Numerical Methods in Soil Mechanics 17.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 "PARALLEL PIPES AND TRENCHES" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 17-1 Pipe-clad soil column between parallel pipes showing minimum section AA, which must support part of the surface live load W plus the dead load, shown cross-hatched ©2000 CRC Press LLC CHAPTER 17 PARALLEL PIPES AND TRENCHES When buried pipes are installed in parallel, principles of analysis for single pipes still apply Soil cover must be greater than minimum However, the design of parallel buried pipes requires an additional analysis for heavy surface loads Consider a freebody-diagram of the pipe-clad soil column between two parallel pipes See Figure 17-1 Section AA is the minimum cross section This column must support the full weight of the soil mass, shown cross-hatched, plus part of the surface load W shown as a live load pressure diagram The soil column is critical at its minimum section AA at the spring lines For design, the strength of the column at section AA must be greater than the vertical load horizontal σx = Kσ y Px = P = σ x + u where u is the water pressure at Section AA STRESSES If bond between the soil and the pipe wall could be assured, the column would be analyzed as a reinforced concrete column based on an equivalent (transformed) section But bond between soil and pipe cannot be assured because of fluctuations in temperature, moisture, and loads, all of which tend to break down bond It is assumed that bond is zero Therefore, stresses in the pipe and soil are each calculated independently STRENGTHS DESIGN OF PIPE Performance limits of the column are either: ring compression strength of the pipe wall; or compressive strength (vertical passive resistance) of the soil at Section AA Ring compression strength of the pipe wall is yield strength σ f For steel pipes, σ f is usually 36 or 42 ksi For rigid pipes, σf is crushing strength of the wall For plastic pipes, σf depends on temp-erature, s tress, and service life Manufacturers publish values Strength of the soil is found as follows Assume that the embedment is granular and compacted Soil strength is vertical stress, σy, at slip Horizontal soil stress is provided by the pipe walls Approximate soil strength may be found from triaxial soil tests in which interchamber pressure is equal to the horizontal pressure, Px, of the pipe against the soil For circular, flexible pipes at soil slip, Px = Pd = γ H Live load pressure, P1, has no effect because the live load is not directly above the pipe If there should be a water table above Section AA, compressive soil strength at failure would be the effective vertical σy confined by ©2000 CRC Press LLC Before the soil column is analyzed, the pipe must be adequate See Chapter Design starts with the ring compression equation, P(OD)/2A = σf /sf, where OD = A = σf = sf = P = outside diameter of the pipe, pipe wall area per unit length of pipe, ring compression strength of the wall, safety factor, maximum vertical soil pressure on top of the pipe For worst-case ring compression, live load W is directly above the pipe where P = P1 + Pd The live load effect, P1, can be found by Boussinesq or Newmark If W is assumed to be a point load, according to Boussinesq, P1 = 0.477W/H2 See Chapter If live load W is assumed to be a distributed surface pressure, the Newmark integration can be used Soil cover must be greater than minimum by the the pyramid/cone analysis of Chapter 13 In the following it is assumed that the pipe is adequate DESIGN OF SOIL COLUMN The following analysis is for flexible pipes Rigid pipes require modification of the procedure See Figure 17-1 The vertical load supported by the two flexible pipe walls at section AA is no less than 2PD/2 = PD So, in the design of the soil column, it is assumed, conservatively, that the pipe wall cladding takes a vertical load of PD But this is only part of the total load The remainder must be supported by the soil The greatest load occurs when the heavy live load W is centered above section AA — not over the top of the pipe At this location, not only is the live load pressure maximum, but the portion supported by the pipe wall cladding is minimum Pipe walls carry dead load, PdD = γ HD Live load, Pl, on the pipes is small enough to be neglected It is already supported by