Numerical Methods in Soil Mechanics 01.PDF

Numerical Methods in Soil Mechanics 01.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 "INTRODUCTION" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 CHAPTER INTRODUCTION Buried conduits existed in prehistory when caves were protective habitat, and ganats (tunnels back under mountains) were dug for water The value of pipes is found in life forms As life evolved, the more complex the organism, the more vital and complex were the piping systems In a phenomenon as complex as the soil-structure interaction of buried pipes, all three sources must be utilized There are too many variables; the interaction is too complex (statically indeterminate to the infinite degree); and the properties of soil are too imprecise to rely on any one source of information The earthworm lives in buried tunnels His is a higher order of life than the amoeba because he has developed a gut — a pipe — for food processing and waste disposal Buried structures have been in use from antiquity The ancients had only experience as a source of knowledge Nevertheless, many of their catacombs, ganats, sewers, etc., are still in existence But they are neither efficient nor economical, nor we have any idea as to how many failed before artisans learned how to construct them The Hominid, a higher order of life than the earthworm, is a magnificent piping plant The human piping system comprises vacuum pipes, pressure pipes, rigid pipes, flexible pipes — all grown into place in such a way that flow is optimum and stresses are minimum in the pipes and between the pipes and the materials in which they are buried Consider a community A termite hill contains an intricate maze of pipes for transportation, ventilation, and habitation But, despite its elegance, the termite piping system can't compare with the piping systems of a community of people The average city dweller takes for granted the services provided by city piping systems, and refuses to contemplate the consequences if services were disrupted Cities can be made better only to the extent that piping systems are made better Improvement is slow because buried pipes are out-of-sight, and, therefore, out-ofmind to sources of funding for the infrastructure Engineering design requires knowledge of: performance, and limits of performance Three general sources of knowledge are: SOURCES OF KNOWLEDGE Experience Experimentation Principles ©2000 CRC Press LLC (Pragmatism) (Empiricism) (Rationalism) The other two sources of knowledge are recent Experimentation and principles required the development of soil mechanics in the twentieth century Both experience and experimentation are needed to verify principles, but principles are the basic tools for design of buried pipes Complex soil-structure interactions are still analyzed by experimentation But even experimentation is most effective when based on principles — i.e., principles of experimentation This text is a compendium of basic principles proven to be useful in structural design of buried pipes Because the primary objective is design, the first principle is the principle of design DESIGN OF BURIED PIPES To design a buried pipe is to devise plans and specifications for the pipe-soil system such that performance does not reach the limits of performance Any performance requirement is equated to its limit divided by a safety factor, sf, i.e.: Figure 1-1 Bar graph of maximum peak daily pressures in a water supply pipeline over a period of 1002 days with its corresponding normal distribution curve shown directly below the bar graph ©2000 CRC Press LLC Performance = Performance Limit Safety Factor Examples: Stress = Strength/sf Deformation = Deformation Limit/sf Expenditures = Income/sf; etc If performance were exactly equal to the performance limit, half of all installations would fail A safety factor, sf, is required Designers must allow for imperfections such as less-than-perfect construction, overloads, flawed materials, etc At present, safety factors are experience factors Future safety factors must include probability of failure, and the cost of failure — including risk and liability Until then, a safety factor of two is often used In order to find probability of failure, enough failures are needed to calculate the standard deviation of normal distribution of data NORMAL DISTRIBUTION