Numerical Methods in Soil Mechanics 05.PDF

Numerical Methods in Soil Mechanics 05.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 "PIPE MECHANICS" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 46 STRUCTURAL MECHANICS OF BURIED PIPES Figure 5-1 Nomenclature used in the ring analysis of buried pipes ©2000 CRC Press LLC CHAPTER PIPE MECHANICS Theoretical mechanics is the analysis of forces and their effects on materials In the case of buried pipes, forces are statically indeterminate, and are often indeterminable because the soil is not uniform Internal pressures, if any, may also be indeterminable Unknown soil loads are mitigated by the ability of soil to arch over the pipe and relieve the pipe of some load The effect of force on material is deformation Traditionally, force per unit area is stress, and deformation per unit length is strain Design is the analysis of stresses or strains to make sure they not exceed the maximum allowable Maximum allowable occurs at performance limits In the case of buried pipes, performance limit is usually excessive deformation; i.e., that deformation beyond which performance is not acceptable Excessive deformations include: buckling, collapsing, cracking, and tearing, as well as excessive deformation of the pipe Most useful, then, is the analysis of deformation Some deformations can be related to stresses such that classical stress theories can be used Stress theories are more responsive to loads than are strain theories But strain and strain energy theories are more responsive to deformation performance limits Traditional stress theories are presented in this text wherever they contribute to understanding In general, stresses are analyzed by theories of elasticity Clearly, performance of pipes is not limited to the range of elasticity The following comprises theoretical analyses of stresses, strains, and deformations Some basic simplifications are justified because of inevitable imprecisions such as deviations of the geometry, non-uniformities of the soil and indeterminable loads Combined stress analysis is not justified Therefore, longitudinal analysis, and ring analysis are each considered independently of the other Concentrated loads are the worst case loads, because loads are, in fact, distributed over a finite area Ring instability is the worst case of collapse analysis because instability is reduced by the interaction of ring stiffness and longitudinal stiffness ©2000 CRC Press LLC LONGITUDINAL ANALYSIS The two basic longitudinal analyses are axial and flexural Axial analysis considers the longitudinal effects of temperature changes, catenary tension, thrust at valves and elbows, and the Poisson effect of radial pressure Flexural analysis considers the longitudinal effect of beam bending Longitudinal beam analysis of buried pipes follows classical procedures Depending on the loads (weight of the pipe and its contents plus soil loads) and the reactions (high points or hard spots in the bedding), bending moment diagrams can be drawn, and deformations, strains, and stresses can be evaluated Longitudinal analysis is discussed in Chapter 14 For most buried pipes, either the manufacturer provides adequate longitudinal strength, or the pipe is so flexible longitudinally that it relieves itself of stress Corrugated pipes, for example, relieve themselves of longitudinal stresses by changing length and by beam bending that conforms with uneven beddings Lengths of pipe sections are limited by manufacturers in order to prevent longitudinal failure RING ANALYSIS Ring analysis considers stress, strain, deformation, and stability of the cross section (ring) cut by a plane perpendicular to the axis of the pipe See Figure 51 Stress Stress theory provides an acceptable analysis for rigid rings Deformation and strain theories provide better analyses for flexible rings Circumferential stresses comprise: hoop or ring compression stress, and moment stress or its equivalent ring deformation stress Circumferential stress analysis is analogous to the stress analysis Figure 5-2 Comparison of stress analyses of a short column and a pipe ring ©2000 CRC Press LLC of an eccentrically-loaded short column, see Figure 5-2, for