Technical Paper by R.W.I. Brachman and R.P. Krushelnitzky STRESS CONCENTRATIONS AROUND CIRCULAR HOLES IN PERFORATED DRAINAGE PIPES

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1 Technical Paper by R.W.I. Brachman and R.P. Krushelnitzky STRESS CONCENTRATIONS AROUND CIRCULAR HOLES IN PERFORATED DRAINAGE PIPES ABSTRACT: Stress concentrations around circular holes in perforated buried pipes obtained from three-dimensional elastic finite element analysis are presented. The maximum stresses around the perforation were found to depend on the thickness and diameter of the pipe, the diameter of the perforation, the circumferential location of the perforation, and the axial and circumferential spacing between perforations. It was found that a stress concentration of 3.0, based on the simple hole in a plate solution, may be conservatively used for most perforated pipes. Better estimates of stress concentration factors derived from finite element analysis are presented. Guidelines are provided for specifying the location, size, and spacing of perforations that minimize the stress concentrations. Two example calculations are presented to illustrate the proposed procedure for estimating the maximum stresses around the perforations. KEYWORDS: Pipe, Perforation, Stress concentration, Leachate collection, Landfill, Heap-leach pad. AUTHORS: R.W.I. Brachman, Assistant Professor, GeoEngineering Centre at Queen s-rmc, Department of Civil Engineering, Queen s University, Kingston, Ontario, Canada, K7L 3N6, Telephone: 1/ , Telefax: 1/ , E- mail: brachman@civil.queensu.ca; and R.P. Krushelnitzky, Graduate Student, GeoEngineering Centre at Queen s-rmc, Department of Civil Engineering, Queen s University, Kingston, Ontario, Canada, K7L 3N6. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 1801 County Road B West, Roseville, Minnesota , USA, Telephone: 1/ , Telefax: 1/ Geosynthetics International is registered under ISSN DATE: Original manuscript submitted 11 October 2001, revised version received 18 April 2002, and accepted 12 June Discussion open until 1 February REFERENCE: Brachman, R.W.I. and Krushelnitzky, R.P., 2002, Stress Concentrations Around Circular Holes in Perforated Drainage Pipes, Geosynthetics International, Vol. 9, No. 2, pp GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

2 1 INTRODUCTION 1.1 Objectives Perforated pipes are used in a variety of drainage applications for both industrial and municipal infrastructure. They are commonly used to provide drainage for foundations, retaining walls, road bases, and septic systems. In most of these cases, the earth pressures acting on the pipe are small and consequently the stresses in the pipe are often ignored. Perforated pipes are also used as drains for leachate collection systems either in solid waste landfills or mining heap-leach pads, or as part of drainage galleries beneath large earth dams. For such cases, the earth pressures acting on the pipes can be enormous, thus, it is important to ensure that the stresses in the pipe remain below allowable levels. The stresses in buried pipes without perforations can be calculated using existing design procedures (e.g., Moore 2001, pp ). However, no method currently exists to calculate the stresses around perforations in a buried pipe. The objective of this paper is to quantify the effects of circular perforations on the stresses within a buried pipe. Results from three-dimensional elastic finite element analysis are presented to assess the impact of perforation size, location and spacing, as well as pipe diameter and thickness on the stresses around perforations. Guidelines for specifying the location, size, and spacing of perforations to minimize the stress concentrations in buried pipes are provided. Two examples are presented to illustrate the proposed procedure for estimating the maximum stresses around the perforation. 1.2 Design Considerations for Perforations in Buried Pipes A perforated buried pipe containing four circular perforations of diameter a located at the quarter points of the pipe (i.e., shoulders θ = 45 and 315, and haunches θ = 135 and 225 ) is shown in Figure 1. Alternative perforation patterns are commonly used and may feature more perforations around the circumference and at different locations. Current design of perforation size and pattern is arbitrary. In general, the perforations should be: 1. large enough to provide adequate flow, 2. small enough to minimize movement of fines from the backfill, 3. large enough to limit the extent of biologically induced clogging (in the case of landfill applications), and 4. small enough to limit increase of stress in the pipe. Panu and Filice (1992) developed a technique to estimate flow rates in perforated pipes, which allows the number and spacing of the perforations to be selected based on flow requirements. To reduce the potential for loss of fines, Cedergreen (1989) suggested the maximum opening of a circular perforation should be less than ½ D 85 of the backfill soil, where D 85 is the particle size of which 85% of the backfill soil material is finer than by mass. With respect to clogging, Fleming et al. (1999) have reported that 8 mm-diameter circular holes were mostly clogged after only one to four years of exposure to landfill leachate. Thus, in landfill leachate collection systems, it is probably 190 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

3 Figure 1. Perforated drainage pipe with four circular holes of diameter α equally spaced around the pipe circumference. desirable to use fewer large diameter circular perforations instead of many small diameter circular holes or rectangular slots in order to minimize clogging of the openings. However, little is known about the influence of perforations on the structural performance of the buried pipe. Guidance on the maximum size, ideal location, and minimum spacing of perforations that limit the stresses in the pipe are required. 2 STRESSES AROUND CIRCULAR HOLES IN A THIN PLATE 2.1 Single Hole in a Plate The simple plane-stress solution for a hole in a thin plate is a useful starting point to consider the stress concentrations around circular perforations. For a circular hole with radius, A, in a thin plate subject to compressive boundary stresses p x and p y distant from the hole (Figure 2a), the radial, σ r, and tangential, σ α, stresses are given by (Poulos and Davis 1974): 1 A 2 3A 4 4A 2 1 A 2 3A 4 4A 2 σ r = --p (1a) 2 x r r r 2 cos2α + --p 2 y r r r 2 cos2α 1 A 2 3A 4 1 A 2 σ -- α p (1b) 2 x r r 4 cos2α -- p 2 y A 4 = + r r 4 cos2α where all other parameters are defined in Figure 2a and compressive stresses are positive. The stresses around a single hole in a plate depend solely on the relative magnitude GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