the ring stiffness required for minimum cover What cannot be neglected is the Boussinesq live load on section AA Soil stress, σy, must be less than strength S' Vertical stress is soil load divided by the crosssectional area σy = Q'/X = S'/sf where σy = S' = X = sf = Q' = γ H D Q = = = = (17.1) vertical soil stress on section AA, vertical soil compression strength, width of section AA between pipes, safety factor, Q - γ HD = load supported by the soil at section AA = total load reduced by the load that is supported by the pipe walls, unit weight of soil, height of soil cover, diameter of the pipe = 2r, vertical load on section AA = wd + w Per unit length of pipe, Q is the sum of the dead weight of the cross-hatched soil mass w d, and that portion w of the surface live load W that reaches section AA See Figure 17-1 The dead load wd per unit length (1) of pipe is soil unit weight times the cross-hatched area; i.e., w d = (1)[(X+2r)(H+r) - πr2/2]γ ©2000 CRC Press LLC (17.2) The live load w is the volume under the live load pressure diagram of Figure 17-1 at section AA It is calculated by means of Boussinesq or Newmark as described in Chapter The pyramid/cone punch-through stress analysis does not apply because the cover is not less than minimum If Boussinesq is justified, the live load w per unit length is w = 0.477WX/(H+r)2 (17.3) Example What is the vertical soil stress at section AA of Figure 17-1? The pipes are corrugated steel, 72inch diameter, 2-2/3 by 1/2 corrugations, t = 0.0598, separated by X = 1.0 ft of soil, with 1.5 ft of soil cover at unit weight of 120 pcf A surface wheel load of W = 20 kips is anticipated From a ring compression analysis, the 20-kip load can pass over each pipe without exceeding yield stress of the pipe Soil cover is greater than minimum In order to evaluate soil stress at section AA, from Equation 17.2, dead load on section AA is wd = 2.08 kips The live load, w l, can be evaluated by Boussinesq Equation 17.3, or by Newmark Figure 4-6 If the dual-wheel print is ft by ft, based on Newmark, w1 = 4MWX(1 ft)/2ft2, where, X = width of section AA = ft w1 = total live load on section AA, W = wheel load on the surface, M = f[(L/B), (B/H)] = coefficient from the Newmark chart Figure 4-6, for each quarter area of surface load W, where, B = 0.5 ft, L = 1.0 ft, H = 4.5 ft = the Newmark H, which is the depth to section AA = 1.5ft + ft The Newmark denominator, ft2, is the area of surface load W Substituting values, L/B = 2, and B/H = 0.111, the Newmark M = 0.012, and w1 = 0.48 kips The total load on section AA is, Q = 2.08 + 0.48 = 2.56 kips The load supported by the soil alone is, Q' = Q - γ HD = 2.56 - 1.08 = 1.48 kips γ HD is the load supported by the pipe walls Vertical soil stress on section AA is, σy = 1.48/(1ft)(1ft) = 1480 psf Could Boussinesq have been used without significant error? From Chapter 4, if H/B is greater than 3, Boussinesq is adequate In this case, H/B is 4.5/0.5 = Let's see if it's adequate From Equation 17.4, w = 0.477(20 kips)(1ft)(1ft)/(4.5ft)2 = 0.47 kips compared to 0.48 kips by Newmark There is no question that Boussinesq is adequate Rigid Pipes: Unlike flexible pipes, rigid pipes not exert pressure, Px = P, against the soil Total load, Q, is supported by the pipe walls in ring compression and the soil in vertical passive resistance It is possible to analyze the equivalent section by column design See texts on reinforced concrete There is a great difference between the modulus of elasticity of the pipe wall and the modulus of elasticity (compressibility) of the soil Safety Factors: Analyses of the soil column with pipe wall cladding, are conservative Longitudinal resistance of the pipes and soil cover is neglected Also the arching action of the soil cover is neglected Safety factors can be small Example What is the vertical soil strength at section AA for the parallel pipes of the example above? The soil friction angle is φ = 30o The horizontal pressure of the pipe wall against the soil at section AA is σx = γ H = 180 psf The vertical strength of the soil at slip is σxK where K = (1+sinφ)/(1-sinφ) = Vertical soil strength is, S' = 180(3) = 540 psf The vertical soil stress from Example is 1480 psf — much greater than the soil strength, 540 psf It would be necessary to: triple the space between the parallel pipes, or place concrete between