Numerical Methods in Soil Mechanics 20.PDF

Numerical Methods in Soil Mechanics 20.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 "PLASTIC PIPES" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 20-1 Nomenclature for a plastic pipe buried in a select embedment within a trench and showing ring deflection, d = ∆/D due to the vertical pressure of a landfill Figure 20-2 Typical tensile strength regression line for PVC at 70°F showing how strength decreases linearly with time on a log-log plot ©2000 CRC Press LLC CHAPTER 20 PLASTIC PIPES The use of buried plastic pipes is widespread and increasing Plastic pipes are attractive for use in aggressive environments, and for collection and transmission of liquids that are abrasive and/or corrosive Most small-diameter gas dstribution pipes are plastic Corrugated plastic pipes are used extensively for land drainage and culverts Plastics are relatively resistant to corrosion by aggressive chemicals and to abrasion by sediment in fluid flow Plastic pipes are flexible, and conform with soil in a favorable pipe-soil interaction that reduces stress concentrations Plastics can tolerate deformation Plastics have high tolerance for fatigue stress (cyclic loading) Plastics relax under constant deformation Plastic pipes are light weight They have low flow resistance They have long service life They are easily fabricated and easily joined in the field It is possible to manufacture plastic pipes in various shapes, and with corrugations, ribs, fillets, etc On the other hand, compared to traditional pipe materials, plastics have low strength, and low modulus of elasticity They may be sensitive to temperature — both too hot and too cold Plastics have high Poisson ratios and high coefficients of thermal expansion Questions arise as to the ability of plastic pipes to perform under high pressure, both internal and external, and under heavy loads These questions are poignant because of the urgent need for corrosion/abrasion-resistant piping for drainage under high landfills (such as sanitary landfills) and in mines, where leachate is aggressive and sediment is abrasive Following is a discussion of the conditions under which plastic pipes can resist internal pressures and external loads, and the effect of time on pipe performance Most of the performance data are empirical; i.e., provided by tests and by pipes in service As always, much useful data comes from failures See Figure 20-1 for nomenclature ©2000 CRC Press LLC UNIQUE PROPERTIES OF PLASTICS A Under Constant Stress: Plastics Creep Creep is time-dependent strain under persistent stress Plastic creep progresses gradually to equilibrium or fracture Persistent stress is constant or recurring stress Strength Regresses Strength is stress at fracture The time to fracture is roughly exponential, as shown in the log-log plot of Figure 20-2 B Under Constant Deformation (Strain): Stresses Relax Over time stresses relax, at a decreasing rate, to a state of stress equilibrium in which stresses remain essentially constant, but less than the initial stresses Strains are constant because deformation is held constant Strength Is Not Reduced The ability to resist pressures or loads is not reduced over time under constant deformation (strain) Only under persistent stress and the resulting creep does strength regress C Stress-strain Relationship Properties of Plastics Remain Pristine For pipe design, plastics are non-degradable Therefore, properties of material remain pristine (non-regressive) over time The most pertinent properties are strength, modulus of elasticity, and Poisson ratio — all of which remain pristine even after long-term persistent pressure and/or constant deformation The only exceptions occur under extreme temperatures, degradation of the plastics, and strength regression under long-term persistent stress Figure 20-3 Typical strength regression of plastics, showing short-term (pristine) strength that prevails up to the point of long-term failure Figure 20-4 Typical stress-strain diagram for plastics, showing the effect due to creep over the long term, and the resulting virtual modulus after the long term ©2000 CRC Press LLC Rate of Stress Relaxation Exceeds Rate of Strength Regression Under constant strain, stresses relax at a faster rate than the strength regresses Therefore, failure does not occur If a stressed plastic pipe is subjected to constant deformation, such as a pressed fit in an encasement; and if the pipe does not fail at the time it is squeezed into the encasement; then stresses in the pipe will relax and the pipe will not fail over the long term Gross deformations, such as clamped-off ends of plastic gas pipes, are evidence of the stress "relaxability" of plastic pipes Example Figure 20-3 shows typical strength regression of a plastic pipe If stress in the pipe is σ = S1, failure will occur in one year If σ = S50, failure will occur in 50 years In this example, a pipe, under