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

CHAPTER 5 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 do 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

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

5-1

Stress Stress theory provides an acceptable analysis for rigid rings Deformation and strain theories provide better analyses for flexible rings

Circumferential stresses comprise: 1 hoop or ring compression stress, and 2 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.

of an eccentrically-loaded short column, see Figure

5-2, for which, within the elastic limit,

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

For a pipe ring, by theory of elasticity,

s = Pr/A + Mc/I (5.1)

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

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

Example

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 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,

s = Pr/A + Ecdq (5.2)

where

dq = q - q' = 1/r - 1/r',

E = modulus of elasticity,

c = distance from NS to the most remote fiber

For a plain (bare) pipe, Equation 5.2 becomes,

s = Pm + (E/m) (r'-r)/2r' (5.3) where

m = r/t = wall flexibility,

r = mean radius,

t = wall thickness

Strain Within the elastic limit, strain is e = s /E Therefore, Equation 5.2 can be written as,

e = Pr/AE + cdq (5.4)

where

e = circumferential strain in the surfaces of the pipe wall,

dq = 1/r - 1/r'

For a plain pipe with wall thickness, t,

e = Pm/E + (r'-r)/2mr' (5.5)

Deformation

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

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

Ring deflection from circle to ellipse decreases radius of curvature at B by, dq = 1/rx-1/r

But from Figure 3-2,

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

Substituting, and neglecting higher orders of d,

for ellipse, by elastic analysis at spring lines

s = Pr/A + (Ec/r)3d/(1-2d) (5.6)

comp deformation

term term

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

analyses are available For example, a non-constrained, circular, flexible ring will collapse catastrophically under pressure if,

Pr3(1-n2)/EI = 3, or PD3(1-n2)/EI = 24

where n = Poisson ratio For most pipe design, third-dimensional effects enter in such that the effect of

n 2 is reduced and may be neglected Conservatively,

Pr3/EI = 3 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/r3 = 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/r3 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

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

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

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

Corrugated Pipes:

Figure 5-8 is the wall of a corrugated pipe Equation 5.10 may be used as demonstrated in the following example

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?

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 reinfor ced 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

Values of section modulus, I/c, per length of the pipe,

are listed in industry manuals

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

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 = 4 mm, wall thickness,

OD = 500 mm, over the ribs,

E = 3.5 GPa, modulus of elasticity

Ribs are 4 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 = 4 mm, but without ribs? (4.8 kPa)

Figure 5-9 Wall cross section of an externally ribbed PVC pipe

C−