Numerical Methods in Soil Mechanics 26.PDF Numerical Methods in Geotechnical Engineering contains the proceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE 2014, Delft, The Netherlands, 18-20 June 2014). It is the eighth in a series of conferences organised by the European Regional Technical Committee ERTC7 under the auspices of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The first conference was held in 1986 in Stuttgart, Germany and the series has continued every four years (Santander, Spain 1990; Manchester, United Kingdom 1994; Udine, Italy 1998; Paris, France 2002; Graz, Austria 2006; Trondheim, Norway 2010). Numerical Methods in Geotechnical Engineering presents the latest developments relating to the use of numerical methods in geotechnical engineering, including scientific achievements, innovations and engineering applications related to, or employing, numerical methods. Topics include: constitutive modelling, parameter determination in field and laboratory tests, finite element related numerical methods, other numerical methods, probabilistic methods and neural networks, ground improvement and reinforcement, dams, embankments and slopes, shallow and deep foundations, excavations and retaining walls, tunnels, infrastructure, groundwater flow, thermal and coupled analysis, dynamic applications, offshore applications and cyclic loading models. The book is aimed at academics, researchers and practitioners in geotechnical engineering and geomechanics.
Anderson, Loren Runar et al "NON-CIRCULAR LININGS AND COATINGS" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 26-1 Two important deformations of buried flexible pipes Figure 26-2 Crack at the invert of the lining caused by increase in radius of curvature ©2000 CRC Press LLC CHAPTER 26 NON-CIRCULAR LININGS AND COATINGS Linings and coatings are not always circular in cross section Non-circular pipes are discussed in Chapter Circular linings (encased in host pipes) are discussed in Chapter 11 Following is a discussion of the effects of non-circularity on linings and coatings Coatings prevent external corrosion of the host pipe and stiffen the ring The performance limit is excessive cracking Linings prevent leaks and stiffen the ring Two basic performance limits of linings are cracking and collapse (inversion) Following are two analyses of non-circular linings and coatings, rigid and flexible RIGID LININGS AND COATINGS Rigid linings and coatings are applied to thin-wall s teel pipes that are to be buried The linings and coatings are of high-strength Portland cement mortar Purposes of the linings and coatings are: to provide adequate stiffness of the pipe in order to handle and transport the pipe, and to install and backfill; and, to protect the steel against corrosion and against abrasion caused by sediment in the fluid flow and possibly by cavitation For design, the first two performance limits are hoop strength against internal pressure and ring compression strength against external pressure A third performance limit is ring stiffness The pipe must be stiff enough to be handled and installed A fourth performance limit is excessive cracking of the mortar Pipe engineers consider performance limit to be crack widths greater than 1/16 inch This is a rule of thumb that is intended to avoid the break-out of shards in the lining and to prevent circulation of corrosive fluid through the crack to the steel Shards are usually caused by a dent or inversion in the pipe wall Cracks in the coating that are wider than 1/16 inch might allow electrically conductive ©2000 CRC Press LLC groundwater and transient electric al currents to get to the steel Some pipe engineers still hold to the very conservative 0.01-inch crack as performance limit Compared with the 1/16 crack (0.0625) the safety factor is more than Clearly, a broad safety zone is available for mitigation A paradox arises between ring stiffness and mortar cracking Because the mortar is thicker than the steel, ring stiffness is affected primarily by thickness of the mortar However, the thicker the mortar, the wider are the cracks at any given ring deflection The optimum coating thickness between desirable stiffness and undesirable crack width comes from experience Both the minimum stiffness and the maximum crack width are related to ring deformation Of course, ring deformation also depends upon loads and embedment soil Ring deformations in the following analyses are either: a) ellipse, or b) D-bedding See Figure 26-1 The ellipse is the basic deformation under vertical soil pressure The D-bedding (so-called by rigid pipe industries) is a worst-case deformation (usually labeled "impermissible") The pressure, P, can be greater than the soil prism load when surface live loads pass over, or when a deflected pipe is rerounded by internal pressure Notation r = radius to the neutral surface of circular pipe, r' = maximum (or minimum) radius, d = ring deflection = D/D where D = 2r, D = vertical decrease in diameter of deflected ring, t = thickness of each lamina in the wall, w = width of a crack in the mortar (See Figure 26-2), c = distance from the neutral surface of the wall to the exposed mortar surface For worstcase analysis, this is the thickness of mortar, either lining or coating It is sufficiently accurate to assume that the steel is the neutral surface Whether a radius to the neutral surface is to the inside or the outside surface of the steel makes little difference The minimum radius of curvature for the ellipse is at