Numerical Methods in Soil Mechanics B.PDF

Numerical Methods in Soil Mechanics B.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 "RECONCILIATION OF FORMULAS FOR RING DEFLECTION"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

APPENDIX B RECONCILIATION OF FORMULAS FOR RING DEFLECTION

M G Spangler was the first to predic t the ring

deflection of buried flexible circular pipes His Iowa

Formula is:

Dx = Df KsWcr3/(EI+0.061E'r3)

where (See Figure B-1.)

Dx = horizontal increase in diameter due to

vertical soil pressure P,

P = vertical soil pressure on top of pipe,

D = v e r t i c a l d e c r e a s e in diameter which

Spangler assumed to be equal to Dx,

d = ring deflection = D/D ,

D = mean circular diameter of the pipe,

r = D/2 = mean radius of the pipe,

Df = deflection lag factor which can be

incorporated into a time-dependent soil

modulus and can be ignored (Spangler

assumed Df = 1.5.),

Ks = bedding factor Spangler found it can vary

from 0.083 to 0.110 depending upon the

bedding angle, a (A reasonable assumption

is Ks = 0.1.)

Figure B-1 Assumed loads on a flexible ring in

Spangler's Iowa Formula

Wc = Marston load per unit length,

(Wc PD for most flexible pipes.)

e = vertical soil strain, (compression) in sidefill

soil due to pressure P,

EI = stiffness of the pipe wall per unit length

(See Figure B-2.) E' = soil modulus (Spangler defines this as

horizontal modulus of soil reaction, E", at

the spring lines It is more relevant to relate

ring deflection to vertical soil modulus from

confined compression tests.)

Rs = E'/(EI/D3) = stiffness ratio = ratio of soil

stiffness E' to ring stiffness EI/D3

Rs = 53.77R's R's = E'/(F/D) = stiffness ratio based on pipe

stiffness, F/D

The Iowa Formula can be written in the following form:

d/e = Rs/(80+0.061Rs) IOWA FORMULA

This is the relationship between two dimensionless variables: d/e = ring deflection term, and Rs = stiffness ratio This relationship is shown on Figure B-3 The graph approaches a horizontal asymptote

at d/e = 1.64 But empirical ring deflections do not exceed vertical compression of the sidefill soil Therefore, d/e does not exceed unity More

reasonable is the empirical graph of Figure B-3

which is expressed by the formula:

d/e = Rs/(30+Rs) EMPIRICAL FORMULA The empirical formula is an upper 90 percentile of ring deflections from tests at USU in the 1960s

Figure B-2 Pertinent notation for calculating the ring deflection of buried flexible pipes.

Figure B-3 Comparison of the Iowa Formula and an Upper Limit Empirical graph for predicting ring

deflection of buried flexible pipes

Iowa Formula

The Iowa Formula was derived to predict horizontal

ring deflection of buried flexible pipes Figure B-1

is the basis for the derivation Bedding angle, a ,

could vary from 0 to 180o Horizontal passive soil

resistance acted over a parabola subtended by an

arc of 100o If an arc of 90o seemed reasonable,

100o must be conservative The passive resistance

was better represented by a sine curve throughout

the height of the ring, but a sine curve appeared to

be more difficult to integrate Integrations were

performed using virtual work to solve the two

deformation equations required for analysis Three

equations were available from static equilibrium

With load, Wc, and soil resistance, h, known, five

unknowns can be solved The mathematical

analysis by Spangler was elegant and correct after

a modification of definition of h A critical variable

was the soil resistance, h, assumed to be a function

of horizontal soil modulus of elasticity, E-prime (E')

E' was assumed to be constant for any soil and

density (compaction) For details see text by

Spangler, M.G and Handy, R.L., Soil Engineering,

3rd Ed., IEP, New York 1975

E-prime (E')

