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A NUMERICAL INVESTIGATION INTO THE BEHAVIOUR OF CRACKS IN WATER PIPES AM Cassa, JE van Zyl Corresponding author: A.M. Cassa, Department of Civil Engineering Science, University of Johannesburg PO Box 524, Auckland Park, 2006, South Africa Tel: +27 11 559 2479, Fax: +27 11 559 2395, E-mail: [email protected] JE van Zyl University of Cape Town, Email: [email protected] Abstract Studies show that pressure has a major impact on the rate of leakage from leak openings in pipes, and that the leakage exponents in a real network can be substantially larger than the theoretical orifice exponent of 0.5. The most important reason for this behaviour is that leak areas are not fixed, but increase with increasing pressure. This study used Finite Element Analysis to study the relationship between pressure and leak area in different pipes with longitudinal, circumferential and spiral cracks. It was found that the longitudinal, circumferential and spiral crack areas increase linearly with pressure. The slope of this linear relationship depends on various parameters, including the loading state, pipe dimensions and pipe material properties. The impact of the different parameters on the pressure-area slope was investigated, and an attempt will be made to find an expression to express the pressure-area slope as a function of the parameters. If such a relationship can be established, it will be possible to predict the behaviour of different types of leak openings in different pipes and pipe materials. Keywords Leakage, pipes, crack behaviour, pressure, water distribution systems, uPVC pipes INTRODUCTION Studies show that pressure has a major effect on the rate of leakage from leak openings in pipes and that the leakage exponents in networks can be significantly different from the theoretical orifice exponent of 0.5. A possible reason for this is that leaks are not fixed, but that they rather increase with increasing pressure. This means that leakage is more sensitive to pressure than previously believed and can have serious implications for pressure and water loss management. Using the methods of Finite Element Analysis this study aims to find a relationship between pressure and the leak area in pipes with various leak openings. The study focuses on three types of cracks (longitudinal, circumferential and spiral cracks) and their behaviour under certain conditions. It will concentrate on predicting the effect of various parameters of the pipe and crack, and finding a mathematical relationship to describe this behaviour. Once this relationship is established it will be possible to predict the leakage from a given pipe under various circumstances. BACKGROUND Research conducted previously (1) has shown that leakage exponent in pipes with cracks can differ substantially from the theoretical value of 0.5 as stated in the orifice equation. In that study pipes were modelled using finite element analysis to determine the leakage exponents in pipes with circumferential and longitudinal cracks. This was done with the use of the Torricelli (2) equation for discharge, given by equation [1]
[1] Where Q is the leakage flow rate, Cd the coefficient of discharge, A the orifice area, g the acceleration due to gravity and h the pressure head. The results of that study showed that the loading state of the pipe played a significant role in the leakage exponent for pipes with circumferential cracks. It showed that in the absence of longitudinal stresses the leakage exponent stayed close to 0.5 for short cracks and then dropped below 0.5 for longer cracks, however with the longitudinal stresses present the leakage exponent increased significantly as the length of the cracks increased. The loading state did not play a role in the effect of the leakage exponent for the longitudinal cracks but the exponents did exhibit similar increases to that of the circumferential cracks with longitudinal stresses present. Figure 1 shows the relationship between the leakage exponent and the length of the crack for the two openings.
