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On thermoplastic piping stress analysis
Journal Pre-proof On thermoplastic piping stress analysis Nikola Jaćimović, Fabio D'Agaro, Zdravko Ivančić, Mirjana Stamenić PII:
S0308-0161(19)30369-2
DOI:
https://doi.org/10.1016/j.ijpvp.2019.104010
Reference:
IPVP 104010
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 26 August 2019 Accepted Date: 27 October 2019
Please cite this article as: Jaćimović N, D'Agaro F, Ivančić Z, Stamenić M, On thermoplastic piping stress analysis, International Journal of Pressure Vessels and Piping (2019), doi: https://doi.org/10.1016/ j.ijpvp.2019.104010. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
ON THERMOPLASTIC PIPING STRESS ANALYSIS Nikola Jaćimovića*, Fabio D’Agaroa, Zdravko Ivančićb, Mirjana Stamenićc a
Danieli & C. Officine Meccaniche S.p.A., Via Nazionale 41, 33042 Buttrio (UD), Italy
b
Numikon Ltd., Buzinski prilaz 10, 10010 Zagreb, Croatia
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia c
*Corresponding author Address: Danieli & C. Officine Meccaniche S.p.A., Via Nazionale 41, 33042 Buttrio (UD), Italy Phone: +39 345 9723691 E-mail: [email protected] ABSTRACT Thermoplastic piping is becoming increasingly important in the general industry applications. Owing mainly to their corrosion resistance, light weight and low maintenance requirements, plastic pipes can be found in many industry branches with their use becoming ever more widespread. Despite this fact, up to date there are no piping codes or other sound references which would clearly and univocally define the procedure and parameters for thermoplastic pipe stress analysis. Paper discusses stress analysis and design of thermoplastic piping systems and proposes a dedicated procedure for their calculation. Presuming isotropic and perfectly plastic material properties, stress intensification factors, which do not exist in open literature, have been determined using finite element and regression analysis. Results presented in the paper show that the new equations result in a much better fit and much lower mean deviation than existing equations for metallic piping. Although stress intensification factors determined based on the finite element analysis can be found even in some of the most advanced piping codes in use today, it could be recommended that in future dedicated tests on physical specimens be made in order to confirm the presented findings or suggest better equations. KEY WORDS Thermoplastics, Piping, Stress analysis, Stress intensification factor INTRODUCTION Analysis of thermoplastic systems presents a set of specific issues which need to be addressed in order to design a successful piping system. There are no literature references or international design codes which specifically address stress analysis of thermoplastic piping systems (although there is an ASME standard under development which should address the issue of thermoplastic 1
piping analysis in the future). Considering the ever-growing application of thermoplastic materials in industrial applications, it is important to set clear rules and recommendations for design of these systems. It should be mentioned that, according the authors’ experience, there are two different design approaches for plastic piping systems. First approach is based on the completely anchored pipe (i.e. placing fixed points all along the piping, thus effectively completely restraining pipe expansion). This approach, although seemingly simple, has two major problems: it is difficult to execute properly in site and it may result in high compressive stresses in the pipe itself. Second approach is to support the pipes as “normal” metallic piping system (i.e. providing fixed points, guides, axial stops, pipe loops, etc.). This approach is much simpler to execute and, if executed properly, will lead to a better, lighter and very predictable piping design. Major problem with the second approach is complete lack of design method and applicable equations. Aim of this paper is to provide the said method which would guarantee safe piping system design while limiting conservatism. Method described in this paper is limited to the plastic materials which are isotropic in nature. Materials which are not isotropic (e.g. plastic lined FRP pipes) should be calculated in accordance with dedicated standards, such as [1] or [2]. Analysis performed and method presented in this paper aim to set some basic requirements for stress analysis of thermoplastic piping systems in the absence or more applicable data. CALCULATION PROCEDURE Thermoplastic materials are viscoelastic, meaning that the properties, such as strength or deformation, are significantly affected by both time (i.e. load duration) and temperature. Stressstrain response of plastic materials is non-linear, as described in [3] and unlike metals they do not have elastic constants (elastic modulus or proportional limit) or clearly defined yield points. However, most equations which are developed for stress analysis of metallic piping, can still be used provided that correct values of strength and stiffness can be established. Discussion on allowable strength and applicable elastic moduli (albeit using the term elastic or modulus may be regarded as erroneous since thermoplastic materials do not have true elastic-plastic properties) is provided in dedicated sections in further text. For further understanding of viscoelastic properties, definition of various moduli (creep, stress-relaxation), stress-rupture behavior and other phenomena related to plastic materials, reader should refer to [3]. All further discussion shall be based on calculation methodology from [4]. Although there is a dedicated section in [4] discussing non-metallic piping, it does not provide detailed guidelines on the calculation principles of thermoplastic piping systems. Sustained stresses In order to estimate the effects of long-term sustained loads, first of all it is necessary to extrapolate the effective material properties for applicable load duration. Data in open literature are sufficient 2
to estimate allowable stress and moduli for long-term design life and will be discussed in further text in dedicated sections. In evaluating sustained loads, another major difference with respect to metallic piping, besides material properties, is the effect of the pressure on the plastic piping. Namely, in traditional metallic piping analysis pressure is regarded as primary load and is verified only by taking into account the axial and hoop stresses due to pressure. However, in thermoplastic piping systems pressure will produce also elongation and bending stresses which need to be taken into consideration. Thermal expansion stresses Typically, thermal expansion is regarded as secondary, or self-limiting stress, and is normally verified for low cycle fatigue. Considering that the thermoplastic materials do not have yield strength, a dedicated evaluation of fatigue in accordance with [4] requirements has little practical sense. Also, considering the viscoelastic material properties which are greatly dependent on time, the concept of self-limiting stresses is not really applicable, since there will always be some amount of progressive deformation. In order to evaluate the effect of temperature, its effect needs to be calculated together with sustained stresses and the maximum stress magnitude needs to be limited to a safe value. Operating stresses Based on the above, it can be concluded that using the conventional verification of sustained stresses and expansion stress range (i.e. separate verification of primary and secondary stresses) in case of plastic materials has no practical significance. Since no major piping codes which are based on beam theory and design by formulae approach (e.g. [4], [5], [6]) allow the possibility to estimate the joint effect of pressure and thermal expansion (i.e. primary and secondary stresses), it is necessary to define additional load scenarios in the pipe stress analysis software in order to verify these effects acting contemporarily. Typically, it can be suggested to calculate both stresses due to primary loads (i.e. pressure and weight) and stresses due to the joint effect of primary and secondary loads (i.e. pressure, weight and temperature) and compare both effects against a single allowable stress. In effect, it may be sufficient to calculate only operating stresses and compare them against allowable stress. Calculation of purely sustained stresses (i.e. due to pressure and weight), although useful, is not necessary for evaluating system acceptability. Another important issue with thermoplastic piping systems, which is often overlooked, is correct placing of supports. Problem arises when less experienced designers use rule-of-thumb analysis for metallic piping to place the supports in thermoplastic piping system. Using the same approach as for metallic piping (which is much stronger than plastic piping) will surely result in under-designed, or under-supported piping system. Therefore, it is always necessary to use specific codes and evaluate the used piping spans using realistic project data (such as [31]). Once preliminary 3
supports are placed throughout the system it is always recommended to perform piping stress analysis in order to determine realistic piping stresses and equipment/support loads. Recurring short-term loading Since allowable ultimate strength and ultimate strain of thermoplastic pipes greatly depends on load duration, these pipes can withstand (significantly) higher temporary loading than foreseen by using their long-term strength. If this short-term loading is infrequent, or is not recurring in such manner to generate fatigue damage to piping, then it may be sufficient to use the information on the allowable short-term overloading from various guidelines. For example, some information regarding the allowable short-term overpressure is published in [7], [8], [9], [10] or [11]. In cases in which expected short-term overloading is frequent, it may be necessary to perform analysis considering pipe fatigue resistance capabilities. In this case, actual S-N curves should be used. Some limited information on fatigue capability of PVC piping can be found in [12]. Considering that the available S-N data for thermoplastic piping materials is very limited, all subsequent discussion will be limited to the effects which do not cause fatigue failure. ALLOWABLE STRESS For plastic piping, allowable stress should be based in hydrostatic strength of a material for sufficiently long design life and for the maximum design temperature. Typical design life for thermoplastic piping systems is 10÷50 years, but other values can be used. Hydrostatic strength curves (such as the example shown in Figure 1) can be found in various standards, such as [13], [14] or [15]. Typical allowable stress, besides the effects of design life and maximum design temperature, should include the following effects: -
effect of the transported fluid on the material; joining method; minimum design temperature; safety factor.