constant internal pressure of σ = S50, is doomed to fail in 50 years However, suppose that in 30 years, it is suddenly subjected to increased internal pressure What strength is available to resist this stress? From Figure 20-3, the strength is S0 — pristine initial strength Creep Depends Upon a Virtual Modulus Some analysts use a virtual long-term modulus of elasticity This is not the true modulus The true modulus remains pristine A serviceable definition of modulus of elasticity in a plastic material is stiffness — the slope of a secant on the stressstrain diagram An example is shown in Figure 20-4 Virtual modulus, E', can be used to predict longterm strain (creep) under persistent stress It does not apply to collapse Collapse is sudden — a function of pristine stiffness of the plastic Example A section of plastic pipe is a beam supporting a constant load What is the deflection in the long term? Under constant load (persistent stress), plastic creeps, and deflection increases gradually Deflection of a beam is inversely proportional to the modulus of elasticity, which is the slope of the ©2000 CRC Press LLC secant on the stress-strain diagram Figure 20-4 shows the true modulus, E, in the short-term In the long term, the virtual modulus is less Virtual modulus can be used to predict long-term deflection of the beam under constant load Plastic Has a Memory for Distress Analysts refer to a plastic's "memory for a load." Once plastic is distressed by a load, subsequent reloading will tend to re-distress the plastic in the same pattern as the original distress This is not a change in the pristine properties of the plastic It is the consequence of prestress that has become set by creep over a period of time The prestress is reversed when the load is removed and the plastic tries to return to its original shape Time of Recovery Equals Time of Creep Deformed plastics tend to return to their original shape after the deformation is released Some analysts use a rule of thumb that the time required for plastic to return to its original shape after deformation is equal to the time of deformation when stress relaxation was in progress This implies that the distress memory fades in a period of time more or less equal to the time of distress The concept is faulty in that the plastic does not recover completely Performance Limits Are Different for Tension and Compression Under constant tension, plastics creep (elongate) and decrease in cross-sectional area They fail at yield stress Under constant compres sion, plastics creep and increase in cross-sectional area such that resistance to failure is increased They not fail at tensile yield stress Therefore, allowable compression strength is ultra-conservative when based on tensile yield strength PLASTIC PIPE NOMENCLATURE Some distinctive notation and nomenclature have arisen out of the development of plastic pipes DR = dimension ratio = OD/t, SDR = standard dimension ratio, t = OD ID D r d ∆ S P' P p wall thickness or the equivalent area per unit length of pipe, = outside diameter, = inside diameter, = mean diameter (to neutral surface) = (OD - t) for plain pipe, = mean radius = D/2, = ring deflection = ∆/D , = vertical decrease in diameter, = allowable stress in the plastic, = internal pressure, = external soil pressure, = internal vacuum plus external hydrostatic pressure Some of the first commercially available plastic pipes were extruded with a constant OD The wall thickness was adjusted for various pipe "series" (pressure ratings) by adjusting the ID Blowmolding, spiral welding, and other manufacturing techniques have been developed, but the fixed OD is still the basis for the ratings DR = OD/t The dimension ratio was more precisely defined as the mean OD divided by the minimum t For analysis, correction factors were introduced: (Odmin /ODmax)3, and (tmin /tmax)3 These correction factors are not accurate, but introduce a margin of safety, and alert pipeliners to the need for precise pipe dimensions William Allman of DuPont found that the exponent, 3, is low His tests indicated that 4.6 is more accurate Improvement in the control of plastic pipe dimensions has virtually eliminated the need for correction factors Based on the above notation, design equations for plastic pipes are rewritten in the following forms The Barlow formula for internal pressure is changed from σ = P'(ID)/2t, to, P' = 2S/(DR-1) (20.1) ALLOWABLE INTERNAL PRESSURE A safety factor is included in allowable stress, S The ring compression formula for external soil pressure is changed from σ = P(OD)/2t, to, ©2000 CRC Press LLC P = 2S/(DR) (20.2) ALLOWABLE EXTERNAL SOIL PRESSURE A safety factor, sf, is included in allowable stress, S, but is probably unnecessary Compressive yield stress is greater than tensile yield stress which is found from tensile tests The formula for vacuum at collapse is changed from pD3/EI = 24, to, p = 2E/(DR-1)3sf (20.3) ALLOWABLE INTERNAL VACUUM