spring lines (9:00 and 3:00 o'clock) where, from Figure 26-1a, r' = rx and, r'/r = (1-d)2/(1+d) Crack Width From the geometry of Figure 26-3, assuming conservatively that the crack penetrates to the steel, c is the coating mortar thickness, and the crack width is w = cDq where Dq = (1/r'-1/r)inch Substituting with r in inches, w = c(1/r - 1/r')inch2 (26.1) Radius of Curvature The maximum radius of curvature, r', is at 6:00 o'clock as follows: Ellipse, r'/r = (1+d)2/(1-d) (26.2) D-bedding, r/r' = - 5d (26.3) For a derivation of Equation 26.3, see Figure 26-1a From Appendix A, the maximum moment is at the invert, B, and is, M = 0.5872Pr2 From mechanics of solids, M/EI = 1/r - 1/r' where I = mom of inertia, E = mod of elasticity, r = circular radius, r' = radius at B Substituting for M, and solving, r/r' = - 0.5872(Pr3/EI) From Appendix A, d = 0.116(Pr3/EI) where d = ring deflection In terms of d, eliminating (Pr3/EI), r/r' = - 5d ©2000 CRC Press LLC (26.4) The minimum radius, rx, for the D-bedding does not differ significantly from an ellipse CRACKS IN MORTAR LINING Structural cracks in the lining can be performance limits if shards break out Potential shards can be detected by inspection and by rapping on the lining inside of the pipe The lining is usually high-quality mortar Density is achieved by centrifugally spinning the sand-cement slurry into place Water is squeezed out The inside of the pipe is a satinsmooth, cement-rich surface Hair cracks develop if the lining dries out For storing and transporting pipe sections, the ends are often sealed with sheets of plastic to reduce drying Hair cracks may develop during backfilling because of ring deflection Pressure in the pipe rerounds the pipe and closes deflection cracks Cracks narrower than 1/16 inch are closed by autogenous healing when the mortar is wet Autogenous healing is the formation of silica gel by hydration Small cracks in the lining usually don't penetrate to the steel Width of Cracks in Mortar Lining Cracks open when the ring is deflected Typically the widest cracks are inside the pipe at 6:00 and 12:00 o'clock See Figure 26-2 It is reasonable to assume that the inside surface of the steel is the neutral surface In the following analyses, two deformations are compared: elliptical, and Dbedding From Equations 26.1, 26.2, and 26.3, the relationship between width of crack, w, and ring deflection, d, can be found If the ring is deformed, the relationship between w and r’ may be more dependable From inside the pipe, r’ can be found by laying a horizontal cord (straightedge) at B, Figure 26-3 Free-body-diagram of a section of pipe wall at spring line The notation shown is used in analysis of crack width, w, in the mortar coating w = cDq Figure 26-4 Exposed cracks in a coating that is deflected into an ellipse, shown exaggerated ©2000 CRC Press LLC and measuring the vertical middle ordinate from the cord to the lining surface Then r' = L2/8e where L is cord length, and e is the middle ordinate from cord to pipe surface It is assumed that radius, r', is circular the outside surface of the steel that the coating is entirely in tension It is noteworthy that the deflected radius at spring line is roughly the same for the ellipse and the D-bedding The ellipse is analyzed in the following Caveat Once the lining is cracked, a plastic hinge may form in the steel with a radius smaller than the mean radius subtended by the cord The plastic hinge can be analyzed from yield stress of the steel and the radius of curvature A plastic hinge is recognized visually as the crimped rim around a dent Example: What is the width of a crack in the 1.5-inch mortar coating on the following pipe ID = 72 inches = nominal diameter, D = 74 inches = outside diameter of steel (assumed neutral surface, NS) w = width of crack, r = 37 = initial circular radius of the NS, r' = deflected radius of elliptical NS at spring lines, c = 1.5 from NS to the outside surface of coating (coating thickness), d = ring deflection (percent) of elliptical cross section Example: What are the widths of cracks in the lining of a pipe at the invert? The pipe is nominal 42-inch (inside) diameter r = 21.5 inches to the neutral surface (assumed to be at the inside surface of the steel), c = 0.5 inch = thickness of the lining From Equations 26.1, 26.2, and 26.3, in terms of ring deflection, d, the maximum radii of curvature, r', and corresponding widths of crack, w, are listed in Table 26-I CRACKS IN MORTAR COATING The coating on some steel pipes is tape Improvements in the protective qualities have made tape attractive However, if ring stiffness is of concern, or if increased weight of pipe is desired, then mortar coating is used From Figure 26-3, w = cDq where Dq = q'-q = (1/r'-1/r)inch w = c(1/r'-1/r)inch (26.5) where r' = rx in Figure 26-1a This r' is approximately the same for both the ellipse and the Dbedding of Figure 26-1 For an ellipse, the deflected radius, r', is, r'/r = (1-d)2/(1+d) (26.4) Substituting values into Equation 26.1 and neglecting the small value of d2, Excessive cracking is performance limit Excessive cracking allows corrosion of the steel from aggressive groundwater and electrical currents in the soil As with linings, cracks are caused either by ring deflection or by flat spots (dents) where K is a ring deflection factor in brackets, [ ] See Table 26-II for values Ring Deflection of Ellipse and D-Bedding: As a result of ring deflection, the widest cracks in the coating occur near spring lines Figure 26-4 is an