appropriate value for^{
}the local soil (for buckling check) 
Winkler spring coefficient, k_{0} 
^{*} 
^{*} 
^{*}^{ }Appropriate value for
the local soil (for buckling check) 
First
citation in article
Table 2. Maximum temperature change
^{} 
S. No. 
D mm (in) 
H_{c} mm (in) 
R_{b} m (ft) 
(Deg)^{ } 
D/t 
p kPa (psi) 
G_{f} 
Maximum temperature change °C (°F)

1 
300 (12) 
1750 (70) 
15^{ }(50) 
20 
50 
1034 (150) 
0 
30.14 (54.26) 
2 
600 (24) 
900 (36) 
15^{ }(50) 
20 
50 
1034 (150) 
0 
12.72 (22.89) 
3 
600 (24) 
1500 (60) 
15^{ }(50) 
20 
50 
1034 (150) 
0 
20.34 (36.61) 
4 
1050 (42) 
900 (36) 
15^{ }(50) 
20 
50 
1034 (150) 
0 
13.91 (25.04) 
5 
1050 (42) 
1500 (60) 
15^{ }(50) 
20 
50 
1034 (150) 
0 
20.71 (37.27) 
6 
1500 (60) 
900 (36) 
15^{ }(50) 
20 
50 
1034 (150) 
0 
15.68 (28.22) 
7 
300 (12) 
300 (12) 
90 (300)^{ } 
20 
50 
1034 (150) 
0 
22.77 (40.99) 
8 
300 (12) 
750 (30) 
90 (300) 
20^{ } 
50 
1034 (150) 
0 
61.75 (111.15) 
9 
600 (24) 
750 (30) 
90 (300) 
20 
50^{ } 
1034 (150) 
0 
32.3 (58.14) 
10 
600 (24) 
425 (17) 
210 (700) 
20 
50 
1034 (150)^{ } 
0 
39.52 (71.13) 
11 
600 (24) 
525 (21) 
210 (700) 
20 
50 
1034 (150) 
0^{ } 
47.78 (86.01) 
12 
1500 (60) 
1500 (60) 
15 (50) 
15 
50 
1034 (150) 
0 
25.48 (45.86)^{ } 
13 
300 (12) 
300 (12) 
90 (300) 
15 
50 
1034 (150) 
0 
22.49 (40.48) 
14^{ } 
1050 (42) 
750 (30) 
210 (700) 
15 
50 
1034 (150) 
0 
42.68 (76.82) 
15 
1050 (42)^{ } 
900 (36) 
210 (700) 
15 
50 
1034 (150) 
0 
50.09 (90.16) 
16 
1050 (42) 
1050 (42)^{ } 
90 (300) 
8 
50 
1034 (150) 
0 
31.8 (57.24) 
17 
1050 (42) 
1500 (60) 
90 (300)^{ } 
8 
50 
1034 (150) 
0 
45.24 (81.43) 
18 
1500 (60) 
900 (36) 
90 (300) 
8^{ } 
50 
1034 (150) 
0 
26.77 (48.19) 
19 
1500 (60) 
1050 (42) 
90 (300) 
8 
50^{ } 
1034 (150) 
0 
30.54 (54.98) 
20 
1500 (60) 
1500 (60) 
90 (300) 
8 
50 
1034 (150)^{ } 
0 
41.63 (74.93) 
21 
300 (12) 
250 (10) 
210 (700) 
8 
50 
1034 (150) 
0^{ } 
35.71 (64.27) 
22 
300 (12) 
375 (15) 
210 (700) 
8 
50 
1034 (150) 
0 
54.23 (97.61)^{ } 
23 
600 (24) 
900 (36) 
15 (50) 
20 
100 
1034 (150) 
0 
19.33 (34.8) 
24^{ } 
600 (24) 
1500 (60) 
15 (50) 
20 
100 
1034 (150) 
0 
33.32 (59.97) 
25 
1050 (42)^{ } 
900 (36) 
15 (50) 
20 
100 
1034 (150) 
0 
20.1 (36.18) 
26 
1500 (60) 
1500 (60)^{ } 
15 (50) 
20 
100 
1034 (150) 
0 
32.99 (59.39) 
27 
600 (24) 
375 (15) 
90 (300)^{ } 
20 
100 