Typical equation for establishing thermoplastic piping allowable strength, given in [16] is =
∙
∙ ∙
(1)
in which: -
, MPa, allowable stress; , MPa, hydrostatic strength at design life and temperature; , weld joint factor ( = 0.4 ÷ 0.8, depending on material and welding procedure); , safety factor (typically 1.25÷2); , reduction factor for transported fluid;
4
, reduction factor for specific strength.
-
Figure 1 Change of PP-H hydrostatic strength in time at design temperature of 60°C [14] Allowable stress, as defined above, can be used to verify the combined effect of pressure, weight and temperature (i.e. operating stress). Other possibility to determine the allowable stress is defined in [17], which states that the design stress at design life (, years) and design temperature (t, °C) shall be equal to
=
,
(2)
in which: -
!",# , MPa, categorized required strength (as defined in [17]); , design coefficient taking into account the effects of specific product requirements, time and temperature (with minimum value defined in in [17]).
ELASTIC MODULUS Using correct elastic moduli for analysis is of paramount importance when evaluating thermoplastic piping systems. In general, elastic modulus of thermoplastic material decreases with time, as shown in Figure 2. 5
Figure 2 Change of PP-H elastic modulus in time (extrapolated from [18]) Using short-term elastic modulus will typically result in higher equipment loads. On the other hand, using lower value of long-term elastic modulus (commonly denoted as creep modulus) under the same design loads will typically result in greater movement/sagging. Obviously, greatest elastic modulus will occur for design life of 0 years. However, using this value in the analysis may result in very conservative design, considering that the elastic modulus value will sharply drop within a very short timeframe and that the temperature difference in piping system requires time to result in changes in length. Therefore, it is suggested in [19] to use a value of elastic modulus after 100 minutes in order to determine anchor loads. This value was successfully used by the author in multiple projects and has proved to be a good trade-off between conservative design and limiting unnecessary system modifications. Therefore, it may be suggested to perform two sets of analyses: 1. using high value of short-elastic modulus (i.e. at design life at 100 minutes) for calculating equipment loads and stresses; 2. using low value of long-term elastic modulus (i.e. at long-term design life) for calculating stresses and displacements. It should be noted that the effects of any possible short-term loading, such as pressure surge or instantaneous short-term negative pressure in piping system, should be, as a minimum, calculated using short-term elastic modulus. This approach will generally result in higher support forces, while the calculated stresses may be underestimated with respect to the values obtained using long-term modulus. Designer should carefully evaluate if a dedicated load case for short-term loading should also be made using the long-term modulus in order to estimate the effect of the 6
modulus on the stress distribution under short-term loading. This approach, although generally recommended, may result in overly conservative results and unnecessary piping modifications. Calculation of the creep modulus used in the analysis should include factors for transported fluid and safety factor. Permissible creep modulus ($% , MPa) can be calculated using the following equation $% =
& ∙
(3)
in which E, MPa, denotes creep modulus from applicable diagram and for applicable design parameters. Based on all the above discussion, it will typically be necessary to define different load cases for pipe stress analysis. As a general recommendation, load cases shown in Table 1 should be established. Table 1 Recommended minimum load cases to be verified Acting loads Operating weight Design pressure Design temperature Operating weight Design pressure Operating weight Design pressure Operating weight Design pressure Design temperature Operating weight Design pressure Design temperature Water filled weight Hydrostatic test pressure
Applicable modulus
Verified variables
100 minute modulus
Equipment loads Support loads
Short-term modulus
Stress
Long-term modulus
Stress
Short-term modulus
Stress
Long-term modulus
Stress
Short-term modulus
Stress Equipment loads Support loads
STRESS INTENSIFICATION FACTORS Major issue with calculation of stresses of thermoplastic piping systems is the application of correct stress intensification factors, or SIFs. These factors account for the existence of discontinuity (e.g. pipe bend or tee) inside a piping system, and account for the increase of stresses due to its existence. Up to date, the only two sources for plastic pipe stress intensification factors (i.e. SIFs) are [20] and [21]. Both of these sources define constant SIF values regardless of the type or geometry of 7
discontinuity. While [20] is based on old manufacturer’s standards and defines a constant SIF value of 2.3, [21] simply defines all SIFs as unity. There are many obvious shortcomings of a constant SIF approach, such as complete independence on the type of discontinuity (elbow, tee, etc.), geometry (diameter, thickness, radii, etc.), etc. Other possibility is to use the existing equations for metallic piping. However, considering that the metallic piping has significantly lower thickness and different material properties than thermoplastic piping, application of these equations is questionable, to say the least. Therefore, a numerical study has been done in order to estimate the realistic SIF values for most common piping elements: bends, straight and reducing tees. Analysis of SIFs is based on numerical modeling (i.e. finite element analysis) of bends and tees. Analysis software is NozzlePRO (version 12) developed by PRG. Material properties used in the analysis are presented in Table 2. Table 2 Material properties used in the analysis PP-H
PE
11
10.4
Short term modulus, MPa
1200
800
Allowable stress, MPa
4.22
5.2
Density, kg/m3
910
960
Poisson's coefficient
0.38
0.45
Linear expansion coefficient, mm/(km∙°C)
150
180
Long term hydrostatic strength, MPa
Majority of the above data have been taken from various international standards ([13], [14], [16], [18] and [22]), while the values not defined in standards have been taken directly from the catalogues of renowned plastic pipe manufacturers. All of the above data (moduli of elasticity, hydrostatic strength, etc.) have been taken at the ambient temperature (20°C). However, multiple runs have been made with data taken at higher temperatures and design life (i.e. short and long term) and they resulted in virtually significant variation of results. Total number of analyzed models is: -
73 bends (37 in PP-H and 36 in PE100); 102 equal tees (37 in PP-H and 65 in PE100); 190 reducing tees (128 in PP-H and 62 in PE100).
