Numerical Methods in Soil Mechanics 07.PDF

Numerical Methods in Soil Mechanics 07.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 "RING DEFLECTION"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 7-1 Deflected ring showing notation for dimensions and for ring deflection analysis.

Figure 7-2 Segmented ring deflection showing the relationship between ring deflection, d, width, w, of the crack, and wall thickness, t

CHAPTER 7 RING DEFLECTION

Ring deflection is defined as the ratio of change in

vertical diameter to the original diameter, d = /D

See Figure 7-1 Diameter D is the diameter to the

neutral surfaces of the cross section of the wall For

mos t pipe analyses, it is sufficiently accurate to use

the mean diameter, (OD+ID)/2 The error of using

mean diameter increases for reinforced concrete

pipes, pipes with ribs or stiffener rings, etc Ring

deflection is the result of: 1 inflation or deflation of

the pipe, 2 flexing of the ring, 3 cracking of the ring

into segments, and 4 plastic hinging (or crushing) of

the pipe walls Ring deflections of rigid and flexible

pipes are two different phenomena Each is

analyzed separately

RIGID RING

Typical rigid pipes are concrete and vitrified clay

pipes For rigid pipes, two basic modes of ring

deflection are elastic and segmented Most rigid

pipes are brittle The limit of elastic deflection is

reached when the pipe cracks into segments as

shown in Figure 7-2 Because there is no such thing

as a perfectly rigid pipe, the question arises, how

much elastic ring deflection occurs in the rigid ring?

For most rigid rings, elastic deflection is small

enough to be neglected Hairline cracks are not

critical Reduction in flow capacity is negligible

Elastic ring deflection is calculated in the same way

for both rigid and flexible pipes Composite

(reinforced) pipe walls require a transformed section

for analysis This is true for reinforced concrete

pipes, but may also be true for materials with

different properties in compression and tension

Elastic ring deflections for various load conditions

are listed in Appendix A

Example 1

A concrete pipe has ID = 36 and OD = 42 inches

with double cages of 1/4 inch steel reinforcing rods

spaced at 2 inches and located 0.6 inch from the

inside and outside surfaces of the pipe See Figure

7-3

What is the elastic ring deflection at yield stress

of 1000 psi in the concrete for each of the three different loading conditions shown? Cracks open if tensile stress is greater than 1000 psi From the transformed section, EI/r3 = 1651 psi; and from deflection equations in Appendix A, the corresponding ring deflections are calculated and summarized in Figure 7-3 None of these elastic ring deflections is greater than 0.1%

Segmented ring deflection is the result of cracks opening at spring lines, crown and invert See Figure 7-2 It is assumed that the segments are rigid Ring deflection can be calculated in terms of crack width

w If the wall thickness is t and the neutral surface

is at mid-thickness of the wall, the ring deflection is

d = w/t Allowing for some deflection of the segments, and allowing for the possibility that neutral surfaces are further from the pipe surfaces than t/2, the lower limit of ring deflection is greater than w/2t Therefore,

td < w < 2td (7.1)

which shows a range of widths of the crack as a function of wall thickness and segmented ring deflection The relationship is not precise because cracks are undependable For example, two parallel cracks may open where only one is expected Example 2

Consider the same 36 inch ID reinforced concrete pipe with three inch thick walls If 0.01 inch wide cracks open inside the pipe at the invert and crown, from Equation 7.1, ring deflection is between d = 0.17% and d = 0.33% Because of balanced placement of steel, actual ring deflection may be closer to the upper limit, say, d = 0.3%

Ring deflection is more the result of cracking than it

is the result of elas tic deformation of the ring The small ring deflections justify design by rigid pipe theories See Chapter 12

Figure 7-3 Ring deflection at incipient cracking of a reinforced concrete pipe under three different loading conditions Note that the maximum deflection is d = 0.1%

