Numerical Methods in Soil Mechanics 07.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 "RING DEFLECTION" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 7-1 Deflected ring showing notation for dimensions and for ring deflection analysis Figure 7-2 Segmented ring deflection showing the relationship between ring deflection, d, width, w, of the crack, and wall thickness, t ©2000 CRC Press LLC CHAPTER RING DEFLECTION Ring deflection is defined as the ratio of change in vertical diameter to the original diameter, d = /D See Figure 7-1 Diameter D is the diameter to the neutral surfaces of the cross section of the wall For mos t pipe analyses, it is sufficiently accurate to use the mean diameter, (OD+ID)/2 The error of using mean diameter increases for reinforced concrete pipes, pipes with ribs or stiffener rings, etc Ring deflection is the result of: inflation or deflation of the pipe, flexing of the ring, cracking of the ring into segments, and plastic hinging (or crushing) of the pipe walls Ring deflections of rigid and flexible pipes are two different phenomena Each is analyzed separately RIGID RING Typical rigid pipes are concrete and vitrified clay pipes For rigid pipes, two basic modes of ring deflection are elastic and segmented Most rigid pipes are brittle The limit of elastic deflection is reached when the pipe cracks into segments as shown in Figure 7-2 Because there is no such thing as a perfectly rigid pipe, the question arises, how much elastic ring deflection occurs in the rigid ring? For most rigid rings, elastic deflection is small enough to be neglected Hairline cracks are not critical Reduction in flow capacity is negligible Elastic ring deflection is calculated in the same way for both rigid and flexible pipes Composite (reinforced) pipe walls require a transformed section for analysis This is true for reinforced concrete pipes, but may also be true for materials with different properties in compression and tension Elastic ring deflections for various load conditions are listed in Appendix A Example A concrete pipe has ID = 36 and OD = 42 inches with double cages of 1/4 inch steel reinforcing rods spaced at inches and located 0.6 inch from the inside and outside surfaces of the pipe See Figure 7-3 ©2000 CRC Press LLC What is the elastic ring deflection at yield stress of 1000 psi in the concrete for each of the three different loading conditions shown? Cracks open if tensile stress is greater than 1000 psi From the transformed section, EI/r3 = 1651 psi; and from deflection equations in Appendix A, the corresponding ring deflections are calculated and summarized in Figure 7-3 None of these elastic ring deflections is greater than 0.1% Segmented ring deflection is the result of cracks opening at spring lines, crown and invert See Figure 7-2 It is assumed that the segments are rigid Ring deflection can be calculated in terms of crack width w If the wall thickness is t and the neutral surface is at mid-thickness of the wall, the ring deflection is d = w/t Allowing for some deflection of the segments, and allowing for the possibility that neutral surfaces are further from the pipe surfaces than t/2, the lower limit of ring deflection is greater than w/2t Therefore, td < w < 2td (7.1) which shows a range of widths of the crack as a function of wall thickness and segmented ring deflection The relationship is not precise because cracks are undependable For example, two parallel cracks may open where only one is expected Example Consider the same 36 inch ID reinforced concrete pipe with three inch thick walls If 0.01 inch wide cracks open inside the pipe at the invert and crown, from Equation 7.1, ring deflection is between d = 0.17% and d = 0.33% Because of balanced placement of steel, actual ring deflection may be closer to the upper limit, say, d = 0.3% Ring deflection is more the result of cracking than it is the result of elas tic deformation of the ring The small ring deflections justify design by rigid pipe theories See Chapter 12 Figure 7-3 Ring deflection at incipient cracking of a reinforced concrete pipe under three different loading conditions Note that the maximum deflection is d = 0.1% ©2000 CRC Press LLC FLEXIBLE RING As soil and surface loads are placed over a buried flexible pipe, the ring tends to deflect — primarily into an ellipse with a decrease in vertical diameter and an almost equal (slightly