the pipes, or specify stiffer pipes Tank spacing: For multiple parallel tanks, the following are minimum spacings which must be increased as needed to accommodate deadmen or anchor slabs Refer to Chapter 21 on tank anchors If sufficient clearance must be allowed for deadmen to be set outside of the tank shadow; Spacing between parallel tanks should be no less than one-fourth tank diameter The live load is assumed to be HS20 dual-wheel load with minimum soil cover of 2.25 ft ©2000 CRC Press LLC PARALLEL TRENCH Buried flexible pipes depend on the embedment for stability Compacted soil at the sides supports and stiffens the top arch What happens to a buried flexible pipe when a trench is excavated parallel to it? What is the stability of the trench? At what minimum separation between the pipe and the parallel trench will the pipe collapse? What are the variables that influence collapse? Answers to these questions were the objectives of an experiment at USU in 1968 In order to reduce the number of variables, ring stiffness was assumed to be zero Results were conservative because no pipe has zero ring stiffness For the most flexible plain steel pipes, D/t is less than 300 For the test pipes, D/t was 600 in an attempt to approach zero stiffness It was necessary to hold the pipes in shape on mandrels during placement of the backfill Vertical Trench Walls Figure 17-2 is the cross section of a buried, flexible pipe with an open cut vertical trench wall parallel to it If trench wall AB is cut back closer and closer to the buried pipe, side cover X decreases to the point where the sidefill soil is no longer able to Figure 17-2 Vertical trench wall parallel to a buried flexible pipe showing the soil wedge and shear planes that form as the pipe collapses Figure 17-3 Formation of a soil prism on the pipe during ring deflection as the soil wedge is thrust into the trench ©2000 CRC Press LLC provide the lateral support required to retain the flexible ring The ring deflects, thrusting out a soil wedge as indicated in Figure 17-3 As the ring deflects, a soil prism breaks loose directly over the ring The soil prism collapses the flexible ring In order to write pi-terms to investigate this phenomenon, the pertinent fundamental variables must be identified Ring stiffness is ignored because the ring is flexible In fact, at zero ring deflection, the ring stiffness has no effect anyway The remaining fundamental variables are: Fundamental Variables X = D H = = Z = Basic Dimensions minimum side cover (minimum horizontal separation between pipe and trench at collapse), pipe diameter, height of soil cover over the top of the pipe, critical depth of trench in vertical cut (vertical sidewalls) L L L L Critical depth, Z, is a convenient measure of soil strength It is defined as the maximum depth of a trench at which the walls stand in vertical cut At greater depths the trench walls slip or cave in Critical depth Z may be determined by excavation in the field, or it may be calculated from the dimensionless stability number, Zγ /C See Figure 17-6 2C/γ Z = tan(45 - ϕ /2) o where Z = γ = C = ϕ = (17.4) critical depth of trench in vertical cut, unit weight of soil (pcf), soil cohesion (psf), soil friction angle of the trench wall Z can be found from Equation 17.4 if soil properties, γ , C, and ϕ , are provided by laboratory tests To investigate the four fundamental variables, ©2000 CRC Press LLC three pi-terms are required One possible set is (X/D), (H/D), and (H/Z) Tests show that (H/D) is not pertinent Only (X/D) and (H/Z) remain as pertinent pi-terms Results of the tests are as follows For a vertical trench wall excavated parallel to a flexible pipe, Failure is sudden and complete collapse The ring collapses under a free-standing prism of soil that breaks loose on top of the pipe If the ring has some stiffness, and if soil cover H is not great enough to collapse the ring, soil may slough off the pipe into the trench This is not considered failure because the soil can be replaced during backfilling Test data are plotted in Figure 17-4, which shows (X/D) as a function of (H/Z) The best fit straight line equation is, X/D = 1.4(H/Z) The probable error in X/D is plus or minus 0.1, so probable error in side soil cover, X, is roughly plus or minus D/10 Because field conditions may be less reliable than laboratory conditions, the