FLEXIBLE RING

As soil and surface loads are placed over a buried

flexible pipe, the ring tends to deflect — primarily

into an ellipse with a decrease in vertical diameter

and an almost equal (slightly less) increase in

horizontal diameter Any deviation from elliptical

cross section is a secondary deformation which may

be the result of non-uniform soil pressure The

increase in horizontal diameter develops lateral soil

support which increases the load-carrying capac ity

of the ring The decrease in vertical diameter

partially relieves the ring of load The soil above the

pipe takes more of the load in arching action over

the pipe — like a masonry arch Both the increase

in strength of the ring and the soil arching action

contribute to structural integrity Although some ring

deflection is beneficial, it cannot exceed a practical

performance limit Therefore the prediction of ring

deflection of buried flexible pipes is essential Ring

deflection is elastic up to the formation of cracks or

permanent ring deformations Clearly, the ring can

perform with permanent deformations — and even

with small crac ks, under some circumstances

Performance can surpass yield stress to the

determination at which the ring becomes unstable

Instability is explained in Chapter 10

The following analyses of ring deflection are based

on elastic theory for which the pertinent pi-terms

are:

d = ring deflection,

e = average sidefill soil settlement,

d/e = ring deflection term,

Rs = stiffness ratio = E'D3/EI,

= ratio of soil stiffness E' to ring stiffness,

EI/D3; or to pipe stiffness, F/D , where

F/D = 53.77 EI/D3

Notation:

d = D /D = ring deflection,

= vertical decrease in ring diameter,

D = original diameter of the flexible ring (more

precisely, the diameter to the neutral surfaces of

the wall cross section),

D = vertical soil strain due to the anticipated

vertical soil pressure at the pipe springlines,

E' = soil modulus = slope of a secant on the stress-strain diagram from the point of initial vertical effective soil pressure to the point of maximum vertical effective soil pressure,

E = modulus of elasticity of the pipe wall,

I = centroidal moment of inertia of the pipe wall cross section per unit length of the pipe

Figure 7-4 is a graph of the ring deflection term as a function of stiffness ratio From the graph, ring deflection can be found as follows Enter Figure 7-4 with a stiffness ratio, either Rs or Rs' and read out the ring deflection term, d/e If the vertical soil strain e is known, ring deflection follows directly from d/e Figure 7-4 represents tests and field data for buried flexible pipes

Vertical soil strain e is predicted from laboratory compression tests data such as the stress-strain graphs of Figure 7-5 for cohesionless siltly sand Soil stiffness E' is the slope of a secant to the anticipated soil pressure P on the stress-strain diagram for a specific soil density Graphs can be provided by soil test laboratories for the specific embedment to be used, and at the density to be specified

Ring stiffness contributes significant resistance to ring deflection if Rs is less than about 300 (or Rs' is less than 6); i.e low soil stiffness and high ring stiffness For flexible pipes buried in good soil, stiffness ratio is usually greater than 300 Therefore,

For design, ring deflection of flexible pipes buried in good soil is equal to (no greater than) the vertical strain (compression) of the sidefill soil.

Circumstances aris e under which the above rule is not accurate Equations for ring deflection are listed

in Appendix A for a few loadings on rings of uniform wall thickness that are initially circular If not, one set of approximate adjustment factors is:

D to be multiplied by Dmax /Dmin

Figure 7-4 Ring deflection term as a function of stiffness ratio The graph is a summary of 140 tests plotted

at 90 percent level of confidence; i.e., 90 percent of test data fall below the graph

Figure 7-5 Stress-strain relationship for typical cohesionless soil (silty sand) Ninety percent of all strains fall

to the left of the graphs Vertical pressure is effective (intergranular) soil pressure

Soil stiffness, E', is the slope of the secant from initial to ultimate effective soil pressures

Rs = (SOIL STIFFNESS)/(RING STIFFNESS)

t to be multiplied by tmin /tmax

Equations have been proposed for predicting ring

deflection of flexible pipes One of these is the Iowa

Formula derived by M G Spangler The Iowa

Formula is elegant and correctly derived, but

depends upon a number of factors which may be

difficult to evaluate Such questionable factors

include a deflection lag factor, bedding factor, the

horizontal soil modulus, and the assumptions on

which the Marston load is based See Appendix B

and Spangler (1973)

