The effect of pressure on leakage in longitudinally cracked pipes

Stefano de Miranda1, Luisa Molari1, Giulia Scalet1, Francesco Ubertini11DICAM, University of Bologna, ItalyE-mail: stefano.demiranda@unibo.it, luisa.molari@unibo.it, giulia.scalet2@unibo.it,francesco.ubertini@unibo.it

Keywords: Beam model, leakage, longitudinal split.

SUMMARY. Losses from water distribution systems are reaching alarming levels in many citiesthroughout the world. This paper presents a simple model based on a beam with elastic constraintsto evaluate the effect of pressure on the opening area of the crack and to estimate leakage in longi-tudinally cracked pressurized pipes. The model is calibrated on the results of a three-dimensionalfinite element analysis and then validated by experimental results.

1 INTRODUCTIONThe growing attention to a careful, sustainable, and economically efficient use of water makes

the problem of water loss and the need of an efficient management of water systems very important.The international literature [1] agrees that leakage constitutes the prevailing source of loss due toaging and deterioration of water systems. Pressure has been verified to assume a key-role in waterloss management. The orifice equation, widely accepted in practice to quantify water loss, assumesthe water flow proportional to the square root of the pressure head. However, according to recentstudies, this equation can not accurately estimate loss due to leakage. The water loss is, in fact, moresensitive to pressure. One of the main reason is that the orifice equation does not account for thepipe deformation, even though many studies demonstrate that the leakage exponent depends on it. Inparticular, this equation gives good results for round holes, but not for circular cracks nor longitudinalsplits. Extensive experimental investigations, highlighting the effect of pipe deformation on leakage,can be found in [2]-[3]-[4]. This paper aims to estimate leakage in longitudinally cracked pressurizedpipes and to evaluate the effect of pressure on the opening area of the crack through the introductionof a simple model based on a beam with continuous elastic restraints. The behaviour of the crackedpipe is considered linear-elastic and the related water flow is evaluated from the deformation of thepipe, which depends on pressure. The beam model is calibrated through a comparison with theresults of a refined three-dimensional finite element analysis. Then its validity is assessed througha comparison with several experimental investigations [2]-[3]-[4]. The beam model allows, despiteits simplicity, to describe realistic situations that could occur in a longitudinally cracked pressurizedpipe, to provide reliable results for the study of water loss, and to aid Water Authorities in leakagemanagement.

2 BEAM-BASED PIPE MODELConsider a cylinder pipe of medium radius R and thickness s, that works at an internal pressure

p. The pipe has a crack of length 2L, assumed as an idealized longitudinal wall crack (Fig. 1(a)).The longitudinal strip of the pipe, containing the crack, is modelled as an elastic beam with

continuous elastic restraints that account for the effect of the remainder part of the pipe on the strip.Naturally, the beam-based model of the part of the strip belonging to the sound pipe is different fromthat of the part of the strip containing the crack. In the following the two models are described.

1

(a)

(b) (c)

Figure 1: Longitudinally cracked pressure pipe.

Hereinafter the quantities related to the sound part and the cracked part of the pressure pipe aremarked by the apexes s and c, respectively.

2.1 Sound pressure pipeIn Fig. 1(b) the sound part of the pressure pipe is cross-sectioned and the coordinate system is

represented.Due to the axial-symmetry, the only displacement of the sound part of the pipe is the radial

displacement ws. According to this, the sound part of the pressure pipe is modeled, classically, asa longitudinal beam over Winkler foundation [5]. In particular, w s is considered as the deflectionof a longitudinal prismatic Euler-Bernoulli beam of cross-section (2b× s) subjected to a distributedload p, which rests on a continuous elastic foundation created by the remainder of the cylinder (Fig.2(a)).

The beam equilibrium equation in terms of deflection is:

d4ws

d(zs)4+ 4α4ws = q, zs ≥ L, zs ≤ −L (1)

where q = p2b/EJsy , α4 = β/4EJs

y , and Jsy = 2bs3/12(1− ν2), being ν the Poisson’s ratio, E the

Young’s modulus, and β = 2bEs/R2 the stiffness to the radial displacement of a transverse closedannular element (Fig. 1(b)). J s

y is the moment of inertia of the longitudinal beam cross-section withrespect to the ys-axis corrected by the factor (1− ν 2) because of the presence of the remaining partof the cylinder.

The solution of Eq. (1) is:

ws = Ae−αzs

cos(αzs) +Be−αzs

sin(αzs) +pR2

Es, (2)

2

(a) (b)

Figure 2: Longitudinal beam of the: (a) sound part; (b) cracked part.

sum of the solution of the corresponding homogeneous equation and the solution related to theparticular loading condition.