exaggerated sketch For worst-case analysis, it is assumed that the neutral surface is near enough to In Table 26-II, e is the middle ordinate from a twelve inch cord on the inside of the pipe to the inside surface of the pipe See Figure 26-5 The neutral surface (NS) is approximately at the outside surface of the steel where r = 37 inches If the ©2000 CRC Press LLC w = (r/c)[3d/(1-2d)] = K(r/c) (26.6) Table 26-1 LINING CRACK WIDTHS AS FUNCTIONS OF RING DEFLECTION at the invert of 42D pipe with 0.5-inch mortar lining Elliptical deformation r' (inch) w (in) 22.154 0.0007 22.825 0.0014 23.515 0.0020 24.223 0.0026 24.951 0.0032 25.700 0.0038 26.468 0.0044 27.258 0.0049 28.070 0.0054 d (%) D-bedding deformation r'(inch) w (in) 22.632 0.0012 23.889 0.0023 25.294 0.0035 26.875 0.0047 28.667 0.0058 30.714 0.0070 33.077 0.0081 35.833 0.0093 39.091 0.0105 Cracks in the D-bedding deformation are wider by roughly two to one A 0.01-inch crack in D-bedding would occur at ring deflection of d = 8.6% A more dependable equivalent is the long radius of roughly 37 found by measuring middle ordinate from a cord (straightedge) at the invert Table 26-2 COATING CRACK WIDTHS — ELLIPSE DEFLECTION at the spring line, of 72D pipe, 1.5-inch mortar coating d (%) 7* 25** K 0.0306 0.0625 0.0957 0.1304 0.1667 0.2045 0.2442 1.5000 w (inch) 0.0000 0.0012 0.0025 0.0039 0.0053 0.0068 0.0083 0.0099 0.0608 r' (inch) 37.00 35.90 34.84 33.80 32.79 31.80 30.84 29.91 16.65 e (inch) 0.500 0.516 0.532 0.549 0.566 0.584 0.603 0.623 1.150 * w = 0.01-inch crack ** w = 1/16-inch crack More precisely, d = 25.34%, but such precision is not justified The inaccuracies of these approximate ellipses increase as ring deflections increase Moreover, the assumption of ellipse is questionable — especially after the mortar becomes cracked at large ring deflections K = 3d/(1-2d) = ring deflection factor for ellipse r' = radius at springline e = middle ordinate at the invert from a 12-inch cord to the lining ©2000 CRC Press LLC thickness of steel plus lining is 1.0 inch, the inside radius is r'-1 inch for this 72D pipe with 1.5-inchthick coating In this example, from geometry, in terms of the middle ordinate, e, from a 12-inch cord to the lining, pipe Thereafter the soil holds the pipe in its nearlycircular shape Consequently, service life is not shortened by cycles of dewatering Flat Spots r'-1 = 18inch /e (26.7) Equation 26.7 provides a means from inside the pipe to estimate the width of cracks in the coating For this example, if inside the pipe, e = 0.625, from Equation 26.7, r' = 29.8 in, which is then substituted into Equation 26.5 to find that w = 0.01 inch A list of values appears in Table II for this particular example The procedure applies to other mortarcoated pipes Excessive Crack Widths What is excessive crack width? For linings, experienced pipeline engineers say "any crack wider than 1/16 inch." A more conservative specification is the 0.01-inch crack Water in the pipe causes the lining to expand and close cracks Autogenous healing (hydration of silicates in the mortar) seals small cracks For coatings, also, the 0.01-inch crack is conservative For 1/16 inch coating cracks, conditions for corrosion are affected by: electrical ground current, the water table, and water quality Cracks in coatings not penetrate to the steel if the only loads are soil pressure It might be argued that when the pipe is pressurized internally, the steel stretches, and cracks in the coating widen and penetrate to the steel However, pressure in the pipe rerounds the pipe So cracks in the coating are narrowed by the pressure Dents in the pipe are often called "flat spots" even though they are not flat Inside the pipe it is possible to measure the radius of curvature of the "flat spot." Then from Equation 26.1 the width of cracks in the mortar can be estimated An alternate rationale is Figure 26-6 which is a grossly exaggerated sketch of a flat spot in a coating The crack at B in the middle of the flat spot is not a problem because it opens to steel — not to groundwater The two cracks at the ends of the flat spot each open at roughly half the middle crack width Therefore, if the lining is half as thick as the coating, the crack widths, w, in the coating are roughly equal to the crack width measured in the lining at the middle of the flat spot If the flat spot is circular rather than long, cracks in the lining appear as a starburst Of course, crack widths cannot be predicted precisely The above rationale is conservative Flexible Linings and Coatings Development of flexible linings and coatings, such as epoxies and tapes, is significant Not only is corrosion resistance achieved, but resistance to impacts and pipe deformations is impressive A backhoe tooth can dent a mortar-lined steel pipe leaving a starburst in the mortar lining, but without damage to the tape coating Inversion Analysis It is noteworthy that cracks are widest when the pipe is emptied after hydrotests In service, with pressure in the line, cracks are narrow Cycles of dewatering cause pipe-soil interaction to stabilize at crack widths that are less than they were immediately after hydrotests Over time, the soil migrates into place against the pressure-stiffened ©2000 CRC Press LLC One cause of inversion is a concentrated reaction, Q, (hard spot) See Figure 26-7 From Appendix A, the moment at B is, MB = 0.3Qr For worst-case analysis, mortar-to-steel bond is discounted For inversion analysis, see Figure 26-8 In the example that follows, the data are: Figure 26-5 Procedure for finding ring deflection of an elliptical pipe by laying a 12-inch cord, at the spring line and measuring e Figure 26-6 Exaggerated sketch of flat spot in a mortar coating ©2000 CRC Press LLC Figure 26-8 Equivalent wall section in mortar for stiffness analysis of the mortar coating at invert, B ©2000 CRC Press LLC Q P W sf n Es Em tl ts tc rl rs rc = = = = = = = = = = = = = P(OD) + W per unit length, soil pressure on the pipe, wt (pipe + contents) per unit length, 1200 psi = yield strength of mortar, 7.5 = Es /Em, 30(106) psi = mod/elast, steel, 4(106) psi = mod/elast, mortar, 0.75 = thickness of the lining, 0.32 = thickness of the steel, 1.50 = thickness of the coating, 36.38 = radius to center of lining, 36.91 = radius to center of steel, 37.82 = radius to center of coating Example: What is the line reaction, Q, at inversion? First, find the moment resisting ring stiffness of the critical mortar coating as a fraction of the total ring stiffness which must resist moment, MB = Qr/4 See Figure 26-7 and Appendix A Total ring stiffness = S Emb(t/r)3/12 Lining Steel Coating b 7.5 t r 0.75 36.38 0.32 36.91 1.50 37.82 TOTAL Emb(t/r)3 35 20 250 305 The last column, Emb(t/r)3, is ring stiffness, 12EI/r3, where moment of inertia is I = bt3/12 For ratios, the 12 is factored out Moment in the Coating: (Coating is critical) Mc = (250/305)MB = 0.82MB = Qr/4 Coating yields (i.e., cracks) at Mc = s f (tc)2/6 = 450 lbin/in Since r = 37.82 for the coating, Q = 570 lb/ft Q is the critical line load that cracks the coating The weight of pipe and contents is W = 2370 lb/ft Clearly, the reaction must not be the concentrated Q-reaction of Figure 26-7 The pipe must be supported by soil under the haunches As soon as the coating cracks at the invert, the lining and steel let go, and the Qreaction is distributed over an area ©2000 CRC Press LLC Remedies Coating Ideally, wherever cracks are excessive in the mortar coating, the pipe could be uncovered and re-rounded by internal pressure Then the embedment could be carefully re-compacted while the pipe is circular If it is not practical to pressurize the pipe, it may be possible to reround the pipe by carefully monitoring ring deflection during re-compaction of sidefill soil Lining Linings tend to expand in a moist environment thereby reducing crack widths One basic concern is cracks so wide (>1/16 inch) that water could circulate through cracks to the steel Another basic concern is shards of lining that might break out Break-out would require suction such as the Bernoulli effect Loose shards can be detected by the flat, hollow sound when the lining is rapped with a hard object at locations of multiple cracks, parallel or starburst Wide cracks may be associated with inversions (Q-reactions) or flat spots (dents) If the pipe is not uniformly bedded, but is propped up on high spots, and if ring deflection is excessive during installation, hydrotests tend to reround the pipe by forcing it down against the high spots The result could be inversion at the high spots, and the potential for shards to form General Uniform bedding is essential Ring deflection during installation should be limited by specification Good embedment should be placed with care The reason for an embedment is remembered as P5 — packing for placing, positioning and protecting the pipe In fact, the embedment is part of the conduit — not just pressure on a pipe Example A tape-coated, mortar-lined steel pipe is installed and hydrotested Then it is inspected and found to have excessive ring deflection and flat spots at various locations under the haunches Maximum allowable ring deflection was conservatively specified as 3% The embedment had been placed by covering the pipe with sand, then jetting down to the level of the invert with high-pressure water jets that flush soil under the haunches What caused excessive ring deflection and flat spots? What can be done to remedy or mitigate it? Class pipe: 150 to 200 D = 43 inches to neutral surface of the wall, t = 0.175 = thickness of the steel, tl = 0.500 = thickness of the lining, E = 30(106) psi = mod/elast, steel, El = 3(106) psi = mod/elast, mortar lining, n = 0.3 = Poisson ratio, rs = 21.59 = mean radius of steel, rm = 21.25 = mean radius of mortar lining, r = 21.5 = approximate radius to NS, t = 0.2645 = wall thickness of equivalent unlined steel pipe crack excessively Small cracks (less than 1/16inch-wide) close in time by autogenous healing (hydration of the silicates in the cement.) Cracks should be no wider than 1/16 inch Figure 26-2 shows a crack in the lining at the invert Figure 26-9 shows crack width, w, as a function of ring deflection, d, when the pipe is deflected into an ellipse The ordinate at right is the ratio of radii, maximum ry to mean circular, r The width of crack, w, can be found either as a function of the ratio of radii, ry /r; or as a function of ring deflection, d The maximum radius, ry, of a flat spot or inversion in the pipe can be found by measuring the ordinate to the pipe wall from the middle of a cord of known length In this pipe (r = 21.5, t = 0.59 lining to center of the steel), for a 1/16- inch crack to open, the pipe wall must invert to a radius of ry = -17 in See Figure 26-10 For cracks to occur in the steel, radius ry must be less than 2.5t Manufacturers recommend a lower limit of ry = 7t Hoop tension stress under internal pressure is not excessive Longitudinal stresses are caused by temperature change, internal pressure change, and