8
The standard dimension ratios (SDR) considered were SDR11, SDR17, SDR 17.6 and SDR33 for all examined cases. These were selected based on the vendor catalogues and represent the values typically available on the market. It is important to note that the thickness of tees considered in the analysis equals the thickness of pipe. However, typically tees made of plastic materials have greater thickness than the straight pipe. Seen as this over-thickness depends on the manufacturer, it cannot be estimated with sufficient accuracy. In any case, the approach used in this paper will result in conservative results. Comparison is given in the later text with some of the sources for piping stress intensification factors ([4], [23] and [24]). It must be noted that all of the aforementioned references are valid for metallic piping. As expected, the major difference between thicknesses and material properties of metallic and plastic piping make the existing equations for metallic piping SIFs of little use. It should be mentioned that the SIFs are calculated as “peak” SIFs, which correspond to the values typically used in the piping codes. Since typical SIFs used in piping codes typically represent approximately one half of the actual stress intensification (due to the presence of weld joint, as described in [25]), and considering that the secondary stress evaluation typically used in piping stress analysis is invalid, it might seem reasonable to consider twice the calculated value for the analysis. However, considering the fact that the allowable stress, as defined in equation (1) above, already considers the effect of weld joint (f=0.4÷0.8), it is sufficient to consider the peak stress intensification factors for engineering practice. Aside from hoop stress calculation, weld quality is typically not considered for stress analysis of metallic piping (ref. [4], [5], [6]), and it is a known fact that the allowable expansion stress range in major piping codes (such as [4]) is reduced by an approximate factor of 2 (or more accurately 1.56, as shown in [25]) to take into consideration presence of butt-welds in piping system and the fact that the calculated stress intensification factor represent approximately one half of the real value. Although above mentioned factor f is essentially different from the weld stress intensification factor, it yields almost identical result, decreasing the allowable stress by a factor of 1.25÷2.5 (as defined in [16] f = 0.4÷0.8, depending on material and welding process). This approach can be further reinforced considering the statement from [26]: “These (plastic) materials remain ductile for a very long time and exhibit high resistance to slow crack growth, and thus are much less prone to failure from localized stress intensifications which can occur during normal operation of a plastic piping system”. All subsequent analysis and stress intensification factor comparison is based on the presumption that the finite element results represent realistic values. This approach is in line with the work done in the past in which experimental data had been used as a reference for any comparison ([23], [24]). Today we can treat finite element analysis as a very precise and widely accepted analysis method. Regression analysis Regression analysis of the finite element model results was performed using the least square analysis. In order to verify the results, the parameters shown in the Annex of this paper are used. 9
Both maximal positive error and maximal negative error have been taken into account in order to ensure that the proposed equations result in conservative design while limiting excessive overdesign of a piping system. Bends Regression analysis has been used to find equations which provide a good fit for all the analyzed cases. These equations are: 1. In plane stress intensification factor ('() ) '() =
*.+ ../ ,-
=
*.+
1∙2
../
0 4 5 3 ,-
(4)
2. Out plane stress intensification factor ('67# ) '67# =
*.89 ../ :;
=
*.89
../ 1∙2 < = 3:;
(5)
in which: -
r, mm, bend radius of curvature; T, mm, bend (pipe) wall thickness;
ℎ() , flexibility characteristic for in-plane moment (ℎ() = !() , mm, pipe internal radius (!() = (B − 2 ∙ E)/2);
?∙@
4 ,-
);
ℎ67# , flexibility characteristic for out-plane moment (ℎ67# = !67# , mm, pipe external radius (!67# = B/2); B, mm, pipe outside diameter.
?∙@ 4 ); :;
Nomenclature used in equations (4) and (5) is shown in Figure 3. Using existing equations published in piping codes for metallic piping ([4], [6] or [23] would result in overestimated SIFs by as much as 30%. As such, they are suitable for use, but may result in overly conservative design. Using a fixed stress intensification factor of 2.3, as suggested in [20], would result in an overestimate of in-plane SIFs by as much as 100% and of out-plane SIFs by as much as 150%. Finally, using the proposed equations still yields conservative results, but limits the maximum deviation to less than 10%. Dependence of in-plane and out-plane stress intensification factors of flexibility characteristic (i.e. on bend geometry) is shown in Figure 4.
10
Figure 3 – Nomenclature for equations (4) and (5)
Figure 4 – Plastic bend SIFs Proposed equations for tees (unreinforced straight and reducing) Below equations have been determined as the best fit for all examined tees. Due to the complexity of these equations and the dependence of stress intensification factors on multiple variables, it is impossible to show these results in graphic form. However, statistical parameters shown in Table 3 can be used to estimate the quality of the proposed equations.
11
For run pipe (main header) *.8K
'(? = 0.77 ∙ I J @
*.PK
'6? = 0.53 ∙ I J @
*.9M
'#? = 0.58 ∙ I@ J
For branch pipe *.TT
'(R = 0.6 ∙ I J @
*.KQ
∙I J L
*.Q8
∙ IL J
*.U8
∙I J L
*.9K
'6R = 0.69 ∙ I J @
*.MK
∙I J L
*.99
'#R = 0.57 ∙ I J @
M.UP
∙I J L
(6)
# *.KQ
∙I J @
(7)
# *.Q8
∙ I@ J
(8)
# *.U8
∙I J @
M.QQ
∙I J L
# *.MK
∙I J @
(9)
# WM.QQ
∙I J @
(10)
# M.UK
∙I J @
(11)
or alternatively (better fit) for in-plane and out-plane branch stress intensification factors *.K8
'(R = 1.6 + 0.3 ∙ ZI J @ *.Q
'6R = 6.61 ∙ ZI J @
8
# 8
*.K8
+ I J ∙ I J − 2.5[ + 0.11 ∙ ZI J L @ @ *.M
+ 0.5 ∙ I J L
# *.M
∙I J @
− 1.7[
8
# 8
+ I J ∙ I J − 2.5[ L @
U
(12) (13)
in which R, mm, denotes mean header radius, as shown in Figure 5, and can be calculated as !=
LW@ U
(14)
Figure 5 – Nomenclature for equations (6)÷(13) Equations (6)÷(13) are based on the same form as used in [23] and [24]. However, it has been found that a different form of equations can result in a much better for in-plane and out-plane SIFs 12
of branch pipe (equations (12) and (13)). Above equations yield statistical parameters as shown in Table 3. Table 3 Statistical parameters of equations (6)÷(13) Equation number
minRE-, % maxRE+, % meanRE, % SD, % CR
(4)
8.9
1.5
-2.9
3.8
0.9841
(5)
5.2
1.6
-1.9
2.9
0.9805
(6)
14.2
16.1
1.0
9.0
0.9462
(7)
2.8
8.6
1.6
2.0
0.9968
(8)
18.7
24.7
-0.9
11.0
0.9756
(9)
21.0
25.9
-0.3
10.8
0.9621
(10)
11.7
49.7
1.7
8.2
0.9793
(11)
7.5
28.4
2.6
7.5
0.9973
(12)
8.3
16.8
-0.4
4.0
0.9946
(13)
11.1
24.7
-0.6
7.3
0.9775
Comparison of relative errors (expressed in % using only equation for maximal positive error) between various sources are shown in Table 4. Table 4 Comparison of relative error ranges between various sources [4]
Fixed 2.3 [20]
[23]
[24]
7.3÷96.7
-36.8÷103.5
-0.5÷69.2
23.1÷85.5
137.4÷269.6
-22.3÷130.0
-51.8÷69.8
-44.8÷34.8
Torsion
-
-
-23.5÷35.9
-18.3÷43.6
In plane
-14.2÷51.2
-56.2÷56.5
-38.6÷-17.9
-28.3÷20.9
Out plane
-22.3÷45.3
-67.9÷27.1
-17.5÷23.3
-38.8÷92.2
-
-
-88.4÷10.1
-59.6÷6.2
In plane Run pipe
Branch pipe
Out plane
Torsion
Maximum negative deviation (underestimation of SIFs) for the proposed equations is ~20%, while other equations applied to metallic piping go up to ~90%. This means that using conventional equations may result in a largely underestimated stress. 13
On the other hand, the maximum positive deviation (overestimate of SIFs) for the proposed equations is ~50%. Again, using conventional equations typically used for metallic piping would result in an overestimated stress by as much as 270%. This means that using conventional equations may result in a very conservative system resulting in unnecessary modifications and complex routing. If fixed SIF value of 2.3 is used, the relative error compared to FEA solutions ranges from -70% (underestimate) up to +130% (overestimate). It is clear that using a fix value cannot cover all the variables connected to the geometrical characteristics of a discontinuity. Therefore, using this approach can only be regarded as a last resort if absolutely no other data is available. Flexibility factors Flexibility factors for all plastic piping fittings can be taken as 1. This was confirmed by the results of finite element analysis, which has revealed that the flexibility factors were lower than 1 for almost all cases. There were only a handful of cases which have had flexibility factors slightly greater than 1. However, considering that the flexibility factor depends on fitting thickness (i.e. greater the thickness, lower the flexibility factor) and the fact that all plastic fittings typically have greater thickness than straight pipe (which was the presumption used in the finite element analysis performed in the paper), using flexibility factors of 1 for all cases should be sufficiently conservative. This is also in line with the recommendation indicated in [20]. CONCLUSION Due to their many advantages, such as chemical resistance properties, lower density (i.e. weight), negligible costs of external protection (painting, galvanizing, etc.) and high deformation capacity, plastic pipe market is ever increasing in the industrial applications. In fact, as indicated in [3], even during the 1990s, the amount of meters of thermoplastic piping exceeded the amount of all other piping materials combined. Considering the continuing growth of plastic pipe market ([27] and [28]) it can only be concluded that plastic materials will continue to develop and gain ever more ground in general industry applications. However, major problem with thermoplastic piping is the relatively low strength capacity and lack of applicable standards which can be used in stress analysis and design. Method and equation presented in this paper are developed with the aim to establish some basic rules and principles for pipe stress and flexibility analysis of thermoplastic piping systems. Although there are international guidelines on establishing pressure ratings, allowable stresses and moduli for the analysis, literature review has revealed complete lack of stress intensification factors, which present an important parameter for piping system design. Consequently, finite element analysis has been used (much like in [23] and [24]) in order to establish stress intensification factors which can be applied to all plastic piping systems. Analysis has revealed (based on comparison between polyethylene and polypropylene materials) that the differences in material 14
properties between various thermoplastic materials are negligible and that single set of equations can be used regardless of the material. Comparison between the finite element analysis results has shown that using the existing references may yield an underestimated stress by as much as 90% and thus result in an unsafe design. On the other hand, the maximum overestimate of stresses using the same references may reach 270%, resulting in an overly conservative design. Considering that the industry trend in to optimize piping diameters in order to reduce the overall costs ([29] and [30]), and the effect that stress intensification factors may have on pipe wall thickness, it is only logical to try and design the piping without excessive conservatism, while remaining in the safe zone. Using the proposed equations for stress intensification factors the deviation from finite element analysis results ranges from -20% (underestimate) to +50% (overestimate), while keeping the mean error typically below 10%. Therefore, the possibility of under-designing a thermoplastic piping system is minimized, while keeping the level of conservatism within a reasonable margin. LITERATURE [1]
International Standard ISO 14692-3, Petroleum and Natural Gas Industries –Glass-reinforced Plastics (GRP) Piping – Part3: System Design, 2nd Edition, ISO, Switzerland, 2017.
