Numerical Methods in Soil Mechanics 04.PDF

Numerical Methods in Soil Mechanics 04.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 "SOIL MECHANICS"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 4-1 Vertical soil pressure under one pair of dual wheels of a single axle HS-20 truck load, acting on

a pipe buried at depth of soil cover, H, in soil of 100 pcf unit weight Pressure is minimum at 5 or 6 ft of cover

CHAPTER 4 SOIL MECHANICS

An elementary knowledge of basic principles of soil

stresses is essential to understanding the structural

performance of buried pipes These principles are

explained in standard texts on soil mechanics A

few are reviewed in the following paragraphs

because of their special application to buried pipes

VERTICAL SOIL PRESSURE P

For the analysis and design of buried pipes, external

soil pressures on the pipes must be known Vertical

soil pressure at the top of the pipe is caused by: 1

dead load, Pd , the weight of soil at the top of the

pipe; and 2 live load, Pl , the effect of surface live

loads at the the top of the pipe Figure 4-1 shows

these vertical soil pressures at the top of the pipe as

functions of height of soil cover, H, for an HS=20

truck axle load of 32 kips, and soil unit weight of 100

pcf Similar graphs are found in pipe handbooks

such as the A ISI Handbook of Steel Drainage and

Highway Construction Products Soil unit weight

can be modified as necessary Also other factors

must be considered What if a water table rises

above the top of the pipe, or the pipe deflects, or the

soil is not compacted, or is overcompacted? For

these and other special cases, the following

fundamentals of soil mechanics may be useful

If the embedment about a buried pipe is densely

compacted, vertical soil pressure at the top of the

pipe is reduced by arching action of the soil over the

pipe, like a masonry arch, that helps to support the

load To be conservative, arching action is usually

ignored However, soil arching provides an added

margin of safety If the soil embedment is loose,

vertical soil pressure at the top of the pipe may be

increased by pressure concentrations due to the

relatively noncompressible area within the ring in

loose, compressible soil Pressure concentrations

due to loose embedment cannot be ignored For

design, either a pressure

concentration factor is needed, or minimum soil density should be specified Over a long period of time, pressure concentrations on the pipe may be reduced by creep in the pipe wall (plastic pipes), earth vibrations, freeze-thaw cycles, wet-dry cycles, etc The most rational soil load for design is vertical soil pressure at the top of the pipe due to dead weight of soil plus the effect of live load with a specification that the soil embedment be denser than critical void ratio Critical void ratio, roughly 85% soil density (AASHTO T-99), is the void ratio

at such density that the volume of the soil skeleton does not decrease due to disturbance of soil particles

For design, the total vertic al soil pressure at the top

of the pipe is:

where (see Figure 4-2)

P = total vertical soil pressure at the level of

the top of the pipe

Pd = dead load pressure due to weight of the

soil (and water content)

Pl = vertical live load pressure at the level of

the top of the pipe due to surface loads This is a useful concept in the analysis of buried pipes Even rigid pipes are designed on this basis if

a load factor is included See Chapter 12

In fact, P is only one of the soil stresses At a given point in a soil mass, a precise stress analysis would consider three (triaxial) direct stresses, three shearing stresses, direct and shearing moduli in three directions and three Poisson's ratios — with the additional condition that soil may not be elastic The imprecisions of soil placement and soil compaction obfuscate the arguments for such rigor Elastic analysis may be adequate under some few circumstances Superposition is usually adequate without concern for a combined stress analysis involving triaxial stresses and Poisson ratio Basic soil mechanics serves best

©2000 CRC Press LLC

Figure 4-2 Vertical soil pressure P at the level of the top of a buried pipe where P = Pl + Pd , showing live load pressure Pl and dead load pressure Pd superimposed

Figure 4-3 A single stratum of saturated soil with water table at the top (buoyant case) showing vertical stress

at depth H

Dead Load Vertical Soil Pressure Pd

Dead load is vertical pressure due to the weight of

soil at a given depth H In the design of buried

pipes, H is the height of soil cover over a pipe Total

pressure Pd is the weight of soil, including its water

content, per unit area See Figure 4-3

Intergranular (or effective) pressure Pd is the

pressure felt by the soil skeleton when immersed in

water The total and intergranular vertical stresses

at the bottom of a submerged stratum can be related

by the following stress equation:

