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Ameron Calculation Manual for Bondstrand® GRE Pipe Systems

INDEX 1.

Introduction

2.

Pipewall Thickness based on internal pressure

2.1

Wall thickness

2.2

Diameter

2.3

Dimensional pipe properties

3.

Trust force due to temperature and variation in length

3.1

Length Change

3.2

Thrust

4.

Support and guide spacing

5.

Pipe bending radius

6.

Collapse resistance for liquid

7.

Pipe-ring stiffness

8.

Waterhammer and surge

9.

Headloss or pressure drop for liquid flow

10.

Literature

11.

Legenda

Fiberglass-Composite Pipe Group division Europe P.O. Box 6 - 4191 CA Geldermalsen - Holland tel. +31 345 587 587 - fax +31 345 587 561 email: [email protected]

Calculation Manual for Bondstrand® GRE Pipe Systems

1. Introduction

In this Technical Bulletin an overview is given of commonly used formulas in relation with Glassfibre Reinforced Epoxy piping.

2. Pipe wall thickness

The minimum required wallthickness of the pipe is based on design codes as ASME and ANSI. To most products an inferior liner is added, consisting of C-veil and resin.

3. Trust forces due to temperature, pressure and variation in length

On many occasions the pipe is fabricated to pressure as well as a varying temperature of the medium. Pressure variation will cause a length change if the product is unrestrained and due to the Poisson effect an increase in pressure will shorten the pipe. This is alos mathematically explained. Expension and contraction due to temperature variations and internal pressure will either combined or individual result in thrust forces on the anchoring points

4. Support and Guide spacing

The formulas for the calculation of the optimal distance between two supports or guide spacings for single, partial and continuous spans are given. The calculations take into account density of the liquid and the weight of the pipe.

5. Bending radius

A slight gradual change in direction or deviation of the pipe may be obtained by using the flexibility of the pipe. In that case the allowable bending radius of the glass reinforced epoxy pipe can be calculated

6. Collapse resistance for liquid

When the external pressure on the pipe may exceed the internal pressure one has to take into account the collapse resistance of the pipe. This is ruled by equations which differs from those for internal pressure.

7. Pipe-ring stiffness

To make calculations for earth and wheel-loads on buried pipe, values have to be used like STIS (= Specific Tangential Initial Stiffness), STES (= Specific Tangential End Stiffness) and other values, as used in the U.S.A., Stiffness Factor and Pipe Stiffness.

8. Waterhammer and surge

Changes in velocity of fluids cause changes in pressure. Especially when these velocity changes are sudden, they can result in high forces, which may harm the piping system

9. Head loss or pressure drop for liquid flow

Head loss or pressure drop can be calculated by using the Hazen-Williams equation for water and the Darcy-Weisbach for laminar flows, e.g. for oil. Head loss in fittings are calculated by defining a corresponding pipe length.

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Calculation Manual for Bondstrand® GRE Pipe Systems

1. INTRODUCTION This manual presents the calculations, used by Ameron to calculate the various aspects related to glass reinforced epoxy (= GRE) pipe. This will help the reader to understand the equations which govern certain common engineering cases of GRE pipesystems. Also these equations can be used to make the required calculations. When making these calculations the input data should be based on the physical mechanical properties, diameter and wallthickness of Ameron products by: The spreadsheet presented by Ameron in its documentation gives these values. 2. PIPEWALL THICKNESS BASED ON INTERNAL PRESSURE 2.1 Wall Thickness

The minimum pipewall thickness is calculated with the formula according to ASME / ANSI B31.3 [1] (Paragraph A304.1.2):

ts =

Dp 2sF + p

(1)

ASTM D-2992 [2] uses the same type of formula to calculate the hoop stress as follows:

σ=p

(D a - t s ) 2t s

(2)

The above mentioned formula has been rearranged to induce the internal liner and is used by Ameron to calculate the minimum reinforced wall thickness of Bondstrand pipe as follows: Minimum reinforced wall thickness in [m]:

ts =

p(d + 2t l ) 2σst s ⇔p= 2σs − p d + t s + 2t l

(3)

Minimum total wall thickness in [m]:

t = ts + tl + ta 2.2 Diameter

(4)

Minimum outside diameter of pipe in [m]:

D = d + 2t

(5)

Mean pipe wall diameter in [m]:

Dm = d + t

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Calculation Manual for Bondstrand® GRE Pipe Systems

2.3 Dimensional pipe properties

2

Cross section area of minimum pipe wall in [m ] :

Aw = π (d + t)t

(7) 2

Area of pipe bore in [m ]:

A b = 0,25πd 2

(8)

2

Cross section area of min. structural wall in [m ]:

(

)

A s = π ( d + 2t l ) + t s t s

(9)

2

Cross section area of inner liner in [m ]:

Al = π (d + tl )tl,

(10) 2

Weight of pipe per unit length in [kg/m ]:

w p = A s ρs + A l ρl

(11) 1

Weight of fluid per unit pipe length in [kg/m ]:

w f = 0,25πd 2 ρ f

(12) 4

Linear moment of inertia of the pipe [3] in [m ]:

I l = I s + I lin

(13) 4

Linear moment of inertia of the structural wall in [m ]:

Is =

(

π (d + 2t l + 2t s )4 − (d + 2t l )4 64

)

(14)

4

Linear moment of inertia of the inner liner in [m ]:

I lin =

(

π (d + 2t l )4 − d 64

)

4

(15)

Note! In case of calculating with the moment of inertia of the total wall thickness and the elasticity modulus of structural wall, the moment of inertia may be multiplied by 0,25, which is the approximate ratio between the modulus of elasticity of the liner and the structural wall. The stiffness factor IE = Is Es + Ilin El = Is Es + Ilin Es 0,25 so I = Is + Ilin 0,25. 3. TRUST FORCE DUE TO TEMPERATURE AND VARIATION IN LENGTH 3.1 Length change

Like in other types of pipe material, in unrestrained condition, Bondstrand fiberglass reinforced pipe changes its length with temperature change. Tests have shown that the amount of expansion varies linearly with temperature, in other words the coefficient of thermal expansion of Bondstrand pipe is constant [4, 5, 6].

