Numerical Methods in Soil Mechanics 13.PDF

Numerical Methods in Soil Mechanics 13.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 "Frontmatter"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 13-1 Soil stress models for minimum soil cover — free-body-diagrams of truncated pyramid and cone showing shear planes at "punch-through" that direct the live load onto the pipe

Figure 13-2 Truncated pyramid showing how a surface load, W, punches out a pyramid, which spreads the load over a base area of (B+H)(L+H) at depth H

CHAPTER 13 MINIMUM SOIL COVER

As the soil cover H decreases, the live load effect

on the buried pipe increases There exist minimum

heights of soil cover H, less than which the surface

live load may damage the pipe Less evident is a

minimum height of soil cover for dead load, weight

of soil only, on buried pipes Each of these cases is

discussed for rigid and flexible rings in this chapter

Only cohesionless soil is considered Vehicles are

generally unable to maneuver on poor soil such as

wet cohesive soil They get stuck in the mud

In many of the following analyses, the effect of the

surface live load on the pipe is based on the

pyramid/cone model The Boussinesq and

Newmark procedures for calculating live load

pressure on the pipe are based on the assumption

that the soil is elastic The assumption is not

adequate for failures of buried pipes Failure of

pipes due to surface loads on less-than-minimum soil

cover is punch-through The pipe is not subjected to

failure load until the soil cover fails in shear as

wheels punch through the soil cover and then

fracture or distort the buried pipe Based on such a

model, the soil stress on a buried pipe is referred to

as the pyramid/cone soil stress

PYRAMID/CONE SOIL STRESS

Figure 13-1 shows the truncated pyramid/cone soil

stress models A surface live load can damage the

buried pipe only after it punches through the soil

cover If the loaded surface area is circular, a

truncated cone of soil is punched out If the loaded

surface area is a rectangle, a truncated pyramid of

soil is punched out The truncated pyramid is

imperfect because its sharp edges do not form

Nevertheless, by slight adjustment of the pyramid

angle, the analysis can be made applicable The tire

print of dual wheels is more nearly rectangular than

circular Therefore the following analyses are based

on the pyramid

Figure 13-2 represents a dual-wheel tire print over a

rectangular area of width B and length L If the surface load is great enough to punch through granular soil and damage the pipe, then shear planes must form in the soil isolating a truncated pyramid which, like a pedestal, supports the load

The total load on the pipe is the surface load W plus the weight of the pyramid of soil The weight of the pyramid is ignored because it is small compared to any surface load great enough to punch through The vertical soil pressure on the pipe is load W divided by the base area of the pyramid The angle

θ which the shear planes make with the vertical is the pyramid angle θ = 45o - ϕ/2 where ϕ is the soil friction angle At depth H, the base area over which the load is spread is (B + 2Htanθ)(L + 2Htanθ) From tests on cohesionless soil, the pyramid angle is roughly 35o for which the base area is approximately (B+H)(L+H) The precision is as good as can be justified for typical installations The results are conservative Refinements may be forthcoming It follows that at soil failure, the pressure on the pipe is the pressure on the base of the pyramid; i.e.,

P = W/(B+H)(L+H) (13.1)

For HS-20 dual-wheel load on a firm surface, B = 7 and L = 22 inches with tire pressure of 105 psi NOTATION

P = vertical soil pressure at the level of the top

of the pipe due to a surface load uniformly distributed over a rectangular area,

W = weight of the surface load,

γ = unit weight of soil,

σy = stress at yield point,

σ = ring compression stress,

D = mean diameter of the pipe,

r = mean radius of the pipe,

c = distance from neutral surface of the pipe

wall cross section to the most remote fiber

on the surface

Figure 13-3 Sketch of a surface live wheel load W passing over a pipe buried in loose granular soil

Figure 13-4 Flexible ring in the process of collapse under minimum dead load soil cover showing the load wedges advancing against the ring and the lighter restraint wedges being lifted

A = cross sectional area of the pipe wall per

unit length of the pipe,

I = centroidal moment of inertia of the pipe

wall cross section per unit length of pipe,

M = moment in the wall of the ring due to ring

deformation,

T = circumferential thrust in the ring,

S = compressive strength of the pipe wall,

E = modulus of elasticity of the pipe material,

H = installed height of cover of the soil,

H' = rutted height of soil cover,

H" = depth of the rut (See Figure 13-3),

ρ = soil density in percent Standard Proctor

(AASHTO T-99) for the granular soil

cover and the embedment

For a dual truck wheel on a compacted soil surface,

the tire print area is about 7 inches by 22-inc hes

based on typical tire pressures as follows

Dual Load (kips) 5.5 7 9 16

Tire Pres (psi) 36 45 58 104

The loaded surface area can be adjusted for

different loads and different tire pressures For a

truck on pavement, the 7 x 22-inch contact area is in

reasonable agreement with observations

MINIMUM HEIGHT OF SOIL COVER

Minimum height of soil cover can be found by

solving Equation 13.1 for H if the surface load W is

known and if the allowable pressure P on the pipe

can be evaluated for any given pipe and for any

given performance limit, such as inversion or ring

compression at yield The next problem is evaluation

of the allowable pressure P This must include ring

compressive strength, ring stiffness, and the critical

location of the load The remainder of the chapter

is devoted to this problem

Example

Consider a perfectly flexible ring

Suppose that load W on a highway truck dual is 10

kips Vertical pressure P on the pipe at inversion is 0.8 ksf If B and L for the dual tire print are 7 and

