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FE model for simulating wire-wrapping during prestressing of an embedded prestressed concrete cylinder pipe
Simulation Modelling Practice and Theory 18 (2010) 624–636
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FE model for simulating wire-wrapping during prestressing of an embedded prestressed concrete cylinder pipe Huan Xiong, Penghui Li *, Qingbin Li State Key Laboratory of Hydroscience and Engineering, Tsinghua University, 100084 Beijing, China
a r t i c l e
i n f o
Article history: Received 31 August 2009 Received in revised form 4 November 2009 Accepted 7 January 2010 Available online 18 January 2010 Keywords: Prestressed concrete cylinder pipe Prestressing Wire-wrapping Finite element simulation
a b s t r a c t A new 3D FE model, called wire-wrapping model, is proposed to simulate the process of wrapping wire for the prestressing of an embedded prestressed concrete cylinder pipe. This model is described in detail for its presentation and applied to a practical project for the prestressing simulation. The results obtained using this model are analyzed and compared with those obtained using the equivalent external radial pressure method and those using the AWWA C304 standard for its verification. Finally, the characteristics and contributions of this model are specified. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Prestressed concrete cylinder pipe (PCCP) was widely used in the world during the last few decades for water conveyance in many areas like municipal, industrial, plant piping systems, etc. Two types of PCCP are produced: lined and embedded. In recent years, some significant projects such as the Great Man Made River (GMMR) in Libya and the South-to-North Water Diversion (SNWD) in China required extremely large diameter embedded PCCP for water transmission. So, most research has been focused on the performance of embedded PCCP for their larger size compared with the lined type. This type of PCCP is compressed by a helical high-strength wire. The wire is wrapped under high tension around a core composed of a steel cylinder encased in concrete. A mortar coating is applied to the exterior of the pipe to protect the wires from corrosion. As we all know, the wire in the PCCP is used to produce compressive prestress in the core that offsets the tensile stresses from loads and pressure. So, the wire should bean important load-bearing component and has a significant influence on the structural integrity of a PCCP. Lots of investigations showed that most failures of PCCPs could be attributed to the loss of prestress due to wire breaking [1]. Recently, FE analysis has been used to study the performance of a PCCP subjected to a combination of many types of loads and pressure, although it is well known that FEA is a difficult method to implement because of the composite structure of the PCCP. Generally, the effects of the wire component on the results of the FE analysis must be considered since the wire is an essential component of the structure as mentioned above. In addition, the simulation of prestressing is pivotal and complicated and can significantly affect the results obtained for the PCCP structure. Currently, some traditional equivalent methods such as those that employ an equivalent external radial pressure, equivalent temperature descent, initial stress, or initial strain are adopted in simulating the prestressing; however, each has its disadvantage. The equivalent external radial pressure method is popular in engineering and is a conventional approach for simulating the prestressing of a PCCP. The method was proposed by Zarghamee et al. [2–4], Diab and Bonierbale [5], Gomez et al. [6], * Corresponding author. Tel.: +86 010 62785550. E-mail address: [email protected] (P. Li). 1569-190X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2010.01.007
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and Lotfi et al. [7] among others. This method considers a thin layer of wire so that the stiffness of the wire contributing to the pipe structure is not taken into account and the wire cannot bear any load. Using this method, the composition of the wire and the relation between the stress of the wire and the prestress of the PCCP cannot be considered, and neither can the interaction between the wire and concrete. Studies by Zhang [8] and Lin [9] included the wire component in the pipe model and used the equivalent temperature descent method for applying prestress, which considers the contribution of the wire stiffness, but the results did not agree well with what is seen in practice. Although the stress of the wire after the temperature descent meets the demands of prestressing, the strain of the wire can be compressive, which is not the case in practice, and thus the wire–concrete interaction is not correctly addressed. Lin [9] also employed the initial strain method to simulate the prestressing by prescribing the initial strain corresponding to the prestress of the wire. However, this method is suitable only for analysis of part of the cross section rather than the entire cross section. The initial stress method is an intuitive approach for applying prestress and is easily understood by researchers. However, a calculation of the equilibrium iteration should be conducted behind the definition of the initial stress since the initial stress state of the wire and concrete core cannot be an exact equilibrium state for the FE model. In the framework of elasticity, it is easy to achieve an equilibrium state, whereas when the nonlinearity of materials and the interaction is considered, the equilibrium calculation is too difficult to complete. The objective of this paper is to propose a new 3D FE model using ABAQUS software, called the wire-wrapping model, which is used to simulate the process of wrapping wire for the prestressing of an embedded PCCP. This model is applied to an embedded PCCP with a diameter of 4 m used in the SNWD project for the prestressing simulation. The results obtained using the wire-wrapping model are analyzed and compared with those obtained using the equivalent external radial pressure method and those using the AWWA C304 [10] standard for its verification. 2. Model presentation 2.1. General framework The general idea of the wire-wrapping model is to simulate the process of wrapping wire for the prestressing of an embedded PCCP from the viewpoint of manufacturing, which is a dynamic procedure. This model is based on the principle of different speeds and the assumption of uniform wire tension. A kinematic coupling constraint method is introduced to simulate the rotation of the pipe core and a contact model is given to reveal the mechanism of the interaction between the wire and concrete core during wrapping. The nonlinear properties of materials are used to determine the validity of the wire-wrapping model. Geometrical nonlinearity is addressed throughout the analysis procedure to consider the large deformation of the wire. The explicit central-difference time integration rule is used for the quasi-static solution of this dynamic wrapping procedure. All these described above are integrated forming the framework of the wire-wrapping model as shown in Fig. 1. 2.2. Principle of different-speed wire-wrapping In the practical manufacturing of a PCCP, the wire is wrapped around the concrete core using the principle of different speeds. Fig. 2 shows the photograph of different-speed wire-wrapping at a construction site. The wrapping stress of the wire
Fig. 1. Diagram of model framework.
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rsw is produced by the difference between the linear rotational speed of the stress-generation device v2 and that of the rotary workbench v1. Assuming the wire length released by the stress-generation device is l2 and that of the wrapping around the concrete core is l1, the wrapping stress of the wire can be written as
rsw ¼ Es es ¼ Es ðl1 l2 Þ=l ¼ Es ðv 1 v 2 Þ=v 2 ;
ð1Þ
where es and Es are the strain and elastic modulus of the wire respectively. Therefore, the whole wrapping wire process is divided into two stages in the wire-wrapping model: stretching the wire and wrapping the wire. 2.3. Assumption of uniform wire tension Analyzing a complete segment of the PCCP is problematic owing to the huge cost of computation; hence, it is recommended to analyze only part of a segment since the behaviour of the PCCP along the pipe axis does not change. According to the mechanism of loop prestress [11], there are two effects of the wrapping wire on the concrete core: the external radial contact pressure pcs and the tangential friction scs between the interface of the wire and concrete core, which are given as
pcs ¼ rsw As =Rsw ; c s
s
ð2Þ
As @ rsw ¼ ; Rsw @h
ð3Þ
where h is the circumferential coordinate, As is the cross sectional area of the wire and Rsw is the outer radius of the concrete core, which is referred to as the wire-wrapping radius. In design, the wrapping stress is 75% of the specified tensile strength of the wire fsu as a constant value. In practice, wire tension during the prestressing operation is continuously recorded and the mean tension is specified to produce the required stress in the wire, which results in the tension not deviating from a mean value by more than ±10%. Therefore, here we assume uniform wire tension and neglect the tangential friction of the wire during wrapping, which essentially makes the external radial pressure even. 2.4. Rotation of the pipe core The kinematic coupling constraint is enforced here to rotate the pipe core in the wire-wrapping model, which is the theoretical premise of the wrapping wire. A group of slave nodes in the domain of the pipe core C can be constrained to a master reference point R in the shaft of the pipe for the same translation and rotation. Combinations of all the slave node degrees of freedom except for the radial translational degree of freedom are selected for the constraint as shown in Fig. 3.