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Calculation Manual for Bondstrand® GRE Pipe Systems

Change in length due to thermal expansion in [m]:

∆LT = λL∆T

(16)

Subjected to an internal pressure, a free Bondstrand pipeline will expand its length due to thrust force at the ends of the pipeline. The amount of change in the pipeline is a function of pressure, pipe wall thickness, diameter, Poisson's ratio and the effective moduli of elasticity in both, axial and circumferential direction at the operating temperature. Change in length due to pressure in [m]:

pd 2 pd 2 E l pd 2 ∆Lp = L − µc = L (1 − 2 µc ) (17) E c 2tD m E c 4tD m E l 4tD m E l The total length change is the sum of the change due to temperature and due to pressure. The above shown equation for length change due to pressure, compared to the general equation:

π 2 p 4 d pd 2 Pbf (18) = L ∆L = L = L 4tD m E l A w El πtD m E l shows that, the length increase due to the bulkhead force is considerably reduced by the Poisson’s effect. The reduction may amount to 50%, subject to the value 2µcEl / Ec , e.g. for Series 2000: 2 x 0,56 x 11000 / 25200 = 0,49 (at 21°C). 3.2 Thrust

Thrust due to temperature is principally independent of pipe length. In practice, the largest compressive thrust is normally developed on the first positive temperature cycle. Subsequently the pipe develops both, compressive and tensile loads as it is subjected to temperature cycles. Neither, compressive nor tensile loads, are expected to exceed the thrust on the first cycle, irregardless the range of the temperature changes. In a fully restrained and blocked or anchored Bondstrand pipe, length changes induced by temperature change are resisted by the anchors and converted to thrust [4, 5, 6]. Thrust due to temperature in [N]:

FT = λ∆TA w E l = λ∆T(πD m t ) E l

(19)

The theory of thrust due to internal pressure in a restrained pipeline is rather complicated. This is because in straight, restrained pipelines with rigid joints, the Poisson's effect produces considerable tension in the pipe wall. As internal pressure is applied, the pipe expands circumferentially and at the same time tries to contract

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Calculation Manual for Bondstrand® GRE Pipe Systems

longitudinally. This creates a considerable tensile force in the pipe wall, and acts to reduce the hydrostatic thrust on the anchors. In pipelines with elbows, closed valves, reducers or closed ends, the internal pressure works on the cross sectional area of the ends. This thrust may be twice the effect of pressure on the pipe wall. The thrust is independent of the run length or support spacing. Thrust due to pressure in [N]:

E Fp = pA b 1 − 2 µc l Ec

(20)

The concurrent effects of pressure and temperature must be combined for the design of anchors. Similarly, on multiple pipe runs, thrusts developed in all runs must be added for the total effect on the anchors. Thrust due to temperature and pressure in [N]:

FTp = FT + Fp

(21)

Resulting force due to thrust from two pipelines meeting at an elbow or turn in the pipeline in [N]:

δ Fe = 2sin F 2

(22)

Force at a reduction in a straight run by the larger diameter in [N]:

E Fr = λ∆TE l ( A bl − A bs ) + p( A wl − A ws )1 − 2 µ c l (23) Ec In a blocked or anchored pipe system the Poisson's effect causes tension in the pipe wall which counteracts the pipe thrust due to temperature. The tension in the pipe wall may be positive or negative, subject to the direction of the temperature / pressure change. Thrust or tension in pipewall in a restrained blocked or anchored pipeline due to temperature change and pressure in [N]:

πpd 2 E l Fw = λ∆TA w E l − µ c 2E c

(24)

Fp Fp ∆L σ =ε = = = L E l A w E l πD m E l t

(25)

Equation 20 is valid, and

∆L pd 2 El = 1 − 2 µc L 4tD m E l Ec

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Calculation Manual for Bondstrand® GRE Pipe Systems

As can be seen substituting and re-arranging gives equation 20.

The product from equation 20

( pA )−2 µ b

c

El pd 2 E l ⇔ ( − µ c ) Ec 2 Ec

(27)

gives a negative sign, which is tension in the pipe wall caused by the Poisson’s effect, which counteracts the bulkhead force exercised on the blocking or anchoring. 4. SUPPORT AND GUIDE SPACING The distance between two succeeding supports, depends on the parameters as load, moment of inertia and elasticity, as well as the layout of the system. Local loads, such as heavy fittings heavy flange arrangements, valves vertical runs etc. as well as changes in horizontal directions, may also affect the support distances. A long term deflection of 0,0127 m, is normally acceptable for appearance and sufficient for drainage. Distance between supports for partial run [3, 7] in [m]:

E I L p = 1,24 b l w

0,25

(28)

(Note! For imperial input resulting in inches this equation can be used if 1,24 is replaced by 0,258). For continuous span Lp may be increased by 20%, for single spans Lp should be decreased by 20%, giving the same results as when using following equations: Distance between supports for continuous run in [m]:

E I L c = 1,486 b l w

0,25

(29)

Distance between supports for single run in [m]:

E I L s = 0,994 b l w

0,25

(30)

Note! For imperial input resulting in inches these equations can be used if 1,486 is replaced by 0,31 and 0,994 by 0,207. Explanation of partial span The general equations to calculate the maximum deflection of a tubular body are for continuous span

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Calculation Manual for Bondstrand® GRE Pipe Systems

fw =

5wL4 384EI

(31)

and for single span

wL4 fw = 384EI

(32)

Taking 0,0127 m as deflection “fw”, these equations may be rearranged to calculate the span as equation 29 and equation 30. The partial span equation 28 is the average between the equations 29 and 30, giving the same result as equation 29 if increased by 20% or equation 30 if decreased by 20%. If anchors are used at both ends of a pipeline, to restrict axial movements, until a method of controls must be designed in order to prevent excessive lateral deflection or buckling of pipe due to compressive load. Guides may be required in conjunction with expansion joints or expansion loops, to control excessive deflection. The guide spacing can be calculated by using the rearranged Euler equation multiplied by 75%. Distance between guides in [m]: 0,5

ElIl Lg = 0,75π El λ∆TA w E l + pA b 1 - 2 µ c Ec

(33)

The above equation solves for the maximum stable length of a pipe column when fixed ends are assumed, which is reduced by 25% to develop the original portion of curves, now seen only in the smaller diameters, and to allow for non-Euler behaviour near the origin of the curve. Not only to resist buckling of the pipe as a column or "snaking" but to also adjust guide spacing to prevent excessive vertical deflections due to weight, the length calculated by the Euler equation, should be checked by using it in the following equation of [4]. Vertical deflection in [m]:

− wLg kLg kLg y= − tan 2kPw 4 4

0,5

P where k = w E bIl

(34)

If "y" is less than -0,0127 m, the guide distance, Lg obtained from the Euler equation is the recommended guide spacing. If "y" is greater than -0,0127m, a shorter length Lg should be chosen and used in the Roark equation until by trial and error a final length, Lg, is determined that closely approximates a "y" of -0,0127 m. Bending moments in the pipe due to deflection or buckling (using [4]) in

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Calculation Manual for Bondstrand® GRE Pipe Systems

[Nm]:

kLg −w 2 − M = 2 1 − kLg k tan 2

and

kL g (35 −w 2 +M= 2 − 1 k kLg tan 2

5. PIPE BENDING RADIUS Knowledge about the bending radius is required for buried pipe systems, in order to know if the pipeline can follow an existing or projected curved pipe track. The minimum allowable bending radius depends on temperature and pressure. Minimum allowable bending radius [7] in [m]:

RB =

0,5t E b D σl − σp

(36)