[2]
British Standard BS 7159, Code of Practice for Design and Construction of Glass Reinforced Plastics (GRP) Piping Systems for Individual Plants or Sites, BSI, UK, 1989.
[3]
McGrath, T. J., Mruk, S. A., Chapter D1 – Thermoplastics Piping, in Nayyar, M. L., Piping Handbook, 7th Edition, McGraw-Hill, USA, 2000.
[4]
ASME B31.3, Process Piping, American Society of Mechanical Engineers, New York, USA
[5]
ASME B31.1, Power Piping, American Society of Mechanical Engineers, New York, USA
[6]
EN 13480-3 Metallic Industrial Piping – Part 3: Design and Calculations, European Committee for Standardization (CEN), Belgium, 2017.
[7]
ANSI/AWWA Standard C900-07, Polyvinyl Chloride (PVC) Pressure Pipe and Fabricated Fittings 4 In. Through 12 In. (100 mm Through 300 mm), For Water Transmission and Distribution, American Water Works Association, USA, 2007.
[8]
ANSI/AWWA Standard C901-08, Polyethylene (PE) Pressure Pipe and Tubing, ½ In. (13 mm) Through 3 In. (76 mm), for Water Service, American Water Works Association, USA, 2008.
[9]
ANSI/AWWA Standard C905-10, Polyvinyl Chloride (PVC) Pressure Pipe and Fabricated Fittings 14 In. Through 48 In. (350 mm Through 1,200 mm), American Water Works Association, USA, 2010.
15
[10]
ANSI/AWWA Standard C906-07, Polyethylene (PE) Pressure Pipe and Tubing, 4 In. (100 mm) Through 63 In. (1,600 mm), for Water Distribution and Transmission, American Water Works Association, USA, 2007.
[11]
AWWA Manual M23, PVC Pipe – Design and Installation, 2nd Edition, American Water Works Association, USA, 2002.
[12]
Technical Report, Fatigue of Plastic Water Pipe: A Technical Review with Recommendations for PE4710 Pipe Design Fatigue, Jana Laboratories Inc., USA, 2012.
[13]
Deutsche Norm DIN 8075, Polyethylene (PE) Pipes – General Quality Requirements and Testing, DIN, Germany, 1999.
[14]
Deutsche Norm DIN 8078, Polypropylene (PP) Pipes – PP-H, PP-B, PP-R, PP-RCT – General Quality Requirements and Testing, DIN, Germany, 2008.
[15]
International Standard ISO 15494, Plastics Piping Systems for Industrial Applications – Polybutene (PB), Polyethylene (PE) and Polypropylene (PP) – Specifications for Components and the System – Metric Series, 1st Edition (corrected version), ISO, Switzerland, 2004.
[16]
Directive DVS 2205-1, Design Calculations for Containers and Apparatus Made From Thermoplastics; Characteristic Values, DVS-Verlag GmbH, Germany, 2002.
[17]
International Standard ISO 12162, Thermoplastics Materials for Pipes and Fittings for Pressure Applications – Classification, Designation and Design Coefficient, 2nd Edition, ISO, Switzerland, 2009.
[18]
Directive DVS 2205-2, Calculation of Thermoplastic Tanks and Apparatus – Vertical Cylindrical Non-Pressurized Tanks, DVS-Verlag GmbH, Germany, 1997.
[19]
SIMONA Technical Manual, Engineering Manual for Piping Systems, SIMONA AG, Germany, 2010.
[20]
Plastic Pipe Modeling, COADE Mechanical Engineering News, COADE, USA, April 1991.
[21]
GOST Standard 32388:2013, Process Piping - Standard for the Stress, Vibration and Seismic Analysis, Interstate Council for Standardization, Metrology and Certification (ISC), Russia, 2014.
[22]
Deutsche Norm DIN 8077, Polypropylene (PP) Pipes – PP-H, PP-B, PP-R, PP-RCT – Dimensions, DIN, Germany, 2008.
[23]
ASME B31J, Stress Intensification Factors (i-Factors), Flexibility Factors (k-Factors), and Their Determination for Metallic Piping Components, American Society of Mechanical Engineers, New York, USA
16
[24]
Jacimovic, N., Analysis of Piping Stress Intensification Factors Based on Numerical Models, International Journal of Pressure Vessels and Piping, Vol. 163, 2018
[25]
Jacimovic, N., Uncertanties in Expansion Stress Evaluation Criteria in Piping Codes, International Journal of Pressure Vessels and Piping, Vol. 169, 2019
[26]
Boros, S., Long-Term Hydrostatic Strength and Design of Thermoplastic Piping Compounds, Journal of ASTM International, Vol.8, No. 9, 2011.
[27]
Plastic Growth to Exceed Other Materials in Pipe Market, Plastic News, 2012.
[28]
Plastic pipe market, Global demand to reach 11.2 bn metres by 2017 / 8.5% annual growth / Increase in construction spending plays key role / Freedonia study, https://www.plasteurope.com/news/PLASTIC_PIPE_MARKET_t225210/, 2013.
[29]
Genic S., Jaćimovic B., A Shortcut to Optimize Pipe Diameters by Economic Criteria, Chemical Engineering, Vol. 116 (12), pp 50-53, 2018.
[30]
Genic, S., Jacimovic, B., Shortcut Optimization of Steam Pipeline Diameter for Complete Turbulence, Chemical Engineering Progress, Vol. 114 (8), 2018.
[31]
Directive DVS 2210-1, Industrial Pipelines Made of Thermoplastics –Planning and Execution Above-Ground Piping Systems, DVS-Verlag GmbH, Germany, 1997.
Appendix - Correlation parameters The comparison of experimental (ym,i) and correlated (yc,i) parameter can be done by certain number of statistical parameters: 1. standard deviation (SD), which is defined as B =
\
∑`(aM <
^_,( − ^%,( U ^_,( = b
2. mean relative error (meanRE), which is defined as
` ^ 1 _,( − ^%,( ∙g b ^_,( (aM 3. maximal positive error (maxRE+), which is defined as ^_,( − ^%,( ceh!$ i = max 0 5 ^_,(
cdef!$ =
4. maximal negative error (maxRE-), which is defined as 17
^%,( − ^_,( ceh!$ W = max 0 5 ^_,( 5. Correlation ratio (CR), which is defined as ! = n1 −
∑`(aM(^_,( − ^%,( )U ∑`(aM(^_,( − ^_,o )U
in which ^_,o is the average value ^_ of for the complete set of z experimental data.