which, within the elastic limit, For a plain (bare) pipe, Equation 5.2 becomes, s = Pm + (E/m) (r'-r)/2r' (5.3) s = F/A + Mc/I where s = maximum stress in the most remote fibers, F = compressive load on the column, M = moment acting on the cut section, I/c = section modulus of wall where m = r/t = wall flexibility, r = mean radius, t = wall thickness Strain For a pipe ring, by theory of elasticity, s = Pr/A + Mc/I (5.1) Within the elastic limit, strain is e = s /E Therefore, Equation 5.2 can be written as, where P = radial pressure, r = mean radius of the pipe, A = wall cross-sectional area per unit length, M = moment acting on the wall cross-section, I/c = section modulus of the wall per unit length e = Pr/AE + cdq For rigid rings, Equation 5.1 applies Thrust, T, (= Pr) and moment, M, are functions of the soil loading See Appendix A for values of T and M For a plain pipe with wall thickness, t, Example Deformation Find stres, s , at spring line of a ring loaded as shown in Figure 5-6a From Appendix A, T = Pr and M = Pr2/4 Let m = r/t = ring flexibility Substituting into Equation 5.1, s = Pm(1 + 3m/2) For a flexible ring, deliberate control of ring deformation is usually a better option than control of soil pressure The best control is specification of maximum allowable ring deformation For flexible rings, Equation 5.1 is more useful if flexural stress Mc/I is written in terms of change in radius of the ring From theory of elasticity, M/EI = dq = 1/r - 1/r' where dq is change in radius of curvature See Figure 5-3 Solving for M and substituting into Equation 5.1, Where it is necessary to predict ring deformation, the basic ring deformation of a buried circular pipe is from circle to ellipse See Figure 5-4 s = Pr/A + Ecdq But from Figure 3-2, (5.2) where dq = q - q' = 1/r - 1/r', E = modulus of elasticity, c = distance from NS to the most remote fiber ©2000 CRC Press LLC (5.4) where e = circumferential strain in the surfaces of the pipe wall, dq = 1/r - 1/r' e = Pm/E + (r'-r)/2mr' (5.5) Ring deflection from circle to ellipse decreases radius of curvature at B by, dq = 1/rx-1/r rx = r(1-d)2/(1+d) for small ring deflections — say less than 10% Figure 5-4 First mode ring deflection from a circle to an ellipse Ring deflection is a function of the vertical soil strain (compression) in the sidefill ©2000 CRC Press LLC Substituting, and neglecting higher orders of d, for ellipse, by elastic analysis at spring lines s = Pr/A + (Ec/r)3d/(1-2d) comp deformation term term (5.6) Pr3(1-n2)/EI = 3, or PD3(1-n2)/EI = 24 where n = Poisson ratio For most pipe design, thirddimensional effects enter in such that the effect of n is reduced and may be neglected Conservatively, where d = D/D = ring deflection =Dy /D ~ ~ Dx/D For homogeneous plain pipe, wall thickness t, and mean radius r; m = r/t = wall flexibility Stress is, s = Pm + 3Ed/2m(1-2d) (5.7) It is noteworthy from Equation 5.6 that the deformation term is insignificant at small values of d (when maximum ring deflection is specified) If the pipe wall can yield without fracture (such as metals and plastics), wall buckling or crushing does not occur until ring compression stress reaches yield strength The only exception is instability caused by external pressure when the ring is not constrained to nearly circular shape For flexible pipes, stability analysis is stiffness analysis — not stress analysis Stability Ring stability is resistance to progressive (runaway) deformation due to persistent loads The persistent loads may be caus ed by internal pressure, beam loading, or external pressure Failure is usually sudden and catastrophic Failure due to internal pressure is runaway rupture because, at yield stress, the diameter of the ring increases and wall thickness decreases Failure due to beam loading is fracture or buckling of the pipe wherever the bending moment is excessive Failure due to external pressure is collapse The loading for progressive deformation must be persistent; i.e., the load must bear against the pipe even as the pipe deforms away from the load Persistent loads include constant or intermittent internal pressure or vacuum, and gravity loads that are not relieved by soil arching The term, instability, most