2.2 Cracked pressure pipeIn Fig. 1(c) the cracked part of the pressure pipe is cross-sectioned and the coordinate system is

represented. The cracked part of the pipe is modeled by considering a longitudinal beam of cross-section (b× s).

The behaviour of the beam is described by the radial displacement w c, by the tangential dis-placement uc, and by the torsional rotation θc. The effect of the remaining part of the cylinder on thelongitudinal element can be accounted for by considering continuous elastic restraints correspondingto each of the displacements, wc and uc, and θc (Fig. 2(b)).

From equilibrium, compatibility, and elastic constitutive laws, the equations of the deflection inthe xc-zc plan and in the yc-zc plan, and the equation of the rotation along the z c-axis can be put inthe forms:

EJcy

d4wc

d(zc)4+ kwww

c + kwuuc + kwθθ

c = P, −L ≤ zc ≤ L (3)

EJcx

d4uc

d(zc)4+ kwuw

c + kuuuc + kuθθ

c = Q, −L ≤ zc ≤ L (4)

GJct

d2θc

d(zc)2− kwθw

c − kuθuc − kθθθ

c = −C, −L ≤ zc ≤ L (5)

where G = E/2(1 + ν) is the tangential modulus, J cy = bs3/12(1 − ν2) and J c

x = sb3/12 arethe moments of inertia of the longitudinal element cross-section with respect to the y c-axis and thexc-axis, respectively, J c

t = bs3/3 is the torsional stiffness of the cross-section, and P , Q, and C arethe generalized loadings on the beam element. The evaluations of the stiffness coefficients and ofthe generalized loadings are detailed in [6].

2.3 Boundary and interface conditionsTwelve conditions, both boundary and interface conditions, are necessary to integrate the system

of differential equations (1)-(3)-(4)-(5).

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The kinematic boundary conditions imposed by symmetry at z c = 0 are:

dwc

dzc

∣∣∣∣0

= 0,duc

dzc

∣∣∣∣0

= 0,dθc

dzc

∣∣∣∣0

= 0, (6)

while the static ones are:d3wc

d(zc)3

∣∣∣∣0

= 0,d3uc

d(zc)3

∣∣∣∣0

= 0. (7)

In addition, there are four kinematic interface conditions at z c = zs = L determined by consid-erations of compatibility between the cracked and the sound part of the pipe:

wc(L) = ws(L),dwc

dzc

∣∣∣∣L

=dws

dzs

∣∣∣∣L

, uc(L) = 0, θc(L) = 0. (8)

The first two conditions impose the continuity of radial displacement and its derivative at the inter-face, while the last two conditions impose the absence of tangential and torsional effects, accordingto the axial symmetry conditions that the sound part has to satisfy.

Finally, there are three static interface conditions. The first reads as:

d2uc

d(zc)2

∣∣∣∣L

= 0, (9)

and corresponds to assume a cylindrical hinge at the interface.In order to obtain the last two conditions it is necessary to focus the attention on the equilibrium

of the element of the pipe of infinitesimal length connecting the cracked and the sound part (Fig. 3).

Figure 3: Equilibrium between the two parts.

Moment equilibrium around a tangential axis reads 2M cy(L) = M s

y (L), which can be expressedin terms of displacements in the form:

2EJcy

d2wc

d(zc)2

∣∣∣∣L

= EJsy

d2ws

d(zs)2

∣∣∣∣L

, (10)

that constitutes the eleventh condition.The last condition is obtained by imposing the equilibrium of the forces in the radial direction:

−2T cx(L) + T s

x(L) + 3Mc

z

b = 0, which, expressed in terms of generalized displacements, becomes:

2EJcy

d3wc

d(zc)3

∣∣∣∣L

− EJsy

d3ws

d(zs)3

∣∣∣∣L

+3

bGJc

t

dθc

dzc

∣∣∣∣L

= 0. (11)

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The simple beam model includes Eq. (1) to describe the behaviour of the sound pipe and Eqs.(3)-(4)-(5) to describe the cracked zone, completed by the twelve boundary-interface conditionsexpressed by Eqs. (6)-(7)-(8)-(9)-(10)-(11). This system can be easily solved analytically with thehelp of a standard software tool to handle ordinary differential equations [7].

3 MODEL CALIBRATIONIn order to calibrate the beam model, several finite element models of longitudinally cracked

pipes have been developed, with different properties chosen according to the experimental investi-gations reported in [2]-[3]-[4]. The pipes considered in this calibration process work at a pressurep = 1 m= 9.806·10−3 MPa and are made of PVC (Young’s modulusE = 2900 MPa, and Poisson’sratio ν = 0.38). The pipes present three different external radii R (R 1 = 27.5 mm, R2 = 55 mm,and R3 = 110 mm) and four different thicknesses s (s1 = 1.5 mm, s2 = 3 mm, s3 = 6 mm, ands4 = 12 mm). Moreover six crack lengths have been considered: 2L 1 = 50 mm, 2L2 = 100 mm,2L3 = 150 mm, 2L4 = 200 mm, 2L5 = 250 mm, and 2L6 = 300 mm.