4 (a) (b) (c) Figure 2. Holes in flat plates subject to boundary stresses, p y > p x : (a) single circular hole in a plate; (b) multiple circular holes in a plate perpendicular to the direction of the larger boundary stress; (c) multiple circular holes in a plate parallel to the direction of the larger boundary stress. of the distant boundary stresses. Table 1 summarizes the tangential stresses at the edge of the hole (i.e., σ α r=a ) for various combinations of boundary stresses. For a hole in a uniformly stressed plate with p x = p y (Case ii), the stresses around the hole are radially symmetric and have a maximum stress concentration of 2.0. When one of the boundary stresses is much larger than the other (either Cases iii or iv), the maximum stress around the hole occurs tangential to the hole in the direction of the larger boundary stress. For example, with p x >> p y the maximum stress occurs tangential to the hole at α = 0 with magnitude 3p x, whereas with p y >> p x the maximum stress of 3p y occurs tangential to the hole at α = 90. For these two cases, tensile stresses of p x or p y act at 90 from the maximum stress. Table 1. stresses. Tangential stresses, σ α, for a hole in a plate subject to different boundary Case Boundary stresses σ α at r = a/2 α = 0 α = 90 i p x p y 3p x p y 3p y p x ii p x = p y = p 2p 2p iii p x >> p y 3p x p x iv p x << p y p y 3p y v p x = 0.5 p y 0.5p y 2.5p y vi p x = 0.2 p y 0.4p y 2.8p y 192 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

5 The distant boundary stresses acting on the plate may be considered analogous to the internal circumferential σ θ and axial σ z stresses in a perforated buried pipe (Figure 1). When subject to earth pressures, the circumferential stresses in the pipe are larger than the axial stresses. Stresses around the perforation are therefore expected to increase at α = 90 (Point B in Figure 1) and decrease at α = 0 (Point A). Cases v and vi in Table 1 show that small amounts of stress in the x direction result in smaller stresses around the hole relative to Case iv. This is similar to the axial stresses that occur in a pipe since (for axial plane strain conditions) the axial pipe stresses on the inside surface of the pipe are equal to Poisson s ratio times the circumferential stresses. Based on such solutions, stress concentration factors of 2.4 and 3.0 have been recommended for circular holes in buried perforated pipes subject to bending and uniform tension, respectively (Chambers and McGrath 1981). 2.2 Multiple Holes in a Plate Stress concentrations around a series of holes in a plate may be larger or smaller than those for a single perforation, depending on the orientation of the line of holes with respect to the direction of the larger boundary stress. Consider the case of a plate with p y > p x as shown in Figure 2. For a single hole in a plate, there is a zone of higher stresses at α = 90 and a zone of lower stresses at α = 0 as a result of the hole (Figure 2a). If a series of holes are oriented perpendicular to the direction of the larger boundary stress (Figure 2b), the zones of higher stress between adjacent perforations can interact and result in larger stress concentrations relative to a single hole (Griffel 1966; Peterson 1974). If the series of holes are oriented parallel to the direction of the larger boundary stress (Figure 2c), the zones of higher stress do not occur between adjacent perforations and do not interact. Rather the lower zones of stress occur between adjacent perforations and theses lower zones may interact as the spacing between perforations becomes smaller, resulting in smaller stress concentrations around the edge of the hole (Griffel 1966; Peterson 1974). Since the circumferential stresses in a buried pipe are larger than the axial stresses, it is expected that closely spaced holes in the circumferential direction of a buried pipe will result in smaller stress concentrations, and closely spaced holes in the axial direction of a buried pipe will result in larger stress concentrations. However, spacings that minimize stress concentrations and the influence of closely spaced holes on pipe deflections are currently unknown. The available solutions for circular holes in plates differ from the case of holes in pipes subjected to earth pressures. They do not properly represent the stresses that act in the buried pipe or include the effects of pipe curvature and thickness. Three dimensional finite element analysis that can fully capture the geometry and boundary conditions will be used to solve the problem of a buried perforated pipe. GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