Figure 1 – Relationship between the leakage exponent and the crack length From Figure 1 it can be seen that the leakage exponents differ substantially from the orifice equation for both longitudinal and circumferential cracks. It can also be shown that longitudinal cracks are the most sensitive to pressure increases. This study concluded that material behaviour may explain the sensitivity of pressure to leakage and that it may have implications on water loss management. FINITE ELEMENT ANALYSIS The finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. This method first introduced in the 1950s and was originally designed to study stresses in complex airframe structures, but has been adapted to a wide field of continuum mechanics (3). Engineering problems of mechanics, solid or fluid have been tackled in the past by deriving differential equations relating the variables of interest. The basic principles used in describing the behaviour of the problems included Newton’s second law of motion, conservation of mass, conservation of energy, and others. Finite elements are used by creating different geometric regions, establishing separate approximating functions in each region and then joining them
together. Thus a finite element model is made up a many small interconnected subregions of elements. A finite element model of a problem gives a piecewise approximation to the governing equations. Thus the finite element method shows that a solution region can be analytically modelled or approximated by replacing it with a grouping of discrete elements. These elements can be put together in a variety of ways, and can be used to represent exceedingly complex shapes. The ABAQUS® Finite Element software was used in this investigation as well as the software program Solidworks where the pipes are modelled according to the dimensions required. In this investigation three types of leak openings were modelled, namely longitudinal, circumferential and spiral cracks. The study is based on a 110 mm diameter class 6 uPVC pipe with a wall thickness of 3 mm the dimensions and properties of which are given in Table 1. Table 1 – Dimensions and properties of a class 6 uPVC pipe Variable Value Pressure head, h 600 Young’s modulus, E 3 Material Properties Poisson’s ratio, ν 0.4 Longitudinal stress, σ l 5.2 900 Length of pipe, Lpipe Geometry of pipe Wall thickness, t 3 Internal Diameter, ID 104 60 Length of crack, Lcrack Crack geometry Width of crack, W crack 1
Unit kPa GPa – MPa mm mm mm mm mm
The pipes that were modelled in this study assumed orifice hydraulics and elastic behaviour of the pipe material. Ten-noded quadratic tetrahedron elements were used throughout the pipes. Sensitivity analyses were done for each configuration to determine the optimal element size. Generally the sizes of the elements in the region around the cracks were 1 mm and 2 mm for all three types of cracks while the rest of the pipe had element sizes of 5 mm. The boundary conditions of the pipe consisted of clamping the pipe along an internal line furthest away from the crack, as well as a point on the outside of the pipe adjacent to the internal line. The pipes were loaded with an internal pressure from the water in the pipe and an applied stress at the ends of the pipe to simulate the longitudinal stresses within the pipe material. The longitudinal stresses are calculated using equation [2], for longitudinal stress for cylindrical pressure vessels (4). [2] Where r is the inner radius of the pipe, P the internal pressure, t the thickness of the pipe wall and σ the longitudinal stress. LEAKAGE THROUGH CRACKS Understanding what role each variable plays in the behaviour of the crack in relation to pressure is important when trying to predict a mathematical relationship describing this behaviour. Initially the base pipe was studied with only the pressure being varied to study
the basic interaction between pressure and leakage and pressure and the leak area. Simulations were then run where each variable was in turn varied one at a time to study how that variable’s effect on the pressure-leakage relationship of the pipe. Table 2 shows the values for the pipe and leak properties that were used for each of the parameters in the analysis. Table 2 – Variations in dimensions and properties of the pipe Varied Property or dimension Pressure head, h (kPa) 0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 Young’s modulus, E (GPa) 3, 10, 30, 90, 200 Poisson’s ratio, ν (-) 0.17, 0.21, 0.29, 0.4, 0.45, 0.5 Longitudinal stress, σ l (MPa) 0, 1.3, 2.6, 3.9, 5.2 (excluding longitudinal cracks) Wall thickness, t (mm) 2, 2.5, 3, 4, 5 Internal Diameter, ID (mm) 20, 30, 40, 50, 80, 104, 150, 200, 250, 300, 350 Length of crack, Lcrack (mm) 10, 30, 60, 100, 130, 150 Width of crack, W crack (mm) 0.1, 0.5, 1, 2 First step was to investigate the behaviour of longitudinal cracks with changes in pressure. Previous research showed that the effect of longitudinal stresses on pipes with longitudinal cracks were negligible thus only changes in circumferential stresses were investigated for these cracks. The relationship between pressure and the leak area is shown in figure 2 and shows that leak area is directly proportional to the pressure head.
Figure 2 – Relationship between the area of a longitudinal crack and the pressure head The Torricelli discharge equation describes the flow coming out of an orifice, such as a hole or a crack. In order to apply the Torricelli equation to leaks in pipes it can be written in a more general form as: [3]
Where c’ is the leakage coefficient and α the leakage exponent. In order to show the leak area equation [3] can be rewritten as: [4] In figure 2 it was shown that the relationship between pressure head and leak area is a linear function, thus the crack area can be written as: [5] Where Ao is the original area of the crack and m is the slope of the graph in figure 2. Substituting equation [5] into equation [4] and simplifying results in: [6] Looking at equation [6] it can be seen that the area plays a larger role in the overall flow out of a leak and the equation is the sum of two terms with different pressure exponents, i.e. 0.5 and 1.5. Each of the parameters stated in Table 1 were varied and the effect of the slope of the pressure-area curve was investigated. Once the slopes for the particular parameter were calculated they were plotted against the parameter. Figures 3 to 8 show the various parameters and how the slope changes as the value of the parameter is varied in the case of longitudinal cracks.