Normal distribution is a plot of many measurements (observations) of a quantity with coordinates x and y, where, see Figure 1-1, x = abscissa = measurement of the quantity, y = ordinate = number of measurements in any given x-slot A slot contains all measurements that are closer to the given x than to the next higher x or the next lower x On the bar graph of data Figure 1-1, if x = 680 kPa, the 680-slot contains all of x-values from 675 to 685 kPa x) x n w P = the average of all measurements, = 3yx/3y, = total number of measurements = Ey, = deviation, w = x - x) , = probability that measurement will fall between ±w, Pe = probability that a measurement will exceed the failure level of xe (or fall below a minimum level of xe ), ©2000 CRC Press LLC s = standard deviation = deviation within which 68.26 percent of all measurements fall (Ps = 68.26%) P is the ratio of area within +w and the total area Knowing w/x, P can be found from Table 1.1 The standard deviation s is important because: l it is a basis for comparing the precision of sets of measurements, and it can be calculated from actual measurements; i.e., s = %3yw2/(n-1) Standard deviation s is the horizontal radius of gyration of area under the normal distribution curve measured from the centroidal y axis s is a deviation of x with the same dimensions as x and w An important dimensionless variable (pi-term) is w/s Values are listed in Table 1-1 Because probability P is the ratio of area within ±w and the total area, it is also a dimensionless pi-term If the standard deviation can be calculated from test data, the probability that any measurement x will fall within ±w from the average, can be read from Table 1-1 Likewise the probability of a failure, Pe , either greater than an upper limit xe or less than a lower limit, xe , can be read from the table The deviation of failure is needed; i.e., we = x e - x) Because pipesoil interaction is imprecise (large standard deviation), it is prudent to design for a probability of success of 90% (10% probability of failure) and to include a safety factor Probability analysis can be accomplished conveniently by a tabular solution as shown in the following example Example The bursting pressure in a particular type of pipe has been tested 24 times with data shown in Table 1-2 What is the probability that an internal pressure of 0.8 MPa (120 psi or 0.8 MN/m2) will burst the pipe? x = test pressure (MN/m2) at bursting y = number of tests at each x n = Gy = total number of tests Table 1-1 Probability P as a function of w/s that a value of x will fall within +w, and probability Pe a s a function of we /s that a value of x will fall outside of +w e on either the +w e or the -w e w e /s 0.0 0.1 0.2 0.3 0.4 P (%) 0.0 8.0 15.9 23.6 31.1 Pe (%) 50.0 46.0 42.1 38.2 34.5 w e /s 1.5 1.6 1.7 1.8 1.9 P (%) 86.64 89.04 91.08 92.82 94.26 Pe (%) 6.68 5.48 4.46 3.59 2.87 0.5 0.6 0.6745 0.7 38.3 45.1 50.0 51.6 30.9 27.4 25.0 24.2 2.0 2.1 2.2 2.3 95.44 96.42 97.22 97.86 2.28 1.79 1.39 1.07 0.8 0.9 57.6 63.2 21.2 18.4 2.4 2.5 98.36 0.82 98.76 0.62 1.0 1.1 1.2 1.3 1.4 68.26 72.9 78.0 80.6 83.8 15.9 13.6 11.5 9.7 8.1 2.6 2.7 2.8 2.9 3.0 99.06 99.30 99.48 99.62 99.74 0.47 0.35 0.26 0.19 0.135 Table 1-2 Pressure data from identical pipes tested to failure by internal bursting pressure, and a tabular solution of the average bursting pressure and its standard deviation x (Mpa)* 0.9 1.0 1.1 1.2 1.3 1.4 Sums y xy _ (MPa) 1.8 7.0 8.8 4.8 2.6 1.4 24 26.4 n Σxy w (MPa) -0.2 -0.1 0.0 +0.1 +0.2 +0.3 yw (MPa) -0.4 -0.7 0.0 +0.4 +0.4 +0.3 yw (MPa) 0.08 0.07 0.00 0.04 0.08 0.09 0.36 Σyw x = Sxy/n = 1.1 MPa s = [ Syw 2/(n-1)] = 0.125 *MPa is megapascal of pressure where a Pascal is N/m2; i.e., a megapascal is a million Newtons of forc e per square meter of area A Newton = 0.2248 lb A square meter = 10.76 square ft ©2000 CRC Press LLC From the data of Table 1-2, _ x = Σxy/Sy = 26.4/24 = 1.1 s = \ w = _ x - x, so /Syw /(n-1) = \/0.36/23 = 0.125 w e = (0.8 - 1.1) = -0.30 MN/m2 = deviation to failure pressure w e/s = 0.30/0.125 = 2.4 From Table 1-1, interpolating, Pe= 0.82% The probability that a pipe will fail by bursting pressure less than 0.80 MN/m2 is Pe = 0.82 % or one out of every 122 pipe sections Cost accounting of failures then follows The probability that the strength of any pipe section will fall within a deviation of w e = +0.3 