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 1 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

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

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 sight, and, therefore,

out-of-mind to sources of funding for the infrastructure

Engineering design requires knowledge of: 1

performance, and 2 limits of performance Three

general sources of knowledge are:

SOURCES OF KNOWLEDGE

Experimentation (Empiricism)

Principles (Rationalism)

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

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 do we have any idea as to how many failed before artisans learned how to construct them

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

Performance Limit Performance =

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) = the average of all measurements,

x = 3yx/ 3y,

n = total number of measurements = Ey,

w = deviation, w = x - x),

P = 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 ),

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 2 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 pipe-soil 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 +weon either the +we or the -we

we/s P Pe we/s P Pe

(%) (%) (%) (%)

0.0 0.0 50.0 1.5 86.64 6.68

0.1 8.0 46.0 1.6 89.04 5.48

0.2 15.9 42.1 1.7 91.08 4.46

0.3 23.6 38.2 1.8 92.82 3.59

0.4 31.1 34.5 1.9 94.26 2.87

0.5 38.3 30.9 2.0 95.44 2.28

0.6 45.1 27.4 2.1 96.42 1.79

0.6745 50.0 25.0 2.2 97.22 1.39

0.7 51.6 24.2 2.3 97.86 1.07

0.8 57.6 21.2 2.4 98.36 0.82

0.9 63.2 18.4 2.5 98.76 0.62

1.0 68.26 15.9 2.6 99.06 0.47

1.1 72.9 13.6 2.7 99.30 0.35

1.2 78.0 11.5 2.8 99.48 0.26

1.3 80.6 9.7 2.9 99.62 0.19

1.4 83.8 8.1 3.0 99.74 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 y xy w yw yw2

(Mpa)* _ (MPa) (MPa) (MPa) (MPa)2

n Σxy Σyw2

x = Sxy/n = 1.1 MPa

s = [Syw2/(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

From the data of Table 1-2,

x_

= Σxy/Sy = 26.4/24 = 1.1

s = \/ Syw2/(n-1) = \/0.36/23 = 0.125

w = x - x, so

we = (0.8 - 1.1) = -0.30 MN/m2

= deviation to failure pressure

we/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 we = +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

soil mechanics The remainders involve such complex soil-structure interactions that the interr elationships 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 ar e 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 wr2 =

Mrx2

wx2 + Mry2wy2 + 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

1 Number of pi-terms = (number of fundamental variables) minus (number of basic dimensions)

2 All pi-terms are dimensionless

3 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

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

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

Basic Fundamental Variables Dimensions

S = yield strength of the

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

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 = 0 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 2

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 of ten, 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 out-of-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%

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 do 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

A reasonable sequence in the design of buried pipes

is the following:

1 Plans for delivery of the product (distances,

elevations, quantities, and pressures),

2 Hydraulic design of pipe sizes, materials,

3 Structural requirements and design of possible

alternatives,

4 Appurtenances for the alternatives,

5 Economic analysis, costs of alternatives,

6 Revision and iteration of steps 3 to 5,

7 Selection of optimum system

With pipe sizes, pressures, elevations, etc., known

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:

1 Resistance to internal pressure, i.e., strength of materials and minimum wall thickness;

2 Resistance to transportation and installation;

3 Resistance to external pressure and internal vacuum, i.e., ring stiffness and soil strength;

4 Ring deflection, i.e., ring stiffness and soil stiffness;

5 Longitudinal stresses and deflections;

6 Miscellaneous concerns such as flotation of the pipe, construction loads, appurtenances, ins tallation techniques, soil availability, etc

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

PROBLEMS 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

1-3 Verify the standard deviation of Figure 1-1

(s = 27.8 kPa)