including external fluid pressure PLASTIC PIPE PERFORMANCE In the design and analysis of buried plastic pipes, the following are concepts of performance In general, native undisturbed soil is stable It has been in place for a long time, and, except for earthquakes, landslides, etc., is a stable medium for pipe-soil interaction Exceptions include soils that are fluid such that the pipe sinks, or floats, or collapses due to external fluid pressure The best buried pipe installations are those which least disturb the native soil A bored tunnel of precise diameter into which the pipe is squeezed and inserted, would cause the least soil disturbance The pipe then serves as a tunnel liner This concept is ideal, but impractical A more practical installation with little disturbance of the native soil is a narrow trench with only enough side clearance to align the pipe and to permit placement of select soil backfill The backfill develops continuity between the pipe and native soil Pipe-soil interaction is assured The flexible pipe is stabilized Once a plastic pipe is stabilized, its shape remains nearly constant Stresses relax, and the pipe does not fail over the long-term An exception occurs under extreme ring deformation of profile or ribbed walls The highest stresses relax fastest, and shift the neutral surface such that stresses increase on the other side of the neutral surface The consequence could be a long-term crack the pipe — both longitudinal deflections and ring deflections The basic load on plastic pipes is the soil prism load To design a plastic pipe for long-term performance under persistent load, the regressed, long-term strength is used An example is the design of pipe to withstand constant internal pressure The plastic creeps over the long term Therefore, diameter increases and wall thickness decreases The virtual strength decreases The reverse occurs for constant external pressure Compression causes plastic creep over the long term But the wall thickness increases The virtual strength increases Long-term tensile strength regression is not a correct basis (albeit conservative) for ring compression design Long-term (virtual) modulus applies only to long-term deformation (creep) due to persistent load The flexible pipe is held in shape by the soil The stabilized ring itself becomes an arch that supports vertical load Without soil support, the ring would collapse under relatively light load Under quick loading, the strengths and moduli of elasticity are the initial, short-term values This phenomenon is remarkable in that it is independent of previous stress history For example, a pipe s ubjected to internal pressure over a long period of time still maintains its initial quick-load strength and modulus of elasticity Consequently, water hammer analysis is based on initial strength Vacuum collapse is analyzed by using initial modulus of elasticity See Chapter 10 Because plastic pipes are flexible, they conform with the soil Pressure concentrations are relieved by distribution See Figure 20-5 Not only are concentrated stresses in the pipe relieved, but all stresses relax over time The potential for soil slip must be checked See Chapter 10 Arching action of the soil supports part of the load See Figure 20-6 The soil acts as a masonry arch No cement is needed because the pipe confines the soil and retains the soil arch The soil arch protects the pipe But additional loads may be caused by surface vehicles, by differences between unit weights of pipe and soil, and by deflections of ©2000 CRC Press LLC For plastic pipe under constant deformation (good embedment), stresses in the pipe wall relax Because the soil retains the ring in fixed shape, the plastic relaxes over time and relieves itself of some of the stresses in the pipe wall Stresses relax at a faster rate than strengths regress under constant deformation 10 If the pipe is installed in a trench, embedment is a transfer medium completing the intimate fit of the pipe to the native soil The value of the intimate fit is retention of the original pipe shape, and prevention of groundwater flow channels along the outside of the pipe 11 In good embedment, ring deflection is approximately equal to, or less than, the vertical strain (compression) of the sidefill soil DESIGN OF BURIED PLASTIC PIPES Successful performance of buried plastic pipes is based on four performance limits: Circumferential strength (yield stress) — both tensile (hoop) strength and ring compression strength, Deflection — both ring deflection and longitudinal (beam) deflection, Stability — as limited by incipient ring collapse, Pipe-soil continuity — intimate contact between the pipe and soil Circumferential Stress Internal Pressure — For design, use Equation 20.1 If a pipe is subjected to sudden internal pressure, Figure 20-5 Comparison of typical soil loads on a rigid ring and on a flexible ring, showing how the flexible ring deflects just enough to equalize the