MN/m2 is P = 98.36% It is noteworthy that P + 2Pe = 100% From probability data, the standard deviation can be calculated From standard deviation, the zone of +w can be found within which 90% of all measurements fall In this case w/s = w/0.125 for which P = 90% From Table 1-1, interpolating for P = 90%, w/s = 1.64%, and w = 0.206 MPa at 90% probability Errors (three classes) Mistake = blunder — Remedies: double-check, repeat Accuracy = nearness to truth — Remedies: calibrate, repair, correct Precision = degree of refinement — Remedies: normal distribution, safety factor PERFORMANCE Performance in soil-structure interaction is deformation as a function of loads, geometry, and properties of materials Some deformations can be written in the form of equations from principles of ©2000 CRC Press LLC soil mechanics The remainders involve such complex soil-structure interactions that the interrelationships must be found from experience or experimentation It is advantageous to write the relationships in terms of dimensionless pi-terms See Appendix C Pi-terms that have proven to be useful are given names such as Reynold's number in fluid flow in conduits, Mach number in gas flow, influence numbers, stability numbers, etc Pi-terms are independent, dimensionless groups of fundamental variables that are used instead of the original fundamental variables in analysis or experimentation The fundamental variables are combined into pi-terms by a simple process in which three characteristic s of pi-terms must be satisfied The starting point is a complete set of pertinent fundamental variables This requires familiarity with the phenomenon The variables in the set must be interdependent, but no subset of variables can be interdependent For example, force f, mass m, and acceleration a, could not be three of the fundamental variables in a phenomenon which includes other variables because these three are not independent; i.e., f = ma Only two of the three would be included as fundamental variables Once the equation of performance is known, the deviation, w, can be found Suppose r = f(x,y,z, ), then w r2 = Mrx2 w x2 + Mry 2w y + where w is a deviation at the same given probability for all variables, such as standard deviation with probability of 68%; mrx is the tangent to the r-x curve and wx is the deviation at a given value of x The other variables are treated in the same way CHARACTERISTICS OF PI-TERMS Number of pi-terms = (number of fundamental variables) minus (number of basic dimensions) All pi-terms are dimensionless Each pi-term is independent Independence is assured if each pi-term contains a fundamental variable not contained in any other pi-term Figure 1-2 Plot of experimental data for the dimensionless pi-terms (P'/S) and (t/D) used to find the equation for bursting pressure P' in plain pipe Plain (or bare) pipe has smooth cylindrical surfaces with constant wall thickness — not corrugated or ribbed or reinforced Figure 1-3 Performance limits of the soil showing how settlement of the soil backfill leaves a dip in the surface over a flexible (deformed) pipe and a hump and crack in the surface over a rigid (undeformed) pipe ©2000 CRC Press LLC Pi-terms have two distinct advantages: fewer variables to relate, and the elimination of size effect The required number of pi-terms is less than the number of fundamental variables by the number of basic dimensions Because pi-terms are dimensionless, they have no feel for size (or any dimension) and can be investigated by model study Once pi-terms have been determined, their interrelationships can be found either by theory (principles) or by experimentation The results apply generally because the pi-terms are dimensionless Following is an example of a well-designed experiment small scale model study are plotted in Figure 1-2 The plot of data appears to be linear Only the last point to the right may deviate Apparently the pipe is no longer thin-wall So the thin-wall designation only applies if t/D< 0.1 The equation of the plot is the equation of a straight line, y = mx + b where y is the ordinate, x is the abscissa, m is the slope, and b is the y-intercept at x = For the case above, (P'/S) = 2(t/D), from which, solving for bursting pressure, P = 2S/(D/t) This important equation is derived by theoretical principles under "Internal Pressure," Chapter Example Using