Figure 3 – Change in Young’s modulus
Figure 4 – Change in Poisson’s ratio
Figure 5 – Change in internal diameter
Figure 6 – Change in wall thickness
Figure 7 – Change in length of crack
Figure 8 – Change in width of crack
Figures 3 to 8 show that the slope varies significantly with changes in the different parameters. Results that can be gained from these graphs are the behaviour of the different parameters. In figure 3 Young’s modulus shows an inversely proportional relationship with the area slope. In figure 4 Poisson’s ratio shows a linear relationship with the area slope. In figure 5 the internal diameter shows that the area slope is proportional to the internal diameter that is raised to the power of a third. In figure 6 the area slope is inversely proportional to the wall thickness raised to a power of 1.75. In figure 7 the length of crack shows a 3rd order polynomial in relation to the area slope. In figure 8 the width of the crack shows a linear relationship with the area slope. Using the results obtained above, a single equation that will describe the area slope of the crack as a function of the different parameters is currently being sought. The data will be processed using dimensional analysis as well as regression analysis to describe the area of a longitudinal crack with certain parameters. The same method applied to pipes with longitudinal cracks was applied to pipes with circumferential cracks with an additional parameter of the longitudinal stress. Once again the slopes for the various parameters were calculated and plotted against the particular parameter. Figures 9 to 12 show some of the parameters of the circumferential cracks that differ from the longitudinal cracks.
Figure 9 – Change in Poisson’s ratio
Figure 10 – Change in internal diameter
Figure 11 – Change in length of crack
Figure 12 – Change in longitudinal stress
Again the behaviour of the different parameters can be obtained from the graphs in figure 9 to 12. Figure 9 shows Poisson’s ratio as a 4th order polynomial in relation to the area slope. The internal diameter in figure 10 shows a 6th order polynomial in relation to the area slope. The width of the crack shows a linear relationship with the area slope in figure 14 and figure 15 shows a linear relationship with the area slope and also indicates that the crack will tend to pull itself closed in the absence of the longitudinal stresses. Young’s modulus shows an inversely proportional relationship with the area slope which is the same as for the longitudinal crack. The wall thickness also shows similar results to the longitudinal cracks as the area slope is inversely proportional to it raised to a power of 0.3513. The length of crack also remains a 3rd order polynomial in relation to the area slope. For the third set of cracks, spiral cracks, the effect of the various parameters was investigated as was done with the longitudinal and circumferential cracks. The cracks were at a 45° angle to the horizontal and the variables used for the spiral cracks included longitudinal stress as was done for the circumferential cracks. The slopes were then calculated and plotted against the parameters. The results yielded similar trends to that of the circumferential cracks. Figures 13 and 14 show two parameters that differ from circumferential cracks as their slope changes.
Figure 13 – Change in internal diameter
Figure 14 – Change in longitudinal stress
The internal diameter in figure 13 shows a 5th order polynomial in relation to the area slope. Figure 14 shows a linear relationship with the longitudinal stress but is significantly different from the circumferential cracks effect. The other parameters behave in the same way for the circumferential cracks and have similar trends. As for both longitudinal and circumferential cracks the Young’s modulus shows an inversely proportional relationship with the area slope. Poisson’s ratio is a 4th order polynomial in relation to the area slope which ties in with the circumferential crack. The area slope is inversely proportional to the wall thickness raised to a power of 1.682 which is closely related to the longitudinal crack.