Fig. 2. Photograph of different-speed wire-wrapping.
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The rotational degrees of freedom are constrained by
Tð/C Þ ¼ Tð/R Þ;
ð4Þ
where u and T are the rotation and rotational matrix respectively. The translational degrees of freedom are constrained by eliminating the circumferential and axial degrees of freedom at the coupling nodes. Two configurations – are reference configuration and current configuration – are given here. The slave node coordinates xC under the current configuration are derived as
xC ¼ xR þ Tð/R Þ ðXC XR Þ þ ai Tð/R Þ Ei ;
ð5Þ
where xR is the reference point coordinates under the current configuration, XC and XR are the coordinates of the slave node and reference point under the reference configuration, ai is the translational degrees of freedom, and Ei is the reference base vector of the coordinates. 2.5. Properties of materials The prescribed material properties in this section are based on suggestions from the AWWA C304 [10] standard for a PCCP. The materials should have elasticity during wrapping the wire such that they do not yield or soften, as required by design and in practice. This is one of the criteria for checking the validity of the wire-wrapping model that uses nonlinear material properties. 2.5.1. Properties of the concrete core The stress–strain relationship of concrete in compression adopted here refers to the study of Carreira and Chu [12]. The relationship has good agreement with test results for high-strength concrete in compression. It was used by Zarghamee [13,14] to predict the behaviour of a PCCP subjected to combined loads with an analytical multilayered model and it was embodied by an American standard for the design and manufacturing of a PCCP. This stress–strain relationship can be written as
rc fcc
¼
bc eeccc bc ; bc 1 þ eeccc
ð6Þ
where rc and ec are the stress and strain of the concrete respectively, fcc is the compressive strength of the concrete, and ecc is the peak strain corresponding to the compressive strength; that is,
ecc ¼ ð0:71f cc þ 168Þ 105 ;
ð7Þ
in which fcc has units of megapascals. bc is a parameter related to the compressive strength and the elastic modulus of the concrete and is expressed as
b 0:4f cc c fcc bc 1 1 ¼ 0: Ec ecc Ec ecc
ð8Þ
The elastic modulus Ec is given according to the work of Pauw [15] as
Ec ¼ 0:0736q1:51 ðfcc Þ0:3 ; c
ð9Þ
where qc is the concrete density. The ultimate strain of concrete is denoted by ecu as the strain at which the stress is equal to 60% of fcc in the descending branch of stress–strain curve. When ecu is reached, the concrete material is deemed to have lost its load-bearing capability. This stress–strain diagram in compression is shown in Fig. 4.
Fig. 3. Schematic diagram of pipe core rotation.
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Fig. 4. Stress–strain relationship of concrete in compression.
Fig. 5. Stress–strain relationship of concrete in tension.
The behaviour of concrete in tension is described by a trilinear stress–strain curve as shown in Fig. 5. From the work by Raphael [16], Gopalaratnam and Shah [17], the tensile strength fct of concrete is determined from the compressive strength as
pffiffiffiffiffi fct ¼ 0:58 fcc ;
ð10Þ
where fct has units of megapascals. The elastic modulus Ec in the ascending branch of the tensile curve is the same as that for compression, and the peak strain ect in tension is equal to fct/Ec. The modulus in the descending branch of the curve is Ec/10. The cracking strain eck of concrete in tension is equal to 11ect where the stress vanishes. 2.5.2. Properties of the wire The stress–strain relationship of wire used here is based on the ASTM A648 [18] standard, which can be expressed as follows. When es 6 0:75f su =Es ,
rs ¼ es Es ;
ð11aÞ
and when es > 0.75fsu/Es,
n
o
rs ¼ fsu 1 ½1 0:6133ðes Es =fsu Þ2:25 ;
ð11bÞ
where rs and es are the stress and strain of the wire respectively, fsu is the maximum tensile strength of the wire, and Es is the elastic modulus of the wire. The yield stress fsy is 85% of fsu.
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Fig. 6. Schematic diagram of the wire-wrapping contact.
2.5.3. Properties of the cylinder The yield strength and elastic modulus of the cylinder are denoted as fyy and Ey respectively. The perfect elastoplastic stress–strain relationship is used for the steel cylinder on the basis of the AWWA C304 [10] standard. 2.6. Wire–concrete interaction 2.6.1. Decomposition of relative motion The contact interaction between the wire and concrete core must be taken into account and has great influence on the wrapping wire process. On the basis of the work of Anand [19] and Gearing et al. [20], we denote the contact domain of the concrete core as C and that of the wire as S, as shown in Fig. 6. The relative displacement u, relative velocity v, and traction T on the interface can be decomposed in two orthogonal directions represented by two unit column vectors: the normal vector n and the tangential vector t.
v n ¼ h_ ¼ h_ cs n;
c
n : un ¼ h ¼ hs n; t : ut ¼ c ¼ c
c s t;
vt
¼ c_ ¼ c_ cs t;
Tn ¼ T n n ¼ pcs n;
ð12Þ
Tt ¼ T T n n ¼ scs t;
ð13Þ
The tangential relative motion is decomposed into a sticking region cel 6 ccsu (analogous to the elastic regime) and slipping part csl (analogous to the plastic regime), where ccsu is the slip resistance corresponding to the shear resistance scsu . The tangential relative velocity is then decomposed in the same manner. c;pl c ¼ cel þ csl ) ccs t ¼ cc;el s t þ cs t; el
ð14Þ
sl
_ c;pl c_ ¼ c_ þ c_ ) c_ cs t ¼ c_ c;el s t þ cs t:
ð15Þ
2.6.2. Normal contact model c The hard contact model is used to represent the contact pressure–overclosure (pcs hs ) relationship between the wire and concrete core during wrapping, which is shown in Fig. 7 and can be expressed as
(
pcs ¼ 0 c hs
c
if hs < 0;
¼ 0 if
pcs
ð16Þ
> 0; c
where the contact pressure pcs is always positive and the overclosure hs is positive when the interface is closed and negative when open. Thus, when the wire and concrete are in contact, any contact pressure is transmitted between them and they separate if the contact pressure reduces to zero. 2.6.3. Tangential contact model The shear–cap friction contact model is adopted here to give the shear resistance scsu . The model assumes that the shear resistance given by the classic Coulomb model is operative until a suitable threshold value scs;max is reached; that is,
(
scsu ¼
lcs pcs pcs < scs;max =lcs ; ffiffiffiffiffi p scs;max ¼ fcy pcs P scs;max =lcs ;
ð17Þ
where lcs is the friction coefficient and fcy is the yield stress of the concrete under uniaxial compression. The friction shear stress–slip (scs ccs ) relationship between the wire and concrete core is shown in Fig. 8 and is written as
scs ¼
scsu ccs =ccsu scs < scsu ; scsu scs P scsu ;
ð18Þ
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Fig. 7. Contact pressure–overclosure relationship between the wire and concrete core.
