Numerical Methods in Soil Mechanics 09.PDF

Numerical Methods in Soil Mechanics 09.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 "NON-CIRCULAR CROSS SECTIONS"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 9-1 Examples of non-circular cross sections of pipes used commonly in the corrugated steel pipe

industry, as described by AISI Handbook of Steel Drainage & Highway Construction Products.

©2000 CRC Press LLC

CHAPTER 9 NON-CIRCULAR CROSS SECTIONS

If the pipe cross section is not circular, "ring"

analysis must be modified For most buried pipes, a

circular cross section is the most effic ient shape

But even flexible circular rings deflect out-of-round

during installation Morever, a demand exists for

non-circular cross sections Some typical examples

are shown in Figure 9-1 A standing demand exists

in highway departments for culverts with reduced

height of cross section Each inch of height of the

culvert requires an enormous amount of soil to raise

the highway by that amount The pipe arch and low

profile arch are examples of efforts to serve

highway demands for reduced heights of culverts

Multiple culverts serve to reduce heights, but also

increase costs, spread stream beds, and trap trash

A pertinent variable for ring analysis is radius of

curvature, r The basic deflection of a flexible ring

is from circle to ellipse, for which radii of curvature

are shown in Chapter 3 But non-elliptical

deformation could make it necessary to measure

radii of curvature Techniques for measuring radii

are explained in Chapter 3

Figure 9-2 shows the free-body-diagram of an

infinitesimal segment of pipe wall loaded by external

radial pressure P The effects of bending moment

(ring deformation) can be combined by superposition

as discussed in this chapter Reactions are thrust T

in the pipe wall From static equilibrium in the

vertical direction, and noting that for the small

differential angle, sin(dq/2) = (dq/2), the equation of

vertical forces is Prdq = 2Tdq/2; and,

where

T = tangential (circumferential) thrust in the

wall,

P = external radial pressure (plus internal

vacuum),

r = mean radius of curvature (assuming that

the pipe is thin-walled and cylindrical)

Mean radius is sufficiently accurate for thin-wall pipe analyses Outside radius is more accurate — especially for thick-wall pipes

RING COMPRESSION STRESS For non-circular cross section, ring compression stress is simply, s = T/A; or, for plain pipes (smooth cylindrical surfaces, no ribs or corrugations, etc.), s

= T/t "Ring compression" is a misnomer in non-circular pipes Nevertheless, the expression ring compression stress is understood to mean

circumferential stress in the pipe wall

RADIAL SOIL PRESSURE Another pertinent variable is radial soil pressure P on the pipe From Equation 9.1, if thrust T is constant,

P varies inversely as radius r If the ring is flexible, the soil must be able to provide enough pressure P for equilibrium It is conservative to neglect shearing stresses between soil and pipe Shearing stresses reduce radial stresses Moreover, any shearing stresses that develop during installation are easily broken down by earth tremors, variations in temperature, rise and fall of the water table, wetting and drying of the soil, etc Without shearing stress, thrust T is constant around the entire perimeter of the pipe This is evident from Figure 9-3 where, for static equilibrium, T1 = T2 = T = constant thrust around the entire perimeter From Equation 9.1,

P1r1 = P2r2 = Pr = T (9.2) Wherever the radius r is small, the external pressure

P is large This introduces the very important concept that for a flexible non-circular cross section, the external soil-bearing capacity must be increased wherever radii are decreased If the corner plates

on a pipe arch, Figure 9-4, have a radius equal to one-third the top radius, then the external normal pressure (radial soil support) must

Figure 9-2 Free-body-diagram of an infinitesimal segment of pipe from which ring compression thrust in the pipe wall is T = Pr

Figure 9-3 Free-body-diagram of sections of a pipe wall of varying radii of curvature from which the ring compression thrust is constant, T = P1r1 = P2r2 = Pr Shearing stress between the soil and the pipe wall is neglected

©2000 CRC Press LLC

Figure 9-4 Typical cross section of a corrugated steel structural plate pipe arch showing radii of the top plate, corner plates, and bottom plate

Figure 9-5 Elliptical cross section of a flexible ring showing the distribution of external pressure required for equilibrium

r x

be three times as great as the pressure on the top of

the pipe arch The soil against the short-radius

corner plates must have adequate bearing strength

It is noteworthy that only a little spreading of the

corner plates will allow reversal of curvature of the

bottom plate if hydrostatic pressure should act on the

bottom Soil support at the corner plates is

imperative

If a circular cross section is deflected into an ellipse,

then Pxrx = Pyry See Figure 9-5 From Chapter 3,

the ratio of radii is ry /rx = (b/a)3 But (b/a)3 =

(1+d)3/(1-d)3, approximately Therefore,

Px = Py(b/a)3 = Py(1+d)3/(1-d)3 (9.3)

where

a = r(1-d) = minimum semi-diameter

b = r(1+d) = maximum semi-diameter

d = /D = ring deflection

An accurate solution, from Chapter 3, is,

1 + 3d + 4d2 + 4d3 + )

Px = Py (9.4)

1 - 3d + 4d2 - 4d3 + )

The accuracy of Equation 9.4 is seldom justified

The following example illustrates the point

Example 1

Assume that ring deflection of the elliptical cross

section is d = 10% What is Px in terms of Py?