1034 (150) 
0 
30.81 (55.45) 
28 
600 (24) 
600 (24) 
90 (300) 
20^{ } 
100 
1034 (150) 
0 
46.54 (83.77) 
29 
1500 (60) 
300 (12) 
210 (700) 
20 
100^{ } 
1034 (150) 
0 
29.07 (52.32) 
30 
1500 (60) 
450 (18) 
210 (700) 
20 
100 
1034 (150)^{ } 
0 
39.08 (70.35) 
31 
1050 (42) 
600 (24) 
90 (300) 
18 
100 
1034 (150) 
0^{ } 
28.11 (50.59) 
32 
1050 (42) 
900 (36) 
90 (300) 
15 
100 
1034 (150) 
0 
40.24 (72.44)^{ } 
33 
1500 (60) 
600 (24) 
210 (700) 
15 
100 
1034 (150) 
0 
44.45 (80.01) 
34^{ } 
1050 (42) 
600 (24) 
90 (300) 
11 
100 
1034 (150) 
0 
27.09 (48.77)^{ } 
35 
600 (24) 
750 (30) 
15 (50) 
8 
100 
1034 (150) 
0 
29 (52.2) 
36^{ } 
1050 (42) 
600 (24) 
210 (700) 
8 
100 
1034 (150) 
0 
51.57 (92.83) 
37 
1500 (60)^{ } 
600 (24) 
210 (700) 
8 
100 
1034 (150) 
0 
40.09 (72.16) 
38 
1500 (60) 
900 (36)^{ } 
210 (700) 
8 
100 
1034 (150) 
0 
59 (106.2) 
39 
600 (24) 
900 (36) 
15 (50)^{ } 
20 
150 
1034 (150) 
0 
24.66 (44.39) 
40 
600 (24) 
1050 (42) 
15 (50) 
20^{ } 
150 
1034 (150) 
0 
29.52 (53.14) 
41 
1050 (42) 
900 (36) 
15 (50) 
20 
150^{ } 
1034 (150) 
0 
23.92 (43.06) 
42 
600 (24) 
375 (15) 
90 (300) 
15 
150 
1034 (150)^{ } 
0 
34.79 (62.63) 
43 
600 (24) 
450 (18) 
90 (300) 
15 
150 
1034 (150) 
0^{ } 
42.26 (76.06) 
44 
1050 (42) 
600 (24) 
90 (300) 
15 
150 
1034 (150) 
0 
35.26 (63.47)^{ } 
45 
1500 (60) 
450 (18) 
210 (700) 
15 
150 
1034 (150) 
0 
41.29 (74.33) 
46^{ } 
1050 (42) 
600 (24) 
90 (300) 
11 
150 
1034 (150) 
0 
35.63 (64.13) 
47 
600 (24)^{ } 
600 (24) 
15 (50) 
8 
150 
1034 (150) 
0 
28.29 (50.93) 
48 
1050 (42) 
700 (28)^{ } 
15 (50) 
8 
150 
1034 (150) 
0 
35.66 (64.19) 
49 
1500 (60) 
700 (28) 
15 (50)^{ } 
8 
150 
1034 (150) 
0 
38.67 (69.6) 
50 
1500 (60) 
375 (15) 
210 (700) 
8^{ } 
150 
1034 (150) 
0 
31.24 (56.23) 
51 
1500 (60) 
450 (18) 
210 (700) 
8 
150^{ } 
1034 (150) 
0 
36.06 (64.91) 
52 
1500 (60) 
900 (36) 
210 (700) 
20 
50 
4309 (625)^{ } 
0 
34.53 (62.15) 
53 
600 (24) 
1050 (42) 
90 (300) 
15 
50 
4309 (625) 
0^{ } 
36.07 (64.92) 
54 
1500 (60) 
900 (36) 
15 (50) 
8 
50 
4309 (625) 
0 
19.8 (35.64)^{ } 
55 
1500 (60) 
600 (24) 
15 (50) 
8 
100 
4309 (625) 
0 
13.89 (25) 
56^{ } 
1500 (60) 
600 (24) 
210 (700) 
20 
150 
4309 (625) 
0 
59.82 (107.67) 
57 
300 (12)^{ } 
1750 (70) 
15 (50) 
20 
50 
5516 (800) 
0 
21.74 (39.14) 
58 
300 (12) 
3500 (140)^{ } 
15 (50) 
20 
50 
5516 (800) 