less) increase in horizontal diameter Any deviation from elliptical cross section is a secondary deformation which may be the result of non-uniform soil pressure The increase in horizontal diameter develops lateral soil support which increases the load-carrying capac ity of the ring The decrease in vertical diameter partially relieves the ring of load The soil above the pipe takes more of the load in arching action over the pipe — like a masonry arch Both the increase in strength of the ring and the soil arching action contribute to structural integrity Although some ring deflection is beneficial, it cannot exceed a practical performance limit Therefore the prediction of ring deflection of buried flexible pipes is essential Ring deflection is elastic up to the formation of cracks or permanent ring deformations Clearly, the ring can perform with permanent deformations — and even with small crac ks, under some circumstances Performance can surpass yield stress to the determination at which the ring becomes unstable Instability is explained in Chapter 10 E' E I = soil modulus = slope of a secant on the stressstrain diagram from the point of initial vertical effective soil pressure to the point of maximum vertical effective soil pressure, = modulus of elasticity of the pipe wall, = centroidal moment of inertia of the pipe wall cross section per unit length of the pipe Figure 7-4 is a graph of the ring deflection term as a function of stiffness ratio From the graph, ring deflection can be found as follows Enter Figure 7-4 with a stiffness ratio, either Rs or Rs' and read out the ring deflection term, d/e If the vertical soil strain e is known, ring deflection follows directly from d/e Figure 7-4 represents tests and field data for buried flexible pipes Vertical soil strain e is predicted from laboratory compression tests data such as the stress-strain graphs of Figure 7-5 for cohesionless siltly sand Soil stiffness E' is the slope of a secant to the anticipated soil pressure P on the stress-strain diagram for a specific soil density Graphs can be provided by soil test laboratories for the specific embedment to be used, and at the density to be specified The following analyses of ring deflection are based on elastic theory for which the pertinent pi-terms are: d = ring deflection, e = average sidefill soil settlement, d/e = ring deflection term, Rs = stiffness ratio = E'D3/EI, = ratio of soil stiffness E' to ring stiffness, EI/D3; or to pipe stiffness, F/D , where F/D = 53.77 EI/D3 Notation: d = D /D = ring deflection, = vertical decrease in ring diameter, D = original diameter of the flexible ring (more precisely, the diameter to the neutral surfaces of the wall cross section), For design, ring deflection of flexible pipes buried in good soil is equal to (no greater than) the vertical strain (compression) of the sidefill soil D D to be multiplied by Dmax /Dmin = vertical soil strain due to the anticipated vertical soil pressure at the pipe springlines, ©2000 CRC Press LLC Ring stiffness contributes significant resistance to ring deflection if Rs is less than about 300 (or Rs' is less than 6); i.e low soil stiffness and high ring stiffness For flexible pipes buried in good soil, stiffness ratio is usually greater than 300 Therefore, Circumstances aris e under which the above rule is not accurate Equations for ring deflection are listed in Appendix A for a few loadings on rings of uniform wall thickness that are initially circular If not, one set of approximate adjustment factors is: Rs = (SOIL STIFFNESS)/(RING STIFFNESS) Figure 7-4 Ring deflection term as a function of stiffness ratio The graph is a summary of 140 tests plotted at 90 percent level of confidence; i.e., 90 percent of test data fall below the graph Figure 7-5 Stress-strain relationship for typical cohesionless soil (silty sand) Ninety percent of all strains fall to the left of the graphs Vertical pressure is effective (intergranular) soil pressure Soil stiffness, E', is the slope of the secant from initial to ultimate effective soil pressures ©2000 CRC Press LLC t to be multiplied by t /tmax Equations have been proposed for predicting ring deflection of flexible pipes One of these is the Iowa Formula derived by M G Spangler The Iowa Formula is elegant and correctly derived, but depends upon a number of factors which may be difficult to evaluate Such questionable factors