E-prime (E') is the horizontal soil modulus in the

Iowa Formula In the derivation of the formula it

was assumed that all materials are elastic and that

E' is constant for any given soil embedment In fact,

E' is not constant Pipe-soil interaction is not elastic

Flexible pipes are best analyzed by mechanics of

plastics Soil can vary from viscous through plastic

to granular (best analyzed by the mechanics of

particulates) Values of E' were investigated at

USU in 1996 and 1997 E' was found to be a

function of soil depth (confinement), ring stiffness,

and soil type and density Figure B-4 is a graph of

E' in silty sand (Unified Classification SM) Figure

B-5 is a proposal for conservative values of E' for

silty sand Other soils require similar tests to

evaluate E'

EXTERNAL PRESSURE DESIGN BASED ON

RING STIFFNESS (BUCKLING)

Like the Iowa Formula, some ring design equations are based on elastic stiffnesses of the ring and the soil These equations are subject to the same cautions as any equation based on elasticity Two such design equations are considered in the following

One equation is simplified AWWA C950:

P = \/RwEs(EI)/(0.149r3) (B.1) where

P = uniform collapse pressure, Rw

= buoyancy factor

= 1- 0.33(H'/H), where H'<H,

H = height of fill over pipe, H' = height of water table over pipe,

Es = soil stiffness (secant modulus),

E = long-term (virtual) pipe mod,

r = mean radius of pipe = D/2,

I = mom/iner of wall cross section

The other equation is from "Industrial Brochure" by the Large Diameter Pipe Division of PPI It is

similar to Equation 41 in the Uni-Bell Handbook,

P = 1.15 \/E tp (B.2) where

p(1-n2)(D/t)3(r'/r)3 = 2E

p = collapse pressure on circ pipe,

E = modulus at operating temperature,

t = mean wall thickness,

D = mean diameter = 2r,

r = mean circular radius, r' = maximum radius of curvature,

n = Poisson ratio,

Et = soil stiffness (tangent modulus)

In order to compare Equations B.1 and B.2, it is assumed that:

m = r/t = D/2t = ring flexibility,

Rw = 0.67 (Water table at surface),

I = t3/12 (plain pipe), r' = r (circular pipe cross section),

n = 0.38 = Poisson ratio for PVC

Figure B-4 E' for DRY SILTY SAND from USU model studies of 1996 If the soil is saturated, values of E' are reduced to one-half

Figure B-5 Proposed conservative values of Er for silty sand (Unified Classification SM)

Substituting values,

AWWA P = 0.61\/EsE/m3

Uni-Bell P = 0.62 \/EtE/m3

If the secant soil modulus Es is about the same as

the tangent soil modulus Et, Equations B.1 and B.2

are comparable The Uni-Bell equation assumes a

water table at ground surface AWWA disregards

ovality, but includes height of soil cover and water

table

The range of applicability of these two equations is

limited If either of the soil moduli approaches zero,

P approaches zero Not so Clearly poor soil is out

of range

For further comparisons, Equations B.1 and B.2 are

rewritten as service equation,

SERVICE, P = 0.62 \/KEtE/m3

where

P = pressure on circular pipe,

KEt = horizontal tangential soil modulus,

K = (1-sinj)/(1+sinj) = 1/3 if,

j = soil friction angle

= 30o for cohesionless soil,

m = r/t = ratio of mean radius and wall thickness

for plain pipe

If the ring is encased in relatively rigid soil, the soil acts as a casing at yield, S,

ENCASED, P = S/m This is ring compression failure

If the ring is nonconstrained; i.e., not buried, NONCONSTRAINED, P = E/4(1-n 2)m3

This is buckling failure

If the ring is supported by soil in which the cross-section stiffness is calculated from ring deflection tests, then, from an equation for the empirical curve

of Figure B-3, EMPIRICAL, P =0.14E/m3 + 0.27KEt

If ring deflection can be predicted by the Iowa Formula, then effective wall stiffness can be calculated and,

IOWA, P = 0.22E/m3 + 0.16KEt

Figure B-6 is a comparison of values of critical P by the above equations of elasticity

Figure B-6 Comparison of external pressure design procedures for ring compression (ENCASED), top two graphs, and for elastic collapse, bottom graphs