The model calibration involves three parameters: the width b of the beam, the stiffness coefficientkuu, and the stiffness coefficient kuθ . Each parameter is expressed as function of the radius R, thepipe thickness s, and the crack length 2L.

3.1 Calibration of the width b of the beamThe width of the longitudinal beam b is expressed by a bilinear formula in R and L:

b(L,R) = α1R+ α2LR

R0, (12)

where R0 is a reference radius, assumed equal to the intermediate radius R2. In the limiting case ofL equal to zero, b is expected to be different from zero, while in the limiting case of R equal to zero,b is expected to be equal to zero.

3.2 Calibration of the the stiffness coefficients kuu and kuθThe need for calibration of the stiffness coefficients kuu and kuθ is due to the fact that, as ex-

pected, the model is too flexible in the cracked part. In fact, the stiffness as well as the loading ofthe cracked part of the pipe are calculated in the limiting case of a completely opened annular ring.

In order to perform the calibration, kuu and kuθ in Eqs. (4)-(5) are multiplied by the dimen-sionless coefficients a1 and a2, respectively, which are considered dependent upon the radius R, thecrack length 2L, and the thickness s. As it can be easily realized, a1 and a2 must assume a largevalue when the crack is short (reaching infinity in the limiting case of a sound pipe), and must beequal to one in the limiting case of a completely opened pipe.

The following expressions are then adopted for a1 and a2:

a1(L,R, s) = A1 exp(−A2

2LR +4.6)

( s

R

)(−A3)

+

+A4 exp8(−A5

sR+1)

(2L

R

)(−A6)

+ 1,

(13)

a2(L,R, s) = A7 exp(−A8

2LR )

(2L

R

)(−A9) ( s

R

)(−A10)

+ 1, (14)

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where the coefficients Ai (i = 1, ..., 10) are functions of R and can be expressed in the form:

Ai = β1

(R

R0

)2

+ β2R

R0+ β3, i = 1, ..., 10 (15)

being R0 a reference radius (assumed equal to the intermediate radius R 2).

3.3 ResultsGetting along as described above, the parameters α1, α2, β1, β2, and β3 are determined. The

corresponding values are collected in Table 1.

Table 1: Values assumed by α1, α2, β1, β2, and β3.

α1 α2

0.0072 0.232

Ai β1 β2 β3

A1 1.097 -1.562 0.534

A2 -1.081 4.243 -1.625

A3 -1.596 4.395 -0.716

A4 -0.147 -2.445 3.311

A5 7.233 -11.023 5.430

A6 1.860 -5.392 5.889

A7 1.676 -1.144 0.645

A8 -0.694 -1.985 -0.750

A9 1.021 -2.727 1.869

A10 -0.227 0.452 1.085

In Figs. 4(a)-(d) the correlation between the opening area and the ratio 2L/R is shown forvarious thicknesses. As it can be noted there is good agreement between the simple beam modeland the finite element model, particularly for pipes with small-medium radii, thicknesses, and cracklengths.

4 COMPARISON WITH EXPERIMENTAL INVESTIGATIONIn this Section, model’s predictions are verified through a comparison with the results of different

experimental investigations [2]-[3]-[4] conducted on several materials such as PVC, steel, cast iron,and asbestos-cement.

In particular, the correlation between flow Q [m3/s] and pressure head H [m] is described by amodified version of Torricelli equation:

Q = CdA√2gH, (16)

where g is the acceleration due to gravity [m/s2], A the orifice area [m2], and Cd the dischargecoefficient, by assuming the opening area of the hole depending on pressure head as a result ofthe pipe deformation. In particular, the beam model is used to determine the opening area A byintegrating the curves of the tangential displacements. Indeed, since we suppose an elastic-linear

6

(a) (b)

(c) (d)

Figure 4: Opening area A vs. 2L/R for pipes with: (a) s = 1.5 mm; (b) s = 3 mm; (c) s = 6 mm;(d) s = 12 mm.

behaviour of the pipe material and we assume negligible the elastic dilatation of the crack in thelongitudinal direction of the pipe (i.e. the crack length does not vary), a linear relationship betweenA and H is obtained and the modified version of Torricelli equation is:

Q = Cd

√2g(A0H

0.5 +mH1.5), (17)

with A0 the initial area of the crack and m the pressure-area slope which becomes an essentialdescriptive parameter for the behaviour of the leak opening [8].

In the following results it is assumed: A0 = sf2L, with sf the initial thickness of the crack, andCd = 0.75. As it can be noted, accounting for pipe deformation leads to a leakage exponent higherthan the classical 0.5.