Fig. 8. Friction shear stress–slip relationship between the wire and concrete core.
where ccsu is the slip resistance corresponding to scsu . Being similar to the yield function, the slip function Q is given in terms of the tangential traction Tt and slip resistance scsu so as to describe the evolution of the slip with the associated flow rule, which can be expressed in the form
Q ¼ Q ðTt ; scsu Þ ¼ scs scsu ; @Q ; c_ sl ¼ a_ @Tt
ð19Þ ð20Þ
where a_ is a parameter of the slip evolution. The consistent condition can be written as
8 h iT @Q > < @T T_ t ¼ 0 t Q_ ¼ h iT > : @Q T_ þ @Qc @ scsuc p_ c ¼ 0 t s @p @Tt @s su
s
scsu ¼ scs;max ;
ð21Þ
scsu ¼ lcs pcs :
So, we can obtain the formulas of the sticking region and slip part respectively as follows.
( sticking : T_ t ¼
ðscsu =ccsu Þc_ ðs
Tt
c c þ scsu l
c c _ su = su Þ
c _c s ps
scsu ¼ scs;max ; scsu ¼ lcs pcs ;
ð22Þ
0 h iT 1
8 > > c c B@T@Qt @T@Qt C > c c _ > > s = c Þ c ssu =csu @h iT Ac_ ð > su su > @Q @Q > < @Tt @Tt 0 h iT 1 slipping : T_ t ¼ > > @Q @Q > > > ðsc =cc Þc_ sc =cc Bh@Tt i@Tt Cc_ þ @Q lc p_ c > A > T su su su su @ @Tt s s > : @Q @Q @Tt
@Tt
scsu ¼ scs;max ; ð23Þ
scsu ¼ lcs pcs :
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Fig. 9. Definition of the master–slave contact pair between the wire and concrete core.
2.6.4. Enforcement of the contact constraint In this paper, the constraint of the contact between the wire and concrete core is enforced using the penalty method [21] and modelled using the boundary contact algorithm studied by Bathe and Chaudhary [22,23] in the theory of geometric nonlinearity. Fig. 9 illustrates the definition of the contact pair, the behaviour of which is described by a master–slave algorithm. The slave node on the wire surface is in contact with the master surface of the concrete core. The normal relative displacement h can be written as c
h ¼ nhs ¼ XA X1 ;
ð24Þ
where XA are the coordinates of the contact point on the master surface, which can be interpolated from the coordinates of the master nodes. Considering the coordinates of the slave node of the wire yields
N; XA ¼ ½ xA ðn; gÞ yA ðn; gÞ zA ðn; gÞ T ¼ X h¼
c nhs
ð25Þ
¼ XN;
ð26Þ
with
¼ ½ N 2 ðn; gÞ N3 ðn; gÞ N4 ðn; gÞ N5 ðn; gÞ T ; N
¼ ½ X2 X T
N ¼ ½ 1 N2 ðn; gÞ N3 ðn; gÞ N4 ðn; gÞ N5 ðn; gÞ ;
X3
X4
X5 ;
X ¼ ½ X1
X2
X3
X4
X5 ;
ð27Þ
where x, y, and z are the global nodal coordinates and n and g the natural nodal coordinates on the master surface, and Ni (i = 1–5) is a shape function. The tangential unit column vector can be defined by the slip of the contact point as follows.
t¼
dXA dXA ¼ : dccs kdXA k
ð28Þ
By variation, we obtain c
c
dnhs þ ndhs ¼ tdccs þ dXN:
ð29Þ
Eq. (29) can be multiplied by scalars n and t to obtain variations in the normal and tangential relative displacements as c
dhs ¼ nT dXN; c s
T
dc ¼ t dXN:
ð30Þ ð31Þ
2.7. Quasi-static solution Because of geometrical nonlinearity and complicated contact conditions in the wire-wrapping model, the explicit centraldifference time integration rule is used here for the quasi-static solving of the dynamic wrapping procedure. 2.7.1. Steps in analysis The wire is stretched such that the stress reaches rsw in the first step. In the following step, the concrete core is rotated with a velocity of m (revolutions/min). The angle between the wire and cross section of the core is
us ¼ arctanðdw =ð2pRsw ÞÞ;
ð32Þ
where dw is the wire spacing. A smooth amplitude curve is adopted to describe the change in the model status so that the inertial effect is reduced greatly. The curve is expressed as
s ¼ si þ ðsiþ1 si Þf3 ð6f2 15f þ 10Þ;
ð33Þ
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Table 1 The four steps of the wrapping wire process. Time (s)
Description
0–t1 t1–t2 t2–t3 t3–t4
Stretching of the wire The rotational velocity of the core increases from zero to m (revolutions/min) The rotating of the core with a constant velocity m The rotational velocity of the core decreases from m to zero
with
f¼
t ti t iþ1 t i
t i 6 t 6 t iþ1 ;
ð34Þ
where si and si+1 are the model status at times ti and ti+1 respectively. Therefore, there are four steps in the whole analysis of the wrapping wire as shown in Table 1. 2.7.2. Mass scaling The mass scaling method is used here for computational efficiency in quasi-static solving. The minimum stable time increment can be written in the form
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kq q ; Dt ¼ min l ^ k^ þ 2m e
ð35Þ
^ are the effective Lamé’s constants for the where le is the characteristic length of the element, q is the material density, ^ k and m material, and kq is the mass scaling factor determined from the energy criterion. 2.7.3. Energy criterion In the quasi-static solving, the ratio of the kinetic energy except for that of rigid body motion to internal energy should be limited to a maximum of 10% during the whole wrapping wire process except for a short time at the beginning of analysis. Meanwhile, the variation in the history curve of the energy should be smooth, avoiding acute fluctuations. The energy criterion should satisfy
fE ¼
EK ERM 6 10%; EI
ð36Þ
where EK is the total kinetic energy, ERM is the kinetic energy of the rigid body motion, EI is the internal energy, and fE is the energy ratio. ERM is determined as follows.
Z 1 kq qxðtÞ2 La 2p r 3 dr; 2 1 wire : ERM ðtÞ ¼ kq q As Ls xðtÞ2 R2sw : 2 core : ERM ðtÞ ¼
ð37Þ ð38Þ
Here, x is the angular velocity of the pipe core, La is the axial length of the pipe, and Ls is the wire length. 3. Model application and verification The wire-wrapping model presented above is applied here to simulate prestressing of a 4 m diameter embedded PCCP used in the SNWD project in China, which is the most significant water supply project for conveying water from central China to the capital city Beijing. The results are analyzed and compared with those obtained using the equivalent external radial pressure method and the AWWA C304 (2007) standard. 3.1. Parameters of the model The parameters of the wire-wrapping model for this application are given by amplitude values as follows. The FE mesh is shown in Fig. 10. The geometrical parameters are the inner diameter of the pipe Di = 4000 mm, the outer diameter of the cylinder Dy = 4183 mm, the total thickness of the pipe including cylinder hc = 350 mm, the thickness of the cylinder hy = 2 mm, the wire diameter ds = 7 mm, the total sectional area of the wire per meter Ap = 2223 mm2/m, and the wire spacing dw = 14.3 mm. The material parameters are the density of concrete qc = 2.323e–9 ton/mm3, the compressive strength of concrete fcc = 48 MPa, the compressive yield stress of concrete fcy = 39.3 MPa, the wire density qs = 7.8e–9 ton/mm3, the wire modulus Es = 193,050 MPa, the maximum tensile strength of the wire fsu = 1570 MPa, the density of the cylinder qy = 7.8e–9 ton/mm3, the modulus of the cylinder Ey = 206,850 MPa, the yield stress of the cylinder fyy = 227.53 MPa, the infinitesimal slip
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Fig. 10. FE meshing.
Fig. 11. Linear relationship between the energy ratio and mass scaling factor.