longitudinal bending moment Beam action is not anticipated if the pipe is supported by soil bedding and soil under the haunches The longitudinal design is OK Ring compression stress due to external soil pressure is OK Installation and Hydrotesting Design of the Pipe — OK Composite ring stiffness is approximately EI/r3 = 4.59 psi For the steel only, EI/r3 = 1.33 psi For the mortar lining only, EI/r3 = 3.26 psi The composite ring stiffness is the sum, 4.59 psi It is assumed that there is no bond between mortar lining and steel Because there may be bond, and because shrinkage cracks are disregarded, values for ring stiffness are not precise The composite ring stiffness of 4.59 is equivalent to D/t = 163 for unlined steel pipes Steel pipe engineers recommend a maximum D/t = 288, or, with care in installation, up to D/t = 325 Ring stiffness is OK Ring deflection is limited by specification to a maximum of 3% to be sure that the lining will not ©2000 CRC Press LLC The buried pipe-soil interaction has stabilized The sidefill is compacted to 90% standard density Soil arching action has been achieved No increases in deformations or stresses are anticipated The maximum ring deflections and flat spots occurred during installation or hydrotesting Flat spots were discovered in the invert See Figure 26-11 The result could be disbonding and spalling of the lining due the Bernoulli lift and to vibrations (turbulence) in water flow Long flat spots should be rerounded and lining replaced if width of the flat spot is greater than roughly b = inches Circular flat spots should be rerounded and lining replaced if diameter of the flat spot is greater Figure 26-9 Width of crack, w, in the lining at the invert (bottom) of the pipe due to ring deflection from circular radius to invert radius (or maximum measured), and from circle to ellipse For this pipe, r = 21.5, t = 0.59 to center of the steel ©2000 CRC Press LLC Figure 26-10 Inversion of the invert of a pipe showing (top) the circular ring radius, r, and the inverted (negative) radius, ry, of the invert at 1/16-inch crack in the half-inch-thick mortar lining r = 21.5, ry = 17); and showing (bottom) the critical radius of steel at cracking if critical strain is e = 20% ©2000 CRC Press LLC Figure 26-11 Flat spot at the invert of a pipe ©2000 CRC Press LLC than roughly b = inches For different data, especially for pressures greater than 100 psi, values for b must be recalculated Voids were left under the pipe during installation Figure 26-12 shows a void between reactions under the pipe such that the pipe performs as a beam If length of the beam is greater than about L = 58 ft, longitudinal stress in the steel exceeds yield Loose soil under the haunches reduces beam stress Compacted soil under haunches eliminates beam action completely Voids under the pipe contribute to inversion of the pipe invert If loads and reactions are as shown in Figure 26-12, the line reaction is F = wL/LB If soil cover is ft at 110 pcf, and the pipe is full of water, w = 3076 lb/ft With some loose soil support under the haunches at angle of repose, the net load is reduced to 40% (or less), and w = 1.2 k/ft Assuming L/LB = 2, the reactions are k/ft If reactions are line reactions as shown under the pipe cross section at the right of Figure 26-12, inversion occurs at F = 1.4 k/ft Clearly, 2.4 k/ft would invert the pipe If the reaction is distributed over a bedding angle of 90o, inversion occurs at roughly F = k/ft With the more reasonably distributed reaction, inversion would be unlikely — until, that is, the pipe is hydrotested Inversion is possible after the pipe is hydrotested (125% of its designed internal pressure) if ring deflection is significant If ring deflection is 3%, internal pressure rerounds the pipe and increases vertical diameter by 1.3 inch Load on this pipe is increased by an additional k/ft as the pipe tries to lift a soil wedge or as it compresses soil as shown on the top and bottom of the cross section of Figure 2613 Assuming L/LB = 2, the reactions are F = 3.2 k/ft Inversion of the invert occurs at roughly F = k/ft Now inversion is possible \Structural Performance and Performance Limits During Operation Operation causes pipe-soil interaction to come ©2000 CRC Press LLC into equilibrium Water pressure rerounds the pipe and leaves gaps on the sides See Figure 26-13 Turbulent flow of water vibrates the pipe and shakes soil down unto the gaps Cycles of dewatering reduce ring deflection to an equilibrium somewhere between zero and the ring deflection at hydrotest If failures not occur during installation or hydrotesting or as a result of breakout of shards within a short period of operating time, failure will probably not occur The specified ring deflection, d = 3%, is intended to protect the mortar lining of the pipe Ring deflection, per se, is not so much a performance limit as it is a condition that contributes to flat spots and cracking of the lining Remedies for Flat Spots Flat spots should be identified and located In this pipe the critical flat spots are those long flat spots greater than inches wide, and those circular flat spots greater than inches in diameter If voids under the pipe on either side of a flat spot are extensive, they should be filled with low-strength, flowable, soil cement If voids are filled before water (weight and pressure) is in the pipe, the reaction force, F, will be reduced significantly See Figure 26-12 The cause of flat spots will be eliminated Flat spots should be inspected to ascertain