The first comparison concerns the experimental investigations [2]-[3]-[4] conducted on severalclass 6 PVC pipes. The geometrical characteristics of the investigated pipes are listed in Table 2.According to [2] Young’s modulus is set equal to 3000 MPa and Poisson’s ratio to 0.4 for all thepipes.

In Figs. 5(a), 5(b) and 5(c) the correlations between flow and pressure for the proposed model andthe experimental investigations [2]-[3]-[4] are shown. The graphs reveal a good agreement betweenthe beam model and the experimental results. Fig. 5(a) highlights a good agreement at low-mediumpressure for both the crack lengths. The differences for the crack length of 86 mm at medium-highpressure are due to the phenomenon of crack propagation [3] which is not taken into account in thepresent beam model. On the other hand, the results for the crack length of 150 mm are good forthe entire range of pressures. Also the results of Figs. 5(b) and 5(c) show a good agreement for theentire range of pressures.

The other comparisons, here considered, concern different materials. In particular, two asbestos-cement pipes with R = 50 mm, s = 12 mm, and crack lengths 2L of 298 and 324 mm, respectively,

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Table 2: Geometrical characteristics of the investigated class 6 PVC pipes.

Ref. R [mm] s [mm] sf [mm] 2L [mm]

[3] 55 3 0.386150

[4] 55 4 0.5

50100150200

[2] 55 31.2 402.0 601.7 90

are tested in [3]. We assume two values of the Young’s modulus of E 1 = 4000 and E2 = 24000 MPa,which delimit the possible range of values for this kind of material, and a Poisson’s ratio of 0.2. InFig. 6(a) the correlations between flow and pressure obtained by the beam model with the two valuesof the Young’s modulus are shown and compared with the experimental results. As it can be notedthe experimental points are included in the gray zone, bounded by the two curves corresponding toYoung’s moduli E1 and E2.

The results for steel and cast iron pipes, both tested in [4], are shown in Figs. 6(b) and 6(c),respectively. Steel pipes have radius R = 55 mm, thickness s = 4 mm, crack initial thicknesssf = 0.5 mm, and crack lengths 2L of 50, 100, and 150 mm. Moreover, Young’s modulus isassumed equal to 210000 MPa and Poisson’s ratio to 0.3. Cast iron pipes have radius R = 55 mm,thickness s = 4 mm, crack initial thickness sf = 0.5 mm, and crack lengths 2L of 50, 100, 150, and200 mm. Moreover Young’s modulus equal to 170000 MPa and Poisson’s ratio to 0.25 are assumed.The model provides a reliable leakage evaluation in pipes made of different materials too.

5 CONCLUSIONSThis paper presents a simple and effective model, for a reliable evaluation of water loss which

occurs in longitudinally cracked pressurized pipes. Comparisons with the results of the experimentalinvestigations [2]-[3]-[4] demonstrated the ability of the model to estimate the water flow in pipeswith different geometrical and mechanical characteristics for several pressure ranges. It was con-firmed that the leakage exponent is higher than the classical 0.5.

References[1] Farley, M., “Leakage management and control: a best practice training manual,” WHO,

Geneva, Switzerland (2001).

[2] Buckley, R.S., “Theoretical investigation and experimentation into the expansion of roundholes and cracks within pressurized pipes,” Masters Thesis, University of Johannesburg (2007).

[3] Greyvenstein, B., “An experimental investigation into the pressure-leakage relationship ofsome failed water pipes in Johannesburg,” Eng. Final Year Project Report, Rand AfrikaansUniversity (now University of Johannesburg) (2004).

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(a)

(b) (c)

Figure 5: Q vs. H in (a)-(b)-(c) PVC pipes.

[4] Greyvenstein, B., “An experimental investigation into the pressure-leakage relationship of frac-tured water pipes in Johannesburg,” Masters Thesis, University of Johannesburg (2007).

[5] Harvey, J.F., Pressure vessel design: Nuclear and Chemical Applications, D. Van Nostrand Co.Inc., Princeton, N.J. (1963).

[6] de Miranda, S., Molari, L., Scalet, G. and Ubertini, F., “Leakage evaluation in longitudinallycracked pressurized pipes,” in Proc. 4th SEMC, Cape Town, September 6-8, 569-572 (2010).

[7] Heck, A., Introduction to Maple, 3rd edition, Springer-Verlag, New York (2003).

[8] Cassa, A.M., Van Zyl, J.E., and Laubscher, R.F., “A numerical investigation into the effectof pressure on holes and cracks in water supply pipes.” Urban Water Journal, 7(2), 109-120(2010).

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(a)

(b) (c)

Figure 6: Q vs. H in: (a) asbestos-cement pipes; (b) steel pipes; (c) cast iron pipes.

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