Numerical Methods in Soil Mechanics 10.PDF

Numerical Methods in Soil Mechanics 10.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 STABILITY"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 10-1 Top — Elliptical ring with uniform radial pressure P acting on it.

Bottom — Free-body-diagram of half of the elliptical ring showing approximate ring compression thrust in the walls due to pressure P

CHAPTER 10 RING STABILITY

The performance limit of ring stability is instability

Ring instability is a spontaneous deformation that

progresses toward inversion (reversal of curvature)

At worst, instability is ring collapse Buried pipes

can invert only if the ring deflects and the soil slips

at the same time Instability of buried pipes is

analyzed as a soil-structure interaction The

stiffness of the ring resists inversion Soil supports

the ring by holding it in a stable (near circular)

shape Soil resists inversion of the ring

Two basic modes of ring instability are: 1 ring

compression, i.e., wall crushing or buckling at yield

stress; and 2 ring deformation See Figure 10-1

Each is analyzed separately Instabilities of buried

and unburied rings are also analyzed separately

UNBURIED RING COLLAPSE

From Chapters 2 and 5, external pressure at collapse

of an unburied thin-walled, circular, elastic ring is

found from the equations:

Pr3/EI = 3 R I N G D E F O R M A T I O N

COLLAPSE

where

P = external pressure at collapse,

A = wall area per unit length,

t = wall thickness of plain pipe,

m = r/t = ring flexibility,

do = initial ring deflection (ellipse),

area of the wall per unit length,

Pcr = critical pressure on the circular ring

The ring compression collapse equation is a function

of ring flexibility and yield strength

Performance limit is wall crushing The ring deformation collapse equation is a function of ring stiffness, EI/r3 Performance limit is inversion Ring stiffness is related to pipe stiffness; i.e., F /∆ = 53.77EI/D3 Pipe stiffness can be measured by a parallel plate test or three-edge-bearing test The ring deformation collapse equation is based on assumptions that the ring is elastic, and that the pipe

is restrained longitudinally Longitudinal restraint results in a plane stress analysis The Poisson ratio

is not included

In a plane strain analysis, longitudinal stress is zero, the Poisson ratio is included, and the pressure at collapse is,

Pcr = 3EI/r3(1-ν2), where

Pcr = critical pressure, i.e., P at collapse,

ν = Poisson ratio = 0.27 for steel, EI/r3 = ring stiffness

The difference in fluid pressures between top and bottom of the pipe is usually ignored, but may be significant For plane stress analysis of critical pressure at collapse of circular, unburied pipes,

For plain pipes (not coated, lined, rib stiffened, or corrugated), critical pressure at collapse is,

Moment of Inertia, I

In order to evaluate ring stiffness, EI/r3, the moment

of inertia, I, must be known For plain pipes, I =

t3/12 For corrugated pipes, tables of values for I are found in the manuals provided by manufacturers For steel pipes with lining and coating, consider

Figure 10-2 Transformed section of a unit slice of mortar-lined and coated steel pipe wall, transformed into its equivalent section in mortar, for evaluating moment of inertia Shown on the right is the elastic stress distribution

Figure 10-3 Effective T-section , comprising stiffener ring and an effective width of pipe wall — often assumed to be 50t in steel pipes

a unit slice of the wall See Figure 10-2.

Discounting conservatively the bond between mortar

and steel, moment of inertia, I, is the sum of the

separate moments of inertia of steel, lining, and

coating Because the mortar is the critical material,

steel is transformed into its equivalent width, n, in

mortar n = Es /Em For the layers,

Ic = tc3/12

Is = nts3/12

Il = tl3/12,

and I = Ic + Is + Il

For pipes with stiffener rings welded to the pipe, the

moment of inertia is found from the effective

T-s ection See Figure 10-3 The procedure is

described in texts on mechanics of solids For steel

pipes, the T-section comprises the stiffener ring and

an effective width of pipe wall — in steel usually

assumed to be 50t The pipe wall between effective

T-sections is ignored in calculating I

Elliptical Ring Instability

Instability of non-circular, unburied rings is difficult

to analyze However, analysis is available from

texts on mechanics of solids for one important case

— a ring that is initially elliptical with ring deflection

do Vacuum increases ring deflection Stress is

c ritical at the spring lines, B, where the ring is

subjected to both maximum ring compression stress

and maximum flexural stress See Figure 10-1

Vacuum, P, at collapse is found from:

P2 - [σf /m + (1+6mdo)Pcr] P + σ f Pcr /m = 0

(10.3a)

Equation 10.3a is applied by plotting values of P as

a function of m for given values of do and for a

constant σf For design, a simplification is proposed

by Murphy and Langner (1985),

Example 1

A plain 18-inch high-density polyethylene pipe is out-of-round (elliptical) by five percent It is unburied What is the internal vacuum at collapse?