Actual axial stress due to internal pressure for BI-AXIAL loaded 2 systems in [N/mm ]:

p d σp = + 1 4 ts

(37)

Actual axial stress due to internal pressure for UNI-AXIAL 2 loaded systems in [N/mm ]:

p d σp = + 1 8 ts

(38)

Notes: 1. For allowable axial tensile stress 50 % is used of the axial bending strength shown in the pipe data sheets. Since Bondstrand pipe and joints can be loaded bi-axially, consequently most are used in that way. The minimum bending radii shown in the pipe data sheets are based on bi-axial loading for that reason. 6. COLLAPSE RESISTANCE FOR LIQUID Where pipes may be exposed to external pressure, such as in tanks, buoyant systems, divers etc., the resistance against collapse may become determining. 2

Minimum Ultimate collapse pressure [3] in [Pa= N/m ] if pipe is sufficiently long: 1

p c (1 − µ c µ l ) 3 2E c t s pc = ⇔ t = d s 2E c (1 − µ c µ l )d 3 3

(39)

Note! To give sufficient resistance against an external pressure of 1 bar, 0,75 is a well accepted factor. For pipes used in marine environments, such as at bottoms of sea going vessels a factor of 0,3 is used to resist 3 bar or 30 m water column with a sufficient safety.

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Calculation Manual for Bondstrand® GRE Pipe Systems

The effective circumferential moduli of elasticity for external pressure loading, have been determined by a combination of theoretical and empirical data. Temperature Cº 2 ElasticityMN/m Ec

Modulus 2400 based pipe 2 ElasticityMN/m Ec Modulus 3400 based pipe

T

21

66

25200

22100

25300

22000

93

121

7. PIPE - RING STIFFNESS Stiffness data are used in calculations of earth- and wheelloads on buried pipe [8]. STIS and STES according NEN 7037. 2

Specific Tangential Initial Stiffness, STIS [9], in [N/m ]: 3

EI E t STIS = c w3 = c Dm 12 D m

(40) 2

Specific Tangential End Stiffness, STES, in [N/m ]:

STIS = αβ STES

(41)

Pipe Stiffness (acc. ASTM-D2412 test [10]), PS, in [psi]:

PS =

Fpr

(42)

∆y 2

Stiffness Factor (acc.ASTM-D2412 [10]), SF, in [inch .lb/inch]: 3

SF = 0,149rm 3 PS ⇔ E c I w = E c

t 12

(43)

2

Relation between STIS [N/m ] and SF [inch.lb]:

SF = 8,848D m 3 STIS ⇔ STIS = 0,113

SF Dm3

(44)

2

Relation between STIS (N/m ) and PS [psi]:

PS = 475,14STIS ⇔ STIS = 0,002105PS

(45)

8. WATERHAMMER AND SURGE Changes in velocity of fluids cause changes in pressure. Especially when these velocity changes are sudden they can be harmful to the piping system. Velocity changes may be caused by movement of valves, starting and stopping of pumps, closure of check valves, or even pipe rupture elsewhere in the system, and are calculated using the Joukowski equation [11].

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Calculation Manual for Bondstrand® GRE Pipe Systems

Pressure change in meter waterhead:

∆p =

c ∆v g

(46)

The pressure wave velocity in a closed pipe system depends on fluid characteristics, pipe dimensions and the elasticity modulus of the pipe wall. Calculating the pressure wave velocity “c” can be done using the Talbot equation. Velocity of pressure wave in a closed pipe conduit in [m/s]:

c=

1 1 d ρ f + K E t c s

0,5

(47)

The pressure change ∆H, added to the highest occurring working pressure in the system should not be higher than 1,5 times the maximum system design pressure. If a valve is closed within the time of one wave cycle, i.e. from the closed valve to the other end and back, then water hammer should be calculated on the basis of instant valve closure. Time of one pressure wave in [seconds]:

tw =

2Lw c

(48)

As can be seen, increase of tw will decrease c and decrease ∆p subsequently. So the longer the wave cycle, the smaller the pressure shock. Delayed closure time: The hammer pressure rise pv caused by taken into account a valve closure of Tv seconds can be calculated as follows

Lw Tv

(49)

p tot = p v + p

(50)

p v = 2∆p

9. HEADLOSS OR PRESSURE DROP FOR LIQUID FLOW Head loss for liquid flow often can be obtained out of charts and tables. However also there are rather simple ways to calculate the head loss. A very simple method to calculate the head loss was developed by Hazen and Williams. This method may be used for water in a temperature range of 0°C to 37°C (imperial: 31°F to 100°F). Head loss for liquid flow in m of water column / 100 m pipe length (imperial: ft of water/100 ft) using the HAZEN-WILLIAMS equation with a Hazen Williams factor C = 150 for Bondstrand

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Calculation Manual for Bondstrand® GRE Pipe Systems

pipe: For S.I. input:

H l = 0,1007

Q1,852 and for Imperial input: d 4,865

H l = 0,0983

Q1,852 d 4,865

(51)

Another way of calculating this head loss is using the Darcy-Weisbach equation. This method is more complicated than the Hazen-Williams method, but has the advantage that it can also be used for other temperatures and other liquids. The Darcy Friction Factor is variable, subject to the Reynolds Figure, this complicated the use of the method. Head loss for liquid flow, in m of water column / 100 m pipe length (imperial: ft of water/100 ft), using the DARCY-WEISBACH equation:

Hl = f

Lv 2 2dg

(52)

Darcy Friction Factor for laminar flow(Re 4000): 1 6 3 e 10 f = 0,0055 1 + 20000 + d Re

(54)

Reynold's number:

Re =

vd vdρf , and = ν η

(55)

Velocity of liquid flow: Q Q v = = Ab 0 ,2 5π d 2 Temperature Density

Cº/ºF T 3 kg/m r lb/ft -6 2 Absolute 10 N s/m h -6 2 Viscosity10 pdl s/ft 1205 -9

2

Kinematic 10 m /s 1011 -9 2 Viscosity10 ft /s 1931 Temperature Density

Cº/ºF T 3 kg/m r lb/ft -6 2 Absolute 10 N s/m h -6 2 Viscosity10 pdl s/ft 538 -9

2

Kinematic 10 m /s 296 -9 2 Viscosity10 ft /s 865

Technical Bulletin 3 July 1997

0/32 999,87 62,42 1794 1053

4/39.1 1000 62,43 1568 880

ν=η/ρf 1794

10/50 999,72 62,41 1310 678

20/68 998,2 62,16 1009

1568

1310

1687

1410

1088

30/86 995,7 62,16 800 439

40/104 992,2 61,94 653 316

60/140 983,2 61,38 470 191

100/212 958,4 59,83 284

658

478

ν=η/ρf 803 709

514

319

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Calculation Manual for Bondstrand® GRE Pipe Systems

The head loss in joints and fittings can be calculated using the same method as used for pipe after defining the equivalent length. The equivalent length LE can be obtained using the chart for equivalent length of Bondstrand fittings. This chart was developed for fittings with a resistance coefficient "K" of 1 and water as fluid. Subject to the configuration of the fitting, the resistance coefficient "K" varies and can be obtained from the table "Resistance Coefficients for fittings". Multiplying Le by K gives the real equivalent pipe length of the fitting. Real equivalent pipe length of fitting with water as fluid in [m]: (57) L E = K r Le Real equivalent pipe length of fitting with other fluids in [m]:

LE =

Kr d f

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Calculation Manual for Bondstrand® GRE Pipe Systems

10. LITERATURE

. [1]

ASME B31.3 / ANSI, an american national standard ASME code for pressure piping, B31.