The horizontal soil modulus, E', is particularly

troublesome It is based on theory of elasticity

which is questionable E' is not constant In fact, E'

is a function of the depth of burial and the horizontal

compression of the sidefill soil as the pipe expands

into it

The best procedure for design of buried flexible

pipes is to specify the allowable ring deflection, and

then make sure that vertical compression of the

sidefill soil does not exceed allowable ring deflection

The Iowa Formula, and other deflec tion equations,

are approximate, but conservative Some are

compared in Appendix B However, within the

precision justified in most buried pipe analyses, the

procedures described in the following examples are

more relevant and understandable

Example 1

A corrugated plastic drain pipe (flexible) is to be

buried in clean dry sand backfill that falls into place

at 80 percent density (AASHTO T-180) What ring

deflection is anticipated due to 10 ft of soil cover

weighing 120 lb/ft3? Live load is neglected at this

depth of cover The stiffness ratio is larger than R's

= 6, therefore, from the ring deflection graph of

Figure 7-5, ring deflection is not more than about,

d = e = 1%

Example 2

What is the predicted ring deflection? A PVC pipe

of

DR = 14 is to be placed in embedment compacted to

80 percent density (AASHTO T-180) under 24 ft of cohesionless silty sand at dry unit weight of 105 lb/ft3, but with a water table at 9 ft below the surface Saturated unit weight is 132 lb/ft3 From

the Unibell (1882) Handbook of PVC Pipe, page

159, values of PVC pipe stiffness for DR = 14 pipes vary from 815 to 1019 psi Using, conservatively, the lower value, pipe stiffness is F/ = 815 psi The soil stiffness is found from Figure 7-5 Because ring deflection increases from zero at no soil pressure to maximum at ultimate pressure, soil stiffness is the slope of the secant from the origin to the point of ultimate effective soil pressure on the stress-strain diagram Vertical soil strain is a function of

effective soil pressure (intergranular) — not total

pressure Effective soil pressure is,

P = 15ft(132pcf) + 9ft(105pcf)

- 15ft(62.4pcf) = 1.99ksf = 13.8psi

At 80% density, from Figure 7-5, the soil strain at 1.99 ksf is e = 1.85% The soil stiffness is the slope

of the secant from 0 to 2 ksf on the 80% graph; i.e., E' = 13.8psi/0.0185 = 747 psi The resulting stiffness ratio is R's = E'/(F/ ) = 747/815 = 0.92 Entering the graph of Figure 7-4 with R's = 0.92, the corresponding ring deflection term is d/e = 0.48 Ring deflection is 48% of the vertical soil strain e Because soil strain is 1.85%, the predicted ring deflection is, d = 1.85%(0.48) 1.0%

If a straight pipe of elastic material and cir cular cross section is bent into a circular curve, the cross section deforms into an ellipse Ring deflection of the cross section is,

d = 2Z/3 + 71Z2/135 (7.2)

where Z = 1.5(1-n2)D4/16t2R2

d = ring deflection = D/D,

D = decrease in pipe diameter,

n = Poisson ratio,

D = diameter of circular pipe,

t = wall thickness,

R = radius of the bend

C

-C

C C

*Decrease in diameter is in the direction of the

radius of the bend See Chapter 14 for example

REFERENCES

Spangler, M.G (1973) and Handy, R.L Soil

Engineering, IEP, New York

Unibell (1982), Handbook of PVC Pipe

Watkins, R.K (1974), Szpak, E., and Allman, W.B.,

Structural design of polyethylene pipes subjected to

external loads, Eng'rg Expr Sta., USU

PROBLEMS

7-1 What is the ultimate ring deflection of a steel

water pipe, ID = 36 inches and t = 1 inch, buried

under saturated tailings which will rise ultimately to

250 ft? For tailings, G = 2.7 Unconsolidated, e =

0.7 When consolidated under H = 250 ft of tailings,

e = 0.5 Assume a straight line variation of e with

respect to height above the pipe Water table is at

the ground surface (d = 11.8%)

7-2 A corrugated steel storm drain never flows full

Therefore the granular backfill soil is essentially dry

What is the ring deflection? Include HS-20 live load

Soil (granular)

H = 4 ft = height of soil cover,

G = 2.7 = specific gravity,

e = 0.7 = void ratio,

80% density (AASHTO T-180)