Fig. 12. History curves of the energy and energy ratio. (a) Energy. (b) Energy ratio.
resistance ccsu on the contact surface between the wire and concrete core making for rigid plasticity of the tangential contact model, and the friction coefficient lcs = 1e–3 satisfying the uniform wire tension assumption. The multiple-point constraints are imposed on the interface between the cylinder and concrete to harmonize the deformation due to different degrees of freedom. The representative axial length of the pipe La is 20 times the length of the wire spacing. Eight-node linear brick elements are used to model the concrete. Two-node three-dimensional truss elements are used to model the wire. Four-node shell elements are used to model the cylinder. The rotational velocity of the pipe core is 10 revolutions/min. The steps of analysis are defined by t1 = 2, t2 = 4, t3 = 122, and t4 = 124.
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Fig. 13. Graphical description of wrapping process.
Fig. 14. Distribution of the circumferential stress for the concrete outer core (MPa). (a) Wire-wrapping model. (b) Equivalent pressure method.
3.2. Results and evaluation In the following results tension is positive and compression is negative. 3.2.1. Mass scaling factor The mass scaling factor kq is determined by studying the energy variation during the first loop of the wrapping wire. It is found that fE tends to be constant after t3 and increases linearly with an increase in kq. Fig. 11 shows the linear relationship between fE and kq. The optimum mass scaling factor kq is 13.7, corresponding to fE of 10%. 3.2.2. History of energy Fig. 12 shows the history curves of energies and fE. The variations in these curves are not acute, and so the energy criterion is well met. 3.2.3. Process of wrapping wire The results of the wrapping process are shown in Fig. 13. Obviously, the amount of wire wrapped around the core increases linearly with time. The wire spacing is almost equal showing the good geometry of the pipe after wrapping. The magnitude of contact pressure on the concrete core outer surface produced by the pretension of the wire at the end of wrapping is 1.08–1.78 MPa, which is in good agreement with the equivalent external radial pressure determined from Eq. (2) of 1.35 MPa. 3.2.4. Circumferential stress The circumferential stresses obtained using the wire-wrapping model are compared with those obtained using the equivalent external radial pressure method and AWWA C304 (2007) standard. We conclude that the distribution laws of the
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Fig. 15. Distribution of the circumferential stress for the concrete inner core (MPa). (a) Wire-wrapping model. (b) Equivalent pressure method.
Table 2 Comparison of circumferential stresses (MPa). Position
Wire-wrapping model
Equivalent method
AWWA C304
Outer core Inner core Cylinder
7.9 to 8.8 8.7 to 9.1 67.7
8.1 to 8.9 8.9 to 9.2 68.8
8.7 63.5
Fig. 16. History curve of the mean value of the wire stress.
Table 3 Comparison of principle stresses with yield stresses (MPa). Position
Principle stress
Yield stress
Outer core Inner core Cylinder Wire
8.7 9.0 67.7 1282
39.3 39.3 227.5 1334.5
circumferential stresses obtained using different approaches are in good agreement. The distributions are presented in Figs. 14, 15 and Table 2. 3.2.5. Wire stress The history curve of the mean value of the wire stress is shown in Fig. 16 and indicates the satisfaction of the requirement of fluctuations in tension not deviating from the mean by more than ±10%. 3.2.6. Checking of the material status The principle stresses of different components are given in Table 3, and they are compared with the yield stresses for the checking of material statuses. It is confirmed that the status of the materials during the wire-wrapping are in the range of elasticity.
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4. Conclusions Since the conventional FE analysis methods for the prestressing of a PCCP have their intrinsic demerits, this paper focused on the process of wrapping wire in the manufacture of an embedded PCCP and has presented a new FE model, called the wire-wrapping model, to simulate the prestressing of a PCCP. The main conclusions are the following: - The whole process of wrapping wire in the manufacture of an embedded PCCP can be numerically simulated by this FE model. The prestressing of the pipe could be achieved satisfactorily. - The results obtained using the wire-wrapping model, compared with those obtained using the equivalent external radial pressure method and those using the AWWA C304 standard, are proven to be reasonable. That means: the energy criterion could be well met by selecting an appropriate mass scaling factor, the geometry of pipe is fine after wrapping, the distribution of the prestress of the core is regular, the stress of the wire varies smoothly satisfying the requirement of fluctuations in tension, and all materials remain in the elastic domain. - The major characteristics of this FE model, compared with those of traditional equivalent approaches, are that the contribution of the wire stiffness is taken into account in the model, the stress and strain in the wire itself are appropriately considered, and the interaction at the interface between the wire and concrete core is revealed by the contact model. Therefore, the wire-wrapping model has great potential in further FE analysis of a PCCP. Further work will include the study of the performance of a PCCP subjected to the loads and pressures, and to investigate the effects of wire-breaking on the response of a PCCP. In this study, the contribution of wire stiffness and the interaction between the wire and other component must be taken into account, which can be accomplished using the wire-wrapping model. Acknowledgements We acknowledge the support provided by the Chinese National Science & Technology Pillar Program during 11th Five-Year Plan Period, Grant No. 2006BAB04A04-03. References [1] A.E. Romer, G.E.C. Bell, R.D. Ellison, Failure of prestressed concrete cylinder pipe, in: M. Najafi, L. Osborn (Eds.), Proc. Int. Conf. on Advances and Experiences with Trenchless Pipeline Projects, ASCE, Boston, Massachusetts, 2007. [2] M.S. Zarghamee, R.P. Ojdrovic, W.R. Dana, Coating delamination by radial tension in prestressed concrete pipe II: analysis, J. Struct. Eng. 119 (1993) 2720–2732. [3] M.S. Zarghamee, D.W. Eggers, R.P. Ojdrovic, Finite-element modeling of failure of PCCP with broken wires subjected to combined loads, in: G.E. Kurz (Ed.), Proc. Int. Conf. on Beneath Our Feet: Challengers and Solutions, ASCE, Cleveland, Ohio, 2002. [4] M.S. Zarghamee, D.W. Eggers, R.P. Ojdrovic, B. Rose, Risk analysis of prestressed concrete cylinder pipe with broken wires, in: M. Najafi (Ed.), Proc. Int. Conf. on New Pipeline Technologies, Security, and Safety, ASCE, Baltimore, Maryland, 2003 , pp. 599–609. [5] Y.G. Diab, T. Bonierbale, A numerical modeling and a proposal for rehabilitation of PCCP’s, in: J.P. Castronovo (Ed.), Proc. Int. Conf. on Advances in Pipelines Eng. & Constr, ASCE, San Diego, California, 2001. [6] R. Gomez, D. Munoz, R. Vera, J.A. Escobar, Structural model for stress evaluation of prestressed concrete pipe of the Cutzamala system, in: J.J. Galleher, M.T. Stift (Eds.), Proc. Int. Conf. on Pipeline Eng. & Constr.: What’s on the Horizon, ASCE, San Diego, California, 2004. [7] H.R. Lotfi, R.G. Oesterle, J. Roller, Reliability assessment of distressed prestressed concrete cylinder pipe, in: C. Vipulanandan, R. Ortega (Eds.), Proc. Int. Conf. on Optimizing Pipeline Design, Operations, and Maintenance in Today’s Economy, ASCE, Houston, Texas, 2005, pp. 838–852. [8] X.C. Zhang, Finite element analysis of prestressed concrete cylinder pipe, MS thesis, Tianjin Univ., Tianjin, China, 2005. [9] J.H. Lin, Structural analysis and evaluation of pre-stressed concrete cylinder pipe, MS thesis, National Cheng Kung Univ., Tainan, Taiwan, 2008. [10] Standard for design of prestressed concrete cylinder pipe, ANSI/AWWA C304, AWWA, Denver, Colorado, 2007. [11] X.K. Li, Investigation of performance under loads and calculating method of prestressed concrete penstock, PhD thesis, Dalian Univ. of Tech., Dalian, China, 2003. [12] D.J. Carreira, K. Chu, Stress–strain relationship for plain concrete in compression, ACI J. 82 (1985) 797–804. [13] M.S. Zarghamee, A multilayered model for analysis of composite pipes, in: B.A. Bennett (Ed.), Proc. on Pipeline Infrastructure, ASCE, New York, 1988, pp. 136–147. [14] M.S. Zarghamee, K. Fok, Analysis of prestressed concrete cylinder pipe under combined loads, J. Struct. Eng. 116 (1990) 2022–2039. [15] A. Pauw, Static modulus of elasticity of concrete as affected by density, ACI J. 57 (1960) 679–688. [16] J.M. Raphael, Tensile strength of concrete, ACI J. 81 (1984) 158–165. [17] V.S. Gopalaratnam, S.P. Shah, Softening response of plain concrete in direct tension, ACI J. 82 (1985) 310–323. [18] Standard specification for steel wire, hard drawn for prestressing concrete pipe, ASTM A648, ASTM, Philadelphia, PA, 2004. [19] L. Anand, A constitutive model for interface friction, Comput. Mech. 12 (1993) 197–213. [20] B.P. Gearing, H.S. Moon, L. Anand, A plasticity model for interface friction: application to sheet metal forming, Int. J. Plasticity 17 (2001) 237–271. [21] W.R.C. Underhill, M.A. Dokainish, G.E. Oravas, A method for contact problems using virtual elements, Comput. Meth. Appl. Mech. Eng. 143 (1997) 229– 247. [22] K.-J. Bathe, A.B. Chaudhary, A solution method for planar and axisymmetric contact problem, Int. J. Numer. Meth. Eng. 21 (1985) 65–88. [23] A.B. Chaudhary, K.-J. Bathe, A solution method for static and dynamic analysis of three-dimensional contact problems with friction, Comput. Struct. 24 (1986) 855–873.