22 inches, respectively, from Equation 13.1, H = 28.6 inches, which is the minimum granular soil cover for protection of the pipe against collapse This does not include a safety factor, but if it is based on the assumption that the pipe ring is perfectly flexible — like a chainlink watch band — a margin of safety is built in depending on the ring stiffness A perfectly flexible ring is not practical Moreover, longitudinal beam strength of the pipe is ignored

HEIGHT OF SOIL COVER

An unsuspected problem in the minimum cover analysis is the definition of height of soil cover For

a surfaced highway, the height of soil cover remains constant during passes of live loads But during construction, a heavy load crossing a buried pipe leaves ruts See Figure 13-3 In fact, successive passes of the load may increase the depths of the ruts If the depths of ruts approach a limit as the number of passes increases, the pipe-soil system is stable But if the depths of the ruts continue to increase with each pass of the surface load, it is obvious that the pipe feels increasingly adverse loads and may be in the process of inversion by ratcheting; i.e., an additional increment of ring deformation with each pass of the load Whatever the ultimate damage may be, a performance limit has been exceeded So minimum height of soil cover is defined as that soil cover H, less than which the pipe-soil system becomes unstable upon multiple passes of surface load W The height of cover to be used in Equation 13.1 for soil stress on the pipe is H', the height of soil cover after the ruts have reached their maximum depth, H" From Figure 13-3, H = H' + H" From tests in moist, granular, well-graded silty sand (SM classification), following rain (field moisture equivalent), the rut depths for dual wheels are generally not greater than,

H" = 0.315(logW - 0.34)(103.9 - ρ) (13.2)

Figure 13-5 Truncated pyramid punched through the minimum soil cover, H, by an approaching surface wheel load W Shear planes form at a 1:2 slope The inversion arc is 2α Angle α is typically less than 45o

Figure 13-6 Flexible ring in the

process of collapse under minimum

soil cover due to semi-infinite

soil pressure P, showing the load

"wedge" advancing against the ring

and restraint "wedge" being lifted

by the ring

where H" is in inches, and W is in kips Soil density

ρ (in percent) is based on AASHTO T-99 The

following table of values is from field tests from

which roughly 90% of the rut depths were less than

the values listed Equation 13.2 is not dimensionally

homogeneous It was achieved by regression from

plots of the following field data with tire pressures,

p, as indicated

H" = RUT DEPTH (inches)

DEAD LOAD

Minimum cover of cohesionless soil over a buried

pipe exists if the pipe is unable to support the

variation in soil pressures around its perimeter The

concept is shown in Figure 13-4, which shows a top

pressure of γH, but a shoulder pressure greater than

γH If the pipe cannot support the difference in

pressures, the shoulder wedges will slide in against

the pipe, deforming the ring which, in turn, lifts up

the top wedges as shown Collapse of the pipe is

catastrophic If the pipe is rigid (brittle), collapse is

fragmentation If the pipe is flexible, equations of

static equilibrium of the soil wedges provide values

of minimum soil cover H For average granular

backfill, H turns out to be about D/10 Experiments

confirm the above analysis of dead load collapse at

H = D/10 for very flexible pipes under dry granular

backfill A suggested allowable value of D/6 for

minimum cover allows for a margin of conservatism

But this analysis is for a perfectly flexible ring In

fact, pipes have ring stiffness and so provide

resistance to dead load collapse

LIVE LOAD

The minimum cover of cohesionless soil is not based

on a location of live load directly over the crown of the pipe as in Figure 13-1 Rather, the critical location is an approaching load as shown in Figure 13-5 The leading edge of the base area of the truncated pyramid is at the crown The ring tends to deform as indicated Tests to failure of long span corrugated steel arches prove that static surface loads symmetrically located over the crown can be many times greater than the load on one side Static load failure is soil punch-through and ring collapse

For fragile rigid pipes, failure is fracture of the pipe and possible collapse

For flexible pipes, failure is inversion due to deformation as shown in Figure 13-5 The mechanism is downward deformation of the left shoulder of the ring under the load and consequential upward deflection of the right shoulder For granular backfill, the inversion angle is observed to be about

α = 40o For convenience, and to be conservative, the collapse angle is assumed to be α = 45o for the truncated pyramid pressure shown in Figure 13-5 For the semi-infinite surface pressure shown in Figure 13-6, the load wedge contacts a full quadrant

of the ring

Analysis is evaluation of the maximum moment caused by the live load Dead loads are neglected The weights of the wedges and the shear resistance between them are small compared to the live load Moreover, in Figure 13-6, the weight of the load wedge tends to balance the restraint wedge Shear between pipe and soil is neglected The ring is fixed

at both ends of the collapse arch Vertical soil pressure, P, becomes radial P, on a flexible ring See Figure 13-7 Castigliano's equation is used to find the reactions, the maximum moment, M, and thrust, T See Appendix A Maximum M is located

by equating its derivative to zero If wall crushing is critical, thrust T is pertinent If circumferential stress is of interest,