The length of crack is a 3rd order polynomial in relation to the area slope which is the same trend for both the longitudinal and circumferential cracks. The width of the crack shows a 2nd order polynomial with the area slope that again ties in with the trend of the circumferential cracks. DISCUSSION AND CONCLUSION This study investigated the parameters that play a role in the behaviour of cracks in pipes under pressure. Three types of leak openings were discussed, namely longitudinal, circumferential and spiral cracks. It was found that the area of the crack played a larger role in the orifice equation than previously believed and the effect of pressure on the leakage exponent is significant in pipes with cracks. The work shows that the individual properties of the pipe and the geometry of the crack play a more significant role in the area of the crack. The work shows that the Young’s modulus parameter has the same effect regardless of the orientation of the crack in a pipe. It can also be concluded that Poisson’s ratio plays a bigger role when the orientation of the crack is not in the longitudinal direction. The data suggest that the most complicated parameter is possibly the internal diameter of the pipe as it varies significantly across the three types of cracks. The length of the crack as well as the wall thickness has the same effect across the three types of cracks. The work also shows that the width if the crack plays a role in the behaviour of the cracks in pipes and the longitudinal stresses play a role in the circumferential and spiral cracks. ACKNOWLEDGEMENTS This study wishes to gratefully acknowledge the financial support from the NRF programme. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
A.M. Cassa, J. E. van Zyl, R. F. Laubscher, A Numerical Investigation into the Behaviour of Leak Openings in uPVC Pipes under Pressure, WISA2006 The Water Institute of Southern Africa Biennial Conference and Exhibition, South Africa, (2006) A. Chadwick, J. Morfett, Hydraulics in Civil and Environmental Engineering, 3rd ed., E &FN Spon, London, (1999) K. H. Huebner, [et al.], The Finite Element Method for Engineers, 4th ed., John Wiley & Sons, Inc., New York, (2001) J. M. Gere, Mechanics of Materials, 5th ed., Brooks/Cole, United States, (2001) Hibbitt, Karlsson, Sorensen, ABAQUS: Superior Finite Element Analysis Solutions, USA, Copyright 2004 ABAQUS, Inc. S. Timoshenko, J. N. Goodler, Theory of Elasticity, 2nd ed., McGraw-Hill, New York, (1951) M. J. Fagan, Finite Element Analysis Theory and Practice, Pearson Education Limited, Malaysia, (1992) G. E. Dieter, Mechanical Metallurgy, McGraw-Hill, London, (1988)
A NUMERICAL INVESTIGATION INTO THE BEHAVIOUR OF CRACKS IN WATER PIPES AM Cassa, JE van Zyl Corresponding author: A.M. Cassa, Department of Civil Engineering Science, University of Johannesburg PO Box 524, Auckland Park, 2006, South Africa Tel: +27 11 559 2479, Fax: +27 11 559 2395, E-mail: [email protected] JE van Zyl University of Cape Town, Email: [email protected] Abstract Studies show that pressure has a major impact on the rate of leakage from leak openings in pipes, and that the leakage exponents in a real network can be substantially larger than the theoretical orifice exponent of 0.5. The most important reason for this behaviour is that leak areas are not fixed, but increase with increasing pressure. This study used Finite Element Analysis to study the relationship between pressure and leak area in different pipes with longitudinal, circumferential and spiral cracks. It was found that the longitudinal, circumferential and spiral crack areas increase linearly with pressure. The slope of this linear relationship depends on various parameters, including the loading state, pipe dimensions and pipe material properties. The impact of the different parameters on the pressure-area slope was investigated, and an attempt will be made to find an expression to express the pressure-area slope as a function of the parameters. If such a relationship can be established, it will be possible to predict the behaviour of different types of leak openings in different pipes and pipe materials. Keywords Leakage, pipes, crack behaviour, pressure, water distribution systems, uPVC pipes INTRODUCTION Studies show that pressure has a major effect on the rate of leakage from leak openings in pipes and that the leakage exponents in networks can be significantly different from the theoretical orifice exponent of 0.5. A possible reason for this is that leaks are not fixed, but that they rather increase with increasing pressure. This means that leakage is more sensitive to pressure than previously believed and can have serious implications for pressure and water loss management. Using the methods of Finite Element Analysis this study aims to find a relationship between pressure and the leak area in pipes with various leak openings. The study focuses on three types of cracks (longitudinal, circumferential and spiral cracks) and their behaviour under certain conditions. It will concentrate on predicting the effect of various parameters of the pipe and crack, and finding a mathematical relationship to describe this behaviour. Once this relationship is established it will be possible to predict the leakage from a given pipe under various circumstances. BACKGROUND Research conducted previously (1) has shown that leakage exponent in pipes with cracks can differ substantially from the theoretical value of 0.5 as stated in the orifice equation. In that study pipes were modelled using finite element analysis to determine the leakage exponents in pipes with circumferential and longitudinal cracks. This was done with the use of the Torricelli (2) equation for discharge, given by equation [1]
[1] Where Q is the leakage flow rate, Cd the coefficient of discharge, A the orifice area, g the acceleration due to gravity and h the pressure head. The results of that study showed that the loading state of the pipe played a significant role in the leakage exponent for pipes with circumferential cracks. It showed that in the absence of longitudinal stresses the leakage exponent stayed close to 0.5 for short cracks and then dropped below 0.5 for longer cracks, however with the longitudinal stresses present the leakage exponent increased significantly as the length of the cracks increased. The loading state did not play a role in the effect of the leakage exponent for the longitudinal cracks but the exponents did exhibit similar increases to that of the circumferential cracks with longitudinal stresses present. Figure 1 shows the relationship between the leakage exponent and the length of the crack for the two openings.