18
S0308-0161(19)30369-2
DOI:
https://doi.org/10.1016/j.ijpvp.2019.104010
Reference:
IPVP 104010
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 26 August 2019 Accepted Date: 27 October 2019
Please cite this article as: Jaćimović N, D'Agaro F, Ivančić Z, Stamenić M, On thermoplastic piping stress analysis, International Journal of Pressure Vessels and Piping (2019), doi: https://doi.org/10.1016/ j.ijpvp.2019.104010. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
ON THERMOPLASTIC PIPING STRESS ANALYSIS Nikola Jaćimovića*, Fabio D’Agaroa, Zdravko Ivančićb, Mirjana Stamenićc a
Danieli & C. Officine Meccaniche S.p.A., Via Nazionale 41, 33042 Buttrio (UD), Italy
b
Numikon Ltd., Buzinski prilaz 10, 10010 Zagreb, Croatia
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia c
*Corresponding author Address: Danieli & C. Officine Meccaniche S.p.A., Via Nazionale 41, 33042 Buttrio (UD), Italy Phone: +39 345 9723691 E-mail: [email protected] ABSTRACT Thermoplastic piping is becoming increasingly important in the general industry applications. Owing mainly to their corrosion resistance, light weight and low maintenance requirements, plastic pipes can be found in many industry branches with their use becoming ever more widespread. Despite this fact, up to date there are no piping codes or other sound references which would clearly and univocally define the procedure and parameters for thermoplastic pipe stress analysis. Paper discusses stress analysis and design of thermoplastic piping systems and proposes a dedicated procedure for their calculation. Presuming isotropic and perfectly plastic material properties, stress intensification factors, which do not exist in open literature, have been determined using finite element and regression analysis. Results presented in the paper show that the new equations result in a much better fit and much lower mean deviation than existing equations for metallic piping. Although stress intensification factors determined based on the finite element analysis can be found even in some of the most advanced piping codes in use today, it could be recommended that in future dedicated tests on physical specimens be made in order to confirm the presented findings or suggest better equations. KEY WORDS Thermoplastics, Piping, Stress analysis, Stress intensification factor INTRODUCTION Analysis of thermoplastic systems presents a set of specific issues which need to be addressed in order to design a successful piping system. There are no literature references or international design codes which specifically address stress analysis of thermoplastic piping systems (although there is an ASME standard under development which should address the issue of thermoplastic 1
piping analysis in the future). Considering the ever-growing application of thermoplastic materials in industrial applications, it is important to set clear rules and recommendations for design of these systems. It should be mentioned that, according the authors’ experience, there are two different design approaches for plastic piping systems. First approach is based on the completely anchored pipe (i.e. placing fixed points all along the piping, thus effectively completely restraining pipe expansion). This approach, although seemingly simple, has two major problems: it is difficult to execute properly in site and it may result in high compressive stresses in the pipe itself. Second approach is to support the pipes as “normal” metallic piping system (i.e. providing fixed points, guides, axial stops, pipe loops, etc.). This approach is much simpler to execute and, if executed properly, will lead to a better, lighter and very predictable piping design. Major problem with the second approach is complete lack of design method and applicable equations. Aim of this paper is to provide the said method which would guarantee safe piping system design while limiting conservatism. Method described in this paper is limited to the plastic materials which are isotropic in nature. Materials which are not isotropic (e.g. plastic lined FRP pipes) should be calculated in accordance with dedicated standards, such as [1] or [2]. Analysis performed and method presented in this paper aim to set some basic requirements for stress analysis of thermoplastic piping systems in the absence or more applicable data. CALCULATION PROCEDURE Thermoplastic materials are viscoelastic, meaning that the properties, such as strength or deformation, are significantly affected by both time (i.e. load duration) and temperature. Stressstrain response of plastic materials is non-linear, as described in [3] and unlike metals they do not have elastic constants (elastic modulus or proportional limit) or clearly defined yield points. However, most equations which are developed for stress analysis of metallic piping, can still be used provided that correct values of strength and stiffness can be established. Discussion on allowable strength and applicable elastic moduli (albeit using the term elastic or modulus may be regarded as erroneous since thermoplastic materials do not have true elastic-plastic properties) is provided in dedicated sections in further text. For further understanding of viscoelastic properties, definition of various moduli (creep, stress-relaxation), stress-rupture behavior and other phenomena related to plastic materials, reader should refer to [3]. All further discussion shall be based on calculation methodology from [4]. Although there is a dedicated section in [4] discussing non-metallic piping, it does not provide detailed guidelines on the calculation principles of thermoplastic piping systems. Sustained stresses In order to estimate the effects of long-term sustained loads, first of all it is necessary to extrapolate the effective material properties for applicable load duration. Data in open literature are sufficient 2
to estimate allowable stress and moduli for long-term design life and will be discussed in further text in dedicated sections. In evaluating sustained loads, another major difference with respect to metallic piping, besides material properties, is the effect of the pressure on the plastic piping. Namely, in traditional metallic piping analysis pressure is regarded as primary load and is verified only by taking into account the axial and hoop stresses due to pressure. However, in thermoplastic piping systems pressure will produce also elongation and bending stresses which need to be taken into consideration. Thermal expansion stresses Typically, thermal expansion is regarded as secondary, or self-limiting stress, and is normally verified for low cycle fatigue. Considering that the thermoplastic materials do not have yield strength, a dedicated evaluation of fatigue in accordance with [4] requirements has little practical sense. Also, considering the viscoelastic material properties which are greatly dependent on time, the concept of self-limiting stresses is not really applicable, since there will always be some amount of progressive deformation. In order to evaluate the effect of temperature, its effect needs to be calculated together with sustained stresses and the maximum stress magnitude needs to be limited to a safe value. Operating stresses Based on the above, it can be concluded that using the conventional verification of sustained stresses and expansion stress range (i.e. separate verification of primary and secondary stresses) in case of plastic materials has no practical significance. Since no major piping codes which are based on beam theory and design by formulae approach (e.g. [4], [5], [6]) allow the possibility to estimate the joint effect of pressure and thermal expansion (i.e. primary and secondary stresses), it is necessary to define additional load scenarios in the pipe stress analysis software in order to verify these effects acting contemporarily. Typically, it can be suggested to calculate both stresses due to primary loads (i.e. pressure and weight) and stresses due to the joint effect of primary and secondary loads (i.e. pressure, weight and temperature) and compare both effects against a single allowable stress. In effect, it may be sufficient to calculate only operating stresses and compare them against allowable stress. Calculation of purely sustained stresses (i.e. due to pressure and weight), although useful, is not necessary for evaluating system acceptability. Another important issue with thermoplastic piping systems, which is often overlooked, is correct placing of supports. Problem arises when less experienced designers use rule-of-thumb analysis for metallic piping to place the supports in thermoplastic piping system. Using the same approach as for metallic piping (which is much stronger than plastic piping) will surely result in under-designed, or under-supported piping system. Therefore, it is always necessary to use specific codes and evaluate the used piping spans using realistic project data (such as [31]). Once preliminary 3
supports are placed throughout the system it is always recommended to perform piping stress analysis in order to determine realistic piping stresses and equipment/support loads. Recurring short-term loading Since allowable ultimate strength and ultimate strain of thermoplastic pipes greatly depends on load duration, these pipes can withstand (significantly) higher temporary loading than foreseen by using their long-term strength. If this short-term loading is infrequent, or is not recurring in such manner to generate fatigue damage to piping, then it may be sufficient to use the information on the allowable short-term overloading from various guidelines. For example, some information regarding the allowable short-term overpressure is published in [7], [8], [9], [10] or [11]. In cases in which expected short-term overloading is frequent, it may be necessary to perform analysis considering pipe fatigue resistance capabilities. In this case, actual S-N curves should be used. Some limited information on fatigue capability of PVC piping can be found in [12]. Considering that the available S-N data for thermoplastic piping materials is very limited, all subsequent discussion will be limited to the effects which do not cause fatigue failure. ALLOWABLE STRESS For plastic piping, allowable stress should be based in hydrostatic strength of a material for sufficiently long design life and for the maximum design temperature. Typical design life for thermoplastic piping systems is 10÷50 years, but other values can be used. Hydrostatic strength curves (such as the example shown in Figure 1) can be found in various standards, such as [13], [14] or [15]. Typical allowable stress, besides the effects of design life and maximum design temperature, should include the following effects: -
effect of the transported fluid on the material; joining method; minimum design temperature; safety factor.