where

s = intergranular vertical soil stress (felt by the

soil when buoyed up by water),

s = total vertical soil stress = gtH,

u = pore water pressure = gwH,

gt = total unit weight of soil and water,

gw = unit weight of water = 62.4 pcf

Now consider more than one stratum of soil as

shown in Figure 4-4 The total vertical dead load soil

pressure Pd at the bottom of the strata is the sum of

the loads imposed by all of the strata; i.e.,

where

gt = total unit weight (wet weight) of soil

in a given stratum, and

H = height of the same stratum

Values of H for each soil stratum are provided by

soil borings Values of gt are simply the unit

weights of representative soil samples including the

water content If the soil samples are not available,

from soil mechanics,

gt = (G+Se)gw /(1+e) (4.4)

where

G = specific gravity of soil grains, about 2.65,

S = degree of saturation = 1 when saturated,

e = void ratio, from laboratory analysis,

gw = unit weight of water

Table 4-1 is a summary of dead load soil stresses from which dead load pressure Pd can be found and combined with live load pressure Pl Live load pressure is found from techniques described in the paragraphs to follow

Intergranular vertical soil pressure P, at the bottom

of multiple soil strata, is:

P = P - u = P - gwh (4.5) where

P = vertical intergranular soil pressure,

P = total dead plus live load pressures,

h = height of water table above the pipe Total pressure is used to calculate ring compression stress Intergranular soil pressure is used to calcu-late ring deflection which is a function of soil compression As the soil is compressed, so is the pipe compressed — and in direct ratio But soil compression depends only on intergranular stresses See Chapter 7

Live Load Vertical Soil Pressure Pl

Live load soil pressure Pl is the vertical soil pressure

at the top of the buried pipe due to surface loads See Figure 4-5 For a single concentrated load W on the surface, vertical soil pres sure at point A at the top of the pipe is:

where

W = concentrated surface load (dual-wheel)

H = height of soil cover over the top of the pipe

R = horizontal radius to stress, s ,

N = Boussinesq coefficient

from the line of action of load W,

N = Boussinesq coefficient = 3(H/R)5/2p

©2000 CRC Press LLC

_

_

Figure 4-4 Multiple strata

(three strata with the clay

stratum divided into two at

the water table) showing the

total vertical dead load soil

pressure Pd at the bottom

(level of the top of the pipe)

Figure 4-5 Vertical soil pressure at depth H (at the level of the top of a pipe) and at radius R from the line

of action of a concentrated surface load W (After Boussinesq)

Figure 4-6 Chart for evaluating the vertical stress s at a depth H below the corner A of a rectangular surface area loaded with a uniformly distributed pressure q (After Newmark)

For a single wheel (or dual wheel) load, the

maxi-mum stress smax at A occurs when the wheel is

directly over the pipe; i.e R = 0, for which

smax = 0.477 W/H2 (4.7)

Load W can be assumed to be concentrated if depth

H is greater than the maximum diameter or length of

the surface loaded area

For multiple wheel (or dual) loads, the maximum

stress at point A, due to effects of all loads must be

ascertained The trick is to position the wheel loads

so that the combined stress at A is maximum This

can be done by trial

The effect of a uniformly distributed surface load

can be found by dividing the loaded surface area into

infinitesimal areas and integrating to find the sum of

their effects at some point at depth H See Figure

4-6 Newmark performed such an integration and

found the vertical stress s at a depth H below

corner A of a rectangular area of greater length L

and lesser breadth B, loaded with uniform pressure

q His neat solution is:

where M is a coefficient which can be read on the

chart of Figure 4-6 by entering with arguments L/B

and B/H If the stress due to pressure on an area is

desired below some point other than a corner, the

rectangular area can be expanded or subdivided

such that point A is the common corner of a number

of areas The maximum stress under a rectangular

area occurs below the center See Figure 4-7 The

rectangle is subdivided into four identical rectangles

of length L and breadth B as shown The stress at

point A is 4Mq, where M is found from the

Newmark chart, Figure 4-6

Alternatively, the Boussinesq equation can be used

with less than five percent error if the concentrated

load, Q = qBL, for each of the quadrants is assumed

to act at the center of each quadrant For this case,

R = (L2+B2)/2 The resulting stress at A is 4NQ/H2

from Equation 4.6

Example

What is the stress at point A below A' of Figure

4-8? The vertical stress s at depth H below surf a c e point A is, by superposition:

s = s ' - S s'" + s "

where

s ' = stress at corner A due to loaded area

L'xB'

S s'" = sum of stresses at corner A due to

loaded areas L'B" and L"B'

s " = stress at corner A due to loaded area

L"B'

Clearly, s " due to area L"B" was subtracted twice,

so must be added back once

An occlusion in the soil mass, such as a pipe, violates Boussinesq's assumptions of elasticity, continuity, compatibililty, and homogeneity The pipe is a hard spot, a discontinuity Soil is not elastic, nor homoge-neous, nor compatible when shearing planes form Nevertheless, the Boussinesq assumptions are adequate for most present-day installation tech-niques For most buried pipe design, it is sufficient, and conservative, to solve for Pl at the top of the pipe due to a single wheel load W at the surface by using the Boussinesq equation with the radius R = 0 For additional wheel loads, simply add by superposi-tion the influence of other wheel loads at their radii R