[2] ASTM-D-2992, Standard practice for obtaining hydrostatic or pressure design basis for “fiberglass” (glass-fiber-reinforced thermosetting-resin) pipe and fittings. [3] Roark R.J., Formulas for Stress and Strain, Singapore, 1976 [4] Hoa S.V., Analysis for Design of Fiber Reinforced Plastic Vessels and Pipings, Lancaster, Pennsylvania, 1991 [5] Heißler H., Verstärkte Kunststoffe in der Luft- und Raumfarhttechniek. Eyerer von P., Kunststoffe und Elastomere in der Praxis, Kohlhammer, Stuttgart, December 1996. [6] Vinson J.R., Sierakowski R.L., the behaviour of structures composed of composite materials, 1987. [7] Mönch E, Einführungsvorlesung Technische Mechanik, Wien, 1973. [8] Algra E.A.H., Mechanische Aspekte bei drucklos betriebenen, erdverlegten GFK-Rohren, Delft. [9] NEN 7037, Glass reinforced thermosetting plastics pipes for drain and sewer- requirements and test methods. [10] ASTM-D-2412, Standard test method for determination of external loading characteristics of plastic pipe by parallel-plate loading, 1993. [11] Tyler G. Hicks P.E., Hicks D.S., Standard handbook of engineering calculations, United States, 1972 11. LEGENDA α : creeping factor for pipe material β

: altering factor of pipe material

ε

: strain rate 2

3

η : dynamic viscosity Ns/m = 10 centipoise λ

: coefficient of thermal expansion in axial direction in m/m/ºK [in/in/ºF]

∆L : sum of the change in length due to temperature and due to pressure ∆LT: change in length due to thermal expansion [m] ∆Lp : change in length due to pressure ∆p : pressure change µ c : poisson's ratio (contraction in longitudinal direction due to strain in hoop direction)

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Calculation Manual for Bondstrand® GRE Pipe Systems

µl : poisson's ratio for (contraction in hoop direction due to strain in longitudinal direction) ν

2

6

: kinematic viscosity m /s = 10 centistoke 3

3

ρf : density of fluid material in kg/m [lb/inch ] 3

3

ρl : density of the liner material in kg/m [lb/inch ] 3

3

ρs : density of the structural pipe wall in kg/m [lb/inch ] 2

σ : hydrostatic design bases (H.D.B.) in N/m = Pa [psi] σl : allowable axial tensile stress in N/m

2

σp : actual axial stress due to internal pressure in N/m

2

∆T : change in temperature in ºK or ºC; ( ºF) 2

2

A b : cross sectional area of pipe bore m [inch ] 2 2 Abl : cross sectional area of larger pipe bore in m [inch ] 2 2 Abs : cross sectional area of smaller pipe bore in m [inch ] 2 2 Al : cross sectional area of inner liner in m [inch ] 2 2 As : cross sectional area of minimal structural wall m [inch ] : cross sectional area of pipewall in m [inch] Aw 2 2 Awl : cross sectional wall area of larger pipe in m [inch ] 2 2 Aws : cross sectional wall area of smaller pipe in m [inch ] c

: velocity of pressure wave in the pipe conduit in m/s [in/s]

D d Da Dm

: minimum outside diameter of pipe in m : inside diameter of pipe in m : average reinforced outside diameter in m : mean diameter of pipewall in m [inch]

e : absolute roughness of internal pipe wall in m; [ft or inch]. 2 Eb : elastic beam modulus acc. ASTM D-2925 in N/m =Pa; (psi) 2 Ec : circumferential modulus of elasticity in N/m = Pa [psi] 2 El : longitudinal modulus of elasticity in N/m = Pa [psi] 2 EI : stiffness factor per unit length of pipe wall in inch -lbs/inch f fw F Fe

: Darcy Friction Factor - dimensionless : deflection of a tubular body : service (design) factor : resulting force due to thrust from two pipelines meeting at an elbow or turn in the pipeline in [N] Fp : thrust due to pressure 3 Fpr : load applied pipe ring inch.(lbf/m ) Fr : force at a reduction in a straight run by the larger diameter in [N] FT : thrust due to temperature FTp : thrust due to temperature and pressure Fw : thrust or tension in pipewall in a restrained blocked or anchored pipeline due to temperature change and pressure g Hl Il

2

2

: acceleration by gravity in m/s [in/s ] : head loss 4

4

: linear moment of inertia of pipe in m ; [inch ]

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Calculation Manual for Bondstrand® GRE Pipe Systems 4

4

Ilin : linear moment of inertia of the inner liner in m ; [inch ] 4 4 Is : linear moment of inertia of the structural wall in m ; [inch ] 3 : inertia moment of pipe wall in inch Iw k : thrust stiffness relation factor - dimensionless 2 K : bulk modulus of fluid compressibility in N/m Kr : resistance coefficient of fitting - dimensionless L Lc Le LE Lg Lp Lw Ls

: initial length of pipeline in m [inch] : distance between supports for continuous run : equivalent pipe length, obtained from chart in m [ft] : real equivalent pipe length of fitting : distance between guides in m; [inch] : distance between supports for partial run in m; [inch] : length of the closed section of the pipe conduit in m; : distance between supports for single run

M : bending moments in the pipe due to deflection or buckling using [4] in [Nm] p : internal design gage pressure : minimum ultimate collapse pressure pc Pbf : bulkhead force Ptot : total pressure Pv : hammer pressure caused by valve closure Pw :λ ∆TA in meters (inch) 3

3

Q : rate of flow or debit in m /s; [ft /s] RB : minimum allowable bending radius in [m] Re : Reynolds number s

: design stress

t ta tl ts tw T Tv

: total wall thickness in m [inch] : allowance for ring stiffness, external pressure : thickness of internal liner in m [inch] : minimum reinforced wallthicknes in m : time of one pressure wave : temperature : valve closing time [sec.]

v : fluid velocity in m/s; (ft/s) ∆v : change in fluid velocity in m/s w : total uniformly distributed load in N/m; [lb/in] 2 wp : weight of pipe per unit length in [kg/m ] y : vertical deflection ∆y : deflection of inside diameter in inch

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INDEX 1.