horizontal and vertical forces ©2000 CRC Press LLC such as water hammer, the initial (short-term) tensile strength may be used For persistent internal pressure, the long-term tensile strength should be used A safety factor of two is often used for design safety factor, 1.6, is more than adequate for the long term External Pressure — For design use Equation 20.2 If a plastic pipe is in good embedment, the initial (short-term) compressive strength may be used If the pipe does not fail during installation, it is not likely to fail in the long-term During installation, the performance limit is wall crushing at 9:00 and 3:00 o'clock when the ring compression stress reaches yield strength See Figure 20-7 Compression yield strength is higher than tension yield A safety factor of two based on tension yield is excessive, but often used Tests are advisable for compression yield strength Under high landfills, the effect of surface live loads on buried plastic pipes is negligible Longitudinal (beam) deflection is usually of only minor concern With careful placement of the bedding, the pipe does not deflect excessively as a beam Pipe manufacturers specify a minimum longitudinal radius of curvature, R Excessive longitudinal bending could cause a plastic hinge in the beam Under constant deformation over the long term, longitudinal stresses relax Ring deflection causes circumferential flexural s tresses which are also maximum at 9:00 and 3:00 o'clock Flexural stresses are compression on the inside of the pipe and tension on the outside Wall crushing can occur only when the pipe wall is at compression yield stress throughout the entire wall thickness Therefore, flexural stress does not affect wall crushing, but may affect stability and plastic hinging See Stability in the paragraphs to follow Example A polyethylene pipe is buried under a landfill for which the vertical soil pressure on the pipe is 280 psi For this pipe, the dimension ratio is DR = 9.2 Ring deflection is not greater than 10% according to a "bullet" drawn through the pipe What is the safety factor for ring compression stress if the shortterm yield strength of the polyethylene is 2300 psi From Equation 20.2, the ring compression stress is 1417 psi The safety factor is sf = 2300/1417 = 1.6, which is adequate for short-term The yield strength is based on tension tests Moreover, after completion of the landfill, stresses relax if the embedment is good granular soil that holds constant the cross-sectional shape of the pipe Therefore the ©2000 CRC Press LLC Deflection Ring deflection, d = ∆/D , is of greater concern See Figure 20-8 Excessive ring deflection can cause leaks at appurtenances and joints It can contribute to instability of the ring It can obstruct the passage of cleaning equipment and video cameras Ring deflection is usually limited by specification Ring deflection can be predicted by methods described in Chapter Stability Instability is incipient collapse of the ring Incipient means that collapse is not inevitable, but that the ring is at its limit, and further resistance to collapse depends upon soil arching action Practically, enough stiffness is built into the flexible ring to hold it in shape during installation and loading This pipe stiffness contributes to the resistance of the pipe-soil system to collapse See Chapters 10 and 11 The appropriate pipe stiffness is the initial (short-term) pipe stiffness because collapse is a sudden phenomenon Collapse of plastic pipes usually follows the formation of plastic hinges at 9:00 and 3:00 o'clock Not only are plastic hinges a failure mechanism in the pipe, but the very short radius of curvature of the hinge results in such high horizontal soil stresses at the spring lines that the soil may slip Clearly the conditions for stability are assured if good granular soil is carefully compacted in the Figure 20-7 Ring compression of a circular pipe loaded by soil pressure, P, showing the ring compression stress distribution and the flexural (ring deflection) stress distribution across the wall Figure 20-8 Ring deflection, d = ∆D , for a typical plastic pipe cross section deflected into an ellipse due to the typical horizontal and vertical soil pressures shown ©2000 CRC Press LLC sidefills, and if the ring deflection is limited — usually to less than 10% Under conditions where mitigation is sought, the height of soil cover can be restricted deflection to less than d = 5% under 600 ft of soil cover at unit weight of 75 pcf Drain Pipe Caveats Example PVC piping is proposed for drainage under a sanitary landfill to be 600 ft high Unit weight of the landfill is 75 pounds per cubic ft Fifteen years are anticipated to complete the landfill Service life for the piping is to be at least 100 years a) What dimension ratio (DR) is required? DR is an inverse measure of the pipe stiffness Assume that the 15-year compressive yield strength of the PVC is 5000 psi Actually it is greater than the 