6 3 THREE-DIMENSIONAL FINITE ELEMENT ANALYSIS 3.1 Problem Description Three-dimensional finite element analysis of a perforated buried pipe (Figure 1) was performed using the program ANSYS (SAS IP 1998). A pipe with outside diameter D o, inside diameter D i, and thickness t containing four equally spaced circular perforations of diameter a around the circumference of the pipe was modelled. The perforations were located at a centre-to-centre spacing L along the axial direction of the pipe. Results are presented for pipes with different thickness, expressed using the ASTM definition of dimension ratio, DR, which is equal to the outside diameter of the pipe divided by the thickness of the pipe. Perforation diameter was normalized by the mean diameter of the pipe, D, while axial spacing was normalized with respect to the perforation diameter, a. 3.2 Verification of Finite Element Mesh Refinement Figure 3 shows one of the finite element meshes used in the analysis. Only one-quarter of the pipe was modelled since the earth pressures acting on the pipe were assumed to be symmetric about vertical and horizontal planes. The pipe was modelled using tennoded tetrahedron continuum elements. Depending on the thickness of the pipe and axial spacing between perforations, 9,000 to 14,000 elements were used. Smooth rigid boundaries were modelled at the crown and springline with displacements constrained in the circumferential, θ, direction and unconstrained in the axial, z, and radial, r, directions. The axial boundaries were also modelled as smooth and rigid with displacements constrained in the axial, z, direction, but unconstrained in the radial, r, and circumferential, θ, directions. Since no known analytical solution exists for the case of a hole in a buried pipe, adequate refinement of the finite element mesh through the thickness of the pipe was verified by comparison with the solution for a thick-walled cylinder subject to a radial compressive external pressure, while mesh refinement around the perforation was checked against the solution for a single hole in a plate (Poulos and Davis 1974). Figure 4a shows the comparison between results from the finite element analysis and the analytical solution for a thick-walled cylinder. Radial, σ r, and circumferential stresses, σ θ, are shown through the thickness of a 225 mm outside diameter DR = 9 pipe subjected to a uniform external pressure, P o. The finite element solution results are within 0.05% of the analytical solution for the level of mesh refinement that is used for the perforated pipe models. Figure 4b shows the comparison between the finite element analysis and the analytical solution for a hole in a plate. Results are shown for stresses tangential to the hole along a radial line oriented at α = 90 for the case with p y >> p x. The finite element solution in this case converged to within 1% of the analytical solution and accurately estimated the maximum stress concentrations factor of 3.0. The minimum number of elements through the thickness of the pipe and around the hole for the final perforated pipe models was selected based on the above comparisons. 194 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

7 Figure 3. Three-dimensional finite element mesh used to calculate stresses in a perforated pipe. The level of mesh refinement for the final models was also verified since almost doubling the number of elements improved the stress estimates by less than 0.8%. 3.3 Simulation of Earth Pressures Acting on the Pipe The earth pressures that act on a deeply buried pipe depend on the stiffness of the pipe relative to the surrounding soil. Rather than explicitly modeling the stiffness of the backfill soil in the finite element model, the earth pressures acting on the pipe were modelled as applied normal and shear stresses acting on pipe. For the deeply buried pipe shown in Figure 5a, vertical σ v and horizontal σ h earth pressures act at some distance from the pipe. To quantify the stresses that act on the pipe, it is mathematically convenient to consider the distant earth pressures as the sum of two components. Using the notation of Moore (2001, pp ), these components consist of: 1. the mean stress σ m (Figure 5b) where the average of the vertical and horizontal GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

8 (a) (b) Figure 4. Verification of finite element mesh refinement by comparisons with analytical solutions: (a) circumferential, σ θ, and radial stresses, σ r, through a thick-walled cylinder with D o = 225 mm and DR = 9; (b) tangential stresses to a hole in a plate along α = 90 with p x = 0 and p y = 1. earth pressures act distant from the pipe, σ m = ( σ v + σ h ) 2 (2a) 196 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

9 (a) (b) (c) (d) (e) Figure 5. Idealized earth pressures: (a) vertical and horizontal earth pressures; (b) mean component of earth pressures; (c) deviator component of earth pressures; (d) normal stresses acting on a pipe from mean component; (e) normal and shear stresses acting on a pipe from deviator component. 2. the deviator stress σ d (Figure 5c) where one half the difference between the vertical and horizontal stresses act distant from the pipe, σ d = ( σ v σ h ) 2 (2b) The normal and shear stresses acting on the exterior surface of the pipe can be expressed as: σ = σ 0 + σ 2 cos2θ (3a) τ = τ 2 sin2θ where: σ 0 = normal stress on the pipe from the mean boundary stresses (Figure 5d); σ 2 = normal stress on the pipe from the deviator boundary stress (Figure 5e); τ 2 = shear stress on the pipe from the deviator boundary stress (Figure 5e); and θ = angle around the circumference of the pipe with respect to the crown. Existing design procedures for buried pipes (e.g., Moore 2001, pp ) use the elastic arching solution of Hoeg (1968) to calculate the stresses acting on the pipe: σ 0, σ 2, and τ 2. The assumption of symmetric earth pressures acting on the pipe is a useful first approximation and should provide reasonable results for pipes with uniform backfill support. This approach may underestimate the stresses acting at the (3b) GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

10 invert for pipes with poor backfill support at the haunches. Finite element analyses were therefore conducted for the perforated pipe subjected to three loading cases: (1) normal stress σ 0 ; (2) normal stress σ 2 cos2θ; and (3) shear stress τ 2 sin2θ. Addition of the stress concentrations around the perforation for each of these three loading cases will provide an estimate of the stresses in a perforated pipe subject to earth pressures under deep burial. 3.4 Constitutive Model The pipe was modeled as a linear elastic material. Although many pipes are made of polymeric materials that typically exhibit viscoplastic stress-strain behaviour, linear elasticity is a useful first step to solving the local response around holes in buried pipes. Linear elasticity also permits superposition of the three earth pressure cases that act on the pipe, and is compatible with current design procedures for buried polymer pipes (e.g., Moore 2001, pp ). The stress concentration factors calculated using a linear elastic model are expected to be higher than the stress concentration factors that would occur in an actual viscoplastic material. This is because local plastic yielding would likely occur in the viscoplastic material resulting in a reduced stress concentration factors. The stress concentration factors were independent of Young s modulus but depended on the value of Poisson s ratio. A Poisson s ratio of 0.46 was used in the analysis. The results reported in this paper are therefore suitable for high density polyethylene pipes under long-term conditions (Zhang and Moore 1997). Slightly larger stress concentrations were found for smaller values of Poisson s ratio. For example, stresses were 32% higher using a Poisson s ratio of 0, and 8% higher using a Poisson s ratio of 0.30, both relative to a Poisson s ratio of For all practical purposes, the results in this paper can be used for a Poisson s ratio greater than RESULTS FROM FINITE ELEMENT ANALYSIS 4.1 Stresses in the Pipe The specific case of a pipe with DR = 9 containing four circular holes (D/a = 8) located at the quarter points of the pipe (i.e., shoulders and haunches) and an axial hole spacing of L/a = 6 is considered to illustrate the stress distributions in a perforated pipe. Results are presented for stresses in the perforated pipe subject to each of the three boundary stresses simulating earth pressures from deep burial. Circumferential stresses, σ θ, in the pipe caused by the normal stress, σ o, acting on the pipe from the mean boundary earth pressures are shown in Figure 6. The perforation results in a local perturbation in pipe stress with its influence concentrated within one perforation diameter from the hole. Maximum compression occurs directly adjacent to the perforation and toward the inside of the pipe (i.e., near Point B located at α = 90 shown in Figure 1). Opposite Point B on the outside surface of the pipe, the circumferential stresses are slightly smaller because of radial effects through the thick- 198 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