Introduction

2.

Pipewall Thickness based on internal pressure

2.1

Wall thickness

2.2

Diameter

2.3

Dimensional pipe properties

3.

Trust force due to temperature and variation in length

3.1

Length Change

3.2

Thrust

4.

Support and guide spacing

5.

Pipe bending radius

6.

Collapse resistance for liquid

7.

Pipe-ring stiffness

8.

Waterhammer and surge

9.

Headloss or pressure drop for liquid flow

10.

Literature

11.

Legenda

Fiberglass-Composite Pipe Group division Europe P.O. Box 6 - 4191 CA Geldermalsen - Holland tel. +31 345 587 587 - fax +31 345 587 561 email: [email protected]

Calculation Manual for Bondstrand® GRE Pipe Systems

1. Introduction

In this Technical Bulletin an overview is given of commonly used formulas in relation with Glassfibre Reinforced Epoxy piping.

2. Pipe wall thickness

The minimum required wallthickness of the pipe is based on design codes as ASME and ANSI. To most products an inferior liner is added, consisting of C-veil and resin.

3. Trust forces due to temperature, pressure and variation in length

On many occasions the pipe is fabricated to pressure as well as a varying temperature of the medium. Pressure variation will cause a length change if the product is unrestrained and due to the Poisson effect an increase in pressure will shorten the pipe. This is alos mathematically explained. Expension and contraction due to temperature variations and internal pressure will either combined or individual result in thrust forces on the anchoring points

4. Support and Guide spacing

The formulas for the calculation of the optimal distance between two supports or guide spacings for single, partial and continuous spans are given. The calculations take into account density of the liquid and the weight of the pipe.

5. Bending radius

A slight gradual change in direction or deviation of the pipe may be obtained by using the flexibility of the pipe. In that case the allowable bending radius of the glass reinforced epoxy pipe can be calculated

6. Collapse resistance for liquid

When the external pressure on the pipe may exceed the internal pressure one has to take into account the collapse resistance of the pipe. This is ruled by equations which differs from those for internal pressure.

7. Pipe-ring stiffness

To make calculations for earth and wheel-loads on buried pipe, values have to be used like STIS (= Specific Tangential Initial Stiffness), STES (= Specific Tangential End Stiffness) and other values, as used in the U.S.A., Stiffness Factor and Pipe Stiffness.

8. Waterhammer and surge

Changes in velocity of fluids cause changes in pressure. Especially when these velocity changes are sudden, they can result in high forces, which may harm the piping system

9. Head loss or pressure drop for liquid flow

Head loss or pressure drop can be calculated by using the Hazen-Williams equation for water and the Darcy-Weisbach for laminar flows, e.g. for oil. Head loss in fittings are calculated by defining a corresponding pipe length.

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Calculation Manual for Bondstrand® GRE Pipe Systems

1. INTRODUCTION This manual presents the calculations, used by Ameron to calculate the various aspects related to glass reinforced epoxy (= GRE) pipe. This will help the reader to understand the equations which govern certain common engineering cases of GRE pipesystems. Also these equations can be used to make the required calculations. When making these calculations the input data should be based on the physical mechanical properties, diameter and wallthickness of Ameron products by: The spreadsheet presented by Ameron in its documentation gives these values. 2. PIPEWALL THICKNESS BASED ON INTERNAL PRESSURE 2.1 Wall Thickness

The minimum pipewall thickness is calculated with the formula according to ASME / ANSI B31.3 [1] (Paragraph A304.1.2):

ts =

Dp 2sF + p

(1)

ASTM D-2992 [2] uses the same type of formula to calculate the hoop stress as follows:

σ=p

(D a - t s ) 2t s

(2)

The above mentioned formula has been rearranged to induce the internal liner and is used by Ameron to calculate the minimum reinforced wall thickness of Bondstrand pipe as follows: Minimum reinforced wall thickness in [m]:

ts =

p(d + 2t l ) 2σst s ⇔p= 2σs − p d + t s + 2t l

(3)

Minimum total wall thickness in [m]:

t = ts + tl + ta 2.2 Diameter

(4)

Minimum outside diameter of pipe in [m]:

D = d + 2t

(5)

Mean pipe wall diameter in [m]:

Dm = d + t

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Calculation Manual for Bondstrand® GRE Pipe Systems

2.3 Dimensional pipe properties

2

Cross section area of minimum pipe wall in [m ] :

Aw = π (d + t)t

(7) 2

Area of pipe bore in [m ]:

A b = 0,25πd 2

(8)

2

Cross section area of min. structural wall in [m ]:

(

)

A s = π ( d + 2t l ) + t s t s

(9)

2

Cross section area of inner liner in [m ]:

Al = π (d + tl )tl,

(10) 2

Weight of pipe per unit length in [kg/m ]:

w p = A s ρs + A l ρl

(11) 1

Weight of fluid per unit pipe length in [kg/m ]:

w f = 0,25πd 2 ρ f

(12) 4

Linear moment of inertia of the pipe [3] in [m ]:

I l = I s + I lin

(13) 4

Linear moment of inertia of the structural wall in [m ]:

Is =

(

π (d + 2t l + 2t s )4 − (d + 2t l )4 64

)

(14)

4

Linear moment of inertia of the inner liner in [m ]:

I lin =

(

π (d + 2t l )4 − d 64

)

4

(15)

Note! In case of calculating with the moment of inertia of the total wall thickness and the elasticity modulus of structural wall, the moment of inertia may be multiplied by 0,25, which is the approximate ratio between the modulus of elasticity of the liner and the structural wall. The stiffness factor IE = Is Es + Ilin El = Is Es + Ilin Es 0,25 so I = Is + Ilin 0,25. 3. TRUST FORCE DUE TO TEMPERATURE AND VARIATION IN LENGTH 3.1 Length change

Like in other types of pipe material, in unrestrained condition, Bondstrand fiberglass reinforced pipe changes its length with temperature change. Tests have shown that the amount of expansion varies linearly with temperature, in other words the coefficient of thermal expansion of Bondstrand pipe is constant [4, 5, 6].