Typical equation for establishing thermoplastic piping allowable strength, given in [16] is =
∙
∙ ∙
(1)
in which: -
, MPa, allowable stress; , MPa, hydrostatic strength at design life and temperature; , weld joint factor ( = 0.4 ÷ 0.8, depending on material and welding procedure); , safety factor (typically 1.25÷2); , reduction factor for transported fluid;
4
, reduction factor for specific strength.
-
Figure 1 Change of PP-H hydrostatic strength in time at design temperature of 60°C [14] Allowable stress, as defined above, can be used to verify the combined effect of pressure, weight and temperature (i.e. operating stress). Other possibility to determine the allowable stress is defined in [17], which states that the design stress at design life (, years) and design temperature (t, °C) shall be equal to
=
,
(2)
in which: -
!",# , MPa, categorized required strength (as defined in [17]); , design coefficient taking into account the effects of specific product requirements, time and temperature (with minimum value defined in in [17]).
ELASTIC MODULUS Using correct elastic moduli for analysis is of paramount importance when evaluating thermoplastic piping systems. In general, elastic modulus of thermoplastic material decreases with time, as shown in Figure 2. 5
Figure 2 Change of PP-H elastic modulus in time (extrapolated from [18]) Using short-term elastic modulus will typically result in higher equipment loads. On the other hand, using lower value of long-term elastic modulus (commonly denoted as creep modulus) under the same design loads will typically result in greater movement/sagging. Obviously, greatest elastic modulus will occur for design life of 0 years. However, using this value in the analysis may result in very conservative design, considering that the elastic modulus value will sharply drop within a very short timeframe and that the temperature difference in piping system requires time to result in changes in length. Therefore, it is suggested in [19] to use a value of elastic modulus after 100 minutes in order to determine anchor loads. This value was successfully used by the author in multiple projects and has proved to be a good trade-off between conservative design and limiting unnecessary system modifications. Therefore, it may be suggested to perform two sets of analyses: 1. using high value of short-elastic modulus (i.e. at design life at 100 minutes) for calculating equipment loads and stresses; 2. using low value of long-term elastic modulus (i.e. at long-term design life) for calculating stresses and displacements. It should be noted that the effects of any possible short-term loading, such as pressure surge or instantaneous short-term negative pressure in piping system, should be, as a minimum, calculated using short-term elastic modulus. This approach will generally result in higher support forces, while the calculated stresses may be underestimated with respect to the values obtained using long-term modulus. Designer should carefully evaluate if a dedicated load case for short-term loading should also be made using the long-term modulus in order to estimate the effect of the 6
modulus on the stress distribution under short-term loading. This approach, although generally recommended, may result in overly conservative results and unnecessary piping modifications. Calculation of the creep modulus used in the analysis should include factors for transported fluid and safety factor. Permissible creep modulus ($% , MPa) can be calculated using the following equation $% =
& ∙
(3)
in which E, MPa, denotes creep modulus from applicable diagram and for applicable design parameters. Based on all the above discussion, it will typically be necessary to define different load cases for pipe stress analysis. As a general recommendation, load cases shown in Table 1 should be established. Table 1 Recommended minimum load cases to be verified Acting loads Operating weight Design pressure Design temperature Operating weight Design pressure Operating weight Design pressure Operating weight Design pressure Design temperature Operating weight Design pressure Design temperature Water filled weight Hydrostatic test pressure
Applicable modulus
Verified variables
100 minute modulus
Equipment loads Support loads
Short-term modulus
Stress
Long-term modulus
Stress
Short-term modulus
Stress
Long-term modulus
Stress
Short-term modulus
Stress Equipment loads Support loads
STRESS INTENSIFICATION FACTORS Major issue with calculation of stresses of thermoplastic piping systems is the application of correct stress intensification factors, or SIFs. These factors account for the existence of discontinuity (e.g. pipe bend or tee) inside a piping system, and account for the increase of stresses due to its existence. Up to date, the only two sources for plastic pipe stress intensification factors (i.e. SIFs) are [20] and [21]. Both of these sources define constant SIF values regardless of the type or geometry of 7
discontinuity. While [20] is based on old manufacturer’s standards and defines a constant SIF value of 2.3, [21] simply defines all SIFs as unity. There are many obvious shortcomings of a constant SIF approach, such as complete independence on the type of discontinuity (elbow, tee, etc.), geometry (diameter, thickness, radii, etc.), etc. Other possibility is to use the existing equations for metallic piping. However, considering that the metallic piping has significantly lower thickness and different material properties than thermoplastic piping, application of these equations is questionable, to say the least. Therefore, a numerical study has been done in order to estimate the realistic SIF values for most common piping elements: bends, straight and reducing tees. Analysis of SIFs is based on numerical modeling (i.e. finite element analysis) of bends and tees. Analysis software is NozzlePRO (version 12) developed by PRG. Material properties used in the analysis are presented in Table 2. Table 2 Material properties used in the analysis PP-H
PE
11
10.4
Short term modulus, MPa
1200
800
Allowable stress, MPa
4.22
5.2
Density, kg/m3
910
960
Poisson's coefficient
0.38
0.45
Linear expansion coefficient, mm/(km∙°C)
150
180
Long term hydrostatic strength, MPa
Majority of the above data have been taken from various international standards ([13], [14], [16], [18] and [22]), while the values not defined in standards have been taken directly from the catalogues of renowned plastic pipe manufacturers. All of the above data (moduli of elasticity, hydrostatic strength, etc.) have been taken at the ambient temperature (20°C). However, multiple runs have been made with data taken at higher temperatures and design life (i.e. short and long term) and they resulted in virtually significant variation of results. Total number of analyzed models is: -
73 bends (37 in PP-H and 36 in PE100); 102 equal tees (37 in PP-H and 65 in PE100); 190 reducing tees (128 in PP-H and 62 in PE100).
8
The standard dimension ratios (SDR) considered were SDR11, SDR17, SDR 17.6 and SDR33 for all examined cases. These were selected based on the vendor catalogues and represent the values typically available on the market. It is important to note that the thickness of tees considered in the analysis equals the thickness of pipe. However, typically tees made of plastic materials have greater thickness than the straight pipe. Seen as this over-thickness depends on the manufacturer, it cannot be estimated with sufficient accuracy. In any case, the approach used in this paper will result in conservative results. Comparison is given in the later text with some of the sources for piping stress intensification factors ([4], [23] and [24]). It must be noted that all of the aforementioned references are valid for metallic piping. As expected, the major difference between thicknesses and material properties of metallic and plastic piping make the existing equations for metallic piping SIFs of little use. It should be mentioned that the SIFs are calculated as “peak” SIFs, which correspond to the values typically used in the piping codes. Since typical SIFs used in piping codes typically represent approximately one half of the actual stress intensification (due to the presence of weld joint, as described in [25]), and considering that the secondary stress evaluation typically used in piping stress analysis is invalid, it might seem reasonable to consider twice the calculated value for the analysis. However, considering the fact that the allowable stress, as defined in equation (1) above, already considers the effect of weld joint (f=0.4÷0.8), it is sufficient to consider the peak stress intensification factors for engineering practice. Aside from hoop stress calculation, weld quality is typically not considered for stress analysis of metallic piping (ref. [4], [5], [6]), and it is a known fact that the allowable expansion stress range in major piping codes (such as [4]) is reduced by an approximate factor of 2 (or more accurately 1.56, as shown in [25]) to take into consideration presence of butt-welds in piping system and the fact that the calculated stress intensification factor represent approximately one half of the real value. Although above mentioned factor f is essentially different from the weld stress intensification factor, it yields almost identical result, decreasing the allowable stress by a factor of 1.25÷2.5 (as defined in [16] f = 0.4÷0.8, depending on material and welding process). This approach can be further reinforced considering the statement from [26]: “These (plastic) materials remain ductile for a very long time and exhibit high resistance to slow crack growth, and thus are much less prone to failure from localized stress intensifications which can occur during normal operation of a plastic piping system”. All subsequent analysis and stress intensification factor comparison is based on the presumption that the finite element results represent realistic values. This approach is in line with the work done in the past in which experimental data had been used as a reference for any comparison ([23], [24]). Today we can treat finite element analysis as a very precise and widely accepted analysis method. Regression analysis Regression analysis of the finite element model results was performed using the least square analysis. In order to verify the results, the parameters shown in the Annex of this paper are used. 9
Both maximal positive error and maximal negative error have been taken into account in order to ensure that the proposed equations result in conservative design while limiting excessive overdesign of a piping system. Bends Regression analysis has been used to find equations which provide a good fit for all the analyzed cases. These equations are: 1. In plane stress intensification factor ('() ) '() =
*.+ ../ ,-
=
*.+
1∙2
../
0 4 5 3 ,-
(4)
2. Out plane stress intensification factor ('67# ) '67# =
*.89 ../ :;
=
*.89
../ 1∙2 < = 3:;
(5)
in which: -
r, mm, bend radius of curvature; T, mm, bend (pipe) wall thickness;
ℎ() , flexibility characteristic for in-plane moment (ℎ() = !() , mm, pipe internal radius (!() = (B − 2 ∙ E)/2);
?∙@
4 ,-
);
ℎ67# , flexibility characteristic for out-plane moment (ℎ67# = !67# , mm, pipe external radius (!67# = B/2); B, mm, pipe outside diameter.