11 Figure 6. Contours of the circumferential stresses σ θ for a pipe (D o = 225 mm and DR = 9) with four holes (a = 25 mm and L/a = 6) located at the quarter points subject to normal stress σ 0. ness of the pipe. Circumferential stresses around the edge of the perforation (at α = 0 and 180 ) approach zero. As for a non-perforated pipe, circumferential stresses at the springline and crown of the pipe are slightly larger on the inside surface than the outside arising from variations through the pipe thickness. The stresses at the crown and springline near the perforation are slightly smaller than those distant from the perforation as stress is redistributed and concentrated at the perforation. Stresses calculated along the line of symmetry between perforations are not influenced by the perforation (for the axial spacing of L/a = 6 examined) and were consistent with stresses from analytical solutions for non-perforated pipes. Circumferential stresses in the pipe caused by the normal stress, σ 2 cos2θ, and shear stress, τ 2 sin2θ, acting on the pipe from the deviator component of boundary earth pressures are shown in Figures 7 and 8, respectively. The bending stresses within the pipe are evident for both the σ 2 and τ 2 loading cases. Compressive stresses occur at the interior springline and exterior crown; tensile stresses occur at the exterior springline and interior crown these stresses are largely unaffected by the perforation located at the quarter points. The stresses around a perforation located at the quarter points for the σ 2 and τ 2 cases are very small (close to zero) since the shoulder is a point of inflection. If the perforations were moved closer either to the crown or springline, the stresses at the perforation would be much higher, as the stresses in those areas of the pipe are much higher. GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

12 Figure 7. Contours of the circumferential stresses σ θ for a pipe (D o = 225 mm and DR = 9) with four holes (a = 25 mm and L/a = 6) located at the quarter points subject to a normal stress σ 2 cos 2θ. Figure 8. Contours of the circumferential stresses σ θ for a pipe (D o = 225 mm and DR = 9) with four holes (a = 25 mm and L/a = 6) located at the quarter points subject to a shear stress τ 2 sin 2θ. 200 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

13 When all three load cases (σ 0, σ 2, and τ 2 ) are combined, the maximum compressive stress in the pipe usually occurs at the interior springline, while the maximum tensile stress occurs at the interior crown/invert, and there is some stress concentration around the perforation. These values can change however, based on the location and the geometry of the perforation. For example, maximum compression can occur around a perforation if it is located close to the springline. 4.2 Stress Concentration Factors The major factors influencing the stress concentrations around circular perforations in buried pipes are examined in the following sections. Results are presented in the form of stress concentration factors χ σ0, χ σ2, and χ τ2 for each of the three loading cases σ 0, σ 2, and τ 2. Stress concentration factors, χ, were obtained by dividing the stress at the perforation by the stress at the same location in the pipe without the perforation. Based on the analyses performed, the stress concentration factors for circular perforations were found to depend on: (1) the thickness of the pipe; (2) the diameter of the perforation; (3) the circumferential location of the perforation; (4) the axial spacing between perforations; and (5) the circumferential spacing between perforations. The influences of these parameters on the stresses around the perforations are discussed for each of the loading cases in turn. 4.3 Stress Concentration Factors for Mean Boundary Stress The stress concentration factor χ σ0 for the pipe subject to the mean boundary stress σ 0 is plotted in Figure 9. Results are shown for a range of pipe thicknesses and perforation Figure 9. Stress concentration factor χ σ0 as a function of normalized hole diameter D/a and pipe thickness DR for pipes with four circular holes located at the quarter points subject to a normal stress σ 0. GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