From Equation 9.3, Px = 1.826 Py

From Equation 9.4, Px = 1.826 Py

This many significant figures of accuracy is not

justified — either in practice or theory It must be

remembered that: the elliptical cross section is only

a theoretical assumption; shearing stresses are

ignored; the perimeter is assumed to be constant; the

horizontal and vertical ring deflections are assumed

to be equal, etc For an elliptical cross section,

vertical ring deflection is slightly greater than

horizontal ring deflection, but the difference is

negligible if ring deflection is less than about ten percent Equation 9.3 is accurate enough for most ring deflection analyses

Example 2

A flexible pipe is deflected into an approximate ellipse shown in Figure 9-5 Initial ring deflection is

do = 15.9% If pressure on top is P = 1.0 ksf, what

is the required horizontal bearing capacity of the sidefill soil at the spring lines? The horizontal pressure Px at spring lines, from Equation 9.3, is,

Px = Py(ry/rx) = Py(1+d)3/(1-d)3 = 2.617 Py Horizontal bearing capacity of the soil at the spring lines must be greater than 2.62 ksf With a safety factor, specify soil-bearing capacity of 5 ksf Sidefill must be well compacted, otherwise the soil will be at incipient slip, and the ring at incipient collapse Ring compression stress is s = Pyry/A, where A is wall cross-sectional area per unit length

MEASURED CHANGE IN RADIUS

If the ring deflects into an ellipse, all that is needed

to evaluate maximum and minimum radii of curvature is measurement of ring deflection The equations are shown in Figure 9-6

For deformations other than ellipse, the change in radius can be evaluated from changes De in the middle ordinate e of a cord of length L Figure 9-7

shows the analysis, from which the approximate radius of curvature is r = L2/8e for small ratios of e

to L The change in radius from r to r' is found from change in the middle ordinate, De = e - e':

1/r - 1/r' = De/er (9.5)

Procedures for measuring e (either inside or outside the pipe) are described in Chapter 3 Minimum radius is pertinent to soil strength analysis; maximum radius is pertinent to ring stability, Chapter 10 Both are pertinent to circumferential stress analysis

CIRCUMFERENTIAL STRESS

Assume that within cord length L, a pipe is initially

circular and ring compression stress in the pipe wall

is Pr/A Now if the ring is deformed, the change in

radius of curvature causes a change in Pr/A; and

also introduces a flexural stress, (E/m)(r'-r)/2r' See

Equation 5.3 The ring compression stress is

essentially constant around the ring However,

flexural stress is maximum where change in radius

is greatest; i.e., where the change in middle ordinate,

e, is greatest Knowing the change in middle

ordinate, e, the circumferential stress within the

cord length is,

s = Pr/A + ( e/e)(Ec/r) (9.6)

where (See Figure 9-7)

e = middle ordinate for the original circle,

e = L2/8r, or can be measured before the pipe

is deformed,

De = change in middle ordinate due to ring

deformation,

L = length of cord

t = wall thickness

m = r/t = ring flexibility

r = initial radius of curvature at some location

before the pipe is deformed,

r' = radius of curvature at the same location

after the pipe is deformed,

A = cross-sectional area of the pipe wall per

unit length of pipe,

P = radial pressure on the pipe,

E = modulus of elasticity of the pipe,

I/c = section modulus per unit length,

e = change in middle ordinate due to ring

deformation,

c = distance from the neutral surface of the

pipe wall to the most remote surface

For a plain pipe, c = t/2, and (Ec/r) = (E/2m) to be

used in Equation 9.6

From Equation 9.6, for any allowable stress, s , or

strain, e, and external pressure, P; the allowable

change in middle ordinate, De/e, can be found Ring

deformation must then be controlled so that the

measured De/e does not exceed the allowable

Internal pressure P' (no external constraint) causes hoop stress s = P'r/A A non-circular ring tries to re-round and change its radii Equation 9.6 still applies Changes in radii would have to be measured and then used to calculate flexural stresses Or knowing allowable stress, maximum allowable changes in radii, or changes in middle ordinate from a cord can be used by inspectors for control of the pipe shape during installation