if the radius of curvature is so small that the steel could crack and weaken the hoop tensile strength The critical radius for pipe steel is ry = 2.5t See Figure 26-10 (bottom) Steel engineers recommend that radius of curvature be greater than seven times wall thickness Wide cracks (>1/16 inch) or concentrations of cracks that might indicate disbonding and potential breaking out of shards, should be repaired Legal remedies, such as warranties over an extended period of time, could be investigated Figure 26-12 Pipe as a beam on reactions, showing the F-force at the reactions, and showing the moment diagram for a fixed-ended beam Figure 26-13 Soil pressure on pipe due to rerounding by internal pressure ©2000 CRC Press LLC Example What are the critical radius of curvature and the ring deflection in terms of critical crack width? Ring deflection is not exclusively critical, but is a c onvenient basis for relating other conditions which are critical such as critical crack width Critical radius of curvature can cause critical crack width Given: r = ry = d = w = t = Example What is the rerounding of a 42D cement mortar lined pipe due to internal pressure? Data: ID = 42 inches, t = 0.175 = steel thickness, t' = 0.5 = thickness of mortar lining, B = breadth of the flat spot (or diameter), sf = 42 ksi = yield stress of steel, P = 100 psi = internal pressure that rerounds the ring, The moment in the steel at plastic hinging is 3/2 times the moment at yield stress radius of circular pipe, maximum radius of curvature, ring deflection = D/D , width of crack in lining, thickness of lining Find: ry/r = ratio of radii From geometry of an ellipse, ry /r = (1+d)2/(1-d) Find: w = (26.8) For some flat spots the longitudinal length is greater than the circumferential breadth If the length is approximately equal to the breadth, the flat spot is roughly circular For the examples that follow, each is analyzed separately width of crack From the mechanics of pipe ring analysis, w = t(1/r - 1/ry) (26.9) The two equations, 26.8 and 26.9 interrelate the three variables, w, ry, and d Because d applies to elliptical ring deflection, it is limited as a criterion for structural integrity As long as soil support under the pipe is uniform, ring deflection is a relevant criterion for structural integrity Ring deflection should be limited by specifications Pipeline engineers specify and always anticipate that the pipe will be supported uniformly However, in case of flat spots or inversions, the ratio of radii, ry /r, is a more relevant criterion for structural integrity of the pipe than is ring deflection This presumes that support is not uniform under the pipe The reactions are hard spots The ratio of radii, ry /r, is a means of ©2000 CRC Press LLC estimating deficiency of the pipeline in terms of cracks in the lining and reduced service life It is also a basis for estimating the extent of encroachment into the margin of safety (safety factor) by the installer of the pipeline in cases where support under the pipe is not reasonably uniform Long Flat Spot At plastic hinging, (B/d)2 = 3s f /P If P = 100 psi and sf = 42 ksi, B = 6.21 inches If the long flat spot is wider than about 6.21 inches, internal pressure of 100 psi will reround the steel The lining is too brittle and cracked to reround without spalling Therefore, long flat spots wider than about or inches should be rerounded and the lining should be chipped out and reapplied For internal pressure greater than 100 psi, the breadth of the critical flat spot decreases Circular Flat Spot For circular flat spots, the above equation is roughly (B/d)2 = 6sf /P Solving, the critical diameter is B = 8.8 inches Therefore, circular flat spots greater in diameter than about inches should be rerounded and the lining should be chipped out and reapplied Deflections on the model deflections of the prototype are true-to-scale Rationale Rerounding by jacking requires care A line force of 80 lb/in can deflect the ring by about 3% if the pipe is not buried The 100 psi internal pressure required to reround the steel becomes 621 lb/in over the width of the critical 6.21-inch flat spot It might be prudent to apply the rerounding force over a smaller area, or by a ball peen tip, and then peen the flat spot into a circular cross section The reaction end of the jack should be supported over a large area — on compressible material that conforms with, and doesn't concentrate pressure on, the liner This may require longitudinal timber lagging or strong-backs, and possibly, with compressible packing between timber and pipe lining It may be well to rehearse the procedure using 6x6 or 8x8 timbers EGG-SHAPED LINING In order to achieve improved hydraulics in variable flow pipes such as sewer pipes, the cross section of the pipes is egg-shaped See Figure 26-14 Rehabilitation of such pipes results in linings with variable radii Following is a discussion of such linings Tests are recommended because of the many variables in egg-shaped linings It is noteworthy that the following analysis has one serendipitous feature From similitude, the fundamental variables can be combined into dimensionless pi-terms which have no feel for size Therefore, the observations (measurements) on the model apply to any size of "standard egg" if the same materials are used in a test model as in the prototype One size of model can predict performance of many sizes of prototype The conditions for similitude are: Corresponding materials are the same in model and prototype