14 sizes that represent a wide range of configurations in common use. Pipes with DR varying from 9 (very thick pipe) to 35 (very thin pipe) were modelled with perforations diameters D/a between 2.7 (very large holes) to 50 (very small holes). Four perforations were located at the quarter points with an axial spacing of L/a = 6. The influence of a greater number of perforations and closer axial spacings will be discussed in Sections 4.5 and 4.6. The magnitude of χ σ0 is independent of the position of the perforation around the circumference since the compression of the pipe subject to the mean boundary stress σ 0 is radially symmetric. The stress concentration factor χ σ0 increases with both decreases in pipe thickness and increases in perforation diameter. Most values of χ σ0 for D/a > 8 are less than 3.0, except for thin pipes (DR = 35) where very large values of χ σ0 occur with large perforations. When the perforation is idealized as a hole in a plate a stress concentration factor of 2.54 is obtained (taking p x = 0.46p y in Equation 1). However, unlike the hole in the plate solution, the stresses in the perforated pipe depend on pipe thickness and perforation diameter. For a thin pipe with DR = 35, the hole in a plate solution overestimates the value of χ σ0 for very small holes (D/a > 40) and underestimates χ σ0 for holes with D/a < 30. For thick pipes (DR = 9), the simple hole in a plate solution overestimates χ σ0 for most practical perforation sizes (D/a > 8). 4.4 Stress Concentration Factors for Deviator Boundary Stresses The stress concentration factors χ σ2 and χ τ2 for the pipe subject to the deviator boundary stresses σ 2 and τ 2 are plotted in Figures 10a and 10b, respectively. The results in Figure 10 were obtained for a pipe containing four circular perforations at an axial spacing of L/a = 6. These stress concentration factors were insensitive to pipe thickness. Since the boundary stresses σ 2 and τ 2 vary around the pipe, the stresses around the perforations will depend on the circumferential position of the perforation around the pipe. Six cases involving different hole locations were considered (Figure 11): Case 1, perforations located at the quarter points (θ = 45, 135, 225, and 315 ); Case 2a, perforations located at ±30 from each springline; Case 2b, perforations located at ±30 from the crown and invert; Case 3a, perforations located at ±15 from each springline; Case 3b, perforations located at ±15 from the crown and invert; and Case 4, perforations located at the crown, springlines, and invert (θ = 0, 90, 180, and 270 ). Because of symmetry, the stress concentration factors are the same for Cases 2a and 2b; likewise, the stress concentration factors are the same for Cases 3a and 3b. The stresses at the perforations are zero for Case 1 since the holes are located at inflection points for the σ 2 and τ 2 load cases. The stress concentration factors are largest for perforations at the crown, springlines, and invert (Case 4) since the boundary stresses acting on the pipe are largest at these locations. Stress concentration factors increase as the perforation moves closer to either the crown or springline and decrease as the perforation moves closer to the quarter points. The stress concentration factors shown in Figure 10 also depend on the size of the perforation. The analysis indicates that for the σ 2 and τ 2 components with constant pipe diameter, larger holes result in smaller stress concentration factors. This is differ- 202 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

15 (a) (b) Figure 10. Stress concentration factors as a function of normalized hole diameter D/a and position around the pipe: (a) χ σ2 with a normal stress σ 2 cos 2θ ; (b) χ τ2 with a shear stress τ 2 sin 2θ. ent from the observation for the stress concentration factor for the mean boundary stress where larger holes resulted in larger values of χ σ0. The nature of the deviator boundary stress distribution is the main factor resulting in smaller stress concentration factors χ σ2 and χ τ2 for larger holes. This can be explained by returning to the simple case of a single hole in a plate (Figure 2a) as an GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

16 Figure 11. Location of perforations analysed for Cases 1 to 4. analogy. It was noted that the stress concentrations depended on the relative magnitudes of the distant boundary stresses acting on the plate. With p y > p x, a decrease in p y with p x constant will result in a smaller maximum stress concentration factor at α = 90. For the perforated pipe, as the perforation becomes larger, the circumferential stresses in the pipe σ θ (similar influence as the distant stresses acting on the plate) decrease since the edge of the perforation is closer to the low-stress region at the quarter points of the pipe. This decrease in σ θ that acts near the perforation results in smaller values of χ σ2 and χ τ2 for larger holes. As the holes become smaller (i.e., larger D/a value), the stress concentration factors increase toward a limiting value for each case. For the holes at the crown and springline, χ σ2 and χ τ2 tend toward a maximum value of Based on the results in Figure 10, it is recommended that whenever possible, the location of the perforations be as close to the quarter points as possible to minimize the stress concentrations around the holes. Perforations located at the crown, springlines, or invert will result in a concentration of the already large stresses at these locations these stresses should be kept below the maximum design stress of the pipe material. Example calculations of stresses in a perforated pipe with holes at the quarter points (minimum stresses at hole) and with holes at the crown, springlines, and invert (maximum stresses at hole) are presented in Section Axial Spacing Between Perforations The results reported in Sections 4.3 and 4.4 were obtained for perforations with a centreto-centre axial spacing L/a = 6 between perforations. Perforations that are too closely 204 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

17 spaced in the axial direction of the pipe can influence each other and result in larger stress concentrations. Specifying a perforation pattern with an axial spacing for which the holes no longer influence each other would be one method to minimize the stress concentrations. Analyses were performed on a series of pipes with DR = 9 with perforations located at the quarter points with varying axial spacings. Figure 12 shows the influence of axial spacing on the stress concentration factor χ σ0. The stresses caused by σ 2 and τ 2 are negligible for perforations located at the quarter points and are not presented. The results in Figure 12 show that stresses around the perforation increase as the axial spacing between perforations decreases. This occurs because the axial rows of perforations are oriented perpendicular to the direction of the larger (i.e., circumferential) stress in the pipe, as described in reference to Figure 2b. As the axial spacing decreases, the areas of maximum stress around the perforation (at α = 90 ) approach and interact with each other, resulting in larger stresses. For axial spacings larger than L/a = 4, the stress concentration factor χ σ0 is essentially independent of axial spacing. Based on these results, it is recommended that the perforation have an axial spacing of at least L/a = 4 to minimize the stress concentrations around the holes. 4.6 Circumferential Spacing Between Perforations The stress concentration factors, χ σ0, in Figure 9 were obtained for four equally spaced perforations around the circumference of the pipe. Since some perforation patterns may involve multiple, closely spaced holes in the circumferential direction, it is Figure 12. Influence of axial perforation spacing on stress concentration factor χ σ0 for a DR = 9 pipe with four holes located at the quarter points subject to a normal stress σ 0. GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