safety factor should be two Therefore, the minimum side cover, X, might be specified as, X/D = 3H/Z (17.5) Of interest in Figure 17-4 are the data points indicated by squares These not represent collapse The ring stiffness for the test pipes was great enough that part of the shallow soil cover merely sloughed off the pipes after the soil wedge fell into the trenc h If ring stiffness were to be included as a fundamental variable, ring deflection would have to be included Then the coefficient of friction between pipe and soil should also be included as a fundamental variable If the pipe has significant ring stiffness, the height of soil cover that it can support without collapse can be found for uniform vertical pressure with Figure 17-4 Cover term X/D as a function of soil strength term H/Z for a vertical trench wall excavated parallel to a very flexible buried pipe Figure 17-5 Trench wall sloped at angle of repose for which the soil is stable, but the flexible ring requires either some ring stiffness or a minimum cover ©2000 CRC Press LLC no side support See Appendix A, from which moment = Pr2/4 = σ I/c For plain steel pipes based on elastic theory, at yield stress, P = 16σf I/cD2 where P = I = D c = = t σf m = = = (17.7) Based on plastic theory (plastic hinging), P = 24σf I/cD2 (17.9) Vertical ring deflection at plastic hinging is, d = 0.01PD3/EI (17.6) vertical soil pressure at collapse, moment of inertia of the wall cross section, pipe diameter = 2r, half the distance to wall surface from the neutral surface = t/2 for plain pipes, wall thickness of plain pipes, yield strength of the pipe, D/t = ring flexibility for plain pipes P = (8σf /3m2) for plain pipes P = 4σf /m2 for plain pipes (17.8) where d = ∆ = P = D = EI = ∆/D = ring deflection, decrease in vertical diameter, vertical pressure on the ring, circular pipe diameter, wall stiffness per unit length of pipe Sloped Trench Walls Figure 17-5 shows a flexible pipe in cohesionless soil for which the slope is angle of repose ~ ~ ϕ Pressure distribution on the ring is triangular as shown Maximum moment at A can be found by Castigliano's equation However, it is sufficiently accurate to find equivalent moment MA = Pr 2/4 for average uniform pressure, Px = rγ See Figure 17-6 Rationale for finding critical depth, Z, of a vertical open cut in a trench wall in brittle soil with cohesion, C, and soil friction angle, ϕ ; 2C/γ Z = tan(45o - ϕ/2) ©2000 CRC Press LLC Appendix A The required section modulus is, I/c = MA(sf)/σ f where σf is yield stress EXCAVATION Depth of the excavation must include "overexcavation" required to remove unstable subbase material It should be replaced by approved bedding material Some tank manufacturers consider soil to be unstable if the cohesion is less than C = 750 psf bas ed on unconfined compression test, or if the bearing capacity is less than 3500 psf In the field, bearing capacity is adequate if an employee can walk on the excavation floor without leaving footprints A muddy excavation floor can be choked with gravel until it is stable These are conservative criteria for soil stability Of greater concern are OSHA safety requirements for retaining or sloping the walls of the trench Excavations for tanks are usually short enough that OSHA trench requirements leave a significant margin of safety Longi-tudinal, horizontal soil arching action is significant the slope of the failure plane is (45o+ϕ /2) For a two-dimensional trench analysis, the infinitesimal soil cube, O, is subjected to vertical stress, γ Z, where, γ = soil unit weight, Z = critical depth of vertical trench wall, ϕ= soil friction angle, C = soil cohesion The Mohr circle is shown in Figure 17-6 (right) The orientation diagram (x-z) of planes on which stresses act, is superimposed, showing the location of the origin, O The strength envelope slopes at soil friction angle ϕ from the cohesive strength, C At soil slip, the Mohr stress circle is tangent to the strength envelope From trigonometry, tan(45o - ϕ/2) = 2C/γ Z This is the critical depth Equation 17.4 From tests, Equation 17.4 provides a reasonable analysis for brittle soil If the soil is plastic, soil slip does not occur until shearing stresses reach shearing strength C Consequently, in plastic soil, the critical depth equation is 2c/γ Z = Below the water table, critical depth is essentially