include a deflection lag factor, bedding factor, the horizontal soil modulus, and the assumptions on which the Marston load is based See Appendix B and Spangler (1973) The horizontal soil modulus, E', is particularly troublesome It is based on theory of elasticity which is questionable E' is not constant In fact, E' is a function of the depth of burial and the horizontal compression of the sidefill soil as the pipe expands into it The best procedure for design of buried flexible pipes is to specify the allowable ring deflection, and then make sure that vertical compression of the sidefill soil does not exceed allowable ring deflection The Iowa Formula, and other deflec tion equations, are approximate, but conservative Some are compared in Appendix B However, within the precision justified in most buried pipe analyses, the procedures described in the following examples are more relevant and understandable Example A corrugated plastic drain pipe (flexible) is to be buried in clean dry sand backfill that falls into place at 80 percent density (AASHTO T-180) What ring deflection is anticipated due to 10 ft of soil cover weighing 120 lb/ft3? Live load is neglected at this depth of cover The stiffness ratio is larger than R' s = 6, therefore, from the ring deflection graph of Figure 7-5, ring deflection is not more than about, - = 1% d=C e Example What is the predicted ring deflection? A PVC pipe of ©2000 CRC Press LLC DR = 14 is to be placed in embedment compacted to 80 percent density (AASHTO T-180) under 24 ft of cohesionless silty sand at dry unit weight of 105 lb/ft3, but with a water table at ft below the surface Saturated unit weight is 132 lb/ft3 From the Unibell (1882) Handbook of PVC Pipe, page 159, values of PVC pipe stiffness for DR = 14 pipes vary from 815 to 1019 psi Using, conservatively, the lower value, pipe stiffness is F/ = 815 psi The soil stiffness is found from Figure 7-5 Because ring deflection increases from zero at no soil pressure to maximum at ultimate pressure, soil stiffness is the slope of the secant from the origin to the point of ultimate effective soil pressure on the stress-strain diagram Vertical soil strain is a function of effective soil pressure (intergranular) — not total pressure Effective soil pressure is, P = 15ft(132pcf) + 9ft(105pcf) - 15ft(62.4pcf) = 1.99ksf = 13.8psi At 80% density, from Figure 7-5, the soil strain at 1.99 ksf is C e = 1.85% The soil stiffness is the slope of the secant from to ksf on the 80% graph; i.e., E' = 13.8psi/0.0185 = 747 psi The resulting stiffness ratio is R' s = E'/(F/ ) = 747/815 = 0.92 Entering the graph of Figure 7-4 with R' s = 0.92, the corresponding ring deflection term is d/C e = 0.48 Ring deflection is 48% of the vertical soil strain C e Because soil strain is 1.85%, the predicted ring deflection is, d = 1.85%(0.48) 1.0% If a straight pipe of elastic material and circular cross section is bent into a circular curve, the cross section deforms into an ellipse Ring deflection of the cross section is, d = 2Z/3 + 71Z2/135 where Z = 1.5(1-n2)D4/16t2R2 d = ring deflection = D /D, D = decrease in pipe diameter, n = Poisson ratio, D = diameter of circular pipe, t = wall thickness, R = radius of the bend (7.2) *Decrease in diameter is in the direction of the radius of the bend See Chapter 14 for example 7-4 What is the probable ring deflection of an unreinforced concrete pipe, ID = 30 inch and wall thickness = 3.5 inches if a video from inside the pipe reveals a 0.1-inch-wide crack in the crown? REFERENCES Spangler, M.G (1973) and Handy, R.L Soil Engineering, IEP, New York Unibell (1982), Handbook of PVC Pipe Watkins, R.K (1974), Szpak, E., and Allman, W.B., Structural design of polyethylene pipes subjected to external loads, Eng'rg Expr Sta., USU PROBLEMS 7-1 What is the ultimate ring deflection of a steel water pipe, ID = 36 inches and t = inch, buried under saturated tailings which will rise ultimately to 250 ft? For tailings, G = 2.7 Unconsolidated, e = 0.7 When consolidated under H = 250 ft of tailings, e = 0.5 Assume a straight line variation of e with respect to height above the pipe Water table is at the ground surface (d = 11.8%) 7-2 A corrugated steel storm drain never flows full Therefore the granular backfill soil is essentially dry What is the ring deflection? Include HS-20 live load Given; (0.8) Soil (granular) H = ft = height of soil cover, G = 2.7 = specific gravity, e = 0.7 = void ratio, 80% density (AASHTO T-180) Steel pipe (corrugations 2/3 x 1/2) D = 48 inches = diameter, I = 0.0180 in 4/ft (t = 0.052), E = 30(106) psi = modulus of elasticity 7-3 What is the change in ring deflection of Problem 7-2 if the soil cover is increased from ft to 26 ft using the same soil, same density? (d = 1.6%) ©2000 CRC Press LLC 7-5 What is the ring deflection of a steel pipe, OD = 26 inches and ID = 24 inches, if E = 30(106) ps i and the soil cover is H = 40 ft.? The soil unit weight is 100 lb/ft3 at 80% density (AASHTO T-180) (Rs = 5.425; d = 0.064%) 7-6 Predict ring deflection of a plain steel pipe if: D = 10 ft, t = 0.5 inch, E = 30(106) psi Soil is granular, 90% dense (AASHTO T-180) g = 120 lb/ft3, H = 30 ft 7-7 If the neutral surface is at the geometrical center of the wall of Figure 7-2, prove that the width of the crack is approximately w = td; where w = width of crack t = wall thickness d = segmented ring deflection = decrease in vertical diameter D = diameter to neutral surface (NS) It is assumed that the wall crushes in compression on one side of the neutral surface just as much as it stretches in tension on the other side before the cracks open This is not true for all materials 7-8 If the ring of Figure 7-2 is vitrified clay or unreinforced concrete, both of which are many times stronger in compression than in tension, the compression crushing zones in the wall are very small As a worst case, assume no wall crushing and find the segmented ring deflection d if the cracks open 0.01 inch 7-9 Assume that the ring of Figure 7-2 is reinforced concrete with a single steel wire cage in the center of the wall The wire is 1/4-inch diameter spaced at inches What is the vertical diameter to the neutral surfaces? (34.7 in) t = inches Es = 30(10 ) psi ID = 30 inches Ec = 3(106) psi 7-10 What is the horizontal diameter to the neutral surfaces of Problem 7-9? What would be the difference if the cracks were caused by uniform soil load on top and bottom (no sidefill)? 7-11 A high-density polyethylene (HDPE) pipe has a minimum outside diameter (OD) of 6.60 inch and a maximum OD of 6.66 inch The wall thickness varies from 0.83 maximum to 0.80 minimum What is predicted ring deflection under the loading shown in Figure 7-3c if P = 5.0 kips per square ft? Assume that long-term virtual modulus of elasticity for HDPE is 85 ksi 7-12 What is the approximate ring deflection of the reinforced concrete pipe of Figure 7-6 when the width of crack at the crown is w = 0.06 inch? 7-13 If the load in Figure 7-6 is an F-load (parallel plate load), where is the neutral surface at the spring lines? 7-14 Are the cracks in Problem 7-13 exactly equal on the inside at crown and invert, and on the outside at spring lines? Explain 7-15 What is the maximum limit of ring deflection due to an F-load on a plastic pipe if permanent strain damage occurs at 1.75 percent strain on the outside surface? See Appendix A for deflection due to Floading (d = 12.78 13%) D = 0.5 meter, t = 16 mm, E = 300 ksi (2.07 GN/m2) 7-16 How high can plastic pipes be stacked if maximum allowable ring deflection is 10 percent? OD = 4.24 inches = outside diameter, ID = 3.92 inches = inside diameter, w = 1.08 lb/ft = pipe weight, E = 200 ksi = short term modulus of elasticity 7-17 A flexible plastic pipe, DR = 31, for which F/D = 90 psi, is buried in cohesionless soil with unit weight of 110 pcf at 80 percent density (AASHTO T-180) If the height of cover is 25 ft, what is the predicted ring deflection? Figure 7-6 Cross section of the wall of a reinforced concrete pipe, showing the transformed section in concrete, and showing the procedure for finding the neutral surface (NS) It is assumed in this case that concrete can take no tension ©2000 CRC Press LLC ... relieves the ring of load The soil above the pipe takes more of the load in arching action over the pipe — like a masonry arch Both the increase in strength of the ring and the soil arching action... which the ring becomes unstable Instability is explained in Chapter 10 E' E I = soil modulus = slope of a secant on the stressstrain diagram from the point of initial vertical effective soil pressure... at the spring lines? 7-14 Are the cracks in Problem 7-13 exactly equal on the inside at crown and invert, and on the outside at spring lines? Explain 7-15 What is the maximum limit of ring deflection