Corresponding lengths are to scale, angles equal The model is true-to-scale Corresponding pressures are equal ©2000 CRC Press LLC When external pressure is applied to an egg-shaped lining, the lining feels circumferential compression The circumference shortens, and a gap forms between the lining and the host pipe at the location of maximum radius of curvature — in this case, at one of the long-radius sides Pressure in the gap "blows" the rest of the lining into a tight fit against the host pipe The radius of lining in the gap increases, ring compression increases, and pressure of lining against host pipe increases Failure is reversal of curvature of lining in the gap If external pressure persists, the lining crumples Physical Tests Physical tests are important especially if the lining is plastic and creeps Beam deflection is approximate and usually does not occur until collapse is in process This should be watched for in test programs Compressive yield stress in the longradius arc is a pertinent condition for collapse Stress is a function of deflection of the arc which is a function of shortening of the circumference of the lining which, in turn, is a function of external pressure and modulus of elasticity Theory of elasticity is only approximate for plastics and requires physical tests Performance limits can be identified by tests It is not incontrovertible that deflection of the long radius of the egg lining is the only performance limit for design In test linings, snap-through occurs at yield stress after a period of time Snap-through is the onset of collapse The primary performance limit is excessive deformation The most pertinent fundamental variable is deformation Failure is excessive deformation — reversal of curvature and collapse Figure 26-14 Cross section of the standard egg host pipe in a lining test (taken from laboratory notes) Figure 26-15 Critical, long-radius are BCD, showing the basic geometry of the center plane of the lining Lengths are in inches ©2000 CRC Press LLC Figure 26-16 Critical are BCD, showing the gap between dotted and solid lines caused by external pressure Lengths apply to the center plane of the lining ©2000 CRC Press LLC Excessive deformations include leaks in the lining, and wall crushing when circumferential compression stress exceeds yield (based on yield strength), arc snap-through (based on the modulus of elasticity), and, maybe, beam failure (based on both yield strength and modulus of elasticity) Other pertinent fundamental variables are from geometry and pressure The three categories of pertinent variables are: geometry, loads, and properties of materials (modulus, strength, and virtual long-term modulus) Poisson ratio is only of minor concern in plastics Time is a function of loads (pressure) and creep (geometry) Time is a pertinent fundamental variable because plastics creep and change shape under persistent pressure The effect of time is reduction of the virtual modulus of elasticity Data are available on the relationship of time and virtual modulus Therefore, the effect of time can be included in analytical models by using a "virtual modulus." The virtual modulus is of value only in calculating the shortening of the circumference due to creep over the long term Persistent pressure includes intermittent pressure Of course, the rate of reduction of virtual modulus is slowed if pressures are intermittent The plastic tends to rebound partially between applications of the intermittent pressures The lining is cylindrical — no elbows or other special sections There is no circumferential shearing stress on the lining This is a worst-case assumption that is conservative, but is as accurate as can be justified External pressure on the egg-shaped lining is perpendicular to the surface In service there may be spots where the lining bonds to the host pipe, but bond tends to break down in time The circumference of the lining shortens under external pressure A gap forms between lining and pipe at the location of greatest radius of curvature The lining conforms with the host pipe except at the gap See Figures 26-15 and 26-16 The circumferential thrust, T, in the wall of the lining is constant all around the circumference of the lining, and is, T = PR = Piri = constant, The properties of plastics remain pristine The true modulus for snap-through, is short-term modulus — not virtual modulus where, at any point on the lining, T = circumferential thrust per unit length, P = external pressure, r = radius of the center plane of the lining where the gap will form (greatest radius), R = radius of lining at the gap after gap has formed, t = wall thickness, s = PR/t = ring compression stress The precision of models must be taken into consideration Classical equations are usually based on elasticity Linings are plastic — not elastic Decrease in circumference is a function of circumferential strain (stress divided by virtual modulus of elasticity) It is time dependent Analysis of the Standard Egg Lining Following is the rationale for performance of a close-fit plastic lining encased in a rigid standard egg host pipe when pressure is applied between lining and host pipe Worst-case assumptions are: ©2000 CRC Press LLC Because the material is plastic — not elastic — virtual modulus is the slope of the secant on the stress-strain diagram over time (allowing for creep) from zero to the compression stress, s = PR/t This is approximate and conservative because stressstrain diagrams are in tension The lining is in compression