ID = 16.217 and t = 0.857,

σf = 3.2 ksi = yield strength',

E = 110 ksi = modulus of elasticity,

Pcr = 27.5 psi = E/4m3 from Equation 10.2,

do = 0.05 = initial ring deflection

Substituting into Equation 10.3a, internal vacuum at collapse is, P = 21.5 psi From Equation 10.3b, P = 9.2 psi with ample safety factor included

Example 2

Calculate the vacuum at collapse of a mortar-lined and coated steel pipe, for which:

r = 25.5 inch (D = 51 inch for steel),

Em = 4(106) psi,

Es = 30(106) psi,

n = 7.5 = Es /Em,

σf = 10 ksi for mortar (critical),

I = 0.04893 in3,

d = 0 = ring deflection (negligible)

From Equation 10.1, Pcr = 3Σ(EmI/r3) Assuming the mean radius of the steel is,

rs = 25.5, then rc = 25.9625 and rl = 25.1625 Substituting values, Pcr = 32 psi

Example 3

What does Equation 10.3a reduce to if Pcr = P? It is easily shown that all of the terms cancel except 6mdoP, which must be zero The only way this term can be zero is if initial ring deflection is do = 0 As expected, for a circular ring, P = Pcr

BURIED RING COLLAPSE

Stress Analysis of Elliptical Ring

Figure 10-1 is a half ring free-body-diagram At B

the ring compression stress is,

σc = P(OD)(1+d)/2A (10.4)

and the ring deformation stress (flexural stress) is,

σd = Ec(1/r'x - 1/rx) (10.5)

Notation:

P = vertical external pressure,

OD = outside diameter of the circular ring,

do = initial ring deflection,

d = ring deflection after P is applied,

r'x = radius of curvature at B due to initial ring

deflection do,

rx = radius of curvature at B after vacuum P is

applied,

c = distance from the neutral surface of the

wall to the most remote fiber (t/2 for a

plain pipe),

σd = flexural stress caused by ring deformation

Substituting in values of radii of curvature for the

ellipse from Chapter 3, the equation for flexural (ring

deflection) stress becomes:

σd = (Ec/r) 3(d-do) / (1-2d-2do) (10.6)

The maximum stress is the sum of Equations 10.4

and 10.6; i.e., σ = σc + σ f The maximum stress at B

in a buried plain pipe is,

σ = Pm(1+d) + (E/2m)3(d-do) / (1-2d-2d (10.7)o)

where m = r/t = ring flexibility of a plain pipe

Equation 10.7 can be solved to find vacuum P at

yield stress for brittle (rigid) pipes

But yield stress is not failure for plastics or

elasto-plastics (metals), for which wall crushing can occur only after ring compression stress (not flexural stress) reaches yield strength See Chapter 5 Therefore, Equation 10-7 is limited Initial ring deflection, do, depends upon compression of sidefill which requires analysis of pipe-soil interaction

Equations 10.4 to 10.7 are based on elastic theory Under some circumstances, plastic theory is justified For plain pipes and corrugated pipes, the plastic moment (at plastic hinging) is 3/2 times the moment at yield stress by elastic theory

Ring Deformation Collapse of Buried Pipes

For the following analyses, vacuum is negative

pressure, p, inside the pipe plus positive external hydrostatic pressure, u Both affect ring collapse Rigid Pipes

Because ring deflection of rigid pipes is negligible, rigid pipes are analyzed by ring compression except that vertical pressure on the pipe is P+p

Ring compression stress is,

where

σ = ring compression stress in the pipe wall,

P = total soil pressure at the top of the pipe,

including water pressure, u,

OD = outside diameter of the pipe,

A = wall area per unit length of pipe

Area A is a transformed section if the wall is composite such as concrete reinforced with steel bars For design, the ring compression stress, σ , from Equation 10.8 is equated to the strength of the pipe wall, σf, reduced by a safety factor

In the case of a very large diameter pipe, it may be necessary to consider the change in pressure of liquids (both inside and outside) throughout the depth