0 
51.97 (93.55) 
59 
300 (12) 
750 (30) 
90 (300)^{ } 
15 
50 
5516 (800) 
0 
48.98 (88.17) 
60 
1500 (60) 
900 (36) 
210 (700) 
20^{ } 
50 
7584 (1100) 
0 
29.26 (52.67) 
61 
1050 (42) 
600 (24) 
90 (300) 
20 
150^{ } 
7584 (1100) 
0 
20.87 (37.56) 
62 
1500 (60) 
750 (30) 
210 (700) 
20 
150 
7584 (1100)^{ } 
0 
51.55 (92.79) 
63 
525 (21) 
450 (18) 
198 (660) 
10 
85 
689 (100) 
0^{ } 
62.16 (111.89) 
64 
1050 (42) 
500 (20) 
202.5 (675) 
19 
115 
1379 (200) 
0^{ } 
50.87 (91.56) 
65 
1375 (55) 
2000 (80) 
30 (100) 
18 
75 
5516 (800) 
0 
27.04 (48.67)^{ } 
66 
675 (27) 
500 (20) 
113.4 (378) 
19 
90 
1586 (230) 
0 
39.97 (71.95)^{ } 
67 
450 (18) 
1750 (70) 
30 (100) 
18 
75 
1551 (225) 
0 
51 (91.8) 
68^{ } 
650 (26) 
2000 (80) 
30 (100) 
18 
75 
1551 (225) 
0 
46.35 (83.43) 
69 
900 (36)^{ } 
1750 (70) 
30 (100) 
18 
75 
1551 (225) 
0 
35.64 (64.15) 
70 
1250 (50) 
2000 (80)^{ } 
30 (100) 
18 
75 
3447 (500) 
0 
33.41 (60.13) 
71 
650 (26) 
2000 (80) 
30 (100)^{ } 
18 
75 
2413 (350) 
0 
43.92 (79.06) 
72 
450 (18) 
250 (10) 
190.5 (635)^{ } 
16 
135 
2068 (300) 
0 
36.12 (65.02) 
73 
1200 (48) 
1750 (70) 
30 (100) 
18^{ } 
75 
3103 (450) 
1 
31.14 (56.06) 
74 
1500 (60) 
900 (36) 
210 (700) 
20 
50^{ } 
1034 (150) 
1 
46.25 (83.25) 
75 
1500 (60) 
900 (36) 
15 (50) 
8 
50 
1034 (150)^{ } 
1 
31.12 (56.01) 
76 
1500 (60) 
450 (18) 
210 (700) 
20 
100 
1034 (150) 
1^{ } 
50.03 (90.05) 
77 
1500 (60) 
450 (18) 
15 (50) 
8 
100 
1034 (150) 
1 
30.52 (54.94)^{ } 
78 
1500 (60) 
900 (36) 
90 (300) 
8 
100 
1034 (150) 
1 
48.36 (87.05) 
79^{ } 
600 (24) 
300 (12) 
210 (700) 
20 
150 
4309 (625) 
1 
24.7 (44.46) 
80 
1050 (42)^{ } 
60 (24) 
90 (300) 
18 
150 
4309 (625) 
1 
39.95 (71.91) 
81 
1050 (42) 
375^{ }(15) 
210 (700) 
15 
150 
4309 (625) 
1 
45.97 (82.75)^{ } 
82 
1050 (42) 
450 (18) 
210 (700) 
15 
100 
7584^{ }(1100) 
1 
32.01 (57.62) 
83 
1500 (60) 
900 (36) 
90^{ }(300) 
8 
100 
7584 (1100) 
1 
24.41 (43.93) 
84 
600^{ }(24) 
450 (18) 
90 (300) 
15 
150 
7584 (1100) 
1^{ } 
36.82 (66.28) 
First
citation in article
Table 3. Generated models for the ultimate
change in temperature and^{ }depth of cover for pipes with
vertical bends 
Bend Radius (ft) 
Pipe Diameter (in.)^{ } 
Generated Model 
R^{2} 
Signifi cance level 
50 
All 
T = 71.5294 + 0.2184 D/t
+ 0.9088 H_{c} 28.6915*ln()0.0496 p + 19.2352
G_{f} 
0.8877 
0.000 