doubled Example These criteria for bearing capacity and cohesion are equivalent to a vertical trench wall over 20 ft deep Bearing capacity of 3500 psf can support more than 29 ft of vertical trench depth at soil unit weight of 120 pcf Cohesion of 750 psf can support a vertical open cut trench wall that is more than 20 ft deep Critical Depth of Vertical Trench Wall Granular soil with no cohesion cannot stand in vertical cut Much of the native soil in which pipes and tanks are buried have cohesion Therefore, the wall of the excavation can stand in vertical open cut to some critical depth, Z See Figure 17-6 (left) Greater depth will result in a "cave-in" starting at the bottom corner, O, where ©2000 CRC Press LLC What is the critical depth, Z, of a vertical, opencut, trench wall if, C = 750 lbs/ft2, γ = 120 lbs/ft3, ϕ = 30o? Substituting into Equation 17-4, Z = 22 ft This is a lower limit if the soil has some plasticity (is not brittle) Excavations for tanks are almost never greater than 20 feet Example Suppose that a sloped trench wall exposes a pipe as shown in Figure 17-5 Pressure, Px, must be resisted by ring stiffness What is the required wall thickness for a 72-inch plain steel pipe? Assume the soil is granular with unit weight of 120 pcf 17-2 Reconsider Problem 17-1 if the soil is cohesionless; i.e., c = From Appendix A, the maximum moment in the ring is M = Pxr2/4 Ring resistance by elastic theory must be I/c = t2/6 = M/σf Yield stress is σf = 42 ksi Allowable stress is reduced from 42 ksito 21 ksi From Chapter 16, P x = γ r Substituting values, t = 0.48 inch, for which D/t = 150 17-3 A two meter plain steel pipe with wall thickness t = 10 mm, is buried in compacted fine sand with a soil cover of meter and with water table at the surface on occasions If a trench can be exc avated with vertical sidewalls to a maximum depth of meters in the soil at high water table, what is the minimum separation X of a trench excavated parallel to the pipe? (X = 1.5 m) A little ring stiffness makes a big difference in the stability of a flexible ring on a sloped trench wall (sidehill) In the case of steel pipes, mortar lining significantly increases the ring stiffness The above rationale is based on an infinite slope For a sloped trench wall, pressures on the ring are considerably less For most pipes, the ring stiffness required for installation is adquate if it complies with the familiar rule of thumb according to which minimum cover is D/2 For a pipe parallel to a sloped trench wall in cohesionless soil, minimum cover is half a diameter to the sloped surface of the trench wall 17-4 Find vertical pressure at collapse of the pipe of Problem 17-3? Elastic yield σf = 290 MPa (19.3 kPa) 17-5 What would be the maximum soil cover if the pipe of Problem 17-3 is not to collapse, X is minimum 1.5 m, and P = 19.3 kPa? (H = 2.9 ft) 17-6 Prove that for a plain pipe, the vertical soil pressure at plastic hinging is 3/2 times the vertical soil pressure at elastic limit Assume yield strengths are the same for both elastic and plastic analyses Assume that the elastic limit is yield strength PROBLEMS 17-1 What is the minimum allowable separation (side cover X) between a buried flexible plain steel pipe and a parallel trench if the trench walls are vertical and if: ϕ = 30 degrees = soil friction angle C = 400 psf = soil cohesion D = ft = pipe diameter t = 3/8 inch = pipe wall thickness H = ft = soil cover γ = 125 pcf = soil unit weight (X = 6.5 ft) ©2000 CRC Press LLC 17-7 Two parallel 14-ft diameter flexible steel pipes are buried under H = ft of dune sand, γ = 120 pcf and ϕ = 25o, separated by X = ft What is the maximum allowable wheel load? DEAD LOAD SOIL SLIP So compact soil, ϕ = 38o, and increase X = ft (W = 70 kips) ... deflects, thrusting out a soil wedge as indicated in Figure 17-3 As the ring deflects, a soil prism breaks loose directly over the ring The soil prism collapses the flexible ring In order to write... of a soil prism on the pipe during ring deflection as the soil wedge is thrust into the trench ©2000 CRC Press LLC provide the lateral support required to retain the flexible ring The ring deflects,... of a flexible ring on a sloped trench wall (sidehill) In the case of steel pipes, mortar lining significantly increases the ring stiffness The above rationale is based on an infinite slope For