Internal pressure in a pipe with external constraint is sometimes analyzed by neglecting any changes in radii of curvature The presumption is that the soil

is rigid But, if the ring is constrained by compressible soil, or by concentrated point loads and reactions, further analysis may be necessary A finite element analysis may be a good option

For flexible pipes — even non-circular rings — flexural stresses due to change in radius, are not performance limits in general Brittle linings may pose an exception Many common pipe materials can yield without fracture Consequently, the flexural stress term can be neglected The ring simply sustains permanent deformation without fracture or inversion

In summary, the circumferential stress analysis of non-circular pipes is based on Equation 9.6 which always includes ring compression stress and, possibly, flexural stress

Pr/A = ring compression stress (or hoop tension),

(De/e)(Ec/r) = (De/e)(E/2m) = (Mc/I) = flexural stress

where (E/2m) = (Ec/r) = arc modulus — used if either change in radius of curvature or De is known (I/c) = section modulus — used if the moment M is known The moment M can be evaluated from circumferential strains measured by electrical resistance strain gages positioned both inside and outside the pipe at locations where critical moment

is anticipated Below yield, both thrust and moment can be found from these strains

©2000 CRC Press LLC

Circumferential stress due to ring deformation is

dependent upon either the section modulus or the arc

modulus

For brittle pipes — the ring compres sion stress and

ring deformation stress must be combined for

analysis Steel pipes with brittle linings or coatings

are not brittle pipes Small cracks are not serious

For flexible pipes — deformation is caused by the

soil Changes in wall thickness make little

difference If soil is placed such that ring deflection

is constant, the performance limit is wall crushing at

yield stress due to ring compression stress, Pr/A

This is an important basis for design See Chapter

6 On the other hand, if ring deformation stress

exceeds yield, the deformation is permanent But

deformation is not a performance limit until it

becomes excessive For analysis of flexible buried

pipes, the stress due to ring deformation is not an

appropriate basis for design Design by ring

compression stress and design by ring deflection are

the two basic design procedures

For corrugated and profile wall pipes, the

com-bination of ring compression stress and ring

deformation stress may result in dimpling of

corrugations, or plastic hinging But for buried pipes,

dimpling and incipient hinging are not collapse

PROBLEMS

9-1 Derive Equation 9.6 Remember that flexural

stress is Mc/I and that M/EI = 1/r - 1/r' for a circle

w here r is the original radius and r' is the deformed

radius

9-2 In order to check the assumption that a flexible

pipe, ID = 42, with ring deflection of d = 10%, is an

ellipse, what should be the middle ordinate inside the

pipe to the spring line from a vertical cord (straight

edge) that is 10 inches long? (e = 0.83)

9-3 Figure 9-8 shows the cross section of a corrugated steel culvert comprising circular panels with radii as follows:

ry = 82.50 inches for top and bottom panels,

rx = 26.25 inches = radius of side panels

This culvert is to be installed under a highway with the major diameter horizontal Soil cover is 2 ft including an asphalt concrete pavement which is assumed to be flexible enough that the load is not spread by the pavement The road is designed for HS-20 truck loads Unit weight of soil is 135 pcf Because the pipe is a drainage culvert, the water table is never more than a few inches above the invert of the pipe What is the minimum sidefill soil-bearing capacity required at the spring lines if safety factor is to be 2.0?

(Px = 12.2 ksf) 9-4 In Problem 9-3, if the sidefill soil is cohesionless with a soil friction angle of f = 35o, what is the safety factor against soil shear failure?

(sf = 0.4 — incipient collapse)

9-5 The elliptical culvert of Problem 9-3 is rotated

90o to serve as a livestock underpass See Figure

9-9 At what soil friction angle of sidefill at the spring lines would the culvert collapse by reversal of curvature of the 82.5-inch-radius side panels? Neglect H-20 surface live load Why?

9-6 In cohesionless soil what soil friction angle is required to assure that shearing failure does not occur in the embedment of a pipe arch for which:

rc = 18 inches at corner plates,

rt = 60 inches at top plate,

rb = 180 inches at bottom plate, Corrugations are 2 2/3 x ½ Assume high fill Neglect shearing stresses between the pipe and the

9-7 In Problem 9-6, what is the minimum rc in terms

of rt if the soil friction angle is 30o?

(rc = rt /3)

Figure 9-8 Horizontal "ellipse" of corrugated steel plate to be used as a culvert and for which rx = 26.25 inches and ry = 82.5 inches and showing the radial soil pressure acting on it

Figure 9-9 Corrugated steel plate underpass for

livestock with H=2 feet of soil cover

Figure 9-10 Cross section of a flexible circular ring that has been deformed into an ellipse during installation such that d=D/D=16% and for which the required horizontal soil pressure is greater than the vertical soil pressure at the crown

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