18 useful to consider the stresses around the perforations as they become closer to each other in the circumferential direction. Perforated pipes with an outside diameter of 225 mm and DR = 9 subject to a uniform external pressure σ o were considered. A constant axial spacing of L/a = 6 was used to minimize any increases in stress due to axial interaction of perforations. The diameter of the perforations was kept constant at a = 15 mm. Increasing the number of equally spaced perforations around the circumference of the pipe decreased the circumferential spacing(s) between perforations. Figure 13 illustrates that increasing the number of equally spaced holes in the circumferential direction results in a decrease in χ σ0. Thus, closely spaced perforations in the circumferential direction lead to smaller stress concentrations. This is different from the effect observed when holes are more closely spaced in the axial direction. For the case of closely spaced holes in the circumferential direction, the perforations are more closely spaced in the direction parallel to the direction of the larger (i.e., circumferential) stress in the pipe. As described in Section 2.2 with reference to Figure 2c, perforations more closely spaced in the circumferential direction will result in smaller stress concentrations around the perforation. As the perforations become further apart in the circumferential direction, the stress concentration factor increases until it converges on a constant value, as the case with four holes is nearly the same as for two holes. Provided that the axial spacing between perforations, L/a, is greater than 6, the results in Figure 9 obtained for a pipe with four equally spaced holes will conservatively overestimate the stress concentrations for a pipe with more than four equally spaced holes around the circumference. Figure 13. Influence of circumferential spacing, s, between circular perforations (a = 15 mm, L/a = 6) on stress concentration factor χ σ0 for a pipe with D o = 225 mm and DR = GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

19 It appears that it is beneficial from the perspective of minimizing stress concentrations to use many holes rather than just four holes at the quarter points. This is true when considering only the σ 0 component. When also taking into account σ 2 and τ 2, however, the decrease in χ from the closer spacing may be offset by the increase in stress since the perforations are located closer to the crown, springlines, and invert (Figure 10). 5 PROPOSED PROCEDURE FOR ESTIMATING STRESSES AROUND CIRCULAR PERFORATIONS IN A BURIED PIPE 5.1 Calculating Stresses in Buried Pipes The first step to estimate the stresses acting around the perforation is to calculate the earth pressures acting on the pipe (σ 0, σ 2, and τ 2 ) using the equations given by Moore (2001, pp ): σ 0 σ 2 = = A m σ m A dσ σ d (4a) (4b) τ 2 = A dτ σ d (4c) where: A m, A dσ, and A dτ are factors used to provide stresses acting on the pipe in terms of mean and deviatoric field stresses and are defined as: A m = 21 ( v s ) [ 1 + C( 1 2v s )] (4d) A dσ = 41 ( v s )( 4 + 3C( 1 2v s ) 2F) for a bonded interface A dσ = 12( 1 v s ) ( 2F + 5 6v s ) for a smooth interface A dτ = 16( 1 v s )( F + 1) for a bonded interface A dτ = 0 for a smooth interface = C( 1 2v s )( 5 6v s + 2F) + 2F( 3 2v s ) + 43 ( 4v s ) (4e) (4f) (4g) (4h) (4i) C = ( E s D) [ 21 ( + v s )( 1 2v s )E p A p ] (4j) F = E s D 3 [ 48( 1 + v s )E p I p ] (4k) where: E s = Young s modulus of the soil; ν s = Poisson ratio of the soil; E p = Young s modulus of the pipe, A p = cross-section area per unit length of the pipe; and I p = second moment of area per unit length of the pipe. The second step is to calculate the circumferential stresses σ θ at any location around the pipe, θ, using thin shell theory for each of the three loading cases: σ θ ( θ) = ( σ θ ) σ0 + ( σ θ ) σ2 +( σ θ ) τ2 (5a) GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

20 ( ) σ0 = ( σ 0 D) ( 2t) σ θ D ( σ θ ) σ2 = σ ± t 2t 2 cos2θ (+) for pipe exterior, (-) for pipe interior D 2 (5b) (5c) D D 2 ( σ θ ) τ2 = τ ± (5d) 3t 4t 2 cos2θ (+) for pipe exterior, (-) for pipe interior where (σ θ ) σ0, (σ θ ) σ2, and (σ θ ) τ2 are the circumferential stresses in the pipe from the σ 0, σ 2, and τ 2 loading cases, respectively. The maximum stress acting around the perforation can then be calculated by multiplying the stresses in the pipe where the perforation is to be located from Equations 5b, 5c, and 5d by the appropriate stress concentration factors for the perforation (χ σ0, χ σ2 and χ τ2 ) from Figures 9 and 10, viz: σ p = ( χ σ0 )( σ θ ) σ0 + ( χ σ2 )( σ θ ) σ2 + ( χ τ2 )( σ θ ) τ2 (6) 5.2 Sample Calculations Sample Problem Parameters and Stresses Acting Far From the Perforations Example calculations of stresses in a perforated plastic pipe are presented in order to illustrate the use of the results in this paper. A 225 mm outside diameter, DR = 9, highdensity polyethylene (HDPE) pipe with four perforations equally spaced around the pipe circumference is examined for two cases. The first case is for perforations located at the quarter points and the second case is for perforations located at the crown, springlines, and invert. The HDPE pipe is assumed to have: Young s Modulus of 150 MPa and Poisson s ratio equal to Perforations with a diameter of a = 25 mm are considered (D/a = 8). The axial centre-to-centre spacing between perforations is selected to be L/a = 6 (150 mm apart) to minimize the stress concentrations around the hole. The backfill material around the pipe is assumed to have: Young s Modulus of 20 MPa, Poisson s ratio equal to 0.29, and lateral earth pressure coefficient of 0.3. A vertical surcharge of 550 kpa acts above the pipe, and the water table is assumed to be well below the pipe. The boundary stresses acting on the pipe were calculated to be: σ 0 = MPa, σ 2 = MPa, and τ 2 = MPa, using Equations 4a to 4k. Using these boundary stresses, the resulting circumferential stresses in a non-perforated pipe calculated using Equations 5b, 5c, and 5d are: ( σ θ ) σ2 = 4.05cos ( σ θ ) σ0 = 1.53 MPa 2θ MPa (at pipe exterior) (7a) (7b) ( σ θ ) σ2 = 4.40cos2θ MPa (at pipe interior) (7c) 208 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