H_{c} = 1/0.9088*(T + 71.5294 +
0.2184 *D/t28.6915*ln()0.0496* p +
19.2352*G_{f})^{ } 


300 
24 
T = 32.3662188.241 0t + 2.6496
H_{c} + 11.7831 ln()0.0306 p + 12.6470
G_{f} 
0.8837 
0.000 


H_{c} = 1/2.6496*(T32.3662 + 188.241
*t 11.7831*ln() + 0.0306*
p12.647 *G_{f}) 


300 
42 
T = 1/(0.0191 + 0.0223 t0.0004
H_{c} 0.0148(1/) + 1.02 *E5*
p0.0076 G_{f}) 
0.9067^{ } 
0.000 


H_{c} = 1/0.0004*(1/T + 0.0191 + 0.0223
*t 0.0148*(1/) + 0.0000102
*p0.0076 *G_{f}) 


300 
60 
T = exp(3.68720.5906 ln(t) +
0.0216H_{c} 0.2022 ln()9.49 *E4*p +
0.4650 G_{f}) 
0.9323 
0.000 


H_{c} = 1/0.0216*(ln(T) +
3.68720.5906 *ln(t)0.2022*ln()0.000949 *p + 0.465*
G_{f}) 


700 
24^{ } 
T = exp(3.06770.5615 ln(t) +
0.0676 H_{c} + 0.1169 ln()4.85*E3*p +
0.8480 G_{f}) 
0.9453 
0.000 


H_{c} = 1/0.0676*(ln(T) + 3.0677 0.5615*ln(t) +
0.1169*ln() 0.00485 *p + 0.848*
G_{f}) 


700 
42 
T = 20.961229.7225 ln(t) +
2.2437 H_{c} + 11.3463 ln()0.0280 p + 12.4599
G_{f} 
0.9064 
0.000 


H_{c} = 1/2.2437*(T20.961229.7225* ln(t) +
11.3463*ln()0.028 *p + 12.4599
*G_{f})^{ } 


700 
60 
T = 41.128466.1520 t + 2.0727
H_{c} + 10.8649 ln()0.0123 p + 33.8879
G_{f} 
0.8894 
0.000 


H_{c} = 1/2.0727*(T + 41.128466.152 *t +
10.8649*ln()0.0123 *p + 33.8879 *
G_{f}) 

T=ultimate change in^{
}temperature, °F =angle of bend, ° t=pipe wall
thickness,^{ }in p=internal pressure,
psi H_{c}=depth of cover, in
^{}G_{f}=carried material specific gravity
To convert from °F to^{ }°C: T(°C) = [T(°F)]5/9 To convert from in. to mm:
H_{c} (mm) = [H_{c}
(in.)]25.4 
First
citation in article
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