of the pipe For example, if the pipe is empty, but

the water table is above the top of the pipe, it may

be prudent to apply Equation 10.8 to the bottom of

the pipe where total pressure P, acting up on the

bottom, is greater than prismatic soil pressure on top

by the increase in hydrostatic pressure between the

top and bottom Of course, water inside the pipe will

negate any increase in external hydrostatic pressure

It is noteworthy that internal vacuum and external

hydrostatic pressure have little effect on the opening

of cracks in rigid pipes The 0.01-inch crack is not

a suitable performance limit

Flexible Pipes

Collapse of buried flexible pipes is either: 1 wall

crushing (ring compression) or 2 inversion (ring

deformation) Collapse due to longitudinal bending

is not included in this analysis Bending deforms the

pipe cross section into an ellipse with the short

diameter in the plane of the bend Bending strength

is decreased Bending failure is collapse Following

are procedures for evaluating the vacuum at which

a buried flexible ring collapses

If the ring could be held circular, analysis would be

simple ring compression — the same as for a rigid

pipe But flexible ring analysis anticipates ring

deflection, do, before the vacuum is applied Ring

deflection depends upon ring stiffness and stiffness

of the embedment soil It is assumed that pipes are

initially circular and empty, and that coefficient of

friction between the pipe and the backfill is zero

because of the inevitable breakdown of shearing

stresses due to earth tremors and changes in

temperature, moisture, and pressures The

embedment is assumed to be granular The flexible

pipe is often assumed to be thin-walled; i.e., OD =

ID = D = mean diameter of the pipe

Performance limit is collapse which occurs if the

ring either crushes due to ring compression, or

inverts due to sidefill soil slip at B See Figure 10-4

COLLAPSE BY WALL CRUSHING where

σf = ring compression stress at yield stress in

the pipe wall at B,

A = wall area per unit length of pipe,

t = wall thickness = A for plain pipes,

OD = outside diameter,

P = external pressure at top of pipe,

d = ∆D = ring deflection

Area, A, is used for transformed composite sections,

or ribbed, or ring-stiffened or corrugated

At ring deformation collapse the soil must slip in order for the ring to deflect See Figure 10-4 In the left sketch, the vertical pressure includes soil pressure and vacuum; i.e PA = P+p Before it is buried, the ring is circular, but as backfill is placed, the ring deflects into an ellipse

If the ring is flexible, and if shearing stresses between pipe and soil are neligible, vertical and horizontal soil pressures are related as follows:

PArA = PBrB = Pr = constant where Pr is the product of pressure and radius of curvature at any point on the circumference of the ring For an ellipse,

rA /rB = (1+d)3/(1-d)3 Therefore,

PB = PA(1+d)3/(1-d)3 = PArr (10.10) where

rA = mean radius of curvature at the top A,

rB = mean radius of curvature at the side B,

PA = pressure on the pipe at A,

PB = pressure on the pipe at B,

d = /D = initial ring deflection,

rr = (1+d)3/(1-d)3 = ratio of radii

Figure 10-4 UNSATURATED SOIL — (left) Free-body-diagram of an infinitesimal cube at spring line, B, showing the stresses at incipient soil slip (right) Vertical soil pressure, Po, supported by the pipe due to ring stiffness

Figure 10-5 Free-body-diagrams for finding the vertical deflection of point B by means of the Castigliano theorem

p

But for any pipe stiffness, F/∆ (or equivalent ring

stiffness, 53.77 EI/D3), the ring itself is able to

support part of the vertical pressure as it deflects

That part of the vertical pressure supported by ring

stiffness is Po shown in the sketch on the right of

Figure 10-4 For a given ring deflection, Po can be

calculated by the Castigliano theorem from the

free-body-diagram in Figure 10-5 From the sketch on

the left, the moment M at point B can be evaluated

by noting that the slope at B does not change during

deflection θBA = 0 Knowing M, Castigliano can be

applied again using the sketch on the right from

which the vertical deflection, YB, is evaluated for the

virtual load p due to the forces on the ring quadrant

Knowing YB, the ring deflection, d, can be found as

a function of EI and Po, and from this relationship, Po

can be found as a function of d The result is:

Po = 96(EI/D3)d/(1-2d)

For small ring deflections, it is conservative to

disregard 2d in the denominator; whereupon,

Po = Ed/m3 = 96(EI/D3)d

where

that can be supported by ring

stiffness,

Ed/m3 = Po for a plain pipe,

EI/D3 = ring stiffness = 0.0186 F/∆ ,

unit length of pipe = t3/12 for plain

pipe,

Pressure Against Soil at Spring Lines

The horizontal pressure of the pipe against the soil at

B is reduced by Po; i.e.,

PB = (PA+p-Po)rr - p See Figure 10-6 for free- body-diagram and assumptions For a plain pipe, substituting in Po,

PB = (PA+uA+p-Ed/m3)rr - p (10.11) where

PB = horizontal pressure of pipe on soil,

PA = vertical external soil pressure at A,

EI/D3 = ring stiffness, F/ = 53.77(EI/D3) = pipe stiffness,

D = mean diameter of the circular ring,

r = mean radius of the circular ring = D/2,

t = thickness of the plain pipe wall,

m = r/t = ring flexibility,

d = ∆/D = initial ring deflection,

rr = (1+d)3/(1-d)3 = ratio of vertical and

horizontal radii (maximum and minimum radii of the ellipse)

If soil at B does not have adequate strength, the soil slips, and the ring inverts

Strength of Soil at Spring Lines

Because most embedment is granular, the following

is analysis of strength for granular (cohesionless) sidefill See Figure 10-6 The horizontal strength of soil at point B, at soil slip, is soil passive resistance,

σx = Kσ y

where

σx = horizontal effective soil stress at B,

y = vertical effective soil stress at B,

K = ratio of horizontal to vertical effective stresses at soil slip (ring collapse),

K = (1+sinφ)/(1-sinφ),

φ = friction angle of the embedment, for which values can be obtained from tests

σy can be evaluated at the spring lines by methods

of Chapter 4

µA = rwh for floods at level h

σ _ _

_

Figure 10-6 SATURATED SOIL — Free-body-diagram of an infinitesimal soil cube at B, showing the stresses acting on it at incipient soil slip, and showing the shear planes at soil slip

Figure 10-7 Pressure diagram for analyzing critical hydrostatic pressure on the bottom of the pipe at inversion

of the ring from the bottom rr = ry /rx