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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
FE model for simulating wire-wrapping during prestressing of an embedded prestressed concrete cylinder pipe Huan Xiong, Penghui Li *, Qingbin Li State Key Laboratory of Hydroscience and Engineering, Tsinghua University, 100084 Beijing, China
a r t i c l e
i n f o
Article history: Received 31 August 2009 Received in revised form 4 November 2009 Accepted 7 January 2010 Available online 18 January 2010 Keywords: Prestressed concrete cylinder pipe Prestressing Wire-wrapping Finite element simulation
a b s t r a c t A new 3D FE model, called wire-wrapping model, is proposed to simulate the process of wrapping wire for the prestressing of an embedded prestressed concrete cylinder pipe. This model is described in detail for its presentation and applied to a practical project for the prestressing simulation. The results obtained using this model are analyzed and compared with those obtained using the equivalent external radial pressure method and those using the AWWA C304 standard for its verification. Finally, the characteristics and contributions of this model are specified. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Prestressed concrete cylinder pipe (PCCP) was widely used in the world during the last few decades for water conveyance in many areas like municipal, industrial, plant piping systems, etc. Two types of PCCP are produced: lined and embedded. In recent years, some significant projects such as the Great Man Made River (GMMR) in Libya and the South-to-North Water Diversion (SNWD) in China required extremely large diameter embedded PCCP for water transmission. So, most research has been focused on the performance of embedded PCCP for their larger size compared with the lined type. This type of PCCP is compressed by a helical high-strength wire. The wire is wrapped under high tension around a core composed of a steel cylinder encased in concrete. A mortar coating is applied to the exterior of the pipe to protect the wires from corrosion. As we all know, the wire in the PCCP is used to produce compressive prestress in the core that offsets the tensile stresses from loads and pressure. So, the wire should bean important load-bearing component and has a significant influence on the structural integrity of a PCCP. Lots of investigations showed that most failures of PCCPs could be attributed to the loss of prestress due to wire breaking [1]. Recently, FE analysis has been used to study the performance of a PCCP subjected to a combination of many types of loads and pressure, although it is well known that FEA is a difficult method to implement because of the composite structure of the PCCP. Generally, the effects of the wire component on the results of the FE analysis must be considered since the wire is an essential component of the structure as mentioned above. In addition, the simulation of prestressing is pivotal and complicated and can significantly affect the results obtained for the PCCP structure. Currently, some traditional equivalent methods such as those that employ an equivalent external radial pressure, equivalent temperature descent, initial stress, or initial strain are adopted in simulating the prestressing; however, each has its disadvantage. The equivalent external radial pressure method is popular in engineering and is a conventional approach for simulating the prestressing of a PCCP. The method was proposed by Zarghamee et al. [2–4], Diab and Bonierbale [5], Gomez et al. [6], * Corresponding author. Tel.: +86 010 62785550. E-mail address: [email protected] (P. Li). 1569-190X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2010.01.007
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and Lotfi et al. [7] among others. This method considers a thin layer of wire so that the stiffness of the wire contributing to the pipe structure is not taken into account and the wire cannot bear any load. Using this method, the composition of the wire and the relation between the stress of the wire and the prestress of the PCCP cannot be considered, and neither can the interaction between the wire and concrete. Studies by Zhang [8] and Lin [9] included the wire component in the pipe model and used the equivalent temperature descent method for applying prestress, which considers the contribution of the wire stiffness, but the results did not agree well with what is seen in practice. Although the stress of the wire after the temperature descent meets the demands of prestressing, the strain of the wire can be compressive, which is not the case in practice, and thus the wire–concrete interaction is not correctly addressed. Lin [9] also employed the initial strain method to simulate the prestressing by prescribing the initial strain corresponding to the prestress of the wire. However, this method is suitable only for analysis of part of the cross section rather than the entire cross section. The initial stress method is an intuitive approach for applying prestress and is easily understood by researchers. However, a calculation of the equilibrium iteration should be conducted behind the definition of the initial stress since the initial stress state of the wire and concrete core cannot be an exact equilibrium state for the FE model. In the framework of elasticity, it is easy to achieve an equilibrium state, whereas when the nonlinearity of materials and the interaction is considered, the equilibrium calculation is too difficult to complete. The objective of this paper is to propose a new 3D FE model using ABAQUS software, called the wire-wrapping model, which is used to simulate the process of wrapping wire for the prestressing of an embedded PCCP. This model is applied to an embedded PCCP with a diameter of 4 m used in the SNWD project for the prestressing simulation. The results obtained using the wire-wrapping model are analyzed and compared with those obtained using the equivalent external radial pressure method and those using the AWWA C304 [10] standard for its verification. 2. Model presentation 2.1. General framework The general idea of the wire-wrapping model is to simulate the process of wrapping wire for the prestressing of an embedded PCCP from the viewpoint of manufacturing, which is a dynamic procedure. This model is based on the principle of different speeds and the assumption of uniform wire tension. A kinematic coupling constraint method is introduced to simulate the rotation of the pipe core and a contact model is given to reveal the mechanism of the interaction between the wire and concrete core during wrapping. The nonlinear properties of materials are used to determine the validity of the wire-wrapping model. Geometrical nonlinearity is addressed throughout the analysis procedure to consider the large deformation of the wire. The explicit central-difference time integration rule is used for the quasi-static solution of this dynamic wrapping procedure. All these described above are integrated forming the framework of the wire-wrapping model as shown in Fig. 1. 2.2. Principle of different-speed wire-wrapping In the practical manufacturing of a PCCP, the wire is wrapped around the concrete core using the principle of different speeds. Fig. 2 shows the photograph of different-speed wire-wrapping at a construction site. The wrapping stress of the wire
Fig. 1. Diagram of model framework.