?∙@ 4 ); :;
Nomenclature used in equations (4) and (5) is shown in Figure 3. Using existing equations published in piping codes for metallic piping ([4], [6] or [23] would result in overestimated SIFs by as much as 30%. As such, they are suitable for use, but may result in overly conservative design. Using a fixed stress intensification factor of 2.3, as suggested in [20], would result in an overestimate of in-plane SIFs by as much as 100% and of out-plane SIFs by as much as 150%. Finally, using the proposed equations still yields conservative results, but limits the maximum deviation to less than 10%. Dependence of in-plane and out-plane stress intensification factors of flexibility characteristic (i.e. on bend geometry) is shown in Figure 4.
10
Figure 3 – Nomenclature for equations (4) and (5)
Figure 4 – Plastic bend SIFs Proposed equations for tees (unreinforced straight and reducing) Below equations have been determined as the best fit for all examined tees. Due to the complexity of these equations and the dependence of stress intensification factors on multiple variables, it is impossible to show these results in graphic form. However, statistical parameters shown in Table 3 can be used to estimate the quality of the proposed equations.
11
For run pipe (main header) *.8K
'(? = 0.77 ∙ I J @
*.PK
'6? = 0.53 ∙ I J @
*.9M
'#? = 0.58 ∙ I@ J
For branch pipe *.TT
'(R = 0.6 ∙ I J @
*.KQ
∙I J L
*.Q8
∙ IL J
*.U8
∙I J L
*.9K
'6R = 0.69 ∙ I J @
*.MK
∙I J L
*.99
'#R = 0.57 ∙ I J @
M.UP
∙I J L
(6)
# *.KQ
∙I J @
(7)
# *.Q8
∙ I@ J
(8)
# *.U8
∙I J @
M.QQ
∙I J L
# *.MK
∙I J @
(9)
# WM.QQ
∙I J @
(10)
# M.UK
∙I J @
(11)
or alternatively (better fit) for in-plane and out-plane branch stress intensification factors *.K8
'(R = 1.6 + 0.3 ∙ ZI J @ *.Q
'6R = 6.61 ∙ ZI J @
8
# 8
*.K8
+ I J ∙ I J − 2.5[ + 0.11 ∙ ZI J L @ @ *.M
+ 0.5 ∙ I J L
# *.M
∙I J @
− 1.7[
8
# 8
+ I J ∙ I J − 2.5[ L @
U
(12) (13)
in which R, mm, denotes mean header radius, as shown in Figure 5, and can be calculated as !=
LW@ U
(14)
Figure 5 – Nomenclature for equations (6)÷(13) Equations (6)÷(13) are based on the same form as used in [23] and [24]. However, it has been found that a different form of equations can result in a much better for in-plane and out-plane SIFs 12
of branch pipe (equations (12) and (13)). Above equations yield statistical parameters as shown in Table 3. Table 3 Statistical parameters of equations (6)÷(13) Equation number
minRE-, % maxRE+, % meanRE, % SD, % CR
(4)
8.9
1.5
-2.9
3.8
0.9841
(5)
5.2
1.6
-1.9
2.9
0.9805
(6)
14.2
16.1
1.0
9.0
0.9462
(7)
2.8
8.6
1.6
2.0
0.9968
(8)
18.7
24.7
-0.9
11.0
0.9756
(9)
21.0
25.9
-0.3
10.8
0.9621
(10)
11.7
49.7
1.7
8.2
0.9793
(11)
7.5
28.4
2.6
7.5
0.9973
(12)
8.3
16.8
-0.4
4.0
0.9946
(13)
11.1
24.7
-0.6
7.3
0.9775
Comparison of relative errors (expressed in % using only equation for maximal positive error) between various sources are shown in Table 4. Table 4 Comparison of relative error ranges between various sources [4]
Fixed 2.3 [20]
[23]
[24]
7.3÷96.7
-36.8÷103.5
-0.5÷69.2
23.1÷85.5
137.4÷269.6
-22.3÷130.0
-51.8÷69.8
-44.8÷34.8
Torsion
-
-
-23.5÷35.9
-18.3÷43.6
In plane
-14.2÷51.2
-56.2÷56.5
-38.6÷-17.9
-28.3÷20.9
Out plane
-22.3÷45.3
-67.9÷27.1
-17.5÷23.3
-38.8÷92.2
-
-
-88.4÷10.1
-59.6÷6.2
In plane Run pipe
Branch pipe
Out plane
Torsion
Maximum negative deviation (underestimation of SIFs) for the proposed equations is ~20%, while other equations applied to metallic piping go up to ~90%. This means that using conventional equations may result in a largely underestimated stress. 13
On the other hand, the maximum positive deviation (overestimate of SIFs) for the proposed equations is ~50%. Again, using conventional equations typically used for metallic piping would result in an overestimated stress by as much as 270%. This means that using conventional equations may result in a very conservative system resulting in unnecessary modifications and complex routing. If fixed SIF value of 2.3 is used, the relative error compared to FEA solutions ranges from -70% (underestimate) up to +130% (overestimate). It is clear that using a fix value cannot cover all the variables connected to the geometrical characteristics of a discontinuity. Therefore, using this approach can only be regarded as a last resort if absolutely no other data is available. Flexibility factors Flexibility factors for all plastic piping fittings can be taken as 1. This was confirmed by the results of finite element analysis, which has revealed that the flexibility factors were lower than 1 for almost all cases. There were only a handful of cases which have had flexibility factors slightly greater than 1. However, considering that the flexibility factor depends on fitting thickness (i.e. greater the thickness, lower the flexibility factor) and the fact that all plastic fittings typically have greater thickness than straight pipe (which was the presumption used in the finite element analysis performed in the paper), using flexibility factors of 1 for all cases should be sufficiently conservative. This is also in line with the recommendation indicated in [20]. CONCLUSION Due to their many advantages, such as chemical resistance properties, lower density (i.e. weight), negligible costs of external protection (painting, galvanizing, etc.) and high deformation capacity, plastic pipe market is ever increasing in the industrial applications. In fact, as indicated in [3], even during the 1990s, the amount of meters of thermoplastic piping exceeded the amount of all other piping materials combined. Considering the continuing growth of plastic pipe market ([27] and [28]) it can only be concluded that plastic materials will continue to develop and gain ever more ground in general industry applications. However, major problem with thermoplastic piping is the relatively low strength capacity and lack of applicable standards which can be used in stress analysis and design. Method and equation presented in this paper are developed with the aim to establish some basic rules and principles for pipe stress and flexibility analysis of thermoplastic piping systems. Although there are international guidelines on establishing pressure ratings, allowable stresses and moduli for the analysis, literature review has revealed complete lack of stress intensification factors, which present an important parameter for piping system design. Consequently, finite element analysis has been used (much like in [23] and [24]) in order to establish stress intensification factors which can be applied to all plastic piping systems. Analysis has revealed (based on comparison between polyethylene and polypropylene materials) that the differences in material 14
properties between various thermoplastic materials are negligible and that single set of equations can be used regardless of the material. Comparison between the finite element analysis results has shown that using the existing references may yield an underestimated stress by as much as 90% and thus result in an unsafe design. On the other hand, the maximum overestimate of stresses using the same references may reach 270%, resulting in an overly conservative design. Considering that the industry trend in to optimize piping diameters in order to reduce the overall costs ([29] and [30]), and the effect that stress intensification factors may have on pipe wall thickness, it is only logical to try and design the piping without excessive conservatism, while remaining in the safe zone. Using the proposed equations for stress intensification factors the deviation from finite element analysis results ranges from -20% (underestimate) to +50% (overestimate), while keeping the mean error typically below 10%. Therefore, the possibility of under-designing a thermoplastic piping system is minimized, while keeping the level of conservatism within a reasonable margin. LITERATURE [1]
International Standard ISO 14692-3, Petroleum and Natural Gas Industries –Glass-reinforced Plastics (GRP) Piping – Part3: System Design, 2nd Edition, ISO, Switzerland, 2017.