5000 psi which is based on tensile yield In good granular embedment, ring deflection is constant, and longterm stresses relax Use a safety factor of sf = 1.5 From Equation 20.2, P = 312.5 psi, and ring compression stress is, σ = 0.5 P(1+d)(DR)sf For a long-term sanitary landfill, it is prudent to hold ring deflection to near zero by compaction of sidefill If d 0, the equation above yields DR = 21.3 A good selection is PVC pipe SDR 21(200) ASTM D 2241 SDR = OD/t = standard dimension ratio b) What is maximum allowable ring deflection if the embedment is loose (φ = 25°)? From the Uni-Bell Handbook of PVC Pipe, published by the Uni-Bell PVC Pipe Association, for SDR = 21, pipe stiffness is F/∆ = 234 psi for E = 400,000 psi, and F/∆ = 292 psi for E = 500,000 psi Using 234 (to be conservative), EI/r3 = (F/∆) /6.72, and P/(EI/r3) = Entering the graph, Figure 10-9, with φ = 25o, and P/(EI/r3) = 9, ring deflection at incipient collapse is about d = 20% Clearly, if ring deflection is held to less than d = 10%, the safety factor is greater than two Ring deflection can be controlled by the quality and density of the sidefill From laboratory tests, select crushed stone compacted to 95% density AASHTO T-99 (70% relative density) will hold ring ©2000 CRC Press LLC The embedment must not dissolve, decompose, or plug up flow for 100 years Limestone, for example, is dissolved by leachate in sanitary landfills, and can be redeposited inside the pipe Another example is loss of sidefill by migration of fine soil particles into the drain pipe through the slots or perforations in the drain pipe Either the soil must be designed as a filter with well-graded soil particle size, or geotextiles must be used to prevent soil particle migration In the installation of corrugated high-density polyethylene pipes (HDPE), if the pipe is stretched longitudinally, pipe stiffness is reduced and could result in instability Manufacturers caution that increase in length during installation of more than 10% is to be avoided Corrugated polyethylene pipes have the capability of short longitudinal radius of curvature, R The corrugated pipe can be transported on a spool It can be installed down through a chute directly behind the excavator buckets of a trenching machine The corrugated pipe can bend around the short radius of a shoe, and into the trench where it is immediately backfilled But the pipe must not bind in the shoe and stretch Pipe-soil Continuity Intimate contact between pipe and embedment helps to assure position and alignment of the pipe If groundwater flow erodes channels along the pipe, intimate pipe-soil contact and soil density could be lost Soil contact retains the circular cross section of the pipe By fixing the shape of the ring, stresses relax over the long term Intimate soil contact is assured by using select granular embedment placed under the haunches of Figure 20-9 Four shapes selected at random from many variations of plastic pipe shapes ©2000 CRC Press LLC the pipe by shovel-slicing, J-barring, flushing, etc It is noteworthy that for structural integrity of pipes with negligible ring deflection and significant ring stiffness, intimate soil contact may not be absolutely essential For example, in the case of plastic pipes of low DR in good embedment, the sidefill can support the soil load by arching action with or without contribution from the pipe The phenomenon is tantamount to boring a tunnel under the landfill and inserting the pipe The pipe only has to support the talus that would slip against it if the pipe could deflect slightly Consequently, the structural importance of intimate soil contact can be mitigated under some conditions However, good sidefill is essential to soil arching action Temperature The properties of plastics are affected by temperature Pipe manufacturers can provide the necessary design data For example, if PVC pipe is to be used at a temperature of 120oF, it may be prudent to assume a modulus of elasticity of E = 400 ksi rather than E = 500 ksi, and yield strength of ksi rather than ksi Decomposition of the biomass in sanitary landfills generates heat Temperatures of 120oF have been reported, but are not common Similar adjustments apply to other plastics where temperature is of concern Example Two samples of high-density polyethylene pipes (HDPE-SDR11), were heated to temperatures over 140oF when buried under 12.7 ksf of vertical soil pressure Under this pressure, ring deflection was 7% and 9% for the two tests The embedment was compacted fine sand There was no significant change in ring deflection due to the increase in temperature The ring deflection of one pipe increased a fraction of 1% and the other decreased a similar amount ©2000 CRC Press LLC PLASTIC PIPE SPECIALTIES A wide variety of special plastic pipes are becoming available for underground use A random sample of four of them is shown in Figure 20-9 For design, each special pipe may require modifications of