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Calculation Manual for Bondstrand® GRE Pipe Systems

Change in length due to thermal expansion in [m]:

∆LT = λL∆T

(16)

Subjected to an internal pressure, a free Bondstrand pipeline will expand its length due to thrust force at the ends of the pipeline. The amount of change in the pipeline is a function of pressure, pipe wall thickness, diameter, Poisson's ratio and the effective moduli of elasticity in both, axial and circumferential direction at the operating temperature. Change in length due to pressure in [m]:

pd 2 pd 2 E l pd 2 ∆Lp = L − µc = L (1 − 2 µc ) (17) E c 2tD m E c 4tD m E l 4tD m E l The total length change is the sum of the change due to temperature and due to pressure. The above shown equation for length change due to pressure, compared to the general equation:

π 2 p 4 d pd 2 Pbf (18) = L ∆L = L = L 4tD m E l A w El πtD m E l shows that, the length increase due to the bulkhead force is considerably reduced by the Poisson’s effect. The reduction may amount to 50%, subject to the value 2µcEl / Ec , e.g. for Series 2000: 2 x 0,56 x 11000 / 25200 = 0,49 (at 21°C). 3.2 Thrust

Thrust due to temperature is principally independent of pipe length. In practice, the largest compressive thrust is normally developed on the first positive temperature cycle. Subsequently the pipe develops both, compressive and tensile loads as it is subjected to temperature cycles. Neither, compressive nor tensile loads, are expected to exceed the thrust on the first cycle, irregardless the range of the temperature changes. In a fully restrained and blocked or anchored Bondstrand pipe, length changes induced by temperature change are resisted by the anchors and converted to thrust [4, 5, 6]. Thrust due to temperature in [N]:

FT = λ∆TA w E l = λ∆T(πD m t ) E l

(19)

The theory of thrust due to internal pressure in a restrained pipeline is rather complicated. This is because in straight, restrained pipelines with rigid joints, the Poisson's effect produces considerable tension in the pipe wall. As internal pressure is applied, the pipe expands circumferentially and at the same time tries to contract

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Calculation Manual for Bondstrand® GRE Pipe Systems

longitudinally. This creates a considerable tensile force in the pipe wall, and acts to reduce the hydrostatic thrust on the anchors. In pipelines with elbows, closed valves, reducers or closed ends, the internal pressure works on the cross sectional area of the ends. This thrust may be twice the effect of pressure on the pipe wall. The thrust is independent of the run length or support spacing. Thrust due to pressure in [N]:

E Fp = pA b 1 − 2 µc l Ec

(20)

The concurrent effects of pressure and temperature must be combined for the design of anchors. Similarly, on multiple pipe runs, thrusts developed in all runs must be added for the total effect on the anchors. Thrust due to temperature and pressure in [N]:

FTp = FT + Fp

(21)

Resulting force due to thrust from two pipelines meeting at an elbow or turn in the pipeline in [N]:

δ Fe = 2sin F 2

(22)

Force at a reduction in a straight run by the larger diameter in [N]:

E Fr = λ∆TE l ( A bl − A bs ) + p( A wl − A ws )1 − 2 µ c l (23) Ec In a blocked or anchored pipe system the Poisson's effect causes tension in the pipe wall which counteracts the pipe thrust due to temperature. The tension in the pipe wall may be positive or negative, subject to the direction of the temperature / pressure change. Thrust or tension in pipewall in a restrained blocked or anchored pipeline due to temperature change and pressure in [N]:

πpd 2 E l Fw = λ∆TA w E l − µ c 2E c

(24)

Fp Fp ∆L σ =ε = = = L E l A w E l πD m E l t

(25)

Equation 20 is valid, and

∆L pd 2 El = 1 − 2 µc L 4tD m E l Ec

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Calculation Manual for Bondstrand® GRE Pipe Systems

As can be seen substituting and re-arranging gives equation 20.

The product from equation 20

( pA )−2 µ b

c

El pd 2 E l ⇔ ( − µ c ) Ec 2 Ec

(27)

gives a negative sign, which is tension in the pipe wall caused by the Poisson’s effect, which counteracts the bulkhead force exercised on the blocking or anchoring. 4. SUPPORT AND GUIDE SPACING The distance between two succeeding supports, depends on the parameters as load, moment of inertia and elasticity, as well as the layout of the system. Local loads, such as heavy fittings heavy flange arrangements, valves vertical runs etc. as well as changes in horizontal directions, may also affect the support distances. A long term deflection of 0,0127 m, is normally acceptable for appearance and sufficient for drainage. Distance between supports for partial run [3, 7] in [m]:

E I L p = 1,24 b l w

0,25

(28)

(Note! For imperial input resulting in inches this equation can be used if 1,24 is replaced by 0,258). For continuous span Lp may be increased by 20%, for single spans Lp should be decreased by 20%, giving the same results as when using following equations: Distance between supports for continuous run in [m]:

E I L c = 1,486 b l w

0,25

(29)

Distance between supports for single run in [m]:

E I L s = 0,994 b l w

0,25

(30)

Note! For imperial input resulting in inches these equations can be used if 1,486 is replaced by 0,31 and 0,994 by 0,207. Explanation of partial span The general equations to calculate the maximum deflection of a tubular body are for continuous span

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Calculation Manual for Bondstrand® GRE Pipe Systems

fw =

5wL4 384EI

(31)

and for single span

wL4 fw = 384EI

(32)

Taking 0,0127 m as deflection “fw”, these equations may be rearranged to calculate the span as equation 29 and equation 30. The partial span equation 28 is the average between the equations 29 and 30, giving the same result as equation 29 if increased by 20% or equation 30 if decreased by 20%. If anchors are used at both ends of a pipeline, to restrict axial movements, until a method of controls must be designed in order to prevent excessive lateral deflection or buckling of pipe due to compressive load. Guides may be required in conjunction with expansion joints or expansion loops, to control excessive deflection. The guide spacing can be calculated by using the rearranged Euler equation multiplied by 75%. Distance between guides in [m]: 0,5

ElIl Lg = 0,75π El λ∆TA w E l + pA b 1 - 2 µ c Ec

(33)

The above equation solves for the maximum stable length of a pipe column when fixed ends are assumed, which is reduced by 25% to develop the original portion of curves, now seen only in the smaller diameters, and to allow for non-Euler behaviour near the origin of the curve. Not only to resist buckling of the pipe as a column or "snaking" but to also adjust guide spacing to prevent excessive vertical deflections due to weight, the length calculated by the Euler equation, should be checked by using it in the following equation of [4]. Vertical deflection in [m]:

− wLg kLg kLg y= − tan 2kPw 4 4

0,5

P where k = w E bIl

(34)

If "y" is less than -0,0127 m, the guide distance, Lg obtained from the Euler equation is the recommended guide spacing. If "y" is greater than -0,0127m, a shorter length Lg should be chosen and used in the Roark equation until by trial and error a final length, Lg, is determined that closely approximates a "y" of -0,0127 m. Bending moments in the pipe due to deflection or buckling (using [4]) in

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Calculation Manual for Bondstrand® GRE Pipe Systems

[Nm]:

kLg −w 2 − M = 2 1 − kLg k tan 2

and

kL g (35 −w 2 +M= 2 − 1 k kLg tan 2

5. PIPE BENDING RADIUS Knowledge about the bending radius is required for buried pipe systems, in order to know if the pipeline can follow an existing or projected curved pipe track. The minimum allowable bending radius depends on temperature and pressure. Minimum allowable bending radius [7] in [m]:

RB =

0,5t E b D σl − σp

(36)