The plastic lining "creeps" over a period of time when subjected to persistent pressure But creep reduces the circumference which increases the radius of curvature and the width of the gap It is a progressive sequence that converges as shown in the following example Example 12 mm LINING UNDER FT OF HEAD OF GROUNDWATER What is the increase in radius of curvature from the original r to deflected R due to external pressure on the lining? What is the gap width, De , of the following standard egg-shaped lining? Notation: H = 36 inches = inside height of the standard egg host pipe, B = 24 inches = inside width of the standard egg host pipe, P = psi = pressure in the gap = ft of head at 62.4 pcf, t = 0.4724 = 12 mm = wall thickness of the lining, C = 93 inches = mean circumference of the lining (92.91), ro = 36 = outside radius of the long-radius sides of the lining, r = 35.7638 = ro - t/2 = mean radius (to center plane) of long-radius sides, R = deflected mean radius of the long radius side at the gap, sf = 5000 psi = yield stress, E = 500 ksi = mod/elast (short-term), E' = 250 ksi = virtual mod/elast (longterm), y = 36.1388o = See Figure 26-14, s = 229 psi = Pr o /t = circumferential stress in the wall, Arc BC = 11.2789 = r(y )/2 = mean arc length before deflection, Cord BC = 11.0928 = rsin(18.0694 o ) = cord length, See Figure 26-15 e = mid ordinate from the cord to the arc, De = width of th e g a p = er - eR See Figure 26-16 Arc BCD is assumed to be circular ©2000 CRC Press LLC Find: R for deflected ArcBCD — SHORT TERM E = 500 ksi First trial r = 35.7638 DC = ProC/Et = 0.0425 inch ArcBC = 11.2789 - 0.0425/2 = 11.2576 sin q = 11.0928/R q = ArcBC/R = 11.2576/R (1) (2) Solving by iteration (values of q must be equal.), Trial R 50 40 38 R = 38 sin q (2) 0.2233 0.2777 0.2919 sin q (1) 0.2219 0.2773 0.2919 Second trial r = 38; ro = 38.2362 DC = ProC/Et = 0.0452 ArcBC = 11.2789 - 0.0452/2 = 11.2562 sin q = 11.0928/R q = ArcBC/R = 11.256 Trial R sin q (2) 40 0.2777 39 0.2846 38.5 0.2882 R = 38.7, MAX SHORT TERM (1) (2) sin q(1) 0.2773 0.2844 0.2881 Find: GAP, De See Figure 26.16 De = er - eR De = 61.5251[(1/r) - (1/R)] = 0.13 (3) De = 1/8 in where: (cordBC) = 11.0928 er = (cordBC) 2/2r = 61.5251/r r = 35.7638 eR = (cordBC) 2/2R = 61.5251/R R = 38.7 Find: s = P(R+t/2)/t = 3(38.7+0.2362)/0.4724 s = 247 psi The safety factor is 20 based on yield strength of ksi Find: P at arc snap-through (reversal of curvature) From texts on mechanics of materials, P = (k2 - 1)(EI/R3) where (See Appendix A), k = 15 at q= 17o, o 17.2 15 , 8.62 30o, I = t /12, and q = sin -1(cordBC/R) q = sin -1(11.0928/38.7) = 16.66o = 17o Substituting values, P = 16.98 psi, but only approximately P = 17 psi The safety factor is 5.7 based on pressure P = psi Find: R for deflected ArcBCD — LONG TERM E = 250 ksi First trial r = 35.7638 DC = ProC/Et = 0.0850 inch ArcBC = 11.2789 - 0.0850/2 = 11.2364 sin q = 11.0928/R q = 11.2364/R P = 14 psi This is equivalent to a head of 32 ft The long-term safety factor is 4.6 based on a head of ft Closure The rationale described above is approximate, but conservative Overlooked are such contributions to structural integrity of the lining as the following Longitudinal action of the cylindrical lining helps to reduce deflections The above rationale considers only two dimensions — the cross section Ring stiffness of the lining helps to reduce deflections Ring stiffness begins to affect the deflection as soon as deflection begins Before deflection, stiffness has no effect on performance The moduli of elasticity are based on tension tests Compression moduli are greater than tension moduli (1) (2) Continuing the solution by iteration (Values of q must be equal.), R = 41.5, MAX LONG TERM The yield strength is based on tension tests Compression yield strength is greater PROBLEMS 26-1 Find width of crack in the coating of the example in CRACKS IN MORTAR LINING De = 1/4 inch s = 265 psi Because yield strength is 5ksi, the safety factor is 19 for circumferential (ring compression) stress in the wall Find: P at arc snap-through (reversal of curvature) P = (k2 - 1)(EI/R3) where k = 15, roughly, at q = 15.5o q = sin -1(11.0928/41.5) = 8.5o I = t3/12, t = 0.4724, R = 41.5, R/t = 87.85, E = 250 ksi, Substituting values, P = 13.77 psi ©2000 CRC Press LLC 26-2 Find width of crack in the lining of the example in CRACKS IN MORTAR COATING 26-3 What is the radius of curvature of a mortarlined steel pipe at the invert where a single 0.01inch crack opens in the mortar? The mortar is 1.0 inch thick and the ID of the thin-wall steel pipe is 36 inches The pipe is subjected to external hydrostatic pressure, but not to internal pressure 26-4 The mortar-lined pipe of Problem 26-1 leaks It is to be lined with a close-fit high-density polyethylene (HDPE) lining that is 0.9 inch thick The long-radius arc is 60o What is the persistent external pressure on the HDPE lining at snapthrough in 50 years? ... non-circular linings and coatings, rigid and flexible RIGID LININGS AND COATINGS Rigid linings and coatings are applied to thin-wall s teel pipes that are to be buried The linings and coatings are... pipe The radius of lining in the gap increases, ring compression increases, and pressure of lining against host pipe increases Failure is reversal of curvature of lining in the gap If external... Crack at the invert of the lining caused by increase in radius of curvature ©2000 CRC Press LLC CHAPTER 26 NON-CIRCULAR LININGS AND COATINGS Linings and coatings are not always circular in cross