Figure 1 – Relationship between the leakage exponent and the crack length From Figure 1 it can be seen that the leakage exponents differ substantially from the orifice equation for both longitudinal and circumferential cracks. It can also be shown that longitudinal cracks are the most sensitive to pressure increases. This study concluded that material behaviour may explain the sensitivity of pressure to leakage and that it may have implications on water loss management. FINITE ELEMENT ANALYSIS The finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. This method first introduced in the 1950s and was originally designed to study stresses in complex airframe structures, but has been adapted to a wide field of continuum mechanics (3). Engineering problems of mechanics, solid or fluid have been tackled in the past by deriving differential equations relating the variables of interest. The basic principles used in describing the behaviour of the problems included Newton’s second law of motion, conservation of mass, conservation of energy, and others. Finite elements are used by creating different geometric regions, establishing separate approximating functions in each region and then joining them
together. Thus a finite element model is made up a many small interconnected subregions of elements. A finite element model of a problem gives a piecewise approximation to the governing equations. Thus the finite element method shows that a solution region can be analytically modelled or approximated by replacing it with a grouping of discrete elements. These elements can be put together in a variety of ways, and can be used to represent exceedingly complex shapes. The ABAQUS® Finite Element software was used in this investigation as well as the software program Solidworks where the pipes are modelled according to the dimensions required. In this investigation three types of leak openings were modelled, namely longitudinal, circumferential and spiral cracks. The study is based on a 110 mm diameter class 6 uPVC pipe with a wall thickness of 3 mm the dimensions and properties of which are given in Table 1. Table 1 – Dimensions and properties of a class 6 uPVC pipe Variable Value Pressure head, h 600 Young’s modulus, E 3 Material Properties Poisson’s ratio, ν 0.4 Longitudinal stress, σ l 5.2 900 Length of pipe, Lpipe Geometry of pipe Wall thickness, t 3 Internal Diameter, ID 104 60 Length of crack, Lcrack Crack geometry Width of crack, W crack 1
Unit kPa GPa – MPa mm mm mm mm mm
The pipes that were modelled in this study assumed orifice hydraulics and elastic behaviour of the pipe material. Ten-noded quadratic tetrahedron elements were used throughout the pipes. Sensitivity analyses were done for each configuration to determine the optimal element size. Generally the sizes of the elements in the region around the cracks were 1 mm and 2 mm for all three types of cracks while the rest of the pipe had element sizes of 5 mm. The boundary conditions of the pipe consisted of clamping the pipe along an internal line furthest away from the crack, as well as a point on the outside of the pipe adjacent to the internal line. The pipes were loaded with an internal pressure from the water in the pipe and an applied stress at the ends of the pipe to simulate the longitudinal stresses within the pipe material. The longitudinal stresses are calculated using equation [2], for longitudinal stress for cylindrical pressure vessels (4). [2] Where r is the inner radius of the pipe, P the internal pressure, t the thickness of the pipe wall and σ the longitudinal stress. LEAKAGE THROUGH CRACKS Understanding what role each variable plays in the behaviour of the crack in relation to pressure is important when trying to predict a mathematical relationship describing this behaviour. Initially the base pipe was studied with only the pressure being varied to study