experimental techniques, find the equation for internal bursting pressure, P', for a thin-wall pipe Start by writing the set of pertinent fundamental variables together with their basic dimensions, force F and length L Fundamental Variables P' t D S = = = = internal pressure wall thickness inside diameter of ring yield strength of the pipe wall material Basic Dimensions FL-2 L L FL-2 These four fundamental variables can be reduced to two pi-terms such as (P'/S) and (t/D) The pi-terms were written by inspection keeping in mind the three characteristics of pi-terms The number of pi-terms is the number of fundamental variables, 4, minus the number of basic dimensions, 2, i.e., F and L The two pi-terms are dimensionless Both are independent because each contains a fundamental variable not contained in the other Conditions for bursting can be investigated by relating only two variables, the pi-terms, rather than interrelating the original four fundamental variables Moreover, the investigation can be performed on pipes of any convenient size because the pi-terms are dimensionless Test results of a ©2000 CRC Press LLC PERFORMANCE LIMITS Performance limit for a buried pipe is basically a deformation rather than a stress In some cases it is possible to relate a deformation limit to a stress (such as the stress at which a crack opens), but such a relationship only accommodates the designer for whom the stress theory of failure is familiar In reality, performance limit is that deformation beyond which the pipe-soil system can no longer serve the purpose for which it was intended The performance limit could be a deformation in the soil, such as a dip or hump or crack in the soil surface over the pipe, if such a deformation is unacceptable The dip or hump would depend on the relative settlement of the soil directly over the pipe and the soil on either side See Figure 1-3 But more often, the performance limit is excessive deformation of the pipe whic h could cause leaks or could restrict flow capacity If the pipe collapses due to internal vacuum or external hydrostatic pressure, the restriction of flow is obvious If, on the other hand, the deformation of the ring is slightly outof-round, the restriction to flow is usually not significant For example, if the pipe cross section deflects into an ellipse such that the decrease of the minor diameter is 10% of the original circular diameter, the decrease in cross-sectional area is only 1% Figure 1-4 Typical performance limits of buried pipe rings due to external soil pressure ©2000 CRC Press LLC The more common performance limit for the pipe is that deformation beyond which the pipe cannot resist any increase in load The obvious case is bursting of the pipe due to internal pressure Less obvious and more complicated is the deformation due to external soil pressure Typical examples of performance limits for the pipe are shown in Figure 1-4 These performance limits not imply collapse or failure The soil generally picks up any increase in load by arching action over the pipe, thus protecting the pipe from total collapse The pipe may even continue to serve, but most engineers would prefer not to depend on soil alone to maintain the conduit cross section This condition is considered to be a performance limit The pipe is designed to withstand all external pressures Any contribution of the soil toward withstanding external pressure by arching action is just that much greater margin of safety The soil does contribute soil strength On inspection, many buried pipes have been found in service even though the pipe itself has "failed." The soil holds broken clay pipes in shape for continued service The inverts of steel culverts have been corroded or eroded away without failure Cast iron bells have been found cracked Cracked concrete pipes are still in service, etc The mitigating factor is the embedment soil which supports the conduit the structural design of the pipe can proceed in six steps as follows STEPS IN THE STRUCTURAL DESIGN OF BURIED PIPES In order of importance: Resistance to internal pressure, i.e., strength of materials and minimum wall thickness; Resistance to transportation and installation; Resistance to external pressure and internal vacuum, i.e., ring stiffness and soil strength; Ring deflection, i.e., ring stiffness and soil stiffness; Longitudinal