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rsw is produced by the difference between the linear rotational speed of the stress-generation device v2 and that of the rotary workbench v1. Assuming the wire length released by the stress-generation device is l2 and that of the wrapping around the concrete core is l1, the wrapping stress of the wire can be written as
rsw ¼ Es es ¼ Es ðl1 l2 Þ=l ¼ Es ðv 1 v 2 Þ=v 2 ;
ð1Þ
where es and Es are the strain and elastic modulus of the wire respectively. Therefore, the whole wrapping wire process is divided into two stages in the wire-wrapping model: stretching the wire and wrapping the wire. 2.3. Assumption of uniform wire tension Analyzing a complete segment of the PCCP is problematic owing to the huge cost of computation; hence, it is recommended to analyze only part of a segment since the behaviour of the PCCP along the pipe axis does not change. According to the mechanism of loop prestress [11], there are two effects of the wrapping wire on the concrete core: the external radial contact pressure pcs and the tangential friction scs between the interface of the wire and concrete core, which are given as
pcs ¼ rsw As =Rsw ; c s
s
ð2Þ
As @ rsw ¼ ; Rsw @h
ð3Þ
where h is the circumferential coordinate, As is the cross sectional area of the wire and Rsw is the outer radius of the concrete core, which is referred to as the wire-wrapping radius. In design, the wrapping stress is 75% of the specified tensile strength of the wire fsu as a constant value. In practice, wire tension during the prestressing operation is continuously recorded and the mean tension is specified to produce the required stress in the wire, which results in the tension not deviating from a mean value by more than ±10%. Therefore, here we assume uniform wire tension and neglect the tangential friction of the wire during wrapping, which essentially makes the external radial pressure even. 2.4. Rotation of the pipe core The kinematic coupling constraint is enforced here to rotate the pipe core in the wire-wrapping model, which is the theoretical premise of the wrapping wire. A group of slave nodes in the domain of the pipe core C can be constrained to a master reference point R in the shaft of the pipe for the same translation and rotation. Combinations of all the slave node degrees of freedom except for the radial translational degree of freedom are selected for the constraint as shown in Fig. 3.
Fig. 2. Photograph of different-speed wire-wrapping.
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The rotational degrees of freedom are constrained by
Tð/C Þ ¼ Tð/R Þ;
ð4Þ
where u and T are the rotation and rotational matrix respectively. The translational degrees of freedom are constrained by eliminating the circumferential and axial degrees of freedom at the coupling nodes. Two configurations – are reference configuration and current configuration – are given here. The slave node coordinates xC under the current configuration are derived as
xC ¼ xR þ Tð/R Þ ðXC XR Þ þ ai Tð/R Þ Ei ;
ð5Þ
where xR is the reference point coordinates under the current configuration, XC and XR are the coordinates of the slave node and reference point under the reference configuration, ai is the translational degrees of freedom, and Ei is the reference base vector of the coordinates. 2.5. Properties of materials The prescribed material properties in this section are based on suggestions from the AWWA C304 [10] standard for a PCCP. The materials should have elasticity during wrapping the wire such that they do not yield or soften, as required by design and in practice. This is one of the criteria for checking the validity of the wire-wrapping model that uses nonlinear material properties. 2.5.1. Properties of the concrete core The stress–strain relationship of concrete in compression adopted here refers to the study of Carreira and Chu [12]. The relationship has good agreement with test results for high-strength concrete in compression. It was used by Zarghamee [13,14] to predict the behaviour of a PCCP subjected to combined loads with an analytical multilayered model and it was embodied by an American standard for the design and manufacturing of a PCCP. This stress–strain relationship can be written as
rc fcc
¼
bc eeccc bc ; bc 1 þ eeccc
ð6Þ
where rc and ec are the stress and strain of the concrete respectively, fcc is the compressive strength of the concrete, and ecc is the peak strain corresponding to the compressive strength; that is,
ecc ¼ ð0:71f cc þ 168Þ 105 ;
ð7Þ
in which fcc has units of megapascals. bc is a parameter related to the compressive strength and the elastic modulus of the concrete and is expressed as
b 0:4f cc c fcc bc 1 1 ¼ 0: Ec ecc Ec ecc
ð8Þ
The elastic modulus Ec is given according to the work of Pauw [15] as
Ec ¼ 0:0736q1:51 ðfcc Þ0:3 ; c
ð9Þ
where qc is the concrete density. The ultimate strain of concrete is denoted by ecu as the strain at which the stress is equal to 60% of fcc in the descending branch of stress–strain curve. When ecu is reached, the concrete material is deemed to have lost its load-bearing capability. This stress–strain diagram in compression is shown in Fig. 4.
Fig. 3. Schematic diagram of pipe core rotation.
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Fig. 4. Stress–strain relationship of concrete in compression.
Fig. 5. Stress–strain relationship of concrete in tension.
The behaviour of concrete in tension is described by a trilinear stress–strain curve as shown in Fig. 5. From the work by Raphael [16], Gopalaratnam and Shah [17], the tensile strength fct of concrete is determined from the compressive strength as
pffiffiffiffiffi fct ¼ 0:58 fcc ;
ð10Þ
where fct has units of megapascals. The elastic modulus Ec in the ascending branch of the tensile curve is the same as that for compression, and the peak strain ect in tension is equal to fct/Ec. The modulus in the descending branch of the curve is Ec/10. The cracking strain eck of concrete in tension is equal to 11ect where the stress vanishes. 2.5.2. Properties of the wire The stress–strain relationship of wire used here is based on the ASTM A648 [18] standard, which can be expressed as follows. When es 6 0:75f su =Es ,
rs ¼ es Es ;
ð11aÞ
and when es > 0.75fsu/Es,
n
o
rs ¼ fsu 1 ½1 0:6133ðes Es =fsu Þ2:25 ;
ð11bÞ
where rs and es are the stress and strain of the wire respectively, fsu is the maximum tensile strength of the wire, and Es is the elastic modulus of the wire. The yield stress fsy is 85% of fsu.
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Fig. 6. Schematic diagram of the wire-wrapping contact.
2.5.3. Properties of the cylinder The yield strength and elastic modulus of the cylinder are denoted as fyy and Ey respectively. The perfect elastoplastic stress–strain relationship is used for the steel cylinder on the basis of the AWWA C304 [10] standard. 2.6. Wire–concrete interaction 2.6.1. Decomposition of relative motion The contact interaction between the wire and concrete core must be taken into account and has great influence on the wrapping wire process. On the basis of the work of Anand [19] and Gearing et al. [20], we denote the contact domain of the concrete core as C and that of the wire as S, as shown in Fig. 6. The relative displacement u, relative velocity v, and traction T on the interface can be decomposed in two orthogonal directions represented by two unit column vectors: the normal vector n and the tangential vector t.