the design principles Example Figure 20-9(a) shows an edge drain Edge drains are narrow, cylindrical, pipes that are placed vertically along the edges of paved surface slabs (such as highways and airport runways) to intercept and remove water that would otherwise saturate the subbase soil and reduce the strength of the soil that supports the slab In this case, the edge drain is 12 inches high with inches of soil cover It is located along the edge of a highway slab on which the heaviest load is an HS-20 dual-wheel load What is the test force for which this edge drain must be designed? Figure 20-10(a) shows the worst probable loading For worst-case analysis, it is assumed that the soil is loose enough that the dual-wheel load punches out a truncated pyramid as shown in Figure 20-10(b) Vertical soil pressure as a function of depth is, Py = 2W/[2L+y)(B+y)] where y = Py = W = L = B = (20.4) depth from soil surface, vertical soil pressure at depth y, 16 kip load on rectangular tire print, width of the dual tire print, breath of dual tire print Equation 20.4 is based on the assumption that the soil is granular and that the slopes of the pyramid are about 2v:1h At any depth, y, the horizontal (active) soil pressure on the edge drain is, Figure 20-10 Worst-case load on an edge drain installed along a highway slab Figure 20-11 Pressure distribution and the resultant forces acting on an edge drain ©2000 CRC Press LLC Px = Py /K (20.5) where K = (1+sin φ)/(1-sin φ) = if, φ = soil friction angle = 30o (assumed for this soil), Px = horizontal pressure on the edge drain, Q = resultant horizontal force on the edge drain assuming that the pressure distribution is trapezoidal The Px diagram is shown dotted on Figure 20-11 (left) If the coefficient of friction of soil on edge drain is assumed to be φ' = 30o, the vertical shearing force on the edge drain is V = Qtanφ' The resultant of Q and V is F as shown in Figure 20-11 (right) A reasonable test for strength of the edge drain would be the application of load F on a sample of edge drain "sandwiched" between two parallel blocks with sandpaper surfaces F is inclined at angle φ' and is located a distance, e, from the center of the edge drain, such that e locates the centroid of the trapezoidal pressure diagram The maximum test force F on the edge drain must be greater than the HS-20 force F for which the edge drain is designed backfill at which the pipe will collapse? Assume for worst case that the pipe is empty (37.5 ft) 20-2 A 6D HDPE pipe SDR9 is to be installed under a river by directional boring The pipe will be 80 feet below river water surface The river banks are 10 feet above the river water surface The bored hole is retained up to bank level by drillers mud (liquid) with unit weight of 80 pcf The pipe is filled with water and pulled through the bored hole in the drillers mud Then the drillers mud is displaced by grout that has a unit weight of 95 lb per cubic ft After the grout sets, the water in the pipe is pumped out leaving hydrostatic pressure of the river on the encased pipe through cracks in the grout If the pipe stiffness is F/∆ = 600 psi, what is the safety factor against collapse? a) during installation? b) during grouting? c) due to external pressure of river water? 20-3 What is the ratio of longitudinal radius of curvature, R, to pipe radius, r, for corrugated HDPE pipe if maximum allowable elongation and contraction are each limited to 10%? (R/r = 10) PROBLEMS 20-1 The measured pipe stiffness of an 8D corrugated polyethylene pipe is F/∆ = 70 psi The pipe is to be used under a sanitary landfill of unit weight γ = 75 lb per cubic ft It is to be installed and backfilled below the water table Therefore, for an instant, it may be subjected to hydrostatic external pressure of saturated sand at 120 lb per cubic ft What is the depth of liquefied sand ©2000 CRC Press LLC 20-4 From parallel block tests on the 12-inch edge drain of the example above, test strength is F = 395 lb/inch What is the safety factor against failure if the edge drain is to be placed under inches of soil cover with HS-20 load, W = 16 kips on a 22 x inch tire print, as shown in Figure 20-10(a)? Friction angles for the soil, φ, and for soil on the edge drain, φ', are both 30o Assume trapezoidal pressure diagram as shown in Figure 20-11(a) (F = 240 lb/inch; sf = 1.65) ... Arching action of the soil supports part of the load See Figure 20-6 The soil acts as a masonry arch No cement is needed because the pipe confines the soil and retains the soil arch The soil. .. strength and ring compression strength, Deflection — both ring deflection and longitudinal (beam) deflection, Stability — as limited by incipient ring collapse, Pipe -soil continuity — intimate contact... deflection can be predicted by methods described in Chapter Stability Instability is incipient collapse of the ring Incipient means that collapse is not inevitable, but that the ring is at its limit, and