21 ( σ θ ) τ2 = 5.07cos2θ MPa (at pipe exterior) ( σ θ ) τ2 = 7.10cos 2θ MPa (at pipe interior) (7d) (7e) The maximum stresses for a non-perforated pipe, are obtained using Equation 5a by adding the stresses from the three loading cases. For the non-perforated pipe, the maximum compressive stress σ Cmax is at the interior springline and the maximum tensile stress σ Tmax is at the exterior crown, and for this case they are equal to: σ Cmax = σ θ (interior springline) = 1.53 MPa 4.40 MPa MPa = 4.23 MPa (8a) σ Tmax = σ θ (interior crown) = 1.53 MPa MPa 7.10 MPa = 1.17 MPa (8b) Case 1 Perforations Located at Quarter Points Stress concentration factors for a pipe with four perforations located at the quarter points (DR = 9, D/a = 8, and L/a = 6) are equal to: χ σ0 = 2.4 (Figure 9) χ σ2 = 0 (Figure 10a) (9a) (9b) χ τ2 = 0 (Figure 10b) (9c) The maximum stresses at the perforation are given by Equation 6 and are equal to: σ p = 2.4( 1.53 MPa) = 3.67 MPa (10) For this case where the perforations are located at the quarter points, the maximum compressive stress that would govern pipe design still occurs at the interior springline. This example illustrates the benefit of placing the perforations at the quarter points Case 2 Perforations Located at Crown, Springlines, and Invert For perforations located at the crown, springlines, and invert of the pipe, the stress concentration factors are equal to: χ σ0 = 2.4 (Figure 9) χ σ2 = 1.9 (Figure 10a) (11a) (11b) χ τ2 = 2.0 (Figure 10b) (11c) The maximum stress at the perforation located at the springline (θ = 90 ) is equal to: σ p = 2.4( 1.53 MPa) + 1.9( 4.40 MPa) + 2.0( 7.10 MPa) = 9.51 MPa = σ Cmax (12) while for the perforation at the crown (θ = 0 ), the maximum stress at the perforation is equal to: GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

22 Results from three-dimensional elastic finite element analysis were presented to clarify the effects of circular perforations on the stresses within a buried pipe. The principal effect of perforations on the structural performance of the pipe is a local perturbation in stresses concentrated around the perforation. Stress concentration factors (χ σ0, χ σ2, and χ τ2 ) were presented for three load cases of pressures acting on the pipe (σ 0, σ 2, cos2θ, and τ 2 sin2θ). Addition of the results from the three cases provides an estimate of the stresses in a perforated pipe under deep burial. The maximum stresses around the perforation were found to depend on: (i) the thickness and diameter of the pipe, (ii) the diameter of the perforation, (iii) the circumferential location of the perforation, (iv) the axial spacing between perforations, and (v) the circumferential spacing between perforations. Larger stress concentrations χ σ0 were found for thinner pipes when the pipe was subjected to the mean component of normal stress on the pipe σ 0. Stress concentration factors χ σ2 and χ τ2 were independent of pipe thickness. Stress concentration factor χ σ0 increased as the perforation diameter was increased, whereas stress concentration factors χ σ2 and χ τ2 decreased as the diameter of the perforations was increased. For the practical case with perforations located at the quarterpoints, increasing perforation size will increase the stress concentrations. For this case, whenever possible, the holes should not be greater than 1/20 of the pipe diameter to minimize the stress concentrations. Larger holes can be used, however, if larger holes are required thicker pipes should be used, as the stress concentrations are smaller for thicker pipes. Stress concentrations are largest for perforations placed at the crown, invert, and springlines and are smallest for perforations located at the quarter points (shoulders and haunches). Decreases in axial spacing between perforations resulted in larger stress concentrations, whereas decreases in circumferential spacing between perforations (obtained by increasing the number of perforations around the pipe) resulted in smaller stress concentrations. This difference occurs because of the alignment of the line of holes relative to the direction of the circumferential stresses in the pipe. An axial centre-toσ p = 2.4( 1.53 MPa) + 1.9( 4.40 MPa) + 2.0( 7.10 MPa) = 2.17 MPa = σ Tmax (13) For this case with perforations at the crown, springlines, and invert, the maximum compressive and tensile stresses both occur at the perforations. If maximum allowable stresses for polyethylene are 9 and 4.4 MPa for compression and tension, respectively, the stresses for the pipe in Case 2 exceed the allowable compressive stresses, and the design would have to be modified. Using a thicker pipe could decrease the stresses in the pipe, but if that decrease were not great enough then the maximum overburden pressure would likely have to be reduced for this pipe. This example illustrates that the influence of perforations should not be neglected if they are located near the crown or springline. 6 CONCLUSIONS 210 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