σ = T/A + Mc/I (13.3)

ELASTIC LIMIT

The thrust term is usually so small compared to the

moment term, that it can be neglected

It is much more likely that the critical performance

limit is inversion Inversion is the result of plastic

hinging that ultimately generates a three link

mechanism See Figure 13-7 The thrust term is

relatively small enough to be neglected Plastic

hinging is a function of the Mc/I term, except that M

is at plastic limit — not elastic limit

For plain pipes and corrugated pipes, the moment at

plastic hinging is approximately 3/2 times the elastic

moment at yield stress Therefore,

PLASTIC HINGING

Truncated Pyramid Load (Surface Wheel Load)

For truncated pyramid pressure, the free-body-diagram is a fixed-ended 90o arch See Figure 13-7 Dead load soil pressure is neglected Live load soil pressure is constant radial pressure, P, over 45o left

of the crown, point A From Castigliano's equation, the maximum moment occurs at the point of minimum radius of curvature, about 12o to the right

of the crown, A, and is:

M = 0.022 Pr2 (13.5) WHEEL LOAD

The circumferential thrust is T = γHr due to the dead weight of soil cover on the right side of the crown For design of the pipe based on yield stress,

σf, the minimum required section modulus, I/c, is;

Figure 13-7 Free-body-diagram of the inversion arch for finding the maximum moment, M in terms of pressure P due to a surface dual-wheel load W approaching a pipe under minimum soil cover H The locations of potential plastic hinges are shown as circles starting at the location of maximum moment

I/c = (0.022Pr2)sf/σf (13.6)

ELASTIC LIMIT

I/c = (0.015Pr2)sf/σf (13.7)

PLASTIC HINGING

where sf is safety factor Tests show that I/c from

these equations is conservative A safety factor of

1.5 is usually adequate, and does not need to be

greater than 2 for highway culverts

With M and T known, Equation 13.3 can be solved

for the maximum stress whenever stress (or strain)

is of interest, as in the case of bonded linings

Semi-infinite Surface Pressure

For the semi-infinite uniform surface pressure of

Figure 13-6, the collapse arc under the load wedge

is 90o Even though the ring is flexible, the

assumption of constant radial pressure is

conservative From Castigliano's equation, the

maximum moment is,

M = 0.08 Pr2 (13.8)

SEMI-INFINITE SURFACE LOAD

and is located at about 12o to the right of the crown

T = γHr for use in Equation 13.3

But because ring compression stress T/A is usually

small compared to the Mc/I stress, it can be

neglected Setting Mc/I = yield strength/safety

factor, and solving for section modulus I/c,

I/c = (0.08Pr2)sf/σf (13.9)

ELASTIC LIMIT

I/c = (0.05Pr2)sf/σf (13.10)

PLASTIC HINGING

I/c is the required section modulus which can be

found from tables of values for corrugated metal

pipes and can be calculated for other pipes The

equations for I/c are conservative, but are

responsive to the inversion of very flexible pipes with minimum soil cover when subjected to heavy surface loads It is noteworthy that H does not appear in Equations 13.7 and 13.8 It is presumed that the height of cover is already minimum as calculated by Equation 13.1 for punch-through of a truncated pyramid It is presumed that for a semi-infinite surface pressure, at the level of the top of the pipe,

P is equal to surface pressure

Example 1 Assume that a pipe buried under minimum cover is

s ubjected to a semi-infinite surface pressure suc h that the edge of the base area of the load is at the crown of a buried pipe See Figure 13-6 What section modulus I/c is required for the pipe wall? The pipe is to be a 6-ft diameter corrugated steel pipe with granular soil cover of 2 ft and semi-infinite surface pressure of P = 900 psf The yield strength

of the pipe is 36 ksi Because the steel can yield, assume performance limit to be the formation of plastic hinges starting at the point of maximum moment Equation 13.10 applies Assume a safety factor of 2 Substituting values, the required I/c = 0.27 in3/ft From the AISI Handbook of Steel Drainage and Highway Construction Products,

a 3x1 corrugated steel pipe of 0.109-inch-thick steel

is adequate For this 3x1, the I/c is listed as 0.3358

in3/ft

Example 2 Consider Example 1 again, but for plain steel water pipe The section modulus I/c can be transformed into required wall thickness from the relationships I

= t3/12 and c = t/2 But from example 1, I/c = 0.27

in3/ft The required wall thickness is t = 0.367 inch Specify wall thickness of 0.375 inch

In the case of heavy surface wheel loads, it is often more economical to increase the height of soil cover

in order to reduce the possibility of inverting the pipe Under some circumstances, stiffener rings can be attached to the ring to increase the section modulus See Chapter 21

Figure 13-8

AASHTO shandard H trucks

Figure 13-9 Load-deflection diagrams from tests on 18D, HDPE corrugated pipe under 7 inches of granular soil cover at 85% density (AASHTO T-99), showing permanent ring deflection d’ and rebound ring deflection d”, and showing the zone of instability