often implies collapse due to external pressure, P See Figure 5-5 Classical ©2000 CRC Press LLC analyses are available For example, a nonconstrained, circular, flexible ring will collapse catastrophically under pressure if, Pr3/EI = and PD3/EI = 24 (5.8) where Pr3/EI = ring stability number, P = critical uniform external pressure, r = mean radius = D/2, EI = wall stiffness per unit length of pipe, EI/r = ring stiffness, F/D = pipe stiffness, S = strength F/D = 53.77 EI/D3 = 6.72 EI/r3 (5.9) where F/D, called pipe stiffness by the plastic pipe industries, is the slope of the load-deflection diagram from a parallel plate test See Figure 5-5 The deflected cross section is not an ellipse Ring stiffness, EI/r3, is that property of a circular ring which resists collapse caused by external pressure EI/r is related to elasticity E — not to strength S In that respect, it differs from section modulus and arc modulus, which are related to strength, SI/c Ring stiffness can either be calculated or measured from a parallel plate test in which a plot of F vs provides the slope F/D, called pipe stiffness, from which EI/r3 = 0.149 F/D Classical unburied analysis is not responsive to buried pipe performance If the pipe is buried (constrained), soil support has a major effect on stability Pressure on the pipe is not uniform Moreover, the buried pipe will be out-of-round It may even have initial out-of-roundness, called ovality For these reasons, stability is considered further in Chapter 10 Figure 5-5 Notation used in deriving the equation for external pressure, P, at collapse of a flexible, circular ring, based on pipe stiffness, F/D , from a parallel plate test (or three-edge bearing test) Figure 5-6 Two soil loading assumptions for the analysis of rigid pipes ©2000 CRC Press LLC Example Corrugated Pipes: A steel pipe has a 51-inch mean diameter Wall thickness is 0.187 inch, E = 30,000 ksi, and yield strength is 42 ksi Neglecting Pm in Equation 5.3, what is the deformed radius of curvature r' at tensile yield stress on the inside surface? From Equation 5.3, s = E(r'-r)/2mr' Solving, r' = 41.25 inches Figure 5-8 is the wall of a corrugated pipe Equation 5.10 may be used as demonstrated in the following example What is r' at tensile yield on the outside surface? Equation 5.3 now becomes s = E(r-r')/2mr' Solving, r' = 18.45 inches Example Figure 5-8 is a typical 6x2 or 3x1 corrugation Values of section modulus are listed in industry manuals — but are all based on elastic theory What is the relationship of plastic theory to elastic theory for this corrugation? Plastic Performance Limits The limit of normal stress, s , is strength S For design, s = S/sf Performance limit is yield stress for: internal pressure, ring compression, and longitudinal stress However, for instability, the performance limit is ring collapse, which is a function of ring stiffness Ring stiffness, EI/r3, is derived from the theory of elasticity It is conservative When mitigation or failure analysis is needed, plastic theory may be more appropriate Plastic theory can be related to elastic theory by moment resistance as follows See Figure 5-7 In the center is a cross section (cross-hatched) of an element of pipe wall of thickness t and of unit length along the pipe, located at the top of the pipe, point A On the left is the elastic stress distribution due to ring deflection The resisting moment is Me = SI/c, where, I/c = section modulus, and S = yield stress On the right is the plastic stress distribution The resisting moment is Mp = 3SI/2c Elastic moment, Me, at surface yield stress, is not collapse Once the surface starts to yield, stresses within the wall thickness increase to the yield strength as shown at the right of Figure 5-7 Performance limit is the idealized plastic moment, Mp = 3Me /2 (5.10) The ring is now buckling (plastic flow) Collapse is in process ©2000 CRC Press LLC For a single corrugation, section modulus, I/c, is not changed if the corrugation is compressed horizontally as shown to the right of the corrugation But the section modulus for the compressed corrugation is essentially the same as the rectangular equivalent section for which Mp /Me = 1.5 From exact