v n ¼ h_ ¼ h_ cs n;
c
n : un ¼ h ¼ hs n; t : ut ¼ c ¼ c
c s t;
vt
¼ c_ ¼ c_ cs t;
Tn ¼ T n n ¼ pcs n;
ð12Þ
Tt ¼ T T n n ¼ scs t;
ð13Þ
The tangential relative motion is decomposed into a sticking region cel 6 ccsu (analogous to the elastic regime) and slipping part csl (analogous to the plastic regime), where ccsu is the slip resistance corresponding to the shear resistance scsu . The tangential relative velocity is then decomposed in the same manner. c;pl c ¼ cel þ csl ) ccs t ¼ cc;el s t þ cs t; el
ð14Þ
sl
_ c;pl c_ ¼ c_ þ c_ ) c_ cs t ¼ c_ c;el s t þ cs t:
ð15Þ
2.6.2. Normal contact model c The hard contact model is used to represent the contact pressure–overclosure (pcs hs ) relationship between the wire and concrete core during wrapping, which is shown in Fig. 7 and can be expressed as
(
pcs ¼ 0 c hs
c
if hs < 0;
¼ 0 if
pcs
ð16Þ
> 0; c
where the contact pressure pcs is always positive and the overclosure hs is positive when the interface is closed and negative when open. Thus, when the wire and concrete are in contact, any contact pressure is transmitted between them and they separate if the contact pressure reduces to zero. 2.6.3. Tangential contact model The shear–cap friction contact model is adopted here to give the shear resistance scsu . The model assumes that the shear resistance given by the classic Coulomb model is operative until a suitable threshold value scs;max is reached; that is,
(
scsu ¼
lcs pcs pcs < scs;max =lcs ; ffiffiffiffiffi p scs;max ¼ fcy pcs P scs;max =lcs ;
ð17Þ
where lcs is the friction coefficient and fcy is the yield stress of the concrete under uniaxial compression. The friction shear stress–slip (scs ccs ) relationship between the wire and concrete core is shown in Fig. 8 and is written as
scs ¼
scsu ccs =ccsu scs < scsu ; scsu scs P scsu ;
ð18Þ
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Fig. 7. Contact pressure–overclosure relationship between the wire and concrete core.
Fig. 8. Friction shear stress–slip relationship between the wire and concrete core.
where ccsu is the slip resistance corresponding to scsu . Being similar to the yield function, the slip function Q is given in terms of the tangential traction Tt and slip resistance scsu so as to describe the evolution of the slip with the associated flow rule, which can be expressed in the form
Q ¼ Q ðTt ; scsu Þ ¼ scs scsu ; @Q ; c_ sl ¼ a_ @Tt
ð19Þ ð20Þ
where a_ is a parameter of the slip evolution. The consistent condition can be written as
8 h iT @Q > < @T T_ t ¼ 0 t Q_ ¼ h iT > : @Q T_ þ @Qc @ scsuc p_ c ¼ 0 t s @p @Tt @s su
s
scsu ¼ scs;max ;
ð21Þ
scsu ¼ lcs pcs :
So, we can obtain the formulas of the sticking region and slip part respectively as follows.
( sticking : T_ t ¼
ðscsu =ccsu Þc_ ðs
Tt
c c þ scsu l
c c _ su = su Þ
c _c s ps
scsu ¼ scs;max ; scsu ¼ lcs pcs ;
ð22Þ
0 h iT 1
8 > > c c B@T@Qt @T@Qt C > c c _ > > s = c Þ c ssu =csu @h iT Ac_ ð > su su > @Q @Q > < @Tt @Tt 0 h iT 1 slipping : T_ t ¼ > > @Q @Q > > > ðsc =cc Þc_ sc =cc Bh@Tt i@Tt Cc_ þ @Q lc p_ c > A > T su su su su @ @Tt s s > : @Q @Q @Tt
@Tt
scsu ¼ scs;max ; ð23Þ
scsu ¼ lcs pcs :
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Fig. 9. Definition of the master–slave contact pair between the wire and concrete core.
2.6.4. Enforcement of the contact constraint In this paper, the constraint of the contact between the wire and concrete core is enforced using the penalty method [21] and modelled using the boundary contact algorithm studied by Bathe and Chaudhary [22,23] in the theory of geometric nonlinearity. Fig. 9 illustrates the definition of the contact pair, the behaviour of which is described by a master–slave algorithm. The slave node on the wire surface is in contact with the master surface of the concrete core. The normal relative displacement h can be written as c
h ¼ nhs ¼ XA X1 ;
ð24Þ
where XA are the coordinates of the contact point on the master surface, which can be interpolated from the coordinates of the master nodes. Considering the coordinates of the slave node of the wire yields
N; XA ¼ ½ xA ðn; gÞ yA ðn; gÞ zA ðn; gÞ T ¼ X h¼
c nhs
ð25Þ
¼ XN;
ð26Þ
with
¼ ½ N 2 ðn; gÞ N3 ðn; gÞ N4 ðn; gÞ N5 ðn; gÞ T ; N
¼ ½ X2 X T
N ¼ ½ 1 N2 ðn; gÞ N3 ðn; gÞ N4 ðn; gÞ N5 ðn; gÞ ;
X3
X4
X5 ;
X ¼ ½ X1
X2
X3
X4
X5 ;
ð27Þ
where x, y, and z are the global nodal coordinates and n and g the natural nodal coordinates on the master surface, and Ni (i = 1–5) is a shape function. The tangential unit column vector can be defined by the slip of the contact point as follows.
t¼
dXA dXA ¼ : dccs kdXA k
ð28Þ
By variation, we obtain c
c
dnhs þ ndhs ¼ tdccs þ dXN:
ð29Þ
Eq. (29) can be multiplied by scalars n and t to obtain variations in the normal and tangential relative displacements as c
dhs ¼ nT dXN; c s
T
dc ¼ t dXN:
ð30Þ ð31Þ
2.7. Quasi-static solution Because of geometrical nonlinearity and complicated contact conditions in the wire-wrapping model, the explicit centraldifference time integration rule is used here for the quasi-static solving of the dynamic wrapping procedure. 2.7.1. Steps in analysis The wire is stretched such that the stress reaches rsw in the first step. In the following step, the concrete core is rotated with a velocity of m (revolutions/min). The angle between the wire and cross section of the core is
us ¼ arctanðdw =ð2pRsw ÞÞ;
ð32Þ
where dw is the wire spacing. A smooth amplitude curve is adopted to describe the change in the model status so that the inertial effect is reduced greatly. The curve is expressed as
s ¼ si þ ðsiþ1 si Þf3 ð6f2 15f þ 10Þ;
ð33Þ
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Table 1 The four steps of the wrapping wire process. Time (s)
Description
0–t1 t1–t2 t2–t3 t3–t4
Stretching of the wire The rotational velocity of the core increases from zero to m (revolutions/min) The rotating of the core with a constant velocity m The rotational velocity of the core decreases from m to zero
with
f¼
t ti t iþ1 t i
t i 6 t 6 t iþ1 ;
ð34Þ
where si and si+1 are the model status at times ti and ti+1 respectively. Therefore, there are four steps in the whole analysis of the wrapping wire as shown in Table 1. 2.7.2. Mass scaling The mass scaling method is used here for computational efficiency in quasi-static solving. The minimum stable time increment can be written in the form
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kq q ; Dt ¼ min l ^ k^ þ 2m e
ð35Þ
^ are the effective Lamé’s constants for the where le is the characteristic length of the element, q is the material density, ^ k and m material, and kq is the mass scaling factor determined from the energy criterion. 2.7.3. Energy criterion In the quasi-static solving, the ratio of the kinetic energy except for that of rigid body motion to internal energy should be limited to a maximum of 10% during the whole wrapping wire process except for a short time at the beginning of analysis. Meanwhile, the variation in the history curve of the energy should be smooth, avoiding acute fluctuations. The energy criterion should satisfy
fE ¼
EK ERM 6 10%; EI
ð36Þ
where EK is the total kinetic energy, ERM is the kinetic energy of the rigid body motion, EI is the internal energy, and fE is the energy ratio. ERM is determined as follows.