Steel pipe (corrugations 2 2/3 x 1/2)

D = 48 inches = diameter,

I = 0.0180 in4/ft (t = 0.052),

E = 30(106) psi = modulus of elasticity

7-3 What is the change in ring deflection of

Prob-lem 7-2 if the soil cover is increased from 4 ft to 26

ft using the same soil, same density? (d = 1.6%)

7-4 What is the probable ring deflection of an unreinforced concrete pipe, ID = 30 inch and wall thickness = 3.5 inches if a video from inside the pipe reveals a 0.1-inch-wide crack in the crown? 7-5 What is the ring deflection of a steel pipe, OD

= 26 inches and ID = 24 inches, if E = 30(106) ps i and the soil cover is H = 40 ft.? The soil unit weight

is 100 lb/ft3 at 80% density (AASHTO T-180)

(Rs = 5.425; d = 0.064%) 7-6 Predict ring deflection of a plain steel pipe if:

D = 10 ft,

t = 0.5 inch,

E = 30(106) psi

Soil is granular, 90% dense (AASHTO T-180)

g = 120 lb/ft3,

H = 30 ft

7-7 If the neutral surface is at the geometrical center of the wall of Figure 7-2, prove that the width

of the crack is approximately w = td; where w

= width of crack

t = wall thickness

d = segmented ring deflection = decrease in vertical diameter

D = diameter to neutral surface (NS)

It is assumed that the wall crushes in compression

on one side of the neutral surface just as much as it stretches in tension on the other side before the cracks open This is not true for all materials 7-8 If the ring of Figure 7-2 is vitrified clay or unreinforced concrete, both of which are many times stronger in compression than in tension, the compression crushing zones in the wall are very small As a worst case, assume no wall crushing and find the segmented ring deflection d if the cracks open 0.01 inch

7-9 Assume that the ring of Figure 7-2 is reinforced concrete with a single steel wire cage in the center

of the wall The wire is 1/4-inch diameter spaced at

2 inches What is the vertical diameter to the neutral

t = 3 inches Es = 30(106) psi

ID = 30 inches Ec = 3(106) psi

7-10 What is the horizontal diameter to the neutral

surfaces of Problem 7-9? What would be the

difference if the cracks were caused by uniform soil

load on top and bottom (no sidefill)?

7-11 A high-density polyethylene (HDPE) pipe has

a minimum outside diameter (OD) of 6.60 inch and

a maximum OD of 6.66 inch The wall thickness

varies from 0.83 maximum to 0.80 minimum What

is predicted ring deflection under the loading shown

in Figure 7-3c if P = 5.0 kips per square ft? Assume

that long-term virtual modulus of elasticity for HDPE

is 85 ksi

7-12 What is the approximate ring deflection of the

reinforced concrete pipe of Figure 7-6 when the

width of crack at the crown is w = 0.06 inch?

7-13 If the load in Figure 7-6 is an F-load (parallel

plate load), where is the neutral surface at the spring

lines?

7-14 Are the cracks in Problem 7-13 exactly equal

on the inside at crown and invert, and on the outside

at spring lines? Explain

7-15 What is the maximum limit of ring deflection due to an F-load on a plastic pipe if permanent strain damage occurs at 1.75 percent strain on the outside surface? See Appendix A for deflection due to F-loading (d = 12.78 13%)

D = 0.5 meter,

t = 16 mm,

E = 300 ksi (2.07 GN/m2)

7-16 How high can plastic pipes be stacked if maximum allowable ring deflection is 10 percent?

OD = 4.24 inches = outside diameter,

ID = 3.92 inches = inside diameter,

w = 1.08 lb/ft = pipe weight,

E = 200 ksi = short term modulus of elasticity

7-17 A flexible plastic pipe, DR = 31, for which F/D

= 90 psi, is buried in cohesionless soil with unit weight of 110 pcf at 80 percent density (AASHTO T-180) If the height of cover is 25 ft, what is the predicted ring deflection?

Figure 7-6 Cross section of the wall of a reinforced concrete pipe, showing the transformed section in concrete, and showing the procedure for finding the neutral surface (NS) It is assumed in this case that concrete can take no tension