Actual axial stress due to internal pressure for BI-AXIAL loaded 2 systems in [N/mm ]:

p d σp = + 1 4 ts

(37)

Actual axial stress due to internal pressure for UNI-AXIAL 2 loaded systems in [N/mm ]:

p d σp = + 1 8 ts

(38)

Notes: 1. For allowable axial tensile stress 50 % is used of the axial bending strength shown in the pipe data sheets. Since Bondstrand pipe and joints can be loaded bi-axially, consequently most are used in that way. The minimum bending radii shown in the pipe data sheets are based on bi-axial loading for that reason. 6. COLLAPSE RESISTANCE FOR LIQUID Where pipes may be exposed to external pressure, such as in tanks, buoyant systems, divers etc., the resistance against collapse may become determining. 2

Minimum Ultimate collapse pressure [3] in [Pa= N/m ] if pipe is sufficiently long: 1

p c (1 − µ c µ l ) 3 2E c t s pc = ⇔ t = d s 2E c (1 − µ c µ l )d 3 3

(39)

Note! To give sufficient resistance against an external pressure of 1 bar, 0,75 is a well accepted factor. For pipes used in marine environments, such as at bottoms of sea going vessels a factor of 0,3 is used to resist 3 bar or 30 m water column with a sufficient safety.

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Calculation Manual for Bondstrand® GRE Pipe Systems

The effective circumferential moduli of elasticity for external pressure loading, have been determined by a combination of theoretical and empirical data. Temperature Cº 2 ElasticityMN/m Ec

Modulus 2400 based pipe 2 ElasticityMN/m Ec Modulus 3400 based pipe

T

21

66

25200

22100

25300

22000

93

121

7. PIPE - RING STIFFNESS Stiffness data are used in calculations of earth- and wheelloads on buried pipe [8]. STIS and STES according NEN 7037. 2

Specific Tangential Initial Stiffness, STIS [9], in [N/m ]: 3

EI E t STIS = c w3 = c Dm 12 D m

(40) 2

Specific Tangential End Stiffness, STES, in [N/m ]:

STIS = αβ STES

(41)

Pipe Stiffness (acc. ASTM-D2412 test [10]), PS, in [psi]:

PS =

Fpr

(42)

∆y 2

Stiffness Factor (acc.ASTM-D2412 [10]), SF, in [inch .lb/inch]: 3

SF = 0,149rm 3 PS ⇔ E c I w = E c

t 12

(43)

2

Relation between STIS [N/m ] and SF [inch.lb]:

SF = 8,848D m 3 STIS ⇔ STIS = 0,113

SF Dm3

(44)

2

Relation between STIS (N/m ) and PS [psi]:

PS = 475,14STIS ⇔ STIS = 0,002105PS

(45)

8. WATERHAMMER AND SURGE Changes in velocity of fluids cause changes in pressure. Especially when these velocity changes are sudden they can be harmful to the piping system. Velocity changes may be caused by movement of valves, starting and stopping of pumps, closure of check valves, or even pipe rupture elsewhere in the system, and are calculated using the Joukowski equation [11].

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Calculation Manual for Bondstrand® GRE Pipe Systems

Pressure change in meter waterhead:

∆p =

c ∆v g

(46)

The pressure wave velocity in a closed pipe system depends on fluid characteristics, pipe dimensions and the elasticity modulus of the pipe wall. Calculating the pressure wave velocity “c” can be done using the Talbot equation. Velocity of pressure wave in a closed pipe conduit in [m/s]:

c=

1 1 d ρ f + K E t c s

0,5

(47)

The pressure change ∆H, added to the highest occurring working pressure in the system should not be higher than 1,5 times the maximum system design pressure. If a valve is closed within the time of one wave cycle, i.e. from the closed valve to the other end and back, then water hammer should be calculated on the basis of instant valve closure. Time of one pressure wave in [seconds]:

tw =

2Lw c

(48)

As can be seen, increase of tw will decrease c and decrease ∆p subsequently. So the longer the wave cycle, the smaller the pressure shock. Delayed closure time: The hammer pressure rise pv caused by taken into account a valve closure of Tv seconds can be calculated as follows

Lw Tv

(49)

p tot = p v + p

(50)

p v = 2∆p

9. HEADLOSS OR PRESSURE DROP FOR LIQUID FLOW Head loss for liquid flow often can be obtained out of charts and tables. However also there are rather simple ways to calculate the head loss. A very simple method to calculate the head loss was developed by Hazen and Williams. This method may be used for water in a temperature range of 0°C to 37°C (imperial: 31°F to 100°F). Head loss for liquid flow in m of water column / 100 m pipe length (imperial: ft of water/100 ft) using the HAZEN-WILLIAMS equation with a Hazen Williams factor C = 150 for Bondstrand

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Calculation Manual for Bondstrand® GRE Pipe Systems

pipe: For S.I. input:

H l = 0,1007

Q1,852 and for Imperial input: d 4,865

H l = 0,0983

Q1,852 d 4,865

(51)

Another way of calculating this head loss is using the Darcy-Weisbach equation. This method is more complicated than the Hazen-Williams method, but has the advantage that it can also be used for other temperatures and other liquids. The Darcy Friction Factor is variable, subject to the Reynolds Figure, this complicated the use of the method. Head loss for liquid flow, in m of water column / 100 m pipe length (imperial: ft of water/100 ft), using the DARCY-WEISBACH equation:

Hl = f

Lv 2 2dg

(52)

Darcy Friction Factor for laminar flow(Re 4000): 1 6 3 e 10 f = 0,0055 1 + 20000 + d Re

(54)

Reynold's number:

Re =

vd vdρf , and = ν η

(55)

Velocity of liquid flow: Q Q v = = Ab 0 ,2 5π d 2 Temperature Density

Cº/ºF T 3 kg/m r lb/ft -6 2 Absolute 10 N s/m h -6 2 Viscosity10 pdl s/ft 1205 -9

2

Kinematic 10 m /s 1011 -9 2 Viscosity10 ft /s 1931 Temperature Density

Cº/ºF T 3 kg/m r lb/ft -6 2 Absolute 10 N s/m h -6 2 Viscosity10 pdl s/ft 538 -9

2

Kinematic 10 m /s 296 -9 2 Viscosity10 ft /s 865

Technical Bulletin 3 July 1997

0/32 999,87 62,42 1794 1053

4/39.1 1000 62,43 1568 880

ν=η/ρf 1794

10/50 999,72 62,41 1310 678

20/68 998,2 62,16 1009

1568

1310

1687

1410

1088

30/86 995,7 62,16 800 439

40/104 992,2 61,94 653 316

60/140 983,2 61,38 470 191

100/212 958,4 59,83 284

658

478

ν=η/ρf 803 709

514

319

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Calculation Manual for Bondstrand® GRE Pipe Systems

The head loss in joints and fittings can be calculated using the same method as used for pipe after defining the equivalent length. The equivalent length LE can be obtained using the chart for equivalent length of Bondstrand fittings. This chart was developed for fittings with a resistance coefficient "K" of 1 and water as fluid. Subject to the configuration of the fitting, the resistance coefficient "K" varies and can be obtained from the table "Resistance Coefficients for fittings". Multiplying Le by K gives the real equivalent pipe length of the fitting. Real equivalent pipe length of fitting with water as fluid in [m]: (57) L E = K r Le Real equivalent pipe length of fitting with other fluids in [m]:

LE =

Kr d f

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Calculation Manual for Bondstrand® GRE Pipe Systems

10. LITERATURE

. [1]

ASME B31.3 / ANSI, an american national standard ASME code for pressure piping, B31.