Z 1 kq qxðtÞ2 La 2p r 3 dr; 2 1 wire : ERM ðtÞ ¼ kq q As Ls xðtÞ2 R2sw : 2 core : ERM ðtÞ ¼
ð37Þ ð38Þ
Here, x is the angular velocity of the pipe core, La is the axial length of the pipe, and Ls is the wire length. 3. Model application and verification The wire-wrapping model presented above is applied here to simulate prestressing of a 4 m diameter embedded PCCP used in the SNWD project in China, which is the most significant water supply project for conveying water from central China to the capital city Beijing. The results are analyzed and compared with those obtained using the equivalent external radial pressure method and the AWWA C304 (2007) standard. 3.1. Parameters of the model The parameters of the wire-wrapping model for this application are given by amplitude values as follows. The FE mesh is shown in Fig. 10. The geometrical parameters are the inner diameter of the pipe Di = 4000 mm, the outer diameter of the cylinder Dy = 4183 mm, the total thickness of the pipe including cylinder hc = 350 mm, the thickness of the cylinder hy = 2 mm, the wire diameter ds = 7 mm, the total sectional area of the wire per meter Ap = 2223 mm2/m, and the wire spacing dw = 14.3 mm. The material parameters are the density of concrete qc = 2.323e–9 ton/mm3, the compressive strength of concrete fcc = 48 MPa, the compressive yield stress of concrete fcy = 39.3 MPa, the wire density qs = 7.8e–9 ton/mm3, the wire modulus Es = 193,050 MPa, the maximum tensile strength of the wire fsu = 1570 MPa, the density of the cylinder qy = 7.8e–9 ton/mm3, the modulus of the cylinder Ey = 206,850 MPa, the yield stress of the cylinder fyy = 227.53 MPa, the infinitesimal slip
H. Xiong et al. / Simulation Modelling Practice and Theory 18 (2010) 624–636
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Fig. 10. FE meshing.
Fig. 11. Linear relationship between the energy ratio and mass scaling factor.
Fig. 12. History curves of the energy and energy ratio. (a) Energy. (b) Energy ratio.
resistance ccsu on the contact surface between the wire and concrete core making for rigid plasticity of the tangential contact model, and the friction coefficient lcs = 1e–3 satisfying the uniform wire tension assumption. The multiple-point constraints are imposed on the interface between the cylinder and concrete to harmonize the deformation due to different degrees of freedom. The representative axial length of the pipe La is 20 times the length of the wire spacing. Eight-node linear brick elements are used to model the concrete. Two-node three-dimensional truss elements are used to model the wire. Four-node shell elements are used to model the cylinder. The rotational velocity of the pipe core is 10 revolutions/min. The steps of analysis are defined by t1 = 2, t2 = 4, t3 = 122, and t4 = 124.
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Fig. 13. Graphical description of wrapping process.
Fig. 14. Distribution of the circumferential stress for the concrete outer core (MPa). (a) Wire-wrapping model. (b) Equivalent pressure method.
3.2. Results and evaluation In the following results tension is positive and compression is negative. 3.2.1. Mass scaling factor The mass scaling factor kq is determined by studying the energy variation during the first loop of the wrapping wire. It is found that fE tends to be constant after t3 and increases linearly with an increase in kq. Fig. 11 shows the linear relationship between fE and kq. The optimum mass scaling factor kq is 13.7, corresponding to fE of 10%. 3.2.2. History of energy Fig. 12 shows the history curves of energies and fE. The variations in these curves are not acute, and so the energy criterion is well met. 3.2.3. Process of wrapping wire The results of the wrapping process are shown in Fig. 13. Obviously, the amount of wire wrapped around the core increases linearly with time. The wire spacing is almost equal showing the good geometry of the pipe after wrapping. The magnitude of contact pressure on the concrete core outer surface produced by the pretension of the wire at the end of wrapping is 1.08–1.78 MPa, which is in good agreement with the equivalent external radial pressure determined from Eq. (2) of 1.35 MPa. 3.2.4. Circumferential stress The circumferential stresses obtained using the wire-wrapping model are compared with those obtained using the equivalent external radial pressure method and AWWA C304 (2007) standard. We conclude that the distribution laws of the
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Fig. 15. Distribution of the circumferential stress for the concrete inner core (MPa). (a) Wire-wrapping model. (b) Equivalent pressure method.
Table 2 Comparison of circumferential stresses (MPa). Position
Wire-wrapping model
Equivalent method
AWWA C304
Outer core Inner core Cylinder
7.9 to 8.8 8.7 to 9.1 67.7
8.1 to 8.9 8.9 to 9.2 68.8
8.7 63.5
Fig. 16. History curve of the mean value of the wire stress.
Table 3 Comparison of principle stresses with yield stresses (MPa). Position
Principle stress
Yield stress
Outer core Inner core Cylinder Wire
8.7 9.0 67.7 1282
39.3 39.3 227.5 1334.5
circumferential stresses obtained using different approaches are in good agreement. The distributions are presented in Figs. 14, 15 and Table 2. 3.2.5. Wire stress The history curve of the mean value of the wire stress is shown in Fig. 16 and indicates the satisfaction of the requirement of fluctuations in tension not deviating from the mean by more than ±10%. 3.2.6. Checking of the material status The principle stresses of different components are given in Table 3, and they are compared with the yield stresses for the checking of material statuses. It is confirmed that the status of the materials during the wire-wrapping are in the range of elasticity.
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4. Conclusions Since the conventional FE analysis methods for the prestressing of a PCCP have their intrinsic demerits, this paper focused on the process of wrapping wire in the manufacture of an embedded PCCP and has presented a new FE model, called the wire-wrapping model, to simulate the prestressing of a PCCP. The main conclusions are the following: - The whole process of wrapping wire in the manufacture of an embedded PCCP can be numerically simulated by this FE model. The prestressing of the pipe could be achieved satisfactorily. - The results obtained using the wire-wrapping model, compared with those obtained using the equivalent external radial pressure method and those using the AWWA C304 standard, are proven to be reasonable. That means: the energy criterion could be well met by selecting an appropriate mass scaling factor, the geometry of pipe is fine after wrapping, the distribution of the prestress of the core is regular, the stress of the wire varies smoothly satisfying the requirement of fluctuations in tension, and all materials remain in the elastic domain. - The major characteristics of this FE model, compared with those of traditional equivalent approaches, are that the contribution of the wire stiffness is taken into account in the model, the stress and strain in the wire itself are appropriately considered, and the interaction at the interface between the wire and concrete core is revealed by the contact model. Therefore, the wire-wrapping model has great potential in further FE analysis of a PCCP. Further work will include the study of the performance of a PCCP subjected to the loads and pressures, and to investigate the effects of wire-breaking on the response of a PCCP. In this study, the contribution of wire stiffness and the interaction between the wire and other component must be taken into account, which can be accomplished using the wire-wrapping model. Acknowledgements We acknowledge the support provided by the Chinese National Science & Technology Pillar Program during 11th Five-Year Plan Period, Grant No. 2006BAB04A04-03. References [1] A.E. Romer, G.E.C. Bell, R.D. Ellison, Failure of prestressed concrete cylinder pipe, in: M. Najafi, L. Osborn (Eds.), Proc. Int. Conf. on Advances and Experiences with Trenchless Pipeline Projects, ASCE, Boston, Massachusetts, 2007. [2] M.S. Zarghamee, R.P. Ojdrovic, W.R. Dana, Coating delamination by radial tension in prestressed concrete pipe II: analysis, J. Struct. Eng. 119 (1993) 2720–2732. [3] M.S. Zarghamee, D.W. Eggers, R.P. Ojdrovic, Finite-element modeling of failure of PCCP with broken wires subjected to combined loads, in: G.E. Kurz (Ed.), Proc. Int. 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