analyses, the moment ratio can be as much as Mp /Me = 1.7, but for design, it is conservative to hold to a ratio of 1.5 Ribbed and Reinforced Pipes: Plastic analysis of ribbed pipe walls follows the same procedure as the plain walls of Figure 5-7, but requires location of the neutral surface, NS, and evaluation of moment of inertia, I, of the cross section See texts on mechanics of solids Plastic analysis of reinforced pipe walls can be related to elastic analysis by transforming the pipe wall cross section into its equivalent section in one material or the other Procedure is then the same as for ribbed pipe walls The procedure for transformation to equivalent section is described in texts on solid mechanics and on reinforced concrete design However, reinforced concrete pipes comprise steel which is somewhat plastic, and concrete, which is not plastic Therefore, plastic analysis of reinforced concrete pipes is of questionable value In general, rigid pipes should be designed by theories of elasticity, not theories of plasticity Figure 5-7 Flexural stresses on a longitudinal section (cross-hatched) of pipe wall at A showing maximum elastic stress distribution to the left, and maximum plastic stress distribution to the right The plastic resisting moment is 1.5 times the elastic resisting moment Figure 5-8 Cross section of corrugated pipe wall, showing how it can be compressed horizontally to an equivalent rectangular section for evaluating section modulus I/c ©2000 CRC Press LLC Values of section modulus, I/c, per length of the pipe, are listed in industry manuals PROBLEMS 5-1 A thin-wall pipe is initially an ellipse with ring deflection, d What is the maximum moment in the ring due to rerounding? 5-2 Find the moment at B on the rigid pipe of Figure 5-6b if the vertical soil pressure is P and the horizontal soil pressure is P/K; i.e., active horizontal soil pressure (Appendix A) 5-3 If t = D/10 in Figure 5-6a, where and what is the maximum tangential normal stress? Include ring compression as well as flexural stresses (40 P at A, 43 P at B) 5-4 For a diametral line load F on a rigid pipe of wall thickness t < D/10, what and where is the maximum tangential normal stress ? Include ring compression (s = 9.55F/t at location of load F) 5-5 From Equation 5.8 and the parallel plate load of Appendix A, show that critical pressure on a flexible circular ring is P = 0.446F/D, where F/D is the slope of the F/-D diagram from a parallel plate test 5-6 What is the maximum strain in a pipe ring if D/t = 20 and the ring is deflected from a circle into an ellipse with ring deflection of d = 10%? Neglect the ring compression strain Consider only flexural − =1.9%) strains (C 5-7 Find ring deflection at yield stress in a steel pipe if the ring deflects into an ellipse Assume that ring compression stress is negligible (d = 9.9%) Given: D = 51 inch t = 0.187 inch E = 30,000 ksi Sy = 42 ksi What can be said about ring deflection at plastic hinging? Unstable? Indeterminable? 5-8 What is the external pressure on the pipe of Problem 5-7 at collapse, if the pipe is not buried? (2.9 psi) 5-9 What is external pressure at collapse of an unburied PVC pipe with ribs? The pipe is stiffened by external ribs, Figure 5-9 (215 kPa) Given: ID = 450 mm, smooth bore, t = mm, wall thickness, OD = 500 mm, over the ribs, E = 3.5 GPa, modulus of elasticity Ribs are mm thick spaced at 50 mm 5-10 What would be the external pressure at collapse of the PVC pipe of Problem 5-9 if ID = 450 mm and t = mm, but without ribs? (4.8 kPa) e Figure 5-9 Wall cross section of an externally ribbed PVC pipe ©2000 CRC Press LLC ... In the case of buried pipes, forces are statically indeterminate, and are often indeterminable because the soil is not uniform Internal pressures, if any, may also be indeterminable Unknown soil. .. pipe, are listed in industry manuals PROBLEMS 5-1 A thin-wall pipe is initially an ellipse with ring deflection, d What is the maximum moment in the ring due to rerounding? 5-2 Find the moment... loads are, in fact, distributed over a finite area Ring instability is the worst case of collapse analysis because instability is reduced by the interaction of ring stiffness and longitudinal stiffness