23 centre spacing of at least four times the perforation diameter between perforations is suggested to minimize the stress concentration around the hole. A stress concentration factor of 3.0 obtained from the simple hole in a plate solution can be conservatively used for most perforated pipes, provided that the holes are spaced at least 4a apart on centers. However, the hole in a plate solution is not conservative for very thin pipes (DR 35) with large holes (D/a < 12). Better estimates of stress concentration factors derived from the finite element analysis were presented and can be used to design various sized pipes with circular perforations of varying diameter, axial location, circumferential location, and circumferential spacing. Careful inspection is required of both the fabrication of perforations (obtained by drilling holes in the pipe) and the placement of pipes in the field to ensure the perforations are at the intended locations. The authors are aware of cases where perforations intended to go at the quarter points were located too close to the springline because of machining errors or improper orientation of the pipe during installation. Both of these cases would have resulted in overstressing of the pipe if the improper perforation locations were not detected and corrected. ACKNOWLEDGMENTS This research was funded by the Natural Sciences and Engineering Research Council of Canada. REFERENCES Cedergreen, H.G., 1989, Seepage, drainage, and flow nets, Third Edition, John Wiley & Sons, New York, New York, USA, 465 p. Chambers, R.E. and McGrath, T.J., 1981, Structural Design of Buried Plastic Pipe, International Conference on Underground Plastic Pipe, Schrock, B.J., Editor, ASCE, Proceedings of the International Conference on Underground Plastic Pipe: March /April 1981, New Orleans, Louisiana, USA, pp Fleming, I.R., Rowe, R.K., and Cullimore, D.R. 1999, Field Observations of Clogging in a Landfill Leachate Collection System, Canadian Geotechnical Journal, Vol. 36, No. 4, pp Griffel, W., 1966, Handbook of formulas for stress and strain, Frederick Ungar Publishing Company, New York, New York, USA, 418 p. Hoeg, K., 1968, Stresses Against Underground Structural Cylinders, Journal of the Soil Mechanics and Foundations Division, SM4, Proceedings of the American Society of Civil Engineers, pp Moore, I.D., 2001, Buried Pipes and Culverts, Chapter 18, Geotechnical and Geoenvironmental Engineering Handbook, Rowe, R.K., Editor, Kluwer Academic Publishing, Norwell, Georgia, USA, 1088 p. Panu, U.S. and Filice, A., 1992, Techniques of Flow Rates into Draintubes with Cir- GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

24 cular Perforations, Journal of Hydrology, Vol. 137, pp Peterson, R.E., 1974, Stress concentration factors: charts and relations useful in making strength calculations for machine parts and structural elements, Wiley, New York, New York, USA, 317 p. Poulos, H.G. and Davis, E.H., 1974, Elastic Solutions for Soil and Rock Mechanics, Wiley, New York, New York, USA, 411 p. SAS IP, Inc., 1998, ANSYS Analysis and Procedures Guides, Third Edition, Release 5.5, Canonsburg, Pennsylvania, USA. Zhang, C. and Moore, I.D., 1997, Nonlinear Mechanical Response of High Density Polyethylene. Part I: Experimental Investigation and Model Evaluation, Polymer Engineering and Science, Vol. 37, No. 2, pp NOTATIONS A = radius of circular hole in plate (m) A m = mean field stress factor (dimensionless) A dσ = normal deviatoric field stress factor (dimensionless) A dτ = shear deviatoric field stress factor (dimensionless) A p = cross section area per unit length of pipe (m) a = diameter of circular perforation in pipe (m) C = relative circumferential stiffness (dimensionless) D = mean diameter of pipe (m) D i = inside diameter of pipe (m) D o = outside diameter of pipe (m) D 85 = particle size of which 85% of material is finer than by mass (m) DR = dimension ratio of pipe (dimensionless) E p = Young s modulus of pipe (Pa) E s = Young s modulus of soil (Pa) F = relative bending stiffness (dimensionless) I p = second moment of area per unit length of pipe (m 3 ) L = axial spacing between centres of perforations (m) P o = uniform external pressure acting on pipe (Pa) p x = boundary pressure acting on plate (Pa) p y = boundary pressure acting on plate (Pa) 212 GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO. 2

25 r = cylindrical coordinate in radial direction from centre of pipe or centre of hole in plate (m) s = circumferential spacing between centres of perforations (m) t = thickness of pipe (m) z = cylindrical coordinate in axial direction along pipe (m) α = cylindrical coordinate in circumferential direction around perforation ( ) θ = cylindrical coordinate in circumferential direction around pipe with respect to the crown ( ) ν p = Poisson s ratio of pipe (dimensionless) ν s = Poisson s ratio of soil (dimensionless) σ = normal stress on the pipe (Pa) σ C max = maximum compressive stress (Pa) σ d = deviator stress (Pa) σ h = horizontal earth pressure (Pa) σ m = mean stress (Pa) σ P = maximum stress around perforation (Pa) σ r = radial stress in pipe (Pa) σ T max = maximum tensile stress (Pa) σ v = vertical earth pressure (Pa) σ z = axial stress in pipe (Pa) σ θ = circumferential stress in pipe (Pa) (σ θ ) σ0 = circumferential stress in pipe from σ 0 component (Pa) (σ θ ) σ2 = circumferential stress in pipe from σ 2 component (Pa) (σ θ ) τ2 = circumferential stress in pipe from τ 2 component (Pa) σ 0 = normal stress on the pipe from the mean boundary stress (Pa) σ 2 = normal stress on the pipe from the deviator boundary stress (Pa) τ = shear stress on the pipe (Pa) τ 2 = shear stress on the pipe from the deviator boundary stress (Pa) χ σ0 = stress concentration factor for σ 0 component (dimensionless) χ σ2 = stress concentration factor for σ 2 component (dimensionless) χ τ2 = stress concentration factor for τ 2 component (dimensionless) GEOSYNTHETICS INTERNATIONAL 2002, VOL. 9, NO

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