Example What is the maximum vertical soil stress at a depth

of 30 inches due to the live load of a single axle

HS-20 truck? Neglect surface paving See Figures 4-9

and 4-15 By trial, it can be shown that the point of maximum stress is point A under the center of one tire print The rectangular tire prints are subdivided

as shown for establishing a common corner A' The effects of the left tire print and the right tire print are analyzed separately, then combined The length L and breadth B of each tire print are based on 104 psi tire pressure Use Newmark because H is less than 3L

©2000 CRC Press LLC

Figure 4-7 Procedure for subdividing a rectangular surface area such that the stress below the center at depth

H is the sum of the stresses below the common corners A' of the four quadrants

Figure 4-8 Subdivision of the loaded surface area, LxB, for evaluation of vertical stress, P, at depth, H, under point A

Figure 4-9 Single axle HS-20 truck load showing typical tire prints for tire pressure of 104 psi, and showing the Newmark subdivision for evaluating vertical soil stress under the center of one tire print

©2000 CRC Press LLC

Figure 4-10 Infinitesimal soil cube B and the corresponding Mohr circle which provides stresses on any plane through B Note the stresses sq and tq shown on the q-plane At soil slip, the circle is tangent to the strength envelopes described below

Figure 4-11 Shearing stress t as a function of normal stress s , showing a series of Mohr circles at soil slip, and the strength envelopes tangent to the Mohr circles

W = 32 k for HS-20 truck load (single axle),

q = 104 psi,

B = 7 inches,

L = 22 inches,

H = 30 inches

For the left tire print:

sL = 4Mq,

L/B = 11/3.5 = 3.14,

B/H = 3.5/30 = 0.12,

From Figure 4.6, M = 0.018 and sL = 1.078 ksf

For the right tire print:

sR = 2(M'-M")q, where,

L'/B' = 83/3.5 = 23.7,

B'/H' = 3.5/30 = 0.12,

From Figure 4.6, M' = 0.02 (extrapolated)

L"/B" = 61/3.5 = 17.4,

B"/H" = 3.5/30 = 0.12,

From Figure 4.6, M" = 0.02 (extrapolated)

sR = r(M'- M")q = 0: sR = 0 ksf

At point A, s = sL + s R; s = 1.08 ksf

A rough check by Boussinesq is of interest because

the results are conservatively higher and are more

easily solved

Given:

W = 16 kips at the center of each tire print,

H = 2.5 ft,

RL = 0,

RR = 6 ft,

From Figure 4.5, N = 0.004

sL = 0.477 W/H2 = 1.22 ksf

sR = NW/H2 = 0.01 ksf

At point A, s = sL + s R ; s = 1.23 ksf

The Boussinesq solution is in error by 13.9%, but on

the high (conservative) side Of interest is the small

(negligible) effect of the right wheel load

SOIL STRENGTH

Failure of a buried pipe is generally associated with failure of the soil in which the pipe is buried The classical, two-dimensional, shear-strength soil model

is useful for analysis Analysis starts with an infini-tesimal soil cube on which stresses are known and the orientation is given The model comprises three elements, the Mohr stress circle, orientation diagram, and strength envelopes

Mohr Stress Circle

The Mohr stress circle is a plot of shearing stress, t ,

as a function of normal stress, s , on all planes at angle q through an infinitesimal soil cube B See

Figures 4-10 and 4-11 The sign convention is compressive normal stress positive ( + ) and counterclockwise shearing stress positive ( + ) The center of the cir cle is always on the s -axis Two additional points are needed to determine the circle They are (sx,t xy) on a y-plane and (s y,ty x)

on an x-plane These are known stresses on cube

B An origin of planes always falls on the circle

txy = -ty x from standard texts on solid mechanics.

Any plane from the origin intersects the Mohr circle at the stress coordinates acting on that plane— which is correctly oriented if the following

procedure is followed

Orientation Diagram

Figure 4-10 shows infinitesimal cube B with the x-plane and y-x-plane identified and with the soil stresses acting on each plane Cube B and its axes of orientation can be superimposed on the Mohr circle such that stress coordinates (where each plane inter-sects the Mohr circle) are the stresses on that plane With cube B located on the Mohr circle as the origin

of axes, and with the axes correctly oriented, any plane through B will intersect the Mohr circle at the point whose stress coordinates are the stresses

acting on that plane, and all planes are correctly

oriented with respect to the original soil cube B.

©2000 CRC Press LLC