[2] ASTM-D-2992, Standard practice for obtaining hydrostatic or pressure design basis for “fiberglass” (glass-fiber-reinforced thermosetting-resin) pipe and fittings. [3] Roark R.J., Formulas for Stress and Strain, Singapore, 1976 [4] Hoa S.V., Analysis for Design of Fiber Reinforced Plastic Vessels and Pipings, Lancaster, Pennsylvania, 1991 [5] Heißler H., Verstärkte Kunststoffe in der Luft- und Raumfarhttechniek. Eyerer von P., Kunststoffe und Elastomere in der Praxis, Kohlhammer, Stuttgart, December 1996. [6] Vinson J.R., Sierakowski R.L., the behaviour of structures composed of composite materials, 1987. [7] Mönch E, Einführungsvorlesung Technische Mechanik, Wien, 1973. [8] Algra E.A.H., Mechanische Aspekte bei drucklos betriebenen, erdverlegten GFK-Rohren, Delft. [9] NEN 7037, Glass reinforced thermosetting plastics pipes for drain and sewer- requirements and test methods. [10] ASTM-D-2412, Standard test method for determination of external loading characteristics of plastic pipe by parallel-plate loading, 1993. [11] Tyler G. Hicks P.E., Hicks D.S., Standard handbook of engineering calculations, United States, 1972 11. LEGENDA α : creeping factor for pipe material β

: altering factor of pipe material

ε

: strain rate 2

3

η : dynamic viscosity Ns/m = 10 centipoise λ

: coefficient of thermal expansion in axial direction in m/m/ºK [in/in/ºF]

∆L : sum of the change in length due to temperature and due to pressure ∆LT: change in length due to thermal expansion [m] ∆Lp : change in length due to pressure ∆p : pressure change µ c : poisson's ratio (contraction in longitudinal direction due to strain in hoop direction)

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Calculation Manual for Bondstrand® GRE Pipe Systems

µl : poisson's ratio for (contraction in hoop direction due to strain in longitudinal direction) ν

2

6

: kinematic viscosity m /s = 10 centistoke 3

3

ρf : density of fluid material in kg/m [lb/inch ] 3

3

ρl : density of the liner material in kg/m [lb/inch ] 3

3

ρs : density of the structural pipe wall in kg/m [lb/inch ] 2

σ : hydrostatic design bases (H.D.B.) in N/m = Pa [psi] σl : allowable axial tensile stress in N/m

2

σp : actual axial stress due to internal pressure in N/m

2

∆T : change in temperature in ºK or ºC; ( ºF) 2

2

A b : cross sectional area of pipe bore m [inch ] 2 2 Abl : cross sectional area of larger pipe bore in m [inch ] 2 2 Abs : cross sectional area of smaller pipe bore in m [inch ] 2 2 Al : cross sectional area of inner liner in m [inch ] 2 2 As : cross sectional area of minimal structural wall m [inch ] : cross sectional area of pipewall in m [inch] Aw 2 2 Awl : cross sectional wall area of larger pipe in m [inch ] 2 2 Aws : cross sectional wall area of smaller pipe in m [inch ] c

: velocity of pressure wave in the pipe conduit in m/s [in/s]

D d Da Dm

: minimum outside diameter of pipe in m : inside diameter of pipe in m : average reinforced outside diameter in m : mean diameter of pipewall in m [inch]

e : absolute roughness of internal pipe wall in m; [ft or inch]. 2 Eb : elastic beam modulus acc. ASTM D-2925 in N/m =Pa; (psi) 2 Ec : circumferential modulus of elasticity in N/m = Pa [psi] 2 El : longitudinal modulus of elasticity in N/m = Pa [psi] 2 EI : stiffness factor per unit length of pipe wall in inch -lbs/inch f fw F Fe

: Darcy Friction Factor - dimensionless : deflection of a tubular body : service (design) factor : resulting force due to thrust from two pipelines meeting at an elbow or turn in the pipeline in [N] Fp : thrust due to pressure 3 Fpr : load applied pipe ring inch.(lbf/m ) Fr : force at a reduction in a straight run by the larger diameter in [N] FT : thrust due to temperature FTp : thrust due to temperature and pressure Fw : thrust or tension in pipewall in a restrained blocked or anchored pipeline due to temperature change and pressure g Hl Il

2

2

: acceleration by gravity in m/s [in/s ] : head loss 4

4

: linear moment of inertia of pipe in m ; [inch ]

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Calculation Manual for Bondstrand® GRE Pipe Systems 4

4

Ilin : linear moment of inertia of the inner liner in m ; [inch ] 4 4 Is : linear moment of inertia of the structural wall in m ; [inch ] 3 : inertia moment of pipe wall in inch Iw k : thrust stiffness relation factor - dimensionless 2 K : bulk modulus of fluid compressibility in N/m Kr : resistance coefficient of fitting - dimensionless L Lc Le LE Lg Lp Lw Ls

: initial length of pipeline in m [inch] : distance between supports for continuous run : equivalent pipe length, obtained from chart in m [ft] : real equivalent pipe length of fitting : distance between guides in m; [inch] : distance between supports for partial run in m; [inch] : length of the closed section of the pipe conduit in m; : distance between supports for single run

M : bending moments in the pipe due to deflection or buckling using [4] in [Nm] p : internal design gage pressure : minimum ultimate collapse pressure pc Pbf : bulkhead force Ptot : total pressure Pv : hammer pressure caused by valve closure Pw :λ ∆TA in meters (inch) 3

3

Q : rate of flow or debit in m /s; [ft /s] RB : minimum allowable bending radius in [m] Re : Reynolds number s

: design stress

t ta tl ts tw T Tv

: total wall thickness in m [inch] : allowance for ring stiffness, external pressure : thickness of internal liner in m [inch] : minimum reinforced wallthicknes in m : time of one pressure wave : temperature : valve closing time [sec.]

v : fluid velocity in m/s; (ft/s) ∆v : change in fluid velocity in m/s w : total uniformly distributed load in N/m; [lb/in] 2 wp : weight of pipe per unit length in [kg/m ] y : vertical deflection ∆y : deflection of inside diameter in inch

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