[2]
British Standard BS 7159, Code of Practice for Design and Construction of Glass Reinforced Plastics (GRP) Piping Systems for Individual Plants or Sites, BSI, UK, 1989.
[3]
McGrath, T. J., Mruk, S. A., Chapter D1 – Thermoplastics Piping, in Nayyar, M. L., Piping Handbook, 7th Edition, McGraw-Hill, USA, 2000.
[4]
ASME B31.3, Process Piping, American Society of Mechanical Engineers, New York, USA
[5]
ASME B31.1, Power Piping, American Society of Mechanical Engineers, New York, USA
[6]
EN 13480-3 Metallic Industrial Piping – Part 3: Design and Calculations, European Committee for Standardization (CEN), Belgium, 2017.
[7]
ANSI/AWWA Standard C900-07, Polyvinyl Chloride (PVC) Pressure Pipe and Fabricated Fittings 4 In. Through 12 In. (100 mm Through 300 mm), For Water Transmission and Distribution, American Water Works Association, USA, 2007.
[8]
ANSI/AWWA Standard C901-08, Polyethylene (PE) Pressure Pipe and Tubing, ½ In. (13 mm) Through 3 In. (76 mm), for Water Service, American Water Works Association, USA, 2008.
[9]
ANSI/AWWA Standard C905-10, Polyvinyl Chloride (PVC) Pressure Pipe and Fabricated Fittings 14 In. Through 48 In. (350 mm Through 1,200 mm), American Water Works Association, USA, 2010.
15
[10]
ANSI/AWWA Standard C906-07, Polyethylene (PE) Pressure Pipe and Tubing, 4 In. (100 mm) Through 63 In. (1,600 mm), for Water Distribution and Transmission, American Water Works Association, USA, 2007.
[11]
AWWA Manual M23, PVC Pipe – Design and Installation, 2nd Edition, American Water Works Association, USA, 2002.
[12]
Technical Report, Fatigue of Plastic Water Pipe: A Technical Review with Recommendations for PE4710 Pipe Design Fatigue, Jana Laboratories Inc., USA, 2012.
[13]
Deutsche Norm DIN 8075, Polyethylene (PE) Pipes – General Quality Requirements and Testing, DIN, Germany, 1999.
[14]
Deutsche Norm DIN 8078, Polypropylene (PP) Pipes – PP-H, PP-B, PP-R, PP-RCT – General Quality Requirements and Testing, DIN, Germany, 2008.
[15]
International Standard ISO 15494, Plastics Piping Systems for Industrial Applications – Polybutene (PB), Polyethylene (PE) and Polypropylene (PP) – Specifications for Components and the System – Metric Series, 1st Edition (corrected version), ISO, Switzerland, 2004.
[16]
Directive DVS 2205-1, Design Calculations for Containers and Apparatus Made From Thermoplastics; Characteristic Values, DVS-Verlag GmbH, Germany, 2002.
[17]
International Standard ISO 12162, Thermoplastics Materials for Pipes and Fittings for Pressure Applications – Classification, Designation and Design Coefficient, 2nd Edition, ISO, Switzerland, 2009.
[18]
Directive DVS 2205-2, Calculation of Thermoplastic Tanks and Apparatus – Vertical Cylindrical Non-Pressurized Tanks, DVS-Verlag GmbH, Germany, 1997.
[19]
SIMONA Technical Manual, Engineering Manual for Piping Systems, SIMONA AG, Germany, 2010.
[20]
Plastic Pipe Modeling, COADE Mechanical Engineering News, COADE, USA, April 1991.
[21]
GOST Standard 32388:2013, Process Piping - Standard for the Stress, Vibration and Seismic Analysis, Interstate Council for Standardization, Metrology and Certification (ISC), Russia, 2014.
[22]
Deutsche Norm DIN 8077, Polypropylene (PP) Pipes – PP-H, PP-B, PP-R, PP-RCT – Dimensions, DIN, Germany, 2008.
[23]
ASME B31J, Stress Intensification Factors (i-Factors), Flexibility Factors (k-Factors), and Their Determination for Metallic Piping Components, American Society of Mechanical Engineers, New York, USA
16
[24]
Jacimovic, N., Analysis of Piping Stress Intensification Factors Based on Numerical Models, International Journal of Pressure Vessels and Piping, Vol. 163, 2018
[25]
Jacimovic, N., Uncertanties in Expansion Stress Evaluation Criteria in Piping Codes, International Journal of Pressure Vessels and Piping, Vol. 169, 2019
[26]
Boros, S., Long-Term Hydrostatic Strength and Design of Thermoplastic Piping Compounds, Journal of ASTM International, Vol.8, No. 9, 2011.
[27]
Plastic Growth to Exceed Other Materials in Pipe Market, Plastic News, 2012.
[28]
Plastic pipe market, Global demand to reach 11.2 bn metres by 2017 / 8.5% annual growth / Increase in construction spending plays key role / Freedonia study, https://www.plasteurope.com/news/PLASTIC_PIPE_MARKET_t225210/, 2013.
[29]
Genic S., Jaćimovic B., A Shortcut to Optimize Pipe Diameters by Economic Criteria, Chemical Engineering, Vol. 116 (12), pp 50-53, 2018.
[30]
Genic, S., Jacimovic, B., Shortcut Optimization of Steam Pipeline Diameter for Complete Turbulence, Chemical Engineering Progress, Vol. 114 (8), 2018.
[31]
Directive DVS 2210-1, Industrial Pipelines Made of Thermoplastics –Planning and Execution Above-Ground Piping Systems, DVS-Verlag GmbH, Germany, 1997.
Appendix - Correlation parameters The comparison of experimental (ym,i) and correlated (yc,i) parameter can be done by certain number of statistical parameters: 1. standard deviation (SD), which is defined as B =
\
∑`(aM <
^_,( − ^%,( U ^_,( = b
2. mean relative error (meanRE), which is defined as
` ^ 1 _,( − ^%,( ∙g b ^_,( (aM 3. maximal positive error (maxRE+), which is defined as ^_,( − ^%,( ceh!$ i = max 0 5 ^_,(
cdef!$ =
4. maximal negative error (maxRE-), which is defined as 17
^%,( − ^_,( ceh!$ W = max 0 5 ^_,( 5. Correlation ratio (CR), which is defined as ! = n1 −
∑`(aM(^_,( − ^%,( )U ∑`(aM(^_,( − ^_,o )U
in which ^_,o is the average value ^_ of for the complete set of z experimental data.
18