the basic interaction between pressure and leakage and pressure and the leak area. Simulations were then run where each variable was in turn varied one at a time to study how that variable’s effect on the pressure-leakage relationship of the pipe. Table 2 shows the values for the pipe and leak properties that were used for each of the parameters in the analysis. Table 2 – Variations in dimensions and properties of the pipe Varied Property or dimension Pressure head, h (kPa) 0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 Young’s modulus, E (GPa) 3, 10, 30, 90, 200 Poisson’s ratio, ν (-) 0.17, 0.21, 0.29, 0.4, 0.45, 0.5 Longitudinal stress, σ l (MPa) 0, 1.3, 2.6, 3.9, 5.2 (excluding longitudinal cracks) Wall thickness, t (mm) 2, 2.5, 3, 4, 5 Internal Diameter, ID (mm) 20, 30, 40, 50, 80, 104, 150, 200, 250, 300, 350 Length of crack, Lcrack (mm) 10, 30, 60, 100, 130, 150 Width of crack, W crack (mm) 0.1, 0.5, 1, 2 First step was to investigate the behaviour of longitudinal cracks with changes in pressure. Previous research showed that the effect of longitudinal stresses on pipes with longitudinal cracks were negligible thus only changes in circumferential stresses were investigated for these cracks. The relationship between pressure and the leak area is shown in figure 2 and shows that leak area is directly proportional to the pressure head.
Figure 2 – Relationship between the area of a longitudinal crack and the pressure head The Torricelli discharge equation describes the flow coming out of an orifice, such as a hole or a crack. In order to apply the Torricelli equation to leaks in pipes it can be written in a more general form as: [3]
Where c’ is the leakage coefficient and α the leakage exponent. In order to show the leak area equation [3] can be rewritten as: [4] In figure 2 it was shown that the relationship between pressure head and leak area is a linear function, thus the crack area can be written as: [5] Where Ao is the original area of the crack and m is the slope of the graph in figure 2. Substituting equation [5] into equation [4] and simplifying results in: [6] Looking at equation [6] it can be seen that the area plays a larger role in the overall flow out of a leak and the equation is the sum of two terms with different pressure exponents, i.e. 0.5 and 1.5. Each of the parameters stated in Table 1 were varied and the effect of the slope of the pressure-area curve was investigated. Once the slopes for the particular parameter were calculated they were plotted against the parameter. Figures 3 to 8 show the various parameters and how the slope changes as the value of the parameter is varied in the case of longitudinal cracks.
Figure 3 – Change in Young’s modulus
Figure 4 – Change in Poisson’s ratio
Figure 5 – Change in internal diameter
Figure 6 – Change in wall thickness
Figure 7 – Change in length of crack
Figure 8 – Change in width of crack
Figures 3 to 8 show that the slope varies significantly with changes in the different parameters. Results that can be gained from these graphs are the behaviour of the different parameters. In figure 3 Young’s modulus shows an inversely proportional relationship with the area slope. In figure 4 Poisson’s ratio shows a linear relationship with the area slope. In figure 5 the internal diameter shows that the area slope is proportional to the internal diameter that is raised to the power of a third. In figure 6 the area slope is inversely proportional to the wall thickness raised to a power of 1.75. In figure 7 the length of crack shows a 3rd order polynomial in relation to the area slope. In figure 8 the width of the crack shows a linear relationship with the area slope. Using the results obtained above, a single equation that will describe the area slope of the crack as a function of the different parameters is currently being sought. The data will be processed using dimensional analysis as well as regression analysis to describe the area of a longitudinal crack with certain parameters. The same method applied to pipes with longitudinal cracks was applied to pipes with circumferential cracks with an additional parameter of the longitudinal stress. Once again the slopes for the various parameters were calculated and plotted against the particular parameter. Figures 9 to 12 show some of the parameters of the circumferential cracks that differ from the longitudinal cracks.