stresses and deflections; Miscellaneous concerns such as flotation of the pipe, construction loads, appurtenances, ins tallation techniques, soil availability, etc A reasonable sequence in the design of buried pipes is the following: Environment, aesthetics, risks, and costs must be considered Public relations and social impact cannot be ignored However, this text deals only with structural design of the buried pipe Plans for delivery of the product (distances, elevations, quantities, and pressures), PROBLEMS Hydraulic design of pipe sizes, materials, Structural requirements and design of possible alternatives, Appurtenances for the alternatives, Economic analysis, costs of alternatives, Revision and iteration of steps to 5, Selection of optimum system With pipe sizes, pressures, elevations, etc., known ©2000 CRC Press LLC 1-1 Fluid pressure in a pipe is 14 inches of mercury as measured by a manometer Find pressure in pounds per square inch (psi) and in Pascals (Newtons per square meter)? Specific gravity of mercury is 13.546 (6.85 psi)(47.2 kPa) 1-2 A 100 cc laboratory sample of soil weighs 187.4 grams mass What is the unit weight of the soil in pounds per cubic ft? (117 pcf) 1-3 Verify the standard deviation of Figure 1-1 (s = 27.8 kPa) 1-4 From Figure 1-1, what is the probability that any maximum daily pressure will exceed 784.5 kPa? (Pe = 0.62%) 1-5 Figure 1-5 shows bar graph for internal vacuum at collapse of a sample of 58 thin-walled plastic pipes x = collapse pressure in Pascals, Pa (Least increment is Pa.) y = number that collapsed at each value of x (a) What is the average vacuum at collapse? (75.0 Pa) (b) What is the standard deviation? (c) What is the probable error? (8.38 Pa) (+5.65 Pa) 1-6 Eleven 30 inch ID, non-reinforced concrete pipes, Class 1, were tested in three-edge-bearing (TEB) test with results as follows: x = ultimate load in pounds per lineal ft x w w2 (lb/ft) (lb/ft) 3562 3125 4375 3438 4188 3688 3750 4188 4125 3625 2938 (a) What is the average load, x, at failure? (x = 3727.5 lb/ft) (b) What is the standard deviation? (s = 459.5 lb/ft) (c) What is the probability that the load, x, at failure is less than the minimum specified strength of 3000 lb/ft (pounds per linear ft)? (Pe = 5.68%) ©2000 CRC Press LLC Figure 1-5 Bar graphs of internal vacuum at collapse of thin-walled plastic pipes 1-7 Fiberglass reinforced plastic (FRP) tanks were designed for a vacuum of inches of mercury (4inHg) They were tested by internal vacuum for which the normal distribution of the results is shown as Series A in Figure 1-6 Two of 79 tanks failed at less than 4inHg In Series B, the percent of fiberglas was increased The normal distribution curve has the same shape as Series A, but is shifted 1inHg to the right What is the predicted probability of failure of Series B at or below in Hg? (Pe = 0.17 % or one tank in every 590) 1-8 What is the probability that the vertical ring deflection d = y/D of a buried culvert will exceed 10% if the following measurements were made on 23 culverts under identical conditions? Measured values of d (%) 6 7 5 (0.24 %) 1-9 The pipe stiffness is measured for many samples of a particular plastic pipe the average is 24 with a standard deviation of a) What is the probability that the pipe stiffness will be less than 20? (Pe = 9.17 %) b) What standard deviation is required if the probability of a stiffness less than 20 is to be reduced to half its present value; i.e., less than 4.585%? (s = 2.37) 1-10 A sidehill slope of cohesionless soil dips at angle Write pi-terms for critical slope when saturated 1-11 Design a physical model for problem 1-10 Figure 1-6 Normal distribution diagrams for fiberglass tanks designed for 4inHg vacuum ©2000 CRC Press LLC ... deformations can be written in the form of equations from principles of ©2000 CRC Press LLC soil mechanics The remainders involve such complex soil- structure interactions that the interrelationships must... are independent because each contains a fundamental variable not contained in the other Conditions for bursting can be investigated by relating only two variables, the pi-terms, rather than interrelating... (t/D) used to find the equation for bursting pressure P'' in plain pipe Plain (or bare) pipe has smooth cylindrical surfaces with constant wall thickness — not corrugated or ribbed or reinforced Figure