Figure 9 – Change in Poisson’s ratio
Figure 10 – Change in internal diameter
Figure 11 – Change in length of crack
Figure 12 – Change in longitudinal stress
Again the behaviour of the different parameters can be obtained from the graphs in figure 9 to 12. Figure 9 shows Poisson’s ratio as a 4th order polynomial in relation to the area slope. The internal diameter in figure 10 shows a 6th order polynomial in relation to the area slope. The width of the crack shows a linear relationship with the area slope in figure 14 and figure 15 shows a linear relationship with the area slope and also indicates that the crack will tend to pull itself closed in the absence of the longitudinal stresses. Young’s modulus shows an inversely proportional relationship with the area slope which is the same as for the longitudinal crack. The wall thickness also shows similar results to the longitudinal cracks as the area slope is inversely proportional to it raised to a power of 0.3513. The length of crack also remains a 3rd order polynomial in relation to the area slope. For the third set of cracks, spiral cracks, the effect of the various parameters was investigated as was done with the longitudinal and circumferential cracks. The cracks were at a 45° angle to the horizontal and the variables used for the spiral cracks included longitudinal stress as was done for the circumferential cracks. The slopes were then calculated and plotted against the parameters. The results yielded similar trends to that of the circumferential cracks. Figures 13 and 14 show two parameters that differ from circumferential cracks as their slope changes.
Figure 13 – Change in internal diameter
Figure 14 – Change in longitudinal stress
The internal diameter in figure 13 shows a 5th order polynomial in relation to the area slope. Figure 14 shows a linear relationship with the longitudinal stress but is significantly different from the circumferential cracks effect. The other parameters behave in the same way for the circumferential cracks and have similar trends. As for both longitudinal and circumferential cracks the Young’s modulus shows an inversely proportional relationship with the area slope. Poisson’s ratio is a 4th order polynomial in relation to the area slope which ties in with the circumferential crack. The area slope is inversely proportional to the wall thickness raised to a power of 1.682 which is closely related to the longitudinal crack.
The length of crack is a 3rd order polynomial in relation to the area slope which is the same trend for both the longitudinal and circumferential cracks. The width of the crack shows a 2nd order polynomial with the area slope that again ties in with the trend of the circumferential cracks. DISCUSSION AND CONCLUSION This study investigated the parameters that play a role in the behaviour of cracks in pipes under pressure. Three types of leak openings were discussed, namely longitudinal, circumferential and spiral cracks. It was found that the area of the crack played a larger role in the orifice equation than previously believed and the effect of pressure on the leakage exponent is significant in pipes with cracks. The work shows that the individual properties of the pipe and the geometry of the crack play a more significant role in the area of the crack. The work shows that the Young’s modulus parameter has the same effect regardless of the orientation of the crack in a pipe. It can also be concluded that Poisson’s ratio plays a bigger role when the orientation of the crack is not in the longitudinal direction. The data suggest that the most complicated parameter is possibly the internal diameter of the pipe as it varies significantly across the three types of cracks. The length of the crack as well as the wall thickness has the same effect across the three types of cracks. The work also shows that the width if the crack plays a role in the behaviour of the cracks in pipes and the longitudinal stresses play a role in the circumferential and spiral cracks. ACKNOWLEDGEMENTS This study wishes to gratefully acknowledge the financial support from the NRF programme. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
A.M. Cassa, J. E. van Zyl, R. F. Laubscher, A Numerical Investigation into the Behaviour of Leak Openings in uPVC Pipes under Pressure, WISA2006 The Water Institute of Southern Africa Biennial Conference and Exhibition, South Africa, (2006) A. Chadwick, J. Morfett, Hydraulics in Civil and Environmental Engineering, 3rd ed., E &FN Spon, London, (1999) K. H. Huebner, [et al.], The Finite Element Method for Engineers, 4th ed., John Wiley & Sons, Inc., New York, (2001) J. M. Gere, Mechanics of Materials, 5th ed., Brooks/Cole, United States, (2001) Hibbitt, Karlsson, Sorensen, ABAQUS: Superior Finite Element Analysis Solutions, USA, Copyright 2004 ABAQUS, Inc. S. Timoshenko, J. N. Goodler, Theory of Elasticity, 2nd ed., McGraw-Hill, New York, (1951) M. J. Fagan, Finite Element Analysis Theory and Practice, Pearson Education Limited, Malaysia, (1992) G. E. Dieter, Mechanical Metallurgy, McGraw-Hill, London, (1988)
